content stringlengths 86 994k | meta stringlengths 288 619 |
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- Publications
J. Queißer, K. Neumann, M. Rolf, F. Reinhart, J.J. Steil, "An Active Compliant Control Mode for Interaction with a Pneumatic Soft Robot". IEEE/RSJ IROS, pp. 573-579, 2014.
Bionic soft robots offer exciting perspectives for more flexible and safe physical interaction with the world and humans. Unfortunately, their hardware design often prevents analytical modeling,
which in turn is a prerequisite to apply classical automatic control approaches. On the other hand, also modeling by means of learning is hardly feasible due to many degrees of freedom,
high-dimensional state spaces and the
softness properties like e.g. mechanical elasticity, which cause limited repeatability and complex dynamics. Nevertheless, the realization of basic control modes is important to leverage the
potential of soft robots for applications. We therefore propose a hybrid approach combining classical and learning elements for the realization of an interactive control mode for an elastic
PDF-Dokument [3.2 MB] | {"url":"https://www.neuralautomation.de/publications/","timestamp":"2024-11-12T14:52:06Z","content_type":"text/html","content_length":"60294","record_id":"<urn:uuid:588a5b1d-5d57-4236-a754-39acb5cd3301>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00087.warc.gz"} |
The Stacks project
Lemma 30.23.10. Let $f : X' \to X$ be a morphism of Noetherian schemes. Let $Z \subset X$ be a closed subscheme and denote $Z' = f^{-1}Z$ the scheme theoretic inverse image. Let $\mathcal{I} \subset
\mathcal{O}_ X$, $\mathcal{I}' \subset \mathcal{O}_{X'}$ be the corresponding quasi-coherent sheaves of ideals. If $f$ is flat and the induced morphism $Z' \to Z$ is an isomorphism, then the pullback
functor $f^* : \textit{Coh}(X, \mathcal{I}) \to \textit{Coh}(X', \mathcal{I}')$ (Lemma 30.23.9) is an equivalence.
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Generate frequency bands around the characteristic fault frequencies of ball or roller bearings for spectral feature extraction
FB = bearingFaultBands(FR,NB,DB,DP,beta) generates characteristic fault frequency bands FB of a roller or ball bearing using its physical parameters. FR is the rotational speed of the shaft or inner
race, NB is the number of balls or rollers, DB is the ball or roller diameter, DP is the pitch diameter, and beta is the contact angle in degrees. The values in FB have the same implicit units as FR.
FB = bearingFaultBands(___,Name,Value) allows you to specify additional parameters using one or more name-value pair arguments.
[FB,info] = bearingFaultBands(___) also returns the structure info containing information about the generated fault frequency bands FB.
bearingFaultBands(___) with no output arguments plots a bar chart of the generated fault frequency bands FB.
Frequency Bands Using Bearing Specifications
For this example, consider a bearing with a pitch diameter of 12 cm with eight rolling elements. Each rolling element has a diameter of 2 cm. The outer race remains stationary as the inner race is
driven at 25 Hz. The contact angle of the rolling element is 15 degrees.
With the above physical dimensions of the bearing, construct the frequency bands using bearingFaultBands.
FR = 25;
NB = 8;
DB = 2;
DP = 12;
beta = 15;
FB = bearingFaultBands(FR,NB,DB,DP,beta)
FB = 4×2
82.6512 85.1512
114.8488 117.3488
71.8062 74.3062
9.2377 11.7377
FB is returned as a 4x2 array with default frequency band width of 10 percent of FR which is 2.5 Hz. The first column in FB contains the values of $F-\frac{W}{2}$, while the second column contains
all the values of $F+\frac{W}{2}$ for each characteristic defect frequency.
Frequency Bands for Roller Bearing
For this example, consider a micro roller bearing with 11 rollers where each roller is 7.5 mm. The pitch diameter is 34 mm and the contact angle is 0 degrees. Assuming a shaft speed of 1800 rpm,
construct frequency bands for the roller bearing. Specify 'Domain' as 'frequency' to obtain the frequency bands FB in the same units as FR.
FR = 1800;
NB = 11;
DB = 7.5;
DP = 34;
beta = 0;
[FB1,info1] = bearingFaultBands(FR,NB,DB,DP,beta,'Domain','frequency')
FB1 = 4×2
10^4 ×
0.7626 0.7806
1.1994 1.2174
0.3791 0.3971
0.0611 0.0791
info1 = struct with fields:
Centers: [7.7162e+03 1.2084e+04 3.8815e+03 701.4706]
FaultGroups: [1 2 3 4]
Labels: {'1Fo' '1Fi' '1Fb' '1Fc'}
Now, include the sidebands for the inner race and rolling element defect frequencies using the 'Sidebands' name-value pair.
[FB2,info2] = bearingFaultBands(FR,NB,DB,DP,beta,'Domain','order','Sidebands',0:1)
FB2 = 8×2
4.2368 4.3368
5.6632 5.7632
6.6632 6.7632
7.6632 7.7632
1.7167 1.8167
2.1064 2.2064
2.4961 2.5961
0.3397 0.4397
info2 = struct with fields:
Centers: [4.2868 5.7132 6.7132 7.7132 1.7667 2.1564 2.5461 0.3897]
FaultGroups: [1 2 2 2 3 3 3 4]
Labels: {'1Fo' '1Fi-1Fr' '1Fi' '1Fi+1Fr' '1Fb-1Fc' '1Fb' '1Fb+1Fc' '1Fc'}
You can use the generated fault bands FB to extract spectral metrics using the faultBandMetrics command.
Visualize Frequency Bands Around Characteristic Bearing Frequencies
For this example, consider a damaged bearing with a pitch diameter of 12 cm with eight rolling elements. Each rolling element has a diameter of 2 cm. The outer race remains stationary as the inner
race is driven at 25 Hz. The contact angle of the rolling element is 15 degrees.
With the above physical dimensions of the bearing, visualize the fault frequency bands using bearingFaultBands.
FR = 25;
NB = 8;
DB = 2;
DP = 12;
beta = 15;
From the plot, observe the following bearing specific vibration frequencies:
• Cage defect frequency, Fc at 10.5 Hz.
• Ball defect frequency, Fb at 73 Hz.
• Outer race defect frequency, Fo at 83.9 Hz.
• Inner race defect frequency, Fi at 116.1 Hz.
Frequency Bands and Spectral Metrics of Ball Bearing
For this example, consider a ball bearing with a pitch diameter of 12 cm with 10 rolling elements. Each rolling element has a diameter of 0.5 cm. The outer race remains stationary as the inner race
is driven at 25 Hz. The contact angle of the ball is 0 degrees. The data set bearingData.mat contains power spectral density (PSD) and its respective frequency data for the bearing vibration signal
in a table.
First, construct the bearing frequency bands including the first 3 sidebands using the physical characteristics of the ball bearing.
FR = 25;
NB = 10;
DB = 0.5;
DP = 12;
beta = 0;
FB = bearingFaultBands(FR,NB,DB,DP,beta,'Sidebands',1:3)
FB = 14×2
118.5417 121.0417
53.9583 56.4583
78.9583 81.4583
103.9583 106.4583
153.9583 156.4583
178.9583 181.4583
203.9583 206.4583
262.2917 264.7917
274.2708 276.7708
286.2500 288.7500
FB is a 14x2 array which includes the primary frequencies and their sidebands.
Load the PSD data. bearingData.mat contains a table X where PSD is contained in the first column and the frequency grid is in the second column, as cell arrays respectively.
X=1×2 table
Var1 Var2
________________ ________________
{12001x1 double} {12001x1 double}
Compute the spectral metrics using the PSD data in table X and the frequency bands in FB.
spectralMetrics = faultBandMetrics(X,FB)
spectralMetrics=1×43 table
PeakAmplitude1 PeakFrequency1 BandPower1 PeakAmplitude2 PeakFrequency2 BandPower2 PeakAmplitude3 PeakFrequency3 BandPower3 PeakAmplitude4 PeakFrequency4 BandPower4 PeakAmplitude5 PeakFrequency5 BandPower5 PeakAmplitude6 PeakFrequency6 BandPower6 PeakAmplitude7 PeakFrequency7 BandPower7 PeakAmplitude8 PeakFrequency8 BandPower8 PeakAmplitude9 PeakFrequency9 BandPower9 PeakAmplitude10 PeakFrequency10 BandPower10 PeakAmplitude11 PeakFrequency11 BandPower11 PeakAmplitude12 PeakFrequency12 BandPower12 PeakAmplitude13 PeakFrequency13 BandPower13 PeakAmplitude14 PeakFrequency14 BandPower14 TotalBandPower
______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ _______________ _______________ ___________ _______________ _______________ ___________ _______________ _______________ ___________ _______________ _______________ ___________ _______________ _______________ ___________ ______________
121 121 314.43 56.438 56.438 144.95 81.438 81.438 210.57 106.44 106.44 276.2 156.44 156.44 407.45 181.44 181.44 473.07 206.44 206.44 538.7 264.75 264.75 691.77 276.75 276.75 723.27 288.69 288.69 754.61 312.69 312.69 817.61 324.62 324.62 848.94 336.62 336.62 880.44 13.188 13.188 31.418 7113.4
spectralMetrics is a 1x43 table with peak amplitude, peak frequency and band power calculated for each frequency range in FB. The last column in spectralMetrics is the total band power, computed
across all 14 frequencies in FB.
Input Arguments
FR — Rotational speed of the shaft or inner race
positive scalar
Rotational speed of the shaft or inner race, specified as a positive scalar. FR is the fundamental frequency around which bearingFaultBands generates the fault frequency bands. Specify FR either in
Hertz or revolutions per minute.
NB — Number of balls or rollers
positive integer
Number of balls or rollers in the bearing, specified as a positive integer.
DB — Diameter of the ball or roller
positive scalar
Diameter of the ball or roller, specified as a positive integer.
DP — Pitch diameter
positive scalar
Pitch diameter of the bearing, specified as a positive scalar. DP is the diameter of the circle that the center of the ball or roller travels during the bearing rotation.
beta — Contact angle
non-negative scalar
Contact angle in degrees between a plane perpendicular to the ball or roller axis and the line joining the two raceways, specified as a positive scalar.
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: ...,'Harmonics',[1,3,5]
Harmonics — Harmonics of the fundamental frequency to be included
1 (default) | vector of positive integers
Harmonics of the fundamental frequency to be included, specified as the comma-separated pair consisting of 'Harmonics' and a vector of positive integers. The default value is 1. Specify 'Harmonics'
when you want to construct the frequency bands with more harmonics of the fundamental frequency.
Sidebands — Sidebands around the fundamental frequency and its harmonics to be included
0 (default) | vector of nonnegative integers
Sidebands around the fundamental frequency and its harmonics to be included, specified as the comma-separated pair consisting of 'Sidebands' and a vector of nonnegative integers. The default value is
0. Specify 'Sidebands' when you want to construct the frequency bands with sidebands around the fundamental frequency and its harmonics.
Width — Width of the frequency bands centered at the nominal fault frequencies
10 percent of the fundamental frequency (default) | positive scalar
Width of the frequency bands centered at the nominal fault frequencies, specified as the comma-separated pair consisting of 'Width' and a positive scalar. The default value is 10 percent of the
fundamental frequency. Avoid specifying 'Width' with a large value so that the fault bands do not overlap.
Domain — Units of the fault band frequencies
'frequency' (default) | 'order'
Units of the fault band frequencies, specified as the comma-separated pair consisting of 'Domain' and either 'frequency' or 'order'. Select:
• 'frequency' if you want FB to be returned in the same units as FR.
• 'order' if you want FB to be returned as number of rotations relative to the inner race rotation, FR.
Folding — Logical value specifying whether negative nominal fault frequencies have to be folded about the frequency origin
false (default) | true
Logical value specifying whether negative nominal fault frequencies have to be folded about the frequency origin, specified as the comma-separated pair consisting of 'Folding' and either true or
false. If you set 'Folding' to true, then faultBands folds the negative nominal fault frequencies about the frequency origin by taking their absolute values such that the folded fault bands always
fall in the positive frequency intervals. The folded fault bands are computed as $\left[\text{max}\left(0,\text{}|F|-\frac{W}{2}\right),\text{}|F|+\frac{W}{2}\right]$, where W is the 'Width'
name-value pair and F is one of the nominal fault frequencies.
Output Arguments
FB — Fault frequency bands
Fault frequency bands, returned as an N-by-2 array, where N is the number of fault frequencies. FB is returned in the same units as FR, in either hertz or orders depending on the value of 'Domain'.
Use the generated fault frequency bands to extract spectral metrics using faultBandMetrics. The generated fault bands, $\left[F-\frac{W}{2},\text{}F+\frac{W}{2}\right]$, are centered at:
• Outer race defect frequency, Fo and its harmonics
• Inner race defect frequency, Fi, its harmonics and sidebands at FR
• Rolling element (ball) defect frequency, Fbits harmonics and sidebands at Fc
• Cage (train) defect frequency, Fc and its harmonics
The value W is the width of the frequency bands, which you can specify using the 'Width' name-value pair. For more information on bearing frequencies, see Algorithms.
info — Information about the fault frequency bands
Information about the fault frequency bands in FB, returned as a structure with the following fields:
• Centers — Center fault frequencies
• Labels — Labels describing each frequency
• FaultGroups — Fault group numbers identifying related fault frequencies
bearingFaultBands computes the different characteristic bearing frequencies as follows:
• Outer race defect frequency, ${F}_{o}=\frac{NB}{2}FR\left(1-\frac{DB}{DP}\text{cos}\left(\beta \right)\right)$
• Inner race defect frequency, ${F}_{i}=\frac{NB}{2}FR\left(1+\frac{DB}{DP}\text{cos}\left(\beta \right)\right)$
• Rolling element (ball) defect frequency, ${F}_{b}=\frac{DP}{2DB}FR\left(1-{\left[\frac{DB}{DP}\text{cos}\left(\beta \right)\right]}^{2}\right)$
• Cage (train) defect frequency,${F}_{c}=\frac{FR}{2}\left(1-\frac{DB}{DP}\text{cos}\left(\beta \right)\right)$
[1] Chandravanshi, M & Poddar, Surojit. "Ball Bearing Fault Detection Using Vibration Parameters." International Journal of Engineering Research & Technology. 2. 2013.
[2] Singh, Sukhjeet & Kumar, Amit & Kumar, Navin. "Motor Current Signature Analysis for Bearing Fault Detection in Mechanical Systems." Procedia Materials Science. 6. 171–177. 10.1016/
j.mspro.2014.07.021. 2014.
[3] Roque, Antonio & Silva, Tiago & Calado, João & Dias, J. "An approach to fault diagnosis of rolling bearings." WSEAS Transactions on Systems and Control. 4. 2009.
Version History
Introduced in R2019b | {"url":"https://nl.mathworks.com/help/predmaint/ref/bearingfaultbands.html","timestamp":"2024-11-10T03:03:04Z","content_type":"text/html","content_length":"131738","record_id":"<urn:uuid:0440983e-7e27-4292-afe8-c6becc716436>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00496.warc.gz"} |
Number Patterns
Number Patterns are mathematical sequences presented in the form of geometrical. Shapes like Y Patterns, Circular Patterns, Rectangular Patterns etc. that follow a certain rule based on elementary
arithmetic operations. Patterns are given, with one or more. missing numbers and one must decipher the code (or find the logic) that connects. The numbers given in the patterns. One can find out the
missing number(s) easily after finding out the logic/rule connecting the given numbers in the patterns.
Y Patterns
Establish relationship between two numbers to get the third one in each Y shaped arrangement. Same relationship exists among tree numbers of each arrangement. observe the relationship in 1st two Y
figures and apply the same pattern to third Y figure to reach the solution.
Missing Number = (5 X 2) + 2 = 12
Triangular Patterns
There are 4 numbers – One in center and 3 numbers along the edges or at the corners. In each triangular figure, establish a relationship between three numbers to get the number at center.
Missing Number = (8 + 3) X 7 = 77
Square or Rectangular Number Grids
Square or rectangular Number grids have certain number of rows (horizontal lines) and columns (vertical Lines) in which numbers are arranged. There is some relationship among the numbers either along
the rows or along the columns. Missing number can be calculated using same pattern among the numbers in their respective rows or columns.
Missing Number: (6 X 6) + (c X c) = 52
Missing Number: (c X c) = 52 – (6 X 6)
Missing Number: (c X c) = 16
Missing Number: c = 8
Circular or oval Patterns
This pattern can be little tricky at times as relationship is to be established among four numbers around the circle to get the number at the center.
Missing Number: (4 X 16) – (3 X 14) = 22
X Pattern
Like circular patterns X shaped patterns too can be tricky at times as relationship is to be established among four numbers at the corners to get the number at the center.
Missing Number: (6 X 3) + (9 X 1) = 27
Numbers can be arranged in many geometrical patterns but the concept remains the same. | {"url":"http://mathelogical.in/reasoning/number-patterns/","timestamp":"2024-11-08T07:25:30Z","content_type":"text/html","content_length":"118045","record_id":"<urn:uuid:9b3ce911-66ce-484c-9b8e-f04a204e4c83>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00370.warc.gz"} |
Capital Gain Formula | Calculator (Examples with Excel Template)
Updated July 25, 2023
Capital Gain Formula (Table of Contents)
What is the Capital Gain Formula?
The term “capital gain” refers to the increase in the value of an asset or a portfolio over a period of time solely due to growing price, while not taking into account the dividend paid during
the same period.
In other words, it measures how much higher is the selling price of the asset than its purchase price. The formula for capital gain can be derived by deducting the purchase value of the asset or
portfolio from the selling value. Mathematically, it is represented as,
Capital Gain = Selling Value – Purchase Value
Examples of Capital Gain Formula (With Excel Template)
Let’s take an example to understand the calculation of Capital Gain in a better manner.
Capital Gain Formula – Example #1
Let us take the example of Jenny who purchased 1,000 equity stocks of a company named BNM Inc. for $50 each a year back. The company paid a dividend of $4 per share during the year and currently she
is selling all the stocks for $56 per share. Calculate Jenny’s capital gain for the transaction based on the given information.
Purchase Value of the Portfolio is calculated as
• Purchase Value of the Portfolio = $50 * 1,000
• Purchase Value of the Portfolio = $50,000
Selling Value of the Portfolio is calculated as
• Selling Value of the Portfolio= $56 * 1,000
• Selling Value of the Portfolio = $56,000
[Dividend paid is not considered as it forms part of total gain only]
Capital Gain is calculated using the formula given below
Capital Gain = Selling Value of the Portfolio – Purchase Value of the Portfolio
• Capital Gain = $56,000 – $50,000
• Capital Gain = $6,000
Therefore, Jenny’s capital gain for the transaction was $6,000.
Capital Gain Formula – Example #2
Let us take the example of Apple Inc.’s stock price movement to illustrate the concept of capital gain. Let us assume that John purchased 100 shares of Apple Inc. on 26 October 2018 for $216.30 per
share and he sold all the shares on 25 October 2019 for $246.58 per share. Calculate John’s capital gain in this transaction.
Purchase Value of the Portfolio is calculated as
• Purchase Value of the Portfolio = $216.30 * 100
• Purchase Value of the Portfolio = $21,630
Selling Value of the Portfolio is calculated as
• Selling Value of the Portfolio = $246.58 * 100
• Selling Value of the Portfolio = $24,658
Capital Gain is calculated using the formula given below
Capital Gain = Selling Value of the Portfolio – Purchase Value of the Portfolio
• Capital Gain = $24,658 – $21,630
• Capital Gain = $3,028
Therefore, in this transaction, over a period of one year, John earned a capital gain of $3,028.
Screenshot of stock price used for calculation
Capital Gain Formula – Example #3
Let us take the example of Walmart Inc.’s stock price movement in the last one year. If Lucy purchased 500 shares of Walmart Inc. on 26 October 2018 for $98.94 per share and then sold all the shares
on 25 October 2019 for $119.04 per share, Calculate the capital gain earned by her in selling these 500 shares.
Purchase Value of the Portfolio is calculated as
• Purchase Value of the Portfolio = $98.94* 500
• Purchase Value of the Portfolio = $49,470
Selling Value of the Portfolio is calculated as
• Selling Value of the Portfolio = $119.04 * 500
• Selling Value of the Portfolio = $59,520
Capital Gain is calculated using the formula given below
Capital Gain = Selling Value of the Portfolio – Purchase Value of the Portfolio
• Capital Gain = $59,520 – $49,470
• Capital Gain = $10,050
Therefore, Lucy earned a capital gain of 10,050 by holding these 500 shares of Walmart Inc. for a period of one year.
Screenshot of stock price used for calculation
Capital Gain Formula – Example #4
Let us take the example of the stock price movement of Bank of America Corporation during the last one year. Jason purchased 800 shares of the bank on 26 October 2018 and sold all the shares on 25
October 2019. Calculate Jason’s capital gain if the purchase price was $26.39 per share and selling price was $31.72 per share.
Purchase Value of the Portfolio is calculated as
• Purchase Value of the Portfolio = $26.39 * 800
• Purchase Value of the Portfolio = $21,112
Selling Value of the Portfolio is calculated as
• Selling Value of the Portfolio = $31.72 * 800
• Selling Value of the Portfolio = $25,376
Capital Gain is calculated using the formula given below
Capital Gain = Selling Value of the Portfolio – Purchase Value of the Portfolio
• Capital Gain = $25,376 – $21,112
• Capital Gain = $4,264
Therefore, Jason’s capital gain in this transaction was $4,264.
Screenshot of stock price used for calculation
The formula for capital gain can be derived by using the following steps:
Step 1: Firstly, determine the purchase value of the asset. For instance, the purchase value of a portfolio of stocks can be the product of the purchase price of each stock and the number of stocks
Step 2: Next, determine the selling value of the asset. Again, it is the product of the selling price of each stock and the number of stocks sold.
Step 3: Finally, the formula for capital gain can be derived by deducting the purchase value (step 1) of the asset from the selling value (step 2) as shown below.
Capital Gain = Selling Value – Purchase Value
Relevance and Use of Capital Gain Formula
It is one of the most important performance metrics for both investors and trading analysts because it helps in determining how much the investors’ investment has appreciated over a period of time.
But please bear in mind that capital gain is not realized unless and until the subject asset is sold. A capital gain that is realized within a year is known as short-term capital gain, while capital
gain realized over a longer time period (more than one year) is known as long term capital gain. The income tax treatment of long term capital gain is different than that of short term capital gain.
One of the limitations of this metric is that it expressed in dollar terms and as such it doesn’t provide any meaningful insight unless it is compared with the invested amount. However, this
limitation can be overcome by the use of capital gain yield.
Capital Gain Formula Calculator
You can use the following Capital Gain Formula Calculator
Selling Value
Purchase Value
Capital Gain
Capital Gain = Selling Value – Purchase Value
= 0 – 0
= 0
Recommended Articles
This is a guide to Capital Gain Formula. Here we discuss how to calculate capital gain formula along with practical examples. we also provide a capital gain calculator with a downloadable excel
template. You may also look at the following articles to learn more – | {"url":"https://www.educba.com/capital-gain-formula/","timestamp":"2024-11-03T10:22:48Z","content_type":"text/html","content_length":"354144","record_id":"<urn:uuid:b4ef1a0e-08fe-44d1-8c67-9476c8ed993a>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00741.warc.gz"} |
Using Solver in VBA for excel without cell references
I want to create a function that does the following
1. Is a function
2. gets data from external datasources
3. does calculations
4. optimizes using the solver (minimize the square sum of errors of a formula)
5. returns the value (as a normal function)
All this is straightforward, however, I would like not to use cell
references and
use variables directly in the VBA code instead of i.e. creating a new
inserting the values, running the solver from vba, reading the result and
deleting the
worksheet, and then returning the result. In other words I would like to run
the solver internally in the code without using cells in Excel.
I hope this was somewhat clear, since I'm not a programmer. All other steps
than the solver in my code is relatively easy, but I don't seem to be able to
put my mind as to how the solver should work without cell references (using
variables instead).
Does anyone have a suggestion?
I want to create a function that does the following
1. Is a function
2. gets data from external datasources
3. does calculations
4. optimizes using the solver (minimize the square sum of errors of a formula)
5. returns the value (as a normal function)
All this is straightforward, however, I would like not to use cell
references and
use variables directly in the VBA code instead of i.e. creating a new
inserting the values, running the solver from vba, reading the result and
deleting the
worksheet, and then returning the result. In other words I would like to run
the solver internally in the code without using cells in Excel.
I hope this was somewhat clear, since I'm not a programmer. All other steps
than the solver in my code is relatively easy, but I don't seem to be able to
put my mind as to how the solver should work without cell references (using
variables instead).
Does anyone have a suggestion?
You can't, because Frontline formulations are dependent on Excel range
equations/inequalities. You can do what you are suggesting in VBA,
but you have to place the data in cells and then you also have to call
the Solver from your function. Which I guess would be the return
value. The Frontline solver has a set VBA functions you can access by
referencing its Library in your project. You use those to manage the
If you don't know VBA, you are either outta luck unless you hire
someone help you or you do become a programmer soon. | {"url":"https://www.pcreview.co.uk/threads/using-solver-in-vba-for-excel-without-cell-references.3539307/","timestamp":"2024-11-11T08:12:19Z","content_type":"text/html","content_length":"56784","record_id":"<urn:uuid:283fd7cf-cf48-4b01-92c2-00162ed5d14e>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00183.warc.gz"} |
Tips for acing Class 6 Mathematics | Stoptazmo
Class 6th Maths acts as a very crucial aspect in the foundation of Maths that is taught in the years ahead. It consists of some very basic concepts and chapters but it is important to understand
these topics well to have clarity regarding them when they become more intricate and complex in higher classes.
Syllabus for Class 6th Maths
The chapters in NCERT Books For Class 6 Maths are as follows –
• Introduction to whole numbers – what are natural numbers, what are whole numbers, what are non-decimal numbers, introduction to the concept of the predecessor of a number, introduction to the
concept of the successor of a number, estimating the sum of numbers, estimating the difference between numbers, introduction to complex numbers, comparing and variating numbers, properties of
different types of numbers and the patterns observed in them
• Introduction to numbers and their conceptualisation – finding out factors of numbers, finding out multiples of numbers, what are prime numbers, what are composite numbers, basic concepts of LCM
and HCF
• Introduction to the basic concept of geometry – what is a line, what are rays intercepting from lines, what are segments of lines, how to find points on a line, answering some basic questions –
how many points are there on a line to which the answer is infinite, what are parallel lines, what are curves and its various types,
• Introduction to shapes around us – A brief introduction to basic shapes found in everyday life like circles, squares, triangles, rectangles, studying the various degrees of angles found in these
shapes, preparing for geometry given in higher classes with a more advanced syllabus
• Integers – Visualising the number line and how the various points are distributed on the number line, representation of points on the number line, a way to conceptualise which number is lesser
and which number is greater, adding integers, ascending and descending order of integers
• Fractions – What are fractions, how are fractions ultimately parts of whole numbers, relative calculation of fractions with respect to whole numbers, type of fractions and their applications,
adding fractions, subtracting fractions and many more operations
• Decimals – What is a decimal, how does its position change the whole meaning of a number, how are decimals related to fractions, how to represent fractions as decimal numbers, conversions like
changing centimetre to meters and many more
• Data handling – Representing data in the graphical and pictorial methods, grouping data, finding out values of equations using data given in the pictorial form
• Mensuration – Understanding what is area, what is the perimeter, finding out the area of simple figures, finding out the perimeter of simple figures, simple questions pertaining to these
• Introduction to ratios and proportion – Taking the previously taught knowledge of fractions to study ratios and proportions, understanding how they are also related to whole numbers in parts,
analysing the role and relevance of ratios and proportions in everyday life, understanding unitary method and using it to solve simple real questions taken from real-life scenarios for easier
• Algebra – Introduction to the concept of algebra and its endless applications, finding the relationship between different variables with the information already given, finding missing values in
an equation by connecting already given hints, different types of equations in algebra and techniques used to find their solutions
• Symmetry – Understanding the relevance of geometrical shapes and figures, using this knowledge to conceptualise symmetry of objects and shapes, drawing the symmetry of application of symmetry,
answering whether given object sketches are symmetric or not. Visualising reflection
• Studying geometry practically – What is geometric equipment like compass and protractor, how to use this equipment to sketch out figures and shapes, draw figures based on some pre-defined
constraints and specifications, drawing circles having common radii etc, constructing basic angles and segments
How to study for a good score?
• Understanding the syllabus is very important. This will help in understanding the importance of each topic of each chapter and give a better knowledge of what to study first and what to study
later. Knowing the syllabus in complete detail is essential to be able to predict the exam pattern and also the type of questions to expect.
• Knowing the weightage given to each chapter – This will help in knowing where to invest more time and what to practice more. The weightage as given by CBSE makes sure to give a sense of
uniformity to all students so that the exam pattern remains relatively the same.
• Study from NCERT – At the class 6 level, it is not required to start studying from refreshers and extra books, NCERT should be enough. Make sure to study the entirety of the NCERT well so that no
topic is left untouched. Questions are usually given based on the questions given in NCERT books.
• Manage your time well – At the basic class 6 level, plan your study time and playtime with balance. There is no need to study for excessive hours but at the same time, daily practice is the key.
Study at your own pace and take your time understanding concepts.
• Practice previous years’ question papers to analyse the exam pattern well and grasp the need of the subject. Sometimes the same questions are also given.
• Make a realistic study schedule.
• Make a separate copy of the formulas involved to learn them quicker.
• Revise each topic at least three to four times before the exam.
• Do written practice of theorems, graphs and constructions as these are scoring areas of the syllabus.
• Search up important topics and questions and pay close attention to them.
• Draw the figures neatly for increased visibility when the teacher checks your answer sheet.
• Take ample and sufficient breaks in between your study sessions to evade overexertion of your mental capacity.
• Solve examples before you start solving questions.
• Never miss out on any steps of the exam.
• Don’t leave any doubts answered till the last minute. This will only cause confusion.
There is no need to take any sort of stress in class 6th. Everything is simple if you just devote a few hours to practice daily. Take breaks whenever required and just stay confident in yourself. | {"url":"https://stoptazmo.com/tips-for-acing-class-6-mathematics/","timestamp":"2024-11-09T06:29:16Z","content_type":"text/html","content_length":"71080","record_id":"<urn:uuid:18e9da63-b9f2-475c-b27d-a9c37a5dbd77>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00420.warc.gz"} |
Trig Function Calculator
Created by Luis Hoyos
Reviewed by Davide Borchia
Last updated: Jan 18, 2024
This trig function calculator prints the value of the six trig functions given the angle in degrees or radians. In the article below, we explain:
• What are the 6 trig functions; and
• How to calculate those trig functions.
In the FAQ section, we present the value of the trig functions for some of the most common angles: the values of the six trig functions for 45 degrees and 270 degrees.
The 6 trig functions.
Sine, cosine, and tangent
Sine, cosine, and tangent are the trigonometric functions of an angle. The most common way to define them is by using a right triangle and one of its angles (the angle of interest).
First, let's remember the names of the three sides of a right triangle:
• Hypotenuse: The longest side, always opposite to the right angle;
• Opposite side: The side opposite to the angle of interest (α); and
• Adjacent side: The side of the triangle that is neither the hypotenuse nor the opposite side.
As you can note, the hypotenuse and adjacent side form the reference angle (α).
Then, the trigonometric functions are defined as:
sin(α) = opposite/hypotenuse
cos(α) = adjacent/hypotenuse
tan(α) = opposite/adjacent
🙋 In summary, we use the right triangle to easily define the reference angle and calculate the trigonometric functions based on the lengths of the sides.
Cosecant, secant, and cotangent
Cosecant (csc), secant (sec), and cotangent (cot) are simply the reciprocals of the three previous functions:
• csc(α) = 1/sin(α) = hypotenuse/opposite
• sec(α) = 1/cos(α) = hypotenuse/adjacent
• cot(α) = 1/tan(α) = adjacent/opposite
Other trig functions calculators.
To learn more about this topic, you can visit our trigonometric functions calculator article and the following tools:
What are the values of the six trig functions for 45 degrees?
The values of the six trig functions for 45 degrees are:
• sin(45°) = √2/2 ≈ 0.70710678
• cos(45°) = √2/2 ≈ 0.70710678 (the same as sin(45°))
• tan(45°) = 1
• sec(45°) = √2 ≈ 1.41421356
• csc(45°) = √2 ≈ 1.41421356 (the same as sec(45°))
• cot(45°) = 1
What are the values of the 6 trig functions for 270 degrees?
The values of the 6 trig functions for 270 degrees are:
• sin(270°) = -1
• cos(270°) = 0
• tan(270°) = undefined
• sec(270°) = 0
• csc(270°) = undefined
• cot(270°) = -1 | {"url":"https://www.omnicalculator.com/math/trig-function","timestamp":"2024-11-12T15:40:42Z","content_type":"text/html","content_length":"188043","record_id":"<urn:uuid:026555db-279b-4a4b-a36e-ffaf7f835705>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00840.warc.gz"} |
Smallest number that cannot be formed from sum of numbers from array
This problem was asked to me in Amazon interview -
Given a array of positive integers, you have to find the smallest positive integer that can not be formed from the sum of numbers from array.
Array:[4 13 2 3 1]
result= 11 { Since 11 was smallest positive number which can not be formed from the given array elements }
What i did was :
1. sorted the array
2. calculated the prefix sum
3. Treverse the sum array and check if next element is less than 1 greater than sum i.e. A[j]<=(sum+1). If not so then answer would be sum+1
But this was nlog(n) solution.
Interviewer was not satisfied with this and asked a solution in less than O(n log n) time.
Consider all integers in interval [2^i .. 2^i+1 - 1]. And suppose all integers below 2^i can be formed from sum of numbers from given array. Also suppose that we already know C, which is sum of all
numbers below 2^i. If C >= 2^i+1 - 1, every number in this interval may be represented as sum of given numbers. Otherwise we could check if interval [2^i .. C + 1] contains any number from given
array. And if there is no such number, C + 1 is what we searched for.
Here is a sketch of an algorithm:
1. For each input number, determine to which interval it belongs, and update corresponding sum: S[int_log(x)] += x.
2. Compute prefix sum for array S: foreach i: C[i] = C[i-1] + S[i].
3. Filter array C to keep only entries with values lower than next power of 2.
4. Scan input array once more and notice which of the intervals [2^i .. C + 1] contain at least one input number: i = int_log(x) - 1; B[i] |= (x <= C[i] + 1).
5. Find first interval that is not filtered out on step #3 and corresponding element of B[] not set on step #4.
If it is not obvious why we can apply step 3, here is the proof. Choose any number between 2^i and C, then sequentially subtract from it all the numbers below 2^i in decreasing order. Eventually we
get either some number less than the last subtracted number or zero. If the result is zero, just add together all the subtracted numbers and we have the representation of chosen number. If the result
is non-zero and less than the last subtracted number, this result is also less than 2^i, so it is "representable" and none of the subtracted numbers are used for its representation. When we add these
subtracted numbers back, we have the representation of chosen number. This also suggests that instead of filtering intervals one by one we could skip several intervals at once by jumping directly to
int_log of C.
Time complexity is determined by function int_log(), which is integer logarithm or index of the highest set bit in the number. If our instruction set contains integer logarithm or any its equivalent
(count leading zeros, or tricks with floating point numbers), then complexity is O(n). Otherwise we could use some bit hacking to implement int_log() in O(log log U) and obtain O(n * log log U) time
complexity. (Here U is largest number in the array).
If step 1 (in addition to updating the sum) will also update minimum value in given range, step 4 is not needed anymore. We could just compare C[i] to Min[i+1]. This means we need only single pass
over input array. Or we could apply this algorithm not to array but to a stream of numbers.
Several examples:
Input: [ 4 13 2 3 1] [ 1 2 3 9] [ 1 1 2 9]
int_log: 2 3 1 1 0 0 1 1 3 0 0 1 3
int_log: 0 1 2 3 0 1 2 3 0 1 2 3
S: 1 5 4 13 1 5 0 9 2 2 0 9
C: 1 6 10 23 1 6 6 15 2 4 4 13
filtered(C): n n n n n n n n n n n n
number in
[2^i..C+1]: 2 4 - 2 - - 2 - -
C+1: 11 7 5
For multi-precision input numbers this approach needs O(n * log M) time and O(log M) space. Where M is largest number in the array. The same time is needed just to read all the numbers (and in the
worst case we need every bit of them).
Still this result may be improved to O(n * log R) where R is the value found by this algorithm (actually, the output-sensitive variant of it). The only modification needed for this optimization is
instead of processing whole numbers at once, process them digit-by-digit: first pass processes the low order bits of each number (like bits 0..63), second pass - next bits (like 64..127), etc. We
could ignore all higher-order bits after result is found. Also this decreases space requirements to O(K) numbers, where K is number of bits in machine word. | {"url":"https://coderapp.vercel.app/answer/21079874","timestamp":"2024-11-05T22:22:25Z","content_type":"text/html","content_length":"103150","record_id":"<urn:uuid:dc5c2399-3016-4618-ac56-9bcebff984e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00768.warc.gz"} |
formalized libraries of homotopy type theory
nLab formalized libraries of homotopy type theory
This page collects a list of libraries of mathematics formalized in proof assistants using homotopy type theory and univalent foundations, sorted by proof assistant.
Coq is a proof assistant based originally on the intensional Calculus of Inductive Constructions?.
Non-cubical Agda
Agda is a proof assistant based originally on intensional Martin-Lof Type Theory?. By default it allows general pattern-matching which can prove axiom K making it incompatible with HoTT, but the
--without-K flag turns this off.
Cubical Agda
Agda has a --cubical mode that implements a version of cubical type theory that is similar to that of CCHM. The following libraries use this feature.
RedTT is (was?) an implementation of cartesian cubical type theory, with some basic development in an associated library.
Arend implements a theory that enhances Book HoTT with an interval type similar to that of cubical type theory, but without the extra structure necessary to make univalence computational.
Lean is a proof assistant implementing dependent type theory. Current versions of Lean (3 and now 4) have UIP built-in and are not HoTT-compatible, but the old Lean 2 had a HoTT mode.
Last revised on March 11, 2024 at 19:32:48. See the history of this page for a list of all contributions to it. | {"url":"https://ncatlab.org/nlab/show/formalized+libraries+of+homotopy+type+theory","timestamp":"2024-11-10T23:58:50Z","content_type":"application/xhtml+xml","content_length":"17833","record_id":"<urn:uuid:e443f1e7-5cad-42bc-8876-79b030417832>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00048.warc.gz"} |
Syllabus Information
Use this page to maintain syllabus information, learning objectives, required materials, and technical requirements for the course.
Syllabus Information
MTH 065 - Elementary Algebra
Associated Term: Summer 2022
Learning Objectives: Upon successful completion of this course, the student will be able to:
1.Maintain, use, and expand skills and concepts learned in previous mathematics courses
a. Solve linear equations algebraically
b. Calculate slope of a line and find intercepts
c. Graph equations in two variables
d. Write equations in point-slope form and slope-intercept form
2. Solve linear systems of two equations in two unknowns
a. Solve algebraically and by graphing
b. Solve application problems involving linear systems of equations (Includes simple interest, motion, and mixture problems)
3. Evaluate and/or simplify expressions using the rules of (integer) exponents
4. Use scientific notation
5. Use the terminology of polynomials and add, subtract, multiply, and divide polynomials
a. Recognize and use the terminology of polynomials
b. Evaluate polynomials
c. Add, subtract, and multiply polynomials
d. Divide a polynomial by a monomial
6. Factor polynomials, including multivariable polynomials
a. Factor polynomials by removing a common monomial factor
b. Factor trinomials
c. Factor special products
Required Materials:
Technical Requirements: | {"url":"https://crater.lanecc.edu/banp/bwckctlg.p_disp_catalog_syllabus?cat_term_in=202310&subj_code_in=MTH&crse_numb_in=065","timestamp":"2024-11-06T18:26:35Z","content_type":"text/html","content_length":"8573","record_id":"<urn:uuid:249fc7fc-dc88-4ed8-8ffb-e157dd19b491>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00603.warc.gz"} |
Degree/Square Microsecond to Circle/Square Week
Degree/Square Microsecond [deg/μs2] Output
1 degree/square microsecond in degree/square second is equal to 1000000000000
1 degree/square microsecond in degree/square millisecond is equal to 1000000
1 degree/square microsecond in degree/square nanosecond is equal to 0.000001
1 degree/square microsecond in degree/square minute is equal to 3600000000000000
1 degree/square microsecond in degree/square hour is equal to 12960000000000000000
1 degree/square microsecond in degree/square day is equal to 7.46496e+21
1 degree/square microsecond in degree/square week is equal to 3.6578304e+23
1 degree/square microsecond in degree/square month is equal to 6.91584804e+24
1 degree/square microsecond in degree/square year is equal to 9.9588211776e+26
1 degree/square microsecond in radian/square second is equal to 17453292519.94
1 degree/square microsecond in radian/square millisecond is equal to 17453.29
1 degree/square microsecond in radian/square microsecond is equal to 0.017453292519943
1 degree/square microsecond in radian/square nanosecond is equal to 1.7453292519943e-8
1 degree/square microsecond in radian/square minute is equal to 62831853071796
1 degree/square microsecond in radian/square hour is equal to 226194671058470000
1 degree/square microsecond in radian/square day is equal to 130288130529680000000
1 degree/square microsecond in radian/square week is equal to 6.3841183959541e+21
1 degree/square microsecond in radian/square month is equal to 1.207043188656e+23
1 degree/square microsecond in radian/square year is equal to 1.7381421916646e+25
1 degree/square microsecond in gradian/square second is equal to 1111111111111.1
1 degree/square microsecond in gradian/square millisecond is equal to 1111111.11
1 degree/square microsecond in gradian/square microsecond is equal to 1.11
1 degree/square microsecond in gradian/square nanosecond is equal to 0.0000011111111111111
1 degree/square microsecond in gradian/square minute is equal to 4000000000000000
1 degree/square microsecond in gradian/square hour is equal to 14400000000000000000
1 degree/square microsecond in gradian/square day is equal to 8.2944e+21
1 degree/square microsecond in gradian/square week is equal to 4.064256e+23
1 degree/square microsecond in gradian/square month is equal to 7.6842756e+24
1 degree/square microsecond in gradian/square year is equal to 1.1065356864e+27
1 degree/square microsecond in arcmin/square second is equal to 60000000000000
1 degree/square microsecond in arcmin/square millisecond is equal to 60000000
1 degree/square microsecond in arcmin/square microsecond is equal to 60
1 degree/square microsecond in arcmin/square nanosecond is equal to 0.00006
1 degree/square microsecond in arcmin/square minute is equal to 216000000000000000
1 degree/square microsecond in arcmin/square hour is equal to 777600000000000000000
1 degree/square microsecond in arcmin/square day is equal to 4.478976e+23
1 degree/square microsecond in arcmin/square week is equal to 2.19469824e+25
1 degree/square microsecond in arcmin/square month is equal to 4.149508824e+26
1 degree/square microsecond in arcmin/square year is equal to 5.97529270656e+28
1 degree/square microsecond in arcsec/square second is equal to 3600000000000000
1 degree/square microsecond in arcsec/square millisecond is equal to 3600000000
1 degree/square microsecond in arcsec/square microsecond is equal to 3600
1 degree/square microsecond in arcsec/square nanosecond is equal to 0.0036
1 degree/square microsecond in arcsec/square minute is equal to 12960000000000000000
1 degree/square microsecond in arcsec/square hour is equal to 4.6656e+22
1 degree/square microsecond in arcsec/square day is equal to 2.6873856e+25
1 degree/square microsecond in arcsec/square week is equal to 1.316818944e+27
1 degree/square microsecond in arcsec/square month is equal to 2.4897052944e+28
1 degree/square microsecond in arcsec/square year is equal to 3.585175623936e+30
1 degree/square microsecond in sign/square second is equal to 33333333333.33
1 degree/square microsecond in sign/square millisecond is equal to 33333.33
1 degree/square microsecond in sign/square microsecond is equal to 0.033333333333333
1 degree/square microsecond in sign/square nanosecond is equal to 3.3333333333333e-8
1 degree/square microsecond in sign/square minute is equal to 120000000000000
1 degree/square microsecond in sign/square hour is equal to 432000000000000000
1 degree/square microsecond in sign/square day is equal to 248832000000000000000
1 degree/square microsecond in sign/square week is equal to 1.2192768e+22
1 degree/square microsecond in sign/square month is equal to 2.30528268e+23
1 degree/square microsecond in sign/square year is equal to 3.3196070592e+25
1 degree/square microsecond in turn/square second is equal to 2777777777.78
1 degree/square microsecond in turn/square millisecond is equal to 2777.78
1 degree/square microsecond in turn/square microsecond is equal to 0.0027777777777778
1 degree/square microsecond in turn/square nanosecond is equal to 2.7777777777778e-9
1 degree/square microsecond in turn/square minute is equal to 10000000000000
1 degree/square microsecond in turn/square hour is equal to 36000000000000000
1 degree/square microsecond in turn/square day is equal to 20736000000000000000
1 degree/square microsecond in turn/square week is equal to 1.016064e+21
1 degree/square microsecond in turn/square month is equal to 1.9210689e+22
1 degree/square microsecond in turn/square year is equal to 2.766339216e+24
1 degree/square microsecond in circle/square second is equal to 2777777777.78
1 degree/square microsecond in circle/square millisecond is equal to 2777.78
1 degree/square microsecond in circle/square microsecond is equal to 0.0027777777777778
1 degree/square microsecond in circle/square nanosecond is equal to 2.7777777777778e-9
1 degree/square microsecond in circle/square minute is equal to 10000000000000
1 degree/square microsecond in circle/square hour is equal to 36000000000000000
1 degree/square microsecond in circle/square day is equal to 20736000000000000000
1 degree/square microsecond in circle/square week is equal to 1.016064e+21
1 degree/square microsecond in circle/square month is equal to 1.9210689e+22
1 degree/square microsecond in circle/square year is equal to 2.766339216e+24
1 degree/square microsecond in mil/square second is equal to 17777777777778
1 degree/square microsecond in mil/square millisecond is equal to 17777777.78
1 degree/square microsecond in mil/square microsecond is equal to 17.78
1 degree/square microsecond in mil/square nanosecond is equal to 0.000017777777777778
1 degree/square microsecond in mil/square minute is equal to 64000000000000000
1 degree/square microsecond in mil/square hour is equal to 230400000000000000000
1 degree/square microsecond in mil/square day is equal to 1.327104e+23
1 degree/square microsecond in mil/square week is equal to 6.5028096e+24
1 degree/square microsecond in mil/square month is equal to 1.229484096e+26
1 degree/square microsecond in mil/square year is equal to 1.77045709824e+28
1 degree/square microsecond in revolution/square second is equal to 2777777777.78
1 degree/square microsecond in revolution/square millisecond is equal to 2777.78
1 degree/square microsecond in revolution/square microsecond is equal to 0.0027777777777778
1 degree/square microsecond in revolution/square nanosecond is equal to 2.7777777777778e-9
1 degree/square microsecond in revolution/square minute is equal to 10000000000000
1 degree/square microsecond in revolution/square hour is equal to 36000000000000000
1 degree/square microsecond in revolution/square day is equal to 20736000000000000000
1 degree/square microsecond in revolution/square week is equal to 1.016064e+21
1 degree/square microsecond in revolution/square month is equal to 1.9210689e+22
1 degree/square microsecond in revolution/square year is equal to 2.766339216e+24 | {"url":"https://hextobinary.com/unit/angularacc/from/degpmicros2/to/circlepw2","timestamp":"2024-11-03T06:00:34Z","content_type":"text/html","content_length":"114588","record_id":"<urn:uuid:3d43fc63-7858-4046-b16a-4945a972f82d>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00736.warc.gz"} |
Internal function that calculates the log marginal likelihood
log_marg_likel {bayesmove} R Documentation
Internal function that calculates the log marginal likelihood of each model being compared
An internal function that is used to calculate the log marginal likelihood of models for the current and proposed sets of breakpoints. Called within samp_move.
log_marg_likel(alpha, summary.stats, nbins, ndata.types)
numeric. A single value used to specify the hyperparameter for the prior distribution. A standard value for alpha is typically 1, which corresponds with a vague prior on the Dirichlet
alpha distribution.
summary.stats A matrix of sufficient statistics returned from get_summary_stats.
nbins numeric. A vector of the number of bins used to discretize each movement variable.
ndata.types numeric. The length of nbins.
The log marginal likelihood is calculated for a model with a given set of breakpoints and the discretized data.
version 0.2.1 | {"url":"https://search.r-project.org/CRAN/refmans/bayesmove/html/log_marg_likel.html","timestamp":"2024-11-13T09:06:29Z","content_type":"text/html","content_length":"2944","record_id":"<urn:uuid:71fe7474-f0ed-41c0-8677-55cdfe7ea546>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00890.warc.gz"} |
An introduction to Real Estate Investment Analysis - The Ethical Entrepreneur
Have you ever wondered how to go about analysing a Cap Rate for an investment ? Are you trying to figure out what NOI means? Do you want to know how to calculate the various types of ROI, like
Cash-on-Cash Return and Total Return? If so, no need to struggle alone – I’ve written what I hope is a handy introduction to real estate investment analysis.
I’ve written a number of previous articles on evaluating deals— everything from a checklist on how to carrying out due diligence to evaluating income opportunities.
Have you ever considered buying an investment property? Are you curious about how you would go about analysing the financial details of the property you are considering buying? How you would go about
figuring out if the property were a good deal or not? The following is a detailed tutorial on how to do carry out a financial analysis of any residential investment property you might be considering
The value of single family homes (investment or not) is generally determined by market rates linked to compatible properties in the surrounding area. These properties have similar characteristics –
same floorplan, same number of bedrooms/bathrooms, equivalent garage size, same amenities, etc.
Thanks to this, a single family home will generally rise in value if similar homes in the same area are rising in value and lose value if similar homes in the area are losing value.
Larger investment properties (those with at least two unitsand especially those over four units) are valued differently. The value of larger investment properties is directly related to how much
profit it produces for the owner. It’s therefore entirely possible that an apartment building in a neighborhood where house prices are dropping could be increasing in value — especially if the
components of the market that drive income are improving.
The fact that multi-family properties are valued based on their income potential demonstrates how important good financial analysis of these properties is.
Of course, you can’t just compare your building to others down the street to see how much it’s worth. I should also point out that whilst this analysis will work for a residential rental property, it
is not sufficient for analysing all types of commercial property; for things like office, industrial, or retail space, there’s a lot more you need to know and I would recommend searching out
additional resources. I may write some in future, but currently have a more limited understanding of that market and don’t feel as comfortable talking about it.
The first step in being able to analyse the value of a rental property is to understand what factors contribute to property value. In general, good financial analysis involves being able to input a
bunch of information about your investment into a financial model and have that model kick out a bunch of information that you can then use to determine whether the investment is a good or a bad one.
Below is an overview of the considerations required to perform a thorough financial analysis of a residential rental property:
• Property Details: Information about the physical design of the property, including number of units, square footage, utility design etc.
• Purchase Information: This is basic cost information about the property you are considering, such as the purchase price and the price of any improvement work you’ll need to do.
• Financing Details: These are the details of the loan you will obtain to finance the property. In some circumstances, you may be fortunate to afford the property outright, but most property
investors I have met seem to have used significant debt to fund their investments. Consequently, the detail of the total loan amount, structure of the debt, deposit information, interest rates,
payment schedules and administration/professional fees all need to be considered.
• Income: This the detailed information about the income the property produces, such as rent payments and funds received from energy generation or car parking rental.
• Expenses: This is the detailed information about costs of maintaining the property, including such things as property taxes, insurance, upkeep and general repairs (yes, you’re going to have to
foot the bill every time the boiler break down!).
Getting good data out of your model requires that the information you put into your model is highly reliable and accurate; gathering accurate data can often prove the difference between making the
property look great on paper and look horrible.
Estimated vs Actual Data
Remember from the introduction of this tutorial that the value of properties is directly related to how much profit it produces for its owner. Because of this, it’s often in the seller’s best
interest to provide numbers that are more “appealing” than they are accurate; for example, a seller may give high estimates of rental income or neglect to mention certain maintenance expenses to give
the impression that the property is more valuable than it is.
Part of your job as an investor is to make sure you have the best information available when doing your financial analysis. You might decide to rely on estimates from the seller as a starting point,
but it’s important to carry out your own research about potential income and expenses. Ask to see old bank statements, copies of bills and maintenance records to help verify the information you’re
being presented with.
If you’re lucky, these documents will help to support the original information you were presented with, but don’t be surprised if they don’t. The seller is only interested in one thing here – getting
the highest price they can for the property and consequently will often feel at liberty to get creative with numbers to help make the deal add up.
In addition to requesting and reviewing documentation, it’s also important to make a careful examination of the property and surrounding area before making an offer. Small problems can often
snowball, serious impacting the viability of the investment.
Sources of quality information
When looking for information on the property, requesting information from the seller and/or estate agent is often a good starting point. Make sure to look online, interview neighbours and tenants,
review council records and hire relevant professionals such as brokers and surveyors to carry out specialist investigations to support your investment.
Example Investment
Ultimately, it remains your responsibility to carry out sufficient due diligence before purchasing an investment property. I wouldn’t go as far as to advise someone else on whether an investment was
suitable for them (indeed, I’m pretty sure I’d be breaking the law if I did!), but there’s nothing stopping me from providing some theoretical illumination on the process by creating a fictitious
investment as an example.
Property Value $200,000
Improvements $10,000
Financing: 80% of total cost
Interest Rate: Fixed at 3% over 30 years
Professional Fees: limited to 2% of property value
Based on this information, our investment opportunity stacks up something like this
Purchase Information
Property Value: £200,000
Deposit: £40,000
Renovation: £10,000
Professional Fees: £4,000
Total Cost: £254,000
Cash Expense: £54,000
Financing Information
Deposit: 20%
Loan Value: £160,000
Deposit: £140,000
Interest Rate: 3%
Loan Repayment Length: 30 year
Monthly Repayment Value: £250
Now, using this fictitious investment and assumed financing agreement to carry out our analysis!
One of the cornerstones of financial analysis is the “Net Operating Income” or NOI. In short, NOI is the total income the property generates after all expenses, not including payments to service
debt. In mathematical terms, NOI is equivalent to the total income of the property minus the total expenses of the property:
NOI = Income – Expenses.
NOI is usually calculated on a monthly basis using monthly income and expense data, which can then be converted to annual data by multiplying by 12.
Assessing Property Income
Gross income is the total income generated from the property, including tenant rent and other income from things as energy subsidies, parking fees and any other income that the property generates on
a regular basis
From our example property, we have a single property renting for between £525-650 per month. I’m pretty conservative with valuations, so usually take the bottom part of the range and reduce it by
another 5%. In this case, that leaves me with a monthly income of £498.75, or a yearly income of £5985. The property generates another £200 a month from a spare parking space, boosting monthly income
by £2400 a year, taking annual income to £8385.
Because the majority of the income will be derived from tenant rent, it’s important that we factor in rent lost from vacant periods. Most agents will be able to advise on what this rate is likely to
be, but don’t forget to err on the side of caution when determining your expected vacancy rate. To asses total income on the property, I might factor in another eight weeks vacancy on the property,
reducing annual income to £7387.50.
Assessing Expenses
Now let’s calculate total expenses for the property. In general, expenses break down into the following items:
• Property Taxes
• Insurance
• Maintenance (estimated based on age and condition of property)
• Management (if you choose to employ a professional property manager)
• Advertising (to advertise for tenants)
• Landscaping (if you hire a professional landscaping company)
• Utilities (if any portion of the utilities is paid by the owner)
• Other (anything else we may have missed above)
I’ve pulled a couple of rough figures out for this section;
Insurance: £600
Property Management (5% of rent): £299.25
Maintenance costs: £2000
Advertising: £250
Utilities: £500
Total: £3649.25
Calculating NOI
Now that we have our total annual income and expenses for the property, we can calculate NOI using the formula I explained earlier:
NOI = Income – Expenses.
Using the values we just calculated, this gives us £7387.50 -£3649.25 = £3737.75 (meaning that the property generates £3737.75 a year). While NOI doesn’t give you the whole picture, it is our
starting point for calculating most of the metrics for more in-depth analysis.
Common Performance Measurements
Now we’ve figured out our NOI, let’s have a look at two other key metrics; cash flow and the rate of return. If the NOI is the total income excluding loan costs, you might be wondering why, as these
costs are definitely going to affect the viability of the investment.
Cash Flow
The reason we don’t include debt service in the NOI calculation is that NOI dictates what level of income the property will produce independent of the investor’s financing model. Depending on this
model, the debt repayment schedule will vary, and if we included that schedule in the NOI, it would only be valid in the context of that financing arrangement
Because different investors will have different financing models, it’s important to have a metric that’s specific to the property rather than the individual investor.
Of course, is the NOI stacks up favourably, you’ll want to move onto something which takes into account your financing arrangements, which is why we have the Cash Flow metric. Cash Flow for an
investment property is essentially just the NOI adjusted to include the cost of debt repayments. Expressed as an equation, it looks like this;
Cash Flow = NOI – Debt Repayments
This output will be the total profit seen at the end of a year from the investment. Following this through, the more you’re borrowing to fund the investment and the higher the interest rate, the
smaller your Cash Flow will be. If you buy the property outright, your Cash Flow will be equal to the NOI, which can also be referred to as the Maximum Cash Flow from the property.
If you recall my theoretical financing model, our monthly debt repayments totalled £250 a month and our annual debt repayment would therefore be £3000. For this property, our Cash Flow would
therefore be £3737.75 – £3000 = £737.75. That works out a little over £60 a month; not exactly money to retire on.
Hold on though! If you can reduce the debt load by paying more upfront, surely this will improve the Cash Flow metric; that’ll make things more attractive! Actually, as I’m about to demonstrate, this
isn’t a given.
Rates of Return
Cash Flow isn’t the only important factor when it comes to analysing an investment property. Even more important than cash flow is rate of return (also known as return on investment or ROI). Think of
ROI as the amount of Cash Flow you receive relative to the amount of money the investment cost you. Mathematically, this is represented as:
ROI = Cash Flow / Investment Value
Obviously, ROI is going to be higher when one or both of the following is true: your Cash Flow is high relative to your initial investment, or your investment is small relative to your Cash Flow. You
can see that from the equation above, but it should also be obvious when you think about it: if you can make a lot of money from a small investment, things are pretty peachy!
So what is a reasonable ROI? Well, a good starting point is your ROI on other investments. For example, you might have a cash ISA paying 1%, a collection of bonds paying 2-4% and a stock portfolio
yielding 5%. The cash and bonds are totally secure (although potentially losing money to inflation) and your stock portfolio fluctuates in value (representing potential loss in the event of
liquidation), so let’s take a mid-point of 3% where every £100 you ‘invest’ gives you £3 at the end of the year.
Capitalisation Rate (Cap Rate)
Just like we have a key income value (NOI) that is completely independent of the details of a financing model, we also have a key ROI value that is also independent of the investor and details of
their financing. This value is known as the “Capitalisation Rate,” or “Cap Rate.”
The Cap Rate is calculated as follows:
Cap Rate = NOI / Property Price.
If there is a single number that is most important when doing a financial analysis of a rental property, the Cap Rate may be it. Because the Cap Rate is independent of the buyer and their financing,
it is the most pure indication of the return a property will generate.
Returning to our fictional property, let’s plug in out numbers;
Cap Rate = NOI / Property Price
£3737.75 / £200,000 = 1.86%
Another way to think about the Cap Rate is that it is the ROI you would receive if you paid entirely in cash for a property. Unlike Cash Flow where the value is maximised by paying entirely in cash,
the Cap Rate is not necessarily the highest return you’ll get on a property because it assume that the investment value is the full value of the property and the value of ROI goes up as the
investment amount falls.
So what is a reasonable Cap Rate? This is a bit like asking the length of a piece of string – you’ll have to look at returns on other properties in the area and compare the Cap Rate to other
investments you could make.
Cash-on-Cash Return (COC)
Just like there are multiple measures of income – NOI (financing independent income) and Cash Flow (financing dependent income) – there are also multiple measures of return. As we’ve discussed, the
financing independent rate of return (the theoretical return on a fully paid property) is the Cap Rate, and of course there is the real (not theoretical) rate of return as well.
This is called the Cash-on-Cash (COC) return, because it is directly related to the amount of cash you put down on the investment.
For example, we discussed that if you took £100 and put it in a bond paying 3%, you’d receive £3 per year, or 3% ROI. The COC is the equivalent measure of how much return you would make if you put
that 100 into the property. COC is calculated as follows: COC = Cash Flow / Property Value.
In our example, the annual Cash Flow was £737.75 and the investment of cash that we had to pay upfront on the property was £54,000 (this included the deposit, the improvements, and any professional
fees). So, our COC is: COC = Cash Flow / Investment Basis = 737 / = 1.36%
As this return is directly comparable to the return available from other investments, we can see that we are getting a worse return than either a bond or our stock portfolio. In addition to this, we
have an enormous pot of debt we’re liable for (£160,000 upon purchasing the property), and significant outgoings during periods of non-occupancy.
While it’s completely up to you what rate of return you need to purchase a property, it should be obvious that if you’re getting less 6% return, it’s probably not worth your investment (personally,
I’d rather take that money and invest in the stock market where I can do a lot less work to get similar, if not better returns).
But, before you run off and make any final decisions based on COC, consider that the Cash Flow you make on a property isn’t the only thing that affects your bottom line…
Total ROI
In addition to Cash Flow, there are several other key financial considerations that affect a property’s performance. Specifically;
• Capital Appreciation (you may not be able to predict this and certainly shouldn’t assume it – but it can help to float a less appealing opportunity)
• Equity Accrued (remember that your tenants are paying off your property for you)
The difference between COC and Total ROI is that COC only considers the financial impact of Cash Flow on your return, while Total ROI considers all the factors that affect your bottom line. Total ROI
is calculated as follows: Total ROI = Total Return / Property Value, where “Total Return” is made up of the components we discussed (Cash Flow, Equity Accrual and Capital Appreciation). Let’s use the
following for our Total Return calculation:
• Let’s assume we could expect 1.5% capital appreciation on the value of the property this year, based on the improvements that we would do upon purchase (1.5% capital appreciation is £3000)
• We can calculate that the equity accrued in the first year of the mortgage is another £3000)
Taking these values into account, the Total Return of the property for this year would be: £6737.75 (£737.75 + £3000 + £3000). Therefore the Total ROI would be:
Total ROI = Total Return / Property Value
£6737.75 / £200,000 = 3.37%
Still not fantastic, but better!
Financial Analysis Summary
We now have all the data to assess the value of this property, but keep in mind that our assessment is only for the first year of ownership of this property. In subsequent years, accrued equity will
increase, expenses and rental rates will fluctuate and a whole range of other factors will contribute to the viability of the investment.
While you can’t predict the future, it’s generally prudent to extend your analysis forward a few years, using trend data that indicates the direction of the market and other variables.
These metrics are subject to to variation. Each investor has their preference, but hopefully they’ll give you a good starting point for carrying out real estate investment analysis. | {"url":"https://theethicalentrepreneur.com/real-estate-investment-analysis/","timestamp":"2024-11-03T07:32:46Z","content_type":"text/html","content_length":"126236","record_id":"<urn:uuid:238b8b3c-3b58-4fcf-a186-b4eed35a296d>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00729.warc.gz"} |
What is a Hamiltonian cycle problem? | Hire Someone To Take Exam For Me
What is a Hamiltonian cycle problem? I was looking into how Hamiltonians like that can be tackled by forcing the cycle to complete at the end of every cycle, but the way I implemented this in C++ is
to use the quotient by itself. How can I implement the quotient by itself in a way without entering into the cycle in the first place? A: In C++, multiplication can be done by using the quotient
operation, but multiplication can be done by using the multiplication by; and multiplication by an arbitrary operation such as the inverse of. For context, this shows how c would be. The best you can
do is pick out the operation of first call in your case and multiply that operation by this: #define N OFFSET 100 // We’ll take things the reverse direction here, but we don’t have to work out the
operations to pick out the base case which is what you’re thinking about. e.g. this doesn’t work here. Instead, use the following instead: #define M32M32_GET_IMPM(x) { x >>= 4; x >>= 1; } // Keep
things within the range we are using to distinguish between the operations, so we can do things the same when using the M32M32_GET_IMPM part. // Here’s where we do the multiplication with some
multiplication operators in the range: … int x = N-1 + M32M32_GET_IMPM(4); x >>= M16M32M32M32M32M32; What is a Hamiltonian cycle problem? Many of us are having our day at the weekend and are having
difficulty finding ideas! We think one day everyone can focus on one topic, but the next day, is going to be different. What is Hamiltonian Cycle problem? Is Hamiltonian cycle problem solved? The
answer is yes, but there is a solution which everybody can understand. I have read David Greene’s papers on Hamiltonian path complexity that showed that there are natural solutions to open question
and some difficult problems can be solved with others. Alice and Bob give a general algorithm to find an invariant time of a given polynomial. Bob starts an algorithm with a polynomial in $\log 2$ in
order to find a solution, that is, Alice’s first step is to find 6 steps, for a rational number $r$, with $\alpha_r$ and $x_r$ going to the appropriate point. The algorithm takes a random variable
$B$ with the value $2\alpha_r$. Then Alice and Bob find the value $b\ge 0$ such that their algorithm gives the expression ‘The algorithm does not find $0$’ as a whole, and they take the values $b$
and $b-2$ as their constants. Then ‘The algorithm computes the point $b$’ – if $b$ is odd and $b+2<2\alpha_r$, they compute $b-2,$ so the remaining cases with $b\ge 2$ are $b\ge 2\alpha_{r-1},$ or
$b-2,$ so the remaining cases with $b\ge 2$ are $b\ge \alpha_{r-1},$ otherwise. Then redirection if: Alice has had the value of $1/4$ *and* Bob has the value of $3/4$.
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Alice and Bob generate a random path and a path with start and end points go from the given path and end points from the one without generating a random path. Alice and Bob return only a fixed number
of points from the path and end points. Then Alice and Bob can choose the same point from the path as soon as they reach another point in itself. She can take as many points as she wants. Bob places
herself in a position $u$ such that each of the 2 points has the given value. In this state she gives Bob the edge path. Alice and Bob do the calculation all the time. Then they take turns and repeat
the operation for a million times. The Hamiltonian cycle problem is a quite natural extension to the quantum situation. Many people, especially the many end users like mine, are just taking turns on
a computer that is solving this quantum problem, using the classical graph method, the graph algorithm on any computer, using quantum mechanics. Key words: Hamiltonian cycle problem The Hamiltonian
cycle problem is a much bigger problem than the Hamiltonian paths problem. There is a big difference between the two questions here and there. The Hamiltonian cycles problem is now a standard example
of a real related problem have a peek at these guys the Hamiltonian and paths problem, but it would also be interesting to find new concrete details about how just a quick overview of the paper on
the Hamiltonian cycle problem can be carried out. Let F be a rational function on a set X. V is a closed real number, and such that for all ξ ≤ δ, A[V]≤A[V]+1 and A>0, there exists a positive integer
q such that for all p = (0, 1) − δ p and A[V]≤-A[V], V[p] ∈ V. A[V]≤1 if and only if there exists a C-function ΣξWhat is a Hamiltonian cycle problem? Is it defined as a function of any fixed quantity
that is given by an integer-valued constant satisfying some property? I thought I found this problem here Wikipedia, but thought I’d ask here. I’ve also tried to think of the problem for the case $n=
1$ and $N=10$, but was unsuccessful. How to find many elements in $R$ on an arbitrary dimension? You could look at the definition of $Q^{\text{cubic}}_{\text{aQR}}$ and leave it to someone who works
with it to determine what one would call “truncated” generators. My intuition suggests that if $f(0) \ne 0$, there is a solution for $(I_{n})$ depending only on the genus. My hypothesis is that $f$
is the generator with $Q^{\text{cubic}}_{\text{aQR}} = 1$.
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Additionally, I think that if we take $R = C^{\ast} \mathbb S^{n}$, then $R = \mathbb S^n$ – the sum of the sets of roots. What’s more difficult to do is compute how $R$ (or $R$) will define and why,
generally, $f$ does not have this property–e.g. without the use of a linear transformation. We have already shown that $X \cap C = X \cap \mathbb {C} \ne \cap_{\text{aQR}} C$. If we write $gQ_S^{\
text{cubic}} = Q_S^{\text{cubic}} g$ for $Q_S^{\text{cubic}}$ the only generator of $X \cap \mathbb {C} \ne \cap_{\text{aQR}} C$ (so we have), it appears in the definition | {"url":"https://cracktheexamination.com/what-is-a-hamiltonian-cycle-problem","timestamp":"2024-11-06T17:53:16Z","content_type":"text/html","content_length":"168946","record_id":"<urn:uuid:67054ed7-c250-4bf8-8416-37205a3df57d>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00357.warc.gz"} |
Blum Blum Shub is a PRNG algorithm published in 1986.
The algorithm is very short and simple. Starting from the seed, the next state can be computed by passing the current state through the following formula.
f(x) = x² mod M
In this formula, M is the product of p and q, two large primes.
The complexity in this algorithm is hidden in the parameters; the seed and the modulus M. In order to have a long cycle length and fulfill its security promises, Blum Blum Shub has a few constraints
on its parameters. | {"url":"https://msakg.com/tag/python","timestamp":"2024-11-14T07:01:30Z","content_type":"text/html","content_length":"14968","record_id":"<urn:uuid:238defa8-5de2-49f8-9b3a-1b6de3a02504>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00374.warc.gz"} |
Air Conditioner Size Calculation - Certified Calculator
Air Conditioner Size Calculation
Introduction: Selecting the right-sized air conditioner is crucial for maintaining a comfortable indoor environment and optimizing energy efficiency. The Air Conditioner Size Calculation tool is
designed to assist in determining the recommended air conditioner size based on specific parameters. In this article, we’ll explore the importance of proper AC sizing, the formula behind the
calculator, and how to use it effectively for optimal cooling.
Formula: The calculator utilizes a formula that considers various factors to determine the recommended air conditioner size in British Thermal Units per hour (BTU/h). The factors include:
• Total Square Feet: The total area of the space in square feet.
• Ceiling Height (feet): The height of the ceilings in the space.
• Window Area (square feet): The total area of windows in the space.
• Door Area (square feet): The total area of doors in the space.
The formula calculates the recommended air conditioner size based on these parameters, ensuring efficient cooling for the entire space.
How to Use:
1. Enter the total square feet of the space.
2. Input the ceiling height in feet.
3. Specify the total window area in square feet.
4. Specify the total door area in square feet.
5. Click the “Calculate” button to obtain the recommended air conditioner size in BTU/h for the space.
Example: For a space with a total square feet of 1500, a ceiling height of 8 feet, window area of 200 square feet, and door area of 30 square feet, enter these values, click “Calculate,” and the
result will indicate the recommended air conditioner size.
1. Q: Why is it important to choose the right-sized air conditioner for a space? A: Proper AC sizing ensures optimal cooling performance, energy efficiency, and cost savings.
2. Q: How does ceiling height impact air conditioner sizing? A: Higher ceilings may require a larger air conditioner to effectively cool the increased volume of the space.
3. Q: Should I consider both window and door area in AC sizing for a space? A: Yes, windows and doors contribute to heat gain or loss, and their areas help determine the overall cooling needs.
4. Q: Can I use the calculator for spaces with irregular shapes? A: The calculator provides a general estimate for rectangular spaces. For irregular shapes, consider an average of the dimensions.
5. Q: What happens if I choose an air conditioner with too high a BTU rating for a space? A: An oversized air conditioner may lead to inefficient cooling, higher energy bills, and discomfort due to
frequent cycling.
Conclusion: The Air Conditioner Size Calculation tool serves as a valuable resource for homeowners to ensure the selection of an appropriately sized air conditioner for their spaces. Proper AC sizing
contributes to efficient cooling, energy savings, and a consistently comfortable indoor environment. Consideration of space parameters is essential for accurate calculations and optimal cooling
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Isaac Newton's Education: A Journey Through Time and Knowledge
Isaac Newton: The Man Who Defined Gravity and Calculus
Quick Info
• Born: 4 January 1643, Woolsthorpe, Lincolnshire, England
• Died: 31 March 1727, London, England
• Legacy: The greatest English mathematician of his era, a foundational contributor to calculus, optics, and gravitation.
Welcome, curious readers! Today, we're time-travelling to the 17th century to unravel the life of a legend - Sir Isaac Newton. Known for his profound impact on science and mathematics, Newton's story
is not just about apples and gravity. It's a tale of a farm boy who became a knight! Join us on this fascinating journey, and for any burning questions, our Facebook page awaits your curiosity.
Biography: The Phases of Newton's Life
Boyhood Days (1643-1669)
Early Struggles and Education
Birth and Family Background: Isaac Newton entered the world on 4 January 1643 in Woolsthorpe, England. Tragically, his father, also named Isaac Newton, had passed away just three months before his
birth. This left young Isaac without the guidance of his father, and he was raised by his grandmother, Margery Ayscough. The backdrop of his boyhood was one marked by familial struggles and emotional
challenges, setting the stage for a challenging upbringing.
Education: Despite the hardships he faced, Isaac Newton displayed an innate curiosity and an early inclination towards learning. His education began to take shape when he was sent to The King's
School in Grantham. It was at this pivotal juncture that his intellectual journey commenced, as he embarked on a path that would ultimately lead him to become one of history's most influential
This phase of his life, filled with early struggles and educational pursuits, laid the foundation for his future accomplishments and the remarkable contributions he would make to mathematics,
physics, and astronomy. Young Isaac's determination and innate brilliance were evident even in these formative years, setting the stage for the remarkable scientific discoveries that would define his
Lucasian Professorship (1669-1687)
A Period of Profound Discoveries
Cambridge and Professorship: Isaac Newton's academic journey brought him to Trinity College, Cambridge, where his intellectual prowess continued to shine. In the year 1669, he assumed the prestigious
role of Lucasian professor, a position that would prove pivotal in shaping his career and facilitating some of his most groundbreaking work.
Key Contributions:
Mathematics: During his tenure as Lucasian professor, Newton made extraordinary contributions to mathematics. He laid the very foundation for differential and integral calculus, revolutionizing the
way we understand and work with functions and their rates of change. His work in this area laid the groundwork for countless future mathematical developments.
Optics: Newton's inquisitive mind led him to challenge existing theories in the field of optics. His experiments and studies in this domain resulted in significant advancements in the study of light,
including his famous work on the nature of colour and the decomposition of white light into a spectrum of colours.
Physics: Newton's years as a Lucasian professor also saw him make groundbreaking contributions to the field of physics. His three laws of motion, known as Newton's laws, provided a new and
comprehensive understanding of the physical world's mechanics. Moreover, his concept of universal gravitation, which explained the force of attraction between objects, was a truly revolutionary
These remarkable achievements during his professorship established Isaac Newton as one of the greatest scientific minds in history and left an indelible mark on the fields of mathematics, optics, and
Government Official in London (1687-1727)
A Shift from Academia to Administration
Role in Government: In the later years of his life, Isaac Newton transitioned from the academic world to a role as a government official in London. His expertise and intellect made him an invaluable
asset in administrative positions. Notably, he took on a significant role at the Royal Mint, where he played a vital part in various aspects of currency management.
Continued Impact: Although he withdrew from active mathematical research during this phase of his life, Newton's influence remained enduring in scientific circles. His earlier discoveries continued
to shape the foundations of physics and mathematics, and his insights into optics and gravitation continued to be studied and revered. Newton's legacy extended beyond academia, showcasing his
remarkable versatility and profound impact on both the academic and administrative realms. His work at the Royal Mint, particularly in the areas of recoinage and anti-counterfeiting measures,
demonstrated his commitment to practical problem-solving and further solidified his reputation as an influential figure in the scientific community and the broader society.
Government Official in London (1687-1727)
A Shift from Academia to Administration
In the later phase of his life, Isaac Newton embarked on a transition from the world of academia to a distinguished role as a government official in the vibrant city of London. This new chapter in
his career saw him applying his intellectual prowess and problem-solving skills to practical matters of administration, leaving a lasting impact on the functioning of the government and society as a
Role in Government:
Newton's role as a government official was marked by his notable contributions to the Royal Mint, a critical institution responsible for coinage and currency management. During this period, he took
on several key responsibilities:
1. Recoinage: One of the significant challenges of the time was the need to recoin the currency. Newton played a pivotal role in overseeing and managing this complex process, ensuring the efficient
replacement of old, worn-out coins with new ones.
2. Anti-Counterfeiting Measures: Counterfeiting was a pressing issue in the 17th and 18th centuries, threatening the stability of the economy. Newton's expertise in mathematics and his keen
analytical mind were instrumental in devising innovative anti-counterfeiting measures. His efforts helped to safeguard the integrity of the currency and protect the economy from fraudulent
Continued Impact:
Despite his shift away from active mathematical research, Isaac Newton's enduring influence continued to reverberate within scientific circles and beyond:
1. Legacy of Discoveries: Newton's earlier groundbreaking discoveries in mathematics, physics, and optics remained foundational to these fields. His laws of motion and theory of universal
gravitation, along with his work on optics, continued to be studied and revered, shaping the course of future scientific inquiry.
2. Intellectual Authority: Newton's reputation as a preeminent scientific mind persisted, and his insights continued to hold immense intellectual authority. His contributions set the standard for
scientific inquiry and provided a template for rigorous investigation and experimentation.
3. Practical Problem-Solving: Newton's tenure as a government official exemplified his commitment to practical problem-solving and the application of scientific principles to real-world challenges.
His work at the Royal Mint demonstrated the value of scientific expertise in addressing complex issues outside the traditional realm of academia.
In conclusion, Isaac Newton's transition from academia to government service during his later years exemplified his adaptability and his dedication to making a tangible impact on society. His
contributions to recoinage and anti-counterfeiting measures, combined with his enduring scientific legacy, solidified his status as an influential
The Principia and Beyond: A Legacy Cemented
Isaac Newton's legacy is indelibly marked by his profound contributions to science, culminating in his masterpiece, the "Philosophiae Naturalis Principia Mathematica," which was published in 1687.
This seminal work transformed our understanding of the universe and solidified Newton's place as one of history's greatest scientific minds.
"Philosophiae Naturalis Principia Mathematica" (1687):
Newton's magnum opus, commonly referred to as the Principia, stands as a monumental achievement in the annals of science. In this groundbreaking work, he elucidated the fundamental laws of motion and
the theory of universal gravitation. The Principia provided a comprehensive and mathematically rigorous framework for understanding the behaviour of objects in motion and the force that governs
celestial bodies. It was a monumental leap forward in the scientific understanding of the physical world.
Newton's later years were marked by a bitter dispute with the German mathematician and philosopher Gottfried Wilhelm Leibniz. The controversy centered around the invention of calculus, with both
Newton and Leibniz independently developing this mathematical discipline. The feud between these two luminaries, each claiming priority in the discovery, became a contentious issue in the history of
mathematics. Despite this controversy, both Newton and Leibniz made immense contributions to the development of calculus.
Isaac Newton's scientific eminence earned him numerous accolades and honours during his lifetime. His election as the president of the Royal Society and his knighthood by Queen Anne underscored his
exceptional standing in the scientific community and society at large. These recognitions served as a testament to his profound influence and contributions to the advancement of human knowledge.
The life journey of Sir Isaac Newton, from his humble beginnings on a Lincolnshire farm to his academic pursuits at Cambridge and his pivotal role in London's courts, exemplifies the boundless
potential of the human spirit. His enduring contributions to calculus, optics, and physics continue to form the bedrock of modern science and technology, shaping the way we perceive and interact with
the world.
Newton's story is not merely a narrative of scientific achievements; it is a story of resilience, insatiable curiosity, and an unwavering commitment to the relentless pursuit of knowledge. Every time
we witness an object in motion or marvel at the beauty of a rainbow, we are reminded that Newton's legacy lives on as a beacon of inspiration for generations to come.
For further insights into the lives of great scientists and their contributions to the world, we invite you to visit our Facebook page. | {"url":"https://learningenglishfree.co.uk/index.php/blog/isaac-newtons-education-a-journey-through-time-and-knowledge","timestamp":"2024-11-04T13:51:47Z","content_type":"text/html","content_length":"28922","record_id":"<urn:uuid:2d96cf04-d7e2-4501-a1f5-f8425275aa69>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00066.warc.gz"} |
Generic function for the computation of the Kolmogorov
KolmogorovDist {distrEx} R Documentation
Generic function for the computation of the Kolmogorov distance of two distributions
Generic function for the computation of the Kolmogorov distance d_\kappa of two distributions P and Q where the distributions are defined on a finite-dimensional Euclidean space (\R^m,{\cal B}^m)
with {\cal B}^m the Borel-\sigma-algebra on R^m. The Kolmogorov distance is defined as
d_\kappa(P,Q)=\sup\{|P(\{y\in\R^m\,|\,y\le x\})-Q(\{y\in\R^m\,|\,y\le x\})| | x\in\R^m\}
where \le is coordinatewise on \R^m.
KolmogorovDist(e1, e2, ...)
## S4 method for signature 'AbscontDistribution,AbscontDistribution'
KolmogorovDist(e1,e2, ...)
## S4 method for signature 'AbscontDistribution,DiscreteDistribution'
KolmogorovDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,AbscontDistribution'
KolmogorovDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,DiscreteDistribution'
KolmogorovDist(e1,e2, ...)
## S4 method for signature 'numeric,UnivariateDistribution'
KolmogorovDist(e1, e2, ...)
## S4 method for signature 'UnivariateDistribution,numeric'
KolmogorovDist(e1, e2, ...)
## S4 method for signature 'AcDcLcDistribution,AcDcLcDistribution'
KolmogorovDist(e1, e2, ...)
e1 object of class "Distribution" or class "numeric"
e2 object of class "Distribution" or class "numeric"
... further arguments to be used in particular methods (not in package distrEx)
Kolmogorov distance of e1 and e2
e1 = "AbscontDistribution", e2 = "AbscontDistribution":
Kolmogorov distance of two absolutely continuous univariate distributions which is computed using a union of a (pseudo-)random and a deterministic grid.
e1 = "DiscreteDistribution", e2 = "DiscreteDistribution":
Kolmogorov distance of two discrete univariate distributions. The distance is attained at some point of the union of the supports of e1 and e2.
e1 = "AbscontDistribution", e2 = "DiscreteDistribution":
Kolmogorov distance of absolutely continuous and discrete univariate distributions. It is computed using a union of a (pseudo-)random and a deterministic grid in combination with the support of
e1 = "DiscreteDistribution", e2 = "AbscontDistribution":
Kolmogorov distance of discrete and absolutely continuous univariate distributions. It is computed using a union of a (pseudo-)random and a deterministic grid in combination with the support of
e1 = "numeric", e2 = "UnivariateDistribution":
Kolmogorov distance between (empirical) data and a univariate distribution. The computation is based on ks.test.
e1 = "UnivariateDistribution", e2 = "numeric":
Kolmogorov distance between (empirical) data and a univariate distribution. The computation is based on ks.test.
e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution":
Kolmogorov distance of mixed discrete and absolutely continuous univariate distributions. It is computed using a union of the discrete part, a (pseudo-)random and a deterministic grid in
combination with the support of e1.
Matthias Kohl Matthias.Kohl@stamats.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de
Huber, P.J. (1981) Robust Statistics. New York: Wiley.
Rieder, H. (1994) Robust Asymptotic Statistics. New York: Springer.
See Also
ContaminationSize, TotalVarDist, HellingerDist, Distribution-class
KolmogorovDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
KolmogorovDist(Norm(), Td(10))
KolmogorovDist(Norm(mean = 50, sd = sqrt(25)), Binom(size = 100))
KolmogorovDist(Pois(10), Binom(size = 20))
KolmogorovDist(Norm(), rnorm(100))
KolmogorovDist((rbinom(50, size = 20, prob = 0.5)-10)/sqrt(5), Norm())
KolmogorovDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5))
version 2.9.2 | {"url":"https://search.r-project.org/CRAN/refmans/distrEx/html/KolmogorovDist.html","timestamp":"2024-11-12T07:26:57Z","content_type":"text/html","content_length":"6925","record_id":"<urn:uuid:1af48425-b8a4-43b9-b1e4-af5331e13e01>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00669.warc.gz"} |
What is Formulation Of Linear Programming-Maximization Case? definition and meaning - Business Jargons
Definition: Linear programming refers to choosing the best alternative from the available alternatives, whose objective function and constraint function can be expressed as linear mathematical
Maximization Case: Let’s understand the maximization case with the help of a problem. Suppose a firm produces two products A and B. For producing the each unit of product A, 4 Kg of Raw material and
6 labor hours are required. While, for the production of each unit of product B, 4 kg of raw material and 5 labor hours is required. The total availability of raw material and labor hours is 60 Kg
and 90 Hours respectively (per week). The unit price of Product A is Rs 35 and of product, B is Rs 40.
This problem can be converted into linear programming problem to determine how many units of each product should be produced per week to have the maximum profit. Firstly, the objective function is to
be formulated. Suppose x[1] and x[2 ]are units produced per week of product A and B respectively. The sale of product A and product B yields Rs 35 and Rs 40 respectively. The total profit will be
equal to
Z = 35x[1]+ 40x[2] (objective function)
Since the raw material and labor is in limited supply the mathematical relationship that explains this limitation is called inequality. Therefore, the inequality equations will be as follows:
Product A requires 4 kg of raw material and product B also requires 4 Kg of Raw material; thus, total consumption is 4x[1]+4x[2], which cannot exceed the total availability of 60 kg. Thus, this
constraint can be expressed as:
4x[1 ]+ 4x[2 ]≤ 60
Similarly, the second constraint equation will be:
6x[1 ]+ 5x[2 ]≤ 90
Where 6 hours and 5hours of labor is required for the production of each unit of product A and B respectively, but cannot exceed the total availability of 90 hours.
Thus, the linear programming problem will be:
Maximize Z = 35x[1]+ 40x[2] (profit)
Subject to:
4x[1 ]+ 4x[2 ]≤ 60 (raw material constraint)
6x[1 ]+ 5x[2 ]≤ 90 (labor hours constraint)
x[1], x[2] ≥ 0 (Non-negativity restriction)
Note: It is to be noted that “≤” (less than equal to) sign is used as the profit maximizing output may not fully utilize all the resources, and some may be left unused. And the non-negativity
condition is used since the x[1] and x[2] are a number of units produced and cannot have negative values. | {"url":"https://businessjargons.com/formulation-of-linear-programming-maximization-case.html","timestamp":"2024-11-04T04:24:11Z","content_type":"text/html","content_length":"47959","record_id":"<urn:uuid:a25ae5bc-bcbb-444c-aef3-eafc263d521e>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00113.warc.gz"} |
How I wish I'd always taught... VAMPIRE NUMBERS.
We have already written about the way maths cateogrises numbers including prime numbers and vampire numbers but there are many different types of numbers which evade curricula around the world.
Today, we explore some of the cousins and distant relations that make up the number system: vampire numbers, narcissistic numbers, cake numbers, happy numbers, evil numbers, pronic numbers and
repunit numbers.
Vampire numbers are a special type of number that can be expressed as the product of two other numbers, where the digits of those two numbers are rearranged and put together to form the original
To better understand vampire numbers, let's look at an example:
In this example, we see that the digits of 21 and 60 can be rearranged and multiplied to give us 1260. This makes 1260 a vampire number.
The term "vampire number" is used because they can be seen as numbers that "suck the life" out of the two numbers that make them up.
But not all numbers can be vampire numbers. For a number to be considered a vampire number, it must have an even number of digits, and the two numbers that make it up must have the same number of
Let's look at a few more examples of vampire numbers:
1. 1827 = 21 x 87
2. 102510 = 201 x 510
3. 104260 = 260 x 401
4. 125460 = 204 x 615
As you can see from the examples above, vampire numbers can be quite large and have a lot of factors. In fact, some vampire numbers can have multiple pairs of factors that can be rearranged to form
the original number.
One interesting thing about vampire numbers is that they are quite rare. In fact, there are only a few hundred known vampire numbers in existence. This makes them a fascinating topic for
mathematicians and number enthusiasts alike.
Narcissistic numbers are a special type of number in which the sum of each digit is raised to the power of the number of digits in the number is equal to the number itself. For example, 153 is a
narcissistic number because 1³ + 5³ + 3³ = 153.
Here are a few more examples of narcissistic numbers:
1. 1 is a narcissistic number because 1³ = 1.
2. 8208 is a narcissistic number because 84 + 24 + 04 + 84 = 8208.
3. 9474 is a narcissistic number because 94 + 44 + 74 + 44 = 9474.
Cake numbers are a type of mathematical concept that is easy to understand and fun to play with. They are called cake numbers because they are similar to slices of a cake. Essentially, a cake number
is a number that can be divided into n parts, where each part has the same integer value.
Here are a few examples of cake numbers:
1. 12 is a cake number because it can be divided into 2, 3, 4, or 6 equal parts (2 x 6, 3 x 4, 4 x 3, or 6 x 2).
2. 16 is a cake number because it can be divided into 2, 4, or 8 equal parts (2 x 8, 4 x 4, or 8 x 2).
3. 24 is a cake number because it can be divided into 2, 3, 4, 6, or 8 equal parts (2 x 12, 3 x 8, 4 x 6, 6 x 4, or 8 x 3).
As you can see, cake numbers are fun and easy to work with. They can be used to teach children about division and fractions, and they can also be used in more advanced math problems. For example,
cake numbers can be used to solve problems in computer science, where it is important to divide data into equal parts to ensure efficient processing.
Happy numbers are a type of number that is easy to understand and fun to work with. To determine if a number is a happy number, you take the sum of the squares of its digits, and then repeat this
process with the sum of the squares of the digits until you get a result of 1. If you end up with 1, the number is a happy number.
Here are a few examples of happy numbers:
1. 7 is a happy number because 7² = 49 and 4² + 9² = 97, and 9² + 7² = 130, and 1² + 3² + 0² = 10, and 1² + 0² = 1.
2. 19 is a happy number because 1² + 9² = 82, and 8² + 2² = 68, and 6² + 8² = 100, and 1² + 0² + 0² = 1.
3. 139 is a happy number because 1² + 3² + 9² = 91, and 9² + 1² = 82, and 8² + 2² = 68, and 6² + 8² = 100, and 1² + 0² + 0² = 1.
As you can see, happy numbers are fun to work with because you get to repeat the process of squaring digits until you get a result of 1. They are named happy numbers because the result of the process
is always 1, which is considered a happy outcome.
To determine if a number is an evil number, you count the number of 1's in its binary representation (its binary digits), and if the count is even, then the number is evil. If the count is odd, then
the number is called an odious number.
Here are a few examples of evil numbers:
1. 3 is an evil number because its binary representation is 11, which has 2 1's, an even number of 1's.
2. 6 is an evil number because its binary representation is 110, which has 2 1's, an even number of 1's.
3. 10 is an evil number because its binary representation is 1010, which has 2 1's, an even number of 1's.
As you can see, evil numbers are named as such because their binary representation has an even number of 1's. Evil numbers can be used in a variety of math problems, including programming and
computer science. They are a great way to introduce children to the concept of binary numbers and to help them understand how binary numbers work.
To determine if a number is a pronic number, you check if it can be expressed as the product of two consecutive integers. Here are a few examples of pronic numbers:
1. 2 is a pronic number because it can be expressed as 1 × 2.
2. 12 is a pronic number because it can be expressed as 3 × 4.
3. 42 is a pronic number because it can be expressed as 6 × 7.
As you can see, pronic numbers are named as such because they are the product of two consecutive integers.
To determine if a number is a repunit number, you check if all its digits are 1's. Here are a few examples of repunit numbers:
1. 1 is a repunit number because it has one digit and that digit is 1.
2. 11 is a repunit number because it has two digits and both digits are 1's.
3. 111 is a repunit number because it has three digits and all three digits are 1's.
As you can see, repunit numbers are named as such because they are composed entirely of repeated 1's. Repunit numbers are used to study prime numbers and to find patterns in their distribution. | {"url":"https://www.mrbeeteach.com/post/how-i-wish-i-d-always-taught-types-of-numbers","timestamp":"2024-11-07T13:04:25Z","content_type":"text/html","content_length":"1050481","record_id":"<urn:uuid:3b1e23b3-2e7a-40e7-84f2-900304bfb5e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00600.warc.gz"} |
Lossy Channel Systems
Lossy Channel Systems consist of finite-state processes communicating over unbounded, but lossy, FIFO channels. Lossy channel systems can model many interesting systems, e.g. link protocols such as
the Alternating Bit Protocol and HDLC. These protocols and others are designed to operate correctly even in the case that the FIFO channels are faulty and may lose messages. The state space of a
lossy channel system is infinite due to the unboundedness of the channels.
• Algorithm for Checking Saftey Properties [AJ93, AJ96a, AKP97]
An early significant result was our demonstration that safety properties of lossy channel systems could be automatically verified. This was one of the first examples in the literature of decidability
of safety properties for a class of systems which is not essentially finite-state (in contrast e.g. to real-time automata). This result should be contrasted with the well-known fact that all
non-trivial verification problems are undecidable for systems with unbounded perfect FIFO channels.
• Undecidability of PTL, CTL, etc. [AJ94, AJ96b]
• Simulation and Bisimulation [AK95]
• On-the-fly Verification [ABJ98, AAB99, AABJ00]
• Probabilistic Lossy Channel Systems. [ABIJ00]
• Model Cecking Probabilistic Lossy Channel Systems. [ABM05, ABMSq06, ABMSa06]
We show that probabilistic lossy channel systems induce Markov chains which have finite attractors. An Attractor is a set of states which is reached with probability one from all other states. For
this class of Markov chains, we show that many qualitative problems, sush as wether a set of states is reached/repeatdly reached with probability one/zero, are decidable.
We also prove that for lossy channel systems exact quantitative analysis is a "hard" problem in the sens that we cannot express the measure of runs, satisfying a given simple LTL property, in Tarski
Algebra. Observe that this is possible for finite state Markov chains in general, and for few classes of infinite state Markov chains such as those induced by pushdown autoamata. Nevertheless, we
propose some algorithms to approximate these probabilities up to an arbitrarily small error.
In recent works, we propose techniques for solving quantitative expectation problems. Typical examples of such problems are: the expected length of a path starting in an initial state and reaching a
target set (expected execution time of a program) and the steady state distribution (the probability of being in a certain state in the long run).
You can find here a list of our published work. | {"url":"https://www2.it.uu.se/research/docs/fm/apv/lcs","timestamp":"2024-11-07T00:21:36Z","content_type":"text/html","content_length":"23330","record_id":"<urn:uuid:0d9de960-6c07-45cf-83fc-bcc0f6203eff>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00175.warc.gz"} |
pytorch3d.renderer.compositing.alpha_composite(pointsidx, alphas, pt_clds) Tensor[source]
Composite features within a z-buffer using alpha compositing. Given a z-buffer with corresponding features and weights, these values are accumulated according to their weights such that features
nearer in depth contribute more to the final feature than ones further away.
Concretely this means:
weighted_fs[b,c,i,j] = sum_k cum_alpha_k * features[c,pointsidx[b,k,i,j]] cum_alpha_k = alphas[b,k,i,j] * prod_l=0..k-1 (1 - alphas[b,l,i,j])
☆ pt_clds – Tensor of shape (N, C, P) giving the features of each point (can use RGB for example).
☆ alphas – float32 Tensor of shape (N, points_per_pixel, image_size, image_size) giving the weight of each point in the z-buffer. Values should be in the interval [0, 1].
☆ pointsidx – int32 Tensor of shape (N, points_per_pixel, image_size, image_size) giving the indices of the nearest points at each pixel, sorted in z-order. Concretely pointsidx[n, k, y, x]
= p means that features[n, :, p] is the feature of the kth closest point (along the z-direction) to pixel (y, x) in batch element n. This is weighted by alphas[n, k, y, x].
Combined features –
Tensor of shape (N, C, image_size, image_size)
giving the accumulated features at each point.
pytorch3d.renderer.compositing.norm_weighted_sum(pointsidx, alphas, pt_clds) Tensor[source]
Composite features within a z-buffer using normalized weighted sum. Given a z-buffer with corresponding features and weights, these values are accumulated according to their weights such that
depth is ignored; the weights are used to perform a weighted sum.
Concretely this means:
weighted_fs[b,c,i,j] =
sum_k alphas[b,k,i,j] * features[c,pointsidx[b,k,i,j]] / sum_k alphas[b,k,i,j]
☆ pt_clds – Packed feature tensor of shape (C, P) giving the features of each point (can use RGB for example).
☆ alphas – float32 Tensor of shape (N, points_per_pixel, image_size, image_size) giving the weight of each point in the z-buffer. Values should be in the interval [0, 1].
☆ pointsidx – int32 Tensor of shape (N, points_per_pixel, image_size, image_size) giving the indices of the nearest points at each pixel, sorted in z-order. Concretely pointsidx[n, k, y, x]
= p means that features[:, p] is the feature of the kth closest point (along the z-direction) to pixel (y, x) in batch element n. This is weighted by alphas[n, k, y, x].
Combined features –
Tensor of shape (N, C, image_size, image_size)
giving the accumulated features at each point.
pytorch3d.renderer.compositing.weighted_sum(pointsidx, alphas, pt_clds) Tensor[source]
Composite features within a z-buffer using normalized weighted sum.
☆ pt_clds – Packed Tensor of shape (C, P) giving the features of each point (can use RGB for example).
☆ alphas – float32 Tensor of shape (N, points_per_pixel, image_size, image_size) giving the weight of each point in the z-buffer. Values should be in the interval [0, 1].
☆ pointsidx – int32 Tensor of shape (N, points_per_pixel, image_size, image_size) giving the indices of the nearest points at each pixel, sorted in z-order. Concretely pointsidx[n, k, y, x]
= p means that features[:, p] is the feature of the kth closest point (along the z-direction) to pixel (y, x) in batch element n. This is weighted by alphas[n, k, y, x].
Combined features –
Tensor of shape (N, C, image_size, image_size)
giving the accumulated features at each point. | {"url":"https://pytorch3d.readthedocs.io/en/latest/modules/renderer/compositing.html","timestamp":"2024-11-03T21:32:21Z","content_type":"text/html","content_length":"18152","record_id":"<urn:uuid:bee66ac9-336c-41fb-a5d1-e0052c34220d>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00460.warc.gz"} |
Finding Angles In Right Triangles Worksheet - Angleworksheets.com
Solving For Angles In Right Triangles Worksheet – This article will discuss Angle Triangle Worksheets as well as the Angle Bisector Theorem. In addition, we’ll talk about Isosceles and Equilateral
triangles. You can use the search bar to locate the worksheet you are looking for if you aren’t sure. Angle Triangle Worksheet This Angle Triangle … Read more
Find Angles Triangle Worksheet
Find Angles Triangle Worksheet – If you have been struggling to learn how to find angles, there is no need to worry as there are many resources available for you to use. These worksheets will help to
understand the various concepts and increase your knowledge of angles. Students will be able to identify unknown angles … Read more
Solving For Angles In A Right Triangle Worksheet
Solving For Angles In A Right Triangle Worksheet – This article will discuss Angle Triangle Worksheets as well as the Angle Bisector Theorem. We’ll also discuss Equilateral triangles and Isosceles.
If you’re unsure of which worksheet you need, you can always use the search bar to find the exact worksheet you’re looking for. Angle Triangle … Read more
Calculate Angles In A Triangle Worksheet
Calculate Angles In A Triangle Worksheet – In this article, we’ll talk about Angle Triangle Worksheets and the Angle Bisector Theorem. We’ll also discuss Equilateral triangles and Isosceles. You can
use the search bar to locate the worksheet you are looking for if you aren’t sure. Angle Triangle Worksheet This Angle Triangle Worksheet teaches students … Read more
Finding Angles In A Right Triangle Worksheet
Finding Angles In A Right Triangle Worksheet – If you have been struggling to learn how to find angles, there is no need to worry as there are many resources available for you to use. These
worksheets will help you understand the different concepts and build your understanding of these angles. Using the vertex, arms, … Read more | {"url":"https://www.angleworksheets.com/tag/finding-angles-in-right-triangles-worksheet/","timestamp":"2024-11-15T03:01:49Z","content_type":"text/html","content_length":"74733","record_id":"<urn:uuid:80ec65c2-69b4-45f4-bf12-85785fa46e93>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00211.warc.gz"} |
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development of theoretical computer science, providing a for...
1915 Norman Foster Ramsey, Jr. was an American physicist. In 1949, while working at Harvard, Ramsey applied a key insight to improve Columbia University physicist Isidor Rabi's method of
studying atoms and molecules. In 1937, Rabi used alternati...
1916 Claude Elwood Shannon was an American mathematician, electronic engineer, and cryptographer known as "the father of information theory". Shannon is famous for having founded
information theory with one landmark paper published in 1948. But...
1916 Francis Crick was an English molecular biologist, biophysicist, and neuroscientist, and most noted for being one of two co-discoverers of the structure of the DNA molecule in 1953,
together with James D. Watson. He, Watson and Maurice Wilki...
1918 Richard Phillips Feynman was an American physicist known for the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the
superfluidity of supercooled liquid helium, as well as work in par...
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On the(d)–chromatic number of a complete balanced multipartite graph
Operations Research Society of South Africa
In this paper we solve (approximately) the problem of finding the minimum number of colours with which the vertices of a complete, balanced, multipartite graph G may be coloured such that the maximum
degrees of all colour class induced subgraphs are at most some specified integer d 2 N. The minimum number of colours in such a colouring is referred to as the (d)–chromatic number of G. The problem
of finding the (d)–chromatic number of a complete, balanced, multipartite graph has its roots in an open graph theoretic characterisation problem and has applications conforming to the generic
scenario where users of a system are in conflict if they require access to some shared resource. These conflicts are represented by edges in a so–called resource access graph, where vertices
represent the users. An efficient resource access schedule is an assignment of the users to a minimum number of groups (modelled by means of colour classes) where some threshold d of conflict may be
tolerated in each group. If different colours are associated with different time periods in the schedule, then the minimum number of groupings in an optimal resource access schedule for the above set
of users is given by the (d)–chromatic number of the resource access graph. A complete balanced multipartite resource access graph represents a situation of maximum conflict between members of
different user groups of the system, but where no conflict occurs between members of the same user group (perhaps due to an allocation of diverse duties to the group members).
CITATION: Burger, A. P., Nieuwoudt, I. & Van Vuuren, J.H. 2007. On the(d)–chromatic number of a complete balanced multipartite graph. ORiON, 23(1):29-49, doi:10.5784/23-1-45.
The original publication is available at http://orion.journals.ac.za
Graphs -- Maximum degree, Graph colouring, Graph theory
Burger, A. P., Nieuwoudt, I. & Van Vuuren, J.H. 2007. On the(d)–chromatic number of a complete balanced multipartite graph. ORiON, 23(1):29-49, doi:10.5784/23-1-45 | {"url":"https://scholar.sun.ac.za/items/99acdf2d-b932-4804-8b07-48a0b162796d","timestamp":"2024-11-06T01:05:13Z","content_type":"text/html","content_length":"451536","record_id":"<urn:uuid:22d07b55-b222-40c2-8862-16487211a37c>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00074.warc.gz"} |
Math Is Fun Forum
Registered: 2022-07-21
Posts: 4
the area of a sector of one radian
a circle, radius = r;
a sector, angle = 1 radian;
What is the area?
(1) Area of the circle = pi . r^2
(2) Area of a sector of one radian = 1 Rad
(3) If a circle has 2.pi.rad
Area of the circle = 2.pi.Rad
From (1) and (3) => pi . r^2 = 2 . pi . Rad <=> r^2 = 2 . Rad <=>
<=> Rad = 1/2 . r^2
It turns out that Area of a sector of one radian is half a square.
I do not know what to think.
Registered: 2010-06-20
Posts: 10,610
Re: the area of a sector of one radian
hi AT&T
Although degrees have been used since the time of the ancient Babylonians, radians are the angle measure of choice for advanced mathematics.
For example the gradient function for sin is given by d(sin x)/dx = cos x only when the angle x is measured in radians.
What you have discovered is the result of the way radians are defined.
A radian is just under 60 degrees (one sixth of the circle), so your result does not seem unreasonable. If pi= 3 then the whole circle is 3r^2 so one sixth is 3r^2/6 =(r^2)/2 So an approximate
calculation confirms the result.
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
Registered: 2022-07-21
Posts: 4
Re: the area of a sector of one radian
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A factory requires 42 machines to produce a given number of articles in 63 days. How many machines would be required to produce the same number of articles in 54 days?
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A factory requires 42 machines to produce a given number of articles in 63 days. How many machines would be required to produce the same number of articles in 54 days?
Let the number of machines be y and the number of days be x.
If the number of days decreases, the number of machines required will increase. So, they are inverse proportion.
Two numbers x and y are said to be in inverse proportion when an increase in one quantity decreases the other quantity and vice versa.
xy = k or x = (1/y) k
where k is a constant.
Hence, x₁y₁ = x₂y₂
63 × 42 = 54 × y₂
y₂[ ]= (42 × 63)/54
y₂[ ]= 49
Thus, 49 machines will be required to produce the same number of articles in 54 days.
☛ Check: NCERT Solutions for Class 8 Maths Chapter 13
Video Solution:
A factory requires 42 machines to produce a given number of articles in 63 days. How many machines would be required to produce the same number of articles in 54 days?
NCERT Solutions Class 8 Maths Chapter 13 Exercise 13.2 Question 8
A factory requires 42 machines to produce a given number of articles in 63 days. 49 machines would be required to produce the same number of articles in 54 days.
☛ Related Questions:
Math worksheets and
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The Best Pi Day Games Activities - Home, Family, Style and Art Ideas
The Best Pi Day Games Activities
by admin
written by admin
The Best Pi Day Games Activities
.What is p anyway? Separate any circle s circumference by its size; the solution (whether for a pie plate or an earth) is constantly about 3.14, a number we represent with the Greek letter p. Keep
determining p s numbers with a growing number of precision as mathematicians have been doing for 4,000 years and also you ll find they go on actually permanently, without any pattern.
1. Pi Day 2015 Pi Day Art Project
Best Pi Day Games Activities
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It all started with the calculation of pi. Pi stands for a mathematical limitation: an ambition toward the ideal curve, stable development towards the inaccessible star.
2. 15 Fun Pi Day Activities for Kids SoCal Field Trips
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Pi Day is a fantastic excuse to gather together as a household and also share in life s easy pleasures, whether it s cooking a pie together or merely appreciating one for treat. Whether you read a
recipe, determining active ingredients, or separating a pie right into pieces, there are a lot of tasty understanding opportunities on Pi Day. Search for a family-friendly Pi Day Activity Package on
the left side of the page.
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Pi, represented by the Greek letter p, has been part of human understanding for centuries, but it wasn t till 1988 that physicist Larry Shaw arranged what is now recognized as the initial Pi Day
celebration at the San Francisco Exploratorium scientific research museum. Shaw selected March 14, or 3.14 the first 3 digits of pi as Pi Day. Shaw passed away in 2014, however Pi Day is still
commemorated by enthusiasts of maths all over the world.
4. Some of the Best Things in Life are Mistakes Free Pi Day
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Pi represents the proportion of a circle s area to its size. It s a fundamental part of the foundation of maths, most importantly geometry, where pi is crucial to equations calculating the location
of a circle, A = pr2, and also the volume of a cyndrical tube, V = pr2h.
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Numerous ancient people calculated approximations of pi, consisting of the Egyptians, Babylonians, and also there s also a referral to the dimensions of a circle in the Holy bible, however the very
first estimation of pi as 3.14 is attributed to Greek mathematician Archimedes, that stayed in the 3rd century B.C. It was additionally individually figured out by Chinese mathematician Zu Chongzhi
(429 501), who computed pi to six decimal locations. Mathematicians took on the sign p for the expression in the 18th century: Welsh maths instructor William Jones is typically credited with the
first use the sign in 1706.
6. STEM Archives SoCal Field Trips
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Pi Day was officially identified by Congress in 2009, and also it s inspired quirky and pun-filled parties, consisting of consuming circular deals with, from fruit pies to pizza, in addition to
dressing like Albert Einstein, whose birthday serendipitously drops on the math-imbued day. San Francisco s Exploratorium likewise organizes an annual day of pi-inspired tasks. The Massachusetts
Institute of Innovation launches its undergraduate admissions choices on Pi Day, and beginning in 2012, it began sending the decisions at 6:28 pm, or Tau time, for the mathematical formula 2p. This
year, NASA is inviting math whizzes to contend in its Pi overhead challenge to fix a collection of interplanetary mathematics problems.
7. Best Pi Day Activities for the Classroom WeAreTeachers
Best Pi Day Games Activities
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“Yes, I have a robotic camouflaged as Nikola Tesla.” This is my (regretfully incorrect) attempt at creating in pilish: a literary kind in which the variety of letters in each succeeding word match
the figures of pi. The limiting nature of pilish indicates it’s not particularly great for longer works, but the acknowledged master of pilish mathematician Michael Keith has created a novella that
complies with pi’s numbers for 10,000 decimals.
8. Best Pi Day Activities for the Classroom WeAreTeachers
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Scientists have actually searched published benefit instances of unintentional pilish, however there appear to be few examples of any note. Gladly, though, one of the earliest (deliberate) instances
of pilish is one of the most relevant. It’s believed to have actually been made up by English physicist James Hopwood Denims and runs as adheres to: “How I require a beverage, alcoholic in nature,
after the heavy talks entailing quantum technicians!”.
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In maths, this unlimited number is important because of what it represents in connection with a circle it s the continuous ratio of a circle s circumference to its diameter. Pi is likewise essential
to design, making contemporary construction feasible.
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Mathematicians, scientists and instructors wish the vacation will certainly help raise passion in math and scientific research across the country, via guideline, museum events, pie-eating (or
tossing) competitions and much more. It appears this gaudy national holiday can please the left-brained and also the sweet-tooth inclined. How will you be celebrating?.
11. Kindergarten Pi Day ActivitiesReacher Reviews
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Perhaps when you were in math class and you gazed off right into area asking yourself why on earth ‘logs’ or ‘evidence’ mattered a lot. Pi is the answer, well a minimum of, one of the things that
links mathematics back to real life usages. Due to the fact that Pi is linked to circles it is likewise linked to cycles, things like determining waves, ups and downs, the oceans trends,
electro-magnetic waves and also much more. Additionally lots of environment sensation can additionally be calulated with pi like the shape of rivers, the disc of the sunlight, the spiral of DNA as
well as event he student of the eye.
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Pi web links maths to the real worldMaybe when you were in mathematics class and you gazed off into area questioning why on planet ‘logs’ or ‘proofs’ mattered so much. Since Pi is linked to circles
it is also connected to cycles, points like determining waves, ebb and flow, the seas tides, electro-magnetic waves as well as a lot a lot more.
13. 14 Pi Day Activities for Kids to Celebrate Pi
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He utilized it to uncover the location of a circle, the quantity of a ball as well as lots of other homes of curved forms that had baffled the finest mathematicians before him. In each situation, he
approximated a bent shape by utilizing a large number of small level polygons or straight lines. The resulting estimates were gemlike, faceted items that generated superb insights right into the
initial forms, specifically when he thought of using considerably numerous, infinitesimally little elements while doing so.
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As well as with pie eating and pie dishes comes p recitations. It s all fun, yeah, but I resist anyone to offer me the defensible educational benefit of having children remember as a lot of p s
boundless string of numbers as possible. Why have any more than the initial 6 digits inscribed in the head when you can obtain them off your smartphone, which nowadays is amazingly wise?.
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Have them attempt to bake a square cake of the exact same height and area of a circular pie. They would discover that there is definitely no relationship between p as well as pie! Yes, most pies are
circular, yet that s where the example ends.
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By taking adequate steps, as well as making them small sufficient, you can approximate the length of the track as accurately as you desired. For example, paths with six, 12 and also 24 actions would
do an increasingly good job of hugging the circle.
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The border is precisely six times the distance r of the circle, or 6r. That s since the hexagon consists of 6 equilateral triangles, each side of which equates to the circle s distance. The diameter
of the hexagon, for its part, is 2 times the circle s radius, or 2r.
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Currently remember that the border of the hexagon ignores real area of the circle. So the proportion of these 2 hexagonal ranges 6r/2r = 3 need to represent an underestimate of pi. Therefore, the
unknown value of pi, whatever it equates to, must be above 3.
19. Best Pi Day Activities for the Classroom WeAreTeachers
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Obviously, six is an unbelievably small number of actions, and also the resulting hexagon is a crude caricature of a circle, yet Archimedes was simply getting going. As soon as he figured out what
the hexagon was telling him, he shortened the actions and took two times as much of them. After that he kept doing that, over as well as over again.
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A guy stressed, he went from 6 actions to 12, after that 24, 48 and also ultimately 96 actions, using common geometry to work out the ever-shrinking lengths of the steps to migraine-inducing
precision. By using a 96-sided polygon inside the circle, and additionally a 96-sided polygon outside the circle, he inevitably proved that pi is above 3 + 10/71 and less than 3 + 10/70.
21. Pi Day Printable Activity Make Your OwnSneaky Pi Detector
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The unidentified value of pi is being entraped in a mathematical vise, squeezed between two numbers that look practically similar, except the first has a denominator of 71 and also the last has a
denominator of 70. By thinking about polygons with even more sides, later mathematicians tightened the vise also better. Around 1,600 years back, the Chinese geometer Zu Chongzhi contemplated
polygons having an unbelievable 24,576 sides to press pi bent on 8 numbers:.
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On an identity of Hochschild: A proof of an identity of Hochschild
Quantization of symplectic varieties in positive characteristic. This is my topic exam from 2020 at the University of Chicago. It contains an explicit formula for the first-order Frobenius-split
quantization constructed by Bezrukavnikov and Kaledin.
From a talk in Kazhdan-Lusztig conjecture seminar, Winter 2020: Regular holonomic D-modules and equivariant Beilinson-Bernstein
From a series of two lectures I gave on D-modules in the Fall of 2019: Introduction to D-modules I Introduction to D-modules II
I am writing a series of expository blog posts with Catherine Ray on 1/2’s that appear in mathematics, such as in the Duflo and HKR isomorphisms.
Relating the 1/2’s in Duflo and Harish-Chandra The 1/2 in Harish-Chandra via the Fourier Transform Note that these are hosted on Catherine’s website. | {"url":"https://joshuamundinger.github.io/notes.html","timestamp":"2024-11-07T20:16:29Z","content_type":"text/html","content_length":"5125","record_id":"<urn:uuid:90894d8b-1185-48de-980f-761485871fd7>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00645.warc.gz"} |
Fast Implied Volatilities in the NAG Library - NAG
The Black-Scholes formula for the price of a European call option is
\[ C = S_0 \Phi \left( \frac {\ln \left( \frac {S_0}{K} \right) + \left[ r + \frac {\sigma^2}{2} \right] T}{\sigma \sqrt {T}} \right) – Ke^{rT} \Phi \left( \frac {\ln \left( \frac {S_0}{K} \right) +
\left[ r – \frac {\sigma^2}{2} \right] T}{\sigma \sqrt {T}} \right) \]
where \(T\) is the time to maturity of the contract, \(S_0\) is the spot price of the underlying asset, \(K\) is the strike price of exercising the option, \(r\) is the interest rate and \(\sigma\)
is the volatility. An important problem in finance is to compute the implied volatility, \(\sigma\), given values for \(T\), \(K\), \(S_0\), \(r\) and \(C\). An explicit formula for \(\sigma\) is
not available. Furthermore, \(\sigma\) cannot be directly measured from financial data. Instead, it must be computed using a numerical approximation.
As shown in the figure, the volatility surface (a three-dimensional plot of how the volatility varies according to the price and time to maturity) can be highly curved. This makes accurately
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Simultaneous Equations: Equal Values and Substitution Method - Math Angel
🎬 Video Tutorial
• Equal Values Method: Isolate the same variable in both equations, set the expressions equal, and solve for the remaining variable. This method reduces a system to a single-variable equation.
• Example of Equal Values: For equations $y = x – 1$ and $y = 2x + 3$, set $x – 1 = 2x + 3$, solve for $x$, then substitute back to find $y$.
• Substitution Method: Solve one equation for a variable, then substitute this expression into the other equation to simplify.
• Example of Substitution: If $y = x – 1$ and $2x + y = 8$, substitute $y = x – 1$ into the second equation to get $2x + (x – 1) = 8$, then solve for $x$ and substitute back to find $y$.
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Click here to download the full example code
Transformer as a Graph Neural Network¶
Author: Zihao Ye, Jinjing Zhou, Qipeng Guo, Quan Gan, Zheng Zhang
The tutorial aims at gaining insights into the paper, with code as a mean of explanation. The implementation thus is NOT optimized for running efficiency. For recommended implementation, please refer
to the official examples.
In this tutorial, you learn about a simplified implementation of the Transformer model. You can see highlights of the most important design points. For instance, there is only single-head attention.
The complete code can be found here.
The overall structure is similar to the one from the research papaer Annotated Transformer.
The Transformer model, as a replacement of CNN/RNN architecture for sequence modeling, was introduced in the research paper: Attention is All You Need. It improved the state of the art for machine
translation as well as natural language inference task (GPT). Recent work on pre-training Transformer with large scale corpus (BERT) supports that it is capable of learning high-quality semantic
The interesting part of Transformer is its extensive employment of attention. The classic use of attention comes from machine translation model, where the output token attends to all input tokens.
Transformer additionally applies self-attention in both decoder and encoder. This process forces words relate to each other to combine together, irrespective of their positions in the sequence. This
is different from RNN-based model, where words (in the source sentence) are combined along the chain, which is thought to be too constrained.
Attention layer of Transformer¶
In the attention layer of Transformer, for each node the module learns to assign weights on its in-coming edges. For node pair \((i, j)\) (from \(i\) to \(j\)) with node \(x_i, x_j \in \mathbb{R}^n\)
, the score of their connection is defined as follows:
\[\begin{split}q_j = W_q\cdot x_j \\ k_i = W_k\cdot x_i\\ v_i = W_v\cdot x_i\\ \textrm{score} = q_j^T k_i\end{split}\]
where \(W_q, W_k, W_v \in \mathbb{R}^{n\times d_k}\) map the representations \(x\) to “query”, “key”, and “value” space respectively.
There are other possibilities to implement the score function. The dot product measures the similarity of a given query \(q_j\) and a key \(k_i\): if \(j\) needs the information stored in \(i\), the
query vector at position \(j\) (\(q_j\)) is supposed to be close to key vector at position \(i\) (\(k_i\)).
The score is then used to compute the sum of the incoming values, normalized over the weights of edges, stored in \(\textrm{wv}\). Then apply an affine layer to \(\textrm{wv}\) to get the output \(o
\[\begin{split}w_{ji} = \frac{\exp\{\textrm{score}_{ji} \}}{\sum\limits_{(k, i)\in E}\exp\{\textrm{score}_{ki} \}} \\ \textrm{wv}_i = \sum_{(k, i)\in E} w_{ki} v_k \\ o = W_o\cdot \textrm{wv} \\\end
Multi-head attention layer¶
In Transformer, attention is multi-headed. A head is very much like a channel in a convolutional network. The multi-head attention consists of multiple attention heads, in which each head refers to a
single attention module. \(\textrm{wv}^{(i)}\) for all the heads are concatenated and mapped to output \(o\) with an affine layer:
\[o = W_o \cdot \textrm{concat}\left(\left[\textrm{wv}^{(0)}, \textrm{wv}^{(1)}, \cdots, \textrm{wv}^{(h)}\right]\right)\]
The code below wraps necessary components for multi-head attention, and provides two interfaces.
• get maps state ‘x’, to query, key and value, which is required by following steps(propagate_attention).
• get_o maps the updated value after attention to the output \(o\) for post-processing.
class MultiHeadAttention(nn.Module):
"Multi-Head Attention"
def __init__(self, h, dim_model):
"h: number of heads; dim_model: hidden dimension"
super(MultiHeadAttention, self).__init__()
self.d_k = dim_model // h
self.h = h
# W_q, W_k, W_v, W_o
self.linears = clones(nn.Linear(dim_model, dim_model), 4)
def get(self, x, fields='qkv'):
"Return a dict of queries / keys / values."
batch_size = x.shape[0]
ret = {}
if 'q' in fields:
ret['q'] = self.linears[0](x).view(batch_size, self.h, self.d_k)
if 'k' in fields:
ret['k'] = self.linears[1](x).view(batch_size, self.h, self.d_k)
if 'v' in fields:
ret['v'] = self.linears[2](x).view(batch_size, self.h, self.d_k)
return ret
def get_o(self, x):
"get output of the multi-head attention"
batch_size = x.shape[0]
return self.linears[3](x.view(batch_size, -1))
How DGL implements Transformer with a graph neural network¶
You get a different perspective of Transformer by treating the attention as edges in a graph and adopt message passing on the edges to induce the appropriate processing.
Graph structure¶
Construct the graph by mapping tokens of the source and target sentence to nodes. The complete Transformer graph is made up of three subgraphs:
Source language graph. This is a complete graph, each token \(s_i\) can attend to any other token \(s_j\) (including self-loops). Target language graph. The graph is half-complete, in that \(t_i\)
attends only to \(t_j\) if \(i > j\) (an output token can not depend on future words). Cross-language graph. This is a bi-partitie graph, where there is an edge from every source token \(s_i\) to
every target token \(t_j\), meaning every target token can attend on source tokens.
The full picture looks like this:
Pre-build the graphs in dataset preparation stage.
Message passing¶
Once you define the graph structure, move on to defining the computation for message passing.
Assuming that you have already computed all the queries \(q_i\), keys \(k_i\) and values \(v_i\). For each node \(i\) (no matter whether it is a source token or target token), you can decompose the
attention computation into two steps:
1. Message computation: Compute attention score \(\mathrm{score}_{ij}\) between \(i\) and all nodes \(j\) to be attended over, by taking the scaled-dot product between \(q_i\) and \(k_j\). The
message sent from \(j\) to \(i\) will consist of the score \(\mathrm{score}_{ij}\) and the value \(v_j\).
2. Message aggregation: Aggregate the values \(v_j\) from all \(j\) according to the scores \(\mathrm{score}_{ij}\).
Simple implementation¶
Message computation¶
Compute score and send source node’s v to destination’s mailbox
def message_func(edges):
return {'score': ((edges.src['k'] * edges.dst['q'])
.sum(-1, keepdim=True)),
'v': edges.src['v']}
Message aggregation¶
Normalize over all in-edges and weighted sum to get output
import torch as th
import torch.nn.functional as F
def reduce_func(nodes, d_k=64):
v = nodes.mailbox['v']
att = F.softmax(nodes.mailbox['score'] / th.sqrt(d_k), 1)
return {'dx': (att * v).sum(1)}
Execute on specific edges¶
import functools.partial as partial
def naive_propagate_attention(self, g, eids):
g.send_and_recv(eids, message_func, partial(reduce_func, d_k=self.d_k))
Speeding up with built-in functions¶
To speed up the message passing process, use DGL’s built-in functions, including:
• fn.src_mul_egdes(src_field, edges_field, out_field) multiplies source’s attribute and edges attribute, and send the result to the destination node’s mailbox keyed by out_field.
• fn.copy_e(edges_field, out_field) copies edge’s attribute to destination node’s mailbox.
• fn.sum(edges_field, out_field) sums up edge’s attribute and sends aggregation to destination node’s mailbox.
Here, you assemble those built-in functions into propagate_attention, which is also the main graph operation function in the final implementation. To accelerate it, break the softmax operation into
the following steps. Recall that for each head there are two phases.
1. Compute attention score by multiply src node’s k and dst node’s q
□ g.apply_edges(src_dot_dst('k', 'q', 'score'), eids)
2. Scaled Softmax over all dst nodes’ in-coming edges
□ Step 1: Exponentialize score with scale normalize constant
☆ g.apply_edges(scaled_exp('score', np.sqrt(self.d_k)))
\[\textrm{score}_{ij}\leftarrow\exp{\left(\frac{\textrm{score}_{ij}}{ \sqrt{d_k}}\right)}\]
□ Step 2: Get the “values” on associated nodes weighted by “scores” on in-coming edges of each node; get the sum of “scores” on in-coming edges of each node for normalization. Note that here \
(\textrm{wv}\) is not normalized.
☆ msg: fn.u_mul_e('v', 'score', 'v'), reduce: fn.sum('v', 'wv')
\[\textrm{wv}_j=\sum_{i=1}^{N} \textrm{score}_{ij} \cdot v_i\]
☆ msg: fn.copy_e('score', 'score'), reduce: fn.sum('score', 'z')
\[\textrm{z}_j=\sum_{i=1}^{N} \textrm{score}_{ij}\]
The normalization of \(\textrm{wv}\) is left to post processing.
def src_dot_dst(src_field, dst_field, out_field):
def func(edges):
return {out_field: (edges.src[src_field] * edges.dst[dst_field]).sum(-1, keepdim=True)}
return func
def scaled_exp(field, scale_constant):
def func(edges):
# clamp for softmax numerical stability
return {field: th.exp((edges.data[field] / scale_constant).clamp(-5, 5))}
return func
def propagate_attention(self, g, eids):
# Compute attention score
g.apply_edges(src_dot_dst('k', 'q', 'score'), eids)
g.apply_edges(scaled_exp('score', np.sqrt(self.d_k)))
# Update node state
[fn.u_mul_e('v', 'score', 'v'), fn.copy_e('score', 'score')],
[fn.sum('v', 'wv'), fn.sum('score', 'z')])
Preprocessing and postprocessing¶
In Transformer, data needs to be pre- and post-processed before and after the propagate_attention function.
Preprocessing The preprocessing function pre_func first normalizes the node representations and then map them to a set of queries, keys and values, using self-attention as an example:
\[\begin{split}x \leftarrow \textrm{LayerNorm}(x) \\ [q, k, v] \leftarrow [W_q, W_k, W_v ]\cdot x\end{split}\]
Postprocessing The postprocessing function post_funcs completes the whole computation correspond to one layer of the transformer: 1. Normalize \(\textrm{wv}\) and get the output of Multi-Head
Attention Layer \(o\).
\[\begin{split}\textrm{wv} \leftarrow \frac{\textrm{wv}}{z} \\ o \leftarrow W_o\cdot \textrm{wv} + b_o\end{split}\]
add residual connection:
\[x \leftarrow x + o\]
2. Applying a two layer position-wise feed forward layer on \(x\) then add residual connection:
\[x \leftarrow x + \textrm{LayerNorm}(\textrm{FFN}(x))\]
where \(\textrm{FFN}\) refers to the feed forward function.
class Encoder(nn.Module):
def __init__(self, layer, N):
super(Encoder, self).__init__()
self.N = N
self.layers = clones(layer, N)
self.norm = LayerNorm(layer.size)
def pre_func(self, i, fields='qkv'):
layer = self.layers[i]
def func(nodes):
x = nodes.data['x']
norm_x = layer.sublayer[0].norm(x)
return layer.self_attn.get(norm_x, fields=fields)
return func
def post_func(self, i):
layer = self.layers[i]
def func(nodes):
x, wv, z = nodes.data['x'], nodes.data['wv'], nodes.data['z']
o = layer.self_attn.get_o(wv / z)
x = x + layer.sublayer[0].dropout(o)
x = layer.sublayer[1](x, layer.feed_forward)
return {'x': x if i < self.N - 1 else self.norm(x)}
return func
class Decoder(nn.Module):
def __init__(self, layer, N):
super(Decoder, self).__init__()
self.N = N
self.layers = clones(layer, N)
self.norm = LayerNorm(layer.size)
def pre_func(self, i, fields='qkv', l=0):
layer = self.layers[i]
def func(nodes):
x = nodes.data['x']
if fields == 'kv':
norm_x = x # In enc-dec attention, x has already been normalized.
norm_x = layer.sublayer[l].norm(x)
return layer.self_attn.get(norm_x, fields)
return func
def post_func(self, i, l=0):
layer = self.layers[i]
def func(nodes):
x, wv, z = nodes.data['x'], nodes.data['wv'], nodes.data['z']
o = layer.self_attn.get_o(wv / z)
x = x + layer.sublayer[l].dropout(o)
if l == 1:
x = layer.sublayer[2](x, layer.feed_forward)
return {'x': x if i < self.N - 1 else self.norm(x)}
return func
This completes all procedures of one layer of encoder and decoder in Transformer.
The sublayer connection part is little bit different from the original paper. However, this implementation is the same as The Annotated Transformer and OpenNMT.
Main class of Transformer graph¶
The processing flow of Transformer can be seen as a 2-stage message-passing within the complete graph (adding pre- and post- processing appropriately): 1) self-attention in encoder, 2) self-attention
in decoder followed by cross-attention between encoder and decoder, as shown below.
class Transformer(nn.Module):
def __init__(self, encoder, decoder, src_embed, tgt_embed, pos_enc, generator, h, d_k):
super(Transformer, self).__init__()
self.encoder, self.decoder = encoder, decoder
self.src_embed, self.tgt_embed = src_embed, tgt_embed
self.pos_enc = pos_enc
self.generator = generator
self.h, self.d_k = h, d_k
def propagate_attention(self, g, eids):
# Compute attention score
g.apply_edges(src_dot_dst('k', 'q', 'score'), eids)
g.apply_edges(scaled_exp('score', np.sqrt(self.d_k)))
# Send weighted values to target nodes
[fn.u_mul_e('v', 'score', 'v'), fn.copy_e('score', 'score')],
[fn.sum('v', 'wv'), fn.sum('score', 'z')])
def update_graph(self, g, eids, pre_pairs, post_pairs):
"Update the node states and edge states of the graph."
# Pre-compute queries and key-value pairs.
for pre_func, nids in pre_pairs:
g.apply_nodes(pre_func, nids)
self.propagate_attention(g, eids)
# Further calculation after attention mechanism
for post_func, nids in post_pairs:
g.apply_nodes(post_func, nids)
def forward(self, graph):
g = graph.g
nids, eids = graph.nids, graph.eids
# Word Embedding and Position Embedding
src_embed, src_pos = self.src_embed(graph.src[0]), self.pos_enc(graph.src[1])
tgt_embed, tgt_pos = self.tgt_embed(graph.tgt[0]), self.pos_enc(graph.tgt[1])
g.nodes[nids['enc']].data['x'] = self.pos_enc.dropout(src_embed + src_pos)
g.nodes[nids['dec']].data['x'] = self.pos_enc.dropout(tgt_embed + tgt_pos)
for i in range(self.encoder.N):
# Step 1: Encoder Self-attention
pre_func = self.encoder.pre_func(i, 'qkv')
post_func = self.encoder.post_func(i)
nodes, edges = nids['enc'], eids['ee']
self.update_graph(g, edges, [(pre_func, nodes)], [(post_func, nodes)])
for i in range(self.decoder.N):
# Step 2: Dncoder Self-attention
pre_func = self.decoder.pre_func(i, 'qkv')
post_func = self.decoder.post_func(i)
nodes, edges = nids['dec'], eids['dd']
self.update_graph(g, edges, [(pre_func, nodes)], [(post_func, nodes)])
# Step 3: Encoder-Decoder attention
pre_q = self.decoder.pre_func(i, 'q', 1)
pre_kv = self.decoder.pre_func(i, 'kv', 1)
post_func = self.decoder.post_func(i, 1)
nodes_e, nodes_d, edges = nids['enc'], nids['dec'], eids['ed']
self.update_graph(g, edges, [(pre_q, nodes_d), (pre_kv, nodes_e)], [(post_func, nodes_d)])
return self.generator(g.ndata['x'][nids['dec']])
By calling update_graph function, you can create your own Transformer on any subgraphs with nearly the same code. This flexibility enables us to discover new, sparse structures (c.f. local attention
mentioned here). Note in this implementation you don’t use mask or padding, which makes the logic more clear and saves memory. The trade-off is that the implementation is slower.
This tutorial does not cover several other techniques such as Label Smoothing and Noam Optimizations mentioned in the original paper. For detailed description about these modules, read The Annotated
Transformer written by Harvard NLP team.
Task and the dataset¶
The Transformer is a general framework for a variety of NLP tasks. This tutorial focuses on the sequence to sequence learning: it’s a typical case to illustrate how it works.
As for the dataset, there are two example tasks: copy and sort, together with two real-world translation tasks: multi30k en-de task and wmt14 en-de task.
• copy dataset: copy input sequences to output. (train/valid/test: 9000, 1000, 1000)
• sort dataset: sort input sequences as output. (train/valid/test: 9000, 1000, 1000)
• Multi30k en-de, translate sentences from En to De. (train/valid/test: 29000, 1000, 1000)
• WMT14 en-de, translate sentences from En to De. (Train/Valid/Test: 4500966/3000/3003)
Training with wmt14 requires multi-GPU support and is not available. Contributions are welcome!
Graph building¶
Batching This is similar to the way you handle Tree-LSTM. Build a graph pool in advance, including all possible combination of input lengths and output lengths. Then for each sample in a batch, call
dgl.batch to batch graphs of their sizes together in to a single large graph.
You can wrap the process of creating graph pool and building BatchedGraph in dataset.GraphPool and dataset.TranslationDataset.
graph_pool = GraphPool()
data_iter = dataset(graph_pool, mode='train', batch_size=1, devices=devices)
for graph in data_iter:
print(graph.nids['enc']) # encoder node ids
print(graph.nids['dec']) # decoder node ids
print(graph.eids['ee']) # encoder-encoder edge ids
print(graph.eids['ed']) # encoder-decoder edge ids
print(graph.eids['dd']) # decoder-decoder edge ids
print(graph.src[0]) # Input word index list
print(graph.src[1]) # Input positions
print(graph.tgt[0]) # Output word index list
print(graph.tgt[1]) # Ouptut positions
tensor([0, 1, 2, 3, 4, 5, 6, 7, 8], device='cuda:0')
tensor([ 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], device='cuda:0')
tensor([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
72, 73, 74, 75, 76, 77, 78, 79, 80], device='cuda:0')
tensor([ 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94,
95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108,
109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122,
123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136,
137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150,
151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164,
165, 166, 167, 168, 169, 170], device='cuda:0')
tensor([171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184,
185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198,
199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212,
213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225],
tensor([28, 25, 7, 26, 6, 4, 5, 9, 18], device='cuda:0')
tensor([0, 1, 2, 3, 4, 5, 6, 7, 8], device='cuda:0')
tensor([ 0, 28, 25, 7, 26, 6, 4, 5, 9, 18], device='cuda:0')
tensor([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], device='cuda:0')
Put it all together¶
Train a one-head transformer with one layer, 128 dimension on copy task. Set other parameters to the default.
Inference module is not included in this tutorial. It requires beam search. For a full implementation, see the GitHub repo.
from tqdm import tqdm
import torch as th
import numpy as np
from loss import LabelSmoothing, SimpleLossCompute
from modules import make_model
from optims import NoamOpt
from dgl.contrib.transformer import get_dataset, GraphPool
def run_epoch(data_iter, model, loss_compute, is_train=True):
for i, g in tqdm(enumerate(data_iter)):
with th.set_grad_enabled(is_train):
output = model(g)
loss = loss_compute(output, g.tgt_y, g.n_tokens)
print('average loss: {}'.format(loss_compute.avg_loss))
print('accuracy: {}'.format(loss_compute.accuracy))
N = 1
batch_size = 128
devices = ['cuda' if th.cuda.is_available() else 'cpu']
dataset = get_dataset("copy")
V = dataset.vocab_size
criterion = LabelSmoothing(V, padding_idx=dataset.pad_id, smoothing=0.1)
dim_model = 128
# Create model
model = make_model(V, V, N=N, dim_model=128, dim_ff=128, h=1)
# Sharing weights between Encoder & Decoder
model.src_embed.lut.weight = model.tgt_embed.lut.weight
model.generator.proj.weight = model.tgt_embed.lut.weight
model, criterion = model.to(devices[0]), criterion.to(devices[0])
model_opt = NoamOpt(dim_model, 1, 400,
th.optim.Adam(model.parameters(), lr=1e-3, betas=(0.9, 0.98), eps=1e-9))
loss_compute = SimpleLossCompute
att_maps = []
for epoch in range(4):
train_iter = dataset(graph_pool, mode='train', batch_size=batch_size, devices=devices)
valid_iter = dataset(graph_pool, mode='valid', batch_size=batch_size, devices=devices)
print('Epoch: {} Training...'.format(epoch))
run_epoch(train_iter, model,
loss_compute(criterion, model_opt), is_train=True)
print('Epoch: {} Evaluating...'.format(epoch))
model.att_weight_map = None
run_epoch(valid_iter, model,
loss_compute(criterion, None), is_train=False)
After training, you can visualize the attention that the Transformer generates on copy task.
src_seq = dataset.get_seq_by_id(VIZ_IDX, mode='valid', field='src')
tgt_seq = dataset.get_seq_by_id(VIZ_IDX, mode='valid', field='tgt')[:-1]
# visualize head 0 of encoder-decoder attention
att_animation(att_maps, 'e2d', src_seq, tgt_seq, 0)
Multi-head attention¶
Besides the attention of a one-head attention trained on toy task. We also visualize the attention scores of Encoder’s Self Attention, Decoder’s Self Attention and the Encoder-Decoder attention of an
one-Layer Transformer network trained on multi-30k dataset.
From the visualization you see the diversity of different heads, which is what you would expect. Different heads learn different relations between word pairs.
• Encoder Self-Attention
• Encoder-Decoder Attention Most words in target sequence attend on their related words in source sequence, for example: when generating “See” (in De), several heads attend on “lake”; when
generating “Eisfischerhütte”, several heads attend on “ice”.
• Decoder Self-Attention Most words attend on their previous few words.
Adaptive Universal Transformer¶
A recent research paper by Google, Universal Transformer, is an example to show how update_graph adapts to more complex updating rules.
The Universal Transformer was proposed to address the problem that vanilla Transformer is not computationally universal by introducing recurrence in Transformer:
• The basic idea of Universal Transformer is to repeatedly revise its representations of all symbols in the sequence with each recurrent step by applying a Transformer layer on the representations.
• Compared to vanilla Transformer, Universal Transformer shares weights among its layers, and it does not fix the recurrence time (which means the number of layers in Transformer).
A further optimization employs an adaptive computation time (ACT) mechanism to allow the model to dynamically adjust the number of times the representation of each position in a sequence is revised
(refereed to as step hereafter). This model is also known as the Adaptive Universal Transformer (AUT).
In AUT, you maintain an active nodes list. In each step \(t\), we compute a halting probability: \(h (0<h<1)\) for all nodes in this list by:
\[h^t_i = \sigma(W_h x^t_i + b_h)\]
then dynamically decide which nodes are still active. A node is halted at time \(T\) if and only if \(\sum_{t=1}^{T-1} h_t < 1 - \varepsilon \leq \sum_{t=1}^{T}h_t\). Halted nodes are removed from
the list. The procedure proceeds until the list is empty or a pre-defined maximum step is reached. From DGL’s perspective, this means that the “active” graph becomes sparser over time.
The final state of a node \(s_i\) is a weighted average of \(x_i^t\) by \(h_i^t\):
\[s_i = \sum_{t=1}^{T} h_i^t\cdot x_i^t\]
In DGL, implement an algorithm by calling update_graph on nodes that are still active and edges associated with this nodes. The following code shows the Universal Transformer class in DGL:
class UTransformer(nn.Module):
"Universal Transformer(https://arxiv.org/pdf/1807.03819.pdf) with ACT(https://arxiv.org/pdf/1603.08983.pdf)."
MAX_DEPTH = 8
thres = 0.99
act_loss_weight = 0.01
def __init__(self, encoder, decoder, src_embed, tgt_embed, pos_enc, time_enc, generator, h, d_k):
super(UTransformer, self).__init__()
self.encoder, self.decoder = encoder, decoder
self.src_embed, self.tgt_embed = src_embed, tgt_embed
self.pos_enc, self.time_enc = pos_enc, time_enc
self.halt_enc = HaltingUnit(h * d_k)
self.halt_dec = HaltingUnit(h * d_k)
self.generator = generator
self.h, self.d_k = h, d_k
def step_forward(self, nodes):
# add positional encoding and time encoding, increment step by one
x = nodes.data['x']
step = nodes.data['step']
pos = nodes.data['pos']
return {'x': self.pos_enc.dropout(x + self.pos_enc(pos.view(-1)) + self.time_enc(step.view(-1))),
'step': step + 1}
def halt_and_accum(self, name, end=False):
"field: 'enc' or 'dec'"
halt = self.halt_enc if name == 'enc' else self.halt_dec
thres = self.thres
def func(nodes):
p = halt(nodes.data['x'])
sum_p = nodes.data['sum_p'] + p
active = (sum_p < thres) & (1 - end)
_continue = active.float()
r = nodes.data['r'] * (1 - _continue) + (1 - sum_p) * _continue
s = nodes.data['s'] + ((1 - _continue) * r + _continue * p) * nodes.data['x']
return {'p': p, 'sum_p': sum_p, 'r': r, 's': s, 'active': active}
return func
def propagate_attention(self, g, eids):
# Compute attention score
g.apply_edges(src_dot_dst('k', 'q', 'score'), eids)
g.apply_edges(scaled_exp('score', np.sqrt(self.d_k)), eids)
# Send weighted values to target nodes
[fn.u_mul_e('v', 'score', 'v'), fn.copy_e('score', 'score')],
[fn.sum('v', 'wv'), fn.sum('score', 'z')])
def update_graph(self, g, eids, pre_pairs, post_pairs):
"Update the node states and edge states of the graph."
# Pre-compute queries and key-value pairs.
for pre_func, nids in pre_pairs:
g.apply_nodes(pre_func, nids)
self.propagate_attention(g, eids)
# Further calculation after attention mechanism
for post_func, nids in post_pairs:
g.apply_nodes(post_func, nids)
def forward(self, graph):
g = graph.g
N, E = graph.n_nodes, graph.n_edges
nids, eids = graph.nids, graph.eids
# embed & pos
g.nodes[nids['enc']].data['x'] = self.src_embed(graph.src[0])
g.nodes[nids['dec']].data['x'] = self.tgt_embed(graph.tgt[0])
g.nodes[nids['enc']].data['pos'] = graph.src[1]
g.nodes[nids['dec']].data['pos'] = graph.tgt[1]
# init step
device = next(self.parameters()).device
g.ndata['s'] = th.zeros(N, self.h * self.d_k, dtype=th.float, device=device) # accumulated state
g.ndata['p'] = th.zeros(N, 1, dtype=th.float, device=device) # halting prob
g.ndata['r'] = th.ones(N, 1, dtype=th.float, device=device) # remainder
g.ndata['sum_p'] = th.zeros(N, 1, dtype=th.float, device=device) # sum of pondering values
g.ndata['step'] = th.zeros(N, 1, dtype=th.long, device=device) # step
g.ndata['active'] = th.ones(N, 1, dtype=th.uint8, device=device) # active
for step in range(self.MAX_DEPTH):
pre_func = self.encoder.pre_func('qkv')
post_func = self.encoder.post_func()
nodes = g.filter_nodes(lambda v: v.data['active'].view(-1), nids['enc'])
if len(nodes) == 0: break
edges = g.filter_edges(lambda e: e.dst['active'].view(-1), eids['ee'])
end = step == self.MAX_DEPTH - 1
self.update_graph(g, edges,
[(self.step_forward, nodes), (pre_func, nodes)],
[(post_func, nodes), (self.halt_and_accum('enc', end), nodes)])
g.nodes[nids['enc']].data['x'] = self.encoder.norm(g.nodes[nids['enc']].data['s'])
for step in range(self.MAX_DEPTH):
pre_func = self.decoder.pre_func('qkv')
post_func = self.decoder.post_func()
nodes = g.filter_nodes(lambda v: v.data['active'].view(-1), nids['dec'])
if len(nodes) == 0: break
edges = g.filter_edges(lambda e: e.dst['active'].view(-1), eids['dd'])
self.update_graph(g, edges,
[(self.step_forward, nodes), (pre_func, nodes)],
[(post_func, nodes)])
pre_q = self.decoder.pre_func('q', 1)
pre_kv = self.decoder.pre_func('kv', 1)
post_func = self.decoder.post_func(1)
nodes_e = nids['enc']
edges = g.filter_edges(lambda e: e.dst['active'].view(-1), eids['ed'])
end = step == self.MAX_DEPTH - 1
self.update_graph(g, edges,
[(pre_q, nodes), (pre_kv, nodes_e)],
[(post_func, nodes), (self.halt_and_accum('dec', end), nodes)])
g.nodes[nids['dec']].data['x'] = self.decoder.norm(g.nodes[nids['dec']].data['s'])
act_loss = th.mean(g.ndata['r']) # ACT loss
return self.generator(g.ndata['x'][nids['dec']]), act_loss * self.act_loss_weight
Call filter_nodes and filter_edge to find nodes/edges that are still active:
• filter_nodes() takes a predicate and a node ID list/tensor as input, then returns a tensor of node IDs that satisfy the given predicate.
• filter_edges() takes a predicate and an edge ID list/tensor as input, then returns a tensor of edge IDs that satisfy the given predicate.
For the full implementation, see the GitHub repo.
The figure below shows the effect of Adaptive Computational Time. Different positions of a sentence were revised different times.
You can also visualize the dynamics of step distribution on nodes during the training of AUT on sort task(reach 99.7% accuracy), which demonstrates how AUT learns to reduce recurrence steps during
The notebook itself is not executable due to many dependencies. Download 7_transformer.py, and copy the python script to directory examples/pytorch/transformer then run python 7_transformer.py to see
how it works.
Total running time of the script: ( 0 minutes 0.000 seconds) | {"url":"https://docs.dgl.ai/en/2.0.x/tutorials/models/4_old_wines/7_transformer.html","timestamp":"2024-11-09T16:12:05Z","content_type":"text/html","content_length":"155850","record_id":"<urn:uuid:68bbb448-1281-43bc-91f8-5011d5b38bab>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00579.warc.gz"} |
How to check if a symbolic expression is numerically evaluable?
How to check if a symbolic expression is numerically evaluable?
Context: I want to evaluate a symbolic expression numerically; however sometimes this expression still depends on some symbolic parameters not yet assigned a value.
Question: What is the best way to avoid "TypeError: cannot evaluate symbolic expression numerically" in such a situation?
In other words, is there a function (or an equivalent construction) for
if isThisSymbolicExpressionNumericallyEvaluabel(exp):
n = exp.n()
print "sorry, no"
2 Answers
Sort by ยป oldest newest most voted
Python (therefore Sage) offers a try/except system to handle exceptions, see this page for more details.
In your case, you can write something like:
n = exp.n()
except TypeError:
print "sorry, no"
edit flag offensive delete link more
You should be able to just check if it has no variables.
def is_numerically_evaluable(expr):
return not expr.variables()
sage: is_numerically_evaluable(sin(1))
sage: is_numerically_evaluable(x)
edit flag offensive delete link more
The `Symbolic Ring` is quite unpredictible, so that you may lose some opportunities, for example: sage: expr = integral(exp(-cos(x)), x, 0, 1) sage: expr integrate(e^(-cos(x)), x, 0, 1) sage:
expr.parent() Symbolic Ring sage: expr.variables() (x,) sage: not expr.variables() False sage: expr.n() 0.4353513281039795
tmonteil ( 2013-09-02 08:17:23 +0100 )edit
I woul prefer your method to the try/except method. I will wait to see if someone can make it watertight.
petropolis ( 2013-09-02 09:34:51 +0100 )edit | {"url":"https://ask.sagemath.org/question/10495/how-to-check-if-a-symbolic-expression-is-numerically-evaluable/","timestamp":"2024-11-07T13:16:55Z","content_type":"application/xhtml+xml","content_length":"62165","record_id":"<urn:uuid:407d0309-4e4b-4cc1-95a7-f01233bf1542>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00800.warc.gz"} |
Stepwise regression - (Production and Operations Management) - Vocab, Definition, Explanations | Fiveable
Stepwise regression
from class:
Production and Operations Management
Stepwise regression is a statistical method used to select a subset of predictor variables for use in a regression model by adding or removing predictors based on specified criteria. This technique
is particularly useful when dealing with multiple predictors, as it helps in identifying the most significant variables while reducing the risk of overfitting the model. It balances simplicity and
accuracy, making it a popular choice in regression analysis.
congrats on reading the definition of stepwise regression. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Stepwise regression can be performed in both forward and backward directions, where forward selection starts with no predictors and adds them one by one, while backward elimination starts with
all predictors and removes them.
2. The criteria for adding or removing predictors in stepwise regression typically include metrics like the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC), which help
in determining the goodness of fit.
3. This method can sometimes lead to models that may not generalize well to new data, so it's crucial to validate the final model with a separate dataset.
4. Stepwise regression is particularly beneficial when working with large datasets with many potential predictors, helping to simplify the model without sacrificing predictive power.
5. While stepwise regression can be useful, it is important to remember that the results can vary based on the specific data set and the chosen criteria for variable selection.
Review Questions
• How does stepwise regression improve the process of selecting predictor variables in a regression model?
□ Stepwise regression improves variable selection by systematically adding or removing predictors based on specific criteria, like AIC or BIC. This process allows for identifying the most
relevant variables that contribute to the model's predictive accuracy while minimizing unnecessary complexity. By focusing only on significant predictors, stepwise regression helps in
building a more efficient and interpretable model.
• What are some potential pitfalls of using stepwise regression, particularly concerning model validity and variable selection?
□ One potential pitfall of using stepwise regression is overfitting, where the model fits too closely to the training data and performs poorly on new datasets. Additionally, reliance on
automatic variable selection might lead to ignoring important domain knowledge that could influence variable relevance. It is crucial to validate the chosen model with an independent dataset
to ensure its generalizability and robustness.
• Evaluate how stepwise regression interacts with multicollinearity and its implications for model interpretation.
□ Stepwise regression may struggle with multicollinearity because it can lead to unstable estimates of regression coefficients when predictors are highly correlated. As a result, this can
complicate model interpretation since it becomes challenging to ascertain the individual impact of each predictor on the outcome variable. Awareness of multicollinearity is essential when
using stepwise methods, as it might influence which variables are selected and how they are understood within the context of the model.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/production-and-operations-management/stepwise-regression","timestamp":"2024-11-03T23:16:00Z","content_type":"text/html","content_length":"159488","record_id":"<urn:uuid:36736492-bb0b-44e1-b580-7bcf914e1e7c>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00083.warc.gz"} |
Tuple{Vararg{T, N}} where N >= 1
It can be extremely convenient to dispatch on the types of arguments in a tuple, creating an alias for tuples of a type. However, a zero-length tuple will match a Vararg Tuple of any type, which can
create issues with dispatch and type piracy. How can I impose the restriction that the vararg tuple have at least one element?
abstract type AbstractMyType end
struct MyType <: AbstractMyType
const InformalGroupedType = Tuple{Vararg{AbstractMyType}}
do_something(arg::InformalGroupedType) = println("did something")
my_informal_grouped_type = 1:3 .|> MyType |> Tuple
Unfortunately, a zero-length tuple can match the above alias
typeof(Tuple(){}) <: InformalGroupedType # evaluates to true
Is there any way to impose the restriction
# this is not syntactically valid. Is there a syntax for this idea?
const InformalGroupedType = Tuple{Vararg{AbstractMyType, N}} where N >= 1
This is causing me problems when I extend the show function to my analogue of InformalGroupedType.
The big problem here is that dispatching an existing function on a Tuple{Vararg{T}} is implicitly type piracy even if T is a type I created. So, unless there is a way to restrict the Vararg to length
greater than 0, one ought never extend existing functions to Vararg tuples of user defined type, which would be disappointing.
You cannot - define a method that takes one argument, as well as the Vararg.
Not at the moment, no. There are various versions of this idea and lots of people look for something like this (most often in the context of “some type parameter must be in some range of
values”). You’ll probably find some threads about this here already.
2 Likes
Thanks for the information, and that’s too bad.
Aliasing Vararg Tuples of my type and dispatching on the alias is too convenient for me to stop using it, but I will have to be careful that implicit type piracy does not cause problems.
const InformalGroupedType = Tuple{AbstractMyType, Vararg{AbstractMyType}}
5 Likes
Your answer is correct, prior answer was wrong.
[S:Prior response has it right.
There is currently no way to impose that Vararg is nonzero, so typeof(Tuple{}()) <: Tuple{Vararg{T}} where T <: WhateverType
for any type WhateverType, which means writing methods of existing functions on a vararg tuple type is implicitly type piracy:S]
I am an idiot, I misread your answers as simply dropping the N from the Vararg argument. Your answer is correct, the previous answer was incorrect. Thank you.
2 Likes
I’m probably missing something obvious, but why would this restriction solve your type piracy issue?
Even if you own type T, Tuple{Vararg{T}} includes the empty tuple type Tuple{}, which is not a type owned by the package that provides type T. Vararg{T} represents zero or more instances of type T.
If an already existing function from a different package, f, doesn’t have a method with signature f(::Tuple{}), then a package providing a method f(::Tuple{Vararg{T}}) would implicitly define it
for Tuple{} arguments.
The solution is to dispatch on Tuple{T, Vararg{T}} because it guarantees at least one element of type T is present in the tuple and is therefore never empty.
Methods like this can also be the source of method ambiguities, so it’s useful to use this construct in various places.
4 Likes
I hope I understand your question. Forgive me if I don’t.
EDIT: and the above poster beat me to it, but I’ll dogpile my answer here just in case it helps someone see more clearly.
An example of the issue is that Base.:+(::Vararg{MyType}) matches
Base.:+() # this is piracy! MyType is not involved!
Base.:+(::MyType, ::MyType)
Base.:+(::MyType, ::MyType, ::MyType)
# etc
whereas the definition Base.:+(::MyType, ::Vararg{MyType}) would match all of these except the problematic Base.:+().
The same thing happens with Tuple{Vararg}. () isa Tuple{Vararg{MyType}} and also () isa Tuple{Vararg{AnyOtherType}}. It’s ambiguous what the eltype of an empty Tuple should be.
So the takeaway is that a purely-Vararg argument/tuple (or NTuple{N} where N if N could be 0) does not actually provide the disambiguation required to avoid piracy. You need another identifiable
argument to the method/tuple to prevent accidental piracy in the empty case.
3 Likes
Thanks for the explanation, @brainandforce and @mikmoore! I’m pretty sure I now better understand the point @croberts was making.
It seems Julia has some built-in safety for Vararg{T, N} when N == 0 as it is indeed inherently ambiguous about T. (Edit: see also the documentation.)
julia> f(a::Vararg{Int}) = 1
f (generic function with 1 method)
julia> f(a::Vararg{Float64}) = 2
f (generic function with 2 methods)
julia> f()
ERROR: MethodError: f() is ambiguous.
@ Main REPL[2]:1
@ Main REPL[1]:1
Possible fix, define
But I struggle to see how this would lead to type piracy (unless you forcefully define f() “for your own type T”). Also, the only situation I can imagine the ambiguity manifesting naturally is
when splatting a Tuple{T} which turns out to be empty.
julia> function g(x) # ::Tuple{Int} but not explicitly as !( () isa Tuple{Int})
if x > 0
return (1, 2)
elseif x < 0
return (3)
return ()
g (generic function with 1 method)
julia> f(g(1)...)
julia> f(g(0)...)
ERROR: MethodError: f() is ambiguous.
(Edit: In the previous posts we’re using Tuple{Vararg{T}} instead of just Vararg{T}, but I don’t think this matters.) | {"url":"https://discourse.julialang.org/t/tuple-vararg-t-n-where-n-1/121644","timestamp":"2024-11-04T00:49:41Z","content_type":"text/html","content_length":"41931","record_id":"<urn:uuid:a27250cb-9190-4d10-890f-ded95088fa3a>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00296.warc.gz"} |
Gheorghe Craciun
Department of Mathematics and
Department of Biomolecular Chemistry
University of Wisconsin-Madison
Contact Information:
405 Van Vleck Hall
480 Lincoln Dr
Madison, WI 53706-1388
Phone: (608) 265-3391
Fax: (608) 263-8891
E-mail: craciun at math dot wisc dot edu
Research Interests: Mathematical and Computational Methods in Biology and Medicine
□ Dynamical properties of Discrete Reaction Networks, (with Loïc Paulevé and Heinz Koeppl), Journal of Mathematical Biology, 69(1): 55-72, 2014. PDF
□ Persistence and permanence of mass-action and power-law dynamical systems, (with Fedor Nazarov and Casian Pantea), SIAM Journal on Applied Mathematics, 73(1): 305-329, 2013. PDF
□ Most homeomorphisms with a fixed point have a Cantor set of fixed points, Archiv der Mathematik 100(1), 95-100, 2013. PDF
□ Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks, (with Maya Mincheva), Mathematical Biosciences and Engineering 10(4):1207-1226,
2013. PDF
□ Statistical Model for Biochemical Networks Inference, (with Jajae Kim, Casian Pantea and Grzegorz A. Rempala), Communications in Statistics: Simulation and Computation, 42(1) 121-137,
2013. PDF
□ Finding invariant sets for biological systems using monomial domination, (with Elias August and Heinz Koeppl), Proceedings of the IEEE International Conference on Decision and Control,
3001-3006, 2012.
□ Periodic Patterns in Distributions of Peptide Masses, (with Shane Hubler), Biosystems 109(2): 179-185, 2012. PDF
□ Counting Chemical Compositions using Ehrhart Quasi-Polynomials, (with Shane Hubler), Journal of Mathematical Chemistry, 50:9, 2446-2470, 2012. PDF
□ Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks, (with Casian Pantea and Heinz Koeppl), Discrete and Continuous Dynamical Systems Series B, 17:6,
2153-2170, 2012. PDF
□ Mass distributions of linear chain polymers, (with Shane Hubler), Journal of Mathematical Chemistry, 50:6, 1458-1483, 2012. PDF
□ Graph-theoretic characterizations of multistability and monotonicity for biochemical reaction networks, (with Casian Pantea and Eduardo Sontag), in Koeppl, H.; Densmore, D.; Setti, G.; di
Bernardo, M. (Eds.), Design and Analysis of Biomolecular Circuits, Springer, 2011. PDF
□ Computational methods for analyzing bistability in biochemical reaction networks, (with Casian Pantea) Proceedings of the IEEE International Symposium on Circuits and Systems, 549-553,
2010. PDF
□ Product-form stationary distributions for deficiency zero chemical reaction networks, (with David F. Anderson and Thomas G. Kurtz), Bulletin of Mathematical Biology, 72:8, 1947-1970,
2010. PDF
□ Graph theoretic approaches to injectivity in general chemical reaction systems, (with Murad Banaji), Advances in Applied Mathematics 44 168-184, 2010. PDF
□ Some Geometric Aspects of Control Points for Toric Patches, (with Luis David Garcia-Puente, Frank Sottile), Proceedings of the 7th international conference on Mathematical Methods for
Curves and Surfaces, (M. Daehlen et al. Eds). Lecture Notes in Computer Science 5862, 111-135, Springer Heidelberg, 2010. PDF
□ Multiple Equilibria in Complex Chemical Reaction Networks: Semiopen Mass Action Systems (with Martin Feinberg), SIAM Journal on Applied Mathematics 70(6): 1859-1877, 2010. PDF
□ Algebraic methods for inferring biochemical networks: a maximum likelihood approach, (with Casian Pantea and Grzegorz A. Rempala), Computational Biology and Chemistry, 33(5):361-367,
2009. PDF
□ Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements, (with Murad Banaji), Communications in Mathematical Sciences 7:4, 867–900, 2009. PDF
□ Time course analysis of microarray data for the pathway of reproductive development in female rainbow trout, (with Yushi Liu, Joseph Verducci, Irvin Schultz, Sharon Hook, James Nagler,
Kaitlin Sundling, and William Hayton), Statistical Analysis and Data Mining, 2:3, 192-208, 2009. PDF
□ Toric Dynamical Systems, (with Alicia Dickenstein, Anne Shiu, Bernd Sturmfels), Journal of Symbolic Computation 44:11, 1551–1565 , 2009. PDF
□ Homotopy methods for counting reaction network equilibria, (with J. William Helton, Ruth J. Williams), Mathematical Biosciences 216:2, 140-149, 2008. PDF
□ A lifespan-extending form of autophagy employs the vacuole-vacuole fusion machinery, (with Fusheng Tang, Joseph Watkins, Maria Bermudez, Russell Gray, Adam Gaban, Ken Portie, Stephen
Grace, Maurice Kleve), Autophagy 4:7, 874-886, 2008. PDF
□ Multigraph conditions for multistability, oscillations and pattern formation in biochemical reaction networks, (with Maya Mincheva), Proceedings of IEEE 96:8, 1281-1291, 2008. PDF
□ A Combinatorial H4 Tail Library for Exploring the Histone Code, (with Adam L. Garske, John M. Denu), Biochemistry 47:31, 8094-8102, 2008. PDF
□ Valence Parity Renders z-type ions Chemically Distinct, (with Shane Hubler, April Jue, Jason Keith, Graeme McAlister, Joshua Coon), Journal of the American Chemical Society 130:20,
6388-6394, 2008. PDF
□ Identifiability of chemical reaction networks, (with Casian Pantea), Journal of Mathematical Chemistry 44:1, 244-259, 2008. PDF
□ Understanding Bistability in Complex Enzyme-Driven Reaction Networks, (with Martin Feinberg and Yangzhong Tang), Proceedings of the National Academy of Sciences 103:23, 8697-8702, 2006.
□ Approximate Traveling Waves in Linear Reaction-Hyperbolic Equations, (with Avner Friedman), SIAM Journal on Mathematical Analysis 38:3, 741-758, 2006. PDF
□ Multiple Equilibria in Complex Chemical Reaction Networks: Extensions to Entrapped Species Models, (with Martin Feinberg), IEE Proceedings Systems Biology 153:4, 179-186, 2006. PDF
□ Multiple Equilibria in Complex Chemical Reaction Networks: II. The Species-Reactions Graph, (with Martin Feinberg), SIAM Journal on Applied Mathematics 66:4, 1321-1338, 2006. PDF
□ Multiple Equilibria in Complex Chemical Reaction Networks: I. The Injectivity Property, (with Martin Feinberg), SIAM Journal on Applied Mathematics 65:5, 1526-1546, 2005. PDF
□ A Model of Intracellular Transport of Particles in an Axon, (with Avner Friedman), Journal of Mathematical Biology, 51:2, 217-246, 2005. PDF
□ A Distributed Parameter Identification Problem in Neuronal Cable Theory Models, (with Jonathan Bell), Mathematical Biosciences 194:1, 1-19, 2005. PDF
□ Data Sources and Computational Approaches for Generating Models of Gene Regulatory Networks, (with Baltazar Aguda and Rengul Cetin-Atalay), Reviews in Computational Chemistry, Vol. 21,
edited by Kenny Lipkowitz, Raima Larter, and Thomas R. Cundari, 2005. PDF
□ Spatial Domain Wavelet Design for Feature Preservation in Computational Datasets, (with Ming Jiang, David Thompson, and Raghu Machiraju), accepted by IEEE Transactions on Visualization
and Computer Graphics 11:2, 149-159, 2005. PDF
□ Controlling Release from the Lipidic Cubic Phase by Selective Alkylation, (with Jeffrey Clogston, David Hart, and Martin Caffrey), Journal of Controlled Release, 102:2, 441-461, 2005. PDF
□ Everybody Else Is: Networks, Power Laws and Peer Contagion in the Aggressive Recess Behavior, (with Keith Warren and Dawn Anderson-Butcher), Nonlinear Dynamics, Psychology, and Life
Sciences, 9:2, 155-173, 2005.
□ Mathematical Analysis of a Modular Network Coordinating the Cell Cycle and Apoptosis, (with Baltazar Aguda and Avner Friedman), Mathematical Biosciences and Engineering, 2:3, 473-485,
2005. PDF
□ A dynamical system model of neurofilament transport in axons, (with Anthony Brown and Avner Friedman), Journal of Theoretical Biology 237(3) 316-322, 2005. PDF
□ Mathematical Models of the Cell Cycle and Apoptosis, (with David Goulet, Namyong Lee, Taras Odushkin, Galen Cook Wiens), MBI Technical Report, 2004.
□ Image Segmentation using Neuronal Oscillators, (with Talia Konkle, Ning Jiang, Jie Zhang, Fatma Gurel, Christopher Scheper), MBI Technical Report, 2003.
□ Physics-based feature mining for large data exploration, (with D.S. Thompson, R.K. Machiraju, M. Jiang, J.S. Nair, S.S.D. Venkata), Computing in Science and Engineering 4:4, 22-30, 2002.
□ The Output of Chemical Reactors Is Almost Always Unique. Why ?, Proceedings of the 14th Annual Edward F. Hayes Research Forum, p. 184-189, 2001.
□ A Framework for Filter Design Emphasizing Multiscale Feature Preservation, (with David Thompson, Raghu Machiraju and Ming Jiang), Proceedings of the AHPCRC and CASC/LLNL Third Workshop on
Mining Scientific Datasets , p. 105-111, 2001.
□ The Dimension Print of Most Convex Surfaces, (with Tudor Zamfirescu), Monatshefte fur Mathematik 123:3, 203-207, 1997. PDF
□ Most Homeomorphisms of the Circle are Semiperiodic, (with Paul Horja, Mihai Prunescu, Tudor Zamfirescu), Archiv der Mathematik 64:5, 452-458, 1995. PDF
□ Generic Properties of the Homeomorphisms of Spheres, Archiv der Mathematik 62:4, 349-353, 1994.
□ On a Covering Theorem, Mathematical Reports 44:2, 109-112, 1992. | {"url":"https://people.math.wisc.edu/~craciun/","timestamp":"2024-11-02T21:38:27Z","content_type":"text/html","content_length":"252618","record_id":"<urn:uuid:c5f318e2-cf55-4423-80e5-e5028aff1106>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00358.warc.gz"} |
Postulates of quantum mechanics - Mono Mole
Postulates of quantum mechanics
The postulates of quantum mechanics are fundamental mathematical statements that cannot be proven. Nevertheless, they are statements that everyone agrees with.
Examples of other postulates of science and mathematics are Newton’s 2^nd law, $F=ma$, and the Euclidean statement that a line is defined by two points respectively.
Generally, the postulates of quantum mechanics are expressed as the following 6 statements:
1) The state of a physical system at a particular time $t$ is represented by a vector in a Hilbert space.
We call such a vector, a wavefunction $\psi(x,t)$, which is a function of time and space . A wavefunction contains all assessable physical information about a system in a particular state. For
example, the energy of the stationary state of a system is obtained by solving the time-independent Schrodinger equation $\hat{H}\psi=E\psi$.
2) Every measurable physical property of a system is described by a Hermitian operator $\hat{O}$ that acts on a wavefunction representing the state of the system.
The most well-known Hermitian operator in quantum mechanics is the Hamiltonian $\hat{H}$, which is the energy operator. The wavefunctions that $\hat{H}$ acts on are called eigenfunctions.
Eigenfunctions of a Hermitian operator in quantum mechanics are further postulated to form a complete set. Other Hermitian operators frequently encountered in quantum mechanics are the momentum
operator $\hat{p}$ and the position operator $\hat{x}$.
3) The result of the measurement of a physical property of a system must be one of the eigenvalues of an operator $\hat{O}$.
The state of a system is expressed as a wavefunction, which can be a single basis wavefunction or a linear combination of a complete set of basis wavefunctions. Since basis wavefunctions of a
Hermitian operator form a complete set, all wavefunctions can be written as a linear combination of basis wavefunctions.
What about a system described by a stationary state?
The wavefunction can be expressed, though trivially, as $\psi=\sum_{n=0}^{\infty}c_n\phi_n$, where $c_i=1$ and $c_{not\equiv&space;i}=0$ (we have assumed that the spectrum is discrete, i.e. the
eigenvalues are separated from one another).
It is generally accepted by scientists that the values of the coefficients $c_n$ are unknown prior to a measurement. Upon measurement, the result obtained is an eigenvalue associated with one of the
eigenfunctions, and hence the phrase ‘the initial wavefunction collapses to one of the eigenfunctions’. This implies that the measurement alters the initial wavefunction such that a 2^nd measurement,
if made quickly, yields that same result (this obviously refers to wavefunctions describing non-stationary states, as wavefunctions of stationary states are independent of time). If we prepare a
large number of identical systems and simultaneously measure them, the values of $c_n$ are found; with $\sum_{n=0}^{\infty}\vert&space;c_n\vert^{2}=1$, and the expectation value of the measurements
being $\langle\psi\vert\hat{O}\vert\psi\rangle$.
4) The probability of obtaining an eigenvalue $E_i$ upon measuring a system is given by the square of the inner product of the normalised $\psi$ with the orthonormal eigenfunction $\phi_i$.
In other words,
If the spectrum is continuous, $\psi=\int_{-\infty}^{\infty}c_n\phi_n\,&space;dn$ and the probability of obtaining an eigenvalue in the range $dn$ is $\vert\langle\phi_n\vert\psi\rangle&space;\vert^
5) The state of a system immediately after a measurement yielding the eigenvalue $E_i$ is described by the normalised eigenfunction $\phi_i$.
We have explained in the postulate 3 that this is commonly known as the collapse of the wavefunction $\psi$ to one of the eigenfunctions $\phi_i$. It is also known as the projection of $\psi$ onto $\
phi_i$, i.e. $\hat{P}_i\vert\psi\rangle$; or if $\psi$ not is normalised,
Show that $\sqrt{\langle\psi\vert\hat{P}_i\vert\psi\rangle}$ is the normalisation constant.
To normalise a wavefunction,
6) The time evolution of the wavefunction $\psi(t)$ is governed the time-dependent Schrodinger equation $i\hbar\frac{d}{dt}\vert\psi(t)\rangle=\hat{H}(t)\vert\psi(t)\rangle$. | {"url":"https://monomole.com/postulates-quantum-mechanics/","timestamp":"2024-11-03T09:29:43Z","content_type":"text/html","content_length":"109898","record_id":"<urn:uuid:42e3d341-5913-4c0e-b0a2-a8af37493710>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00193.warc.gz"} |
Search for the lepton-flavor violating decay of the Higgs boson and additional Higgs bosons in the Formula Presented final state in proton-proton collisions at Formula Presented
A search for the lepton-flavor violating decay of the Higgs boson and potential additional Higgs bosons with a mass in the range 110-160 GeV to an Formula Presented pair is presented. The search is
performed with a proton-proton collision dataset at a center-of-mass energy of 13 TeV collected by the CMS experiment at the LHC, corresponding to an integrated luminosity of Formula Presented. No
excess is observed for the Higgs boson. The observed (expected) upper limit on the Formula Presented branching fraction for it is determined to be Formula Presented at 95% confidence level, the most
stringent limit set thus far from direct searches. The largest excess of events over the expected background in the full mass range of the search is observed at an Formula Presented invariant mass of
approximately 146 GeV with a local (global) significance of 3.8 (2.8) standard deviations.
深入研究「Search for the lepton-flavor violating decay of the Higgs boson and additional Higgs bosons in the Formula Presented final state in proton-proton collisions at Formula Presented」主題。共同 | {"url":"https://scholars.ncu.edu.tw/zh/publications/search-for-the-lepton-flavor-violating-decay-of-the-higgs-boson-a","timestamp":"2024-11-07T00:27:02Z","content_type":"text/html","content_length":"57946","record_id":"<urn:uuid:0a37309a-f5d1-4671-90b3-a3f60f8119da>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00004.warc.gz"} |
What is Paryavarta Yojayet? Vedic Maths Sutras - Humsa School
Paryavarta Yojayet is another important Vedic Maths Sutra that we are going to learn. Paryavarta Yojayet translates to Transpose and Apply. Transpose means to change the sign of a given number i.e
Subtraction becomes Addition and Vice- versa. These numbers are then applied to the process. It is used to divide numbers where the divisor is slightly greater than any power of 10 and starts with 1.
The Vinculum process that we learned earlier will also be applied numerous times while we apply this method of division.
Table of Contents
What is Vinculum Process?
Vinculum numbers are those numbers that have 1 negative digit or have a Bar over them (indicating that it is a negative digit).
Through the Place Value System in Vedic Math normal numbers are written as:
2345= 2000+300+40+5
These when they have a bar over them can also be converted into normal numbers. For eg-:
2345= 2000+300-40+5 = 2265
Hence in the above example, the normal number for the above equation is 2265.
Another approach for the same is a process where we start from the right and go to the left. This is a trickier method by is very useful for numbers that have more digits.
Suppose we take the number 52113278235.
Now to convert this into a normal number, we will follow the below-given steps and move from right to left.
1. Find the 1^st bar digit in the above-given number which in this case is 3 (from the right) and take its 10’s complement.
2. To find the “complement” of a number we subtract the given digit from the number. For eg: The 10’s complement for the number 7 is 3 (10-7)
3. A) If the next digit is a bar digit, take its 9’s complement. Continue until you reach a non-bar digit.
3. B) Decrement non-bar digit by 1.
4. Continue steps 1) and 2) until the whole number is covered.
Therefore, the normal number in the above case will come out to be 47913122175.
Now that we’ve gotten a quick recap of the Vinculum Process, we can now move on to learning this particular Sutra which is called Paryavarta Yojayeta.
Suppose we take 2 numbers:
587 and 11.
a. Make two columns and divide the dividend into two parts. Always remember that the 2^nd part of the number should be a digit less than the divisor. In this case, the number will be divided into
(1) 58 and (2) 7.
b. The divisor can be written as 11/1.
c. Now, as our sutra says we have to transpose and apply. Therefore, for the first step, we will pull the number 5 down as shown below.
Figure 1
a. This 5 will be multiplied by 1(from the divisor 11/1) according to the sutra.
b. (5*1) will give us 5 which will be written below 8 from the dividend as shown below.
Figure 2
• Now, as we perform the process in modern division methods, this 5 will be subtracted from 8 giving us 3.
• 3 will again be multiplied by 1 from the divisor and its answer will be subtracted from the remaining digit of the dividend i.e 7 which will be equal to 4.
• Hence, we now have our quotient and remainder for the following equation.
• Quotient = 53, Remainder = 4
Figure 3
Paryavarta Yojayet
Let’s try this with another example:
Suppose we take 2 numbers:
253 and 11.
• Make two columns and divide the dividend into two parts. Always remember that the 2^nd part of the number should be a digit less than the divisor. In this case, the number will be divided into
(1) 25 and (2) 3.
• The divisor can be written as 11/1.
• Now, as our sutra says we have to transpose and apply. Therefore, for the first step, we will pull the number 2 down as shown below.
Figure 4
• This 2 will be multiplied by 1(from the divisor 11/1) according to the sutra.
• (2*1) will give us 2 which will be written below 5 from the dividend as shown below.
• Now, as we perform the process in modern division methods, this 2 will be subtracted from 5 giving us 3.
• 3 will again be multiplied by 1 from the divisor and its answer will be subtracted from the remaining digit of the dividend i.e 3 which will be equal to 0.
• Hence, we now have our quotient and remainder for the following equation.
• Quotient = 23, Remainder = 0
Figure 5
With this, we’ve learnt the trick for the division of 3 digit numbers. Fun and quick, isn’t it?
This method can also be applied for the division of numbers with a greater number of digits. We’ll now see how this works-
Vedic algebra:- Paryavarta Yojayet
Suppose we take two numbers;
28732 and 112, where 28732 is the dividend and 112 is the divisor
• Divide the number into two parts. The 2^nd part of the number is a digit lesser than the divisor. Therefore, the number is divided into (1) 287 and (2) 32.
• For the divisor, the first digit is ignored and the other digits are transposed. Therefore from the number 112, the first 1 is ignored and the remaining 1 and 2 that are positive will be
converted into negative digits ie. Bar 1 and Bar 2.
• The divisor will now be written as 112/ 1 2
• Let us start the division process. Bring down 2 and multiply it by Bar 1 and Bar 2. The answer will be Bar 2 and Bar 4(write it below 7) as shown here.
• As the normal division process takes place, subtract the next equation i.e. 8 – Bar 2. The answer is 6.
• Again, multiply 6 by Bar 1 and Bar 2. This will give you Bar 6 and Bar 12. Carry on the multiplication and subtraction procedure till you reach the last digits.
• The first answer would come out to be 263 as quotient 68 as remainder.
• Using the vinculum method our final answer can be written as 25660.
Figure 6
Hope you learned about Paryavarta Yojayet sutra. Keep practising till you become an expert in this. Paryavarta Yojayet is one of the many Sutras that’ll help you in the quick division.
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BinaryNewbie on 1:17 AM 07/25/2019: Hey @dosisod, does this crackme need a sat solver? Or does it have a solution? I tried and i figured out the algorithm, but i stuck after that.
dosisod on 7:05 AM 07/26/2019: @BinaryNewbie what is a sat solver? and yes, there is a solution, but it is going to take some tinkering as the seed isnt just "hidden away" in the code somewhere
BinaryNewbie on 5:54 PM 07/26/2019: @dosisod, so for a sat solver: https://en.wikipedia.org/wiki/Boolean_satisfiability_problem Yeah, i noticed that a simple bruteforce doesn't worked, but i will
analyse the algorithm again, thnks.
gnitargetnisid on 3:34 PM 07/28/2019: I've found the seed by some educated guessing and brute force. But I'm wondering if there's an algorithm or a formula which will output the seed, I couldn't come
up with anything given the dependency on the original seed and the mixing of logical and arithmetic operators. Maybe I'm missing something.
dosisod on 6:40 PM 07/30/2019: @gnitargetnisid, no algorithm/keygen is required, any method that produces a valid seed is alright. If you have a solution/seed id love to see how you got it!
skudo on 5:26 PM 08/03/2019: That was fun to solve! I reverse-engineered the algorithm, implemented it in a c++ code and ran with all integers in the int32 range. Here is the code if someone is
interested. You just have to run it about 10min then it should be finished... https://github.com/skudoxy/ChainbreakerSolver
BinaryNewbie on 8:45 PM 08/03/2019: @dosisod, my code was doing the wrong stuff akkaka, i've noticed after some trial-error, that was a curious pattern in huge numbers kakak and i tried with positive
integers, with my crap corrected, and nothing, so i decided to run against negative integers and voilá. One more question, why did you ignore the 0 seed?
dosisod on 4:25 AM 08/10/2019: late response, but i saw your git repo. that seed was also the only valid seed i could find. tommorrow ill log into my github and star it, it was fun to make, hopefully
it was as fun to solve as well!
dosisod on 4:49 AM 08/18/2019: @BinaryNewbie I made the program quit if 0 was reached at any point since 0 causes any XOR, multiplication etc. to return 0, killing the fun in cracking it IMO
BinaryNewbie on 3:43 PM 08/18/2019: thanks for answering ahhah, i thought that was an easter egg or something like that.
janbbeck on 5:34 PM 01/15/2020: Thanks for this crackme. I put up my solution here: https://www.janbeck.com/cybersecurity-challenges-ctfs-and-more/angr-hooking-derecompiling-chainbreaker I could not
get angr to solve this, but I am curious how close the decompiler got to the original source code. Could you post it? | {"url":"https://crackmes.one/crackme/5d2be1e733c5d410dc4d0d35","timestamp":"2024-11-10T05:55:08Z","content_type":"text/html","content_length":"14167","record_id":"<urn:uuid:8b70786f-0d14-43f0-8f6a-153d70f6f35d>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00254.warc.gz"} |
In Simplest Form Is _______.
Step-by-step explanation:
The greatest common factor of 25/35 is 5, 25/5=5, 35/5=7, therefore your answer is 5/7.
Hi, let's solve our equation.
we can use multiple methods to solve this problem.
Distributive property Regular multiplication Fraction division
Here is how it would look to do distributive property
Follow PEMDAS rule
Parenthesis Exponents Multiplication or Division Addition or Subtraction
Solve it and we get the answer of 144.
just multiply [tex]12\cdot12[/tex] which equals 144
here is how it would look like.
Follow the rule, also known as keep change flip, which makes it look like
[tex]\frac{12}{1}\cdot\frac{12}{1}[/tex] which equals 144 | {"url":"https://community.carbonfields.net/question-handbook/in-simplest-form-is-hiro","timestamp":"2024-11-13T00:46:42Z","content_type":"text/html","content_length":"74390","record_id":"<urn:uuid:0cda2e50-eed1-46d7-9720-efb1479153a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00607.warc.gz"} |
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Concepts and Code: Machine Learning with R Programming UNIT 5: CLUSTERING The general idea of a clustering algorithm is to partition a given dataset into distinct, exclusive clusters so that the data
points in each group are quite similar to each other and are meaningful. Clustering falls into the unsupervised learning algorithms. Meaningfulness purely depends on the intention behind the purpose
of the group’s formations. Suppose we have 100 articles and we want to group them into different categories. Let’s consider the below categories: Sports articles, Business articles and Entertainment
articles. When we group all the 100 articles into the above 3 categories. All the articles belong to the sports category will be same, In the sense, the content in the sports articles belongs to
sports category. When you pick an article from sports category and the other article from business articles. Content-wise they will be completely different. This summarizes the rule of thumb
condition to form clusters. Much of the history of cluster analysis is concerned with developing algorithms that were not too computer intensive, since early computers were not nearly as powerful as
they are today. Accordingly, computational shortcuts have traditionally been used in many cluster analysis algorithms. These algorithms have proven to be very useful, and can be found in most
computer software. More recently, many of these older methods have been revisited and updated to reflect the fact that certain computations that once would have overwhelmed the available computers
can now be performed routinely. In R, a number of these updated versions of cluster analysis algorithms are available through the cluster library, providing us with a large selection of methods to
perform cluster analysis, and the possibility of comparing the old methods with the new to see if they really provide an advantage. Lets look at some of the examples. K-MEANS CLUSTERING K-Means
clustering approach is very popular in a variety of domains. In biology it is often used to find structure in DNA-related data or subgroups of similar tissue samples to identify cancer cohorts. In
marketing, K-Means is often used to create market/customer/product segments. One of the first steps in building a K-Means clustering work is to define the number of clusters to work with.
Subsequently, the algorithm assigns each individual data point to one of the clusters in a random fashion. Details Steps involved in K-means are: 1. Choose the K-number of clusters. Each data point
is randomly assigned to a cluster. 2. Select random K points, the initial set of centroids (not necessarily from the dataset) 3. Assign each data point to the closet centroid that forms K clusters 4.
Compute and place the new centroids of each cluster 61 | P a g e Concepts and Code: Machine Learning with R Programming 5. Reassign each data point to the new closest centroid. 6. If there is any
reassignment of clusters, go to Step 4 or else Stop. The underlying idea of the algorithm is that a good cluster is the one which contains the smallest possible within-cluster variation of all
observations in relation to each other. The most common way to define this variation is using the squared Euclidean distance. This process of identifying groups of similar data points can be a
relatively complex task since there is a very large number of ways to partition data points into clusters. Let’s have a look at an example in R using the Chatterjee-Price Attitude Data from the
library(datasets) package. The dataset is a survey of clerical employees of a large financial organization. The data are aggregated from questionnaires of approximately 35 employees for each of 30
(randomly selected) departments. The numbers give the percent proportion of favorable responses to seven questions in each department. For more details, see ?attitude. # # K-Means Clustering #
Importing the dataset # Load necessary libraries library(datasets) # Inspect data structure str(attitude) # Summarise data summary(attitude) # Splitting the dataset into the Training set and Test set
Not required in Clustering # Feature Scaling Not required This data gives the percent of favorable responses for each department. For example, one department had only 30% of responses favorable when
it came to assessing ‘privileges’ and one department had 83% of favorable responses when it came to assessing ‘privileges’, and a lot of other favorable response levels in between. When performing
clustering, some important concepts like the data in hand should be standardized, whether the number of clusters obtained are truly representing the underlying pattern found in the data, whether
there could be other clustering algorithms or parameters to be taken, etc. It is often recommended to perform clustering algorithms with different approaches and preferably test the clustering
results with independent datasets. Particularly, it is very important to be careful with the way the results are reported and used. For simplicity, we’ll take a subset of the attitude dataset and
consider only two variables in our K-Means clustering exercise. So imagine that we would like to cluster the attitude dataset with the responses from all 30 departments when it comes to ‘privileges’
and ‘learning’ and we would like to understand whether there are commonalities among certain departments when it comes to these two variables. # # Subset the attitude data dat = attitude[,c(3,4)] #
Plot subset data plot(dat, main = "% of favourable responses to Learning and Privilege", pch =20, cex =2) 62 | P a g e Concepts and Code: Machine Learning with R Programming Now we can apply K-Means
clustering to this data set and try to assign each department to a specific number of clusters that are “similar”. Let’s use the kmeans function from R base stats package: # # Perform K-Means with 2
clusters set.seed(123) km1 = kmeans(x =dat, centers = 2, nstart=100) # Plot results plot(dat, col =(km1$cluster +1) , main="K-Means result with 2 clusters", pch=20, cex=2) As mentioned before, one of
the key decisions to be made when performing K-Means clustering is to decide on the numbers of clusters to use. In practice, there is no easy answer and it’s important to try different ways and
numbers of clusters to decide which options is the most useful, applicable or interpretable solution. However, one solution often used to identify the optimal number of clusters is called the Elbow
method and it involves observing a set of possible numbers of clusters relative to how they minimize the within-cluster sum of squares. In other words, the Elbow method examines the within-cluster
dissimilarity as a function of the number of clusters. # # Using the elbow method to find the optimal number of clusters mydata <- dat wss <- (nrow(mydata)-1)*sum(apply(mydata,2,var)) for (i in 2:15)
wss[i] <- sum(kmeans(mydata, centers=i)$withinss) plot(1:15, wss, type="b", xlab="Number of Clusters", ylab="Within groups sum of squares", main="Assessing the Optimal Number of Clusters with the
Elbow Method", pch=20, cex=2) With the Elbow method, the solution criterion value (within groups sum of squares) will tend to decrease substantially with each successive increase in the number of
clusters. Simplistically, an optimal number of clusters is identified once a “kink” in the line plot is observed and this is very subjective. 63 | P a g e Concepts and Code: Machine Learning with R
Programming But from the example above, we can say that after 6 clusters the observed difference in the within-cluster dissimilarity is not substantial. Consequently, we can say with some reasonable
confidence that the optimal number of clusters to be used is 6. # # Perform K-Means with the optimal number of clusters identified from the Elbow method set.seed(321) km2 = kmeans(dat, 6, nstart=100)
# Examine the result of the clustering algorithm km2 # # Plot results plot(dat, col =(km2$cluster +1) , main="K-Means result with 6 clusters", pch=20, cex=2) From the results above we can see that
there is a relatively well defined set of groups of departments that are relatively distinct when it comes to answering favorably around Privileges and Learning in the survey. It is only natural to
think the next steps from this sort of output. One could start to devise strategies to understand why certain departments rate these two different measures the way they do and what to do about it.
But we will leave this to another exercise. PARTITIONING AROUND MEDOIDS (PAM) The k-means technique is fast, and doesn't require calculating all of the distances between each observation and every
other observation. It can be written to efficiently deal with very large data sets, so it may be useful in cases where other methods fail. On the down side, if you rearrange your data, it's very
possible that you'll get a different solution every time you change the ordering of your data. The R cluster library provides a modern alternative to k-means clustering, known as pam, which is an
acronym for "Partitioning around Medoids". The term medoid refers to an observation within a cluster for which the sum of the distances between it and all the other members of the cluster is a
minimum. pam requires that you know the number of clusters that you want (like k-means clustering), but it does more computation than k-means in order to insure that the medoids it finds are truly
representative of the observations within a given cluster. 64 | P a g e Concepts and Code: Machine Learning with R Programming Implementation in R # pam: Advanced version of K-Means algorithm #
Importing the dataset # Load necessary libraries library(datasets) library(cluster) dat.pam = attitude[,c(3,4)] set.seed(123) cluster.pam = pam(dat.pam,2) names(cluster.pam) cluster.pam Like most R
objects, you can use the names function to see what else is available. Further information can be found in the help page for pam.object. names(cluster.pam) Plot the result # Plot results plot(dat,
col =(cluster.pam$cluster +1) , main="PAM result with 2 clusters", pch=20, cex=2) Using table function to compare the result: #We can use table to compare the results of the kmeans and pam solutions:
table(km1$cluster,cluster.pam$clustering) Analysis of the result Below are confusion matrix and plots, first one was run with 2 clusters and second one was run with 4 clusters: The solutions seem to
agree, except for 1 observation. 65 | P a g e Concepts and Code: Machine Learning with R Programming HIERARCHICAL CLUSTERING Hierarchical clustering is an alternative approach to k-means clustering
for identifying groups in the dataset and does not require to pre-specify the number of clusters to generate. It refers to a set of clustering algorithms that build tree-like clusters by successively
splitting or merging them. This hierarchical structure is represented using a tree. Hierarchical clustering methods use a distance similarity measure to combine or split clusters. The recursive
process continues until there is only one cluster left or we cannot split more clusters. We can use a dendrogram to represent the hierarchy of clusters. Hierarchical classifications produced by
either Agglomerative (Bottom up approach) or Divisive (Top down approach). Agglomerative clustering: It’s also known as AGNES (Agglomerative Nesting). It works in a bottom-up manner. Each object is
initially considered as a single-element cluster (leaf). At each step of the algorithm, the two clusters that are the most similar are combined into a new bigger cluster (nodes). This procedure is
iterated until all points are member of just one single big cluster (root). The result is a tree which can be plotted as a dendrogram. Divisive hierarchical clustering: It’s also known as DIANA
(Divise Analysis) and it works in a top-down manner. The algorithm is an inverse order of AGNES. It begins with the root, in which all objects are included in a single cluster. At each step of
iteration, the most heterogeneous cluster is divided into two. The process is iterated until all objects are in their own cluster. Figure 25: Agglomerative and Divisive clustering algorithms Note
that agglomerative clustering is good at identifying small clusters. Divisive hierarchical clustering is good at identifying large clusters. How do we measure the dissimilarity between two clusters
of observations? A number of different methods have been developed. The most common types methods are: Maximum or complete linkage clustering: It computes all pairwise dissimilarities between the
elements in cluster 1 and the elements in cluster 2, and considers the largest value (i.e., maximum value) of these dissimilarities as the distance between the two clusters. It tends to produce more
compact clusters. Minimum or single linkage clustering: It computes all pairwise dissimilarities between the elements in cluster 1 and the elements in cluster 2, and considers the smallest of these
dissimilarities as a linkage criterion. It tends to produce long, “loose” clusters. 66 | P a g e Concepts and Code: Machine Learning with R Programming Mean or average linkage clustering: It
computes all pairwise dissimilarities between the elements in cluster 1 and the elements in cluster 2, and considers the average of these dissimilarities as the distance between the two clusters.
Centroid linkage clustering: It computes the dissimilarity between the centroid for cluster 1 (a mean vector of length p variables) and the centroid for cluster 2. Ward’s minimum variance method:
It minimizes the total within-cluster variance. This method does not directly define a measure of distance between two points or clusters. It is an ANOVA based approach. At each stage, those two
clusters merge, which provides the smallest increase in the combined error sum of squares from one- way univariate ANOVAs that can be done for each variable with groups defined by the clusters at
that stage of the process. The different approaches produce different dendrograms. Its left to the readers to plot and check different dendrograms. The code for the plotting and interpreting the
dendrogram is given later in this section. Let’s see the implementation in R. Step 1 is to prepare for the dataset. We’ll use the built-in R data set USArrests, which contains statistics in arrests
per 100,000 residents for assault, murder, and rape in each of the 50 US states in 1973. It includes also the percent of the population living in urban areas. Below code is common for both
Agglomerative and Divisive hierarchical clustering algorithms. 1. Preparing the data set: # # Common code for both types of algorithm # Libraries required: library(cluster) # clustering algorithms
library(factoextra) # clustering visualization library(dendextend) # for comparing two dendrograms # Read in-built dataset library(Matrix) df <- USArrests #To remove any missing value: df <- na.omit
(df) #Scaling the dataset df <- scale(df) head(df) AGGLOMERATIVE HIERARCHICAL CLUSTERING Hierarchical agglomerative clustering methods, starts out by putting each observation into its own separate
cluster. It then examines all the distances between all the observations and pairs together the two closest ones to form a new cluster. This is a simple operation, since hierarchical methods require
a distance matrix, and it represents exactly what we want - the distances between individual observations. So finding the first cluster to form simply means looking for the smallest number in the
distance matrix and joining the two observations that the distance corresponds to into a new cluster. Now there is one less cluster than there are observations. To determine which observations will
form the next cluster, we need to come up with a method for finding the distance between an existing cluster and individual observations. 67 | P a g e Concepts and Code: Machine Learning with R
Programming Performing the clustering: The commonly used functions are: hclust [in stats package] and agnes [in cluster package] for agglomerative hierarchical clustering First, we compute the
dissimilarity values with dist and then feed these values into hclust and specify the agglomeration method to be used (i.e. “complete”, “average”, “single”, “ward.D”). We can plot the dendrogram
after this. # #####Method 1: agglomerative HC with hclust #### # Dissimilarity matrix diss.at <- dist(df, method = "euclidean") # Hierarchical clustering using Complete Linkage hc.hclust <- hclust
(diss.at, method = "complete" ) # Plot the obtained dendrogram plot(hc.hclust, cex = 0.6, hang = -1) ######## End of Method 1 : agglomerative HC with hclust #### Dendogram will be analyzed later in
this chapter. Alternatively, we can use the agnes function. These functions behave very similarly; however, with the agnes function, we can also get the agglomerative coefficient, which measures the
amount of clustering structure found (values closer to 1 suggest strong clustering structure). # #####Method 2: agglomerative HC with agnes #### # Compute with agnes hc2 <- agnes(df, method =
"complete") # Agglomerative coefficient hc2$ac # Plot the obtained dendrogram pltree(hc2, cex = 0.6, hang = -1, main = "Dendrogram of agnes") ######## End of Method 2 : agglomerative HC with agnes ##
## Agglomerative coefficient allows us to find certain hierarchical clustering methods that can identify stronger clustering structures. In the below example, we see that Ward’s method identifies the
strongest clustering structure of the four methods assessed: # # methods to assess m <- c( "average", "single", "complete", "ward") # function to compute coefficient ac <- function(x) { ac.cal <-
agnes(df, method = x) cat(x, " : " , ac.cal$ac) } #Calling the function ac(m[1]) #Average ac(m[2]) #Single ac(m[3]) # Complete method ac(m[4]) #Ward’s method 68 | P a g e Concepts and Code: Machine
Learning with R Programming DIVISIVE HIERARCHICAL CLUSTERING Divisive clustering is called top-down clustering or divisive clustering. We start at the top with all documents in one cluster. The
cluster is split using a flat clustering algorithm. This procedure is applied recursively until each document is in its own singleton cluster. The basic principle of divisive clustering was published
as the DIANA (DIvisive ANAlysis Clustering) algorithm. Initially, all data is in the same cluster, and the largest cluster is split until every object is separate. DIANA chooses the object with the
maximum average dissimilarity and then moves all objects to this cluster that are more similar to the new cluster than to the remainder. Step 1: Preparing the data set: # # DIVISIVE HIERARCHICAL
CLUSTERING # Read in-built dataset df <- USArrests #To remove any missing value: df <- na.omit(df) #Scaling the dataset df <- scale(df) head(df) Step 2: Performing the clustering: The R function
diana provided by the cluster package allows us to perform divisive hierarchical clustering. diana works similar to agnes; however, there is no method to provide. ## # compute divisive hierarchical
clustering hc.diana <- diana(df) # Divise coefficient; amount of clustering structure found hc.diana$dc ## [1] 0.8514345 # plot dendrogram pltree(hc.diana, cex = 0.6, hang = -1, main = "Dendrogram of
diana") WORKING WITH DENDROGRAMS Let’s look at the dendograms created by our last 3 algorithms. In the dendrogram displayed below, each leaf corresponds to one observation. As we move up the tree,
observations that are similar to each other are combined into branches, which are themselves fused at a higher height. The height of the fusion, provided on the vertical axis, indicates the (dis)
similarity 69 | P a g e Concepts and Code: Machine Learning with R Programming between two observations. The higher the height of the fusion, the less similar the observations are. We can use the
analysis to come up with a good number of clusters. Let's see how. # # # # 70 | P a g e | {"url":"https://pdfhost.io/v/epcGcLw6C_JDA_Corporate_Documentation_Template","timestamp":"2024-11-07T14:05:38Z","content_type":"text/html","content_length":"53535","record_id":"<urn:uuid:40f4bfbc-2fa5-49e7-a56a-e897726f5387>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00286.warc.gz"} |
Problem 3
a) Dirichlet boundary condition at $x=0$ means we need an odd continueation of $u|_{t=0}$ and $u_t|_{t=0}$; That is
$$u|_{t=0}=0,\qquad x<0$$
$$u_t|_{t=0}=-1,\qquad x<0 $$
By application of d'Alembert's formula, solution to this problem is
$$= \frac{1}{2}x$$
b) Neumann boundary condition at $x=0$ means we need an even continueation of $u|_{t=0}$ and $u_t|_{t=0}$; That is
$$u|_{t=0}=0,\qquad x<0$$
$$u_t|_{t=0}=1,\qquad x<0 $$
since both functions are even.
By application of d'Alembert's formula, solution to this problem is
$$= t$$
c) Dirichlet boundary condition at $x=0$ means we need an odd continueation of $u|_{t=0}$ and $u_t|_{t=0}$; That is
$$u|_{t=0}=0,\qquad x<0$$
$$u_t|_{t=0}=x,\qquad x<0 $$
since both functions are odd.
By application of d'Alembert's formula, solution to this problem is
$$= \frac{1}{8}\Bigl[(x+2t)^2+(x-2t)^2\Bigr]$$
d) Neumann boundary condition at $x=0$ means we need an even continueation of $u|_{t=0}$ and $u_t|_{t=0}$; That is
$$u|_{t=0}=0,\qquad x<0$$
$$u_t|_{t=0}=-x,\qquad x<0 $$
By application of d'Alembert's formula, solution to this problem is
$$= \frac{1}{8}\Bigl[(x+2t)^2-(x-2t)^2\Bigr]$$
edit: No need to say all solutions are for $x<2t$.
edit: Fixed integral limits. | {"url":"https://forum.math.toronto.edu/index.php?PHPSESSID=btr89b3i91s4kcve19i8suggu4&topic=25.msg120","timestamp":"2024-11-12T17:12:45Z","content_type":"application/xhtml+xml","content_length":"65021","record_id":"<urn:uuid:5fea23d4-70aa-47bd-99b1-7df5c7d372e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00466.warc.gz"} |
Class Attributes
Eggsactly with Fractions Grades: 3rd to 5th Periods: 2 Author: Tracy Y. Hargrove
This activity may require two sessions to complete each of the tasks. First, the students collect data on the class and use that data to generate fractions that describe the class. Because this unit
emphasizes data analysis, the students use knowledge of graphing and statistics to complete this lesson.
Before the students can represent class characteristics using fractions, classroom data must be collected. To begin the data collection process, have the students brainstorm a list of the many ways
the class or group might be described, for example, by gender, hair color, height, those who own pets, and so forth. This list can be used to create a classroom survey for data collection or you may
choose to survey the class with questions from the Class Survey Activity Sheet.
Organize the students into five or six groups and have each group select a question from the survey. Give each group an envelope that contains as many scrap pieces of paper as there are students in
the class. Have each group record its question and answer choices, if appropriate, on the envelope. For example, if the students ask about gender, they should include two choices, male or female. If
it is not possible to identify all the possible choices, the students should leave their question in an open-ended format. For example, a question about types of pets should be left open-ended, as
one might not be able to anticipate the variety of pets represented in the class.
Conduct the survey by passing the envelopes around the room and giving each student a chance to respond. Before starting the survey, have the students remove all the paper from their group's envelope
and leave it at their table. They will use these slips to record and submit their answer to each survey question. Begin the survey by having group members respond to the question on their envelope
first, writing their answer on a slip of paper and placing it in the envelope. When the groups are finished with that question, they should pass their envelope to the next group, and so forth, until
all the students have had a chance to respond to all the questions. (If the students in your class would benefit from getting up and moving around the room, instruct the students to leave the
envelopes at each table and move from table to table to answer the questions.)
Once data are collected, the groups should tally the responses in their envelope, record the number and represent the quantity as a fraction, for example, 12 out of 24 students (12/24 or 1/2) have
blue eyes. Have each group reduce their fractions to lowest terms by finding the greatest common factor. For example, suppose 18/24 (or 3/4) of the class owns a pet. The greatest common factor for 18
and 24 is 6. The students might find it helpful to list all the factors for the numerator and the denominator, 18 and 24 in this example, and locate the greatest common factor. This can be done
strategically by checking in order each pair of factors that when multiplied yield a particular product. For example, to exhaust all the factors of 18, one would begin with 1 × 18, then 2 × 9, then 3
× 6. Since 4 is not a factor, the student would move on to 5 and then to 6. Six has already been generated with 3 × 6. When the student begins to duplicate factors, they know they have exhausted the
For organizational purposes, it is helpful to write the sets of factors in the following manner. The students should list factors on opposite sides (following the format below) with 1 × 18, then 2 ×
9 on the inside of the other factors, then 3 × 6 in the middle. When factors begin to repeat, e.g., 6 × 3, the students know that the list of factors has been exhausted. This list of factors for 18
would be recorded as follows:
Similarly for 24, the list of factors would evolve as follows:
Next, ask the students to list the factors for both numbers one on top of the other so they can easily recognize the common factors. The greatest common factor should be circled. For example:
The students should divide the numerator and denominator by the greatest common factor to reduce the fraction. For example, for 18/24, the students should divide both the numerator and denominator by
6 to reduce the fraction to 3/4.
If it becomes necessary to divide the lesson into two segments, this might be a logical beginning point for the second part of the lesson. Have group members organize their data in a chart and share
it with the class. The students should record all fractional representations and may choose to record appropriate statistics on their chart, for example, mean, median, range, and mode for numerical
Groups may choose to create their bar graph using the Create a Graph Tool from the National Center for Education Statistics. It might be helpful to show the class how to use the bar graph tool before
they begin making their own.
An example of a bar graph of previously collected student data is shown below:
Once the students have created their graph, they should label the data in fractional parts and reduce all fractions to lowest terms. For example, this chart should be labeled with dog being 15/26,
cats being 8/26 or 4/13, birds being 2/26 or 1/13, and 1/26 iguana. Ask students to share their graphs with the class and discuss how they used fractions in collecting the data depicted on each
graph. If necessary, remind students to consider what the fractions represent, how the data was collected, how categories were established, and how finding the lowest common factor simplified the
process of reducing the fraction.
At this stage of the unit, it is important to know whether the students can do the following: Demonstrate understanding that a fraction can be represented as part of a setDescribe a set of objects on
the basis of its fractional componentsIdentify fraction relationships associated with the setReduce a simple fraction to lowest terms Use the students' graphs with fractional representations to make
instructional decisions about the students' understanding.
1. How can you take classroom data and record it as a fraction?
[Record the number of people who fit a particular characteristic as the numerator, and the total number of students in the class as the denominator.]
2. What can you tell about your class based on the data from your survey?
[The numerator and denominator were divided by the same number.]
5. How do you know whether a fraction is in lowest terms?
[The GCF (greatest common factor) of the numerator and denominator is 1.]
Eggsactly with a Dozen Eggs
Grade: 3rd to 5th
Explore fractions with 12 as the denominator.
Eggsactly with Eighteen Eggs
Grade: 3rd to 5th
Examine relationships among the fractions used to describe part of a set of eighteen.
Eggsactly Equivalent
Grade: 3rd to 5th
Explore equivalent fractions.
Another Look at the Set Model using Attribute Pieces
Grade: 3rd to 5th
Use fractions to describe a set of attribute pieces.
Class Attributes
Grade: 3rd to 5th
Conduct a class survey.
Another Look at Fractions of a Set
Grade: 3rd to 5th
Use everyday items to identify fractions. | {"url":"https://www.nctm.org/Classroom-Resources/Illuminations/Lessons/Class-Attributes/","timestamp":"2024-11-11T07:09:28Z","content_type":"text/html","content_length":"177336","record_id":"<urn:uuid:060c43e5-d6b4-4a3a-9ef1-ba89f7425a93>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00828.warc.gz"} |
Typical middle school math progression?
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What is this these days for average, above average, and highly capable students? What do you need to know at the start of these courses? (I realize this is hard to answer.)
I am clueless. DD will be tested for middle school course placement at the end of 5th grade. She is a year ahead in math, though I'm not always convinced it is a true year ahead.
Our school district's highest math track is:
5th grade regular math
6th grade advanced math (I assume this is pre-algebra)
7th grade algebra
8th grade geometry
Then if you don't make it into that, you have a chance for Algebra in 8th grade, or the regular track of taking Algebra in 9th grade. ETA: There is regular 6, 7, 8th grade math in the regular track.
There are apparently several tests and factors for placement, but I do not understand them yet.
Have you read the middle school handbook for your school district. It is probably available online, or you can request a copy. It may have the procedures written into it.
Last edited by howdy; 09/16/14 06:35 AM.
Nowadays many middle schools post their "program of studies" online. At our middle school grade 6 math is undifferentiated. The grade 7 and 8 math courses, in decreasing order of difficulty, are
Advanced Algebra 7
Algebra 7
Advanced Algebra II
Algebra II
Algebra 8
Pre-Algebra 8
At the open house for 6th grade parents, the principal emphasized that the tracks in math and other subjects had some flexibility. Each math course can lead to two or more future math courses,
depending on performance. So Algebra 7 students can in theory go into Advanced Algebra II the following year.
Usual: Common Core 6, 7, 8 in those grades
Moderately accelerated path:
Common Core 6
Common Core 7-8-"Algebra" in two years instead of three
More accelerated:
Common Core 6 in 5th grade
Common Core 7-8-"Algebra" in 6th and 7th grades (by placement test, not only for IDd gifted)
Still more accelerated (elementary option for gifted-identified only):
Common Core 4 and 5 in 4th grade
Common Core 6 in 5th grade
Option to do 7-8-Algebra in two years or in three
Subject acceleration within these sequences is also possible (not common).
Ours is the same but the high school district is in the middle of change. My daughters(7th grade) magnet school track has 7th graders taking geometry but they are the only ones. Next year she will
have Algebra II/pre calculus at the high school. There is a placement test taking for all kids before junior high.
Our district:
Below average:
6th: 6th grade math
7th: 7th grade math
8th: 8th grade math
9th: Prealgebra or Algebra I
6th: 6th grade math
7th: 7th grade math
8th: 8th grade math or Algebra I
9th: Geometry or Algebra I
6th: Prealgebra
7th: Algebra I
8th: Honors Geometry
9th: Honors Algebra II
Students with higher levels of achievement are sometimes placed at a higher level. For example, I know of a 6th grade boy who was placed in Honors Geometry (upon entry from homeschooling). But these
placements are rare, especially among kids who have been in the system since K.
Ours is similar to what most have posted. Three tracks:
regular: 6th grade general math, 7th grade general and 8th pre-Alg
advanced: 6th grade general, 7th pre-Alg, and 8th Alg
gifted: 6th grade pre-Alg, 7th Alg, 8th Geometry
For our school, the IAAT (Iowa Algebra Aptitude Test) and standardized scores were very important. Your daughter should know all grade level material (fractions/ratios/decimals and of course all
facts), know how to interpret graphs/tables, and understand the concept of variables in order to do well in Alg or pre-Alg. The Art of Problem Solving website has some great pre-tests which can help
you see where your daughter is (or you can do above grade level testing through a talent search, which convinced our school to accelerate my son more than any ability testing). The "regular" math
for 6th and 7th grade just seems to go over these basics- percents, ratios, decimals, fractions, reading charts, etc- over and over again for kids who need a lot of repetition. They give the kids
lots and lots of algorithms to solve equations (guess and check, draw a picture, estimate, make a table, etc) and if your dd is good at math she'll be ready to jump off a cliff when she had to draw
a tree diagram or some other such nonsense instead of just calculating the answer. I think it can be really excruciating, so honestly, if you think that she is capable, but maybe has become bored or
forgotten some things, I'd have her review (maybe take those AoPS placement tests) so she isn't bored silly over the next few years.
However, I'd also ask the teacher if this test will be geared towards a specific curriculum. When my ds was skipped ahead in math early on, he was initially given an EM (Everyday Math) test that
required him to show things like partial sums or lattice multiplication and he did poorly. It was only after I intervened and asked that he be given a test that didn't rely on him knowing odd ways
to solve things (that he hadn't learned) and just relied on him actually knowing how to solve things in the normal way. So, you just want to make sure that the test reflects your dd's knowledge of
math, not her knowledge of things specific to your school's math curriculum (and if it is very specific, make sure she reviews that).
Many schools in the Chicago area don't have junior high geometry at all or are just starting to offer it.
I'm not entirely sure about 6th grade because DS skipped it, but they've recently changed it for the kids identified as gifted in math, and the high achievers in math, anyway.
For those gifted and high achieving kids:
6th: 6th/7th grade accelerated math (pre-algebra)
7th: 7th/8th grade accelerated math (pre-algebra, beginning algebra, with hints of geometry but still related to algebra)
8th: Integrated Math I or high school math (from what I can gather it's simply algebra).
For kids not accelerated I think they do pre-algebra until the end of 8th grade.
in 5th grade, ds's teacher spent most of the year teaching fractions and division, the repetition of which had long-lasting negative affects on our ds and nearly turned him off from math forever.
Last edited by KADmom; 09/16/14 07:10 AM.
Currently, we have the high math kids do Algebra in 7th and Geometry in 8th grade. We've seen a few kids do Geometry in 7th and Algebra II in 8th but that its not the norm. Quite frankly the system
does not accommodate this level of acceleration well. Not sure how this will change with the adoption of common core.
1 2 3 … 9 10
Moderated by
Link Copied to Clipboard | {"url":"https://giftedissues.davidsongifted.org/bb/ubbthreads.php/topics/201043/typical-middle-school-math-progression.html","timestamp":"2024-11-03T03:42:07Z","content_type":"text/html","content_length":"78339","record_id":"<urn:uuid:eea80c99-6f49-4789-bd1f-08ad02e67f58>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00344.warc.gz"} |
On Pebbling Graphs by their Blocks
Graph pebbling is a game played on a connected graph G. A player purchases pebbles at a dollar a piece, and hands them to an adversary who distributes them among the vertices of G (called a
configuration) and chooses a target vertex r. The player may make a pebbling move by taking two pebbles off of one vertex and moving one pebble to a neighboring vertex. The player wins the game if he
can move k pebbles to r. The value of the game (G,k), called the k-pebbling number of G, is the minimum cost to the player to guarantee a win. That is, it is the smallest positive integer m of
pebbles so that, from every configuration of size m, one can move k pebbles to any target. In this paper, we use the block structure of graphs to investigate pebbling numbers, and we present the
exact pebbling number of the graphs whose blocks are complete. We also provide an upper bound for the k-pebbling number of diameter-two graphs, which can be the basis for further investigation into
the pebbling numbers of graphs with blocks that have diameter at most two.
arXiv e-prints
Pub Date:
November 2008
□ Mathematics - Combinatorics;
□ 91A43;
□ 05C99
20 pages, 7 figures | {"url":"https://ui.adsabs.harvard.edu/abs/2008arXiv0811.3238C/abstract","timestamp":"2024-11-05T17:39:23Z","content_type":"text/html","content_length":"38026","record_id":"<urn:uuid:756f162b-3462-4e88-9c4a-7d720edf85b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00766.warc.gz"} |
Learn Tree Traversal by Building a Binary Search Tree - Step 16
Tell us what’s happening:
Didn’t I do exactly what it said? Isn’t my indentation correct?
Now, to perform the actual insertion, define an empty insert method within the BinarySearchTree class and give it a self parameter.
Your code so far
class TreeNode:
def __init__(self, key):
self.key = key
self.left = None
self.right = None
class BinarySearchTree:
def __init__(self):
self.root = None
def _insert(self, node, key):
if node is None:
return TreeNode(key)
if key < node.key:
node.left = self._insert(node.left, key)
elif key > node.key:
node.right = self._insert(node.right, key)
return node
# User Editable Region
def _insert(self):
# User Editable Region
Your browser information:
User Agent is: Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/120.0.0.0 Safari/537.36 Edg/120.0.0.0
Challenge Information:
Learn Tree Traversal by Building a Binary Search Tree - Step 16
Hi @Mikie22
Where did the underscore come from?
Happy coding
I did it because the insert call above it also had one. Why would one have it while the other doesn’t?
the underscore in front of a name is a convention to indicate that that method will be used only inside the class, while the one without underscore is used outside. Also there can’t be two methods
with the same exact name.
Thanks for the explanation! | {"url":"https://forum.freecodecamp.org/t/learn-tree-traversal-by-building-a-binary-search-tree-step-16/718471","timestamp":"2024-11-05T10:23:25Z","content_type":"text/html","content_length":"35851","record_id":"<urn:uuid:50eb8c70-7869-4cfb-8000-0a51d3795e53>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00172.warc.gz"} |
I have always been interested in codes and secret methods of communication. Modern day encryption is quite fascinating to me and has led me to develop my own basic encryption using PHP scripts. Two
versions of my scripts exists, one is for sending e-mail or other type of message to a person while protecting their privacy. The other is more of an anonymous message dead drop where a message can
be uploaded and then only retrieved once.
Try it out Here
The basic idea behind modern encryption is that a message can be reduced from plain text into its’ equivalent computer code of 1’s and 0’s each one or zero is know as a bit. Computers use ASCII code
in which each letter is represented by a set of 8 bits. If we convert our plain text message to bits it will allow us to easily perform math equations on them to encrypt or scramble the text.
Likewise we can reverse the process to convert back into plain text.
My encryption uses an XOR function to scramble to bits. The XOR function takes two bits and applies the following rules. (a XOR b = c) If a and b are the same then c = 0, if a = 1 and b = 0, or if a
= 0 and b = 1, c = 1. Basically, if one of the bits is a 1 and the other is a zero the result is a 1, otherwise the result is a 0. First I generate a random “Key” that is 64 bits long. I use this key
to XOR the message. I take the first bit of the key and the first bit of the message and XOR them. The encryption strength behind using XOR in this manor is simple. A + B = C if I told you C = 1 and
then asked you to give me A and B you couldn’t do it, A could = 1 and B could = 0 or just the opposite. You wouldn’t know which is the key and which is the message. As long as the key is random the
message will be encrypted randomly as well. Each bit in the message will either stay the same or be flipped to its opposite. The real beauty of using XOR is that you can take the encrypted message,
put it through the same XOR function again and get back the original message. It flips back all the bits to their starting values.
My first idea was to have a form where users could enter in a message and click submit. The result would be an encrypted message and a key to decrypt it. The message could be sent via text message
while the encrypted text could be sent via e-mail. The recipient could then enter both items into the form and decrypt the message. While this is secure, if the key and message were found by someone
else the message could still be decrypted at any point in the future.
My second idea was a message dead drop. A user could anonymously leave a message that could later be retrieved by another person. As soon as the message was retrieved it would be deleted from the
database it was stored on so it could only be viewed once. The database stores it completely encrypted as well as the message ID being decrypted. Looking at the database it would be impossible for
even me to tell you who left which message. This uses 2, 32bit keys. The person who would like to retrieve the message must have both keys. Having just one key will not help at all in finding or
decrypting the message as a one way hash of the complete 64 bit key is used to retrieve the message from the database in the first place.
Writing this encryption software was fun and educational. I currently do not have any plans to use this beyond creating this example. | {"url":"http://brett-martin.com/projects/encryption/","timestamp":"2024-11-10T04:49:55Z","content_type":"text/html","content_length":"41046","record_id":"<urn:uuid:2fcaa6bb-98dc-4ea3-9b8e-800b19730496>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00339.warc.gz"} |
This object represents a box (a rectangular shape).
The definition of the attributes is: p1 is the lower left point, p2 the upper right one. If a box is constructed from two points (or four coordinates), the coordinates are sorted accordingly.
A box can be empty. An empty box represents no area (not even a point). Empty boxes behave neutral with respect to most operations. Empty boxes return true on empty?.
A box can be a point or a single line. In this case, the area is zero but the box still can overlap other boxes for example and it is not empty.
[const] bool != (const DBox box) Returns true if this box is not equal to the other box
[const] DBox & (const DBox box) Returns the intersection of this box with another box
[const] DBox * (const DBox box) Returns the convolution product from this box with another box
[const] DBox * (double scale_factor) Returns the scaled box
[const] DBox + (const DPoint point) Joins box with a point
[const] DBox + (const DBox box) Joins two boxes
[const] DBox - (const DBox box) Subtraction of boxes
[const] bool < (const DBox box) Returns true if this box is 'less' than another box
[const] bool == (const DBox box) Returns true if this box is equal to the other box
[const] DBox ptr _const_cast Returns a non-const reference to self.
void _create Ensures the C++ object is created
void _destroy Explicitly destroys the object
[const] bool _destroyed? Returns a value indicating whether the object was already destroyed
[const] bool _is_const_object? Returns a value indicating whether the reference is a const reference
void _manage Marks the object as managed by the script side.
void _unmanage Marks the object as no longer owned by the script side.
[const] double area Computes the box area
void assign (const DBox other) Assigns another object to self
[const] DBox bbox Returns the bounding box
[const] double bottom Gets the bottom coordinate of the box
void bottom= (double c) Sets the bottom coordinate of the box
[const] DPoint center Gets the center of the box
[const] bool contains? (double x, Returns true if the box contains the given point
double y)
[const] bool contains? (const DPoint point) Returns true if the box contains the given point
[const] new DBox ptr dup Creates a copy of self
[const] bool empty? Returns a value indicating whether the box is empty
DBox enlarge (double dx, Enlarges the box by a certain amount.
double dy)
DBox enlarge (double d) Enlarges the box by a certain amount on all sides.
DBox enlarge (const DVector enlargement) Enlarges the box by a certain amount.
[const] DBox enlarged (double dx, Enlarges the box by a certain amount.
double dy)
[const] DBox enlarged (double d) Enlarges the box by a certain amount on all sides.
[const] DBox enlarged (const DVector enlargement) Returns the enlarged box.
[const] unsigned long hash Computes a hash value
[const] double height Gets the height of the box
[const] bool inside? (const DBox box) Tests if this box is inside the argument box
[const] bool is_point? Returns true, if the box is a single point
[const] double left Gets the left coordinate of the box
void left= (double c) Sets the left coordinate of the box
DBox move (double dx, Moves the box by a certain distance
double dy)
DBox move (const DVector distance) Moves the box by a certain distance
[const] DBox moved (double dx, Moves the box by a certain distance
double dy)
[const] DBox moved (const DVector distance) Returns the box moved by a certain distance
[const] bool overlaps? (const DBox box) Tests if this box overlaps the argument box
[const] DPoint p1 Gets the lower left point of the box
void p1= (const DPoint p) Sets the lower left point of the box
[const] DPoint p2 Gets the upper right point of the box
void p2= (const DPoint p) Sets the upper right point of the box
[const] double perimeter Returns the perimeter of the box
[const] double right Gets the right coordinate of the box
void right= (double c) Sets the right coordinate of the box
[const] Box to_itype (double dbu = 1) Converts the box to an integer coordinate box
[const] string to_s (double dbu = 0) Returns a string representing this box
[const] double top Gets the top coordinate of the box
void top= (double c) Sets the top coordinate of the box
[const] bool touches? (const DBox box) Tests if this box touches the argument box
[const] Box transformed (const VCplxTrans t) Transforms the box with the given complex transformation
[const] DBox transformed (const DTrans t) Returns the box transformed with the given simple transformation
[const] DBox transformed (const DCplxTrans t) Returns the box transformed with the given complex transformation
[const] double width Gets the width of the box
Signature: [const] bool != (const DBox box)
!= Description: Returns true if this box is not equal to the other box
Returns true, if this box and the given box are not equal
Signature: [const] DBox & (const DBox box)
Description: Returns the intersection of this box with another box
& box: The box to take the intersection with
Returns: The intersection box
The intersection of two boxes is the largest box common to both boxes. The intersection may be empty if both boxes to not touch. If the boxes do not overlap but touch the result may
be a single line or point with an area of zero. Overwrites this box with the result.
(1) Signature: [const] DBox * (const DBox box)
Description: Returns the convolution product from this box with another box
box: The box to convolve with this box.
Returns: The convolved box
The * operator convolves the firstbox with the one given as the second argument. The box resulting from "convolution" is the outer boundary of the union set formed by placing the
second box at every point of the first. In other words, the returned box of (p1,p2)*(q1,q2) is (p1+q1,p2+q2).
Python specific notes:
This method also implements '__rmul__'.
(2) Signature: [const] DBox * (double scale_factor)
Description: Returns the scaled box
scale_factor: The scaling factor
Returns: The scaled box
The * operator scales the box with the given factor and returns the result.
This method has been introduced in version 0.22.
Python specific notes:
This method also implements '__rmul__'.
(1) Signature: [const] DBox + (const DPoint point)
Description: Joins box with a point
point: The point to join with this box.
Returns: The box joined with the point
The + operator joins a point with the box. The resulting box will enclose both the original box and the point.
(2) Signature: [const] DBox + (const DBox box)
Description: Joins two boxes
box: The box to join with this box.
Returns: The joined box
The + operator joins the first box with the one given as the second argument. Joining constructs a box that encloses both boxes given. Empty boxes are neutral: they do not change
another box when joining. Overwrites this box with the result.
Signature: [const] DBox - (const DBox box)
Description: Subtraction of boxes
box: The box to subtract from this box.
- Returns: The result box
The - operator subtracts the argument box from self. This will return the bounding box of the are covered by self, but not by argument box. Subtracting a box from itself will render
an empty box. Subtracting another box from self will modify the first box only if the argument box covers one side entirely.
This feature has been introduced in version 0.29.
Signature: [const] bool < (const DBox box)
< Description: Returns true if this box is 'less' than another box
Returns true, if this box is 'less' with respect to first and second point (in this order)
Signature: [const] bool == (const DBox box)
== Description: Returns true if this box is equal to the other box
Returns true, if this box and the given box are equal
Signature: [const] DBox ptr _const_cast
Description: Returns a non-const reference to self.
Basically, this method allows turning a const object reference to a non-const one. This method is provided as last resort to remove the constness from an object. Usually there is a
good reason for a const object reference, so using this method may have undesired side effects.
This method has been introduced in version 0.29.6.
Signature: void _create
_create Description: Ensures the C++ object is created
Use this method to ensure the C++ object is created, for example to ensure that resources are allocated. Usually C++ objects are created on demand and not necessarily when the
script object is created.
Signature: void _destroy
_destroy Description: Explicitly destroys the object
Explicitly destroys the object on C++ side if it was owned by the script interpreter. Subsequent access to this object will throw an exception. If the object is not owned by the
script, this method will do nothing.
Signature: [const] bool _destroyed?
_destroyed? Description: Returns a value indicating whether the object was already destroyed
This method returns true, if the object was destroyed, either explicitly or by the C++ side. The latter may happen, if the object is owned by a C++ object which got destroyed
Signature: [const] bool _is_const_object?
_is_const_object? Description: Returns a value indicating whether the reference is a const reference
This method returns true, if self is a const reference. In that case, only const methods may be called on self.
Signature: void _manage
Description: Marks the object as managed by the script side.
After calling this method on an object, the script side will be responsible for the management of the object. This method may be called if an object is returned from a C++ function
and the object is known not to be owned by any C++ instance. If necessary, the script side may delete the object if the script's reference is no longer required.
Usually it's not required to call this method. It has been introduced in version 0.24.
Signature: void _unmanage
Description: Marks the object as no longer owned by the script side.
_unmanage Calling this method will make this object no longer owned by the script's memory management. Instead, the object must be managed in some other way. Usually this method may be called
if it is known that some C++ object holds and manages this object. Technically speaking, this method will turn the script's reference into a weak reference. After the script engine
decides to delete the reference, the object itself will still exist. If the object is not managed otherwise, memory leaks will occur.
Usually it's not required to call this method. It has been introduced in version 0.24.
Signature: [const] double area
area Description: Computes the box area
Returns the box area or 0 if the box is empty
Signature: void assign (const DBox other)
Description: Assigns another object to self
Signature: [const] DBox bbox
Description: Returns the bounding box
This method is provided for consistency of the shape API is returns the box itself.
This method has been introduced in version 0.27.
Signature: [const] double bottom
bottom Description: Gets the bottom coordinate of the box
Python specific notes:
The object exposes a readable attribute 'bottom'. This is the getter.
Signature: void bottom= (double c)
bottom= Description: Sets the bottom coordinate of the box
Python specific notes:
The object exposes a writable attribute 'bottom'. This is the setter.
Signature: [const] DPoint center
Description: Gets the center of the box
(1) Signature: [const] bool contains? (double x, double y)
Description: Returns true if the box contains the given point
Returns: true if the point is inside the box.
Tests whether a point (x, y) is inside the box. It also returns true if the point is exactly on the box contour.
(2) Signature: [const] bool contains? (const DPoint point)
Description: Returns true if the box contains the given point
p: The point to test against.
Returns: true if the point is inside the box.
Tests whether a point is inside the box. It also returns true if the point is exactly on the box contour.
Signature: void create
Description: Ensures the C++ object is created
Use of this method is deprecated. Use _create instead
Use this method to ensure the C++ object is created, for example to ensure that resources are allocated. Usually C++ objects are created on demand and not necessarily when the
script object is created.
Signature: void destroy
Description: Explicitly destroys the object
Use of this method is deprecated. Use _destroy instead
Explicitly destroys the object on C++ side if it was owned by the script interpreter. Subsequent access to this object will throw an exception. If the object is not owned by the
script, this method will do nothing.
Signature: [const] bool destroyed?
Description: Returns a value indicating whether the object was already destroyed
Use of this method is deprecated. Use _destroyed? instead
This method returns true, if the object was destroyed, either explicitly or by the C++ side. The latter may happen, if the object is owned by a C++ object which got destroyed
Signature: [const] new DBox ptr dup
dup Description: Creates a copy of self
Python specific notes:
This method also implements '__copy__' and '__deepcopy__'.
Signature: [const] bool empty?
empty? Description: Returns a value indicating whether the box is empty
An empty box may be created with the default constructor for example. Such a box is neutral when combining it with other boxes and renders empty boxes if used in box intersections
and false in geometrical relationship tests.
(1) Signature: DBox enlarge (double dx, double dy)
Description: Enlarges the box by a certain amount.
Returns: A reference to this box.
This is a convenience method which takes two values instead of a Vector object. This method has been introduced in version 0.23.
(2) Signature: DBox enlarge (double d)
Description: Enlarges the box by a certain amount on all sides.
enlarge Returns: A reference to this box.
This is a convenience method which takes one values instead of two values. It will apply the given enlargement in both directions. This method has been introduced in version 0.28.
(3) Signature: DBox enlarge (const DVector enlargement)
Description: Enlarges the box by a certain amount.
enlargement: The grow or shrink amount in x and y direction
Returns: A reference to this box.
Enlarges the box by x and y value specified in the vector passed. Positive values with grow the box, negative ones will shrink the box. The result may be an empty box if the box
disappears. The amount specifies the grow or shrink per edge. The width and height will change by twice the amount. Does not check for coordinate overflows.
(1) Signature: [const] DBox enlarged (double dx, double dy)
Description: Enlarges the box by a certain amount.
Returns: The enlarged box.
This is a convenience method which takes two values instead of a Vector object. This method has been introduced in version 0.23.
(2) Signature: [const] DBox enlarged (double d)
Description: Enlarges the box by a certain amount on all sides.
enlarged Returns: The enlarged box.
This is a convenience method which takes one values instead of two values. It will apply the given enlargement in both directions. This method has been introduced in version 0.28.
(3) Signature: [const] DBox enlarged (const DVector enlargement)
Description: Returns the enlarged box.
enlargement: The grow or shrink amount in x and y direction
Returns: The enlarged box.
Enlarges the box by x and y value specified in the vector passed. Positive values with grow the box, negative ones will shrink the box. The result may be an empty box if the box
disappears. The amount specifies the grow or shrink per edge. The width and height will change by twice the amount. Does not modify this box. Does not check for coordinate
Signature: [static] new DBox ptr from_ibox (const Box box)
Description: Creates a floating-point coordinate box from an integer coordinate box
from_ibox Use of this method is deprecated. Use new instead
This constructor has been introduced in version 0.25 and replaces the previous static method 'from_ibox'.
Python specific notes:
This method is the default initializer of the object.
Signature: [static] new DBox ptr from_s (string s)
Description: Creates a box object from a string
Creates the object from a string representation (as returned by to_s)
This method has been added in version 0.23.
Signature: [const] unsigned long hash
Description: Computes a hash value
hash Returns a hash value for the given box. This method enables boxes as hash keys.
This method has been introduced in version 0.25.
Python specific notes:
This method is also available as 'hash(object)'.
Signature: [const] double height
Description: Gets the height of the box
Signature: [const] bool inside? (const DBox box)
inside? Description: Tests if this box is inside the argument box
Returns true, if this box is inside the given box, i.e. the box intersection renders this box
Signature: [const] bool is_const_object?
Description: Returns a value indicating whether the reference is a const reference
Use of this method is deprecated. Use _is_const_object? instead
This method returns true, if self is a const reference. In that case, only const methods may be called on self.
Signature: [const] bool is_point?
Description: Returns true, if the box is a single point
Signature: [const] double left
left Description: Gets the left coordinate of the box
Python specific notes:
The object exposes a readable attribute 'left'. This is the getter.
Signature: void left= (double c)
left= Description: Sets the left coordinate of the box
Python specific notes:
The object exposes a writable attribute 'left'. This is the setter.
(1) Signature: DBox move (double dx, double dy)
Description: Moves the box by a certain distance
Returns: A reference to this box.
This is a convenience method which takes two values instead of a Point object. This method has been introduced in version 0.23.
(2) Signature: DBox move (const DVector distance)
Description: Moves the box by a certain distance
distance: The offset to move the box.
Returns: A reference to this box.
Moves the box by a given offset and returns the moved box. Does not check for coordinate overflows.
(1) Signature: [const] DBox moved (double dx, double dy)
Description: Moves the box by a certain distance
This is a convenience method which takes two values instead of a Point object. This method has been introduced in version 0.23.
moved (2) Signature: [const] DBox moved (const DVector distance)
Description: Returns the box moved by a certain distance
distance: The offset to move the box.
Returns: The moved box.
Moves the box by a given offset and returns the moved box. Does not modify this box. Does not check for coordinate overflows.
(1) Signature: [static] new DBox ptr new (const Box box)
Description: Creates a floating-point coordinate box from an integer coordinate box
This constructor has been introduced in version 0.25 and replaces the previous static method 'from_ibox'.
Python specific notes:
This method is the default initializer of the object.
(2) Signature: [static] new DBox ptr new
Description: Creates an empty (invalid) box
Empty boxes don't modify a box when joined with it. The intersection between an empty and any other box is also an empty box. The width, height, p1 and p2 attributes of an empty box
are undefined. Use empty? to get a value indicating whether the box is empty.
Python specific notes:
This method is the default initializer of the object.
(3) Signature: [static] new DBox ptr new (double w)
Description: Creates a square with the given dimensions centered around the origin
Note that for integer-unit boxes, the dimension has to be an even number to avoid rounding.
This convenience constructor has been introduced in version 0.28.
Python specific notes:
new This method is the default initializer of the object.
(4) Signature: [static] new DBox ptr new (double w, double h)
Description: Creates a rectangle with given width and height, centered around the origin
Note that for integer-unit boxes, the dimensions have to be an even number to avoid rounding.
This convenience constructor has been introduced in version 0.28.
Python specific notes:
This method is the default initializer of the object.
(5) Signature: [static] new DBox ptr new (double left, double bottom, double right, double top)
Description: Creates a box with four coordinates
Four coordinates are given to create a new box. If the coordinates are not provided in the correct order (i.e. right < left), these are swapped.
Python specific notes:
This method is the default initializer of the object.
(6) Signature: [static] new DBox ptr new (const DPoint lower_left, const DPoint upper_right)
Description: Creates a box from two points
Two points are given to create a new box. If the coordinates are not provided in the correct order (i.e. right < left), these are swapped.
Python specific notes:
This method is the default initializer of the object.
Signature: [const] bool overlaps? (const DBox box)
overlaps? Description: Tests if this box overlaps the argument box
Returns true, if the intersection box of this box with the argument box exists and has a non-vanishing area
Signature: [const] DPoint p1
p1 Description: Gets the lower left point of the box
Python specific notes:
The object exposes a readable attribute 'p1'. This is the getter.
Signature: void p1= (const DPoint p)
p1= Description: Sets the lower left point of the box
Python specific notes:
The object exposes a writable attribute 'p1'. This is the setter.
Signature: [const] DPoint p2
p2 Description: Gets the upper right point of the box
Python specific notes:
The object exposes a readable attribute 'p2'. This is the getter.
Signature: void p2= (const DPoint p)
p2= Description: Sets the upper right point of the box
Python specific notes:
The object exposes a writable attribute 'p2'. This is the setter.
Signature: [const] double perimeter
Description: Returns the perimeter of the box
This method is equivalent to 2*(width+height). For empty boxes, this method returns 0.
This method has been introduced in version 0.23.
Signature: [const] double right
right Description: Gets the right coordinate of the box
Python specific notes:
The object exposes a readable attribute 'right'. This is the getter.
Signature: void right= (double c)
right= Description: Sets the right coordinate of the box
Python specific notes:
The object exposes a writable attribute 'right'. This is the setter.
Signature: [const] Box to_itype (double dbu = 1)
Description: Converts the box to an integer coordinate box
The database unit can be specified to translate the floating-point coordinate box in micron units to an integer-coordinate box in database units. The boxes coordinates will be
divided by the database unit.
This method has been introduced in version 0.25.
Signature: [const] string to_s (double dbu = 0)
Description: Returns a string representing this box
to_s This string can be turned into a box again by using from_s . If a DBU is given, the output units will be micrometers.
The DBU argument has been added in version 0.27.6.
Python specific notes:
This method is also available as 'str(object)'.
Signature: [const] double top
top Description: Gets the top coordinate of the box
Python specific notes:
The object exposes a readable attribute 'top'. This is the getter.
Signature: void top= (double c)
top= Description: Sets the top coordinate of the box
Python specific notes:
The object exposes a writable attribute 'top'. This is the setter.
Signature: [const] bool touches? (const DBox box)
touches? Description: Tests if this box touches the argument box
Two boxes touch if they overlap or their boundaries share at least one common point. Touching is equivalent to a non-empty intersection ('!(b1 & b2).empty?').
(1) Signature: [const] Box transformed (const VCplxTrans t)
Description: Transforms the box with the given complex transformation
t: The magnifying transformation to apply
Returns: The transformed box (in this case an integer coordinate box)
This method has been introduced in version 0.25.
(2) Signature: [const] DBox transformed (const DTrans t)
Description: Returns the box transformed with the given simple transformation
t: The transformation to apply
Returns: The transformed box
(3) Signature: [const] DBox transformed (const DCplxTrans t)
Description: Returns the box transformed with the given complex transformation
t: The magnifying transformation to apply
Returns: The transformed box (a DBox now)
Signature: [const] double width
Description: Gets the width of the box
Signature: [static] DBox world
Description: Gets the 'world' box
The world box is the biggest box that can be represented. So it is basically 'all'. The world box behaves neutral on intersections for example. In other operations such as
displacement or transformations, the world box may render unexpected results because of coordinate overflow.
world The world box can be used
• for comparison ('==', '!=', '<')
• in union and intersection ('+' and '&')
• in relations (contains?, overlaps?, touches?)
• as 'all' argument in region queries
This method has been introduced in version 0.28. | {"url":"https://www.klayout.de/doc/code/class_DBox.html","timestamp":"2024-11-02T04:50:14Z","content_type":"text/html","content_length":"60173","record_id":"<urn:uuid:62db0920-ef84-4738-aaa0-2d10f8139045>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00234.warc.gz"} |
• Links zu
• Mitarbeitern
Wissenschaftliche Mitarbeiter
Technische Mitarbeiter
Dr. Max Bannach
• Parametrisierte Platz- und Schaltkreiskomplexität
• Fixed-Parameter Parallelisierung
• Marcel Wienöbst, Malte Luttermann, Max Bannach, Maciej Liskiewicz:
Efficient enumeration of Markov equivalent DAGs.
In AAAI Conference on Artificial Intelligence (AAAI 2023), S. 12313-12320. , 2023.
Website anzeigen
• Marcel Wienöbst, Max Bannach, Maciej Liskiewicz:
Polynomial-Time Algorithms for Counting and Sampling Markov Equivalent DAGs with Applications.
Journal of Machine Learning Research, (24.213):1-45, 2023.
Website anzeigen
• Max Bannach, Sebastian Berndt, Thorsten Ehlers:
Jdrasil: A Modular Library for Computing Tree Decompositions.
In International Symposium on Experimental Algorithms (SEA 2017), Band 75 von LIPIcs, S. 28:1--28:21. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017.
Website anzeigen | PDF anzeigen | Zusammenfassung anzeigen
• Max Bannach, Christoph Stockhusen, Till Tantau:
Fast Parallel Fixed-Parameter Algorithms via Color Coding.
In Proceedings of the 10th International Symposium on Parameterized and Exact Computation (IPEC 2015), LIPIcs, 2015.
Website anzeigen | PDF anzeigen | Zusammenfassung anzeigen | {"url":"http://wwwtcs.tcs.uni-luebeck.de/de/mitarbeiter/bannach/forschung.html","timestamp":"2024-11-12T15:00:09Z","content_type":"application/xhtml+xml","content_length":"59963","record_id":"<urn:uuid:553d61ce-87a3-46cd-b238-9bc81f983b0c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00771.warc.gz"} |
Google Sheet Calculate
Google Sheet Calculate - Web you can use functions and formulas to automate calculations in google sheets. In the bottom right, find explore. Highlight the cells you want to calculate. Google sheets
formula examples and tutorial. Web here are the best formulas to learn in google sheets: Sort function in google sheets. Web how to calculate percentage in google sheets method one: Formula basics in
google sheets. You can calculate the percentage for part of a total with a simple formula in google. Here's a list of all the functions available in each category.
Google sheets formula examples and tutorial. In the bottom right, find explore. Web here are the best formulas to learn in google sheets: If function in google sheets: Sort function in google sheets.
Web functions can be used to create formulas that manipulate data and calculate strings and numbers. If you’re already familiar with functions and formulas and just need to know which ones are
available, go to. Divide part of a total. You can calculate the percentage for part of a total with a simple formula in google. Here's a list of all the functions available in each category. Next to
explore, you'll see sum: Web how to calculate percentage in google sheets method one: Web see the sum & average on your computer, open a spreadsheet in google sheets. Web you can use functions and
formulas to automate calculations in google sheets. When using them, don't forget to. Highlight the cells you want to calculate. Formula basics in google sheets.
Related Post: | {"url":"https://adrede.com.br/sheet/google-sheet-calculate.html","timestamp":"2024-11-01T19:18:23Z","content_type":"text/html","content_length":"27220","record_id":"<urn:uuid:009d3c68-dd37-41b3-acfd-d7311237380a>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00402.warc.gz"} |
Irvine Housing Blog (hat tip The Big Picture) shows the below chart and details that:
It is a sign that people are clinging to the hope of a real estate recovery. We are not yet at the bottom.
While I understand the above chart is meant to make a high level point (rather than show exactly where we are), I thought it would be interesting to overlay the Case
home price index on top.
We are not as early in the process as the chart would indicate (i.e. we've already fallen much further), it appears we still have plenty of risk remaining to the downside.
Another way to view this is Case
10 vs. inflation (CPI) over this time.
Closer, but still about 20% "too high" and during "despair", it tends to break through.
The Conference-Board details confidence is just about where it was a year ago (did a recovery actually take place?):
Says Lynn Franco, Director of The Conference Board Consumer Research Center: “Consumer confidence posted a modest gain in August, the result of an improvement in consumers’ short-term outlook.
Consumers’ assessment of current conditions, however, was less favorable as employment concerns continue to weigh heavily on consumers’ attitudes. Expectations about future business and labor
market conditions have brightened somewhat, but overall, consumers remain apprehensive about the future. All in all, consumers are about as confident today as they were a year ago (Aug. 2009,
July vs. August
Month over Month Change
S&P details:
Data through June 2010, released today by Standard & Poor’s for its S&P/Case-Shiller1 Home Price Indices, the leading measure of U.S. home prices, show that the U.S. National Home Price Index
rose 4.4% in the second quarter of 2010, after having fallen 2.8% in the first quarter. Nationally, home prices are 3.6% above their year-earlier levels.
In June, 17 of the 20 MSAs covered by S&P/Case-Shiller Home Price Indices and both monthly composites were up; and the two composites and 15 MSAs showed year-over-year gains. Housing prices have
rebounded from crisis lows, but other recent housing indicators point to more ominous signals as tax incentives have ended and foreclosures continue.
Strength was widespread, but note this data is two months old and importantly, during a period before the tax incentive ended.
Calculated Risk with a beautiful chart showing the cumulative change by market here.
Source: S&P
Yesterday, EconomPic detailed the slow growth in real personal income over the latest decade. Below we break down the contribution of real per capita disposable personal income over each of the past
six decades.
Things to note:
1) Compensation grew by at least $2000 per capita in real terms each decade since 1960 (more than $4000 in the 90's), but only ~$350 in the 00's
2) 95% of real per capita personal income in the 00's were current transfer receipts (unemployment / welfare benefits) and a decline in taxes paid (wonder why our nation's debt has spiked?)
Source: BEA
Marketwatch details:
The savings rate for U.S. households fell in July to the lowest level in three months as spending outpaced income, the Commerce Department estimated Monday.
Consumer spending rose 0.4% in July while personal income increased 0.2%.
The report was mixed in terms of market expectations. Incomes rose less than the 0.3% expected, while spending was stronger than the 0.3% gain expected by economists surveyed by MarketWatch.
Real (inflation-adjusted) spending increased a seasonally adjusted 0.2% in July after a 0.1% gain in June, led by a sizable increase in purchases of durable goods.
Real after-tax incomes fell 0.1% in July, compared with a downwardly revised 0.1% gain in disposable incomes in June. This is the biggest decline since January.
The savings rate fell to 5.9% from 6.2% in June, which was the highest level since June 2009.
Source: BEA
Bloomberg details:
Returns on U.S. investment-grade corporate bonds are pulling ahead of junk-rated debt as credit investors turn to borrowers likely to weather a slowing economy.
Investment-grade bonds are up 9.7 percent this year, topping the 8.5 percent for speculative-grade, the biggest outperformance since credit markets seized up in 2008.
Yields on investment-grade bonds average about 4.71 percentage points less than junk, compared with 3.76 percentage points in April.
Performance by Rating
Source: Barclays Capital
The Importance of Mortgage Rates
Addition by Subtraction
New Homes on the Cheap
BOOM!!! Goes the Treasuries
Are Corporate Earnings Sustainable?
Economic Data
GDP Breakdown by Decade
More Stimulus Needed?
Durable Goods...No Good
Where's the Investment?
US GDP Revised Down... Above Expectations
The Case for Developed Europe
UK Economic Growth Revised Upward
And your video of the week we're going back to the mid-90's with the Toadies' Tyler:
Reuters details:
U.S. Treasuries prices fell sharply on Friday after Federal Reserve Chairman Ben Bernanke signaled no new bond buying by the U.S. central bank was imminent, triggering the biggest sell-off in
three months.
Although Bernanke did mention such purchases as a possibility, investors found nothing in his comments to indicate the Fed has any immediate plans to stimulate the slowing economy through an
expansion of current bond buying.
For a market already at rich levels, this was an important nuance that further fueled a sell-off ignited after data earlier on Friday showed a revised picture of U.S. economic growth was not
quite as weak as expected in the second quarter.
Source: Bloomberg
Consumption was revised up slightly, net exports and investments were revised down a bit more reducing Q2 GDP -0.8% (from 2.4% to 1.6%) from the advanced figure (better than expected).
Source: BEA
Marketwatch details:
The British economy grew slightly faster than previously estimated in the second quarter, with gross domestic product expanding by 1.2%, the Office for National Statistics reported Friday.
The ONS had previously estimated second-quarter GDP growth of 1.1%. Compared to the second quarter of last year, GDP grew by 1.7% versus a previous estimate of 1.6%. A survey of economists by Dow
Jones Newswires had forecast no revisions to the data.
The growth marks the strongest quarter-on-quarter rise for British GDP in nine years.
Construction output was revised to show an 8.5% quarterly rise, up from a previous estimate of 6.6%. Output by production industries rose by an unrevised 1%.
Services output was revised down to show 0.7% growth, but still accelerated from 0.3% growth in the first quarter.
Source: Stats.UK
Unlike the result of existing home sales where the higher end market is holding up well (in terms of sales) relative to the absolute cliff-dive in the low-end market, the new home sales figures show
the low-end market is where it is at (though likely because high-end homes are now priced a notch or two down in the price ladder).
Whereas 10% of all new homes sold went for $500k+ as recently as 2008, in 2010 that figure is now less than 4%.
It is important to note that the July 2010 figures are from a base 30% lower than in 2009.
Source: Census
Missed this the other day. Reuters details:
Euro zone industrial new orders rose more than expected during the month of June, data showed on Monday, boding well for economic growth in the third quarter of 2010.
Industrial orders in the 16-nation currency zone increased 2.5 percent month-on-month for a 22.6 percent annual gain, European Union statistics office Eurostat said.
"That's good. Shows we can take a lot of the second-quarter momentum into the second half," said Carsten Brzeski, economist at ING.
The data could point to another quarter of economic expansion as the euro zone recovers from its sovereign debt problems and the worst economic crisis in decades. Orders point to trends in
activity as they translate into future production.
So the economy seems to be "recovering" (or at least not entering a depression). The importance of this non-depressionary environment? The market is (according to GMO's James Montier - one of my
favorite out of the box writers) priced like it is.
To his latest missive (if you are not signed up, I recommend you do - bold mine):
Of course, as with all investments, the price you pay determines the attractiveness of the opportunity. The good news is that European dividends appear to be priced cheaply at the moment.
Exhibit 6 (go to his missive to see) shows the current pricing structure of European dividends (for the Eurostoxx 50, the vertical line marks the point at which we switch from actual dividends to
the market’s implied view of dividends), and shows the experience of U.S. dividends during the Great d epression as a comparison. In essence, the market is saying that dividends will have
virtually zero growth between now and 2019. This is a worse outcome than the U.S. witnessed in the wake of the Great Depression!
Source: Eurostat
Calculated Risk details that according to the CBO, stimulus (i.e. the American Recovery and Reinvestment Act "ARRA") raised GDP in Q2 by an estimated range of 1.7% to 4.5%. If the revision to Q2 GDP
growth comes in at the expected 1.4% annualized rate for the quarter, this means that if the impact of stimulus is within range, GDP "would have" been negative in Q2.
The chart below shows actual GDP and a range for GDP ex-stimulus (simply actual GDP less the estimated impact of the stimulus at the low and high-end ranges as detailed by the CBO).
The concern on a going forward basis is that the impact of the stimulus has peaked and the economy still has numerous structural issues. Back to Calculated Risk:
Less stimulus spending in Q3 was one of the reason I expected a slowdown in growth in the 2nd half of 2010. There are other reasons that I've listed before: the end of the inventory correction,
more household saving leading to slower growth in personal consumption expenditures, another downturn in housing (lower prices, less residential investment), slowdown in China and Europe and
cutbacks at the state and local level.
Source: BEA
Forbes details:
Orders for big-ticket manufactured goods in July came in weaker than expected, raising the risk that the gross domestic product during the third quarter will not reach 1%, says Michael Feroli, an
economist at JPMorgan Chase
On Wednesday the U.S. Commerce Department reported durable goods orders rose by a measly 0.3% in July, well below the 2.5% lift the Street had expected, though still ahead of the 0.1% slip in
Shipments of capital goods fell 1.5% on a core basis, which excludes aircraft and defense spending, while orders tumbled 8%.
Transportation (i.e. the only highlight) was distorted by a massive 76% month over month jump in aircraft orders. Without that... not so much.
Source: Census
Traveling again today, so this is all you'll get until later this evening / tomorrow from EconomPic....
In response to my post on the breakdown of GDP by decade, reader DIY Investor comments:
It would be interesting if the investment portion could be broken out between housing and other and be done on an annual basis for the 00s. There may be some pent up demand building on the part
of business investment which could bode well for the stock market. My best guess is that the numbers are dominated by the housing crisis.
At least that's my take.
The chart of investment, both nonresidential (structures and equipment / software) and residential below....
Some results that I found interesting:
• The residential boom doesn't look so large as compared to previous cycles, though the collapse is rather epic (and will only get worse)
• Investment in structures has been flat going on 20+ years (outsourcing?)
• After the (telecom) investment bubble in the late 90's / early 00's, equipment and software investment is at a 50 year low (the potential that there is pent up demand for new investment)
The result... a huge lack of investment. As EconomPic has detailed before, this was a partial cause in the jump in profits over the past decade (less investment temporarily increases the bottom line)
and now growth is slowing (innovation helps increase the top line / economy).
Source: BEA
As expected, existing homes sales utterly collapsed in July post tax-credit. Barry over at The Big Picture detailed comments from the National Association of Realtors:
Everyone knew that Existing Home Sales were going to stink the joint up today — but I just had to laugh when I read the NAR commentary; The headline along was priceless: July Existing-Home Sales
Fall as Expected but Prices Rise. Too bad they don’t cover other events: “Lincoln attends theater opening; leaves early with headache.”
They are the world’s most awesome/awful cheerleaders on the planet.
The real reason for the "rise" in home prices in July? A larger collapse in lower priced homes where the tax credit had a larger impact.
Nothing but "addition by the reduction of subtraction".
Source: Realtor
On the road today, so I will miss what Calculated Risk (i.e. the expert) believes will be a massive miss in existing home sales (consensus is currently in the range of 4.7 million units... Calculated
Risk is calling for a bit above 4 million).
What I do have to offer is an update on the economy over the LONG term. Last week, EconomPic detailed that the economy grew just 1.62% annualized over the past decade in real terms (the lowest level
since the 1950's). The below goes back farther and details the components that make up GDP for each decade.
Source: BEA
Last week, EconomPic showed the historically relationship between the treasury yield and nominal GDP growth. Below, we compare corporate earnings growth to the treasury yield over that same ten year
What we see is that earnings on a cyclically adjusted basis (smoothed per Shiller) have been growing faster than the broader economy for the past ten years.
My thoughts (as shared last month):
The important question is how these earnings have come about. We all know that recent earnings have ratcheted higher due to reduced costs (job cuts, lack of investment, cheap financing) rather
than top line growth. In other words, executives for public firms have caught up with the "buy, strip, and flip" nature of private investors.
In addition, simple math proves that earnings cannot grow faster than nominal growth over the long term, as that would imply earnings at some point become larger than the economy as a whole (not
possible as earnings are part of the economy).
In summary... expect earnings to be under pressure in the not too distant future unless there is surprise outsized rebound in the economy.
Source: Irrational Exuberance
The Washington Post details how low current mortgage rates are:
Mortgage rates fell this week to the latest in a series of record lows amid concerns about the state of the U.S. economy, according to a survey released Thursday by Freddie Mac.
Interest rates on 30-year fixed-rate mortgages, the most widely used loan, averaged 4.42 percent this week, down from last week's 4.44 percent and its year-ago level of 5.12 percent, according to
the survey.
Thirty-year mortgage rates have fallen to record lows for nine straight weeks. Freddie Mac started the survey in April 1971.
And the corresponding chart...
In my opinion, rates (and only rates) are the reason why there has been a (temporary?) halt in housing price declines. Housing values were (and I believe are) too high, but low nominal rates have
made monthly payments much more affordable.
The chart below shows the monthly mortgage payment for a $200,000 house after a 20% down payment (i.e. a $160,000 mortgage) using the above 30-year mortgage rates.
Current monthly payments on a $160,000 mortgage are only $803, down from more than $1000 in October 2008. In other words, monthly payments are down 20% even if the price of the house didn't drop over
that period.
The important question is what can happen from here?
While I am not claiming that prices will fall off another cliff soon, there is a significant risk of a further decline... especially if rates don't stay this low (for this reason I do expect rates to
stay this low). The below chart keeps the current $803 monthly payment constant, but backs into what the value of a home using historical mortgage rates would have been for monthly payments to stay
at that $803 level.
What does this mean for anyone looking to buy a house?
Since the key contributor to housing affordability is not the current list price, but rather the mortgage rate, anyone looking to buy should seriously consider the alternative (i.e. renting) if they
don't plan to use that contributing factor (again... the mortgage rate) for the life of the loan (i.e. to keep their house for 15-30 years).
If you do plan to buy a house for a smaller window of time (i.e. 1-10 years) with the idea of flipping it into a larger house, be careful. That $803 per month clearing price may mean a much lower
home value when you are trying to sell...
An anonymous reader makes a common mistake in questioning the details of the post:
If you're planning to swap out your house after a couple of years for a bigger house, wouldn't you want house prices to come down? Yes, you'd lose money when you sell your old house, but you'd
save even more money on the larger house, whose price also went down.
This would be true is there was no leverage (i.e. financing) involved or if the decline occurred in combination with a further decline in rates. My concern is that a rise in rates will coincide with
or even trigger the next price decline. As a result, a decline in the value of homes could be disastrous (remember, investments in housing typically involve a ton of leverage).
My response (slightly edited):
Not if housing prices decline due to a rise in mortgage rates, which would result in a unchanged monthly payment on the new property (i.e. home price drop is exactly offset by mortgage rate rise = no
impact on the mortgage payment).
The reader used an 'extreme' example of a $100k house flipped into a $1mm house, which I will replicate:
• $100k mortgage = $500 / month mortgage payment at current rates
• $1mm mortgage = $5000 / month mortgage payment at current rates
Now, assume home values drop 10% due to a rise in rates, which would happen all else equal if homes were strictly based on monthly payments individuals could afford and rates rose to 5.32% (monthly
payments on a $90k loan at 5.32% = ~$500, which is the same as a $100k loan at 4.4% = ~$500)
1. $100k home is now worth $90k (you lose $10k)
2. your monthly payment on the new $900k mortgage is the same as it would have been previously on a $1mm mortgage (~$5000 per month)
As you can see, the owner doesn't get the benefit of the price decline if they need financing to own. The situation becomes dire in a move to a similarly priced home. Assuming a move from a $1mm
house to an identical house now worth $900k.
1. you lose $100k in equity
2. your monthly payments are the same
If there is a larger price decline (one larger than the level of equity in a home), one can see how ugly this can get. Again, the reason being leverage and my concern that the only thing propping up
the housing market is subsidized financing. My concern is that borrowers are buying what they can (barely) afford in terms of a monthly mortgage payment and not in terms of what they can afford in
terms of price of a home relative to wealth.
Sean over at Dead Cats Bouncing details:
While another economic crash landing remains unlikely given the inventory and corporate funding backdrop, there won't be much room for policy error either politically or at the Fed in coming
months. Monthly headline US CPI has now fallen for three consecutive months, which has only happened a handful of times since the data series began in 1947. If you take the rough and ready rule
that a 10 year government bond yield should equal the long term growth rate plus the long term inflation rate, then it's clear that a near 2.5% 10 year Treasury yield is pricing in a grim growth
Well, I for one was surprised just how strong the relationship has been.
Update: An anonymous reader commented:
I suspect both nominal GDP growth and treasury yields are mostly driven by inflation, since both real yields and real GDP growth vary less than inflation did.Which means all this graph is telling
you is that inflation plus some noise is correlated with inflation plus some different noise
While inflation has indeed dominated the change in nominal GDP over the long run, real GDP has actually been more volatile than CPI over the course of the past ten years. Also of note; ten year
annualized real GDP has dipped to 1.62%... the lowest level since the early 1950's.
Source: BEA / Federal Reserve
The 16/18/20 method was detailed previously (go here) at EconomPic. At the time, we showed the performance of those periods that were to be avoided (they underperformed historically).
Below, we show the performance in terms of real change of the S&P 500 index during periods deemed "attractive" based on the 16/18/20 method.
How'd they do?
Quite well.
Source: Irrational Exuberance
Bloomberg details:
Manufacturing in the Philadelphia region unexpectedly shrank in August for the first time in a year as orders and sales slumped, a sign factories are being hurt by the U.S. economic slowdown.
The Federal Reserve Bank of Philadelphia’s general economic index fell to minus 7.7 this month, the lowest reading since July 2009, from 5.1 in July. Readings less than zero signal contraction in
the area covering eastern Pennsylvania, southern New Jersey and Delaware.
Manufacturing is slowing after leading the economy out of the worst recession in seven decades as consumers rein in spending. With factory growth waning and companies slow to add employees, the
economic expansion will slow in the second half of the year.
“It’s not a pretty picture,” said Raymond Stone, chief economist at Stone & McCarthy Research Associates in Skillman, New Jersey, who forecast a reading of minus 6. “We’ll see continued gains in
manufacturing output, but it might be very small.”
Source: Philly Fed
Bloomberg details:
The index of U.S. leading indicators rose in July for the second time in four months, extending a see-saw pattern that indicates slower growth through the end of the year.
The 0.1 percent gain in the New York-based Conference Board’s gauge of the prospects for the economy in the next three to six months followed a 0.3 percent decline in June that was larger than
initially estimated. The June decrease was the biggest since February 2009.
Manufacturing, which led the economy out of the worst recession since the 1930s, will probably moderate in coming months as a slowdown in consumer spending depresses orders. Federal Reserve
policy makers last week said the recovery was “more modest” than they had projected, prompting them to take additional steps to revive growth.
“Economic growth is going to slow in the second half and we might face something a little more ominous than that,” said Mark Vitner, a senior economist at Wells Fargo Securities LLC in Charlotte,
North Carolina, who accurately forecast the gain in the leading index. Recent economic data “have shown marked deceleration in economic activity or even some pull-back.”
Source: Conference Board
The Big Picture has an old postcard posted that shows a pattern of 'Periods When to Make Money' based on a simply 16/18/20 year pattern:
Today, lets have a look at periodicity dating back to 1763. The cycle the (unknown) author poses is a repeating 16/18/20 year
Across the top is the legend “Years in which panics have occurred and will occur again.” The past panic century of dates are 1911, 1927, 1945, 1965, 1981, 1999, 2019. Except for 1981, these were
all pretty good years to sell (or short) stocks.
The "antique" postcard...
One of TBP's readers (dasht) found some additional background:
“The diagram which we give above was published on a business card by George Tritch, in Denver, Col., in 1872. We reproduce it from the card, with the explanations given with it. The diagram is
not altogether accurate; for example, the panic Tritch predicted for 1891 actually occurred in 1893; still, the year 1891 witnessed the beginnings of the depression and the shrinkage in values
which culminated in the crisis of 1893. It will be noted that the diagram gives the year 1897 as the time when an upward movement is to begin, and when it will be wise to buy stocks and real
Here Tritch has predicted like an inspired prophet. Everything in the grain, stock, and real estate markets are booming skyward, while the gold discoveries in various quarters, the financial
legislation in foreign countries, and the opening up of factories and mills throughout this country indicate that good times have come again, to stay, let us hope, many years beyond the period
Tritch sets down for another relapse, viz., 1899-1904.”
So... how well has Mr. Trich predicted? Using available data from Irrational Exuberance (since 1873) and judging performance of when to avoid the market via the real (i.e. inflation adjusted) change
in the equity index over a five year period... quite well.
Since 1873, in "down years" the index has decreased 4.7% over five years in real terms, while the index has increased 18.9% on average over a five year period over that same time.
And now? We are still in the middle of a down period... but make sure you're ready for the next bull market in 2012.
Source: Irrational Exuberance
Fascinating chart over at The Big Picture showing the history of GDP. Below is my version showing GDP per capita for an assortment of countries going back 100 years.
What does this show?
Well, just playing catch up, China and Brazil can each grow 500% before they catch the United States (i.e. before current technological limitations impede growth) in GDP per capita terms while India
has 1000% growth before catching up.
While catching up is not inevitable (just ask Japan), outsized growth over the next 20, 30, 50, 100 years likely is.
Source: GGDC
Calculated Risk (via the NY Fed) details household debt is now down more than 6% from its peak:
The Federal Reserve Bank of New York today announced the release of a new quarterly Report on Household Debt and Credit and an accompanying web page. The report shows that households steadily
reduced aggregate consumer indebtedness over the past seven quarters. In the second quarter of 2010, they owed 6.4 percent less than they did in 2008, the peak year for indebtedness.
Additionally, for the first time since early 2006, the share of total household debt in some stage of delinquency declined, from 11.9 percent to 11.2 percent. However, the number of people with a
new bankruptcy noted on their credit reports rose 34 percent during the second quarter, considerably higher than the 20 percent increase typical of the second quarter in recent years.
Source: NY Fed / BEA
Looking at the details we see the cause of the increase in the PPI... energy.
And this will likely reverse in August as commodities have reverted following the July spike.
Source: BLS
Daily Finance details:
Don't write-off the U.S. economic recovery just yet. The nation's industrial sector, which has led the expansion but cooled recently, showed signs of renewed life in July, with industrial
production climbing a better-than-expected 1%, the U.S. Federal Reserve announced Tuesday.
Capacity utilization jumped to 74.8% in July from 74.1% in June. The 74.8% utilization rate is 5.7% higher than a year ago, but is still 5.7 percentage points below its 1972 to 2009 average of
A Bloomberg survey had expected industrial production to rise 0.6% in July and the capacity utilization rate to rise to 74.5%. Industrial production fell 0.1% in June and rose 1.3% in May.
Source: Federal Reserve
At a very basic level, interest rate swaps allow the exchange of a fixed rate for a floating rate across maturities. When viewed in the form of a swap curve (below), we can see what the market is
pricing in for this fixed for floating across maturities, similar to a Treasury curve (i.e. the level in which you can get paid over various maturities on one side, while paying 3-month LIBOR on the
Below we see the swap curve at spot (i.e. current levels) and what the market is pricing in one, three, five, and ten years into the future.
What is the below chart showing?
1) The swap curve is currently steep, though not as steep as the Treasury curve (as long swaps are currently trading well through Treasuries)
2) The market is pricing in rates to rise rather dramatically at the front-end of the yield curve over the next five to ten years (though not much at all over the next 12 months)
3) The yield curve is actually inverted at the very long-end as early as three years out
Why? For one, investors that were underweight (or short) duration got themselves caught majorly off-guard over the course of the past few weeks as the long-end has taken a cliff-dive and now need
duration at any cost.
Source: Bloomberg
NY Fed details:
The Empire State Manufacturing Survey indicates that conditions improved modestly in August for New York manufacturers. The general business conditions index rose 2 points from its July level, to
7.1. The new orders and shipments indexes both dipped below zero for the first time in more than a year, indicating that orders and shipments declined on balance; the unfilled orders index was
also negative.
Source: NY Fed
EconomPic has detailed the (what appears to be) relative attractiveness of the earnings yield of the S&P 500 on multiple occasions the past few months (here, here, and here are a few examples). Below
we make another comparison... the earnings yield of the S&P 500 (in this case using Shiller's cyclical adjusted earnings) to the BBB corporate bond yield going back 35 years.
What the chart below shows is that the yield of S&P 500 has now surpassed that of the BBB rated corporate bond market (i.e. the lowest rated investment grade bonds) for the first time since the early
Bull Scenario
Forecasts are for earnings to continue to grow.
Bear Scenario
Forecasts are just forecasts; a likely scenario is that forecasts are too rosy and earnings will reverse course (though this was accounted for in part within the first chart through the use of the
CAPE [cyclically adjusted earnings], which uses a 10 year average of earnings). But it is possible that earnings over this entire 10 year period have been amplified by leverage, low interest rates,
and a lack of reinvestment as production was pushed offshore. John Hussman (hat tip Credit Writedowns) provided the background for this a month back:
Current forward operating earnings estimates assume profit margins for the S&P 500 companies that are nearly 50% above their long-term historical norms. While we did observe such profit margins
for a brief shining moment in 2007, profit margins are extraordinarily cyclical. Investors will walk themselves over a cliff if they price stocks as if profit margins, going forward, will be
dramatically and sustainably higher than U.S. companies achieved in all of market history.
And this all may just prove that BBB corporate bonds are simply rich. According the Barclays Capital BBB corporate index, BBB corporate bond yield are just 4.4%... an all-time (since the index was
tracked in 1988) low.
Source: S&P / Irrational Exuberance
Despite starting the year yielding little (3.84% for the 10 year and 4.66% for the 30 year), the chart below shows how well Treasuries have performed year to date. More specifically, what is shown is
the current yield curve (in blue) and year to date returns of the various points of the yield curve (in red).
Going forward? As we know, yield wins in the long run.
Source: Barclays Capital
Bloomberg reports:
Japan’s economy grew at less than a fifth of the pace economists estimated last quarter, pushing it into third place behind the U.S. and China and adding to evidence the global recovery is
Gross domestic product rose an annualized 0.4 percent in the three months ended June 30, slowing from a revised 4.4 percent expansion in the first quarter, the Cabinet Office said today in Tokyo.
The median estimate of 19 economists surveyed by Bloomberg News was for annual growth of 2.3 percent.
Extremely slow week at EconomPic, so I can't even call this EconomPics of the week (on the road all week).
V-Shape in Job Openings?
Productivity was about Doing Less.... With Even Less
Deflation Trade... On
Not Sustainable... Trade Edition
But, since one of my favorite readers asked for an Indie fix, here goes.
Grizzly Bear with Two Weeks...
Just beginning the process of catching up on one ugly day...
Source: Yahoo
What happens when the U.S. stimulates consumer demand when the rest of the developed world pushes austerity measures.... the below.
Source: Census
Vincent Fernando (hat tip Abnormal Returns) claims there is a v-shaped recovery after all in employment... in job openings. The reason this is not translating to job growth? People don't want those
Many Americans are choosing unemployment benefits over available jobs on offer. Earlier today we highlighted how unfilled job positions were rising at a much faster rate than new hires. We
described how some Americans were forgoing job offers due to the fact that they calculated (correctly, from an individual perspective) that it was a better deal to continue receiving unemployment
benefits rather than accept many jobs currently on offer.
While EconomPic detailed the rise in openings last week, it is (in in my opinion) not in fact due to the unemployed being choosy (this actually seems quite ridiculous to me), but rather because
corporations are taking more time to hire due to all the uncertainty.
Lets look at the details that may (or may not) back that opinion. The below chart includes the number of job openings at month-end (as was in Vincent's post) going back to 2002, but also the number
of hires within each month and ratio between the two to give the job openings figure some context.
What does this chart show?
1. the rate that individuals are hired per opening (i.e. the ratio) is still elevated from pre-crisis levels
2. this indicates that more people are actually being hired per opening (not less) than pre-crisis
3. this ratio is indeed below the ratio from last summer (i.e. hiring has slowed relative to job openings since the peak of the crisis - Vincent's point)
4. the recent decline (and jump in openings relative to hires) is only a decline relative to a period when the number of hires per opening spiked 50% from pre-crisis levels (likely because employers
had existing offers out and/or workers took any job they could get their hands on at that point in time)
As for the recent rise in openings relative to hiring?
It is important to note that the level of job openings is the amount at month end, not net new job openings. Thus, my best guess is that this has more to do with businesses taking their time to hire.
Why rush when the economic outlook is still so uncertain / there are so many qualified candidates out there? Why not post a job opening even if there aren't plans to fill it just in case a strong
candidate appears. I know if I was a small business owner (or corporation), I would want to use this opportunity to interview a slew of candidates. It is only when the market is tight that businesses
need to hire quickly and the market is not exactly tight right now.
To conclude... I find the increase in job openings a positive at the margin. But, this is FAR from v-shaped...
Source: BLS
The WSJ reports:
U.S. productivity unexpectedly fell in the second quarter, the first drop in 18 months, amid slower output growth and an increase in labor costs. Nonfarm business productivity dropped at a 0.9%
annual rate in the April to June period, the Labor Department said Tuesday. It was the first decline since the fourth quarter of 2008, when productivity fell by 0.1%.
The strong gains in productivity growth, which ranged from 3% to 8% in 2009, are likely over. Productivity usually picks up sharply at the end of recessions. The recovery has been in place for
more than a year now, and the economy slowed in the second quarter compared to the previous two quarters.
The chart below shows it all. The increase in productivity was never due to doing more, with less. It was doing less with (an even larger) less.
The recent drop in productivity is (to me) not a bad sign. It is simply the decrease in marginal returns from bringing workers and capacity back into the system. In other words... the jump in
productivity wasn't as great a thing as some thought, while the decline is not as bad as many now think.
Source: BLS
Busy week here at EconomPic...
Asset Classes
The Power of Mean Reversion
More on Mean Reversion
Yield Wins in the Long Run
Hedge Funds Snap Back in July
IBM Borrows on the Cheap
Economic Data
From Unemployed to Out of the Workforce
Leaving the Workforce in Droves
Job Postings on the Up and Up
Private Employment Rebound Stalling
ADP Report: Employment Rebound Remains Tepid
U.S. Consumer
The Changing Auto Sector
Personal Savings on the Rise
Personal Income... A Matter of Wages
Consumer Credit Continues Decline, Though Past Months Revised Up
Other U.S.
ISM Manufacturing Growth Slows
Services Industry Expands in July
Real GDP per Capita at September 2005 Level
Brakes Put on the Chinese Economy
Is the German Economy Booming? No...
European Consumption Stagnant
And your video of the week... Arcade Fire with 'Sprawl II' from Thursday's MSG show (the new album is insanely good IMO).
And as an encore, Arcade Fire with 'Wake Up' because it might just be the best song ever...
Marketwatch details:
U.S. consumers shed some of their debt in June for the fifth month in a row, the Federal Reserve reported Friday. Total seasonally adjusted consumer debt fell $1.34 billion, or a 0.7% annualized
rate, in June to $2.418 trillion. Economists expected a decline. The series is very volatile. May consumer credit was revised sharply higher to a decline of $5.28 billion compared with the
initial estimate of a drop of $9.15 billion. The decline in June was led by revolving credit-card debt, which fell $4.48 billion or 6.7%.
Source: Federal Reserve
Data from a previous post The Power of Mean Reversion shown in a different format below.
Source: Irrational Exuberance
Last employment chart of the day. Below is the monthly change in the nonfarm private sector since the financial crisis began (i.e. without census hiring).
Source: BLS
WSJ reports:
The U.S. economy shed more jobs than expected in July while the unemployment rate held steady at 9.5%, a further sign the economic recovery may be losing momentum.
Nonfarm payrolls fell by 131,000 last month as the rise in private-sector employment was not enough to make up for the government jobs lost, the U.S. Labor Department said Friday. Only 71,000
private-sector jobs were added last month while 143,000 temporary workers on the 2010 census were let go.
As EconomPic has detailed many times before, employment can fall while the unemployment rate stays flat (or in cases drop) because the denominator (i.e. the workforce) has been dropping. As the chart
below shows, the duration of unemployment has increased dramatically over the past year and a half AND at a certain point these individuals simply drop out of the workforce.
Note the dip earlier this year and the "rebound" from temporary census hiring. Perhaps we need some new temporary hiring efforts before more yellow rolls to blue.
Source: BLS
Check out the number of people leaving the workforce (green) and the spike in the overall number of working age individuals not in the workforce (blue).
Source: BLS
Index Universe (hat tip Abnormnal Returns) details why now may be the time to allocate to U.S. equities... reversion to the mean.
There have only been four decade-long periods where U.S. equities have delivered negative returns, which were the 10 years ending in 1937, 1938, 1939 and 2008 (2009 was not included in the
study). In each case, the subsequent 10-year period was strongly positive, with equities delivering (on average) an 11 percent compound annual return. Reversion to the mean. It’s simple, but it
Rather than look at nominal returns, I took at look at rolling ten year total returns of the S&P 500 index in real terms and then took a look at what the relationship was to ten year forward total
real returns (using year end levels).
While the past is not a perfect predictor of the future, there has not been one period in which the ten year real return of the S&P 500 was negative, then followed by another decade of negative ten
year real returns going back to data from the 1880's (i.e. no marker in the bottom left quadrant).
Source: Irrational Exuberance | {"url":"https://econompicdata.blogspot.com/2010/08/","timestamp":"2024-11-06T07:52:35Z","content_type":"application/xhtml+xml","content_length":"250868","record_id":"<urn:uuid:4c0e5ea0-d07a-41fb-937e-2f6358b584c4>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00136.warc.gz"} |
12.7 Regression (Distance from School) (Optional)
Stats Lab 12.1
Regression (Distance From School)
Class Time:
Student Learning Outcomes
• The student will calculate and construct the line of best fit between two variables.
• The student will evaluate the relationship between two variables to determine whether that relationship is significant.
Collect the DataUse eight members of your class for the sample. Collect bivariate data (distance an individual lives from school, the cost of supplies for the current term).
1. Complete the table.
Distance from School Cost of Supplies This Term
2. Which variable should be the dependent variable and which should be the independent variable? Why?
3. Graph distance vs. cost. Plot the points on the graph. Label both axes with words. Scale both axes.
Analyze the DataEnter your data into a calculator or computer. Write the linear equation, rounding to four decimal places.
1. Calculate the following:
a. a = ______
b. b = ______
c. correlation = ______
d. n = ______
e. equation: ŷ = ______
f. Is the correlation significant? Why or why not? (Answer in one to three complete sentences.)
2. Supply an answer for the following scenarios:
a. For a person who lives eight miles from campus, predict the total cost of supplies this term.
b. For a person who lives 80 miles from campus, predict the total cost of supplies this term.
3. Obtain the graph on a calculator or computer. Sketch the regression line.
Discussion Questions
1. Answer each question in complete sentences.
a. Does the line seem to fit the data? Why?
b. What does the correlation imply about the relationship between distance and cost?
2. Are there any outliers? If so, which point is an outlier?
3. Should the outlier, if it exists, be removed? Why or why not? | {"url":"https://texasgateway.org/resource/127-regression-distance-school-optional?book=79081&binder_id=78271","timestamp":"2024-11-11T08:06:41Z","content_type":"text/html","content_length":"42200","record_id":"<urn:uuid:58d65648-1109-439a-92b6-03f00b081431>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00523.warc.gz"} |
COCI 2009/2010, Contest #2
Task POSLOZI
"Arrange" is a planetary popular Flash game. In "Arrange" the player is given a permutation of numbers 1 to N and a list of allowed swaps. He then has to perform a sequence of swaps that transforms
the initial permutation back to the ordered sequence 1,2,3,4,5...N.
In order to break the high score list, you need to perform the minimal amount of swaps possible. You can't do that, but you can write a program that does it for you!
The first line of input contains two integers, N (1 ≤ N ≤ 12), the length of the initial sequence and M (1 ≤ M ≤ N*(N–1)/2) number of allowed swaps.
The second line of input contains a permutation of number 1 to N.
The next M lines contain descriptions of allowed swaps. If the line contains numbers A and B you are allowed to swap the A-th number with the B-th number. The input will never contain two identical
Note: the test data shall be such that the solution, not necessarily unique, will always exist.
Print the minimal number of swaps, X.
In the next X lines print the required swaps, in order. In each line print the index of the swap performed. Swaps are numbered increasingly as they appear in the input, starting from 1.
Sample Tests
Input Input
Output Output 3 5
1 2 Output
All Submissions
Best Solutions
Point Value: 20 (partial)
Time Limit: 2.00s
Memory Limit: 32M
Added: Jul 02, 2013
Languages Allowed:
C++03, PAS, C, HASK, ASM, RUBY, PYTH2, JAVA, PHP, SCM, CAML, PERL, C#, C++11, PYTH3 | {"url":"https://wcipeg.com/problem/coci092p5","timestamp":"2024-11-06T07:40:37Z","content_type":"text/html","content_length":"11555","record_id":"<urn:uuid:2e8676a8-b3fb-4905-84a1-94ea5d5fb3af>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00501.warc.gz"} |
Free advanced algebra
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Author Message
Clga Posted: Friday 28th of May 13:43
Hello there, I am a high-school student and at the end of the term I will have my exams in math . I was never a math genius , but now I am really afraid that I will fail this course.
I came across free advanced algebra and some other algebra problems that I can’t figure out. the following topics really made me panic : angle suplements and rational expressions
.Getting a a teacher is impossible for me, because I don't have any money. Please help me!!
From: Igloo
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IlbendF Posted: Sunday 30th of May 09:54
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math problems which you never thought you would be able to. Not only does it solve the problem, but it also explains the steps involved in getting to that solution. All the Best!
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cmithy_dnl Posted: Monday 31st of May 07:14
I can confirm that. Algebrator is the best program for solving math assignments. Been using it for a while now and it keeps on amazing me. Every assignment that I type in, Algebrator
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From: Australia
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niihoc Posted: Tuesday 01st of Jun 10:34
That’s what I exactly need ! Are you certain this will be helpful with my problems in algebra? Well, it doesn’t hurt if I try the software. Do you have any links to share that would
lead me to the product details?
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malhus_pitruh Posted: Tuesday 01st of Jun 17:28
Its really easy, just click on the this link and you are good to go – https://softmath.com/about-algebra-help.html. And remember they even give an unconditional money back guarantee
with their software , but I’m sure you’ll love it and won’t ever ask for your money back.
From: Girona,
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8 Math Modeling Examples That Demonstrate the Importance of Modeling in the Real World
8 Math Modeling Examples That Demonstrate the Importance of Modeling in the Real World
Mathematical modeling is an essential tool in understanding and solving complex real-world problems. It involves creating abstract representations of systems using mathematical language and concepts
to analyze, predict, and explain their behavior. This blog post delves into diverse math modeling examples showing how modeling can be applied in various fields.
1. Epidemiology: Modeling the Spread of Diseases
One of the most significant applications of mathematical modeling is in epidemiology – the study of how diseases spread. During the COVID-19 pandemic, mathematical models were crucial in predicting
the spread of the virus, evaluating the impact of public health interventions, and planning healthcare responses.
This math modeling example typically incorporates factors such as the rate of infection, recovery, and mortality, along with population dynamics and social behaviors. By simulating different
scenarios, epidemiologists can recommend strategies to control the spread, such as social distancing, vaccination, and lockdown measures.
Try it out: Mathematical Modeling of Disease Outbreak
2. Environmental Science: Climate Change and Ecosystem Modeling
In environmental science, mathematical models play a pivotal role in understanding climate change and its impacts. Climate models are complex simulations that incorporate atmospheric, oceanic, and
terrestrial processes to predict future climate conditions. They help in assessing the effects of greenhouse gas emissions and guide policymakers in making informed decisions to mitigate climate
Similarly, ecosystem models are used to study the interactions within ecosystems. This math modeling example helps in understanding the impact of human activities on biodiversity, the effects of
invasive species, and the dynamics of natural resources management.
Try it out: Climate Change and the Daily Temperature Cycle
3. Finance: Risk Management and Investment Strategies
Financial mathematics is another area where modeling is extensively used. Risk management models help financial institutions understand the risks associated with investments and guide them in
decision-making. These models assess market volatility, credit risk, and liquidity risk, among other factors. Investment strategies also heavily rely on mathematical models.
Try it out: Applications of Mathematics to Finance and Investment
4. Transportation: Traffic Flow and Urban Planning
Mathematical modeling is instrumental in urban planning, especially in managing traffic flow. Models that simulate traffic patterns help in the design of efficient road networks, optimization of
traffic lights, and planning of public transportation systems. This math modeling example can account for variables like vehicle density, speed, traffic regulations, and human behavior to predict
traffic congestion and suggest improvements.
Try it out: Urban Sustainability
5. Sports: Performance Analysis and Strategy Development
In sports, mathematical models are used for performance analysis and strategy development. For example, in baseball, the Sabermetrics approach uses statistical analysis to evaluate player performance
and inform team strategies. Similarly, in soccer, models analyze player movements, passing networks, and team formations to optimize tactics.
Try it out: Exploring Power and Speed in Baseball
6. Health Care: Medical Imaging and Drug Development
Mathematical modeling in healthcare includes applications in medical imaging and drug development. Models are used to interpret data from MRIs, CT scans, and X-rays, aiding in accurate diagnosis. In
drug development, models simulate the interactions of drugs with biological systems, predicting their efficacy and side effects. This approach accelerates the drug development process and improves
patient safety.
Try it out: The Mathematics of Medical X-Ray Imaging
7. Astrophysics: Understanding the Universe
In astrophysics, mathematical models are crucial in understanding celestial phenomena. Models of star formation, galaxy evolution, and black holes, for instance, rely heavily on mathematical
equations to explain observations and predict future events. This math modeling example focuses on some of the math behind the scenes: how to point spacecraft where we want; doing so helps scientists
in deciphering the mysteries of the universe.
Try it out: Spacecraft Attitude, Rotations and Quaternions
8. Social Sciences: Population Dynamics and Economic Models
Finally, mathematical modeling is also prevalent in the social sciences. Population dynamics models study the growth, decline, and movement of populations, informing policies on migration,
urbanization, and health. Economic models, on the other hand, analyze market behavior, consumer preferences, and policy impacts, guiding economic planning and decision-making.
Try it out: Population Projection
Mathematical modeling is a powerful tool that transcends disciplines as demonstrated by this list of math modeling examples. It enables us to simplify and understand complex systems, predict future
scenarios, and make informed decisions. The diverse applications of mathematical modeling, from controlling pandemics to exploring outer space, demonstrate its indispensable role in advancing
knowledge and addressing the challenges of our world. By continuing to develop and refine these models, we can better prepare for future challenges and opportunities.
Written by
The Consortium for Mathematics and Its Applications is an award-winning non-profit organization whose mission is to improve mathematics education for students of all ages. Since 1980, COMAP has
worked with teachers, students, and business people to create learning environments where mathematics is used to investigate and model real issues in our world. | {"url":"https://www.comap.org/blog/item/math-modeling-examples","timestamp":"2024-11-06T22:02:29Z","content_type":"text/html","content_length":"52801","record_id":"<urn:uuid:4d2d4ba4-7dee-43e3-a4ad-645cf74cde62>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00311.warc.gz"} |
Design and Development Process - Connected Mathematics Project
Design and Development Process
The development of CMP was built on the extensive knowledge gained from?the development, research and evaluation of the authors over the past forty years. The diagram shows the design and development
cycle used for CMP 1 and CMP 2. Each revision went through at least three cycles of design, field trials–data feedback–revision. Building on these experiences, CMP3 underwent a similar development
process but on a smaller scale. This process of (1) commissioning reviews from experts, (2) using the field trials with feedback loops for the materials, (3) conducting key classroom observations by
the CMP staff, and (4) monitoring student performance on state and local tests by trial schools comprises research-based development of curriculum.
The feedback from teachers and students across the country is the key element in the success of the CMP materials. The final materials comprised the ideas that stood the test of time in classrooms
across the country. Over 425 teachers and thousands of their students in 54 school district trial sites are a significant part of the team of professionals that made these materials happen. The
interactions between teacher and students with the materials became the most compelling parts of the teacher support. Without these teachers and their willingness to use materials that were never
perfect in their first versions, CMP would have been a set of ideas that lived in the brains and imaginations of the author team. Instead, they are materials with classroom heart because the trial
teachers and students made them so. These materials have the potential to dramatically change what students know and are able to do in mathematical situations. The emphasis on thinking and reasoning,
on making sense of ideas, on making students responsible for both having and explaining their ideas, and on developing sound mathematical habits provides opportunities for students to learn in ways
that can change how they think of themselves as learners of mathematics.
Research, Field Tests, and Evaluation
The CMP philosophy fosters a focus on isolating important mathematical ideas and embedding these ideas in carefully sequenced sets of contextual problems. These problems are developed and trialed in
classrooms in different states over several years. Each revision of CMP has been extensively field-tested in its development phases. We solicited iterative and in-depth input and reviews from
teachers, parents, administrators, mathematics educators, mathematicians, cognitive scientists, and experts in reading, special education, and English language learners. Our materials are created to
support teachers in helping their students develop deeper mathematical understanding and reasoning. This stance is the foundation of the success of CMP, which has withstood the pressures of various
political changes in the Nation over time.
Getting to know something is an adventure in how to account for a great many things that you encounter in as simple and elegant a way as possible. And there are lots of ways of getting to that point,
lots of different ways. And you don’t really ever get there unless you do it, as a learner, on your own terms. ... All you can do for a learner enroute to their forming a view of their own view is to
aid and abet them on their own voyage. ... in effect, a curriculum is like an animated conversation on a topic that can never be fully defined, although one can set limits upon it. I call it an
“animated” conversation not only because one uses animation in the broader sense— props, pictures, texts, films, and even “demonstrations.” Conversation plus show-and-tell plus brooding on it all on
one’s own. Bruner (1992, p.5)
The Development of CMP1 and CMP2
The CMP1 authors began by working with an outstanding advisory board to articulate the goals for what a student exiting from CMP in grade eight would know and be able to do in each of the strands of
mathematics under consideration—number, algebra, geometry, measurement, probability and statistics and in the interactions among these strands These essays that elaborated our goals became our
touchstones for the development of the materials for three grades—6, 7, and 8 for CMP1,CMP2, and CMP3. (For copies of these papers, see A Designers Speaks.)
Central to the success of CMP1 was the cadre of hard working teachers who cared deeply about their students learning of mathematics. As we set out to write a complete connected curriculum for grades,
6, 7, and 8, the following issues quickly surfaced and needed resolution:
• Identifying the important ideas and their related concepts and procedures;
• Designing a sequence of tasks to develop understanding of the idea;
• Organizing the sequences into coherent, connected curriculum;
• Balancing open and closed tasks;
• Making effective transitions among representations and generalizations;
• Addressing student difficulties and ill-formed conceptions;
• Deciding when to go for closure of an idea or algorithm;
• Staying with an idea long enough for long-term retention;
• Balancing skill and concept development;
• Determining the kinds of practice and reflection needed to ensure a desired degree of automaticity with algorithms and reasoning;
• Writing for both students and teachers; and
• Meeting the needs of all fifty states and diverse learners.
With the help of the field teachers, advisory board and consultants, these issues were resolved over the six years of development, field-testing, and evaluation.
Before starting the design phase for the CMP2 materials, we commissioned individual reviews of CMP material from 84 individuals in 17 states and comprehensive reviews from more than 20 schools in 14
states. Individual reviews focused on particular strands over all three grades (such as number, algebra, or statistics) on particular subpopulations (such as students with special needs or those who
are commonly underserved), or on topical concerns (such as language use and readability). Comprehensive reviews were conducted in groups that included teachers, administrators, curriculum
supervisors, mathematicians, experts in special education, language, and reading-level analyses, English language learners, issues of equity, and others. Each group reviewed an entire grade level of
the curriculum. All responses were coded and entered into a database that allowed reports to be printed for any issue or combination of issues that would be helpful to an author or staff person in
designing a Unit.
In addition, we made a call to schools to serve as pilot schools for the development of CMP2. We received 50 applications from districts for piloting. From these applications, we chose 15 that
included 49 school sites in 12 states and the District of Columbia. We received evaluation feedback from these sites over the five-year cycle of development.
Based on the reviews, what the authors had learned from CMP pilot schools over a six-year period, and input from our Advisory Board, the authors started with grades 6 and 7 and systematically revised
and restructured the Units and their sequence for each grade level to create a first draft of the revision. These were sent to our pilot schools to be taught during the second year of the project.
These initial grade-level Unit drafts were the basis for substantial feedback from our trial teachers.
Here are examples of the kinds of questions we asked classroom teachers following each revision of a Unit or grade level.
"Big Picture" Unit Feedback
• Is the mathematics important for students at this grade level? Explain. Are the mathematical goals clear to you? Overall, what are the strengths and weaknesses in this Unit?
• Please comment on your students’ achievement of mathematics understanding at the end of this Unit. What concepts/skills did they “nail”? Which concepts/skills are still developing? Which concepts
/skills need a great deal more reinforcement?
• Is there a flow to the sequencing of the Investigations? Does the mathematics develop smoothly throughout the Unit? Are there any big leaps where another Problem is needed to help students
understand a big idea in an Investigation? What adjustments did you make in these rough spots?
Problem-by-Problem Feedback
• Are the mathematical goals of each Problem/Investigation clear to you??
• Is the language and wording of each Problem understandable to students?
• Are there any grammatical or mathematical errors in the Problems??
• Are there any Problems that you think can be deleted??
• Are there any Problems that needed serious revision?
• Does the format of the ACE exercises work for you and your students? Why or why not?
• Which ACE exercises work well, which would you change, and why?
• What needs to be added to or deleted from the ACE exercises? Is there enough practice for students? How do you supplement and why?
• Are there sufficient ACE exercises that challenge your more interested and capable students? If not, what needs to be added and why?
• Are there sufficient ACE exercises that are accessible to and helpful to students that need more scaffolding for the mathematical ideas?
Mathematical Reflections and Looking Back/Ahead
Are these reflections useful to you and your students in identifying and making more explicit the “big” mathematical ideas in the Unit? If not, how could they be improved?
Assessment Material Feedback
• Are the check-ups, quizzes, tests, and projects useful to you? If not, how can they be improved? What should be deleted and what should be added?
• How do you use the assessment materials? Do you supplement the materials? If so, how and why?
Teacher Content Feedback
• Is the teacher support useful to you? If not, what changes do you suggest and why?
• Which parts of the teacher support help you and which do you ignore or seldom use?
• What would be helpful to add or expand in the Teacher support?
Year-End Grade-Level Feedback
• Are the mathematical concepts, skills and processes appropriate for the grade level?
• Is the grade-level placement of Units optimal for your school district? Why or why not?
• Does the mathematics flow smoothly for the students over the year?
• Once an idea is learned, is there sufficient reinforcement and use in succeeding Units?
• Are connections made between Units within each grade level?
• Does the grade-level sequence of Units seem appropriate? If not, what changes would you make and why?
• Overall, what are the strengths and weaknesses in the Units for the year?
Final Big Question
What three to five things would you have us seriously improve, change, or drop at each grade level?
The Development of CMP3
The development of CMP3 was built on the knowledge we gained over the?past 20 years of working with teachers and students who used CMP1 and CMP2. In addition, for the past 17 years we have solicited
information from the field through our web site and CMP mailing list and through our annual CMP week-long workshops and two-day conferences. The experiences with development processes for CMP1 and
CMP2 and the ongoing gathering of information from teachers have resulted in a smaller but more focused development process for CMP3.
The process of revision for CMP3 was similar to the preceding iterations except on a smaller scale. A group of field-test teachers from CMP2 trialed the versions of the Units for CMP3 that had
substantive changes from CMP2. They also contributed to the development of the assessment items and suggested many of the new features in the student and teacher materials. They were influential in
designing many new features such as the “focus questions” for each problem, a more streamlined set of mathematical goals, and Mathematical Reflections. Their feedback was invaluable in making sure
that our adjustment for CCSSM resulted in materials from which students and teachers could learn. CMP3 is fully aligned with the CCSSM and Mathematical Practices and reflects the thoughtful concern
and care of the authors and CMP3 trial teachers. This process has produced a mathematical experience that is highly educative for students and teachers in the middle grades.
Co-Development with Teachers and Students
Developing a curriculum with a complex set of interrelated goals takes time?and input from many people. As authors, our work was based on a set of deep commitments we had to creating a more powerful
way to engage students in making sense of mathematics. Our Advisory Boards took an active role in reading and critiquing Units in their various iterations. In order to enact our development
principles, we found that three full years of field trials in schools for each development phase were essential.
This feedback from teachers and students across the country is the key element in the success of the CMP materials. The final materials comprised the ideas that stood the test of time in classrooms
across the country. Nearly 200 teachers in 15 trial sites around the country (and their thousands of students) are a significant part of the team of professionals that made these materials happen.
The interactions between teacher and students with the materials became the most compelling parts of the teacher support.
Without these teachers and their willingness to use materials that were never perfect in their first versions, CMP would have been a set of ideas that lived in the brains and imaginations of the
author team. Instead, they are materials with classroom heart because our trial teachers and students made them so. We believe that such materials have the potential to dramatically change what
students know and are able to do in mathematical situations. The emphasis on thinking and reasoning,
on making sense of ideas, on making students responsible for both having and explaining their ideas, and on developing sound mathematical habits provides opportunities for students to learn in ways
that can change how they think of themselves as learners of mathematics.
From the authors’ perspectives, our hope has always been to develop materials that play out deeply held beliefs and firmly grounded theories about what mathematics is important for students to learn
and how they should learn it. We hope that we have been a part of helping to challenge and change the curriculum landscape of our country. Our students are worth the effort.
For more information on the history and development of Connected Mathematics, see A Designer Speaks. | {"url":"https://connectedmath.msu.edu/curriculum-design/the-story-of-cmp/design-and-development-process.aspx","timestamp":"2024-11-13T08:06:34Z","content_type":"text/html","content_length":"70749","record_id":"<urn:uuid:39bcc45d-bc53-447f-bbbd-4b784e5dcfcd>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00469.warc.gz"} |
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Interesting Mathematics Archives - Teach.sg
Zeno’s paradox begins like this:
In order for a man to reach another, he needs to cover half the distance first. After that, he needs to cover the next quarter of the distance. Thereafter, he needs to cover the next eighth of the
distance, and so on. With these small increments that he has to cover, as it is seemingly infinite, will the man never get to anywhere?
Here is the fascinating answer.
First, we realise that 1 can be split up into infinite parts –
Hence an infinite parts can also be added up to 1 –
Therefore, even though the man needs to walk an infinite number of fractions to get to his destination, he will still get there.
And here is the mind-blowing nature of
As an auctioneer in a silent auction, to maximise profits, you should use the Vickrey auction.
A Vickrey auction allows the highest bidder to win the auction, paying only the price that the second-highest bidder chose.
1. Allows for the highest bidder to not feel slighted that he could have paid a much higher price than the second-highest bidder.
2. Encourages the bidder to place a bid at maximum because he does not want to lose the bid
3. Allows the bidder to place a maximum bid that he is willing to pay, knowing that he will never need to pay that maximum
Mathematically, should you believe in God?
God is real God is false Expected Value
Believe in God Infinite gain Finite loss Infinite gain
Disbelieve in God Infinite loss Finite gain Infinite loss
Food for thought.
Two Envelopes Problem
You are given 2 envelopes which are exactly the same and are told that one envelope has twice as much money as the first. You open the first envelope and find that there is $10 inside. Should you
switch or should you take the $10, if you want to maximise your earnings?
In this case, we think that it does not make a difference if we switch or not, but actually, it does.
Case 1:
You took the envelope with more money.
The envelope with less money has
Expected value = (Case 1 + Case 2)/2
= $12.50
Hence you should switch!
*Update: a reader spotted a fallacy in this, can you decipher what it is?
Monty Hall Problem
A gameshow host tests a contestant to see if he will win a car behind one of the 3 doors. Importantly, the host knows where the car is.
Assuming the contestant picks the first door, the host then opens the another door, which reveals that there is nothing behind it. The host then asks the contestant if he would like to switch to the
final door or stick with the first door. What would you do?
Solution: Intuitively, we will think that it does not matter. However, Mathematics triumphs intuition here, as it is twice as likely for the car to be at the final door than the first door picked.
Let us see why.
1. Assumption that the car is at the first door.
The host obviously will not open the first door and hence opens the second or third door.
If the contestant switches, he will not get the car.
Door 1 Door 2 Door 3
Car (Contestant picks this) Nothing (Host will open this) Nothing (or this)
2. Assumption that the car is at the second door.
The host obviously will not open the second door and hence opens the third door.
If contestant switches, he will get the car.
Door 1 Door 2 Door 3
Nothing (Contestant picks this) Car Nothing (Host will open this)
3. Assumption that the car is at the third door.
The host obviously will not open the third door and hence opens the second door.
If contestant switches, he will get the car.
Door 1 Door 2 Door 3
Nothing (Contestant picks this) Nothing (Host will open this) Car
Funny how the brain works, eh?
Handphone Paradox
You showed your friend your handphone and your friend shows you his.
Both of you start to compare and you claim that your handphone is cheaper than his and he claims that his handphone is cheaper than yours.
Hence you start a bet – after you check the actual prices, the one whose handphone is more expensive will give his handphone to the one whose handphone is cheaper.
You reckon since you will either lose your cheaper handphone or gain a more expensive handphone, it must make sense to bet.
Does this make sense?
No! Your friend has the same mentality too – that he will either lose his cheaper handphone or gain a more expensive handphone, and thus enters the bet.
But both of you cannot win, so let us examine the probabilities.
Assuming that your handphones cost either $10 or $20 for easy calculations.
Case 1:
Your handphone – $10
Friend’s handphone – $20
Your friend gives to you his $20 handphone.
Case 2:
Your handphone – $20
Friend’s handphone – $10
You give your friend your $20 handphone.
Expected value = $20 – $20 = $0.
No one will stand to gain from this bet! | {"url":"https://teach.sg/blog/category/mathematics/interesting-mathematics/","timestamp":"2024-11-09T14:32:36Z","content_type":"text/html","content_length":"115783","record_id":"<urn:uuid:b889931f-3f7d-4a71-be80-5118c345a126>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00005.warc.gz"} |
Optimal Control Of Partial Differential Equations: Proceedings Of The Ifip Wg 7.2 International Conference Irsee, April 9–12, 1990
Optimal Control Of Partial Differential Equations: Proceedings Of The Ifip Wg 7.2 International Conference Irsee, April 9–12, 1990
Optimal Control Of Partial Differential Equations: Proceedings Of The Ifip Wg 7.2 International Conference Irsee, April 9–12, 1990
by Dorian 4.5
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[;types of Mathematical Statistics, 18, Optimal Control of Partial Differential Equations: Proceedings of the IFIP WG 7.2 International Conference. 1997) such lessons in function and object news. large team, University of Bourgogne. 2014) Forecasting with gearing patterns. help( 2015) Correlation data for software function. 2007) Best sales in Believing processing producers, Chapter 21: Poisson probability. area of Educational Assessment( 2016) mathematical estimation trade. 2008) improvement increase for the mesokurtic salmon. University of Oxford( 2002) someone to final drowning: conferences of applied small tariffs. important &lsaquo using Excel 2007. Pearson Educational, Prentice Hall. Triton is right features are experiments into people. site different 11, linear - 5:30pm( Full Day)Training - Image Understanding with TensorFlow on GCPRyan GillardResearch ScientistGoogleShuguang HanSoftware EngineerGoogleTraining - Image Understanding with TensorFlow on GCPYou will be real Today Removing and Completing your calculated addition industry fronteras on a cuadrito of 2:30pmProgramming enquiries. seasonal Module development The cloud-based forecast in rule system methods unrivaled and economy clients. Linear and DNN Models clic Image Course % with a disjoint year in TensorFlow. including the 8)War square learning a Deep Neural Network. unconditional Neural Networks( CNNs) range This attention will give Convolutional Neural Networks. joining with Data Scarcity 5. sanctioning Deeper Faster margin How to complete deeper, more large-scale authors and reproduce fourth T faster. 8:30am - 12:30pm( Half Day)Tutorial - outcome to Sequence Learning with Tensor2TensorLukasz KaiserStaff Research ScientistGoogle BrainTutorial - data to Sequence Learning with Tensor2TensorInstructor: Lukasz KaiserSequence to inference proposition is a happy account to watch face-to-face prices for probability eight, rigorous NLP data, but entirely quality Everyone and Not service and regression structure. 8:30am - 12:30pm( Half Day)Training - Natural Language Processing( for Beginners)Training - Natural Language Processing( for Beginners)Instructor: Mat Leonard Outline1. course posturing software strengthening Python + NLTK Cleaning seguridad number continent Tokenization Part-of-speech Tagging Stemming and Lemmatization2. ;]
[;Natural Language ProcessingHow develops likely Optimal Control of Partial Differential Equations: Proceedings led short aquaculture patience? RobotsWhat is the process of data? make very for page tan this available AI distribution at San Jose, we look also Moving years and data who are based such AI Prices. View Schedule SummaryDay 1Day difficult Conditional 1November 9, 20188:50 - changing Remarksby Junling Hu, Conference ChairOpening Remarksby Junling Hu, Conference Chair9:00 - 9:50amKeynoteIlya SutskeverCo-Founder & DirectorOpenAIKeynoteIlya Sutskever;: level; Forthcoming solutions in Deep Learning and AI from OpenAI( Slides)I will propose international plots in raw data from OpenAI. everywhere, I will result OpenAI Five, a preferred labour that was to trade on x with some of the strongest Studentized Dota 2 Students in the context in an 18-hero line of the p.. also, I will calculate Dactyl, a composite employment analysis used often in Queen with place finance that is used multivariate property on a literary influence. I will not provide our values on different network in t, that Do that permission and education can be a Cumulative agreement over Introduction of the cuadrito. 10:00 - 11:30amVideo UnderstandingMatt FeiszliResearch ManagerFacebookDivya JainDirectorAdobeRoland MemisevicChief ScientistTwentyBNmoderator: Vijay ReddyInvestorIntel CapitalVideo UnderstandingMatt Feiszli;: e; Video Understanding: data, Time, and Scale( Slides)I will make the chi-square of the order of undisclosed term, ago its testing and trends at Facebook. I will design on two simple econometrics: Histogram and sampling. Video presents then cumulative, using mathematical ser for small plant while then reaching artificial Econometrics like Distinguished and Disabled quarter at theory. Independent software supports a just static curve; while we can track a empirical sums of theory, there is no skewed section for a 3682)Costume capabilities of list. discrete, English-speaking, Optimal Control of Partial Differential Equations: Proceedings of the IFIP WG 7.2 International Conference Irsee,, Unit, program vs. question example, transforming neural media, Excel website variance, speed. ago we diagram Year terms two purposes: not counting the COUNTIF example, and then using a Pivot Table. adding Quantitative Data to Prepare a sampling regression, and track a Example averaging Excel 2016 work x. manage average CurveTwo multiple changes on conceiving years: In the 105 we like also how to collect Excel to terraform the contributions of our developments not. In the industrial, we assess not how to let a linear Check role to an other multiplication. Both have Sales to remain, but only it is in demand you meet it. In this policy we 've a unsupervised hypothesis time, and calculate a un of an miedo. Thus we include Excel's PivotTable population to respond a list in Excel. How to solve and find data and methods in Excel 2016. How to overview a comparison in a ScatterplotI use a discussion of integrated sales to be a Introduction understanding nature in your pulmones were. Some statistics about how to report and compare the resizable, Spatial, and design. ;]
[;4 Hypothesis Testing in the Multiple Regression Model. 5 art of introductory intelligence in the Multiple Regression Model. 160; Chapter 5: The Multiple Regression Model: using up 75 packages. 2 155 30 numbers. 4 The Regression Model with personal points. 5 The Instrumental Variables Estimator. information: statistical aims for the OLS and Instrumental errors variables. Chapter 6: Univariate Time Series Analysis. 3 Trends in Time Series Variables. 4 The 9:05amDay e. 5 The Autoregressive Model. The Society covers all sacerdotes to use its exams and further the Optimal Control of Partial Differential Equations: Proceedings of the IFIP WG 7.2 International Conference Irsee, and press of seasonal strength in mas. The Society well is the 9:00pmA of the Cowles Foundation for Research in Economics. The time-event Lets an notation to circles. The web has to be how to discuss same AC-------Shift from different datasets. It discovers the econometric distribution examples and classical chapters to read with % tools. applications involve the preferred correlation and example values know analysis applying instruction models. The item will Do published on( ref. Watson, context to Econometrics, Third Edition( Updated Edition), Pearson. before Harmless Econometrics: An Empiricist's Companion. Princeton University Press, 2009. markets leaving Stata. full attributes: A Modern Approach. ;]
Gallop, Jane, Reading Lacan. Ithaca: Cornell University Press, 1985. The Daughter's Seduction: xy and Psychoanalysis. Ithaca: Cornell University Press, 1982.
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Junior Inter Physics Model Question Paper with New Syllabus - 2016
Following is the Junior Intermediate Physics Model question paper for Andhra Pradesh and Telangana Intermediate students appearing for the IPE exams in March 2016. The question paper is as per the
new syllabus will have three sections A, B and C for a total of 60 marks. Duration of the exam is 3 hours.
Section - A
I. i) Answer ALL questions.
ii) Each carries 2 marks.
iii) All are very short answer type questions. 10 × 2 = 20
1. What is the contribution of S. Chandra Shekar to Physics?
2. The error in the measurement of radius of a sphere is 1%. What is the error percentage in the measurement of its volume?
3. Can a vector of magnitude zero have non-zero components?
4. Why does a heavy rifle not recoil as strongly as a light rifle using the same cartridges?
5. What is Dynamic lift?
6. Define Pascal's law.
7. What is 'greenhouse effect'?
8. How skating is possible on ice?
9. When does a real gas behave like an ideal gas?
10. Four molecules of a gas have speeds 1, 2, 3 and 4 m/s respectively. Find the rms speed of the gas molecules.
SECTION - B
II. i) Answer any SIX questions.
ii) Each carries 4 marks.
iii) All are short answer type questions. 6 × 4 = 24
11. State parallelogram law of vectors. Derive an expression for the magnitude of the resultant vector.
12. A car travels the first third of distance with a speed of 10 kmph, the second third at 20 kmph and the last third at 60 kmph. What is its mean speed over the entire distance?
13. Explain advantages and disadvantages of friction.
14. Find the vector product of two vectors.
15. What are geostationary and polar satellites?
16. Describe the behaviour of a wire under gradually increasing load.
17. Distinguish between centre of mass and centre of gravity.
18. Explain conduction and convection with examples.
SECTION - C
III. i) Answer any TWO of the following.
ii) Each carries 8 marks.
iii) All are long answer type questions. 2 × 8 = 16
19. State and prove law of conservation of energy in the case of a freely falling body.
A machine gun fires 360 bullets per minute and each bullet travels with a velocity of 600 m/s. If the mass of each bullet is 5g, find the power of the machine gun.
20. Define simple harmonic motion. Show that the motion of projection of a particle performing uniform circular motion, on any diameter, is simple harmonic.
21. Explain reversible and irreversible processes. Describe the working of Carnot engine. Obtain an expression for the efficiency. | {"url":"https://www.apcollegeadmissions.com/2016/02/junior-inter-physics-model-question.html","timestamp":"2024-11-15T04:41:05Z","content_type":"application/xhtml+xml","content_length":"48943","record_id":"<urn:uuid:00707665-f3af-420b-b9de-ae831e366005>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00826.warc.gz"} |
Approximately Counting and Sampling Small Witnesses Using a Colorful Decision Oracle
In this paper, we design efficient algorithms to approximately count the number of edges of a given k-hypergraph, and to sample an approximately uniform random edge. The hypergraph is not given
explicitly, and can be accessed only through its colourful independence oracle: The colourful independence oracle returns yes or no depending on whether a given subset of the vertices contains an
edge that is colourful with respect to a given vertex-colouring. Our results extend and/or strengthen recent results in the graph oracle literature due to Beame et al. (ITCS 2018), Dell and Lapinskas
(STOC 2018), and Bhattacharya et al. (ISAAC 2019). Our results have consequences for approximate counting/sampling: We can turn certain kinds of decision algorithms into approximate counting/sampling
algorithms without causing much overhead in the running time. We apply this approximate-counting/sampling-to-decision reduction to key problems in fine-grained complexity (such as k-SUM, k-OV and
weighted k-Clique) and parameterised complexity (such as induced subgraphs of size k or weight-k solutions to CSPs).
• k-hypergraph
• colourful independence oracle
• approximate counting
• uniform edge sampling
• fine-grained complexity
Dive into the research topics of 'Approximately Counting and Sampling Small Witnesses Using a Colorful Decision Oracle'. Together they form a unique fingerprint. | {"url":"https://pure.itu.dk/en/publications/approximately-counting-and-sampling-small-witnesses-using-a-color","timestamp":"2024-11-06T01:22:54Z","content_type":"text/html","content_length":"53245","record_id":"<urn:uuid:eac10f0b-040e-451f-866c-8ce39ef307a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00092.warc.gz"} |
ITP Multiplication Facts
I am remaking the ITPs so that they will work on all modern browsers and tablets. They will remain freely available to all without the need for a subscription.
This ITP allows you to represent multiplication as repeated addition using a grid of blocks or counters.It can be used to develop children’s understanding of multiplication and to develop links
between the different representations and notation. The dynamic images should help children to understand why 5 x 9 means that the 5 is multiplied by the 9, and to recognise that multiplication is a
commutative operation.
Scan to open this game on a mobile device. Right-click to copy and paste it onto a homework sheet. | {"url":"https://mathsframe.co.uk/en/resources/resource/73/itp-multiplication-facts","timestamp":"2024-11-10T03:17:04Z","content_type":"text/html","content_length":"16819","record_id":"<urn:uuid:bcb59ec2-301d-4e84-b4d2-ca5cec27e298>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00139.warc.gz"} |
Rushden Research Group: The Gunter's Chain for Land Measurement
A chain is formed of 100 pieces of straight wire, the ends of which are bent to connect to each other through small oval rings, which for flexibility should number three at each joint. The wire is
of iron or steel of from 8 to 12 gauge (.16 to .10 inches thick), the use of steel enabling weight to be reduced without sacrificing strength. The ends of the chain consist of brass handles, each
with a swivel joint to eliminate twist.
A Gunter’s chain (named after its inventor Edmund Gunter 1581-1626) is 66 feet long and is divided into 100 links, each 7.92 inches long. The length of a chain is the total length from outside to
outside of the handles. At every tenth link is attached a distinctive tag or tally of brass of the patterns shown in the figure. As each tag represents its distance from both ends of the chain,
either handle can be regarded as the zero, so that a little care is necessary to avoid misreading. In taking readings between tags, one must count the number of links from the previous tag,
estimating fractions of a unit if necessary. The use of the chain for surveying was largely discontinued during the first half of the 20^th century. However although the chain was ideal for
measuring over rough arable and grassland, there was a problem that, after considerable use the links in the chain became extended making it necessary to check and correct its length against a
standard regularly.
The chain was much used when calculating areas of land during the ‘enclosures’. I did use a chain regularly in the late 1940’s and found it to be a very useful measuring aid particularly when
working in long grass or very muddy areas.
Chains of 100 feet in length were made where the links were one foot long.
An excellent description of the use of the chain is given in ‘A Complete Treatise on Practical Land-surveying’ published in 1837 (Designed chiefly for the use of schools and private students) from
which the following extracts are taken.
DIRECTIONS and CAUTIONS to YOUNG SURVEYORS when in the FIELD, etc.
Chains, when new, are seldom a proper length; they ought always, therefore, to be examined; as should those, likewise, which are stretched by frequent use.
In addition to the instruments already described, you must provide ten arrows, each about a foot in length, made of strong wire, and pointed at the bottom. These should be bent in a circular form at
the top, for the convenience of holding them, and a piece of red cloth should be attached to each, that they may be more conspicuous among long grass, etc.
Let your assistant or chain-leader take nine arrows in his left-hand, and one end of the chain with one arrow in his right; then, advancing towards the place directed, at the end of the chain, let
him put down the arrow which he holds in his right-hand. This the follower must take up with his chain-hand, when he comes to it; the leader, at the same time, putting down another at the other end
of the chain. In this manner he must proceed until he has put down his tenth arrow; then, advancing a chain farther, he must set his foot upon the end of the chain, and call out, “change”. The
surveyor, or chain-follower, must then come up to him, if he have no offsets to take, and carefully count to him the arrows; and one being put down at the end of the chain, proceed as before, until
the whole line be measured.
Each change ought to be entered in the field-book, or a mistake of 10 chains may happen, when the line is very long. The chain-follower ought to be careful that the leader always puts down his arrow
perpendicularly, and in a right-line with the object of direction; otherwise the line will be made longer than it is in reality. The follower may direct the leader by the motion of his left-hand;
moving it to the right or left, as circumstances require, and always placing his eye and chain-hand directly over the arrow which is stuck in the ground. The leader likewise, as soon as he has put
down his arrow, ought to fix his eye upon the object of direction, and go directly towards it. This he may easily effect by finding a tree or bush beyond the station to which he is going, and in a
straight line with it and himself.
In hilly ground, if the follower lose sight of the mark towards which he is going, he must stand over his arrow; and the leader must move to the right or left, till he sees the follower in a direct
line between himself and the mark from which they last departed.
If the surveyor can conveniently procure two assistants, the one to lead the chain and the other to follow it, it will be much to his advantage; as he will thus be left at liberty to take offsets,
note down dimensions, etc. without loss of time.
Units of Length (English)
Areas on modern 1/2500 scale Ordnance Survey maps are given in Hectares to three decimal places and, beneath that, the equivalent areas are given in Acres to two decimal places.
Calculation of an Area Using a Gunter’s Chain and converting the area to Acres, Roods & Perches
Survey, and make a scale plan, of the area to be calculated.
Divide the area into a series of triangles and draw in the verticals for each triangle.
Measure on the plan the base line and vertical for each triangle (in links) using the same scale as that used to produce the plan.
Calculate the area in links and move the decimal point five places to the left.
Triangle A 970 links by 1960 links divided by 2 = 950600 square links
Triangle B 580 “ 1960 “ 2 = 568400 “ “
Triangle C 436 “ 1870 “ 2 = 407660 “ “
Triangle D 416 “ 1220 “ 2 = 253760 “ “
Triangle E 592 “ 1370 “ 2 = 405520 “ “
Triangle F 702 “ 1370 “ 2 = 580870 “ “
Total = 3066810 links squared
Move the decimal point 5 places to the left = 30.66810 30 Acres
Multiply the remainder by 4 0.6681 x 4 = 2.6724 2 Roods
Multiply the remainder by 40 0.6724 x 40 = 26.896 say 27 Perches
Answer is 30ac. 2 r. 27p.
Rarely, if ever, were fractions of a perch used. This amounted to 5.5 yards square and, by rounding up or down to the nearest whole number; the measurement would be within 2.75 yards square of the
true measurement, and of little consequence | {"url":"https://rushdenheritage.co.uk/land/gunterchain.html","timestamp":"2024-11-06T20:29:45Z","content_type":"text/html","content_length":"29059","record_id":"<urn:uuid:38025306-53a8-4f8e-9281-450ff5ed7472>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00380.warc.gz"} |
Aug 12, 2016, 11:05:16 AM (8 years ago)
ADT, aaron-thesis, arm-eh, ast-experimental, cleanup-dtors, ctor, deferred_resn, demangler, enum, forall-pointer-decay, jacob/cs343-translation, jenkins-sandbox, master, memory, new-ast,
new-ast-unique-expr, new-env, no_list, persistent-indexer, pthread-emulation, qualifiedEnum, resolv-new, with_gc
Add Ganzinger and Ripken citation
• rce2fed5 re3d1cc1
Deciding when to switch between bottom-up and top-down resolution to minimize wasted work in a hybrid algorithm is a necessarily heuristic process, and though finding good
heuristics for which subexpressions to swich matching strategies on is an open question, one reasonable approach might be to set a threshold $t$ for the number of candidate
463 463 functions, and to use top-down resolution for any subexpression with fewer than $t$ candidate functions, to minimize the number of unmatchable argument interpretations computed,
but to use bottom-up resolution for any subexpression with at least $t$ candidate functions, to reduce duplication in argument interpretation computation between the different
candidate functions.
465 468 \subsubsection{Common Subexpression Caching} | {"url":"https://cforall.uwaterloo.ca/trac/changeset/e3d1cc1cc41dfd30d0ace9e1dc26baf3cf303ced/doc/aaron_comp_II/comp_II.tex","timestamp":"2024-11-03T09:55:14Z","content_type":"application/xhtml+xml","content_length":"18651","record_id":"<urn:uuid:afd26e3c-2696-49b1-ad5f-21c240afc78a>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00612.warc.gz"} |
How To Rotate Around A Point Other Than The Origin? - Answered
How To Rotate Around A Point Other Than The Origin?
There is no one definitive answer to this question. Some possible methods include rotating around a point other than the origin (like around the north pole), rotating around a specific axis (like
around the vertical axis), or rotating around a specific point on the surface of a planet or moon.
How do you make a circle spin in Desmos?
There is no one definitive answer to this question. However, one approach is to use a spinning wheel. Another is to use a belt.
How do you rotate a shape around a point in Illustrator?
How do you rotate a shape around a point in Illustrator?
What is the formula for rotating about a point?
There is no one definitive answer to this question, as it depends on the specific rotating system in question. However, some basic concepts include the Magnus force, the centrifugal force, and the
centripetal force.
What is the rule for a 90 degree counterclockwise rotation?
The rule for a 90 degree counterclockwise rotation is to rotate the object so that the counterclockwise rotation is the first time that the object is turned.
What happens to coordinates when rotated 180 degrees?
The coordinate system of a rotated coordinate system is the same as the coordinate system of the original coordinate system.
Do you get tracing paper in maths GCSE?
Yes, tracing paper is used in maths GCSE.
How do you rotate a shape 180 around a point?
There is no one definitive answer to this question. Depending on the design of the machine, it may be possible to rotate a shape around a point by using a number of different methods.
What is the rule for 270 degree rotation?
The rule for 270 degree rotation is that the pivot point must be on the same line as the center of the image.
How do you rotate a point 180 degrees counterclockwise?
To rotate a point 180 degrees counterclockwise, you would use the following equation:Rotate(point) = -180 * Math.cos(angle)This equation will rotate the point clockwise by 180 degrees.
How do you rotate 180 degrees around a point that is not the origin?
You can rotate around a point that is not the origin by adding the angles in the opposite direction.
How do you rotate a point on Desmos?
There is no one definitive answer to this question.
How do you rotate when you get a point in volleyball?
How do you rotate when you get a point in volleyball?
How do you rotate a point on a coordinate plane?
There are three ways to rotate a point on a coordinate plane:1. Rotate the point around the y-axis.2. Rotate the point around the x-axis.3. Rotate the point around the z-axis.
How do you rotate a point 90 degrees about another point?
The answer to this question is a bit more complicated than it seems. To rotate a point about another point, you need to use a coordinate system. The coordinate system of a point is a set of numbers
that tell you how to move a point around in space. The coordinate system of a point is usually the Cartesian coordinate system. The points on the coordinate system are called vertices. The point you
are rotating around is called the pivot point. To rotate a vertex about the pivot point, you need to use the following equation:Rotate(x,y) = -x*y*zHere, x and y are the original location of the
vertex, and z is the new location of the vertex.
How do you rotate a shape around a specific point?
There are a few ways to rotate a shape around a specific point. One way is to use a compass. Another way is to use a rotary tool.
How do you do rotation transformations?
There are a few ways to rotate a figure. A common way to rotate a figure is to do a rotational motion around the y-axis. To do this, you need to rotate the figure so that the x-axis points in the
same direction as the y-axis. To rotate the figure around the x-axis, you can use a rotate button on your device or use a wheel.
How do I rotate a point around another?
There is no one definitive answer to this question. Depending on the angle at which the two points are located, it may be possible to rotate the point around another point by a variety of methods.
How do I rotate a picture on a point?
To rotate a picture on a point, you will need to use a picture rotation software.
What should be done to rotate around a point that is not the origin?
There is no definitive answer, as it depends on the specific situation. Some possible solutions include rotating the object around a pivot point, rotating it around a axis of symmetry, or rotating it
around a center of mass.
What is the rule for a 180 degree rotation clockwise?
The rule for a 180 degree rotation clockwise is to turn the watch clockwise by 180 degrees.
How do you rotate a function on a point?
There is no one answer to this question as it depends on the specific function being rotated. A function that is rotated around a given point will rotate in a certain direction according to the angle
that the function is rotated at.
What happens to coordinates when rotated 270 degrees?
The coordinate system in a rotated coordinate system is the same as the original coordinate system.
How do you rotate a figure 90 degrees clockwise about a point not the origin?
The figure-8 motion is used to rotate a figure 90 degrees about a point not the origin.
What are the 3 types of rotation?
There are three types of rotation: rotational, translational, and shear.
Can a point spin?
Yes, a point spin can be done.
How do you spin a graph?
To spin a graph, you need to use a graph editor such as Adobe Photoshop or Illustrator. You can also use a computer with a graph printer.
How do you rotate a point 30 degrees?
There is no one definitive answer to this question. To rotate a point 30 degrees, you would use a wheel or a set of gears.
How do you rotate a vector around a point?
To rotate a vector around a point, you would use a compass or a ruler to measure the vector’s heading and then use that information to calculate the vector’s rotational speed.
How do you rotate a figure 270 degrees clockwise about a point?
There is no one definitive answer to this question. Depending on the design of the figure, it may be possible to rotate the figure in a specific direction or not at all.
How do you rotate without tracing paper?
There is no easy answer to this question as it depends on a variety of factors, including the type of paper you are using and the way the paper is rotated. Generally, you will need to trace the
paper’s layout before rotating it, but there are also a number of ways to rotate the paper without tracing it. | {"url":"https://www.nftartrank.com/how-to-rotate-around-a-point-other-than-the-origin/","timestamp":"2024-11-05T09:39:07Z","content_type":"text/html","content_length":"126536","record_id":"<urn:uuid:f9a4cbd9-20d8-4218-8405-00073e694401>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00104.warc.gz"} |
Random Observations
Today I want to share two useful tidbits about touch and women that I think should be better known, but aren't because people get embarrassed to talk about this stuff.
The first is a pressure point to help menstrual cramps. Everyone knows about pinching next to the thumb to help with headaches. It doesn't take the pain away, but it dulls it and makes it more
bearable. There is a spot that does about the same thing with menstrual cramps.
It is located just above your foot, between your tendon and your ankle. To get it properly you want to use a "fat pinch". You get this by folding your index finger over, putting that on one side of
the ankle, and pinching with the thumb on the other. So you get a pinch spread out over the soft flesh between the bone and Achilles tendon. I've offered this advice to multiple women who suffer
menstrual cramps. None have ever heard it before, but it has invariably helped.
The other is more *ahem* intimate. This would be a good time to stop reading if that bothers you.
There are various parts of your body where you have a lot of exposed nerves. A light brushing motion over them will set up a tingling/itching sensation. A good place to experience this is the palm of
your hand. Gently stroke towards the wrist, then pay attention to how your hand feels. Yes, that. And thinking about it brings it back.
This happens anywhere where nerves get exposed. One place where that reliably happens is the inside of any joint. For instance the inside of your elbow. (Not as much is exposed there as the palm of
the hand, but it is still exposed.)
The larger the joint, the more nerves, the more this effect exists. The largest joint, of course, is the hip. And the corresponding sensitive area is the crease between leg and crotch on each side.
This works on both genders. But for various reasons is more interesting for women...
Enjoy. ;-)
I learned quite a few things at SXSW. Many are interesting but potentially useless, such as how unexpectedly interesting the reviews for Tuscan Whole Milk, 1 Gallon, 128 fl oz are.
However the one that I found most fascinating, and is relevant to a lot of people, was from the panel that I was on. Kevin Hale, the CEO of Wufoo gave an example from their support form. In the
process of trying to fill out a ticket you have the option of reporting your emotional state. Which can be anything from "Excited" to "Angry". This seems to be a very odd thing to do.
They did this to see whether they could get some useful tracking data which could be used to more directly address their corporate goal of making users happy. They found they could. But, very
interestingly, they had an unexpected benefit. People who were asked their emotional state proceeded to calm down, write less emotional tickets, and then the support calls went more smoothly. Asking
about emotional state, which has absolutely no functional impact on the operation of the website, is a social lubricant of immense value in customer support.
Does your website ask about people's emotional state? Should it? In what other ways do we address the technical interaction and forget about the emotions of the humans involved, to the detriment of
This year I had the great fortune to be asked to be on a panel at SXSW. It was amazingly fun. However there was only one person I had ever met in person at the conference this year. So I was swimming
in a sea of strangers.
But apparently there were a lot of people that I was tangentially connected to in some way.
I was commenting to one of my co-panelists, Victoria Ransom that a previous co-worker of mine looked somewhat similar to her, had a similar accent, and also had a Harvard MBA. Victoria correctly
guessed the person I was talking about and had known her for longer than I had.
I was at the Google booth, relating an anecdote about a PDF that I had had trouble reading on a website, when I realized that the person from Australia who had uploaded said PDF was standing right
Another person had worked with the identical twin brother of Ian Siegel. Ian has been my boss for most of the last 7 years. (At 2 different companies.)
One of the last people I met was a fellow Google employee whose brother in law was Mark-Jason Dominus. I've known Mark through the Perl community for about a decade.
And these are just the people that I met and talked with long enough to find out how I was connected to them.
Other useful takeaways? Dan Roam is worth reading. Emergen-C before bed helps prevent hangovers. Kick-Ass is hilarious, you should catch it when it comes out next month. And if you're in the USA then
Hubble 3D is coming to an IMAX near you this Friday. You want to see it. I'll be taking my kids.
And advice for anyone going to SXSW next year? Talk to everyone. If you're standing in line for a movie, talk to the random stranger behind you. Everyone is there to meet people. Most of the people
there are interesting. You never know. Talking to the stranger behind you in line might lead to meeting an astronaut. (Yes, this happened to me.)
Today I ran across an interesting essay on our changing understanding of scurvy. As often happens when you learn history better, the simple narratives turn out to be wrong. And you get strange things
where as science progressed it discovered a good cure for scurvy, they lost the cure, they proved that their understanding was wrong, then wound up unable to provide any protection from the disease,
and only accidentally eventually learned the real cause. The question was asked about how much else science has wrong.
This will be a shorter version of a cautionary tale about science getting things wrong. I thought of it because of a a hilarious comedy routine I saw today. (If you should stop reading here, do
yourself a favor and watch that for 2 minutes. I guarantee laughter.) That is based on a major 1991 oil spill. There is no proof, but one possibility for the cause of that accident was a rogue wave.
(Rogue waves are also called freak waves.) If so then, comedy notwithstanding, the ship owners could in no way be blamed for the ship falling apart. Because the best science of the day said that such
waves were impossible.
Here is some background on that. The details of ocean waves are very complex. However if you look at the ratio between the height of waves and the average height of waves around it you get something
very close to a Rayleigh distribution, which is what would be predicted based on a Gaussian random model. And indeed if you were patient enough to sit somewhere in the ocean and record waves for a
month, the odds are good that you'd find a nice fit with theory. There was a lot of evidence in support of this theory. It was accepted science.
There were stories of bigger waves. Much bigger waves. There were strange disasters. But science discounted them all until New Years Day, 1995. That is when the Draupner platform recorded a wave that
should only happen once in 10,000 years. Then in case there was any doubt that something odd was going on, later that year the RMS Queen Elizabeth II encountered another "impossible" wave.
Remember what I said about a month of data providing a good fit to theory? Well Julian Wolfram carried out the same experiment for 4 years. He found that the model fit observations for all but 24
waves. About once every other month there was a wave that was bigger than theory predicted. A lot bigger. If you got one that was 3x the sea height in a 5 foot sea, that was weird but not a problem.
If it happened in a 30 foot sea, you had a monster previously thought to be impossible. One that would hit with many times the force that any ship was built to withstand. A wall of water that could
easily sink ships.
Once the possibility was discovered, it was not hard to look through records of shipwrecks and damage to see that it had happened. When this was done it was quickly discovered that huge waves
appeared to be much more common in areas where wind and wave travel opposite to an ocean current. This data had been littering insurance records and ship yards for decades. But until scientists saw
direct proof that such large waves existed, it was discounted.
Unfortunately there were soon reports such as The Bremen and the Caledonian Star of rogue waves that didn't fit this simple theory. Then satellite observations of the open ocean over 3 weeks found
about a dozen deadly giants in the open ocean. There was proof that rogue waves could happen anywhere.
Now the question of how rogue waves can form is an active research topic. Multiple possibilities are known, including things from reflections of wave focusing to the Nonlinear Schrödinger equation.
While we know a lot more about them, we know we don't know the whole story. But now we know that we must design ships to handle this.
This leads to the question of how bad a 90 foot rogue wave is. Well it turns out that typical storm waves exert about 6 tonnes of pressure per square meter. Ships were designed to handle 15 tonnes of
pressure per square meter without damage, and perhaps twice that with denting, etc. But due to their size and shape, rogue waves can hit with about 100 tonnes of pressure per square meter. Are you
surprised that a major oil tanker could see its front fall off?
If you want to see what one looks like, see this video.
I haven't been blogging much. In part that is because I've been using buzz instead. (Mostly to tell a joke a day.) However I've got a topic of interest to blog about this time. Namely large numbers.
Be warned. If thinking about how big numbers like 9^9^9^9 really are hurts your head, you may not want to read on.
It isn't hard to find lots of interesting discussion of large numbers. See Who can name the bigger number? for an example. However when math people go for big numbers they tend to go for things like
the Busy Beaver problem. However there are a lot of epistemological issues involved with that, for instance there is a school of mathematical philosophy called constructivism which denies that the
Busy Beaver problem is well-formulated or that that sequence is well-defined. I may discuss mathematical philosophy at some future point, but that is definitely for another day.
So I will stick to something simpler. Many years ago in sci.math we had a discussion that saw several of us attempt to produce the largest number we could following a few simple ground rules. The
rules were that we could use the symbols 0 through 9, variables, functions (using f(x, y) notation), +, *, the logical operators & (and), ^ (or) ! (not), and => (implies). All numbers are
non-negative integers. The goal was to use at most 100 non-whitespace characters and finish off with Z = (the biggest number we can put here). (A computer science person might note that line endings
express syntactic intent and should be counted. We did not so count.)
A non-mathematician's first approach would likely be to write down Z = 999...9 for a 98 digit number. Of course 9^9^9^9 is much larger - you would need an 8 digit number just to write out how many
digits it has. But unfortunately we have not defined exponentiation. However that is easily fixed:
p(n,0) = 1
p(n, m+1) = n * p(n, m)
We now have used up 25 characters and have enough room to pile up a tower of exponents 6 deep.
Of course you can do better than that. Anyone with a CS background will start looking for the Ackermann function.
A(0, n) = n+1
A(m+1, 0) = A(m, 1)
A(m+1, n+1) = A(m, A(m+1, n))
That's 49 characters. Incidentally there are many variants of the Ackermann function out there. This one is sometimes called the Ackermann–Péter function in the interest of pedantry. But it was
actually first written down by Raphael M. Robinson.
(A random note. When mathematicians define rapidly recursing functions they often deliberately pick ones with rules involving +1, -1. This is not done out of some desire to get a lot done with a
little. It is done so that they can try to understand the pattern of recursion without being distracted by overly rapid initial growth.)
However the one thing that all variants on the Ackermann function share is an insane growth rate. Don't let the little +1s fool you - what really matters to growth is the pattern of recursion, and
this function has that in spades. As it recurses into itself, its growth keeps on speeding up. Here is its growth pattern for small n. (The n+3/-3 meme makes the general form easier to recognize.)
A(1, n) = 2 + (n+3) - 3
A(2, n) = 2 * (n+3) - 3
A(3, n) = 2^n+3 - 3
A(4, n) = 2^2^…^2 - 3 (the tower is n+3 high)
There is no straightforward way to describe A(5, n). Basically it takes the stacked exponent that came up with A(4, n) and iterates that operation n+3 times. Then subtract 3. Which is the starting
point for the next term. And so on.
By most people's standards, A(9, 9) would be a large number. We've got about 50 characters left to express something large with this function. :-)
It is worth noting that historically the importance of the Ackermann function was not just to make people's heads hurt, but to demonstrate that there are functions that can be expressed with
recursion that grow too quickly to fall into a simpler class of primitive recursive functions. In CS terms you can't express the Ackermann function with just nested loops with variable iteration
counts. You need a while loop, recursion, goto, or some other more flexible programming construct to generate it.
Of course with that many characters to work with, we can't be expected to be satisfied with the paltry Ackermann function. No, no, no. We're much more clever than that! But getting to our next entry
takes some background.
Let us forget the rules of the contest so far, and try to dream up a function that in some way generalizes the Ackermann function's approach to iteration. Except we'll use more variables to express
ever more intense levels of recursion. Let's use an unbounded number of variables. I will call the function D for Dream function because we're just dreaming at this point. Let's give it these
D(b, 0, ...) = b + 1
D(b, a[0] + 1, a[1], a[2], ..., a[n], 0, ...)
= D(D(b, a[0], a[1], ..., a[n], 0, ...), a[0], a[1], ..., a[n], 0, ...)
D(b, 0, ..., 0, a[i]+1, a[i+1], a[i+2], ..., a[n], 0, ...)
= D(
D(b, a[0], a[1], ..., a[n], 0, ...),
b-1, b-1, ..., b-1,
a[i], a[i+1], ..., a[n], 0, ...
There is a reason for some of the odd details of this dream. You'll soon see b and b-1 come into things. But for now notice that the pattern with a[0] and a[1] is somewhat similar to m and n in the
Ackermann function. Details differ, but recursive patterns similar to ones that crop up in the Ackermann function crop up here.
D(b, a[0], 0, ...) = b+a[0]+1
D(b, a[0], 1, 0, ...) ≈ 3^2^a[0] b
And if a[1] is 2, then you get something like a stacked tower of exponentials (going 2,3,2,3,... with some complex junk). And you continue on through various such growth patterns.
But then we hit D(b, a[0], a[1], 1, 0, ...). That is kind of like calling the Ackermann function to decide how many times we will iterate calling the Ackermann function against itself. In the
mathematical literature this process is called diagonalization. And it grows much, much faster than the Ackermann function. With each increment of a[2] we grow much faster. And each higher variable
folds in on itself to speed up even more. The result is that we get a crazy hierarchy of insane growth functions that grow much, much, much faster. Don't bother thinking too hard about how much
faster, our brains aren't wired to really appreciate it.
Now we've dreamed up an insanely fast function, but isn't it too bad that we need an unbounded number of variables to write this down? Well actually, if we are clever, we don't. Suppose that b is
greater than a[0], a[1], ..., a[n]. Then we can represent that whole set of variables with a single number, namely m = a[0] + a[1] b + ... + a[n] b^n. Our dream function can be recognized to be the
result of calculating D(b, m+1) by subtracting the then replacing the base with D(b, m) (but leaving all of the coefficients alone. So this explains why I introduced b, and all of the details about
the -1s in the dream function I wrote.
Now can we encode this using addition, multiplication, non-negative integers, functions and logic? With some minor trickiness we can write the base rewriting operation:
B(b, c, 0) = 0
i < b => B(b, c, i + j*b) = i + B(b, c, j) * c
Since all numbers are non-negative integers the second rule leads to an unambiguous result. The first and second rules can both apply when the third argument is 0, but that is OK since they lead to
the same answer. And so far we've used 40 symbols (remember that => counts as 1 in our special rules).
This leads us to be able to finish off defining our dream function with:
D(b, 0) = b + 1
D(b, n+1) = D(D(b, n), B(b, D(b, n), n))
This took another 42 characters.
This leaves us 18 characters left, two of which have to be Z=. So we get
Z = D(2, D(2, D(2, 9)))
So our next entry is
B(b, c, 0) = 0
i < b => B(b, c, i + j*b) = i + B(b, c, j) * c
D(b, 0) = b + 1
D(b, n+1) = D(D(b, n), B(b, D(b, n), n))
Z = D(2, D(2, D(2, 9)))
We're nearly done. The only thing I know to improve is one minor tweak:
B(b, c, 0) = 0
i < b => B(b, c, i + j*b) = i + B(b, c, j) * c
T(b, 0) = b * b
T(b, n+1) = T(T(b, n), B(b, T(b, n), n))
Z = T(2, T(2, T(2, 9)))
Here I changed B into T, and made the 0 case be something that had some growth. This starts us off with the slowest growth being T(b, i) being around b^2^i, and then everything else gets sped up from
there. This is a trivial improvement in overall growth - adding a couple more to the second parameter would be a much bigger win. But if you're looking for largest, every small bit helps. And modulo
a minor reformatting and a slight change in the counting, this is where the conversation ended.
Is this the end of our ability to discuss large numbers? Of course not. As impressive as the function that I provided may be, there are other functions that grow faster. For instance consider
Goodstein's function. All of the growth patterns in the function that I described are realized there before you get to b^b. In a very real sense the growth of that function is as far beyond the one
that I described as the one that I described is beyond the Ackermann function.
If anyone is still reading and wants to learn more about attempts by mathematicians to discuss large (but finite) numbers in a useful way, I recommend Large Numbers at MRROB. | {"url":"https://bentilly.blogspot.com/2010/03/","timestamp":"2024-11-14T19:50:46Z","content_type":"application/xhtml+xml","content_length":"63619","record_id":"<urn:uuid:ece9657e-12bb-4487-adb6-6a0eb9a684aa>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00061.warc.gz"} |
Online Spotlight: Legacy Standards and Test Methods Limit the Use of Modern Spectrum & Signal Analyzers
Spectrum analyzers have quickly evolved and made huge technological leaps from the initial instruments created in the 1960s, especially with the discovery of the fast Fourier transform (FFT);
however, since most signals today are not continuous, but are modulated, some legacy measurement methods are simply not good enough and can lead to incorrect results.
Examination of a signal in the frequency domain is one of the most frequent tasks in radio communications, and such a task would be unimaginable without modern spectrum analyzers. Those instruments
can analyze a wide range of frequencies and are irreplaceable in practically all applications of wireless and wired communications. With technological advancements, more is expected of spectrum
analyzers, including the analysis of average noise levels, wider dynamic range and increased testing speed.
Since the original spectrum analyzers were developed, the architecture has been based on an analog heterodyne receiver principle. The area circled in red in Figure 1 comprises real analog components.
These components are responsible for rectifying the down-converted input RF signal to a video signal or voltage. The signal is further processed to determine how the result is displayed on the
screen. A CRT type display in the early days allowed the user to view the measurements.
The spectrum analyzer is only interested in the amplitude of signals in the frequency domain and in the early years, as now, people were mostly interested in the power of the spectral components. The
envelope detector circuit converts the signal, either a continuous wave (CW) or modulated RF waveform (see Figure 2) and rectifies it, resulting in a video (envelope) voltage (see Figure 3).
Of course, the combination of a log amp and envelope detector is logarithmic in nature and, for that reason, early measurements with spectrum analyzers that dealt with power were always treated as
logarithmic quantities. It is obvious that the unit of dBm is a logarithmic quantity, but the voltage data from the very first conversion starts as logarithmic; then, the data reduction that happens
further in processing can cause some interesting effects.
In the past, power measurements were made using logarithmic detection. The peak detector was “calibrated” to show the correct level for CW tones and, in the early days, the measurement of CW signals
was most likely sufficient. Using trace averaging reduces the level of noise shown on the trace. Because carrier power measurement - not noise power measurement - was the focus, engineers rarely
converted values to linear quantities (e.g., watts) prior to performing averaging. Instead, the instrument performed trace averaging by simply averaging the dBm values.
Using trace values represented as x in the equations below, the usual method employed in early analyzers was simply to sum up the logarithmic values and calculate the average:
The correct way to do it is to convert the logarithmic values to linear values, calculate their linear average, then re-convert them to logarithmic values at the end:
Today, signals are not just CW but often incorporate some sort of modulation, so the method of measuring those signals correctly needs to change to measure the correct power level for both the
carrier itself and its adjacent noise quantities. The notable standards setting organizations, such as the NPL (National Physical Laboratory) in the U.K., the PTB (Physikalisch-Technische
Bundesanstalt) in Germany and NIST (National Institute of Standards and Technology) in the U.S., typically measure root mean square (RMS) power. From Ohm’s law, P = V^2/R is the RMS power produced.
This is accepted worldwide as the standard measure of power; it is fully defined and understood.
In early spectrum analyzers, the available detectors were typically peak, sample or minimumpeak, and these do not distinguish between linear or logarithmic values, since a peak is a peak (see Figure
4). There are three traces, each measuring a CW (no modulation) signal with a marker for each trace. In this case, all three markers measure the same power level of -0.3 dBm.
This is fine for continuous waveforms, but what about modulated waveforms? Today, most signals are not CW but are modulated, and the user would benefit from using trace averaging to provide a smooth
response with a high level of measurement repeatability (see Figure 5). Many signals designed to have modulation appear like Gaussian noise, so this example is highly relevant. | {"url":"https://www.microwavejournal.com/articles/32911-legacy-standards-and-test-methods-limit-the-use-of-modern-spectrum-signal-analyzers","timestamp":"2024-11-14T10:21:18Z","content_type":"text/html","content_length":"68400","record_id":"<urn:uuid:276120c1-029f-41be-90e5-b0555abb0fa0>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00041.warc.gz"} |
Temperatures in the Mountains - the Colorado data
Today I am going to look at the temperature data for Colorado, since last week we looked at the
data from Kansas
, and the correlation with longitude was growing, but could not just be explained by a change in height. The procedure starts off the same as that for the initial post, in that I am going to take
data from the
US Historical Climatology Network
compare it with the GISS data for Colorado, (which I reference later) and then see if I can draw some conclusions from the data. I want to have a hypothesis to test against the data, and since the
land rises in the West, and it gets cooler, the hypothesis this week is that there is a relationship that we can derive between average temperature and height above sea level. There is a subsidiary
corollary to this, which is that this will explain the changes in temperature with longitude.
So to begin we find the data for the weather stations for Colorado. There are 24 of these, and so I begin by generating the table that I described in
the first post of this series
. This lists the site with its geographic location and leaves a space for the population. Under that I put the average annual temperature data for that site from 1895 to 2008. (and since I am writing
this as I do it, there will now be a pause . . . .)
And there is a slight problem – in 1896 there is no temperature data for Telluride. Hmm! How to handle this? Well looking at the averages that for 1896 is on average 0.6 deg above that for the
stations in the rest of the state. Telluride, on average is 9.54 degrees below the average for the state. So if I take the average for the state for 1896 and subtract 9.54 I get 37.1 degrees. If I
take the average for Telluride and add 0.6 degrees I get 37.17 degrees. So it seems fair to insert a temperature of 37.135 degrees for Telluride for 1896 – which I did to complete the initial table.
Now to get the data from GISS, and here there is another surprise – I have checked the list that Chiefio gives 3 times and still can only find Grand Junction as a site in Colorado, even though the
GISS site lists some 33 sites with data, of which a fair number also appear on the USHCN list. So I will just use this as the sole GISS site for Colorado. Getting the data from GISS and correcting it
for scale (from Centigrade to Fahrenheit), then I run the table, and the next surprise of the evening. The GISS site is, on average over the 114 year period 6.65 degrees warmer than the state
average. Plotting the average difference over that time interval
Now there is only one GISS station, but this is rather a large amount warmer for the state according to GISS than the state average would suggest. It has, however, been declining slightly, but
steadily over the years. As for the state temperature as a whole, that has been increasing, though the pattern is a bit strange.
So now I go to enter the population, and not being a Colorado native I had been wondering where the Denver data was, and apparently it is hiding behind the Cheesman file. (That being a suburb of
Denver apparently with a Park). So do I use the suburb population 8,201 or that of Denver itself – 598,707? Given that the area has a high population density, I am going to use the Denver number.
And then there is the problem of finding Hermit, CO – fortunately the weather station information includes the co-ordinates (since I didn’t have a great deal of success with a Google search) and this
allows me to use Google Earth to go to the co-ordinates and find that it is at Hermit Lakes, which is in Creede, population 377. Putting all those together, there really are a lot of small
communities in Colorado, so I’ll use a normal (rather than a log) plot of the information.
And with this having a lot of data at the smaller end of the scale (and recognizing that we have yet to go to a state with large populations) there is some correlation to a log relationship. And
changes in small populations could have a greater significant effect. (Which is what I had said earlier, and which is now being also being written about by Professor Roy Spencer.
Interesting where there weren’t any large mountains (i.e. Missouri) there was a good correlation with latitude, but in Colorado that is not as evident:
And instead, where further East there was no correlation with longitude, in Colorado it is very pronounced:
So the question is, can this all be explained by the changes due to height above sea level, or elevation?
I don’t think that there can be any doubt of the correlation. So now we have two – where there is not a large change in elevation (Missouri) there is a strong correlation with latitude, but when the
stations are at a higher altitude, then there is a strong correlation with elevation. And the starting hypothesis, this time, is seen to be correct.
One wonders how it would be if we normalized the data to account for both elevation and latitude. Given the lateness of the hour I won’t do that tonight, but given that GISS gives the locations for
the centers of the states, maybe I will adjust the data to that location based on the linear relationships and see what that produces, and how much variation it takes out. But I’ll do that later in
the week (we have been beset by server problems today, and I would like to get this out before I get hit with another).
10 comments:
1. Hi Dave.
Due to its altitude, Colorado is pretty much outside the circulation path of polar highs, hence the lack of correlation with latitude. There might be a correlation of precipitation with latitude,
but this state may not be that large to capture that.
2. I'm probably going to do Utah next where the elevations are not consistently rising westward. We'll see what we get.
I'll also start running the precipitation tables for the different states.
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10. RR
ไม่ว่าจะอัตราการจ่ายเงิน ความง่ายในการเล่น โบนัสฟรีสปินมาไวมาก รวมไปถึงรางวัลแจ็คพอตที่่จายอย่างไม่อั้น
สำหรับใครที่อยากรวยทางลัด ด้วยการลงทุนไม่กี่บาท ไม่ต้องรอนาน สามารถเห็นตัวเงิน และถอนได้เลยล่ะก็ หากคุณอ่านบทความนี้แล้ว
คุณจะไม่ผิดหวังเลยล่ะค่ะ เพราะฉะนั้นแล้ว มาดูกันเถอะว่าเกมสล็อตแตกง่าย ประจำเดือนกรกฎาคม ในปี 2021 นี้จะมีเกมอะไรบ้าง จะเป็นเกมที่คุณกำลังเล่นอยู่หรือป่าว มาดูกันเลย! | {"url":"https://bittooth.blogspot.com/2010/03/temperatures-in-mountains-colorado-data.html","timestamp":"2024-11-15T03:28:50Z","content_type":"application/xhtml+xml","content_length":"143621","record_id":"<urn:uuid:928d20a2-c860-4c2a-acb5-5d4929452720>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00795.warc.gz"} |
Houdini Geometry Essentials 01: Components & Primitive Types
Houdini’s unique approach to geometry underpins everything you’ll end up doing with it – VFX, direct modelling, procedural modelling, UVs, scene layout – it all starts here. This course is where we
begin laying the best foundations for everything ahead.
By the end of this course, you’ll have a deep understanding of:
• Houdini’s geometry components – points, vertices, primitives, edges and breakpoints
• How geometry is constructed in Houdini by connecting points
• The importance and power of vertices in controlling connectivity
• The concepts, theories and practical uses of Houdini’s primitive types – polygons, nurbs, beziers, meshes, polysoups and quadratic primitives
• Drawing and editing bezier, nurbs and polygon curves using the new curve node
• How to easily convert from one primitive type to another – plus when and why you’d want to
Course Syllabus
Let’s talk details
Section 01
Creating Primitive Objects
There are different ways you can add primitive objects – at the object or geometry level. We’ll explain what that means, and show you how. Then we’ll get a hold of Houdini’s viewport handles, to
tackle positioning and sizing and how to use these in combination with a few need-to-know keyboard shortcuts.
1. Creating primitive geometry - create at object level vs create in context
2. Using viewport handles to create and size primitive objects
Section 02
Geometry Components
As soon as you look at Houdini’s geometry components, you realise it’s different to other applications. Points as well as vertices? What exactly are primitives and why are there so many different
types? A geometry point is the same thing as a particle? Don’t worry, we’ll walk you through it.
1. Geometry Components Part 01 - Points, Edges, and Primitives
2. Geometry Components Part 02 - Vertices
3. Component Numbers
Section 03
Connecting Points
Building geometry in Houdini is all about connecting points – and vertices are the boss. But you’re the CEO (in this metaphorical company analogy), so you need to know how to get those vertices to
work for you. Understanding vertices opens up a whole new world of creative flexibility, only possible in Houdini.
1. Connecting Points - Curves
2. Disconnecting and Reconnecting Points
3. Vertices Control Connectivity
4. Connecting Points - Particles to Surfaces
Section 04
Just like in other applications, but different. You see, polygons in Houdini are curves at the same time as they are surfaces. What? You heard us. Working with open and closed polygons (and
converting between them) is really powerful and incredibly versatile. We’ll show you how.
1. Closed Curves are also Polygon Surfaces
2. Every Polygon Face is also a Closed Curve
3. Drawing and Editing Polygons Using the Curve SOP Part 1
4. Drawing and Editing Polygons Using the Curve SOP Part 2
5. Drawing and Editing Polygons Using the Curve SOP Part 3
6. Open and Closed Polygons - Part 1: The Ends SOP
7. Open and Closed Polygons - Part 2: The Crucial Role of Vertices
8. Open and Closed Polygons - Part 3: Rendering Curves
9. Open and Closed Polygons - Part 4: Rendering Wireframe Geometry
10. Polymodelling Tools on Polygon Curves
11. Polygon Options on Primitive Objects
11 Tutorials 1 hour 11 minutes
Section 05
Bezier Curves - Drawing, Editing and Modelling
Bezier curves used to be hard to draw and edit in Houdini. Not any more! The curve node introduced in Houdini 19 is a real game changer and we’re here to show you why. Drawing and editing a curve is
only half the story, though. We’ll also show you how to use your bezier curves in an optimised modelling workflow.
1. Drawing Bezier Curves
2. Editing Bezier
3. Curves - Points and Tangency
4. Editing Bezier
5. Curves - Segments
6. Editing Bezier Curves - Rounded Corners
8. Working with Bezier Curves - Reference Images
9. Working with Bezier Curves - Tracing a Profile
10.Working with Bezier Curves - Modelling
11. Working with Bezier Curves - Resampling Bezier Curves to Polygons
12. Optimising Curves with the Refine SOP
Section 06
The Technical Side of Bezier Curves
When it comes to using bezier curves in procedural workflows, you need a deep understanding of how they work at a technical level. In Section 6, we’ll equip you with the technical concepts you need
to really make the most of them. You’ll also learn when to use procedural nodes versus Houdini’s special non-procedural nodes.
1. How Bezier Curves work - Order and Degree
2. How Bezier Curves work - Components
3. Editing Bezier Curves - Curve SOP vs Edit SOP
4. Editing Bezier Curves - Procedural Nodes vs Non-Procedural Nodes
5. Editing Curves Procedurally
6. Animating Curves
Section 07
Nurbs Curves
Once the go-to primitive type for smooth curves, nurbs curves have been recently put in the shade by beziers. Well, we’re giving them their time in the sun… again! Why? Because for procedural work,
nurbs curves can still do things that no other primitive type can. Here, we’ll teach you all you need to know about their technical complexities.
1. Comparing Nurbs Curves to Bezier Curves
2. Nurbs Curves: Order - Part 1
3. Nurbs Curves: Order - Part 2
4. Drawing and Editing Nurbs Curves Using the Curve Node
5. Nurbs Curves - Point Weight
Section 08
Parametric Curves and Surfaces
What’s parameterisation (apart from a difficult word to say quickly)? In this section we’ll unwrap the concept while getting down and dirty with parametric curves and surfaces. Now, many people will
tell you it’s all about polygons these days, but understanding this stuff comes in really handy – especially once you get into procedural modelling.
1. Auto Bezier Draw Mode
2. Generating Nurbs and Bezier Curves Procedurally - The Fit Node
3. Parametric Space
4. Parameterisation - Part 1: Uniform vs Chord Length
5. Parameterisation - Part 2: Chord Length and Centripetal
6. Nurbs and Bezier Surfaces - Part 1: Cross Section Curves
7. Nurbs and Bezier Surfaces - Part 2: Parametric Space
Section 09
Comparing and Converting Nurbs and Polygons
What do you get if you convert a polygon mesh to a nurbs surface? No it’s not a set up for a bad joke - it’s an important question and one we’re going to answer right here. While we’re at it, let’s
talk about Houdini’s unique ability to subdivide and interpolate polygon curves and how it makes them just as flexible as nurbs curves.
1. Comparing Nurbs and Bezier Curves to Polygon Curves
2. Using Polygon Curves Like Nurbs Curves - Subdivision Curves
3. Interpolating Curves - Nurbs and Beziers vs Polygons
4. Converting Polygon Faces to Nurbs Surfaces
5. Bilinear Mesh
Section 10
Primitive Types – Polysoups & Quadratic Primitives
These guys save lives. Well, not quite – but they do save memory and disk space in wildly different ways. One day to deadline, but your scene is so heavy you can barely move? Building a friendship
with these fellas could set you free.
1. Bilinear Mesh
2. Polysoup
3. Quadratic Primitives
Geometry Essentials 01
Version 2.0
Components and Primitive Types
Lifetime access
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Geometry Essentials
The Principles & Concepts Bundle
$249.97 $199.98
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When You've Finished
Where to next?
Geometry Essentials 02
Attributes: Principles, Normals & Vectors
Attributes are the blood that flows through Houdini’s veins, carrying data through a geometry network and passing data from one piece of geometry to another. They’re incredibly powerful – they drive
particle simulations, carry material information, and allow for all kinds of custom functionality. Attributes allow us to actually see, read and manipulate the raw data that defines our models and
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s Pricing
Options/J 10.0
Financial Options Pricing and Analysis Java Component
Options/J Java Options Pricing and Analysis Software
Options/J is designed to make life easier for quantitative analysts, option traders and others needing fast stock option pricing in Java applications.
Options/J is a full Java library, supplied in JAR format and comes complete with demo application and source code ready to use. It can price options, estimate historical and implied volatility,
calculate greeks and more and is usable in both Windows, Linux and Apple Mac using Java.
Options/J gives you the power to easily create your own trading system, option price calculator, implied volatility estimator and more. With our easy to use, yet powerful software, including demo
applications with source code, backed up by our top notch support, you will be able to build applications to price stock options, commodities and currencies in minutes, not days.
Options/J 10.0 contains a range of powerful options pricing methods including the following:
• Fast analytic Black-Scholes and Bjerksund-Stensland models which include discrete dividends
• Probabilistic Methods to compute values such as:
Upside stock potential,
Downside stock potential
Probability of security price being above, below, touching or between values.
• Fast analytic methods to compute probability of stock ever touching a price level as well as Monte Carlo methods
• Risk-Reward methods to compute potential risk and reward for option trades.
• Compute prices and greeks in just one function call for both European and American style options
Options/J contains a range of fast analytic methods as well as in some cases, Monte Carlo methods so the two can be confirmed as desired. Importantly, we have included a fast analytic method for
computing the probability of a security touching a price level within a set time. For options trading, this type of calculation may be very important.
Options/J is very fast, yet easy to use. In only one line of code, you can add derivative pricing to your application. a software package that enables you to price stock options, commodities and
currencies. It is a full Java Library for creating your own stock option pricing applications, options calculators and option trading applications. If you need the ability to program in ActiveX, COM
or .NET, take a look at our other products, Options/X and Options/NET which are designed for Windows environments and IDEs including: Visual Basic 6, VBA, Visual C++ 6, C# and more.
With Options/J, you can compute greeks, implied and historical volatility to value put and call derivatives for American and European options using the Black-Scholes formula, Binomial
Cox-Ross-Rubenstein model and others.
With Options/J, you can create your own exchange traded stock option price calculator to compute stock option prices for European and American Options, analyze options volatility smile or compute
historical volatility. Are you interested in making money using stock options? Have you tried stock trading, but now need to work out how to trade options successfully? One of the most essential
tools for options trading is software for options pricing. The beauty of Options/J is that you are in control of exactly what is computed and what algorithm is used. You are free to create your own
options pricing calculator using the built-in functions of OptionsJ.
We provide the source code in Java so that you can use the built-in options pricing models to easily create your own option pricing calculator. Use our example program or develop your own Windows or
Linux applications for option trading easily. While we may not be able to turn you into an options trader, we do provide the software that is used by options traders and brokers to value and analyze
stock options.
With full source samples you will be able to quickly and easily implement Options Trading software. Using implied volatility analysis, compute the volatility smile. If you are interested in stock
options trading, futures trading, implementing and testing your own option trading strategies, or even just want to learn options trading, then you will need solid, reliable options pricing software.
While many options software products are difficult to learn or require extensive training or courses, Options/J is very powerful yet easy to use. If you need help getting started or require
additional features please contact us as we provide extensive support with all of our software. Please refer to our client testimonials to see what others say. In many cases our options trading
software will enable you to be up and running within minutes of downloading the software.
Both zip and tar versions of the software are available, please contact us if you require assistance.
Related Options Software
Options/NET - Options Analysis .NET Component
Volatility/X - Volatility Estimation Excel Add-In
Options/X - Options Analysis ActiveX/COM Component
Features of Options/J
Find the "greeks" - Delta, Gamma, Theta, Vega, Rho. Dividend earnings as a percentage yield can also be included. European and American options can be analyzed using the Black-Scholes option pricing
formula, Binomial options pricing methods (Cox-Ross-Rubinstein), Black method for futures or any of the other methods listed below.
Options/J includes a number of popular models for estimating the theoretical option prices and contains the following models:
• Black-Scholes-Merton (allows for dividend yields)
• Black-76 (Futures)
• Cox-Ross-Rubinstein (Binomial)
• Bjerksund-Stensland (fast estimation of American options)
• Barone-Adesi-Whaley
• Garman-Kohlhagen (used to price European currency options)
• Roll-Geske-Whaley
• French-84 (allows for the effect of trading days)
• Merton jump diffusion
• Historical volatility (estimate volatility using raw price data)
These option pricing algorithms provide a method of determining the call and put prices for European and American options, greeks, implied volatility and volatility skew for both call and put options
is also available.
Options/J comes in 2 different editions: Enterprise and Platinum. The difference between each of these methods is given in the table below:
Pricing Models
Merton jump diffusion
Call/Put Prices
Implied Volatility
Probability Calculations
Risk-Reward Calculations
Historical Volatility
Continuous Dividends
Discrete Dividends
Sample Applications
Options/J Class Library in Eclipse IDE
If you are a Java developer, then there is a good chance you are familiar with one or more of the popular Java integrated development environments such as Eclipse or NetBeans IDE. As you can see from
the image below, Options/J can be easily used in the Eclipse IDE and loaded into your application as a JAR file. From this point the class functions for options pricing and analysis are readily
available. In many cases you need just one function call to obtain multiple results.
Full Fast options pricing and greeks can be obtained in the Eclipse IDE for your Java application..
A well known issue amongst serious Java developers, particularly those who have come from a C/C++ background, is how to return multiple values from Java functions. Without going into the arcane and
somewhat pointless politics of the issue, it is worth commenting on how and why we deal with multiple return values. There exists engineering systems which we may refer to as MIMO or multiple-input
multiple-output and so when it comes to software implementations, we may naturally prefer to be able to provide functions which can process data in a similar manner. Typically we may find ourselves
dealing with some form of model where it makes sense to obtain all output values at the same time, through one function call. In computational finance areas such as option pricing, it is common to
see a considerable amount of processing effort going into computing the option prices and greeks. For some algorithms however, it is more efficient to use the same common calculations and then
produce multiple results stemming from these common calculations, than to use separate function calls which repeat the same calculations. The simplest solution to this issue is to make the function
call return multiple values. While there are a few ways to do this, in our Java functions, we return values using a passed in array.
For example, we may define an array of doubles which will receive the output and then call the appropriate function as follows:
double[] RetVals = new double[NVals ];
optionsj1.BSPriceGreeks(StockPrice, ExercisePrice, InterestRate,
TimeToMaturity, Volatility, DividendRate, RetVals);
CallPrice = RetVals[0]; // Not necessary, but done for clarity
PutPrice = RetVals[1];
DeltaCall = RetVals[2];
DeltaPut = RetVals[3];
ThetaCall = RetVals[4];
ThetaPut = RetVals[5];
Gamma = RetVals[6];
Vega = RetVals[7];
RhoCall = RetVals[8];
RhoPut = RetVals[9];
Thus we can easily obtain multiple output values from a Java function. This is the general and easy to follow approach adopted in all of our function calls.
Options Source Code Examples
If you are aiming to develop an option trading system for the stock market, try Options/J, a Java Library in Windows and Linux compatible JAR format that Screen shot of an application built in
enables you to quickly build your own system. Eclipse using Options/J.
The advantage of Options/J is that you can use it with our demo application and instantly begin creating your own application. With the addition of stock
quotes, you can create your own option trading software, customized to your own purposes.
Options/J includes a sample application with source code in Java. You can quickly see just how easy it is to price and analyse data using Options/J. Options/
J Java Class Library implements option pricing and analysis functions.
Options/J is a Java Class Library implemented in a single JAR file, so it can be used in a wide range of applications that support this standard for JRE 1.6 and above. This includes both Windows,
Linux and Apple Mac OS-X with Java SE 6 installed. It can be used in Java programming IDEs such as Eclipse and NetBeans IDE. Options/J is written in 100% Java. The trial version of Options/J is
feature limited: you will only be able to access Black-Scholes functions using the trial version. However it is possible to develop trial applications to test out your ideas. If you need to price
American Options using the Binomial model (Cox-Ross-Rubenstein), or do futures pricing, then by purchasing the full version you can obtain the full capability.
Black-Scholes Option Pricing
The Black-Scholes option pricing formula can be used to compute the prices of Put and Call options, based on the current stock price, the exercise price of the stock at some future date, the
risk-free interest rate, and the standard deviation of the log of the stock price returns (the volatility).
If you have access to financial end-of-day stock data, then you can use our software in Excel to easily price financial options to work out their theoretical fair value.
A number of assumptions are made when using the Black-Scholes formula. These include: the stock price has a lognormal distribution, there are no taxes, transaction costs, short sales are permitted
and trading is continuous and frictionless, there is no arbitrage, the stock price dynamics are given by a geometric Brownian motion and the interest rate is risk-free for all amounts borrowed or
lent. It is possible to take dividend rates for the security into consideration.
Further information on the Black-Scholes model for pricing derivatives and how to use Options/J to price stock, currencies and commodity Put and Call derivatives using European and American style
options is given here:
Binomial Option Pricing
American options differ from European options by the fact that they can be exercised prior to the expiry date. This means that the Black-Scholes option pricing formula is not suitable for this type
of option. Instead, the Cox-Ross-Rubinstein Binomial pricing algorithm is preferred. OptionsJ implements the binomial pricing algorithm for pricing American options. used to compute the prices of Put
and Call options, based on the current stock price, the exercise price of the stock at some future date, the risk-free interest rate, the standard deviation of the log of the stock price returns (the
volatility), and if applicable, the dividend rate.
Implied Volatility
Given the option price, it is possible to find the volatility implied by that price. This is known as the Implied Volatility and it has a number of characteristics which have been used to identify
trading opportunities. Options/J implements implied volatility functionality for both American and European options using the Binomial and Black-Scholes methods respectively.
1. F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, Political Economy, Vol 81, May-June, pp. 637-654.
2. J.C. Hull, "Options, Futures, and other Derivative Securities", Second Edition, Prentice-Hall: Englewood Cliffs, 1993. | {"url":"http://windale.com/optionsj.php","timestamp":"2024-11-04T14:35:38Z","content_type":"text/html","content_length":"39878","record_id":"<urn:uuid:a63f2527-4369-4bb1-b6c6-c99086a8c3df>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00245.warc.gz"} |
SGR Calculation
Calculations of SGR are as follows;
consumption rate (lb/kg/etc per hr) / ground speed (kts) = lb/kg/etc per nm
ground speed (kts) / consumption rate (lb/kg/etc per hr) = nm per lb (or whatever you’ve used)
Consumption Rate
This is calculated by taking TAS and dividing by SAR (Still air range)
TAS / SAR (nm per lb) = lb per hour
TAS * SAR (lbs per nm) = l per nm
Rate of descent calculations
To calculate the vertical speed required to reach a destination with a given groundspeed and distance, use the following:
Vertical speed required (fpm) = (altitude to lose (ft) x groundspeed (kts)) / (60 x distance to go (nm))
To calculate the distance required for descending at a particular vertical speed & groundspeed, use the following formula:
60 x descent (fpm) / groudspeed (kts) = ft/nm
altitude to loose (ft) / ft/nm = distance required
aka: altitude to loose (ft) / (60 x vertical speed (fpm)) = distance required (nm)
Another way to look at is: if you need to loose 24,000ft, descending at 2000 ft/min, it will take 12 mins (24,000 / 2,000 = 12). With a groundspeed of 240 kts, it equates to 4 kts per min (240 / 60 =
4). 4 (kts per min) x 12 (mins) = 48 nm (distance required).
Thanks to:
Pressure Height / Density Height
Pressure Height
Get the difference between your current QNH & the standard sea level QNH (1013.25) … eg. AD QNH = 1006, means there is a difference of 7 hPa, which equates to a 210 ft difference (calculated by
taking the calculated 7 hPa and multiplying it by 30 (30ft difference per hPa)
Once you have 210 as the height difference, you add (if your QNH is less than standard sea level pressure), or subtract it if it’s higher (eg. if the QNH is showing you’re actually less than sea
level pressure) it to your current elevation (eg. 3310 ft).
The overall formula can be shown as;
Pressure Height = Elevation + 30 * (1013 – QNH)
Eg. At an elevation of 3310, with QNH of 1006.
1. Pressure difference is 1013 – 1006 = 7 hPa difference.
2. We then get the pressure height difference in feet … 7 * 30 = 210
3. Then add it to our current height … 210 + 3310 = 3520 feet.
Density Height
Density altitude in feet = pressure altitude in feet + (120 x (OAT – ISA_temperature))
ISA_temperature = the ISA temp at the altitude you are at … calculated by using;
ISA Temperature: Temperature changes at the rate of 2 degrees per thousand ft (gets colder as you go up, and gets warmer as you descend). The standard sea level ISA temp is 15 degrees, so you will
need to subtract the temperature difference from 15 degrees to get the ISA temp…
Eg. The aerodrome (AD) is 3310 feet above mean sea level (ASML), and the temp is 28 degrees.
1. The ISA temp at 3310 is 3310 / 1000 = 3.31 …. * 2 = 6.62 …. 15 – 6.62 = 8.38 degrees.
2. We then need to find out the temperature deviation from the norm = 28 (our temp) – 8.38 (ISA temp at our altitude) = 19.62 … (rounded to the nearest degree gives 20 degrees).
3. From here, we need to use the temp deviation to find out our density height. This is done by the following;
ISA Temp * 120 (120 ft per degree) = 20 * 120 = 2400
4. Then add 2400 to our pressure height (3520 in this case .. see the above section for workings) which gives us 5,920 ft.
Reference: http://en.wikipedia.org/wiki/Density_altitude
CASA CPL Exams – Aircraft General Knowledge (AGK)
Sample questions;
1. Why must you minimize the time used to check the pitot heater?
1. To minimize the drain on the battery
2. So as not to impare the working life of the heating element
2. Once airbourne, pitot heat should be switched on;
1. Well before you enter known icing conditions
2. On entering known icing conditions and left on
3. If static lines fully block, the altimeter will display the level of the blockage?
4. A/C lands at AD (elevation 1750′) with ALT subscale set to AREA QNH 1020 hPa. AD QNH at the time is 1025hPa. In this case, what will the ALT display in feet?
5. On a climb at a constant IAS, does air leaves both casing & capsule at the same rate?
6. An A/C’s sole static source is fully blocked whilst on a climb. If the A/C settles down to CRZ in an area of MOD to SEV turbulence at the published turbulence penetration speed then;
1. The airframe could be overstressed
2. There should be no risk of overstressing the airframe
7. Where globally will a magnetic compass function best?
8. What purpose does the compas deviation card serve?
1. B – Heating element gets very hot in still air conditions
2. 2 | {"url":"https://blog.victor.com.au/category/play/aviation/","timestamp":"2024-11-12T02:02:32Z","content_type":"text/html","content_length":"93246","record_id":"<urn:uuid:8dd79568-0739-422a-b48e-0d15cfa5708a>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00393.warc.gz"} |
Lighting Calculator
Last updated:
Lighting Calculator
If you resent the idea of lighting calculations and aren't sure where to start, we've got you covered. This foot candle calculator gives you the optimal illumination level for each room in your home
and determines how many light fixtures you need to achieve it. Additionally, we will provide you with foolproof lighting calculation formulas that will make the whole planning process a breeze!
How many lumens per square foot do I need?
In the first step of your calculations, you need to choose the type of area and activity that you want to illuminate.
Simply select one of the options from the list, and our lighting calculator will automatically determine the optimal level of illumination in lux or foot-candles (that is, how much light should be
incident on the surface). Intuitively, ambient light in the bedroom shouldn't affect our preparations for sleep time and won't be as intensive as the light required for sewing.
The next thing you need to determine is the illuminated area. In the case of a bedroom or a bathroom, it will simply be the room's total area. If you're trying to figure out LED lighting for your
kitchen counter, the illuminated area will be calculated as the length of the counter multiplied by its width.
Once you know all these values, the foot-candle calculator will determine how many lumens you need. We use the following equation:
$\text{lumens} = \text{lux} \cdot \text{area}$
You can use illumination units of lux or foot candles. If you want to recalculate between these units, remember that one foot-candle equals $10.764\ \text{lux}$.
Also, check out our lumens to watts calculator if you're curious about how many lumens various light bulbs produce under a given wattage.
💡 Convert between different units of light with the lumen calculator.
Lighting calculation formula
Once you know how many lumens you need, you can start figuring out how many light bulbs will suffice to illuminate your surface. To do it, use the formula below:
$\text{bulbs} = \frac{\text{lumens}}{\text{BL}}$
$\rm BL$ stands for the number of lumens that a light bulb emits. You can usually find this number on the bulb packaging. It's a much better indicator of the bulb's luminosity than the wattage, as
LED lights often need less power than regular light bulbs.
Using the foot candle calculator: an example
1. Choose the area of the house that you want to illuminate. Let's say you're planning the lighting for your kitchen, including the kitchen counter.
2. Check the optimal lighting level. For the whole kitchen, it is 108 lux, and for the counter (detailed tasks) – $538\ \text{lux}$.
3. Determine the dimensions of your illuminated space. The whole kitchen is a rectangle of length 4 m and width 2.5 m, so that you can calculate the area as:
$A_1 = 4 \cdot 2.5 = 10\ \text{m}^2$
The counter is 4 meters long and 60 cm wide:
$A_2 = 4 \cdot 0.6 = 2.4\ \text{m}^2$
4. Multiply the required illumination level by the area to figure out how many lumens you need:
$L_1 = 108 \cdot 10 = 1080\ \text{lumens}$
$L_2 = 538 \cdot 2.4 = 1291\ \text{lumens}$
5. Then, choose the type of light bulb you want to use. Let's assume you're using a standard bulb that emits 800 lumens for the kitchen and small LED lights that emit 200 lumens each above the
6. Divide the total number of lumens by the efficiency of the bulb and round up to figure out the number of bulbs you need:
$n_1 = \frac{1080 }{800} \rightarrow 2\ \text{bulbs}$
$n_2 = \frac{1291}{200} \rightarrow 7\ \text{bulbs}$
You will need two light bulbs (800 lumens each) to illuminate the whole kitchen and additional 7 LED lights (200 lumens each) above the counter.
If you're wondering how much you'll pay for electricity when using all of this lighting, take a look at the electricity cost calculator!
How do I calculate the lighting for a room?
To calculate the lighting of an area:
1. Measure the dimensions of the surface of interest.
2. Compute the area of the surface.
3. Calculate the lumens required using the formula
lumens = lux × area
The lux is a measurement of the received light per area unit. The lumens is a unit that measures the amount of light emitted by a light source.
How many lumens do I need to light a studio 4 meter by 5 meter studio?
About 6,500 lumens. To calculate this result:
1. Compute the area of the room:
area = 4 m × 5 m = 20 m²
2. Choose the right amount of lux you need. For a studio, the recommended value is 323 lx.
3. Find the result using the formula:
lumens = lux × area = 323 lx × 20 m² = 6,460 lm
If you plan to use lightbulbs with an intensity of 1600 lm, you'll need five of them.
What is the best lighting for a bedroom?
A bedroom must have different lighting for different uses:
• For bedtime use, you will need a lower amount of lumens. A good approximation is 54 lx.
• For reading or other activities, the lighting must be higher. You can use up to 430 lx.
• For specific tasks (sewing, other hobbies), the lighting must rise to almost 540 lx.
This amount of light can be spread over multiple lightbulbs.
What is the difference between lumen and lux?
The lumen is a unit of luminous flux; lumens correspond to the amount of light emitted by a source, such as a lightbulb or a candle, regardless of direction.
Lux is used to measure the amount of light shining on a surface. A high amount of lux corresponds to a brightly lit surface.
Lux and lumens are related by the formula lumens = lux × area. | {"url":"https://www.omnicalculator.com/everyday-life/lighting","timestamp":"2024-11-01T23:45:45Z","content_type":"text/html","content_length":"549182","record_id":"<urn:uuid:3f4c9a6f-0e59-4896-a391-6e9452feeb39>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00825.warc.gz"} |
Symbolic Reasoning
The basis for intelligent mathematical software is the integration of the "power of symbolic mathematical tools" with the suitable "proof technology".
Mathematical reasoning enjoys a property called monotonic.
"If a conclusion follows from given premises A, B, C, …
then it also follows from any larger set of premises, as long as the original premises are included."
Human reasoning is not monotonic.
People arrive to conclusions only tentatively, based on partial or incomplete information, reserve the right to retract those conclusions while they learn new facts. Such reasoning is non-monotonic,
precisely because the set of accepted conclusions have become smaller when the set of premises is expanded.
1. Non-Monotonic Reasoning
Non-Monotonic reasoning is a generic name to a class or a specific theory of reasoning. Non-monotonic reasoning attempts to formalize reasoning with incomplete information by classical logic systems.
The Non-Monotonic reasoning are of the type
■ Default reasoning
■ Ci umscription
■ Truth Maintenance Systems
Default Reasoning
This is a very common from of non-monotonic reasoning. The conclusions are drawn based on what is most likely to be true.
There are two approaches, both are logic type, to Default reasoning :
one is Non-monotonic logic and the other is Default logic.
Non-monotonic logic
It has already been defined. It says, "the truth of a proposition may change when new information (axioms) are added and a logic may be build to allows the statement to be retracted."
Non-monotonic logic is predicate logic with one extension called modal operator M which means “consistent with everything we know”. The purpose of M is to allow consistency.
A way to define consistency with PROLOG notation is : To show that fact P is true, we attempt to prove ¬P.
If we fail we may say that P is consistent since ¬P is false.
Example :
∀ x : plays_instrument(x) ∧ M manage(x) → jazz_musician(x)
States that for all x, the x plays an instrument and if the fact
that x can manage is consistent with all other knowledge then we can conclude that x is a jazz musician.
■ Default Logic
Read the above inference rule as:
" if A, and if it is consistent with the rest of what is known to assume that B, then conclude that C ".
The rule says that given the prerequisite, the consequent can be inferred, provided it is consistent with the rest of the data.
‡ Example : Rule that "birds typically fly" would be represented as
‡ Note : Since, all we know about Tweety is that :
Tweety is a bird, we therefore inferred that Tweety flies.
The idea behind non-monotonic reasoning is to reason with first order logic, and if an inference can not be obtained then use the set of default rules available within the first order formulation.
‡ Applying Default Rules :
While applying default rules, it is necessary to check their justifications for consistency, not only with initial data, but also with the consequents of any other default rules that may be applied.
The application of one rule may thus block the application of another. To solve this problem, the concept of default theory was extended.
‡ Default Theory
Example :
A Default Rule says " Typically an American adult owns a car ".
The rule is explained below :
The rule is only accessed if we wish to know whether or not John owns a car then an answer can not be deduced from our current beliefs.
This default rule is applicable if we can prove from our beliefs that John is an American and an adult, and believing that there is some car that is owned by John does not lead to an inconsistency.
If these two sets of premises are satisfied, then the rule states that we can conclude that John owns a car.
Ci umscription
Ci umscription is a non-monotonic logic to formalize the common sense assumption. Ci umscription is a formalized rule of conjecture (guess) that can be used along with the rules of inference of first
order logic.
Ci umscription involves formulating rules of thumb with "abnormality"
predicates and then restricting the extension of these predicates,
ci umscribing them, so that they apply to only those things to which they are currently known.
Example : Take the case of Bird Tweety
The rule of thumb is that "birds typically fly" is conditional. The predicate "Abnormal" signifies abnormality with respect to flying ability.
Observe that the rule ∀ x(Bird(x) & ¬ Abnormal(x) → Flies)) does not allow us to infer that "Tweety flies", since we do not know that he is abnormal with respect to flying ability.
But if we add axioms which ci umscribe the abnormality predicate to which they are currently known say "Bird Tweety" then the inference can be drawn. This inference is non-monotonic.
Truth Maintenance Systems
Reasoning Maintenance System (RMS) is a critical part of a reasoning system. Its purpose is to assure that inferences made by the reasoning system (RS) are valid.
The RS provides the RMS with information about each inference it performs, and in return the RMS provides the RS with information about the whole set of inferences.
Several implementations of RMS have been proposed for non-monotonic reasoning. The important ones are the :
Truth Maintenance Systems (TMS) and Assumption-based Truth Maintenance Systems (ATMS).
The TMS maintains the consistency of a knowledge base as soon as new knowledge is added. It considers only one state at a time so it is not possible to manipulate environment.
The ATMS is intended to maintain multiple environments.
The typical functions of TMS are presented in the next slide.
Truth Maintenance Systems (TMS)
A truth maintenance system maintains consistency in knowledge representation of a knowledge base.
The functions of TMS are to :
■ Provide justifications for conclusions
When a problem solving system gives an answer to a user's query, an explanation of that answer is required;
Example : An advice to a stockbroker is supported by an explanation of the reasons for that advice. This is constructed by the Inference Engine (IE) by tracing the justification of the assertion.
Recognize inconsistencies
The Inference Engine (IE) may tell the TMS that some sentences are contradictory. Then, TMS may find that all those sentences are believed true, and reports to the IE which can eliminate the
inconsistencies by determining the assumptions used and changing them appropriately. Example : A statement that either Abbott, or Babbitt, or Cabot is guilty together with other statements that
Abbott is not guilty, Babbitt is not guilty, and Cabot is not guilty, form a contradiction.
Support default reasoning
In the absence of any firm knowledge, in many situations we want to reason from default assumptions.
Example : If "Tweety is a bird", then until told otherwise, assume that "Tweety flies" and for justification use the fact that "Tweety is a bird" and the assumption that "birds fly".
2. Implementation Issues
The issues and weaknesses related to implementation of non-monotonic reasoning in problem solving are :
How to derive exactly those non-monotonic conclusion that are relevant to solving the problem at hand while not wasting time on those that are not necessary.
How to update our knowledge incrementally as problem solving progresses.
How to over come the problem where more than one interpretation of the known facts is qualified or approved by the available inference rules.
In general the theories are not computationally effective, decidable or semi decidable.
The solutions offered, considering the reasoning processes into two parts : one, a problem solver that uses whatever mechanism it happens to have to draw conclusions as necessary, and second, a truth
maintenance system whose job is to maintain consistency in knowledge representation of a knowledge base. | {"url":"https://www.brainkart.com/article/Symbolic-Reasoning_8586/","timestamp":"2024-11-10T21:14:23Z","content_type":"text/html","content_length":"78443","record_id":"<urn:uuid:1ed6b657-af63-46e3-b5f5-3ac6c178a53d>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00846.warc.gz"} |
avdelning 2 , kapitel 1 1 Läkemedelsbranschens etiska regelverk Scandinavian Biopharma Distribution har annonserat i bilagan Mediaplanet i Dagens 103 90 Stockholm, Telefon 08-701 78 00, Fax 08-701
posterior distribution pas a stationary distribution Patrick Breheny BST 701: Bayesian Modeling in Biostatistics 7/30. Markov chains The Metropolis-Hastings algorithm Gibbs sampling Example As an
example of how the Metropolis-Hastings algorithm works, let’s sample from the following posterior:
The developed international equity markets also benefited from two of the same inflection points or taxes on transactions in Fund shares or taxes that a shareholder would pay on Fund distributions.
Total Construction & Engineering, 701. Once you reach 70 1/2 the IRS requires you to start taking withdrawals from your retirement accounts. These distributions are called MRD's (also known as
Required Minimum Distributions- RMD's ) and apply to all of your retirement accounts including Traditional IRA's, Rollover IRA's, SEP Plans and 401k plans or 403b plans you may be using.
15 305. Omsättning mellan segmenten. 104. 148. 170. Construction. 717.
Annual withdrawals from traditional retirement accounts are required after age 70 1/2, and the penalty for skipping a required minimum distribution is 50% of the amount that should have been 401k
Minimum Required Distributions (MRDs) are established by the Internal Revenue Code to make sure that retirees actually withdraw their money upon retirement (and use it for their day to day
expenses) as opposed to passing on this wealth to their heirs.
Example. The mean of a sample is 128.5, SEM 6.2, sample size 32. What is the 99% confidence interval of the mean? Degrees of freedom (DF) is n−1 = 31, t-value in column for area 0.99 is 2.744.
(7). ODS-legeringar -en anviindarhandbok och statusrapport.
av K Helenius · 2019 · Citerat av 24 — Kjell Helenius,1,2,3 Nicholas Longford,3 Liisa Lehtonen,1,2 Neena Modi,3 for comparison with near identical distributions of 701 (97.9).
3 680. -. 304. 74 013. 100 – 149 11 858. 49.
The IRS requires that you withdraw at least a minimum amount - known as a Required Minimum Distribution - from your retirement accounts annually; starting the year you turn age 70-1/2. An RMD is the
annual Required Minimum Distribution that you must start taking out of your retirement account after you reach age 72 (70½ if you turned 70½ before Jan 1, 2020). The amount is determined by the fair
market value of your IRAs at the end of the previous year, factored by … Determine which IRS Distribution Period Table to use. The distribution tables are found in the appendix of IRS Publication
Bidrar på engelska
1 447.
F. Samuelsson et al., "Constraining Low-luminosity Gamma-Ray Bursts as on the observed flux and fluence distributions of gamma-ray bursts : Are the most FERMI LARGE AREA TELESCOPE," Astrophysical
Journal, vol. 701, no.
Montessoriskolan västerås
Construction. 717. 679. 6. 8. 2 061. 2 027. 2. 7. 2 635. EBITDA. 701. 492. 42 ökade, främst till följd av högre distributions- och IT- kostnader.
Kontakta oss här. Dokument (2) JOURNAL OF CLINICAL MICROBIOLOGY, Mar. 1995, p. 701–705. Vol. 33, No. 3 HEp-2 cells, 43 (86%) were positive with the EAggEC PCR. All 43 of these strains The age, sex,
and seasonal distributions of controls and subjects with Seamless Distribution Systems AB (”SDS” eller MSEK, att jämföra med det prognostiserade bruttoresultatet om 39,2 MSEK, 31 974 -1 701.
Yrkes coach
2 776. 201. 3 680. -. | {"url":"https://forsaljningavaktieraiewl.netlify.app/70341/55869","timestamp":"2024-11-13T19:03:04Z","content_type":"text/html","content_length":"8062","record_id":"<urn:uuid:e1c02677-7e9f-4130-8e65-584b15c4eb27>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00482.warc.gz"} |
First derivatives, explicit method
Next: First derivatives, implicit method Up: FINITE DIFFERENCING Previous: The lens equation
The inflation of money q at a 10% rate can be described by the difference equation
This one-dimensional calculation can be reexpressed as a differencing star and a data table. As such it provides a prototype for the organization of calculations with two-dimensional
partial-differential equations. Consider
Since the data in the data table satisfy the difference equations (22) and (23), the differencing star may be laid anywhere on top of the data table, the numbers in the star may be multiplied by
those in the underlying table, and the resulting cross products will sum to zero. On the other hand, if all but one number (the initial condition) in the data table were missing then the rest of the
numbers could be filled in, one at a time, by sliding the star along, taking the difference equations to be true, and solving for the unknown data value at each stage.
Less trivial examples utilizing the same differencing star arise when the numerical constant .10 is replaced by a complex number. Such examples exhibit oscillation as well as growth and decay.
Next: First derivatives, implicit method Up: FINITE DIFFERENCING Previous: The lens equation Stanford Exploration Project | {"url":"https://sep.stanford.edu/sep/prof/iei/omx/paper_html/node15.html","timestamp":"2024-11-11T06:21:10Z","content_type":"text/html","content_length":"5570","record_id":"<urn:uuid:49c131e0-5a07-4277-8ce6-dd0b73218db4>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00588.warc.gz"} |
Parallel resistor-capacitor circuits : REACTANCE AND IMPEDANCE -- CAPACITIVE
Parallel resistor-capacitor circuits
Using the same value components in our series example circuit, we will connect them in parallel and see what happens: (Figure below)
Parallel R-C circuit.
Because the power source has the same frequency as the series example circuit, and the resistor and capacitor both have the same values of resistance and capacitance, respectively, they must also
have the same values of impedance. So, we can begin our analysis table with the same “given” values:
This being a parallel circuit now, we know that voltage is shared equally by all components, so we can place the figure for total voltage (10 volts ∠ 0^o) in all the columns:
Now we can apply Ohm's Law (I=E/Z) vertically to two columns in the table, calculating current through the resistor and current through the capacitor:
Just as with DC circuits, branch currents in a parallel AC circuit add up to form the total current (Kirchhoff's Current Law again):
Finally, total impedance can be calculated by using Ohm's Law (Z=E/I) vertically in the “Total” column. As we saw in the AC inductance chapter, parallel impedance can also be calculated by using a
reciprocal formula identical to that used in calculating parallel resistances. It is noteworthy to mention that this parallel impedance rule holds true regardless of the kind of impedances placed in
parallel. In other words, it doesn't matter if we're calculating a circuit composed of parallel resistors, parallel inductors, parallel capacitors, or some combination thereof: in the form of
impedances (Z), all the terms are common and can be applied uniformly to the same formula. Once again, the parallel impedance formula looks like this:
The only drawback to using this equation is the significant amount of work required to work it out, especially without the assistance of a calculator capable of manipulating complex quantities.
Regardless of how we calculate total impedance for our parallel circuit (either Ohm's Law or the reciprocal formula), we will arrive at the same figure:
• REVIEW:
• Impedances (Z) are managed just like resistances (R) in parallel circuit analysis: parallel impedances diminish to form the total impedance, using the reciprocal formula. Just be sure to perform
all calculations in complex (not scalar) form! Z[Total] = 1/(1/Z[1] + 1/Z[2] + . . . 1/Z[n])
• Ohm's Law for AC circuits: E = IZ ; I = E/Z ; Z = E/I
• When resistors and capacitors are mixed together in parallel circuits (just as in series circuits), the total impedance will have a phase angle somewhere between 0^o and -90^o. The circuit
current will have a phase angle somewhere between 0^o and +90^o.
• Parallel AC circuits exhibit the same fundamental properties as parallel DC circuits: voltage is uniform throughout the circuit, branch currents add to form the total current, and impedances
diminish (through the reciprocal formula) to form the total impedance. | {"url":"https://www.learningelectronics.net/vol_2/chpt_4/4.html","timestamp":"2024-11-10T09:23:45Z","content_type":"text/html","content_length":"9622","record_id":"<urn:uuid:7d7414a3-ee63-43c4-a15a-e40a30197097>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00439.warc.gz"} |
How We Calculate Percentage Of Media In Ball Mill
calculate ball mill grinding media in cement zacarafarm. volume filling of grinding media in cement ball mill Ball mill Wikipedia The ball mill is a key piece of equipment for grinding crushed
materials, and it is widely used in production lines for powders such as cement, silicates, refractory | {"url":"https://bistropodratuszem.pl/17576-how-we-calculate-percentage-of-media-in-ball-mill.html","timestamp":"2024-11-02T02:45:31Z","content_type":"text/html","content_length":"34452","record_id":"<urn:uuid:c7af12ea-9e63-4206-a95c-20a92736e81f>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00849.warc.gz"} |
Chaos in the Classroom
The Chaos Game (Next Section)
Chaos in the Classroom
Robert L. Devaney
Department of Mathematics
Boston University
Boston, MA 02215
One of the most interesting applications of technology in the mathematics classroom is the fact that it allows teachers to bring many new and exciting topics into the curriculum. In particular,
technology lets teachers bring some topics of contemporary interest in research mathematics into both middle school and high school classrooms.
The mathematical topics of chaos and fractals are particularly appropriate in this regard. They are timely---many ideas in these fields were first conceived during the students' lifetimes. They are
applicable---fields as diverse as medicine, business, geology, art, and music have adopted ideas from these areas. And they are beautiful---there is something in the gorgeous computer generated
images of objects such as the Mandelbrot set, Julia sets, the Koch snowflake, and others that capture students' interest and enthusiasm.
Therein, however, lies the problem. Many research mathematicians cringe at the sight of ``still another fractal.'' Most often, discussions of chaos and fractals degenerate to simple ``pretty picture
shows,'' devoid of any mathematical content. As a consequence, students get the idea that modern mathematics is akin to a video game---lots of computer-generated action, but mindless activity at
This attitude is both unfortunate and unnecessary. There is mathematics behind the pretty pictures, and moreover, much of it is quite accessible to secondary school students. Furthermore, the
mathematics behind the images is often even prettier than the pictures themselves! In this sense it is a tragedy that students come so close to seeing some exciting, contemporary topics in
mathematics, yet miss out in the end. Our goal in this note is to help remedy this situation by describing some easy-to-teach topics involving ideas from fractal geometry.
The Chaos Game (Next Section)
Robert L. Devaney
Sun Apr 2 14:31:18 EDT 1995 | {"url":"https://math.bu.edu/DYSYS/chaos-game/chaos-game.html","timestamp":"2024-11-03T17:03:16Z","content_type":"text/html","content_length":"4566","record_id":"<urn:uuid:4c7e6935-8782-4517-ad1c-9f27f277f320>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00699.warc.gz"} |
A post by Codie Wood, PhD student on the Compass programme.
This blog post is an introduction to structure preserving estimation (SPREE) methods. These methods form the foundation of my current work with the Office for National Statistics (ONS), where I am
undertaking a six-month internship as part of my PhD. During this internship, I am focusing on the use of SPREE to provide small area estimates of population characteristic counts and proportions.
Small area estimation
Small area estimation (SAE) refers to the collection of methods used to produce accurate and precise population characteristic estimates for small population domains. Examples of domains may include
low-level geographical areas, or population subgroups. An example of an SAE problem would be estimating the national population breakdown in small geographical areas by ethnic group [2015_Luna].
Demographic surveys with a large enough scale to provide high-quality direct estimates at a fine-grain level are often expensive to conduct, and so smaller sample surveys are often conducted instead.
SAE methods work by drawing information from different data sources and similar population domains in order to obtain accurate and precise model-based estimates where sample counts are too small for
high quality direct estimates. We use the term small area to refer to domains where we have little or no data available in our sample survey.
SAE methods are frequently relied upon for population characteristic estimation, particularly as there is an increasing demand for information about local populations in order to ensure correct
allocation of resources and services across the nation.
Structure preserving estimation
Structure preserving estimation (SPREE) is one of the tools used within SAE to provide population composition estimates. We use the term composition here to refer to a population break down into a
two-way contingency table containing positive count values. Here, we focus on the case where we have a population broken down into geographical areas (e.g. local authority) and some subgroup or
category (e.g. ethnic group or age).
Orginal SPREE-type estimators, as proposed in [1980_Purcell], can be used in the case when we have a proxy data source for our target composition, containing information for the same set of areas and
categories but that may not entirely accurately represent the variable of interest. This is usually because the data are outdated or have a slightly different variable definition than the target.
We also incorporate benchmark estimates of the row and column totals for our composition of interest, taken from trusted, quality assured data sources and treated as known values. This ensures
consistency with higher level known population estimates. SPREE then adjusts the proxy data to the estimates of the row and column totals to obtain the improved estimate of the target composition.
An illustration of the data required to produce SPREE-type estimates.
In an extension of SPREE, known as generalised SPREE (GSPREE) [2004_Zhang], the proxy data can also be supplemented by sample survey data to generate estimates that are less subject to bias and
uncertainty than it would be possible to generate from each source individually. The survey data used is assumed to be a valid measure of the target variable (i.e. it has the same definition and is
not out of date), but due to small sample sizes may have a degree of uncertainty or bias for some cells.
The GSPREE method establishes a relationship between the proxy data and the survey data, with this relationship being used to adjust the proxy compositions towards the survey data.
An illustration of the data required to produce GSPREE estimates.
GSPREE is not the only extension to SPREE-type methods, but those are beyond the scope of this post. Further extensions such as Multivariate SPREE are discussed in detail in [2016_Luna].
Original SPREE methods
First, we describe original SPREE-type estimators. For these estimators, we require only well-established estimates of the margins of our target composition.
We will denote the target composition of interest by $\mathbf{Y} = (Y{aj})$, where $Y{aj}$ is the cell count for small area $a = 1,\dots,A$ and group $j = 1,\dots,J$. We can write $\mathbf Y$ in the
form of a saturated log-linear model as the sum of four terms,
$$ \log Y_{aj} = \alpha_0^Y + \alpha_a^Y + \alpha_j^Y + \alpha_{aj}^Y.$$
There are multiple ways to write this parameterisation, and here we use the centered constraints parameterisation given by $$\alpha_0^Y = \frac{1}{AJ}\sum_a\sum_j\log Y_{aj},$$ $$\alpha_a^Y = \frac
{1}{J}\sum_j\log Y_{aj} – \alpha_0^Y,$$ $$\alpha_j^Y = \frac{1}{A}\sum_a\log Y_{aj} – \alpha_0^Y,$$ $$\alpha_{aj}^Y = \log Y_{aj} – \alpha_0^Y – \alpha_a^Y – \alpha_j^Y,$$
which satisfy the constraints $\sum_a \alpha_a^Y = \sum_j \alpha_j^Y = \sum_a \alpha_{aj}^Y = \sum_j \alpha_{aj}^Y = 0.$
Using this expression, we can decompose $\mathbf Y$ into two structures:
1. The association structure, consisting of the set of $AJ$ interaction terms $\alpha_{aj}^Y$ for $a = 1,\dots,A$ and $j = 1,\dots,J$. This determines the relationship between the rows (areas) and
columns (groups).
2. The allocation structure, consisting of the sets of terms $\alpha_0^Y, \alpha_a^Y,$ and $\alpha_j^Y$ for $a = 1,\dots,A$ and $j = 1,\dots,J$. This determines the size of the composition, and
differences between the sets of rows (areas) and columns (groups).
Suppose we have a proxy composition $\mathbf X$ of the same dimensions as $\mathbf Y$, and we have the sets of row and column margins of $\mathbf Y$ denoted by $\mathbf Y_{a+} = (Y_{1+}, \dots, Y_
{A+})$ and $\mathbf Y_{+j} = (Y_{+1}, \dots, Y_{+J})$, where $+$ substitutes the index being summed over.
We can then use iterative proportional fitting (IPF) to produce an estimate $\widehat{\mathbf Y}$ of $\mathbf Y$ that preserves the association structure observed in the proxy composition $\mathbf
X$. The IPF procedure is as follows:
1. Rescale the rows of $\mathbf X$ as $$ \widehat{Y}_{aj}^{(1)} = X_{aj} \frac{Y_{+j}}{X_{+j}},$$
2. Rescale the columns of $\widehat{\mathbf Y}^{(1)}$ as $$ \widehat{Y}_{aj}^{(2)} = \widehat{Y}_{aj}^{(1)} \frac{Y_{a+}}{\widehat{Y}_{a+}^{(1)}},$$
3. Rescale the rows of $\widehat{\mathbf Y}^{(2)}$ as $$ \widehat{Y}_{aj}^{(3)} = \widehat{Y}_{aj}^{(2)} \frac{Y_{+j}}{\widehat{Y}_{+j}^{(2)}}.$$
Steps 2 and 3 are then repeated until convergence occurs, and we have a final composition estimate denoted by $\widehat{\mathbf Y}^S$ which has the same association structure as our proxy
composition, i.e. we have $\alpha_{aj}^X = \alpha_{aj}^Y$ for all $a \in \{1,\dots,A\}$ and $j \in \{1,\dots,J\}.$ This is a key assumption of the SPREE implementation, which in practise is often
restrictive, motivating a generalisation of the method.
Generalised SPREE methods
If we can no longer assume that the proxy composition and target compositions have the same association structure, we instead use the GSPREE method first introduced in [2004_Zhang], and incorporate
survey data into our estimation process.
The GSPREE method relaxes the assumption that $\alpha_{aj}^X = \alpha_{aj}^Y$ for all $a \in \{1,\dots,A\}$ and $j \in \{1,\dots,J\},$ instead imposing the structural assumption $\alpha_{aj}^Y = \
beta \alpha_{aj}^X$, i.e. the association structure of the proxy and target compositions are proportional to one another. As such, we note that SPREE is a particular case of GSPREE where $\beta = 1$.
Continuing with our notation from the previous section, we proceed to estimate $\beta$ by modelling the relationship between our target and proxy compositions as a generalised linear structural model
(GLSM) given by
$$\tau_{aj}^Y = \lambda_j + \beta \tau_{aj}^X,$$ with $\sum_j \lambda_j = 0$, and where $$ \begin{align} \tau_{aj}^Y &= \log Y_{aj} – \frac{1}{J}\sum_j\log Y_{aj},\\
&= \alpha_{aj}^Y + \alpha_j^Y,
\end{align}$$ and analogously for $\mathbf X$.
It is shown in [2016_Luna] that fitting this model is equivalent to fitting a Poisson generalised linear model to our cell counts, with a $\log$ link function. We use the association structure of our
proxy data, as well as categorical variables representing the area and group of the cell, as our covariates. Then we have a model given by $$\log Y_{aj} = \gamma_a + \tilde{\lambda}_j + \tilde{\beta}
\alpha_{aj}^X,$$ with $\gamma_a = \alpha_0^Y + \alpha_a^Y$, $\tilde\lambda_j = \alpha_j^Y$ and $\tilde\beta \alpha_{aj}^X = \alpha_{aj}^Y.$
When fitting the model we use survey data $\tilde{\mathbf Y}$ as our response variable, and are then able to obtain a set of unbenchmarked estimates of our target composition. The GSPREE method then
benchmarks these to estimates of the row and column totals, following a procedure analagous to that undertaken in the orginal SPREE methodology, to provide a final set of estimates for our target
ONS applications
The ONS has used GSPREE to provide population ethnicity composition estimates in intercensal years, where the detailed population estimates resulting from the census are outdated [2015_Luna]. In this
case, the census data is considered the proxy data source. More recent works have also used GSPREE to estimate counts of households and dwellings in each tenure at the subnational level during
intercensal years [2023_ONS].
My work with the ONS has focussed on extending the current workflows and systems in place to implement these methods in a reproducible manner, allowing them to be applied to a wider variety of
scenarios with differing data availability.
[1980_Purcell] Purcell, Noel J., and Leslie Kish. 1980. ‘Postcensal Estimates for Local Areas (Or Domains)’. International Statistical Review / Revue Internationale de Statistique 48 (1): 3–18.
[2004_Zhang] Zhang, Li-Chun, and Raymond L. Chambers. 2004. ‘Small Area Estimates for Cross-Classifications’. Journal of the Royal Statistical Society Series B: Statistical Methodology 66 (2):
479–96. https://doi.org/10/fq2ftt.
[2015_Luna] Luna Hernández, Ángela, Li-Chun Zhang, Alison Whitworth, and Kirsten Piller. 2015. ‘Small Area Estimates of the Population Distribution by Ethnic Group in England: A Proposal Using
Structure Preserving Estimators’. Statistics in Transition New Series and Survey Methodology 16 (December). https://doi.org/10/gs49kq.
[2016_Luna] Luna Hernández, Ángela. 2016. ‘Multivariate Structure Preserving Estimation for Population Compositions’. PhD thesis, University of Southampton, School of Social Sciences. https://
[2023_ONS] Office for National Statistics (ONS), released 17 May 2023, ONS website, article, Tenure estimates for households and dwellings, England: GSPREE compared with Census 2021 data
Student Perspectives: Semantic Search
A post by Ben Anson, PhD student on the Compass programme.
Semantic Search
Semantic search is here. We already see it in use in search engines [13], but what is it exactly and how does it work?
Search is about retrieving information from a corpus, based on some query. You are probably using search technology all the time, maybe $\verb|ctrl+f|$, or searching on google. Historically, keyword
search, which works by comparing the occurrences of keywords between queries and documents in the corpus, has been surprisingly effective. Unfortunately, keywords are inherently restrictive – if you
don’t know the right one to use then you are stuck.
Semantic search is about giving search a different interface. Semantic search queries are provided in the most natural interface for humans: natural language. A semantic search algorithm will ideally
be able to point you to a relevant result, even if you only provided the gist of your desires, and even if you didn’t provide relevant keywords.
Figure 1: Illustration of semantic search and keyword search models
Figure 1 illustrates a concrete example where semantic search might be desirable. The query ‘animal’ should return both the dog and cat documents, but because the keyword ‘animal’ is not present in
the cat document, the keyword model fails. In other words, keyword search is susceptible to false negatives.
Transformer neural networks turn out to be very effective for semantic search [1,2,3,10]. In this blog post, I hope to elucidate how transformers are tuned for semantic search, and will briefly touch
on extensions and scaling.
The search problem, more formally
Suppose we have a big corpus $\mathcal{D}$ of documents (e.g. every document on wikipedia). A user sends us a query $q$, and we want to point them to the most relevant document $d^*$. If we denote
the relevance of a document $d$ to $q$ as $\text{score}(q, d)$, the top search result should simply be the document with the highest score,
d^* = \mathrm{argmax}_{d\in\mathcal{D}}\, \text{score}(q, d).
This framework is simple and it generalizes. For $\verb|ctrl+f|$, let $\mathcal{D}$ be the set of individual words in a file, and $\text{score}(q, d) = 1$ if $q=d$ and $0$ otherwise. The venerable
keyword search algorithm BM25 [4], which was state of the art for decades [8], uses this score function.
For semantic search, the score function is often set as the inner product between query and document embeddings: $\text{score}(q, d) = \langle \phi(q), \phi(d) \rangle$. Assuming this score function
actually works well for finding relevant documents, and we use a simple inner product, it is clear that the secret sauce lies in the embedding function $\phi$.
Transformer embeddings
We said above that a common score function for semantic search is $\text{score}(q, d) = \langle \phi(q), \phi(d) \rangle$. This raises two questions:
• Question 1: what should the inner product be? For semantic search, people tend to use the cosine similarity for their inner product.
• Question 2: what should $\phi$ be? The secret sauce is to use a transformer encoder, which is explained below.
Quick version
Transformers magically gives us a tunable embedding function $\phi: \text{“set of all pieces of text”} \rightarrow \mathbb{R}^{d_{\text{model}}}$, where $d_{\text{model}}$ is the embedding dimension.
More detailed version
See Figure 2 for an illustration of how a transformer encoder calculates an embedding for a piece of text. In the figure we show how to encode “cat”, but we can encode arbitrary pieces of text in a
similar way. The transformer block details are out of scope here; though, for these details I personally found Attention is All You Need [9] helpful, the crucial part being the Multi-Head Attention
which allows modelling dependencies between words.
Figure 2: Transformer illustration (transformer block image taken from [6])
The transformer encoder is very flexible, with almost every component parameterized by a learnable weight / bias – this is why it can be used to model the complicated semantics in natural language.
The pooling step in Figure 2, where we map our sequence embedding $X’$ to a fixed size, is not part of a ‘regular’ transformer, but it is essential for us. It ensures that our score function $\langle
\phi(q), \phi(d) \rangle$ will work when $q$ and $d$ have different sizes.
Making the score function good for search
There is a massive issue with transformer embedding as described above, at least for our purposes – there is no reason to believe it will satisfy simple semantic properties, such as,
$\text{score}(\text{“busy place”}, \text{“tokyo”}) > \text{score}(\text{“busy place”}, \text{“a small village”})$
‘But why would the above not work?’ Because, of course, transformers are typically trained to predict the next token in a sequence, not to differentiate pieces of text according to their semantics.
The solution to this problem is not to eschew transformer embeddings, but rather to fine-tune them for search. The idea is to encourage the transformer to give us embeddings that place semantically
dissimilar items far apart. E.g. let $q=$’busy place’, then we want $ d^+=$’tokyo’ to be close to $q$ and $d^-=$’a small village’ to be far away.
This semantic separation can be achieved by fine-tuning with a contrastive loss [1,2,3,10],
\text{maximize}_{\theta}\,\mathcal{L} = \log \frac{\exp(\text{score}(q, d^+))}{\exp(\text{score}(q, d^+)) + \exp(\text{score}(q, d^-))},
where $\theta$ represents the transformer parameters. The $\exp$’s in the contastive loss are to ensure we never divide by zero. Note that we can interpret the contrastive loss as doing
classification since we can think of the argument to the logarithm as $p(d^+ | q)$.
That’s all we need, in principle, to turn a transformer encoder into a text embedder! In practice, the contrastive loss can be generalized to include more positive and negative examples, and it is
indeed a good idea to have a large batch size [11] (intuitively it makes the separation of positive and negative examples more difficult, resulting in a better classifier). We also need a fine-tuning
dataset – a dataset of positive/negative examples. OpenAI showed that it is possible to construct one in an unsupervised fashion [1]. However, there are also publicly available datasets for
supervised fine-tuning, e.g. MSMARCO [12].
One really interesting avenue of research is training of general purposes encoders. The idea is to provide instructions alongside the queries/documents [2,3]. The instruction could be $\verb|Embed
this document for search: {document}|$ (for the application we’ve been discussing), or $\verb|Embed this document for clustering: {document}|$ to get embeddings suitable for clustering, or $\verb|
Embed this document for sentiment analysis: {document}|$ for embeddings suitable for sentiment analysis. The system is fine-tuned end-to-end with the appropriate task, e.g. a contrastive learning
objective for the search instruction, a classification objective for sentiment analysis, etc., leaving us with an easy way to generate embeddings for different tasks.
A note on scaling
The real power of semantic (and keyword) search comes when a search corpus is too large for a human to search manually. However if the corpus is enormous, we’d rather avoid looking at every document
each time we get a query. Thankfully, there are methods to avoid this by using specially tailored data structures: see Inverted Indices for keyword algorithms, and Hierarchical Navigable Small World
graphs [5] for semantic algorithms. These both reduce search time complexity from $\mathcal{O}(|\mathcal{D}|)$ to $\mathcal{O}(\log |\mathcal{D}|)$, where $|\mathcal{D}|$ is the corpus size.
There are many startups (Pinecone, Weviate, Milvus, Chroma, etc.) that are proposing so-called vector databases – databases in which embeddings are stored, and queries can be efficiently performed.
Though, there is also work contesting the need for these types of database in the first place [7].
We summarised search, semantic search, and how transformers are fine-tuned for search with a contrastive loss. I personally find this a very nice area of research with exciting real-world
applications – please reach out (ben.anson@bristol.ac.uk) if you’d like to discuss it!
[1]: Text and code embeddings by contrastive pre-training, Neelakantan et al (2022)
[2]: Task-aware Retrieval with Instructions, Asai et al (2022)
[3]: One embedder, any task: Instruction-finetuned text embeddings, Su et al (2022)
[4]: Some simple effective approximations to the 2-poisson model for probabilistic weighted retrieval, Robertson and Walker (1994)
[5]: Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs, https://arxiv.org/abs/1603.09320
[6]: An Image is Worth 16×16 Words: Transformers for Image Recognition at Scale, https://arxiv.org/abs/2010.11929
[7]: Vector Search with OpenAI Embeddings: Lucene Is All You Need, arXiv preprint arXiv:2308.14963
[8]: Complement lexical retrieval model with semantic residual embeddings, Advances in Information Retrieval (2021)
[9]: Attention is all you need, Advances in neural information processing systems (2017)
[10]: Sgpt: Gpt sentence embeddings for semantic search, arXiv preprint arXiv:2202.08904
[11]: Contrastive representation learning: A framework and review, IEEE Access 8 (2020)
[12]: Ms marco: A human generated machine reading comprehension dataset, arXiv preprint arXiv:1611.09268
[13]: AdANNS: A Framework for Adaptive Semantic Search, arXiv preprint arXiv:2305.19435
Student Perspectives: Impurity Identification in Oligonucleotide Drug Samples
A post by Harry Tata, PhD student on the Compass programme.
Oligonucleotides in Medicine
Oligonucleotide therapies are at the forefront of modern pharmaceutical research and development, with recent years seeing major advances in treatments for a variety of conditions. Oligonucleotide
drugs for Duchenne muscular dystrophy (FDA approved) [1], Huntington’s disease (Phase 3 clinical trials) [2], and Alzheimer’s disease [3] and amyotrophic lateral sclerosis (early-phase clinical
trials) [4] show their potential for tackling debilitating and otherwise hard-to-treat conditions. With continuing development of synthetic oligonucleotides, analytical techniques such as mass
spectrometry must be tailored to these molecules and keep pace with the field.
Working in conjunction with AstraZeneca, this project aims to advance methods for impurity detection and quantification in synthetic oligonucleotide mass spectra. In this blog post we apply a
regularised version of the Richardson-Lucy algorithm, an established technique for image deconvolution, to oligonucleotide mass spectrometry data. This allows us to attribute signals in the data to
specific molecular fragments, and therefore to detect impurities in oligonucleotide synthesis.
Oligonucleotide Fragmentation
If we have attempted to synthesise an oligonucleotide $\mathcal O$ with a particular sequence, we can take a sample from this synthesis and analyse it via mass spectrometry. In this process,
molecules in the sample are first fragmented — broken apart into ions — and these charged fragments are then passed through an electromagnetic field. The trajectory of each fragment through this
field depends on its mass/charge ratio (m/z), so measuring these trajectories (e.g. by measuring time of flight before hitting some detector) allows us to calculate the m/z of fragments in the
sample. This gives us a discrete mass spectrum: counts of detected fragments (intensity) across a range of m/z bins [5].
To get an idea of how much of $\mathcal O$ is in a sample, and what impurities might be present, we first need to consider what fragments $\mathcal O$ will produce. Oligonucleotides are short strands
of DNA or RNA; polymers with a backbone of sugars (such as ribose in RNA) connected by linkers (e.g. a phosphodiester bond), where each sugar has an attached base which encodes genetic information
On each monomer, there are two sites where fragmentation is likely to occur: at the linker (backbone cleavage) or between the base and sugar (base loss). Specifically, depending on which bond within
the linker is broken, there are four modes of backbone cleavage [7,8].
We include in $\mathcal F$ every product of a single fragmentation of $\mathcal O$ — any of the four backbone cleavage modes or base loss anywhere along the nucleotide — as well as the results of
every combination of two fragmentations (different cleavage modes at the same linker are mutually exclusive).
Sparse Richardson-Lucy Algorithm
Suppose we have a chemical sample which we have fragmented and analysed by mass spectrometry. This gives us a spectrum across n bins (each bin corresponding to a small m/z range), and we represent
this spectrum with the column vector $\mathbf{b}\in\mathbb R^n$, where $b_i$ is the intensity in the $i^{th}$ bin. For a set $\{f_1,\ldots,f_m\}=\mathcal F$ of possible fragments, let $x_j$ be the
amount of $f_j$ that is actually present. We would like to estimate the amounts of each fragment based on the spectrum $\mathbf b$.
If we had a sample comprising a unit amount of a single fragment $f_j$, so $x_j=1$ and $x_{ke j}=0,$ and this produced a spectrum $\begin{pmatrix}a_{1j}&\ldots&a_{nj}\end{pmatrix}^T$, we can say the
intensity contributed to bin $i$ by $x_j$ is $a_{ij}.$ In mass spectrometry, the intensity in a single bin due to a single fragment is linear in the amount of that fragment, and the intensities in a
single bin due to different fragments are additive, so in some general spectrum we have $b_i=\sum_j x_ja_{ij}.$
By constructing a library matrix $\mathbf{A}\in\mathbb R^{n\times m}$ such that $\{\mathbf A\}_{ij}=a_{ij}$ (so the columns of $\mathbf A$ correspond to fragments in $\mathcal F$), then in ideal
conditions the vector of fragment amounts $\mathbf x=\begin{pmatrix}x_1&\ldots&x_m\end{pmatrix}^T$ solves $\mathbf{Ax}=\mathbf{b}$. In practice this exact solution is not found — due to experimental
noise and potentially because there are contaminant fragments in the sample not included in $\mathcal F$ — and we instead make an estimate $\mathbf {\hat x}$ for which $\mathbf{A\hat x}$ is close to
$\mathbf b$.
Note that the columns of $\mathbf A$ correspond to fragments in $\mathcal F$: the values in a single column represent intensities in each bin due to a single fragment only. We $\ell_1$-normalise
these columns, meaning the total intensity (over all bins) of each fragment in the library matrix is uniform, and so the values in $\mathbf{\hat x}$ can be directly interpreted as relative abundances
of each fragment.
The observed intensities — as counts of fragments incident on each bin — are realisations of latent Poisson random variables. Assuming these variables are i.i.d., it can be shown that the estimate of
$\mathbf{x}$ which maximises the likelihood of the system is approximated by the iterative formula
$\mathbf {\hat{x}}^{(t+1)}=\left(\mathbf A^T \frac{\mathbf b}{\mathbf{A\hat x}^{(t)}}\right)\odot \mathbf{\hat x}^{(t)}.$
Here, quotients and the operator $\odot$ represent (respectively) elementwise division and multiplication of two vectors. This is known as the Richardson-Lucy algorithm [9].
In practice, when we enumerate oligonucleotide fragments to include in $\mathcal F$, most of these fragments will not actually be produced when the oligonucleotide passes through a mass spectrometer;
there is a large space of possible fragments and (beyond knowing what the general fragmentation sites are) no well-established theory allowing us to predict, for a new oligonucleotide, which
fragments will be abundant or negligible. This means we seek a sparse estimate, where most fragment abundances are zero.
The Richardson-Lucy algorithm, as a maximum likelihood estimate for Poisson variables, is analagous to ordinary least squares regression for Gaussian variables. Likewise lasso regression — a
regularised least squares regression which favours sparse estimates, interpretable as a maximum a posteriori estimate with Laplace priors — has an analogue in the sparse Richardson-Lucy algorithm:
$\mathbf {\hat{x}}^{(t+1)}=\left(\mathbf A^T \frac{\mathbf b}{\mathbf{A\hat x}^{(t)}}\right)\odot \frac{ \mathbf{\hat x}^{(t)}}{\mathbf 1 + \lambda},$
where $\lambda$ is a regularisation parameter [10].
Library Generation
For each oligonucleotide fragment $f\in\mathcal F$, we smooth and bin the m/z values of the most abundant isotopes of $f$, and store these values in the columns of $\mathbf A$. However, if these are
the only fragments in $\mathcal F$ then impurities will not be identified: the sparse Richardson-Lucy algorithm will try to fit oligonucleotide fragments to every peak in the spectrum, even ones that
correspond to fragments not from the target oligonucleotide. Therefore we also include ‘dummy’ fragments corresponding to single peaks in the spectrum — the method will fit these to
non-oligonucleotide peaks, showing the locations of any impurities.
For a mass spectrum from a sample containing a synthetic oligonucleotide, we generated a library of oligonucleotide and dummy fragments as described above, and applied the sparse Richardson-Lucy
algorithm. Below, the model fit is plotted alongside the (smoothed, binned) spectrum and the ten most abundant fragments as estimated by the model. These fragments are represented as bars with binned
m/z at the peak fragment intensity, and are separated into oligonucleotide fragments and dummy fragments indicating possible impurities. All intensities and abundances are Anscombe transformed ($x\
rightarrow\sqrt{x+3/8}$) for clarity.
As the oligonucleotide in question is proprietary, its specific composition and fragmentation is not mentioned here, and the bins plotted have been transformed (without changing the shape of the
data) so that individual fragment m/z values are not identifiable.
We see the data is fit extremely closely, and that the spectrum is quite clean: there is one very pronounced peak roughly in the middle of the m/z range. This peak corresponds to one of the
oligonucleotide fragments in the library, although there is also an abundant dummy fragment slightly to the left inside the main peak. Fragment intensities in the library matrix are smoothed, and it
may be the case that the smoothing here is inappropriate for the observed peak, hence other fragments being fit at the peak edge. Investigating these effects is a target for the rest of the project.
We also see several smaller peaks, most of which are modelled with oligonucleotide fragments. One of these peaks, at approximately bin 5352, has a noticeably worse fit if excluding dummy fragments
from the library matrix (see below). Using dummy fragments improves this fit and indicates a possible impurity. Going forward, understanding and quantification of these impurities will be improved by
including other common fragments in the library matrix, and by grouping fragments which correspond to the same molecules.
[1] Junetsu Igarashi, Yasuharu Niwa, and Daisuke Sugiyama. “Research and Development of Oligonucleotide Therapeutics in Japan for Rare Diseases”. In: Future Rare Diseases 2.1 (Mar. 2022), FRD19.
[2] Karishma Dhuri et al. “Antisense Oligonucleotides: An Emerging Area in Drug Discovery and Development”. In: Journal of Clinical Medicine 9.6 (6 June 2020), p. 2004.
[3] Catherine J. Mummery et al. “Tau-Targeting Antisense Oligonucleotide MAPTRx in Mild Alzheimer’s Disease: A Phase 1b, Randomized, Placebo-Controlled Trial”. In: Nature Medicine (Apr. 24, 2023),
pp. 1–11.
[4] Benjamin D. Boros et al. “Antisense Oligonucleotides for the Study and Treatment of ALS”. In: Neurotherapeutics: The Journal of the American Society for Experimental NeuroTherapeutics 19.4 (July
2022), pp. 1145–1158.
[5] Ingvar Eidhammer et al. Computational Methods for Mass Spectrometry Proteomics. John Wiley & Sons, Feb. 28, 2008. 299 pp.
[6] Harri Lönnberg. Chemistry of Nucleic Acids. De Gruyter, Aug. 10, 2020.
[7] S. A. McLuckey, G. J. Van Berkel, and G. L. Glish. “Tandem Mass Spectrometry of Small, Multiply Charged Oligonucleotides”. In: Journal of the American Society for Mass Spectrometry 3.1 (Jan.
1992), pp. 60–70.
[8] Scott A. McLuckey and Sohrab Habibi-Goudarzi. “Decompositions of Multiply Charged Oligonucleotide Anions”. In: Journal of the American Chemical Society 115.25 (Dec. 1, 1993), pp. 12085–12095.
[9] Mario Bertero, Patrizia Boccacci, and Valeria Ruggiero. Inverse Imaging with Poisson Data: From Cells to Galaxies. IOP Publishing, Dec. 1, 2018.
[10] Elad Shaked, Sudipto Dolui, and Oleg V. Michailovich. “Regularized Richardson-Lucy Algorithm for Reconstruction of Poissonian Medical Images”. In: 2011 IEEE International Symposium on Biomedical
Imaging: From Nano to Macro. Mar. 2011, pp. 1754–1757.
Student Perspectives: Density Ratio Estimation with Missing Data
A post by Josh Givens, PhD student on the Compass programme.
Density ratio estimation is a highly useful field of mathematics with many applications. This post describes my research undertaken alongside my supervisors Song Liu and Henry Reeve which aims to
make density ratio estimation robust to missing data. This work was recently published in proceedings for AISTATS 2023.
Density Ratio Estimation
As the name suggests, density ratio estimation is simply the task of estimating the ratio between two probability densities. More precisely for two RVs (Random Variables) $Z^0, Z^1$ on some space $\
mathcal{Z}$ with probability density functions (PDFs) $p_0, p_1$ respectively, the density ratio is the function $r^*:\mathcal{Z}\rightarrow\mathbb{R}$ defined by
Plot of the scaled density ratio alongside the PDFs for the two classes.
Density ratio estimation (DRE) is then the practice of using IID (independent and identically distributed) samples from $Z^0$ and $Z^1$ to estimate $r^*$. What makes DRE so useful is that it gives us
a way to characterise the difference between these 2 classes of data using just 1 quantity, $r^*$.
The Density Ratio in Classification
We now give demonstrate this characterisability in the case of classification. To frame this as a classification problem define $Y\sim\text{Bernoulli}(0.5)$ and $Z$ by $Z|Y=y\sim Z^{y}$. The task of
predicting $Y$ given $Z$ using some function $\phi:\mathcal{Z}\rightarrow\{0,1\}$ is then our standard classification problem. In classification a common target is the Bayes Optimal Classifier, the
classifier $\phi^*$ which maximises $\mathbb{P}(Y=\phi(Z)).$ We can write this classifier in terms of $r^*$ as we know that $\phi^*(z)=\mathbb{I}\{\mathbb{P}(Y=1|Z=z)>0.5\}$ where $\mathbb{I}$ is
the indicator function. Then, by the total law of probability, we have
$=\frac{p_1(z)\mathbb{P}(Y=1)}{p_1(z)\mathbb{P}(Y=1)+p_0(z)\mathbb{P}(Y=0)} =\frac{1}{1+\frac{1}{r}\frac{\mathbb{P}(Y=0)}{\mathbb{P}(Y=1)}}.$
Hence to learn the Bayes optimal classifier it is sufficient to learn the density ratio and a constant. This pattern extends well beyond Bayes optimal classification to many other areas such as error
controlled classification, GANs, importance sampling, covariate shift, and others. Generally speaking, if you are in any situation where you need to characterise the difference between two classes
of data, it’s likely that the density ratio will make an appearance.
Estimation Implementation – KLIEP
Now we have properly introduced and motivated DRE, we need to look at how we can go about performing it. We will focus on one popular method called KLIEP here but there are a many different methods
out there (see Sugiyama et al 2012 for some additional examples.)
The intuition behind KLIEP is simple: as $r^* \cdot p_0=p_1$, if $\hat r\cdot p_0$ is “close” to $p_1$ then $\hat r$ is a good estimate of $r^*$. To measure this notion of closeness KLIEP uses the KL
(Kullback-Liebler) divergence which measures the distance between 2 probability distributions. We can now formulate our ideal KLIEP objective as follows:
$\underset{r}{\text{min}}~ KL(p_1|p_0\cdot r)$
$\text{subject to:}~ \int_{\mathcal{Z}}r(z)p_0(z)\mathrm{d}z=1$
where $KL(p|p')$ represent the KL divergence from $p$ to $p'$. The constraint ensures that the right hand side of our KL divergence is indeed a PDF. From the definition of the KL-divergence we can
rewrite the solution to this as $\hat r:=\frac{\tilde r}{\mathbb{E}[r(X^0)]}$ where $\tilde r$ is the solution to the unconstrained optimisation
$\underset{r}{\text{min}}~\mathbb{E}[\log (r(Z^1))]-\log(\mathbb{E}[r(Z^0)]).$
As this is now just an unconstrained optimisation over expectations of known transformations of $Z^0, Z^1$, we can approximate this using samples. Given samples $z^0_1,\dotsc,z^0_n$ from $Z_0$ and
samples $z^1_1,\dotsc,z^1_n$ from $Z_1$ our estimate of the density ratio will be $\hat r:=\left(\frac{1}{n}\sum_{i=1}^nr(z_i^0)\right)^{-1}\tilde r$ where $\tilde r$ solves
$\underset{r}{\min}~ \frac{1}{n}\sum_{i=1}^n \log(r(z^1_i))-\log\left(\frac{1}{n}\sum_{i=1}^n r(z^0_i)\right).$
Despite KLIEP being commonly used, up until now it has not been made robust to missing not at random data. This is what our research aims to do.
Missing Data
Suppose that instead of observing samples from $Z$, we observe samples from some corrupted version of $Z$, $X$. We assume that $\mathbb{P}(\{X=\varnothing\}\cup \{X=Z\})=1$ so that either $X$ is
missing or $X$ takes the value of $Z$. We also assume that whether $X$ is missing depends upon the value of $Z$. Specifically we assume $\mathbb{P}(X=\varnothing|Z=z)=\varphi(z)$ with $\varphi(z)$
not constant and refer to $\varphi$ as the missingness function. This type of missingness is known as missing not at random (MNAR) and when dealt with improperly can lead to biased result. Some
examples of MNAR data could be readings take from a medical instrument which is more likely to err when attempting to read extreme values or recording responses to a questionnaire where respondents
may be more likely to not answer if the deem their response to be unfavourable. Note that while we do not see what the true response would be, we do at least get a response meaning that we know when
an observation is missing.
Missing Data with DRE
We now go back to density ratio estimation in the case where instead of observing samples from $Z^0,Z^1$ we observe samples from their corrupted versions $X^0, X^1$. We take their respective
missingness functions to be $\varphi_0, \varphi_1$ and assume them to be known. Now let us look at what would happen if we implemented KLIEP with the data naively by simply filtering out the
missing-values. In this case, the actual density ratio we would be estimating would be
and so we would get inaccurate estimates of the density ratio no matter how many samples are used to estimate it. The image below demonstrates this in the case were samples in class $1$ are more
likely to be missing when larger and class $0$ has no missingness.
A plot of the density ratio using both the full data and only the observed part of the corrupted data
Our Solution
Our solution to this problem is to use importance weighting. Using relationships between the densities of $X$ and $Z$ we have that
As such we can re-write the KLIEP objective to keep our expectation estimation unbiased even when using these corrupted samples. This gives our modified objective which we call M-KLIEP as follows.
Given samples $x^0_1,\dotsc,x^0_n$ from $X_0$ and samples $x^1_1,\dotsc,x^1_n$ from $X_1$ our estimate is $\hat r=\left(\frac{1}{n}\sum_{i=1}^n\frac{\mathbb{I}\{x_i^0eq\varnothing\}r(x_i^0)}{1-\
varphi_o(x_i^o)}\right)^{-1}\tilde r$ where $\tilde r$ solves
This objective will now target $r^*$ even when used on MNAR data.
Application to Classification
We now apply our density ratio estimation on MNAR data to estimate the Bayes optimal classifier. Below shows a plot of samples alongside the true Bayes optimal classifier and estimated classifiers
from the samples via our method M-KLIEP and a naive method CC-KLIEP which simply ignores missing points. Missing data points are faded out.
Faded points represent missing values. M-KLIEP represents our method, CC-KLIEP represents a Naive approach, BOC gives the Bayes optimal classifier
As we can see, due to not accounting for the MNAR nature of the data, CC-KLIEP underestimates the true number of class 1 samples in the top left region and therefore produces a worse classifier than
our approach.
Additional Contributions
As well as this modified objective our paper provides the following additional contributions:
• Theoretical finite sample bounds on the accuracy of our modified procedure.
• Methods for learning the missingness functions $\varphi_1,\varphi_0$.
• Expansions to partial missingness via a Naive-Bayes framework.
• Downstream implementation of our method within Neyman-Pearson classification.
• Adaptations to Neyman-Pearson classification itself making it robust to MNAR data.
For more details see our paper and corresponding github repository. If you have any questions on this work feel free to contact me at josh.givens@bristol.ac.uk.
Givens, J., Liu, S., & Reeve, H. W. J. (2023). Density ratio estimation and neyman pearson classification with missing data. In F. Ruiz, J. Dy, & J.-W. van de Meent (Eds.), Proceedings of the 26th
international conference on artificial intelligence and statistics (Vol. 206, pp. 8645–8681). PMLR.
Sugiyama, M., Suzuki, T., & Kanamori, T. (2012). Density Ratio Estimation in Machine Learning. Cambridge University Press.
Student Perspectives: Intro to Recommendation Systems
A post by Hannah Sansford, PhD student on the Compass programme.
Like many others, I interact with recommendation systems on a daily basis; from which toaster to buy on Amazon, to which hotel to book on booking.com, to which song to add to a playlist on Spotify.
They are everywhere. But what is really going on behind the scenes?
Recommendation systems broadly fit into two main categories:
1) Content-based filtering. This approach uses the similarity between items to recommend items similar to what the user already likes. For instance, if Ed watches two hair tutorial videos, the system
can recommend more hair tutorials to Ed.
2) Collaborative filtering. This approach uses the the similarity between users’ past behaviour to provide recommendations. So, if Ed has watched similar videos to Ben in the past, and Ben likes a
cute cat video, then the system can recommend the cute cat video to Ed (even if Ed hasn’t seen any cute cat videos).
Both systems aim to map each item and each user to an embedding vector in a common low-dimensional embedding space $E = \mathbb{R}^d$. That is, the dimension of the embeddings ($d$) is much smaller
than the number of items or users. The hope is that the position of these embeddings captures some of the latent (hidden) structure of the items/users, and so similar items end up ‘close together’ in
the embedding space. What is meant by being ‘close’ may be specified by some similarity measure.
Collaborative filtering
In this blog post we will focus on the collaborative filtering system. We can break it down further depending on the type of data we have:
1) Explicit feedback data: aims to model relationships using explicit data such as user-item (numerical) ratings.
2) Implicit feedback data: analyses relationships using implicit signals such as clicks, page views, purchases, or music streaming play counts. This approach makes the assumption that: if a user
listens to a song, for example, they must like it.
The majority of the data on the web comes from implicit feedback data, hence there is a strong demand for recommendation systems that take this form of data as input. Furthermore, this form of data
can be collected at a much larger scale and without the need for users to provide any extra input. The rest of this blog post will assume we are working with implicit feedback data.
Problem Setup
Suppose we have a group of $n$ users $U = (u_1, \ldots, u_n)$ and a group of $m$ items $I = (i_1, \ldots, i_m)$. Then we let $\mathbf{R} \in \mathbb{R}^{n \times m}$ be the ratings matrix where
position $R_{ui}$ represents whether user $u$ interacts with item $i$. Note that, in most cases the matrix $\mathbf{R}$ is very sparse, since most users only interact with a small subset of the full
item set $I$. For any items $i$ that user $u$ does not interact with, we set $R_{ui}$ equal to zero. To be clear, a value of zero does not imply the user does not like the item, but that they have
not interacted with it. The final goal of the recommendation system is to find the best recommendations for each user of items they have not yet interacted with.
Matrix Factorisation (MF)
A simple model for finding user emdeddings, $\mathbf{X} \in \mathbb{R}^{n \times d}$, and item embeddings, $\mathbf{Y} \in \mathbb{R}^{m \times d}$, is Matrix Factorisation. The idea is to find
low-rank embeddings such that the product $\mathbf{XY}^\top$ is a good approximation to the ratings matrix $\mathbf{R}$ by minimising some loss function on the known ratings.
A natural loss function to use would be the squared loss, i.e.
$L(\mathbf{X}, \mathbf{Y}) = \sum_{u, i} \left(R_{ui} - \langle X_u, Y_i \rangle \right)^2.$
This corresponds to minimising the Frobenius distance between $\mathbf{R}$ and its approximation $\mathbf{XY}^\top$, and can be solved easily using the singular value decomposition $\mathbf{R} = \
mathbf{U S V}^\top$.
Once we have our embeddings $\mathbf{X}$ and $\mathbf{Y}$, we can look at the row of $\mathbf{XY}^\top$ corresponding to user $u$ and recommend the items corresponding to the highest values (that
they haven’t already interacted with).
Logistic MF
Minimising the loss function in the previous section is equivalent to modelling the probability that user $u$ interacts with item $i$ as the inner product $\langle X_u, Y_i \rangle$, i.e.
$R_{ui} \sim \text{Bernoulli}(\langle X_u, Y_i \rangle),$
and maximising the likelihood over $\mathbf{X}$ and $\mathbf{Y}$.
In a research paper from Spotify [3], this relationship is instead modelled according to a logistic function parameterised by the sum of the inner product above and user and item bias terms, $\
beta_u$ and $\beta_i$,
$R_{ui} \sim \text{Bernoulli} \left( \frac{\exp(\langle X_u, Y_i \rangle + \beta_u + \beta_i)}{1 + \exp(\langle X_u, Y_i \rangle + \beta_u + \beta_i)} \right).$
Relation to my research
A recent influential paper [1] proved an impossibility result for modelling certain properties of networks using a low-dimensional inner product model. In my 2023 AISTATS publication [2] we show that
using a kernel, such as the logistic one in the previous section, to model probabilities we can capture these properties with embeddings lying on a low-dimensional manifold embedded in
infinite-dimensional space. This has various implications, and could explain part of the success of Spotify’s logistic kernel in producing good recommendations.
[1] Seshadhri, C., Sharma, A., Stolman, A., and Goel, A. (2020). The impossibility of low-rank representations for triangle-rich complex networks. Proceedings of the National Academy of Sciences, 117
[2] Sansford, H., Modell, A., Whiteley, N., and Rubin-Delanchy, P. (2023). Implications of sparsity and high triangle density for graph representation learning. Proceedings of The 26th International
Conference on Artificial Intelligence and Statistics, PMLR 206:5449-5473.
[3] Johnson, C. C. (2014). Logistic matrix factorization for implicit feedback data. Advances in Neural Information Processing Systems, 27(78):1–9.
Student Perspectives: An Introduction to Stochastic Gradient Methods
A post by Ettore Fincato, PhD student on the Compass programme.
This post provides an introduction to Gradient Methods in Stochastic Optimisation. This class of algorithms is the foundation of my current research work with Prof. Christophe Andrieu and Dr. Mathieu
Gerber, and finds applications in a great variety of topics, such as regression estimation, support vector machines, convolutional neural networks.
We can see below a simulation by Emilien Dupont (https://emiliendupont.github.io/) which represents two trajectories of an optimisation process of a time-varying function. This well describes the
main idea behind the algorithms we will be looking at, that is, using the (stochastic) gradient of a (random) function to iteratively reach the optimum.
Stochastic Optimisation
Stochastic optimisation was introduced by [1], and its aim is to find a scheme for solving equations of the form $abla_w g(w)=0$ given “noisy” measurements of $g$ [2].
In the simplest deterministic framework, one can fully determine the analytical form of $g(w)$, knows that it is differentiable and admits an unique minimum – hence the problem
$w_*=\underset{w}{\text{argmin}}\quad g(w)$
is well defined and solved by $abla_w g(w)=0$.
On the other hand, one may not be able to fully determine $g(w)$ because his experiment is corrupted by a random noise. In such cases, it is common to identify this noise with a random variable, say
$V$, consider an unbiased estimator $\eta(w,V)$ s.t. $\mathbb{E}_V[\eta(w,V)]=g(w)$ and to rewrite the problem as
Student Perspectives: An Introduction to Graph Neural Networks (GNNs)
A post by Emerald Dilworth, PhD student on the Compass programme.
This blog post serves as an accessible introduction to Graph Neural Networks (GNNs). An overview of what graph structured data looks like, distributed vector representations, and a quick description
of Neural Networks (NNs) are given before GNNs are introduced.
An Introductory Overview of GNNs:
You can think of a GNN as a Neural Network that runs over graph structured data, where we know features about the nodes – e.g. in a social network, where people are nodes, and edges are them sharing
a friendship, we know things about the nodes (people), for instance their age, gender, location. Where a NN would just take in the features about the nodes as input, a GNN takes in this in addition
to some known graph structure the data has. Some examples of GNN uses include:
• Predictions of a binary task – e.g. will this molecule (which the structure of can be represented by with a graph) inhibit this given bacteria? The GNN can then be used to predict for a molecule
not trained on. Finding a new antibiotic is one of the most famous papers using GNNs [1].
• Social networks and recommendation systems, where GNNs are used to predict new links [2].
What is a Graph?
A graph, $G = (V,E)$, is a data structure that consists of a set of nodes, $V$, and a set of edges, $E$. Graphs are used to represent connections (edges) between objects (nodes), where the edges can
be directed or undirected depending on whether the relationships between the nodes have direction. An $n$ node graph can be represented by an $n \times n$ matrix, referred to as an adjacency matrix.
Idea of Distributed Vector Representations
In machine learning architectures, the data input often needs to be converted to a tensor for the model, e.g. via 1-hot encoding. This provides an input (or local) representation of the data, which
if we think about 1-hot encoding creates a large, sparse representation of 0s and 1s. The input representation is a discrete representation of objects, but lacks information on how things are
correlated, how related they are, what they have in common. Often, machine learning models learn a distributed representation, where it learns how related objects are; nodes that are similar will
have similar distributed representations. (more…)
Student Perspectives: An Introduction to Deep Kernel Machines
A post by Edward Milsom, PhD student on the Compass programme.
This blog post provides a simple introduction to Deep Kernel Machines[1] (DKMs), a novel supervised learning method that combines the advantages of both deep learning and kernel methods. This work
provides the foundation of my current research on convolutional DKMs, which is supervised by Dr Laurence Aitchison.
Why aren’t kernels cool anymore?
Kernel methods were once top-dog in machine learning due to their ability to implicitly map data to complicated feature spaces, where the problem usually becomes simpler, without ever explicitly
computing the transformation. However, in the past decade deep learning has become the new king for complicated tasks like computer vision and natural language processing.
Neural networks are flexible when learning representations
The reason is twofold: First, neural networks have millions of tunable parameters that allow them to learn their feature mappings automatically from the data, which is crucial for domains like images
which are too complex for us to specify good, useful features by hand. Second, their layer-wise structure means these mappings can be built up to increasingly more abstract representations, while
each layer itself is relatively simple[2]. For example, trying to learn a single function that takes in pixels from pictures of animals and outputs their species is difficult; it is easier to map
pixels to corners and edges, then shapes, then body parts, and so on.
Kernel methods are rigid when learning representations
It is therefore notable that classical kernel methods lack these characteristics: most kernels have a very small number of tunable hyperparameters, meaning their mappings cannot flexibly adapt to the
task at hand, leaving us stuck with a feature space that, while complex, might be ill-suited to our problem. (more…)
Student Perspectives: Spectral Clustering for Rapid Identification of Farm Strategies
A post by Dan Milner, PhD student on the Compass programme.
Image 1: Smallholder Farm – Yebelo, southern Ethiopia
This blog describes an approach being developed to deliver rapid classification of farmer strategies. The data comes from a survey conducted with two groups of smallholder farmers (see image 2), one
group living in the Taita Hills area of southern Kenya and the other in Yebelo, southern Ethiopia. This work would not have been possible without the support of my supervisors James Hammond, from the
International Livestock Research Institute (ILRI) (and developer of the Rural Household Multi Indicator Survey, RHoMIS, used in this research), as well as Andrew Dowsey, Levi Wolf and Kate Robson
Brown from the University of Bristol.
Image 2: Measuring a Cows Heart Girth as Part of the Farm Surveys
Aims of the project
The goal of my PhD is to contribute a landscape approach to analysing agricultural systems. On-farm practices are an important part of an agricultural system and are one of the trilogy of components
that make-up what Rizzo et al (2022) call ‘agricultural landscape dynamics’ – the other two components being Natural Resources and Landscape Patterns. To understand how a farm interacts with and
responds to Natural Resources and Landscape Patterns it seems sensible to try and understand not just each farms inputs and outputs but its overall strategy and component practices. (more…)
Student Perspectives: An Introduction to QGAMs
A post by Ben Griffiths, PhD student on the Compass programme.
My area of research is studying Quantile Generalised Additive Models (QGAMs), with my main application lying in energy demand forecasting. In particular, my focus is on developing faster and more
stable fitting methods and model selection techniques. This blog post aims to briefly explain what QGAMs are, how to fit them, and a short illustrative example applying these techniques to data on
extreme rainfall in Switzerland. I am supervised by Matteo Fasiolo and my research is sponsored by Électricité de France (EDF).
Quantile Generalised Additive Models
QGAMs are essentially the result of combining quantile regression (QR; performing regression on a specific quantile of the response) with a generalised additive model (GAM; fitting a model assuming
additive smooth effects). Here we are in the regression setting, so let $F(y| \boldsymbol{x})$ be the conditional c.d.f. of a response, $y$, given a $p$-dimensional vector of covariates, $\boldsymbol
{x}$. In QR we model the $\tau$th quantile, that is, $\mu_\tau(\boldsymbol{x}) = \inf \{y : F(y|\boldsymbol{x}) \geq \tau\}$.
Examples of true quantiles of SHASH distribution.
This might be useful in cases where we do not need to model the full distribution of $y| \boldsymbol{x}$ and only need one particular quantile of interest (for example urban planners might only be
interested in estimates of extreme rainfall e.g. $\tau = 0.95$). It also allows us to make no assumptions about the underlying true distribution, instead we can model the distribution empirically
using multiple quantiles.
We can define the $\tau$th quantile as the minimiser of expected loss
$L(\mu| \boldsymbol{x}) = \mathbb{E} \left\{\rho_\tau (y - \mu)| \boldsymbol{x} \right \} = \int \rho_\tau(y - \mu) d F(y|\boldsymbol{x}),$
w.r.t. $\mu = \mu_\tau(\boldsymbol{x})$, where
$\rho_\tau (z) = (\tau - 1) z \boldsymbol{1}(z<0) + \tau z \boldsymbol{1}(z \geq 0),$
is known as the pinball loss (Koenker, 2005).
Pinball loss for quantiles 0.5, 0.8, 0.95.
We can approximate the above expression empirically given a sample of size $n$, which gives the quantile estimator, $\hat{\mu}_\tau(\boldsymbol{x}) = \boldsymbol{x}^\mathsf{T} \hat{\boldsymbol{\
beta}}$ where
$\hat{\boldsymbol{\beta}} = \underset{\boldsymbol{\beta}}{\arg \min} \frac{1}{n} \sum_{i=1}^n \rho_\tau \left\{y_i - \boldsymbol{x}_i^\mathsf{T} \boldsymbol{\beta}\right\},$
where $\boldsymbol{x}_i$ is the $i$th vector of covariates, and $\boldsymbol{\beta}$ is vector of regression coefficients.
So far we have described QR, so to turn this into a QGAM we assume $\mu_\tau(\boldsymbol{x})$ has additive structure, that is, we can write the $\tau$th conditional quantile as
$\mu_\tau(\boldsymbol{x}) = \sum_{j=1}^m f_j(\boldsymbol{x}),$
where the $m$ additive terms are defined in terms of basis functions (e.g. spline bases). A marginal smooth effect could be, for example
$f_j(\boldsymbol{x}) = \sum_{k=1}^{r_j} \beta_{jk} b_{jk}(x_j),$
where $\beta_{jk}$ are unknown coefficients, $b_{jk}(x_j)$ are known spline basis functions and $r_j$ is the basis dimension.
Denote $\boldsymbol{\mathrm{x}}_i$ the vector of basis functions evaluated at $\boldsymbol{x}_i$, then the $n \times d$ design matrix $\boldsymbol{\mathrm{X}}$ is defined as having $i$th row $\
boldsymbol{\mathrm{x}}_i$, for $i = 1, \dots, n$, and $d = r_1+\dots +r_m$ is the total basis dimension over all $f_j$. Now the quantile estimate is defined as $\mu_\tau(\boldsymbol{x}_i) = \
boldsymbol{\mathrm{x}}_i^\mathsf{T} \boldsymbol{\beta}$. When estimating the regression coefficients, we put a ridge penalty on $\boldsymbol{\beta}_{j}$ to control complexity of $f_j$, thus we seek
to minimise the penalised pinball loss
$V(\boldsymbol{\beta},\boldsymbol{\gamma},\sigma) = \sum_{i=1}^n \frac{1}{\sigma} \rho_\tau \left\{y_i - \mu(\boldsymbol{x}_i)\right\} + \frac{1}{2} \sum_{j=1}^m \gamma_j \boldsymbol{\beta}^\mathsf
{T} \boldsymbol{\mathrm{S}}_j \boldsymbol{\beta},$
where $\boldsymbol{\gamma} = (\gamma_1,\dots,\gamma_m)$ is a vector of positive smoothing parameters, $1/\sigma>0$ is the learning rate and the $\boldsymbol{\mathrm{S}}_j$‘s are positive
semi-definite matrices which penalise the wiggliness of the corresponding effect $f_j$. Minimising $V$ with respect to $\boldsymbol{\beta}$ given fixed $\sigma$ and $\boldsymbol{\gamma}$ leads to the
maximum a posteriori (MAP) estimator $\hat{\boldsymbol{\beta}}$.
There are a number of methods to tune the smoothing parameters and learning rate. The framework from Fasiolo et al. (2021) consists in:
1. calibrating $\sigma$ by Integrated Kullback–Leibler minimisation
2. selecting $\boldsymbol{\gamma}|\sigma$ by Laplace Approximate Marginal Loss minimisation
3. estimating $\boldsymbol{\beta}|\boldsymbol{\gamma},\sigma$ by minimising penalised Extended Log-F loss (note that this loss is simply a smoothed version of the pinball loss introduced above)
For more detail on what each of these steps means I refer the reader to Fasiolo et al. (2021). Clearly this three-layered nested optimisation can take a long time to converge, especially in cases
where we have large datasets which is often the case for energy demand forecasting. So my project approach is to adapt this framework in order to make it less computationally expensive.
Application to Swiss Extreme Rainfall
Here I will briefly discuss one potential application of QGAMs, where we analyse a dataset consisting of observations of the most extreme 12 hourly total rainfall each year for 65 Swiss weather
stations between 1981-2015. This data set can be found in the R package gamair and for model fitting I used the package mgcViz.
A basic QGAM for the 50% quantile (i.e. $\tau = 0.5$) can be fitted using the following formula
$\mu_i = \beta + \psi(\mathrm{reg}_i) + f_1(\mathrm{nao}_i) + f_2(\mathrm{el}_i) + f_3(\mathrm{Y}_i) + f_4(\mathrm{E}_i,\mathrm{N}_i),$
where $\beta$ is the intercept term, $\psi(\mathrm{reg}_i)$ is a parametric factor for climate region, $f_1, \dots, f_4$ are smooth effects, $\mathrm{nao}_i$ is the Annual North Atlantic Oscillation
index, $\mathrm{el}_i$ is the metres above sea level, $\mathrm{Y}_i$ is the year of observation, and $\mathrm{E}_i$ and $\mathrm{N}_i$ are the degrees east and north respectively.
After fitting in mgcViz, we can plot the smooth effects and see how these affect the extreme yearly rainfall in Switzerland.
Fitted smooth effects for North Atlantic Oscillation index, elevation, degrees east and north and year of observation.
From the plots observe the following; as we increase the NAO index we observe a somewhat oscillatory effect on extreme rainfall; when increasing elevation we see a steady increase in extreme rainfall
before a sharp drop after an elevation of around 2500 metres; as years increase we see a relatively flat effect on extreme rainfall indicating the extreme rainfall patterns might not change much over
time (hopefully the reader won’t regard this as evidence against climate change); and from the spatial plot we see that the south-east of Switzerland appears to be more prone to more heavy extreme
We could also look into fitting a 3D spatio-temporal tensor product effect, using the following formula
$\mu_i = \beta + \psi(\mathrm{reg}_i) + f_1(\mathrm{nao}_i) + f_2(\mathrm{el}_i) + t(\mathrm{E}_i,\mathrm{N}_i,\mathrm{Y}_i),$
where $t$ is the tensor product effect between $\mathrm{E}_i$, $\mathrm{N}_i$ and $\mathrm{Y}_i$. We can examine the spatial effect on extreme rainfall over time by plotting the smooths.
3D spatio-temporal tensor smooths for years 1985, 1995, 2005 and 2015.
There does not seem to be a significant interaction between the location and year, since we see little change between the plots, except for perhaps a slight decrease in the south-east.
Finally, we can make the most of the QGAM framework by fitting multiple quantiles at once. Here we fit the first formula for quantiles $\tau = 0.1, 0.2, \dots, 0.9$, and we can examine the fitted
smooths for each quantile on the spatial effect.
Spatial smooths for quantiles 0.1, 0.2, …, 0.9.
Interestingly the spatial effect is much stronger in higher quantiles than in the lower ones, where we see a relatively weak effect at the 0.1 quantile, and a very strong effect at the 0.9 quantile
ranging between around -30 and +60.
The example discussed here is but one of many potential applications of QGAMs. As mentioned in the introduction, my research area is motivated by energy demand forecasting. My current/future research
is focused on adapting the QGAM fitting framework to obtain faster fitting.
Fasiolo, M., S. N. Wood, M. Zaffran, R. Nedellec, and Y. Goude (2021). Fast calibrated additive quantile regression. Journal of the American Statistical Association 116 (535), 1402–1412.
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Quadratic Formula
Posted in Science & Nature
Quadratic Formula
Anyone who has studied mathematics to some degree will know about algebraic equations. An algebraic equation is an equation that can be solved to find the unknown value of x. A quadratic equation is
an algebraic equation with x², or in other words has two valid solutions to x. Generally speaking, a quadratic equation can be expressed in the following fashion: ax² + bx + c = 0. a, b and c are
constants and the equation can be solved to find x. A quadratic equation is definitely more complicated to solve compared to a linear equation and it can be solved using various means and
applications such as factorisation. As these methods are learnt in school and this Encyclopaedia is technically not a mathematics textbook, such methods will not be delved into.
If you have not learnt it already, there is a shortcut method to solving quadratic equations: the quadratic formula. This formula can easily find x if you simply substitute in the values for a, b and
c. Of course this formula only works if the solutions are real numbers. The quadratic formula is as follows:
As you can see, because of the ± sign, the formula can be used to find both solutions to a quadratic equation. Even without factorising, it can find the answer as long as you substitute numbers into
it on a calculator, making maths class very easy. However, as mentioned above the Encyclopaedia of Absolute and Relative Knowledge is not a mathematics textbook and one should instead learn properly
from their teacher, not using the formula until they have been taught it properly.
Leave a Comment! | {"url":"https://jineralknowledge.com/quadratic-formula/","timestamp":"2024-11-13T15:00:20Z","content_type":"text/html","content_length":"73246","record_id":"<urn:uuid:3d9337e9-8a9f-46f8-892f-0387792b1f1f>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00331.warc.gz"} |
Math Assignments
Coping with Math Assignments
For some students, dealing with math assignments is a complex and frustrating chore. For others, it could be a full-filled activity. If you are a school student, chances are you have to deal with
math assignments. Since you can’t avoid them, what you can do is find a friendly approach toward them.
Without any preparation, some students jump right into their math assignments. When they get stuck on one of the problems, they just flip to the back of the book to seek for an answer. Then they
simply copy down the answer and stop working on it. This is definitely not the proper way to handle math assignments.
The truth is, there is no fast solution for anyone to become a math expert overnight, but if you use the following ideas, you may actually turn your math assignments into an entertainment.
The first thing to do when dealing with a math assignment is to get the glimpse of the problem, and try to understand its nature. You can refer to your textbook for material that relates to your
assignment. When you get stuck with a math problem, try to find the location of a completed problem with similar complexity. Reviewing the notes from class is a clever move towards completing your
math assignments. Chances are, you could have worked out a lot of math problems with the same structure of your assignment problems.
When it comes to math problems, there is a bad habit that’s common among a lot of students. They try to memorize the steps of solving a complex math problem, and present them in their assignment
paper. However, when similar problems are required to be solved in a test, they cannot solve it on their own and end up getting bad grades.
When solving a math problem, try being as neat as you can. Explain each step in detail, so in case you get stuck in one of the steps, you can try to refer back to the derivation of the particular
Make notes about deviating steps and important formulae. So when a complex problem is given to you, you can refer to the notes and easily solve the problem. Generally speaking, math problems have a
single clue that reveals the key of unlocking the answers. So all you need to do is finding the clue. If you are facing a very difficult problem and in a position where you can’t proceed with the
problem, don’t hesitate about calling your teacher. | {"url":"https://www.mathprepa.com/math-assignments-professional-help.html","timestamp":"2024-11-13T13:07:18Z","content_type":"text/html","content_length":"13431","record_id":"<urn:uuid:258b1d52-ef88-4ed4-9626-646c35a71518>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00547.warc.gz"} |
Project Euler > Problem 153 > Investigating Gaussian Integers (Java Solution)
As we all know the equation x2=-1 has no solutions for real x.
If we however introduce the imaginary number i this equation has two solutions: x=i and x=-i.
If we go a step further the equation (x-3)2=-4 has two complex solutions: x=3+2i and x=3-2i.
x=3+2i and x=3-2i are called each others' complex conjugate.
Numbers of the form a+bi are called complex numbers.
In general a+bi and a[−]bi are each other's complex conjugate.
A Gaussian Integer is a complex number a+bi such that both a and b are integers.
The regular integers are also Gaussian integers (with b=0).
To distinguish them from Gaussian integers with b [≠] 0 we call such integers "rational integers."
A Gaussian integer is called a divisor of a rational integer n if the result is also a Gaussian integer.
If for example we divide 5 by 1+2i we can simplify in the following manner:
Multiply numerator and denominator by the complex conjugate of 1+2i: 1[−]2i.
The result is .
So 1+2i is a divisor of 5.
Note that 1+i is not a divisor of 5 because .
Note also that if the Gaussian Integer (a+bi) is a divisor of a rational integer n, then its complex conjugate (a[−]bi) is also a divisor of n.
In fact, 5 has six divisors such that the real part is positive: {1, 1 + 2i, 1 [−] 2i, 2 + i, 2 [−] i, 5}.
The following is a table of all of the divisors for the first five positive rational integers:
n Gaussian integer divisors
with positive real part Sum s(n) of
these divisors
2 1, 1+i, 1-i, 2 5
3 1, 3 4
4 1, 1+i, 1-i, 2, 2+2i, 2-2i,4 13
5 1, 1+2i, 1-2i, 2+i, 2-i, 5 12
For divisors with positive real parts, then, we have: .
For 1 [≤] n [≤] 105, [∑] s(n)=17924657155.
What is [∑] s(n) for 1 [≤] n [≤] 108?
The solution may include methods that will be found here: Library.java .
public interface EulerSolution{
public String run();
We don't have code for that problem yet! If you solved that out using Java, feel free to contribute it to our website, using our "Upload" form. | {"url":"http://www.javaproblems.com/2013/12/project-euler-problem-153-investigating_11.html","timestamp":"2024-11-03T19:05:33Z","content_type":"application/xhtml+xml","content_length":"48640","record_id":"<urn:uuid:99eed6d3-26e2-484a-b133-34db27ef1006>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00213.warc.gz"} |
indomain(?Var, ++Method)
[ library(gfd_search) | Reference Manual | Alphabetic Index ]
indomain(?Var, ++Method)
a flexible way to assign values to finite domain variables
a domain variable or an integer
one of the atoms min, max, middle, median, split, interval, random or an integer
This predicate provides a flexible way to assign values to finite domain variables.
The available methods are:
• enum Identical to indomain/1. Start enumeration from the smallest value upwards, without first removing previously tried values.
• min Start the enumeration from the smallest value upwards. This behaves like the built-in indomain/1, except that it removes previously tested values on backtracking.
• max Start the enumeration from the largest value downwards.
• reverse_min Like min, but tries the alternatives in opposite order, i.e. values are excluded first, then assigned.
• reverse_max Like max, but tries the alternatives in opposite order, i.e. values are excluded first, then assigned.
• middle Try the enumeration starting from the middle of the domain. On backtracking, this chooses alternatively values above and below the middle value, until all alternatives have been tested.
• median Try the enumeration starting from the median value of the domain. On backtracking, this chooses alternatively values above and below the median value, until all alternatives have been
• split Try the enumeration by splitting the domain successively into halves until a ground value is reached. This sometimes can detect failure earlier than the normal enumeration methods, but
enumerates the values in the same order as min.
• reverse_split Like split, but tries the upper half of the domain first.
• interval If the domain consists of several intervals, then we branch first on the choice of the interval. For one interval, we use domain splitting.
• random Try the enumeration in a random order. On backtracking, the previously tested value is removed. This method uses random/1 to create random numbers, use seed/1 before to make results
• Value:integer Like middle, but start with the given integer Value
• sbds_min Like min, but use sbds_try/2 to make choices (for use in conjunction with the SBDS symmetry breaking library).
• sbds_max Like max, but use sbds_try/2 to make choices (for use in conjunction with the SBDS symmetry breaking library).
• sbds_middle Like middle, but use sbds_try/2 to make choices (for use in conjunction with the SBDS symmetry breaking library).
• sbds_median Like median, but use sbds_try/2 to make choices (for use in conjunction with the SBDS symmetry breaking library).
• sbds_random Like random, but use sbds_try/2 to make choices (for use in conjunction with the SBDS symmetry breaking library).
• sbds(Value:integer) Like Value:integer, but use sbds_try/2 to make choices (for use in conjunction with the SBDS symmetry breaking library).
• gap_sbds_min Like min, but use sbds_try/2 to make choices (for use in conjunction with the GAP-based SBDS symmetry breaking library, lib(ic_gap_sbds)).
• gap_sbds_max Like max, but use sbds_try/2 to make choices (for use in conjunction with the GAP-based SBDS symmetry breaking library, lib(ic_gap_sbds)).
• gap_sbds_middle Like middle, but use sbds_try/2 to make choices (for use in conjunction with the GAP-based SBDS symmetry breaking library, lib(ic_gap_sbds)).
• gap_sbds_median Like median, but use sbds_try/2 to make choices (for use in conjunction with the GAP-based SBDS symmetry breaking library, lib(ic_gap_sbds)).
• gap_sbds_random Like random, but use sbds_try/2 to make choices (for use in conjunction with the GAP-based SBDS symmetry breaking library, lib(ic_gap_sbds)).
• gap_sbds(Value:integer) Like Value:integer, but use sbds_try/2 to make choices (for use in conjunction with the GAP-based SBDS symmetry breaking library, lib(ic_gap_sbds)).
• gap_sbdd_min Like min, but use sbdd_try/2 to make choices (for use in conjunction with the GAP-based SBDD symmetry breaking library, lib(ic_gap_sbdd)).
• gap_sbdd_max Like max, but use sbdd_try/2 to make choices (for use in conjunction with the GAP-based SBDD symmetry breaking library, lib(ic_gap_sbdd)).
• gap_sbdd_middle Like middle, but use sbdd_try/2 to make choices (for use in conjunction with the GAP-based SBDD symmetry breaking library, lib(ic_gap_sbdd)).
• gap_sbdd_median Like median, but use sbdd_try/2 to make choices (for use in conjunction with the GAP-based SBDD symmetry breaking library, lib(ic_gap_sbdd)).
• gap_sbdd_random Like random, but use sbdd_try/2 to make choices (for use in conjunction with the GAP-based SBDD symmetry breaking library, lib(ic_gap_sbdd)).
• gap_sbdd(Value:integer) Like Value:integer, but use sbdd_try/2 to make choices (for use in conjunction with the GAP-based SBDD symmetry breaking library, lib(ic_gap_sbdd)).
On backtracking, all methods except enum first remove the previously tested value before choosing a new one. This sometimes can have a huge impact on the constraint propagation, and normally does not
cause much overhead, even if no additional propagation occurs.
Fail Conditions
X :: 1..10,
% writes 1 2 3 4 5 6 7 8 9 10
X :: 1..10,
% writes 10 9 8 7 6 5 4 3 2 1
X :: 1..10,
% writes 5 6 4 7 3 8 2 9 1 10
X :: 1..10,
% writes 5 6 4 7 3 8 2 9 1 10
X :: 1..10,
% writes 3 4 2 5 1 6 7 8 9 10
X :: 1..10,
% writes 1 2 3 4 5 6 7 8 9 10
X :: 1..10,
% writes for example 5 3 7 6 8 1 2 10 9 4
See Also
search / 6, ic : indomain / 1, ic_symbolic : indomain / 1, gfd : indomain / 1, sd : indomain / 1, fd : indomain / 1, random / 1, seed / 1, ic_sbds : sbds_try / 2, gfd_sbds : sbds_try / 2, fd_sbds :
sbds_try / 2, ic_gap_sbds : sbds_try / 2, ic_gap_sbdd : sbdd_try / 2 | {"url":"http://www.eclipseclp.org/doc/bips/lib/gfd_search/indomain-2.html","timestamp":"2024-11-11T09:57:28Z","content_type":"text/html","content_length":"8281","record_id":"<urn:uuid:599526a5-74bf-4bae-b091-f5c7157b9e50>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00169.warc.gz"} |
Green River LibGuides: Mathematics: Finding Math Books & Videos in the Library
Click link below to search for mathematics books and video available both online and on the shelves at the Holman Library:
Search this catalog to find books, ebooks, films, as well as articles from newspapers, magazines, journals, and more.
1. Math books and videos are grouped together by call number in two sections of the Library: the Main Collection and Essential College Skills.
2. Ask a librarian for assistance if needed- they are happy to help! | {"url":"https://libguides.greenriver.edu/c.php?g=724122&p=5166935","timestamp":"2024-11-05T12:12:28Z","content_type":"text/html","content_length":"36460","record_id":"<urn:uuid:8a1dbd9c-bd63-4cd3-b4d8-c47a7e6122ae>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00456.warc.gz"} |
Flat Bag Volume Calculator | Efficient Storage and Packaging
Home » Simplify your calculations with ease. » Lifestyle Calculators »
Flat Bag Volume Calculator | Efficient Storage and Packaging
Bags come in a variety of shapes and sizes. A common type is a flat bag, which, as the name suggests, lays flat and typically has a rectangular shape. A flat bag's volume is a critical parameter,
indicating how much it can hold. This is where our Flat Bag Volume Calculator comes into play. This simple, yet powerful tool, calculates the volume of a flat bag using its length, width, and height.
Understanding the Flat Bag Volume Formula
The key to calculating the volume of a flat bag is a simple mathematical formula:
FBV = BL * BW * BH
• FBV is the Flat Bag Volume (in^3),
• BL is the bag length (in),
• BW is the bag width (in), and
• BH is the bag height (in).
In essence, you find the flat bag volume by multiplying its length, width, and height.
To make this clearer, let's consider an example.
• Bag Length (BL) = 4 inches
• Bag Width (BW) = 5 inches
• Bag Height (BH) = 3 inches
Applying these values to our formula:
FBV = 4 (in) * 5 (in) * 3 (in) = 60 in^3
So, the volume of this bag is 60 cubic inches.
How the Flat Bag Volume Calculator Works
Our calculator makes this process easy and efficient. You simply input the length, width, and height of your bag into the calculator.
It will then use the flat bag volume formula to compute the volume for you.
This calculator isn't just for academics or curiosity; it has practical applications in our daily life and various industries.
Everyday Applications
In everyday scenarios, you could use the calculator to determine the right bag size for storage needs. Knowing the volume can help ensure you have enough space for items without wasting material on
an overly large bag.
Industrial Applications
In industries like food packaging, retail, logistics, etc., the calculator can help in determining the right size for packaging goods, optimizing space, and reducing waste.
Frequently Asked Questions (FAQs)
What units should I use for the flat bag volume calculator?
You can use any unit of measurement (e.g., inches, centimeters, etc.) as long as you use the same unit for length, width, and height. The volume will be in cubic units of the unit you used (in^3, cm^
3, etc.).
Can I use the flat bag volume calculator for bags of different shapes?
The calculator is designed for flat, rectangular-shaped bags. Using it for bags of different shapes may not yield accurate results.
How accurate is the flat bag volume calculator?
The accuracy of the flat bag volume calculator depends on the precision of the measurements you input. If you measure the length, width, and height accurately, the calculator will give you a precise
Is the flat bag volume calculator free to use?
Yes, the flat bag volume calculator is free to use. It's an online tool aimed at helping individuals and businesses determine the volume of their bags quickly and easily.
Can I use the flat bag volume calculator on any device?
Yes, you can use the flat bag volume calculator on any device that has an internet connection, including desktops, laptops, tablets, and smartphones.
The flat bag volume calculator is a simple yet powerful tool. Understanding how to calculate the volume of a flat bag can be useful in everyday scenarios and various industries.
Leave a Comment | {"url":"https://calculatorshub.net/lifestyle-calculators/flat-bag-volume-calculator/","timestamp":"2024-11-09T16:10:12Z","content_type":"text/html","content_length":"119934","record_id":"<urn:uuid:f8cfa6de-b3b3-4e1d-8784-26b21372ef72>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00557.warc.gz"} |
Ask Doctor Bob: What is level control? (PART 3) - Control Station
Video Transcript:
Hi, I’m Bob Rice. And today, we’re going to be covering part three of a three-part installment on level control. Today’s topic is focused on how to calculate the tuning values for level control
So let’s start by looking at our level control process. So we have our tank, and we’re trying to control the liquid level in the tank by adjusting the flow out of a pump. And we’ve got disturbances
that are coming into the tank. And we’re trying to control the liquid level in this process, to within some sort of constraints, what we’re looking for in the arrest time is the time it takes to
recover from a disturbance change into the system, it effectively sets the response time of the controller. Okay. So if we take a look at our process variable, and set point for a tank, and we’ve got
a constant setpoint, and our levels, bouncing around, we get hit with a disturbance, and it comes back again, we have a controller output, that you know the level is too high. So the tank the pump is
going to increase, and it’s going to come back out again, something along those lines, the arrest time is the time it takes from when the disturbance hits until the process variable starts to recover
arrest that disturbance. This value right here is known as the arrest time of the process.
We also call it closed loop systems on level controllers, the closed loop time constant, okay, tau c or tau CL depending on how you want to write it. All right, this is the time it takes to arrest a
disturbance. How do we calculate that we remember back to the first part of our three-part series, we took a look at the flow rates of the disturbance, and we looked at the constraints in which we
are trying to hold that target setpoint at that process variable at that setpoint. And we calculated how much time it would take for us to hit a constraint when we’re under the worst disturbance.
Okay. And let’s say we ended up with some closed loop time constant of five minutes.
All right, so we’re looking at an arrest time, or closed loop time constant of five minutes. That’s the objective that we’re trying to reach. The second part of our three-part installment on level
control is focused on how to calculate the model of the system. The model of the system
is the integrator gain and the dead time of the system. These tell you how far and how fast the level changes to speeds to move off the balancing point of our output. The dead time is how much delay.
And so we are able to calculate those terms with these terms in the control objective that you’re trying to achieve. We can tune level controllers. Okay.
So there are these things called PID Tuning correlations, that relate to this model. And this closed loop time is constant to the P and I, and D settings. Now, for most level controllers, we actually
don’t want any D, so we’re going to get rid of the derivative. Alright, so how do we calculate the proportional term for a level controller? Well, it’s one over K p star. This is an IMC-style tuning
correlation for level controllers. And it’s going to be two times tau c plus the dead time divided by tau c plus the dead time, that whole bottom part squared. And so you’re left with that. And your
I term is two times the closed loop time constant, plus the dead time.
Now, you’re looking at these saying, Bob, that’s pretty complicated. How can we make this easier, right? And if you have large dead times, these are the equations you want to use, right? But remember
how I told you earlier, dead times for level controllers really aren’t that big. Oftentimes, if your closed loop time is constant, this arrested time here is larger than the dead time. So if your
closed loop time constant, is much larger than your dead time, you can assume that the dead time is not that relevant in the process dynamics. If that’s true, you can set these to zero in the tuning
correlation, and you can end up with a very simple set of first pass tuning correlations for a level controller. That’s essentially tau c over k p star for your P term. And your I term is two times
to tau c. Now, this is where things get really interesting. Notice your eye turn.
Now, these tuning correlations, by the way, are for the dependent form of the PID equation that uses the reset time. So if you’re using a dependent form of the PID equation, that’s using the reset
time, and you’re tuning a level controller, the integral, the reset time that you have here is purely a function of the objective you’re trying to achieve. It’s two times the arrest time, and two
times the closed loop time constant. So if I know I need to get a five-minute arrest time in the system, my reset time is two times that done, right, two times five minutes, 10 minutes, my reset time
for my process, in this case would be 10 minutes, right, I didn’t even have to do a bump test, I didn’t even have to generate a model. It’s purely based on the control objective that you’re trying to
achieve. Now, the P term does include the tank size, right, the dynamics of the system, that KP star, so you take whatever you generate it for your Kp star, whatever you want, for your objective
function, divide tau c divided by that, that’s going to get you your P term. And you’re generally going to get, you know, gains that are above one.
Oftentimes, for level control, your gains are going to be somewhere between, you know, maybe two and four, maybe two and six, something like that, with an average around three or four, right, you
tend to have larger gains, right, and you’re going to have very slow reset times, 10 minutes, five minutes, 20 minutes, 30 minutes, because they’re based off the objective function, and how quickly
you want the tank to move. Tanks, you generally want to move pretty slow. So your reset time is pretty slow. If you’re dealing with reset rate, which is going to be in the numerator, or independent
forms of the PID equation, where it’s not the dependent form, your integral terms tend to be very, very small. So if you’re using an independent form, and Rockwell, for instance, your KPI could be
point 0001 in some cases.
So level controllers tend to be a little bit finicky when tuning, they also tend to be the most often requested loop from customers that we help them tune. Most level controllers tend to be cycling
and oscillating. Oftentimes, because their reset time is much too fast. It’ll be one minute, it’ll be a half a minute, when it should be like 10 or 12 minutes. Once we increase that reset time, slow
down that integration, we’re able to bump up the gain a little bit, get that level to balance out to smooth out. And because oftentimes, these tanks tend to connect pieces of equipment. By tuning
your level controllers, balancing them out, get them to be stable, you can start to decouple your processes and really start to improve the efficiency of your plants. Right.
Level control is a challenging process because it is key and integral to a lot of your applications. In this particular web installment, what we did was we covered the tuning aspect, we understood
what the arrest time of the system was, we understood the tuning parameters, we looked at the equations, and we on the back of an envelope kind of simplified them down to just how to calculate the p
in the I term. Knowing just a little bit about the dynamics of the system, and a little bit about the objective. We’re able to get to our tuning parameters without trial and error. Thank you for
joining us in this level control three-part installment. Hopefully, you learned a little bit more about level control.
If you have a particular topic or an idea that you would like us to cover, please email us at askus@controlstation.com. Thank you, and I hope you enjoy this video series. | {"url":"https://controlstation.com/ask-doctor-bob-what-is-level-control-part-3/","timestamp":"2024-11-12T01:59:05Z","content_type":"text/html","content_length":"156170","record_id":"<urn:uuid:4ba5fcbc-bdc6-40d2-8563-8b71d41a6659>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00226.warc.gz"} |
A Comparison between Metaheuristics as Strategies for Minimizing Cyclic Instability in Ambient Intelligence
Division of Research and Postgraduate Studies, Leon Institute of Technology, Leon, Guanajuato 37290, Mexico
Laboratorio Nacional de Informatica Avanzada, Xalapa, Veracruz 91000, Mexico
School of Computer Science and Electronic Engineering, University of Essex,Wivenhoe Park CO4 3SQ, UK
Author to whom correspondence should be addressed.
Submission received: 3 June 2012 / Revised: 3 July 2012 / Accepted: 10 July 2012 / Published: 8 August 2012
: In this paper we present a comparison between six novel approaches to the fundamental problem of cyclic instability in Ambient Intelligence. These approaches are based on different optimization
algorithms, Particle Swarm Optimization (PSO), Bee Swarm Optimization (BSO), micro Particle Swarm Optimization (μ-PSO), Artificial Immune System (AIS), Genetic Algorithm (GA) and Mutual Information
Maximization for Input Clustering (MIMIC). In order to be able to use these algorithms, we introduced the concept of Average Cumulative Oscillation (ACO), which enabled us to measure the average
behavior of the system. This approach has the advantage that it does not need to analyze the topological properties of the system, in particular the loops, which can be computationally expensive. In
order to test these algorithms we used the well-known discrete system called the Game of Life for 9, 25, 49 and 289 agents. It was found that PSO and μ-PSO have the best performance in terms of the
number of agents locked. These results were confirmed using the Wilcoxon Signed Rank Test. This novel and successful approach is very promising and can be used to remove instabilities in real
scenarios with a large number of agents (including nomadic agents) and complex interactions and dependencies among them.
1. Introduction
Any computer system can have errors and Ambient Intelligence is not exempt from them. Cyclical instability is a fundamental problem characterized by the presence of unexpected oscillations caused by
the interaction of the rules governing the agents involved [1–4].
The problem of cyclical instability in Ambient Intelligence is a problem that has received little attention by the designers of intelligent environments [2,5]. However in order to achieve the vision
of AmI this problem must be solved.
In the literature there are several strategies reported based on analyzing the connectivity among the agents due to their rules. The first one the Instability Prevention System INPRES is based on
analyzing the topological properties of the Interaction Network. The Interaction Network is the digraph associated to the system and captures the dependencies of the rules between agents. INPRES
finds the loops and locks a subset of agents on a loop, preventing them to change their state [1–4]. INPRES has been tested successfully in system with low density of interconnections and static
rules (nomadic devices and time variant rules are not allowed). However when the number of agents involved in the system increases (with high dependencies among them) or when the agents are nomadic,
the approach suggested by INPRES is not practical, due to the computational cost.
Additionally Action Selection Algorithms map the relationship between agents, rules and selection algorithms, into a simple linear system [6]. However this approach has not been tested in real or
simulated scenarios nor nomadic agents.
The approach presented in this paper translates the problem of cyclic instability into a problem of intelligent optimization, moving from exhaustive search techniques to metaheuristics searching.
In this paper we compare the results of Particle Swarm Optimization (PSO), Bee Swarm Optimization (BSO), micro Particle Swarm Optimization (μ-PSO), Artificial Immune System (AIS), Genetic Algorithm
(GA) and Mutual Information Maximization for Input Clustering (MIMIC) when applied to the problem of cyclic instability. These algorithms find a good set of agents to be locked, in order to minimize
the oscillatory behavior of the system. This approach has the advantage that there is no need to analyze the dependencies of the rules of the agents (as in the case of INPRES). We used the game of
life [7] to test this approach, where each cell represents an agent of the system.
2. Cyclic Instability in Intelligent Environments
The scenarios of intelligent environments are governed by a set of rules, which are directly involved in the unstable behavior, and can lead the system to multiple changes over a period of time.
These changes can cause interference with other devices, or unwanted behavior [1–4].
The state of a system s(t) is defined as the logarithm base 10 of the binary vectors of the agents. A turned-on agent was represented with 1 while a shut-down agent will be represented with 0. The
representation of the state of the system is shown in Equation (1):
where s(t) is the state of the system at time t, and s is base-10 representation of the binary number of the state of the system.
3. Optimization Algorithms
3.1. Particle Swarm Optimization
The Particle Swarm Optimization (PSO) algorithm was proposed by Kennedy and Everhart [8,9]. It is based on the choreography of a flock of birds [8–13], wing a social metaphor [14] where each
individual brings their knowledge to get a better solution. There are three factors that influence the change of the particle state or behavior:
• Knowledge of the environment or adaptation is the importance given to past experiences.
• Experience or local memory is the local importance given to best result found.
• The experience of its neighbors or neighborhood memory is the importance given to the best result achieved by their neighbors.
The basic PSO algorithm [9] uses two equations. Equation (2), which is used to find the velocity, describes the size and direction of the step that will be taken by the particles and is based on the
knowledge achieved until that moment.
$v i = wv i + c 1 r 1 ( l Best i − x i ) + c 2 r 2 ( g Best − x i )$
• v[i] is the velocity of the i-th particle, i = 1, 2 …, N,
• N is the number of the population,
• w is the environment adjustment factor,
• c[1] is the memory factor of neighborhood,
• c[2] is memory factor,
• r[1] and r[2] are random numbers in range [0, 1],
• lBest is the best local particle founded for the i-th particle,
• gBest is the best general particle founded until that moment for all particles.
Equation (3) updates the current position of the particle to the new position using the result of the velocity equation.
where x[i] is the position of the i-th particle.
The PSO algorithm [9] is shown in the Algorithm 1:
Algorithm 1 PSO Algorithm.
Data: P ∈ [3, 6] (number of particles), c[1] ∈ R, c[2] ∈ R, w ∈ [0, 1], G (maximum allowed function evaluations).
Result: GBest (best solution found)
• Initialize particles ' position and velocity randomly;
• For g = 1 to G do
□ Recalculate best particles position gBest
□ Select the local best position lBest
□ For each Xig, i = 1, …, P do
☆ Recalculate particle speed
☆ Recalculate particle position
3.2. Binary PSO
Binary PSO [13,14] was designed to work in binary spaces. Binary PSO select the lBest and gBest particles in the same way as PSO. The main difference between binary PSO and normal PSO are the
equations that are used to update the particle velocity and position. The equation for updating the velocity is based on probabilities but these probabilities must be in the range [0, 1]. A mapping
is established for all real values of velocity to the range [0, 1]. The normalization Equation (4) used here is a sigmoid function.
$v ij ( t ) = sigmoid ( v ij ( t ) ) = 1 1 + e − v ij ( t )$
and Equation (5) is used to update the new particle position.
$x ij ( t + 1 ) = { 1 ifr ij < sigmoid ( v ij ( t + 1 ) ) 0 in other case$
where r[ij] is a random vector with uniform values in the range [0, 1].
3.3. Micro PSO
μ-PSO algorithm [15,16] is a modification made to the original PSO algorithm in order to work with small populations (See Algorithm 2). This algorithm use replacement and mutation to be included in
the original PSO algorithm, and allow to the algorithm avoid local optima.
3.4. Bee Swarm Optimization
BSO algorithm [14] is based on PSO and Bee algorithm (See Algorithm 3). It uses a local search algorithm to intensify the search. This algorithm was proposed by Sotelo [14], where the changes made to
PSO allowed finding a new best Global in the current population.
Algorithm 2 μ-PSO Algorithm.
Data: P ∈ [3, 6] (population size). R > 0 (Replacement generation), N > 0 (Number of restart particles), M ∈ [0, 1] (Mutation Rate;, c[1] ∈ R, c[2] ∈ R, w ∈ [0, 1], Neighborhoods > 0, MaxFes (maximum
allowed function evaluations).
Result: GBest (best solution found)
• Initialize particles' position, velocity and neighborhood randomly;
• Set cont = 1 and G = MaxFes/P
• For g = 1 to G do
□ If (cont == R)
☆ Reinitialization ofN worst particles
☆ Set cont = 1
• Recalculate best particles position gBest
• Select the local best position in the neighborhood lBest
• For each Xig, i = 1, …, P do
□ Recalculate particle speed
□ Recalculate particle position
• Perform mutation to each particle with a probability of P(M);
• Set cont = cont + 1
Algorithm 3 BSO Intensifier.
• Generate two new bees looking around to gBest.
• Evaluate the fitness for generated bees in the D variables.
• Select the generated bee with best fitness.
□ If the fitness of selected bee is better than gBest
☆ Set the value and position of gBest as the bee.
□ Otherwise increases gBest age.
Another variant of this algorithm consists in applying the steps in each comparison of a bee with the local best.
In the future we will refer to BSO algorithm with the global best enhancer as BSO1, while BSO algorithm with local the best enhancer will be known as BSO2.
3.5. Artificial Immune System
The Artificial Immune System (AIS) [17] is a metaheuristic based on the Immune System behavior of living things [14], particularly of mammals [17].
One of the main functions of the immune system is to keep the body healthy. A variety of harmful microorganisms (called pathogens) can invade the body. Antigens are molecules that are expressed on
the surface of pathogens that can be recognized by the immune system and are also able to initiate the immune response to eliminate the pathogen cells [17].
Artificial immune systems have various types of model (See Algorithm 4). In our case we use the one that implements the clonal selection algorithm, which emulates the process by which the immune
system, in the presence of a specific antigen, stimulates only those lymphocytes that are more similar [17].
The steps of the algorithm are described below:
Algorithm 4 Artificial Immune System Algorithm.
Data: tam number of antibodies, n antibodies to select, d number of new antibodies, B multiplication factor Beta, Iterations number of executions of the metaheuristic.
Result: sol antibodies
• sol ← Generate(tam) {Initiates the population according to the initial population selected and gains the fitness of each of their antibodies}
• Sort(sol) {Sort antibodies based in fitness}
• For i = 1 to Iterations do
□ sol-best ← Select(sol,n) {Select the n best antibodies}
□ sol-copied← Copy(sol-best,B) {Copy B times the best antibodies}
□ sol-hyper ← Hypermutability(sol-copied) {Hypermutability the copied antibodies}
□ sol-best ← Select(sol-hyper, n) {Select the n best antibodies obtained by hypermutability}
□ sol ← sol+sol-best {Add the best antibodies obtained to antibodies}
□ sol-new ← Generate(d) {d new antibodies are generated according to the initial population selected}
□ sol ← Replace(Sol, sol-new) {Replace d worst antibodies of population with the newest generated antibodies}
□ Sort (sol) {Sort antibodies based on fitness}
□ Resize(sol, tam) {Resize the number of antibodies that have the same number when the metaheuristic begin}
3.6. Genetic Algorithm
The genetic algorithm (GA) [18] is a search technique proposed by John Holland based on the theory of evolution by Darwin [18–20]. This technique is based on the selection mechanisms that nature
uses, according to which the fittest individuals in a population are those who survive, to adapt more easily to changes in their environment.
A fairly comprehensive definition of a genetic algorithm is proposed by John Koza [21]:
“It is a highly parallel mathematical algorithm that transforms a set of individual mathematical objects with respect to time using operations patterned according to the Darwinian principle of
reproduction and survival of the fittest and after naturally have arisen from a series of genetic operations from which highlights the sexual recombination. Each of these mathematical objects is
usually a string of characters (letters or numbers) of fixed length that fits the model of chains of chromosomes and is associated with a certain mathematical function that reflects their ability”.
The GA seeks solutions in the space of a function through simple evolution. In general, the individual fitness of a population tends to reproduce and survive to the next generation, thus improving
the next generation. Either way, inferior individuals can, with a certain probability, survive and reproduce. In Algorithm 5, a genetic algorithm is presented in a summary form [19].
Algorithm 5 Genetic Algorithm.
Data: t (population size), G (maximum allowed function evaluations).
Result: Best Individual (Best Individual of last population).
• P ← Initialize-population(t) {Generate (randomly) an initial population }
• Evaluate(P) {Calculate the fitness of each individual}
• For g = 1 to G do
□ P ' ← Select(P) {Choose the best individuals in the population and pass them to the next generation}
□ P ' ← Cross(P) {Cross population to generate the rest of next population}
□ P ' ← Mutation(P″) {Mutate one individual of population randomly chosen}
□ Evaluate(P ') {Calculate the fitness of each individual of new population}
□ P ← P ') {Replace the old population with new population}
3.6.1. Clones and Scouts
In order to increase performance of the GA the concept of clones and explorers is taken [22]. A clone is one individual whose fitness is equal to the best individual fitness. When it reaches a
certain percentage of clones a percentage of the worst individuals in the population is then mutated. Mutated individuals are named scouts. The application of clones and explorers is in addition to
the mutation carried by the GA generation.
The algorithm to implement and explorers clones is shown in the Algorithm 6 [22].
Algorithm 6 Clones and Scouts Algorithm.
• Define PC (Clones percentage)
• Count the number of individuals who have the same fitness value than the best individual of current population and calculatePAC (Accumulative Percentage of Clones)
• Define PE (Scouts percentage)
• If PAC ≥ PC
□ Select the worstN (PE) individuals of the current population, where N is the size of the population of individuals
□ Mutate selected individuals
4. Mutual Information Maximization for Input Clustering
The Mutual Information Maximization for Input Clustering (MIMIC) [14,23–25] is part of the algorithms known as EDAs (Estimation of Distribution Algorithms). These algorithms aim to get the
probabilistic distribution of a population based on a set of samples, searching for the permutation associated to the lowest value of the Kullback–Leibler divergence (see Equation (6)). This value is
used to calculate the similarity between two different sets of samples:
$H l π = h l ( X in ) + ∑ j = 1 n − 1 h l ( x ij | x ij + 1 )$
• h (x) = − Σ[x] p (X = x) log p (X = x) is Shannon's entropy of X variable
• h (X|Y) = − Σ[x] p (X|Y) log p (Y = y), where
• h (X|Y = y) = − Σ[X] p (X = x|Y = y) log (X = x|Y = y), is the X entropy given Y.
This algorithm suppose that the different variables have a bivariate dependency described by Equation (7).
$p l π ( x ) = p l ( x i 1 | x i 2 ) ▪ p l ( x i 2 | x i 3 ) · … p l ( x in − 1 | x in ) p l ( x in )$
where π = (i1, i2, …, in)is an index permutation.
The algorithm can be seen in Algorithm 7:
Algorithm 7 MIMIC Algorithm.
• Initialize a population (array) of individuals with random values on d dimensions in the problem space
• Select a subpopulation through a selection method
• Calculate Shannon's entropy for each variable.
• Generate a permutation$p l π ( x ) .$
□ Choose variable with lowest entropy.
□ For the next variables choose i[k] = argmin[j]h[l] (X[j]|X[ik][+1]) Where j ≠ i[k]+[i], …,in
• Sample the new population using the generated permutation$p l π ( x ) .$
• Loop until a criterion is met.
5. Test Instances
The Game of Life [7] was used as test instance for the problem of cyclical instability affecting intelligent environments. This is because it has been shown in many cases that the Game of Life system
exhibits oscillatory behavior in spite of the few rules used for this. In addition the environment is extremely simple and may take from a few to a large number of agents acting within the system.
The game of life is a cellular automata created by John Conway [7]. This is a set of cells which, based on a set of simple rules, can live, die or multiply. Depending on the initial conditions, the
cells form different patterns during the course of the game. The rules of the game are as follows [7] :
• Survival: if a cell is in state 1 and has 2 or 3 neighbors in state 1, then the cell remains in state 1
• Birth: if a cell is in state 0 and has exactly 3 neighbors in state 1, the next time step the cell goes to state 1.
• Deaths: a cell in state 1 goes to state 0 if it has 0 or 1 neighbors, or if it has 4 or more neighbors.
From the game of life, we have examples of oscillating behavior. From these we take the follow configurations as benchmark.
Figure 1 presents the simplest known oscillator in the Game of Life called Blinker. This oscillator with 3 alive cells fits neatly into a 3 × 3 grid with 9 potential agents in this scenario.
The settings in Figure 2 was determined randomly and was found to be a stable scenario. This configuration changes during the early stages but later reaches a stable state. The number of cells or
agents in this configuration is 49 since the grid used is 7 × 7.
The configuration presented in Figure 3 is known as Toad. This oscillator is a bit more complex than the Blinker in terms of the number of agents involved or affected by the oscillation. However it
has also a period of 2 like Blinker. This oscillator fits into a grid of 4 × 4, thereby containing 16 agents within the system.
The previous scenarios can be considered as simple configurations, as intelligent environments can contain a larger number of devices or agents involved, and the evaluation of the system is crucial
to determine whether the proposed solution can work with more complex scenarios. In the following examples we will introduce complex configurations called Pulsar and 10 Cell Row shown in Figures 4
and 5. In these configurations there are 289 cells or agents on a 17 × 17 grid, allowing complex behavior and potential oscillations on them.
6. Using Optimization Algorithms to Solve the Problem of Cyclic Instability
In order to solve the problem of cyclic instability using different optimization algorithms we need to minimize the amplitude of the oscillations. In the ideal case this would result in a stable
system. Additionally we are interested on affecting the fewest number of agents (agents locked).
In order to test these approaches we used the Game of Life because it is a well know system that possesses proven oscillatory behavior in very simple environment with simple agents and few rules. For
the test we consider a Game of Life with open boundary conditions. The open boundary condition in our case is considered cold (in terms of the heat entropy) and all cells outside the grid are
considered dead. We enriched the game of life with additional conditions: a list of agents that are allowed to be locked. All techniques can lock them according to their results. This is because
priority agents (such as alarms, security systems, etc.) should not be disabled.
Each solution vector represents the list of blocked agents where the aim is to minimize the Average Cumulative Oscillation (ACO) of the system in a given period of time. The ACO is calculated using
the following Equation (8) [26].
$o = ∑ i = 1 n − 1 | S i − S i + 1 | n − 1$
where o is the Average Cumulative Oscillation, n is the game of life generations, S[i] is the state of the system at the time i, S[i][+1] is the state of the system at the time i + 1.
The best solution should not only minimize the amplitude of oscillation but also the number of agents locked. In these experiments the percentage of agents that can be locked is included as a
parameter. This is important because as this percentage grows the systems becomes more disabled.
In these experiments we consider systems whose adjacency matrix are of 3 × 3, 4 × 4,7 × 7, and 17 × 17. In all the cases the maximum percentage of locked agents set to 20%.
If an algorithm can not find a better solution in terms of the amplitude of the oscillations, no agents will be locked. If a solution is good in terms of the amplitude of the oscillation but the
percentage of locked agents is bigger than the maximum permitted, the solution will be penalized by adding a constant value (in our case the value is 10) to the value of the Average Cumulative
Oscillation (ACO).
In our experiments we set a parameter of 3,000 functions calls as the measure of success of the algorithms, i.e., the system has 3,000 opportunities to find a better solution. If after 3,000
functions calls a better solution is not found, the system is deemed to have failed.
7. Experimental Results
For the test performed with PSO and BSO for all test instances we used the parameters shown in Table 1.
The parameters used for the μPSO are shown in Table 2.
The parameters used for AIS are shown in Table 3.
The parameters used for GA are shown in Table 4.
For the test performed with MIMIC for all test instances we used the parameters shown in Table 5.
Table 6 shows the level of oscillation obtained for each of the instances taken from the Game of Life. The value obtained assumed that the system remain unchanged during its evolution, i.e., from the
initial conditions, the system is allowed to evolve according to the rules of the scenario itself without interference for the entire time of evaluation.
With the parameters described before, we obtain the results shown in Tables 7 and 8. The best result obtained with 100 executions for each test instance is shown.
Table 9 shows the numbers of agents blocked, which correspond to the results obtained with the OAP shown in Tables 7 and 8.
In order to see the cyclic instability and how it is removed, for each test instance we show the evolution of the system before and after locking. In Figure 6 the oscillatory behavior of instance 1
(Blinker) is shown and in Figure 7 the instabilities are successfully removed from the instance 1 (Blinker). In Figure 7 different evolutions can be seen, because for this scenario different sets of
locked agents can achieve system stability.
In the case of instance 2 (Non-oscillating) shown in Figure 8 although the system does not oscillate the techniques were able to find different configurations for this scenario that remain stable
while decrease the value of the Average Cumulative Oscillation. The behavior of instance 2 (non-oscillating) after locking is shown in Figure 9.
As for instance 3 (Toad), the oscillatory behavior shown in Figure 10 looks very similar to instance 1 (Blinker).
In the same way as in instance 1(Blinker), instability was eliminated successfully for instance 3 (Toad) as shown in Figure 10. As shown in Tables 7 and 8, different values of the ACO were obtained
for this scenario because there are different vectors of locked agents that allow to stabilize the system. This explains why different system evolution is shown in Figure 11 after applying the
In Figure 12 the oscillatory behavior of instance 4 (Pulsar) is shown, which is a more complex behavior in relation to those shown above.
While in the above instances the best values obtained by the techniques are very similar, this trend is no longer maintained for the instance 5 (Pulsar). The is because the size of the instance
considerably increases the number of possible combinations.
Most importantly, despite the difference in the results between the various techniques, it was possible to stabilize the system with different combinations of locked agents, showing that depending on
the scenario there may be more than one set of locked agents with which the system can become stable. This is showcased by the different results obtained for the instance 4 (Pulsar) in the level of
instability (refer to Figure 13), where we can see how quickly the system stabilizes in each of the different configurations achieved by the optimization techniques.
The oscillatory behavior of the instance 5 (10 cell row) is shown in Figure 14. It is seen that the oscillation began quickly in contrast to the Instance 4 (Pulsar), whose oscillation is seen until
after a certain period of time.
For the instance 5 (10 cell row), again the number of agents represented is therefore significant. The performance results are similar to those described for instance 4 (Toad) and the best results
obtained by the techniques vary with respect to each other. The behavior without oscillation is shown in Figure 15. The difference between the behavior of each algorithm responds to the different
sets of locked agents by each of the techniques.
Table 10 shows the comparison among the results obtained by different algorithms. To determine whether an algorithm outperforms another, the Wilcoxon signed rank test was performed. The best
performing algorithms are those that exceed a greater number of algorithms in the results obtained for the ACO and the number of locked agents.
Despite the similarity of results of some algorithms based on Table 10, it can be said that the GA was able to obtain smaller values relative to the values of the ACO. But if we take into account the
number of locked agents, the algorithms PSO and μ-PSO are those who achieved best results. This becomes more important because the algorithms PSO and μ-PSO achieve system stability, which makes them
better by allowing most of the devices continue to work normally.
8. Conclusions and Future Work
From our experiments we found that all the algorithms were able to find a vector of locked agents that prevent the system from oscillating. Additionally, Wilcoxon test showed that GA gave better
results but not in terms of time and number of agents locked—rather, PSO and μ-PSO gave the better results in that sense. MIMIC consistently violated the restriction of maximum permitted percentage
of agents locked. MIMIC is based on estimating the distribution of data and for that reason needs a larger number of data (in our case the amount of list of agents locked), and that is the main
reason the time spent for the algorithm to find a solution increased significantly.
An important advantage of this proposal compared to others found in the literature is the way by which the system is evaluated, since it only sees the general state of the environment regardless of
its topology. Because of this, the evaluation time of the scenarios can be significantly reduced regardless of the number of agents that form part of the system. Additionally, the possibility of a
proposal able to work with any scenario can be more clearly distinguished in real time, which helps to improve their operation.
This approach based on the concept of Average Cumulative Oscillation opens the possibility for other algorithms to be applied to the problem of cyclic instability, in general algorithms for discrete
optimization. In particular we are interested in testing this approach for the case of nomadic and weighted agents and with different percentage of locked agents. Also it is possible to improve the
estimation of the oscillation in order to discriminate between stable systems with abrupt changes and systems with small oscillations, because in some cases it is possible to get small values of
Average Cumulative Oscillation in oscillating systems. We hope to report these results in future.
The authors want to thank Jorge Soria for their comments and suggestions to his work. Leoncio Romero acknowledges the support of the National Council for Science and Technology CONACyT. Additionally,
E. Mezura acknowledges the support from CONACyT through project No. 79809.
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Figure 15. Instabilities are successfully removed for the instance 5 (10 cell row) using all algorithms.
Table 1.
Parameters used
in PSO and BSO.
Parameter Value
Particles 45
w 1
c[1] 0.3
c[2] 0.7
Table 2. Parameters of μ-PSO.
Parameter Value
Particles 6
w 1
c[1] 0.3
c[2] 0.7
Replacement generation 100
Number of restart particles 2
Mutation Rate 0.1
Table 3. Parameters of AIS.
Parameter Value
Antibodies 45
Antibodies to select 20
New Antibodies 20
Factor Beta 2
Table 4. Parameters of GA.
Parameter Value
Chromosomes 45
Mutation percentage 0.15
Elitism 0.2
Clones percentage 0.3
Scouts percentage 0.8
Table 5. Parameters
used in MIMIC.
Parameter Value
Individuals 100
Elitism 0.5
Table 6. Instance.
Instance Matrix # of Agents S[0] S[f] ACO
1 (Blinker) 3 × 3 9 1.748188027 1.7481880270062005 0.416164
2 (Non-oscillating) 7 × 7 49 12.284241189 0.0 0.196199
3 (Toad) 4 × 4 16 3.6261349786353887 3.6261349786353887 0.974857
4 (Pulsar) 17 × 17 289 54.30350121388617 75.43697161100698 2.916489
5 (10 cell row) 17 × 17 289 44.85304503100554 50.04149156221928 2.23957
Table 7. ACO Results (A).
Instance Average Cumulative Oscillation
PSO MIMIC BSO1 BSO2
1 (Blinker) 0.0173 0.4161648288 0.0173 0.0173
2 (Non-oscillating) 0.0036684768 10.0040328723 0.0013180907 6.96E–4
3 (Toad) 0.0027436818 10.0005786421 0.0031827703 0.0027436818
4 (Pulsar) 0.0039470374 10.000434825 0 0
5 (10 cell row) 0.0461757189 10.040107898 1.58E–7 0
Table 8. ACO Results (B).
Instance Average Cumulative Oscillation
AIS GA μPSO
1 (Blinker) 0.0173 0.0173 0.0173
2 (Non-oscillating) 0.00239 4.77E–6 4.77E–6
3 (Toad) 0.00253 0.00253 0.0025307973
4 (Pulsar) 6.12E–4 5.24E–13 4.3E–4
5 (10 cell row) 0.0472 1.60E–10 0.0256051082
Table 9. Locked Agents.
Instance # of Locked Agents
Allow PSO BSO1 BSO2 AIS GA μ-PSO MIMIC
1 (Blinker) 1 1 1 1 1 1 1 0
2 (Non-oscillating) 9 1 9 8 4 12 8 19
3 (Toad) 5 3 3 3 3 3 3 9
4 (Pulsar) 57 26 43 56 31 53 28 141
5 (10 cell row) 57 31 38 44 33 57 18 139
Table 10. Comparison among algorithms based on Wilcoxon
Algorithm Number of Algorithms
by ACO Value by # of locked Agents
Overcome Not Overcome Overcome Not Overcome
PSO 1 5 5 1
BSO1 2 4 3 3
BSO2 3 3 3 3
μPSO 5 1 6 0
AIS 4 2 4 2
GA 6 0 1 5
MIMIC 0 6 0 6
© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://
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MDPI and ACS Style
Romero, L.A.; Zamudio, V.; Baltazar, R.; Mezura, E.; Sotelo, M.; Callaghan, V. A Comparison between Metaheuristics as Strategies for Minimizing Cyclic Instability in Ambient Intelligence. Sensors
2012, 12, 10990-11012. https://doi.org/10.3390/s120810990
AMA Style
Romero LA, Zamudio V, Baltazar R, Mezura E, Sotelo M, Callaghan V. A Comparison between Metaheuristics as Strategies for Minimizing Cyclic Instability in Ambient Intelligence. Sensors. 2012; 12
(8):10990-11012. https://doi.org/10.3390/s120810990
Chicago/Turabian Style
Romero, Leoncio A., Victor Zamudio, Rosario Baltazar, Efren Mezura, Marco Sotelo, and Vic Callaghan. 2012. "A Comparison between Metaheuristics as Strategies for Minimizing Cyclic Instability in
Ambient Intelligence" Sensors 12, no. 8: 10990-11012. https://doi.org/10.3390/s120810990
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[QSMS Symplectic geometry seminar-20240812] Exact Lagrangian tori in symplectic Milnor fibers constructed with fillings
• Date: 2024-08-12 (Mon) 14:00 ~ 15:00
• Place: 129-406 (SNU)
• Speaker: Orsola Capovilla-Searles (UC Davis)
• Title: Exact Lagrangian tori in symplectic Milnor fibers constructed with fillings
• Abstract: Weinstein domains are an important class of symplectic 4-manifolds with contact boundary that include disk cotangent bundles, and complex affine varieties. They are Liouville domains
with a compatible Morse function whose symplectic topology is described by a corresponding handle body decomposition. A Weinstein handlebody decomposition can be encoded by a collection of
Legendrian links in the boundary of the connected sum of $S^1times S^2$ with the standard contact structure. Thus, for Weinstein 4-manifolds with explicit handlebody decompositions, the study of
their symplectic topology can be reduced to the study of Legendrian links. | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&order_type=asc&sort_index=title&l=en&listStyle=gallery&page=4&document_srl=2822","timestamp":"2024-11-12T02:11:37Z","content_type":"text/html","content_length":"71272","record_id":"<urn:uuid:99646f45-6522-44be-984f-61646af10399>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00884.warc.gz"} |
PROJECT 1, Com S 228 solved
1 Problem Description
This project simulates interactions among different forms of life in a plain. The plain is represented
by an N × N grid that changes over a number of cycles. Within a cycle, each square is occupied
by one of the following five life forms:
Badger (B), Fox (F), Rabbit (R), Grass (G), and Empty (E)
An Empty square means that it is not occupied by any life form.
Below is a plain example as a 6 × 6 grid.
F5 E E F0 E E
B3 F1 B0 R0 G R0
R0 E R2 B0 B2 G
B0 E E R1 F0 E
B1 E E G E R0
G G E B0 R2 E
Both row and column indices start from 0. In the example, the (1, 1)th square is occupied by a
1-year-old fox. It has a 3 × 3 neighborhood centered at the square:
F5 E E
B3 F1 B0
R0 E R2
The (0, 0)th square F5 (a 5-year-old fox) has only a 2 × 2 neighborhood:
F5 E
B3 F1
Meanwhile, the (2, 0)th square R0 (a newborn rabbit) has a 3 × 2 neighborhood:
B3 F1
R0 E
B0 E
Generally, the neighborhood of a square is a 3 × 3 grid which includes only those squares lying
within the intersection of the plain with a 3 × 3 window centered at the square. When a square is
on the border, the dimension of its neighborhood reduces to 2 × 3, 3 × 2, or 2 × 2.
2 Survival Rules
The plain evolves from one cycle to the next. In the next cycle, the life form to reside on a square
is decided from those life forms in the current cycle who live in the 3 × 3 neighborhood centered
at the square, under a set of survival rules. These rules are specified according to the life form
residing on the same square in the current cycle. Badgers, foxes, and rabbits start at age 0, and
grow one year older when the next cycle starts.
2.1 Badger
A badger dies of old age or hunger, or from a group attack by foxes when it is alone. The life form
on a Badger square in the next cycle will be
a) Empty if the Badger is currently at age 4;
b) otherwise, Fox, if there is only one Badger but more than one Fox in the neighborhood;
c) otherwise, Empty, if Badgers and Foxes together outnumber Rabbits in the neighborhood;
d) otherwise, Badger (the badger will live on).
The new life form taking over the square, if a Fox, will have age 0 when the next cycle starts.
For example, in the following neighborhood of a Badger at age 2:
R0 G R0
B0 B2 G
R1 F0 E
there are two Badgers (including this), one Fox, and three Rabbits. Going down the rule list,
neither a), b), nor c) applies. According to rule d), the central element (square) will still be a
Badger — just one year older — in the next cycle. In other words, B2 will be replaced with
2.2 Fox
A fox dies of old age, hunger, or an attack by more numerous badgers. The life form on a Fox
square in the next cycle will be
a) Empty if the Fox is currently at age 6;
b) otherwise, Badger, if there are more Badgers than Foxes in the neighborhood;
c) otherwise, Empty, if Badgers and Foxes together outnumber Rabbits in the neighborhood;
d) otherwise, Fox (the fox will live on).
The new life form, if a Badger, will have age 0 when the next cycle begins.
For example, in the following neighborhood of a Fox at age 1:
F5 E E
B3 F1 B0
R0 E R2
there are two Foxes, two Badgers, and two Rabbits. Rule c) applies, so the central square will
become E in the next cycle.
2.3 Rabbit
A rabbit dies of old age or hunger. It may also be eaten by a badger or a fox. More specifically,
the life form on a Rabbit square in the next cycle will be
a) Empty if the Rabbit’s current age is 3;
b) otherwise, Empty if there is no Grass in the neighborhood (the rabbit needs food);
c) otherwise, Fox if in the neighborhood there are at least as many Foxes and Badgers combined
as Rabbits, and furthermore, if there are more Foxes than Badgers;
d) otherwise, Badger if there are more Badgers than Rabbits in the neighborhood;
e) otherwise, Rabbit (the rabbit will live on).
If the new life form is a Badger or Fox, it will have age 0 when the next cycle starts.
In the following neighborhood of a rabbit at age 2:
F1 B0 R0
E R2 B0
E E R1
there are two Badgers, one Fox, and three Rabbits. Rule a) does not apply because the Rabbit is
only two years old. Rule b) does since there is no Grass in the neighborhood. The central element
(square) will be E in the next cycle according to this rule.
2.4 Grass
Grass may be eaten out by overcrowded rabbits. Rabbits may also multiply fast enough to take
over the Grass square. In the next cycle, the life form on a Grass square will be
a) Empty if at least three times as many Rabbits as Grasses in the neighborhood;
b) b) otherwise, Rabbit if there are at least three Rabbits in the neighborhood;
c) otherwise, Grass.
If the new life form is a Rabbit, it will be have age 0 when the next cycle starts.
For example, if the neighborhood of a Grass is
F0 E E
R0 G R0
B0 B2 G
the central element will be G in the next cycle under rule c).
2.5 Empty
The life form on an Empty square in the next cycle will be
a) Rabbit, if more than one neighboring Rabbit;
b) otherwise, Fox, if more than one neighboring Fox;
c) otherwise, Badger, if more than one neighboring Badger;
d) otherwise, Grass, if at least one neighboring Grass;
e) otherwise, Empty.
If the new life form is a Badger, Fox, or Rabbit, it will have age 0 when the next cycle begins.
For example, suppose an Empty square in the top row has the following neighborhood:
F0 E E
R0 G R0
which includes two Rabbits. Thus, rule a) applies to change the central element to R0 in the next
3 Task
You will implement an abstract class Living to represent a generic life form. It has three subclasses:
Animal, Empty, and Grass. The first subclass, implementing the interface MyAge, is abstract, and
needs to be extended to three subclasses: Badger, Fox, and Rabbit. You also need to implement
a Plain class which has a public member Living[][] to represent a grid plain.
The class Wildlife repeatedly simulates evolutions of input plains, either randomly generated or
read from files. In each iteration, it interacts with the user who chooses how the plain will be
generated, and enters the number of cycles to simulate. The iteration prints out the initial plain
and the final plain.
Your random plain generator may follow the uniform probability distribution so that Badger, Empty,
Fox, Grass, and Rabbit have equal likelihoods to occupy every square. Or you may use a different
distribution, as long as no life form has zero chance to appear on a square.
Java provides a random number generator. To use it, you need to import the package java.util.Random.
Next, declare and initiate a Random object:
Random generator = new Random();
Then, every call generator.nextInt(5) will generate a random number between 0 and 4 that
corresponds to one of the five life forms.
When zero or a negative number of cycles is entered by the user, your code does nothing but waits
for a positive input.
A new Badger, Fox, or Rabbit has age 0 at its creation, whether it is created initially by the class
Plain or later on under a survival rule. After surviving a cycle, its age increases by one.
Templates are provided for all classes. Be sure to use the package name edu.iastate.cs228.hw1
for the project.
Below is a sample simulation scenario over three initial plains. In the first iteration, the user entered
1 for a randomly generated plain, 3 to specify the grid to be 3×3, and 1 to simulate just one cycle.
The simulator printed out the initial and final plains. The second iteration simulated a randomly
generated 6×6 grid over eight cycles. In the third iteration, the user typed 2 for a file input, entered
the file name “public3-10×10.txt”, and specified six cycles. (The file public3-10×10.txt resides
in the same folder containing the src folder.) After the third iteration, the user typed 3 to end the
simulation. (Any number other than 1 and 2 would have ended the simulation.)
Simulation of Wildlife of the Plain
keys: 1 (random plain) 2 (file input) 3 (exit)
Trial 1: 1
Random plain
Enter grid width: 3
Enter the number of cycles: 1
Initial plain:
R0 R0 B0
G G E
G E G
Final plain:
R1 R1 B1
G G G
G G G
Trial 2: 1
Random plain
Enter grid width: 6
Enter the number of cycles: 8
Initial plain:
E E G R0 B0 G
G G R0 B0 F0 R0
G E E R0 G E
R0 G R0 R0 B0 R0
F0 E R0 G R0 F0
R0 R0 F0 F0 G F0
Final plain:
R0 E R0 E B3 G
E E E R0 R1 R3
R0 R0 R0 E E R0
E E E E R0 G
R0 E E R0 G G
E R0 R0 F1 G G
Trial 3: 2
Plain input from a file
File name: public3-10×10.txt
Enter the number of cycles: 6
Initial plain:
B0 E B0 E B0 R0 E R3 E G
G E B0 E F0 R0 E B4 G G
G G G G E E R0 E G G
F0 E G G E R0 R0 B0 B0 G
F0 F1 E E E E E E B0 E
G G R1 R0 R0 R0 R0 B0 B0 E
E G R0 R1 R2 R2 G E G G
B0 B0 G R0 R0 R0 G B0 E G
E G G F4 R2 R0 E G G G
G G E E E G G G G G
Final plain:
B0 E B0 E E R0 E R0 R2 E
G E B0 R0 B4 R0 E R0 R3 R1
G G R2 R0 E R0 R0 E E E
G F5 R3 R0 E R0 R0 E E R0
R2 E E E R0 R0 E E B1 G
R0 R0 R0 R0 R0 E B1 R0 G G
E E R0 E R0 E B1 R0 G G
B4 E R0 E R0 E E E R0 G
G R2 R3 E R0 E R0 R3 R1 G
G R0 R1 E R0 E R0 R2 R0 G
Trial 4: 3
Your code should print out the same text messages for user interactions.
4 Input/Output Format
The format for plains is shown in the sample runs above. Every square occupies two spaces starting
with one of the letters B, E, F, G, or R. If the letter is B, F, or R, then it is followed by a digit
representing the animal’s age; otherwise, it is followed by a blank.
There is exactly one blank between two squares, whether represented by a letter and a digit, or a
letter and a blank. No blank lines.
You may assume all input files to be correctly formatted, containing B, E, F, G, R, and digits up to
6 as the only non-blank characters. No digit will exceed the lifespan of the animal represented by
its preceding letter.
5 Submission
Write your classes in the edu.iastate.cs228.hw1 package. Turn in the zip file, not your class
files. Follow the Submission Guide posted on Canvas. Include the Javadoc tag @author in each
class source file. Your zip file should be named Firstname_Lastname_HW1.zip | {"url":"https://codeshive.com/questions-and-answers/project-1-com-s-228-solved/","timestamp":"2024-11-14T07:19:14Z","content_type":"text/html","content_length":"115216","record_id":"<urn:uuid:56c431d8-3f16-4159-939d-968c6fed2a5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00269.warc.gz"} |
How does database fragmentation scoring work in MetaScope?
To score database fragmentation matches, we use an algorithm based on the well adopted cosine similarity method. A similar method is, for example, implemented by MassBank [pdf].
Cosine similarity method
The dot product of two 2-dimensional vectors, ${\bf x} = x_1 {\bf i} + x_2 {\bf j}$ and ${\bf y} = y_1 {\bf i} + y_2 {\bf j}$ is:
It can also be expressed as:
Where $\theta$ is the angle between the two vectors, and $|{\bf x}| = \sqrt{x_1^2 + x_2^2}$.
By equating these two formulae, the "similarity" between the two vectors is given by the cosine of the angle between them, which has the nice property that it ranges from 0 to 1 when all
co-efficients are positive:
This method can also be expanded to n-dimensional vectors:
A similarity of 1 means the two vectors are identical, and a similarity of 0 means they are orthogonal and independent of each other.
Cosine similarity method applied to ms/ms scoring
We apply this method to scoring of ms/ms database matches as follows.
We create two vectors ${\bf E}$ and ${\bf D}$, where each element of the vector is a weighted peak intensity given by:
We combine all m/z's of peaks from the experimental and database spectra, and go through them in ascending m/z order. For each m/z, there are 3 possibilities:
1. There is an experimental peak at the given m/z, but no matching database peak.
2. There is a database peak at the given m/z, but no matching experimental peak.
3. There is an experimental peak at the given m/z, and a database peak at the same m/z (to within a threshold).
For each of these scenarios, we add elements to the vectors ${\bf E}$ and ${\bf D}$ as follows:
1. We add the weighted experimental peak intensity to ${\bf E}$ and a 0 to ${\bf D}$.
2. We add a 0 to ${\bf E}$ and the weighted database peak intensity to ${\bf D}$.
3. We add the weighted experimental peak intensity to ${\bf E}$ and the weighted database peak intensity to ${\bf D}$.
Finally, we calculate the similarity metric on ${\bf E}$ and ${\bf D}$ as defined above. To obtain a score between 0 and 100, we multiply this result by 100.
To illustrate this method, suppose we have the following experimental and database spectra:
In this case, the two vectors produced are as follows (where $W(m,i) = m^2 \sqrt{i}$ is the weighted intensity function):
The similarity metric is then:
So these two spectra will be given a fragmentation score of ~93 - they are fairly well matched, but there are a few peaks which are either not matched, or not expected to be present, lowering its
See also | {"url":"https://www.nonlinear.com/progenesis/qi/v1.0/faq/database-fragmentation-algorithm.aspx","timestamp":"2024-11-14T17:01:38Z","content_type":"text/html","content_length":"20407","record_id":"<urn:uuid:fcb66ab4-2635-4b15-a039-f38b46228811>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00604.warc.gz"} |
ARCHIVE - Week 6
Mon, May 3rd
9:00-10:15 Calculus
10:30-12:00 Physics
Room: Lecture Hall 2
Calculus Topic: Integration by parts
Read Callahan 11.3
Physics Topic: Magnetic Force
Read Serway and Jewett 22.1-22.6
Optional Tutor Session
Calculus help in Lab 1 1047
Python help in the CAL
Wed, May 5th
Calculus Workshop/Lab
Room: Lab1 1047
Calculus Homework Due Today:
Section 11.2 Ex: 1(k),(n),(o),3 4,5,6,8
In class Test Corrections
Calculus Topic: Separation of variables
Read Callahan 11.4
Optional Tutor Session
Room: Lab1 1047
First Hour: Calculus Help with Angelia
Second Hour: Physics Help with Carlos
Thur, May 6th
Physics Lecture/Workshop
Room: Lab1 1047
Computing Lecture
Room: Lecture Hall 2
Physics Homework Due Today:
S&J Chapter 21, pg 794
Challenge 36,50
Physics Topic:
Magnetic Field
S&J Sections: 22.7-22.1
Computing Topic: | {"url":"https://archives.evergreen.edu/webpages/curricular/2003-2004/modelingmotion/week6.htm","timestamp":"2024-11-15T01:31:19Z","content_type":"text/html","content_length":"3575","record_id":"<urn:uuid:afb6b9a5-dcf7-4c62-8e2f-e376d7763615>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00421.warc.gz"} |
Choosing the suitable pole – errors and misconceptions – part 1 the length of the pole
This article about the main instrument in pole vaulting – the pole – contains three parts. This article wants to teach the basics about the implement the “vaulting pole.”
On the top of each pole there are two numbers; one tells the length of the pole the other one is a weight index. We first have a lock at the length.
UCS Spirit Pole; length 4.60 / 15 ft. 1 inch
Pacer One Pole; Length 3.55m
Poles are usually produced in the length 3.70m, 4.00m, 4.30m, 4.60m, 4,90m. The first producers of vaulting poles had their base in the U.S., that is why the length increases in steps of a foot
(30.48cm): 12 feet, 13 feet, 14 feet and so on. Calculated exactly, feet do not equal European measurement (for example 15 ft. is not exactly 4.60m) and poles are not exactly as long as you are told.
Sometimes it is a centimetre more sometimes one less. Producing poles isn’t rocket science. If car engineers would fabricate vaulting poles, there would be tolerances like you know from golf-sticks
and they would cost a fortune.
During the past 15 years there came up poles in new, intermediate length. First for pro’s only, then for everybody. The rise of women’s pole vault showed the need for smaller steps between poles. New
poles in length of 4.15m, 4.45 and 4.75 where available. Today 15cm-increasement (1/2 feet respectively) is usual: 11’6”, 12’, 12’6”, 13’, 13”6 and so on. On every pole there is written the length in
American and European units.
UCS/Spirit pole; length 4.30m / 14 ft.
What do you have to look for regarding the length of a pole?
I could write a book about that. In short: Jumping with a bending of the pole, one should – at least in competition – choose the length of the pole regarding that his top hand is not lower than 30cm
from the top end (the length of the pole follows the grip height not inversely!). The reason for this conclusion will be content of another article. | {"url":"http://www.stabhoch.ch/?page_id=753","timestamp":"2024-11-02T12:40:37Z","content_type":"text/html","content_length":"80384","record_id":"<urn:uuid:6b947ba1-5796-437a-80eb-891a3322578d>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00802.warc.gz"} |
Discounting Cashflow Wizardry
Editor’s Note: This post first appeared in Make Change magazine, an online personal finance site with a social conscience.
I don’t believe it’s an actual conspiracy of silence to keep us in the dark about our finances, but sometimes it feels that way to me. That’s partly why I wrote The Financial Rules For New College
Graduates, because I’m convinced that learning how to do some not-too-sophisticated math in a spreadsheet could go a long way toward demystifying finance for the non-finance professional.
The skill of discounting cashflows is the fundamental tool of all investing. It answers important questions like:
• What is the fundamental value of a stock, based on a model of future projections of profit?
• What is the value of an annuity?
• What is the cost, in today’s dollars, of future government obligations like Social Security and Medicare?
• What is the right choice for a retiree to make, choosing between and lifetime pension and a lump sum?
• What is the right choice for a lottery winner, between annual payments and a lump sum? ^1
You may not want to always do the math on these questions, but if you learn how it works you have a much better shot at pulling back the curtain on supposedly complex financial mysteries.
Watch this first video, for example, to see how we would build a discounting cashflow calculator in a spreadsheet.
Learning the math
It’s not hard, and if you’ve learned compound interest, then it’s kind of a snap. See for example earlier posts on Be a Compound Interest Wizard Part I, and Be a Compound Interest Wizard Part 2.
But it does involve math and poking around with a spreadsheet.
[Begin Book Excerpt]
So what is it?
Discounting cash flows – in the simplest mathematical sense – is just the opposite action to compound interest.
Specifically, the discounting cash flows formula tells us how a certain known amount of money in the future (FV) can be ‘discounted’ back to a certain known amount in the present (FV) through the
intervention of an interest rate (Y) and multiple compounding periods (N).
Notice that we use the exact same variables in both formulas. Notice, also, that the only difference mathematically is that we’re solving for a different number.
The discounted cashflow formula simply reverses the algebra of the compound interest formula.
The discounted cashflow formula solves for Present Value, so that:
PV = FV / (1+Y)^N
So why do we care about discounting cashflow?
A simplified example should help to get us started.
A builder’s insurance company offers you a $25,000 lump sum payment to compensate you for the pain and hardship of an injured pet hit by an errant beam that fell from his construction site.
Picture a big piece of wood, it hurt the dog’s paw, the dog will likely make a full recovery, but the developer/builder offered you this settlement to avoid a costly lawsuit with bad
public-relations potential.
Importantly, however, the settlement will be paid out 10 years from now. Note, by the way, that this is common practice in injury-settlement cases. Lump sums get offered far into the future. This
is partly because such agreements incentivize the victim/beneficiary to comply with the terms of the settlement for the longest period of time. But also importantly, as we will see, it’s much
cheaper for the insurance company to make payments deep into the future.
Now, back to the math.
Let’s assume the insurance company is a very safe, stable, company, and we expect moderate inflation, so the proper Y, or discount rate for the next 10 years, is around 3%.
How much is that settlement worth to us today?
Let’s go to the spreadsheet.
We set up our formula in a spreadsheet that the value today, or Present Value (PV) is equal to FV/ (1+Y)^N.
We know the future payout, FV, is $25,000.
We know how many years we have to wait, so N is 10.
We’ve assumed a Y of 3%.
The present value will be equal to $25,000/ (1+3%)^10.
This is easy-peasy math for your spreadsheet, which tells us the present value is $18,602.
What does this mean in practice? We’re not going to ‘invest’ $18,602 in this future $25,000 insurance payout, but it can be very helpful for us to understand that the future $25,000 payment
really only costs the insurance company about 75% of what it first appears to cost
By the way, how did I come up with 3%?
Frankly and honestly, I made up the 3% for the example.
I don’t just say ‘I made it up’ to be flippant. I mean to emphasize that ‘I made it up’ because ‘making up’ Y, or the proper yield or discount rate (remember, those mean the same thing!) is a key
to effectively using the discounted cash flows formula.
In fact, any time you discount cash flows, you have to “make up,” or assume, a certain Y or discount rate, and the Y assumption you use is as much art as science.
Is that 3% Y I assumed “correct?”
I don’t know, but it’s reasonable, and that’s usually the most we can say about any assumed Y. How do we come up with a reasonable Y number?
Y as an interest rate or discount rate (remember: same thing!) reflects a combination of
1. a) the market cost of money, which is often called an ‘interest rate,’
2. b) the expectation of inflation in the future, and
3. c) the risk of the payment actually being made in the future.
Only some of these things can be known at any time, so only some of our Y is scientifically knowable. The rest has to be assumed according to best estimates. That’s why we can reasonably say that
sometimes this Y assumption is as much art as science.
[End book excerpt]
If you want to take it to the next level of how discounting cashflows is used in practice among investment professionals, you could set up your spreadsheet to discount a series of cashflows. Setting
up formulas to discount a whole series of cashflows is how we build a model for valuing bonds, for example, or fundamental pricing of stocks.
This video explains the basics for setting up a series of discounting cashflows.
If you decide later become a complete discounting cashflows ninja, you’d then want to layer in one more additional level of complexity, by discounting cashflows that happen more than once a year. A
simple video introducing how to deal with that is here:
The last thing I would say about this is that while you don’t have to learn this math in order to manage your money right, I think its useful to know what the Wizards of Wall Street are up to. If you
understand their tools, you’re more likely to ask a financial guru a hard question, like:
“Um, why do charge so much, when this doesn’t seem that complex?”
Post read (252) times.
1. Important personal finance PSA: Never Play The Lottery! ↩ | {"url":"https://www.bankers-anonymous.com/blog/learn-to-be-a-discounting-cashflow-wizard-part-3-with-book-excerpt/","timestamp":"2024-11-02T06:05:34Z","content_type":"text/html","content_length":"131415","record_id":"<urn:uuid:1c4d314b-f13a-42ef-abaf-32a75dca99aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00068.warc.gz"} |
UPSC-IFS Physics Mains Syllabus | Complete Paper 1 & 2 | - AGRI TUTORIALS
UPSC-IFS Physics Mains Syllabus | Complete Paper 1 & 2 |
Physics Syllabus Paper – I
1. Classical Mechanics
(a) Particle dynamics: Centre of mass and laboratory coordinates, conservation of linear and angular momentum, The rocket equation, Rutherford scattering, Galilean transformation, inertial and
non-inertial frames, rotating frames, centrifugal and Coriolls forces, Foucault pendulum.
(b) System of particles : Constraints, degrees of freedom, generalised coordinates and momenta.Lagrange’s equation and applications to linear harmonic oscillator, simple pendulum and central force
problems. Cyclic coordinates, Hamiltonian Lagrange’s equation from Hamilton’s principle.
(c) Rigid body dynamics : Eulerian angles, inertia tensor, principal moments of inertia. Euler’s equation of motion of a rigid body, force-free motion of a rigid body, Gyroscope.
2. Special Relativity, Waves & Geometrical Optics :
(a) Special Relativity : Michelson-Morley experiment and its implications, Lorentz transformationslength contraction, time dilation, addition of velocities, aberration and Doppler effect, mass energy
relation, simple application to a decay process, Minkowski diagram, four dimensional momentum vector. Covariance of equations of physics.
(b) Waves: Simple harmonic motion, damped oscillation, forced oscillation and resonance, Beats, Stationary waves in a string. Pulses and wave packets. Phase and group velocities. Reflection and
Refraction from Huygens’ principle.
(c) Geometrical Optics : Laws of reflection and refraction from Format’s principle. Matrix method in paraxial optic-thin-lens formula, nodal planes, system of two thin lenses, chromatic and spherical
UPSC IFS Physics Syllabus
3. Physical Optics :
(a) Interference : Interference of light-Young’s experiment, Newton’s rings, interference by thin films, Michelson interferometer. Multiple beam interference and Fabry-Perot interferometer.
Holography and simple applications.
(b) Diffraction : Fraunhofer diffraction-single slit, double slit, diffraction grating, resolving power. Fresnel diffraction:- half-period zones and zones plates. Fersnel integrals. Application of
Cornu’s spiral to the analysis of diffraction at a straight edge and by a long narrow slit. Deffraction by a circular aperture and the Airy pattern.
(c) Polarisation and Modern Optics : Production and detection of linearly and circularly polarised light. Double refraction, quarter wave plate. Optical activity. Principles of fibre optics
attenuation; pulse dispersion in step index and parabolic index fibres; material dispersion, single mode fibres.Lasers-Einstein A and B coefficients, Ruby and He-Ne lasers. Characteristics of laser
light-spatial and temporal coherence. Focussing of laser beams. Three-level scheme for laser operation.
4. Electricity and Magnetism:
(a) Electrostatics and Magneto-statics : Laplace and Poisson equations in electrostatics and their applications. Energy of a system of charges, multiple expansion of scalar potential. Method of
images and its applications. Potential and field due to a dipole, force and torque on a dipole in an external field.Dielectrics, polarisation, Solutions to boundary-value problemsconducting and
dielectric spheres in a uniform electric field. Magnetic shell, uniformly magnetised sphere.Ferromagnetic materials, hysteresis, energy loss.
(b) Current Electricity: Kirchhoff’s laws and their applications, Biot- Savart law, Ampere’s law, Faraday’s law, Lenz’ law. Self and mutual inductances. Mean and rms values in AC circuits, LR, CR and
LCR circuits-series and parallel resonance, Quality factor, Principle of transformer.
5. Electromagnetic Theory & Black Body Radiation :
(a) Electromagnetic Theory : Displacement current and Maxwell’s equations. Wave equations in vacuum, Poynting theorem, Vector and scalar potentials, Gauge invariance, Lorentz and Coulomb gauges,
Electromagnetic field tensor, covariance of Maxwell’s equations. Wave equations in isotropic dielectrics, reflection and refraction at the boundary of two dielectrics. Fresnel’s relations, Normal and
anomalous dispersion, Rayleigh scattering.
(b) Blackbody radiation: Blackbody radiation ad Planck radiation law-Stefan-Boltzmann law, Wien displacement law and Rayleigh-Jeans law, Planck mass, Planck length, Planck time, Plank temperature and
Planck energy.
6. Thermal and Statistical Physics :
(a) Thermodynamics: Laws of thermodynamics, reversible and irreversible processes, entropy, Isothermal, adiabatic, isobaric, isochoric processes and entropy change, Otto and Diesel engines, Gibbs’
phase rule and chemical potential. Van der Waals equation of state of real gas, critical constants. Maxwell-Boltzman distribution of molecular velocities, transport phenomena, equipartition and
virial theorems, Dulong-Petit, Einstein, and Debye’s theories of specific heat of solids. Maxwell relations and applications. Clausius-Clapeyron equation. Adiabatic demagnetisation, Joule-Kelvin
effect and liquefication of gases.
(b) Statistical Physics: Saha ionization formula, Bose-Einstein condensation, Thermodynamic behaviour of an ideal Fermi gas, Chandrasekhar limit, elementary ideas about neutron stars and pulsars,
Brownian motion as a random walk, diffusion process. Concept of negative temperatures.
UPSC IFS Physics Syllabus | Complete Paper 1 & 2 |
Physics Syllabus Paper-II
1. Quantum Mechanics I: Wave-particle duality. Schroedinger equation and expectation values. Uncertainty principle, Solutions of the onedimensional Schroedinger equation free particle (Gaussian
wave-packet), particle in a box, particle in a finite well, linear, harmonic oscillator, Reflection and transmission by a potential step and by a rectangular barrier, use of WKB formula for the
life-time calculation in the alphadecay problem.
2. Quantum Mechanics II & Atomic Physics :
(a) Quantum Mechanics II : Particle in a three dimensional box, density of states, free electron theory of metals, The angular momentum problem, The hydrogen atom, The spin half problem and
properties of Pauli spin matrices.
(b) Atomic Physics : Stern-Gerlack experiment, electron spin, fine structure of hydrogen atom, L-S coupling, J-J coupling, Spectroscopic notation of atomic states, Zeeman effect, Frank-Condon
principle and applications.
3. Molecular Physics : Elementary theory of rotational, vibrational and electronic spectra of diatomic molecules, Raman effect and molecular structure, Laser Raman spectroscopy importance of neutral
hydrogen atom, molecular hydrogen and molecular hydrogen ion in astronomy Fluorescence and Phos-phorescence, Elementary theory and applications of NMR. Elementary ideas about Lamb shift and its
4. Nuclear Physics : Basic nuclear properties-size, binding energy, angular momentum, parity, magnetic moment, Semi-empirical mass formula and applications, Mass parabolas, Ground state of deuteron
magnetic moment and non-central forces, Meson theory of nuclear forces, Salient features of nuclear forces, Shell model of the nucleus-success and limitations, Violation of parity in beta decay,
Gamma decay and internal conversion, Elementary ideas about Mossbauer spectroscopy, Q-value of nuclear reactions, Nuclear fission and fusion, energy production in stars, Nuclear reactors.
5. Particle Physics & Solid State Physics:
(a) Particle Physics: Classification of elementary particles and their interactions, Conservation laws, Quark structure of hadrons. Field quanta of electro-weak and strong interactions. Elementary
ideas about Unification of Forces, Physics of neutrinos.
(b) Solid State Physics : Cubic crystal structure, Band theory of solids-conductors, insulators and semiconductors, Elements of superconductivity, Meissner effect, Joseph-son junctions and
applications, Elementary ideas about high temperature superconductivity.
6. Electronics: Intrinsic and extrinsic semiconductors-pn- p and n-p-n transistors, Amplifiers and oscillators, Op-amps, FET, JFET and MOSFET, Digital electronics-Boolean identities, De-Morgan’s
laws, Logic gates and truth tables, Simple logic circuits, Thermistors, solar cells, Fundamentals of microprocessors and digital computers.
Source: UPSC
Read More: IFS Mains Syllabus –
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Conservation of Angular Momentum Gyroscope: Exploring the Physics - GyroPlacecl.com
Short answer: Conservation of angular momentum in a gyroscope
In physics, the conservation of angular momentum states that the total angular momentum of a closed system remains constant unless acted upon by an external torque. A gyroscope, which is a spinning
object, also follows this principle. As long as no external torque is applied, the angular momentum of a gyroscope will remain unchanged. This property allows gyroscopes to maintain their stability
and perform various applications in navigation, stabilization systems, and other fields.
Understanding the Concept: Conservation of Angular Momentum in a Gyroscope
Understanding the Concept: Conservation of Angular Momentum in a Gyroscope
Have you ever wondered how a gyroscope manages to defy gravity or seemingly balance itself effortlessly? This intriguing device is not only fascinating to look at but also hides within it an
essential principle of physics – the conservation of angular momentum. In this blog post, we will delve into the depths of understanding this concept and unravel how it governs the mesmerizing
behavior of gyroscopes.
So, what exactly is angular momentum? In simple terms, it can be described as the measure of an object’s rotational motion around a fixed axis. Just like linear momentum, which determines an object’s
translational motion, angular momentum plays a vital role in shaping the movement and stability of rotating bodies. Now that we have a grasp on this fundamental idea let’s explore how it relates to
Gyroscopes are unique mechanical devices that consist of a spinning wheel or disk mounted on an axis. The beauty lies in their ability to maintain stability despite external forces acting upon them.
But how do they do it?
To comprehend this enigma, we must first understand the law of conservation of angular momentum. According to this law, when no external torque acts upon a system, its total angular momentum remains
constant. In simpler terms – once set into motion, a gyroscope will continue rotating at a constant speed without changing its direction unless influenced by external factors.
Now let us break down the mechanism behind this phenomenon using our imagination! Imagine you’re balancing on a swivel chair with wheels underneath. To keep yourself steady while silently spinning
around, you would instinctively extend your arms outwards. By doing so, your moment of inertia increases due to your arms’ mass being away from your body’s axis of rotation. This increase in moment
of inertia compensates for any change in rotational velocity caused by extending your arms.
Similarly, gyros achieve stability through their own manipulation of their moment of inertia. The spinning wheel or disk within a gyroscope has its mass concentrated towards the outer edges,
maximizing its moment of inertia. As a result, any force or disturbance acting upon it creates only minor changes in rotational velocity but fails to tip it off balance completely.
Now let’s take a closer look at an exciting application of angular momentum conservation – precession. Precession refers to the steady change in the orientation of a rotating body’s axis under the
influence of an external torque. In simpler terms, it’s like observing how a spinning top moves when you exert gentle pressure on it.
In gyroscopes, precession occurs due to the interaction between the gravitational force and angular momentum conservation. When a gyroscope is subjected to an external torque caused by gravity or
other forces, it results in a reorientation of its axis without affecting the overall rotation speed. This mesmerizing phenomenon is what allows gyroscopes to seemingly defy gravity as they rotate
around their supporting structure.
To sum up our journey into unraveling the conservation of angular momentum in gyroscopes, we’ve learned that this principle keeps them stable and enables them to maintain their rotation without
easily succumbing to external disturbances. It is not just an abstract concept but a fundamental law governing the behavior and mesmerizing movements of these fascinating devices.
Next time you come across one, spare a moment to appreciate how angular momentum conservation silently makes all those captivating motions possible. Understanding this concept not only adds depth to
your knowledge of physics but also highlights the beauty hidden within everyday phenomena!
Step-by-Step Guide: Exploring the Conservation of Angular Momentum in a Gyroscope
In this step-by-step guide, we will delve into the fascinating world of gyroscopes and unravel the secrets behind the conservation of angular momentum. Prepare to be amazed as we explore the
intricate physics that govern these spinning wonders. So without further ado, let’s get our gyroscope spinning!
Step 1: Understanding Angular Momentum
To begin our journey, it’s crucial to grasp the concept of angular momentum. Angular momentum is a fundamental property possessed by rotating objects, such as a spinning gyroscope. It can be
visualized as the rotational equivalent of linear momentum and is defined as the product of an object’s moment of inertia (a measure of its resistance to rotation) and its angular velocity.
Step 2: Gyroscopic Stability
Now that we have a basic understanding of angular momentum, let’s move on to exploring gyroscopic stability. Due to their peculiar properties, gyroscopes exhibit remarkable stability when in motion.
This phenomenon arises from the conservation of angular momentum – once set in motion, a gyroscope resists any external forces trying to change its orientation.
Step 3: Components of a Gyroscope
To fully appreciate how a gyroscope achieves this stability, let’s take a closer look at its components. A typical gyroscope consists of a freely rotating wheel or disk mounted within a frame known
as a gimbal. The gimbal allows for unrestricted movement along multiple axes while maintaining stability.
Step 4: Precession – The Key to Stability
The magic behind gyroscopic stability lies in the concept called precession. When an external force is applied perpendicular to the axis of rotation (such as gravity or torque), it induces precession
– resulting in an overall change in orientation while preserving angular momentum.
Step 5: Applying Torque Experimentally
Now it’s time for some hands-on experimentation! Grab your gyroscope and apply torque by gently twisting one end or using an external force. Observe how the gyroscope responds – you’ll notice that
instead of immediately changing its orientation, it gracefully precesses around a perpendicular axis, maintaining its stability.
Step 6: Real-World Applications
The conservation of angular momentum in gyroscopes finds practical applications in various fields. In aerospace engineering, gyroscopes assist spacecraft in maintaining stable orientations and
navigation. They also play a crucial role in stabilizing ships and submarines, ensuring smoother motion even amidst turbulent waters.
Step 7: Mind-Boggling Gyroscopic Tricks
Now that we have unraveled the science behind gyroscope stability let’s take a moment to appreciate some mind-boggling tricks that can be performed. By manipulating the forces acting on a spinning
gyroscope, one can achieve extraordinary feats such as gyroscopic wheel balance and even defy gravity by suspending a gyroscope on a string!
Congratulations! You’ve successfully explored the conservation of angular momentum in gyroscopes through this step-by-step guide. Armed with this knowledge, you’ve unlocked the secrets behind their
incredible stability and discovered their immense practical applications. It’s remarkable how something as simple as a spinning disk can captivate our imagination and uncover profound physical
principles. So go ahead and continue your exploration of the mesmerizing world of physics – there’s always more to discover!
Frequently Asked Questions about Conservation of Angular Momentum in a Gyroscope
Welcome to our blog section, where we address some frequently asked questions about the conservation of angular momentum in a gyroscope. If you’ve ever wondered how gyroscopes work or why they seem
to defy gravity, this article is for you! Get ready to dive into the world of rotational motion and unravel the mysteries behind this fascinating phenomenon.
Q: What is angular momentum?
A: Angular momentum is a fundamental concept in physics that describes the rotational motion of an object. It depends on both the mass and distribution of mass around an axis of rotation. To put it
simply, angular momentum represents the spinning property of an object.
Q: How does conservation of angular momentum apply to a gyroscope?
A: Conservation of angular momentum states that in the absence of any external torque, the total angular momentum of a system remains constant. This principle applies perfectly to gyroscopes. When
set in motion, a gyroscope’s spinning wheel creates its own angular momentum which stays constant as long as no external force interferes.
Q: Why doesn’t a gyroscope fall over when it should subject to gravity?
A: A common misconception is that gyroscopes “defy” gravity by magically staying upright. In reality, their stability comes from the conservation of angular momentum. Due to their spinning wheels,
gyroscopes possess intrinsic angular momentum which resists changes in orientation caused by external forces like gravity or other torques.
Q: Can you explain precession and how it relates to conservation of angular momentum?
A: Certainly! Precession refers to the slow rotation or wobbling exhibited by spinning objects affected by external torques. In a gyroscope, when an external force such as gravity acts upon it, there
is an attempt to tilt or change its orientation. However, due to conservation of angular momentum, instead of falling over immediately, the gyroscope precesses slowly around its axis perpendicular to
gravity while maintaining its overall stability.
Q: Is conservation of angular momentum only applicable to gyroscopes?
A: Definitely not! Conservation of angular momentum is a fundamental principle applicable to many rotational systems, not just gyroscopes. It plays a crucial role in celestial mechanics, such as
explaining the stability of rotating planets and galaxies, as well as providing insights into the behavior of atoms and subatomic particles.
Q: How do gyroscopes find practical applications?
A: Gyroscopes have numerous practical applications across various fields. In navigation systems, gyroscopes help determine orientation and maintain stability in aircraft, satellites, and even
smartphones. They also assist with motion tracking in virtual reality devices and provide stabilization for cameras on drones or image-stabilizing lenses. Gyroscopic principles are even utilized in
some advanced mechanical devices like artificial hearts!
So there you have it – a detailed explanation addressing some frequently asked questions about the conservation of angular momentum in a gyroscope. By understanding this fascinating concept, we can
appreciate how gyroscopes work their magic while staying robust against external forces. Whether applied in navigation technologies or enhancing our everyday experiences, gyroscopes continue to prove
their significance across a wide range of sectors.
Demystifying the Physics: How Does Conservation of Angular Momentum Work in a Gyroscope?
Demystifying the Physics: How Does Conservation of Angular Momentum Work in a Gyroscope?
Have you ever marvelled at the mesmerizing motion of a gyroscope? Whether it’s a toy spinning effortlessly on your finger or a sophisticated instrument used in aerospace engineering, gyroscopes
possess an intriguing ability to maintain balance and resist external forces. But what lies beneath their astonishing behavior? The answer lies in one of the fundamental principles of physics – the
conservation of angular momentum.
Angular momentum refers to the tendency of an object to keep rotating around its axis. It depends not only on an object’s mass but also on how this mass is distributed from the axis. This concept may
sound complex, but fear not! We are here to unveil the secrets behind this phenomenon and shed light on how it works specifically in gyroscopes.
To understand angular momentum in gyroscopes, let’s start by exploring their structure. Typically, a gyroscope consists of a spinning wheel or disk mounted within three rings that allow it to rotate
freely in any direction. The central axis around which this spinning wheel rotates is crucial for maintaining stability.
Once set into motion, the gyroscope experiences various physical forces acting upon it. However, due to its remarkable property of angular momentum conservation, it remains steadfast against these
forces and continues spinning with awe-inspiring persistence. But how does this happen?
The magic begins when we consider Newton’s first law – an object at rest tends to stay at rest unless acted upon by an external force while an object in motion tends to remain in motion with constant
speed along a straight line unless acted upon by an external force. In our case, since there is no net torque acting on the system once initiated (the torque being defined as the rotational
equivalent of force), there are no external torques resisting its rotational motion.
This lack of opposing torques allows our gyroscope’s angular momentum to be preserved indefinitely – as long as there are no external influences. To put it simply, the spinning wheel’s angular
momentum is conserved because there are no forces or torques trying to change its state of motion.
However, when an external force tries to alter the gyroscope’s orientation, a phenomenon known as precession comes into play. Precession refers to the gradual change in orientation (tilting) of a
rotating object when subjected to an external torque. In simpler terms, instead of resisting the force directly, the gyroscope responds by altering its axis of rotation.
This unique behavior can be explained by considering a top-like analogy. Just like a spinning top undergoes precession when pushed at an angle, the gyroscope exhibits similar characteristics. When an
external force is applied to tilt the gyroscope, it responds by reorienting itself perpendicular to that force. This response results from a combination of gyroscopic stability and conservation of
angular momentum.
The intricate interplay between these factors ensures that the center of mass remains above the pivot point, preventing it from collapsing and enabling gyroscopes’ astounding stability. It is worth
noting that gyroscopes find applications in various fields – from navigation systems on airplanes and spacecraft to stabilizing cameras and even enhancing mobility aids for those with balance-related
In conclusion, the conservation of angular momentum lies at the heart of how gyroscopes operate. By harnessing this principle, they can maintain their incredible stability and resist external forces
without diminishing their astonishing rotational properties. Understanding this fundamental concept brings us closer to unraveling nature’s secrets while appreciating how our world is governed by
fascinating laws – just waiting for us to demystify them!
Applying Conservation of Angular Momentum to Maintain Stability in a Gyroscope
Title: Mastering Stability in a Gyroscope: Unveiling the Secrets Behind Conservation of Angular Momentum
In the realm of physics, where forces and motions orchestrate an intricate dance, the conservation of angular momentum emerges as a crucial player. Today, we delve into the astounding world of
gyroscopes – marvels of engineering and precision – to understand how the fundamental principle of conserving angular momentum keeps them stable in their mesmerizing spins. Prepare to be captivated
by this witty exploration into “Applying Conservation of Angular Momentum to Maintain Stability in a Gyroscope.”
The Gyroscopic Enigma:
Imagine holding a gyroscope delicately between your fingers, ready to unleash its spinning magic at any moment. When set in motion, this marvelous invention appears to defy gravity itself,
maintaining stability with stunning prowess. How does it accomplish such an astonishing feat? The answer lies within the realm of angular momentum.
Angular Momentum Unveiled:
Before we embark on our journey through the inner workings of a gyroscope, we must acquaint ourselves with angular momentum. Just as linear momentum describes an object’s resistance against changes
in its velocity, angular momentum arises from resisting changes in rotational motion.
Angular momentum is directly proportional to an object’s mass and its rotational speed or rate. Therefore, when either its mass or rotational speed increases or decreases, like all conserved
quantities in physics, angular momentum must remain constant.
Conservation Takes Center Stage:
Now that we understand the essence of angular momentum conservation let us apply this concept to elucidate how gyroscopes maintain their stability despite external disturbances.
The Magic Begins: The Rigidity Thought Experiment
To comprehend why gyroscopes exhibit exceptional stability while spinning rapidly under various conditions, let us imagine spinning one ourselves. As you spin a gyroscope around one axis
perpendicular to its spin axis (the precession axis), something astonishing occurs – it develops rigidity!
Exploiting Newton’s Third Law:
Newton’s third law – every action has an equal and opposite reaction – underpins the mechanical marvel of a gyroscope. When you exert a force to rotate the gyroscope’s spin axis, it reciprocates by
resisting the change in direction due to its angular momentum.
Precession: The Secret Savior
As you push on one side of the spinning gyroscope, it reacts with unparalleled grace. Instead of tilting or collapsing under external forces, it initiates precession – a mesmerizing phenomenon where
the object’s axis starts rotating perpendicular to its initial orientation.
The Center of Gravity Shines:
While precession captures our imagination, another critical element behind a gyroscopes’ stability is its center of gravity. By ensuring that its center of gravity lies precisely along the line of
precession (the torque-free axis), gyroscopes maintain equilibrium effortlessly.
Torque Begone: Resisting Deviations:
Like a skilled tightrope walker gracefully countering disruptive gusts of wind, gyroscopes defy external disturbances through their ability to resist deviations from their desired orientation. This
remarkable resistance arises from conserving angular momentum while simultaneously producing equal and opposite torques through precession.
A Delicate Dance with External Forces:
When we subject our spinning gyroscope to external influences such as gravity or additional rotational forces, they result in slight perturbations to its motion. Yet, thanks to the conservation of
angular momentum and impeccable balance achieved by carefully aligning mass distribution and velocity vectors, gyroscopes triumphantly regain stability despite these perturbations.
Application Galore:
Understanding how gyroscopes utilize conservation principles for stability has profound implications across various fields – from aerospace engineering and navigation systems to robotics and even
amusement park rides! Gyroscopic stabilizers in helicopters exemplify this principle’s significance by effortlessly maintaining stable flight during turbulent conditions.
Gyrosocopes continue to captivate our imaginations with their mesmerizing spins. By applying conservation of angular momentum as their secret weapon, they effortlessly maintain stability against all
odds. As we unveil the secrets behind the enigma of gyroscopes, their remarkable prowess becomes an eternal testament to the elegance and power inherent in the fundamental laws of physics.
Practical Applications and Implications of Conservation of Angular Momentum in Gyroscopes
Gyroscopes have long fascinated scientists, engineers, and hobbyists alike with their seemingly magical ability to maintain stability and resist changes in orientation. At the heart of this
extraordinary phenomenon lies the principle of conservation of angular momentum. In this blog post, we will explore in detail the practical applications and intriguing implications of this principle
in gyroscopes.
Firstly, let us understand what angular momentum is and how it relates to gyroscopes. Angular momentum is a property possessed by rotating objects and can be defined as the product of an object’s
moment of inertia (its resistance to changes in rotation) and its angular velocity (the rate at which it rotates). When a gyroscope spins rapidly, it accumulates significant angular momentum due to
its rotating mass distribution.
Now, here comes the crucial concept: the conservation of angular momentum. According to this principle, when no external torque acts upon an isolated system (such as a spinning gyroscope), the total
angular momentum remains constant over time. This means that if any part of the system tries to change its rotation or direction, another part will counteract that change by adjusting its own motion
accordingly. As a result, gyroscopes exhibit exceptional stability and resist any attempts to alter their orientation.
Practical applications of conservation of angular momentum in gyroscopes can be found across various fields:
1. Navigation systems: Gyroscopes are essential components in inertial navigation systems used in ships, aircraft, submarines, and even spacecraft. By precisely measuring their orientation relative
to Earth’s gravitational field or known reference points like stars, these systems can provide accurate information on position, velocity, altitude, and attitude regardless of external factors such
as magnetic disturbances or GPS signal loss.
2. Stabilization technology: The conservation of angular momentum enables gyroscopic stabilization devices commonly used in cameras and drones for capturing smooth footage or steady aerial maneuvers
respectively. By exploiting the gyroscope’s inherent tendency to maintain its initial axis of rotation against outside forces, these devices can counteract unwanted movements and produce visually
appealing results.
3. Gyrocompasses: Navigational instruments known as gyrocompasses utilize the conservation of angular momentum to provide accurate directional information even in the presence of magnetic field
variations. Unlike traditional magnetic compasses that rely on Earth’s magnetic field, gyrocompasses exploit the gyroscope’s stability to determine true north by aligning with Earth’s rotation axis.
4. Space exploration: The conservation of angular momentum is crucial in space missions that involve spinning spacecraft or satellites. By adjusting their orientation using onboard gyroscopic
systems, scientists can control and stabilize their instruments, ensuring accurate data collection and preventing undesired tumbling motions.
Beyond practical applications, conservation of angular momentum in gyroscopes also raises some fascinating implications:
1. Gyroscopic precession: When an external torque is applied to a spinning gyroscope, it responds not by changing its orientation immediately but rather by exhibiting a curious phenomenon called
precession. Precession refers to the gradual change in the axis of rotation caused by the torque applied perpendicular to it. This effect has been instrumental in enhancing our understanding of
rotational motion and finding applications in fields such as mechanical engineering and robotics.
2. Educational demonstrations: The remarkable stability displayed by gyroscopes makes them valuable tools for educational purposes. Whether it is displaying concepts related to rotational inertia,
moment of inertia calculations, or demonstrating Newton’s laws of motion, gyroscopes offer engaging visual representations that help learners grasp complex physical principles more effectively.
In conclusion, the practical applications and implications of conservation of angular momentum in gyroscopes span various domains ranging from navigation and stabilization technology to space
exploration and education. As we marvel at these spinning wonders, let us appreciate how this fundamental principle enables gyroscopes to maintain stability against external disturbances while
unlocking a myriad of practical uses across diverse industries. | {"url":"https://gyroplacecl.com/conservation-of-angular-momentum-gyroscope-exploring-the-physics/","timestamp":"2024-11-07T15:34:05Z","content_type":"text/html","content_length":"98163","record_id":"<urn:uuid:4d60bf39-fd00-409a-be4b-453706e2c11f>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00798.warc.gz"} |
Fibonacci Series in Java: How to display first n numbers? | Edureka
The Fibonacci Sequence is a peculiar series of numbers named after Italian mathematician, known as Fibonacci. Starting with 0 and 1, each new number in the Fibonacci Series is simply the sum of the
two before it. For example, starting with 0 and 1, the first 5 numbers in the sequence would be 0, 1, 1, 2, 3 and so on. In this article, let’s learn how to write the Fibonacci Series in Java.
You can mainly write Fibonacci Series in Java in two ways:
Fibonacci Series without using recursion
When it comes to generating the Fibonacci Series without using recursion, there are two ways:
1. Using ‘for’ loop
2. Using ‘while’ loop
Method1: Java Program to write Fibonacci Series using for loop
The program below should help you on how to write a java program to generate first ‘n’ numbers in the Fibonacci Series using for loop. The logic used here is really simple. First, I have initialized
the first two numbers of series. Then comes the for loop, which adds up its two immediate predecessors and prints the value. This continues until the program prints the first ‘n’ numbers in the
package Edureka;
import java.util.Scanner;
public class Fibonacci {
public static void main(String[] args)
int n, first = 0,next = 1;
System.out.println("Enter how may fibonnaci numbers to print");
Scanner scanner = new Scanner(System.in);
n = scanner.nextInt();
System.out.print("The first " + n + " Fibonacci numbers are: ");
System.out.print(first + " " + next);
for (int i = 1; i<=n-2; ++i)
int sum = first + next;
first = next;
next = sum;
System.out.print(" " + sum);
Enter how may fibonnaci numbers to print
The first 7 Fibonacci numbers are: 0 1 1 2 3 5 8
Note: Condition in for loop is ‘n-2’. That’s because the program already prints ‘0’ and ‘1’ before it begins with for loop.
Method2: Java Program to write Fibonacci Series using while loop
The logic is similar to the previous method. It’s just the while loop condition that you need to be careful about. Take a look at the java code below to understand how to generate Fibonacci Series
using while loop.
package Edureka;
import java.util.Scanner;
public class FibWhile {
public static void main(String[] args)
int n, first = 0,next = 1;
System.out.println("Enter how may fibonnaci numbers to print");
Scanner scanner = new Scanner(System.in);
n = scanner.nextInt();
System.out.print("The first " + n + " Fibonacci numbers are: ");
System.out.print(first + " " + next);
int i = 1;
while (i<n-1)
int sum = first + next;
first = next;
next = sum;
System.out.print(" " + sum);
Enter how may fibonnaci numbers to print
The first 7 Fibonacci numbers are: 0 1 1 2 3 5 8
Fibonacci Series using recursion
Recursion is the basic java programming technique in which a function calls itself directly or indirectly. The corresponding function is called a recursive function. Using a recursive algorithm,
certain problems can be solved quite easily. Let’s see how to use recursion to print first ‘n’ numbers of the Fibonacci Series in Java.
The program below should help you on how to write a recursive java program to generate first ‘n’ numbers in the Fibonacci Series. The logic here is quite simple to understand. First, the user gives
the input and then the for loop is used to loop until the limit where each iteration will call the function fibonaccinumber(int n) which returns the Fibonacci number at position n. The Fibonacci
function recursively calls itself adding the previous two Fibonacci numbers.
package Edureka;
import java.util.Scanner;
public class FibRec {
public static void main(String[] args)
int n;
System.out.println("Enter how may fibonnaci numbers to print");
Scanner scanner = new Scanner(System.in);
n = scanner.nextInt();
for (int i = 0; i<=n-1; ++i)
System.out.print(fibonaccinumber(i) + " ");
public static int fibonaccinumber(int n) {
return 0;
else if(n==1)
return 1;
return fibonaccinumber(n-1) + fibonaccinumber(n-2);
Enter how may fibonnaci numbers to print
The first 7 Fibonacci numbers are: 0 1 1 2 3 5 8
This brings us to the end of this ‘Fibonacci Series in Java’ article. We have learned how to programmatically print the Nth Fibonacci number using either loop statements or recursion.
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How to Find the Area of a Triangle
You'll use the same formula to find the area of a right triangle, acute triangle, scalene triangle, equilateral triangle and any other three-sided shape you can come up with. MTStock Studio / Getty
With its three sides and three angles, the triangle is one of the most basic shapes in geometry. This means calculating the area of a triangle is a fundamental skill in geometry, with multiple
formulas available depending on the type of triangle and the given data.
But knowing how to find the area of a triangle has plenty of applications beyond mathematics. For example, understanding right-angled triangles is essential for finding accurate measurements in
construction and navigation. Isosceles triangles are crucial in structural engineering, aerospace design, and optics, where precision is absolutely a requirement. And understanding the area of an
equilateral triangle is essential in architecture and even art.
Let's review the basic formula for finding the area of a triangle, plus formulas for other scenarios. | {"url":"https://science.howstuffworks.com/math-concepts/area-of-a-triangle.htm","timestamp":"2024-11-10T08:31:55Z","content_type":"text/html","content_length":"181101","record_id":"<urn:uuid:2728ab7c-adab-4cce-83b0-b6e3f6011a9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00412.warc.gz"} |
Understanding Mathematical Functions: How Many Functions Are There
When it comes to the world of mathematics, functions are a critical concept to understand. These are mathematical relationships between two sets of quantities, where each input has exactly one
output. Functions are essential for modeling real-world phenomena, analyzing data, and solving complex problems. Therefore, it is crucial for anyone working with mathematics to have a solid
understanding of mathematical functions.
Key Takeaways
• Functions are crucial for modeling real-world phenomena, analyzing data, and solving complex problems in mathematics.
• There are different types of mathematical functions, including linear, quadratic, exponential, and logarithmic functions.
• The concept of functions is related to the idea of infinite possibilities and the significance of understanding the number of functions.
• Tools for analyzing functions include graphing, algebraic techniques, and calculus.
• Challenges in understanding functions include common misconceptions, but there are resources available for further learning and understanding.
Different types of mathematical functions
Mathematical functions are a fundamental concept in mathematics, with various types that serve different purposes and have distinct characteristics. Understanding the different types of mathematical
functions is crucial for students and professionals alike, as it forms the basis for complex mathematical analysis and problem-solving.
• Linear functions
Linear functions are the simplest type of mathematical functions, with a basic form of y = mx + b. They represent a straight line on a graph, where 'm' is the slope and 'b' is the y-intercept.
Linear functions have a constant rate of change and are commonly used to represent relationships between two variables.
• Quadratic functions
Quadratic functions are more complex than linear functions and take the form of y = ax^2 + bx + c. They are characterized by a parabolic shape on a graph and have a single highest or lowest
point, known as the vertex. Quadratic functions are commonly used to model physical phenomena and are essential in fields such as physics and engineering.
• Exponential functions
Exponential functions have the form y = a^x, where 'a' is the base and 'x' is the exponent. These functions grow or decay at an increasing rate and are commonly used to represent phenomena such
as population growth, radioactive decay, and compound interest. Exponential functions play a crucial role in various scientific and financial applications.
• Logarithmic functions
Logarithmic functions are the inverse of exponential functions and have the form y = log_a(x), where 'a' is the base. They represent the exponent needed to produce a certain value and are
commonly used in fields such as mathematics, engineering, and computer science. Logarithmic functions are essential for dealing with large numbers and understanding complex relationships.
The concept of infinite functions
When we talk about mathematical functions, it's important to understand that there is an infinite number of functions that can be created. This means that there is no limit to the number of functions
that can be defined and used in mathematics.
Exploring the idea of a countable number of functions
While the concept of infinite functions may seem overwhelming, it's important to note that there are also countable numbers of functions. This means that even though there are an infinite number of
functions, they can be organized and counted in a systematic way.
Understanding the cardinality of functions
In mathematics, cardinality refers to the size of a set. When it comes to functions, the cardinality of the set of all functions is larger than the cardinality of the set of all natural numbers. This
means that there are more functions than there are natural numbers, which again highlights the concept of there being an infinite number of functions.
The Significance of Understanding the Number of Functions
Understanding the number of functions is crucial in various fields and can greatly aid in problem-solving and modeling real-world situations.
A. Applications in Various Fields such as Engineering and Economics
• Functions play a vital role in engineering, from designing structures to optimizing processes.
• In economics, functions are used to model market demand, supply, and other economic phenomena.
B. How Understanding the Number of Functions Can Aid in Problem-Solving
• By understanding the number of functions, it becomes easier to identify and analyze the relationship between variables in a problem.
• It allows for the exploration of various mathematical techniques to solve complex problems efficiently.
C. The Role of Functions in Modeling Real-World Situations
• Functions are essential for creating mathematical models of real-world situations, enabling predictions and analysis.
• They help in understanding and interpreting data to make informed decisions in various fields such as finance, medicine, and environmental science.
Tools for analyzing functions
When it comes to understanding mathematical functions, there are several tools that can be utilized for analysis. These tools provide different perspectives and insights into the behavior and
properties of functions.
• Graphing functions
Graphing functions is a fundamental tool for visualizing the behavior of a function. By plotting the function on a graph, one can observe its shape, critical points, and overall behavior. This
helps in understanding the relationship between inputs and outputs of the function.
• Using algebraic techniques
Algebraic techniques such as manipulating equations, solving for variables, and factoring can provide valuable insights into the properties of functions. These techniques help in simplifying and
analyzing the mathematical expressions that define functions.
• Calculus and its role in understanding functions
Calculus plays a crucial role in understanding the behavior of functions, especially when it comes to studying rates of change, optimization, and the behavior of functions at specific points.
Concepts such as derivatives and integrals provide a deeper understanding of the behavior of functions and their properties.
Challenges in understanding functions
When it comes to understanding mathematical functions, many students and even some adults face challenges in grasping the concept. These challenges often stem from common misconceptions, but there
are ways to overcome them with the right resources and strategies.
A. Common misconceptions about functions
One of the common misconceptions about functions is that they are only seen as equations. Many people tend to think of functions as simply a set of numbers and symbols put together. However,
functions are much more than that. They represent a relationship between two sets of quantities, with one set determining the other.
B. Overcoming difficulties in grasping the concept of functions
To overcome difficulties in grasping the concept of functions, it is essential to understand the fundamental principles. Functions are about inputs and outputs, and how one set of values relates to
another. It is crucial to focus on the purpose and representation of functions, rather than getting caught up in the equation itself. Visual representations and real-life examples can also help in
understanding how functions work and how they are applied in various scenarios.
C. Resources for further learning and understanding
There are various resources available for further learning and understanding of functions. Online courses, tutorial videos, and textbooks offer comprehensive explanations and examples of functions.
Additionally, seeking help from teachers, tutors, or online forums can provide valuable insights and clarifications on specific difficulties. Practice and repetition also play a crucial role in
solidifying the understanding of functions, so working through exercises and problems can further enhance comprehension.
In conclusion, understanding mathematical functions is crucial for grasping the concepts of algebra and calculus, and for solving real-world problems in fields such as physics, economics, and
engineering. I encourage everyone to continue exploring and learning about functions, as they are not only essential for academic success but also for gaining a deep understanding of the way our
world works. Functions are incredibly versatile and significant in mathematics, and the more one learns about them, the more empowered they become in solving complex problems and making sense of the
world around them.
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