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COUNTIF question for pulling only particular year
Closed 167
In-Progress 22
Cancelled 1
=COUNTIF({Data Set 1}, "Closed")
Hello All,
I want to be able to pull/report number for a particular year. For the COUNTIF function above, it is currently pulling all "Closed" items, but would like to be able to change the function so that it
only would pull from dates in 2020, 2021, 2022, etc. Thank you.
Best Answer
• @JayTeeDee It would depend on how your data is setup - if the column you are searching against is the date data type, and you want to evaluate by year (or across multiple years) then you can try
=COUNTIFS({Data Set 1},"Closed",{Date Data Set},IFERROR(YEAR(@cell),0) >= 2021)
This example would return the number of records that are status Closed and with a date that has a year that is greater than or equal to 2021.
If your data is date type and you want to evaluate based on a specific date value (as you had in your example of ">=1/1/2022" then you would need to call the DATE() function to feed the formula
the date value to evaluate. As example:
=COUNTIFS({Data Set 1},"Closed",{Date Data Set},>= DATE(2022,1,1))
• If I am reading your post correctly, it sounds like you might benefit from the COUNTIFS() function. This function allows for a record to be counted if all the criteria (multiple criteria) are
For example:
=COUNTIFS({Data Set 1},"Closed",{Year Data Set},2022)
In the example the records would be counted if they had a status of Closed and were in the year 2022.
Hope that helps,
More infor on COUNTIFS: https://help.smartsheet.com/function/countifs
• @William Meixner What if I wanted it to be a range of dates for anything in 2022 instead of just 2022? This is the function I came up with but not reporting anything.
=COUNTIFS({Data Set 1},"Closed",{Year Data Set}, ">=1/1/2022")
• @JayTeeDee It would depend on how your data is setup - if the column you are searching against is the date data type, and you want to evaluate by year (or across multiple years) then you can try
=COUNTIFS({Data Set 1},"Closed",{Date Data Set},IFERROR(YEAR(@cell),0) >= 2021)
This example would return the number of records that are status Closed and with a date that has a year that is greater than or equal to 2021.
If your data is date type and you want to evaluate based on a specific date value (as you had in your example of ">=1/1/2022" then you would need to call the DATE() function to feed the formula
the date value to evaluate. As example:
=COUNTIFS({Data Set 1},"Closed",{Date Data Set},>= DATE(2022,1,1))
Help Article Resources
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PHP Exercise: Check whether three given lengths of three sides form a right triangle - w3resource
PHP Exercises: Check whether three given lengths of three sides form a right triangle
PHP: Exercise-48 with Solution
Write a PHP program to check whether three given lengths (integers) of three sides form a right triangle. Print "Yes" if the given sides form a right triangle otherwise print "No".
Integers separated by a single space.
1 ≤ length of the side ≤ 1,000
Pictorial Presentation:
Pictorial Presentation:
Sample Solution:
PHP Code:
// Assign values to variables representing the sides of a triangle
$a = 5;
$b = 3;
$c = 4;
// Square each side by multiplying it by itself
$a *= $a;
$b *= $b;
$c *= $c;
// Check if the sum of the squares of two sides equals the square of the third side
if ($a + $b == $c || $a + $c == $b || $b + $c == $a) {
// Print "YES" if the condition is true
echo "YES\n";
} else {
// Print "NO" if the condition is false
echo "NO\n";
• Variable Assignment:
□ Three variables, $a, $b, and $c, are assigned values representing the sides of a triangle: $a = 5, $b = 3, and $c = 4.
• Squaring the Sides:
□ Each side is squared by multiplying it by itself:
☆ $a *= $a; (Now $a is 25)
☆ $b *= $b; (Now $b is 9)
☆ $c *= $c; (Now $c is 16)
• Checking the Pythagorean Theorem:
□ The code checks if the sum of the squares of any two sides equals the square of the third side using the condition:
☆ if ($a + $b == $c || $a + $c == $b || $b + $c == $a)
□ This condition evaluates whether the triangle with the given sides is a right triangle.
• Printing the Result:
□ If any of the conditions are true, it prints "YES\n", indicating that the triangle can be classified as a right triangle.
□ If none of the conditions are met, it prints "NO\n", indicating that the triangle is not a right triangle.
PHP Code Editor:
Have another way to solve this solution? Contribute your code (and comments) through Disqus.
Previous: Write a PHP program to compute the digit number of sum of two given integers.
Next: Write a PHP program which solve the equation. Print the values of x, y where a, b, c, d, e and f are given.
What is the difficulty level of this exercise?
Test your Programming skills with w3resource's quiz.
It will be nice if you may share this link in any developer community or anywhere else, from where other developers may find this content. Thanks.
• Weekly Trends and Language Statistics
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3D Signpost - Pre-Analysis & Start-Up
Pre-Analysis & Start-Up
This problem requires a relatively straightforward application of linearly superposed solutions from individual loadings. A simple spreadsheet can be prepared to give the results for the stresses
associated with the separate loadings experienced by the signpost. An example is given here for the case of a solid post with a diameter of 1.12 feet:
Note that the formula for the moment about the x-axis is highlighted and shown in the formula bar above the spreadsheet. Not surprisingly, the stresses are quite low as solid posts are almost never
used in practice. You may wish to begin with this case of an over-designed signpost. The tutorial contains geometry files for both solid and hollow poles. Then you will want to consider hollow poles
and compare results as you attempt to optimize the post's load-carrying capacity:
You will want to continue and re-design lighter hollow posts which sustain higher stresses, but remain in the elastic regime.
Launch ANSYS Workbench and start a "Static Structural" analysis in the project page as shown in the video below.
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Worksheet on solving polynomials of a single variable
worksheet on solving polynomials of a single variable Related topics: equivelent ratios for five ninths
solving 4 equation 4 unknowns
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Author Message
Btirxeih Posted: Wednesday 22nd of Apr 08:41
Hi guys I’m really stuck on worksheet on solving polynomials of a single variable and would sure desperately need guidance to get me started with equation properties, dividing
fractions and inequalities. My math assignment is due soon. I have even thought of hiring a algebra tutor, but they are so costly . So any suggestion would be greatly valued .
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kfir Posted: Wednesday 22nd of Apr 10:25
Don’t fret my friend. It’s just a matter of time before you’ll have no trouble in answering those problems in worksheet on solving polynomials of a single variable. I have the exact
solution for your math problems, it’s called Algebrator. It’s quite new but I guarantee you that it would be perfect in helping you in your algebra problems. It’s a piece of software
where you can answer any kind of math problems easily. It’s also user friendly and displays a lot of useful data that makes you learn the subject matter fully.
From: egypt
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Dnexiam Posted: Wednesday 22nd of Apr 20:47
1.Hey mate, you are on the mark about Algebrator! It is absolutely fab! I downloaded it recently from https://softmath.com/algebra-software-guarantee.html after a magazine suggested
it to me. Now, all I do is type in the problem assigned by my teacher and click on Solve. Bingo! I get a step-by-step solution to my math problem. It’s almost like a tutor is
explaining it to you. I have been using it for three weeks and so far, haven’t come across any problem that Algebrator can’t solve. I have learnt so much from it!
From: City 17
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Doz Posted: Friday 24th of Apr 10:43
Hello Friends, Thanks a ton for all your answers. I shall surely give Algebrator at https://softmath.com/links-to-algebra.html a try and would keep you posted with my experience. The
only thing I am particular about is the fact that the program should offer required aid on Algebra 2 which in turn would help me to complete my assignment before the deadline .
From: Yorkshire,
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Svizes Posted: Saturday 25th of Apr 19:36
There you go https://softmath.com/.
From: Slovenia
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Dxi_Sysdech Posted: Monday 27th of Apr 19:06
I remember having often faced difficulties with linear algebra, quadratic formula and adding fractions. A really great piece of math program is Algebrator software. By simply typing
in a problem homework a step by step solution would appear by a click on Solve. I have used it through many math classes – Intermediate algebra, Algebra 2 and Basic Math. I greatly
recommend the program.
From: Right
here, can't you
see me?
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rq(stack.loss ~ stack.x,.5) #median (l1) regression fit for the stackloss data.
rq(stack.loss ~ stack.x,.25) #the 1st quartile,
#note that 8 of the 21 points lie exactly on this plane in 4-space!
rq(stack.loss ~ stack.x, tau=-1) #this returns the full rq process
rq(rnorm(50) ~ 1, ci=FALSE) #ordinary sample median --no rank inversion ci
rq(rnorm(50) ~ 1, weights=runif(50),ci=FALSE) #weighted sample median
#plot of engel data and some rq lines see KB(1982) for references to data
plot(income,foodexp,xlab="Household Income",ylab="Food Expenditure",type = "n", cex=.5)
taus <- c(.05,.1,.25,.75,.9,.95)
xx <- seq(min(income),max(income),100)
f <- coef(rq((foodexp)~(income),tau=taus))
yy <- cbind(1,xx)%*%f
for(i in 1:length(taus)){
lines(xx,yy[,i],col = "gray")
abline(lm(foodexp ~ income),col="red",lty = 2)
abline(rq(foodexp ~ income), col="blue")
legend(3000,500,c("mean (LSE) fit", "median (LAE) fit"),
col = c("red","blue"),lty = c(2,1))
#Example of plotting of coefficients and their confidence bands
plot(summary(rq(foodexp~income,tau = 1:49/50,data=engel)))
#Example to illustrate inequality constrained fitting
n <- 100
p <- 5
X <- matrix(rnorm(n*p),n,p)
y <- .95*apply(X,1,sum)+rnorm(n)
#constrain slope coefficients to lie between zero and one
R <- cbind(0,rbind(diag(p),-diag(p)))
r <- c(rep(0,p),-rep(1,p))
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Calculating energy expectation in infinite DMRG
Hi Miles,
I am trying to calculate something like <psi|U^\dagger H U|psi> in infinite dmrg. As a first step I want to calculate <psi|H|psi> and compare it with the dmrg output. I tried the following code:
auto E = overlap(psi, H, psi);
and it doesn't work. I think it might have something to do with the "edge vectors" of the Hamiltonian as well as the MPS. What's the right way of handling this? Thanks!
Hi Chengshu,
Sorry about the slow reply.
The overlap(psi,H,psi) method is intended for finite, open boundary MPS and finite, open boundary MPOs.
To get the energy from infinite DMRG, one way is to get the energy from the result object:
auto res = idmrg(...);
To actually evaluate the energy from an MPO takes more explanation and it's hard to type it all out here. But basically the kind of MPOs that idmrg expects have left and right boundary vectors stored
in H.A(0) and H.A(N+1) (where N = H.N() is the number of sites of the MPO).
The MPS returned from idmrg (meaning the value of psi which is passed by reference after idmrg returns) is in a right-orthogonal gauge, meaning all of it's tensors obey the right orthogonality
condition. The tensor psi.A(0) contains the "center matrix" of the MPS, so to construct part of the infinite wavefunction you must multiply psi.A(0)* psi.A(1) to get the orthogonality center tensor
on site 1. See the sample/idmrg.cc sample code to see an example of this. Then one can extend the MPS by continuing to multiply by psi.A(2), psi.A(3), ..., psi.(N), psi.A(1), psi.A(2), etc for as
many unit cells as needed e.g. to compute correlation functions and or matrix elements of part of the Hamiltonian or an MPO.
Finally, in the result object the idmrg algorithm returns, there are two tensors res.HL and res.HR which are the left and right "Hamiltonian environment" tensors computed as the idmrg algorithm grows
the system longer and longer. These are the MPO projected into the semi-infinite "wings" or left and right basis of the infinite MPS. You can use these together with one unit cell of the MPO and MPS
to compute the same energy that idmrg reports.
At some point I plan to write a detailed documentation with figures about the idmrg algorithm in ITensor. There is a draft of one written by a couple students but it may have some inaccuracies. It is
a fairly complex algorithm to explain, but it follows pretty closely to the algorithm as explained by Schollwock in his review article. I hope what I wrote above gives you the information you need.
Also I'd encourage you to read through the idmrg code itself and draw diagrams and/or take notes on how it works for yourself and you can learn a lot this way about its inner workings and the
Hi Miles,
Thank you very much for the very informative reply! I think I know how to perform the calculations now. I agree that reading the code+review article will be very helpful for understaning the idmrg
algorithm and also the data structure of infinite MPS/MPO.
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ASA Community
Hi everyone,
I am working on a power calculation for a mixed effects model and am not sure whether I have formulated the model correctly. I will share what I have in the hope that someone on this list could
confirm whether I am going about this in an appropriate way.
The power calculation concerns a study where we know the following:
1. A small number of subjects will be recruited in the study (perhaps 20 or 30) - these will actually be volunteers;
2. All workers are shift workers, who have 4 types of shifts: day, evening, night and off.
3. A sequence of 35 calendar days will be selected for the study (which will cover a combination of the 4 shifts), during which the workers will be administered a cognition test, whose outcomes will
be a) completion time and b) total test score.
4. During each shift, the workers will be asked to take the cognition test at the beginning and end of the shift (at least), though the exact times when the test is taken may differ among workers.
5. Other information collected on each worker includes: Age, Gender, Stress Level.
6. It is expected that cognitive function will decline as the shift progresses. Interest lies in testing differences between shifts as well as differences between genders with respect to the rate of
decline in cognitive function.
In considering the above, it seems to me that this is a 3-level mixed effects model (though I am not sure, as I don't work with mixed effects models all that often). Is what I am proposing below
Level 3 Subject 1 Subject n
Level 2 Day 1 Day 2 .... Day 35 Day 1 Day 2 ... Day 35
Level 1 T1 T2 T1 T2 ... T1 T2 T1 T2 T1 T2 ... T1 T2
Test occasions (denoted by T1 and T2) would represent the first level of nesting, followed by calendar days (the second level of nesting), followed by subject (the third level of nesting). T1 and T2
could perhaps be represented as hour of day (?) or "Beginning" and "End" (?).
Calendar Day would be treated as a random factor and "Shift Type" (factor with 4 levels) would be treated as a level-2 predictor (?). But does it make sense to treat calendar day as a random factors
when the days are chosen to that a particular sequence of day, afternoon, evening and off shifts are captured? On the other hand, with only 20-30 subjects, maybe this is not unreasonable.
Subject would be treated as a random factor (?). If the subjects are volunteers, how reasonable is this? Age, Gender and Stress Level would be treated as level-3 predictors (?).
Subject and Calendar Day would be treated as crossed factors (?), as the same set of Calendar Days is used for each subject.
For power calculations, would it be reasonable to just consider the simplest model possible - say, one which includes a random effect for subject and no interactions?
Thank you in advance for any insights you may be able to share.
Isabella Ghement
Ghement Statistical Consulting Company Ltd.
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Implement constraints correctly
I am struggling to implement constraints correctly into my model.
I am looking into carbon capture and storage as shipboard application. As a vessel sails it will capture its own emission and store it in tanks onboard. Related to this I want to create a simple
model that will choose at which nodes the ship will stop to unload its stored CO2 in order to minimize the cost of the system. The size of the CO2 tank onboard the vessel generates a cost to the
system that increase linearly with the distance sailed since last reception point. It will however only be the largest distance traveled that will define this cost contribution. There is also a cost
related to establishing reception points/infrastructure in given ports and and nodes, these will be different for different nodes.
I have a set of nodes that are ports, that has to be visited, and a set of nodes that are reception points for CO2, that can be visited if this generates a cheaper route.
The error code that appears is KeyError (0,0) for 'Constraint5'. But I am also a bit unsure about the rest of my constraints as well.
Let me know if anything needs clarification in my model!
import pandas as pd
from pandas import ExcelFile
import numpy as np
import matplotlib.pyplot as plt
import gurobipy as gp
from gurobipy import *
#Generating coordinates for nodes
xc = np.array([1,3,5,8,13,3,6,7,8,9,10])
yc = np.array([1,8,4,9,5,5,8,3,6,8,6])
Splitted into set of ports and reception points and drawn with matplotlib to:
P = [0]+[1]+[2]+[3]+[4]
RP = [5]+[6]+[7]+[8]+[9]+[10]
N = P + RP
#Set of arcs for all nodes:
A = [(i,j) for i in N for j in N if i!=j]
#Distance between nodes
D = {(i,j): np.hypot(xc[i]-xc[j],yc[i]-yc[j]) for i,j in A if j!=0 and np.hypot(xc[i]-xc[j],yc[i]-yc[j])<=10}
#Making the data easier to work with
data = []
for i,j in D:
#Arcs beginning at source node
for i in data:
ifi[0] == 0:
#Arcs ending in sink node
for i in data:
ifi[1] == len(P)-1:
#Cost of generating reception point in each node, including ports
cost = [70,70,70,70,70,90,150,70,500,150,90]
#Lost opportunity cost of reduced cargo capacity of the vessel, is to be multiplied by the longest arc. This will define the tank size on the vessel, and related cost
costCCS = 1500
Initializing Gurobi model:
f = gp.Model(name = "SimpleModel")
#Creating decision variables
#x = 1 if RP in 1, = 0 otherwise
x = f.addVars(N, name = 'x', vtype = GRB.BINARY)
#y = 1 if vessel sails between i and j, = 0 otherwise
y = f.addVars(D, name = 'y', vtype = GRB.BINARY)
#Auxiliary variable to help model the minimum of a max funtion
aux = f.addVar(name = 'aux', vtype = GRB.CONTINUOUS)
#Two constraints that define that if an arc (eks: from node 1 to port node 2) is chosen, there has to be an established RP in node 1. And vice versa.
c1 = f.addConstrs((y[i,j]<=x[i] for i,j in D), name='Constraint1')
c2 = f.addConstrs((y[i,j]<=x[j] for i,j in D), name='Constraint2')
#Constraints that force vessel to travel from port 1 and end in port 5
c3 = f.addConstr((sum(y[i[0],i[1]] for i in startres)==1), name='Constraint3')
c4 = f.addConstr((sum(y[i[0],i[1]] for i in ends)==1), name='Constraint4')
#Constraint that force vessel to visit all Ports
for i in N:
(sum(y[i,j] for j in P) == 1), 'Constraint5')
#Flow conservation constraint, excluding arcs with start node in port 1 and end node in port 5
c6 = f.addConstrs((y.sum('*',j) - y.sum(j,'*') >= 0 for j in N if j != 0 and j != len(P)-1), name = 'Constraint6')
#Max constraint that holds the maximum distance we want to minimize.
c7 = f.addConstrs((y[i]*D[i] <= aux for i in D), name='MaxConstraint')
##Objective funtion
obj = sum(x[i] * cost[i] for i in N) + (costCCS * aux)
f.setObjective(obj, GRB.MINIMIZE)
• You define your \(y\) variables over \(D\). However, the set \(D\) does not contain the keys \((0,0),(0,3)\) and more. The error occurs, because you build constraint 5 over the sets \(N\) and \(P
\) of which, the combinations \((0,0),(0,3)\) and other are not present in \(D\). The probably easiest way to fix this would be to add an additional \(\texttt{if}\) check when constructing
constraint 5
#Constraint that force vessel to visit all Ports
for i in N:
(sum(y[i,j] for j in P if (i,j) in D) == 1), 'Constraint5')
Best regards,
Please sign in to leave a comment.
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Seminar, 8 May 2013, C. Shonkwiler
8 May 2013, 16:15
Abbeanum, lecture hall 4
The geometry of random polygons
Clayton Shonkwiler, PhD (Dept. of Mathematics, University of Georgia, USA)
Here is a natural question in statistical physics: what is the expected shape of a ring polymer with n monomers in solution? For example, what is the expected radius of gyration or total curvature?
What is the likelihood of knotting? Numerical experiments are essential in this field, but pose some interesting geometric challenges since the space of closed n-gons in 3-space is a nonlinear
submanifold of the larger space of open n-gons.
I will describe a natural probability measure on n-gons of total length 2 which is pushed forward from the standard measure on the Stiefel manifold of 2-frames in complex n-space using methods from
algebraic geometry. We can directly sample the Stiefel manifold in O(n) time, which gives us a fast, direct sampling algorithm for closed n-gons via the pushforward map. We can also explicitly
compute the expected radius of gyration and expected total curvature and even recover some topological information. This talk describes joint work primarily with Jason Cantarella (University of
Georgia) and Tetsuo Deguchi (Ochanomizu University).
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: Problem of the week - 20/02/2012
I am a little bit late today, so this is not my original question, but a borrowed one (I will give source next week). Besides, we will venture into a simple organic chemistry for a change.
Achiral organic compound A has a molar mass of 100 g/mol. From other trials we know that the compound is free of geometric isomerism and doesn't contain quaternary carbon. Complete combustion of 100
mg of the compound gives 372 mg of mixture of H[2]O and CO[2]. Gentle oxidation of the compound A with potassium dichromate gives compound B with molar mass of 98. Compound A doesn't react with
bromine dissolved in carbon tetrachloride.
Give structural formulas of both compounds A and B.
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7.1 The Central Limit Theorem for Sample Means (Averages)
33 7.1 The Central Limit Theorem for Sample Means (Averages)
Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution).
Using a subscript that matches the random variable, suppose:
1. μ[X] = the mean of X
2. σ[X] = the standard deviation of X
If you draw random samples of size n, then as n increases, the random samples [latex]\displaystyle\overline{{X}}[/latex] which consists of sample means, tend to be normally distributed.
[latex]\displaystyle\overline{{X}}[/latex] ~ N ([latex]\displaystyle{\mu}_{x}[/latex], [latex]\displaystyle\frac{{\sigma_{x}}}{{\sqrt{n}}}[/latex])
The central limit theorem for sample means says that if you keep drawing larger and larger samples (such as rolling one, two, five, and finally, ten dice) and calculating their means, the sample
means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided
by, the sample size. The variable n is the number of values that are averaged together, not the number of times the experiment is done.
To put it more formally, if you draw random samples of size n, the distribution of the random variable [latex]\displaystyle\overline{{X}}[/latex], which consists of sample means, is called the
sampling distribution of the mean. The sampling distribution of the mean approaches a normal distribution as the sample size n increases.
The random variable [latex]\displaystyle\overline{{X}}[/latex] has a different z-score associated with it from that of the random variable X. The mean [latex]\displaystyle\overline{x}[/latex] is the
value of [latex]\displaystyle\overline{X}[/latex] in one sample.
z = [latex]\displaystyle\frac{{\overline{x}-{\mu}_{x}}}{{\frac{{{\sigma}_{x}}}{{\sqrt{n}}}}}[/latex]
[latex]\displaystyle{\mu}_{x}[/latex] = [latex]\displaystyle{\mu}_{\overline{x}}[/latex] (mean of X = mean of [latex]\displaystyle\overline{X}[/latex]. )
[latex]\displaystyle{\sigma}_{\overline{x}} = {{\frac{{{\sigma}_{x}}}{{\sqrt{n}}}}}[/latex] = standard deviation of [latex]\displaystyle\overline{{X}}[/latex] and is called the standard error of the
Guide for TI-Calculator:
To find probabilities for means on the TI-calculator, follow these steps:
• “2nd”
• “DISTR”
• “2: normalcdf”
• [latex]\displaystyle\text{ normalcdf ( lower value, upper value, mean,} \frac{\text{standard deviation}}{\sqrt{\text{sample size}}} \text{)} \[/latex]
where: mean is the mean of the original distribution, standard deviation is the standard deviation of the original distribution sample size
Example 1
An unknown distribution has a mean of 90 and a standard deviation of 15. Samples of size n = 25 are drawn randomly from the population.
1. Find the probability that the sample mean is between 85 and 92.
2. Find the value that is two standard deviations above the expected value, 90, of the sample mean.
lower value = 85, upper value = 92, mean [latex]{\mu}[/latex] = 90, std dev [latex]{\sigma}[/latex] = 15, sample size = 25.
1. In this example, the probability that the sample mean is btw 85 and 92 = area btw 85 and 92.
Since 25 samples are drawn, [latex]\displaystyle{\mu}_{x}[/latex] = mean = 90, [latex]\displaystyle{\sigma}_{x}[/latex] = [latex]\displaystyle\frac{{\text{std dev}}}{{\text{sample size}}}[/latex]
= [latex]\displaystyle\frac{{15}}{{\sqrt{25}}}[/latex].By using TI-83/84, normalcdf(85, 92, 90, [latex]\displaystyle\frac{{15}}{{\sqrt{25}}}[/latex] ) = 0.0697
2. To find the value that is two standard deviations above the expected value 90, use the formula:
value = [latex]\displaystyle{\mu}_{x}[/latex] + (# of STD DEV)[latex]\displaystyle\left(\frac{{{\sigma}_{x}}}{{\sqrt{n}}}\right)[/latex]
Value that is 2 std dev above 90
= 90 + 2 ([latex]\displaystyle\frac{{15}}{{\sqrt{25}}}[/latex] )
= 96
The value that is two standard deviations above the expected value is 96.
(Note: The standard error of the mean is [latex]\displaystyle\frac{{\sigma}}{{\sqrt{n}}}[/latex] = [latex]\displaystyle\frac{{15}}{{\sqrt{25}}}[/latex] = 3. )
Recall that the standard error of the mean ( [latex]\displaystyle\frac{{\sigma}}{{\sqrt{n}}}[/latex]) is a description of how far (on average) that the sample mean will be from the population mean in
repeated simple random samples of size n.
Try It
An unknown distribution has a mean of 45 and a standard deviation of 8. Samples of size n = 30 are drawn randomly from the population. Find the probability that the sample mean is between 42 and 50.
[practice-area rows=”2″][/practice-area]
Show Answer
TI-Calculator: normalcdf(42, 50, 45, [latex]\displaystyle\frac{{8}}{{\sqrt{30}}}[/latex])
P(42 < [latex]\displaystyle\overline{x}[/latex] < 50) = 0.9797
Example 2
The length of time, in hours, it takes an “over 40” group of people to play one soccer match is normally distributed with a mean of 2 hours and a standard deviation of 0.5 hours. A sample of size n =
50 is drawn randomly from the population. Find the probability that the sample mean is between 1.8 hours and 2.3 hours.
In this example, mean [latex]{\mu}[/latex] = 2, std dev [latex]{\sigma}[/latex] = 0.5, sample size n = 50
Let [latex]\displaystyle\overline{X}[/latex] = the mean time, in hours, it takes to play one soccer match.
We are looking for P(1.8 < [latex]\displaystyle\overline{x}[/latex] < 2.3).
TI-Calculator: normalcdf (1.8,2.3, 2, [latex]\displaystyle\frac{0.5}{\sqrt{50}}[/latex])
The probability that the mean time is between 1.8 hours and 2.3 hours
= P(1.8 < [latex]\displaystyle\overline{x}[/latex] < 2.3)
= 0.9977.
Try It
The length of time taken on the SAT for a group of students is normally distributed with a mean of 2.5 hours and a standard deviation of 0.25 hours. A sample size of n = 60 is drawn randomly from the
population. Find the probability that the sample mean is between two hours and three hours.
[practice-area rows=”1″][/practice-area]
Show Answer
normalcdf( 2, 3, 2.5, [latex]\displaystyle\frac{{0.25}}{{\sqrt{60}}}[/latex]
P(2 < [latex]\displaystyle\overline{x}[/latex] < 3) = 1
Guide for TI-Calculator:
To find percentiles for means on the calculator, follow these steps.
• 2nd DIStR
• 3:invNorm
• k = invNorm (area to the LEFT of k, mean, [latex]\displaystyle{\frac{{\text{standard deviation}}}{{\sqrt{\text{sample size}}}}}[/latex])
where: k = the k^th percentile, mean is the mean of the original distribution, standard deviation is the standard deviation of the original distribution, sample size = n
Example 3
In a recent study reported Oct. 29, 2012 on the Flurry Blog, the mean age of tablet users is 34 years. Suppose the standard deviation is 15 years. Take a sample of size n = 100.
1. What are the mean and standard deviation for the sample mean ages of tablet users?
2. What does the distribution look like?
3. Find the probability that the sample mean age is more than 30 years (the reported mean age of tablet users in this particular study).
4. Find the 95th percentile for the sample mean age (to one decimal place).
In this example, mean [latex]{\mu}[/latex] = 34 years, std dev [latex]{\sigma}[/latex] = 15 years, sample size n = 100
1. Since the sample mean tends to target the population mean, the mean for the sample mean ages of tablet users μ[X] = μ = 34.
The sample standard deviation for the sample mean ages is given by [latex]\displaystyle\frac{{\sigma}}{{\sqrt{n}}}=\frac{{15}}{{\sqrt{100}}}=\frac{{15}}{{10}}={1.5}[/latex]
2. The central limit theorem states that for large sample sizes(n), the sampling distribution will be approximately normal.
3. TI-Calculator: normalcdf(30,1E99,34,1.5)
The probability that the sample mean age is more than 30 = P(Χ > 30) = 0.9962
4. Let k = the 95th percentile.
TI-Calculator: invNorm(0.95, 34, [latex]\displaystyle\frac{{15}}{{\sqrt{100}}}[/latex])
k = 95^th percentile = 36.5.
Try It
In an article on Flurry Blog, a gaming marketing gap for men between the ages of 30 and 40 is identified. You are researching a startup game targeted at the 35-year-old demographic. Your idea is to
develop a strategy game that can be played by men from their late 20s through their late 30s. Based on the article’s data, industry research shows that the average strategy player is 28 years old
with a standard deviation of 4.8 years. You take a sample of 100 randomly selected gamers. If your target market is 29- to 35-year-olds, should you continue with your development strategy?
Show Answer
You need to determine the probability for men whose mean age is between 29 and 35 years of age wanting to play a strategy game (also known as P(29 < [latex]\displaystyle\overline{x}[/latex] < 35). )
TI-Calculator: normalcdf = 0.0186 (29, 35, 28, [latex]\displaystyle\frac{{4.8}}{{\sqrt{100}}}[/latex])
P(29 < [latex]\displaystyle\overline{x}[/latex] < 35) = 0.0186
You can conclude there is approximately a 2% chance that your game will be played by men whose mean age is between 29 and 35.
Example 4
The mean number of minutes for app engagement by a tablet user is 8.2 minutes. Suppose the standard deviation is one minute. Take a sample of 60.
1. What are the mean and standard deviation for the sample mean number of app engagement by a tablet user?
2. What is the standard error of the mean?
3. Find the 90th percentile for the sample mean time for app engagement for a tablet user. Interpret this value in a complete sentence.
4. Find the probability that the sample mean is between eight minutes and 8.5 minutes.
In this example, mean [latex]{\mu}[/latex] = 8.2, std dev [latex]{\sigma}[/latex] = 1, sample size n = 60
1. The mean for the sample mean number of app engagement by a tablet user = [latex]\displaystyle{\mu}_{\overline{x}}={\mu}=8.2[/latex].
2. The std dev for the sample mean number of app engagement by a tablet user = [latex]\displaystyle{\sigma}_{\overline{x}}=\frac{{\sigma}}{{\sqrt{n}}}=\frac{{1}}{{\sqrt{60}}} = 0.13[/latex]
This allows us to calculate the probability of sample means of a particular distance from the mean, in repeated samples of size 60.
3. Let k = the 90^th percentile.
TI-Calculator: invNorm(0.9, 8.2, [latex]\displaystyle\frac{{1}}{{\sqrt{60}}}[/latex])
k = the 90^th percentile = 8.37.
90 percent of the average app engagement time for table users is less than 8.37 minutes.
4. TI-Calculator: normalcdf(8, 8.5, 8.2[latex]\displaystyle\frac{{1}}{{\sqrt{60}}}[/latex])
P(8 < [latex]\displaystyle\overline{x}[/latex] < 8.5) = 0.9293
Try It
Cans of a cola beverage claim to contain 16 ounces. The amounts in a sample are measured and the statistics are n = 34,[latex]\displaystyle\overline{x}[/latex] = 16.01 ounces. If the cans are filled
so that μ = 16.00 ounces (as labeled) and σ= 0.143 ounces, find the probability that a sample of 34 cans will have an average amount greater than 16.01 ounces. Do the results suggest that cans are
filled with an amount greater than 16 ounces?
Show Answer
Ti-Calculator: normalcdf(16.01, 1E99, 16, [latex]\displaystyle\frac{{0.143}}{{\sqrt{34}}}[/latex] )
P([latex]\displaystyle\overline{x}[/latex] > 16.01) = 0.3417
Since there is a 34.17% probability that the average sample weight is greater than 16.01 ounces, we should be skeptical of the company’s claimed volume. If I am a consumer, I should be glad that I am
probably receiving free cola. If I am the manufacturer, I need to determine if my bottling processes are outside of acceptable limits.
Baran, Daya. “20 Percent of Americans Have Never Used Email.”WebGuild, 2010. Available online at http://www.webguild.org/20080519/20-percent-of-americans-have-never-used-email (accessed May 17,
Data from The Flurry Blog, 2013. Available online at http://blog.flurry.com (accessed May 17, 2013).
Data from the United States Department of Agriculture.
Concept Review
In a population whose distribution may be known or unknown, if the size ( n) of samples is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the
sample means will equal the population mean. The standard deviation of the distribution of the sample means, called the standard error of the mean, is equal to the population standard deviation
divided by the square root of the sample size (n).
Formula Review
The Central Limit Theorem for Sample Means:[latex]\displaystyle\overline{X}{\sim}{N}({\mu}_{x},\frac{{{\sigma}_{x}}}{{\sqrt{n}}})[/latex]
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3,637 research outputs found
We investigate the evaluation of the six-fold integral representation for the second order exchange contribution to the self energy of a three dimensional electron gas at the Fermi surface.Comment: 6
This paper gives a momentum-space representation of the Argonne V18 potential as an expansion in products of spin-isospin operators with scalar coefficient functions of the momentum transfer. Two
representations of the scalar coefficient functions for the strong part of the interaction are given. One is as an expansion in an orthonormal basis of rational functions and the other as an
expansion in Chebyshev polynomials on different intervals. Both provide practical and efficient representations for computing the momentum-space potential that do not require integration or
interpolation. Programs based on both expansions are available as supplementary material. Analytic expressions are given for the scalar coefficient functions of the Fourier transform of the
electromagnetic part of the Argonne V18. A simple method for computing the partial-wave projections of these interactions from the operator expressions is also given.Comment: 61 pages. 26 figure
We solve exactly the scalar box integral using the Mellin-Barnes representation. Firstly we recognize the hypergeometric functions resumming the series coming from the scalar integrals, then we
perform an analytic continuation before applying the Laurent expansion in^2 = (d !' 4)=2 of the result.Comment: 13 pages, no figure
We generalize the discrete quantum walk on the line using a time dependent unitary coin operator. We find an analytical relation between the long-time behaviors of the standard deviation and the coin
operator. Selecting the coin time sequence allows to obtain a variety of predetermined asymptotic wave-function spreadings: ballistic, sub-ballistic, diffusive, sub-diffusive and localized.Comment: 6
pages, 3 figures, appendix added. to appear in PR
We solve the path integral in momentum space for a particle in the field of the Coulomb potential in one dimension in the framework of quantum mechanics with the minimal length given by $(\Delta X)_
{0}=\hbar \sqrt{\beta}$, where $\beta$ is a small positive parameter. From the spectral decomposition of the fixed energy transition amplitude we obtain the exact energy eigenvalues and momentum
space eigenfunctions
Here, we study the effects of stochastic nuclear motions on the electron transport in doped polymer fibers assuming the conducting state of the material. We treat conducting polymers as granular
metals and apply the quantum theory of conduction in mesoscopic systems to describe the electron transport between the metalliclike granules. To analyze the effects of nuclear motions we mimic them
by the phonon bath, and we include the electron-phonon interactions in consideration. Our results show that the phonon bath plays a crucial part in the intergrain electron transport at moderately low
and room temperatures suppressing the original intermediate state for the resonance electron tunneling, and producing new states which support the electron transport.Comment: 6 pages, 4 figures,
minor changes are made in the Fig. 3, accepted for publication in J. of Chem. Phys
The stationary points of the Hamiltonian H of the classical XY chain with power-law pair interactions (i.e., decaying like r^{-{\alpha}} with the distance) are analyzed. For a class of
"spinwave-type" stationary points, the asymptotic behavior of the Hessian determinant of H is computed analytically in the limit of large system size. The computation is based on the Toeplitz
property of the Hessian and makes use of a Szeg\"o-type theorem. The results serve to illustrate a recently discovered relation between phase transitions and the properties of stationary points of
classical many-body Hamiltonian functions. In agreement with this relation, the exact phase transition energy of the model can be read off from the behavior of the Hessian determinant for exponents
{\alpha} between zero and one. For {\alpha} between one and two, the phase transition is not manifest in the behavior of the determinant, and it might be necessary to consider larger classes of
stationary points.Comment: 9 pages, 6 figure
We study non-Gaussianities in the primordial perturbations in single field inflation where there is radiation era prior to inflation. Inflation takes place when the energy density of radiation drops
below the value of the potential of a coherent scalar field. We compute the thermal average of the two, three and four point correlation functions of inflaton fluctuations. The three point function
is proportional to the slow roll parameters and there is an amplification in $f_{NL}$ by a factor of 65 to 90 due to the contribution of the thermal bath, and we conclude that the bispectrum is in
the range of detectability with the 21-cm anisotropy measurements. The four point function on the other hand appears in this case due to the thermal averaging and the fact that thermal averaging of
four-point correlation is not the same as the square of the thermal averaging of the two-point function. Due to this fact $\tau_{NL}$ is not proportional to the slow-roll parameters and can be as
large as -42. The non-Gaussianities in the four point correlation of the order 10 can also be detected by 21-cm background observations. We conclude that a signature of thermal inflatons is a large
trispectrum non-Gaussianity compared to the bispectrum non-Gaussianity.Comment: 17 RevTeX4 pages, 2 figures, One paragraph added in Introduction, No further changes made, Accepted for publication in
We derive the spectrum in the broken phase of a $\lambda\phi^4$ theory, in the limit $\lambda\to\infty$, showing that this goes as even integers of a renormalized mass in agreement with recent
lattice computations.Comment: 4 pages, 1 figure. Accepted for publication in International Journal of Modern Physics
Effects of a Kekule distortion on exciton instability in single-layer graphene are discussed. In the framework of quantum electrodynamics the mass of the electron generated dynamically is worked out
using a Schwinger-Dyson equation. For homogeneous lattice distortion it is shown that the generated mass is independent of the amplitude of the lattice distortion at the one-loop approximation.
Formation of excitons induced by the homogeneous Kekule distortion could appear only through direct dependence of the lattice distortion.Comment: 6 pages, 1 figur
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ESSAY How many ways can 1 computer science major be chosen?How many ways can 1 computer science major be chosen? - AcademiaElites
How many ways can 1 computer science major be chosen?
Posted: March 1st, 2022
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Each answer must have all the steps that you used to arrive at that answer.
Your presentation must be very neat and clear with one step under the one before it. There is only two questions I need answered and that’s it. Question 1. 27. You decide to invest in stock of a
particular type of company and set the guideline that
you will only buy stock on companies that are ranked in the 80th percentile or above in
terms of dividends paid in the previous year. You are looking at a company that ranked
78 of 345 companies that paid dividends in 2019. a. Will this company qualify for your portfolio?
b. If you had the data on the total dividends paid by each of the 345 companies, what would the mean and the median each tell you and how would you compute each measure? Question 2. In a class of 18
students there are 11 math majors and 7 computer science majors. Four students are randomly picked to prepare a demonstrate on the use of a graphing calculator. 1. If one person in the class is
chosen at random to draw the names out of a hat, what is the probability that the person drawing the names is a math major? 2. How many ways can the group of students be formed if there are no
restrictions on composition? 3. How many ways can three math majors be chosen? 4. How many ways can 1 computer science major be chosen? 5. What is the probability that the random selection of the
four-person group will result in three math majors and 1 computer science major?
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Compression and encryption algorithms
• ZLIB algorithm (modified by Davide Moretti dave@rimini.com)
• BZIP algorithm (by Julian Seward, jseward@acm.org.)
• PPM algorithm, variant G (by Dmitry Shkarin, shkarin@arstel.ru)
ECL has 3 levels for each of above algorithms: Fastest, Normal and Max.
Level Meaning
eclNone No compression
zlibFastest ZLib algorithm, Fastest compression
zlibNormal ZLib algorithm, Normal balance between speed and compression rate
zlibMax ZLib algorithm, Maximum compression, rather slow speed
ppmFastest PPM algorithm, Fastest compression
ppmNormal PPM algorithm, Normal balance between speed and compression rate
ppmMax PPM algorithm, Maximum compression, rather slow speed
bzipFastest BZIP algorithm, Fastest compression
bzipNormal BZIP algorithm, Normal balance between speed and compression rate
bzipMax BZIP algorithm, Maximum compression, rather slow speed
Typically ZLib is the fastest algorithm, BZIP is fast and has a good rate, PPM provides the maximum rate with rather low speed.
But it also depends on compression level: i.e. for some files bzipNormal could give you both better compression rate and higher speed than zlibMax.
So we strongly recommend you to test the above compression levels with your application data, to make an optimal choice.
The most advanced users of ECL editions with source code could also tune the parameters of compression algorithms to achieve the best result for their specific tasks.
Encryption algorithm
Password protection is provided by the Rijndael encryption algorithm (AES), 128/256 bit key.Hash algorithm is RipeMD-128 / RipeMD-256.
Implementation of encryption routines is provided by well-known cryptography expert Hagen Reddmann (HaReddmann@AOL.COM), DEC Part I.
Users of the Pro version can switch between a 128-bit and 256-bit key encryption.
See Tuning ECL for more details.
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Neural Networks Principal Component Analysis for estimating the generative multifactor model of returns under a statistical approach to the Arbitrage Pricing Theory. Evidence from the Mexican Stock Exchange
Neural Networks Principal Component Analysis for estimating the generative multifactor model of returns under a statistical approach to the Arbitrage Pricing Theory. Evidence from the Mexican Stock
A nonlinear principal component analysis (NLPCA) represents an extension of the standard principal component analysis (PCA) that overcomes the limitation of the PCA’s assumption about the linearity
of the model. The NLPCA belongs to the family of nonlinear versions of dimension reduction or the extraction techniques of underlying features, including nonlinear factor analysis and nonlinear
independent component analysis, where the principal components are generalized from straight lines to curves. The NLPCA can be achieved via an artificial neural network specification where the PCA
classic model is generalized to a nonlinear mode, namely, Neural Networks Principal Component Analysis (NNPCA). In order to extract a set of nonlinear underlying systematic risk factors, we estimate
the generative multifactor model of returns in a statistical version of the Arbitrage Pricing Theory (APT), in the context of the Mexican Stock Exchange. We used an auto-associative multilayer
perceptron neural network or autoencoder, where the ‘bottleneck’ layer represented the nonlinear principal components, or in our context, the scores of the underlying factors of systematic risk. This
neural network represents a powerful technique capable of performing a nonlinear transformation of the observed variables into the nonlinear principal components, and to execute a nonlinear mapping
that reproduces the original variables. We propose a network architecture capable of generating a loading matrix that enables us to make a first approach to the interpretation of the extracted latent
risk factors. In addition, we used a two stage methodology for the econometric contrast of the APT involving first, a simultaneous estimation of the system of equations via Seemingly Unrelated
Regression (SUR), and secondly, a cross-section estimation via Ordinary Least Squared corrected by heteroskedasticity and autocorrelation by means of the Newey-West heteroskedasticity and
autocorrelation consistent covariances estimates (HEC). The evidence found shows that the reproductions of the observed returns using the estimated components via NNPCA are suitable in almost all
cases; nevertheless, the results in an econometric contrast lead us to a partial acceptance of the APT in the samples and periods studied.
Extraction of underlying risk factors, nonlinear principal component analysis, arbitrage pricing theory, mexican stock exchange
Full Text:
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Infinite Analysis Seminar Tokyo
Seminar information archive ～11/09｜Next seminar｜Future seminars 11/10～
Date, time & place Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.)
Seminar information archive
15:30-16:30 Room # (Graduate School of Math. Sci. Bldg.)
Shin'ichi Arita (The University of Tokyo)
Dirac作用素に対するRellich型の定理について (日本語)
15:30-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Davide Dal Martello
(Rikkyo University)
Convolutions, factorizations, and clusters from Painlevé VI (English)
[ Abstract ]
The Painlevé VI equation governs the isomonodromic deformation problem of both 2-dimensional Fuchsian and 3-dimensional Birkhoff systems. Through duality, this feature identifies the two systems. We
prove this bijection admits a more transparent middle convolution formulation, which unlocks a monodromic translation involving the Killing factorization. Moreover, exploiting a higher Teichmüller
parametrization of the monodromy group, Okamoto's birational map of PVI is given a new realization as a cluster transformation. Time permitting, we conclude with a taste of the quantum version of
these constructions.
10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)
Chiara Franceschini
(University of Modena and Reggio Emilia)
Harmonic models out of equilibrium: duality relations and invariant measure (ENGLISH)
[ Abstract ]
Zero-range interacting systems of Harmonic type have been recently introduced by Frassek, Giardinà and Kurchan [JSP 2020] from the integrable XXX Hamiltonian with non compact spins. In this talk I
will introduce this one parameter family of models on a one dimensional lattice with open boundary whose dynamics describes redistribution of energy or jump of particles between nearest neighbor
sites. These models belong to the same macroscopic class of the KMP model, introduced in 1982 by Kipnis Marchioro and Presutti. First, I will show their similar algebraic structure as well as their
duality relations. Second, I will present how to explicitly characterize the invariant measure out of equilibrium, a task that is, in general, quite difficult in this context and it has been achieved
in very few cases, e.g. the well known exclusion process. As an application, thanks to this characterization, it is possible to compute formulas predicted by macroscopic fluctuation theory.
This is from joint works with: Gioia Carinci, Rouven Frassek, Davide Gabrielli, Cirstian Giarinà, Frank Redig and Dimitrios Tsagkarogiannis.
10:30-12:00 Room #056 (Graduate School of Math. Sci. Bldg.)
John Alex Cruz Morales
(National University of Colombia)
What would be equivariant mirror symmetry for Hitchin systems? (ENGLISH)
[ Abstract ]
In some recent works Aganagic has introduced the idea of equivariant mirror symmetry for certain kind of hyperkahler manifolds. In this talk, after reviewing Aganagic's proposal, we will discuss how
some parts of this framework could be used to study mirror symmetry of Hitchin systems. This is based on work in progress with O. Dumitrescu and M. Mulase.
13:00-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Laszlo Feher
(University of Szeged, Hungary)
Bi-Hamiltonian structures of integrable many-body models from Poisson reduction (ENGLISH)
[ Abstract ]
We review our results on bi-Hamiltonian structures of trigonometric spin Sutherland models
built on collective spin variables.
Our basic observation was that the cotangent bundle $T^*\mathrm{U}(n)$ and its holomorphic analogue $T^* \mathrm{GL}(n,{\mathbb C})$,
as well as $T^*\mathrm{GL}(n,{\mathbb C})_{\mathbb R}$, carry a natural quadratic Poisson bracket,
which is compatible with the canonical linear one. The quadratic bracket arises by change of variables and analytic continuation
from an associated Heisenberg double.
Then, the reductions of $T^*{\mathrm{U}}(n)$ and $T^*{\mathrm{GL}}(n,{\mathbb C})$ by the conjugation actions of the
corresponding groups lead to the real and holomorphic spin Sutherland models, respectively, equipped
with a bi-Hamiltonian structure. The reduction of $T^*{\mathrm{GL}}(n,{\mathbb C})_{\mathbb R}$ by the group $\mathrm{U}(n) \times \mathrm{U}(n)$ gives
a generalized Sutherland model coupled to two ${\mathfrak u}(n)^*$-valued spins.
We also show that
a bi-Hamiltonian structure on the associative algebra ${\mathfrak{gl}}(n,{\mathbb R})$ that appeared in the context
of Toda models can be interpreted as the quotient of compatible Poisson brackets on $T^*{\mathrm{GL}}(n,{\mathbb R})$.
Before our work, all these reductions were studied using the canonical Poisson structures of the cotangent bundles,
without realizing the bi-Hamiltonian aspect.
Finally, if time permits, the degenerate integrability of some of the reduced systems
will be explained as well.
[1] L. Feher, Reduction of a bi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$
to spin Ruijsenaars--Sutherland models, Lett. Math. Phys. 110, 1057-1079 (2020).
[2] L. Feher, Bi-Hamiltonian structure of spin Sutherland models: the holomorphic case, Ann. Henri Poincar\'e 22, 4063-4085 (2021).
[3] L. Feher, Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n)^*$-valued spins from Poisson reduction,
Nonlinearity 35, 2971-3003 (2022).
[4] L. Feher and B. Juhasz,
A note on quadratic Poisson brackets on $\mathfrak{gl}(n,\mathbb{R})$ related to Toda lattices,
Lett. Math. Phys. 112:45 (2022).
[5] L. Feher,
Notes on the degenerate integrability of reduced systems obtained from the master systems of free motion on cotangent bundles of
compact Lie groups, arXiv:2309.16245
13:00-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Misha Feigin
(University of Glasgow)
This seminar has been cancelled.
Flat coordinates of algebraic Frobenius manifolds (ENGLISH)
[ Abstract ]
This seminar has been cancelled.
Orbit spaces of the reflection representation of finite irreducible Coxeter groups provide Frobenius manifolds with polynomial prepotentials. Flat coordinates of the corresponding flat metric, known
as Saito metric, are distinguished basic invariants of the Coxeter group. They have applications in representations of Cherednik algebras. Frobenius manifolds with algebraic prepotentials remain not
classified and they are typically related to quasi-Coxeter conjugacy classes in finite Coxeter groups. We obtain flat coordinates for the majority of known examples of algebraic Frobenius manifolds
in dimensions up to 4. In all the cases, flat coordinates appear to be some algebraic functions on the orbit space of the Coxeter group. This is a joint work with Daniele Valeri and Johan Wright.
16:00-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)
Takahiko Nobukawa
(Kobe University )
Euler type integral formulas and hypergeometric solutions for
variants of the $q$ hypergeometric equations.
[ Abstract ]
We know that Papperitz's differential equation is essentially obtained from
Gauss' hypergeometric equation by applying a Moebius transformation,
implying that we have Euler type integral formulas or hypergeometric solutions.
The variants of the $q$ hypergeometric equations, introduced by
Hatano-Matsunawa-Sato-Takemura (Funkcial. Ekvac.,2022), are second order
$q$-difference systems which can be regarded as $q$ analoges of Papperitz's equation.
This motivates us for deriving Euler type integral formulas and hypergeometric solutions
for the pertinent $q$-difference systems. If time admits, I explain
the relation with $q$-analogues of Kummer's 24 solutions,
or the variants of multivariate $q$-hypergeometric functions.
This talk is based on the collaboration with Taikei Fujii.
15:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Vincent Pasquier
(IPhT Saclay)
Hydrodynamics of a one dimensional lattice gas.
[ Abstract ]
The simplest boxball model is a one dimensional lattice gas obtained as
a certain (cristal) limit of the six vertex model where the evolution
determined by the transfer matrix becomes deterministic. One can
study its thermodynamics in and out of equilibrium and we shall present
preliminary results in this direction.
Collaboration with Atsuo Kuniba and Grégoire Misguich.
15:00-16:00 Room #駒場国際教育研究棟（旧６号館）108 (Graduate School of Math. Sci. Bldg.)
Ryo Ohkawa
(Waseda University)
(-2) blow-up formula (JAPANESE)
[ Abstract ]
In this talk, we will consider the moduli of ADHM data
corresponding to the affine A_1 Dynkin diagram.
It is a moduli of framed sheaves on the (-2) curve or the projective
plane with a group action.
Each of these two types of moduli integrals has a combinatorial
description. In particular, the Hirota derivative of the Nekrasov
function can be obtained on the (-2) curve.
We introduce equalities among these two integrals and the
corresponding functional equations in some cases.
This is similar to the blow-up formula by Nakajima-Yoshioka.
I would also like to talk about relationships with the study of the
Painleve tau function by Bershtein-Shchechkin.
16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Takashi Takebe
(National Research University Higher School of Economics (Moscow))
Q-operators for generalised eight vertex models associated
to the higher spin representations of the Sklyanin algebra. (ENGLISH)
[ Abstract ]
The Q-operator was first introduced by Baxter in 1972 as a
tool to solve the eight vertex model and recently attracts
attention from the representation theoretical viewpoint. In
this talk, we show that Baxter's apparently quite ad hoc and
technical construction can be generalised to the model
associated to the higher spin representations of the
Sklyanin algebra. If everybody in the audience understands Japanese, the talk
will be in Japanese.
16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Francesco Ravanini
(University of Bologna)
Integrability and TBA in non-equilibrium emergent hydrodynamics (ENGLISH)
[ Abstract ]
The paradigm of investigating non-equilibrium phenomena by considering stationary states of emergent hydrodynamics has attracted a lot of attention in the last years. Recent proposals of an exact
approach in integrable cases, making use of TBA techniques, are presented and discussed.
16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Andrew Kels
(Graduate School of Arts and Sciences, University of Tokyo)
Integrable quad equations derived from the quantum Yang-Baxter
equation. (ENGLISH)
[ Abstract ]
I will give an overview of an explicit correspondence that exists between
two different types of integrable equations; 1) the quantum Yang-Baxter
equation in its star-triangle relation (STR) form, and 2) the classical
3D-consistent quad equations in the Adler-Bobenko-Suris (ABS)
classification. The fundamental aspect of this correspondence is that the
equation of the critical point of a STR is equivalent to an ABS quad
equation. The STR's considered here are in fact equivalent to
hypergeometric integral transformation formulas. For example, a STR for
$H1_{(\varepsilon=0)}$ corresponds to the Euler Beta function, a STR for
$Q1_{(\delta=0)}$ corresponds to the $n=1$ Selberg integral, and STR's for
$H2_{\varepsilon=0,1}$, $H1_{(\varepsilon=1)}$, correspond to different
hypergeometric integral formulas of Barnes. I will discuss some of these
examples and some directions for future research.
16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Nobutaka Nakazono
(Aoyama Gakuin University Department of Physics and Mathematics)
Classification of quad-equations on a cuboctahedron (JAPANESE)
[ Abstract ]
Adelr-Bobenko-Suris (2003, 2009) and Boll (2011) classified quad-equations on a cube using a consistency around a cube. By use of this consistency, we can define integrable two-dimensional partial
difference equations called ABS equations. A major example of ABS equation is the lattice modified KdV equation, which is a discrete analogue of the modified KdV equation. It is known that Lax
representations and Backlund transformations of ABS equations can be constructed by using the consistency around a cube, and ABS equations can be reduced to differential and difference Painlevé
equations via periodically reductions.
In this talk, we show a classification of quad-equations on a cuboctahedron using a consistency around a cuboctahedron and the relation between a resulting partial difference equation and a discrete
Painlevé equation.
This work has been done in collaboration with Prof Nalini Joshi (The University of Sydney).
16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Valerii Sopin:
(Higher School of Economics (Moscow))
Operator algebra for statistical model of square ladder (ENGLISH)
[ Abstract ]
In this talk we will define operator algebra for square ladder on the basis
of semi-infinite forms.
Keywords: hard-square model, square ladder, operator algebra, semi-infinite
forms, fermions, quadratic algebra, cohomology, Demazure modules,
Heisenberg algebra.
15:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Naoki Genra
(RIMS, Kyoto U.)
Screening Operators and Parabolic inductions for W-algebras
[ Abstract ]
(Affine) W-algebras are the family of vertex algebras defined by
Drinfeld-Sokolov reductions. We introduce the free field realizations of
W-algebras by the Wakimoto representations of affine Lie algebras, which
we call the Wakimoto representations of W-algebras. Then W-algebras may be
described as the intersections of the kernels of the screening operators.
As applications, the parabolic inductions for W-algebras are obtained.
This is motivated by results of Premet and Losev on finite W-algebras. In
A-types, this becomes a chiralization of coproducts by Brundan-Kleshchev.
In BCD-types, we also have analogue results in special cases.
15:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Yoshiki Fukusumi
(The University of Tokyo, The Institute for Solid State Physics)
Schramm-Loewner evolutions and Liouville field theory (JAPANESE)
[ Abstract ]
Schramm-Loewner evolutions (SLEs) are stochastic processes driven by Brownian motions which preserves conformal invariance. They describe the cluster boundaries associated with the minimal models of
the conformal field theory, including the Ising model and the percolation as typical examples. The correlation functions of such models remarkably satisfy the martingale condition. We briefly review
some known results. Then we analyse the time reversing procedure of Schramm Loewner evolutions and its relation to Liouville field theory or 2d pure gravity. We can get martingale observables by the
calculation of the correlation functions of Liouville field theory without matter.
15:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Hosho Katsura
(Department of Physics, Graduate School of Science, The Univeristy of Tokyo ) 15:00-16:00
Sine-square deformation of one-dimensional critical systems (ENGLISH)
[ Abstract ]
Sine-square deformation (SSD) is one example of smooth boundary conditions that have significantly smaller finite-size effects than open boundary conditions. In a one-dimensional system with SSD, the
interaction strength varies smoothly from the center to the edges according to the sine-square function. This means that the Hamiltonian of the system is inhomogeneous, as it lacks translational
symmetry. Nevertheless, previous studies have revealed that the SSD leaves the ground state of the uniform chain with periodic boundary conditions (PBC) almost unchanged for critical systems. In
particular, I showed in [1,2,3] that the correspondence is exact for critical XY and quantum Ising chains. The same correspondence between SSD and PBC holds for Dirac fermions in 1+1 dimension and a
family of more general conformal field theories. If time permits, I will also introduce more recent results [4,5] and discuss the excited states of the SSD systems.
[1] H. Katsura, J. Phys. A: Math. Theor. 44, 252001 (2011).
[2] H. Katsura, J. Phys. A: Math. Theor. 45, 115003 (2012).
[3] I. Maruyama, H. Katsura, T. Hikihara, Phys. Rev. B 84, 165132 (2011).
[4] K. Okunishi and H. Katsura, J. Phys. A: Math. Theor. 48, 445208 (2015).
[5] S. Tamura and H. Katsura, Prog. Theor. Exp. Phys 2017, 113A01 (2017).
Ryo Sato
(Graduate School of Mathematical Sciences, The University of Tokyo) 16:30-17:30
Modular invariant representations of the $N=2$ vertex operator superalgebra (ENGLISH)
[ Abstract ]
One of the most remarkable features in representation theory of a (``good'') vertex operator superalgebra (VOSA) is the modular invariance property of the characters. As an application of the
property, M. Wakimoto and D. Adamovic proved that all the fusion rules for the simple $N=2$ VOSA of central charge $c_{p,1}=3(1-2/p)$ are computed from the modular $S$-matrix by the so-called
Verlinde formula. In this talk, we present a new ``modular invariant'' family of irreducible highest weight modules over the simple $N=2$ VOSA of central charge $c_{p,p'}:=3(1-2p'/p)$. Here $(p,p')$
is a pair of coprime integers such that $p,p'>1$. In addition, we will discuss some generalization of the Verlinde formula in the spirit of Creutzig--Ridout.
17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)
Fabio Novaes
(International Institute of Physics (UFRN))
Chern-Simons, gravity and integrable systems. (ENGLISH)
[ Abstract ]
It is known since the 80's that pure three-dimensional gravity is classically equivalent to a Chern-Simons theory with gauge group SL(2,R) x SL(2,R). For a given set of boundary conditions, the
asymptotic classical phase space has a central extension in terms of two copies of Virasoro algebra. In particular, this gives a conformal field theory representation of black hole solutions in 3d
gravity, also known as BTZ black holes. The BTZ black hole entropy can then be recovered using CFT. In this talk, we review this story and discuss recent results on how to relax the BTZ boundary
conditions to obtain the KdV hierarchy at the boundary. More generally, this shows that Chern-Simons theory can represent virtually any integrable system at the boundary, given some consistency
conditions. We also briefly discuss how this formulation can be useful to describe non-relativistic systems.
[ Reference URL ]
17:30-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Soichi Okada
(Graduate School of Mathematics, Nagoya University)
$Q$-functions associated to the root system of type $C$ (JAPANESE)
[ Abstract ]
Schur $Q$-functions are a family of symmetric functions introduced
by Schur in his study of projective representations of symmetric
groups. They are obtained by putting $t=-1$ in the Hall-Littlewood
functions associated to the root system of type $A$. (Schur
functions are the $t=0$ specialization.) This talk concerns
symplectic $Q$-functions, which are obtained by putting $t=-1$
in the Hall-Littlewood functions associated to the root system
of type $C$. We discuss several Pfaffian identities as well
as a combinatorial description for them. Also we present some
positivity conjectures.
14:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Yuta Nozaki
(Graduate School of Mathematical Sciences, the University of Tokyo) 14:00-15:30
Homology cobordisms over a surface of genus one (JAPANESE)
[ Abstract ]
Morimoto showed that some lens spaces have no genus one fibered knot,
and Baker completely determined such lens spaces.
In this talk, we introduce our results for the corresponding problem
formulated in terms of homology cobordisms.
The Chebotarev density theorem and binary quadratic forms play a key
role in the proof.
Shunsuke Tsuchioka
(Graduate School of Mathematical Sciences, the University of Tokyo) 16:00-17:30
Generalization of Schur partition theorem (JAPANESE)
[ Abstract ]
The celebrated Rogers-Ramanujan partition theorem (RRPT) claims that
the number of partitions of n whose parts are ¥pm1 modulo 5
is equinumerous to the number of partitions of n whose successive
differences are
at least 2. Schur found a mod 6 analog of RRPT in 1926.
We will report a generalization for odd $p¥geq 3$ via representation
theory of quantum groups.
At p=3, it is Schur's theorem. The statement for p=5 was conjectured by
Andrews in 1970s
in a course of his 3 parameter generalization of RRPT and proved in 1994
by Andrews-Bessenrodt-Olsson with an aid of computer.
This is a joint work with Masaki Watanabe (arXiv:1609.01905).
15:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Yohei Kashima
(Graduate School of Mathematical Scineces, The University of Tokyo)
Superconducting phase in the BCS model with imaginary
magnetic field (JAPANESE)
[ Abstract ]
We prove that in the BCS model with an imaginary magnetic field
at positive temperature a spontaneous symmetry breaking (SSB) and
an off-diagonal long range order (ODLRO) occur. Here the BCS model
is meant to be a self-adjoint operator on the Fermionic Fock space,
consisting of a free part describing the electrons' nearest neighbor
hopping and a quartic interacting part describing a long range
interaction between Cooper pairs. The interaction with the imaginary
magnetic field is given by the z-component of the spin operator
multiplied by a pure imaginary parameter. The SSB and the ODLRO are
shown in the infinite-volume limit of the thermal average over the
full Fermionic Fock space. The insertion of the imaginary magnetic
field changes the gap equation. Consequently the SSB and the ODLRO
are shown in high temperature, weak coupling regimes where these
phenomena do not take place in the conventional BCS model. The proof
is based on the method of Grassmann integration.
15:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Ryou Sato
(Graduate School of Mathematical Scineces, The University of Tokyo)
Non-unitary highest-weight modules over the $N=2$ superconformal algebra (JAPANESE)
[ Abstract ]
The $N=2$ superconformal algebra is a generalization of the Virasoro algebra having the super symmetry.
The character formulas associated with the unitary highest weight representations
are expressed in terms of the classical theta functions, and have the remarkable
modular invariance. Based on the method of the $W$-algebras,
Kac and Wakimoto, on the other hand, showed that the
characters for a certain class of non-unitary highest weight representations
can be written in terms of the mock theta functions associated with the affine ${sl}_{2|1}$.
Then they found a way to identify these formulas with
real analytic modular forms by using the correction terms given by Zwegers.
In this seminar, we explain a way to construct the above mentioned
non-unitary representations from the representations of the algebra affine ${sl}_{2}$,
based on the Kazama-Suzuki coset construction, namely not from the $W$-algebra method.
We also investigate the relations between the mock theta functions and the ordinary
theta functions, appearing in this method.
13:30-15:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Vincent Pasquier
(IPhT Saclay)
q-Bosons, Toda lattice and Baxter Q-Operator (ENGLISH)
[ Abstract ]
I will use the Pieri rules of the Hall Littlewood polynomials to construct some
lattice models, namely the q-Boson model and the Toda Lattice Q matrix.
I will identify the semi infinite chain transfer matrix of these models with well known
half vertex operators. Finally, I will explain how the Gaudin determinant appears in the evaluation
of the semi infine chain scalar products for an arbitrary spin and is related to the Macdonald polynomials.
14:00-15:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Simon Wood
(The Australian National University)
Classifying simple modules at admissible levels through
symmetric polynomials (ENGLISH)
[ Abstract ]
From infinite dimensional Lie algebras such as the Virasoro
algebra or affine Lie (super)algebras one can construct universal
vertex operator algebras. These vertex operator algebras are simple at
generic central charges or levels and only contain proper ideals at so
called admissible levels. The simple quotient vertex operator algebras
at these admissible levels are called minimal model algebras. In this
talk I will present free field realisations of the universal vertex
operator algebras and show how they allow one to elegantly classify
the simple modules over the simple quotient vertex operator algebras
by using a deep connection to symmetric polynomials.
14:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Simon Wood
(The Australian National University)
Classifying simple modules at admissible levels through symmetric polynomials (ENGLISH)
[ Abstract ]
From infinite dimensional Lie algebras such as the Virasoro
algebra or affine Lie (super)algebras one can construct universal
vertex operator algebras. These vertex operator algebras are simple at
generic central charges or levels and only contain proper ideals at so
called admissible levels. The simple quotient vertex operator algebras
at these admissible levels are called minimal model algebras. In this
talk I will present free field realisations of the universal vertex
operator algebras and show how they allow one to elegantly classify
the simple modules over the simple quotient vertex operator algebras
by using a deep connection to symmetric polynomials.
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Lesson 1
Lines, Angles, and Curves
Lesson Narrative
Students begin this lesson by discussing what they notice and wonder about several images of circles that contain different kinds of line segments. Then, they are introduced to the vocabulary terms
chord (a segment whose endpoints are on a circle), central angle (an angle formed by 2 rays whose endpoints are the center of the same circle), and arc (the portion of a circle between 2 endpoints).
Students write definitions of these terms based on examples and non-examples. Then, they prove a property of congruent chords.
The definitions developed in this lesson are foundational for the rest of the unit. For example, students will use their knowledge of chords when analyzing inscribed angles. They’ll use central angle
measurements to find areas of sectors and lengths of arcs, and they will define the radian measure of a central angle as the ratio of the length of the arc it defines to the radius of the circle.
Students attend to precision (MP6) as they write careful definitions of vocabulary terms.
Learning Goals
Teacher Facing
• Comprehend (in spoken and written language) the definitions of chord, arc, and central angle.
Student Facing
• Let’s define some line segments and angles related to circles.
Student Facing
• I know what chords, arcs, and central angles are.
CCSS Standards
Building On
Building Towards
Glossary Entries
• arc
The part of a circle lying between two points on the circle.
• central angle
An angle formed by two rays whose endpoints are the center of a circle.
• chord
A chord of a circle is a line segment both of whose endpoints are on the circle.
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Magnetic: The Game of Games is a puzzle game set on a deserted tropical island. The game was developed by small Australian Developer Mulawa Dreaming, and is the follow-up to the previous game,
. The subsequent game by the same developer is
Magicama: Beyond Words
. You wander around the island and play against the computer AI in 16 different games. At the beginning of the game you choose out of six companions to accompany you around the island and help
explain the puzzles. A new version of the game has also been released,
Magnetic Revisited
There are 16 games created for you on the island, and the first is revealed by a storm at the outset. Each of the other games is "guarded" by a puzzle that must first be solved. Once you have
completed 3 puzzles and games, the Transporter will be available from the top left corner of the screen. This allows you to jump quickly to puzzles and games you have previously seen.
You do not have to complete the game in the order outlined below; it depends on which directions you travel as you explore the island. The first paragraph below each of the puzzle/game pairs gives
directions on finding them (in terms of movements from your starting position on the beach).
2D Nim
Right twice, then forward.
This isn't actually a puzzle, it is just the introduction to the game.
The game starts with a random number of metallic pieces on the beach, and a lever at the bottom right. The aim of the game is to remove the last piece - with each turn between 1 and 3 pieces can be
removed. The technique to win is to always leave the computer with a multiple of 4 pieces to choose from.
Press the button in the middle of the lever to start the game. Once the pieces are displayed, if a multiple of 4 is present, move the handle up to the left (to make the computer go first); if not,
move it down to the right. When you have removed as many pieces each turn as you want (to leave a multiple of 4), push the lever to signify the end of your turn. Win the game a number of times to
light up all the indicators on the lever.
Queens and 3D Nim
Right 3 times, forward 10 times, left twice, forward 4 times and right.
You must place one queen in each row of squares so that no 2 queens exist on any column or diagonal. Reading from top to bottom, for the 4x4 grid they should be placed in squares 2, 4, 1, 3. For the
6x6 grid, place them in 2, 4, 6, 1, 3, 5. Finally, for the 8x8 grid, place the queens in squares 2, 5, 7, 4, 1, 8, 6, 3. Now go forward twice to reach the game.
The aim here is the same as for the game in 2D Nim - you must remove the final piece. This time you can remove as many pieces as you want as long as they are of the same color. The way to win is to
put the opponent in a position where there are 2 colors remaining and they have an equal number of pieces. Other positions you should be able to win from are 1, 2 and 3 pieces of the three colors,
and similarly 1, 4 and 5.
Coins and Mancala
Left 3 times, forward 8 times, right, forward twice, left, forward 7 times (to a "no fishing" sign), and forward 3 more times.
There are sixteen coins of various types, with two already placed in position. The coins represent the different pieces in a game of chess, and need to be arranged as if the game is about to begin.
Make sure the back row (rook, knight, bishop, queen, king, bishop, knight, rook) is formed with the coins touching each other, and space the pawns out in front equally.
The object of this game is to collect more pieces in your goal on the right than your opponent does on the left. You choose one of your bowls when it is your turn, and distribute the number of
objects one at a time into the following bowls in an anti-clockwise direction (e.g. 3 pieces would be spread out over the following 3 bowls). If the last piece is placed in your goal, you get another
turn. If the last piece is placed in an empty space and there are pieces in the opposite bowl, all are deposited in your goal. If you or your opponent clears all pieces on your side, the other player
gets to claim all remaining pieces as their own.
One winning series of moves from the start is to move your pieces from bowls 3, 6, 5, 6, 1, 2. The rest of the game should be straight-forward.
Music and XOX
Right 6 times, forward 5 times, left and forward 8 times.
Behind the rock are 5 pressure plates. Pressing these causes music to play. Once you can remember which piece of music goes with each plate, pull the lever. As each sample of music is played, press
the corresponding plate. Get all 5 correct to proceed.
Two robots are in a room playing tic-tac-toe against each other (must get 3 pieces of the same color in any row, column or diagonal). You don't have direct control over where the left robot places
your gold pieces. After losing 5 times, click on your robot and place the various options in the following order from left to right:
• 3 blue
• 2 blue
• 1 blue
• 1 blue, 1 gold
• 2 blue, 1 gold
• 1 blue, 2 gold
• (blank)
• 1 gold
• 2 gold
• 3 gold
Make sure the 3 blue are to the very far left, and the 3 gold are to the very far right, but keep all the others relatively close to the middle. Your robot will still lose but should eventually
achieve 4 consecutive draws.
Map and Checkers
Right 3 times, forward 8 times and left.
There is no actual puzzle here; you just need to see the map to reveal the hotspot to trigger the game. Once you have seen the map, go right, back 8 times, right 7 times, forward twice, left, forward
and left again. Click on the dark area near the base of the trees.
This is a simple game of checkers/draughts. Your white pieces can move on diagonals only, and can only go forward until they reach the far side and become "queens", after which point they can also
move backward. You take the opponent's pieces by jumping diagonally over them. The aim is to remove your opponent's pieces. The best opening moves are to move your 2 corner pieces towards the middle
of the board. After this it is not too difficult to win.
Numbers and Fitchneal
Right 3 times, forward 10 times, left twice and forward 3 times. Pick up the small tin on the ground.
The tin contains the controls to make equations consisting of the numbers 1-4 and the standard arithmetic symbols. Your job is to make equations that result in the numbers 0 through 7 - each equation
must use each of the numbers 1 through 4, and each operator only once. Note that multiply and divide take precedence over add and subtract. There are many possible solutions, but here is a working
• 0 = 1 + 4 / 2 - 3
• 1 = 2 + 3 - 4 x 1
• 2 = 2 x 3 - 4 / 1
• 3 = 4 x 1 - 3 + 2
• 4 = 4 / 2 - 1 + 3
• 5 = 4 x 1 + 3 - 2
• 6 = 4 x 2 - 3 + 1
• 7 = 4 / 2 x 3 + 1
You can choose to either defend (easier) where you must make the king (central piece) escape to the edge of the board, or attack (harder) where you must surround the king on all four sides. Pieces
can be removed from the board by surrounding them on two opposite sides. The defending player always goes first, and pieces can be moved as rooks in a game of chess (as far as you want either
horizontally or vertically). If you label the columns A-I from left to right, and the rows 1-9 from top to bottom, the following is a winning series of moves for the defending player that will work
most of the time: E3-C3, E4-C4, D5-D4, F5-F4, E5-E3, C3-C1, C4-C2, E3-B3, B3-B1.
Shapes and Hexagons
Left 3 times, forward 8 times, right, forward twice, left, forward 7 times (to a "no fishing" sign), left and forward.
Click on the turtle and you will see 2 patterns displayed on the rocks. The aim is to make the left pattern exactly the same as the right one. The four numbers control the appearance of the left
image. The first number changes the color; it cycles through red, yellow, dark blue, orange, green, purple, white and light blue. The second number changes the shape; it cycles through octagon,
hexagon, pentagon, square, triangle, star and straight line. The third number changes the number of repetitions. The fourth number changes the speed of rotation (1 = fast to the left, 4 = none, 7 =
fast to the right).
Click on the turtle and change the numbers until you have matched the pattern correctly. Repeat this twice more and the game will be revealed. The requested patterns change every time, so specific
answers cannot be given here.
The aim is to create a continuous line of orange hexagons from left to right across the board before the opponent forms a blue line from top to bottom. The basic strategy is to always leave yourself
two options. If you want to block the opponent, place a piece not directly adjacent to his, but two steps away. Win 8 times to complete the game.
Balls and Boxes
Left 3 times, forward 8 times, right, forward twice, left, forward 7 times (to a "no fishing" sign), forward and left.
This puzzle consists of a grid of 25 spaces that can be either gold or blank. There is also a hint book just to the right of the grid. The aim is to turn all of the spaces gold. When you click on a
space, all of the surrounding spaces also change their status. The way to complete the puzzle is to start from the second row and work down - click on spaces in the second row to change all of the
first row to gold, then move down a row and continue. When you reach the bottom, the spaces will be in one of the configurations shown in the hint book. Note that in this book, there will be one or
two spaces circled in the picture corresponding to your current configuration. Click on these two spaces, then work down from the second row again and you will complete the puzzle. You must complete
it several times to open the passage to the game.
This game involves capturing as many squares on the board as possible; this is achieved by completing all four sides of the square (at which point its color changes to orange). Each turn involves
drawing in one line - if you complete a square you get another turn. The basic strategy is to avoid completing 3 sides of a square if at all possible, because the opponent could then complete it with
his turn. Start the game by completing any squares with 3 sides completed, then play a series of safe moves. Eventually someone will have to make 3 sides of a square, so if you must, try to do this
in a region where you will only give away a run of a few squares. The player with the most squares at the end of the round wins. Win 8 rounds to finish the game.
Colors and Pinball
Left 3 times, forward twice, left, forward, left twice and forward 3 times. Look in the fallen tree to see a scrap of paper containing an important color clue. Now go back 3 times, right twice, back,
right, back twice, left 4 times, forward 5 times and right.
Click on the Wazzidor sign and the letters will change color. Click on each of the letters so that the first two match the first color from your clue, then second two match the next color and so on.
Once you are done, go left to see the game.
You must get 8 balls to the left side of the screen before the opponent gets 8 to the right. On each turn, you can drop a ball from one of four positions at the top; balls will stop when they hit
platforms, and will disturb the platforms when the hit the attached levers. If you manage to clear the play field with your ball, you get another go (unless the field was empty when you started). It
is important to note that two balls cannot balance on top of each other. Simply win this 8 times to continue.
Cube and Rectangles
Left 3 times and forward 5 times.
Build a cube from the bottom up. The correct order of pieces is purple, light blue, pink, dark blue, yellow, orange and green.
The aim here is to place your horizontal 2x1 blocks so that you get the last move. In the 3x3 grid you must start by placing a piece in the middle row. In the 4x4 grid you should start with a piece
in the middle of the 2nd row, and place your next piece offset to this one row below. In the 5x5 grid up to the 8x8 grid, always start in the second row and remember to try to create spaces where the
opponent's vertical pieces cannot fit.
Anagrams and Crosswords
Left 3 times, forward 8 times, right, forward twice, left, forward 7 times (to a "no fishing" sign), right and forward.
This puzzle involves anagrams. You must rearrange the letters provided to form a word (click on each letter in turn). To scramble the letters, click just to their left. To get a new set of letters,
click just to their right. You must solve a series of 6 anagrams correctly to reveal the passage down towards the game. There is a hint given by your companion that the first letter is always A-D.
Probably the easiest game of the lot - simple scrabble variant where every letter is worth 1 point, you only have 1 letter to place per turn, and the board is only 5x5 in size. Just score more points
than the opponent to win the game.
Dissections and Bowling
Right 3 times, forward 11 times and left once.
Slide the oddly-shaped pieces into the square to the bottom left. The large triangle must go at the bottom left corner and the square in the middle (at an angle). Once this is complete, click on the
tiles and you will see a black screen. Click the mouse to turn on a flashlight and move the mouse to reveal a faded photograph of a beach scene. Note the small fractal image in the bottom left. Go to
this location by heading right, then backward 11 times. Click where the fractal was to trigger the game.
The goal is to bowl over the last pin, after taking it in turns with the computer to knock down either one or two at a time. The major strategy is to create a "symmetrical" pattern of remaining pins
(paired groups of 1-7 pins) and then mirror whatever move the computer makes with a similar move of your own. In this way, there will always be 2 pins left at the end, with the computer knocking down
one, and then you knocking down the other.
Sequences and Chess
Left 3 times, forward 8 times, right, forward twice and left twice.
Pick up the calculator, and it will start displaying a sequence of numbers. All you have to do is give the next number in the sequence, then click the "Enter" button. The following are the correct
• Whole Numbers - 1, 2, 3, 4, 5 - 6
• Odd Numbers - 1, 3, 5, 7, 9 - 11
• Even Numbers - 2, 4, 6, 8, 10 - 12
• Primes - 2, 3, 5, 7, 11 - 13
• Squares - 1, 4, 9, 16, 25 - 36
• Powers of Two - 1, 2, 4, 8, 16 - 32
• Fibonacci - 1, 1, 2, 3, 5 - 8
What follows is a simplified version of chess played on a 5x5 board. You might like to experiment and win this by yourself, otherwise here is a series of moves that will always win (numbers go
upwards 1-5, letters go across A-E). Note that if the opponent does not play the moves indicated in brackets, you will have to change strategy slightly, but always start with this opening move and
things should be relatively easy.
• E2 - E3 (D4 x E3)
• D2 x E3 (A4 - A3)
• B2 x A3 (B4 - B3)
• A2 x B3 (C4 x B3)
• C2 x B3 (D5 x E3)
• A1 - A2 (E3 x D1)
• E1 x D1 (B5 - E2)
• C1 - D2 (E2 x D2)
• D1 x D2 (A5 - B5)
• B3 - B4 (C5 - E3)
• B1 - B3 (E5 - D5)
• D2 x D5 (E3 - C5)
• A3 - A4
Tessellation and Pawns
Left 5 times.
Look at the tessellation mat, which is an almost blank area with a stack of hexagons, squares and triangles. Starting from the right edge, simply slot the pieces into place to form a repeating
pattern across the mat (the pattern is a hexagon surrounded by 6 squares and 6 triangles - it will become obvious as you start).
In this game, the pawns cannot take each other, than can only move forward and block. The winner is the person to make the last move (before all moves are blocked). Pawns on the first row can move 1
or two spaces, all others only one. The way to win is to conserve any pawns on your first row until right at the end, so that their move determines who goes last.
Keypad and Links
Left 3 times, forward 9 times.
Pick up the crumpled piece of paper, which contains the following sequence - 1, 2, 6, 24, 120, 720. Now go back, right, forward twice, left, forward twice and right. Enter the next number in the
sequence (5040) to pass the keypad.
Once again, the aim is to have the last move. The green square is the current position, and the black squares are places where you can move (the red ones are blocked, and you cannot go back over your
blue path). The secret to winning is always going to an area where there is an odd number of black squares remaining.
Dominoes and Paint Cans
Left 3 times, forward 8 times, right, forward twice, left, forward twice and left.
Pull the lever to turn on the power, then turn left. This puzzle is essentially "Dominoes Patience", where pairs of dominoes can be removed when their adjacent colors match. Always try to plan your
moves so that two more dominoes are brought together that match in color and this puzzle is quite simple.
The goal is to fill the container on the right with 27 gallons of blue liquid. The cans contain 5, 4, 3, 2 and 1 gallon each from front to back. The correct technique to win is to match the
computer's move with one to total 6 each time, until you get the opportunity to make the total 21 - in this way you can always take the last move and win the game.
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Performs type conversion from an arbitrary input type to a type that is expressed by a QuantizedType. More...
Performs type conversion from an arbitrary input type to a type that is expressed by a QuantizedType.
This handles cases where the inputType is a supported primitive type (i.e. f32, bf16, etc) or a vector/tensor type based on a supported elemental type.
Since conversion often involves introspecting some attributes of the input type in order to determine how to represent it, this is a two step process.
Definition at line 34 of file UniformSupport.h.
const Type mlir::quant::ExpressedToQuantizedConverter::expressedType
Supported, elemental expressed type (i.e.
f32). Will be nullptr if conversion is not supported.
Definition at line 51 of file UniformSupport.h.
Referenced by convert(), and operator bool().
const Type mlir::quant::ExpressedToQuantizedConverter::inputType
The input type that is being converted from.
This may be an elemental or composite type.
Definition at line 47 of file UniformSupport.h.
Referenced by convert(), and forInputType().
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zkalc: a cryptographic calculator | EF Cryptography Research
zkalc: a cryptographic calculator
George Kadianakis, Michele Orrù | January 16, 2023
zkalc helps you calculate how much time cryptographic operations take on a real computer.
zkalc was created to instantly answer questions like "How quickly can I run an MSM of size $2^{18}$ and compute $120$ pairings on an AWS C5 machine?" or "Which curve can perform DH in less than $10$
Tune in and play with it!
Cryptographers tend to be good at cryptography but they can be quite bad at estimating the time it takes a computer to run their schemes.
We hope that zkalc can help shorten the gap between cryptography and practice:
• Cryptographers can use the simple UX to learn how fast their new fancy scheme runs on various machines without wasting computer cycles and CO2;
• Protocol designers can easily tune the parameters of their protocol depending on their requirements
We designed zkalc to be easy to use but also easy to extend. Writing new types of benchmarks, or adding fresh data to the zkalc website is easy.
How does zkalc work?
Let's now go over our benchmarking pipeline and how we derive our results. In short:
1. For each supported operation, we run benchmarks that measure its performance. We use criterion.rs to take multiple sample (at least 10, ever for large computations like MSMs), and then select the
2. We collect benchmark results inside the perf/data/ directory and make them freely available for anyone to use.
3. For each operation, we fit a function to its benchmark results. We use linear interpolation inside the benchmark bounds and least squares regression outside the benchmarking bounds.
4. When a user queries zkalc for an operation of size $n$, we estimate the its running time using the produced function.
In this blog post we will go deeper into the above process. We will mainly focus on the function fitting, but if you are interested in the entire story of how our benchmarks work, or if you want to
see the interactive version of the graphs below, please visit the zkalc methodology page.
Running benchmarks
For every supported operation, we write benchmarks and run them in multiple platforms. We then store the results in the perf/ directory of zkalc.
Answering user queries
Now we have benchmark data for every operation in the perf/ directory. The next step is to fit a function $f(x)$ to every operation, so that when a user queries us for an operation with arbitrary
size $n$, we can answer it by evaluating $f(n)$.
For simple operations like basic scalar multiplication and field addition (which are not amortized) we consider them to be sequential computations. That is, if a single scalar multiplication takes
$x$ seconds, $n$ such operations will take $n \cdot x$ seconds. That results in a simple linear function $f(x) = n \cdot x$.
More complicated operations like MSMs and pairing products are amortized and their performance doesn't follow a simple linear curve.
For such operations, we collect benchmark data for various sizes. For example, consider the figure below which displays the benchmark data from a $\mathbb G_1$ MSM operation for sizes from $2$ to $2^
{21}$ (both axis are in log scale):
To answer user queries within the benchmark range, we perform polynomial interpolation over the benchmark data.
That is, for each pair of benchmark data $(x_i, f(x_i))$ and $(x_{i+1}, f(x_{i+1}))$ we trace the line that goes through both points. We end up with a piecewise function that covers the entire
benchmark range, as we can see in the figure below:
For user queries outside of the benchmarking range we extrapolate via non-linear least squares. To make things more exciting we decided that it should be done... in Javascript inside your browser.
In the specific case of MSMs, Pippenger's complexity is well known to be asymptotically $O({n} / {\log n})$. Hence, we use least squares to fit the data set to a function $h(x) = \frac{a x + b}{\log
x}$ solving for $a, b$.
Here is an illustration of the extrapolation behavior of $\mathbb G_1$ MSM outside of the benchmarking range (that is, after $2^{21}$):
We do not expect extrapolation to faithfully follow the benchmarks. We believe however that the estimates provide a rough idea of how long an algorithm will take.
In the end of this process, we end up with a piecewise function for each operation that we can query inside and outside the benchmarking range to answer user queries.
Do give zkalc a try and let us know what you think!
Visualizing crypto performance with zkalc
In the zkalc website, you will also find the zcharts corner where we visualize all the raw benchmark data we used in the above section.
We hope that this visual approach will help you grok the benchmark data that zkalc is based on, but also acquire a better understanding of the performance variations between different implementations
A call for help
zkalc can be only as useful as the data it provides,and there is lots of room for additional benchmarks. Can you run benchmarks on a large cloud provider? We would love to get in touch and gather
benchmarks for zkalc. Do you have access to a beefy GPU prover? We would love to help you run zkalc. Did you just design a new elliptic curve? Benchmark it with zkalc. Are you working on a new crypto
library? You guessed it. Adding benchmarks to zkalc is actually not hard; check our website for instructions!
In the future, we also want to expand zkalc to support higher level primitives. From FFTs, to IPAs, to various polynomial commitment and lookup argument schemes. If you want to write benchmarks for
any of these, check out our TODO file and please get in touch! :)
Many thanks to Patrick Armino and Jonathan Xu for their help with the UX.
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How Many Sides Does A Rhombus Have? | Templatesz234.com
How Many Sides Does A Rhombus Have?
Rhombus Math\/Science\/English Homeschool\/Afterschool from doctemplates.us
How Many Sides Does A Rhombus Have?
Are you wondering how many sides a rhombus has? A rhombus is a type of parallelogram that has four sides of equal length. The angles of the sides are not always equal, so the rhombus is not always a
perfect square. But all rhombuses have four sides.
What is a Rhombus?
A rhombus is a type of quadrilateral. It has four sides that all have equal length, and the angles of the sides are not always equal. A rhombus is not a regular polygon, which means that it is not
always a perfect square.
What is the Difference Between a Rhombus and a Square?
The main difference between a rhombus and a square is that a rhombus has four sides of equal length, while a square has four sides of equal length and four right angles. The angles of the sides of a
rhombus can be different, while all the angles of a square are right angles.
What is the Formula for Finding the Number of Sides of a Rhombus?
The formula for finding the number of sides of a rhombus is simple: the rhombus has four sides. All rhombuses have four sides of equal length and the angles of the sides may be different.
What are Some Other Facts About Rhombuses?
Rhombuses are also known as Diamonds, due to their shape. A rhombus can also be inscribed in a circle, meaning that it is possible to draw a circle that touches all four sides of a rhombus.
Additionally, rhombuses are symmetrical, meaning that the opposite sides and angles are equal.
In conclusion, a rhombus has four sides of equal length. The angles of the sides are not always equal, so it is not always a perfect square. Additionally, a rhombus can be inscribed in a circle, and
it is symmetrical.
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Copulas Primer | TensorFlow Probability
import numpy as np
import matplotlib.pyplot as plt
import tensorflow.compat.v2 as tf
import tensorflow_probability as tfp
tfd = tfp.distributions
tfb = tfp.bijectors
A [copula](https://en.wikipedia.org/wiki/Copula_(probability_theory%29) is a classical approach for capturing the dependence between random variables. More formally, a copula is a multivariate
distribution \(C(U_1, U_2, ...., U_n)\) such that marginalizing gives \(U_i \sim \text{Uniform}(0, 1)\).
Copulas are interesting because we can use them to create multivariate distributions with arbitrary marginals. This is the recipe:
• Using the Probability Integral Transform turns an arbitrary continuous R.V. \(X\) into a uniform one \(F_X(X)\), where \(F_X\) is the CDF of \(X\).
• Given a copula (say bivariate) \(C(U, V)\), we have that \(U\) and \(V\) have uniform marginal distributions.
• Now given our R.V's of interest \(X, Y\), create a new distribution \(C'(X, Y) = C(F_X(X), F_Y(Y))\). The marginals for \(X\) and \(Y\) are the ones we desired.
Marginals are univariate and thus may be easier to measure and/or model. A copula enables starting from marginals yet also achieving arbitrary correlation between dimensions.
Gaussian Copula
To illustrate how copulas are constructed, consider the case of capturing dependence according to multivariate Gaussian correlations. A Gaussian Copula is one given by \(C(u_1, u_2, ...u_n) = \Phi_\
Sigma(\Phi^{-1}(u_1), \Phi^{-1}(u_2), ... \Phi^{-1}(u_n))\) where \(\Phi_\Sigma\) represents the CDF of a MultivariateNormal, with covariance \(\Sigma\) and mean 0, and \(\Phi^{-1}\) is the inverse
CDF for the standard normal.
Applying the normal's inverse CDF warps the uniform dimensions to be normally distributed. Applying the multivariate normal's CDF then squashes the distribution to be marginally uniform and with
Gaussian correlations.
Thus, what we get is that the Gaussian Copula is a distribution over the unit hypercube \([0, 1]^n\) with uniform marginals.
Defined as such, the Gaussian Copula can be implemented with tfd.TransformedDistribution and appropriate Bijector. That is, we are transforming a MultivariateNormal, via the use of the Normal
distribution's inverse CDF, implemented by the tfb.NormalCDF bijector.
Below, we implement a Gaussian Copula with one simplifying assumption: that the covariance is parameterized by a Cholesky factor (hence a covariance for MultivariateNormalTriL). (One could use other
tf.linalg.LinearOperators to encode different matrix-free assumptions.).
class GaussianCopulaTriL(tfd.TransformedDistribution):
"""Takes a location, and lower triangular matrix for the Cholesky factor."""
def __init__(self, loc, scale_tril):
super(GaussianCopulaTriL, self).__init__(
# Plot an example of this.
unit_interval = np.linspace(0.01, 0.99, num=200, dtype=np.float32)
x_grid, y_grid = np.meshgrid(unit_interval, unit_interval)
coordinates = np.concatenate(
[x_grid[..., np.newaxis],
y_grid[..., np.newaxis]], axis=-1)
pdf = GaussianCopulaTriL(
loc=[0., 0.],
scale_tril=[[1., 0.8], [0., 0.6]],
# Plot its density.
plt.contour(x_grid, y_grid, pdf, 100, cmap=plt.cm.jet);
The power, however, from such a model is using the Probability Integral Transform, to use the copula on arbitrary R.V.s. In this way, we can specify arbitrary marginals, and use the copula to stitch
them together.
We start with a model:
\[\begin{align*} X &\sim \text{Kumaraswamy}(a, b) \\ Y &\sim \text{Gumbel}(\mu, \beta) \end{align*}\]
and use the copula to get a bivariate R.V. \(Z\), which has marginals Kumaraswamy and Gumbel.
We'll start by plotting the product distribution generated by those two R.V.s. This is just to serve as a comparison point to when we apply the Copula.
a = 2.0
b = 2.0
gloc = 0.
gscale = 1.
x = tfd.Kumaraswamy(a, b)
y = tfd.Gumbel(loc=gloc, scale=gscale)
# Plot the distributions, assuming independence
x_axis_interval = np.linspace(0.01, 0.99, num=200, dtype=np.float32)
y_axis_interval = np.linspace(-2., 3., num=200, dtype=np.float32)
x_grid, y_grid = np.meshgrid(x_axis_interval, y_axis_interval)
pdf = x.prob(x_grid) * y.prob(y_grid)
# Plot its density
plt.contour(x_grid, y_grid, pdf, 100, cmap=plt.cm.jet);
Joint Distribution with Different Marginals
Now we use a Gaussian copula to couple the distributions together, and plot that. Again our tool of choice is TransformedDistribution applying the appropriate Bijector to obtain the chosen marginals.
Specifically, we use a Blockwise bijector which applies different bijectors at different parts of the vector (which is still a bijective transformation).
Now we can define the Copula we want. Given a list of target marginals (encoded as bijectors), we can easily construct a new distribution that uses the copula and has the specified marginals.
class WarpedGaussianCopula(tfd.TransformedDistribution):
"""Application of a Gaussian Copula on a list of target marginals.
This implements an application of a Gaussian Copula. Given [x_0, ... x_n]
which are distributed marginally (with CDF) [F_0, ... F_n],
`GaussianCopula` represents an application of the Copula, such that the
resulting multivariate distribution has the above specified marginals.
The marginals are specified by `marginal_bijectors`: These are
bijectors whose `inverse` encodes the CDF and `forward` the inverse CDF.
block_sizes is a 1-D Tensor to determine splits for `marginal_bijectors`
length should be same as length of `marginal_bijectors`.
See tfb.Blockwise for details
def __init__(self, loc, scale_tril, marginal_bijectors, block_sizes=None):
super(WarpedGaussianCopula, self).__init__(
distribution=GaussianCopulaTriL(loc=loc, scale_tril=scale_tril),
Finally, let's actually use this Gaussian Copula. We'll use a Cholesky of \(\begin{bmatrix}1 & 0\\\rho & \sqrt{(1-\rho^2)}\end{bmatrix}\), which will correspond to variances 1, and correlation \(\rho
\) for the multivariate normal.
We'll look at a few cases:
# Create our coordinates:
coordinates = np.concatenate(
[x_grid[..., np.newaxis], y_grid[..., np.newaxis]], -1)
def create_gaussian_copula(correlation):
# Use Gaussian Copula to add dependence.
return WarpedGaussianCopula(
loc=[0., 0.],
scale_tril=[[1., 0.], [correlation, tf.sqrt(1. - correlation ** 2)]],
# These encode the marginals we want. In this case we want X_0 has
# Kumaraswamy marginal, and X_1 has Gumbel marginal.
tfb.Invert(tfb.KumaraswamyCDF(a, b)),
tfb.Invert(tfb.GumbelCDF(loc=0., scale=1.))])
# Note that the zero case will correspond to independent marginals!
correlations = [0., -0.8, 0.8]
copulas = []
probs = []
for correlation in correlations:
copula = create_gaussian_copula(correlation)
# Plot it's density
for correlation, copula_prob in zip(correlations, probs):
plt.contour(x_grid, y_grid, copula_prob, 100, cmap=plt.cm.jet)
plt.title('Correlation {}'.format(correlation))
Finally, let's verify that we actually get the marginals we want.
def kumaraswamy_pdf(x):
return tfd.Kumaraswamy(a, b).prob(np.float32(x))
def gumbel_pdf(x):
return tfd.Gumbel(gloc, gscale).prob(np.float32(x))
copula_samples = []
for copula in copulas:
plot_rows = len(correlations)
plot_cols = 2 # for 2 densities [kumarswamy, gumbel]
fig, axes = plt.subplots(plot_rows, plot_cols, sharex='col', figsize=(18,12))
# Let's marginalize out on each, and plot the samples.
for i, (correlation, copula_sample) in enumerate(zip(correlations, copula_samples)):
k = copula_sample[..., 0].numpy()
g = copula_sample[..., 1].numpy()
_, bins, _ = axes[i, 0].hist(k, bins=100, density=True)
axes[i, 0].plot(bins, kumaraswamy_pdf(bins), 'r--')
axes[i, 0].set_title('Kumaraswamy from Copula with correlation {}'.format(correlation))
_, bins, _ = axes[i, 1].hist(g, bins=100, density=True)
axes[i, 1].plot(bins, gumbel_pdf(bins), 'r--')
axes[i, 1].set_title('Gumbel from Copula with correlation {}'.format(correlation))
And there we go! We've demonstrated that we can construct Gaussian Copulas using the Bijector API.
More generally, writing bijectors using the Bijector API and composing them with a distribution, can create rich families of distributions for flexible modelling.
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ORA-04006: START WITH cannot be less than MINVALUE
ORA-04006: START WITH cannot be less than MINVALUE error occurs if the sequence starts with a value less than minimum value of the sequence. You have created a sequence that starts with a value less
than the sequence’s minimum value.
The minimum value of the sequence is defined as the lowest value that the sequence can create. The sequence shall start with a value that is more than or equal to the minimum value. If the sequence
begins with a value smaller than the sequence’s minimum value, the sequence will throw an error ORA-04006: START WITH cannot be less than MINVALUE.
the given starting value is less than MINVALUE
make sure that the starting value is >= MINVALUE
The Problem
The sequence is designed to generate values ranging from the smallest to the largest. The descending order sequence will begin with the highest value and finish with the lowest value. It is not
required to begin the sequence with the smallest value. The START WITH keyword can be used to define the starting value of a sequence.
Configuring the starting value to be less than the minimum value has less relevance. The ascending order sequence’s start value cannot be less than the minimum value.
create sequence mysequence start with 5 MINVALUE 10 increment by 1;
Error starting at line : 11 in command -
create sequence mysequence start with 5 MINVALUE 10 increment by 1
Error report -
ORA-04006: START WITH cannot be less than MINVALUE
04006. 00000 - "START WITH cannot be less than MINVALUE"
*Cause: the given starting value is less than MINVALUE
*Action: make sure that the starting value is >= MINVALUE
Solution 1
Change the sequence’s start value, minimum value, or both if sequence starts with a value less than the minimum value. The sequence’s start value should be larger than the sequence’s minimum value.
create sequence mysequence start with 50 MINVALUE 10 increment by 1;
Solution 2
The sequence details are important for comprehending the sequence’s present configurations. To decide how to fix the error, the sequence minimum value, maximum value, increment value, and so on must
be necessary. The sequence’s start value may be found in the LAST NAME column of the user sequences table. The select query below will provide you all the information you need about the sequence.
select * from user_sequences where sequence_name = 'MYSEQUENCE';
Solution 3
By default, when an ascending order sequence is created, the minimum value is set to 1. The ascending order sequence must start with a value higher than or equal to 1. The following example
illustrates the error since it starts with 0 and increases by 1.
The ascending order sequence’s default minimum and maximum values are 1 and 999999999999999999999999999. The ascending order sequence’s start value should be greater than or equal to 1, or the
minimum value should be decreased.
create sequence mysequence start with 0 increment by 1;
Error starting at line : 2 in command -
create sequence mysequence start with 0 increment by 1
Error report -
ORA-04006: START WITH cannot be less than MINVALUE
04006. 00000 - "START WITH cannot be less than MINVALUE"
*Cause: the given starting value is less than MINVALUE
*Action: make sure that the starting value is >= MINVALUE
create sequence mysequence increment by 1;
The sequence minimum and maximum values are 1 and 999999999999999999999999999. The default start value of the sequence is 1.
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When the target variable that we’re trying to predict is continuous, such as in our housing example, we call the learning problem a regression problem.
Cost Function
How to fit the best possible line?
We can measure the accuracy of our hypothesis function by using a cost function.
This takes an average difference (actually a fancier version of an average) of all the results of the hypothesis with inputs from x's and the actual output y's.
$$ J(\theta_0, \theta_1) = \dfrac {1}{2m} \displaystyle \sum _{i=1}^m \left ( \hat{y}_{i}- y_{i} \right)^2 = \dfrac {1}{2m} \displaystyle \sum _{i=1}^m \left (h_\theta (x_{i}) - y_{i} \right)^2 $$
The mean is halved (1/2) as a convenience for the computation of the gradient descent, as the derivative term of the square function will cancel out the 1/2 term.
Parameter Learning
Gradient Descent
So we have our hypothesis function and we have a way of measuring how well it fits into the data. Now we need to estimate the parameters in the hypothesis function. That's where gradient descent
comes in.
Gradient descent algorithm starts with initial θ0 and θ1 then updates the values try to find the lowest point.
repeat until convergence:
$$ \theta_j := \theta_j - \alpha \frac{\partial}{\partial \theta_j} J(\theta_0, \theta_1) $$
where j = 0,1 represents the feature index number.
But it's really important to keep that in mind that we have to update θ0 and θ1 simultaneously.
At each iteration j, one should simultaneously update the parameters θ1,θ2,...,θn Updating a specific parameter prior to calculating another one on the $j^{(th)}$ iteration would yield to a wrong
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Teaching Multiplication
Multiplication is an operation that requires you to add another number to itself a certain number of times as indicated in the multiplication equation.
When students first start learning the concept of multiplication, it is more simple as time goes on for kids to learn. Memorizing multiplication facts works for some students but not for all! Some
students need to learn by using different models and representations. When students have a conceptual understanding of multiplication and realize that it is connected to the real world, they tend to
perform better on assessments. If a child is only ever taught isolated facts or memorized facts, they risk the chance of not understanding the meaning behind the objects they are multiplying. Knowing
a variety of ways to solve multiplication problems will allow a student to figure out which strategy works best for them.
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How to use the AVERAGE Function | Encyclopedia-Excel
The AVERAGE function returns the arithmetic mean of any set of numbers given to it.
= AVERAGE(number1, [number2],...)
number1 - the first number to averaged
[number2] - optional arguments, additional numbers to be averaged
The AVERAGE function is part of the "Statistical" group of functions within Excel.
The AVERAGE function in Excel works by adding up a range of numerical values and then dividing the sum by the count of those values, giving you the arithmetic mean.
For example, if you have a range of values from A1 to A5, and you use the formula:
Excel will add up the values in cells A1 through A5, then divide that sum by 5 (the count of values in the range) to calculate the average.
Only cells that contain numerical values can be used. If a cell in the range contains text, logical values (TRUE/FALSE), or errors (#DIV/0!, #VALUE!, etc.), those values will be ignored in the
Different Types of Averages In Excel
The most common types of averages in Excel are:
AVERAGE: Calculates the arithmetic mean of a range of values. Used to summarize and analyze data that is normally distributed.
MEDIAN: Calculates the median of a range of values. Summarize and analyze data that is not normally distributed, or that contains outliers.
MODE: Calculates the mode of a range of values. Useful to analyze data that has multiple peaks or modes.
GEOMEAN: Calculates the geometric mean of a range of values. It is used to summarize data that is exponential or logarithmic in nature.
HARMEAN: Calculates the harmonic mean of a range of values. Good for data that is rate-based, such as speeds or rates of change.
How to Average a Range of Cells in Excel
In this example, we'll demonstrate how to take the average of a range of cells in Excel.
Using the AVERAGE function, you can quickly take the average of a row of numbers, a column of numbers, or even an array. Just type in the AVERAGE function and select the numbers that you would like
to average.
How to take the Average of a column:
How to take the Average of an array:
How to Take the Average the Results of Another Formula
The average function can also be used to average the results of other formulas.
In this table, we have a list of customers and sales across 2022 and 2023. If we wanted to know what the average sales we were making across these two years, we could use the following formula:
= AVERAGE(SUM(B2:B11),SUM(C2:C11))
In this formula, we are individually summing up each year's sales, and taking the resulting average of those two numbers.
The sum of 2022 sales is $1,250, and the sum of 2023 sales is $1,270. So, our formula would actually look like this once the SUM was calculated:
Once the average is taken of these two numbers, we are left with the correct average of $1,260 sales per year.
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al a
Activity report
Inria teams are typically groups of researchers working on the definition of a common project, and objectives, with the goal to arrive at the creation of a project-team. Such project-teams may
include other partners (universities or research institutions).
RNSR: 200718255S
Team name:
Advanced 3D Numerical Modeling in Geophysics
Digital Health, Biology and Earth
Earth, Environmental and Energy Sciences
Creation of the Project-Team: 2007 July 01
• A6. Modeling, simulation and control
• A6.1. Methods in mathematical modeling
• A6.1.1. Continuous Modeling (PDE, ODE)
• A6.1.4. Multiscale modeling
• A6.1.5. Multiphysics modeling
• A6.2. Scientific computing, Numerical Analysis & Optimization
• A6.2.1. Numerical analysis of PDE and ODE
• A6.2.7. High performance computing
• A6.3.1. Inverse problems
• A6.5. Mathematical modeling for physical sciences
• A6.5.1. Solid mechanics
• A6.5.4. Waves
• B3. Environment and planet
• B3.3. Geosciences
• B3.3.1. Earth and subsoil
• B4. Energy
• B4.1. Fossile energy production (oil, gas)
• B5.2. Design and manufacturing
• B5.5. Materials
• B5.7. 3D printing
• B9.2.1. Music, sound
• B9.5.2. Mathematics
• B9.5.3. Physics
1 Team members, visitors, external collaborators
Research Scientists
• Hélène Barucq [Team leader, Inria, Senior Researcher, HDR]
• Juliette Chabassier [Inria, Researcher]
• Julien Diaz [Inria, Senior Researcher, HDR]
• Augustin Ernoult [Inria, Researcher, from Oct 2020]
• Titly Farhana Faisal [Inria, Starting Research Position]
• Ha Howard Faucher [Inria, Researcher]
• Papa Mangane [Inria, Starting Research Position, until Mar 2020]
• Yder Masson [Inria, Starting Research Position, until Oct 2020]
Faculty Members
• Marc Duruflé [Institut National Polytechnique de Bordeaux, Associate Professor]
• Sébastien Tordeux [Univ de Pau et des pays de l'Adour, Associate Professor, HDR]
Post-Doctoral Fellows
• Augustin Ernoult [Inria, until Sep 2020, funded by Conseil Régional de Nouvelle-Aquitaine]
• Tobias Van Baarsel [Inria, from Feb 2020]
PhD Students
• Guillaume Castera [Inria, from Oct 2020]
• Stefano Frambati [TOTAL-Pau]
• Alexandre Gras [Institut d'optique graduate school, until Sep 2020]
• Pierre Jacquet [Inria]
• Victor Martins Gomes [Univ de Pau et des pays de l'Adour, funded by E2S UPPA]
• Rose Cloe Meyer [Univ de Pau et des pays de l'Adour, funded by E2S UPPA]
• Nathan Rouxelin [Univ de Pau et des pays de l'Adour, funded by E2S UPPA]
• Chengyi Shen [Univ de Pau et des pays de l'Adour, until May 2020]
• Margot Sirdey [ONERA]
• Alexis Thibault [Univ de Pau et des pays de l'Adour, from Sep 2020]
• Vinduja Vasanthan [Inria]
Technical Staff
• Aurelien Citrain [Inria, Engineer]
• Olivier Geber [Inria, Engineer, from Oct 2020]
• Chengyi Shen [Inria, Engineer, from Jun 2020, funded by FEDER-POCTEFA]
Interns and Apprentices
• Anais Binet [Inria, until Jan 2020]
• Yolan Levrero [Ministère de l'Education Nationale, Feb 2020]
• Alexis Thibault [École Normale Supérieure de Paris, until Feb 2020]
Visiting Scientist
• Mounir Tlemcani [Université des Sciences et de la Technologie d'Oran - Mohamed Boudiaf, Mar 2020]
2 Overall objectives
Numerical geosciences encompass a large variety of scientific activities tackling societal challenges like water resources, energy supply, climate change, etc. They are based upon observations,
physical modeling and accurate mathematical formulations. The tremendous progresses of scientific computing have allowed the addition of extensive numerical simulations which provide tools based on
wave measurements to study and possibly monitor complex environments that are otherwise difficult to probe and even fathomless e.g. the subsurface or the interior of stars.
Bridging the gap between experimental measurements and numerical simulations is an important objective of Magique-3D, which pursues a balance between accuracy and efficiency depending on the
application domains in consideration. A common strategy will be to develop frugal models using mathematical methods (asymptotical methods, artificial boundary conditions, reduction methods…), and
efficient numerical schemes (in both time and harmonic domains, using analytical and high order numerical methods).
Magique-3D research program is to develop numerical software packages for retrieving shapes and/or physical properties of complex media with a particular focus on the Earth and its natural
reservoirs. An outstanding goal will consist in coupling seismic wave propagation with other physics in order to improve the knowledge of natural reservoirs which complex definition requires using
high-resolution imaging techniques. The underlying models involves a genuinely larger number of parameters, and in order to take them into account, it is necessary to build models that are simplified
but just as accurate, however easier to solve numerically. For this, Magique-3D collaborates with experimental geophysicists who help to assess the impact of parameters on the wave propagation.
In addition to geophysical setting, Magique-3D research program has enlarged its application range by addressing two other topics: solar imaging and musical acoustics. For solar imaging, modeling is
of great importance and requires working with new equations in an equally new mathematical formalism. This also leads to the need of developing simulation codes with a long-term view to solve inverse
problems. Given the similarities that exist between seismic and solar imaging methods, software development is carried out in-house using many of the skills acquired by the team in geophysical
imaging. Regarding modeling of musical instruments, the size of the objects and the wavelengths considered are different from geophysical or solar contexts, but similar physical principles and
theoretical aspects of models and numerical methods are applicable. Last but not least, parameter reduction and great precision required in the simulation and the possibility to easily compare
numerical and experimental data make them an ideal topic to develop new research related to modeling and simulating wave propagation.
To address the above research agenda, Magique-3D gathers applied mathematicians and acousticians who have long working experience in wave propagation. The team is jointly shared by the University of
Pau and Pays de l’Adour (UPPA) and Inria. The majority of Magique-3D members are located in Pau. The team is therefore attached to LMAP (Mathematics and Applications Laboratory in Pau, UMR CNRS
5142). However, some members of the team are located in Talence, in the Inria building of the Bordeaux campus. The choice of Magique-3D's principal location in Pau is fully justified by the long-term
involvement of the city of Pau in Geosciences, which offers an important network of companies working in the geo-resources sector. In particular, the company Total is our main industrial partner with
whom we aim at developing new numerical methods for energy transition.
Magique-3D relies on strong collaborations and partnerships with various institutions including (a) local industry (TOTAL, RealTimeSeismic), (b) national research centers (ONERA), and (c)
international academic partnerships (e.g. Interdisciplinary Research Institute for the Sciences (IRIS) at California State University, Northridge, USA; University of Pays Basque and Basque Center of
Applied Mathematics at Bilbao, Spain; University of California at Berkeley, Lawrence Berkeley National Laboratory, Max Planck Institute at Göttingen).
3 Research program
Magique-3D organizes its research program from in-house accurate solution methodologies for simulating wave propagation in realistic scenarios to various applications involving transdisciplinary
efforts. Performing simulations of real-world phenomena is an ultimate endeavor by all numerical scientists. To achieve this, one needs real data and advanced mathematical models and high-order
numerical schemes that are compatible with high-performance computing architectures.
To obtain real data, in addition to its current collaborations with scientists both from Academia and Industry, Magique-3D is developing a new branch of research activities by carrying out its own
laboratory measurements. The desire to carry out its own measurements is motivated by the need to solve problems whose increasing complexity involves a large number of physical parameters that need
to be calibrated. For instance, in order to take into account porosity, parameters such as viscosity, attenuation, thermodynamic effects, etc., must be integrated, and their impact must be properly
analyzed before considering using them to characterize the propagation media. This constitutes a clear step ahead for Magique-3D , and opens up new prospects of contributing to the characterization
of very complex media based on wave field measurements.
Regarding the development of numerical schemes, Magique-3D is developing high-order Discontinuous Galerkin (DG) methods and high-order time schemes. Recently, the team has launched a new research
project on space-time integration for seismic waves, in partnership with Total. The coupling of DG methods with other techniques of discretization is also under consideration. Trefftz-DG and
Hybridizable DG methods are currently developed both for poro-elastic waves and electromagnetic waves. HDG and HDG+ formulations are also under study for helioseismology.
The research activities of members of Magique-3D share a common theme of using numerically computed wavefield measurements to reconstruct the propagation medium they passed through before recording.
The medium can be reconstructed by identifying either the physical parameters or the geometrical parameters that characterize it. In each case, the next step is to solve an inverse problem that is
non-linear and ill-posed. To solve it, Magique-3D is focusing on the Full Waveform Inversion (FWI), which is a high-definition imaging method widely used in the field of geophysics.
4 Application domains
Magique-3D research program is organized around three principal domains of applications: geophysical exploration, solar imaging, and music. Each of them requires a relevant panel of significant
contributions requiring achievements in laboratory measurements, modeling, mathematical analysis, advanced numerical schemes and massively parallel software development. Experimental research is a
new activity that will ensure the team to have its own set of real data in addition to those provided by its partners. Magique-3D's application domains can be regrouped into a long-standing activity
dedicated to subsurface imaging, and two more recent activities dedicated to solar imaging and the development of numerical wind instruments. Each field of application is not compartmentalized in the
methodological sense of the term: equations, numerical schemes and programming practices can be shared and then adapted to the application in question.
4.1 Geophysical exploration
Geophysical exploration is a historical field for the team (see e.g 36, 40, 41, 44.Geophysical exploration has been driven for a very long time by the goal of finding hydrocarbons. Today, it is
evolving towards a very proactive direction in favor of renewable energies and Magique-3D commits part of its research activities in this direction. As a powerful tool for mapping the subsurface,
seismic imaging is very useful in many applications like geothermal energy and injection of CO2.
These applications share the Full Waveform Inversion (FWI) as a solution methodology for reconstructing quantitatively the physical parameters from observed data. FWI can be carried out in
time-domain 39, 62, 73, 74 or in frequency domain 67, 66, 65. Its main feature is to avoid the formation of the large Jacobian matrix by computing the gradient of the misfit functional using the
adjoint-state method 46. A detailed review of FWI for geophysical applications can be found in 64.
4.1.1 Deep geothermal energy
Obtaining accurate images of natural reservoirs is still critical for their management and exploitation and seismic imaging is an efficient tool (see 61, 59 and their references therein. One example
is with deep geothermal energy which requires precise imaging of deep fractured reservoirs filled with geothermal fluids. Standard seismic imaging is based upon inverting mechanical waves which have
difficulties to detect them, whereas electromagnetic waves are more sensitive. We see here a clear interest of coupling seismic with electromagnetic methods and this is what Magique-3D is developing
with the CHICkPEA project. This is a multidisciplinary project involving experimental geophysicists from UPPA, members of the LFCR (Laboratory of Complex Fluids and their Reservoirs) and Steve Pride,
professor at the University of Berkeley, who developed the theory describing the coupling between seismic waves and electromagnetic waves called seismoelectric effects 68. CHICkPEA started in 2018
and it was scheduled to be completed at the end of 2021. However, beginning of 2021, the team will be involved in a new project SEE4GEO funded by ADEME, in the framework of Geothermica call http://
www.geothermica.eu. Hence CHICkPEA project will continue as long as SEE4GEO, i.e. until the end of 2023. The project belongs with E2S (Energy Environment Solutions) program of UPPA which steers the
actions of the University labeled I-Site within the framework of the Investment Plan for the Future.
4.1.2 Shallow geothermal energy
Regarding shallow geothermal energy, Magique-3D has started a collaboration with RealTimeSeismic SME in the framework of the FEDER-Poctefa Pixil project in order to use surface waves for a better
imaging of shallow reservoirs. This project goes further in the work carried out on the inversion of seismic waves. Surface waves have long been considered as noise in seismograms because they were
used to study the subsurface at depths. In shallow geothermal energy, surface waves contain interesting information on the first layers of the subsurface. Inverting them is a real problem because the
surface waves are of high amplitude while propagating slowly. They therefore pose difficulties for multi-frequency optimization methods. The analysis of surface wave properties derives from the
analysis of elastic wave propagation in horizontally stratified media 75, 57, 60. Consequently, current surface inversion derives 1D property profiles from dispersion curves picked in the
frequency-wavenumber domain 72. Most of the available methods are limited to only inverting the fundamental modes, while lateral variations are difficult to handle with this approach. To overcome
these limitations, the academia community has recently started to apply Full Waveform Inversion to this specific problem 70, 63.
4.1.3 CO2 injection
The reduction of greenhouse gases in the atmosphere is a societal topic of the utmost importance, with the Paris Agreement setting ambitious goals for many countries. One fundamental pillar of
greenhouse emission management is Carbon Capture Utilisation and Storage (CCUS) 76. With this strategy, carbon dioxide produced on- or off-site is sequestered and injected into depleted reservoirs,
thus offsetting an important portion of current CO2 emissions. The successful and safe implementation of this strategy requires the prediction, monitoring and surveillance of stored CO2 over long
periods, which presents significant challenges in terms of seismic acquisition, seismic inversion and numerical simulation. These tools, coupled with state-of-the-art flow simulations, are vital in
order to support the injection operations with vital real-time and long-term information. Moreover, specific challenges related to the physics of injected CO2, such as viscosity, temperature and
multi-phase fluid conditions push to the limits our current numerical models, and require ambitious new multi-physics simulations to support safe and cost-effective CO2 injection operations. For
example, some recent publications like 71, 79 have shown that the combination of CO2-brine flow with wave propagation provides efficient simulations for the monitoring of sequestered CO2. Magique-3D
proposes to develop numerical methods for this new application, in collaboration with Total, in the framework of the research agreement DIP (Depth Imaging Partnership).
Figure 1: Experimental device and numerical simulation of waves in porous media, CHICkPEA, E2S
4.2 Solar imaging
The Sun sustains various types of waves which are driven by near-surface turbulent convection. These movements can be observed at the surface by the Dopplergrams given by ground-based or
satellite-borne observatories. In recent years, methods for understanding Earth subsurfaces have opened up new ways to study the interior of the Sun as in the case with helioseismology and the
interior of stars with aesteroseismology from oscillation observed at their surface. Techniques in helioseisomolgy is generally divided into global and local helioseismology. The first approach
studies frequencies of oscillations modes, cf. 42, 47. This is also the current strategy of asteroseismology, cf 58, 34, 35. On the other hand, local helioseimology which adapts techniques of
geophysical seismic interferometry studies, measure local wave propagation and works with the full 3D observed wavefield, and is thus more adapted to study additional features such large-scale flows
in active region, sun spots and plage, cf. 54, 53.
With its long-run expertise in numerical tools for imaging Earth subsurfaces, Magique-3D is extending its activity on terrestrial seismology to studying the Sun, for the latter offers a vast wealth
of problems to be explored both for direct modeling as well as inversion. In this development, associated team ANTS (Advance Numerical meThods for helioSeismology was created in 2019 to formalize a
collaboration with MPS (Max Planck institute for Solar research, https://www.mps.mpg.de/institute/organization). In a first step, one can study acoustic waves which are identified with p-modes on the
power spectrum, and acoustic waves at low frequencies can be adequately described by a scalar equation which allows for convection. The stochastic nature is described by random right-hand source
term, and in using statistical analysis, under appropriate assumptions (e.g. the convenient source assumption), power spectrums and time-distance diagrams can obtained from the deterministic Green
kernel of modelling wave equation, cf. 52. In this approach, the Green kernel becomes a crucial object in local helioseismology, and its accurate and efficient computation is the main goal of forward
As a first result of the collaboration of Magique-3D with MPS, in 52, a new computational framework based on the scalar equation was developed and produces solar-like power spectra and
time-distance diagrams under appropriate assumption of source excitation. A second topic under active research is boundary conditions that allow waves to propagate. This also plays a crucial point in
forward modeling as well as inversion. There are two on-going directions to proceed.
• In a goal to create power spectrum and time-distance diagram that are closer to real observable, we need to include important physical effects such as gravity, magnetic and rotation forces. This
requires extending the computation framework for scalar equation to vector equation. In including the effect of gravity, one hopes to find g-modes on simulated power spectrum (currently missing
from that associated with the scalar equation). These above physics are also needed in order to study active regions of the sun such as Sunspots. Shallow layers of the Sun will be probed with
more accuracy, which will be useful for the study of supergranulation. This is topic, Magique-3D can benefit from discussion and collaboration with Inria teams such as TONUS team with their
experience in simulating tokamak plasma for thermonuclear reaction (http://schnaps.gforge.inria.fr/).
• On the other hand, with the current established framework for the scalar equation, the next step is to the inverse problem, in particular with time-distance helioseismology 45, 50 and holography
80, 55. Current state-of-the art tools in these references is linear inversion using Born approximation. In additional they are carried out in 1D or 2D. It is thus interesting to apply nonlinear
inversion such as Full Waveform Inversion cf. 49 to these problems.
4.3 Musical acoustics
This field of application is a subject of study for which the team is willing to take risks. We propose using a mix of experimental and numerical approach in order to study and design musical
instruments. Makers have designed wind musical instruments (as flutes, trumpets, clarinets, bassoons...) in the past through “trial and error” procedures by performing a geometrical calibration of
musical instruments in order to improve their accuracy, tone, homogeneity and even their sound volume, ergonomics, and robustness. During the past few decades, musical acoustics has been in a process
of rationalizing the empiric understanding of instrument makers in order to formulate a scientific approach to future evolution. Our research proposal is along this axis of research by proposing new
mathematical models based on our solid experience in terms of wave propagation in media with interfaces that can significantly change the sound. As was done in geophysical exploration, we propose to
assist the modelling process with laboratory experiments. Direct comparison between simulations and experiments will allow to assess the model error. For this purpose, an experimental device has been
developed in collaboration with I2M, Mechanics Laboratory of the University of Bordeaux and Humeau Factory, Montpon- Ménestérol, and is currently in use.
4.3.1 Modeling
Although the playing context should always be the final reference, some aspects of the behavior of a wind instrument can be firstly characterized by its entry impedance which quantifies the
Dirichlet-to-Neumann map of the wave propagation in the pipe in the harmonic domain. This impedance can be both measured 51, 48 and computed with simulations based on accurate and concise models of
the pipe 69, 4377,13. A more realistic approach accounts for the embouchure 56, 43, 37, 38, 78, which is modeled as a nonlinear oscillator coupled with the pressure and acoustic velocity at the
entry of the pipe, allowing to predict the sound qualities. Mathematical properties of the underlying models are not yet totally understood, and adequate models still need to be developed. This is
particularly true when accounting for dissipation phenomena, junctions of pipes, pipe porosity and rugosity, embouchures...
To reproduce the sound of instruments, time-dependent models are more suitable. Here, nonlinear lumped elements induce an “auto-oscillatory” behavior of the instrument. The models currently available
in the literature are meant to reproduce viscothermal effects, pipe junctions, pipe radiation, lips oscillation, etc. They do not necessarily possess adequate mathematical properties to ensure stable
simulations and they should be improved using asymptotic analysis methods or Lagrangian formalism.
4.3.2 Numerical methods
As far as numerical developments are concerned, the accuracy of the calculations is essential. Indeed, for some aspects like the sounding frequency, a deviation of 1% between the predictions and the
observations is unacceptable. Moreover, contrary to what the team is used to do for geophysics or astrophysics thanks to HPC, numerical methods for acoustical musics must be frugal to be run on
personal computers by acousticians and makers. Magique-3D has a wide range of numerical methods that have been implemented in its codes for linear problems. New numerical schemes will have to be
implemented to take into account the non-linearities of time-dependent models.
4.3.3 Virtual workshop
Beyond the idea of mathematically modeling musical instruments, Magique-3D wishes to develop a virtual workshop whose vocation will be twofold: (i) support the manufacturers to design new
instruments; (ii) recreate the sound of old and historical instruments. To implement this idea, we propose to elaborate optimization techniques that are well-known in the team to define optimal
geometries to meet given specifications. This can be used to reconstruct existing instruments from acoustic measurement or to design new instruments by fixing relevant quantitative objective which is
a research activity by its own 14. Behind the idea of the virtual workshop is also the intention to hear the instruments, from the knowledge of their shape and playing regime. For that purpose,
time-domain models are essential.
5 Social and environmental responsibility
5.1 Footprint of research activities
Since March 2020, the team no longer travels abroad. However, it has not slowed down its international collaborations. It is therefore a forced and interesting experience that shows us that we can,
in the future, consider selecting our missions abroad to reduce our carbon footprint. We also continue to believe that it is important to meet our foreign collaborators to create real ties and solve
certain problems that require us to put ourselves in front of a board. The year 2020 will have helped us begin this change in attitude.
All team members living within 10km of their workplace prefer to bike or walk to work. Others, when possible, prefer public transportation. It should be noted that the team members hosted in Pau do
not benefit from the same conditions as those hosted in Talence: no shower at the workplace, the public transportation network is not very developed outside the city of Pau.
5.2 Impact of research results
The team has been working with Total since its creation. With them, it has developed algorithms to improve the imaging of the subsurface. For more than 10 years, subsurface imagery was carried out in
support of geophysical exploration campaigns deployed to find oil and gas reservoirs. Today, the Total Group is engaged in a process of transformation towards green and renewable energies. With
Total, the team is carrying out new algorithmic developments that are designed to be applied to CO2 injection and storage as well as geothermal energy.
Solving wave equations in complex media to simulate the propagation of several thousand sources is a classical problem in geophysical exploration. To achieve these direct simulations, it is still
absolutely necessary to have a high computational power. Thus, even using the most advanced clusters, it is mandatory to reduce the computational loads to consider solving inverse geophysical
problems. The team has therefore been working for a long time on the development of accurate numerical schemes, compatible with modern computer architectures and capable of pushing back the limits of
simulations already carried out. Terrestrial and solar imaging is still awaiting advances in this field and the team will continue to contribute to this. However, the team has forged new partnerships
for which access to supercomputers is not always possible, due to insufficient financial resources, or desired, because potential users only have a personal computer. This means that the team is also
committed to an approach that promotes the use of reasonably energy-efficient computing resources.
Regarding the domain of wind musical instruments, an effort is made on model reduction in order to achieve frugal models that can be implemented on regular, not demanding infrastructure. The impact
of this research is twofold. First it focuses on the development of a virtual workshop in order to help instrument makers quantify the consequence of a geometrical change without needing a prototype,
therefore preventing tool making and material loss. Second, it aims at computing the sound of heritage musical instruments in a cultural purpose.
6 Highlights of the year
The team was privileged to recruit Augustin Ernoult, acoustician and wind instrument specialist as Inria's research fellow.
The team has formed a new partnership with Lawrence Livermore National Laboratory with whom it proposed a European project See4Geo to the call Geothermica 2020. This project has been accepted for a
start date of January 1st, 2021.
We have made a major release of our openwind software.
6.1 Awards
Augustin Ernoult received the "Prix Yves Rocard 2020: prix jeune chercheur de la société française d'acoustique".
6.2 Covid impact on research activity
The year 2020 was marked by the covid crisis and its impact on society and its overall activity. The world of research was also greatly affected:
Faculty members have seen their teaching load increase significantly;
PhD students and post-docs have often had to deal with a worsening of their working conditions, as well as with reduced interactions with their supervisors and colleagues;
Most scientific collaborations have been greatly affected, with many international activities cancelled or postponed to dates still to be defined.
7 New software and platforms
7.1 New software
7.1.1 Hou10ni
• Keywords: 2D, 3D, Elastodynamic equations, Acoustic equation, Elastoacoustic, Frequency Domain, Time Domain, Discontinuous Galerkin
• Scientific Description: Hou10ni simulates acoustic and elastic wave propagation in time domain and in harmonic domain, in 2D and in 3D. It is also able to model elasto acoustic coupling. It is
based on the second order formulation of the wave equation and the space discretization is achieved using Interior Penalty Discontinuous Galerkin Method. Recently, the harmonic domain solver has
been extended to handle Hybridizable Discontinuous Galerkin Methods.
• Functional Description: This software simulates the propagation of waves in heterogeneous 2D and 3D media in time-domain and in frequency domain. It is based on an Interior Penalty Discontinuous
Galerkin Method (IPDGM) and allows for the use of meshes composed of cells of various order (p-adaptivity in space).
• News of the Year: In 2020, we have implemented the 3D poroelastic equations and the coupling porelastic equations (poroelastic+electromagnetic) for the HDG formulation.
• URL: https://team.inria.fr/magique3d/software/hou10ni/
• Publications: hal-01513597, hal-01957131, hal-01388195, hal-01972134, hal-01957147, hal-02152117, hal-02486942, hal-02408315, hal-02911686, tel-03014772, hal-01656440, hal-01662677, hal-01623953,
hal-01623952, hal-01513597, hal-01519168, hal-01254194, hal-01400663, hal-01400656, hal-01400643, hal-01313013, hal-01303391, hal-01408981, tel-01304349, hal-01184090, hal-01223344, hal-01207897,
hal-01184111, hal-01184110, hal-01184107, hal-01207906, hal-01184104, hal-01207886, hal-01176854, hal-01408705, hal-01408700, tel-01292824, hal-01656440, hal-00931852, hal-01096390, hal-01096392,
hal-01096385, hal-01096324, hal-01096318, tel-01133713, tel-00880628
• Authors: Julien Diaz, Elodie Estecahandy, Marie Bonnasse, Marc Fuentes, Rose-Cloé Meyer, Vinduja Vasanthan, Lionel Boillot, Conrad Hillairet
• Contact: Julien Diaz
• Participants: Conrad Hillairet, Elodie Estecahandy, Julien Diaz, Lionel Boillot, Marie Bonnasse, Marc Fuentes, Rose-Cloé Meyer, Vinduja Vasanthan
7.1.2 MONTJOIE
• Keywords: High order finite elements, Edge elements, Aeroacoustics, High order time schemes
• Scientific Description: Montjoie is designed for the efficient solution of time-domain and time-harmonic linear partial differential equations using high-order finite element methods. This code
is mainly written for quadrilateral/hexahedral finite elements, partial implementations of triangular/tetrahedral elements are provided. The equations solved by this code, come from the ”wave
propagation” problems, particularly acoustic, electromagnetic, aeroacoustic, elastodynamic problems.
• Functional Description: Montjoie is a code that provides a C++ framework for solving partial differential equations on unstructured meshes with finite element-like methods (continuous finite
element, discontinuous Galerkin formulation, edge elements and facet elements). The handling of mixed elements (tetrahedra, prisms, pyramids and hexahedra) has been implemented for these
different types of finite elements methods. Several applications are currently available : wave equation, elastodynamics, aeroacoustics, Maxwell's equations.
• URL: https://gitlab.inria.fr/durufle/montjoie
• Authors: Marc Durufle, Gary Cohen
• Contact: Marc Durufle
• Participants: Juliette Chabassier, Marc Durufle, Morgane Bergot
7.1.3 OpenWind
• Name: Open Wind Instrument Design
• Keywords: Wave propagation, Inverse problem, Experimental mechanics, Time Domain, Physical simulation
• Scientific Description: implementation of first order finite elements for wind musical instrument simulation. implementation of the Full Waveform inversion method for wind musical instrument
inversion. implementation of energy consistent numerical schemes for time domain simulation of valve-type wind musical instrument.
• Functional Description: Simulation and inversion of wind musical instruments using one-dimensional finite element method with tonholes and fingering chart. The software has three
functionnalities. First, the software takes the shape of a wind instrument and computes the acoustical response (answer to a given frequential excitation). Second, the software takes the
instrument shape and the control parameters of a musician, and computes the produced sound and the time evolution of many acoustical quantities. Last, the software takes a measured acoustical
response and computes the corresponding instrument geometry (inner bore and tone holes parameters).
• Release Contributions: Inversion module and temporal module
• URL: https://openwind.gitlabpages.inria.fr/web/
• Publications: hal-02984478, hal-02996142, hal-03132474, hal-02917351, hal-02432750, hal-02019515, hal-01963674
• Authors: Robin Tournemenne, Juliette Chabassier, Alexis Thibault, Augustin Ernoult, Guillaume Castera, Tobias Van Baarsel, Olivier Geber
• Contacts: Juliette Chabassier, Alexis Thibault, Augustin Ernoult, Olivier Geber
• Participants: Juliette Chabassier, Augustin Ernoult, Alexis Thibault, Robin Tournemenne, Olivier Geber, Guillaume Castera, Tobias Van Baarsel
7.1.4 Gar6more2D
• Keywords: Validation, Wave propagation
• Functional Description: This code computes the analytical solution of problems of waves propagation in two layered 3D media such as- acoustic/acoustic- acoustic/elastodynamic- acoustic/porous-
porous/porous,based on the Cagniard-de Hoop method.
• News of the Year: In 2020, we have added the elasto/poroelastic coupling.
• URL: https://gitlab.inria.fr/jdiaz/gar6more2d
• Publications: inria-00274136, inria-00404224, inria-00305395
• Contacts: Julien Diaz, Abdelaâziz Ezziani
• Participants: Abdelaâziz Ezziani, Julien Diaz
• Partner: Université de Pau et des Pays de l'Adour
7.1.5 Utmodeling
• Name: Time-domain Wave-equation Modeling App
• Keywords: 2D, 3D, Elastoacoustic, Elastodynamic equations, Discontinuous Galerkin, Time Domain
• Scientific Description: tmodeling-DG simulate acoustic and elastic wave propagation in 2D and in 3D, using Discontinuous Galerkin Methods. The space discretization is based on two kind of basis
functions, using Lagrange or Jacobi polynomials. Different kinds of fluxes (upwind and centered) are implemented, coupled with RK2 and RK4 time schemes.
• Functional Description: tmodelling-DG is the follow up to DIVA-DG that we develop in collaboration with our partner Total. Its purpose is more general than DIVA-DG and should contains various DG
schemes, basis functions and time schemes. It models wave propagation in acoustic media, elastic (isotropic and TTI) media and elasto-acoustic media, in two and three dimensions.
• News of the Year: In 2020, the major addition comes from the implementation of the Spectral Element method to simulate wave propagation in acoustic media meshed with structured quadrilateral
meshes. We had in particular to adapt the existing spectral element codes (including the one developed in Aurélien Citrain's PhD thesis) to the structure of the utModeling code developed mainly
for Discontinuous Galerking method.
• Contacts: Julien Diaz, Hélène Barucq
• Participants: Julien Diaz, Lionel Boillot, Simon Ettouati, Hélène Barucq, Aurelien Citrain
• Partner: TOTAL
7.1.6 utFWI
• Name: Unstructured Time Domain Full Waveform Inversion
• Keywords: Discontinuous Galerkin, Inverse problem, Acoustic equation, 2D, Time Domain
• Functional Description: utFWI is developed in collaboration with Total within the framework of DIP (Depth Imaging Partnership). The aim is to solve the problem of seismic imaging using the FWI
(Full Waveform Inversion) method in the time domain on unstructured meshes, in particular using Galerkin Discontinuous methods. The objective is to characterize the physical properties of the
considered medium by iteratively solving a minimization problem. The direct problem is simulated using Galerkin Discontinuous elements. The reconstruction of the physical parameters is carried
out using gradient descent methods. The Solver works with the acoustic equation.
• News of the Year: In 2020, we have fully validated the 2D case. We have implemented the 3D case, and completed the validation on toy examples. The validation on realistic model is on going. We
have implemented new features : WADG (Weight Adjusted Discontinuous Galerkin methods) and adaptive meshing.
• Contacts: Julien Diaz, Pierre Jacquet, Hélène Barucq
• Participants: Pierre Jacquet, Julien Diaz, Hélène Barucq
8 New results
8.1 Analytical and experimental solutions for validation
8.1.1 Analytical solutions for elasto/poroelastic coupling
Participants: Julien Diaz.
Our software Gar6more computes the analytical solution of waves propagation problems in 2D homogeneous or bilayered media, based on the Cagniard-de Hoop method. In the bilayered case, we had
implemented the following coupling: acoustic/acoustic, acoustic/elastic, acoustic poroelastic, elastic/elastic, poroelastic/poroelastic. In the frame work of collaboration with Peter Moczo (Comenius
University Bratislava and Slovak Academy of Sciences) and David Gregor (Comenius University Bratislava), we have implemented the coupling between elastic and poroelastic media.
8.1.2 Experimental solutions for seismic/electromagnetic coupling
Participants: Hélène Barucq, Victor Gomes Martins.
Obtaining accurate images of water, mineral and energy sources deep below the surface is critical for their management and exploitation. Seismic imaging allows for obtaining maps of the Earth’s
interior and can be improved by coupling seismic and electromagnetic waves. Seismo-electric effects have been highlighted by Pride theory which yields equations combing Biot’s equations describing
waves in porous media with Maxwell equations. The question of reproducing seismo-electric coupling in laboratory is very interesting for two main reasons: (1) measurements the co-seismic and
converted wave are used for confirming the theory; (2) measurements in laboratory consist of a set of data that can be used in numerical experiments and provide a way for validating simulations.
Laboratory experiments are now operational and produce high-resolution data sets of direct and converted seismic and mechanical waves. We have been working on instrumentation for saturated sand
experiments to improve the signal-to-noise ratio. We have also adopted an approach to ensure the repeatability of the measurements. The instrumental and methodological developments open perspectives
for systematic parametric studies (salinity and saturation of the porous medium, thickness and nature of the porous interface) unique and very original. The prospect of quantitatively characterizing
in the laboratory the converted seismo-electric wave in terms of morphology and intensity remains the major objective of the experimental approach with the ultimate goal of applying these advances to
field geophysics. Besides having high-definition data, we aim now at comparing experimental solutions with numerical ones. This work is done in collaboration with Daniel Brito from LFCR, UPPA.
8.1.3 Acoustic Impedance Measurements
Participants: Juliette Chabassier, Augustin Ernoult.
An impedance sensor has been built to give the possibility to compare the simulation results of the wave propagation into wind instruments to experimental data. Several techniques exist to measure
the input impedance defined as the ratio of the acoustic pressure over the acoustic flow in the frequential domain at the entrance of the wind instrument (or any pipes). The sensor developed in
collaboration with Samuel Rodriguez is based on the "two microphones, three calibrations" method. The sensor is composed of a cylindrical pipe along which are placed five microphones, each couple
being associated to different frequency range. At one extremity a loudspeaker emits a chirp, and the measured object is placed at the other extremity. The designed sensor has been built by Augustin
Humeau in its workshop. Different pipes have been used as standards to validate the tools. The measuring bench can measure impedance from 40Hz to 10kHz with a good accuracy in a silent environment as
the one given by the soundproof room recently acquired by BSO-research Center of Inria. Experimental data have been compared to simulation results and are used in full waveform inversion process to
reconstruct the geometry of musical instruments. The presence of the five microphones gives us also the possibility to improve the reconstruction process by using directly the measured signal at the
five observation points without computing the impedance. This work has been done in collaboration with Robin Tournemenne (alumni), Samuel Rodriguez (I2M) and Augustin Humeau (workshop Humeau).
8.2 Mathematical modeling
8.2.1 Nonuniqueness of the quasinormal mode expansion of electromagnetic Lorentz dispersive materials
Participants: Marc Duruflé, Alexandre Gras.
Any optical structure possesses resonance modes, and its response to an excitation can be decomposed onto the quasinormal and numerical modes of a discretized Maxwell operator. In this paper, we
consider a dielectric permittivity that is an N-pole Lorentz function of the frequency. Even for discretized operators, the literature proposes different formulas for the coefficients of the
quasinormal-mode expansion, and this comes as a surprise. We propose a general formalism, based on auxiliary fields, which explains why and evidences that there is, in fact, an infinity of
mathematically sound possible expansion coefficients. The nonuniqueness is due to a choice of the linearization of Maxwell’s equations with respect to frequency and of the choice of the form of the
source term. Numerical results validate the different formulas and compare their accuracy. This work has been done in collaboration with Philippe Lalanne (Bordeaux INP - Institut Polytechnique de
Bordeaux and LP2N - Laboratoire Photonique, Numérique et Nanosciences) and is published in 19.
8.2.2 Improvement of the modal expansion of electromagnetic fields through interpolation
Participants: Marc Duruflé, Alexandre Gras.
We consider optical structures where the dielectric permittivity is a rational function of ω (Lorentz model). Electromagnetic fields can be computed for a large number of frequencies by calculating
the eigenmodes of the optical device and by reconstructing the solution by developing it on these modes. This modal development suffers from many limitations that are detailed in 30. In order to
overcome these limitations, an interpolation procedure is proposed so that the electric field is computed directly for a small number of interpolation points. Numerical experiments in 2-D and 3-D
show the efficiency of this approach. This work has been done in collaboration with Philippe Lalanne (Bordeaux INP - Institut Polytechnique de Bordeaux and LP2N - Laboratoire Photonique, Numérique et
8.2.3 Equivalent multipolar point-source modeling of small spheres for fast and accurate electromagnetic wave scattering computations
Participants: Sébastien Tordeux.
We develop reduced models to approximate the solution of the electromagnetic scattering problem in an unbounded domain which contains a small perfectly conducting sphere. Our approach is based on the
method of matched asymptotic expansions. This method consists in defining an approximate solution using multi-scale expansions over outer and inner fields related in a matching area. We make explicit
the asymptotics up to the second order of approximation for the inner expansion and up to the fifth order for the outer expansion. We validate the results with numerical experiments which illustrate
theoretical orders of convergence for the asymptotic models requiring negligible computational cost. This work has been published in 20 and was done in collaboration with Justine Labat from CEA Cesta
and Victor Péron from LMAP, UPPA.
8.2.4 Extension of the Gunter derivatives to Lipschitz domains and application to the boundary potentials of elastic wavesJournal of Applied Mechanics and Technical Physics
Participants: Sébastien Tordeux.
Regularization techniques for the trace and the traction of elastic waves potentials previously built for regular domains are extended to the Lipschitz case. In particular, this yields an elementary
way to establish the mapping properties of elastic wave potentials from those of the scalar Helmholtz equation without resorting to the more advanced theory for elliptic systems in the Lipschitz
domains. Representations of the Gunter operator and potentials of single and double layers of elastic waves in the two-dimensional case are provided. This work has been published in 12 This is a
joint work with Yuriy Matveevich Volchkov and Abderrahmane Bendali.
8.2.5 Asymptotic behavior of acoustic waves scattered by very small obstacles
Participants: Hélène Barucq, Julien Diaz, Sébastien Tordeux.
The direct numerical simulation of the acoustic wave scattering created by very small obstacles is very expensive, especially in three dimensions and even more so in time domain. The use of
asymptotic models is very efficient and the purpose of this work is to provide a rigorous justification of a new asymptotic model for low-cost numerical simulations. This model is based on asymptotic
near-field and far-field developments that are then matched by a key procedure that we describe and demonstrate. We show that it is enough to focus on the regular part of the wave field to rigorously
establish the complete asymptotic expansion. For that purpose, we provide an error estimate which is set in the whole space, includingthe transition region separating the near-field from the
far-field area. The proof of convergence is established through Kondratiev’s seminal work on the Laplace equation and involves the Mellin transform. Numerical experiments including multiple
scattering illustrate the efficiency of the resulting numerical method by delivering some comparisons with solutions computed with a finite element software. This work has been published in 8. It was
done in collaboration with Vanessa Mattesi from Liège University.
8.2.6 Outgoing solutions for the scalar wave equation in helioseismology
Participants: Hélène Barucq, Ha Pham.
In this work, we study the time-harmonic scalar equation describing the propagation of acoustic waves in the Sun's atmosphere under ideal atmospheric assumptions. We use the Liouville change of
unknown to conjugate the original problem to a Schrödinger equation with a Coulomb-type potential. This transformation makes appear a new wavenumber, k, and the link with the Whittaker's equation. We
consider two different problems: in the first one, with the ideal atmospheric assumptions extended to the whole space, we construct explicitly the Schwartz kernel of the resolvent, starting from a
solution given by Hostler and Pratt in punctured domains, and use this to construct outgoing solutions and radiation conditions. In the second problem, we construct exact Dirichlet-to-Neumann map
using Whittaker functions, and new radiation boundary conditions (RBC), using gauge functions in terms of k. The new approach gives rise to simpler RBC for the same precision compared to existing
ones. The robustness of our new RBC is corroborated by numerical experiments. This work was started in 2019, and results in an article 11 published in 2020. This work was done in collaboration with
Florian Faucher from Vienna University.
8.2.7 Outgoing solution and Radiation boundary condition for spherical Galbrun equation
Participants: Hélène Barucq, Ha Pham.
In this project, we consider the time-harmonic Galbrun’s equation under spherical symmetry in the context of the wave propagation in the Sun without flow and rotation, and neglecting the
perturbations to the gravitational potential. For this equation, we construct the outgoing modal solutions, the 3D Green’s kernel, and radiation boundary conditions. The construction is justified by
indicial and asymptotic analysis of the modal radial ODE. Our asymptotic analysis makes appear the correct wavenumber and the high-order terms of the oscillatory phase function, which we use to
characterize outgoing solutions. The radiation boundary conditions are built for the modal radial ODE and then derived for the initial equation. We approximate them under different hypothesis and
propose some formulations that are independent of the horizontal wavenumber and can thus easily be applied for 3D problems. The results are documented in the Inria report 28. We also prepared an
article and submitted to Journal of Differential Equations.
This works also requires the construction of C2 representations of the background quantities that characterize the interior of the Sun and its atmosphere starting from the data-points of the standard
solar model S. The constructed models are documented in 32. These works were done in collaboration with Florian Faucher from Vienna University, and Damien Fournier and Laurent Gizon from MPS.
8.2.8 Radiation boundary conditions for wave problems based upon Calderon operators
Participants: Hélène Barucq, Ha Pham.
We construct radiation conditions by accurately modeling the propagation of a time-harmonic wave in the vicinity of an interface. This idea is not new and has been exploited in particular to
construct radiation conditions for the Helmholtz equation by solving an Airy equation obtained as an approximation of the Helmholtz operator in the vicinity of the interface. In this project, a
Calderon operator is constructed and exact radiation conditions are constructed. This work is ongoing, in collaboration with Olivier Lafitte from Montréal University.
8.2.9 Modeling and simulation of the piano touch
Participants: Juliette Chabassier, Guillaume Castera.
In this PhD work, we develop physical models for the piano to understand the real influence of the pianist on the sound. Mechanical models [iMMC] will be paired with vibro-acoustical models [INRIA]
to analyse the differences in sound depending on the pianist’s touch. We are currently working on the crucial contact between the hammer and the string(s) which links the piano action to its
vibro-acoustical part. Friction at impact must be taken into account in order to transmit all forces to the string. It could influence the longitudinal vibrations of the string, and then the presence
of some partials in the final sound. We also implement these models in c++ into the MONTJOIE software. We use energy-based numerical schemes to guarantee stability, and auxiliary variables to deal
with nonlinear terms. This PhD is co-supervised with Paul Fisette (Univ. Cath. Louvain, Belgium).
8.2.10 Physical based synthesis of heritage wind musical instruments
Participants: Juliette Chabassier, Augustin Ernoult, Tobias Van Baarsel.
The SYSIMPA project (Synthèse Sonore des Instruments de Musique du PAtrimoine) spans over two years and is carried out by a consortium made up of Inria, la Cité de la Musique-Philharmonie de Paris
(CM-P), l'Institut Technologique Européen des Métiers de la Musique (ITEMM) in Le Mans, and le Centre Culturel et de Restauration des Musées de France (C2RMF). This project aims at studying ancient
music wind instruments (in this case, natural trumpets from around 1900) to estimate their acoustic properties and to synthesise their sound. Also, a copy of one of the instruments will be made by a
instrument maker. The role of Magique-3D is at the same time to coordinate the different teams involved in SYSIMPA, and to deal with the scientific computation and sound synthesis aspects. The CM-P
gives access to the music instruments and provides expertise on conservation and impedance measurements. The C2RMF has the facilities to perform X-ray tomography on ancient instruments. Finally,
ITEMM drives the instrument-making aspect thanks to its partnership with the Institut National des Métiers d'Arts (INMA).
Most of the tools required for the project have been developped during this first year. Ten natural trumpets have been selected for the study, and have been put through x-ray tomography. In the
meantime, all the paper archives (approx.3500 documents) corresponding to the music instruments and their makers have been scanned by the CM-P, and might provide useful information about the
instrument making and/or playing. These documents will eventually be put online and made available to the public. The C2RMF sent us the data from the X-ray tomography. We developped a code to
automatically extract the bore of the instruments from the raw images. The geometry estimation has been compared with a physical measurement of the input impedance performed at the CM-P, and
validates the whole procedure for the most of the instrument. The mouthpiece is still a challenging piece, as the width of the metal in this part is large compared to the inner bore ; a separate
measurement of the mouthpiece is planned in early march 2021 using a silicone mould.
The scientific computation done by Magique-3D consists of two parts : first a frequential computation part aiming at calculating the acoustic characteristic of the resonator through the impedance ;
second a temporal computation that couples the resonator with a vibration model describing the musician's lips. The computation is done using OpenWind, a software currently developped by the INRIA
team. The frequential computation has been compared with impedance measurements done at the CM-P. The comparison shows satisfactory agreement between simulation and experiment. The temporal
simulation is on-going work. The coupling between a source and a resonator is tricky and needs finely tuned parameters in order to reach the sustained oscillation regime. The parameters (i.e.,
stiffness and mass of the lips, mouth pressure, etc.) are constantly adapted by the musician while playing, but cannot be directly measured. We will therefore rather use the dynamic systems theory,
applied to wind instruments, to estimate the right set of parameters. Linear Stability Analysis and Harmonic Balance are some of the techniques being explored. This work on temporal simulation will
allow us to simulate the sound of the studied trumpets, based on physical models and data extracted from x-ray tomography.
This work is done in collaboration with Romain Viala (ITEMM); Clotilde Boust and Elsa Lambert (C2RMF) ; Thierry Maniguet, Marguerite Jossic, Rodolphe Bailly, Cécile Cecconi and Sebastian Kirsch
8.2.11 Modeling and simulation of acoustical and dissipative phenomena in wind instruments
Participants: Juliette Chabassier, Alexis Thibault.
This research has been centered around modeling and simulation of acoustical and dissipative phenomena in wind instruments. It has contributed to the new public version of OpenWInD, released in
december 2020, by implementing a numerical scheme for acoustic wave propagation with viscothermal losses, as well as several other models (spherical waves, transfer matrices, radiation of a pulsating
sphere), and by writing a tutorial for new users of the toolbox. Bibliography around porous effect inside the instrument body is currently under investigation.
8.2.12 Viscothermal models for wind musical instrument
Participants: Juliette Chabassier, Alexis Thibault.
33 is a review of one-dimensional and three-dimensional models of linear acoustic propagation with viscothermal effects, with the intent of applying them to wind instruments. It includes the
derivation of several models from the linearized Navier-Stokes equations. The differences between the models are evaluated numerically and related to the simplifying assumptions used in deriving each
8.2.13 Dissipative time-domain 1D model for viscothermal acoustic propagation in wind instruments
Participants: Juliette Chabassier, Alexis Thibault.
An approximate 1D time-domain model of acoustic propagation with boundary layer losses is proposed, where all the physical parameters of the instrument as the bore shape or the wave celerity are
explicit coefficients. The model depends on absolute tabulated constants which only reflect that the pipe is axisymmetric. It can be seen as a telegrapher's equations augmented by an adjustable
number of auxiliary unknowns. A global energy is dissipated. A variational approximation is proposed along with numerical experiments and comparisons with other models. This work is under review by
the Journal of the Acoustical Society of America.
8.2.14 Time-domain simulation of a dissipative reed instrument
Participants: Juliette Chabassier, Alexis Thibault.
This work focuses on the time-domain models and numerical schemes implemented in the OpenWInD toolbox. Modular components may be assembled and simulated, with stability guaranteed through energy
consistency. This has been communicated during the Forum Acusticum 2020 22.
8.2.15 The virtual workshop OpenWinD : physical modeling assisting wind instrument makers
Participants: Juliette Chabassier, Augustin Ernoult, Olivier Geber, Alexis Thibault, Tobias Van Baarsel.
Our project develops the software OpenWInD for wind instrument making. A first feature is the prediction of the acoustical response of the instrument from the knowledge of its shape (bore and holes).
This can be done in the harmonic (impedance computation) and temporal (sound computation) domains. It can account for various physical situations (non constant temperature, coupling with an
embouchure...). Discretization is done in space with 1D spectral finite elements and in time with energy consistent finite differences. The second feature is the reconstruction of the shape of an
instrument that fulfils a certain objective. This can be used for bore reconstruction, and instrument design. The latter is based on a strong interaction with makers and musicians, aiming at defining
interesting design parameters and objective criteria, from their point of view. After a quantitative transcription of these criteria, under the form of a cost function and a design parameter space,
we implement various gradient-based optimization techniques. More precisely, we exploit the fact that the sound waves inside the instruments are solution to acoustic equations in pipes, which gives
us access to the Full Waveform Inversion technique (FWI) where the gradient is characterized as the solution to another wave equation. The computational framework is flexible (in terms of models,
formulations, coupling terms, objective functions...) and offers the possibility to modify the criterion by the user. The goal is to proceed iteratively between the instrument makers and the
numerical optimisation tool (OpenWInD) in order to achieve, finally, criteria that are representative for the makers. This modeling tool give us the possibility to perform and analyze comparisons
between measurements and simulation on real instruments in order to complete the model until to reach good enough accuracy to help the manufacturers in the design of new instruments. It has been
presented at the conference Forum Acusticum 2020 23. This work has been done in collaboration with Robin Tournemenne (alumni) and Augustin Humeau (workshop Humeau).
8.3 High-order numerical methods for time-dependent problems
8.3.1 Tent Pitcher algorithm for space-time integration of wave problems
Participants: Hélène Barucq, Julien Diaz, Vinduja Vasanthan.
This thesis started on October 1st, 2019. Its objective is to develop a Trefftz-DG-Tent-Pitching formulation equipped with local-time stepping and outgoing boundary conditions in a full parallel
environment constructed on unstructured nD+time meshes. A first formulation constructed on structured meshes and tested for toy examples has been given in E. Shishenina’s thesis. The first year of
the PhD has been mostly devoted to bibliography on :
• Trefftz & boundary element methods,
• Different kinds of variational formulations (i.e Trefftz-Discontinuous Galerkin, Trefftz-Least Squares, Method of Fundamental Solutions, Ultra Weak Variational Formulation, Variational Theory of
Complex Rays, Wave Based Method, etc),
• Different kinds of basis (Generalized Harmonic Polynomials,Plane Waves, fundamental solutions & multipoles, etc),
• Tent Pitcher algorithm.
Based on this, we also derived new formulations with another type of fundamental solutions as basis. This lead us to introduce alternative Absorbing Boundary Conditions applied to our problem.
In parallel, we ported the Matlab code to Fortran.
8.3.2 Construction and convergence analysis of conservative second order local time discretisation for linear wave equations
Participants: Juliette Chabassier.
In this work we present and analyse a time discretisation strategy for linear wave equations based on domain decomposition that aims at using locally in space the most adapted time discretisation
among a family of implicit or explicit centered second order schemes. The proposed family of schemes is adapted to domain decomposition methods such as the mortar element method. They correspond
respectivelyin that case to local implicit schemes and to local time stepping. We show that, if some regularity properties of the source termsolution are satisfied and if the time step verifies a
stability condition, then the family of proposed time discretisations provides, in a strong norm, second order space-time convergence. Finally, we provide 1D numerical illustrations that confirm the
obtained theoretical results and we compare our approach to other existing local time stepping strategies for wave equations. This work is under review 29. It is a collaboration with Sébastien
Impériale from project-team M3DISIM.
8.3.3 High-order locally A-stable implicit schemes for linear ODEs
Participants: Hélène Barucq, Marc Duruflé, Mamadou N'Ddiaye.
Accurate simulations of wave propagation in complex media like Earth subsur-face can be performed with a reasonable computational burden by using hybrid meshes stuffing fine and coarse cells. Locally
implicit time discretizations are then of great interest. They indeed allow using unconditionally stable schemes in the regions of computational domain covered by small cells. The receivable values
of the time step are then increased which reduces the computational costs while limiting the dispersion effects. In this work we construct a method that combines optimized explicit schemes and
implicit schemes to form locally implicit schemes for linear ODEs, including in particular semi-discretized wave problems that are considered herein for numerical experiments. Both the explicit and
implicit schemes used are one-step methods constructed using their stability function. The stability function of the explicit schemes are computed by maximizing the time step that can be chosen. The
implicit schemes used are unconditionally stable. The performance assessment we provide shows a very good level of accuracy for locally implicit schemes. It also shows that locally implicit scheme is
a good compromise between purely explicit and purely implicit schemes in terms of computational time and memory usage. This work has been published in 9.
8.4 High-order numerical methods for time-harmonic problems
8.4.1 A discontinuous Galerkin Trefftz type method for solving the two dimensional Maxwell equations
Participants: Margot Sirdey, Sébastien Tordeux.
Trefftz methods are known to be very efficient to reduce the numerical pollution when associated to plane wave basis. However, these local basis functions are not adapted to the computation of
evanescent modes or corner singularities. In this article, we consider a two dimensional time-harmonic Maxwell system and we propose a formulation which allows to design an electromagnetic Trefftz
formulation associated to local Galerkin basis computed thanks to an auxiliary Nédélec finite element method. The results are illustrated with numerous numerical examples. The considered test cases
reveal that the short range and long range propagation phenomena are both well taken into account. This work has been published in 18 This is a joint work with Håkon Sem Fure and Sébastien Pernet.
8.4.2 Numerical computation of Green function in helioseismology
Participants: Hélène Barucq, Ha Pham.
In this work, we provide an algorithm to compute efficiently and accurately the full outgoing modal Green's kernel for the scalar wave equation in local helioseismology under spherical symmetry. Due
to the high computational cost of a full Green's function, current helioseismic studies rely on single-source computations. However, a more realistic modelization of the helioseismic products
(cross-covariance and power spectrum) requires the full Green's kernel. In the classical approach, the Dirac source is discretized and one simulation gives the Green's function on a line. Here, we
propose a two-step algorithm which, with two simulations, provides the full kernel on the domain. Moreover, our method is more accurate, as the singularity of the solution due to the Dirac source is
described exactly. In addition, it is coupled with the exact Dirichlet-to-Neumann boundary condition, providing optimal accuracy in approximating the outgoing Green's kernel, which we demonstrate in
our experiments. In addition, we show that high-frequency approximations of the nonlocal radiation boundary conditions can represent accurately the helioseismic products. This work results in 81-page
Inria report 10 and article 10.
8.4.3 Low-order absorbing boundary condition for two-dimensional isotropic poroelasticity
Participants: Hélène Barucq, Julien Diaz, Ha Howard Faucher, Rose-Cloé Meyer.
In this work, we construct a low order absorbing boundary condition (ABC) for two-dimensional isotropic poroelasticity in frequency domain. The ABC is obtained for circular geometry by approximating
the behavior of the analytical outgoing wave solution. The ABC is then extended to general non-circular domains and implemented with Hybridizable Discontinuous Galerkin (HDG) method. In circular
symmetry, using the form of the exact solution, the robustness of the ABC is evaluated for the problem of scattering of planewave by a circular obstacle. We also compare the performance of this ABC
with Perfectly Matched Layers, both coupled with HDG method. The results from this work are presented in Inria report 26. It is done in collaboration with Florian Faucher from Vienna University, and
Damien Fournier and Laurent Gizon for MPS.
8.4.4 Hybridizable Discontinuous Galerkin method for time-harmonic anisotropic poroelasticity in two dimensions.
Participants: Hélène Barucq, Julien Diaz, Ha Pham, Rose-Cloé Meyer.
In this work, we apply a Hybridizable Discontinuous Galerkin (HDG) method to numerically solve two-dimensional anisotropic poroelastic wave equations in the frequency domain given by Biot theory. The
motivation for choosing HDG method comes from the complexity of the considered equations and the high number of unknowns. The HDG method possesses all the advantages of Discontinuous Galerkin method
(hp-adaptivity, accuracy, ability to model complex tectonics,...) without a drastic increase in the number of degrees of freedom. We study the accuracy of the proposed method by comparisons with
analytical solutions and a sensitivity analysis of the method as a function of stabilization parameters and frequency. We also show the ability of the method to reproduce the different types of
poroelastic waves including the slow Biot wave on realistic geophysical media. Results on the HDG method for poroelasticity are presented in the research report 25, and in an article in preparation.
8.4.5 HDG methods for the convected Helmholtz Equation.
Participants: Hélène Barucq, Nathan Rouxelin, Sébastien Tordeux.
The need for numerical simulation of harmonic waves propagating in complex flows arises in the context of computational helioseismology. As standard finite element methods do not perform well for
those problems as they usually assume too much regularity on the solution, it seems natural to consider the use of discontinuous Galerkin methods. In order to obtain a method with a reduced numerical
cost, we focus on a particular type of discontinuous Galerkin method : the so-called Hybridizable Discontinuous Galerkin method (HDG). The main feature of this method is the static condensation
process, leading to an elimination of interior degrees of freedom and therefore to a problem posed only on the skeleton of the mesh.
As those kinds of method have never been used for time-harmonic aeroacoustic wave propagation, we have started our work by considering the simplest aeroacoustic model : the convected Helmholtz
equation. Even if this model can only be used in a very limited range of physical configurations, its study is a very important first step. Indeed, in contrary to other models, the natural framework
for the study of the convected Helmholtz equation is clear and standard. We can therefore perform the numerical analysis of the method, leading to results on the local and global solvability of the
method, as well as a detailed analysis of the convergence rate.
It is important to notice that in the process of designing a HDG method for the convected Helmholtz equation, we had to make some choices on both the unknowns of the method and the approximation
spaces. We therefore chose to work on the three most natural choices to understand their different properties.
In the future, we hope to generalize our results and construct HDG methods for more realistic aeroacoustic models such as Galbrun’s or Goldstein’s equations.
8.4.6 Isogeometric analysis of sharp boundaries in full waveform inversion
Participants: Hélène Barucq, Julien Diaz, Stefano Frambati.
Efficient seismic full-waveform inversion simultaneously demands a high efficiency per degree of freedom in the solution of the PDEs, and the accurate reproduction of the geometry of sharp contrasts
and boundaries. Moreover, it has been shown that the stability constant of the FWI minimization grows exponentially with the number of unknowns. Isogeometric analysis has been shown to possess a
higher efficiency per degree of freedom, a better convergence in high energy modes (Helmholtz) and an improved CFL condition in explicit-time wave propagation, and it seems therefore a good candidate
for FWI.
We have first focused on a small-scale one-dimensional problem, namely the inversion over a multi-step velocity model using the Helmholtz equation. By exploiting a relatively little-known connection
between B-splines ad Dirichlet averages, we have added the knot positions as degrees of freedom in the inversion. We have shown that arbitrarily-placed discontinuities in the velocity model can be
recovered using a limited amount of degrees of freedom, as the knots can coalesce at arbitrary positions, obviating the need for a very fine mesh and thus improving the stability of the inversion.
In order to reproduce the same results in two and three dimensions, the usual tensor-product structure of B-splines cannot be used. We have therefore turned our attention to the spaces of
(unstructured) multivariate B-spline bases. In the first part of our work, we have uncovered a connection between unstructured polynomial-reproducing spline spaces and some objects known in
combinatorial geometry as zonotopal tilings. Due to their purely combinatorial character, these spline spaces automatically cover the case of repeated and affinely dependent points, allowing to
construct a broad family of splines with adjustable smoothness, up to (and including) discontinuities. This construction works in any number of space dimensions, and the mathematical properties of
zonotopal tilings can be exploited to devise some practical algorithms for the construction and evaluation of these spline functions, useful in practical applications.
A research article containing these results has been submitted to the journal “Mathematics of Computation” in collaboration with Hélène Barucq, Julien Diaz and Henri Calandra. It is part of a 61-page
long research report 24.
In the second part of our work, we have exploited the flexible regularity of our spline spaces in order to place internal boundaries into our simulation domains, decomposing them into multiple
sub-domains of adjustable size. Adding standard IPDG fluxes between the subdomains then yields a very flexible numerical scheme that contains FEM, DG and (meshless) IGA as extreme cases, and allows
to interpolate between them, offering a unifying perspective. The mass matrix obtained through our approach is block-diagonal, thus realizing a simple but powerful unstructured multi-patch DG-IGA
hybrid, which is especially useful for time-explicit wave propagation simulations. A 2D simulation code was written on this premise, and numerical tests have been realized on synthetic datasets
coming from applications in geoscience, helioseismic and musical instruments. Results have shown that the numerical advantages of IGA, notably the $1/p$ dependence of the CLF timestep on the
polynomial order $p$, are maintained, and good parallelizablility is achieved due to the block-diagonal mass matrix. Furthermore, each IGA domain is allowed to have an arbitrary topology, including
any number of internal holes and boundaries, which can be especially interesting for the simulation and optimization of the acoustic properties of musical instruments.
These numerical results have been presented to the ECCOMAS 2020 congress (https://virtual.wccm-eccomas2020.org/) which was organized online in January 2021.
Our efforts are currently focused on the realization of 3D numerical tests and the implementation of FWI inversion in 2D and 3D. A second journal article on the details of the unstructured
multi-patch DG-IGA is in the works.
A research article has been submitted to Mathematical of Computation in collaboration with Hélène Barucq, Julien Diaz and Henri Calandra. It is part of a 61-page long research report 24. The
numerical results have been presented to ECCOMAS 2020 congress (https://virtual.wccm-eccomas2020.org/) which was organized online.
This work is done in collaboration with Henri Calandra from Total.
8.5 Reconstruction and design using full waveform inversion
8.5.1 Time-Domain Full Waveform Inversion using advanced Discontinuous Galerkin methods
Participants: Hélène Barucq, Julien Diaz, Pierre Jacquet.
In this project, we developed tools for the reconstruction of subsurface media for seismic imaging and reservoir characterization in an industrial context. For that purpose, we used the Full Waveform
Inversion (FWI) method. It is a reconstruction technique using data taken from seismic disturbances and whose behavior reflects the properties of the environment in which they propagate. In the
framework of this thesis, we consider acoustic waves which are simulated thanks to Discontinuous Galerkin methods. These methods offer a very flexible discretization in space allowing to approach
complex models and geometries. Discontinuous Galerkin methods are characterized by the use of fluxes in between each cell. Those fluxes contribute to have low communication costs which are highly
recommended for High Performance Computing. Here, the wave equation is solved in time domain to overcome the memory limitations encountered in frequency domain for the reconstruction of large-scale
3D industrial media.
To reconstruct quantitatively the physical model under study, we wrote the inverse problem as a minimization problem solved by adjoint state method. This method makes it possible to obtain the
gradient of the cost function with respect to the physical parameters for the cost of two simulations; the direct problem and the backward problem also called adjoint problem. The adjoint state will
be the solution of the discretized continuous adjoint problem ("Optimize Then Discretize"). This choice is justified by a 1D comparison with the strategy which consists in "Discretize then Optimize"
completed by an algebraic study in superior dimension. The gradient thus calculated, is a key in the optimization procedure developed and integrated in the industrial environment provided by the
industrial partner, Total.
The propagator is a keystone in solving the inverse problem. Indeed, it is repeated successively and represents the majority of the computation time of the optimization process. It is therefore
important to control the discretization by the Discontinuous Galerkin method as well as possible. In particular, in this thesis, we have considered the idea of using different polynomial bases of
approximation (Legendre or Bernstein-Bézier) as well as the choice of the parameterization, which can either be constant per element or variable thanks to the use of the Weight Adjusted Discontinuous
Galerkin (WADG) method. This last strategy offers the opportunity to enlarge the mesh cells without losing information on the model and thus allows a more advanced use of the $hp$-adaptivity that we
propose to fully exploit thanks to an adaptive mesh that is adjusted to the model meant to evolve with the iterations of the inverse problem.
8.5.2 Full Waveform Inversion on data including surface waves
Participants: Hélène Barucq, Julien Diaz, Chengyi Shen.
This work is a part of the PIXIL project (Pyrenees Imaging eXperience: an InternationaL network). We aim at applying a Fortran HPC imaging tool, named HAWEN and developed by Florian Faucher in
time-harmonic domain featuring the Hybridizable Discontinuous Galerkin method, onto real data. In particular, surface waves will be integrally taken into concern for the following reasons: first, the
PIXIL project focuses on geophysical surveys for geothermal applications, where surface waves carry essential information of near-surfaces especially for shallow geothermal explorations; Second, a
good image of the near-surface can help improve deep imaging. We looked into a 2D synthetic case study in order to establish one or several Multi-level Strategies for FWI on data including surface
waves. A trade-off between robustness and high-resolution is achievable by elaborating suitable strategies such as combining asymptotic methods and FWI with frequency groups. Meanwhile, Bash and
Python programs are created to assist HAWEN for user-friendly concerns as well as data pre/post-processing, for instance, automatization of executions, data processing and visualization. This is a
joint work with Jean-Luc Boelle and Jean-Claude Puech from the SME Real Time Seismic.
8.5.3 Full reciprocity-gap waveform inversion enabling sparse-source acquisition
Participants: Hélène Barucq.
The quantitative reconstruction of subsurface earth properties from the propagation of waves follows an iterative minimization of a misfit functional. In marine seismic exploration, the observed data
usually consist of measurements of the pressure field, but dual-sensor devices also provide the normal velocity. Consequently, a reciprocity-based misfit functional is specifically designed, and it
defines the full reciprocity-gap waveform inversion (FRgWI) method. This misfit functional provides additional features compared to the more traditional least-squares approaches, in particular, in
that the observational and computational acquisitions can be different. Therefore, the positions and wavelets of the sources from which the measurements are acquired are not needed in the
reconstruction procedure and, in fact, the numerical acquisition (for the simulations) can be chosen arbitrarily. Based on 3D experiments, FRgWI is shown to behave better than full-waveform inversion
in the same context. It allows for arbitrary numerical acquisitions in two ways: when few measurements are given, a dense numerical acquisition (compared to the observational one) can be used to
compensate. However, with a dense observational acquisition, a sparse computational one is shown to be sufficient, for instance, with multiple-point sources, hence reducing the numerical cost. FRgWI
displays accurate reconstructions in both situations and appears more robust with respect to crosstalk than least-squares shot stacking. This work has been done in collaboration with Florian Faucher
(Vienna University), Giovanni Alessandrini (Faculty of Mathematics, Vienna), Maarten de Hoop (Rice University,Houston), Romina Gaburro (UL - University of Limerick) and Eva Sincich (University of
Trieste). It has been published in 15.
8.5.4 A priori estimates of attraction basins for velocity model reconstruction by time-harmonic Full Waveform Inversion and Data-Space Reflectivity formulation
Participants: Hélène Barucq.
The determination of background velocity by Full Waveform Inversion (FWI) is known to be hampered by the local minima of the data misfit caused by the phase shifts associated with background
perturbations. Attraction basins for the underlying optimization problems can be computed around any nominal velocity model and guarantee that the misfit functional has only one (global) minimum. The
attraction basins are further associated with tolerable error levels representing the maximal allowed distance between the (observed) data and the simulations (i.e., the acceptable noise level). The
estimates are defined a priori, and only require the computation of (possibly many) first- and second-order directional derivatives of the (model to synthetic) forward map. The geometry of the search
direction and the frequency influence the size of the attraction basins, and complex frequency can be used to enlarge the basins. The size of the attraction basins for the perturbation of background
velocities in the classical FWI (global model parametrization) and the data-space reflectivity reformulation (MBTT) are compared: the MBTT reformulation increases substantially the size of the
attraction basins (by a factor of four to fifteen). Practically, this reformulation compensates for the lack of low-frequency data. Our analysis provides guidelines for a successful implementation of
the MBTT reformulation. This work has been done in collaboration with Florian Faucher (Faculty of Mathematics, Vienna), Guy Chavent (SERENA, Inria) and Henri Calandra (Total E&P) and has been
published in 16.
8.5.5 Eigenvector models for solving the seismic inverse problem for the Helmholtz equation
Participants: Hélène Barucq.
We study the seismic inverse problem for the recovery of subsurface properties in acousticmedia. In order to reduce the ill-posedness of the problem, the heterogeneous wave speed parameter is
represented using a limited number of coefficients associated with a basis of eigenvectors of a diffusion equation, following theregularization by discretization approach. We compare several choices
for the diffusion coefficient in the partial differential equations,which are extracted from the field of image processing. We first investigate their efficiency forimage decomposition (accuracy of
the representation with respect to the number of variables). Next, we implement the method in the quantitative reconstruction procedure for seismic imaging, following the full waveform inversion
method, where the difficulty resides in thatthe basis is defined from an initial model where none of the actual structures is known. In particular, we demonstrate that the method may be relevant for
the reconstruction of media with salt-domes. We use the method in 2-D and 3-D experiments, and show that the eigenvector representation compensates for the lack of low-frequency information, it
eventually serves usto extract guidelines for the implementation of the method. This work has been done with Florian Faucher (Faculty of Mathematics, Vienna) and Otmar Scherzer (University of
Vienna). It has been published in 17.
8.5.6 Bore reconstruction of woodwind instruments using the Full Waveform Inversion
Participants: Juliette Chabassier, Augustin Ernoult.
Several techniques can be used to reconstruct the internal geometry of a wind instrument from acoustics measurements. In this study, the passive linear acoustic response of the instrument is
simulated and an optimization process is used to fit the simulation to the measurements. This technique can be seen as a first step toward the design of wind instruments, where the targeted acoustics
properties come no more longer from measurements but are imposed by the designer. The difficulties of this approach are to find the best acoustic observation allowing the reconstruction (impedance,
reflection function, etc.) but also to have an efficient optimization process. The "full waveform in-version" (FWI) is a technique coming from the seismology community. It uses the knowledge of the
equation modeling the wave propagation into the instrument (here the telegraphist equation) to have an explicit expression of the gradient of the function which is minimized. This gradient is
evaluated with a low computational cost. The FWI methodology, along with 1D spectral finite element discretization in space, applied to the woodwind instruments (with tone holes, losses and
radiation) is presented in this communication. The results obtained for the bore reconstruction with different acoustics observations are then compared and discussed. This work has been presented at
the conference Forum Acusticum 2020 21. subsubsectionAn effective numerical strategy for retrieving all characteristic parameters of an elastic scatterer from its FFP measurements.
Participants: Hélène Barucq, Julien Diaz.
A new computational strategy is proposed for determining all elastic scatterer characteristics including the shape, the material properties (Lamé coefficients and density), and the location from the
knowledge of far-field pattern (FFP) measurements. The proposed numerical approach is a multi-stage procedure in which a carefully designed regularized iterative method plays a central role. The
adopted approach is critical for recognizing that the different nature and scales of the sought-after parameters as well as the frequency regime have different effects on the scattering
observability. Identification results for two-dimensional elastic configurations highlight the performance of the designed solution methodology. This is a joint work with Izar Aspiroz, research
assistant at Vicomtech (Spain) and Rabia Djellouli, professor at CSUN (United States). It has been published in Journal of Computational Physics 7.
9 Bilateral contracts and grants with industry
9.1 Bilateral contracts with industry
• Depth Imaging Partnership (DIP3)
Period: 2019 May - 2021 December, Management: INRIA Bordeaux Sud-Ouest, Amount: 120000 euros/year.
• Tent Pitcher algorithm for space-time integration of wave problems
Period: 2019 November - 2022 October, Management: INRIA Bordeaux Sud-Ouest, Amount: 165000 euros.
• Isogeometric analysis of sharp boundaries in fullwaveform inversion
Period: 2019 January - 2021 December, Management: INRIA Bordeaux Sud-Ouest, Amount: 55000 euros.
• FWI (Full Waveform Inversion) in the time domain based upon hybrid discontinuous numerical methods
Period: 2017 October - 2020 December , Management: INRIA Bordeaux Sud-Ouest, Amount: 180000 euros.
• Petrophysics in pre-salt carbonate rocks
Period: 2019 November - 2021 June, Management: INRIA Bordeaux Sud-Ouest, Amount: 142000 euros.
10 Partnerships and cooperations
10.1 International initiatives
10.1.1 Inria associate team not involved in an IIL
• Title: Advanced Numerical meThods for helioSeismology
• Duration: 2019 - 2022
• Coordinator: Ha Pham
• Partners: Max Plank Institut für Sonnensystemforschung (Germany) – Department Solar and Stellar Interiors – Laurent Gizon.
• Inria contact: Ha Howard Faucher
• Summary: Magique-3D proposes an Associate Team project, Advanced Numerical meThods for helioSeismology (ANTS), with the Max Planck Institute for Solar System Research (MPS), led by Laurent Gizon.
The objective is to develop advanced software for accurate simulation of stellar oscillations and for the reconstruction of the Sun's interior. The novelty and challenge come from working with
convected vector wave equations in the presence of complex flow and gravity, for a more accurate description of the physical phenomenon. The software will use Hybridizable Discontinuous Galerkin
(HDG) approximation and will be developed on the existing platform Montjoie of Magique-3D. The scientific project benefits from the expertise of Magique-3D in seismic imaging, and the expert
knowledge of the MPS group on Solar physics, in order to design accurate and efficient methodology. The project also helps strengthen the on-going collaboration between Magique-3D and MPS, that
started two years ago. ANTS is indispensable to elevate the joint collaboration between Magique-3D and MPS. In addition, ANTS would extend the funds granted to Magique-3D on the theme, obtained
from Université de Pau et des Pays de l'Adour through the E2S consortium, and which includes funding for PhD program and partial travel grants for the students. Finally, ANTS will be an essential
propeller towards the submission of a project in the FETHPC-02-2019 campaign (where Magique-3D is the PI institution).
10.1.2 Inria international partners
Declared Inria international partners
• Title: Advance Modeling in Geophysics
• International Partner (Institution - Laboratory - Researcher):
□ California State University at Northridge (United States) - Department of Mathematics - Djellouli Rabia
• The Associated Team MAGIC was created in January 2006 and renewed in January 2009. At the end of the program in December 2011, the two partners, Magique-3D and the California State University at
Northridge (CSUN) decided to continue their collaboration and obtained the “Inria International Partner” label in 2013.
• See also: https://project.inria.fr/magic/
• The ultimate objective of this research collaboration is to develop efficient solution methodologies for solving inverse problems arising in various applications such as geophysical exploration,
underwater acoustics, and electromagnetics. To this end, the research program will be based upon the following three pillars that are the key ingredients for successfully solving inverse obstacle
problems. 1) The design of efficient methods for solving high-frequency wave problems. 2) The sensitivity analysis of the scattered field to the shape and parameters of heterogeneities/
scatterers. 3) The construction of higher-order Absorbing Boundary Conditions.
10.2 International research visitors
10.2.1 Visits of international scientists
Mounir Tlemcani, from the University of Oran, spent two weeks in Pau in March 2020 .
10.3 European initiatives
10.3.1 FP7 & H2020 Projects
• Title: Multiscale Inversion of Porous Rock Physics using High-Performance Simulators: Bridging the Gap between Mathematics and Geophysics
• Duration: April 2018 - March 2022
• Coordinator: Universidad Del Pais Vasco (EHU UPV)
• Partners:
□ BARCELONA SUPERCOMPUTING CENTER - CENTRO NACIONAL DE SUPERCOMPUTACION (Spain)
□ BCAM - BASQUE CENTER FOR APPLIED MATHEMATICS (Spain)
□ CURTIN UNIVERSITY OF TECHNOLOGY (Australia)
□ PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE (Chile)
□ REPSOL SA (Spain)
□ UNIVERSIDAD CENTRAL DE VENEZUELA (Venezuela)
□ UNIVERSIDAD DE BUENOS AIRES (Argentina)
□ UNIVERSIDAD DEL PAIS VASCO/ EUSKAL HERRIKO UNIBERTSITATEA (Spain)
□ UNIVERSIDAD NACIONAL DE COLOMBIA (Colombia)
□ UNIVERSITAT POLITECNICA DE CATALUNYA (Spain)
• Inria contact: Hélène BARUCQ
• Summary: We will develop and exchange knowledge on applied mathematics, high-performance computing (HPC), and geophysics to better characterize the Earth´s subsurface. We aim to better understand
porous rocks physics in the context of elasto-acoustic wave propagation phenomena. We will develop parallel high-continuity isogeometric analysis (IGA) simulators for geophysics. We will design
and implement fast and robust parallel solvers for linear equations to model multi-physics electromagnetic and elasto-acoustic phenomena. We seek to develop a parallel joint inversion workflow
for electromagnetic and seismic geophysical measurements. To verify and validate these tools and methods, we will apply the results to: characterise hydrocarbon reservoirs, determine optimal
locations for geothermal energy production, analyze earthquake propagation, and jointly invert deep-azimuthal resistivity and elasto-acoustic borehole measurements. Our target computer
architectures for the simulation and inversion software infrastructure consists of distributed-memory parallel machines that incorporate the latest Intel Xeon Phi processors. Thus, we will build
a hybrid OpenMP and MPI software framework. We will widely disseminate our collaborative research results through publications, workshops, postgraduate courses to train new researchers, a
dedicated webpage with regular updates, and visits to companies working in the area. Therefore, we will perform a significant role in technology transfer between the most advanced numerical
methods and mathematics, the latest super-computer architectures, and the area of applied geophysics.
10.3.2 Collaborations in European programs, except FP7 and H2020
• Title: Multiscale Inversion of Porous Rock Physics using High-Performance Simulators: Bridging the Gap between Mathematics and Geophysics
• Duration: September 2019 - April 2022
• Coordinator: BARCELONA SUPERCOMPUTING CENTER - CENTRO NACIONAL DE SUPERCOMPUTACION (Spain)
• Partners:
□ BARCELONA SUPERCOMPUTING CENTER - CENTRO NACIONAL DE SUPERCOMPUTACION (Spain)
□ BCAM - BASQUE CENTER FOR APPLIED MATHEMATICS (Spain)
□ UNIVERSIDAD DEL PAIS VASCO/ EUSKAL HERRIKO UNIBERTSITATEA (Spain)
□ UNIVERSITAT de BARCELONA (Spain)
□ REALTIMESEISMIC (RTS)
□ PÔLE AVENIA
• Inria contact: Julien DIAZ
• Summary: Part of the FEDER Poctefa Program https://www.poctefa.eu/, the PIXIL project is a transnational and multidisciplinary scientific and technological cooperation. Its main goal is to
develop the most advanced tools to analyze the Earth's subsurface, with a special focus on fostering the uptake of geothermal energy in the region. The project will contribute to making the
trans-Pyrenean area a technology hub in subsoil characterization within two years. Its success is expected to boost the wealth and creation of jobs related to the generation and management of
underground natural resources in the area.
• See also: https://pixil-project.eu/en
10.4 National initiatives
10.4.1 Depth Imaging Partnership
Magique-3D maintains active collaborations with Total. In the context of Depth Imaging, Magique-3D coordinates research activities dealing with the development of high-performance numerical methods
for solving wave equations in complex media. This project has involved 2 other Inria Team-Projects (Hiepacs and Nachos) which have complementary skills in mathematics, computing and in geophysics.
DIP is fully funded by Total by the way of an outline agreement with Inria .
The third phase of DIP began in 2019. Aurélien Citrain has been hired as engineer to work on the DIP platform. More than 10 PhD students have defended their PhD since the creation of DIP and most of
them are now post-doctoral researchers or engineers in Europe. DIP is currently employing 3 PhD students.
10.5 Regional initiatives
10.5.1 Project supported by Conseil Régional d'Aquitaine
• title: Revival.
• Coordinator: Juliette Chabassier
• Other partners: Univ Bordeaux, Univ Montreal (Canada), Univ Cath. Louvain (Belgium)
The objective is to develop numerical tools for the virtual restoration of heritage instruments.
This project is supported by the Conseil Régional d'Aquitaine, for a duration of 2 years and has funded the postdoctoral position of Tobias van Baarsel since Feb 2019.
11 Dissemination
11.1 Promoting scientific activities
Reviewer - reviewing activities
Members of Magique 3D have been reviewers for the following journals:
• ESAIM: Mathematical Modelling and Numerical Analysis
• Geophysical Journal International
• International Journal on Geomathematics
• Journal Of Computational Physics
• International Journal for Numerical Methods in Engineering
• SIAM Journal on Scientific Computing
• SIAM Journal on Numerical Analysis
• SIAM journal on Applied Mathematics
• Inverse Problems
• Inverse Problems in Science & Engineering
• Journal of Acoustical Society of America
• Journal of Sound and Vibration
11.1.1 Leadership within the scientific community
• Augustin Ernoult is elected member of the "Groupe spécialisé d'acoustique musicale" (Gsam) of the french acoustical society.
• Hélène Barucq is elected member of the Liaison Committee of SMAI-GAMNI (Society of Applied and Industrial Mathematics - Group for promoting the Numerical Methods for Engineers).
11.1.2 Scientific expertise
• Since 2017, Hélène Barucq is chairwoman of a committee which evaluates research projects in Mathematics, Computer Science, Electronics and Optics to be funded by the Regional Council New
• Since 2018, Hélène Barucq is scientific officer for E2S project. She participates in the evaluation of each E2S call. She is also member of a committee having in charge the recruitment of non
permanent researchers for E2S.
11.1.3 Research administration
• Julien Diaz is elected member of the Inria Technical Committee and of the Inria Administrative Board. He is appointed member of the Bureau du Comité des Projets (BCP) of Inria Bordeaux Sud-Ouest.
Since 2018, he has been the head of the Mescal team of LMAP.
• Juliette Chabassier is member of the Research Position Commission of Inria Bordeaux Sud-Ouest.
• Juliette Chabassier is member of the Center Commitee of Inria Bordeaux Sud-Ouest.
• Rose-Cloé Meyer is elected member of Laboratory Commitee of LMAP.
11.2 Teaching - Supervision - Juries
11.2.1 Teaching
• Licence : Rose-Cloé Meyer, Développements limités, suites et séries, 19.5h Eq. TD, L2, UPPA, France
• Master : Sébastien Tordeux, Outils Mathématiques pour la Mécanique, 49 eq. TD, Master1, UPPA, France
• Master : Margot Sirdey and Sébastien Tordeux, Introduction to wave phenomena, 48 eq. TD, Master, UPPA, France
• Licence : Sébastien Tordeux, Applied Mathematics, 18 eq. TD, L1, UPPA, France
11.2.2 Supervision
• Chengyi Shen, Approches expérimentale et numérique de la propagation d'ondes sismiques dans les roches carbonatées, June 3rd Julien Diaz and Daniel Brito (LFCR).
• PhD in progress : Alexandre Gras, Hybrid resonance for sensing applications, IOGS, October 2017, Philippe Lalanne(IOGS), Marc Duruflé, Hélène Barucq (Magique 3D).
• PhD in progress : Pierre Jacquet, Time domain Full Waveform Inversion involving hybrids numerical method to characterize elasto-acoustic media, October 2017, Hélène Barucq and Julien Diaz.
• PhD in progress: Victor Martins Gomez, Experimental characterization and modeling of seismo-electromagnetic waves, Université de Pau et des Pays de l'Adour, October 2018, Hélène Barucq and Daniel
Brito (LFCR).
• PhD in progress : Rose-Cloé Meyer, Modeling of conducting poro-elastic media using advanced numerical methods , Université de Pau et des Pays de l'Adour, October 2018, Hélène Barucq, Julien Diaz
and Ha Pham.
• PhD in progress : Nathan Rouxelin, Advanced numerical modeling of acoustic waves propagating below the surface of the Sun, Université de Pau et des Pays de l'Adour, October 2018, Hélène Barucq
and Sébastien Tordeux.
• PhD in progress : Margot Sirdey, Méthode de Trefftz pour l'électromagnétisme, October 2019, Sébastien Tordeux and Sébastien Pernet (Onera).
• PhD in progress : Vinduja Vasanthan, Tent Pitcher algorithm for space-time integration of wave problems, October 2019, Hélène Barucq and Julien Diaz.
• PhD in progress : Stefano Frambati, Isogeometric analysis of sharp boundaries in full waveform inversion, January 2019, Hélène Barucq and Julien Diaz.
• PhD in progress : Alexis Thibault, Modeling and simulation of wind musical instruments, October 2020, Juliette Chabassier and Thomas Hélie (IRCAM).
• PhD in progress : Guillaume Castera, Modeling and simulation of the piano touch, October 2020, Juliette Chabassier and Paul Fisette (Louvain Cath. Univ., Belgium).
11.2.3 Juries
• Julien Diaz : Georges Nehmetallah (Université de Nice Côte d'Azur), Méthodes Galerkin discontinues hybrides couplées à des schémas de type explicite/implicite pour les équations de Maxwell
instationnaires, PhD thesis, December 14th 2020, reviewer.
• Hélène Barucq : Pierre Payen (Université Paris 13 et CEA), Modélisation d’un empilement de matériaux dans le domaine fréquentiel par une condition d’impédance d’ordre élevé, December 16th 2020,
• Hélène Barucq: Nourallah Dahmen (ISAE, Université Fédérale Toulouse Midi-Pyrénées), Développement d’un nouveau coeur numérique pour le code de calcul Salammbô de modélisation des ceintures de
radiation terrestres, Reviewer.
• Juliette Chabassier : Erik Alan Petersen (Aix Marseille Université), Propagation d'ondes dans les milieux périodiques appliquée aux instruments à vent à trous latéraux. Comment s'équilibre la
production et le rayonnement ?, Examiner.
11.3 Popularization
11.3.2 Interventions
• Augustin Ernoult participated to "Fête de la Science". He presented his research themes at "Lycée de Bazas" (Bazas high-school)
• Hélène Barucq participated to "Fête de la Science". She gave a talk entitled "Allo la Terre? Ici le Soleil" available online at https://www.youtube.com/watch?v=uySjljEwEOo
11.4 Creation of media or tools for science outreach
Juliette Chabassier co-created with the Ensemble Les Précieuses the show Louis 14.0 that presents how ancient instruments can be restored via numerical techniques, and played in real time in a
musical and theatral show.
12 Scientific production
12.1 Major publications
• 1 article Sparsified discrete wave problem involving radiation condition on a prolate spheroidal surface IMA Journal of Numerical Analysis 2019
• 2 articleEfficient and Accurate Algorithm for the Full Modal Green's Kernel of the Scalar Wave Equation in HelioseismologySIAM Journal on Applied Mathematics806December 2020, 2657-2683
• 3 articleHybridizable discontinuous Galerkin method for the two-dimensional frequency-domain elastic wave equationsGeophysical Journal International2131April 2018, 637--659
• 4 articleWoodwind instrument design optimization based on impedance characteristics with geometric constraintsJournal of the Acoustical Society of America1485November 2020, 2864-2877
• 5 articleNonuniqueness of the quasinormal mode expansion of electromagnetic Lorentz dispersive materialsJournal of the Optical Society of America. A Optics, Image Science, and Vision3772020, 1219
• 6 articleA comparison of a one-dimensional finite element method and the transfer matrix method for the computation of wind music instrument impedanceActa Acustica united with Acustica10552019,
12.2 Publications of the year
International journals
• 7 article An effective numerical strategy for retrieving all characteristic parameters of an elastic scatterer from its FFP measurements Journal of Computational Physics 419 October 2020
• 8 articleAsymptotic behavior of acoustic waves scattered by very small obstaclesESAIM: Mathematical Modelling and Numerical Analysis552021, 705 - 731
• 9 article High-order locally A-stable implicit schemes for linear ODEs Journal of Scientific Computing 85 October 2020
• 10 articleEfficient and Accurate Algorithm for the Full Modal Green's Kernel of the Scalar Wave Equation in HelioseismologySIAM Journal on Applied Mathematics806December 2020, 2657-2683
• 11 article Outgoing solutions and radiation boundary conditions for the ideal atmospheric scalar wave equation in helioseismology ESAIM: Mathematical Modelling and Numerical Analysis February
• 12 articleExtension of the Günter derivatives to Lipschitz domains and application to the boundary potentials of elastic wavesJournal of Applied Mechanics and Technical Physics611January 2020, 21
• 13 article Transfer matrix of a truncated cone with viscothermal losses: application of the WKB method Acta Acustica 4 2 May 2020
• 14 articleWoodwind instrument design optimization based on impedance characteristics with geometric constraintsJournal of the Acoustical Society of America1485November 2020, 2864-2877
• 15 articleFull reciprocity-gap waveform inversion enabling sparse-source acquisitionGeophysics856November 2020, R461-R476
• 16 articleA priori estimates of attraction basins for velocity model reconstruction by time-harmonic Full Waveform Inversion and Data-Space Reflectivity formulationGeophysicsFebruary 2020, 1-126
• 17 articleEigenvector models for solving the seismic inverse problem for the Helmholtz equationGeophysical Journal International2211April 2020, 394-414
• 18 articleA discontinuous Galerkin Trefftz type method for solving the two dimensional Maxwell equationsSN Partial Differential Equations and Applications1232020, 19
• 19 articleNonuniqueness of the quasinormal mode expansion of electromagnetic Lorentz dispersive materialsJournal of the Optical Society of America. A Optics, Image Science, and Vision3772020,
• 20 article Equivalent multipolar point-source modeling of small spheres for fast and accurate electromagnetic wave scattering computations Wave Motion 2020
International peer-reviewed conferences
• 21 inproceedings Bore Reconstruction of Woodwind Instruments Using the Full Waveform Inversion e-Forum Acusticum Lyon / Virtual, France December 2020
• 22 inproceedings Time-domain simulation of a dissipative reed instrument e-Forum Acusticum 2020 Lyon, France December 2020
Conferences without proceedings
• 23 inproceedings The Virtual Workshop OpenWinD : a Python Toolbox Assisting Wind Instrument Makers e-Forum Acusticum Lyon / Virtual, France December 2020
Reports & preprints
• 24 report Multivariate spline bases, oriented matroids and zonotopal tilings Inria; Total E&P June 2020
• 25 report Implementation of HDG method for 2D anisotropic poroelastic first-order harmonic equations Inria Bordeaux Sud-Ouest; UPPA (LMA-Pau) February 2020
• 26 report Low-order absorbing boundary condition for two-dimensional isotropic poroelasticity Inria August 2020
• 27 reportEfficient computation of the modal outgoing Green's kernel for the scalar wave equation in helioseismologyInria Bordeaux Sud-Ouest; Magique 3D; Max-Planck Institute for Solar System
ResearchApril 2020, 1-81
• 28 report On the outgoing solutions and radiation boundary conditions for the vectorial wave equation with ideal atmosphere in helioseismology Inria Bordeaux Sud-Ouest; Magique 3D; Max-Planck
Institute for Solar System Research April 2020
• 29 misc Construction and convergence analysis of conservative second order local time discretisation for linear wave equations October 2020
• 30 report Improvement of the modal expansion of electromagnetic fields through interpolation INRIA Bordeaux - Sud-Ouest December 2020
• 31 report Non-uniqueness of the Quasinormal Mode Expansion of Electromagnetic Lorentz Dispersive Materials INRIA Bordeaux - Sud-Ouest June 2020
• 32 misc C2 representations of the solar background coefficients for the model S-AtmoI October 2020
• 33 report Viscothermal models for wind musical instruments Inria Bordeaux Sud-Ouest August 2020
12.3 Cited publications
• 34 book Asteroseismology Springer Science & Business Media 2010
• 35 article Probing the interior physics of stars through asteroseismology arXiv preprint arXiv:1912.12300 2019
• 36 articleInverse problem for the Helmholtz equation with Cauchy data: reconstruction with conditional well-posedness driven iterative regularizationESAIM: Mathematical Modelling and Numerical
Analysis533May 2019, 1005-1030
• 37 articleThe mechanism producing initial transients on the clarinetThe Journal of the Acoustical Society of America14262017, 3376-3386
• 38 articleQuasistatic nonlinear characteristics of double-reed instrumentsThe Journal of the Acoustical Society of America12112007, 536-546
• 39 articleAbout the stability of the inverse problem in 1-D wave equations—Application to the interpretation of seismic profilesApplied Mathematics and Optimization511979, 1--47
• 40 articleSpace-time Trefftz-DG approximation for elasto-acousticsApplicable Analysis00August 2018, 1 - 16
• 41 articleStability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximationMathematics of Computation863072017, 2129 - 2157
• 42 articleGlobal seismology of the SunLiving Reviews in Solar Physics1312016, 2
• 43 articleDirect Simulation of Reed Wind InstrumentsComputer Music Journal3342009, 43-55
• 44 articleHybridizable discontinuous Galerkin method for the two-dimensional frequency-domain elastic wave equationsGeophysical Journal International2131April 2018, 637--659
• 45 articleInterpretation of helioseismic travel timesSpace Science Reviews1961-42015, 201--219
• 46 inproceedingsIdentification of functional parameters in partial differential equationsJoint Automatic Control Conference121974, 155--156
• 47 articleHelioseismologyReviews of Modern Physics7442002, 1073
• 48 articleImproved precision in measurements of acoustic impedance spectra using resonance-free calibration loads and controlled error distributionThe Journal of the Acoustical Society of America
12132007, 1471-1481
• 49 phdthesis Contributions to Seismic Full Waveform Inversion for Time Harmonic Wave Equations: Stability Estimates, Convergence Analysis, Numerical Experiments involving Large Scale Optimization
Algorithms Université de Pau et des Pays de l'Adour November 2017
• 50 articleSensitivity kernels for time-distance helioseismology-Efficient computation for spherically symmetric solar modelsAstronomy & Astrophysics6162018, A156
• 51 articleAcoustical impedance measurements by the two‐microphone‐three‐calibration (TMTC) methodThe Journal of the Acoustical Society of America8861990, 2533-2545
• 52 articleComputational helioseismology in the frequency domain: acoustic waves in axisymmetric solar models with flowsAstronomy and Astrophysics - A&A600April 2017, A35
• 53 articleLocal helioseismologyLiving Reviews in Solar Physics212005, 6
• 54 articleLocal helioseismology: three-dimensional imaging of the solar interiorAnnual Review of Astronomy and Astrophysics482010, 289--338
• 55 articleSignal and noise in helioseismic holographyAstronomy & Astrophysics6202018, A136
• 56 articleReal-time synthesis of clarinet-like instruments using digital impedance modelsThe Journal of the Acoustical Society of America11812005, 483-494
• 57 articleThe dispersion of surface waves on multilayered mediaBulletin of the seismological Society of America4311953, 17--34
• 58 articleGiant star seismologyThe Astronomy and Astrophysics Review2512017, 1
• 59 inproceedings Seismic imaging of supercritical geothermal reservoir using full-waveform inversion method Proceedings 2019
• 60 book Seismic wave propagation in stratified media ANU Press 2009
• 61 article3-D seismic exploration across the deep geothermal research platform GroßSchönebeck north of Berlin/GermanyGeothermal Energy712019, 1--18
• 62 inproceedingsThe seismic inverse problem as a sequence of before stack migrationsConference on inverse scattering: theory and applicationSiam Philadelphia, PA1983, 206--220
• 63 phdthesis Inversion of surface waves in an oil and gas exploration context Université Grenoble Alpes (ComUE) 2016
• 64 articleA review of the adjoint-state method for computing the gradient of a functional with geophysical applicationsGeophysical Journal International16722006, 495--503
• 65 articleGauss--Newton and full Newton methods in frequency--space seismic waveform inversionGeophysical Journal International13321998, 341--362
• 66 articleTwo-dimensional velocity models from wide-angle seismic data by wavefield inversionGeophysical Journal International12421996, 323--340
• 67 articleINVERSE THEORY APPLIED TO MULTI-SOURCE CROSS-HOLE TOMOGRAPHY. PART 1: ACOUSTIC WAVE-EQUATION METHOD 1Geophysical prospecting3831990, 287--310
• 68 articleSeismic attenuation due to wave-induced flowJ. Geophys. Res.1092004, 681-693
• 69 articleWebster's Horn Equation RevisitedSIAM Journal on Applied Mathematics6562005, 1981-2004
• 70 phdthesis Two-dimensional near-surface seismic imaging with surface waves: alternative methodology for waveform inversion 2013
• 71 articleProperty modelling of a potential CO2 storage site using seismic inversionEGUGA2013, EGU2013--10470
• 72 book Surface wave methods: acquisition, processing and inversion 2003
• 73 book Inverse problem theory and methods for model parameter estimation 89 siam 2005
• 74 book Inverse problem theory: methods for data fitting and model parameter estimation Amsterdam, Netherlands Elsevier Science Publishers 1987
• 75 articleTransmission of elastic waves through a stratified solid mediumJournal of applied Physics2121950, 89--93
• 76 articleDemonstrating storage of CO2 in geological reservoirs: The Sleipner and SACS projectsEnergy299-102004, 1361--1369
• 77 articleA comparison of a one-dimensional finite element method and the transfer matrix method for the computation of wind music instrument impedanceActa Acustica united with Acustica52019, 838
• 78 articleSound production in recorderlike instruments. II. A simulation modelThe Journal of the Acoustical Society of America10151997, 2925-2939
• 79 articleInfluence of fluid displacement patterns on seismic velocity during supercritical CO2 injection: Simulation study for evaluation of the relationship between seismic velocity and CO2
saturationInternational Journal of Greenhouse Gas Control462016, 197--204
• 80 phdthesis Modeling experiments in helioseismic holography Niedersächsische Staats-und Universitätsbibliothek Göttingen 2019
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Charles's law calculator - Infooni
How to use our Charles’s law calculator?
Our Charles’s law calculator helps to calculate the values of the Charles’s law. This law has 4 different values:
• V1 is the initial volume of the gas
• T1 is the initial temperature of the gas
• V2 is the final volume of the gas
• T2 is the final temperature of the gas
When 3 of these values are given in the calculator, it will automatically calculate and display the empty value.
The volume is by default expressed in m³ and the temperature in °K. These units can be changed by clicking on it and selecting another unit.
Table of contents:
Charles’s law
The Charles’s law is one of the laws of the thermodynamics constituting the law of ideal gases. The Charles’s law links the volume and the temperature of an ideal gas at a constant pressure.
The law was named after the French physicist and chemist Jacques Charles (1746 – 1823). The law was published for the first time by Louis Joseph Gay-Lussac (1778 – 1850), but it was discovered by
Jacques Charles in 1787.
The Charles’s law describes the relation between the volume and the temperature of a gas. This law specifies that at constant pressure, the volume of a certain quantity of an ideal gas is
proportional to its temperature.
Illustration of the Charles’s law
This definition means that when the pressure is kept constant, an increase in the temperature of a gas leads to an increase of its volume. On the other hands, when the temperature of the gas
decreases, the volume of the gas decreases.
The graphic of the volume in function of the temperature (at constant pressure) is typical of a directly proportional relation. The mathematical expression of this type of curve is:
V ∝ T or V/T = constant or V = kT
There is no need to know the exact value of the constant to apply the law between two volumes of a gas under different temperature at the same pressure. Indeed, the Charles’s law formula allows to
compare two situations of the same gas when the quantity of gas and the pressure are constant.
Charles’s law formula
The definition of the Charles’s law gives the following formula:
V1 / T1 = V2 / T2
• V1 is the initial volume of the gas in m³
• T1 is the initial temperature of the gas in °K
• V2 is the final volume of the gas in m³
• T2 is the final temperature of the gas in °K
Charles’s law and the absolute zero
The Charles’s law seems to define that the volume of a gas will be 0 at the absolute zero temperature, define as the 0°K (-273.15 °C). But this is not physically correct because gases turn into
liquids at a low temperature.
Limitations of the Charles’s law
The law is not available at high pressure and low temperature.
Daily Life application of the Charles’s law
Daily life has numerous applications of the Charles’s law. One of the best examples is the hot air balloon. Indeed, the air balloon is made of an envelope that stores the air. When the air is heated
in the envelope the temperature increases. So following the Charles’s law the volume also increases. Therefore, the density of the air in the envelope decreases to become lighter than the density of
the surrounding atmosphere. As a result, the balloon gains altitude.
On the opposite, the decrease of the temperature in the balloon brings a decrease in the volume of the gas. Therefore, the density of the air inside the envelope decreases to become heavier than the
surrounding atmosphere. As a result the balloon loses altitude.
Other calculors in this category
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What is Machine Learning, actually?
Computer Science
What is Machine Learning, actually?
A simple, intuitive guide to artificial intelligence
Photo by Ali Shah Lakhani on Unsplash
The terms machine learning or artificial intelligence are thrown around a lot these days. It has become fashionable to say that such and such a device has artificial intelligence. But what does this
My objective is to give a non-mathematical description of how AI algorithms work. Many papers on the topic quickly get bogged down in heavy duty mathematics, which I think obscures from the beauty of
the algorithms.
We start with a ‘normal’ algorithm. An algorithm is a set of instructions that, given an input, will give a determined output. A simple example is an adding algorithm: you input two numbers and the
algorithm spits out the sum of those numbers. Another way an algorithm can be viewed is as a classifier. We can consider the space of all possible inputs and the space of all possible outputs, also
called labels. Then we can see an algorithm as a function assigning each point in input space the appropriate point in output space. We call the classifier h which maps from the input space to the
output space. So in the case of our adding algorithm, the input space is all pairs of numbers and the output space is all the numbers.
Spice things up a bit
In the case of a normal algorithm, everything is determined. The classifier behaves in exactly the way we expect it to work. 2+3 will always come out as 5 (I have a degree in maths so I can say this
with certainty). This algorithm is quite boring and easy to program. A programmer can write an explicit set of instructions on how to add two numbers together. But what if we consider something
slightly more challenging? The classic example of number classification: I give you a hand written digit (0–9) and you have to tell me which one it is.
Hand written digit courtesy of the MNIST Dataset
I defy anyone to write an explicit set of instructions that can classify a hand written digit with more accuracy than just assigning labels at random (which would be right 10% of the time).
Interestingly enough, humans can do this without any issues — are we smarter than computers? This is a question best asked to a philosopher and is a major digression from the topic at hand but is
worth thinking about.
Let’s get learning
This is where machine learning comes in to play. We want an algorithm (or classifier) that gets fed training data and uses that to learn how to classify unseen images. An important feature of our
classifier is that it must generalise the data. We don’t need it to fit the data perfectly but we want it to be able to handle unseen data with ease and accuracy.
The grey line is very accurate when it comes to the training data but does not generalise well. The red line however generalises well and we hope will be able to classify unseen data
The algorithm takes the image as input and will spit out a list of ten numbers between 0 and 1. Each number corresponds to a digit, the higher the number, the more confident the algorithm thinks it
is that digit.
Hand written digit courtesy of the MNIST Dataset — h ‘correctly’ classifies an 8. We will see later that this has quadratic loss of 0.13 (2 d.p.)
The classifier works by taking in the image but also has a bunch of parameters that we can play with to try and classify unseen images correctly.
Training wheels
Training data is images along with what digit it is — that is you already know the correct label. For example it would be an image of the number four along with the list (0,0,0,0,1,0,0,0,0,0) (recall
that our algorithm spits out 10 numbers between 0 and 1 for the certainty that it is that digit). The algorithm then takes in the image and spits out its result. We need to then tell the classifier
how ‘wrong’ it was, we call this the loss. One conventional way of doing this is by adding up the square of the differences, called quadratic loss. Say that the classifier spits out (0.43, 0.13,
0.05, 0.17, 0.57, 0.79, 0.70, 0.38, 0.93, 0.35). Then the loss would be 0.43² + 0.13² + 0.05² + 0.17² + (0.57–1)² + 0.79² + 0.70² + 0.38² + 0.93² + 0.35² giving a loss of 3.86 (2 d.p.). The aim is to
obviously minimize the loss function, as it rewards the algorithm for accurately predicting what the number is but also rewards it for correctly identifying what it isn’t.
Minimize loss
So ideally we want to tweak the parameters of the classifier so that the loss function is zero on every training data. This is problematic for two reasons:
• We could be overfitting our data
• It is actually bloody difficult to do this
So we at least want to get the loss function to be pretty small on average. To do this we will apply gradient descent to the average loss function in parameter space (those of you that have read my
article on dynamical systems will be familiar with gradient descent).
Instead of simply computing the loss for a specific input and set of parameters, we will compute the average loss for a specific set of parameters by averaging over all the training data. Remember
the loss of a specific training data is just a number so taking the average is trivial (provided our data set is finite, mind you). In practice, it is inconvenient to take the average over all
training data so we randomly choose a few pieces each time. So now we will consider the average loss where the input now is the parameters.
We look at the average loss function for every possible combination of parameters. In practice, we will only look at the loss function for parameters near those that are currently being used by our
Everything going downhill
The idea is that we look at how to make small nudges to our parameters that would have the biggest effect of decreasing the loss function. Or the direction of steepest descent of loss. We nudge the
parameters a bit in that direction, then recompute the steepest descent and repeat the process. The analogy to keep in mind is that of a ball rolling down a mountain. At each point, it will go in the
direction of steepest descent to try and minimize its altitude. We can think of the altitude of the ball as the loss — the higher it is the higher the loss, and the latitude and longitude as
parameters. In this case there are only two parameters but in practice there can be hundreds of thousands.
Gradient descent in action
After applying gradient descent to our parameters, we reach a local minima of the loss function. That means we are doing the best we can for all parameters nearby. If the loss function at this minima
turns out to be quite high (that is, it is not classifying inputs accurately) then tough luck. You could choose to start again with wildly different parameters and hope you don’t fall into the same
local minima and also hope that the local minima you fall into is ‘better’ than the previous one. Turns out finding global minima is quite difficult.
Now we hopefully have a well-oiled algorithm that can classify hand written digits fairly confidently, by fairly confidently we mean that the average loss is low.
To recap:
1. We start off with an algorithm that wildly guesses what each image is
2. We feed it an image that we know what the answer is
3. Then we compute how wrong it is
4. According to how wrong it is and other mathematical mumbo-jumbo we slightly adjust the parameters in the algorithm to decrease the wrongness
5. Rinse and repeat steps 2–4
And that is it! That is fundamentally how machine learning works. This is the blandest sort of machine learning algorithm called neural network. We can always spice things up after but this is the
core principles of how most artificial intelligences learn.
The whole structure of how a neural network learns is somewhat similar to how a baby learns. A baby is inquisitive about the world around it, maybe it puts its hand on top of the stove and burns
itself. This is considered a high loss. But the baby then adjusts its brain to tell it not to do that again. Similarly, if the baby does its business in the toilet, it will get a reward — a low loss.
It will then learn to repeat that. And so a human is formed that behaves as expected and can generalise to unfamiliar situations: you don’t have to teach an adult to use the toilet in a house they
have never been to before.
So in many ways AI mimics the way humans learn to try and teach a bunch of computer transistors how to ‘think’. But is this any different to a bunch of neurons thinking?
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Year 7 Scheme of Learning
Understand and use algebraic notation
Term 1 starting in week 3 :: Estimated time: 2 weeks
• Given a numerical input, find the output of a single function machine
• Use inverse operations to find the input given the output
• Use diagrams and letters to generalise number operations
• Use diagrams and letters with single function machines
• Find the function machine given a simple expression
• Substitute values into single operation expressions
• Find numerical inputs and outputs for a series of two function machines
• Use diagrams and letters with a series of two function machines
• Find the function machines given a two-step expression
• Substitute values into two-step expressions
• Generate sequences given an algebraic rule
• Represent one- and two-step functions graphically
This page should remember your ticks from one visit to the next for a period of time. It does this by using Local Storage so the information is saved only on the computer you are working on right
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Word Problem Mistake: Making All Your Word Problems Match Your Content - The Teacher Studio
Over the years I have noticed that students tend to look for routine in math class. If it’s a division unit, they will divide any two numbers they find! If it’s a subtraction unit, they try to
regroup everything! So what does this mean for word problems? It means we need to put students in a position where THEY need to think and decide what operation and strategies to choose.
If we constantly take the thinking away from our students, then even solving word problems becomes computational. For that reason, I try hard to sprinkle in a variety of problems all year that
require students to think and apply what they have learned—perhaps draw a picture or make a table to help . . . but, most importantly, to THINK about math. I hope you find these tips useful! Read on
for more.
Make Sense of Problems and Persevere In Solving Them
Many of you who have followed me for any length of time know I am passionate about the Standards for Mathematical Practice. I am also passionate about helping “unpack” them for teachers because,
unfortunately, they are lengthy and contain SO much information.
Let’s take a quick peek at the first part of the “Make sense of problems and persevere in solving them.” I have noticed that MANY teachers have spent the last years helping their students learn to
PERSEVERE…this has to do with having a growth mindset (click HERE for a blog post) and is SO important. I have also learned that many teachers spend less focus on the first part of the standard.
Check out the first few sentences (and notice how wordy they are as well!)
“Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and
goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.”
Let me simplify it for you.
Mathematicians need to figure out what a problem is asking and find a way to get started. They use the strategies and information they know to make a logical “first try”. Now, I would LOVE for you
to read farther into this standard to realize how complex it is. The reason I am focusing on this is to prove my point that WE CANNOT DO THE THINKING for our students. If we are teaching addition
and only give addition word problems, there is no opportunity to “make sense” of the problem. We have done the work, not them. This is how we create math students who struggle when presented with
novel problems and situations. (Interested in a blog post about explicitly teaching problem solving strategies? CLICK HERE)
We need to give students ample opportunity to solve a variety of problems in a variety of contexts. End of story.
How Do I Find Word Problems To Help?
There are lots of options. Consider “saving” problems in your math series to revisit at later times If you teach a unit early in the year, think about saving some problems to assign later. You can
even take problems and rework them to reuse later. Sometimes problems from earlier in the year can be tweaked with larger numbers–or by adding a second “bonus” part to make them more complex.
Consider writing your OWN problems. You can use your interests (like your pets or other “favorites”. Using students names and hobbies gets them engaged and interested. Over time, you get better
and better at it!
That being said, if you are looking for a lot of problems to work into your instruction, you may need to go to outside sources. Most math series simply do NOT have enough problem solving
experiences, and most elementary teachers have MANY other things on their plates. Because I LOVE to write problems, I have a ton in my store that can help.
Need Some Help?
I wanted to showcase a few sets of task cards today because I want to help you get word problems into your students’ hands. I use word problems so many ways and in so many places in my instruction.
It crushes me when I see so many math series taking one of these two approaches:
1. Many lessons have one or two simple word problems at the end. (And some teachers skip them!)
2. Many chapters/units have one or two lessons that focus on problem solving in isolation.
Students need to be solving problems DAILY, and neither of these options are sufficient. The simple truth is that most of us need to supplement.
If you are interested, these bundles of four sets of 20 problem solving task cards are perfect to use in so many ways:
• as a whole class warm up
• as a cooperative experience
• in small group instruction
• for math stations or centers
• as intervention groups
• or even as a digital learning experience (perfect for distance learning, homework options, or when planning at home!)
These two bundles are divided by informal grade levels–because you know as well as I do that grade leveling resources is a challenge. The 3/4 bundle is geared to be instructional for grade 3 and
independent for MOST grade 4 students. They are review for grade 5. The 4/5 bundle is meant to be instructional for grade 4, independent for grade 5, and intervention for grade 6. Hope this helps!
Task cards can be printed in color or black and white or can be sent digitally.
The Process Is More Important Than the Solution
I have included answers in my resources along with three rubrics to use to help in scoring the Standards for Mathematical Practice. Task cards are included in both color and low-ink, AND DIGITAL
versions for ultimate flexibility. There are blank recording sheets that can be used for students to track their work, or they can simply do their work in a notebook. They are available with a TON of
different math content.
That being said, please know that the ANSWERS to these problems are less important than the process. We want students to learn how to solve problems and make smart solution choices. Do we want them
to add and subtract accurately? YOU BET! But that is addressing the precision standard–not the problem solving strategy. Make sure you keep these two “types” of errors clear when assessing
students. Where are they going wrong? What are they doing well?
Enough for now…but I hope I have given you some food for thought.
Interested in checking out some of my word problems? CLICK HERE for the word problem category in my store.
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Symmetric Cryptography
Symmetric-key ciphers are algorithms that use the same key both to encrypt and decrypt data. The goal is to use short secret keys to securely and efficiently send long messages.
The most famous symmetric-key cipher is Advanced Encryption Standard (AES), standardised in 2001. It's so widespread that modern processors even contain special instruction sets to perform AES
operations. The first series of challenges here guides you through the inner workings of AES, showing you how its separate components work together to make it a secure cipher. By the end you will
have built your own code for doing AES decryption!
We can split symmetric-key ciphers into two types, block ciphers and stream ciphers. Block ciphers break up a plaintext into fixed-length blocks, and send each block through an encryption function
together with a secret key. Stream ciphers meanwhile encrypt one byte of plaintext at a time, by XORing a pseudo-random keystream with the data. AES is a block cipher but can be turned into a stream
cipher using modes of operation such as CTR.
Block ciphers only specify how to encrypt and decrypt individual blocks, and a mode of operation must be used to apply the cipher to longer messages. This is the point where real world
implementations often fail spectacularly, since developers do not understand the subtle implications of using particular modes. The remainder of the challenges see you attacking common misuses of
various modes.
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: Simple stoichiometry problem
If 56.1 L of chlorine and 72.4 L of hydrogen are mixed, what volume of product will form at S.T.P.?
Is this a limiting/excess reagent problem? Do I have to do: 56.1/44.8 and then 72.4/22.4 and cross out the lower of the 2 or something first? I totally forget how to do this kind of problem
Thanks ppl
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Power Symbol Math Copy And Paste: Explain!
Power Symbol Math Copy and Paste: Explain!
Utilizing power symbols in mathematics is crucial for denoting exponents and can be easily accomplished by copying and pasting them from various sources. This is especially useful in digital formats
where typing complex equations is required.
Power symbols, like the superscript numbers and letters, can be seamlessly integrated into documents, spreadsheets, and online content to accurately represent mathematical powers.
Power symbols are often used in mathematical notation to indicate that a number is to be raised to a certain power. For example, the expression “x²” means “x raised to the power of 2”.
Here’s how you can copy and paste power symbols:
Locate a source of power symbols (e.g., online character map, word processor, or Unicode table).
Select the power symbol you need (such as ², ³, etc.). Copy the symbol to your clipboard (Ctrl + C on Windows, Command + C on macOS).
Paste the symbol where needed in your document (Ctrl + V on Windows, Command + V on macOS).
Effortlessly insert exponents in your documents with the simple copy-paste method for power symbols, enhancing clarity and precision.
Key Takeaway
Power symbols are important for denoting exponents in mathematics.
They enhance clarity and precision in mathematical notation.
Power symbols can be copied and pasted from online sources using keyboard shortcuts.
Common power symbols include the caret (^), asterisk (*), and double-asterisk (**), which represent exponentiation and multiplication in math equations.
Common Power Symbols for Copy and Paste
Discussing the common power symbols for copy and paste is essential for understanding their usage in mathematical expressions and technical documentation. The most widely recognized power symbol is
the caret (^), often used to denote exponentiation.
Another common power symbol is the asterisk (*), which represents multiplication and is used in conjunction with a number to indicate exponentiation. In some programming languages, the
double-asterisk (**) is used for exponentiation.
Additionally, the use of the power function (pow) in programming languages such as Python is another important symbol for exponentiation.
Understanding these common power symbols and their respective applications in mathematical operations and programming is crucial for accurately conveying and interpreting numerical expressions and
technical data.
How to Copy and Paste Power Symbols
When working with common power symbols such as the caret, asterisk, and double-asterisk in mathematical expressions and technical documentation, it is important to understand how to efficiently copy
and paste these symbols into various applications and platforms.
To simplify this process, here’s a table showcasing the power symbols along with their corresponding Unicode and keyboard shortcuts:
Symbol Unicode Keyboard Shortcut
Caret (^) U+005E Ctrl + Shift + 6
Asterisk (*) U+002A Shift + 8
Double-Asterisk U+2217 Alt + 42
The power symbols along with their corresponding Unicode and keyboard shortcuts
Using Power Symbols in Math Equations
Continuing from the previous subtopic, power symbols such as the caret (^), asterisk (), and double-asterisk (**) are essential for representing exponentiation and multiplication in math equations.
The caret symbol is commonly used to denote exponentiation, for example, 2^3 represents 2 raised to the power of 3. The asterisk symbol signifies multiplication, as in 54 which equals 20.
Additionally, the double-asterisk symbol is often utilized in programming and some mathematical software to represent exponentiation, for instance, 2**4 signifies 2 raised to the power of 4.
Understanding the proper usage of these power symbols is crucial for accurately expressing mathematical operations. Using the power symbols correctly can make a significant difference in the outcome
of an equation. This is especially important when explaining math operations to others, as a small mistake with a power symbol can completely change the result of a calculation. Therefore, it is
essential for students and professionals alike to have a solid understanding of how to correctly use power symbols in mathematical expressions.
Copy and Paste Shortcuts for Power Symbols
Moving from the discussion of power symbols in mathematical equations, the efficient use of copy and paste shortcuts for these symbols is integral to streamlining mathematical notation.
Utilizing shortcuts can significantly improve productivity and accuracy when working with power symbols.
Here are some popular copy and paste shortcuts for power symbols:
• For squared symbol (^2), use the shortcut “Alt + 0178” on Windows or “Option + 00B2” on Mac.
• For cubed symbol (^3), use the shortcut “Alt + 0179” on Windows or “Option + 00B3” on Mac.
• For other power exponents (x^n), use the Unicode or HTML entity codes for the specific exponent value.
• Consider creating custom keyboard shortcuts or utilizing specialized software for seamless insertion of power symbols.
Efficiently employing these shortcuts can enhance the speed and accuracy of mathematical notation.
Enhancing Documents With Power Symbols
The integration of power symbols into documents enhances the visual representation and clarity of mathematical expressions. Utilizing power symbols such as superscripts and subscripts can
significantly improve the understanding of mathematical concepts within documents.
For instance, when expressing equations or formulas, incorporating power symbols can help to clearly denote exponents, indices, or other important mathematical operations.
This visual enhancement can aid readers in grasping the hierarchical structure of mathematical expressions and the relationships between different elements.
Moreover, the use of power symbols can also contribute to the professional and polished appearance of documents, making them more visually appealing and easier to comprehend.
Therefore, integrating power symbols effectively into documents can elevate the overall quality and communicative power of mathematical content.
In the world of academia, the power of symbols cannot be underestimated. Just as a skilled artist uses a wide array of colors to bring a painting to life, the use of power symbols in math equations
and documents can enhance the clarity and impact of the message being conveyed.
By mastering the art of using power symbols in writing, one can truly unlock the potential for deeper understanding and communication in the academic realm.
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LM 2_5 Addition of velocities Collection
2.5 Addition of velocities by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.
Addition of velocities to describe relative motion
Since absolute motion cannot be unambiguously measured, the only way to describe motion unambiguously is to describe the motion of one object relative to another. Symbolically, we can write `v_(PQ)`
for the velocity of object `P` relative to object `Q`.
Velocities measured with respect to different reference points can be compared by addition. In the figure below, the ball's velocity relative to the couch equals the ball's velocity relative to the
truck plus the truck's velocity relative to the couch:
`v_(BC) =v_(BT) + v_(TC)`
` =5 cm"/"s+10 cm"/"s`
`=15 cm"/"s`
The same equation can be used for any combination of three objects, just by substituting the relevant subscripts for B, T, and C. Just remember to write the equation so that the velocities being
added have the same subscript twice in a row. In this example, if you read off the subscripts going from left to right, you get BC...=...BTTC. The fact that the two “inside” subscripts on the right
are the same means that the equation has been set up correctly. Notice how subscripts on the left look just like the subscripts on the right, but with the two T's eliminated.
Negative velocities in relative motion
My discussion of how to interpret positive and negative signs of velocity may have left you wondering why we should bother. Why not just make velocity positive by definition? The original reason why
negative numbers were invented was that bookkeepers decided it would be convenient to use the negative number concept for payments to distinguish them from receipts. It was just plain easier than
writing receipts in black and payments in red ink. After adding up your month's positive receipts and negative payments, you either got a positive number, indicating profit, or a negative number,
showing a loss. You could then show that total with a high-tech “`+`” or “`-`” sign, instead of looking around for the appropriate bottle of ink.
Nowadays we use positive and negative numbers for all kinds of things, but in every case the point is that it makes sense to add and subtract those things according to the rules you learned in grade
school, such as “minus a minus makes a plus, why this is true we need not discuss.” Adding velocities has the significance of comparing relative motion, and with this interpretation negative and
positive velocities can be used within a consistent framework.
For example, the truck's velocity relative to the couch equals the truck's velocity relative to the ball plus the ball's velocity relative to the couch:
`=- 5 cm"/"s+15 cm"/"s`
`=10 cm"/"s`
If we didn't have the technology of negative numbers, we would have had to remember a complicated set of rules for adding velocities: (1) if the two objects are both moving forward, you add, (2) if
one is moving forward and one is moving backward, you subtract, but (3) if they're both moving backward, you add. What a pain that would have been.
`=>` Solved problem: two dimensions — problem 10
Example 3: Airspeed
On June 1, 2009, Air France flight 447 disappeared without warning over the Atlantic Ocean. All 232 people aboard were killed. Investigators believe the disaster was triggered because the pilots lost
the ability to accurately determine their speed relative to the air. This is done using sensors called Pitot tubes, mounted outside the plane on the wing. Automated radio signals showed that these
sensors gave conflicting readings before the crash, possibly because they iced up. For fuel efficiency, modern passenger jets fly at a very high altitude, but in the thin air they can only fly within
a very narrow range of speeds. If the speed is too low, the plane stalls, and if it's too high, it breaks up. If the pilots can't tell what their airspeed is, they can't keep it in the safe range.
Many people's reaction to this story is to wonder why planes don't just use GPS to measure their speed. One reason is that GPS tells you your speed relative to the ground, not relative to the air.
Letting P be the plane, A the air, and G the ground, we have
where `v_(PG)` (the “true ground speed”) is what GPS would measure, `v_(PA)` (“airspeed”) is what's critical for stable flight, and `v_(AG)` is the velocity of the wind relative to the ground 9000
meters below. Knowing `v_(PG)` isn't enough to determine `v_(PA)` unless `v_(AG)` is also known.
Discussion Questions
A Interpret the general rule `v_(AB) = - v_(BA)` in words.
B Wa-Chuen slips away from her father at the mall and walks up the down escalator, so that she stays in one place. Write this in terms of symbols.
2.5 Addition of velocities by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.
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Musielak-Orlicz Hardy spaces associated with divergence form elliptic operators without weight assumptions
Let L be a divergence form elliptic operator with complexbounded measurable coefficients, let ω be a positive Musielak-Orlicz functionon (0,∞) of uniformly strictly critical lower-type p [ω] ε (0,
1], and let ρ(x, t) =t ^-1/ω-1(x, t ^-1) for x ε R ^n, t ε (0,∞). In this paper, we study the Musielak-Orlicz Hardy space Hω,L(R ^n) and its dual space BMOρ,L* (R ^n), where L*denotes the adjoint
operator of L in L ^2(R ^n). The ρ-Carleson measure characterizationand the John-Nirenberg inequality for the space BMOρ,L(R ^n) are also established. Finally, as applications, we show that the Riesz
transform∇L- ^1/2 and the Littlewood-Paley g-function gL map H [ω,L](R ^n) continuouslyinto L(ω).
Dive into the research topics of 'Musielak-Orlicz Hardy spaces associated with divergence form elliptic operators without weight assumptions'. Together they form a unique fingerprint.
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Parsecs to Inches
Astronomers used trigonometry to calculate the distance to stars long before the term parsec was coined, but the new unit made it easier to conceptualise unfathomable distances.
A parsec is the distance from the sun to an astronomical object which has a parallax angle of one arcsecond (1/3600 of a degree). The parallax angle is found by measuring the parallax motion (or
apparent movement of a star relative to stable, more distant stars) when the star is observed from opposite sides of the Sun (an interval of six months on Earth). The parallax angle is obtained by
halving the angular difference in measurements.
Once the parallax angle is established you can calculate the distance to a star using trigonometry, because we know Earth’s distance from the Sun. The distance from the Sun of a body with a parallax
angle of 1 arcsecond was thus defined as a unit and, thanks to Turner, named the parsec.
With the parsec defined, deriving and describing huge distances became easy, since a distance i
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The Lotus and the Wheel
What is the best deck in the history of Magic? How often does it win? Over the 25-year life of Magic: the Gathering, card and set design have improved and bans and restrictions have kept dominating
cards at bay. For the most part, the game has achieved a healthy balance between the various decks and archetypes across most of its constructed formats. A major part of the game balance is the deck
construction rule that sets the size of the deck at the minimum of 60 cards and limits any one card to a maximum of four copies. But what if those restrictions were lifted?
In the first months after its release in August of 1993, that’s exactly how Magic was played. In this era of Wild Magic, a deck only had to have a minimum of 40 cards and the number of copies of a
single card was not at all limited. Because of no limitations, a number of severely degenerate decks quickly arose. Among those were the Plague Rats deck, the Lightning Bolt deck, and the Timetwister
-Fireball deck. Each of these decks relied on multiple (dozens) of copies of a certain card to overwhelm the opponent with synergy and unmatched consistency (see, for example, The History of Vintage
by Steve Menendian). However, reportedly the most deadly was the deck consisting of a combination of two cards: the Wheel of Fortune and the Black Lotus.
Wheel of Fortune and Black Lotus: the most powerful combo in the history of Magic.
The way the deck functions is extremely simple. The Black Lotus gives the player three mana to cast a Wheel of Fortune. Both players discard their hands and draw seven new cards. With high
probability this gives the player another pair of Lotus and Wheel to continue drawing cards. This goes on until the opponent’s deck runs out of cards (presumably after six draw-sevens), while the
Lotus-and-Wheel deck is constructed to have more cards than the opponent’s. The opponent is then the only player running out of cards and thus loses the game.
The descriptions of the deck I’ve found mention that it was extremely consistent, winning every match almost every time. However, the exact probability is never mentioned, so it’s unknown whether
this anecdote means a win probability of 80 %, 90 %, 95 %, or even more. This is of course dependent on the number of Black Lotuses and Wheel of Fortunes in the deck. Presumably, the number was
optimized, but different sources quote numbers from anywhere between 20 Wheels + 40 Lotuses to 23 Wheels + 20 Lotuses. Another question is then the optimal number of each card in the deck.
Now surely, if it was already done over 25 years ago, the winning probability and the optimal deck composition could be solved with modern computers and a little bit of programming. And that’s
exactly what I set out to do.
Numerical Analysis
Before even starting to write the simulation code, it’s useful to make a rough estimate of the expected result. Since exactly one of each card is needed to spin the Wheel, their number in the deck
should be quite even. Otherwise the probability of getting at least one of each in the starting hand decreases. However, since any Lotuses in excess of the first can be played on the battlefield and
saved for the next hand, the need for drawing Lotuses after the starting hand is statistically lower. Therefore, it’s expected that the optimal deck would have a slightly smaller number of Black
Lotuses than Wheel of Fortunes. Again, the number is probably not much smaller, otherwise the probability of getting none in the starting hand becomes too high.
With that thought experiment in mind as a sanity check, I started working on a short piece of code that is required for the calculation. Although the problem seems mathematically simple enough that
an analytical solution might exist, I opted to save my time in favor of the computer’s, and wrote a simple Monte Carlo simulation. I implemented a gameplay logic that does the following:
1. Starts with a given randomized deck,
2. Draws seven cards,
3. Plays all available Black Lotuses,
4. Uses one of the Black Lotuses to cast a Wheel of Fortune to draw seven cards,
5. Repeats 3. - 4. until more than 40 cards have been drawn,
6. Records the win/loss and repeats 1. - 5. a large number of times to get a statistically significant estimate for the win percentage.
All of the above is then again repeated for different numbers of Lotuses and Wheels in the deck to find the optimal numbers. The IPython notebook that does all of the above can be found here.
Probability to win on turns 1 - 3 with a deck of 43 cards.
After a bit of waiting and number crunching, the results started to come in. It turns out that the 90’s folklore was correct - with the optimal ratio of cards, this was one tremendously consistent
deck. With 43 cards in total, the probability to win on turn 1 was more than [DEL:97 %:DEL] 98 %!
On subsequent turns, the winning probability does not go up. The reason is simply that unless the win happens on the first turn, the deck just runs out of cards. So increasing the number of cards
should push the win percentage even higher. Trying that hypothesis out with a deck of 50 cards in total indeed shows that the win percentage increases on turns 2 and 3, up to and beyond 99 %!
Probability to win on turns 1 - 3 with a deck of 50 cards.
Probability to win on turn 3 with decks of various sizes illustrated with iso-probability contours.
Now, of course the results can be generalized by looking at the fraction of Black Lotuses in the decks of various sizes instead of their raw number. [DEL:For the turn 1 win probability, all the
curves collapse onto a nice master curve (which gives a warm and fuzzy feeling for the statistical physicist). This shows that the turn 1 win percentage is not dependent on the deck size, as long as
the number of Black Lotuses is adjusted accordingly.:DEL] Actually, there is a small dependence on the deck size: the smaller deck has a higher win probability. I explain this in a future post.
Probability to win on turn 1 with decks of various sizes, shown as a function of the fraction of Black Lotuses in the deck.
Finally, as a potential caveat, the simulation naturally assumes a non-responsive opponent. In a real match, of course, the opponent could have an opportunity to disrupt the game plan, particularly
if they take the turn first. However, considering the possible answers to a combo win on turn 1 with the card pool of the fall of ‘93, this point is not a major concern. Basically, the only way to
combat a deck that won on turn 1 with a probability of more than [DEL:97 %:DEL] 98 %, was to play a combo deck of the same caliber.
The Black Lotus and Wheel of Fortune deck of the era of Wild Magic can win on turn one with a probability in excess of [DEL:97 %:DEL] 98 %. This can be achieved with an optimal deck containing
roughly 43 % of Black Lotuses and 57 % of Wheel of Fortunes.
Update 14 January 2019: While working on a follow-up article to this post, I discovered an error in the simulation code. While the effect on the numbers is quite small, I have updated the figures
with correct ones and revised the numbers. I’ve left the old values visible in the post for the sake of honesty and future discussion. I will write a post on the topic with more details about the
analysis in the near future. Also note that later posts were unaffected by the error.
Comments (5)
Was there a Mulligan Rule in place at this time? Can lopsided opening hands get corrected by this?
Ufactor – November 27th, 2018
Good question! I didn’t include any mulligans because in the very early days there were none. But the effect of including one free (non-land) mulligan to seven cards would be basically to
give a free start. Assuming that the card distribution was 50 % - 50 %, the chance of drawing either all Wheels or all Lotuses in the starting hand is 2 * (1/2)^7 = (1/2)^6 ~ 1.6 %. Then
there’s the card drawn at the beginning of turn one that mitigates the chance of stalling at this point by half. So having the chance to mulligan once for free would further increase the turn
one win percentage by roughly 0.8 %. In fact, without the mulligan, a simple back of the envelope estimate for the overall turn 1 win probability would be 1 - ( (1/2) * 2 * (1/2)^7 + 5 * (1/
2)^7 ) ) ~ 0.95 = 95 %. Which is actually pretty close. This is assuming equal card distribution, and always having at least one Lotus remain on the table after the previous Wheel.
Timo – November 28th, 2018
Thanks for the clarification. I tremendously enjoyed this - great read!!
Ufactor – December 1st, 2018
Would love to see you plug thru the most optimal amount of plague rats + dark ritual + Mox Jet & Swamp if its possible.
JamestheIV – December 4th, 2018
The Plague Rat deck is indeed very interesting. The challenge here is that the optimal deck probably depends on what kind of deck we are facing. Assuming a non-reactive opponent and
optimising the damage output so as to reach 20 or more damage as fast as possible can be done. But my worry is that the assuming the opponent cannot slow this down may not be a good enough.
Do you have any suggestions on this point?
Timo – December 6th, 2018
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Stata/Mata - Wikibooks, open books for an open world
Mata is a matrix language for Stata that is separate from the original ado-style programming. It was introduced in version 9. It is an imperative language similar to the C programming language and
was designed to be faster for user program dealing with matrices.
Variables in Mata have both an eltype and an orgtype. The eltype can be any of: real, complex, string, pointer, struct, numeric (real or complex), or transmorphic (any of the previous). All numbers
in stata are stored as 8-byte doubles. The orgtype can be any of: scalar, vector, rowvector, colvector, or matrix.
All functions in Mata are pass-by-reference.
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What our customers say...
Thousands of users are using our software to conquer their algebra homework. Here are some of their experiences:
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Computational Mathematics
Course Description
A modern mathematician needs solid programming and data analysis skills to solve real-world problems in finance, medicine, technology or science. Working with data, solving mathematical problems
computationally, and visualizing results are important skills to combine with mathematical knowledge. Therefore, the aim of this course is to provide a link between mathematics and computer science
for the mathematics students. We will offer an introduction into Python programming language and its applications to mathematics and data analysis. Specifically, we will focus on packages for
scientific computing and linear algebra (NumPy, SciPy), data handling (Pandas), visualisation (Matplotlib) and symbolic computation (SymPy).
Schedule and Venue
The course consists of a lecture (1 SWS) and an exercise class (1 SWS). You can attend the lecture and the exercise class during any of the four available time slots (only choose one lecture and one
exercise class):
Lecture: Tuesdays from 11:00-12:00, 12:00-13:00, Thursdays from 09:00-10:00, 10:00-11:00 (by Ernesto Araya Valdivia)
Exercise Class: Tuesdays 13:00-14:00, 14:00-15:00, Thursdays 11:00-12:00, 12:00-13:00 (by Mariia Seleznova).
Office Hours: Thursdays 16:00-17:00, Akademiestraße 7, Room 512 (by Ernesto), Tuesdays 16:00-17:00, Akademiestraße 7, Room 511 (by Mariia),
The course is targeted at Bachelor students of mathematics and financial mathematics, as well as Lehramt students. Prior knowledge in Analysis I,II and Linear Algebra is recommended. Basic
programming skills (e.g. from Programmieren I lecture) are an advantage. However, we will give an introduction to basic programming concepts and Python syntax in the beginning of the course.
Please register for the course on the moodle page: https://moodle.lmu.de/course/view.php?id=36797
The access key is CoMa25.
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kWh to kW Calculator - calculator
kWh to kW Calculator
This calculator helps you convert kilowatt-hours (kWh) to kilowatts (kW). Kilowatt-hours measure energy consumption, while kilowatts measure power. To find the power in kilowatts, you need to know
the energy consumed in kilowatt-hours and the time period over which it was consumed.
The power P in kilowatts (kW) is equal to the energy E in kilowatt-hours (kWh), divided by the consumption time period t in hours (h):
P(kW) = E(kWh) / t(h)
Energy (E) = kWh
Time (t) = hours
Power (P) = E / t
= kWh / hours
= kW
What is kWh to kW?
kWh (kilowatt-hours) is a unit of energy, while kW (kilowatts) is a unit of power. To convert from kWh to kW, you divide the energy in kilowatt-hours by the time period in hours.
What is a kWh to kW Calculator?
A kWh to kW Calculator is a tool that helps you convert kilowatt-hours to kilowatts. It takes the energy in kilowatt-hours and the time period in hours as inputs, and calculates the power in
How to use a kWh to kW Calculator?
To use a kWh to kW Calculator, simply enter the energy in kilowatt-hours and the time period in hours, then click the "Calculate " button. The calculator will display the power in kilowatts.
What is the formula for a kWh to kW Calculator?
The formula for converting kilowatt-hours to kilowatts is:
P(kW) = E(kWh) / t(h)
P = Power in kilowatts
E = Energy in kilowatt-hours
t = Time period in hours
Advantages and Disadvantages of a kWh to kW Calculator
- Quick and easy way to convert kilowatt-hours to kilowatts
- Useful for understanding power requirements and energy consumption
- Can help in calculating electricity costs
- Requires accurate input of energy and time, which may not always be available
- Does not provide detailed analysis or recommendations for energy efficiency
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n elipdotter::
Function progressive
pub fn progressive<T, L, R, C, M>(
a: L,
b: R,
comparison: C,
matches: M,
minimize_dist_right: Option<fn(_: &T, _: &T) -> usize>,
) -> impl Iterator<Item = Inclusion<T>>
Expand description
Like iter_set::classify but when we get two “equal” from matches, we let one of those stay in the “cache” to match future ones. The last one or the greatest one according to comparison stays.
If minimize_dist_right is Some, the algorithm will only return Inclusion::Both once b is close to a as possible. It should return the distance between the two points (using the same algorithm as
comparison). This is very useful when doing AND NOT operations. Set it to None otherwise.
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When did Isaac Newton discover the theory of gravity?
When did Isaac Newton discover the theory of gravity?
However, he had a large intellect, as shown by his discoveries on gravity, light, motion, mathematics, and more. Legend has it that Isaac Newton came up with gravitational theory in 1665, or 1666,
after watching an apple fall.
Who actually discovered gravity first?
Everyone knows that Isaac Newton came up with the law of gravity after seeing an apple fall from a tree in his mother's garden. Newton himself told the story to several contemporaries, who recorded
it for posterity.
Who discovered gravity in 1666?
Isaac Newton. Most famous for his law of gravitation, English physicist and mathematician Sir Isaac Newton was instrumental in the scientific revolution of the 17th century.
How did Isaac Newton discovered gravity?
The legend is that Newton discovered Gravity when he saw a falling apple while thinking about the forces of nature. Whatever really happened, Newton realized that some force must be acting on falling
objects like apples because otherwise they would not start moving from rest.
Did Newton actually discover gravity?
A genius with dark secrets. Isaac Newton changed the way we understand the Universe. Revered in his own lifetime, he discovered the laws of gravity and motion and invented calculus. ... But Newton's
story is also one of a monstrous ego who believed that he alone was able to understand God's creation.
When did Isaac Newton discover calculus?
1665 Isaac Newton changed the world when he invented Calculus in 1665. We take this for granted today, but what Newton accomplished at the age of 24 is simply astonishing. Calculus has uses in
physics, chemistry, biology, economics, pure mathematics, all branches of engineering, and more.
Where was Isaac Newton when he discovered gravity?
he first thought of his system of gravitation which he hit upon by observing an apple fall from a tree , The incident occurring in the late summer of 1666. In other accounts it is stated that Newton
was sitting in his garden at Woolsthorpe Manor near Grantham in Lincolnshire when the incident occurred.
Was calculus invented or discovered?
Today it is generally believed that calculus was discovered independently in the late 17th century by two great mathematicians: Isaac Newton and Gottfried Leibniz. However, the dispute over who first
discovered calculus became a major scandal around the turn of the 18th century.
How old was Isaac Newton when he discovered gravity?
around 44 years old Isaac Newton was around 44 years old when he discovered gravity. Newton articulated the equations that explained the force of gravity in his Principi...
How did Isaac Newton discover gravity?
The legend is that Newton discovered Gravity when he saw a falling apple while thinking about the forces of nature. Whatever really happened, Newton realized that some force must be acting on falling
objects like apples because otherwise they would not start moving from rest.
How did Sir Isaac Newton discover gravity?
• Sir Isaac Newton discovered gravity around 1665 while he was drinking tea and observed an apple falling from a tree. Newton deduced that the force that caused the apple to fall to the ground also
is the same force that causes the moon to orbit the earth. When he was growing up, Newton spent much of his time on his family farm reading.
When was gravity first discovered?
• Discovered in year : 1666. Gravity is an invisible pulling force between two objects. These two objects can be anything from a grain of rice to a planet in the solar system. Gravity is
everywhere. It is between everything and it is everywhere. It is on Earth, moons, stars, and even in space.
Who is called the father of gravity?
• The English mathematician, astronomer, and physicist, Isaac Newton was born in 1664 according to the old calendar, and as many would say, he is the father of gravity. Newton discovered gravity
with a little help from an apple tree in his childhood...
How many laws of motion did Isaac Newton discover?
• In this work Newton stated the three universal laws of motion that were not to be improved upon for more than two hundred years. He used the Latin word Gravitas (weight) for the effect that would
become known as gravity, and defined the law of universal gravitation.
Relaterade inlägg:
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Use the limit comparison test to determine whether the series converges or diverges.
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.
Use the limit comparison test to determine whether the series converges or diverges.
Use the limit comparison test to determine whether the series converges or diverges.1) ∑_(n=1)^∞▒2/(3+√n)2) ∑_(n=1)^∞▒1/√(n(n+1)(n+2))3) ∑_(n=1)^∞▒n^2/(n^3+1)∑_(n=2)^∞▒1/√(4n^3-5n)
Show more
Homework Categories
Ask a Question
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p43 Snark loop
The p43 Snark loop is a period-43 oscillator made from Snark reflectors. It is the smallest of an infinite family of Snark-based adjustable glider loops that can have any period from 43 up.
There are much smaller glider loops containing four gliders, with a total signal path length of 216+8n and therefore adjustable to any period of the form 54+2n. However, this is the smallest loop
that can contain eight gliders, which is a requirement for universal adjustability.
A 226-glider synthesis of this oscillator was discovered by Jeremy Tan on June 9, 2019.^[1] This was reduced to 196 gliders on October 16, 2020 by MathAndCode and GUYTU6J,^[2] and then to 131 gliders
in April 2021 by MathAndCode and Dave Greene.^[3]^[4]^[5]
See also
External links
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Privacy on the Blockchain | Ethereum Foundation Blog
Blockchains are a powerful technology, as regular readers of the blog already likely agree. They allow for a large number of interactions to be codified and carried out in a way that greatly
increases reliability, removes business and political risks associated with the process being managed by a central entity, and reduces the need for trust. They create a platform on which applications
from different companies and even of different types can run together, allowing for extremely efficient and seamless interaction, and leave an audit trail that anyone can check to make sure that
everything is being processed correctly.
However, when I and others talk to companies about building their applications on a blockchain, two primary issues always come up: scalability and privacy. Scalability is a serious problem; current
blockchains, processing 3-20 transactions per second, are several orders of mangitude away from the amount of processing power needed to run mainstream payment systems or financial markets, much less
decentralized forums or global micropayment platforms for IoT. Fortunately, there are solutions, and we are actively working on implementing a roadmap to making them happen. The other major problem
that blockchains have is privacy. As seductive as a blockchain’s other advantages are, neither companies or individuals are particularly keen on publishing all of their information onto a public
database that can be arbitrarily read without any restrictions by one’s own government, foreign governments, family members, coworkers and business competitors.
Unlike with scalability, the solutions for privacy are in some cases easier to implement (though in other cases much much harder), many of them compatible with currently existing blockchains, but
they are also much less satisfying. It’s much harder to create a “holy grail” technology which allows users to do absolutely everything that they can do right now on a blockchain, but with privacy;
instead, developers will in many cases be forced to contend with partial solutions, heuristics and mechanisms that are designed to bring privacy to specific classes of applications.
The Holy Grail
First, let us start off with the technologies that are holy grails, in that they actually do offer the promise of converting arbitrary applications into fully privacy-preserving applications,
allowing users to benefit from the security of a blockchain, using a decentralized network to process the transactions, but “encrypting” the data in such a way that even though everything is being
computed in plain sight, the underlying “meaning” of the information is completely obfuscated.
The most powerful technology that holds promise in direction is, of course, cryptographically secure obfuscation. In general, obfuscation is a way of turning any program into a “black box” equivalent
of the program, in such a way that the program still has the same “internal logic”, and still gives the same outputs for the same inputs, but it’s impossible to determine any other details about how
the program works.
Think of it as “encrypting” the wires inside of the box in such a way that the encryption cancels itself out and ultimately has no effect on the output, but does have the effect of making it
absolutely impossible to see what is going on inside.
Unfortunately, absolutely perfect black-box obfuscation is mathematically known to be impossible; it turns out that there is always at least something that you can get extract out of a program by
looking at it beyond just the outputs that it gives on a specific set of inputs. However, there is a weaker standard called indistinguishability obfuscation that we can satisfy: essentially, given
two equivalent programs that have been obfuscated using the algorithm (eg. x = (a + b) * c and x = (a * c) + (b * c)), one cannot determine which of the two outputs came from which original source.
To see how this is still powerful enough for our applications, consider the following two programs:
1. y = 0
2. y = sign(privkey, 0) – sign(privkey, 0)
One just returns zero, and the other uses an internally contained private key to cryptographically sign a message, does that same operation another time, subtracts the (obviously identical) results
from each other and returns the result, which is guaranteed to be zero. Even though one program just returns zero, and the other contains and uses a cryptographic private key, if indistinguishability
is satisfied then we know that the two obfuscated programs cannot be distinguished from each other, and so someone in possession of the obfuscated program definitely has no way of extracting the
private key – otherwise, that would be a way of distinguishing the two programs. That’s some pretty powerful obfuscation right there – and for about two years we’ve known how to do it!
So, how do we use this on a blockchain? Here’s one simple approach for a digital token. We create an obfuscated smart contract which contains a private key, and accepts instructions encrypted with
the correponding public key. The contract stores account balances in storage encrypted, and if the contract wants to read the storage it decrypts it internally, and if the contract wants to write to
storage it encrypts the desired result before writing it. If someone wants to read a balance of their account, then they encode that request as a transaction, and simulate it on their own machine;
the obfuscated smart contract code will check the signature on the transaction to see if that user is entitled to read the balance, and if they are entitled to read the balance it will return the
decrypted balance; otherwise the code will return an error, and the user has no way of extracting the information.
However, as with several other technologies of this type, there is one problem: the mechanism for doing this kind of obfuscation is horrendously inefficient. Billion-factor overhead is the norm, and
often even highly optimistic; a recent paper estimates that “executing [a 2-bit multiplication] circuit on the same CPU would take 1.3 * 10^8 years”. Additionally, if you want to prevent reads and
writes to storage from being a data leak vector, you must also set up the contract so that read and write operations always modify large portions of a contract’s entire state – another source of
overhead. When, on top of that, you have the overhead of hundreds of nodes running the code on a blockchain, one can quickly see how this technology is, unfortunately, not going to change anything
any time soon.
Taking A Step Down
However, there are two branches of technology that can get you almost as far as obfuscation, though with important compromises to the security model. The first is secure multi-party computation.
Secure multi-party computation allows for a program (and its state) to be split among N parties in such a way that you need M of them (eg. N = 9, M = 5) to cooperate in order to either complete the
computation or reveal any internal data in the program or the state. Thus, if you can trust the majority of the participants to be honest, the scheme is as good as obfuscation. If you can’t, then
it’s worthless.
The math behind secure multi-party computation is complex, but much simpler than obfuscation; if you are interested in the technical details, then you can read more here (and also the paper of
Enigma, a project that seeks to actually implement the secret sharing DAO concept, here). SMPC is also much more efficient than obfuscation, the point that you can carry out practical computations
with it, but even still the inefficiencies are very large. Addition operations can be processed fairly quickly, but every time an SMPC instance performs some very small fixed number of multiplication
operations it needs to perform a “degree reduction” step involving messages being sent from every node to every node in the network. Recent work reduces the communication overhead from quadratic to
linear, but even still every multiplication operation brings a certain unavoidable level of network latency.
The requirement of trust on the participants is also an onerous one; note that, as is the case with many other applications, the participants have the ability to save the data and then collude to
uncover at any future point in history. Additionally, it is impossible to tell that they have done this, and so it is impossible to incentivize the participants to maintain the system’s privacy; for
this reason, secure multi-party computation is arguably much more suited to private blockchains, where incentives can come from outside the protocol, than public chains.
Another kind of technology that has very powerful properties is zero-knowledge proofs, and specifically the recent developments in “succinct arguments of knowledge” (SNARKs). Zero-knowledge proofs
allow a user to construct a mathematical proof that a given program, when executed on some (possibly hidden) input known by the user, has a particular (publicly known) output, without revealing any
other information. There are many specialized types of zero-knowledge proofs that are fairly easy to implement; for example, you can think of a digital signature as a kind of zero-knowledge proof
showing that you know the value of a private key which, when processed using a standard algorithm, can be converted into a particular public key. ZK-SNARKs, on the other hand, allow you to make such
a proof for any function.
First, let us go through some specific examples. One natural use case for the technology is in identity systems. For example, suppose that you want to prove to a system that you are (i) a citizen of
a given country, and (ii) over 19 years old. Suppose that your government is technologically progressive, and issues cryptographically signed digital passports, which include a person’s name and date
of birth as well as a private and public key. You would construct a function which takes a digital passport and a signature signed by the private key in the passport as input, and outputs 1 if both
(i) the date of birth is before 1996, (ii) the passport was signed with the government’s public key, and (iii) the signature is correct, and outputs 0 otherwise. You would then make a zero-knowledge
proof showing that you have an input that, when passed through this function, returns 1, and sign the proof with another private key that you want to use for your future interactions with this
service. The service would verify the proof, and if the proof is correct it would accept messages signed with your private key as valid.
You could also use the same scheme to verify more complex claims, like “I am a citizen of this country, and my ID number is not in this set of ID numbers that have already been used”, or “I have had
favorable reviews from some merchants after purchasing at least $10,000 worth of products from them”, or “I hold assets worth at least $250,000”.
Another category of use cases for the technology is digital token ownership. In order to have a functioning digital token system, you do not strictly need to have visible accounts and balances; in
fact, all that you need is a way to solve the “double spending” problem – if you have 100 units of an asset, you should be able to spend those 100 units once, but not twice. With zero-knowledge
proofs, we can of course do this; the claim that you would zero-knowledge-prove is something like “I know a secret number behind one of the accounts in this set of accounts that have been created,
and it does not match any of the secret numbers that have already been revealed”. Accounts in this scheme become one-time-use: an “account” is created every time assets are sent, and the sender
account is completely consumed. If you do not want to completely consume a given account, then you must simply create two accounts, one controlled by the recipient and the other with the remaining
“change” controlled by the sender themselves. This is essentially the scheme used by Zcash (see more about how it works here).
For two-party smart contracts (eg. think of something like a financial derivative contract negotiated between two parties), the application of zero-knowledge-proofs is fairly easy to understand. When
the contract is first negotiated, instead of creating a smart contract containing the actual formula by which the funds will eventually be released (eg. in a binary option, the formula would be “if
index I as released by some data source is greater than X, send everything to A, otherwise send everything to B”), create a contract containing the hash of the formula. When the contract is to be
closed, either party can themselves compute the amount that A and B should receive, and provide the result alongside a zero-knowledge-proof that a formula with the correct hash provides that result.
The blockchain finds out how much A and B each put in, and how much they get out, but not why they put in or get out that amount.
This model can be generalized to N-party smart contracts, and the Hawk project is seeking to do exactly that.
Starting from the Other End: Low-Tech Approaches
The other path to take when trying to increase privacy on the blockchain is to start with very low-tech approaches, using no crypto beyond simple hashing, encryption and public key cryptography. This
is the path that Bitcoin started from in 2009; though the level of privacy that it provides in practice is quite difficult to quantify and limited, it still clearly provided some value.
The simplest step that Bitcoin took to somewhat increase privacy is its use of one-time accounts, similar to Zcash, in order to store funds. Just like with Zcash, every transaction must completely
empty one or more accounts, and create one or more new accounts, and it is recommended for users to generate a new private key for every new account that they intend to receive funds into (though it
is possible to have multiple accounts with the same private key). The main benefit that this brings is that a user’s funds are not linked to each other by default: if you receive 50 coins from source
A and 50 coins from source B, there is no way for other users to tell that those funds belong to the same person. Additionally, if you spend 13 coins to someone else’s account C, and thereby create a
fourth account D where you send the remaining 37 coins from one of these accounts as “change”, the other users cannot even tell which of the two outputs of the transaction is the “payment” and which
is the “change”.
However, there is a problem. If, at any point in the future, you make a transaction consuming from two accounts at the same time, then you irrevertibly “link” those accounts, making it obvious to the
world that they come from one user. And, what’s more, these linkages are transitive: if, at any point, you link together A and B, and then at any other point link together A and C, and so forth, then
you’ve created a large amount of evidence by which statistical analysis can link up your entire set of assets.
Bitcoin developer Mike Hearn came up with a mitigation strategy that reduces the likelihood of this happening called merge avoidance: essentially, a fancy term for trying really really hard to
minimize the number of times that you link accounts together by spending from them at the same time. This definitely helps, but even still, privacy inside of the Bitcoin system has proven to be
highly porous and heuristic, with nothing even close to approaching high guarantees.
A somewhat more advanced technique is called CoinJoin. Essentially, the CoinJoin protocol works as follows:
1. N parties come together over some anonymous channel, eg. Tor. They each provide a destination address D[1] … D[N].
2. One of the parties creates a transaction which sends one coin to each destination address.
3. The N parties log out and then separately log in to the channel, and each contribute one coin to the account that the funds will be paid out from.
4. If N coins are paid into the account, they are distributed to the destination addresses, otherwise they are refunded.
If all participants are honest and provide one coin, then everyone will put one coin in and get one coin out, but no one will know which input maps to which output. If at least one participant does
not put one coin in, then the process will fail, the coins will get refunded, and all of the participants can try again. An algorithm similar to this was implemented by Amir Taaki and Pablo Martin
for Bitcoin, and by Gavin Wood and Vlad Gluhovsky for Ethereum.
So far, we have only discussed token anonymization. What about two-party smart contracts? Here, we use the same mechanism as Hawk, except we substitute the cryptography with simpler cryptoeconomics –
namely, the “auditable computation” trick. The participants send their funds into a contract which stores the hash of the code. When it comes time to send out funds, either party can submit the
result. The other party can either send a transaction to agree on the result, allowing the funds to be sent, or it can publish the actual code to the contract, at which point the code will run and
distribute the funds correctly. A security deposit can be used to incentivize the parties to participate honestly. Hence, the system is private by default, and only if there is a dispute does any
information get leaked to the outside world.
A generalization of this technique is called state channels, and also has scalability benefits alongside its improvements in privacy.
Ring Signatures
A technology which is moderately technically complicated, but extremely promising for both token anonymization and identity applications, is ring signatures. A ring signature is essentially a
signature that proves that the signer has a private key corresponding to one of a specific set of public keys, without revealing which one. The two-sentence explanation for how this works
mathematically is that a ring signature algorithm includes a mathematical function which can be computed normally with just a public key, but where knowing the private key allows you to add a seed to
the input to make the output be whatever specific value you want. The signature itself consists of a list of values, where each value is set to the function applied to the previous value (plus some
seed); producing a valid signature requires using knowledge of a private key to “close the loop”, forcing the last value that you compute to equal the first. Given a valid “ring” produced in this
way, anyone can verify that it is indeed a “ring”, so each value is equal to the function computed on the previous value plus the given seed, but there is no way to tell at which “link” in the ring a
private key was used.
There is also an upgraded version of a ring signature called a linkable ring signature, which adds an extra property: if you sign twice with the same private key, that fact can be detected – but no
other information is revealed. In the case of token anonymization, the application is fairly simple: when a user wants to spend a coin, instead of having them provide a regular signature to prove
ownership of their public key directly, we combine public keys together into groups, and ask the user to simply prove membership in the group. Because of the linkability property, a user that has one
public key in a group can only spend from that group once; conflicting signatures are rejected.
Ring signatures can also be used for voting applications: instead of using ring signatures to validate spending from a set of coins, we use them to validate votes. They can also be used for identity
applications: if you want to prove that you belong to a set of authorized users, without revealing which one, ring signatures are well-suited for just that. Ring signatures are more mathematically
involved than simple signatures, but they are quite practical to implement; some sample code for ring signatures on top of Ethereum can be found here.
Secret Sharing and Encryption
Sometimes, blockchain applications are not trying to mediate the transfer of digital assets, or record identity information, or process smart contracts, and are instead being used on more
data-centric applications: timestamping, high-value data storage, proof of existence (or proof of inexistence, as in the case of certificate revocations), etc. A common refrain is the idea of using
blockchains to build systems where “users are in control of their own data”.
In these cases, it is once again important to note that blockchains do NOT solve privacy issues, and are an authenticity solution only. Hence, putting medical records in plaintext onto a blockchain
is a Very Bad Idea. However, they can be combined with other technologies that do offer privacy in order to create a holistic solution for many industries that does accomplish the desired goals, with
blockchains being a vendor-neutral platform where some data can be stored in order to provide authenticity guarantees.
So what are these privacy-preserving technologies? Well, in the case of simple data storage (eg. medical records), we can just use the simplest and oldest one of all: encryption! Documents that are
hashed on the blockchain can first be encrypted, so even if the data is stored on something like IPFS only the user with their own private key can see the documents. If a user wants to grant someone
else the right to view some specific records in decrypted form, but not all of them, one can use something like a deterministic wallet to derive a different key for each document.
Another useful technology is secret sharing (described in more detail here), allowing a user to encrypt a piece of data in such a way that M of a given N users (eg. M = 5, N = 9) can cooperate to
decrypt the data, but no fewer.
The Future of Privacy
There are two major challenges with privacy preserving protocols in blockchains. One of the challenges is statistical: in order for any privacy-preserving scheme to be computationally practical, the
scheme must only alter a small part of the blockchain state with every transaction. However, even if the contents of the alteration are privacy, there will inevitably be some amount of metadata that
is not. Hence, statistical analyses will always be able to figure out something; at the least, they will be able to fish for patterns of when transactions take place, and in many cases they will be
able to narrow down identities and figure out who interacts with whom.
The second challenge is the developer experience challenge. Turing-complete blockchains work very well for developers because they are very friendly to developers that are completely clueless about
the underlying mechanics of decentralization: they create a decentralized “world computer” which looks just like a centralized computer, in effect saying “look, developers, you can code what you were
planning to code already, except that this new layer at the bottom will now make everything magically decentralized for you”. Of course, the abstraction is not perfect: high transaction fees, high
latency, gas and block reorganizations are something new for programmers to contend with, but the barriers are not that large.
With privacy, as we see, there is no such magic bullet. While there are partial solutions for specific use cases, and often these partial solutions offer a high degree of flexibility, the
abstractions that they present are quite different from what developers are used to. It’s not trivial to go from “10-line python script that has some code for subtracting X coins from the sender’s
balance and adding X coins to the recipient’s balance” to “highly anonymized digital token using linkable ring signatures”.
Projects like Hawk are very welcome steps in the right direction: they offer the promise of converting an arbitrary N-party protocol into a zero-knowledge-ified protocol that trusts only the
blockchain for authenticity, and one specific party for privacy: essentially, combining the best of both worlds of a centralized and decentralized approach. Can we go further, and create a protocol
that trusts zero parties for privacy? This is still an active research direction, and we’ll just have to wait and see how far we can get.
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ignal using MATLAB
Digital Communication Lab Experiments
To generate the waveform of Frequency Shift Keying (FSK) signal using MATLAB.
Sep 24 2023
To generate the waveform of Frequency Shift Keying (FSK) signal using MATLAB.
Software required:
1. MATLAB
2. Computer installed with Windows XP or higher Version
Generation of FSK:Frequency-shift keying (FSK) is a frequency modulation scheme in which digital information is transmitted through discrete frequency changes of a carrier wave. The simplest FSK is
binary FSK (BFSK). BFSK uses a pair of discrete frequencies to transmit binary (0s and 1s) information. With this scheme, the "1" is called the mark frequency and the "0" is called the space
In binary FSK system, symbol 1 & 0 are distinguished from each other by transmitting one of the two sinusoidal waves that differ in frequency by a fixed amount.
Si (t) = √2E/Tb cos 2πf1t
0≤ t ≤Tb
0 elsewhere
where i=1, 2 & Eb=Transmitted energy/bit
Transmitted freq= ƒi = (n [c]+i)/T [b], and n = constant (integer), T [b]= bit interval
Symbol 1 is represented by S [1](t) and Symbol 0 is represented by S [0](t)
The input binary sequence is represented in its ON-OFF form, with symbol 1 represented by constant amplitude of √Eb with & symbol 0 represented by zero volts. By using inverter in the lower channel,
we in effect make sure that when symbol 1is at the input, The two frequency f [1]& f [2]are chosen to be equal integer multiples of the bit rate 1/T [b]. By summing the upper & lower channel outputs,
we get BFSK signal as shown in figure 8.1.
FSK modulation:
1. Generate two carriers signal.
2. Start for loop
3. Generate binary data, message signal and inverted message signal
4. Multiply carrier 1 with message signal and carrier 2 with inverted message signal
5. Perform addition to get the FSK modulated signal
6. Plot message signal and FSK modulated signal.
7. End for loop.
8. Plot the binary data and carriers
%FSK Modulation
clear all;
close all;
%Generate Carrier Signal
%generate message signal
for i=1:N
if m(i)>0.5
%plotting the message signal and the modulated signal
axis([0 N -2 2]);
title('message signal');
grid on;
hold on;
title('FSK signal');
grid on;
hold on;
hold off
%Plotting binary data bits and carrier signal
title('binary data bits');
grid on;
title('carrier signal-1');
grid on;
title('carrier signal-2');
grid on;
Observation:The desired BFSK waveforms i.e. binary data, message signal, carrier signal 1&2 and output waveforms are shown in figure 8.2.
Conclusion:The program for binary FSK modulation has been simulated in MATLAB and observed the desired waveforms.
1. Determine the bandwidth and baud for BFSK signal with mark frequency of 49 KHz,a space frequency of 51 KHz, and a bit rate of 2 Kbps.
2. Write a MATLAB program for finding the sum of series 1+ 2+ 3 +……+N.
3. Sketch the FSK waveform for the input (a) 1010110 (b) 1100101.
4. Write the advantages of FSK compared to ASK?
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Topical Language Generation With Transformers
Large-scale transformer-based language models (LMs) demonstrate impressive capabilities in open text generation. However, controlling the generated text’s properties such as the topic, style, and
sentiment is challenging and often requires significant changes to the model architecture or retraining and fine-tuning the model on new supervised data.
We introduce Topical Language Generation (TLG) by combining a pre-trained LM with topic modeling information. We cast the problem using Bayesian probability formulation with topic probabilities as a
prior, LM probabilities as the likelihood, and topical language generation probability as the posterior. In learning the model, we derive the topic probability distribution from the user-provided
document’s natural structure. Furthermore, we extend our model by introducing new parameters and functions to influence the quantity of the topical features presented in the generated text. This
feature would allow us to easily control the topical properties of the generated text.
Language modeling and decoding
The applications of language generation in NLP can be divided into two main categories: directed language generation and open-ended language generation. Directed language generation involves
transforming input to output such as machine translation, summarization, etc. These approaches need some semantic alignment between the inputs and the outputs. On the other hand, open-ended language
generation has much more freedom in the generation process because it does not need to be aligned with any output. The open-ended language generation has applications in conditional story generation,
dialog systems, and predictive response generation. Even though there is more flexibility in choosing the next tokens compared to directed language generation, controlling the top-level features of
the generated text is a desirable property that needs to be addressed and still is a challenging problem.
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Given a sequence of m tokens x_1, .., x_m as the context, the problem of open-ended language generation can be formulated as finding the continuation x_{m+1}, …, x_{m+n} with n tokens. In other
words, if we consider the whole context plus continuation as following:
The language modeling probability can be decomposed using the chain rule as:
The language modeling probability can be used with a decoding strategy to generate the next token for language generation. Finding the optimal continuation can be formulated as:
Solving the above Equation is not tractable so practical decoding strategies use approximations to generate the next tokens. The most famous and widely used decoding strategies are greedy decoding
and beam search methods. Greedy decoding selects the highest probability token at each time step, while the beam search keeps a set of hypotheses and then updates the tokens in the hypotheses as it
goes through and decodes more tokens. These approaches are well suited for directed language generation, but they suffer from repetition, genericness, and degenerate continuations.
Both of these approaches are deterministic in the sense that they do not involve any random selection in their algorithms.
On the other hand, stochastic decoding methods sample from a model-dependent distribution q:
The simplest stochastic sampling consists of sampling from top-k probabilities, the use of constant k is problematic because in some contexts the probability distribution of the next token is flat
which means there are plenty of reasonable next tokens to select from but in some other contexts the distribution is concentrated in a small number of tokens. To solve this problem, (Holtzmanet al.
(2020)) proposed Nucleus Sampling. In this method, a subset of vocabulary is defined which is the smallest set V(p) such that:
Then the resulting distribution which is based on the new vocabulary should be re-scaled to form a probability distribution. Under Nucleus Sampling, the number of plausible next tokens changes
dynamically with the context and generated tokens. In this work, we use Nucleus Sampling as the base decoding technique and propose a new method to take into account topical knowledge about the
Topical Language Modeling
Given a list of K topics t = {1…K}, to control the outputs of the language model to follow a certain topic, at each generation step, we have to model the following probability distribution:
Compared to the previous Equation, the only difference is that it is conditioned on the topic t_j. To create the right-hand side of Equation 6, we change the last layer of the network that creates
the logits.
Here, we adopt the GPT transformer architecture. If S is the last layer we use softmax to get the final probabilities:
We can use the Bayes rule on P(x_i|x<i,t_j) to obtain:
Because in topic modeling, documents are treated as bag of words we can also assume that the probability of the topic for each token is independent of the previously generated tokens. Based on this
assumption we have:
Now, assuming that we have P(t_j|x_i), then using Equation 10 we can prove that the conditions topical language model can be written as:
For complete proof refer to the paper.
Topic modeling
Topic modeling algorithms automatically extract topics from a collection of textual data. They are based on statistical unsupervised models that discover the themes running through documents. We use
two main algorithms in topic modeling.
• LDA (Latent Dirichlet Allocation): The basic idea behind LDA is that in a collection of documents, every document has multiple topics and each topic has a probability distribution. Moreover, each
topic has a distribution over vocabulary. For example, a document can be on the topics of “Football”, “News” and “America” and the topic of “Football” can contain words including “NFL”,
“Football”, “teams” with a higher probability compared to other words. Given a collection of M documents with vocabulary V, we can fix the number of topics to be K. In LDA, the probabilities of
topics per documents and topic for tokens can be summarized in matrix forms, θ_M×K, and φ_K×|V|, respectively. After the learning, we have the distributions of topics for each token and hence we
can write:
• LSI (Latent Semantic Indexing): LSI is the application of the singular value decomposition method to the word-document matrix, with rows and columns representing the words and documents,
respectively. Let X_|V|×M be the token-document matrix such that X_{i,j} is the occurrence of token i in document j, then singular value decomposition can be used to find the low-rank
After the decomposition, U still has the same number of rows as tokens but has fewer columns that represent latent space that is usually interpreted as “topics”. So, normalizing U gives us the scores
of each token per topic. We can use this score for the probability of topic j for each token i in the vocabulary:
Controllable Generation Methods
The conditional topical language model in the equation above gives us a token generation that is conditioned on a specific topic but we cannot control the amount of the influence.
1- Adding topical parameter and logit threshold: adding the term log(P(t_j|x_i)) directly to the actual logit from the model can deteriorate the fluency of generated text in some cases. We propose
two methods to alleviate this problem. We introduce a new parameter γ to control the influence of topical distribution:
Higher values ofγresult in more on-topic text generation because the final probability will be dominated more by log(P(t_j|x_i))than the logit from the base language modeling.
The other approach is to cut the log probabilities of the topic with a threshold. The lower values of S correspond to tokens that the model gives very low probabilities and we do not want to change
them because it introduces unwanted tokens and diminishes the fluency. In Equation above, we only keep log(P(t_j|x_i))for all the values of S that are larger than threshold.
and log prob used in the following equation:
ower values of threshold correlate with more on-topic text generation because we change more tokens from the original model by log(P(t_j|x_i)).
2 -Using α-entmax instead of softmax: The problem with the softmax function is that it gives non-zero probabilities to a lot of unnecessary and implausible tokens. The softmax function is dense
because it is proportional to exp function and can never give exactly zero probabilities at the output. We useα-entmax instead to create more sparse probabilities that are less prone to degenerate
text. α-entmax is defined as
where ∆|V|−1:={p∈IR|V|−1,∑ipi=1}is the probability simplex, and for α≥1, HTα(p) is the Tsallis entropy which defines the family of entropies as follows:
α-entmax is the generalized form of the softmax function. In particular, for α=1 it exactly reduces to the softmax function and as α increases, the sparsity in the output probabilities continuously
increases. Here we are specifically interested in α=2 which results in sparsemax:
Unlike the softmax function, sparsemax can assign zero probabilities.
3-Adding temperature and repetition penalty parameters: We need to make some changes to the base nucleus sampling to control the base distribution flatness and prevent it from generating repetitive
words. We denote the final logit after the above changes as ui. Given a temperature, repetition penalty r and the list of generated tokens g, the final probability distribution for sampling is:
when T→0, the sampling reduces to greedy sampling; while if T→∞ the distribution becomes flatter and more random. The penalized sampling discourages drawing already generated tokens.
Topical Text Generation with Different Topics
One of the biggest benefits of TLG is that it can be used with different language models without any retraining or fine-tuning of the base model, however, to generate topical texts we need to have
topics extracted from a text corpus. For training the topic models, we used Alexa Topical-chat dataset. This data set contains conversations and a knowledge base in a wide variety of topics from
politics and music to sports. We do not use the tags for topics in the dataset but extract them automatically with our LDA and LSI topic models. This unsupervised approach gives us the flexibility to
work with any raw text corpus.
In this experiment, a fixed neutral prompt has been used to make sure the model is not conditioned on the few initial tokens. The results in the table below show that after selecting a topic from the
topic modeling output, the model can create long, coherent, and fluent text continuation without manually injecting extra knowledge from other resources or through training on labeled datasets.
Image by Author
Effects of Hyperparamters on TLG
In our proposed approach, we can useγandthresholdas knob parameters to control the amount of topic influence on the language generation process. More specifically, based on Equation 27higher values
of gamma will result in more on-topic results. Also, lower values of the threshold are correlated with more on-topic language generation. In the limit, if we setγ=0 andthreshold=0TLG reduces to the
original language model without any topic. But, our experiments have shown that changingγvalues are less detrimental to the fluency of the generated text than changing the threshold. This is due to
the fact that thresholding can easily cut off the probabilities that are related to function tokens (like stop words) in the vocabulary which hurts the fluency of the model. Fig below demonstrates
the language generation on a fixed topic (football) with different values ofγandthreshold. To show how much each token accounts for the topic we use color-coding in which stronger colors show more
on-topic words. We skipped the last stage of decoding. This is why the individual tokens from Byte Pair Encoding (BPE) tokenization can be seen.
Image by Author
Now tell me how it works?
The language generation is the task of generating the next token conditioned on the previously generated tokens. The probability distribution of the next token in the base language models is flatter
in some token positions and more peaked at some other token positions. For example, given the prompt of “The issue is that” there are plenty of possible next tokens compared to the next token of a
prompt like “It is focused” which is almost always “on”. This property of language models gives us the flexibility to meddle in the generation process and steer it towards desired tokens when the
probability distribution is flatter.
The concept of flat or peaked distribution can be easily measured in terms of the entropy of the distribution. In Figures a and b we compare the entropy of the base model (token entropy) with the
posterior probability distribution from Equation 20 as the total entropy. Higher entropy for the base model in one position is a sign of its capability to sample from a large set of potential tokens
with almost equal probabilities but in our conditional language modeling, we want to restrict that set to a smaller set that conforms with the chosen topic. Therefore, in almost all cases, the
entropy of the TLG model drops significantly compared to the base model. We can observe the differences are larger for the tokens that represent the topic (like teams, football, culture and, music)
and smaller for function tokens (like stop words that do not play any role in different topics).
Image by Author
Another interesting observation is how the prior distribution that was extracted from topic modeling forces the language model to choose the topical tokens. The top-5 most likely tokens in a
generation process are depicted in Figure 4. For the topic of football, the top-5 candidate tokens chosen by the model are compatible with the chosen topic.
Image by Author
Graphical User Interface
Image by Author
We also provide the GUI as a playground for users to work with the TLG. On the left panel, you can control the dataset, topic model, number of topics, and other generation settings. The playground
gives you a graph plot which is a novel representation of the topics and how they are related to each other. Then you can choose the topic of interest and choose a prompt and finally hit the generate
button to get the topical text.
This article was originally published on Towards Data Science and re-published to TOPBOTS with permission from the author.
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GGR270H1 Quiz: Variance Ratio Test - OneClass
Document Summary
Used to test whether the population (and sample) variances are equal. The answer to this question will determine which version of the t statistic in the two sample. The test uses the f distribution,
which is skewed: hypotheses. H0: no significant difference between the variance of sample one and the variance of sample 2. Therefore, the samples are drawn from the same population. Ha: there is a
significant difference significant difference between the variance of sample 1 and the variance of sample 2. Therefore, the samples are drawn from different populations: critical value. The critical
value for f is read from the f distribution table. In this table, the values are always greater than 1, which implies the ratio of the variances will be greater than 1. The table assumes you divide
the larger sample variance by the smaller one.
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Lesson 10
Using Data Displays to Find Associations
10.1: Sports and Musical Instruments (5 minutes)
The purpose of this warm-up is for students to answer questions about relative frequency of items after finding missing information in a two-way table.
Monitor for students who find the percentages for the final two questions using different strategies to share during the whole-class discussion.
Give students 2 minutes of quiet work time followed by a whole-class discussion.
Student Facing
For a survey, students in a class answered these questions:
• Do you play a sport?
• Do you play a musical instrument?
1. Here is a two-way table that gives some results from the survey. Complete the table, assuming that all students answered both questions.
│ │plays instrument│does not play instrument │total│
│ plays sport │5 │ │16 │
│does not play sport│ │ │ │
│ total │ │15 │25 │
2. To the nearest percentage point, what percentage of students who play a sport don’t play a musical instrument?
3. To the nearest percentage point, what percentage of students who don’t play a sport also don’t play a musical instrument?
Activity Synthesis
Ask students to share the missing information they found for the table. Record and display their responses for all to see.
Select students previously identified to explain how they found the percentages for the final two questions and what that percentage represents.
1. Students who find a percentage using the values given (for example 31% since \(\frac{5}{16} \approx 0.31\)), then subtract from 100% (for example 69% since \(100 - 31 = 69\)) to answer the
2. Students who find the actual values first by subtracting (for example \(16 - 5 = 11\)) then compute the percentage (for example 69% because \(\frac{11}{16}=0.6875\)).
Ask the rest of the class if they agree or disagree with the strategies and give time for any questions they have.
10.2: Sports and Music Association (20 minutes)
Now that students are more familiar with two-way tables showing relative frequency, they are ready to create their own segmented bar graphs. In this activity, students create two segmented bar graphs
based on the same two-way table by considering percentages of the rows and columns separately. After creating the segmented bar graphs, they are analyzed to determine if there is an association
present in the data.
Arrange students in groups of 2. After a brief introduction, give 5–10 minutes of quiet work time. Ask students to compare their answers with their partner and try to resolve any differences. Finish
with a whole-class discussion.
Display the two-way table from the previous lesson's cool-down activity containing the data collected about the class's playing sports and musical instruments. If the data is unavailable, the data
from this lesson's warm-up can be used.
Tell students they should work with their partners to each work on one of the graphs. One student should work on problems 1 and 2 while their partner should work on 3 and 4. After they have completed
their graphs, they should work together to understand their partners graphs and complete the last problem together.
Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example,
present one question at a time. Some students may benefit from a checklist on how to create a segmented bar graph.
Supports accessibility for: Organization; Attention
Student Facing
Your teacher will give you a two-way table with information about the number of people in your class who play sports or musical instruments.
1. Complete this table to make a two-way table for the data from earlier. The table will show relative frequencies by row.
│ │plays instrument│does not play instrument │row total│
│ plays sport │ │ │100% │
│does not play sport│ │ │100% │
2. Make a segmented bar graph for the table. Use one bar of the graph for each row of the table.
3. Complete the table to make a two-way table for the data from earlier. The table will show relative frequencies by column.
│ │plays instrument│does not play instrument │
│ plays sport │ │ │
│does not play sport│ │ │
│ column total │100% │100% │
4. Using the values in the table, make a segmented bar graph. Use one bar of the graph for each column of the table.
5. Based on the two-way tables and segmented bar graphs, do you think there is an association between playing a sport and playing a musical instrument? Explain how you know.
Anticipated Misconceptions
Students may draw the segmented bar graph incorrectly. Most likely, they will accidentally graph frequency instead of relative frequency. They may also graph relative frequencies, but without
stacking them. Both segmented bars should go from 0 to 100.
Activity Synthesis
To clarify how to create and interpret segmented bar graphs, ask:
• “What different information can be seen by the two segmented bar graphs?”
• “Why are the numbers in the top left box in the two tables different? What do they mean?” (In the first table it represents the percentage who also play musical instruments out of all the people
who play sports. In the second table it represents the percentage of people who also play sports out of all the people who play musical instruments.)
• “Is there an association between the two variables? Explain or show your reasoning.” (The answer will depend on class data, but the reasoning should include an analysis of the relative
frequencies within categories. There is an association if the percentages within one category are very different from the percentages in another category.)
If there is an association, ask what the segmented bar graphs would look like if there was no association. If there is not an association, ask what the segmented bar graphs would look like if there
was one.
Writing, Speaking: MLR1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise and refine their response to the last question. Ask each student to meet
with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language (e.g., “Why do you think there is a
(positive/negative) association?”, “How do the relative frequencies help to answer this question?”, “How could you say that another way?”, etc.). Students can borrow ideas and language from each
partner to strengthen the final product. They can return to the first partner and revise and refine their initial response.
Design Principle(s): Optimize output (for explanation)
10.3: Colored Erasers (15 minutes)
This activity provides students less structure for their work in creating segmented bar graphs to determine an association (MP4). In addition, the data in this activity is split into more than two
options. Students work individually to create a segmented bar graph based on either columns or rows and then share their information with a partner who has created the other segmented bar graph.
Together, partners discuss the segmented bar graphs to determine if there is an association between the variables (MP3). In particular, students should notice that there is evidence of an association
is the relative frequencies within a category are very different from the relative frequencies in another category.
As students work, identify groups that use the different segmented bar graphs to explain why there is an association between the color of the eraser and flaws.
Keep students in groups of 2. Give 5 minutes quiet work time followed by 5 minutes of partner discussion and then a whole-class discussion.
Provide students access to colored pencils. Either assign or have partners choose which will make a graph for each row and which will make a graph for each column.
Representation: Access for Perception. Read the directions aloud. Students who both listen to and read the information will benefit from extra processing time. Check for understanding by inviting
students to rephrase directions in their own words.
Supports accessibility for: Language
Student Facing
An eraser factory has five machines. One machine makes the eraser shapes. Then each shape goes through the red machine, blue machine, yellow machine, or green machine to have a side colored.
The manager notices that an uncolored side of some erasers is flawed at the end of the process and wants to know which machine needs to be fixed: the shape machine or some of the color machines. The
manager collected data on the number of flawed and unflawed erasers of each color.
│ │unflawed │flawed│total│
│ red │285 │15 │300 │
│ blue │223 │17 │240 │
│yellow│120 │80 │200 │
│green │195 │65 │260 │
│total │823 │177 │1000 │
1. Work with a partner. Each of you should make one segmented bar graph for the data in the table. One segmented bar graph should have a bar for each row of the table. The other segmented bar graph
should have one bar for each column of the table.
2. Are the flawed erasers associated with certain colors? If so, which colors? Explain your reasoning.
Student Facing
Are you ready for more?
Based on the federal budgets for 2009, the table shows where some of the federal money was expected to go. The values are in billions of U.S. Dollars.
│ │United States │Japan│United Kingdom │
│ defense │718.4 │42.8 │49.2 │
│education│44.9 │47.5 │113.9 │
1. Why would a segmented bar graph be more useful than the table of data to see any associations between the country and where the money is spent?
2. Create a segmented bar graph that represents the data from the table.
3. Is there an association between the country’s budget and their spending in these areas? Explain your reasoning.
Activity Synthesis
The purpose of this discussion is to identify strategies for creating segmented bar graphs and for analyzing them to determine if there is an association among variables.
Ask, “What strategies did you use to create the segmented bar graphs?” (First, we created a new table of the relative frequencies. Then we approximated the heights of the segments based on the
percentages from the table.)
Select previously identified groups to share their explanation for noticing an association.
1. Groups that use the segmented bar graph based on rows.
2. Groups that use the segmented bar graph based on columns.
After both explanations are shared, ask students, "Do you think that noticing the association was easier with one of the graphs?" (Likely the segmented bar graph based on rows is easier since there
are only 2 segments and it is easier to see that the yellow and green erasers are more flawed.)
Finally, ask students, "If there was not an association between color and flaws, what might the segmented bar graph based on the rows look like? What might the segmented bar graph based on the
columns look like?” (The segmented bar graph based on the rows would have each segmented bar look about the same. That is, the line dividing the two segments would be at about the same height in each
bar. The segmented bar graph based on the columns would have segments that are all approximately equal. That is, each segment should represent about 25% of the entire bar.)
Lesson Synthesis
Remind students that we have been looking for associations in categorical data, and that there is evidence of an association if the relative frequencies of some characteristic are very different from
each other in the different groups. Ask:
• “Is it easier to see evidence of an association in a frequency table or a relative frequency table?” (It depends on the data. If the two groups are approximately the same size, it doesn't matter
very much, but when they are different sizes, it is usually easier to compare using relative frequencies.)
• “How can we see evidence of an association in a two-way table of either kind?” (By numerically comparing the proportions between the two groups.)
• “How can we see evidence of an association in a bar graph or segmented bar graph?” (By visually comparing the proportions between the two groups.)
10.4: Cool-down - Class Preferences (5 minutes)
Student Facing
In an earlier lesson, we looked at data on meditation and state of mind in athletes.
Is there an association between meditation and state of mind?
The bar graph shows that more athletes were calm than agitated among the group that meditated, and more athletes were agitated than calm among the group that did not. We can see the proportions of
calm meditators and calm non-meditators from the segmented bar graph, which shows that about 66% of athletes who meditated were calm, whereas only about 27% of those who did not meditate were calm.
This does not necessarily mean that meditation causes calm; it could be the other way around, that calm athletes are more inclined to meditate. But it does suggest that there is an association
between meditating and calmness.
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Vijith Kumar K P, Brijesh Kumar Rai, Tony Jacob
The idea of coded caching was introduced by Maddah-Ali and Niesen who demonstrated the advantages of coding in caching problems. To capture the essence of the problem, they introduced the $(N, K)$
canonical cache network in which $K$ users with independent caches of size $M$ request files from a server that has $N$ files. Among other results, the caching scheme and lower bounds proposed by
them led to a characterization of the exact rate memory tradeoff when $M\geq \frac{N}{K}(K-1)$. These lower bounds along with the caching scheme proposed by Chen et al. led to a characterization of
the exact rate memory tradeoff when $M\leq \frac{1}{K}$. In this paper we focus on small caches where $M\in \left[0,\frac{N}{K}\right]$ and derive new lower bounds. For the case when $\big\lceil\frac
{K+1}{2}\big\rceil\leq N \leq K$ and $M\in \big[\frac{1}{K},\frac{N}{K(N-1)}\big]$, our lower bounds demonstrate that the caching scheme introduced by G{\'o}mez-Vilardeb{\'o} is optimal and thus
extend the characterization of the exact rate memory tradeoff. For the case $1\leq N\leq \big\lceil\frac{K+1}{2}\big\rceil$, we show that the new lower bounds improve upon the previously known lower
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Patterson correlation methods: a review of molecular replacement with CNS
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^aLawrence Berkeley National Laboratory, One Cyclotron Road, Mail Stop 4-230, Berkeley, CA 94720, USA
^*Correspondence e-mail: rwgrosse-kunstleve@lbl.gov
(Received 5 April 2001; accepted 12 June 2001)
This paper presents a review of the principles of molecular replacement with the audience of the CCP4 Study Weekend in mind. A complementary presentation with animated Patterson maps is available
online (https://cci.lbl.gov/~rwgk/ccp4sw2001/ ). The implementation of molecular-replacement methods in the Crystallography and NMR System (CNS) is presented and discussed in some detail. The three
principal components are the direct rotation function, Patterson correlation refinement and the fast translation function. CNS is available online and is free of charge for academic users.
1. Introduction
The method of molecular replacement was pioneered four decades ago by Hoppe (1957 ) and Rossmann & Blow (1962 ). The latter publication marks the beginning of practical application to the solution of
macromolecular crystal structures. The term `molecular replacement' is somewhat misleading because nothing is `replaced' (but it is helpful for remembering the initials of one of the main champions
of the method). The conventional understanding of what molecular replacement encompasses is the placement of one or more known molecular models in the unit cell of the crystal under study. The search
models are often extracted from databases such as the Protein Data Bank (Berman et al., 2000 ) or different crystal forms that were solved previously.
In general, placing a molecular model in a unit cell is a six-dimensional search problem. The six degrees of freedom are most conveniently parameterized as three rotation angles and three
translations along the basis vectors of the coordinate system. Conventionally, an asymmetric unit (a volume of the search space that is unique under symmetry) is sampled on a uniform grid. For a
typical macromolecular unit cell, the product of angular and translational sampling points is usually too large to carry out an exhaustive six-dimensional search in a reasonable time with current
computing resources. However, there have been cases where an exhaustive search has been carried out in spite of the computational cost (Sheriff et al., 1999 ).
Rossmann & Blow (1962 ) showed that it is possible to break up the six-dimensional search into two consecutive three-dimensional searches: a search for the angular orientation of the search molecule
(rotation search) and a subsequent search for the translation (translation search). This greatly reduces the demand for computing resources. The total number of sampling points for the two
three-dimensional searches is roughly proportional to the square root of the number of sampling points for the exhaustive six-dimensional search.
In the Crystallography and NMR System (CNS; Brunger et al., 1998 ), a third powerful procedure is usually inserted between the rotation search and the translation search: Patterson correlation
refinement of the molecular orientation. We will discuss all three stages in the order in which they are typically used.
2. Rotation search
CNS implements two types of rotation search. We will refer to the first type as the `traditional rotation search'. The second type is commonly referred to as the `direct rotation search'. There is a
conceptual distinction between these two types of rotation searches. In the traditional rotation search, two Patterson maps are rotated with respect to each other and then superimposed. This can be
performed either in direct or in reciprocal space (Crowther, 1972 ; Navaza, 1987 ). In contrast, in a direct rotation search the molecular model is rotated directly. The term `directly' is used
because it is the fundamental concept of the rotation search to rotate the model. The following sections explain that rotating maps instead is actually a non-trivial optimization, devised to reduce
the computer time.
2.1. Principles of the rotation search
An animation that illustrates the general ideas behind the traditional and the direct rotation searches is available at https://cci.lbl.gov/~rwgk/ccp4sw2001/ . The fundamental prerequisites for the
understanding of the methods are as follows.
• (i) An observed Patterson map can be directly computed from the experimental diffraction intensities by Fourier transformation.
• (ii) A model Patterson map can be directly computed from the oriented and translated search model and compared with the observed Patterson map by superposition.
• (iii) The peaks in a Patterson map correspond to the interatomic vectors of the crystal structure (Buerger, 1959 ).
To aid the interpretation of a
Patterson map,
the interatomic vectors can be classified as
(within a molecule) and
(between molecules). The intermolecular vectors for a given molecule can in turn be classified as vectors between the following.
• (i) Copies of the molecule arising from lattice translations.
• (ii) Copies of the molecule arising from rotational symmetry operations (note that mirror planes are `improper' rotations and are included).
• (iii) Copies of the molecule arising from non-crystallographic symmetry (i.e. between molecules of the same kind).
• (iv) Other molecules of a different kind.
In the observed
Patterson map,
peaks arising from all of these different types of interatomic vectors are present. However, at the stage of the rotation search this is not true for the model
Patterson map.
Typically, only
search molecule is used at a time [see Tong & Rossmann (1990
) for an alternative procedure that makes use of
non-crystallographic symmetry
at this stage], which eliminates any interatomic vectors arising from
non-crystallographic symmetry
or from other molecules of a different type (types iii and iv in the list above). Furthermore, the three translations that shift the search molecule to the correct location
with respect to the rotational symmetry operations
are unknown and the interatomic vectors arising from this symmetry (type ii in the list above) are best ignored. This is achieved by placing the search molecule in a
unit cell;
in other words, by ignoring the rotational symmetry. Typically, it is computationally most efficient to place the search molecule with its center of gravity at the origin of the unit cell.
In summary, the model Patterson map has only peaks arsing from intramolecular vectors and intermolecular vectors between copies arising from lattice translations (type i in the list above).
Conceptually, both the traditional and the direct rotation search superimpose this `partial' Patterson map with the observed Patterson map. This can be viewed as a pattern-matching procedure. The
model Patterson map is the search pattern. The observed Patterson map contains the search pattern in an unknown angular orientation. In the observed Patterson map, the search pattern is obscured by
other patterns (other types of interatomic vectors) that are not considered in the model Patterson, and noise.
The traditional and the direct rotation search are two implementations of this pattern-matching concept. Both have their advantages and disadvantages.
In the direct rotation search, the model is rotated directly and a structure-factor calculation is carried out for each sampled angular orientation. This has the advantage of avoiding approximations
such as interpolations, but the disadvantage of being computationally expensive.
In the traditional rotation search, the computationally expensive structure-factor calculation is carried out only once to obtain a model Patterson map (the next paragraph explains this in detail)
which is then rotated and superimposed with the observed Patterson map. This has the advantage of being relatively fast, but the disadvantage of involving approximations.
Rotating Patterson maps with respect to each other is not as straightforward as it might seem at first sight. The problem arises from the fact that the two types of interatomic vectors present in the
model Patterson map, the intramolecular vectors and the intermolecular vectors arising from lattice translations, are, in general,spatially intermixed. As the search model is rotated, vectors in the
map that are close to each other rotate around different origins. In order to be able to consistently rotate the search pattern present in the map, the two types of vectors need to be spatially
separated. Fig. 1 explains how this can be achieved by placing the search model in an artificially enlarged unit cell. The intramolecular vectors in the large unit cell are then concentrated around
the origin of the Patterson map and the intermolecular vectors are concentrated around the other lattice points. It is now possible to cut out the spherical region around the origin that contains the
isolated intramolecular vectors, rotate it and superimpose the observed Patterson map in order to find the angular orientation with the best match. For reasons that will become apparent in the next
section (equation 1 ), the spherical region is commonly referred to as the integration sphere. Obviously, the radius of the integration sphere is chosen to be similar to the largest intermolecular
vector. In several implementations of the traditional rotation function, including CNS, a region around the large Patterson origin peak is normally omitted to improve the signal-to-noise ratio. The
resulting actively used region of the model Patterson map is then called the integration shell (represented as a yellow region in Fig. 1 d).
2.2. Rotation search target functions
In CNS, the traditional rotation function is evaluated in real space (Brünger, 1990 ). We will therefore use this term from now on. For the real-space rotation search, the Patterson correlation Rot(Ω
) for a given angular orientation Ω is evaluated as the correlation integral,
P[obs] and P[model] are the observed and model Patterson functions, respectively, and u is a location vector in Patterson space U.
The direct rotation function CC(Ω) for a given angular orientation Ω of the search model is typically evaluated as the standard linear correlation coefficient of the observed and calculated
normalized structure-factor amplitudes |E|^2. The standard linear correlation coefficient is well known and frequently used in statistics to measure the strength of a linear relation of two
variables. The formula for the evaluation of the correlation coefficient is
where X = |E|^2. The summations are computed for all Miller indices H. 〈X〉 denotes the mean of the X[H].
At first sight, (1) and (2) look very different. However, in practice (1) is evaluated as the sum of products. (2) is again a sum of products. The difference is just that in (2) each variable is
centered around its mean (this is achieved by the sub-expressions of the type X − 〈X〉) and the sums in the denominator normalize the coefficient such that it has values in the range from −1 to 1.
In the absence of approximations, the two ways of evaluating the Patterson correlation should give essentially identical results, even though one is evaluated in real space and the other in
reciprocal space. The absolute values will be different because the first expression is not normalized, but the rotation functions should be very similar except for a scaling factor.
2.3. Comparison of rotation searches
The approximate relative CPU times for rotation searches with AMoRe (evaluated in reciprocal space; Navaza, 1987 ), the CNS real space and the CNS direct rotation function are shown in Table 1 .
HyHEL-5 (26–10) Fab–digoxin complex (DeLano & Brünger, 1995 ).
AMoRe 1
CNS real space 20
CNS direct 300
The direct rotation search is more than one order of magnitude slower than the real-space rotation search and AMoRe is yet another order of magnitude faster. What benefit can be expected from the
direct rotation search in return for the large increase in computational expense?
In the previous section we stated that the two ways of evaluating the Patterson correlation should give similar results in the absence of approximations. However, in practice two significant
approximations are made for the real-space rotation function. When the rotated Patterson map is superimposed with the observed Patterson map, grid points do not in general superimpose directly and
interpolation has to be used. The other significant approximation is that the correlation integral (1) is only evaluated for a selected set of Patterson function peaks. Typically, only the highest
3000 peaks in the observed Patterson map are considered in the calculation. In contrast, the direct rotation function is evaluated uniformly for the entire unit cell and does not involve
DeLano & Brünger (1995 ) systematically compared the signal-to-noise ratio for a number of test cases. They define the signal-to-noise ratio of rotation functions as the ratio of the value of the
highest signal point to that of the highest noise point, measured in standard deviations above the mean. `Points' of the rotation function are defined as peaks that are left after reduction by
spatial cluster analysis (DeLano & Brünger, 1995 ). A `signal' is defined by the radius of convergence of Patterson correlation refinement (see § 3). Empirical observation led DeLano & Brünger (1995
) to the conclusion that a rotation-function peak that is within about 15° of one of the correct orientations will, in general, converge to it by Patterson correlation refinement. Rotation-function
peaks that are within the 15° range were thus considered to be a signal. Points outside this range were considered to be noise.
A typical result of DeLano and Brünger's systematic comparisons is shown in Fig. 2 for search models with all atoms, a polyalanine chain and just the C^α atoms. The direct rotation function
consistently has a much better signal-to-noise ratio. Similarly, in Fig. 3 the high-resolution limit is varied. Again, the direct rotation function consistently has a significantly better
signal-to-noise ratio compared with the real-space rotation functions, both with and without removal of the Patterson origin peak.
3. Patterson correlation refinement
The second stage of the CNS molecular-replacement procedure is Patterson correlation (PC) refinement, which is the intervening step between the rotation search and the translation search (Brünger,
1990 ). The goal of PC refinement is to improve the overall orientation of the search model. Typically, the refinement is carried out for rigid bodies such as domains, subdomains or
secondary-structure elements. The major difference from normal crystallographic rigid-body refinement is that PC refinement is conducted without using crystallographic symmetry. The rationale for
this is similar to that for not using the symmetry in the rotation search (see § 2.1). The target function of PC refinement is typically defined as the standard linear correlation between observed
and calculated squared normalized structure-factor amplitudes (|E^2|).
By improving the accuracy of the search model for the correct angular orientation, PC refinement improves the discrimination between correct and incorrect orientations and therefore enables the
location of the correct peak in a noisy rotation function. In general, PC refinement makes the combination of a three-dimensional rotation search with a subsequent three-dimensional translation
search much more robust, so that one does not have to resort to exhaustive six-dimensional searches.
Brünger (1997 ) systematically studied the radius of convergence of rigid-body PC refinement under various conditions. One of the examples is a structure with two domains that are connected by a
linker region. One domain was kept stationary and the other was systematically misaligned. Fig. 4 shows the value of the Patterson correlation coefficient after PC refinement as a function of the
initial misaligned interdomain angle. In this particular case, it is found that the PC refinement converges back to the correct angle if the second domain is misaligned by up to approximately 13°.
Another way to assess the power of PC refinement is shown in Fig. 5 . This figure shows that pre-translation PC refinement has the potential to drastically reduce the number of noise peaks in the
translation function. Owing to this noise reduction it can often become immediately obvious what the correct position of the search molecule is.
4. Translation search
At this stage, the angular orientation of the search molecule is assumed to be known. The remaining problem is to determine the location of this oriented search molecule with respect to the symmetry
elements. The fundamental concept for solving this problem is straightforward: the unit cell is subdivided into a regular grid and the search molecule is moved to each grid point in turn. At each
location, a structure-factor calculation is performed. The agreement between these calculated and the observed structure factors is evaluated by some type of target function. Depending on the
space-group symmetry, for macromolecules the result is a two- or three-dimensional translation function similar to the example shown in Fig. 5 .
The translation-search target functions available in CNS include the standard linear correlation coefficient (equation 2 modified for translation instead of angular orientation) of normalized or
unnormalized structure-factor amplitudes, both squared and unsquared (|E|, |E|^2, |F|, |F|^2) and the crystallographic R factor. The use of the latter is complicated by the fact that a reasonably
accurate estimate of the scale factor between observed and calculated structure factors is required. The literature contains no conclusive evidence that this is a significant disadvantage in
practice. However, a correlation coefficient is the default choice in CNS.
Computing a translation function is relatively time-consuming and optimizations are essential. Fujinaga & Read (1987 ) introduced an efficient method for computing the structure factors for each
sampling point. More recently, Navaza & Vernoslova (1995 ) introduced an ingenious fast Fourier transform based method for computing the final two- or three-dimensional translation function without
explicitly computing the structure factors as intermediate results. The target function for this fast translation function is the correlation coefficient between squared structure-factor amplitudes
Both the more conventional Fujinaga & Read (1987 ) type translation function and the fast translation function are implemented in CNS. Table 2 shows a comparison of the run times of the CNS
conventional translation function (CTF) and the CNS fast translation function (FTF) for a variety of symmetries, unit-cell sizes and resolution ranges. The rightmost column of Table 2 shows the
factor by which the fast translation function is faster than the conventional one. Depending on the symmetry, unit-cell dimensions and resolution range, the fast translation function is 200 to almost
500 times faster than the conventional search.
Space group Unit-cell parameters (Å) d[min ](Å) Time CTF (s) Time FTF (s) Factor
P2[1]2[1]2[1] a = 65.5, b = 72.2, c = 45.0 4 245 0.8 306
C222[1] a = 42.1, b = 97.1, c = 91.9 3 1700 8 210
C222[1] a = 64.1, b = 102.0, c = 187.0 4 3000 13 230
C222 a = 91.9, b = 168.0, c = 137.8 4.5 7850 17 460
P4[3]32 a = 272.8 6 1129644 2400 470
Because of this enormous increase in speed, the fast translation function has also found a use in the automatic heavy-atom search procedure in CNS (Grosse-Kunstleve & Brunger, 1999 ). The search
procedure consists primarily of alternating single-atom translation functions and PC refinements. This strategy is only practical if the fast translation function is used. CNS has been used by
independent groups to automatically locate up to 40 heavy-atom sites in the asymmetric unit (Walsh et al., 2000 ).
5. Summary
The CNS procedures that are presented in the previous sections can be combined into a powerful general strategy for solving difficult molecular-replacement problems.
• (i) Direct rotation searches of model domains. The systematic investigation of DeLano & Brünger (1995 ) shows that the direct rotation search has a high chance of finding the correct solutions
among the highest ranked points in the rotation function (after reduction by spatial cluster analysis) and has the ability to produce a recognizable signal even for relatively small subunits
(Figs. 1 and 2 ).
• (ii) PC refinement of the overall orientation and the interdomain angles. PC refinement enhances the discrimination between correct and incorrect rotation-function points by improving the search
models that are within 10–15° of the correct angular orientation (Fig. 4 ). Another consequence of the improved model quality is that the signal-to-noise ratio in the subsequent translation
function is enhanced (Fig. 5 ).
• (iii) Fast translation function. The implementation of the translation function of Navaza & Vernoslova (1995 ) in CNS is fast enough to be applied to a large number of putative rotation-function
solutions (e.g. testing 100 solutions is entirely feasible for typical macromolecular structures).
For routine molecular-replacement structure solutions, a highly optimized traditional rotation search as is implemented in the
program (Navaza, 1994
) will give the correct answer much faster than the direct rotation search in
. However, for more difficult cases, the unique combination of enhanced signal-to-noise ratio, spatial
cluster analysis
of the rotation function peaks, PC
and the fast translation function is a very attractive and much faster alternative when compared with full six-dimensional searches.
The time needed for the computation of the direct rotation function could be substantially reduced by using a well known optimization employed by several other programs (Castellano et al., 1992 ;
Kissinger et al., 1999 ; Glykos & Kokkinidis, 2000 ). In CNS, a full structure-factor calculation is carried out for each sampling point in the direct rotation search. Alternatively, a fine sampling
of the molecular transform and interpolation in reciprocal space could be employed. We expect that the resulting fast direct rotation search will be at least an order of magnitude faster. Therefore,
the general strategy outlined above will be even more practical.
6. Program availability
CNS is available online at https://cns.csb.yale.edu/ and is free of charge for academic users. The procedures that are discussed in this paper are implemented in the two standard input files
cross_rotation.inp (real-space rotation search and direct rotation search) and translation.inp (PC refinement, conventional translation function and fast translation function).
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© International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For
more information, click here.
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Physics for Problem-Solvers: Top Calculations You Need to Know
Physics plays a crucial role in understanding and solving real-world problems.
It provides us with the tools to analyze complex situations and make informed decisions.
From predicting weather patterns to designing safer vehicles, physics principles guide us in various fields.
Mastering key calculations in physics enhances your problem-solving skills significantly.
These calculations allow you to quantify relationships between different physical quantities.
When you grasp these concepts, you cultivate a deeper understanding of how the world operates.
This blog aims to present the top calculations essential for aspiring problem solvers in physics.
We will focus on fundamental equations that will empower you to tackle a range of challenges.
By engaging with these calculations, you will build confidence in your abilities.
Understanding these calculations can transform your approach to problem-solving.
You will find that physics is not merely about theory; it applies directly to practical situations.
By applying equations, you will begin to see the intricate dance of forces and motions around you.
As you delve into this blog, remember that the goal is to simplify complex problems.
Each top calculation will bring you one step closer to mastery in physics.
With practice, you will gain the analytical skills necessary to approach any physics problem with ease.
Stay engaged as we explore these essential calculations.
Each section will highlight a specific area of physics, providing examples and contexts.
These calculations will serve as your toolkit, empowering you to decipher the laws of nature.
Ultimately, mastering these calculations lays a solid foundation for further exploration in physics.
Whether in academia or industry, these tools prove invaluable.
You will emerge better equipped to challenge the status quo and innovate in your field.
Understanding Basic Concepts in Physics
Physics is a fundamental science that studies matter, energy, and their interactions.
It explores natural phenomena through measurement, experimentation, and mathematical analysis.
Various fields—including engineering, astronomy, and medicine—rely heavily on concepts derived from physics.
This discipline provides a framework for understanding the universe and the principles guiding it.
Relevance of Physics in Various Fields
• Engineering: Engineers apply principles of physics to design structures, machines, and systems.
For instance, structural engineers use mechanics to ensure buildings withstand forces.
• Astronomy: Astronomers study celestial bodies and the universe using physics.
The laws of physics help explain gravitational forces and planetary motion.
• Medicine: Medical physics involves using physics concepts in diagnosing and treating diseases.
Techniques like X-rays and MRIs depend on understanding radiation and electromagnetic waves.
• Environmental Science: Physics aids in understanding energy transfer processes in ecosystems.
This knowledge helps address issues like climate change and energy conservation.
• Technology: Innovations in technology, such as electronics and optics, stem from physical principles.
The development of smartphones illustrates the application of quantum mechanics and electromagnetism.
Importance of Fundamental Concepts
Four fundamental concepts in physics form the backbone of the discipline: force, energy, momentum, and mass.
Understanding these concepts is essential for analyzing physical systems and solving problems.
• Force: A force is an interaction that can change an object’s motion.
It is measured in newtons.
Understanding forces helps analyze how objects react to pushes and pulls.
• Energy: Energy is the capacity to do work or generate heat.
It appears in various forms, such as kinetic, potential, and thermal energy.
Recognizing energy transformations is crucial in problem-solving.
• Momentum: Momentum is the product of an object’s mass and velocity.
It is conserved in isolated systems.
Understanding momentum allows prediction of outcomes in collisions and other interactions.
• Mass: Mass measures the amount of matter in an object.
It influences an object’s resistance to acceleration and gravitational attraction.
A firm grasp of mass is vital for solving numerous physics-related problems.
Interconnection of Concepts
Force, energy, momentum, and mass are interconnected concepts that build a comprehensive understanding of physical systems.
These core principles contribute to problem-solving in physics, allowing for the breakdown and analysis of complex situations.
For example, when analyzing a car’s motion, one must consider the forces acting on it.
These forces cause changes in energy—transforming potential energy into kinetic energy as the car accelerates.
The momentum of the car changes as it speeds up, illustrating how mass and velocity interact.
This example highlights how understanding the interplay of these fundamental concepts allows effective problem-solving.
Applications of Fundamental Concepts in Problem-Solving
Applying the principles of force, energy, momentum, and mass can illuminate various phenomena in real life.
Problem-solving often involves finding solutions through calculations based on these concepts.
Here are some practical applications:
• Projectile Motion: When analyzing the trajectory of a thrown ball, students calculate the forces acting on it, evaluating energy transformations and momentum.
Using equations of motion helps predict the ball’s path.
• Mechanical Systems: In mechanical systems, like levers and pulleys, students apply force and energy concepts to calculate advantages and efficiencies.
Understanding these principles aids in designing effective machines.
• Collisions: In collision problems, momentum conservation principles allow predictions about objects before and after impacts.
Comparing masses and velocities helps determine outcomes, making collision analysis essential in physics.
• Work and Energy: Understanding the relationship between work and energy enables students to analyze forces acting in various situations.
Applications range from calculating energy used by machines to finding energy losses in systems.
Encouraging Problem-Solving Thinking
Learning physics encourages critical thinking and problem-solving skills.
Engaging with fundamental concepts allows students to dissect and analyze a multitude of real-world problems.
The analytical mindset developed through physics can be applied across various disciplines.
Problem-solving in physics fosters abilities such as:
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• Analytical Thinking: Evaluating complex systems holistically allows students to approach problems methodically.
Breaking down problems into manageable parts aids in finding efficient solutions.
• Quantitative Skills: Physics emphasizes measurements and calculations, developing students’ ability to work with data accurately.
Mastery of units, conversions, and formula application is essential.
• Critical Reasoning: Challenging accepted norms through evidence-based analysis is a key component of physics.
This encourages learners to question and explore concepts deeply.
• Collaboration: Physics often involves teamwork to solve problems.
Collaborating with peers enhances learning and fosters diverse approaches to finding solutions.
Understanding basic concepts in physics is pivotal for problem-solving.
The interplay of force, energy, momentum, and mass lays the groundwork for analyzing complex situations.
Mastering these principles boosts proficiency in solving real-world problems.
Physics not only elucidates natural phenomena, but also cultivates essential skills valuable in any field.
Clarity and comprehension in physics empower individuals to navigate the world of science and engineering effectively.
Essential Physics Formulas
Physics is a rich field filled with concepts that fundamentally shape our understanding of the universe.
For problem-solvers, mastering essential physics formulas is crucial.
This section will outline the top physics formulas that every problem solver should know, explaining their variables, units, and applications, along with useful tips for memorization.
Newton’s Second Law
Newton’s Second Law states:
F = ma
• F is the net force applied (in Newtons, N)
• m is the mass of the object (in kilograms, kg)
• a is the acceleration (in meters per second squared, m/s²)
This formula explains how the force acting on an object equals its mass multiplied by its acceleration.
It is foundational for dynamics and motion studies.
Understanding this formula helps in solving problems related to vehicle acceleration, gravitational forces, and many more applications.
Law of Universal Gravitation
The Law of Universal Gravitation is expressed as:
F = G(m₁m₂/r²)
• F is the gravitational force (in Newtons, N)
• G is the gravitational constant (6.674 × 10⁻¹¹ N m²/kg²)
• m₁ and m₂ are the masses of the objects (in kilograms, kg)
• r is the distance between the centers of the two masses (in meters, m)
This formula details the attractive force between two masses.
It finds applications in astrophysics, satellite motion, and understanding planetary orbits.
Kinetic Energy
Kinetic energy is defined by the formula:
KE = 1/2 mv²
• KE is the kinetic energy (in joules, J)
• m is the mass of the object (in kilograms, kg)
• v is the velocity of the object (in meters per second, m/s)
This formula shows that kinetic energy increases with the square of the velocity.
It applies to moving objects, from cars to flying balls.
Potential Energy
The potential energy formula is written as:
PE = mgh
• PE is the potential energy (in joules, J)
• m is the mass (in kilograms, kg)
• g is the acceleration due to gravity (approximately 9.81 m/s²)
• h is the height above ground (in meters, m)
This formula indicates the energy stored by virtue of an object’s position.
It is central in problems related to heights, such as in roller coasters and energy conservation.
Conservation of Energy
The principle of conservation of energy states:
KE_initial + PE_initial = KE_final + PE_final
This formula emphasizes that total mechanical energy in a closed system remains constant.
It applies to systems like pendulums and roller coasters, helping in the analysis of energy transformations.
Ohm’s Law
For electrical contexts, Ohm’s Law is introduced as:
V = IR
• V is the voltage (in volts, V)
• I is the current (in amperes, A)
• R is the resistance (in ohms, Ω)
This law explains the relationship between voltage, current, and resistance in electrical circuits, making it essential for electronics and electrical engineering.
The Ideal Gas Law
The Ideal Gas Law combines several gas laws into one equation:
PV = nRT
• P is the pressure (in pascals, Pa)
• V is the volume (in cubic meters, m³)
• n is the number of moles of gas (in moles, mol)
• R is the universal gas constant (8.314 J/(mol·K))
• T is the temperature (in kelvins, K)
This equation describes the behavior of ideal gases, allowing problem solvers to analyze gas-related phenomena in chemistry and physics.
First Law of Thermodynamics
The First Law of Thermodynamics is depicted as:
ΔU = Q – W
• ΔU is the change in internal energy (in joules, J)
• Q is the heat added to the system (in joules, J)
• W is the work done by the system (in joules, J)
This law presents the principle of energy conservation in thermodynamic systems, guiding problem-solving in heat transfer and engines.
Tips for Memorization
Memorizing formulas can be daunting, but effective techniques can simplify the process:
• Understand, don’t memorize: Grasping the concepts helps retain the formulas more naturally.
• Use flashcards: Create flashcards with the formula on one side and its explanation on the other.
• Practice regularly: Solve problems using these formulas frequently to reinforce their memory.
• Create mnemonics: Develop catchy phrases to remember variable letters in complex formulas.
• Group study: Explain concepts and formulas to a peer to enhance understanding.
These essential physics formulas play a pivotal role in problem-solving.
Whether in theoretical studies or real-world applications, they provide the tools necessary to understand complex interactions within our universe.
As you engage with these formulas, keep practicing, and integrate them into your daily problem-solving scenarios.
With time, they will become second nature, empowering you to tackle any physics problem that comes your way.
Read: 7 Essential Physics Calculations for Engineers in Renewable Energy
Dimensional Analysis: A Problem-Solving Tool
Definition and Importance of Dimensional Analysis
Dimensional analysis serves as a fundamental method in physics, allowing problem-solvers to verify equations and calculations.
It examines the dimensions of physical quantities rather than their numerical values.
This technique enables a deeper understanding of the relationships between different physical variables.
Dimensional analysis offers several benefits:
• It verifies the correctness of equations.
• It helps in converting units from one system to another.
• It allows simplification of complex problems.
• It aids in identifying relationships between variables.
By ensuring dimensional consistency, physicists can validate their work and prevent mistakes in calculations.
A correct dimensional analysis can transform a complicated situation into manageable parts, often leading to insightful solutions.
Step-by-Step Guide on Performing Dimensional Analysis
Performing dimensional analysis involves systematic steps.
Here’s a comprehensive guide to help you carry out this process effectively:
1. Identify the Variables: List all variables involved in the problem.
Write down their units, such as meter (m), kilogram (kg), or second (s).
2. Express Each Variable in Base Units: Convert all variables to base units.
Common base units include length (L), mass (M), and time (T).
3. Set Up the Equation: Write down the equation you want to analyze.
It can involve various physical quantities, like velocity or acceleration.
4. Check Dimensional Consistency: Analyze each term in the equation to ensure that both sides have the same dimensional formula.
Each term should reduce to the same base units.
5. Dimensional Homogeneity: If the dimensions match, the equation verifies dimensionally.
If they do not match, re-evaluate the equation.
Example of Dimensional Analysis
Let’s illustrate dimensional analysis with a practical example.
Consider the equation for kinetic energy:
K.E. = 1/2 mv²
First, identify the variables:
• m (mass) has units of kilograms (kg).
• v (velocity) has units of meters per second (m/s).
Now, express each variable in base units:
Substituting the velocity into the equation gives:
K.E = 1/2 M (L/T)²
Next, simplify:
K.E = 1/2 M (L²/T²)
The dimensional formula for kinetic energy becomes:
[K.E] = M L² T⁻²
Thus, we have successfully verified that the units of kinetic energy match the expected outcome in dimensional analysis, validating the equation’s correctness.
Common Mistakes to Avoid
In dimensional analysis, several common mistakes can lead to incorrect conclusions.
Being aware of these pitfalls can greatly improve your problem-solving success:
• Neglecting Unit Conversions: Always convert units before analysis.
Mismatched units can produce inaccurate results.
• Assuming Dimensionality Equals Numerical Value: Dimensions do not equate to numbers.
Ensure you analyze units, not just numeric coefficients.
• Ignoring Constants: Constants contribute to dimensional analysis but can be overlooked.
Check all components in an equation, including constants.
• Overcomplicating: Simplify when possible.
Including unnecessary dimensions complicates the analysis without adding value.
By avoiding these mistakes, problem-solvers can maintain dimensional consistency, which leads to correct answers.
How Dimensional Consistency Leads to Correct Problem Solving
Dimensional consistency serves as a guiding principle in physics.
It leads to accurate problem-solving in various ways:
• Validation of Theoretical Models: Many theories and models in physics rely on equations’ dimensional consistency.
This technique supports their validity.
• Error Detection: By confirming that dimensions match, you can quickly identify errors in calculations and reasoning.
• Empowering Simplification: Applying dimensional analysis can simplify complex issues.
Analyzing dimensions helps identify which variables matter most.
Incorporating dimensional analysis into your problem-solving toolkit enhances your ability to tackle challenging physics questions.
It minimizes errors, inspires confidence, and strengthens your understanding of fundamental principles.
Ultimately, mastery of dimensional analysis is invaluable for students and professionals alike.
By leveraging this powerful tool, you can navigate the complexities of physics with greater ease.
Embrace dimensional analysis to unlock new levels of problem-solving success and deeper insights into physical phenomena.
Read: Physics Calculations Demystified: Essential Tips for Students
Kinematic Equations in Motion Problems
Understanding kinematics provides essential tools for analyzing motion.
Kinematics deals with the relationships between displacement, velocity, acceleration, and time.
By applying kinematic equations, we can predict an object’s future position based on its initial conditions.
These equations serve as foundational tools in physics, particularly in solving motion-related problems.
Overview of Kinematics and Its Significance
Kinematics focuses on the motion of objects without considering the forces that cause the motion.
This branch of mechanics provides insight into how objects move, which is crucial in various fields.
Whether discussing vehicles on a highway or a ball thrown in the air, kinematic principles apply universally.
The significance of kinematics lies in its ability to simplify complex motion.
By using kinematic equations, we can break down motion into understandable components.
This framework allows us to calculate an object’s path, speed, and acceleration with ease.
Breakdown of the Kinematic Equations
Several key kinematic equations help describe motion in one dimension.
These equations relate the four primary variables: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
Below is a breakdown of the essential kinematic equations:
1. First Equation: v = u + at
2. Second Equation: s = ut + (1/2)at²
3. Third Equation: v² = u² + 2as
4. Fourth Equation: s = (u + v)/2 * t
Each equation can be used individually based on the information available in a given problem.
Understanding how to manipulate these equations is crucial for solving various motion problems.
Equation Explanations
Let’s examine each equation in detail to clarify how they function:
1. v = u + at: This equation allows us to calculate the final velocity of an object.
Here, v is the final velocity, u is the initial velocity, a is acceleration, and t is time.
For example, if a car accelerates from rest (u = 0 m/s) at 2 m/s² for 5 seconds, its final velocity is v = 0 + (2)(5) = 10 m/s.
2. s = ut + (1/2)at²: This equation calculates the displacement of an object when it has constant acceleration.
It combines the initial velocity and the distance covered due to acceleration.
For instance, if an object moves with an initial speed of 3 m/s and accelerates at 1 m/s² for 4 seconds, the displacement is s = (3)(4) + (1/2)(1)(4²) = 12 + 8 = 20 meters.
3. v² = u² + 2as: This equation relates the final velocity, initial velocity, displacement, and acceleration.
It is useful when time is not specified.
Suppose a ball drops from a height with no initial velocity (u = 0) and accelerates at 9.8 m/s².
If the displacement is 20 m, the final velocity can be calculated as v² = 0 + 2(9.8)(20), yielding v = √392 = 19.8 m/s.
4. s = (u + v)/2 * t: This equation finds the average velocity during motion when both initial and final velocities are known.
For example, if a car starts from 20 m/s and slows down to 10 m/s over 10 seconds, the average velocity is (20 + 10)/2. Hence, s = 15 * 10 = 150 meters.
Example Problems
Let’s apply these equations to a few example problems:
Example 1: Free Fall
A rock is dropped from a height of 45 meters.
Calculate the time it takes to hit the ground.
Using the second equation, we know:
s = ut + (1/2)at²
Here, u = 0, s = 45 m, and a = 9.8 m/s²:
45 = 0*t + (1/2)(9.8)t²
45 = 4.9t²
t² = 45/4.9
t² ≈ 9.18
t ≈ 3.03 seconds
Thus, the rock hits the ground in approximately 3.03 seconds.
Example 2: Projectile Motion
A ball is thrown horizontally from a height of 25 meters with an initial velocity of 10 m/s.
Determine how far it travels horizontally before hitting the ground.
First, calculate the time using the free fall equation:
s = (1/2)at²
25 = (1/2)(9.8)t²
25 = 4.9t²
t² = 25/4.9
t² ≈ 5.1
t ≈ 2.26 seconds
Next, calculate the horizontal distance:
horizontal distance = velocity * time
horizontal distance = 10 m/s * 2.26 s = 22.6 meters.
Real-Life Applications
Kinematic equations apply to numerous real-world situations, enhancing our ability to understand motion:
• Projectile Motion: Understanding how objects move when thrown or projected helps fields like sports science and engineering.
• Vehicle Dynamics: Engineers use kinematics to design safer vehicles.
They calculate braking distances and acceleration to ensure optimal performance.
• Sports Analysis: Coaches analyze athletes’ movements in sports.
They leverage kinematic principles to optimize techniques and improve performance.
• Aerospace Engineering: Kinematic equations are crucial in launching rockets and navigating flight paths.
Engineers calculate trajectories to ensure successful missions.
Kinematic equations serve as valuable tools for anyone studying physics or engineering.
Understanding these principles enriches our awareness of motion in our daily lives.
By mastering kinematics, we unlock the secrets of motion that govern our world.
Read: How to Calculate and Solve for Reaction: Lift Falls Freely | Motion
Energy Calculations: Kinetic and Potential Energy
Understanding energy calculations forms a crucial part of physics, particularly in mechanics.
The two primary forms of energy in mechanics are kinetic energy and potential energy.
Both of these energy types help explain how objects move and interact within their environments.
Kinetic Energy
Kinetic energy pertains to the energy of motion.
Anything that moves possesses kinetic energy based on its mass and velocity.
Mathematically, we define kinetic energy (KE) using the formula:
In this formula:
• KE = kinetic energy
• m = mass of the object (in kilograms)
• v = velocity of the object (in meters per second)
Let’s consider a practical example. Imagine a car with a mass of 1,000 kg traveling at a velocity of 20 m/s.
To find the kinetic energy:
• Apply the formula: KE = ½ mv²
• Insert values: KE = ½ (1000 kg)(20 m/s)²
• Calculate: KE = ½ (1000 kg)(400 m²/s²)
• Result: KE = 200,000 J (joules)
This example illustrates how kinetic energy increases exponentially with velocity.
Doubling the speed increases the kinetic energy by a factor of four.
Potential Energy
Potential energy (PE), on the other hand, refers to stored energy based on an object’s position or configuration.
The most common form of potential energy in mechanics is gravitational potential energy.
We can calculate gravitational potential energy using the formula:
PE = mgh
In this equation:
• PE = potential energy
• m = mass of the object (in kilograms)
• g = acceleration due to gravity (approximately 9.81 m/s²)
• h = height above a reference point (in meters)
For example, consider a 10 kg rock perched on a cliff 5 meters high.
To find its gravitational potential energy, use this formula:
• Apply the formula: PE = mgh
• Insert values: PE = (10 kg)(9.81 m/s²)(5 m)
• Calculate: PE = 10 kg * 49.05 m²/s²
• Result: PE = 490.5 J
This scenario shows how potential energy also depends on the height of the object relative to a reference point.
Significance in Mechanics
Both kinetic and potential energy are fundamental concepts in mechanics.
They describe the energy transitions that occur during motion and changes in position.
Understanding these energy forms allows us to analyze real-world situations effectively.
• Kinetic energy is vital for analyzing moving objects, such as vehicles or sports balls.
It helps in predicting outcomes during collisions or throws.
• Potential energy explains how objects store energy due to their position, such as in roller coasters.
This understanding helps in designing thrilling rides safely.
Conservation of Energy Principle
The principle of conservation of energy states that energy cannot be created or destroyed.
It transforms from one form to another but maintains a constant total amount in an isolated system.
This principle is essential in problem-solving throughout physics.
In practical terms, the conservation of energy can manifest in several ways.
Here are some examples:
• When an object falls, gravitational potential energy transforms to kinetic energy as it accelerates.
• A swinging pendulum first demonstrates potential energy at its highest point and then kinetic energy at the lowest point.
Let’s analyze a real-world scenario involving a roller coaster.
At the top of the initial hill, the coaster has maximum potential energy and minimum kinetic energy.
As it descends, potential energy converts to kinetic energy, causing the coaster to speed up and thrill riders.
To understand how to apply conservation principles mathematically, let’s break down a situation:
• Consider a roller coaster with a height of 30 meters.
At the top, its potential energy equals PE = mgh.
• If the coaster has a mass of 500 kg, then:
PE = 500 kg * 9.81 m/s² * 30 m = 147,150 J.
• As the coaster descends, this potential energy transforms into kinetic energy.
At the bottom, potential energy equals zero.
• Thus, the kinetic energy at the lowest point equals 147,150 J, demonstrating energy conservation.
Implications for Problem-Solving
Utilizing energy concepts fosters effective problem-solving in various scenarios.
To solve problems involving energy calculations, follow these general steps:
• Identify the energy forms present: kinetic and potential.
• Apply appropriate formulas for each energy type.
• Use conservation of energy principles to find unknown variables.
• Analyze the results and ensure consistency with physical reality.
This structured approach streamlines problem-solving, providing clarity and accuracy.
Proper mastery of these calculations promotes confidence in tackling physics challenges.
Kinetic and potential energy calculations serve as fundamental tools in mechanics.
Mastery of these concepts empowers you to analyze motion and energy transformations effectively.
Apply these principles consistently, and your problem-solving skills will flourish in physics.
Read: How to Calculate and Solve for Reaction: Lift Moves Down | Motion
Understanding Forces and Their Calculations
In the study of physics, understanding forces is essential for analyzing motion and predicting outcomes in various scenarios.
Forces are interactions that can change the motion of an object.
Here, we explore the different types of forces, how to calculate net force, and apply these concepts through practical examples.
Types of Forces
Forces can be categorized into several types, each playing a distinct role in physical interactions. Here’s an overview of the most common types:
• Gravitational Force: This is the force of attraction that acts between two masses.
It pulls objects toward each other, with the Earth exerting this force on all objects.
• Normal Force: This force acts perpendicular to the surfaces in contact.
It balances gravitational force on objects resting on surfaces.
• Frictional Force: Friction opposes the relative motion between two surfaces.
It acts parallel to the surfaces and can vary in magnitude.
• Tension Force: This force is transmitted through a rope, string, or cable when it is pulled tight.
It acts along the length of the string.
• Applied Force: This is a force that is applied to an object from an external source, such as a person pushing a box.
• Air Resistance: This is the force acting against the motion of an object as it travels through the air.
It increases with speed.
• Spring Force: This force is exerted by a compressed or stretched spring.
It obeys Hooke’s Law, which states that the force is proportional to the displacement.
Calculating Net Force
The net force is the vector sum of all forces acting on an object.
Understanding how to calculate the net force is crucial.
It determines the overall effect of individual forces on an object’s motion.
The net force can be calculated using the following steps:
1. Identify all forces acting on the object. Determine the direction and magnitude of each force involved.
2. Assign a coordinate system. This usually involves defining one direction as positive (usually right or up).
3. Break down forces into components. If forces act at angles, resolve them into horizontal and vertical components.
4. Sum up all the forces. Apply the formula: F[net] = F[1] + F[2] + … + F[n]. Consider the signs based on your coordinate system.
The net force is particularly important because it dictates the acceleration of an object according to Newton’s second law, which states:
F[net] = m * a, where m is mass and a is acceleration.
Example Problems
To solidify the understanding of forces and their calculations, let’s consider some illustrative examples.
Simple Gravitational Force
Imagine a 10 kg mass hanging from a rope.
Calculate the gravitational force acting on it and identify the tension in the rope.
• Start with the formula for gravitational force: F[gravity] = m * g, where g is approximately 9.81 m/s².
• F[gravity] = 10 kg * 9.81 m/s² = 98.1 N (downward).
• Since the mass is at rest, the tension in the rope equals the gravitational force: T = 98.1 N (upward).
Friction on a Surface
Consider a box on a horizontal surface with a mass of 5 kg.
If the coefficient of kinetic friction is 0.3, calculate the frictional force.
• First, calculate the normal force, which equals the gravitational force: F[normal] = m * g = 5 kg * 9.81 m/s² = 49.05 N.
• Next, apply the friction formula: F[friction] = μ * F[normal].
• Thus, F[friction] = 0.3 * 49.05 N = 14.72 N
Net Force with Multiple Forces
A car experiences three forces: a 200 N engine force forward, a 50 N frictional force backward, and a 30 N air resistance backward.
Calculate the net force.
Identify the forces:
• Engine force (F[engine]) = +200 N
• Frictional force (F[friction]) = -50 N
• Air resistance (F[air]) = -30 N
• Now, sum the forces: F[net] = 200 N – 50 N – 30 N = 120 N (forward).
• This tells us the car accelerates forward due to the net force of 120 N.
Understanding forces and their calculations is foundational in physics.
Forces dictate the movement and behavior of objects in the universe.
Whether discussing gravitational pull or forces resulting from friction, mastering these principles empowers problem-solving capabilities.
Familiarity with these concepts enables students and enthusiasts to analyze real-world situations effectively.
By applying these principles to varied scenarios, one can develop a solid grasp of the dynamics governing motion.
Real-World Applications of Physics Calculations
Physics calculations play a critical role in various fields, including engineering, astronomy, and many aspects of daily life.
Understanding these applications enhances our comprehension and appreciation of the universe.
In this section, we will explore how physics impacts these fields and the innovative outcomes resulting from problem-solving.
Furthermore, we will discuss the importance of simulations and modeling in physics.
Applications in Engineering
Engineering heavily relies on physics calculations to design and create structures and machines.
Here are key ways physics influences engineering:
• Structural Engineering: Engineers calculate forces, stresses, and strains to ensure a building’s strength.
They use physics to determine stability under various conditions, such as wind and earthquakes.
• Mechanical Engineering: Understanding motion and forces is crucial in designing moving machines.
Engineers apply Newton’s laws to create efficient and safe machinery.
• Aeronautical Engineering: Physics governs the principles of lift and drag.
Engineers use calculations to design aircraft that can efficiently navigate through the atmosphere.
• Civil Engineering: Physics helps in assessing the integrity of bridges and dams.
Engineers perform calculations to ensure these structures can withstand environmental forces.
Each of these examples illustrates how physics calculations lay the foundation for innovative solutions in engineering projects.
Applications in Astronomy
Astronomy showcases the profound influence of physics calculations on our understanding of the universe.
Here are several applications within the field:
• Orbital Mechanics: Physicists use calculations to predict celestial bodies’ paths.
They apply gravitational laws to determine orbits of planets and satellites.
• Distance Measurement: Astronomers calculate distances to stars using parallax and standard candles.
Physics enables these measuring techniques, enhancing our knowledge of the cosmos.
• Cosmology: Theoretical physicists model the universe’s expansion.
They rely on equations derived from Einstein’s general relativity to understand the universe’s evolution.
• Astrophysics: Physicists study the behavior of matter and energy in space.
They apply principles from thermodynamics and electromagnetism to explain celestial phenomena.
These applications demonstrate how physics calculations broaden our understanding of the universe and lead to groundbreaking discoveries.
Applications in Daily Life
Physics calculations also influence our everyday experiences.
Below are some practical examples:
• Transportation: Cars and public transport systems rely on physics for safe operation.
Engineers calculate acceleration, braking forces, and stability to enhance vehicle safety.
• Sports: Athletes utilize physics to improve their performance.
Understanding angles, velocities, and forces helps improve techniques in sports like basketball, soccer, or swimming.
• Energy Consumption: Understanding thermodynamics helps us manage energy use at home.
Calculating efficiency leads to better appliances and sustainable practices.
• Medical Technology: Devices like MRI machines and ultrasound equipment depend on physics principles.
These technologies use waves and magnetic fields for imaging and diagnosis.
These everyday applications illustrate how physics calculations contribute to enhancing our quality of life.
Case Studies of Breakthrough Innovations
Innovations often arise from applying physics problem-solving techniques in real-world challenges.
Below are a few notable case studies:
• The Internet: The development of fiber optics and telecommunications relies on principles of light and electromagnetism.
Engineers used these physics concepts to revolutionize global communication.
• Renewable Energy Sources: The design of solar panels and wind turbines comes from physics principles.
Understanding energy conversion and efficiency has driven advancements in sustainable energy.
• Medical Imaging: The invention of MRI and CT scans stemmed from applying physics in medicine.
These technologies have immensely improved diagnostic capabilities.
• GPS Technology: The Global Positioning System uses principles of relativity and signal timing.
Engineers calculated satellite positions to provide accurate location services.
These case studies exemplify how physics-driven innovations solve complex problems and improve our lives.
The Role of Simulations and Modeling
Simulations and modeling serve as essential tools in understanding complex physical phenomena.
Here are key aspects of their importance:
• Visual Representation: Simulations allow scientists and engineers to visualize phenomena that are difficult to observe directly, such as fluid dynamics in weather patterns.
• Predictive Capability: Models enable predictions of future states or outcomes based on current data, such as predicting vehicle behavior in crash simulations.
• Experimentation: Simulations provide a safe environment for virtual experimentation.
Researchers can test hypotheses without real-world consequences, which is invaluable in fields like materials science.
• Complex System Analysis: Many systems involve numerous interacting variables.
Models simplify these interactions, allowing for better analysis and understanding.
Through simulations and modeling, physics calculations extend our ability to investigate and understand complex systems efficiently.
In fact, physics calculations have significant applications across various fields.
From engineering and astronomy to daily life, these calculations shape our world.
The resulting innovations improve safety, efficiency, and quality of life.
As we continue to employ simulations and modeling, our understanding of complex physical phenomena will only expand.
Thus, problem-solving through physics remains essential to advancing technology and enhancing human experiences in the modern world.
Mastering key physics calculations is essential for effective problem-solving.
These calculations provide a foundation for understanding complex concepts.
They also enable students and professionals to tackle real-world challenges with confidence.
Practicing these calculations helps reinforce your understanding of physics principles.
Regularly applying them in various situations enhances your analytical skills.
The more you practice, the more intuitive these calculations become.
Consider scenarios like projectile motion or energy conservation.
In each case, mastering calculations allows for accurate predictions and practical applications.
Whether in physics exams or engineering projects, these skills prove invaluable.
Embrace the learning journey in physics.
Each calculation mastered builds upon previous knowledge.
This cumulative learning strengthens your overall grasp of the subject.
Furthermore, the principles of physics extend into everyday life.
Understanding how forces interact or how energy is transformed influences decision-making.
This knowledge shapes our interaction with technology and the environment.
In short, continue practicing and applying physics calculations.
They are more than mere numbers; they represent the language of the universe.
Committing to this practice not only enhances your academic performance but enriches your understanding of the world around you.
Remember, the journey in physics is continuous and rewarding.
Each step taken unveils new insights and fosters curiosity.
Keep exploring, questioning, and solving problems, as these experiences will shape your future endeavors.
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Main Content
Obtain equality constraint arrays from portfolio object
Use the getEquality function with a Portfolio, PortfolioCVaR, or PortfolioMAD object to obtain equality constraint arrays from portfolio objects.
For details on the respective workflows when using these different objects, see Portfolio Object Workflow, PortfolioCVaR Object Workflow, and PortfolioMAD Object Workflow.
Obtain Equality Constraints for a Portfolio Object
Suppose you have a portfolio of five assets and you want to ensure that the first three assets are exactly 50% of your portfolio. Given a Portfolio object p, set the linear equality constraints and
obtain the values for AEquality and bEquality:
A = [ 1 1 1 0 0 ];
b = 0.5;
p = Portfolio;
p = setEquality(p, A, b);
[AEquality, bEquality] = getEquality(p)
AEquality = 1×5
Obtain Equality Constraints for a PortfolioCVaR Object
Suppose you have a portfolio of five assets and you want to ensure that the first three assets are 50% of your portfolio. Given a PortfolioCVaR object p, set the linear equality constraints and
obtain the values for AEquality and bEquality:
A = [ 1 1 1 0 0 ];
b = 0.5;
p = PortfolioCVaR;
p = setEquality(p, A, b);
[AEquality, bEquality] = getEquality(p)
AEquality = 1×5
Obtain Equality Constraints for a PortfolioMAD Object
Suppose you have a portfolio of five assets and you want to ensure that the first three assets are 50% of your portfolio. Given a PortfolioMAD object p, set the linear equality constraints and obtain
the values for AEquality and bEquality:
A = [ 1 1 1 0 0 ];
b = 0.5;
p = PortfolioMAD;
p = setEquality(p, A, b);
[AEquality, bEquality] = getEquality(p)
AEquality = 1×5
Input Arguments
obj — Object for portfolio
Object for portfolio, specified using Portfolio, PortfolioCVaR, or PortfolioMAD object. For more information on creating a portfolio object, see
Data Types: object
Output Arguments
AEquality — Matrix to form linear equality constraints
Matrix to form linear equality constraints, returned as a matrix for a Portfolio, PortfolioCVaR, or PortfolioMAD input object (obj).
bEquality — Vector to form linear equality constraints
Vector to form linear equality constraints, returned as a vector for a Portfolio, PortfolioCVaR, or PortfolioMAD input object (obj).
You can also use dot notation to obtain the equality constraint arrays from portfolio objects.
[AEquality, bEquality] = obj.getEquality;
Version History
Introduced in R2011a
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10 Dollars an Hour Is How Much a Year? (Example Budget Included)
How Much Is 10 Dollars an Hour per Year?
To get the final figure you are making every year, let us assume you are working 40 hours a week. This would mean you are working 2,080 hours every year since 40 multiplied by 52 weeks is 2,080.
Assuming you get 2 weeks of paid leave and are working 2,080 hours every year, you have to multiply your hourly wage by the number of hours you are working.
So, $10 multiplied by 2,080= $20,800 yearly income.
What if you do not have any paid leave?
If you do not have the normal 2 weeks of paid leave, you are working 50 weeks in a year.
So to get to your gross income in a year, multiply 50 weeks by 40 hours first to get your total working hours. The total is 2,000 working hours.
So, your gross income is going to be 2,000 working hours into your hourly wage of $10 which is = $20,000
How many working days in 2020?
Taking the total number of days into account will allow you to calculate your income with more accuracy.
In the year 2020, there are 366 days in total since it is a leap year. In total there are 262 working days out of this.
To calculate your total income, multiply your total working hours in the year by your wage. So in 2020, your number of working hours is 262 multiplied by 8 (hour work day) which is 2,096.
So, your total income is going to be 2,096 multiplied by $10 which is= $20,960
10 Dollars an Hour Is How Much a Year After Taxes?
So if you want to calculate the total net income you have to spend each year after taxes, you are going to need to take your total gross income which in this case is $20,800 and minus any deductibles
(such as the standard deductible of $12,000 or any other itemized deductibles you my have).
After this, you calculate the percentage of tax you are going to need to pay on that income and come to the final amount.
The United States works on a progressive tax system which means the more you earn, the more you will pay in taxes. Depending on your status (single, married, head of household), your tax benefits can
So, what is 10 an hour annually after tax?, here is an example: In the state of Massachusetts, your income after tax would be :
If you make 10 an hour how much is that after taxes? Annual net income after taxes (assuming single status): $17,521 on an annual basis.
What is $10 an hour salary after taxes monthly take-home salary? $1,460 on average.
If you want to calculate your income after taxes quickly, you can use this tool that allows you to input your status (single, married), location, and gross income to give you your net income after
This is a video you can watch if you want to learn more about the U.S tax system and how it works.
Disclosure: This post may contain affiliate links. You can read the full disclosure here.
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10 Dollars an Hour Is How Much a Week?
If you are wondering ‘how much is 10 dollars an hour 40 hours a week’, you are going to need to multiply 40 by 10 which is $400 on average.
This is your average gross income every week if you make $10 an hour weekly. ($400)
10 Dollars an Hour 20 Hours a Week?
To calculate how much you would be earning part-time, multiply 20 by $10 which is $200 every week in income.
How Much Is $10 an Hour per Month?
Your income on a monthly basis can be calculated by multiplying the number of hours you work on a monthly basis by your hourly wage.
In this case, let us assume you are working 40 hours a week. If you have an average of 2,080 working hours a year (52 weeks into 40), divide that by 12 to get your monthly average which is 174 hours
on average.
Now multiply your 174 hours by your salary. You are making an average of $1,740 every month.
How Much Is $10 an Hour per Day?
$10 an hour per day would be around $80 if you are working an 8 hour shift. If you are working a double (16 hours), it would $160 made in one shift.
10 an Hour Is How Much Biweekly?
Assuming you are working 40 hour weeks, two weeks of paid work would include 80 hours.
Multiply the number of hours you have worked with your hourly wage. So, 80 multiplied by $10 is $800.
On average, you will be making $800 biweekly.
10 an Hour Is How Much a Month After Taxes?
We have already calculated your monthly average to be $1,740 if you are making $10 an hour.
So, to get your monthly income after taxes, you can use a tool like this one. For example, your monthly take-home salary after taxes is $1,460 in the state of Massachusetts.
This will differ based on state laws and how much your tax rate is but you can use it as an average.
Is 10 Dollars an Hour Good?
10 dollars an hour is not a wage you can live on in most cases but it is enough for a teenager who needs to make some extra cash or someone who is single and living at home with no rent.
Using the Dave Ramsey Household Budget Percentages, this is an average budget breakdown if you earn $10 an hour. (Take-home pay of $1,460 per month on average).
Category Budget Percent
Total Income $1,460 —
Savings $146 10%
Food $146 10%
Utilities $73 5%
Giving $146 10%
Housing $365 25%
Transport $146 10%
Personal Spending $73 5%
Health $73 5%
Recreation $73 5%
Insurance $146 10%
Misc Expenses $73 5%
You can also create a budget of your own using my free budget planner:
Is $10 an Hour Good for a Teenager?
For a teenager who does not have many bills to pay and lives at home, $10 an hour should be enough for buying most of the additional stuff you may need or want.
This obviously depends on which state you are living in and what your situation is like but on average $10 an hour is fine for a first job and should be enough for a teenager who only needs extra
How to Make $10 an Hour Online?
There are a couple of ways you can make money online and make some extra cash on the side. Some of my favorites would be freelancing and starting a blog.
A couple of other ways you can make money:
What Jobs Pay 10 an Hour?
Depending on which state you live in, your minimum wage is going to differ. In a lot of states, the minimum wage is already more than $10.
To make sure you are earning more than $10 an hour, you can use job boards like Indeed to filter out anything that pays less.
A few jobs that would ideally pay more than $10 would be a customer sales representative, work from home jobs teaching English, data entry clerks, transcribers, etc.
These are also jobs that do not require degrees and ordinarily need little to no work experience.
Can You Live off 10 Dollars an Hour?
This is a pretty tight budget to live off of and you would have to live with roommates or your parents to make it work but it is doable.
I would definitely suggest trying to get a better more high paying job or even a side job to get you some extra cash.
This is doable in the short term but for long term, you need to strive towards a higher income that will allow you to buy a home, save, and invest.
10 Dollars an Hour (Summary Table of Calculations)
Time Period Income
Year (52 weeks) $20,800
Year (50 weeks) $20,000
Year (2020) $20,960
Month $1,740
Biweekly $800
Week (40 hour week) $400
Week (20 hour week) $200
Day (8 hours) $80
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How do you determine whether the function f(x) = 2x^3 + 3x^2 - 432x is concave up or concave down and its intervals? | HIX Tutor
How do you determine whether the function # f(x) = 2x^3 + 3x^2 - 432x # is concave up or concave down and its intervals?
Answer 1
f(x) is concave down on #(-oo;-1/2)# and concave up on #(-1/2;oo)#
You have to investigate the sign of the second derivative around its zero points (inflection points).
#f'(x)=6x^2+6x-432# #f''(x)=12x+6#
#therefore f''(x)=0iffx=-1/2#
#therefore AAx<-1/2=>f''(x)<0# and hence the concavity is down.
#therefore AAx >-1/2=>f''(x)>0# and hence the concavity is up.
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Answer 2
To determine whether the function ( f(x) = 2x^3 + 3x^2 - 432x ) is concave up or concave down and its intervals, you need to find the second derivative of the function.
First, find the first derivative of ( f(x) ), then find the second derivative.
Then, examine the sign of the second derivative:
• If the second derivative is positive for a certain interval, the function is concave up on that interval.
• If the second derivative is negative for a certain interval, the function is concave down on that interval.
The intervals where the concavity changes (i.e., where the second derivative changes sign) are the points of inflection for the function.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Triangles within Triangles
Can you find a rule which connects consecutive triangular numbers?
This diagram shows how the first triangular number can be added to 3 copies of the second triangular number to make the fourth triangular number:
That is: $$ T_1 + 3 \times T_2 = T_4 $$ Here is a diagram showing how the second and third triangular numbers can be combined to make the sixth triangular number:
$$ T_2 + 3 \times T_3 = T_6 $$ Can you generalise this rule?
Can you find a rule in terms of $ T_n $ and $T_{n+1}$?
Getting Started
It is usefult to be able to recreate these patterns on square dotty paper.
Is the way the triangles fit together always going to work with two consecutive triangular numbers?
Yuu might find it useful to look at the problem on
"Sequences and Series"
Student Solutions
Well done Tom, from Finham Park School, for clear use of notation :
Whenever you add 3 triangles ( as in $T_2$ ) together with a triangle one size smaller ( as in $T_1$ ), a new triangle is formed ( $T_4$ ) , twice the height of the triangle which was used three
times ( $T_2$ ) .
The smaller triangle can be called $T_n$ , while the 3 triangles one size up can be called $T_{n+1}$.
One of the $T_{n+1 }$ joins with the $T_n $ to form a square of side length $n+1$ .
The two remaining $T_{n+1}$ fit to that square producing a large triangle that has a height twice that of $T_{n+1}$ ,
So the sum of all four triangles is the triangle $T_{2(n+1)}$
So $T_n $ + $3T_{n+1} =T_{2(n+1)}$ or, if you prefer, $T_{2n+2}$
Teachers' Resources
Why not encourage pupils to discover rules of their own?
By using isometric paper the triangular numbers can be represented as equilateral triangles wich give scope to investigating their connection with hexagonal numbers.
This problem links to "
Triangles within Squares
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Code From Papers
Sampling ratio for multilevel SEM
Here is R code used for Monte Carlo simulations. This relies heavily on the MplusAutomation package in R. An example Mplus input file for part 1: data generation can be found here, while an example
Mplus input file for part 2: data analysis can be found here. For the illustrative example, the doubly latent model can be estimated in both Mplus and R using this data.
Moderated nonlinear factor analysis for SEM
Here is R code used for all steps of conducting moderated nonlinear factor analysis using example data.
Propensity score matching and weighting
This provides R code for a series of Monte Carlo simulations comparing propensity score matching and weighting methods for achieving covcariate balance in observational studies, with an interest on
varying levels of treatment exposure.
Statistical power for cluster randomized trials
Here is R code used to conduct a Monte Carlo simulation of statistical power for cluster randomized controlled trials with clusters of varying size.
Example: Missing data
This provides an example of R code to handle missing data (see the PDF version here , as well as the background reading).
Example: Propensity score matching and weighting
This provides an example of R code using matching and weighting for causal inference (see the PDF version here).
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Monitor The Internet | Is the Internet Down?
Monitor The Internet | Is the internet down?
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Monitoring for the server world requires deep understanding for the use of the server, however luckily for you we have added some of the basics to get you started. Such as, Drive capacity, uptime,
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Android & iOS application monitoring is available. We can monitor your API's and connections. SOAP, and REST are easily checked with custom calls. This includes MySQL monitoring and any other
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Options are financial derivative contracts that give the buyer the right, but not the obligation, to buy or sell an underlying asset at a specific price . Pricing and market data · Derivatives –
commodities · Derivatives – futures exchange-traded · Derivatives – options exchange-traded · Derivatives – volatility. Overview. This course covers the concepts and models underlying the modern
analysis and pricing of financial derivatives. The philosophy of the course is to. The economic requirements, that listed futures and option contracts meet specified criteria, have been a fundamental
tool of Federal regulation of commodity. Futures and options are essentially elementary derivative products mostly traded on exchanges. A futures contract is an agreement between two parties to buy
What Are Options? An Options contract is essentially a type of agreement between two parties, whereby the buyer has the right but not the obligation to buy or. Investing in stocks without owning them
· Types of equity derivatives · Options · Warrants · Futures Contracts · Convertible Bonds · Equity Swaps. Options, Futures, and Other Derivatives by John C. Hull bridges the gap between theory and
practice by providing a current look at the industry, a careful. Page 1. EIGHTH EDITION. OPTIONS, FUTURES,. AND OTHER DERIVATIVES. Page 2. Page 3. EIGHTH EDITION. OPTIONS, FUTURES,. AND OTHER
DERIVATIVES. John C. Hull. Maple. Futures and options are derivative contracts that can be bought and sold in the share market. Futures contract is where the buyer and seller of the contract. A
derivative is a security with a price that is dependent upon or derived from one or more underlying assets. The derivative itself is a contract between two or. Options are considered derivatives
because they derive their value from the price of another asset, called the underlying asset. In the case of options, the. The derivatives market is a financial market dealing with derivatives whose
value is dictated by the underlying asset. As a result, the underlying asset decides. Various types of derivatives include futures, options, swaps, and forwards. Each type has its unique
characteristics and uses. Derivatives markets facilitate. Options are a form of derivative financial instrument in which two parties contractually agree to transact an asset at a specified price
before a future date. Derivative transactions include an assortment of financial contracts, including structured debt obligations and deposits, swaps, futures, options, caps.
The main types of derivatives are futures, forwards, options, and swaps. An example of a derivative security is a convertible bond. Such a bond, at the. An option is a derivative contract that gives
the holder the right, but not the obligation, to buy or sell an asset by a certain date at a specified price. A few examples of derivatives are futures, forwards, options and swaps. The purpose of
these securities is to give producers and manufacturers the possibility. The risk embodied in a derivatives contract can be traded either by trading the contract itself, such as with options, or by
creating a new contract which. What you'll learn · Language of stock options, understanding of the roles and responsibilities of buyers and sellers. · Learn how to deconstruct options. Overview. This
course covers the concepts and models underlying the modern analysis and pricing of financial derivatives. The philosophy of the course is to. In finance, there are four basic types of derivatives:
forward contracts, futures, swaps, and options. In this article, we'll cover the basics of what each. Instead of commodities, financial derivatives are based on stocks, bonds, currencies, interest
rates and indices. Consider the options market: traders write a. CME Group is the world's leading and most diverse derivatives marketplace offering the widest range of futures and options products
for risk management.
What is a Derivative? · Futures Contracts · Equity Options · Derivative Exchanges and Regulations · Example of Commodity Derivative · Benefits of Derivatives. In finance, a derivative is a contract
that derives its value from the performance of an underlying entity. This underlying entity can be an asset, index. Futures and options contracts are traded on Indices and on Single stocks. The
derivatives trading at NSE commenced with futures on the Nifty 50 in June There are various types of derivatives. These all have unique characteristics and are used for different reasons. However,
derivatives like options and futures. There are two types of derivative contracts: futures and options. Since both the seller and the investor forecast the underlying asset's price for a particular.
Investing in stocks without owning them · Types of equity derivatives · Options · Warrants · Futures Contracts · Convertible Bonds · Equity Swaps. Options are called "derivatives" because the value
of the option is "derived" from the underlying asset. Owning an option, in and of itself, does not impart. In the tapestry of modern finance, derivatives such as forwards, futures, options, and swaps
stand out as indispensable instruments. These tools. The course covers the four basic types of derivatives: forward contracts, futures contracts, swaps, and options. Students learn the basic features
of each.
Can You Sell Your Free Stock On Robinhood | 35 A Bra Size
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solutions to higher physics unit 2 practice NAB
Sorry folks, the link I put up here earlier did not have the answers to the practice NAB for unit 2 of Higher Physics. I’ve written my answers for you instead, with mark allocation indicated down
the side in red.
Download a copy using the link below.
2 thoughts on “solutions to higher physics unit 2 practice NAB”
1. For the first answer, it says E=QV, when the databook states E=1/2QV. Is this just a mistake on the answers?
□ Oops! You’re mixing up the relationships for capacitors with those for charged particles in an electric field. If there is a 1/2 in the equation, it’s a capacitor equation. The 1/2 part comes
from a unit 2 experiment where energy stored by the capacitor is found to equal the area under a graph of charge vs voltage, hence E = 1/2 x base x height = 1/2 x Q x V.
In the SQA blue booklet, the capacitor equations are all in a row.
SQA use W=QV for Work Done on the charge by the electric field but in the Int2/Standard Grade section, Ew is used to represent work done (as in Ew=Fd). E=QV is another way of writing this
same relationship. Check your notes and see which way your teacher showed it. It will be at the very start of your unit 2 work.
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When comparing numbers, we want to determine the relationship between two or more numbers. We use comparison symbols to indicate these relationships. The comparison symbols are:
When comparing whole numbers, we look at the value of each digit from left to right. If the digits in the same place value are different, we can determine the relationship between the numbers based
on the value of those digits.
Compare 427 and 593
Since the hundreds place has a 4 in the first number and a 5 in the second number, we know that 427 is less than 593.
When comparing decimal numbers, we follow the same process as with whole numbers. We compare the digits from left to right, and if the digits in the same place value are different, we determine the
relationship based on the value of those digits.
Compare 3.25 and 3.5
Since the tenths place has a 2 in the first number and a 5 in the second number, we know that 3.25 is less than 3.5.
When comparing fractions, we can find a common denominator and then compare the numerators. If the denominators are the same, we can simply compare the numerators to determine the relationship
between the fractions.
Compare 1/4 and 3/8
Since the denominators are different, we find a common denominator, which is 8. Then we compare the numerators: 1 * 2 = 2 and 3 * 1 = 3. So, 1/4 is less than 3/8.
Study Guide
Here are some steps to follow when comparing numbers:
1. Identify the place values of the digits in each number.
2. Compare the digits from left to right, starting with the largest place value.
3. If the digits in the same place value are different, determine the relationship based on the value of those digits.
4. For fractions, find a common denominator if necessary, and then compare the numerators to determine the relationship.
Practice comparing different types of numbers, including whole numbers, decimal numbers, and fractions, to reinforce your understanding of this concept.
Remember to use the comparison symbols (<, >, ≤, ≥, =) to express the relationships between the numbers.
Good luck with your studies!
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[Solved] A family with a monthly income of ₹20,000 had planned... | Filo
A family with a monthly income of had planned the following expenditures per month under various heads:
Draw a bar graph for the data above.
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+ Ask your question
Solution : We draw the bar graph of this data in the following steps. Note that the unit in the second column is thousand rupees. So, '4' against 'grocery' means ₹ 4000 .
1. We represent the Heads (variable) on the horizontal axis choosing any scale, since the width of the bar is not important. But for clarity, we take equal widths for all bars and maintain equal gaps
in between. Let one Head be represented by one unit.
2. We represent the expenditure (value) on the vertical axis. Since the maximum expenditure is ₹ 5000 , we can choose the scale as 1 unit .
3. To represent our first Head, i.e., grocery, we draw a rectangular bar with width 1 unit and height 4 units.
4. Similarly, other Heads are represented leaving a gap of 1 unit in between two consecutive bars.
The bar graph is drawn in Fig. 14.2.
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A family with a monthly income of had planned the following expenditures per month under various heads:
Question Text
Draw a bar graph for the data above.
Topic Statistics
Subject Mathematics
Class Class 9
Answer Type Text solution:1
Upvotes 119
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Derive this Equation:
K[a]K[b] = K[w]
The equation for the dissociation of a weak acid HA in solution is:
HA + H[2]O ⇌ H[3]O^+ + A¯
and its K[a] expression is:
[H[3]O^+] [A¯]
K[a] = –––––––––
The equation for the ionization of a weak base A¯ in solution is:
A¯ + H[2]O ⇌ HA + OH¯
and its K[b] expression is:
[HA] [OH¯]
K[b] = –––––––––
Rearrange the K[a] expression above as follows:
[H[3]O^+] [HA]
–––––– = ––––––
K[a] [A¯]
and substitute the left-hand portion into the K[b] expression from above to obtain:
[H[3]O^+] [OH¯]
K[b] = –––––––––––
Since the equation for the ionization of water is:
K[w] = [H[3]O^+] [OH¯]
by substitution and rearrangement, we obtain:
K[a]K[b] = K[w]
What this means is that, if we know one value for a given conjugate acid base pair, then we can calculate the value for the other member of the pair. For example, if the K[a] for HA = 1.50 x 10¯^5,
then we know that the K[b] for A¯ (the conjugate base) must be 6.67 x 10¯^10. This type of calculation becomes important in doing hydrolysis and buffer calculations.
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KSEEB Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes Ex 5.6
Students can Download Chapter 5 Understanding Elementary Shapes Ex 5.6 Questions and Answers, Notes Pdf, KSEEB Solutions for Class 6 Maths helps you to revise the complete Karnataka State Board
Syllabus and score more marks in your examinations.
Karnataka State Syllabus Class 6 Maths Chapter 5 Understanding Elementary Shapes Ex 5.6
Question 1.
Name the types of following triangles:
a) Triangle with lengths of sides 7 cm,8 cm, and 9 cm.
Scalene triangle
b) ∆ ABC with AB = 8.7 cm, AC = 7 cm, and BC = 6 cm.
Scalene triangle
c) ∆ PQR such that PQ = QR = PR = 5 cm.
Equilateral triangle
d) ∆ DEE with m∠D = 90°
Right – angled triangle
e) ∆ XYZ with m∠Y = 90°and XY = YZ
Right – angled isosceles triangle
f) ∆ LMN with m∠L = 30°, m∠M = 70° and m∠N = 80°
Acute angled triangle.
Question 2.
Match the following :
1 – (e),
2 – (g),
3 – (a),
4 – (f),
5 – (d),
6 – (c),
7 – (b).
Question 3.
Name each of the following triangles in two different ways : (You may judge the nature of the angle by observation)
a) Acute angle and isosceles.
b) Right – angle and scalene
c) Obtuse – angled and isosceles
d) Act Right – angled and isosceles
e) Acute – angles and equilateral
f) Obtuse angled and scalene.
Question 4.
Try to construct triangles using match sticks, some are shown here. Can you make a triangle with
a) 3 matchsticks?
b) 4 matchsticks?
c) 5 matchsticks?
d) 6 matchsticks?
(Remember you have to use all the available matchsticks in each case)
Name the type of triangle in each case. If you cannot make a triangle, think of reasons for it.
a) By using 3 matchsticks. we can from as triangle as
b) By using 4 matchsticks, we cannot form a triangle, of this is because the sum of the lengths
of any two sides of a triangle is always greater than the length of the remaining side of the triangle.
c) By using 5 matchsticks, we can form a triangle as.
d) By using 6 matchsticks, we can from a triangle as.
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Effects of mass transfer on free convective flow of a dissipative, incompressible fluid past an infinite vertical porous plate with suction
Effects of mass transfer on the fully developed two-dimensional free convective flow of an incompressible dissipative viscous fluid (i.e., air) past an infinite vertical porous plate in the presence
of constant suction are analyzed. The term representing viscous dissipative heat is retained in the energy equation. Approximate solutions to the coupled nonlinear equations governing the problem are
obtained, and the resulting velocity and temperature profiles are plotted. Effects of Grashof number, modified Grashof number, Schmidt number, and Eckert number on skin friction and heat transfer
represented by Nusselt number are described quantitatively. The results show that: (1) velocity increases and temperature decreases if foreign matter (e.g., hydrogen, water vapor, ethyl benzene) is
added to the air; (2) skin friction increases for Schmidt numbers above and below unity but decreases for Schmidt numbers of the order of unity due to the addition of foreign matter; and (3) in the
presence of suction and foreign matter, the rate of heat transfer decreases for Schmidt numbers less than unity but increases for Schmidt numbers greater than unity.
Indian Academy of Sciences Proceedings Section
Pub Date:
November 1976
□ Convective Flow;
□ Free Flow;
□ Incompressible Flow;
□ Mass Transfer;
□ Porous Boundary Layer Control;
□ Suction;
□ Viscous Flow;
□ Grashof Number;
□ Heat Flux;
□ Porous Plates;
□ Schmidt Number;
□ Skin Friction;
□ Two Dimensional Flow;
□ Fluid Mechanics and Heat Transfer
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Formulas in Columns with Dependencies?
Is there any way to work around dependencies in columns when entering in formulas?
• Hi Aoaks,
Are you referring to a dependency column in a project sheet (i.e. the column linking task predecessors/successors)?
If so, no there is no way to get around this, as Smartsheet actually uses hidden formulas in project columns to calculate values.
Help Article Resources
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Blocks and Modeling: A Do-It-Yourself Guide to Exploring Geometric Objects - Online Workshop - Natural Math
Blocks and Modeling: A Do-It-Yourself Guide to Exploring Geometric Objects – Online Workshop
• What: An intensive two-day online workshop on creating math puzzles, modeling, and visual-spatial thinking using everyday objects
• Why: Learn how to help your children play with beautiful math, ask questions and pose problems as mathematicians do
• Who: 25 parents, teachers, math circle leaders, and their children (ages 5-12), with Sian Zelbo and Sally Bishop as organizers
• When: Live meetings Thursday & Friday, March 10 & 11, 2016, at 1:00 to 2:30 PM EST (New York)
• Where: Online video-talk software Zoom (similar to Skype); recordings available to participants
• Price: Registration is $20. Add $25 to crowdfund Sian Zelbo’s upcoming book Playing With Blocks, have your name in it, and receive paperback and ebook copies (~Fall 2016). Work-trade stipends are
available upon request.
• Supplies: 12 sticks (toothpicks or popsicle sticks), paper, pencil
We are running an online workshop in two sessions, called “Playing with Blocks,” for 25 parents, math circle leaders, and teachers of elementary age children. The workshop will use open-ended puzzles
to launch a discussion on how to encourage your students and children to approach mathematics with a sense of curiosity and exploration.
Let’s explore!
Mathematics curricula are often structured to help students learn and practice specific concepts and procedures. Children don’t often get the opportunity to ask their own questions, look for
patterns, and make their own mathematics. The “Playing with Blocks” workshop will help parents, math circle leaders, and teachers approach their own mathematics in this way and to encourage their
students to do the same. Once you are asking your own questions and not just answering questions posed by others, you are on the road to thinking like a mathematician. And by modeling a curious,
creative, and playful attitude toward mathematical ideas you will be encouraging your students to enter that same frame of mind and teaching them to be mathematicians too.
Where is mathematics?
The activity you will explore is a joyful and elegant example of mathematics that starts easy, and then takes you far. The areas of math you will touch come from the subjects of geometry, number
theory, and combinatorics. Visual-spatial thinking is taught much less than math with numbers, yet it is crucial to many areas of “grown-up” mathematics, science, engineering, and technology.
You will help your children grow their math eyes, and notice unexpected links between concepts such as angle and area, minimum/maximum and perimeter, combinations that form a sum and types of
triangles (equilateral, scalene, and so on). You will make bridges to other rich math activities, such as pentaminoes and tangrams.
As we talk, you will pick up good math terms for this exploration, and good search phrases for when you want to investigate these topics more. You will also collect questions you can ask about
any problem, to learn deeper, more joyful math. For example:
• What makes Thing One and Thing Two similar or different? (For example, a triangle and a square.)
• What is the largest thing you can make? How do you measure your things? (For example, using a grid to measure areas.)
• What if you change your numbers around? What other things you can change? (For example, 10 toothpicks instead of 12.)
Here’s how it works:
Adults meet twice in a webinar format. During the first workshop, you will become familiar with Natural Math methods and activities and explore the art of creating your own math problems. You’ll be
introduced to a chapter of Sian’s upcoming book and you’ll try it out live.
Between the first and second workshop, you will try out a math activity with your group of children or students. We want you to bring your experience back to the second workshop. Bring the successes,
the challenges, the struggles; whatever happens we want to know!
One goal of our second workshop is to give you the support, feedback and confidence you need to try out even more math puzzling on your own. We will discuss your experiences and share feedback and
ideas with one another. Our final goal is to be sure we have all grown in our confidence and abilities and love of math.
Want to make a difference for your children and see mathematics in a new light? Join this new Natural Math adventure!
Here is a sample of our Math Sparks. The goal is to start thinking about ideas – to spark curiosity. Click to see the full-size PDF and ponder the questions in it.
Sian holds a degree in law from the University of Texas and an MA in math education from Teachers College Columbia University. Sian was a practicing lawyer when she decided to return to school to
explore her latent interest in mathematics. Sian has held various positions in mathematics education including associate director of the Center for Mathematical Talent at the Courant Institute of
Mathematical Sciences, New York University and math specialist at the Speyer Legacy School in New York City. Sian also runs math circles and works with various schools as a math education
consultant. Sian’s interest is in helping young people discover and develop their own interest and ability in mathematics through extracurricular activities that focus on mathematical reasoning and
problem solving. Sian co-authored Camp Logic, a popular math circle and family book published by Natural Math.
Sally Bishop has loved math her entire life. A self-described introvert, she has an innate ability to recognize patterns in nature, ideas and people. She enjoys sharing her enthusiasm for math with
people of all ages, both online and in-person, and is a talented editor with a knack for the big picture and details. Sally started helping others see the inherent beauty of math in homeschool math
cooperatives, where she embraces math learning that is empowering and fun. In her spare time, she reads widely and creates handmade books, often with a dog asleep across her shoulders and a cat in
her lap.
Openness is one of the seven Natural Math main principles, and the focus of this workshop. Openness means you can adapt, collaborate on, and share your math. A single math problem can be made into
diverse others by changing one aspect or another. Which changes make it harder, easier, more interesting or less fun? It’s up to you!
What do you get from Playing With Blocks?
• A highly interactive experience where you make models, talk, and collaborate with other parents and teachers.
• Leading your children or students on playful math adventures.
• The first meeting in a math circle format (you as a student!) to see how to run these activities, and to get inspired.
• A day to try activities with children and friends
• The second meeting to answer your questions, overview other activities, and prepare to do more with children.
• Insight on how to adapt puzzles and problems to facilitate endless math exploration.
• Make an impact on another Natural Math book by sharing your ideas directly with the author
• (The crowdfunding option) Your name as one of crowdfunders in Sian Zelbo’s new book, Playing With Blocks, and the paperback and ebook copies (~Fall 2016)
• Most importantly, you’ll get the confidence in your own ability to do math differently in your family or group!
Questions? Email reach.out@naturalmath.com or ask in comments to this page.
• Connection and devices: you will need fast internet to watch and listen. It’s better to have a microphone and a webcam so you can show and tell as well, but you can also use text chat. We
recommend larger screens rather than phones.
• Software: Zoom is a tool for talking, like Skype or Google Hangouts. Please download and try it, here: https://zoom.us/test The same page has the link to a tech support center, in case you need
• Recording: The meetings will be available to course participants as YouTube videos.
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Worksheet Multiplication By 3
Mathematics, especially multiplication, creates the keystone of various scholastic self-controls and real-world applications. Yet, for several students, understanding multiplication can posture a
challenge. To resolve this obstacle, teachers and moms and dads have actually embraced a powerful device: Worksheet Multiplication By 3.
Introduction to Worksheet Multiplication By 3
Worksheet Multiplication By 3
Worksheet Multiplication By 3 -
These free 3 multiplication table worksheets for printing or downloading in PDF format are specially aimed at primary school students You can also make a multiplication table worksheet yourself using
the worksheet generator
Use the buttons below to print open or download the PDF version of the Multiplying 1 to 12 by 3 100 Questions A math worksheet The size of the PDF file is 63127 bytes Preview images of the first and
second if there is one pages are shown
Value of Multiplication Practice Understanding multiplication is essential, laying a strong foundation for advanced mathematical concepts. Worksheet Multiplication By 3 supply structured and targeted
technique, cultivating a deeper understanding of this essential arithmetic procedure.
Advancement of Worksheet Multiplication By 3
3 Digit Multiplication Worksheets Printable Lexia s Blog
3 Digit Multiplication Worksheets Printable Lexia s Blog
Here is our free generator for multiplication and division worksheets This easy to use generator will create randomly generated multiplication worksheets for you to use Each sheet comes complete with
answers if required The areas the generator covers includes Multiplying with numbers to 5x5 Multiplying with numbers to 10x10
Come and learn here the 3 times table with the 5 step plan Improve with the speed test 3 multiplication table games chart worksheets and get the diploma
From standard pen-and-paper workouts to digitized interactive layouts, Worksheet Multiplication By 3 have actually progressed, dealing with diverse learning styles and preferences.
Sorts Of Worksheet Multiplication By 3
Fundamental Multiplication Sheets Straightforward exercises concentrating on multiplication tables, helping learners develop a strong arithmetic base.
Word Trouble Worksheets
Real-life situations incorporated right into troubles, enhancing important reasoning and application skills.
Timed Multiplication Drills Tests developed to boost speed and accuracy, aiding in quick mental math.
Advantages of Using Worksheet Multiplication By 3
3x2 Multiplication Worksheets Times Tables Worksheets
3x2 Multiplication Worksheets Times Tables Worksheets
Kids completing this third grade math worksheet multiply by 3 to solve each equation and also fill in a multiplication chart for the number 3 Download to complete online or as a printable
Liveworksheets transforms your traditional printable worksheets into self correcting interactive exercises that the students can do online and send to the teacher
Boosted Mathematical Abilities
Consistent technique develops multiplication effectiveness, boosting overall mathematics abilities.
Boosted Problem-Solving Talents
Word issues in worksheets create logical reasoning and technique application.
Self-Paced Learning Advantages
Worksheets suit specific discovering speeds, cultivating a comfy and adaptable discovering atmosphere.
How to Produce Engaging Worksheet Multiplication By 3
Including Visuals and Colors Lively visuals and colors catch attention, making worksheets visually appealing and engaging.
Consisting Of Real-Life Situations
Connecting multiplication to everyday scenarios adds relevance and functionality to workouts.
Customizing Worksheets to Various Ability Levels Personalizing worksheets based on varying effectiveness degrees makes sure comprehensive learning. Interactive and Online Multiplication Resources
Digital Multiplication Equipment and Games Technology-based resources use interactive discovering experiences, making multiplication interesting and delightful. Interactive Internet Sites and
Applications On the internet platforms offer varied and accessible multiplication practice, supplementing traditional worksheets. Personalizing Worksheets for Different Discovering Styles Aesthetic
Learners Aesthetic aids and diagrams help comprehension for students inclined toward aesthetic understanding. Auditory Learners Spoken multiplication problems or mnemonics satisfy students who
comprehend ideas via acoustic means. Kinesthetic Students Hands-on activities and manipulatives support kinesthetic learners in recognizing multiplication. Tips for Effective Implementation in
Understanding Uniformity in Practice Normal method strengthens multiplication abilities, advertising retention and fluency. Balancing Repetition and Variety A mix of repeated exercises and diverse
trouble styles keeps interest and understanding. Supplying Positive Feedback Feedback aids in identifying areas of renovation, urging continued progression. Challenges in Multiplication Method and
Solutions Motivation and Interaction Hurdles Boring drills can lead to uninterest; ingenious methods can reignite inspiration. Getting Over Concern of Math Adverse perceptions around mathematics can
impede progression; developing a positive knowing environment is necessary. Influence of Worksheet Multiplication By 3 on Academic Performance Researches and Research Study Findings Study suggests a
favorable connection in between consistent worksheet use and enhanced math efficiency.
Worksheet Multiplication By 3 emerge as functional tools, promoting mathematical efficiency in students while fitting diverse knowing styles. From basic drills to interactive on-line sources, these
worksheets not only improve multiplication skills yet also advertise crucial thinking and problem-solving abilities.
Free Multiplication Worksheet 3 Digit By 1 Digit Free4Classrooms
Multiplication Brainchimp
Check more of Worksheet Multiplication By 3 below
Free Printable 3 Digit By 3 Digit Multiplication Worksheets 2 Digit multiplication Worksheets
Worksheet On Multiplication Table Of 3 Word Problems On 3 Times Table
Multiplication Practice Sheets Printable Worksheets Multiplication Worksheets Pdf Grade 234
4 Digit Multiplication Worksheets Times Tables Worksheets
Multiplication 3x3 Digit Worksheet Have Fun Teaching
Multiplication Sheets 4th Grade
Multiplying 1 To 12 By 3 100 Questions A Math Drills
Use the buttons below to print open or download the PDF version of the Multiplying 1 to 12 by 3 100 Questions A math worksheet The size of the PDF file is 63127 bytes Preview images of the first and
second if there is one pages are shown
Grade 3 Multiplication Worksheets Free amp Printable K5 Learning
Free 3rd grade multiplication worksheets including the meaning of multiplication multiplication facts and tables multiplying by whole tens and hundreds missing factor problems and multiplication in
columns No login required
Use the buttons below to print open or download the PDF version of the Multiplying 1 to 12 by 3 100 Questions A math worksheet The size of the PDF file is 63127 bytes Preview images of the first and
second if there is one pages are shown
Free 3rd grade multiplication worksheets including the meaning of multiplication multiplication facts and tables multiplying by whole tens and hundreds missing factor problems and multiplication in
columns No login required
4 Digit Multiplication Worksheets Times Tables Worksheets
Worksheet On Multiplication Table Of 3 Word Problems On 3 Times Table
Multiplication 3x3 Digit Worksheet Have Fun Teaching
Multiplication Sheets 4th Grade
Printable 10X10 Multiplication Table Printable Multiplication Flash Cards
Multiplication Grade 2 Math Worksheets
Multiplication Grade 2 Math Worksheets
Multiplication Worksheets For Grade 3
Frequently Asked Questions (Frequently Asked Questions).
Are Worksheet Multiplication By 3 ideal for all age teams?
Yes, worksheets can be tailored to various age and skill levels, making them versatile for numerous students.
How usually should trainees practice using Worksheet Multiplication By 3?
Constant method is vital. Regular sessions, ideally a few times a week, can yield considerable improvement.
Can worksheets alone improve math abilities?
Worksheets are an useful device yet should be supplemented with varied knowing approaches for comprehensive ability development.
Exist on-line systems using totally free Worksheet Multiplication By 3?
Yes, lots of educational internet sites use free access to a variety of Worksheet Multiplication By 3.
Just how can moms and dads sustain their children's multiplication method in the house?
Urging constant technique, providing support, and producing a favorable understanding setting are useful steps.
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R 20 583 income tax calculator 2024 - South Africa - salary after tax
Salary rate
UIF (Unemployment Insurance Fund)
If you make R 246 996 a year living in South Africa, you will be taxed R 31 839. That means that your net pay will be R 215 157 per year, or R 17 930 per month. Your average tax rate is 12.9% and
your marginal tax rate is 26.0%. This marginal tax rate means that your immediate additional income will be taxed at this rate. For instance, an increase of R 100 in your salary will be taxed R 26,
hence, your net pay will only increase by R 74.
Bonus Example
A R 1 000 bonus will generate an extra R 740 of net incomes. A R 5 000 bonus will generate an extra R 3 700 of net incomes.
NOTE* Withholding is calculated based on the tables of South Africa, income tax. For simplification purposes some variables (such as marital status and others) have been assumed. This document does
not represent legal authority and shall be used for approximation purposes only.
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Computer-Based Solving of Partial Differential Equations Using the Method of Nets
Published: 22 October 2024| Version 1 | DOI: 10.17632/rnf86sd695.1
This paper explores numerical methods for solving partial differential equations (PDEs) using the method of nets. The focus is on hyperbolic equations, such as the wave equation, and the application
of net methods in solving problems with boundary conditions. The process of solving these equations using computational tools is illustrated, and the accuracy of the results is analyzed. The
iterative Gauss-Seidel method is applied to solve systems of algebraic equations generated by the net method
Steps to reproduce
Steps to Reproduce the Method of Nets: Problem Setup: Select a PDE: Choose the specific partial differential equation you want to solve, such as the wave equation, heat equation, or Poisson equation.
Define Conditions: Establish the initial conditions (values of the function at the starting time) and boundary conditions (values of the function at the spatial boundaries) relevant to your problem.
Discretize the Domain: Create a Grid: Divide the spatial and temporal domains into discrete points to form a computational grid or mesh. Label Grid Points: Assign indices to each grid point to
represent their positions in space and time, typically using two indices—one for space and one for time. Discretize the Equation: Approximate Derivatives: Replace the continuous partial derivatives
in the PDE with finite difference approximations using values at the grid points. Formulate Algebraic Equations: This substitution transforms the PDE into a system of algebraic equations that relate
the function values at different grid points. Construct the System of Equations: Combine Equations: Assemble the finite difference approximations into a comprehensive system of algebraic equations,
ensuring consistency across the grid. Ensure Correspondence: Each algebraic equation should correspond to a specific grid point in your computational domain. Apply Boundary and Initial Conditions:
Implement Initial Conditions: Set the function values at the initial time based on the problem's initial conditions. Implement Boundary Conditions: Apply the boundary conditions by setting the
function values at the spatial boundaries for all time steps. Solve Iteratively Using the Gauss-Seidel Method: Initial Guess: Start with an initial approximation for the function values at all grid
points. Iterative Update: Use the Gauss-Seidel method to iteratively update the function values. This involves sequentially solving for each variable while using the most recent updates. Convergence
Check: Continue the iterations until the changes between successive updates are smaller than a predetermined tolerance level, indicating that the solution has stabilized. Visualize the Results: Data
Presentation: After obtaining the numerical solution, organize the data for analysis. Graphical Representation: Use software tools (such as MATLAB or Python with Matplotlib) to create visual
representations like plots or surface graphs that illustrate how the function behaves over space and time. Compare with Analytical Solution (If Available): Analytical Benchmark: If an exact
analytical solution exists for the PDE, calculate it for the same conditions. Error Analysis: Compare the numerical results with the analytical solution to evaluate the accuracy of the numerical
method. Assessment: Analyze any discrepancies to understand the limitations or errors introduced by the discretization and numerical methods.
European University
Algebra, Numerical Analysis, Numerical Linear Algebra
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IBDP Mathematics - Modelling
In this topic of IBDP Mathematics, we will be discussing mathematical models, how they are constructed, and different types of mathematical models.
In IBDP Mathematics, we may come across mathematical problems that are more complex than basic counting.
• For example, you can create a model to determine how much time it takes for a laptop to run out of battery.
• Whilst you could simply measure the time it takes for a laptop to run out of battery, this would be very time consuming and not very efficient.
• Instead, you could take smaller measurements, such as how much time it took a laptop to get down to 95%, and use this information to generate a mathematical model.
• You can then use this model to predict when your laptop will run out.
The example above shows the benefits to mathematical modelling. Modelling allows us to not only make predictions, but it can also help explain the relationship between two variables. Thus, it can be
extremely useful in helping us understand real-world problems.
In mathematical modelling, we often have to make assumptions in order to simplify the problem. By making assumptions and removing less important details, it allows us to construct a mathematical
description of the problem which is simple enough for us to work with. This enables the creation of a simpler model that is easier to understand, and we can use the model to approximate the
real-world situation.
• However, it is important to note that these models are not completely accurate - given the assumptions made, there will be some error to the model. But the assumptions have to be made in order
for us to construct a model that can be understood.
1. Pose a real world problem. Make assumptions which simplify the problem without missing key features.
2. Develop a model which represents the problem with mathematics. This may involve a formula or an equation. The model should consider constraints such as the range of possible values each variable
can take in the real world.
3. Test the model by comparing its predictions with known data. If the model is unsatisfactory, return to Step 2.
4. Reflect on your model and apply it to your original problem, interpreting the solution in a real world context.
5. If appropriate, extend your model to make it more general or accurate as needed.
Revisiting the example above, we can follow the steps in the modelling cycle to generate a model for that problem.
• Step 1 - the problem, as mentioned above, is to construct a model to describe the battery level after t minutes in order to help determine the time it takes for the battery to run out.
• In order to simplify the problem, we can make some assumptions, such as 'the battery is used at a constant rate for all laptops'.
□ When we are considering a relationship between two variables, one variable is known as the independent variable (which is placed on the horizontal axis) and the other is known as the
dependent variable (which is placed on the vertical axis).
□ The independent variable is the variable that is used to predict the dependent variable. In this example, given that we want to use time to predict battery level (because we want to determine
the battery level after t minutes), the independent variable is the time, and the dependent variable is battery level.
□ Step 2 - in order to develop a model for this problem, we will need some data. Given the assumption of the model, we can simply record the time it takes for the laptop to lose 1% of battery.
• For the sake of this example, lets say a laptop loses 1% of battery every minute.
• Therefore, at 0 minutes, the laptop will have 100%. At 1 minute, the laptop will have 99%. At 2 minutes, the laptop will have 98%. This can be summarised into the following equation: battery
level = - t + 100.
□ Given this rate, we can construct a linear model (a linear model is used since we assume a constant rate), with battery level on the y-axis and the time on the x-axis.
□ It is important to note that this model will have constraints on the y-axis. Since battery levels only consist of values between 0 and 100, the model should only produce values that are
between 0 and 100.
□ Step 3 - to test the model, you can collect some data on the rate of laptop battery use.
□ If the data does not match the model, use this set of data to refine to model.
• For the sake of this example, we can assume that the model is satisfactory.
• Step 4 - given our equation for the linear model, we can substitute the values to determine the time it takes for the battery level to become 0%.
□ We can substitute 0 for battery level, and solve for t. This should give us t = 100.
□ Interpreting this value, we can say that the laptop will run out of battery after 100 minutes of use.
• Step 5 - In order to extend, we can add in more factors which affect battery level, such as how old the laptop is, the type of activity used on laptop etc.
In IBDP Mathematics, not only will you need to be familiar with constructing models, you are required to be able to use these models, state the assumptions made when constructing the model, and
whether these assumptions are reasonable. You may also be asked to describe whether you think the actual value is larger or smaller than the value predicted by the model. Thus, you must be able to
think critically about these models.
Types of Mathematical Models
• Linear models - these are the models where two variables are linearly related (i.e. they form a straight line on the graph).
• Piecewise models - a model made up of several different straight line segments.
• Non-lines piecewise models - model made of up several segments, some of which are not linear.
This is the end of this topic.
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Base Converter - Exploring Binary
Base Converter
About the Base Converter
This is an arbitrary-precision number base converter, also known as a radix converter. It converts numbers from one base to another; for example: decimal (base 10) to hexadecimal (base 16), binary
(base 2) to hexadecimal, duodecimal (base 12) to decimal, ternary (base 3) to binary, etc. It will convert between any pair of bases, 2 through 36.
This converter can convert fractional as well as integer values. For example, it will convert hexadecimal aaf1.e to binary 1010101011110001.111. It can convert very large and very small numbers (up
to hundreds of digits). It allows you to specify one of several character sets for the digits (the default consists of the digits 0-9 and letters a-z).
How to Use the Base Converter
• From section:
□ Enter a positive or negative number with no commas or spaces, not expressed as a fraction or arithmetic calculation, and not in scientific notation. Fractional values are indicated with a
radix point (‘.’, not ‘,’)
□ Choose the base of the number, from 2 through 36 (if different than the default). The most well-known bases are also described by name: binary (base 2), ternary (base 3), quaternary (base 4),
quinary (base 5), senary (base 6), octal (base 8), decimal (base 10), duodecimal (base 12), hexadecimal (base 16), and vigesimal (base 20).
□ Choose the character set for the digits of the number (if different than the default). You must use a character set with at least as many digits as the value of the base; the first base
characters of the character set are used as the digit symbols. The position of a character in the set correlates to its numeric value (the first character has value 0). For bases requiring
letters, the characters you enter must match the case of the characters in the chosen character set.
There are three character sets specifically for base 12 (duodecimal, or dozenal), although you can use them for any base less than or equal to 12. Each include the digits 0-9, but each has a
different pair of characters for the ten and eleven symbols: X and E, T and E, and * and #. (You can also use a character set that has A and B or a and b serving as the ten and eleven
There are character sets that exclude the letters I (i), L (l), and O (o) — useful for bases 19 and higher — to eliminate confusion with 1 and 0. There are six combinations of these character
sets: one that removes I (and one that removes i), one that removes I and L (and one that removes i and l), and one that removes I, L and O (and one that removes i, l, and o).
There are also character sets with no numeric digits; for example, where ‘a’ serves as 0 (be aware in this case that leading and trailing ‘a’s will be trimmed — they function as numeric
• To section:
□ Choose the base of the number (as described above).
□ Choose the character set for the digits of the number (as described above).
□ Change the number of fractional digits you want displayed in an infinitely repeating fractional result, if different than the default (applies only when converting a fractional value). (These
are places, not significant digits, so leading zeros are counted.)
• Click ‘Convert’ to convert.
• Click ‘Swap From/To’ to do the conversion in the opposite direction. This will swap the bases and character sets, copy the “to” number to the “from” number field, and perform the conversion.
(Truncated “to” values — those ending in ‘…’ — will be copied to the “from” field with the trailing ‘…’ removed.)
• Click ‘Clear’ to reset the form and start from scratch.
If you change the base(es), character set(s), or number of fractional digits, you must click ‘Convert’ (or ‘Swap From/To’) for the conversion to take place.
If you want to convert another number with the same options, just type over the original number and click ‘Convert’.
Besides the converted result, the number of digits in both the original and converted numbers is displayed. For example, when converting decimal 192.25 to binary 11000000.01, the “Num Digits” box
displays ‘3.2 to 8.2’. This means that the decimal input has three digits in its integer part and two digits in its fractional part, and the binary output has eight digits in its integer part and two
digits in its fractional part.
Fractional values that convert to infinite (repeating) fractional values are truncated — not rounded — to the specified number of digits. In this case, an ellipsis (…) is appended to the end of the
converted number, and the number of fractional digits is noted as infinite with the ‘∞’ symbol. Fractional values that terminate are displayed in full precision, regardless of the number of
fractional digits specified.
Exploring Properties of Base Conversion
Use this converter for a deeper understanding of base conversion; for example, to see how the number of digits correspond between different bases. Try converting some large ternary integers to base
20, for example. What is the ratio of the number of base 3 digits to base 20 digits? (Answer: it will approach log[3](20), which is approximately 2.73.)
How about the length of fractional values? Some will convert to an infinite string of digits, and some will terminate. For those that terminate, how do the number of digits correspond? For example,
using the default character set, convert the base 3 number 0.22101220121221102 to base 9, 21, and 27. In base 9 it is 0.835655736 (17 digits to 9 digits, approximately 2:1); in base 21 it is
0.jcgi72edc4i7ff64e (17 digits to 17 digits, 1:1); in base 27 it is 0.p5jnm6 (17 digits to 6 digits, approximately 3:1). Are those ratios consistent across those base pairs? (Answer: Yes.) What
determines whether a fractional is infinite or terminates, and how many digits it has if it terminates? (Answer: it has to do with the prime factors in both bases.)
This converter is implemented in arbitrary-precision decimal arithmetic. The conversion of a fractional value is done through an intermediate base fraction, not a fractional value; this prevents the
intermediate representation from introducing error in the final converted result.
For practical reasons, the size of the inputs — and the number of fractional digits in an infinite fractional result — is limited. If you exceed these limits, you will get an error message. But
within these limits, all results will be accurate (in the case of infinite fractional results, results are accurate through the truncated digit).
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G NOTEs for the TI-83, 83+, 84, 84+ Calculators - Introductory Statistics 2e | OpenStax (2024)
Quick Tips
• represents a button press
• [ ] represents yellow command or green letter behind a key
• < > represents items on the screen
To adjust the contrastPress , then hold to increase the contrast or to decrease the contrast.
To capitalize letters and wordsPress to get one capital letter, or press , then to set all button presses to capital letters.You can return to the top-level button values by pressing again.
To correct a mistakeIf you hit a wrong button, just hit and start again.
To write in scientific notationNumbers in scientific notation are expressed on the TI-83, 83+, 84, and 84+ using E notation, such that...
• 4.321 E 4 = $4.321× 10 4 4.321× 10 4$
• 4.321 E –4 = $4.321× 10 –4 4.321× 10 –4$
To transfer programs or equations from one calculator to another:Both calculators: Insert your respective end of the link cable cableand press , then [LINK].
Calculator receiving information:
1. Use the arrows to navigate to and select <RECEIVE>
2. Press .
Calculator sending information:
1. Press appropriate number or letter.
2. Use up and down arrows to access the appropriate item.
3. Press to select item to transfer.
4. Press right arrow to navigate to and select <TRANSMIT>.
5. Press .
ERROR 35 LINK generally means that the cables have not been inserted far enough.
Both calculators: Insert your respective end of the link cable cableBoth calculators: press , then [QUIT] to exit when done.
Manipulating One-Variable Statistics
These directions are for entering data with the built-in statistical program.
Data Frequency
–2 10
–1 3
Table G1 Sample Data We are manipulating one-variable statistics.
To begin:
1. Turn on the calculator.
2. Access statistics mode.
3. Select <4:ClrList> to clear data from lists, if desired.
4. Enter list [L1] to be cleared.
, [L1] ,
5. Display last instruction.
, [ENTRY]
6. Continue clearing remaining lists in the same fashion, if desired.
, , [L2] ,
7. Access statistics mode.
8. Select <1:Edit . . .>
9. Enter data. Data values go into [L1]. (You may need to arrow over to [L1]).
□ Type in a data value and enter it. (For negative numbers, use the negate (-) key at the bottom of the keypad).
, ,
□ Continue in the same manner until all data values are entered.
10. In [L2], enter the frequencies for each data value in [L1].
□ Type in a frequency and enter it. (If a data value appears only once, the frequency is "1").
□ Continue in the same manner until all data values are entered.
11. Access statistics mode.
12. Navigate to <CALC>.
13. Access <1:1-var Stats>.
14. Indicate that the data is in [L1]...
, [L1] ,
15. ...and indicate that the frequencies are in [L2].
, [L2] ,
16. The statistics should be displayed. You may arrow down to get remaining statistics. Repeat as necessary.
Drawing Histograms
We will assume that the data is already entered.
We will construct two histograms with the built-in STATPLOT application. The first way will use the default ZOOM. The second way will involve customizing a new graph.
1. Access graphing mode.
, [STAT PLOT]
2. Select <1:plot 1> to access plotting - first graph.
3. Use the arrows navigate go to <ON> to turn on Plot 1.
<ON> ,
4. Use the arrows to go to the histogram picture and select the histogram.
5. Use the arrows to navigate to <Xlist>.
6. If "L1" is not selected, select it.
, [L1] ,
7. Use the arrows to navigate to <Freq>.
8. Assign the frequencies to [L2].
, [L2] ,
9. Go back to access other graphs.
, [STAT PLOT]
10. Use the arrows to turn off the remaining plots.
11. Be sure to deselect or clear all equations before graphing.
To deselect equations:
1. Access the list of equations.
2. Select each equal sign (=).
3. Continue, until all equations are deselected.
To clear equations:
1. Access the list of equations.
2. Use the arrow keys to navigate to the right of each equal sign (=) and clear them.
3. Repeat until all equations are deleted.
To draw default histogram:
1. Access the ZOOM menu.
2. Select <9:ZoomStat>.
3. The histogram will show with a window automatically set.
To draw custom histogram:
1. Access window mode to set the graph parameters.
□ $X min =–2.5 X min =–2.5$
□ $X max =3.5 X max =3.5$
□ $X scl =1 X scl =1$ (width of bars)
□ $Y min =0 Y min =0$
□ $Y max =10 Y max =10$
□ $Y scl =1 Y scl =1$ (spacing of tick marks on y-axis)
□ $X res =1 X res =1$
3. Access graphing mode to see the histogram.
To draw box plots:
1. Access graphing mode.
, [STAT PLOT]
2. Select <1:Plot 1> to access the first graph.
3. Use the arrows to select <ON> and turn on Plot 1.
4. Use the arrows to select the box plot picture and enable it.
5. Use the arrows to navigate to <Xlist>.
6. If "L1" is not selected, select it.
, [L1] ,
7. Use the arrows to navigate to <Freq>.
8. Indicate that the frequencies are in [L2].
, [L2] ,
9. Go back to access other graphs.
, [STAT PLOT]
10. Be sure to deselect or clear all equations before graphing using the method mentioned above.
11. View the box plot.
, [STAT PLOT]
Linear Regression
Sample Data
The following data is real. The percent of declared ethnic minority students at De Anza College for selected years from 1970–1995 was:
Year Student Ethnic Minority Percentage
1970 14.13
1973 12.27
1976 14.08
1979 18.16
1982 27.64
1983 28.72
1986 31.86
1989 33.14
1992 45.37
1995 53.1
Table G2 The independent variable is "Year," while the independent variable is "Student Ethnic Minority Percent."
Figure G1 Student Ethnic Minority Percentage By hand, verify the scatterplot above.
The TI-83 has a built-in linear regression feature, which allows the data to be edited.The x-values will be in [L1]; the y-values in [L2].
To enter data and do linear regression:
1. ON Turns calculator on.
2. Before accessing this program, be sure to turn off all plots.
□ Access graphing mode.
, [STAT PLOT]
□ Turn off all plots.
3. Round to three decimal places. To do so:
□ Access the mode menu.
, [STAT PLOT]
□ Navigate to <Float> and then to the right to <3>.
□ All numbers will be rounded to three decimal places until changed.
4. Enter statistics mode and clear lists [L1] and [L2], as describe previously.
5. Enter editing mode to insert values for x and y.
6. Enter each value. Press to continue.
To display the correlation coefficient:
1. Access the catalog.
, [CATALOG]
2. Arrow down and select <DiagnosticOn>
... , ,
3. $rr$ and$r2r2$ will be displayed during regression calculations.
4. Access linear regression.
5. Select the form of y = a + bx.
The display will show:
• y = a + bx
• a = –3176.909
• b = 1.617
• r = 2 0.924
• r = 0.961
This means the Line of Best Fit (Least Squares Line) is:
• y = –3176.909 + 1.617x
• Percent = –3176.909 + 1.617 (year #)
The correlation coefficient r = 0.961
To see the scatter plot:
1. Access graphing mode.
, [STAT PLOT]
2. Select <1:plot 1> To access plotting - first graph.
3. Navigate and select <ON> to turn on Plot 1.
4. Navigate to the first picture.
5. Select the scatter plot.
6. Navigate to <Xlist>.
7. If [L1] is not selected, press , [L1] to select it.
8. Confirm that the data values are in [L1].
9. Navigate to <Ylist>.
10. Select that the frequencies are in [L2].
, [L2] ,
11. Go back to access other graphs.
, [STAT PLOT]
12. Use the arrows to turn off the remaining plots.
13. Access window mode to set the graph parameters.
□ $Xmin=1970Xmin=1970$
□ $Xmax=2000Xmax=2000$
□ $Xscl=10Xscl=10$ (spacing of tick marks on x-axis)
□ $Ymin=−0.05Ymin=−0.05$
□ $Ymax=60Ymax=60$
□ $Yscl=10Yscl=10$ (spacing of tick marks on y-axis)
□ $Xres=1Xres=1$
14. Be sure to deselect or clear all equations before graphing, using the instructions above.
15. Press the graph button to see the scatter plot.
To see the regression graph:
1. Access the equation menu. The regression equation will be put into Y1.
2. Access the vars menu and navigate to <5: Statistics>.
3. Navigate to <EQ>.
4. <1: RegEQ> contains the regression equation which will be entered in Y1.
5. Press the graphing mode button. The regression line will be superimposed over the scatter plot.
To see the residuals and use them to calculate the critical point for an outlier:
1. Access the list. RESID will be an item on the menu. Navigate to it.
, [LIST], <RESID>
2. Confirm twice to view the list of residuals. Use the arrows to select them.
3. The critical point for an outlier is: $1.9VSSEn-21.9VSSEn-2$ where:
□ $nn$ = number of pairs of data
□ $SSESSE$ = sum of the squared errors
□ $Σresidual2Σresidual2$
4. Store the residuals in [L3].
, , [L3] ,
5. Calculate the $Σresidual2n-2Σresidual2n-2$. Note that$n-2=8n-2=8$
, [L3] , , ,
6. Store this value in [L4].
, , [L4] ,
7. Calculate the critical value using the equation above.
, , , , , [V] , , [LIST] , , , , [L4] , , ,
8. Verify that the calculator displays: 7.642669563. This is the critical value.
9. Compare the absolute value of each residual value in [L3] to 7.64. If the absolute value is greater than 7.64, then the (x, y) corresponding point is an outlier. In this case, none of the points
is an outlier.
To obtain estimates of y for various x-values:There are various ways to determine estimates for "y." One way is to substitute values for "x" in the equation. Another way is to use the on the graph of
the regression line.
TI-83, 83+, 84, 84+ instructions for distributions and tests
Access DISTR (for "Distributions").
For technical assistance, visit the Texas Instruments website at http://www.ti.com and enter your calculator model into the "search" box.
Binomial Distribution
• binompdf(n,p,x) corresponds to P(X = x)
• binomcdf(n,p,x) corresponds to P(X ≤ x)
• To see a list of all probabilities for x: 0, 1, . . . , n, leave off the "x" parameter.
Poisson Distribution
• poissonpdf(λ,x) corresponds to P(X = x)
• poissoncdf(λ,x) corresponds to P(X ≤ x)
Continuous Distributions (general)
• $−∞−∞$ uses the value –1EE99 for left bound
• $∞ ∞$ uses the value 1EE99 for right bound
Normal Distribution
• normalpdf(x,μ,σ) yields a probability density function value (only useful to plot the normal curve, in which case "x" is the variable)
• normalcdf(left bound, right bound, μ, σ) corresponds to P(left bound < X < right bound)
• normalcdf(left bound, right bound) corresponds to P(left bound < Z < right bound) – standard normal
• invNorm(p,μ,σ) yields the critical value, k: P(X < k) = p
• invNorm(p) yields the critical value, k: P(Z < k) = p for the standard normal
Student's t-Distribution
• tpdf(x,df) yields the probability density function value (only useful to plot the student-t curve, in which case "x" is the variable)
• tcdf(left bound, right bound, df) corresponds to P(left bound < t < right bound)
Chi-square Distribution
• Χ^2pdf(x,df) yields the probability density function value (only useful to plot the chi^2 curve, in which case "x" is the variable)
• Χ^2cdf(left bound, right bound, df) corresponds to P(left bound < Χ^2 < right bound)
F Distribution
• Fpdf(x,dfnum,dfdenom) yields the probability density function value (only useful to plot the F curve, in which case "x" is the variable)
• Fcdf(left bound,right bound,dfnum,dfdenom) corresponds to P(left bound < F < right bound)
Tests and Confidence Intervals
Access STAT and TESTS.
For the confidence intervals and hypothesis tests, you may enter the data into the appropriate lists and press DATA to have the calculator find the sample means and standard deviations. Or, you may
enter the sample means and sample standard deviations directly by pressing STAT once in the appropriate tests.
Confidence Intervals
• ZInterval is the confidence interval for mean when σ is known.
• TInterval is the confidence interval for mean when σ is unknown; s estimates σ.
• 1-PropZInt is the confidence interval for proportion.
The confidence levels should be given as percents (ex. enter "95" or ".95" for a 95% confidence level).
Hypothesis Tests
• Z-Test is the hypothesis test for single mean when σ is known.
• T-Test is the hypothesis test for single mean when σ is unknown; s estimates σ.
• 2-SampZTest is the hypothesis test for two independent means when both σ's are known.
• 2-SampTTest is the hypothesis test for two independent means when both σ's are unknown.
• 1-PropZTest is the hypothesis test for single proportion.
• 2-PropZTest is the hypothesis test for two proportions.
• Χ^2-Test is the hypothesis test for independence.
• Χ^2GOF-Test is the hypothesis test for goodness-of-fit (TI-84+ only).
• LinRegTTEST is the hypothesis test for Linear Regression (TI-84+ only).
Input the null hypothesis value in the row below "Inpt." For a test of a single mean, "μ∅" represents the null hypothesis. For a test of a single proportion, "p∅" represents the null hypothesis.
Enter the alternate hypothesis on the bottom row.
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PvS - Resources
A few technical resources
Linearly distributive categories provide categorical description of multiplicative linear logic. Even though linear logic has been shown to have applications in many areas of computer science and
quantum, applied category theorists have mostly stayed away from linearly distributive categories mostly because the foundational papers on these categories and their associated structures are highly
technical in nature. The aim of this blog series is to provide an one-stop, accessible, intuitive introduction to linearly distributive categories, functors, transformations, other associated
structures and applications by assimilating information spread across these multiple highly technical articles, and to encourage discussion in this topic.
Cockett and Pastro introduced two-tiered ‘Message Passing Logic’ in order to develop a type theory for concurrent programs with message passing as the communication primitive. The motivation was to
allow one to guarantee certain formal properties of concurrent programs such as deadlock and livelock avoidance, which is not possible using the current programming technologies. The categorical
logic behind the machinery introduced by Cockett and Pastro is based on monoidal categories acting on linearly distributive categories with the proof theory given by multi categories acting on poly
categories, and is called a linear actegory. Chad Nester introduced resource transducers for concurrent process histories which is a toy model of linear actegories. This talk introduces Cockett and
Pastro’s message passing logic and Nester’s resource transducers which is a toy model of this logic.
Dagger monoidal and dagger compact closed categories are the standard settings for Categorical Quantum Mechanics (CQM). These settings of CQM are categorical proof theories of compact dagger linear
logic and are motivated by the interpretation of quantum systems in the category of finite dimensional Hilbert spaces. In this talk, I describe a new non-compact (infinite-dimensional) framework
called Mixed Unitary Categories (MUCs) with examples built on linearly distributive and *-autonomous categories which are categorical proof theories of (non-compact) multiplicative linear logic. This
talk is based on the first part of my thesis.
The notion of complimentary observables lies at the heart of quantum mechanics: two quantum observables A and B are complementary if measuring one increases the uncertainty regarding the value of the
other. In this talk, I show that complementary observables and classical non-linearity are related by proving that every complementary pair of observables can be viewed as the exponential modalities
- ! and ? - of linear logic "compacted" into the unitary core of the MUC, thereby exhibiting a complementary system as arising via the compaction of distinct systems of arbitrary dimensions. The
machinery to arrive at this result involves linear monoids, linear comonoids, linear bialgebras and dagger-exponential modalities.
In physics, a resource theory is used to model physical systems for which certain transformations are considered to be "free of cost". For example, on a hot summer day, cooling down the water
requires energy (used by a refrigerator), hence is not free. However, a glass of chilled water warming up to room temperature is a free transformation. Resources are states of such a system.
Monotones assigns a real number to each resource based on their value or utility. A problem in physics is that, can monotones on one resource theory be extended to another in case the two theories
are related.
Gour and Tomamichel studied the problem of extending monotones using set-theoretical framework when a resource theory embeds fully and faithfully into the larger theory. One can generalize the
problem of computing monotone extensions to scenarios when there exists a functorial transformation of one resource theory to another instead of just a full and faithful inclusion. In this talk, I
show that (point-wise) Kan extensions provide a precise categorical framework to describe and compute such extensions of monotones.
To set up monotone extensions using Kan extensions, we introduce partitioned categories (pCat) as a framework for resource theories and pCat functors to formalize relationship between resource
theories. We describe monotones as pCat functors into ([0,∞],≤), and describe extending monotones along any pCat functor using Kan extensions. We show how our framework works by applying it to extend
entanglement monotones for bipartite pure states to bipartite mixed states, to extend classical divergences to the quantum setting, and to extend a non-uniformity monotone from classical
probabilistic theory to quantum theory.
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Islamic science
The relationship between true science and Islam is a matter of extreme controversy. In the Muslim world, many believe that modern science was first developed in the Muslim world rather than in Europe
and Western countries, that "all the wealth of knowledge in the world has actually emanated from Muslim civilization," and what people call "the scientific method", is actually "the Islamic method."^
[1]^[2] Muslims often cite verse 239 from Surah Al-Baqara —- He has taught you what you did not know. ^[3] —- in support of their view that the Qur'an promotes the acquisition of new knowledge.
In contrast, some people worry that the contemporary Muslim world suffers from a "profound lack of scientific understanding," and lament that, for example, in countries like Pakistan post-graduate
physics students have been known to blame earthquakes on "sinfulness, moral laxity, deviation from the Islamic true path," while "only a couple of muffled voices supported the scientific view that
earthquakes are a natural phenomenon unaffected by human activity."^[4]
The development of scientific thought and knowledge has caused differing reactions among Muslims. In the Muslim world today, most of the focus on the relation between Islam and science involves
scientific interpretations of the Quran (and sometimes the Sunna) that claim to show these sources make prescient statements about the nature of the universe, biological development and other
phenomena later confirmed by scientific research, and proof of the divine origin of the Qur'an. This effort has been criticized by some scientists and philosophers as containing logical fallacies,^
[5] being unscientific, likely to be disproven by evolving scientific theories.^[6]^[7]
Overview[ ]
The religion Islam has its own worldview system including beliefs about "ultimate reality, epistemology, ontology, ethics, purpose, etc."^[8] Muslims believe that the Qur'an is the literal word and
the final revelation of God for the guidance of humankind.
Science in the broadest sense refers to any falsifiable system of knowledge attained by verifiable means,^[9] and in a narrower sense to a system of acquiring knowledge based on empiricism,
experimentation, and methodological naturalism, as well as to the organized body of knowledge humans have gained by such research. Scientists maintain that scientific investigation must adhere to the
scientific method, a process for evaluating empirical knowledge that explains observable events in nature as results of natural causes, rejecting supernatural notions.
One of the most important features of Science, notably Physics is the precise quantitative prediction based on its current level understanding of a phenomenon. In this aspect it differs from many
religious texts where physical phenomena are depicted in a very qualitative way, often by the use of words carrying several meanings.
History[ ]
Classical Islamic science[ ]
Main article: Science in medieval Islam
In the history of science, Islamic science refers to the science developed under Islamic civilization between the 8th and 16th centuries,^[10] during what is known as the Islamic Golden Age.^[11] It
is also known as Arabic science since the majority of texts during this period were written in Arabic, the lingua franca of Islamic civilization. Despite these terms, not all scientists during this
period were Muslim or Arab, as there were a number of notable non-Arab scientists (most notably Persians), as well as some non-Muslim scientists, who contributed to scientific studies in the Islamic
A number of modern scholars such as Fielding H. Garrison,^[12] Bertrand Russell,^[13] Abdus Salam and Hossein Nasr consider modern science and the scientific method to have been greatly influenced by
Muslim scientists who introduced a modern empirical, experimental and quantitative approach to scientific inquiry. Some scholars, notably Donald Routledge Hill, Ahmad Y Hassan,^[14] Abdus Salam,^[15]
and George Saliba,^[16] have referred to their achievements as a Muslim scientific revolution,^[17]^[18] though this does not contradict the traditional view of the Scientific Revolution which is
still supported by most scholars.^[19]^[20]^[21]
According to many historians, science in Islamic civilization flourished during the Middle Ages, but began declining at some time around the 14th^[22] to 16th^[10] centuries. At least some scholars
blame this on the "rise of a clerical faction which froze this same science and withered its progress."^[23] Examples of conflicts with prevailing interpretations of Islam and science - or at least
the fruits of science - thereafter include the demolition of Taqi al-Din's great Istanbul observatory of Taqi al-Din in Galata, "comparable in its technical equipment and its specialist personnel
with that of his celebrated contemporary, the Danish astronomer Tycho Brahe." But while Brahe's observatory "opened the way to a vast new development of astronomical science," Taqi al-Din's was
demolished by a squad of Janissaries, "by order of the sultan, on the recommendation of the Chief Mufti," sometime after 1577 AD.^[24]^[25]
It is believed that it was the empirical attitude of the Qur'an and Sunnah which inspired medieval Muslim scientists, in particular Alhazen (965-1037),^[26]^[27] to develop the scientific method.^
[28]^[29]^[30] It is also known that certain advances made by medieval Muslim astronomers, geographers and mathematicians was motivated by problems presented in Islamic scripture, such as
Al-Khwarizmi's (c. 780-850) development of algebra in order to solve the Islamic inheritance laws,^[31] and developments in astronomy, geography, spherical geometry and spherical trigonometry in
order to determine the direction of the Qibla, the times of Salah prayers, and the dates of the Islamic calendar.^[32]
Other such examples include Ibn al-Nafis (1213-1288), who discovered the pulmonary circulation in 1242 and used his discovery as evidence for the orthodox Islamic doctrine of bodily resurrection.^
[33] Ibn al-Nafis also used Islamic scripture as justification for his rejection of wine as self-medication.^[34] Ali Kuşçu's (1403-1474) support for the Earth's rotation and his rejection of
Aristotelian cosmology (which advocates a stationary Earth) was also motivated by religious opposition to Aristotle by orthodox Islamic theologians such as Al-Ghazali.^[35]^[36] Criticisms against
alchemy and astrology were also motivated by religion, such as the views of astrologers conflicting with orthodox Islam.^[37]
Arrival of modern science in Islamic world[ ]
At the beginning of the nineteenth century, modern science arrived in the Muslim world but it wasn't the science itself that affected Muslim scholars. Rather, it "was the transfer of various
philosophical currents entangled with science that had a profound effect on the minds of Muslim scientists and intellectuals. Schools like Positivism and Darwinism penetrated the Muslim world and
dominated its academic circles and had a noticeable impact on some Islamic theological doctrines." There were different responses to this among the Muslim scholars:^[38] These reactions, in words of
Professor Mehdi Golshani, were the following:
“ 1. Some rejected modern science as corrupt foreign thought, considering it incompatible with Islamic teachings, and in their view, the only remedy for the stagnancy of Islamic societies would be
the strict following of Islamic teachings.^[39]
2. Other thinkers in the Muslim world saw science as the only source of real enlightenment and advocated the complete adoption of modern science. In their view, the only remedy for the
stagnation of Muslim societies would be the mastery of modern science and the replacement of the religious worldview by the scientific worldview.
3. The majority of faithful Muslim scientists tried to adapt Islam to the findings of modern science; they can be categorized in the following subgroups: (a) Some Muslim thinkers attempted to
justify modern science on religious grounds. Their motivation was to encourage Muslim societies to acquire modern knowledge and to safeguard their societies from the criticism of Orientalists
and Muslim intellectuals. (b) Others tried to show that all important scientific discoveries had been predicted in the Qur'an and Islamic tradition and appealed to modern science to explain
various aspects of faith. (c) Yet other scholars advocated a re-interpretation of Islam. In their view, one must try to construct a new theology that can establish a viable relation between
Islam and modern science. The Indian scholar, Sayyid Ahmad Khan, sought a theology of nature through which one could re-interpret the basic principles of Islam in the light of modern science.
(d) Then there were some Muslim scholars who believed that empirical science had reached the same conclusions that prophets had been advocating several thousand years ago. The revelation had
only the privilege of prophecy.
4. Finally, some Muslim philosophers separated the findings of modern science from its philosophical attachments. Thus, while they praised the attempts of Western scientists for the discovery of
the secrets of nature, they warned against various empiricist and materialistic interpretations of scientific findings. Scientific knowledge can reveal certain aspects of the physical world,
but it should not be identified with the alpha and omega of knowledge. Rather, it has to be integrated into a metaphysical framework—consistent with the Muslim worldview—in which higher
levels of knowledge are recognized and the role of science in bringing us closer to God is fulfilled.^[8] ”
Compatibility of Islam and the development of science[ ]
Whether Islamic culture has promoted or hindered scientific advancement is disputed. Islamists such as Sayyid Qutb argue that since "Islam appointed" Muslims "as representatives of God and made them
responsible for learning all the sciences,"^[40] science cannot but prosper in a society of true Muslims. Many "classical and modern [sources] agree that the Qur'an condones, even encourages the
acquisition of science and scientific knowledge, and urges humans to reflect on the natural phenomena as signs of God's creation." Some scientific instruments produced in classical times in the
Islamic world were inscribed with Qur'anic citations. Many Muslims agree that doing science is an act of religious merit, even a collective duty of the Muslim community.^[41]
Others say traditional interpretations of Islam are not compatible with the development of science. Author Rodney Stark, explains Islam's lag behind the West in scientific advancement after (roughly)
1500 AD to opposition by traditional ulema to efforts to formulate systematic explanation of natural phenomenon with "natural laws." They believed such laws were blasphemous because they limit
"Allah's freedom to act" as He wishes. This principle was enshired in aya 14:4: "Allah sendeth whom He will astray, and guideth whom He will," which (they believed) applied to all of creation not
just humanity.^[42]
Decline[ ]
In the early twentieth century ulema forbade the learning of foreign languages and dissection of human bodies in the medical school in Iran.^[43] The ulama at the Islamic university of Al-Azhar in
Cairo taught the Ptolemaic astronomical system (in which the sun circles the earth) until compelled to adopt the Copernican system by the Egyptian government in 1961.^[44]
In recent years, the lagging of the Muslim world in science is manifest in the disproportionately small amount of scientific output as measured by citations of articles published in internationally
circulating science journals, annual expenditures on research and development, and numbers of research scientists and engineers.^[45] Skepticism of science among some Muslims is reflected in issues
such as resistance in Muslim northern Nigeria to polio inoculation, which some believe is "an imaginary thing created in the West or it is a ploy to get us to submit to this evil agenda."^[46]
Qur'an and Science[ ]
Main article: Qur'an and Science
The belief that Qur'an had prophesied scientific theories and discoveries has become a strong and widespread belief in the contemporary Islamic world; these prophecies are often provided as a proof
of the divine origin of the Qur'an.^[47]
The scientific facts claimed to be in the Qur'an exist in different subjects, including creation, astronomy, the animal and vegetables kingdom, and human reproduction.
"a time is fixed for every prophecy; you will come to know in time" (^[Qur'an 6:67]). Islamic scholar Zaghloul El-Naggar thinks that this verse refers to the scientific facts in the Qur'an that would
be discovered by the world in modern time, centuries after the revelation.^[47]
This belief is, however, arguable in the Muslim world, while some support it, other Muslim scholars oppose the belief, claiming that the Qur'an is not a book of science; al-Biruni, one of the most
celebrated Muslim scientists of the classical period, assigned to the Qur'an a separate and autonomous realm of its own and held that the Qur'an "does not interfere in the business of science nor
does it infringe on the realm of science."^[47] These scholars argued for the possibility of multiple scientific explanation of the natural phenomena, and refused to subordinate the Qur'an to an
ever-changing science.^[47]
[ ]
Fossils of ancient humans[ ]
Main article: Islamic creationism
There are three specific verses in the Qur'an concerning human creation:^[48] (^[Qur'an 3:59], ^[Qur'an 4:1], ^[Qur'an 32:7]) According to the first two verses, Adam and Eve were directly created by
God from clay that give sound, they did not descend from any other species as proposed by Charles Darwin and the rest of mankind is the progeny of Adam and Eve. The third verse implies that there
were three stages in their creation, and can be interpreted as follows:^[48]
Conception and inherited characteristics[ ]
The most prominent of the ancient Greek thinkers who wrote on medicine were Hippocrates, Aristotle, and Galen. Hippocrates and Galen, in contrast with Aristotle, wrote that the contribution of
females to children is equal to that of males, and the vehicle for it is a substance similar to the semen of males.^[49] Basim Musallam writes that the ideas of these men were widespread through the
pre-modern Middle East: "Hippocrates, Aristotle, and Galen were as much a part of Middle Eastern Arabic culture as anything else in it."^[49] The sayings in the Quran and those attributed to Muhammad
in the Hadith influenced generations of Muslim scientists by siding with Galen and Hippocrates. Basim Musallam writes: "... the statements about parental contribution to generation in the hadith
paralleled the Hippocratic writings, and the view of fetal development in the Quran agreed in detail with Galen's scientific writings."^[49] He reports that the highly influential medieval Hanbali
scholar Ibn Qayyim, in his book Kitab al-tibyan fi aqsam al-qur'an, cites the following statement of the prophet from the Sahih Muslim:
“ The male semen is white and the female semen is yellowish. When the two meet and the male semen overpowers the female semen, it will be male; when the female semen overpowers the male semen, it
will be female.^[49] „
Ibn Qayyim also quotes a different hadith from the same collection, which is quoted by other Muslim authors as well. Having been asked the question "from what is man created," the Prophet replies:
“ He is created of both, the semen of the man and the semen of the woman. The man's semen is thick and forms the bones and the tendons. The woman's semen is fine and forms the flesh and blood.^[49] „
See also[ ]
References[ ]
1. ↑ Egyptian Muslim geologist Zaghloul El-Naggar quoted in Science and Islam in Conflict Discover magazine 06.21.2007
2. ↑ "Modern Europe's industrial culture did not originate in Europe but in the Islamic universities of Andalusia and of the East. The principle of the experimental method was an offshoot of the
Islamic concept and its explanation of the physical world, its phenomena, its forces and its secrets." From: Qutb, Sayyad, Milestones, p.111
3. ↑ "Islam, Knowledge, and Science - USC MSA Compendium of Muslim Texts". http://www.usc.edu/dept/MSA/introduction/woi_knowledge.html.
4. ↑ "Islam and science – unhappy bedfellows", Pervez Hoodbhoy, 2006. Formerly at http://www.globalagendamagazine.com/2006/Hoodbhoy.asp (dead link can be accessed in the Internet archive)
5. ↑ Cook, Michael, The Koran: A Very Short Introduction, Oxford University Press, (2000), p.30
6. ↑ see also: Ruthven, Malise, A Fury For God, London ; New York : Granta, (2002), p.126
7. ↑ ^8.0 ^8.1 Mehdi Golshani, Can Science Dispense With Religion?
8. ↑ See, e.g., the entry Science in the Oxford English Dictionary ISBN 0-19-522217-2
9. ↑ ^10.0 ^10.1 Ahmad Y Hassan, Factors Behind the Decline of Islamic Science After the Sixteenth Century
10. ↑ Sabra, A. I. (1996). "Situating Arabic Science: Locality versus Essence". Isis 87 (4): 654–670. doi:10.1086/357651. http://links.jstor.org/sici?sici=
""Let us begin with a neutral and innocent definition of Arabic, or what also may be called Islamic, science in terms of time and space: the term Arabic (or Islamic) science the scientific
activities of individuals who lived in a region that might extended chronologically from the eighth century A.D. to the beginning of the modern era, and geographically from the Iberian
Peninsula and north Africa to the Indus valley and from the Southern Arabia to the Caspian Sea—that is, the region covered for most of that period by what we call Islamic Civilization, and in
which the results of the activities referred to were for the most part expressed in the Arabic Language. We need not be concerned over the refinements that obviously need to be introduced
over this seemingly neutral definition.""
11. ↑ Fielding H. Garrison, History of Medicine
12. ↑ Prof. Osman Bakar (Georgetown University), Islam's Contribution to Human Civilization: Science and Culture, CIC's annual Ottawa dinner, October 15, 2001.
13. ↑ Ahmad Y Hassan and Donald Routledge Hill (1986), Islamic Technology: An Illustrated History, p. 282, Cambridge University Press.
14. ↑ Abdus Salam, H. R. Dalafi, Mohamed Hassan (1994). Renaissance of Sciences in Islamic Countries, p. 162. World Scientific, ISBN 9971507137.
15. ↑ George Saliba (1994), A History of Arabic Astronomy: Planetary Theories During the Golden Age of Islam, p. 245, 250, 256-257. New York University Press, ISBN 0814780237.
16. ↑ Abid Ullah Jan (2006), After Fascism: Muslims and the struggle for self-determination, "Islam, the West, and the Question of Dominance", Pragmatic Publishings, ISBN 978-0-9733687-5-8.
17. ↑ Salah Zaimeche (2003), An Introduction to Muslim Science, FSTC.
18. ↑ Grant, Edward. The Foundations of Modern Science in the Middle Ages: Their Religious, Institutional, and Intellectual Contexts. Cambridge: Cambridge Univ. Pr., 1996.
19. ↑ Herbert Butterfield, The Origins of Modern Science, 1300-1800.
20. ↑ Thomas Kuhn, The Copernican Revolution, (Cambridge: Harvard Univ. Pr., 1957), p. 142.
21. ↑ Islam by Alnoor Dhanani in Science and Religion, 2002, p.88
22. ↑ Islamic Technology: An Illustrated History by Ahmad Y. al-Hassan and Donald Hill, Cambridge University Press, 1986, p.282
23. ↑ Aydin Sayili, The Observatory in Islam and its place in the General History of the Observatory (Ankara: 1960), pp. 289 ff.
24. ↑ Islamic Technology: An Illustrated History by Ahmad Y. al-Hassan and Donald Hill, Cambridge University Press, 1986, p.282
25. ↑ Bettany, Laurence (1995), "Ibn al-Haytham: an answer to multicultural science teaching?", Physics Education 30: 247-252 [247])
26. ↑ Steffens, Bradley (2006), Ibn al-Haytham: First Scientist, Morgan Reynolds Publishing, ISBN 1599350246 (cf. Steffens, Bradley, Who Was the First Scientist?, Ezine Articles )
27. ↑ Ahmad, I. A. (June 3, 2002), The Rise and Fall of Islamic Science: The Calendar as a Case Study, Faith and Reason: Convergence and Complementarity, Al Akhawayn University. Retrieved on
28. ↑ C. A. Qadir (1990), Philosophy and Science in the lslumic World, Routledge, London)
29. ↑ Ahmad, I. A. (1995), "The impact of the Qur'anic conception of astronomical phenomena on Islamic civilization", Vistas in Astronomy 39 (4): 395–403, doi:10.1016/0083-6656(95)00033-X
30. ↑ Gandz, Solomon (1938). "The Algebra of Inheritance: A Rehabilitation of Al-Khuwārizmī". Osiris 5: 319–391. doi:10.1086/368492. ISSN 0369–7827.
31. ↑ Gingerich, Owen (April 1986), "Islamic astronomy", Scientific American 254 (10): 74, <http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm>. Retrieved on 2008-05-18
32. ↑ Fancy, Nahyan A. G. (2006), "Pulmonary Transit and Bodily Resurrection: The Interaction of Medicine, Philosophy and Religion in the Works of Ibn al-Nafīs (d. 1288)", Electronic Theses and
Dissertations (University of Notre Dame): 232-3, <http://etd.nd.edu/ETD-db/theses/available/etd-11292006-152615>
33. ↑ Fancy, Nahyan A. G. (2006), "Pulmonary Transit and Bodily Resurrection: The Interaction of Medicine, Philosophy and Religion in the Works of Ibn al-Nafīs (d. 1288)", Electronic Theses and
Dissertations (University of Notre Dame): 49-59 & 232-3, <http://etd.nd.edu/ETD-db/theses/available/etd-11292006-152615>
34. ↑ Ragep, F. Jamil (2001a), "Tusi and Copernicus: The Earth's Motion in Context", Science in Context (Cambridge University Press) 14 (1-2): 145–163
35. ↑ F. Jamil Ragep (2001), "Freeing Astronomy from Philosophy: An Aspect of Islamic Influence on Science", Osiris, 2nd Series, Vol. 16, Science in Theistic Contexts: Cognitive Dimensions, p. 49-64,
36. ↑ Saliba, George (1994), A History of Arabic Astronomy: Planetary Theories During the Golden Age of Islam, New York University Press, 60 & 67-69, ISBN 0814780237
37. ↑ Mehdi Golshani, Does science offer evidence of a transcendent reality and purpose?, June 2003
38. ↑ Mehdi Golshani, Does science offer evidence of a transcendent reality and purpose?, June 2003
39. ↑ Qutb, Sayyid, Milestones, p.112
40. ↑ Qur'an and Science, Encyclopedia of the Qur'an
41. ↑ Stark, Rodney, The Victory of Reason, Random House, 2005, p.20-1
42. ↑ Mackey, The Iranians : Persia, Islam and the Soul of a Nation, 1996, p.179
43. ↑ In the Path of God : Islam and Political Power by Daniel Pipes, c1983 p.113
44. ↑ Abdus Salam, Ideals and Realities: Selected Essays of Abdus Salam (Philadelphia: World Scientific, 1987), p. 109.
45. ↑ Nafiu Baba Ahmed, Secretary General of the Supreme Council for Sharia in Nigeria, telling the BBC his opinion of polio and vaccination. In northern Nigeria "more than 50% of the children have
never been vaccinated against polio," and as of 2006 and more than half the world's polio victims live. Nigeria's struggle to beat polio, BBC News, 31 March 20
46. ↑ ^47.0 ^47.1 ^47.2 ^47.3 Ahmad Dallal, Encyclopedia of the Qur'an, Quran and science
47. ↑ ^48.0 ^48.1 Saleem, Shehzad (May 2000). "The Qur’anic View on Creation". Renaissance 10 (5). ISSN 1606-9382. http://www.renaissance.com.pk/maytitl20.htm. Retrieved 2006-10-11.
48. ↑ ^49.0 ^49.1 ^49.2 ^49.3 ^49.4 Basim Musallam, Sex and Society in Islam. Cambridge University Press. Cite error: Invalid <ref> tag; name "musallam0" defined multiple times with different content
Cite error: Invalid <ref> tag; name "musallam0" defined multiple times with different content Cite error: Invalid <ref> tag; name "musallam0" defined multiple times with different content Cite
error: Invalid <ref> tag; name "musallam0" defined multiple times with different content
External links[ ]
By Professor Mehdi Golshani
By Professor Seyyed Hossein Nasr
Others[ ]
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Volts to Watts Calculator - calculator
Volts to Watts Calculator
Volts to Watts Calculator: This calculator helps you convert volts to watts based on the type of current (DC or AC). You can input the voltage and current to find the power in watts.
What is Volts to Watts?
Volts to Watts is a conversion that calculates the power in watts based on voltage and current. The formula is P(W) = V × I for DC and P(W) = PF × V × I for AC.
What is a Volts to Watts Calculator website?
It is a web
that allows users to input voltage and current values to calculate the corresponding power in watts.
How to use the Volts to Watts Calculator?
Select the type of current, enter the voltage and current values, and click 'Calculate' to see the result.
What is the formula for Volts to Watts Calculator?
For DC: P(W) = V × I; For AC Single Phase: P(W) = PF × V × I; For AC Three Phase: P(W) = √3 × V × I.
What are the advantages of using a Volts to Watts Calculator?
It simplifies power calculations, helps in electrical design, and ensures safety by preventing overloads.
What are the disadvantages of using a Volts to Watts Calculator?
It may not account for all variables in complex electrical systems and can lead to errors if incorrect values are inputted.
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Torsion in Mechanics
Principles of Simple Torsion Theory
Simple torsion theory is a foundational approach for analyzing torsion in slender rods and shafts with circular cross-sections. This theory simplifies the problem by considering only the effects of
torsion, excluding bending or axial forces. It assumes that cross-sections of the shaft remain flat and undistorted after twisting, and that the material is isotropic (having uniform properties in
all directions) and homogeneous (consistent in composition throughout). According to this theory, the angle of twist is directly proportional to the product of the shaft's length and the applied
torque, and inversely proportional to the product of the material's modulus of rigidity and the polar moment of inertia. These simplifications allow for straightforward calculations of torsional
stress and strain in circular shafts.
Exploring the Theory of Pure Torsion
The theory of pure torsion is an idealized concept in mechanical engineering that examines the behavior of cylindrical objects under a twisting moment without the presence of other forces or moments.
This theory is particularly applicable to the design of shafts and other rotational components. It assumes that the material is homogeneous and isotropic, and that circular cross-sections before
torsion remain circular after the application of torque. The shear stress at any point in the material is directly proportional to the radial distance from the center of rotation. This relationship
allows engineers to determine the distribution of shear stress across the cross-section of a shaft experiencing pure torsion.
Torsion Test Theory and Material Strength Assessment
Torsion test theory plays a pivotal role in characterizing the torsional strength and ductility of materials. By applying a controlled torque to a test specimen and measuring its response, such as
the angle of twist and the induced shear stress, material properties like the shear modulus (G), the maximum shear stress (\(\tau_{max}\)), and the angle of twist (\(\theta\)) can be determined.
These parameters are critical for selecting appropriate materials for components that will experience torsional loads. Torsion tests provide insights into the material's behavior under stress and
help predict the potential for failure and the service life of the component.
Advanced Torsion Theories and Practical Applications
Advanced torsion theories extend the analysis to more complex situations, such as non-circular cross-sections, composite materials, and dynamic loading conditions. These theories are essential for
advancing knowledge in disciplines such as physics, mechanical engineering, and computational mechanics. For instance, the torsion pendulum theory in physics describes the oscillatory motion of a
body suspended by a wire and subjected to a torque. In mechanical engineering, torsion theory is applied to the design of components like automotive driveshafts and torsion bars in suspension
systems. Computational tools, such as Finite Element Analysis (FEA), enable the simulation of torsional behavior in structures, improving the ability to predict and analyze their performance under
various loading conditions.
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When the Boss is a Woman
^^ I don't know everything, you empty bucket. I only know that women are mad. Sickening you in the process is a bonus, I suppose.
^^ I am trying to make life smoother for you and your fellow women. But, since you are mad, you will never realise what sort of good service I'm doing you here.
I think we all know when he says "all women are crazy" he means his wife. (Too far? meh.)
Give it a rest man.
NGONGE;751379 wrote:
The Zack is married? MARRIED? So what is all this guff about having his first ever woman boss? Ok, ok, I'll give him the benefit of the doubt and assume he's still a newly wed (or at least under five
years of marriage). It may sound evil but I'm actually smiling now!
Who said I am married or not married? Suuqa sol ha iga xidhin ninyahow. What I said was a hypothetical statement.
I see you are being attacked from left to right LOL.
grasshopper;751412 wrote:
I think we all know when he says "all women are crazy" he means his wife. (Too far? meh.)
Give it a rest man.
Ah you silly hair pin. I mean my wife, my mother, daughter, grandmother, sister and almost every woman on this earth. Don't take it to heart dee. Accept your madness and help me make all the green
and unsuspecting boys understand it.
Naag waalan yaaba heli kara, I am assuming when Ngonge says women are mad, he means wey wada waalanyihiin, it is easy to hang out with a mad woman, I find that to be OK, as everything goes according
to plan if there was a plan already, but it is also true that if a woman is insane, it is much easier to hang with and be friend with.
Ps: For those who befriend or make asxaabo with dumar miyir qabo, I am pretty sure they had tough time, not to mention the lack of progress.
^ Heh.
NG, I can maybe be persuaded that we can occasionally be irrational but I'll never accept inherent madness. No siree. Crazy don't live here.
^^ I rest my case.
(Fair enough Val, I accept your amazing reasoning. You must be in a very good mood today).
I'm starting to see the madness he describes as a term of endearment. If Ng's mother is mad, wax laga xanaaqoba maaha. MashaAllah.
loool all this discussion because Zack is gonna get a female boss, or because NG called all women crazy..is it me or we have too much time in our hands
Hey hey this is a big deal! It is worth more than a discussion, an action LOL. By the way, 16 days to go!
lol@ 16 more days to go..u kinda overthinking the whole thing..
Anyhow, inshallah khyr I will make a dua for you. Everything will work out
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Constant Multiplier Optimization to Reduce Area
This example shows how to perform a design-level area optimization in HDL Coder™ by converting constant multipliers into shifts and adds using canonical signed digit (CSD) techniques. The CSD
representation of multiplier constants for example, in gain coefficients or filter coefficients) significantly reduces the area of the hardware implementation.
Canonical Signed Digit (CSD) Representation
A signed digit (SD) representation is an augmented binary representation with weights 0,1 and -1. -1 is represented in HDL Coder generated code as 1'.
For example, here are a couple of signed digit representations for 93:
Note that the signed digit representation is non-unique. A canonical signed digit (CSD) representation is an SD representation with the minimum number of nonzero elements.
Here are some properties of CSD numbers:
1. No two consecutive bits in a CSD number are nonzero
2. CSD representation uses minimum number of nonzero digits
3. CSD representation of a number is unique
CSD Multiplier
Let us see how a CSD representation can yield an implementation requiring a minimum number of adders.
Let us look at CSD example:
y = 231 * x
= (11100111) * x % 231 in binary form
= (1001'01001') * x % 231 in signed digit form
= (256 - 32 + 8 - 1) * x %
= (x << 8) - (x << 5) + (x << 3) -x % cost of CSD: 3 Adders
HDL Coder CSD Implementation
HDL Coder uses a CSD implementation that differs from the traditional CSD implementation. This implementation preferentially chooses adders over subtractors when using the signed digit
representation. In this representation, sometimes two consecutive bits in a CSD number can be nonzero. However, similar to the CSD implementation, the HDL Coder implementation uses the minimum number
of nonzero digits. For example:
In the traditional CSD implementation, the number 1373 is represented as:
1373 = 0101'01'01'001'01
This implementation does not have two consecutive nonzero digits in the representation. The cost of this implementation is 1 adder and 4 subtractors.
In the HDL Coder CSD implementation, the number 1373 is represented as:
1373 = 00101011001'01
This implementation has two consecutive nonzero digits in the representation but uses the same number of nonzero digits as the previous CSD implementation. The cost of this implementation is 4 adders
and 1 subtractor which shows that adders are preferred to subtractors.
FCSD Multiplier
A combination of factorization and CSD representation of a constant multiplier can lead to further reduction in hardware cost (number of adders).
FCSD can further reduce the number of adders in the above constant multiplier:
y = 231 * x
y = (7 * 33) * x
y_tmp = (x << 5) + x
y = (y_tmp << 3) - y_tmp % cost of FCSD: 2 Adders
CSD/FCSD Costs
This table shows the costs (C) of all 8-bit multipliers.
MATLAB® Design
The MATLAB code used in this example implements a simple FIR filter. The example also shows a MATLAB test bench that exercises the filter.
design_name = 'mlhdlc_csd';
testbench_name = 'mlhdlc_csd_tb';
Simulate the Design
Simulate the design with the test bench prior to code generation to make sure there are no runtime errors.
Create a Fixed-Point Conversion Config Object
To perform fixed-point conversion, you need a 'fixpt' config object.
Create a 'fixpt' config object and specify your test bench name:
close all;
fixptcfg = coder.config('fixpt');
fixptcfg.TestBenchName = 'mlhdlc_csd_tb';
Create an HDL Code Generation Config Object
To generate code, you must create an 'hdl' config object and set your test bench name:
hdlcfg = coder.config('hdl');
hdlcfg.TestBenchName = 'mlhdlc_csd_tb';
Generate Code without Constant Multiplier Optimization
hdlcfg.ConstantMultiplierOptimization = 'None';
Enable the 'Unroll Loops' option to inline multiplier constants.
hdlcfg.LoopOptimization = 'UnrollLoops';
codegen -float2fixed fixptcfg -config hdlcfg mlhdlc_csd
Examine the generated code.
Take a look at the resource report for adder and multiplier usage without the CSD optimization.
Generate Code with CSD Optimization
hdlcfg.ConstantMultiplierOptimization = 'CSD';
Enable the 'Unroll Loops' option to inline multiplier constants.
hdlcfg.LoopOptimization = 'UnrollLoops';
codegen -float2fixed fixptcfg -config hdlcfg mlhdlc_csd
Examine the generated code.
Examine the code with comments that outline the CSD encoding for all the constant multipliers.
Look at the resource report and notice that with the CSD optimization, the number of multipliers is reduced to zero and multipliers are replaced by shifts and adders.
Generate Code with FCSD Optimization
hdlcfg.ConstantMultiplierOptimization = 'FCSD';
Enable the 'Unroll Loops' option to inline multiplier constants.
hdlcfg.LoopOptimization = 'UnrollLoops';
codegen -float2fixed fixptcfg -config hdlcfg mlhdlc_csd
Examine the generated code.
Examine the code with comments that outline the FCSD encoding for all the constant multipliers. In this particular example, the generated code is identical in terms of area resources for the
multiplier constants. However, take a look at the factorizations of the constants in the generated code.
If you choose the 'Auto' option, HDL Coder will automatically choose between the CSD and FCSD options for the best result.
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G. I. Timofeeva's research works | National Academy of Sciences of Belarus and other places
What is this page?
This page lists works of an author who doesn't have a ResearchGate profile or hasn't added the works to their profile yet. It is automatically generated from public (personal) data to further our
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Publications (22)
Pulsed Parametric Oscillator Based on KTP Crystals with a Three-Mirror Ring Cavity: Numerical Simulation
Journal of Applied Spectroscopy
Optical parametric generation in biaxial KTA crystal under pumping by YAG:Nd laser in arbitrary directions
October 2021
15 Reads
1 Citation
Proceedings of the National Academy of Sciences of Belarus Physics and Mathematics Series
Herein, on the basis of expressions for the refractive indices of isonormal waves, the possibility of performing collinear phase matching for optical parametric generation in arbitrary directions of
a biaxial KTA crystal under pumping by radiation of a YAG:Nd laser is analyzed. The tuning curves that determine the tuning range of the signal and idler for type-I and II-type phase-matching and
arbitrary angles θ and φ in cases where the tuning is carried out along the angle θ at a fixed angle φ and vice versa are calculated. The effective nonlinear coefficient is determined. It is shown that
their maximum value is achieved аt a polar angle θ = 90° and type-II phase-matching. For the case of generation of eye-safe radiation the spectral and angular phase matching widths were estimated, as
well as gain widths of KTA-OPO under monochromatic pumping.
Efficiency of Resonance SRS at Different Detunings from Resonance
Journal of Applied Spectroscopy
Dependences of the resonance SRS efficiency on the pump radiation and medium scattering centers were theoretically analyzed. It was shown that this efficiency cannot be significant because the pump
radiation and Stokes components were highly absorbed.
Stimulated Resonance Raman Scattering Considering Radiation Impacts on Energy-Level Populations
September 2019
15 Reads
4 Citations
Journal of Applied Spectroscopy
A system of equations describing stimulated resonance and spontaneous Raman scattering considering radiation impacts on scattering-center level populations was derived. The contribution of the
resonant transition to the Raman gain coefficient was analyzed in the simplest case as a function of the resonance mismatch.
Infrared radiation generation at forced combination scattering
Proceedings of the National Academy of Sciences of Belarus Physics and Mathematics Series
Resonant Two-Photon Transitions
May 2018
80 Reads
2 Citations
Journal of Applied Spectroscopy
We have developed a theory for a two-photon transition when the frequencies of the absorbed or emitted radiation are in resonance with transitions to the same intermediate level in the medium. We
have determined the conditions under which such resonant two-photon transitions can play an important role.
Method for Optimizing a Raman Laser that Generates Several Stokes Components
Journal of Applied Spectroscopy
A system of equations is constructed for the power of the Stokes components generated individually or simultaneously by a steady-state Raman laser. Examples of lasing in one, two, or three components
are used to show that these equations can be used in a simple way to optimize the operation of this type of laser system.
Threshold and Efficiency of an Optial Parametric Generator as Functions of Cavity and Pump Parameters
Journal of Applied Spectroscopy
A simple method for optimizing the parameters of ring optical parametric generator cavities is proposed. Expressions describing the threshold and efficiency of continuous wave generation as functions
of the cavity parameters at the frequencies of the interacting waves are obtained. It is shown that the threshold is proportional to the product of the losses of interacting waves and that the
generation efficiency as a function of the output mirror reflection coefficient for the signal wave has a maximum. The expressions determining the height and position of this maximum are given.
Thermal effects in eye-safe ring optical parametric oscillator based on KTiOPO4 crystalAccounting for Transverse Inhomogeneity of Radiation Beams in Laser Raman Scattering
May 2016
13 Reads
1 Citation
Journal of Applied Spectroscopy
A simple method of accounting for transverse inhomogeneity of the pump and Stokes radiation beams in the description of stimulated Raman scattering (SRS) using intensity-transfer equations for
interacting beams is proposed. Features of the method are illustrated using the calculated dependences of the Raman laser efficiency on the output mirror reflectivity and the pump pulse energy as
Citations (4)
... When a high-intensity laser irradiates a material molecule, photons are scattered into lower-frequency photons by two vibrational dynamic jumps, which drive the rapid growth of Stokes waves
in the medium. At the same time, most of the pump energy is delivered to the Stokes wave [2][3][4]. Higher-order nonlinear effects, including the intra-pulse Raman effect, cannot be ignored for
pulse widths less than 0.1 ps [5,6]. In the anomalous dispersion region of the fiber, optical solitons are generated due to the interaction of dispersion and nonlinear effects. ...
Stimulated Resonance Raman Scattering Considering Radiation Impacts on Energy-Level Populations
• Citing Article
• September 2019
Journal of Applied Spectroscopy
... We start by considering a finite-difference model of an intracavity Raman laser, based on that in [24] and considered in several other works [25]- [29]. The equation relating the rate of
change of the intracavity Stokes intensity I S with time t is ...
Theory of solid state Raman laser stationary generation of visible radiation
Laser Physics Letters
... where ܣ/1 is the normalised overlap integral of the Stokes and fundamental transverse intensity profiles (see e.g. [8,9]) 1 ܣ = ܫ ிܫ ௌܣ݀ ܫ ிܣ݀ ܫ ிܣ݀ . (2.31) ܣ
determines the power flow between the fields, and can be calculated from transverse profiles measured under lasing conditions. ...
Power and lasing threshold for longitudinally pumped lasers with intracavity raman self-conversion
Journal of Applied Spectroscopy
... Specifically, the generation band of the axial Stokes beam was shifted towards the high frequency side with respect to the generation band of the conical Stokes emission [23]. The frequency
shift of the axial Stokes generation was also observed with respect to the Stokes generation excited by a Gaussian beam [24]. ...
Stimulated Raman scattering spectrum of Gaussian and Bessel light beams
• Citing Article
• September 2009
Quantum Electronics
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How do you do two-column geometrical proofs? + Example
How do you do two-column geometrical proofs?
1 Answer
Draw a table with two columns;
in the first column write things you know ("Statements" or "Assertions");
in the second column write the Reason you know the corresponding assertion is true.
This is perhaps best explained with an example:
If a line $A B$ intersects another line $C D$ at a point $M$
prove that $\angle A M D = \angle C M B$
#{: (color(black)("Assertion")," | ",color(black)("Reason")), (bar(color(white)("XXXXXXXXXXXXXXXXXX"))," | ",bar(color(white)("XXXXXXXXXXXX"))), (/_AMD+/_DMB = 180^@," | ","Definition of a
straight line"), (bar(color(white)("XXXXXXXXXXXXXXXXXX"))," | ",bar(color(white)("XXXXXXXXXXXX"))), (/_CMB+/_DMB = 180^@," | ","Definition of a straight line"), (bar(color(white)
("XXXXXXXXXXXXXXXXXX"))," | ",bar(color(white)("XXXXXXXXXXXX"))), (/_AMD+/_DMB = /_CMB+/_DMB," | ","Thing that are equal to the same"), (," | "," thing are equal to each other"), (bar(color
(white)("XXXXXXXXXXXXXXXXXX"))," | ",bar(color(white)("XXXXXXXXXXXX"))), (/_AMD = /_CMB," | ","Subtracting the same amount from"), (," | "," both sides of an equality leaves an equality"), (bar
(color(white)("XXXXXXXXXXXXXXXXXX")),,bar(color(white)("XXXXXXXXXXXX"))) :}#
Impact of this question
3287 views around the world
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Some essential relations for the quaternion quadratic-phase fourier transform.
Mawardi Bahri and Samsul Ariffin Abdul Karim (2023) Some essential relations for the quaternion quadratic-phase fourier transform. MATHEMATICS, 11. p. 1235. ISSN 2227-7390
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Motivated by the fact that the quaternion Fourier transform is a powerful tool in quaternion signal analysis, here, we study the quaternion quadratic-phase Fourier transform, which is a generalized
version of the quaternion Fourier transform. We first give a definition of the quaternion quadratic-phase Fourier transform. We derive in detail some essential properties related to this generalized
transformation. We explore how the quaternion quadratic-phase Fourier transform is related to the quaternion Fourier transform. It is shown that this relation allows us to obtain several versions of
uncertainty principles concerning the quaternion quadratic-phase Fourier transform.
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How To Calculate E Cell
••• AntonioGuillem/iStock/GettyImages
Electrochemical cells tell you about how batteries charge circuits and how electronic devices like cell phones and digital watches are powered. Looking into E cell chemistry, the potential of
electrochemical cells, you'll find chemical reactions powering them that send electric current through their circuits. The potential E of a cell can tell you how these reactions occur.
Calculating E Cell
• Manipulate the half reactions by rearranging them, multiplying them by integer values, flipping the sign of the electrochemical potential, and multiplying the potential. Make sure you follow
rules of reduction and oxidation. Sum the electrochemical potentials for each half reaction in a cell to get the total electrochemical or electromotive potential of a cell.
To calculate the electromotive potential, also known as potential of the electromotive force (EMF), of a galvanic, or voltaic, cell using the E Cell formula when calculating E Cell:
1. Split the equation into half reactions if it isn't already.
2. Determine which equation(s), if any, must be flipped or multiplied by an integer. You can determine this by first figuring out which half reactions are most likely to occur in a spontaneous
reaction. The smaller the magnitude of the electrochemical potential for a reaction, the more likely it is to occur. However, the overall reaction potential must remain positive.
1. For example, a half reaction with electrochemical potential of -.5 V is more likely to occur than one with potential 1 V.
2. When you've determined which reactions are most likely to occur, they will form the basis of the oxidation and reduction used in the electrochemical reaction.
3. Flip equations and multiply both sides of equations by integer numbers until they sum up to the overall electrochemical reaction and the elements on both sides cancel out. For any equation that
you flip, reverse the sign. For any equation you multiply by an integer, multiply the potential by the same integer.
4. Sum up the electrochemical potentials for each reaction while taking into account negative signs.
You can remember the E cell equation cathode anode with the mnemonic "Red Cat An Ox" that tells you reduction occurs at the cathode and the anode oxidizes.
Calculate the Electrode Potentials of the Following Half-Cells
For example, we may have a galvanic cell with a a DC electrical power source. It uses the following equations in a classic AA alkaline battery with corresponding half reaction electrochemical
potentials. Calculating e cell is easy using the E cell equation for the cathode and anode.
1. MnO[2](s) + H[2]O + e^− → MnOOH(s) + OH^-(aq); E^o= +0.382 V
2. Zn (s)+ 2 OH ^-(aq) → Zn(OH)[2](s) + 2e- ; E^o = +1.221 V
In this example, the first equation describes water H[2]O being reduced by losing a proton (H^+) to form OH^- while magnesium oxide MnO[2] is oxidized by gaining a proton (H^+) to form
manganese oxide-hydroxide MnOOH. The second equation describes zinc Zn becoming oxidized with two hydroxide ion OH ^- to form zinc hydroxide Zn(OH)[2] while releasing two electrons.
To form the overall electrochemical equation we want, you first note that equation (1) is more likely to occur than equation (2) because it has a lower magnitude of electrochemical potential. This
equation is a reduction of water H[2]O to form hydroxide OH^-and oxidation of magnesium oxide MnO[2]. This means the corresponding process of the second equation must oxidize hydroxide OH^
-to revert it back to water H[2]O . To achieve this, you must reduce zinc hydroxide Zn(OH)[2]back to zinc Zn.
This means the second equation must be flipped. If you flip it and change the sign of the electrochemical potential, you obtain Zn(OH)[2](s) + 2e- → Zn (s)+ 2 OH ^-(aq) with a corresponding
electrochemical potential E^o = -1.221 V.
Before summing the two equations together, you must multiply each reactant and product of the first equation by the integer 2 to make sure the 2 electrons of the second reaction balance out the
single electron from the first one. This means our first equation becomes 2MnO[2](s) + 2 H[2]O + 2e^− → 2MnOOH(s) + 2OH^-(aq) with an electrochemical potential of E^o= +0.764 V
Add these two equations together and the two electrochemical potentials together to get a combined reaction: 2MnO[2](s) + 2 H[2]O + Zn(OH)[2](s) → Zn (s) + MnOOH(s) with
electrochemical potential -0.457 V. Note that the 2 hydroxide ions and the 2 electrons on both sides cancel out when creating the ECell formula.
E Cell Chemistry
These equations describe the oxidation and reduction processes with a semi-porous membrane separated by a salt bridge. The salt bridge is made of a material such as potassium sulfate that serves as
n inert electrolyte that lets ion diffuse across its surface.
At the cathodes, oxidation, or loss of electrons, occurs, and, at the anodes, reduction, or gain of electrons, occurs. You can remember this with the mnemonic word "OILRIG." It tells you that
"Oxidation Is Loss" ("OIL") and "Reduction Is Gain" ("RIG"). The electrolyte is the liquid that lets ions flow through both of these parts of the cell.
Remember to prioritize equations and reactions that are more likely to occur because they have a lower magnitude of electrochemical potential. These reactions form the basis for galvanic cells and
all their uses, and similar reactions can occur in biological contexts. Cell membranes generate transmembrane electrical potential as ions move across the membrane and through electromotive chemical
For example, the conversion of reduced nicotinamide adenine dinucleotide (NADH) in the presence protons (H^+) and molecular oxygen (O[2]) produces its oxidized counterpart (NAD^+) alongside
water (H[2]O) as part of the electron transport chain. This occurs with a proton electrochemical gradient caused by the potential to let oxidative phosphorylation occur in mitochondria and
produce energy.
Nernst Equation
The Nernst equation lets you calculate the electrochemical potential using the concentrations of products and reactants at equilibrium with cell potential in volts E[cell] as
in which E^-[cell] is the potential for the reduction half reaction, R is the universal gas constant (8.31 J x K−1 mol−1), T is temperature in Kelvins, z is the number of electrons
transferred in the reaction, and Q is the reaction quotient of the overall reaction.
The reaction quotient Q is a ratio involving concentrations of products and reactants. For the hypothetical reaction: aA + bB ⇌ cC + dD with reactants A and B, products C and D, and
corresponding integer values a, b, c, and d, the reaction quotient Q would be Q = [C]^c[D]^d / [A]^a[B]^b with each bracketed value as the concentration, usually in mol/L. For any
example, the reaction measures this ration of products to reactants.
Potential of an Electrolytic Cell
Electrolytic cells differ from galvanic cells in that they use an external battery source, not the natural electrochemical potential, to drive electricity through the circuit. can use electrodes
inside the electrolyte in a nonspontaneous reaction.
These cells also use an aqueous or molten electrolyte in contrast to the salt bridge of galvanic cells. The electrodes match the positive terminal, the anode, and negative terminal, the cathode, of
the battery. While galvanic cells have positive EMF values, electrolytic cells have negative ones which means that, for galvanic cells, the reactions occur spontaneously while electrolytic cells
require an external voltage source.
Similar to the galvanic cells, you can manipulate, flip, multiply, and add the half reaction equations to produce the overall electrolytic cell equation.
• Be sure to balance the two half-reactions before calculating E Cell if there are unequal moles of electrons transferred between reactions.
• If the reactants are not kept at standard conditions (1.0 M), you may wish to use the Nernst equation to convert the E Cell to an adjusted value.
• If working with an actual battery instead of data alone, be sure to wear proper safety equipment. Take necessary precautions to avoid electric shock, such as keeping the circuit away from ionized
About the Author
S. Hussain Ather is a Master's student in Science Communications the University of California, Santa Cruz. After studying physics and philosophy as an undergraduate at Indiana University-Bloomington,
he worked as a scientist at the National Institutes of Health for two years. He primarily performs research in and write about neuroscience and philosophy, however, his interests span ethics, policy,
and other areas relevant to science.
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What is yield function and its examples?What is yield function and its examples? - python in Tamil
n programming, the yield keyword is used in Python to create a generator function. It allows the function to produce a sequence of values one at a time, and each time a value is yielded, the
function’s state is saved, allowing it to be resumed from where it left off.
Here’s an example to illustrate the usage of yield:
def fibonacci():
a, b = 0, 1
while True:
yield a
a, b = b, a + b
# Using the generator
fib_gen = fibonacci()
for _ in range(10):
In this example, the fibonacci function is a generator that generates Fibonacci numbers. It uses an infinite loop to keep generating numbers indefinitely. The yield statement is used to yield the
current Fibonacci number, and the function’s state is saved.
By calling next(fib_gen) in the for loop, we retrieve the next value from the generator. The loop runs 10 times, so it prints the first 10 Fibonacci numbers:
The generator function is paused after each yield statement, allowing you to iterate over the sequence of values without generating all the numbers at once. This is particularly useful when dealing
with large or infinite sequences, as it conserves memory by generating values on the fly.
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How Do You Find Armstrong Number(s) in Python
One of the best things about Python is the ease with which you can perform complex calculations. For example, consider how we could use Python to verify or find Armstrong number patterns. The
following Python tutorial will demonstrate how to leverage Python’s features to quickly and efficiently work with Armstrong numbers.
A Closer Look at Armstrong Numbers
You’re probably familiar with prime numbers. A prime number is a value greater than zero whose only factors are itself and one. An Armstrong number is somewhat similar. The sum of the cubes of the
integer’s digits needs to be equal to the initial value. Essentially, an Armstrong number is an n-digit number equal to a total of the nth powers of the number’s digits.
Armstrong numbers seldom have any practical use. But they can show up in a test or interview in order to see how someone would handle a mathematical challenge. For example, a test might ask you to
develop source code to discover or manipulate Armstrong numbers. And we can, in fact, do so quite easily with Python.
Armstrong Numbers and Python
Testing Armstrong numbers in Python is generally just a matter of feeding values into a formula. You can see how this process operates in the following python example.
originalValue = 153
numLen = len(str(originalValue))
inProgValue = originalValue
addValue = 0
while inProgValue != 0:
x = inProgValue % 10
addValue += x**numLen
inProgValue = inProgValue//10
if addValue == originalValue:
print(‘Not Armstrong’)
We start off by declaring a given number to test. In this case we’ll check whether or not 153 is an Armstrong number. We need to begin by finding the number of digits in originalValue. This is best
done in python by converting the value to a string and then running it through len. Next, we pass that value to a variable called inProgValue which will essentially serve as a placeholder to
The declarations end by creating addValue as a repository for the sum of a calculation. In the next step, we go through each digit in our number, multiply it, and then add it to addValue. Finally, we
check the total in addValue against the original number in originalValue. We can then print out whether or not the original value was an Armstrong number.
Bringing It All Together
The previous example is a good start. But we can build on the ideas presented there to vastly expand on the code’s scope. Consider the following example.
def isArmstrong(originalValue):
numLen = len(str(originalValue))
inProgValue = originalValue
addValue = 0
armstrongResult = 0
while inProgValue != 0:
x = inProgValue % 10
addValue += x**numLen
inProgValue = inProgValue//10
if addValue == originalValue:
armstrongResult = 1
armstrongResult = 0
return armstrongResult
for x in range(0, 501):
if isArmstrong(x):
print(str(x)+” is Armstrong”)
print(str(x)+” is not Armstrong”)
The fact that we’re creating a new function is the biggest change from the original code. We keep most of the original program logic, but it’s contained within a new isArmstrong function. We call the
function by passing a single digit as originalValue. And the function ends by returning either a 1 or 0 to indicate whether the tested number is an Armstrong or not.
By putting the test within a function we now have the freedom to easily loop through it. For example, we could iterate through a python list full of specific numbers. Or we could use a for loop to
iterate and test every individual digit within a set range of values. And we do exactly that within the context of this example. The for loop runs through a range of values while passing every
iteration to isArmstrong. Each number is tested, and we print the result to screen within the if conditional.
How Do You Find Armstrong Number(s) in Python
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Radian per Square Millisecond to Radian per Square Centiseco
Units of measurement use the International System of Units, better known as SI units, which provide a standard for measuring the physical properties of matter. Measurement like angular acceleration
finds its use in a number of places right from education to industrial usage. Be it buying grocery or cooking, units play a vital role in our daily life; and hence their conversions.
unitsconverters.com helps in the conversion of different units of measurement like rad/ms² to rad/cs² through multiplicative conversion factors. When you are converting angular acceleration, you need
a Radian per Square Milliseconds to Radian per Square Centiseconds converter that is elaborate and still easy to use. Converting Radian per Square Millisecond to Radian per Square Centisecond is
easy, for you only have to select the units first and the value you want to convert. If you encounter any issues to convert, this tool is the answer that gives you the exact conversion of units. You
can also get the formula used in Radian per Square Millisecond to Radian per Square Centisecond conversion along with a table representing the entire conversion.
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A patient drives 19 miles one way to a medical facility for treatment. How far does the patient drive round-trip in 22 days of treatment?
Correct Answer : D
Round trip means to and from, which is twice the distance from home to a medical facility.
In one day, the round trip=19+19=38 miles
So, in 22 days, the round trip=38*22=836 miles.
The patient will cover 836 miles in 22 days.
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Related Questions
Correct Answer is B
A dependent variable is one that when another variable changes, it also changes. In our case, the insurance premium changes if the age, model, mileage of car changes. Thus, insurance premium is the
dependent variable while the other three are independent variable.
Correct Answer is D
The mean is the total divided by the number of elements in the data set. From the given data set:
Number of items, N=10
The mean is 26.
Correct Answer is B
The median temperature can be found by organizing the temperature values from the smallest to the largest value as follows:
98.6, 98.7, 99.0, 99.0,99.2, 99.3, 99.7, 100.0
(for an even set of numbers, Median = frac{(frac{n}{2})th observation + (frac{n}{2} + 1) th observation}{2})
From the data set above, there are 8temperature values. The median is the temperature value in the middle position, which falls between the(frac{n}{2} th)and((frac{n}{2} + 1) th) position. Here N=
8and median is found as:
(frac{(frac{n}{2})th + (frac{n}{2} + 1) th}{2} = )(frac{(frac{8}{2})th + (frac{8}{2} + 1) th }{2} = 4.5th position)
The element in the 4.5th position is the average of the 4th and 5th element.
(frac{99.0 + 99.2}{2} = 99.1)
Thus 99.1 is the median temperature.
Correct Answer is A
We are given that 1 teaspoon=4.93 mL, we can interpret it as:
Since we are to find the amount in mL, we look for an option that will cancel teaspoon and remain with mL. The second option is the required conversion, and we proceed as follows:
Therefore, 2.5 teaspoons hold about 12.325 mL.
Correct Answer is A
To find the net force, we choose east direction as positive and west as negative. From this, we can present the tag of war in the diagram below.
So, the force on the dog is -190 N and that of the girl is 165 N.
The net force is the sum of the two forces
The resulting force is negative, meaning it is in the west direction. Thus, the net force is 25 N to the west.
Correct Answer is A
Here we collect like terms together and solve for the unknown value of x.
Add 6 to both sides of the equation
Subtract 3x from both sides of the equation
Divide both sides by 4
x = -5
The value of x = -5
Correct Answer is B
Here we are required to find the area of the square of sides 3.1 m. The square is a four-sided figure with each side equal and opposite sides making 90 degrees.
Area of the square =side*side
Side=3.1 m
Area of the square =3.1 m *3.1 m=9.61 m2
Note: 3.1 + 3.1 = 6.2 which is a wrong answer.
Correct Answer is A
From the given problem,
49.5 pounds of fertilizer is needed to farm 1 acre of land. This can be interpreted as:
Now we are needed to find the acres of land that will be farmed using 2000 pounds of fertilizers. To solve this, we use the second option as follows:
2000 pounds of fertilizers can farm approximately 40 acres of land.
Correct Answer is B
In the simple interest, we utilize the following formula to find the simple interest after a period of time in years.
I is the interest
P=Principal or initial deposit
t=time in years
From the given problem, P=$600, r=6%=6/100=0.06, t=5 years. Then
After 5 years, Pat will earn an interest of $180.
Correct Answer is B
from the provided table, the first option represents the amount of gas in gallons while the second column shows the distance in miles. Thus, (6, 144) will denote that the car can 144 miles by
consuming 6 gallons of a gas.
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How do u find the leap year withput using division and modulo operator in your program??? | Sololearn: Learn to code for FREE!
How do u find the leap year withput using division and modulo operator in your program???
Here are some ways of identifying leap years without using a modulo function. Firstly, letâ s assume (or test) that the year is in the range 1901 - 2099. (a) A leap year expressed as a binary number
will have 00 as the last two digits. So: itâ s a leap year if year & (not 4) == 0 (b) If you have a function available to truncate a real number to an integer then this works: x = trunc(year / 4)
itâ s a leap year if x * 4 == year (c) If you have shift (not circular shift) operators, which Iâ m sure Verilog has then: x = year >> 2 itâ s a leap year if (x << 2) == year If the assumption
about the range being 1901 - 2099 is false then youâ ll need some extra logic to eliminate 1900, 1800, 1700 and 2100, 2200, 2300 and so on. Source: Quora
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Iconic Math
Here’s my dilemma, folks. There’s an order of magnitude more material for this page than on the entire iconicmath site. The application of iconic techniques to logic yields astonishing results. The
Image shows the container representation of the logical concept of equivalence. Iconic logic is unary, there is no true/false duality. When applied, logical results come more efficiently, iconic
deduction is a clearer process. However the newfound clarity requires that we reconsider what is meant by thinking logically. Unit-ensembles and depth-value notation retained a visceral connection to
our current place-value numbers, although iconic arithmetic moves substantively away from the group theoretic concepts of modern algebra. In contrast, iconic logic is radically different from
conventional logic. Conventional logic is conceptual, its ANDs and ORs and NOTs are famously without referents in the actual world. Iconic logic is grounded, it is constructed physically rather than
conceptually. Just like arithmetic, logic is putting stuff into containers and then simplifying the result. Like the Merge and Group operations of iconic numbers, iconic logic’s Cross and Call manage
all facits of simple logical inference.
The iconic patterns of arithmetic and logic are presented below. For logic, assume that • is the same as ( ). One fundamental difference between arithmetic and logic is the units of arithmetic are
full while units of logic are empty. The iconic arithmetic rules, if read from left to right, are Ungroup and Unmerge. Since equality allows both directions of transformation, this change is solely
visual, intended to allow an easier comparison between arithmetic and logic. Similarly Cross and Call have been written with explicit containers and with dots for “units” (the unit for logic is the
value True).
These sets of equations define simple arithmetic and simple logic, and they define the difference between arithmetic and logic. In order to count, arithmetic requires that units accumulate. Logical
truth does not accumulate, the Call rule illustrates what in arithmetic would amount to1+1=1. The Cross rule also assures non-accumulation by defining containment as cancelation rather than grouping.
This visual information is better contemplated than described by words.
Charles Saunders Peirce invented this particular approach to iconic logic in the 1890s, which he called Entitative Graphs. Other iconic approaches during the same time period included Venn’s diagrams
and Frege’s diagrammatic logic. George Spencer Brown published the Laws of Form in 1969 which placed Peirce’s graphs in the algebraic context of equations. Cross and Call are from Laws of Form. My
work elaborates and extends Spencer Brown’s seminal work. To distinguish each system, I’ve called my variations on Spencer Brown’s theme Boundary Logic.
What follows is a dozen articles on iconic logic. I’ve organized them roughly in order of technical complexity, and included short summaries of the content of each. For convenience, and since some
articles are monographs, the length of each is indicated in the title.
Common Sense (5 pages)
The Advantages of Boundary Logic — A Common Sense Approach presents the basic ideas of iconic logic anchored to common sense physical examples.
Class Notes (5 pages)
The Boundary Logic Class Notes also present the basic ideas of iconic logic, in bullet form. This is the version I usually give to students.
From the Beginning (32 pages)
Boundary Logic from the Beginning steps carefully though the origins, concepts, and applications of boundary logic. This is the last paragraph of the article.
Conventional logic has always harbored a contradiction. It is said to be “how we think rationally”, yet symbolic logic is notoriously difficult. Simple problems of inference are known to confuse the
majority of people. Perhaps the most penetrating idea introduced by the iconic approach to deduction is that what we know as rationality is not the formal symbolic manipulation of a complex and
challenging logical formula. Rather it is a simple and inevitable consequence of making marks in empty space.
Take Nothing Seriously (50 pages)
Taking Nothing Seriously: A Foundational Diagrammatic Formalism covers the same ground as From the Beginning, with more of a focus on simple mathematical ideas than on conceptual origins. This
evolutionary document has a fundamental conceptual error, it attributes a meaning, that of sharing, to empty space. The concept of sharing space should have been stated as sharing the same container.
Here is the (uncorrected) abstract.
We explore the consequences of constructing a diagrammatic formalism from scratch. Mathematical interpretation and application of this construction is carefully avoided, in favor of simply pointing
toward potential mathematical anchors for formal diagrammatic structure. Diagrams are drawn in a two-dimensional space, while conventional mathematics emphasizes the one-dimensional linguistic space
of strings. A theme is that the type of REPRESENTATIONAL SPACE interacts with the expressive power of a formal system.
Starting at the simplest beginning, we examine only two aspects of diagrammatic representation: forms that SHARE the representational space, and forms that enclose, or BOUND, portions of that space.
The formalization of SHARING and BOUNDING is called boundary mathematics.
A pure boundary mathematics is a set of formal transformation rules on configurations of spatial boundaries. Due to the desirability of implementing spatial transformation rules by pattern-matching,
we introduce boundary algebra, that is, boundary mathematics with equality defined by valid substitutions. This step connects us firmly to the known mathematical structure of partial orderings,
algebraic monoids, Boolean algebras, and Peirce’s Alpha graphs.
SHARING and BOUNDING do not form an algebraic group, leading to the surprising conclusion that boundary algebra is not isomorphic to Boolean algebra although it is equally expressive. This result is
a formal consequence of Shin’s observation that Alpha graphs have multiple readings for logic, providing formal evidence that, at least for propositional calculus, diagrammatic formalism is more
powerful than linguistic formalism.
Peirce (18 pages)
Boundary Logic and Alpha Existential Graphs reiterates the conceptual foundations of iconic logic found in Taking Nothing Seriously, comparing Spencer Brown’s approach to Peirce’s approach. Here is
the abstract.
Peirce’s Alpha Existential Graphs (AEG) is combined with Spencer Brown’s Laws of Form to create an algebraic diagrammatic formalism called boundary logic. First, the intuitive properties of
configurations of planar nonoverlapping closed curves are viewed as a pure boundary mathematics, without conventional interpretation. Void representational space provides a featureless substrate for
boundary forms. Pattern-equations impose constraints on forms to define semantic interpretation. Patterns emphasize void-equivalence, deletion of structure that is syntactically irrelevant and
semantically inert. Boundary logic maps one-to-many to propositional calculus. However, the three simple pattern-equations of boundary logic provide capabilities that are unavailable in token-based
systems. Void-substitution replaces collection and rearrangement of forms. Patterns incorporate transparent boundaries that ignore the arity and scope of logical connectives. The algebra is
isomorphic to AEG but eliminates difficulties with reading and with use by substituting a purely diagrammatic formalism for the logical mechanisms incorporated within AEG.
Conventional Interpretations (30 pages)
Conventional Interpretations of Boundary Logic Tools again establishes the conceptual foundations and then goes on to describe the algebraic theorems that are of pragmatic importance to the
implementation of boundary logic in software. Here’s the abstract.
The concepts of conventional logic can all be represented in the BL formalism. In so doing, conventional logic becomes simpler and more efficient. Similarly, the concepts of BL can be represented in
conventional logic, but only by adding new tools and perspectives to conventional approaches.
Thus, BL improves and extends the power of conventional logic. Using BL concepts generally and implementing BL concepts in software data structures and algorithms and in semiconductor and other
hardware designs (among other applications of BL) avoids the representational and computational complexities of conventional logic approaches. It is, however, possible to replicate the
transformational rules and mechanisms of BL using a vocabulary of conventional logic. This might have the effect of creating new ideas for conventional logic that appear to be novel and unique, when
they would be, in fact, derivative of BL innovations.
To make the relationship between BL and conventional logic clear, this document includes many comparisons of techniques in representation, in the form of theorems, in proofs, and in hardware
What’s the Difference? (53 pages)
We are now heading into the more technical articles that presume experience with iconic formalisms. What’s the Difference? Contrasting Boundary and Boolean Algebras addresses a fundamental confusion
about Spencer Brown’s work, that the formalism of an iconic logic can be isomorphic (identical up to the choice of symbolic names) with the symbolic system of Boolean algebra. This complete
misinterpretation is obvious, just count the number of ground symbols. That is, compare {TRUE, FALSE} to { O }. Here’s the abstract.
There is a common misconception that boundary algebra is isomorphic with Boolean algebra. The paper describes a dozen deep structural differences that are incompatible with isomorphism. Boundary
algebra does not support functional morphisms of any kind because it does not support functions. It does support a partial order relation. A one-to-many mapping between boundary and Boolean algebras
emphasizes the difference between the two systems. This mapping shows that boundary algebra subsumes Boolean algebra within a formally smaller structure. Boundary algebra applies to n-element Boolean
algebras, and does not have group theoretic structure.
Equality (23 pages)
The main point of Equality Is Not Free is that when an equal sign,=, is interpreted logically as each side of the equation implying the other, then logic and algebra get confused at a foundational
level. This article explores the consequences. Here is the introduction.
Our conceptualization of mathematical expressions, definitions, and proofs is formulated in the language of logic, using AND and NOT and IMPLIES and IF-ANDONLY-IF. This same language maps to boundary
algebra, so that the way we describe and address problems logically can also be formulated as the structural transformation of algebraic boundary forms.
Boundary algebra provides a different way of thinking about deduction and rationality. The language of boundary algebra consists only of SHARING and BOUNDING and EQUALS.
In the sequel, we use boundary algebra tools to analyze and to deconstruct the structure of logic itself, with an emphasis on the relationship between logical connectives and algebraic EQUALS.
Insertion (17 pages)
Generalized Insertion explores an innovation in the concept of logical proof. Like many iconic techniques, it has no parallel in conventional symbolic logic. Both logic proof and Boolean optimization
can be conducted by querying parts of the fact base, without engaging in deduction. This is the opening paragraph of the article.
The Boundary Insertion algorithm is a general proof technique for logic, functionally equivalent to the four other known techniques (truth tables, natural deduction, resolution, and algebraic
substitution). Unlike the other techniques, polynomially bounded Insertion is quite comprehensive.
Complexity (66 pages)
Computational Complexity and Boundary Logic explores the classic P=NP computational problem that a general proof procedure for logic must always, at some point of complexity, make a guess. The
Virtual Insertion technique also encounters this barrier, the article explores when. Here is the Summary.
Logic has evolved over thousands of years within language. Mathematical logic is young, a creation of the 20th century. One of the most outstanding problems for mathematical logic is determining
whether or not there exist tractable algorithms for exponential problems. An answer may lay in one of the simplest exponential problems, that of determining if a given logic expression is a
tautology. Boundary logic (BL) provides a new set of computational tools which are geometric and algebraic, most definitely not conventional logic. These container-based tools are simpler than those
of both mathematical and natural logic. Can boundary logic shed light on tautology identification?
After the nature of algorithmic complexity and BL are introduced, we explore conventional rules of inference and deduction from a BL perspective. Proofs of all but the distributive law are close to
trivial using BL algorithms. Certainly none of the explicit structure of modern logic identifies complexity. We explore significantly difficult tautological problems which can be constructed using
the rules of logic. None of these are complex, although some are non-trivial. It appears that compound logical rules do not identify complex problems, yet logical proof systems rapidly require
exponential effort.
We then describe the central reduction algorithm of BL, called virtual insertion, and apply it to known tractable and intractable problems. The low-degree polynomial virtual insertion algorithm is
not complete, however the tautologies it cannot reduce may be constrained to intractable problems. Thus, virtual insertion is an efficient decision procedure for tractable tautologies. This is
valuable since almost all pragmatic problems are tractable. Using virtual insertion recursively produces a complete decision procedure for elementary logic, but one with the expected exponential
The boundary logic representations and algorithms described herein have been fully implemented in software and applied to computationally difficult practical problems, such as circuit design,
tautology detection, and computer program optimization.
Containment (166 pages)
Iconic and Symbolic Containment in Laws of Form is a recent (2010) monograph that throughly explores the structure of Spencer Brown’s iconic logic and its relationship to other mathematical
approaches. The goal is to decisively demonstrate that Laws of Form is a calculus of one relationship, that of containment. The interpretation of sharing space as a function is an error that violates
a fundamental iconic rule that empty space has no interpretation. This is the overview from the monograph.
At the turn of the twentieth century, the mathematical community adopted a radical plan to put mathematics on a firm foundation. The idea was symbolic formalization, the representation of concepts
using encoded symbols that bear no resemblance to what they mean. At the same time, C.S. Peirce developed Existential Graphs, an iconic representation of logic that is drawn rather than written
[ref]. Iconic forms are images that look like what they mean. In 1967, G. Spencer Brown reintroduced Peirce’s iconic system in a more general algebraic style as Laws of Form [ref]. Today, mathematics
remains symbolic, while other communication media have evolved into visual and interactive experiences. This essay explores the possibility of an iconic mathematics.
Laws of Form (LoF) is an algebraic system expressed in an iconic notation. We interpret LoF as a calculus of containment relations, and develop intuitive, symbolic, and iconic descriptions of the
Contains relation. The representation of the LoF calculus is analyzed from both the iconic and the symbolic formal perspectives, with a focus on the descriptive structures necessary to align the two
representational techniques. An iconic notation resembles its interpretation, which is at variance with modern mathematical conventions that strictly separate syntax from semantics. Iconic form is
displayed in two and three dimensions, while symbolic form unfolds as encoded one-dimensional strings of tokens. Iconic variables stand in place of arbitrary patterns of zero, one, or many objects,
while symbolic variables stand in place of single expressions. Symbols conform to speech, while icons conform to vision.
The iconic mathematics of LoF provides a semantic model for both a relational and a functional calculus. The operation of putting an object into a container forms an algebraic quasigroup. When
further constrained by LoF rules, the functional calculus provides a new perspective on Boolean algebra, an equally expressive calculus that is both non-associative and non-commutative. LoF can be
interpreted as propositional logic; iconic containment as well provides a new, non-symbolic deductive method.
We show four varieties of iconic notation to illustrate the interdependence of representation and meaning. Iconic formal systems extend the ability of symbolic notations to express parallelism,
transformation across nesting, structure sharing, and void-based transformation. The expressive neutrality of several symbolic conventions (labeling, grouping, arity, null objects, variables) is
examined in light of the LoF iconic calculus. Computational examples in both symbolic and iconic notations are compared side-by-side. We show that inducing symbolic representation on essentially
spatial form has led to widespread publication of erroneous information.
Five Liars (6 pages)
Lewis Carroll’s Five Liars Puzzle was considered to be one of the most difficult logic puzzles of its day. It was published in 1897. This article solves the puzzle with iconic techniques.
Interestingly, Carroll got the puzzle wrong, thinking there was only one solution when in fact there are two.
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Class 5 Maths Tenths and Hundredths Worksheet
Read and download free pdf of Class 5 Maths Tenths and Hundredths Worksheet. Download printable Mathematics Class 5 Worksheets in pdf format, CBSE Class 5 Mathematics Math-Magic Chapter 10 Tenths and
Hundredths Worksheet has been prepared as per the latest syllabus and exam pattern issued by CBSE, NCERT and KVS. Also download free pdf Mathematics Class 5 Assignments and practice them daily to get
better marks in tests and exams for Class 5. Free chapter wise worksheets with answers have been designed by Class 5 teachers as per latest examination pattern
Math-Magic Chapter 10 Tenths and Hundredths Mathematics Worksheet for Class 5
Class 5 Mathematics students should refer to the following printable worksheet in Pdf in Class 5. This test paper with questions and solutions for Class 5 Mathematics will be very useful for tests
and exams and help you to score better marks
Class 5 Mathematics Math-Magic Chapter 10 Tenths and Hundredths Worksheet Pdf
Question. What is seven hundreds plus thirteen tens?
(a) 713
(b) 7130
(c) 830
(d) 731
Answer : C
Question. What is Seven hundred plus Seven hundredths?
(a) 1400
(b) 707
(c) 700.7
(d) 700.07
Answer : D
Question. Vessels P and Q contain water as shown below. The amount of water in them is marked in millilitres. If the water in vessels P and Q is completely poured into a third vessel R, how many
litres of water will there be in vessel R?
(a) 0.02 litres
(b) 0.18 litres
(c) 0.2 litres
(d) 20 litres
Answer : C
Question. Rahul is trying to find a long straight branch to break into three equal pieces to use as cricket stumps. A branch of which of these lengths would be most suitable for Rahul to use?
(a) 210 cm
(b) 210 mm
(c) 21 cm
(d) 21 m
Answer : A
Question. Which of the following sets of decimals are in descending order?
(a) 0.88, 88, 8.8
(b) 8.08, 0.88, 80.8
(c) 8.008, 0.888, 0.0888
(d) 0.88, 8.8, 88
Answer : C
Question. 1.2 is the same as:
(a) 10/12
(b) 12 tenths
(c) 12 tens
(d) 10 + 0.2
Answer : B
Question. Which of the following represents 0.3?
Answer : A
Question. 2 tenths is the same as
(a) 2 times 10
(b) 20/100
(c) 10/2
(d) 2/100
Answer : B
Question. If the string shown here is pulled straight, which one of the following will be the closest approximation of its length?
(a) 4 cm
(b) 5 cm
(c) 7 cm
(d) 9 cm
Answer : C
Question. Shorty and his friend Mr. Tallman are shown here.
If Shorty is 80 cm tall, roughly how tall would Mr. Tallman be?
(a) 1 m
(b) 1.5 m
(c) 2 m
(d) 3 m
Answer : C
Question. 10 millimetres = 1 centimetre.
100 centimetres = 1 metre.
From this, we can say that:
(a) 1 metre is 110 times a millimetre.
(b) 1 centimetre is one tenth of a millimetre.
(c) 1 metre is one hundredth of a centimetre.
(d) 1 millimetre is one thousandth of a metre.
Answer : D
Question. Which of the following is closest to 418?
(a) 400
(b) 410
(c) 420
(d) 500
Answer : C
Question. 5/10 + 7/1000 can be written as
(a) 0.5007
(b) 0.0507
(c) 0.507
(d) 7005
Answer : C
Question. Pinky starts drawing a line from P. At which point should she stop if the line is to be 3.5 cm long.
(a) A
(b) B
(c) C
(d) D
Answer : C
Question. Which of the following can be written as 60.07?
(a) 60 + 7/10
(b) 60 + 7/10
(c) 60 + 7/100
(d) 6 + 7/1000
Answer : C
Question. In which division will be the quotient a 3-digit number?
(a) 7620/6
(b) 612/6
(c) 498/6
(d) 348/6
Answer : B
Question. 11/4 is a number between
(a) 1 and 2
(b) 2 and 3
(c) 3 and 4
(d) 11 and 12
Answer : B
Question. These are the two types of 'kadai' that a certain shop has. Shalini visits this shop and buys the steel kadai. However, she changes her mind the next day and comes to take the non-stick one
instead. She pays for the excess amount with a 500 rupee note. What amount should be returned to her?
(a) Rs 363
(b) Rs 265
(c) Rs 137
(d) Rs 128
Answer : A
Question. Which set of digits shown below can be used to form a four-digit EVEN NUMBER that is greater than 7000?
(a) 8, 3, 1, 5
(b) 0, 2, 4, 6
(c) 3, 5, 7, 9
(d) 1, 0, 3, 7
Answer : D
Question. Alok's mother gives him Rs. 200 to spend on his birthday. He decides to choose from the things shown below:
Which set of things should he choose so that he spends the maximum within the 200 rupees that he has?
(a) the book, the T-shirt and the pen
(b) the book, the T-shirt and the chocolates
(c) the book and the T-shirt
(d) the T-shirt, the pen and the chocolates
Answer : C
Question. What is 19 - 18 + 17 - 16 + 15 - 14 +13 - 12?
(a) 124
(b) 48
(c) 4
(d) 1
Answer : C
Question. The number 10 has 4 factors - 1, 2, 5 and 10.
The table below lists the NUMBER OF FACTORS for some numbers.
Question. From this we can say that the number of prime numbers between 520 and 530 is:
(a) 0
(b) 2
(c) 4
(d) cannot be said for sure.
Answer : B
Question. What is the maximum number of 1 metre long pieces that we can get from this rope?
(a) 8
(b) 9
(c) 80
(d) 85
Answer : A
Click on the link below to download Class 5 Maths Tenths and Hundredths Worksheet
CBSE Class 5 Mathematics Math-Magic Chapter 10 Tenths and Hundredths Worksheet
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Worksheet for CBSE Mathematics Class 5 Math-Magic Chapter 10 Tenths and Hundredths
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Inequality Math
Here is a distribution of aeolian sand grain sizes:
Read →
@KhothI was a little bit too fast with my praise of the eeckhout paper (cities).
If you take a closer look at the whole thing, there are more than 20 % of the population missing in the plot. Most (european) countries do not accept "cities" with less than 2 - 5000 people,
cutting off the left 2/3 of the data points, or half the ln size distribution. On the right side, as the paper states, he takes the smaller US size definition, putting LA at 3.7 Mio, instead of 16
Mio with the definition closer to european views, and strong deviation from a log normal or Zipf distribution. That leaves than the transition region for fitting, and you can do this in many ways.
You can even find on the homepage of the author some comment exchange in 2009 with Levy on this. Beyond that, if you look up data on "urbanization", (e.g. CIA, wiki) you will find that over 50 % of
the differences are due to different definitions of "city" (see German wiki "Stadt": DK: 200; JP 50 000)
bottomline: if you want, you can fit the log normal distribution, but many others as well, and it doesn't prove anything about underlying mechanisms.Very typical results for economics / sociology.
Expand full comment
Hah, I thought the firm size graph looked suspiciously neat. I hadn't noticed that the scale on the y-axis was impossible. Nice catch.
Expand full comment
@klothway more complicated.If you look in statistical physics on these self-organized criticality phenomena, pertubation theory and similar stuff, there are also good reasons to cut off at the
right side, when you touch the system size.You can see this, if you know what you are looking for, for example in the "firm size" plot above. Just shift the lin fit line a little up to match the
central points.
Just with the problem, that the plot doesn't pass the smell test.
In the Journal with the highest impact factor of all, "Science", from which that was taken, see the link above, they find it now very often not necessary any more to do any reasonable peer review.
Otherwise they would have seen easily that in a nation of 3e8 people, and probably some 1e7 - 1e8 corporations, you can not have a frequency of less than 3e-9, meaning that at least two points to
the right are garbage.
garbage in, garbage out. Just like New York Times "the physicists does the city" with links to the next "high impact factor" PNAS garbage.
grummel, outspoken arrogance here.
That ordinary people don't know nor understand quantum mechanics, no problem. Scaling theory, of course not, elementary statistic is actually not that difficult and helpful in daily life. But this
is just elementary math, no frequency of 1e-13 possible in a population of 3e8, ad the "Creme de la Creme" doesn't catch it, showing a massive decay of quality in the US in the last 30 years.
Expand full comment
A log-normal distribution is precisely a parabola on a log-log plot.
A regular normal distribution is a parabola on a logy plot.
P = C*exp( (v-u)^2/D ) is the pdf of a normal dist of v. Take the log
log(P) = E + (v-u)^2/Dif y = log(P) and x=v this is the equation of a parabola.
but if v = log(z) then this is the pdf of a log-normally distributed z. Theny = log(P) and x=log(z) is the equation of a parabola.
Expand full comment
I did see from the paper that it was linear regressions. I just didn't notice why anyone would even think of it. It makes sense to do it for the parts that looks basically like straight lines, but
the other ones are just completely meaningless.
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If you read the original paper, the various lines show what linear regression errors you make, if you cut off the complete distribution at various points.From my perspective, you do not only get
increasingly wrong power coefficients, but you miss out on half the dynamic, especially what happens to small and shrinking cities and regions, and how to adjust public investment to that.
Pretty interesting paper !
Expand full comment
A lognormal distribution is not a parabola, unless you are using the term parabola very loosely.
Expand full comment
Not really related to the point, but I can't help wondering why the city size graph has a bunch of lines in it. Do some people have a weird compulsion to try to fit a straight line to anything?
Expand full comment
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Plot up to three curves representing the concentration versus time relationship, each curve representing a different flow. — plotConcTimeSmooth
Plot up to three curves representing the concentration versus time relationship, each curve representing a different flow.
These plots show how the concentration-time relationship is changing over flow.
This plot can also help identify situations where the windowY may be too small. If there are substantial oscillations of some of the curves, then the windowY should be increased. Alternatively,
windowY may be too large. This can be seen when the windowY is reduced (say to 4.0). A good choice of windowY would be a value just great enough to damp out oscillations in the curves.
Although there are a lot of optional arguments to this function, most are set to a logical default.
Data come from named list, which contains a Sample dataframe with the sample data and an INFO dataframe with metadata.
plotConcTimeSmooth(eList, q1, q2, q3, centerDate, yearStart, yearEnd,
qUnit = 2, legendLeft = 0, legendTop = 0, concMax = NA,
concMin = NA, bw = FALSE, printTitle = TRUE, colors = c("black",
"red", "green"), printValues = FALSE, tinyPlot = FALSE, concLab = 1,
monthLab = 1, minNumObs = 100, minNumUncen = 50, windowY = 10,
windowQ = 2, windowS = 0.5, cex.main = 1.1, lwd = 2,
printLegend = TRUE, cex.legend = 1.2, cex = 0.8, cex.axis = 1.1,
customPar = FALSE, lineVal = c(1, 1, 1), logScale = FALSE,
edgeAdjust = TRUE, usgsStyle = FALSE, ...)
named list with at least the Sample and INFO dataframes
numeric This is the discharge value for the first curve to be shown on the plot. It is expressed in units specified by qUnit.
numeric This is the discharge value for the second curve to be shown on the plot. It is expressed in units specified by qUnit. If you don't want a second curve then the argument must be q2=NA
numeric This is the discharge value for the third curve to be shown on the plot. It is expressed in units specified by qUnit. If you don't want a third curve then the argument must be q3=NA
character This is the time of year to be used as the center date for the smoothing. It is expressed as a month and day and must be in the form "mm-dd"
numeric This is the starting year for the graph. The first value plotted for each curve will be at the first instance of centerDate in the year designated by yearStart.
numeric This is the end of the sequence of values plotted on the graph.The last value will be the last instance of centerDate prior to the start of yearEnd. (Note, the number of values plotted on
each curve will be yearEnd-yearStart.)
object of qUnit class. printqUnitCheatSheet, or numeric represented the short code, or character representing the descriptive name.
numeric which represents the left edge of the legend in the units of the plot.
numeric which represents the top edge of the legend in the units of the plot.
numeric value for upper limit on concentration shown on the graph, default = NA (which causes the upper limit to be set automatically, based on the data)
numeric value for lower limit on concentration shown on the vertical log graph, default is NA (which causes the lower limit to be set automatically, based on the data). This value is ignored for
linear scales, using 0 as the minimum value for the concentration axis.
logical if TRUE graph is produced in black and white, default is FALSE (which means it will use color)
logical variable if TRUE title is printed, if FALSE not printed
color vector of lines on plot, see ?par 'Color Specification'. Defaults to c("black","red","green")
logical variable if TRUE the results shown on the graph are printed to the console and returned in a dataframe (this can be useful for quantifying the changes seen visually in the graph), default
is FALSE (not printed)
logical variable, if TRUE plot is designed to be plotted small, as a part of a multipart figure, default is FALSE
object of concUnit class, or numeric represented the short code, or character representing the descriptive name. By default, this argument sets concentration labels to use either Concentration or
Conc (for tiny plots). Units are taken from the eList$INFO$param.units. To use any other words than "Concentration" see vignette(topic = "units", package = "EGRET").
object of monthLabel class, or numeric represented the short code, or character representing the descriptive name.
numeric specifying the miniumum number of observations required to run the weighted regression, default is 100
numeric specifying the minimum number of uncensored observations to run the weighted regression, default is 50
numeric specifying the half-window width in the time dimension, in units of years, default is 10
numeric specifying the half-window width in the discharge dimension, units are natural log units, default is 2
numeric specifying the half-window with in the seasonal dimension, in units of years, default is 0.5
magnification to be used for main titles relative to the current setting of cex
line width, a positive number, defaulting to 2
logical if TRUE, legend is included
number magnification of legend
numerical value giving the amount by which plotting symbols should be magnified
magnification to be used for axis annotation relative to the current setting of cex
logical defaults to FALSE. If TRUE, par() should be set by user before calling this function (for example, adjusting margins with par(mar=c(5,5,5,5))). If customPar FALSE, EGRET chooses the best
margins depending on tinyPlot.
vector of line types. Defaults to c(1,1,1) which is a solid line for each line. Options: 0=blank, 1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash
logical whether or not to use a log scale in the y axis.
logical specifying whether to use the modified method for calculating the windows at the edge of the record. The modified method tends to reduce curvature near the start and end of record.
Default is TRUE.
logical option to use USGS style guidelines. Setting this option to TRUE does NOT guarantee USGS compliance. It will only change automatically generated labels
arbitrary functions sent to the generic plotting function. See ?par for details on possible parameters
q1 <- 1
q2 <- 10
q3 <- 100
centerDate <- "07-01"
yearStart <- 1990
yearEnd <- 2010
eList <- Choptank_eList
plotConcTimeSmooth(eList, q1, q2,q3, centerDate,
yearStart, yearEnd, legendLeft = 1997,
legendTop = 0.44, cex.legend = 0.9)
plotConcTimeSmooth(eList, q1, q2,q3, centerDate, yearStart,
yearEnd, logScale = TRUE, legendLeft = 1994,
legendTop = 0.4, cex.legend = 0.9)
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Three Lesser Known Tips & Tricks For Uni CPO
Think of Uni CPO WooCommerce Product Options and Price Calculation Formulas plugin as a complex system of possibilities for your product configuration, rather than one or several separated tools in a
single package. Thus this system has its bindings or features which are the result of coincidence rather than intention. Still, the following Uni CPO features – I prefer to call them features because
of their importance – are so cool that they had better be invented if never existed.
Trick 1: The order of NOVs matters
I think this is the trick number 1 just because of the fact of widespread use of it during the product configuration. The order in which NOVs are saved matters. To put it simply, it means that you
can use NOVs from top of the list in formulas/matrix for NOVs which are in the bottom of the list. The mentioned above is illustrated in the following screenshot:
NOV ‘{uni_nov_cpo_width_m}‘ is created first. And I am using it in the third NOV called ‘{uni_nov_cpo_width_m_q}‘. The reason why is it possible is that NOVs are getting evaluated one by one from top
to bottom. So, at the moment of evaluation of variable ‘{uni_nov_cpo_width_m_q}‘ the script has already evaluated variable ‘{uni_nov_cpo_width_m}‘, it computed its value and can use it to compute the
value of the third variable.
Trick 2: How to connect NOV matrices with Select/Radio suboptions
Yes, it is tricky because as you may already know that NOV matrices are meant to work with numeric values ONLY. At the same time Suboptions’ slugs, which are the unique names they can be identified
with, can be text only.
So, how to combine them? The answer is the third type of setting which is available for all the suboptions. It is called ‘price/rate’, and this is what we need. The rule is – keep them unique across
suboptions of a specific option. Just like slugs, but numeric. It can be a simple sequence like “1,2,3,4…” or somewhat meaningful series like on the screenshot below:
In the example, my suboptions are “3 mm”, “5 mm” and “10 mm”, so I used “3”, “5” and “10” values. They are unique for my suboptions, and they are not getting repeated. It is crucial! Then I have
chosen this my option as 2nd var in the NOV matrix and used the same values. Note that it could be set as the 1st var and those values would be in the head of the matrix columns. It is not relevant
where do you put them and is totally up to you!
The reason why it works is that all options’ choices are getting transformed into numeric values to use them in the price calculation formula. In the case of Select/Radio Option, such numeric values
can be set in ‘price/rate’ setting. But it does not mean that they will be used anywhere automatically. They will be used only if you decide to use them and where you decide to use them. So, first,
do not fear to use these settings in general. Second, try them in different combinations with other instruments such as NOVs or Formula Conditional Rules.
Trick 3: Maths functions are powerful friends
Sometimes you meet completely non-trivial product configuration and wonder if (and how if possible?) it can be achieved by using Uni CPO. Well, let me give you an example. Hm-m… let’s say you want to
add to the product price some fixed value but a certain number of times which depends on some other option/parameter. To be more precise, let’s pretend we need to add feet to our product “Stand”
according to the following rules:
1. Two pieces initially
2. One extra if the width is more than 3 meters
3. One additional every 2 meters
Width is a custom option.
So, how to achieve this?
First, let’s quickly evaluate what we have got, try to express it mathematically:
• width < 3 = add 2
• width > 3 and width < 5 = add 3 (2+1, I write total here)
• width > 5 and width < 7 = add 4
• width > 7 and width < 9 = add 5
• …
We have got a sequence! 🙂 It can be described with the following formula: ’round({width}/2)+1′. You can see that I have used one of the maths functions. The full list of available ones can be
found in the documentation. How do I get this formula? By trials and errors and deduction. I have just kept looking and trying different ways until I have found the regularity. It might not be
perfectly precise or whatever. At least, only you decide if it is good enough. This is exactly what I have done by testing the formula in a spreadsheet against all possible yet significant values of
width. I think what is more important is that I can do this with Uni CPO. I do not have to write any additional single line of code. You can achieve the same as well. You do not need to hire a
freelancer to do the job. The tool exists already, and it is called Uni CPO WooCommerce Product Options and Price Calculation Formulas plugin 🙂
You do not need to hire a freelancer to do the job. The tool exists already and it is called Uni CPO WooCommerce Product Options and Price Calculation Formulas plugin
The final result (a real-world example)
I have used two NOVs:
The first NOV calculates the number of pieces which should be added based on width parameter (to remind: a width is a custom option; customer sets its value). The second NOV is for a price of 1 piece
of feet. It might not be needed to keep two NOVs. My final part of the product price formula is this ‘…+{uni_nov_cpo_feet_qty}*{uni_nov_cpo_feet_price}‘, but I could keep it like this ‘…+
{uni_nov_cpo_feet_qty}*24}‘. The real reason why I use the second NOV is to achieve wholesale pricing functionality. It is not shown on the screenshot, but yes – wholesale functionality is also
possible with Uni CPO plugin!
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Peak inelastic displacement as a proxy for accumulating seismic structural damage
Roberto Baraschino, Georgios Baltzopoulos, Iunio Iervolino
Ultima modifica: 2022-08-27
Structural reliability assessment for a building accumulating damage in multiple seismic events can play a role in decision-making, for example, during a seismic sequence and/or for prioritizing
post-earthquake repair operations. One of the models required for such an assessment is the probability that the structure will transit from one damage state to another, because of a shock of given
intensity. Such a model is represented by a set of so-called state-dependent fragility functions, which are often evaluated via dynamic analysis of a nonlinear structural model. Developing the
fragility curves in this fashion requires that one or more response measures allow to identify damage accumulation. One typically used seismic demand parameter is the peak transient inelastic
displacement, for example at roof level. Although the practice of associating damage states to inelastic displacement thresholds is well-established for deriving classical fragility models, which
consider the intact building as the initial state and failure the result of only a single transition, past research has indicated that more than one response parameter may be needed to assess
state-dependent fragility. The present paper uses a series of inelastic single-degree-of-freedom systems, each having different natural period of vibration and post-hysteretic behavior, to
investigate the use of supplementary seismic response measures for the definition of transitions between damage states in numerical dynamic analyses. More specifically, the residual displacement and
measures of stiffness and strength deteriorating caused by ground shaking, are considered. The study employs back-to-back incremental dynamic analysis to simulate two consecutive damaging shocks. The
analyses are used to calculate the strength and/or stiffness deterioration that can be associated with traditional ductility demand thresholds per damage state. Results show that damage accumulation
over two shocks, expressed in terms of overall strength and stiffness loss, is hardly effectively represented by the peak inelastic excursion alone. This observation leads to the conclusion that to
adjudicate a damage state transition from numerical analysis results, some response quantity that reflects the change in the dynamic properties of the structure due to the first damaging shock,
should be also considered.
è richiesta l'iscrizione al convegno per poter visualizzare gli interventi.
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Defining a New Operator when Inheriting CouplingMPOModel
I'm new at TeNPy and am trying to create a model that is a CouplingMPOModel, with a new operator Nj.
Code: Select all
class myTFLIMModel(CouplingMPOModel):
def init_sites(self, model_params):
site = SpinHalfSite(conserve=None)
Nj = np.array([[1,0],[0,0]])
site.add_op('Nj', Nj)
return site
def init_terms(self, model_params):
U = model_params.get('U',1.)
self.add_coupling(U, 0, 'Nj',1,'Nj',np.array([0]))
However, when I try to create an object of this class by doing the following:
Code: Select all
model_params = dict(lattice = Ladder(L = 20, site = SpinHalfSite(conserve=None)), L=20, U=1.,bc_MPS='finite', conserve='None')
M = myTFLIMModel(model_params)
I keep receiving an error that claims it has no knowledge of the Nj operator. In particular it says:
Code: Select all
"ValueError: unknown onsite operator 'Nj' for u=0
I don't understand why Nj has not been recognized. Interestingly, I have run a similar code without specifying a particular lattice and it has run without any errors, so I'm assuming something went
wrong there. Any help would be much appreciated.
Re: Defining a New Operator when Inheriting CouplingMPOModel
The issue here is that you directly pass the full Ladder instance in the model_params, which already has the site defined - hence the model doesn't call init_sites anymore.
Try to use
Python: Select all
model_params = dict(
and overwrite init_lattice if you need to adjust creating the lattice in a specialized way.
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Math Tutorials
Math Tutorials
The Gator Success Center staff strives to serve the needs of students enrolled in any math course offered on the LSCO campus. Tutoring for developmental and corequisite math courses, as well as
College Algebra (1314), is available on a walk-in basis. To secure the best qualified tutor for other math courses, it is best to contact us and make an appointment.
Math Study Materials Created by the Gator Success Center
Math Handouts and Study Guides
Word Problem Strategies
Helpful Links:
Paul's Online Math Notes -- This website contains notes for Algebra, Calculus I, Calculus II, Calculus III, Linear Algebra, and Differential Equations. You can also download worksheets.
Virtual Math Lab -- This site, hosted by West Texas A&M University, offers free self-guided tutorials for beginning and intermediate algebra, college algebra, math for the sciences, and test prep for
the math portions of the GRE and college placement tests.
S.O.S. Math -- This site contains tutorials covering algebra, geometry, trigonometry, calculus, differential equations, matrices, and complex variables.
Purple Math -- Elizabeth Stapel's math site is a great resource for all levels of math. In addition, students can find tips on how to study and complete homework assignments.
Khan Academy -- Founded by MIT graduate Salman Khan, this site provides video tutorials for math students of all levels. Scroll down to find lessons in algebra, trigonometry, , precalculus, calculus,
and more.
Patrick JMT (Just Math Tutorials) -- "Patrick", a college/university math instructor, presents videos on a vast range of math topics.
BrownMath.com -- Notes and how-to tutorials for algebra, analytic geometry, trigonometry, statistics, calculus, science and business math topics, as well as tips for teaching/learning math; also
includes how-to's for using TI 83/84/89/92 calculators.
Hawkes TV features video lessons that correspond to Hawkes Learning Systems math textbooks and online course offerings. Subjects include basic math, beginning and intermediate algebra, pre-calculus,
calculus, statistics, business math, and more.
Function Transformations at MathIsFun.com explains how altering different elements of a parent function can reposition and resize the function's graph; self-checking practice problems are included.
More practice with parent functions and transformations is available at Desmos.com.
Graphs of Eight Basic Types of Functions is part of the Algebra Help e-book (browser version) available from MathOnWeb.com. This resource includes graphs of logarithmic and sinusoidal functions.
Harold Toomey's Parent Functions Cheat Sheet (PDF) includes graphs and characteristics for the basic functions students encounter in college algebra and trigonometry courses, along with "graphing
tips" for performing transformations.
Statistics supplements for Hawkes Learning Beginning Statistics textbook -- Formulas and Tables and Statistical Tables.
Free Printable Graph Paper and other math resources from MathBits.com.
Wolfram Alpha can help you answer questions about math as well as some other topics; step-by-step solutions available with paid pro subscription only. A good resource for checking your work.
MOOCulus, offered by Ohio State University, is a collection of lessons for Calculus 1, 2, and 3. Prof. Jim Fowler's calculus lecture videos are available on YouTube.
Understanding Algebra, by James W. Brennan, is a brief overview of pre-algebra and introductory algebra topics, written in an easy-to-understand style. Download in Kindle format from Amazon.com. Read
in Amazon's free Kindle Cloud Reader or get the free Kindle app for your Android, iPhone, iPad, Mac, or PC.
Precalculus, Ver. [π] = 3, Corrected Edition, by Carl Stitz and Jeff Zeager, covers college algebra topics and trigonometry; includes explanations, examples, and exercises with answers. PDF format.
Stitz-Zeager Open Source Mathematics lists links for College Algebra, College Trigonometry, Precalculus, 4th edition, and Chapter 0 Prerequisites textbooks in PDF format, as well as additional
resources, such as downloadable ancillaries (quizzes and Power Points) and solution videos to accompany quizzes.
Applied Finite Mathematics, by Rupinder Sekhon, is available to use online or download (in PDF format) from OpenStax CNX. This free text is used at LSCO in MATH 1324, Math for Business, Spring 2015.
Texas Instruments Calculator Tutorials provides links to online tutorials for graphing, scientific, financial, and elementary calculators.
Finding Your Way Around the TI 83+/84+, including quick reference sheets, for working with algebra, geometry, statistics, trigonometry, pre-calculus, calculus; from MathBits.com.
Montana State University Academic Support Center features a number of TI Graphing Calculator Tutorials, including text and video demonstrating uses of the TI 83/84, 86, 89, & 92 models.
Download TI Graphing Calculator Guidebooks and Software, OS Updates, and Apps from Texas Instruments.
TICalc.org -- Resource for Texas Instruments graphing calculator community news, information, and software.
Graphing Calculator Help--Includes instructions for TI 82, 83, 85,86,89,92 and others.
TI 83/84/89/92 Procedures and Help -- from BrownMath.com; common calculator operations for algebra, trigonometry, statistics, and calculus.
Calculator-1.com -- Basic, simple, scientific, root, and percentage calculator.
GoodCalculators.com -- A collection of online free online calculators for various purposes (statistics, 2D/3D shapes, Grade/GPA, budget, loan, mortgage, conversion, etc.). Works on computers, smart
phones, and tablets. Use the online graphing calculator to plot functions and then save and print an image of the graph.
Desmos.com -- Graph functions, plot tables of data, evaluate equations, explore transformations, and more.
Web2.0calc.com -- Scientific, graphing, programming, equations, and units calculator; register to post questions in the math forum.
Meta-Calculator.com -- Online graphing, scientific, matrix, and statistics calculators, similar to TI-84.
Google Graphing -- In the Google search box, type the word "graph" in front of a function and enter. The first result will be an interactive graph of the function. Try graph y = 3x^3 + 5x^2 - 3x + 7
TI 83 Interactive Calculator -- A free virtual graphing calculator to install on your computer.
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mass to velocity
27 Aug 2024
Title: The Relationship Between Mass and Velocity: A Theoretical Exploration
Abstract: This article delves into the fundamental connection between mass (m) and velocity (v), two essential physical quantities that govern various phenomena in the universe. We examine the
theoretical framework underlying their relationship, highlighting key concepts and mathematical formulations.
Mass and velocity are two fundamental properties of objects in motion. The former is a measure of an object’s resistance to changes in its motion, while the latter represents the rate at which it
moves through space. Understanding the interplay between these two quantities is crucial for grasping various physical phenomena, from the behavior of subatomic particles to the dynamics of celestial
Theoretical Background:
According to Newton’s second law of motion (F = ma), the force (F) applied to an object is equal to its mass (m) multiplied by its acceleration (a). Mathematically, this can be expressed as:
F = m * a
Since acceleration (a) is defined as the rate of change of velocity (v) with respect to time (t), we can rewrite the equation as:
F = m * dv/dt
This equation establishes a direct relationship between force, mass, and velocity.
Kinetic Energy:
The kinetic energy (KE) of an object in motion is directly proportional to its mass and the square of its velocity. Mathematically, this can be expressed as:
KE = 0.5 * m * v^2
This equation highlights the importance of both mass and velocity in determining the kinetic energy of an object.
The relationship between mass and velocity has far-reaching implications for various fields of study, including physics, engineering, and astronomy. Understanding this connection is essential for
designing efficient propulsion systems, predicting the behavior of celestial bodies, and developing new materials with unique properties.
Conclusion: In conclusion, the relationship between mass and velocity is a fundamental aspect of classical mechanics. The mathematical formulations presented in this article provide a theoretical
framework for understanding the interplay between these two essential physical quantities. Further research into this topic will continue to reveal new insights and applications, shaping our
understanding of the universe and its many wonders.
• Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica.
• Feynman, R. P., Leighton, R. B., & Sands, M. L. (1963). The Feynman Lectures on Physics.
Related articles for ‘mass to velocity’ :
• Reading: mass to velocity
Calculators for ‘mass to velocity’
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Directly Proportional – Explanation & Examples
Direct proportion is the relationship between two variables whose ratio is equal to a constant value. In other words, direct proportion is a situation where an increase in one quantity causes a
corresponding increase in the other quantity, or a decrease in one quantity results in a decrease in the other quantity.
Sometimes, the word proportional is used without the word direct, just know that they have a similar meaning.
Directly Proportional Formula
Direct proportion is denoted by the proportional symbol (∝). For example, if two variables x and y are directly proportional to each other, then this statement can be represented as x ∝ y.
When we replace the proportionality sign (∝) with an equal sign (=), the equation changes to:
x = k * y or x/y = k, where k is called non-zero constant of proportionality.
In our day-to-day life, we often encounter situations where a variation in one quantity results in a variation in another quantity. Let’s take a look at some of the real-life examples of directly
proportional concept.
• The cost of the food items is directly proportional to the weight.
• Work done is directly proportional to the number of workers. This means that, more workers, more work and les workers, less work accomplished.
• The fuel consumption of a car is proportional to the distance covered.
Example 1
The fuel consumption of a car is 15 liters of diesel per 100 km. What distance can the car cover with 5 liters of diesel?
• Fuel consumed for every 100 km covered = 15 liters
• Therefore, the car will cover (100/15) km using 1 liter of the fuel
If 1 liter => (100/15) km
• What about 5 liters of diesel
= {(100/15) × 5} km
= 33.3
Therefore, the car can cover 33.3 km using 5 liters of the fuel.
Example 2
The cost of 9 kg of beans is $ 166.50. How many kgs of beans can be bought for $ 259?
• $ 166.50 = > 9 kg of beans
• What about $ 1 => 9/166.50 kg
Therefore the amount of beans purchased for $259 = {(9/166.50) × 259} kg
• =14 kg
Hence, 14 kg of beans can be bought for $259
Example 3
The total wages for 15 men working for 6 days are $ 9450. What is the total wages for 19 men working for 5 days?
Wages of 15 men in 6 days => $ 9450
The wage in 6 days for 1 worker = >$ (9450/15)
The wage in 1 day for 1 worker => $ (9450/15 × 1/6)
Wages of 19 men in a day => $ (9450 × 1/6 × 19)
The total wages of 19 men in 5 days = $ (9450 × 1/6 × 19 × 5)
= $ 9975
Therefore, 19 men earn a total of $ 9975 in 5 days.
Practice Questions
1. If the total daily wages of $7$ women or $5$ men is $\$525$. What will be the total daily wage of $13$ women and $7$ men?
2. Jackie and her sister are going on a road trip. She noted that her car consumes $6.8$ L for every $102$ km. How far can Jackie’s car if its tank contains $30$ L?
3. In Ryan’s construction company, it costs him $\$7, 200$ to pay for the wages of $12$ workers who are working for $6$ days. How much does it cost Ryan’s company to pay for the wages of $18$ staff
working for $5$ days?
4. Alice is known for her handcrafted soaps. It costs her $\$540$ to create $12$ cured bars of soaps each weighing $2.5$ kilograms. How much will it cost her to create $24$ bars of soaps but this
time, weighing $3$ kilograms?
5. Felix is studying a city map that is represented with a scale of $1:30000$ (with units of cm : m). He noticed that two blocks are $8$ cm apart on the map, what is the actual distance between the
two blocks?
6. A $12$-meter flag post casts a shadow of $8$ meters. What is the height of a flag post that casts a shadow of $18$ m?
7. A train takes $8$ hours to cover $600$ kilometers. How long will it take to cover $1500$ kilometers?
8. In a zero-waste store, it costs $\$120$ to refill $12$ jars of body scrub each weighing $800$ g. How much would it cost to refill $36$ jars of body scrub each weighing one kilogram?
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