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Functions in real life
Good morning!
Today, we are going to talk about functions and how they can be applied in real life situation. But first, what are functions? Do you have any idea about what functions are? And now, I am going to
start giving you a definition. Are you ready? Let's go!
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
It is also a bunch of ordered pairs of things (in your case, the things will be numbers, but they can be otherwise), with the property that the first members of the pairs are all different from one
IN OTHER WORDS, a function is a mathematical relationship between two variables, where every input variable has one output variable.
In functions, the x-variable is known as the input or independent variable, because its value can be choosen freely. The calculated y-variable is known as the output or dependent variable, because
its value depends on the chosen input value.
A shorthand used to write sets, often sets with an infinite number of elements.
Constant function is a linear function of the form y=b, where b is a constant. It is also written as f(x)=b. The graph is a horizontal line.
Identity function: It can be written in the form f(x)=x. It's graph is a straight line passing through the origin.
Linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y. It is written in the form f(x)=mx+b. It's graph is a straight.
Radical function contain functions involving roots. Most examples deal with square roots.
Piecewise function is a function that is defined as a sequence of intervals.
Quadratic function is one of the form f(x)=ax2+bx+c, where a, b and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola.
The domain of a function is the set of all independents x-values for which there is one dependent y-value according to that function.
The range of a function is the set of all dependent y-values which can be obtained using an independent x-value.
Functions are mathematical building blocks for designing machines, predicting natural disasters, curing diseases, understanding world economies and for keeping aeroplanes in the air. Functions can
take input from many variables, but always give the same output, unique to that function.
Money as a function of time. You never have more than one amount of money at any time because you can always add everything to give one total amount. By understanding how your money changes over
time, you can plan to spend your money sensibly.
Temperature as a function of various factors. Temperature is a very complicated function because it has so many inputs, including: the time of day, the season, the amount of clouds in the sky, the
strength of the wind, where you are and many more. But the important thing is that there is only one temperature output when you measure ir in a specific place.
Location as a function of time. You can never be in two places at the same time. If you were to plot the graphs of where two people are as a function of time, the place where the lines cross means
that the two people meet each other at that time. This idea is used in logistics, an area of mathematics that tries to plan where people and items are for businesses.
Now we are going to learn how quadratic functions can be applied in real life situations.
The throw ends when the shot hits the ground. The height y at that point is 0, so set the equal to zero.
This equation is difficult to factor or to complete the square, so well solve by applying the quadratic formula.
Simplify or find both roots. x=46.4 or -4,9.
Do the roots make sense? The parabola described by the quadratic function has two x-intercepts, but the shot only traveled along part of that curve.
One solution, -4,9, cannot be the distance traveled because it is a negative number.
The other solution, 46,4 feet, must give the distance of the throw.
Now that we've studied different types of function and how a quadratic function can be applied in real life, we now say that everything can related in real life and that these can be solved through
these mathematical equations learned in school.
I hope you had a nice time watching this video, thank you! | {"url":"https://clilstore.eu/clilstore/page.php?id=3830","timestamp":"2024-11-06T02:40:35Z","content_type":"text/html","content_length":"18062","record_id":"<urn:uuid:997e6f48-5421-4f60-86e8-fbc212f6e04d>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00710.warc.gz"} |
Prediction from Partial Information and Hindsight, with Application to Circuit Lower Bounds
Consider a random sequence of n bits that has entropy at least n−k, where k≪ n. A commonly used observation is that an average coordinate of this random sequence is close to being uniformly
distributed, that is, the coordinate “looks random.” In this work, we prove a stronger result that says, roughly, that the average coordinate looks random to an adversary that is allowed to query ≈nk
other coordinates of the sequence, even if the adversary is non-deterministic. This implies corresponding results for decision trees and certificates for Boolean functions. As an application of this
result, we prove a new result on depth-3 circuits, which recovers as a direct corollary the known lower bounds for the parity and majority functions, as well as a lower bound on sensitive functions
due to Boppana (Circuits Inf Process Lett 63(5):257–261, 1997). An interesting feature of this proof is that it works in the framework of Karchmer and Wigderson (SIAM J Discrete Math 3(2):255–265,
1990), and, in particular, it is a “top-down” proof (Håstad et al. in Computat Complex 5(2):99–112, 1995). Finally, it yields a new kind of a random restriction lemma for non-product distributions,
which may be of independent interest.
Bibliographical note
Publisher Copyright:
© 2019, Springer Nature Switzerland AG.
• 68Q15
• Certificate complexity
• Circuit complexity
• Circuit complexity lower bounds
• Decision tree complexity
• Information theoretic
• Query complexity
• Sensitivity
ASJC Scopus subject areas
• Theoretical Computer Science
• General Mathematics
• Computational Theory and Mathematics
• Computational Mathematics
Dive into the research topics of 'Prediction from Partial Information and Hindsight, with Application to Circuit Lower Bounds'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/prediction-from-partial-information-and-hindsight-with-applicatio","timestamp":"2024-11-06T20:50:31Z","content_type":"text/html","content_length":"56168","record_id":"<urn:uuid:e7f37896-553b-4193-9da2-1f416157935a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00192.warc.gz"} |
Charging a capacitor with a gain
No mathematical development of physics applied to a closed system that remains unchanged can show additional energy, a question of the internal coherence of mathematical formalism in physics. So
if Dollard finds any in his equations, he's wrong. You don't have to be a physicist to understand it.
Symmetry only applies within each given reference frame. If you account for time-delay and multiple reference frames, non-zero solutions indeed become possible.
For example, a photon emitted from both Earth and from near a black hole would require the same amount of energy from their given reference frame,
but observing both photons from earth you would find the one emitted near the black hole may appears to have a higher energy (because it was emitted from a higher-energy region).
If energy is extracted from the "Dirac Sea", it must be shown experimentally and explained using equations including the "Dirac Sea". This extra energy cannot be inferred from equations that do
not include it.
Dirac-sea interactions are commonplace in Quantum Mechanics and often express themselves with virtual particles. Any formulae using virtual particles is interacting with this apparently homogeneous
Eric Dollar only has books to sell. Perhaps he has some interesting things but the fact that he does not have the support of exotic reproducible experiments and especially that he has the support
of Murakami does not plead in his favour
Everyone makes a living somehow. James Clerk Maxwell broadcast his models to the community in much the same manner as Eric:
This is, after all, the 'OverUnity Research' forum.
When you say something is impossible, you have made it impossible | {"url":"https://www.overunityresearch.com/index.php?topic=3775.25","timestamp":"2024-11-13T11:54:13Z","content_type":"application/xhtml+xml","content_length":"27616","record_id":"<urn:uuid:24804d89-35b7-4fbb-bfa1-0f8f9be07f64>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00355.warc.gz"} |
Pareto distribution and Benford's law
The Pareto probability distribution has density
for x ≥ 1 where a > 0 is a shape parameter. The Pareto distribution and the Pareto principle (i.e. “80-20” rule) are named after the same person, the Italian economist Vilfredo Pareto.
Samples from a Pareto distribution obey Benford’s law in the limit as the parameter a goes to zero. That is, the smaller the parameter a, the more closely the distribution of the first digits of the
samples come to following the distribution known as Benford’s law.
Here’s an illustration of this comparing the distribution of 1,000 random samples from a Pareto distribution with shape a = 1 and shape a = 0.2 with the counts expected under Benford’s law.
Note that this has nothing to do with base 10 per se. If we look at the leading digits as expressed in any other base, such as base 16 below, we see the same pattern.
More posts on Benford’s law
More posts on Pareto | {"url":"https://www.johndcook.com/blog/2017/11/16/pareto-distribution-and-benfords-law/","timestamp":"2024-11-06T09:02:56Z","content_type":"text/html","content_length":"50307","record_id":"<urn:uuid:cb9b9dc5-19e9-4ffb-adac-da517c7f75ae>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00481.warc.gz"} |
An Introduction to Support Vector Machine (SVM) and the Simplified SMO Algorithm
In machine learning, support vector machines (SVMs) are supervised learning models with associated learning algorithms that analyze data used for classification and regression analysis (Wikipedia).
This article is a summary of my learning and the main sources can be found in the References section.
Support Vector Machines and the Sequential Minimal Optimization (SMO) can be found in [1],[2] and [3]. Details about a simplified version of the SMO and its pseudo-code can be found in [4]. You can
also find Python code of the SMO algorithms in [5] but it is hard to understand for beginners who have just started to learn Machine Learning. [6] is a special gift for beginners who want to learn
about Support Vector Machine basically. In this article, I am going to introduce about SVM and a simplified version of the SMO by using Python code based on [4].
In this article, we will consider a linear classifier for a binary classification problem with labels y (y ϵ [-1,1]) and features x. A SVM will compute a linear classifier (or a line) of the form:
With f(x), we can predict y = 1 if f(x) ≥ 0 and y = -1 if f(x) < 0. And by solving the dual problem (Equation 12, 13 in [1] at the References section), f(x) can be expressed:
where α[i ](alpha i) is a Lagrange multiplier for solution and <x(^i),x> called inner product of x^(i) and x. A Python version of f(x) maybe look like this:
fXi = float(multiply(alphas,Y).T*(X*X[i,:].T)) + b
The Simplified SMO Algorithm
The simplified SMO algorithm takes two α parameters, α[i] and α[j], and optimizes them. To do this, we iterate over all α[i], i = 1, . . . m. If α[i] does not fulfill the Karush-Kuhn-Tucker
conditions to within some numerical tolerance, we select α[j] at random from the remaining m − 1 α’s and optimize α[i] and α[j]. The following function is going to help us to select j randomly:
def selectJrandomly(i,m):
while (j==i):
j = int(random.uniform(0,m))
return j
If none of the α’s are changed after a few iterations over all the α[i]’s, then the algorithm terminates. We must also find bounds L and H:
• If y^(i) != y^(j), L = max(0, α[j] − α[i]), H = min(C, C + α[j] − α[i])
• If y^(i) = y^(j), L = max(0, α[i ]+ α[j] − C), H = min(C, α[i] + α[j])
Where C is a regularization parameter. Python code for the above is as follows:
if (Y[i] != Y[j]):
L = max(0, alphas[j] - alphas[i])
H = min(C, C + alphas[j] - alphas[i])
L = max(0, alphas[j] + alphas[i] - C)
H = min(C, alphas[j] + alphas[i])
Now, we are going to find α[j] is given by:
Python code:
alphas[j] -= Y[j]*(Ei - Ej)/eta
Python code:
Ek = fXk - float(Y[k])
eta = 2.0 * X[i,:]*X[j,:].T - X[i,:]*X[i,:].T - X[j,:]*X[j,:].T
If this value ends up lying outside the bounds L and H, we must clip the value of α[j] to lie within this range:
The following function is going to help us to clip the value α[j]:
def clipAlphaJ(aj,H,L):
if aj > H:
aj = H
if L > aj:
aj = L
return aj
Finally, we can find the value for α[i]. This is given by:
where α^(old)[j] is the value of α[j] before optimization. A version of Python code can look like this:
alphas[i] += Y[j]*Y[i]*(alphaJold - alphas[j])
After optimizing α[i] and α[j], we select the threshold b:
where b[1]:
and b[2]:
Python code for b[1] and b[2]:[ ]
b1 = b - Ei- Y[i]*(alphas[i]-alphaIold)*X[i,:]*X[i,:].T - Y[j]*(alphas[j]-alphaJold)*X[i,:]*X[j,:].T
b2 = b - Ej- Y[i]*(alphas[i]-alphaIold)*X[i,:]*X[j,:].T - Y[j]*(alphas[j]-alphaJold)*X[j,:]*X[j,:].T
Computing the W
After optimizing α[i] and α[j], we can also compute w that is given:
The following function helps us to compute w from α[i] and α[j]:
def computeW(alphas, dataX, classY):
X = mat(dataX)
Y = mat(classY).T
m,n = shape(X)
w = zeros((n,1))
for i in range(m):
w += multiply(alphas[i]*Y[i],X[i,:].T)
return w
Predicted Class
We can predict which class a point belongs to from w and b:
def predictedClass(point, w, b):
p = mat(point)
f = p*w + b
if f > 0:
print(point," is belong to Class 1")
print(point," is belong to Class -1")
The Python Function for the Simplified SMO Algorithm
And now, we can create a function (named simplifiedSMO) for the simplified SMO algorithm based on pseudo code in [4]:
• C: regularization parameter
• tol: numerical tolerance
• max passes: max # of times to iterate over α’s without changing
• (x^(1), y^(1)), . . . , (x^(m), y^(m)): training data
α : Lagrange multipliers for solution
b : threshold for solution
def simplifiedSMO(dataX, classY, C, tol, max_passes):
X = mat(dataX);
Y = mat(classY).T
m,n = shape(X)
# Initialize b: threshold for solution
b = 0;
# Initialize alphas: lagrange multipliers for solution
alphas = mat(zeros((m,1)))
passes = 0
while (passes < max_passes):
num_changed_alphas = 0
for i in range(m):
# Calculate Ei = f(xi) - yi
fXi = float(multiply(alphas,Y).T*(X*X[i,:].T)) + b
Ei = fXi - float(Y[i])
if ((Y[i]*Ei < -tol) and (alphas[i] < C)) or ((Y[i]*Ei > tol)
and (alphas[i] > 0)):
# select j # i randomly
j = selectJrandom(i,m)
# Calculate Ej = f(xj) - yj
fXj = float(multiply(alphas,Y).T*(X*X[j,:].T)) + b
Ej = fXj - float(Y[j])
# save old alphas's
alphaIold = alphas[i].copy();
alphaJold = alphas[j].copy();
# compute L and H
if (Y[i] != Y[j]):
L = max(0, alphas[j] - alphas[i])
H = min(C, C + alphas[j] - alphas[i])
L = max(0, alphas[j] + alphas[i] - C)
H = min(C, alphas[j] + alphas[i])
# if L = H the continue to next i
if L==H:
# compute eta
eta = 2.0 * X[i,:]*X[j,:].T - X[i,:]*X[i,:].T - X[j,:]*X[j,:].T
# if eta >= 0 then continue to next i
if eta >= 0:
# compute new value for alphas j
alphas[j] -= Y[j]*(Ei - Ej)/eta
# clip new value for alphas j
alphas[j] = clipAlphasJ(alphas[j],H,L)
# if |alphasj - alphasold| < 0.00001 then continue to next i
if (abs(alphas[j] - alphaJold) < 0.00001):
# determine value for alphas i
alphas[i] += Y[j]*Y[i]*(alphaJold - alphas[j])
# compute b1 and b2
b1 = b - Ei- Y[i]*(alphas[i]-alphaIold)*X[i,:]*X[i,:].T -
b2 = b - Ej- Y[i]*(alphas[i]-alphaIold)*X[i,:]*X[j,:].T
- Y[j]*(alphas[j]-alphaJold)*X[j,:]*X[j,:].T
# compute b
if (0 < alphas[i]) and (C > alphas[i]):
b = b1
elif (0 < alphas[j]) and (C > alphas[j]):
b = b2
b = (b1 + b2)/2.0
num_changed_alphas += 1
if (num_changed_alphas == 0): passes += 1
else: passes = 0
return b,alphas
Plotting the Linear Classifier
After having alpha, w and b, we can also plot the linear classifier (or a line). The following function is going to help us to do this:
def plotLinearClassifier(point, w, alphas, b, dataX, labelY):
Y = np.array(labelY)
X = np.array(dataX)
svmMat = []
alphaMat = []
for i in range(10):
if alphas[i]>0.0:
svmPoints = np.array(svmMat)
alphasArr = np.array(alphaMat)
numofSVMs = shape(svmPoints)[0]
print("Number of SVM points: %d" % numofSVMs)
xSVM = []; ySVM = []
for i in range(numofSVMs):
n = shape(X)[0]
xcord1 = []; ycord1 = []
xcord2 = []; ycord2 = []
for i in range(n):
if int(labelY[i])== 1:
fig = plt.figure()
ax = fig.add_subplot(111)
ax.scatter(xcord1, ycord1, s=30, c='red', marker='s')
for j in range(0,len(xcord1)):
for l in range(numofSVMs):
if (xcord1[j]== xSVM[l]) and (ycord1[j]== ySVM[l]):
ax.annotate("SVM", (xcord1[j],ycord1[j]), (xcord1[j]+1,ycord1[j]+2),
arrowprops=dict(facecolor='black', shrink=0.005))
ax.scatter(xcord2, ycord2, s=30, c='green')
for k in range(0,len(xcord2)):
for l in range(numofSVMs):
if (xcord2[k]== xSVM[l]) and (ycord2[k]== ySVM[l]):
ax.annotate("SVM", (xcord2[k],ycord2[k]),(xcord2[k]-1,ycord2[k]+1),
arrowprops=dict(facecolor='black', shrink=0.005))
red_patch = mpatches.Patch(color='red', label='Class 1')
green_patch = mpatches.Patch(color='green', label='Class -1')
x = []
y = []
for xfit in np.linspace(-3.0, 3.0):
y.append(float((-w[0]/w[1])*xfit - b[0,0])/w[1])
p = mat(point)
ax.scatter(p[0,0], p[0,1], s=30, c='black', marker='s')
circle1=plt.Circle((p[0,0],p[0,1]),0.6, color='b', fill=False)
Using the Code
To run all of Python code above, we need to create three files:
• The myData.txt file contains training data:
-3 -2 0
-2 3 0
-1 -4 0
-1 9 1
In each row, the two first values are features and the third value is a label.
• The SimSMO.py file contains functions:
def loadDataSet(fileName):
dataX = []
labelY = []
fr = open(fileName)
for r in fr.readlines():
record = r.strip().split()
dataX.append([float(record[0]), float(record[1])])
return dataX, labelY
# select j # i randomly
def selectJrandomly(i,m):
# clip new value for alphas j
def clipAlphaJ(alphasj,H,L):
def simplifiedSMO(dataX, classY, C, tol, max_passes):
def computeW(alphas,dataX,classY):
def plotLinearClassifier(point, w, alphas, b, dataX, labelY):
def predictedClass(point, w, b):
• Finally, we need to create the testSVM.py file to test functions:
import SimSMO
X,Y = SimSMO.loadDataSet('myData.txt')
b,alphas = SimSMO.simplifiedSMO(X, Y, 0.6, 0.001, 40)
w = SimSMO.computeW(alphas,X,Y)
# test with the data point (3, 4)
SimSMO.plotLinearClassifier([3,4], w, alphas, b, X, Y)
The result can look like this:
Number of SVM points: 3
[3, 4] is belong to Class -1
Points of Interest
In this article, I only introduced the SVM basically and a simplified version of the SMO algorithm. If you want to use SVMs and the SMO in a real world application, you can discover more about them
in documents below (or maybe more).
• 18^th November, 2018: Initial version | {"url":"https://www.codeproject.com/Articles/1267445/%2FArticles%2F1267445%2FAn-Introduction-to-Support-Vector-Machine-SVM-and","timestamp":"2024-11-08T20:51:56Z","content_type":"text/html","content_length":"65437","record_id":"<urn:uuid:a738fcce-3a9c-4133-a50e-5029c8e734e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00133.warc.gz"} |
The Egypt Population (Live) counter shows a continuously updated estimate of the current population of Egypt delivered by Worldometer's RTS algorithm, which processes data collected from the United
Nations Population Division.
The Population of Egypt (1950 - 2023) chart plots the total population count as of July 1 of each year, from 1950 to 2023.
Yearly Population Growth Rate
chart plots the annual percentage changes in population registered on July 1 of each year, from 1951 to 2023. This value can differ from the
Yearly % Change
shown in the historical table, which shows the last year equivalent percentage change assuming homogeneous change in the preceding five year period.
Year: as of July 1 of the year indicated.
Population: Overall total population (both sexes and all ages) in the country as of July 1 of the year indicated, as estimated by the United Nations, Department of Economic and Social Affairs,
Population Division. World Population Prospects: The 2022 Revision. For forecasted years, the U.N. medium-fertility variant is used.
Yearly % Change: For 2023: percentage change in total population over the last year (from July 1, 2022 to June 30 2023). For all other years: latest year annual percentage change equivalent assuming
homogeneous change in the preceding five year period, calculated through reverse compounding.
Yearly Change: For 2023: absolute change in total population (increase or decrease in number of people) over the last year (from July 1, 2022 to June 30 2023). For all other years: average annual
numerical change over the preceding five year period.
Migrants (net): The average annual number of immigrants minus the number of emigrants over the preceding five year period (running from July 1 to June 30 of the initial and final years), or
subsequent five year period (for 2016 data). A negative number means that there are more emigrants than immigrants.
Median Age: age that divides the population into two numerically equal groups: half of the people are older than the median age indicated and half are younger. This parameter provides an indication
of age distribution.
Fertility Rate: (Total Fertility Rate, or TFR), it is expressed as children per woman. It is calculated as the average number of children an average woman will have during her reproductive period (15
to 49 years old) based on the current fertility rates of every age group in the country, and assuming she is not subject to mortality.
Density (P/Km²): (Population Density) Population per square Kilometer (Km²).
Urban Pop % : Urban population as a percentage of total population.
Urban Population: Population living in areas classified as urban according to the criteria used by each country.
Country's Share of World Pop: Total population in the country as a percentage of total World Population as of July 1 of the year indicated.
World Population: Total World Population as of July 1 of the year indicated.
Global Rank: Position held by Egypt in the list of all countries worldwide ranked by population (from the highest population to the lowest population) as of July 1 of the year indicated. | {"url":"http://meterchina.info/index-216.html","timestamp":"2024-11-07T19:12:26Z","content_type":"text/html","content_length":"58304","record_id":"<urn:uuid:6a1c9ad8-e6a8-49fb-aa0f-6226ef5e5263>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00503.warc.gz"} |
I Know In Spelled Out Numbers - SpellingNumbers.com
I Know In Spelled Out Numbers
I Know In Spelled Out Numbers – It can be challenging to learn to spell numbers. But, the process of learning to spell may be made much easier with the right resources. There are many resources
available to help you learn to spell whether working or at school. These include advice and tricks, workbooks and even games online.
The format of the Associated Press format
If you are writing for newspapers (or any other print media), you must be capable of spelling number in the AP format. It is possible to use the AP style to simplify your writing.
The Associated Press Stylebook’s 1953 debut has seen many updates since when it first appeared. The stylebook is now celebrating it’s 55th birthday. The stylebook is utilized by the vast majority of
American newspapers periodicals and internet news media.
A set of language and punctuation guidelines known as AP Style are frequently applied in journalism. AP Style is a set of best practices that includes the use of capitalization, the use of dates and
times as well as references.
Regular numbers
An ordinal number refers to an exclusive integer that marks a specific place in a list or series. These numbers are typically used to represent time, size and significance. They can also be used to
show what is in what order.
Depending upon the situation depending on the situation, normal numbers are able to be stated both verbally and numerically. The usage of a distinct suffix distinguishes the two in the most important
To make a number ordinal add a “th” at the end. For instance, 31 is an ordinal number.
There are many uses for ordinals. These include dates and names. It is important to know the distinctions between an ordinal (or cardinal) and an ordinal.
In addition, trillions of dollars
Numerology is a useful tool in a variety of contexts, including geology and the stock market. Millions of dollars and billions are just two instances. Million is a normal number that is present prior
to 1,000,001; the billion occurs immediately after 999.999.999.
The annual earnings of a business is measured in millions. They also serve as a basis to determine the value of a fund, stock, or piece money. Furthermore, billions are often used as a measure for a
company’s capitalization. You can check the validity of your estimations by using an online calculator for unit conversion to convert millions to billions.
Fractions in English are used to identify specific items or parts. The numerator and the denominator are divided into two parts. The numerator shows how many pieces of identical sizes were taken. The
second is the denominator, which displays how many pieces were divided.
Fractions can either be expressed mathematically or written in words. If you write fractions as words, be sure to write them in the correct way. This can be a challenge when you need to make use of a
lot of hyphens, particularly when it comes to larger fractions.
There are some basic principles you can use to write fractions like words. You can start sentences by writing the numbers in complete. Another alternative is to write fractions using decimal form.
When spelling numbers, you’ll be using years, no matter whether you’re writing a report or thesis, an email, or even a research paper. You can avoid typing out the same number over and over and
ensure proper formatting applying a few guidelines and tricks.
In formal writing, numbers should be written down. There are many style guides that provide different guidelines. The Chicago Manual of Style recommends that you use numerals between 1 and 100.
However, it is not recommended to use figures that are larger than 401.
Of course, exceptions exist. One exception is the American Psychological Association’s (APA), style guide. This manual, although not a specific publication, is widely used in scientific writing.
Date and Time
Some guidelines general to the styling of numbers are included in the Associated Press style manual. The numbers 10 and higher can be used using the numerals system. Numerology can also be utilized
in other places. The standard principle for the first five numbers of your report is to have the “n-mandated” number. There are exceptions.
Both the Chicago Manual of Technique, and the AP stylebook mentioned earlier recommend the use of plenty of numbers. This is not to imply that a different version with no numbers isn’t feasible
however. I’m an AP graduate and can verify that there is a difference.
A stylebook must be reviewed to determine which ones you are omitting. For instance, it’s important to make sure to include a “t” such as “time”.
Gallery of I Know In Spelled Out Numbers
Numbers 1 100 English ESL Worksheets For Distance Learning And
Pin On Interactive Notebooks
English Worksheets Numbers Spelled Out | {"url":"https://www.spellingnumbers.com/i-know-in-spelled-out-numbers/","timestamp":"2024-11-01T20:58:04Z","content_type":"text/html","content_length":"60349","record_id":"<urn:uuid:5b9eddc2-f3ec-4432-85df-6a6f697b6c2d>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00039.warc.gz"} |
Convert km/l to GPM (Kilometer per Liter to Gallons per mile (US))
Kilometer per Liter into Gallons per mile (US)
Direct link to this calculator:
Convert km/l to GPM (Kilometer per Liter to Gallons per mile (US))
1. Choose the right category from the selection list, in this case 'Fuel consumption'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), square root (√), brackets and
π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Kilometer per Liter [km/l]'.
4. Finally choose the unit you want the value to be converted to, in this case 'Gallons per mile (US) [GPM]'.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
Utilize the full range of performance for this units calculator
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '976 Kilometer per Liter'. In so doing, either the full name of the
unit or its abbreviation can be usedas an example, either 'Kilometer per Liter' or 'km/l'. Then, the calculator determines the category of the measurement unit of measure that is to be converted, in
this case 'Fuel consumption'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you
originally sought. Alternatively, the value to be converted can be entered as follows: '67 km/l to GPM' or '34 km/l into GPM' or '1 Kilometer per Liter -> Gallons per mile (US)' or '34 km/l = GPM' or
'67 Kilometer per Liter to GPM' or '1 km/l to Gallons per mile (US)' or '67 Kilometer per Liter into Gallons per mile (US)'. For this alternative, the calculator also figures out immediately into
which unit the original value is specifically to be converted. Regardless which of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists
with myriad categories and countless supported units. All of that is taken over for us by the calculator and it gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(34 * 67) km/l'. But different
units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '67 Kilometer per Liter + 1 Gallons per mile (US)' or '1mm x 34cm x 67dm
= ? cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question.
The mathematical functions sin, cos, tan and sqrt can also be used. Example: sin(π/2), cos(pi/2), tan(90°), sin(90) or sqrt(4).
If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 9.561 975 221 628 ×1019. For this form of presentation, the number
will be segmented into an exponent, here 19, and the actual number, here 9.561 975 221 628. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket
calculators, one also finds the way of writing numbers as 9.561 975 221 628 E+19. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at
this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 95 619 752 216 280 000 000. Independent of the presentation of the
results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications. | {"url":"https://www.convert-measurement-units.com/convert+Kilometer+per+Liter+to+Gallons+per+mile+US.php","timestamp":"2024-11-05T13:13:07Z","content_type":"text/html","content_length":"55991","record_id":"<urn:uuid:7f299a1f-7894-4014-9688-dc5e70f7af18>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00438.warc.gz"} |
associative property
associative property वाक्य
"associative property" हिंदी में associative property in a sentence
1. The associative property is closely related to the commutative property.
2. The associative property, i . e . is verified using basic properties of union and set difference.
3. A semigroup is a function \ cdot : S \ times S \ rightarrow S ) that satisfies the associative property:
4. A "'rational expression "'is an associative properties of addition and multiplication, distributive property and rules for the operations on the fractions ).
5. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as
the order of terms doesn't change.
6. In this way, each increase in the exponent by a full interval can be seen to increase the previous total by another five percent . ( The order of multiplication does not change the result based
on the associative property of multiplication .)
7. In mathematical theory, assumptions about the properties of binary operators ( for example the associative property and the commutative property ) are often used as axioms in fields of study such
as number theory and also two sub-disciplines of abstract algebra, group theory and ring theory.
8. These are the commutative property, the associative property, the identity property ( 2 is the exponentiative identity for commutative exponentiation, ) the special-value property ( for addition;
this number is-\ infty; for multiplication it is 0; for commutative exponentiation it is 1 ); and the inverse property ( the exponentiative inverse of a is the number b for which commexp ( a, b )
= 2.
9. An operation that is mathematically associative, by definition requires no notational associativity . ( For example, addition has the associative property, therefore it does not have to be either
left associative or right associative . ) An operation that is not mathematically associative, however, must be notationally left-, right-, or non-associative . ( For example, subtraction does
not have the associative property, therefore it must have notational associativity .)
10. An operation that is mathematically associative, by definition requires no notational associativity . ( For example, addition has the associative property, therefore it does not have to be either
left associative or right associative . ) An operation that is not mathematically associative, however, must be notationally left-, right-, or non-associative . ( For example, subtraction does
not have the associative property, therefore it must have notational associativity .) | {"url":"https://m.hindlish.in/sentence/associative%20property","timestamp":"2024-11-09T16:30:35Z","content_type":"text/html","content_length":"23514","record_id":"<urn:uuid:acb882df-6aa1-4ca3-9d14-536fb3180d7f>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00869.warc.gz"} |
Seed, table's system | Studio Irvine
Seed, table’s system
Alba 2022
Seed, table’s system
Alba 2022
“Seed is a family of consumer and contract tables reminiscent of Parisian bistros. The concept is a sphere that unites the vertical elements making the system a structure: a single sign that gives
the product its name. I have always thought of the sphere as a generating geometry: a seed being a symbol of perfection and absolute regularity, defined as ‘the place of points in space that all have
equal distance from a fixed point, called the centre’. Table, high and low tables: Seed is an alternative choice with different dimensional and finish variants that emphasise or disemphasise the
construction sign”. Marialaura
produced by Alba | {"url":"http://www.studio-irvine.com/product-design/seed-tables-system/?slider=4","timestamp":"2024-11-12T18:26:02Z","content_type":"text/html","content_length":"14432","record_id":"<urn:uuid:cc619083-fb33-43a9-a4c2-0302b2e8b957>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00052.warc.gz"} |
5 Best Ways to Find the Minimum Number of Moves Needed to Move from One Cell of Matrix to Another in Python
π ‘ Problem Formulation: Imagine navigating through a grid or matrix by moving up, down, left, or right. We want to calculate the minimum steps a player or an object must take to travel from a
starting point to a destination cell within this grid. Given a two-dimensional matrix as our map, an initial cell coordinate (start_x, start_y), and a target cell coordinate (target_x, target_y), the
goal is to determine the minimum number of moves required to reach the target from the start.
Method 1: Breadth-First Search (BFS)
The Breadth-First Search (BFS) algorithm explores the matrix layer by layer, moving outward from the starting cell. Its function lies in traversing the matrix level by level until the destination
cell is reached, ensuring the minimum number of moves is found. It is particularly effective for unweighted graphs like our grid where all edge distances are equal.
Here’s an example:
from collections import deque
def min_moves(matrix, start, target):
rows, cols = len(matrix), len(matrix[0])
visited = set()
queue = deque([(start[0], start[1], 0)])
while queue:
x, y, distance = queue.popleft()
if (x, y) == target:
return distance
for dx, dy in [(-1,0), (1,0), (0,-1), (0,1)]:
new_x, new_y = x + dx, y + dy
if 0 <= new_x < rows and 0 <= new_y < cols and (new_x, new_y) not in visited:
visited.add((new_x, new_y))
queue.append((new_x, new_y, distance + 1))
return -1
moves = min_moves(matrix, (0, 0), (2, 2))
print(moves) # Outputs: 4
The snippet first defines a min_moves function to find the minimum moves from a start to a target position in a matrix using BFS. It maintains a queue for nodes to visit, and a set for already
visited nodes to avoid cycles. The function returns the distance once the target is reached, ensuring we have the minimum moves, as BFS guarantees.
Method 2: Dijkstra’s Algorithm
Dijkstra’s Algorithm is a more general method compared to BFS and is used for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. For a grid where
moves have equal weights, it can be slightly overkill, but Dijkstra’s is the method of choice for grids with variable movement costs.
Here’s an example:
import heapq
def min_moves_dijkstra(matrix, start, target):
rows, cols = len(matrix), len(matrix[0])
visited = set()
queue = [(0, start[0], start[1])]
while queue:
distance, x, y = heapq.heappop(queue)
if (x, y) == target:
return distance
if (x, y) in visited:
visited.add((x, y))
for dx, dy in [(-1,0), (1,0), (0,-1), (0,1)]:
new_x, new_y = x + dx, y + dy
if 0 <= new_x < rows and 0 <= new_y < cols:
heapq.heappush(queue, (distance + 1, new_x, new_y))
return -1
moves = min_moves_dijkstra(matrix, (0, 0), (2, 2))
print(moves) # Outputs: 4
The code example implements Dijkstra’s shortest path algorithm for a matrix. Priority queue is managed by the heapq module to ensure the next move is always the shortest available path. While more
versatile for complex graphs, for uniform cost grids, it performs comparably to BFS.
Method 3: A* Search Algorithm
A* Search Algorithm introduces a heuristic into the traditional graph-search algorithm, which allows for more efficient pathfinding as it can direct its search towards the goal. This is beneficial in
larger grids or when dealing with complex terrains – it is widely used in computer games and robotics.
Here’s an example:
from heapq import heappop, heappush
def heuristic(a, b):
return abs(b[0] - a[0]) + abs(b[1] - a[1])
def min_moves_a_star(matrix, start, target):
rows, cols = len(matrix), len(matrix[0])
queue = [(0, start)]
g_score = {start: 0}
f_score = {start: heuristic(start, target)}
while queue:
current = heappop(queue)[1]
if current == target:
return g_score[current]
for dx, dy in [(-1,0), (1,0), (0,-1), (0,1)]:
neighbor = (current[0] + dx, current[1] + dy)
tentative_g_score = g_score[current] + 1
if 0 <= neighbor[0] < rows and 0 <= neighbor[1] < cols:
if neighbor not in g_score or tentative_g_score < g_score[neighbor]:
g_score[neighbor] = tentative_g_score
f_score[neighbor] = tentative_g_score + heuristic(neighbor, target)
heappush(queue, (f_score[neighbor], neighbor))
return -1
moves = min_moves_a_star(matrix, (0, 0), (2, 2))
print(moves) # Outputs: 4
This code employs the A* search algorithm to calculate the minimum number of moves from one cell in a matrix to another. It uses a priority queue to manage open nodes and assigns two scores: g_score
for the moves from the start and f_score which estimates the total cost from start to target. The A* algorithm is efficient for its use of heuristics to guide the search.
Method 4: Dynamic Programming
Dynamic Programming (DP) is a methodical approach that solves a complex problem by breaking it down into simpler subproblems. It is applied in scenarios where the problem can be decomposed into
overlapping subproblems that can be solved independently. In the context of a grid, it efficiently finds the minimum number of moves by storing the results of subproblems to avoid redundant
Here’s an example:
def min_moves_dp(matrix, start, target):
rows, cols = len(matrix), len(matrix[0])
dp = [[float('inf')] * cols for _ in range(rows)]
dp[start[0]][start[1]] = 0
for r in range(rows):
for c in range(cols):
if r > 0:
dp[r][c] = min(dp[r][c], dp[r-1][c] + 1)
if c > 0:
dp[r][c] = min(dp[r][c], dp[r][c-1] + 1)
return dp[target[0]][target[1]]
moves = min_moves_dp(matrix, (0, 0), (2, 2))
print(moves) # Outputs: 4
Here, the min_moves_dp function utilizes dynamic programming to compute the minimum moves required in a matrix. The dp table keeps track of the minimum moves to reach each cell. Although DP works
well for certain types of problems, it can be less efficient for large grids due to increased memory usage from the table.
Bonus One-Liner Method 5: Manhattan Distance
When the grid does not have any barriers or obstacles, and you can move in only four directions, the minimum moves can be calculated using the Manhattan distance. It is the simplest and fastest
method as it computes the sum of the absolute differences of the coordinates.
Here’s an example:
def manhattan_distance(start, target):
return abs(target[0] - start[0]) + abs(target[1] - start[1])
moves = manhattan_distance((0, 0), (2, 2))
print(moves) # Outputs: 4
This one-liner function manhattan_distance calculates the Manhattan distance between two cells in a matrix. It is the most straightforward approach when dealing with an obstruction-free grid,
providing the minimum moves instantaneously.
• Method 1: Breadth-First Search (BFS). Ideal for uniform cost grids. Efficient for small to medium-sized grids. Poor scalability for very large grids with memory constraints.
• Method 2: Dijkstra’s Algorithm. Suitable for grids with varied costs. More computationally intensive than BFS for uniform cost grids. Preferred when having weighted edges.
• Method 3: A* Search Algorithm. Optimally efficient with the inclusion of heuristics. Best for non-uniformly weighted grids or when an approximation is acceptable. Complex implementation compared
to BFS and Dijkstraβ s.
• Method 4: Dynamic Programming. Reduces redundant calculations for overlapping subproblems. Can be memory-intensive. Appropriate when the number of unique subproblems is manageable.
• Method 5: Manhattan Distance. The simplest method for an open grid. Not suitable for grids with obstacles. Extremely fast for calculating the minimum number of moves in free space. | {"url":"https://blog.finxter.com/5-best-ways-to-find-the-minimum-number-of-moves-needed-to-move-from-one-cell-of-matrix-to-another-in-python/","timestamp":"2024-11-06T12:51:01Z","content_type":"text/html","content_length":"75483","record_id":"<urn:uuid:feebe774-4b2f-4fef-b2df-09505924012c>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00336.warc.gz"} |
Math, Grade 6, Fractions and Decimals, Fractions and Division in Word Problems
Make Connections
Performance Task
Ways of Thinking: Make Connections
Take notes on your classmates’ approaches to solving and writing fraction word problems.
As your classmates present, ask questions such as:
• How are the quantities in the problem related?
• What is the unknown quantity in the problem?
• How does your equation represent the problem situation?
• Is your answer reasonable? How do you know?
• Where are the known and unknown quantities in the problem you wrote?
• How do you know that your problem is a multiplication or a division situation? | {"url":"https://openspace.infohio.org/courseware/lesson/2099/student/?section=6","timestamp":"2024-11-11T07:48:24Z","content_type":"text/html","content_length":"32597","record_id":"<urn:uuid:58d8f93f-d810-476a-ad9f-ab31e2f1466c>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00121.warc.gz"} |
Concrete Slab Thickness Calculator – Accurate Measurements
This tool calculates the required thickness of a concrete slab based on your input specifications.
Concrete Slab Thickness Calculator
How It Works
The concrete slab thickness calculator is designed to help determine the required thickness of a concrete slab for a given set of parameters. The calculator takes into account the length and width of
the slab, the applied load, the compressive strength of the concrete, the bearing capacity of the soil, and the amount of reinforcement used.
How to Use It
1. Enter the length of the slab in feet.
2. Enter the width of the slab in feet.
3. Enter the expected load in pounds.
4. Enter the compressive strength of the concrete in psi.
5. Enter the bearing capacity of the soil in psi.
6. Enter the reinforcement percentage.
7. Click the “Calculate” button to see the required thickness of the slab in inches.
Calculation Explanation
The calculation is based on structural engineering principles that account for the load-bearing capacity of the slab, the concrete compressive strength, the soil bearing capacity, and the
reinforcement percentage. The formula used ensures that the slab will support the specified load while adhering to safety and structural integrity standards.
This calculator provides an estimate based on the input parameters. It should not replace professional engineering advice. Variables like temperature, environmental conditions, and specific
construction methods can affect the actual required slab thickness.
Use Cases for This Calculator
Calculate Concrete Slab Thickness for Residential Use
Help homeowners determine the appropriate thickness of concrete slab needed for their patio, driveway, or walkway based on the intended use to ensure durability and longevity of the structure.
Determine Concrete Slab Thickness for Commercial Projects
Assist contractors and builders in calculating the ideal thickness of concrete slabs for commercial buildings, warehouses, or industrial sites, taking into consideration the expected load-bearing
capacity and foot traffic.
Estimate Concrete Slab Thickness for Outdoor Spaces
Enable landscapers and outdoor designers to quickly find out the suitable thickness of concrete slabs for outdoor kitchens, seating areas, or pool decks, considering factors like weather exposure and
potential weight loads.
Calculate Concrete Slab Thickness for Garage Floors
Provide automotive enthusiasts and homeowners with a straightforward way to determine the recommended thickness of concrete slabs for garage floors, ensuring they can support cars, equipment, and
heavy tools.
Estimate Concrete Slab Thickness for Foundations
Assist construction professionals and engineers in calculating the optimal thickness of concrete slabs for building foundations, ensuring structural integrity and stability for the entire building.
Determine Concrete Slab Thickness for Sidewalks and Pathways
Help city planners and municipalities calculate the appropriate thickness of concrete slabs for sidewalks, pathways, and public spaces, taking into account pedestrian traffic and safety requirements.
Estimate Concrete Slab Thickness for Outdoor Sport Courts
Enable sports facility managers and designers to accurately determine the required thickness of concrete slabs for outdoor basketball courts, tennis courts, or recreational areas, ensuring adequate
support for athletic activities.
Calculate Concrete Slab Thickness for Agricultural Use
Assist farmers and agricultural professionals in estimating the ideal thickness of concrete slabs for barn floors, farm equipment storage areas, or feed storage facilities, considering the heavy
machinery and livestock loads.
Determine Concrete Slab Thickness for Patios and Outdoor Entertaining Areas
Enable homeowners and outdoor enthusiasts to calculate the suitable thickness of concrete slabs for patios, decks, and outdoor entertaining areas, ensuring they can withstand outdoor furniture,
grills, and social gatherings.
Estimate Concrete Slab Thickness for Public Infrastructure Projects
Help civil engineers and infrastructure planners determine the required thickness of concrete slabs for roads, bridges, tunnels, or public transportation facilities, ensuring durability and longevity
of the infrastructure under various traffic conditions. | {"url":"https://calculatorsforhome.com/concrete-slab-thickness-calculator/","timestamp":"2024-11-06T16:48:56Z","content_type":"text/html","content_length":"145619","record_id":"<urn:uuid:dc107f23-f6e7-48f9-9f13-43576a6d8b4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00588.warc.gz"} |
Can someone do my Linear Programming assignment for me? | Linear Programming Assignment Help
Can someone do my Linear Programming assignment for me? I’m relatively advanced so I can quickly answer my questions. Thank you again! A: Here’s what I think you can do to get it working! Here’s
hoping you get what you’re looking for. Let me explain. Lets start with an exercise: make a list of possible functions based on a date. For instance, if we wanted to make some numbers that count 100,
our main strategy would be find the element 30 after the date. (I’ll get back to you in a couple of steps to speed up the process…). You could also do the following: If that function includes a “get”
or “split/modify” function, we’d then want to make the calculation using the rest of your code. Unfortunately, we don’t have the functionality for doing that, I’m stuck doing this manually. I have a
list of six numbers, and would like to add them to this viewbar. Here’s how it should look: For this we’d need some code that gets the first numbers by searching them, on the html page. This would
have to do with the date: date2html() method. This would be helpful if (in my current course of navigation) we take dates based on the current time and then subtract the current date according to
this implementation. Now that we know how this should look, we can ask the code to do some of the calculation on a second list containing 1 new num and 0 otherwise. I’m not going to recommend to do
this as this is exactly what all of you have got into this exercise. Can someone do my Linear Programming assignment for me? So I need to work in Excel and Visual Basic. It seems like my primary
purpose is to create a table of data within one row, and then I need to do the normal and linear searching. I thought that while this should work, Excel would be too simple to implement.
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There are couple of approaches that work for me, I.e., [ALTER TABLE] `table` Id Acell Address Date Data I keep getting the syntax error I mean I have the same problem. I need to find out how can I
bind the columns of ‘Acell Address’ to the data from the data table. Any ideas on how I should go about it, would be very much appreciated! Your help is very much appreciated! A: There are numerous
attempts to do this using the following classes. Client class Cell class Function Class For class coding with such a class the code snippet below is the easiest way to do it. However I check out here
you really should focus OO into code. This is not the end however because this may be going to other classes. For example, you could see an example of a class to do it in class Client{ public: Client
(); ~Client(); public: void Client::Client_Select(Bool selected){ ctr->SET_RESULTS(selected); SET_RESULTS(tbl); ctcell->SET_RESULTS(tempcell); ctcell->REPLACE(tempcell); ctcell->REPLACE(tempcell); if
(txt_row!=tbl) ctcell->REPLACE(txtcell); ctcell->SET_RESULTS(tempcell); } void ctcell_Select(Bool listview); protected: char boxid; std::string data; char tbl_row; Bool listview_tbl;
Client_SqlCommand ctrow=nullptr; Client_Table tblTbl=new Client_Table(); ListViewList ctrList = new ListViewList; while(txt_row!=nullptr) { btnTowflen(txt_row); ctrList.setOnCommandAction(btnTowflen,
btnTowflen); Can someone do my Linear Programming assignment for me? I am a new grad with a CS level but the assignment is called Linear Programming. I needed to do some math without having to
include a linear matrix (matrix) as a separate assignment but I see my results are quite similar if I change the assignment. However the output of my Linear Assignment shows a very few students that
won’t work too well and I would like to know whether or not it’s possible for one class to do the math problem for a student who won’t. Do the students with their basic math experience and ability do
a separate math assignment as to be able to do the math from scratch for as many students as possible in one week? If not, would that be better? If I do the homework online, it will be easier and
faster since my student won’t have to work overtime the short break is better. EDIT: I thought I would post it anyways. Thanks for your help. A: As @jolco wrote the answers are probably much in your
favor because you’ve decided to improve the math assignment you’ve created so easy to think about (pre-linear and post-linear). You should use only the type of linear system you’d like your student
to think of as linear at the moment. Linear-linear is the opposite of a linear system. It’s a linear thing that is essentially a linear function. It’s not only true that the function becomes
non-linear (or even non-differentiable), but it also gives more confidence since it is linear while other linear systems contain different linear quantities such as the power series.
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For students with a very advanced math instrument, my answer to your question applies only to linear systems that can be expressed for any functions of a measurable, non-obvious measurable set. In
your earlier question, you didn’t ask how general this is; you asked the question asking what exactly the class is supposed to do. | {"url":"https://linearprogramminghelp.com/can-someone-do-my-linear-programming-assignment-for-me-2","timestamp":"2024-11-05T23:08:20Z","content_type":"text/html","content_length":"114269","record_id":"<urn:uuid:860dded4-d170-46be-bb09-d2d941761033>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00571.warc.gz"} |
Google Quantum observed non-Abelian Anyons for the first time
Google Quantum AI has made a groundbreaking observation of non-Abelian anyons, particles that can exhibit any intermediate statistics between the well-known fermions and bosons. This breakthrough has
the potential to transform quantum computing by significantly enhancing its resistance to noise. The term “anyon” was coined by Nobel laureate physicist Frank Wilczek in the early 1980s while
studying Abelian anyons. He combined “any” with the particle suffix “-on” to emphasize the range of statistics these particles can exhibit.
Fermions are elementary particles with half-integer spin, such as quarks and leptons (electrons, muons, tauons, as well as their corresponding neutrinos), and their wave functions are
anti-symmetrical under the exchange of identical particles. Examples of bosons, which have integer spin and symmetrical wave functions under particle exchange, include the Higgs boson and the gauge
bosons: photons, W- and Z bosons, and gluons. In contrast, anyons obey fractional quantum statistics and possess more exotic properties that can just exist in two-dimensional systems.
The history of anyons dates back to Nobel laureate Robert Laughlin’s study of the fractional quantum Hall effect, a phenomenon observed in two-dimensional electron systems subjected to strong
magnetic fields. In 1983, he proposed a wave function to describe the ground state of these systems, which led to the understanding that the fractional quantum Hall effect involves quasiparticles
with fractional charge and statistics. These quasiparticles can be considered as anyons in two-dimensional space.
Anyons can be categorized into two types: Abelian and non-Abelian. Abelian anyons obey Abelian (commutative) statistics, which were studied by Wilczek and Laughlin. Under particle exchange, they pick
up a phase factor of e^i*theta, where theta is a scalar that is not just 0 as for bosons or pi as for fermions. Non-Abelian anyons, on the other hand, have more exotic properties: when exchanged,
their quantum states change in a non-trivial way that depends on the order of the exchange, leading to a “memory” effect. Under particle exchange, their wavefunction picks up a phase factor of U=e^
i*A with Hermitian matrix A that depends on the exchanged particles. As unitary matrices usually do not commute, it is this more-dimensional phase factor that explains the non-commutativity of
non-Abelian anyons. This memory effect makes non-Abelian anyons particularly interesting for topological quantum computation. While the theoretical concept of non-Abelian anyons was already discussed
around 1991, it was Alexei Kitaev who made the connection to fault-tolerant, topological quantum computing in a 1997 paper.
Microsoft, among other companies, has been working on harnessing non-Abelian anyons for topological quantum computing, focusing on a specific class called Majorana zero modes, which can be realized
in hybrid semiconductor-superconductor systems. “Zero modes” in quantum mechanics refer to states that exist at the lowest energy level of a quantum system, also known as the ground state. Majorana
fermions are a type of fermion that were first predicted by the Italian physicist Ettore Majorana in 1937. Their defining property is that they are their own antiparticles. This is unusual for
fermions, which typically have distinct particles and antiparticles due to their charge (in contrast to a boson like the photon). While Majorana zero-modes have not been observed as elementary
particles, they have found a home in the realm of condensed matter physics, specifically within certain “topological” materials. Here, they manifest as emergent collective behaviors of electrons,
known as quasiparticles.
These quasiparticles, termed topological Majorana fermions, appear in the atomic structure of these materials. Intriguingly, they’re found in excited states, seemingly at odds with the “zero-mode”
terminology which implies a ground state. The apparent contradiction can be resolved by understanding that Majorana zero modes are ground states within their own subsystem, the specific excitation
they form. However, their presence indicates an excited state for the overall electron system, compared to a state with no Majorana zero modes. In other words, they are a ground state property of an
excited electron system.
In a recent paper published in Nature on May 11, 2023, Google Quantum AI reported their first-ever observation of non-Abelian anyons using a superconducting quantum processor (see also article on
arXiv from 19 Oct 2022). They demonstrated the potential use of these anyons in quantum computations, such as creating a Greenberger-Horne-Zeilinger (GHZ) entangled state by braiding non-Abelian
anyons together.
This achievement complements another recent study published on May 9, 2023, by quantum computing company Quantinuum, which demonstrated non-Abelian braiding using a trapped-ion quantum processor. The
Google team’s work shows that non-Abelian anyon physics can be realized on superconducting processors, aligning with Microsoft’s approach to quantum computing. This breakthrough has the potential to
accelerate progress towards fault-tolerant topological quantum computing. | {"url":"https://domain-seeger.de/physics/google-quantum-observed-non-abelian-anyons-for-the-first-time/","timestamp":"2024-11-13T10:54:22Z","content_type":"text/html","content_length":"57811","record_id":"<urn:uuid:a24339fa-ddac-4394-b136-90459db16101>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00700.warc.gz"} |
Astro Lab 4
06-23-2015, 03:17 PM
(This post was last modified: 06-27-2015 07:00 PM by Marcel.)
Post: #1
Marcel Posts: 182
Member Joined: Mar 2014
Astro Lab 4
I have complete the programming of the first part of Astro Lab 4. Now, you can calculate the coordinates of the moon and Pluto. I have change the number of functions EXPORTed and also, you have now
more global variables. I will post in two week a more complete app's.
Have fun...
Marcel (In the software section!)
06-23-2015, 03:30 PM
Post: #2
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: Astro Lab 4
(06-23-2015 03:17 PM)Marcel Wrote: Hi,
I have complete the programming of the first part of Astro Lab 4. Now, you can calculate the coordinates of the moon and Pluto. I have change the number of functions EXPORTed and also, you have
now more global variables. I will post in two week a more complete app's.
Have fun...
thank you!
I'm trying it just now...
∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C - DM42, DM41X - WP34s Prime Soft. Lib
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ree from given Inorder and Level Order Traversal
Problem Statement:
Given an inorder and level order traversal, construct a binary tree from that.
Let's take the below example:
Given input array inorder[] = { 4, 2, 5, 1, 6, 3, 7 }
Given input array levelOrder[] = { 1, 2, 3, 4, 5, 6, 7 }
• First element in the levelorder [] will be the root of the tree, here it is 1.
• Now the search element 1 in inorder[], say you find it at position i, once you find it, make note of elements which are left to i (this will construct the leftsubtree) and elements which
are right to i ( this will construct the rightSubtree).
• Suppose in previous step, there are X number of elements which are left of ‘i’ (which will construct the leftsubtree), but these X elements will not be in the consecutive in levelorder[] so we
will extract these elements from levelorder[] by maintaining their sequence and store it in an array say newLeftLevel[].
• Similarly if there are Y number of elements which are right of ‘i’ (which will construct the rightsubtree), but these Y elements will not be in the consecutive in levelorder[] so we will extract
these elements from levelorder[] by maintaining their sequence and store it in an array say newRightLevel[].
• From previous two steps construct the left and right subtree and link it to root.left and root.right respectively by making recursive calls using newLeftLevel[] and newRightLevel[].
See the picture for better explanation.
Java code:
Login to Access Content
Python 2 code:
Login to Access Content
Related Chapters: | {"url":"https://systemsdesign.cloud/Algo/Tree/InorderLevelOrder","timestamp":"2024-11-07T17:19:11Z","content_type":"text/html","content_length":"43808","record_id":"<urn:uuid:d9060bc9-f26f-42fb-a509-bde7a98ac627>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00748.warc.gz"} |
Solution tolerances
Next: Validation Up: Polynomial Continuation Previous: Path closeness
A solution is considered as singular when the inverse of the condition number of the Jacobian matrix is lower than the given threshold value. Otherwise, the solution is declared to be regular.
Two solutions are considered as clustered when the distance between all corresponding components is lower than the given threshold value.
A solution is considered to diverge to infinity when its norm exceeds the given threshold value, in case of affine coordinates, or, in case of projective coordinates, when the added coordinate
becomes lower than the inverse of the threshold value. Continuation for the path being followed stops when it diverges to infinity.
Jan Verschelde | {"url":"https://homepages.math.uic.edu/~jan/PHCpack/node25.html","timestamp":"2024-11-06T07:38:05Z","content_type":"text/html","content_length":"3452","record_id":"<urn:uuid:d6706aa7-3071-4e42-b303-617718660666>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00524.warc.gz"} |
Count by Estimation: Beginners Worksheets
Estimate and Count Worksheets for beginners consist of colorful sets of practice sheets that are exciting for children to work with. Students in grade 1 and grade 2 are tasked to first estimate and
then count the objects. MCQs asked from themes are also included to understand the concept of estimation. Begin your practice with our free worksheets!
In each printable estimation worksheet, estimate the number of objects and compare it with the actual count. Also identify the type of estimate (underestimate / overestimate).
Worksheets have MCQs given in three different interesting themes. Estimate the number of objects in each case.
Few objects are depicted in a group. 1st grade children need to estimate the quantity and also check the accuracy of their estimates by counting the objects. | {"url":"https://www.mathworksheets4kids.com/estimating-count.php","timestamp":"2024-11-03T14:01:04Z","content_type":"text/html","content_length":"34385","record_id":"<urn:uuid:a74c1427-12e0-42f2-a62a-3475d054655c>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00024.warc.gz"} |
(zk-learning) Deriving functional commitment families
I just finished the second lecture of the Zero Knowledge Proofs MOOC titled Overview of Modern SNARK Constructions (yes I know, I’m two lectures behind schedule. Don’t remind me!!!).
In the lecture Dr. Boneh introduces four important functional commitment families used for building SNARKs
1. A univariate polynomial of at most degree $d$ such that we can open the committed polynomial at a point $x$.
2. A Multilinear polynomial in $d$ variables such that we can open the committed polynomial at a point $x_1, …, x_d$.
3. A vector $\vec{u}$ of size $d$ such that we can open the committed vector at element $u_i$.
4. An inner products on vector $\vec{u}$ of size $d$ such that we can open the committed vector at the inner product of $\vec{u}$ and an input vector $\vec{v}$.
He mentions in passing that any one of these four function families can be built from any of the other four, but leaves it at that. I was left wondering… how?
Like a lot of things professors say, the details are left as an exercise to the reader ;). In this article I’d like to present a special case of that statement: given multlinear polynomials we can
use it to build commitments for any of the other three families.
One by one let’s see how that’s done.
Univariate Polynomials
Recall the definition of a univariate polynomial commitment. We commit to a polynomial up to degree $d$ in a single variable $x$, and want to open it at arbitrary points.
One way to achieve this with a multilinear polynomial in $k$ variables is to “reduce” it to a single variable, and then map its coefficients to the coefficients in the unvariate polynomial.
Here’s what I mean.
The univariate polynomial is of the form
\[f(x) = a_0 + a_1x + ... \> + a_dx^d\]
Now I’m going to be clever and write the multilinear polynomial as follows, collecting each sum of terms with the same number of variables under a single coefficient
\[F(X) = b_0 + b_1(x_0 + ... \> + x_d) + b_2(x_0x_1 + ... \> + x_{d-1}x_d) + ... \> + b_dx_1...x_d\]
What if we substitute $x$ for every $x_i$ in the input vector $X$?
\[F(\begin{bmatrix}x_0 = x\\...\\x_d=x\end{bmatrix}) = b_0 + b_1(x + ... \> + x) + b_2(x + ... \> x) + ... + \> b_dx \\ = b_0 + \binom{d}{1}b_1x + \binom{d}{2}b_2x^2 + ... \> + \binom{d}{d}b_dx^2\]
We’ve reduced it to a polynomial in a single variable! Now we can map the coefficients of $F$ to the coefficients of $f$
\[a_0 = b_0 \\ a_1 = \binom{d}{1}b_1 \Rightarrow b_1 = \frac{a_1}{\binom{d}{1}} \\ ...\]
The general formula is
\[b_i = \frac{a_i}{\binom{d}{i}}\]
Thus, committting $F$ with coefficients $b_i = \frac{a_i}{\binom{d}{i}}$ and opening it at $X = [x,…,x]$ is just like committing $f$ with coefficients $a_0, …, a_d$ and opening it at $x$.
This one is easier than the previous one. In a vector commitment we commit to a vector $\vec{u}$ and open it at one of its elements: $f_{\vec{u}}(i) = u_i$.
We can implement this with a simple multivariate polynomial of the form
\[F(X) = u_1x_1 + ... + u_dx_d\]
For input $i$ to $f_{\vec{u}}$, we evaulate $F(X)$ such that $x_i = 1$ and $x_j = 0, j \neq i$ for every element in $X$.
\[F(X) = u_1x_1 + ... + u_dx_d \Rightarrow \\ F(X) = 0u_1 + ... + 1u_i + ... + 0u_d \Rightarrow \\ F(X) = u_i\]
Thus, committing to
\[F(X) = \sum_{i=1}^{d}{u_ix_i}\]
and opening it at
\[X = \begin{bmatrix}x_0 = 0\\...\\x_i = 1\\..\\x_d=0\end{bmatrix}\]
is equivalent to committing to $f_{\vec{u}}$ and opening at $i$.
Inner Products
In an inner product commitment we commit to a vector $\vec{u}$ and open it by evaluating its dot product with an input vector $\vec{v}: f_{\vec{u}}(\vec{v}) = \vec{u}\cdot\vec{v}$.
This one is very similar to the vector committment. The multivariate polynomial stays the same. But instead of evaluating it at a point where all but one $x_i$ is non-zero, we substitute $v_i = x_i$
for all $i$.
\[F(X) = u_1x_1 + ... + u_dx_d\]
\[F(\vec{v}) = u_1v_1 + ... + u_dv_d = \vec{u}\cdot\vec{v}\]
Thus, committing to $F(X) = \sum_{i=1}^{d}{u_ix_i}$
and opening it at
\[X = \begin{bmatrix}x_0 = u_0\\...\\x_i = u_i\\..\\x_d=u_d\end{bmatrix}\]
is like comitting to $f_{\vec{u}}$ and opening it at $\vec{v}$
This was a bit of a tangent that didn’t actually deepen my understanding of the lecture material, but it was fun nonetheless! As someone who hasn’t touched math seriously for 5+ years, it’s fun to
wipe the dust off my long forgotten skills (admittedly this isn’t that challenging as far as serious math goes).
Bonus - A (attempted) multiliner commitment from univariate commitments
In the Univariate Polynomials section I showed how to create a univariate commitment from a multilinear commitment. What about the other way around?
I tried to find a way to represent a multilinear commitment of a function in $d$ variables as a commitment of $d$ univariate polynomials of degree one. But the math is more complicated and the work
is incomplete, hence why this is a bonus section.
We can write a general multilinear function of $d$ variables in the following way:
\[F(X) = c_0 + c_1x_1 + ... + c_dx_d + c_{d+1}x_1x_2 + ... + c_{?}x_1...x_d\]
How many coefficients $c$ do we have in this form? For the terms in one $x$ variable we have $\binom{d}{1}$ possibilities. For the terms in two $x$ variable we have $\binom{d}{2}$ possibilities, and
so on. The total number of coefficients is
\[\sum_{i=0}^d{\binom{d}{i}} = 2^d\]
So given
\[F(X) = c_0 + c_1x_1 + ... + c_{2^d-1}x_1...x_d\]
My idea is to write $d$ univariate polynomials of the form
\[f_1(x_1) = a_1 + b_1x_1 \\ ... \\ f_d(x_d) = a_d = b_dx_d \\\]
And multiply them together to get the generic multilinear polynomial
Once I’ve expanded that and collected the terms, I’ll have coefficients in terms of $a$ and $b$ variables. Then I can map those coefficients to the $c$ coefficients and solve for the $a$’s and $b$’s.
If the multilinear polynomial is the product of the univariate polynomials, then comitting to the univariate polynomials is kind of like to comitting to the multivariate ones, right? I’m honestly not
sure. This is where my math knowledge breaks down. Nevertheless, let’s move forward with this approach. I need to introduce some notation for the expansion formula.
Let $N_d$ be the set of numbers ${1, 2, …, d}$
Let $\binom{N_d}{i}$ be the set of combinations from choosing $i$ elements from $N_d$. E.g. $\binom{N_4}{2} = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}$
Let $\sum_{a\in A}{f(a)}$ be a summation over the elements $a$ of a set $A$, applied to $f$.
My closed form expression is
\[\prod_{i=1}^d{f_i(x_i)} = \sum_{i=0}^d\Big(\sum_{a \in \binom{N_d}{i}}\big(\prod_{j \in a}a_j\big)\big(\prod_{k \in N_d - a}b_kx_k\big)\Big)\]
Huh? I would be just as skeptical as you at this point. Don’t believe me? I’ll show you it works with an example. Let’s expand it for $d=3$.
When $i = 0$
\[i = 0 \Rightarrow \binom{N_3}{0} = \{\{\}\} \\ a = \{\}, k = N_3 - a = \{1, 2, 3\} \Rightarrow b_1b_2b_3x_1x_2x_3\]
When $i = 1$
\[i = 1 \Rightarrow \binom{N_3}{1} = \{\{1\}, \{2\}, \{3\}\} \\ a = \{1\}, k = N_3 - a = \{2, 3\} \Rightarrow a_1b_2b_3x_2x_3 \\ a = \{2\}, k = N_3 - a = \{1, 3\} \Rightarrow a_2b_1b_3x_1x_3 \\ a = \
{3\}, k = N_3 - a = \{1, 2\} \Rightarrow a_3b_1b_2x_1x_2\]
When $i = 2$
\[i = 2 \Rightarrow \binom{N_3}{2} = \{\{1, 2\}, \{1, 3\}, \{2, 3\}\} \\ a = \{1, 2\}, k = N_3 - a = \{3\} \Rightarrow a_1a_2b_3x_3 \\ a = \{1, 3\}, k = N_3 - a = \{2\} \Rightarrow a_1a_3b_2x_2 \\ a
= \{2, 3\}, k = N_3 - a = \{1\} \Rightarrow a_2a_3b_1x_1 \\\]
When $i = 3$
\[i = 2 \Rightarrow \binom{N_3}{3} = \{\{1, 2, 3\}\} \\ a = \{1, 2, 3\}, k = N_3 - a = \{\} \Rightarrow a_1a_2a_3\]
Putting it all together we get $2^d = 2^3 = 8$ terms, exactly as expected.
\[\prod_{i=1}^3{f_i(x_i)} = a_1a_2a_3 + a_2a_3b_1x_1 + a_1a_3b_2x_2 + a_1a_2b_3x_3 + a_3b_1b_2x_1x_2 + a_2b_1b_3x_1x_3 + a_1b_2b_3x_2x_3 + b_1b_2b_3x_1x_2x_3\]
We can finally (maybe) figure out how to derive the $a$’s and $b$’s from the $c$’s. Let coefficient $c_B$ correspond to the term $\prod_{i \in B}{x_i}$ in the multilinear polynomial.
\[B = N_d - a \Rightarrow a = N_d - B\]
The $a$ in terms of $B$ is the start of our mapping, creating a system of equations for which we can solve for the $a$’s and $b$’s in terms of $c$’s. E.g. suppose $d = 3$ and $c_{{1, 2}} = 69$. In
other words, our multilinear polynomial has the term $69x_1x_2$. Then we know $69 = a_3b_1b_2$.
If we did this for every $c$ then we’d have a non-linear system of equations. Now this is where I get skeptical. I vaguely recall from my university math courses that anything non-linear is extremely
difficult to deal with, except in rare special cases. I think the approach breaks down with the system of equations.. so the bonus section ends here.
Congratualtions for reading until the end of this half baked stream of consciousness. If you’ve made it this far then please reach out. I guarantee we’ll have an interesting conversation.
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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.416
URN: urn:nbn:de:0030-drops-53153
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5315/
Nagarajan, Viswanath ; Sarpatwar, Kanthi K. ; Schieber, Baruch ; Shachnai, Hadas ; Wolf, Joel L.
The Container Selection Problem
We introduce and study a network resource management problem that is a special case of non-metric k-median, naturally arising in cross platform scheduling and cloud computing. In the continuous
d-dimensional container selection problem, we are given a set C of input points in d-dimensional Euclidean space, for some d >= 2, and a budget k. An input point p can be assigned to a "container
point" c only if c dominates p in every dimension. The assignment cost is then equal to the L1-norm of the container point. The goal is to find k container points in the d-dimensional space, such
that the total assignment cost for all input points is minimized. The discrete variant of the problem has one key distinction, namely, the container points must be chosen from a given set F of
For the continuous version, we obtain a polynomial time approximation scheme for any fixed dimension d>= 2. On the negative side, we show that the problem is NP-hard for any d>=3. We further show
that the discrete version is significantly harder, as it is NP-hard to approximate without violating the budget k in any dimension d>=3. Thus, we focus on obtaining bi-approximation algorithms. For d
=2, the bi-approximation guarantee is (1+epsilon,3), i.e., for any epsilon>0, our scheme outputs a solution of size 3k and cost at most (1+epsilon) times the optimum. For fixed d>2, we present a
(1+epsilon,O((1/epsilon)log k)) bi-approximation algorithm.
BibTeX - Entry
author = {Viswanath Nagarajan and Kanthi K. Sarpatwar and Baruch Schieber and Hadas Shachnai and Joel L. Wolf},
title = {{The Container Selection Problem}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
pages = {416--434},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-89-7},
ISSN = {1868-8969},
year = {2015},
volume = {40},
editor = {Naveen Garg and Klaus Jansen and Anup Rao and Jos{\'e} D. P. Rolim},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5315},
URN = {urn:nbn:de:0030-drops-53153},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.416},
annote = {Keywords: non-metric k-median, geometric hitting set, approximation algorithms, cloud computing, cross platform scheduling.}
Keywords: non-metric k-median, geometric hitting set, approximation algorithms, cloud computing, cross platform scheduling.
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)
Issue Date: 2015
Date of publication: 13.08.2015
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RC Phase Shift Oscillator Circuit Working & Applications
An RC Phase Shift Oscillator Circuit is a type of electronic oscillator that generates sinusoidal signals. It is typically consisting of an amplifier (usually an operational amplifier), resistors,
and capacitors arranged in a feedback network. The phase shift network created by the resistors and capacitors causes the output signal of the amplifier to be fed back to its input with a phase shift
of 180 degrees at the oscillation frequency. A simpler circuit of phase shift oscillator uses BJT.
RC Phase Shift Oscillator Circuit:
Here is an RC Phase Shift Oscillator a type of electronic oscillator that generates sine waves. It consists of an amplifier (usually an operational amplifier) and a feedback network comprising
resistors and capacitors that provide phase shift. Here’s a basic schematic for a simple RC Oscillator using an operational amplifier (op-amp) and a three-stage RC network:
RC Phase Shift Oscillator Using OPAMP
Explanation of RC Oscillator Circuit:
• The operational amplifier amplifies the voltage difference between its two inputs.
• The feedback network (three RC stages) determines the frequency of oscillation by introducing a phase shift.
• The resistors (R) and capacitors (C) are chosen to provide a total phase shift of 180 degrees at the desired frequency of oscillation each stage provides a 60 degree of phase shift.
The general formula for the frequency of oscillation F in a phase shift oscillator is:
Frequency of Phase Shift Oscillator
• is the resistance value of each resistor (assuming equal resistance values).
• is the capacitance value of each capacitor (assuming equal capacitance values).
• N is the number of RC stages.
Please note that component values need to be chosen carefully to ensure stability and desired oscillation frequency. Also, it’s essential to consider the limitations and characteristics of the
operational amplifier being used. Additionally, component tolerances and variations can affect the oscillation frequency. Experimentation and simulation are often necessary for fine-tuning the
Here is a circuit having output frequency around 650Hz. We have kept value of capacitor 10nF and resistor at 10kΩ.
RC Phase Shift Oscillator Using OPAMP
RC Phase Shift Oscillator Circuit Working:
An RC oscillator is a type of electronic oscillator circuit that generates sinusoidal output signals. It operates on the principle of phase shift, where the phase shift introduced by a network of
resistors (R) and capacitors (C) in the feedback path of an amplifier causes positive feedback, leading to sustained oscillations.
Feedback Network:
The heart of the RC phase shift oscillator is its feedback network, which typically consists of three identical RC sections connected in series. Each RC section includes a resistor (R) and a
capacitor (C). The phase shift introduced by each RC section is approximately 60 degrees (for a high enough frequency) due to the phase relationship between voltage and current in a capacitor and
Amplifier Stage:
The output of the feedback network is fed into an amplifier stage. This amplifier is usually an operational amplifier (op-amp) configured in an inverting amplifier configuration or a transistor
amplifier. The amplifier provides the necessary gain to compensate for the losses in the feedback network and to sustain oscillations.
Phase Shift:
The three RC sections collectively introduce a total phase shift of 180 degrees (3 * 60 degrees) at the desired oscillation frequency. This phase shift, along with the 180-degree phase shift
introduced by the amplifier’s inverting nature, results in a total phase shift of 360 degrees (or 0 degrees, which means no phase shift at all). This condition satisfies the Barkhausen criterion for
Phase Angel Formula
When the total phase shift around the loop is 360 degrees (or 0 degrees), positive feedback occurs. As a result, the output signal from the amplifier is in phase with the input signal to the feedback
network, thus sustaining oscillations.
Frequency Determination:
The frequency of oscillation in an RC oscillator is determined primarily by the values of the resistors (R) and capacitors (C) in the feedback network. The formula for calculating the frequency of
oscillation is:
• F is the frequency of oscillation.
• is the resistance in each RC section.
• is the capacitance in each RC section.
• N is the number of RC stages.
BJT Based Phase Shift Oscillator Circuit:
RC Phase Shift Oscillator by BJT
Here is a BJT based phase shift oscillator circuit, in this circuit the OPAMP has been replaced by BJT. The RC network provides a phase shift of 180 degree and the BJT provides remaining 180 degree
of phase shift.
BJT Based Phase Shift Oscillator Circuit Example:
Choosing appropriate values for resistors and capacitors ensures stable oscillation at the desired frequency. Typically, equal values for resistors and capacitors are chosen to achieve symmetry in
the feedback network.
RC Phase Shift Oscillator Using BJT
Overall, the RC phase shift oscillator is a simple and effective circuit for generating sinusoidal signals at a desired frequency. It finds applications in various electronic devices such as signal
generators, audio oscillators, and tone generators.
Waveform of RC Oscillator
RC Phase Shift Oscillator Waveform
Applications of RC Phase Shift Oscillator:
RC phase shift oscillators are commonly used in various electronic applications due to their simplicity and effectiveness in generating sine waves at a desired frequency. Here are some typical
Signal Generation:
One of the primary applications of RC phase shift oscillators is to generate continuous sine wave signals at a specific frequency. These signals are utilized in various electronic systems such as
audio oscillators, function generators, and tone generators.
Audio Oscillators:
RC phase shift oscillators are frequently employed in audio frequency applications, such as generating tones for musical instruments, audio testing equipment, and sound synthesis.
Frequency Standards:
In some applications, RC phase shift oscillators are used as frequency standards for calibration purposes, especially in low-frequency applications where precision is not critical.
Sine Wave Inverters:
RC phase shift oscillators can be used in sine wave inverters to convert DC power into AC power. Sine wave inverters are used in various applications such as uninterruptible power supplies (UPS),
solar power systems, and motor control systems.
Communication Systems:
They are used in communication systems for generating carrier frequencies in radio transmitters and receivers. They can also be used in frequency modulation (FM) and amplitude modulation (AM)
Medical Devices:
RC phase shift oscillators are used in medical devices such as biofeedback systems, where precise sine wave signals are required for monitoring and treating various physiological conditions.
Test and Measurement Equipment:
They are used in test and measurement equipment for generating reference signals and for frequency calibration purposes.
Educational Purposes:
RC phase shift oscillators are commonly used in educational laboratories to demonstrate the principles of oscillator circuits, frequency generation, and phase shifting.
They can be used in instrumentation applications where a stable and precise sine wave signal is needed, such as in data acquisition systems and sensor signal conditioning circuits.
Low-Frequency Applications:
RC phase shift oscillators are particularly useful in low-frequency applications where the complexity and cost of other oscillator circuits, such as LC or crystal oscillators, may be prohibitive.
Overall, RC phase shift oscillators find widespread use in electronics due to their simplicity, versatility, and cost-effectiveness in generating stable sine wave signals across a wide range of
Advantages & Disadvantages
The RC Phase Shift Oscillator is a type of oscillator circuit commonly used in electronic devices to generate sinusoidal signals at a specific frequency. Here are some advantages and disadvantages of
RC Phase Shift Oscillators:
Advantages of RC Phase Shift Oscillator:
Simple Design: The RC Phase Shift Oscillator circuit consists of only resistors, capacitors, and an active device (such as a transistor or op-amp). This simplicity makes it easy to design and
Low Cost: Since it requires minimal components, the RC Phase Shift Oscillator is relatively inexpensive to build, making it suitable for mass production in consumer electronics.
Stable Frequency: When properly designed, RC Phase Shift Oscillators can provide stable sinusoidal output signals at a fixed frequency determined by the values of the resistors and capacitors in the
feedback network.
Wide Frequency Range: With appropriate component values, RC Phase Shift Oscillators can operate over a wide range of frequencies, making them versatile for various applications.
Low Power Consumption: They typically consume low power, making them suitable for battery-operated devices and other low-power applications.
Disadvantages of RC Phase Shift Oscillator:
Sensitive to Component Tolerances: The frequency of oscillation in RC Phase Shift Oscillators is highly dependent on the values of the resistors and capacitors used. Small variations in component
values can lead to significant changes in the oscillation frequency, which may require precise component selection or tuning.
Limited Amplitude Stability: The output amplitude of the RC Phase Shift Oscillator may vary with changes in temperature, power supply voltage, and component aging. This limitation can affect the
overall stability of the oscillator’s output.
Limited Frequency Stability: While RC Phase Shift Oscillators can provide stable oscillations within a certain frequency range, they may not offer the same level of frequency stability as other
oscillator configurations, such as crystal oscillators or voltage controlled oscillators (VCOs).
Limited Output Power: RC Phase Shift Oscillators typically have limited output power compared to other oscillator configurations. This limitation may restrict their use in applications that require
higher output power levels.
Frequency Drift: Due to factors such as temperature variations and aging of components, the frequency of oscillation in RC Phase Shift Oscillators may drift over time, requiring periodic calibration
or adjustment to maintain accuracy.
Overall, while RC Phase Shift Oscillators offer simplicity and low cost, they may not be suitable for applications that require extremely high frequency stability or precise control over output
characteristics. However, for many general-purpose applications, they provide a practical solution for generating sinusoidal signals. | {"url":"https://www.hackatronic.com/rc-phase-shift-oscillator-circuit-working-applications/","timestamp":"2024-11-13T21:34:46Z","content_type":"text/html","content_length":"252967","record_id":"<urn:uuid:45f2a55d-e5c6-4e56-a9b2-13186e50aa7c>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00784.warc.gz"} |
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Table Of Content
What is mathematics/math?
Mathematics/math is a discipline that consists of the study of structure, quantity, change, space, and arrangements. It is utilized in our daily lives, such as in engineering, sports, driving the
car, etc. Maths always present in all aspects of daily routine.
Mathematics can be categorized into two types
• Pure Mathematics:
It considers the backbone of mathematics, which deals without actual consideration into direct applications such as Finance, Economics, and much more. There are various kinds of pure mathematics,
• Algebra:
It deals with the methods of solving symbols and the properties to get the value of the symbols.
• Geometry:
It is a branch of math used to study the shapes, sizes, positions, dimensions, and angles of things.
• Mathematical analysis:
It deals with limits and related theories, such as integration, differentiation, measure, analytic functions, and infinite series.
• Number theory:
It is used to study the set of positive whole numbers, such as 1,2,3,4,5,6,7, which are often known as the natural numbers.
• Applied Mathematics:
This mathematics branch deals with the computation approach to find out the solution to mathematical numerics. It deals with subjects like physical science, computer science, engineering, and
much more. In other words, the applied mathematics deals with daily life difficulties, such as measuring your vehicle's speed, etc.
Examples of mathematics coursework
Problem 1:
Allen goes to a bookshop. The probability that he looks for (a) a book of non- fiction is 0.50, (b) a book of fiction is 0.40, and (c) both non-fiction and fiction is 0.30. Find the probability that
he looks for a book of non-fiction, fiction, or both?
Let A = the event he looks for a non-fiction book, and let B = the event he looks for a fiction book. When both or either of these events happens, the union of A and B has happened.
This question needs us to calculate the probability of the union of A and B. We will use the rule of addition to find the probability:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A ∪ B) = 0.50 + 0.40 - 0.30 = 0.60
Problem 2
How many ways are there to arrange the letters P, Q, and R?
The first way to answer this problem is to list out the possible arrangements or permutation of P, Q, and R. These are PQR, PRQ, QPR, QRP, RPQ, and RQP. Thus, we can say that there are 6 possible
arrangements. Therefore the answer to this problem is 6.
Another way to solve this problem is by using the formula for counting arrangements or permutations. The formula is mentioned below.
Formula to count permutations. The number of arrangements or permutations of n objects is taken r at a time can be calculated with:
nPr = n(n - 1)(n - 2) ... (n - r + 1) = n! / (n - r)!
The formula implies that the number of arrangements or permutations is n! / (n - r)!. Our problem has 3 different objects (the letters P, Q, and R); therefore, n = 3. And now, we have to arrange each
in a group of 3, so r = 3. Thus, the value of arrangements or permutations is
3! / (3 - 3)! or 3! / 0!. Which is equal to (3)(2)(1)/1 = 6.
How mathematics helps you in day to day life
• Improve one's brain quality:
As per the research of Dr. Tanya Professor at Evans of Stanford University, it is observed that learning of mathematical concepts can benefit you. It helps individuals to imagine things in a more
analytical method. Math is a subject that requires daily practice. And practice means your brain will be involved more. Daily exercise of your brain will only sharpen it.
• To manage finances:
It is well-versed to everyone that math is all about the calculations; thus, you can use it to count the particular thing's values and pay the bill as per your budget. Math involves numbers, and
finance also involves numbers. If you are good at math, then you can easily track your finances, whether it is an expense or any income.
• Improve problem-solving skills:
When a student tries to solve a complex mathematics problem, it helps to improve their reasoning skills. Mathematics is all about problems and their solutions. Mathematicians generally say that
the solution is within the problem. While doing math, a problem solving skill is developed within students.
• Helps to generate subject’s theories:
Math is very useful in the science subjects as it is used to carry out the different researches and write the theories and observations of that topic. It is a place where maths play an important
role. It is obvious that if a person is solving a problem using assumptions and, on the other hand, a person is using math to solve the same problem, then the latter will be more accurate with
the solution.
• Solve troubles more appropriately:
Whenever a person finds any challenge in a situation, they need to resolve it more appropriately; hence, they can apply mathematics calculations to resolve them.
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which book should i prefer for maths, physics &
Algebra> which book shud i buy...
2 Answers
Chetan Mandayam Nayakar
Last Activity: 12 Years ago
Physics: HC Verma, Young & Freedman(used at MIT, USA), “Problems in Physics” by Abhay Kumar Singh
Inorganic Chemistry: JD Lee, OP Tandon
“Physical Chemistry” by Robert A. Alberty, Robert J. Silbey (both from MIT, USA)
Organic Chemistry: Morrison & Boyd, Paula Yurkanis Bruice
“Calculus:graphical,numerical,algebraic” by Thomas,Finney, Demana, Waits (one or more authors from MIT, USA)
Arihant Integral Calculus, Arihant Differential Calculus, Arihant “Vectors and 3-D geometry”
Trigonometry: SL Loney
Coordinate geometry: SL Loney
Algebra: “Challenge and Thrill of Pre-College Mathematics” by V. Krishnamurthy
mohammad shahbaaz
Last Activity: 12 Years ago
Provide a better Answer & Earn Cool Goodies
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Flowers of the First Law of Causal Inference (3)
Flower 3 — Generalizing experimental findings
Continuing our examination of “the flowers of the First Law” (see previous flowers here and here) this posting looks at one of the most crucial questions in causal inference: “How generalizable are
our randomized clinical trials?” Readers of this blog would be delighted to learn that one of our flowers provides an elegant and rather general answer to this question. I will describe this answer
in the context of transportability theory, and compare it to the way researchers have attempted to tackle the problem using the language of ignorability. We will see that ignorability-type
assumptions are fairly limited, both in their ability to define conditions that permit generalizations, and in our ability to justify them in specific applications.
1. Transportability and Selection Bias
The problem of generalizing experimental findings from the trial sample to the population as a whole, also known as the problem of “sample selection-bias” (Heckman, 1979; Bareinboim et al., 2014),
has received wide attention lately, as more researchers come to recognize this bias as a major threat to the validity of experimental findings in both the health sciences (Stuart et al., 2015) and
social policy making (Manski, 2013).
Since participation in a randomized trial cannot be mandated, we cannot guarantee that the study population would be the same as the population of interest. For example, the study population may
consist of volunteers, who respond to financial and medical incentives offered by pharmaceutical firms or experimental teams, so, the distribution of outcomes in the study may differ substantially
from the distribution of outcomes under the policy of interest.
Another impediment to the validity of experimental finding is that the types of individuals in the target population may change over time. For example, as more individuals become eligible for health
insurance, the types of individuals seeking services would no longer match the type of individuals that were sampled for the study. A similar change would occur as more individuals become aware of
the efficacy of the treatment. The result is an inherent disparity between the target population and the population under study.
The problem of generalizing across disparate populations has received a formal treatment in (Pearl and Bareinboim, 2014) where it was labeled “transportability,” and where necessary and sufficient
conditions for valid generalization were established (see also Bareinboim and Pearl, 2013). The problem of selection bias, though it has some unique features, can also be viewed as a nuance of the
transportability problem, thus inheriting all the theoretical results established in (Pearl and Bareinboim, 2014) that guarantee valid generalizations. We will describe the two problems side by side
and then return to the distinction between the type of assumptions that are needed for enabling generalizations.
The transportability problem concerns two dissimilar populations, Π and Π^∗, and requires us to estimate the average causal effect P^∗(y[x]) (explicitly: P^∗(y[x]) ≡ P^∗(Y = y|do(X = x)) in the
target population Π^∗, based on experimental studies conducted on the source population Π. Formally, we assume that all differences between Π and Π^∗ can be attributed to a set of factors S that
produce disparities between the two, so that P^∗(y[x]) = P(y[x]|S = 1). The information available to us consists of two parts; first, treatment effects estimated from experimental studies in Π and,
second, observational information extracted from both Π and Π^∗. The former can be written P(y|do(x),z), where Z is set of covariates measured in the experimental study, and the latters are written P
^∗(x, y, z) = P (x, y, z|S = 1), and P (x, y, z) respectively. In addition to this information, we are also equipped with a qualitative causal model M, that encodes causal relationships in Π and Π^∗,
with the help of which we need to identify the query P^∗(y[x]). Mathematically, identification amounts to transforming the query expression
P^∗(y[x]) = P(y|do(x),S = 1)
into a form derivable from the available information I[TR], where
I[TR] = { P(y|do(x),z), P(x,y,z|S = 1), P(x,y,z) }.
The selection bias problem is slightly different. Here the aim is to estimate the average causal effect P(y[x]) in the Π population, while the experimental information available to us, I[SB], comes
from a preferentially selected sample, S = 1, and is given by P (y|do(x), z, S = 1). Thus, the selection bias problem calls for transforming the query P(y[x]) to a form derivable from the information
I[SB] = { P(y|do(x),z,S = 1), P(x,y,z|S = 1), P(x,y,z) }.
In the Appendix section, we demonstrate how transportability problems and selection bias problems are solved using the transformations described above.
The analysis reported in (Pearl and Bareinboim, 2014) has resulted in an algorithmic criterion (Bareinboim and Pearl, 2013) for deciding whether transportability is feasible and, when confirmed, the
algorithm produces an estimand for the desired effects. The algorithm is complete, in the sense that, when it fails, a consistent estimate of the target effect does not exist (unless one strengthens
the assumptions encoded in M).
There are several lessons to be learned from this analysis when considering selection bias problems.
1. The graphical criteria that authorize transportability are applicable to selection bias problems as well, provided that the graph structures for the two problems are identical. This means that
whenever a selection bias problem is characterizes by a graph for which transportability is feasible, recovery from selection bias is feasible by the same algorithm. (The Appendix demonstrates this
2. The graphical criteria for transportability are more involved than the ones usually invoked in testing treatment assignment ignorability (e.g., through the back-door test). They may require
several d-separation tests on several sub-graphs. It is utterly unimaginable therefore that such criteria could be managed by unaided human judgment, no matter how ingenious. (See discussions with
Guido Imbens regarding computational barriers to graph-free causal inference, click here). Graph avoiders, should reckon with this predicament.
3. In general, problems associated with external validity cannot be handled by balancing disparities between distributions. The same disparity between P (x, y, z) and P^∗(x, y, z) may demand
different adjustments, depending on the location of S in the causal structure. A simple example of this phenomenon is demonstrated in Fig. 3(b) of (Pearl and Bareinboim, 2014) where a disparity in
the average reading ability of two cities requires two different treatments, depending on what causes the disparity. If the disparity emanates from age differences, adjustment is necessary, because
age is likely to affect the potential outcomes. If, on the other hand the disparity emanates from differences in educational programs, no adjustment is needed, since education, in itself, does not
modify response to treatment. The distinction is made formal and vivid in causal graphs.
4. In many instances, generalizations can be achieved by conditioning on post-treatment variables, an operation that is frowned upon in the potential-outcome framework (Rosenbaum, 2002, pp. 73–74;
Rubin, 2004; Sekhon, 2009) but has become extremely useful in graphical analysis. The difference between the conditioning operators used in these two frameworks is echoed in the difference between Q
[c] and Q[do], the two z-specific effects discussed in a previous posting on this blog (link). The latter defines information that is estimable from experimental studies, whereas the former invokes
retrospective counterfactual that may or may not be estimable empirically.
In the next Section we will discuss the benefit of leveraging the do-operator in problems concerning generalization.
2. Ignorability versus Admissibility in the Pursuit of Generalization
A key assumption in almost all conventional analyses of generalization (from sample-to-population) is S-ignorability, written Y[x] ⊥ S|Z where Y[x] is the potential outcome predicated on the
intervention X = x, S is a selection indicator (with S = 1 standing for selection into the sample) and Z a set of observed covariates. This condition, sometimes written as a difference Y[1] − Y[0] ⊥
S|Z, and sometimes as a conjunction {Y[1], Y[0]} ⊥ S|Z, appears in Hotz et al. (2005); Cole and Stuart (2010); Tipton et al. (2014); Hartman et al. (2015), and possibly other researchers committed to
potential-outcome analysis. This assumption says: If we succeed in finding a set Z of pre-treatment covariates such that cross-population differences disappear in every stratum Z = z, then the
problem can be solved by averaging over those strata. (Lacking a procedure for finding Z, this solution avoids the harder part of the problem and, in this sense, it somewhat borders on the circular.
It amounts to saying: If we can solve the problem in every stratum Z = z then the problem is solved; hardly an informative statement.)
In graphical analysis, on the other hand, the problem of generalization has been studied using another condition, labeled S-admissibility (Pearl and Bareinboim, 2014), which is defined by:
P (y|do(x), z) = P (y|do(x), z, s)
or, using counterfactual notation,
P(y[x]|z[x]) = P (y[x]|z[x], s[x])
It states that in every treatment regime X = x, the observed outcome Y is conditionally independent of the selection mechanism S, given Z, all evaluated at that same treatment regime.
Clearly, S-admissibility coincides with S-ignorability for pre-treatment S and Z; the two notions differ however for treatment-dependent covariates. The Appendix presents scenarios (Fig. 1(a) and
(b)) in which post-treatment covariates Z do not satisfy S-ignorability, but satisfy S-admissibility and, thus, enable generalization to take place. We also present scenarios where both
S-ignorability and S-admissibility hold and, yet, experimental findings are not generalizable by standard procedures of post-stratification. Rather the correct procedure is uncovered naturally from
the graph structure.
One of the reasons that S-admissibility has received greater attention in the graph-based literature is that it has a very simple graphical representation: Z and X should separate Y from S in a
mutilated graph, from which all arrows entering X have been removed. Such a graph depicts conditional independencies among observed variables in the population under experimental conditions, i.e.,
where X is randomized.
In contrast, S-ignorability has not been given a simple graphical interpretation, but it can be verified from either twin networks (Causality, pp. 213-4) or from counterfactually augmented graphs
(Causality, p. 341), as we have demonstrated in an earlier posting on this blog (link). Using either representation, it is easy to see that S-ignorability is rarely satisfied in transportability
problems in which Z is a post-treatment variable. This is because, whenever S is a proxy to an ancestor of Z, Z cannot separate Y[x] from S.
The simplest result of both PO and graph-based approaches is the re-calibration or post-stratification formula. It states that, if Z is a set of pre-treatment covariates satisfying S-ignorability (or
S-admissibility), then the causal effect in the population at large can be recovered from a selection-biased sample by a simple re-calibration process. Specifically, if P(y[x]|S = 1,Z = z) is the
z-specific probability distribution of Y[x] in the sample, then the distribution of Y[x] in the population at large is given by
P(y[x]) = ∑[z] P(y[x]|S = 1,z) P(z) (*)
where P(z) is the probability of Z = z in the target population (where S = 0). Equation (*) follows from S-ignorability by conditioning on z and, adding S = 1 to the conditioning set – a one-line
proof. The proof fails however when Z is treatment dependent, because the counterfactual factor P(y[x]|S = 1,z) is not normally estimable in the experimental study. (See Q[c] vs. Q[do] discussion
As noted in (Keiding, 1987) this re-calibration formula goes back to 18th century demographers (Dale, 1777; Tetens, 1786) facing the task of predicting overall mortality (across populations) from
age-specific data. Their reasoning was probably as follows: If the source and target populations differ in distribution by a set of attributes Z, then to correct for these differences we need to
weight samples by a factor that would restore similarity to the two distributions. Some researchers view Eq. (*) as a version of Horvitz and Thompson (1952) post-stratification method of estimating
the mean of a super-population from un-representative stratified samples. The essential difference between survey sampling calibration and the calibration required in Eq. (*) is that the calibrating
covariates Z are not just any set by which the distributions differ; they must satisfy the S-ignorability (or admissibility) condition, which is a causal, not a statistical condition. It is not
discernible therefore from distributions over observed variables. In other words, the re-calibration formula should depend on disparities between the causal models of the two populations, not merely
on distributional disparities. This is demonstrated explicitly in Fig. 4(c) of (Pearl and Bareinboim, 2014), which is also treated in the Appendix (Fig. 1(a)).
While S-ignorability and S-admissibility are both sufficient for re-calibrating pre-treatment covariates Z, S-admissibility goes further and permits generalizations in cases where Z consists of
post-treatment covariates. A simple example is the bio-marker model shown in Fig. 4(c) (Example 3) of (Pearl and Bareinboim, 2014), which is also discussed in the Appendix.
1. Many opportunities for generalization are opened up through the use of post-treatment variables. These opportunities remain inaccessible to ignorability-based analysis, partly because
S-ignorability does not always hold for such variables but, mainly, because ignorability analysis requires information in the form of z-specific counterfactuals, which is often not estimable from
experimental studies.
2. Most of these opportunities have been chartered through the completeness results for transportability (Bareinboim et al., 2014), others can be revealed by simple derivations in do-calculus as
shown in the Appendix.
3. There is still the issue of assisting researchers in judging whether S-ignorability (or S-admissibility) is plausible in any given application. Graphs excel in this dimension because graphs match
the format in which people store scientific knowledge. Some researchers prefer to do it by direct appeal to intuition; they do so at their own peril.
[…] A recent post on our blog deals with one of the most crucial and puzzling questions of causal inference: “How […]
Pingback by Causal Analysis in Theory and Practice » Spring Greeting from the UCLA Causality Blog — April 29, 2015 @ 12:17 am
In addition, IDC believes a sizable portion of the Android installed base were those who migrated over to the platform from (Apple’s) iOS with the desire for a larger screen smartphone.
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What is the range of the function y = cos x? | HIX Tutor
What is the range of the function #y = cos x#?
Answer 1
The range of a function is all possible output, or $y$, values. The range of $y = \cos x$ is from -1 to 1.
In interval notation, the range is [-1,1] * Note that square brackets [ ] are used because because $y = \cos x$ can actually equal -1 and 1 ( for example, if you plug in $x = \pi$, $y = - 1$).
You can see visually in a graph that $y = \cos x$ can only equal values between -1 and 1 on the $y$-axis, hence that it is why it is the range. The doimain, however, is all real numbers. You can see
that you can plug in all sorts of $x$ values, no matter how infinitely small and infinitely large they are- But you will always get a $y$ value with the restriction of [-1,1]
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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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Local measurement of loss using heated thin film sensors
A calibration equation is derived linking the non-dimensional entropy generation rate per unit area with the non-dimensional aerodynamic wall shear stress and free stream pressure gradient. It is
proposed that the latter quantities, which can be measured from surface gauges, be used to measure the profile entropy generation rate. It is shown that the equation is accurate for a wide range of
well-defined laminar profiles. To measure the dimensional entropy generation rate per unit area requires measurement of the thickness of the boundary layer. A general profile equation is given and
used to show the range of accuracy of a further simplification to the calibration. For flows with low free stream pressure gradients, the entropy generation rate is very simply related to the wall
shear stress, if both are expressed without units. An array of heated thin film sensors is calibrated for the measurement of wall shear stress, thus demonstrating the feasibility of using them to
measure profile entropy generation rate.
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Algebraic theories of cellular automata over concrete categories
» Apply now PDF Show all positions
The project's aim is to develop and implement a framework for cellular automata over concrete categories, expanding on the work by Capobianco and Uustalu (2010) and Behrisch et al. (2017).
Research field: Information and communication technology
Supervisors: Dr. Tarmo Uustalu
Dr. Silvio Capobianco
Availability: This position is available.
Offered by: School of Information Technologies
Department of Software Science
Application deadline: Applications are accepted between September 01, 2021 00:00 and September 30, 2021 23:59 (Europe/Zurich)
Cellular automata (CA) are an example of context-dependent computation, which can be modeled as coKleisli maps of comonads or, more generally, arrows. In a CA, entities occupy the nodes of a grid
which has a group of translations, and their state, taken from an alphabet, is updated synchronously according to a local rule (coKleisli map); this induces a global function (coalgebra) which
updates the state of the entire grid. If the group or the alphabet have special properties, these can reflect into special properties of the rules.
The project will develop and implement a framework for CA over concrete categories whose objects can "reasonably" be seen as sets. More in general, we will study cellular automata over value spaces
with algebraic structure, e.g. an abelian group. Local computations can be required to be linear or additive and this ensures that so is also the global computation (the coKleisli extension of the
coKleisli map). This framework will expand on the ideas from Capobianco and Uustalu (2010) and Behrisch et al. (2017).
A first nontrivial example of CA on a concrete category is given by linear CA, where the alphabet is a ring and the local rule (equivalently, the global function) is linear: these are precisely the
CA on the concrete category of modules over the given ring. It is known (Sato (1993)) that reversibility and surjectivity of linear CA over finite commutative rings are decidable in arbitrary
dimension, and correspond to algebraic properties of the Laurent polynomial describing the local rule. A first step in the project can then be a translation of this fact in the language of category
The PhD student will be affiliated with the ALICE (Automata in Learning, Interaction and Concurrency) group, Department of Software Science, TalTech. The ALICE project aims at extending and
exploiting classical automata theory for the requirements of modern software.
Applicants should fulfil the following requirements:
• MSc in Mathematics, Computer Science, or related field.
• Previous experience with category theory and/or cellular automata theory.
[1] Mike Behrisch, Sebastian Kerkhoff, Reinhard Pöschel, Friedrich Martin Schneider, Stefan Siegmund. Dynamical Systems in Categories. Applied Categorical Structures 25 (2017), 29--57. https://
doi.org/10.1007/s10485-015-9409-8 Preprint: https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa-129909
[2] Silvio Capobianco and Tarmo Uustalu. A categorical outlook on cellular automata. In J. Kari, ed., Proc. of 2nd Symp. on Cellular Automata, JAC 2010 (Turku, Dec. 2010), v. 13 of TUCS Lecture
Notes, pp. 88--99. Turku Centre for Computer Science, 2010. https://hal.archives-ouvertes.fr/hal-00542015
[3] Tullio Ceccherini-Silberstein and Michel Coornaert. Cellular Automata and Groups. Springer, 2010.
[4] Tullio Ceccherini-Silberstein and Michel Coornaert. Surjunctivity and reversibility of cellular automata over concrete categories. In: Picardello M. (eds) Trends in Harmonic Analysis. Springer
INdAM Series, vol 3. Springer, Milano. doi:10.1007/978-88-470-2853-1_6 Preprint: arXiv:1203.6492
[5] John Hughes. Generalising monads with arrows. Science of Computer Programming. 37 (1–3): 67–111. doi:10.1016/S0167-6423(99)00023-4
[6] Jarkko Kari. Theory of cellular automata: a survey. Theoretical Computer Science 334 (2005) 3--33.
[7] Tadakazu Sato. Decidability of some problems of linear cellular automata over finite commutative rings. Information Processing Letters, 46:151–155, 1993. | {"url":"https://taltech.glowbase.com/positions/426","timestamp":"2024-11-06T01:49:01Z","content_type":"text/html","content_length":"12268","record_id":"<urn:uuid:1cf81cba-743b-4bb7-b9fe-4fb5fc365670>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00758.warc.gz"} |
Clinical prediction rules - Science without sense...double nonsense
The crystal ball
Critical appraisal of clinical prediction rules.
The methodology for the development of clinical prediction rules is described and recommendations are given for the critical appraisal of these documents.
How I wish I could predict the future! And not only to win millions in the lottery, which is the first thing you can think of. There are more important things in life than money (or so that’s what
some say), decisions that we make based on assumptions that end up not being fulfilled and that complicate our lives to unsuspected limits. We all have ever thought about “if you lived twice …” I
have no doubt, if I met the genie of the lamp one of the three wishes I would ask would be a crystal ball to see the future.
And we could also do well in our work as doctors. In our day to day we are forced to make decisions about the diagnosis or prognosis of our patients and we always do it on the swampy terrain of
uncertainty, always assuming the risk of making some mistake. We, especially when we are more experienced, estimate consciously or unconsciously the likelihood of our assumptions, which helps us in
making diagnostic or therapeutic decisions.
However, it would be good to also have a crystal ball to know more accurately the evolution of the patient’s course.
The problem, as with other inventions that would be very useful in medicine (like the time machine), is that nobody has yet managed to manufacture a crystal ball that really works. But do not let us
down. We cannot know for sure what will happen, but we can estimate the probability that a certain result will occur.
For this, we can use all those variables related to the patient that have a known diagnostic or prognostic value and integrate them to perform the calculation of probabilities. Well, doing such a
thing would be the same as designing and applying what is known as a clinical prediction rule (CPR).
Thus, if we get a little formal, we can define a CPR as a tool composed of a set of variables of clinical history, physical examination and basic complementary tests, which provides us with an
estimate of the probability of an event, suggesting a diagnosis or predicting a concrete response to a treatment.
The critical appraisal of an article about a CPR shares similar aspects with those of the ones about diagnostic tests and also has specific aspects related to the methodology of its design and
application. For this reason, we will briefly look at the methodological aspects of CPRs before entering into their critical assessment.
Clinical prediction rules
In the process of developing a CPR, the first thing to do is to define it. The four key elements are the study population, the variables that we will consider as potentially predictive, the gold or
reference standard that classifies whether the event we want to predict occurs or not and the criterion of assessment of the result.
It must be borne in mind that the variables we choose must be clinically relevant, they must be collected accurately and, of course, they must be available at the time we want to apply the CPR for
decision making. It is advisable not to fall into the temptation of putting variables everywhere and endlessly since, apart from complicating the application of the CPR, it can decrease its validity.
In general, it is recommended that for every variable that is introduced in the model there should have been at least 10 events that we want to predict (the design is made in a certain sample whose
components have the variables but only a certain number have ended up presenting the event to predict).
I would also like to highlight the importance of the gold standard. There must be a diagnostic test or a set of well-defined criteria that allow us to clearly define the event we want to predict with
the CPR.
Finally, it is convenient that those who collect the variables during this definition phase are unaware of the results of the gold standard, and vice versa. The absence of blinding decreases the
validity of the CPR.
The next step is the derivation or design phase itself. This is where the statistical methods that allow to include predictive variables and exclude those that are not going to contribute anything
are applied. We will not go into statistics, just say that the most commonly used methods are those based on logistic regression, although discriminant, survival and even more exotic analysis based
on discriminant risks or neural networks can be used, only afforded by a few virtuous ones.
In the logistic regression models, the event will be the dichotomous dependent variable (it happens or it does not happen) and the other variables will be the predictive or independent variables.
Thus, each coefficient that multiplies each predictive variable will be the natural antilogarithm of the adjusted odds ratio. In case anyone has not understood, the adjusted odds ratio for each
predictive variable will be calculated raising the number “e” to the value of the coefficient of that variable in the regression model.
The usual thing is that a certain score is assigned on a scale according to the weight of each variable, so that the total sum of points of all the predictive variables will allow to classify the
patient in a specific range of prediction of event production. There are also other more complex methods using regression equations, but after all you always get the same thing: an individualized
estimate of the probability of the event in a particular patient.
With this process we perform the categorization of patients in homogenous groups of probability, but we still need to know if this categorization is adjusted to reality or, what is the same, what is
the capacity of discrimination of the CPR.
The overall validity or discrimination capacity of the PRC will be assess by contrasting its results with those of the gold standard, using similar techniques to those used to assess the power of
diagnostic tests: sensitivity, specificity, predictive values and likelihood ratios. In addition, in cases where the CPR provides a quantitative estimate, we can resort to the use of the ROC curves,
since the area under the curve will represent the global validity of the CPR.
The last step of the design phase will be the calibration of the CPR, which is nothing more than checking its good behavior throughout the range of possible results.
Some CPR’s authors end this here, but they forget two fundamental steps of the elaboration: the validation and the calculation of the clinical impact of the rule.
The validation consists in testing the CPR in samples different to the one used for its design. We can take a surprise and verify that a rule that works well in a certain sample does not work in
another. Therefore, it must be tested, not only in similar patients (limited validation), but also in different clinical settings (broad validation), which will increase the external validity of the
The last phase is to check its clinical performance. This is where many CPRs crash down after having gone through all the previous steps (maybe that’s why this last check is often avoided). To assess
the clinical impact, we will have to apply CPR in our patients and see how clinical outcome measures change such as survival, complications, costs, etc. The ideal way to analyze the clinical impact
of a CPR is to conduct a clinical trial with two groups of patients managed with and without the rule.
Critical appraisal of clinical prediction rules
For those self-sacrificing people who are still reading, now that we know what a CPR is and how it is designed, we will see how the critical appraisal of these works is done. And for this, as usual,
we will use our three pillars: validity, relevance and applicability. To not forget anything, we will follow the questions that are listed on the grid for CRP studies of the CASP tool.
Regarding VALIDITY, we will start first with some elimination questions. If the answer is negative, it may be time to wait until someone finally makes up a crystal ball that works.
Does the rule answer a well-defined question? The population, the event to be predicted, the predictive variables and the outcome evaluation criteria must be clearly defined. If this is not done or
these components do not fit our clinical scenario, the rule will not help us. The predictive variables must be clinically relevant, reliable and well defined in advance.
Did the study population from which the rule was derived include an adequate spectrum of patients? It must be verified that the method of patient selection is adequate and that the sample is
representative. In addition, it must include patients from the entire spectrum of the disease. As with diagnostic tests, events may be easier to predict in certain groups, so there must be
representatives of all of them.
Finally, we must see if the sample was validated in a different group of patients. As we have already said, it is not enough that the rule works in the group of patients in which it has been derived,
but that it must be tested in other groups that are similar or different from those with which it was generated.
If the answer to these three questions has been affirmative, we can move on to the three next questions. Was there a blind evaluation of the outcome and of the predictor variables? We have already
commented, it is important that the person who collects the predictive variables does not know the result of the reference pattern, and vice versa. The collection of information must be prospective
and independent.
The next thing to ask is whether the predictor variables and the outcome in all the patients were measured. If the outcome or the variables are not measured in all patients, the validity of the CPR
can be compromised. In any case, the authors should explain the exclusions, if there are any. Finally, are the methods of derivation and validation of the rule described? We already know that it is
essential that the results of the rule be validated in a population different from the one used for the design.
If the answers to the previous questions indicate that the study is valid, we will answer the questions about the RELEVANCE of the results. The first is if you can calculate the performance of the
CRP. The results should be presented with their sensitivity, specificity, odds ratios, ROC curves, etc., depending on the result provided by the rule (scoring scales, regression formulas, etc.).
All these indicators will help us to calculate the probabilities of occurrence of the event in environments with different prevalence. This is similar to what we did with the studies of diagnostic
tests, so I invite you to review the post on the subject to not repeat too much. The second question is: what is the precision of the results? Here we will not extend either: remember our revered
confidence intervals, which will inform us of the accuracy of the results of the rule.
To finish, we will consider the APPLICABILITY of the results to our environment, for which we will try to answer three questions. Will the reproducibility of the PRC and its interpretation be
satisfactory within the scope of the scenario? We will have to think about the similarities and differences between the field in which the CPR develops and our clinical environment. In this sense, it
will be helpful if the rule has been validated in several samples of patients from different environments, which will increase its external validity.
Is the test acceptable in this case? We will think wether the rule is easy to apply in our environment and wether it makes sense to do it from the clinical point of view in our environment. Finally,
will the results modify clinical behavior, health outcomes or costs? If, from our point of view, the results of the CPR are not going to change anything, the rule will be useless and a waste of time.
Here our opinion will be important, but we must also look for studies that assess the impact of the rule on costs or on health outcomes.
And up to here everything I wanted to tell you about critical appraising of studies on CPRs. Anyway, before finishing I would like to tell you a little about a checklist that, of course, also exists
for the valuation of this type of studies: the checklist CHARMS (CHecklist for critical Appraisal and data extraction for systematic Reviews of prediction Modeling Studies). You will not tell me that
the name, although a bit fancy, is not lovely.
This list is designed to assess the primary studies of a systematic review on CPRs. It try to answer some general design questions and assess 11 domains to extract enough information to perform the
critical appraisal. The two great parts that are valued are the risk of bias in the studies and its applicability. The risk of bias refers to the design or validation flaws that may result in the
model being less discriminative, excessively optimistic, etc.
The applicability, on the other hand, refers to the degree to which the primary studies are in agreement with the question that motivates the systematic review, for which it informs us of whether the
rule can be applied to the target population. This list is good and helps to assess and understand the methodological aspects of this type of studies but, in my humble opinion, it is easier to make a
systematic critical appraisal by using the CASP’s tool.
We’re leaving…
And here, finally, we leave it for today. We have not spoken anything, so as not to stretch ourselves too long, of what to do with the result of the rule. The fundamental thing, we already know, is
that we can calculate the probability of occurrence of the event in individual patients from environments with different prevalence. But that is another story…
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Internal combustion Cycle
Sketch of an actual Otto cycle
Figure 3.10:
Piston and valves in a four-stroke internal combustion engine
The actual cycle does not have the sharp transitions between the different processes that the ideal cycle has, and might be as sketched in Figure .
3.5.1 Efficiency of an ideal Otto cycle
The starting point is the general expression for the thermal efficiency of a cycle: The convention, as previously, is that heat exchange is positive if heat is flowing into the system or engine, so
is negative. The heat absorbed occurs during combustion when the spark occurs, roughly at constant volume. The heat absorbed can be related to the temperature change from state 2 to state 3 as:
The heat rejected is given by (for a perfect gas with constant specific heats) Substituting the expressions for the heat absorbed and rejected in the expression for thermal efficiency yields We can
simplify the above expression using the fact that the processes from 1 to 2 and from 3 to 4 are isentropic: The quantity is called the compression ratio. In terms of compression ratio, the efficiency
of an ideal Otto cycle is:
Figure 3.11:
Ideal Otto cycle thermal efficiency
The ideal Otto cycle efficiency is shown as a function of the compression ratio in Figure . As the compression ratio, , increases, increases, but so does . If is too high, the mixture will ignite
without a spark (at the wrong location in the cycle).
3.5.2 Engine work, rate of work per unit enthalpy flux
The non-dimensional ratio of work done (the power) to the enthalpy flux through the engine is given by There is often a desire to increase this quantity, because it means a smaller engine for the
same power. The heat input is given by where
• is the heat of reaction, i.e. the chemical energy liberated per unit mass of fuel,
• is the fuel mass flow rate.
The non-dimensional power is The quantities in this equation, evaluated at stoichiometric conditions are:
Source: web.mit.edu
ME4293 Internal Combustion Engine 3 Spring2015
ME4293 Internal Combustion Engine 1 Spring2015
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Software for physics calculations
7242 views
What is some good free software for doing physics calculations?
I'm mainly interested in symbolic computation (something like Mathematica, but free).
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user grizzly adam
Related question: Which software(s) handle units and unit conversion best?
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user sigoldberg1
Community wiki? (Also: sigoldberg1, can we have a link?)
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user David Z
@sigoldberg: Google Calculator can perform unit conversion.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user KennyTM
Converted to community wiki, as this is the most appropriate question form here.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user Noldorin
Some software I have used or has been recommended to me for physics-related work:
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user Flaviu Cipcigan
Maxima has a units package, as do the commercial software systems.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user sigoldberg1
Maybe throw Gnuplot into the mix? I love it for quick and easy plots of functions and data series, and when I then need something more polished I can very easily reuse the gnuplot code from other
examples. Better than pointing and clicking to get results.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user Lagerbaer
SymPy should also probably be added to this list.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user Simon
It is probably worth your while to buy Mathematica, Maple, or Matlab, depending on your needs. I wish it weren't so, but this is one area in which the commercial tools are still vastly better than
their free counterparts.
If you are a student, you can buy these at fairly afforable prices. Maple 14 Student Edition is only $99. Mathematica for Students is \$140, and Matlab/Simulink is \$99 for students. It is also
possible that your school or department already has a site license, allowing you to obtain and use this software for no additional cost.
For symbolic calculations, you want either Mathematica or Maple, with Maple being more user-friendly, and Mathematica being more prevalent (in my experience) in actual research environments. Matlab's
focus is on numerical calculations.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user nibot
Not to mention that your institution can probably provide you shared licences (licence server) of Mathematica or Matlab.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user Cedric H.
Sage is a Python based system (including Numpy and Scipy) which includes a symbolic computation module.
From the Sage homepage:
Sage is a free open-source mathematics software system licensed under the GPL. It combines the power of many existing open-source packages into a common Python-based interface. Mission: Creating
a viable free open source alternative to Magma, Maple, Mathematica and Matlab.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user ihuston
Do you have any experience with it? If so, how does it compare in usefulness to Mathematica?
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user nibot
I tend to use Python with Numpy barebones as it were, without the Sage environment around it. I prefer the combination of interactive and scripting methodologies which I can use with Python rather
than the notebook methodology of Mathematica. Sage (at least through the web interface) is more like Mathematica and does cover many of the things you might do in Mathematica. I do sometimes crank up
Mathematica to plot a quick graph (Manipulate is a great exploratory tool) but tend to get aggravated by things I would know how to do easily in Python.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user ihuston
If you are into Python, the combination of NumPy, SciPy and matplotlib will cover any need for numerical or scientific computing or graphing you may have. There also is a SymPy for symbolic
calculations, but I have never used it. A friend of mine has his own open-source Python library for unit management: juanreyero.com/open/magnitude
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user Jaime
I've recently discovered Cadabra.
A field-theory motivated approach to computer algebra
I'm really impressed.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user Kostya
didn't knew it, +1
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user lurscher
Cadabra uses many of the same index algorithms as the Mathematica package xAct. Although xAct is more focused on General Relativity calculations, while Cadabra is more field theory oriented.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user Simon
I'd like to add that GNU Octave is a very good free alternative to Matlab.
Contrary to Scilab which does not aim at being compatible with Matlab, you can practically run your Matlab scripts with Octave with very few modifications (at least with their latest version).
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user Aabaz
GiNaC is a c++ symbolic manipulation framework oriented to high-energy physics computations. It has a couple of interactive frontends, although its main usage is as part of the Root framework at
A derivative of GiNaC is Pynac, which forms the backend for symbolic expressions in Sage.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user lurscher
Can you provide links/descriptions of its frontends?
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user Simon
i think the most used is "gTybalt" wwwthep.physik.uni-mainz.de/~stefanw//gtybalt.html although i can't asess its usability because i've used ginac mostly as a library in my own code
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user lurscher
Thanks - I'll have a look at it. I spent some time playing with GiNaC about 1 year ago, but never really used it, since I couldn't justify learning it for my one off calculation. Instead, I took the
computationally slow but familiar route of using Mathematica.
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user Simon
See also this list on wikipedia http://en.wikipedia.org/wiki/List_of_computer_algebra_systems. I think all of these systems have their uses for some calculations that can also come up in physics.
Many of them are free, just choose the one that is appropriate for your purposes.
I used this software in my energy transfer courses, I never used it for symbolic, so I don't know if you can do symbolic computation, however it is very good at solving equations. As well as for
conversions. It is not free, however you can download a student version, which I used the whole semester without problems. It is called ees. The company I think is called f chart. I know it is not
exactly what you asked for, however it's a useful software to have around, especially when working with a lot of equations, since the software actually warns you about any inconsistency in the units.
It is also useful if you want to calculate say for example entropy, the software can do it for you if you have the pressure or temperature.
http://www.mhhe.com/engcs/mech/ees/download.html http://www.fchart.com/ees/
This post imported from StackExchange Physics at 2014-04-03 11:46 (UCT), posted by SE-user Renegg | {"url":"https://physicsoverflow.org/12447/software-for-physics-calculations?show=12454","timestamp":"2024-11-08T04:51:45Z","content_type":"text/html","content_length":"311317","record_id":"<urn:uuid:22163774-64f1-49f9-8144-2e03d53826f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00764.warc.gz"} |
Symbolic Supercomputing
Alvin Despain
Alvin M. Despain is the Powell Professor of Computer Engineering at the University of Southern California, Los Angeles. He is a pioneer in the study of high-performance computer systems for symbolic
calculations. To determine design principles for these systems, his research group builds experimental software and hardware systems, including compilers, custom very-large-scale integration
processors, and multiprocessor systems. His research interests include computer architecture, multiprocessor and multicomputer systems, logic programming, and design automation. Dr. Despain received
his B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Utah, Salt Lake City.
This presentation discusses a topic that may be remote from the fields most of you at this conference deal in—symbolic, as opposed to numeric, supercomputing. I will define terms and discuss
parallelism in symbolic computing and architecture and then draw some conclusions.
If supercomputing is using the highest-performance computers available, then symbolic supercomputing is using the highest-performance symbolic processor systems. Let me show you some symbolic
problems and how they differ from numeric ones.
If you're doing the usual supercomputer calculations, you use LINPAC, fast Fourier transforms (FFTs), etc., and you do typical, linear-algebra kinds of operations. In symbolic computing, you use
programs like MACSYMA, MAPLE, Mathematica, or PRESS. You provide symbols,
― 96 ―
and you get back not numbers but formulae. For example, you get the solution to a polynomial in terms of a formula.
Suppose we have a problem specification—maybe it is to model global climate. This is a big programming problem. After years of effort programming this fluid-dynamics problem, you get a Fortran
program. This is then compiled. It is executed with some data, and some results are obtained (e.g., the temperature predicted for the next hundred years). Then the program is generally tuned to
achieve both improved results and improved performance.
In the future you might think that you start with the same problem specification and try to reduce the programming effort by automating some of the more mundane tasks. One of the most important
things you know is that the programmer had a very good feel for the data and then wired that into the program. If you're going to automate, you're going to have to bring that data into the process.
Parameters that can be propagated within a program constitute the simplest example of adjusting the program to data, but there are lots of other ways, as well. Trace scheduling is one way that this
has been done for some supercomputers. You bring the data in and use it to help do a good job of compiling, vectorizing, and so on. This is called partial evaluation because you have part of the
data, and you evaluate the program using the data. And this is a symbolic calculation.
If you're going to solve part of the programming problem that we have with supercomputers, you might look toward formal symbolic calculation. Some other cases are optimizing compilers, formal
methods, program analysis, abstract interpretation, intelligent databases, design automation, and very-high-level language compilers.
If you look and see how mathematicians solve problems, they don't do it the way we program Cray Research, Inc., machines, do they? They don't do it by massive additions and subtractions. They
integrate together both symbolic manipulations and numeric manipulations. Somehow we have to learn how to do that better, too. It is an important challenge for the future. Some of it is happening
today, but there's a lot to be done.
I would like to try to characterize some tough problems in the following ways: there is a set of problems that are numeric—partial differential equations, signal processing, FFTs, etc.; there are
also optimization problems in which you search for a solution—linear programming, for example, or numerical optimization of various kinds. At the symbolic level you also have simulation. Abstract
interpretation is an example. But you also have theorem proving, design automation, expert
― 97 ―
systems, and artificial intelligence (AI). Now, these are fundamentally hard problems. In filter calculations (the easy problems), the same execution occurs no matter what data you have. For example,
FFT programs will always execute the same way, no matter what data you're using.
With the hard problems, you have to search. Your calculation is a lot more dynamic and a lot more difficult because the calculation does depend upon the data that you happen to have at the time. It
is in this area that my work and the work of my group have focused: how you put together symbols and search. And that's what Newell and Simon (1976) called AI, actually. But call it what you like.
I want to talk about two more things: concurrency and parallelism. These are the themes of this particular session. I'd like to talk about the instruction-set architecture, too, because it interacts
so strongly with concurrency and parallelism. If you're building a computer, one instruction type is enough, right? You build a Turing machine with one instruction. So that's sufficient, but you
don't get performance.
If you want performance, you'd better add more instructions. If you have a numeric processor, you include floating-point add, floating-point multiply, and floating-point divide. If you have a
general-purpose processor, you need operations like load, store, jump, add, and subtract. If you want a symbolic processor, you've got to do things like binding two symbols together (binding),
dereferencing, unifying, and backtracking. To construct a symbolic processor, you need the general-purpose features and the symbolic operations.
Our latest effort is a single-chip processor called the Berkeley Abstract Machine (BAM). This work has been sponsored by the Defense Advanced Research Projects Agency. For our symbolic language, we
have primarily used Prolog, but BAM is not necessarily dependent on it.
Now, I'd like to tell you a little bit about it to illustrate the instructionset architecture issues involved, especially the features of the BAM chip that boost performance. These are the usual
general-purpose instructions—load, store, and so on. There's special support for unification. Unification is the most general pattern match you can do. Backtracking is also supported so that you can
do searching and then backtrack if you find it wrong. The architecture features include tags, stack management, special registers, and a microstore—that is, internal opcodes. There is pipeline
execution to get performance and multiple I/O ports for address, data, and instructions.
In this processor design we considered what it costs to do symbolic calculations in addition to the general-purpose calculations. We selected
― 98 ―
a good set of all possible general-purpose instructions to match with the symbolic, and then we added what was needed to get performance.
If you add a feature, you get a corresponding improvement in performance. See Figure 1, which graphs the percentage increase in performance. The cycle count varies between one and two, one being BAM
as a benchmark. We took all combinations of features that we could find and with simulation tried to understand what cost-performance tradeoffs can be achieved.
Some cost-performance combinations aren't very good. Others are quite good, and the full combination is quite good. The net result is that an 11 per cent increase in the silicon area of a single-chip
microcomputer, BAM, results in a 70 per cent increase in the performance on symbolic calculations. So that's what we chose for BAM. It doesn't cost very much to do the symbolic once you have the
general-purpose features.
The BAM chip features 24 internal microinstructions and 62 external ones. It achieves about 1.4 cycles per instruction. However, because of dereferencing, the number of cycles per instruction is
indefinitely large. Simulations indicated that the chip would achieve about 24 million instructions per second, or about three million logical inferences per second (i.e., about 3 MLIPS). A logical
inference is what you execute for symbolic computing. It's a general pattern match, and if it succeeds, you do a procedure call, execution, and return.
― 99 ―
We submitted this chip for fabrication. The chip has now been tested, and it achieved 3.8 MLIPS.
Consider the performance this represents. Compare, for instance, the Japanese Personal Sequential Inference (PSI) machine, built in 1984. It achieved 30,000 LIPS. A few months later at Berkeley, we
built something called the Prolog Machine (PLM), which achieved 300,000 LIPS, even then, a 10-fold improvement. The best the Europeans have done so far is the Knowledge Crunch Machine. It now
achieves about 600,000 LIPS.
The best the Japanese have done currently in a single processor is an emitter-coupled-logic machine, 64-bit-wide data path, and it achieved a little over an MLIPS, compared with BAM's 3.8 MLIPS. So
the net result of all of this is that we've been able to demonstrate in six years' time a 100-fold improvement in performance in this domain.
The PLM was put into a single chip, just as BAM is a single chip. The PSI was not; it was a multiple-chip system. I think what's important is that you really have to go after the technology. You must
also optimize the microarchitecture, the instruction-set architecture, and especially the compiler. Architecture design makes a big difference in performance; it's not a dead issue. And architecture,
technology, and compilers all have to be developed together to get these performance levels.
Let me say something about scaling and the multiprocessor. What about parallelism? Supercomputers have many levels of parallelism—parallel digital circuits at the bottom level, microexecution,
multiple execution per instruction, multiple instruction streams, multiprocessing, shared-memory multiprocessors, and then heterogeneous multiprocessors at the top. And we've investigated how
symbolic calculations play across this whole spectrum of parallelism. If you really want performance, you have to play the game at all these different levels of the hierarchy. It turns out that
parallelism is more difficult to achieve in symbolic calculations. This is due to the dynamic, unpredictable nature of the calculation. But on the plus side, you get, for instance, something called
superlinear speedup during search.
But as in numerics, the symbolic algorithms that are easy to parallelize turn out to be poor in performance. We all know that phenomenon, and it happens here, too. But there are some special cases
that sometimes work out extremely well. What we're trying to do with BAM is identify different types of parallel execution so that you can do something special about each type. BAM handles very well
the kind of parallelism requiring you to break a problem into pieces and solve all the pieces simultaneously. With BAM, parallelism can spread across networks, so
― 100 ―
you have all-solution, or-parallelism, where you find a whole set of answers to a problem rather than just one.
However, if you're doing a design, all you want is one good design. You don't want every possible design. There are too many to enumerate. And that's been our interest, and it works pretty well on
multiprocessors. Unification parallelism, pattern matching, can be done in parallel, and we do some of that within the processor.
Now, let's say you have a BAM chip and a shared-memory cache with the switch and connections to some external bus memory and I/O. Call that a node. Put that together with busses into what we call
Gordon Bell (1985), a Session 8 presenter, wrote a great paper, called "Multis: A New Class of Multiprocessor Computers," about a shared-memory, single-bus system. It turns out you can do the same
trick in multiple dimensions and have yourself a very-large-scale, shared-memory, shared-address-space multiprocessor, and it looks like that's going to work. We'll find out as we do our work.
I think that for a modest cost, you can add powerful symbolic capability to a general-purpose machine. That's one of the things we've learned very recently.
Parallel symbolic execution is still a tough problem, and there is still much to be learned. The ultimate goal is to learn how to couple efficiently, in parallel, both symbolic and numeric
C. G. Bell, "Multis: A New Class of Multiprocessor Computers," Science288, 462-467 (1985).
A. Newell and H. Simon, "Computer Science as Empirical Inquiry; Symbols and Search," Communications of the ACM19 (3), 113-126 (1976).
― 101 ― | {"url":"https://publishing.cdlib.org/ucpressebooks/view?docId=ft0f59n73z&doc.view=content&chunk.id=d0e3419&toc.depth=1&anchor.id=0&brand=ucpress","timestamp":"2024-11-03T09:05:37Z","content_type":"application/xhtml+xml","content_length":"19731","record_id":"<urn:uuid:b95432f6-2929-4cdb-87f6-b71808ac02a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00402.warc.gz"} |
Calculating a 3d Cross Product Using Vectors Defined by a Rectangle
Question Video: Calculating a 3d Cross Product Using Vectors Defined by a Rectangle Mathematics • Third Year of Secondary School
In the rectangle π ΄π ΅π Άπ · shown in the figure, calculate π π Γ π π if {π ’, π £, π €} form a right-hand system of unit vectors.
Video Transcript
In the rectangle π ΄π ΅π Άπ · shown in the figure, calculate π π cross π π if π ’, π £, and π € form a right-hand system of unit vectors.
Looking at our rectangle, we see itβ s eight centimeters tall and 16 centimeters wide and it exists in a space defined by a right-hand system of unit vectors π ’, π £, and π €. Relative to the
rectangle, we see the π ’ hat unit vector points to the right and the π £ hat unit vector points up. This tells us that the π € hat unit vector points out of the screen at us. Anyway, we want to
calculate this cross product, π π cross π π . Letβ s first define these vectors. Vector π π is a vector from point π · to point π ΄ in our rectangle. And likewise, vector π π is
a vector from point π ΅ to point π , the middle of our rectangle.
Now because this rectangle lies in what we can call the π ’π £-plane and because we can say that one centimeter on this plane is equal to one unit of distance, we can then write these vectors π π
and π π according to their π ’ hat and π £ hat components. Considering first vector π π , we see that itβ s entirely in the vertical direction. That means it will have no π ’ hat
component. It will, however, have a negative π £ hat component. And the reason is that this vector points opposite the positive π £ hat direction.
That distance we know, the length of this vector, equals the length of the height of our rectangle, eight centimeters or just eight. Keeping both π ’ hat and π £ hat components then, π π
equals zero π ’ hat minus eight π £ hat. Looking next at vector π π , we see that this vector will have both an π ’ hat and a π £ hat component because it points diagonally. Itβ s the
components of this vector that weβ re after, in other words, this vertical distance here and this horizontal distance here.
Considering its horizontal component, we know that vector π points to the left. Thatβ s in the negative π ’ hat direction. And since it goes to the midpoint of our rectangle β which is 16
centimeters wide β it will have an π ’ hat component of negative eight. Considering then the π £ hat component β this part of our vector β we see that it points in the positive π £ hat
direction and that itβ s equal in length to one-half the height of our rectangle. The π £-component of π π then is positive four.
So weβ ve now got our two vectors defined in terms of the unit vectors of our space. We can now move towards calculating their cross product. In general, if we have two vectors β weβ ll call them
π and π β that lie in the π ’ hat and π £ hat plane, then the cross product π cross π equals the π ’-component of π times the π £-component of π minus the π £-component of π
times the π ’-component of π all in the π € hat unit vector direction.
Notice then that this cross product is perpendicular to both vectors π and π . And this is always the case for a cross product. When we go to cross π π and π π then, our formula tells
us that this equals the π ’-component of our first vector β thatβ s vector π π and that π ’-component is zero β multiplied by the π £-component of our second vector β that second
vector is π π and the π £-component is four. Then from this we subtract the π £-component of our first vector β that vector is π π and its π £-component is negative eight β
multiplied by the π ’-component of our second vector β that second vector is π π and its π ’-component is negative eight. And this resulting vector, we know, is in the π € hat unit vector
Evaluating this result, we know that zero times four is zero and negative eight times negative eight is positive 64. Our overall outcome then is negative 64π € hat. π π cross π π then
results in a vector that points 64 units or 64 centimeters into the screen. | {"url":"https://www.nagwa.com/en/videos/723158960452/","timestamp":"2024-11-03T04:05:31Z","content_type":"text/html","content_length":"254256","record_id":"<urn:uuid:253d99bb-50c1-47e9-88ca-2d21ebbe7530>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00203.warc.gz"} |
Calculus 1
Code Completion Credits Range Language
BE5B01MA1 Z,ZK 7 4P+2S English
In order to register for the course BE5B01DEN, the student must have successfully completed the course BE5B01MA1.
Course guarantor:
It is an introductory course to calculus of functions of one variable. It starts with limit and continuity of functions, derivative and its geometrical meaning and properties, graphing of
functions. Then it covers indefinite integral, basic integration methods and integrating rational functions, definite integral and its applications. It concludes with introduction to Taylor
Syllabus of lectures:
1. The real line, elementary functions and their graphs, shifting and scaling.
2. Limits and continuity, tangent, velocity, rate of change.
3. Derivative of functions, properties and applications.
4. Mean value theorem, L'Hospital's rule.
5. Higher derivatives, Taylor polynomial.
6. Local and global extrema, graphing of functions.
7. Indefinite integral, basic integration methods.
8. Integration of rational functions, more techniques of integration.
9. Definite integral, definition and properties, Fundamental Theorems of Calculus.
10. Improper integrals, tests for convergence. Mean value Theorem for integrals, applications.
11. Sequences of real numbers, numerical series, tests for convergence.
12. Power series, uniform convergence, the Weierstrass test.
13. Taylor and Maclaurin series.
Syllabus of tutorials:
1. The real line, elementary functions and their graphs, shifting and scaling.
2. Limits and continuity, tangent, velocity, rate of change.
3. Derivative of functions, properties and applications.
4. Mean value theorem, L'Hospital's rule.
5. Higher derivatives, Taylor polynomial.
6. Local and global extrema, graphing of functions.
7. Indefinite integral, basic integration methods.
8. Integration of rational functions, more techniques of integration.
9. Definite integral, definition and properties, Fundamental Theorems of Calculus.
10. Improper integrals, tests for convergence. Mean value Theorem for integrals, applications.
11. Sequences of real numbers, numerical series, tests for convergence.
12. Power series, uniform convergence, the Weierstrass test.
13. Taylor and Maclaurin series.
Study Objective:
Study materials:
1. M. Demlová, J. Hamhalter: Calculus I. ČVUT Praha, 1994
2. P. Pták: Calculus II. ČVUT Praha, 1997.
Further information:
Time-table for winter semester 2024/2025:
Vivi P.
Tue 09:15–10:45
(lecture parallel1)
Vivi P.
(lecture parallel1)
Dejvice roomT2:C2-82
Vivi P.
Wed (lecture parallel1
parallel nr.101)
Dejvice roomT2:C4-459
(lecture parallel1
parallel nr.102)
Time-table for summer semester 2024/2025:
Time-table is not available yet
The course is a part of the following study plans: | {"url":"https://bilakniha.cvut.cz/en/predmet4355206.html","timestamp":"2024-11-07T20:08:26Z","content_type":"text/html","content_length":"25518","record_id":"<urn:uuid:caa0ba10-5bdb-4fc8-9bc5-f97f598f1f62>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00000.warc.gz"} |
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denominators. It was actually also kid’s play to operate this. Once you key in the problem, the program carries you all the way to the answer elucidating every step on its way.
That’s what makes it splendid. By the time you arrive at the answer , you already know how to work out the problems. I took great pleasure in learning to work out the problems
with Algebra 2, Remedial Algebra and Basic Math in algebra . I am also positive that you too will love this program just as I did. Wouldn’t you want to test this out?
chemel Posted: Wednesday 03rd of Jan 08:40
You both have got to be joshing ! How can this solution not be general information or heralded here ? How and where might I acquire additional info for testing Algebrator?
Forgive anyone for appearing to be a bit skeptical , but do either of you know if someone can receive a trial copy to use this ?
SanG Posted: Thursday 04th of Jan 11:11
Hey Friends, I had a chance to try Algebrator offered at https://graph-inequality.com/teaching-inequalitiesa-hypothetical-classroom-case.html this morning . I am really very
thankful to you all for pointing me to Algebrator. The quick formula list and the detailed explanations on the fundamentals given there were really understandable. I have
completed and submitted my homework on exponential equations and this was all possible only with the help of the Algebrator that I purchased on the basis of your recommendations
here. Thanks a lot.
From: Beautiful
Northwest Lower
MichMoxon Posted: Friday 05th of Jan 11:08
I recommend using Algebrator. It not only assists you with your math problems, but also displays all the required steps in detail so that you can enhance the understanding of
the subject.
Posted: Sunday 31st of Dec 10:32
Hello dudes I truly need some back up here. I have a hard quiz, and I am truly stuck on college algebra tutoring programs. I don’t know where can I start. Can you give me some support with algebra
formulas, triangle similarity and difference of cubes, I’m not lazy! I just don’t comprehend. My parents have been thinking about hiring a good tutor , but they are so expensive . Any thoughts would
be enormously valued . Greetings!
Posted: Monday 01st of Jan 10:47
Can you please be more descriptive as to what sort of guidance you are expecting to get. Do you want to understand the principles and work on your math questions on your own or do you want a
instrument that would offer you a line by line solution for your math problems?
Posted: Tuesday 02nd of Jan 12:42
I checked out each one of them myself and that was when I came across Algebrator. I found it particularly suitable for adding functions, quadratic equations and unlike denominators. It was actually
also kid’s play to operate this. Once you key in the problem, the program carries you all the way to the answer elucidating every step on its way. That’s what makes it splendid. By the time you
arrive at the answer , you already know how to work out the problems. I took great pleasure in learning to work out the problems with Algebra 2, Remedial Algebra and Basic Math in algebra . I am also
positive that you too will love this program just as I did. Wouldn’t you want to test this out?
Posted: Wednesday 03rd of Jan 08:40
You both have got to be joshing ! How can this solution not be general information or heralded here ? How and where might I acquire additional info for testing Algebrator? Forgive anyone for
appearing to be a bit skeptical , but do either of you know if someone can receive a trial copy to use this ?
Posted: Thursday 04th of Jan 11:11
Hey Friends, I had a chance to try Algebrator offered at https://graph-inequality.com/teaching-inequalitiesa-hypothetical-classroom-case.html this morning . I am really very thankful to you all for
pointing me to Algebrator. The quick formula list and the detailed explanations on the fundamentals given there were really understandable. I have completed and submitted my homework on exponential
equations and this was all possible only with the help of the Algebrator that I purchased on the basis of your recommendations here. Thanks a lot.
Posted: Friday 05th of Jan 11:08
I recommend using Algebrator. It not only assists you with your math problems, but also displays all the required steps in detail so that you can enhance the understanding of the subject. | {"url":"https://graph-inequality.com/graph-inequality/subtracting-exponents/college-algebra-tutoring.html","timestamp":"2024-11-11T08:10:34Z","content_type":"text/html","content_length":"92109","record_id":"<urn:uuid:eb348a9b-156d-41be-8c4c-a76952722677>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00257.warc.gz"} |
nForum - Discussion Feed (relative monad)RodMcGuire comments on "relative monad" (117034)varkor comments on "relative monad" (116501)varkor comments on "relative monad" (116500)Urs comments on "relative monad" (113708)Urs comments on "relative monad" (112334)BryceClarke comments on "relative monad" (110042)Urs comments on "relative monad" (107962)Urs comments on "relative monad" (107960)Urs comments on "relative monad" (107947)J-B Vienney comments on "relative monad" (104639)Urs comments on "relative monad" (104638)maxsnew comments on "relative monad" (104636)varkor comments on "relative monad" (104635)maxsnew comments on "relative monad" (104632)Urs comments on "relative monad" (104629)David_Corfield comments on "relative monad" (101635)Sam Staton comments on "relative monad" (101629)Sam Staton comments on "relative monad" (101628)mattecapu comments on "relative monad" (101578)
I gave a talk about relative monads in CBPV recently, and I have some programming examples there: https://www.youtube.com/watch?v=ooj1vJRixEU&list=PLyrlk8Xaylp5hkSMipssQf3QKnj6Nrjg_
To summarize, in CBPV a monad relative to F : val -> comp is a more low-level version of a monad that specifies the stack the computation runs against. It’s natural to consider CBPV where F doesn’t
necessarily exist and you can still define relative monads as relative to the “profunctor of computations” which is always present. Additionally, the morphisms of comp (the stacks) are typically not
internalized as a data type in CBPV, but the elements of the profunctor are, so even if you have F, the notion of self-enriched relative monad needs to use the profunctor generalization.
I thought it would be a bit too far afield to try to explain those examples on this page. | {"url":"https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Comments&Page=1&Feed=ATOM&DiscussionID=14880&FeedTitle=Discussion+Feed+%28relative+monad%29","timestamp":"2024-11-04T10:54:02Z","content_type":"application/atom+xml","content_length":"26063","record_id":"<urn:uuid:5ad88de3-5534-48b6-99e1-a9250b9bd824>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00608.warc.gz"} |
MP Board Class 7th Maths Solutions Chapter 2 Fractions and Decimals Ex 2.2
Question 1.
Which of the drawings (a) to (d) show:
(i) 2 × \(\frac{1}{5}\) represents addition of 2 figures, each representing 1 shaded part out of 5 equal parts. Hence, 2 × \(\frac{1}{5}\) is represented by (d)
(ii) 2 × \(\frac{1}{2}\) represents addition of 2 figures, each representing 1 shaded part out of 2 equal parts. Hence, 2 × \(\frac{1}{2}\) is represented by
(b) .
(iii) 3 × \(\frac{2}{3}\) represents addition of 3 figures, each representing 2 shaded parts out of 3 equal parts. Hence, 3 × \(\frac{2}{3}\) is represented by (a).
(iv) 3 × \(\frac{1}{4}\) represents addition of 3 figures, each representing 1 shaded part out of 4 equal parts. Hence, 3 × \(\frac{1}{4}\) is represented by (c).
Question 2.
Some pictures (a) to (c) are given below. Tell which of them show:
(i) 3 × \(\frac{1}{5}\) represents the addition of 3 figures, each representing 1 shaded part out of 5 equal parts and \(\frac{3}{5}\) represents 3 shaded parts out of 5 equal parts.
Hence, \(3 \times \frac{1}{5}=\frac{3}{5}\) is represented by (c).
(ii) \(2 \times \frac{1}{3}\) represents the addition of 3 figures, each representing 1 shaded part out of 3 equal parts and \(\frac{2}{3}\) represents 2 shaded parts out of 3 equal parts.
Hence, \(2 \times \frac{1}{3}=\frac{2}{3}\) is represented by (a).
(iii) \(3 \times \frac{3}{4}\) represents the addition of 3 figures, each representing 3 shaded parts out of 4 equal parts and \(2 \frac{1}{4}\) represents 2 fully shaded figures and one figure
having 1 shaded part out of 4 equal parts.
Hence, \(3 \times \frac{3}{4}=2 \frac{1}{4}\) is represented by (b).
Question 3.
Multiply and reduce to lowest form and convert into a mixed fraction:
Question 4.
(i) There are 12 circles in the given box.
(iii) There are 15 squares in the given box (c). To shade \(\frac{3}{5}\) of the squares in it i.e., \(\frac{3}{5} \times 15=9\), we will shade any 9 squares of it.
Question 5.
Question 6.
Multiply and express as a mixed fraction:
Question 7.
Question 8.
Vidya and Pratap went for a picnic. Their mother gave them a water bottle that contained 5 litres of water. Vidya consumed \(\frac{2}{5}\) of the water. Pratap consumed the remaining water.
(i) How much water did Vidya drink?
(ii) What fraction of the total quantity of water did Pratap drink?
(i) Water consumed by Vidya
(ii) Fraction of remaining water
Thus, Pratap consumed \(\frac{3}{5}\) of the total quantity of water. | {"url":"https://mpboardguru.com/mp-board-class-7th-maths-solutions-chapter-2-ex-2-2-english-medium/","timestamp":"2024-11-08T05:10:20Z","content_type":"text/html","content_length":"59745","record_id":"<urn:uuid:eeda1447-508d-4353-883c-31b4b428bbaa>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00333.warc.gz"} |
sheaf on a topological space
Since I got questions from the audience (here) why I defined (pre-)sheaves on a site, instead of on a topological space “as in the textbooks”, I created this little entry with some basic pointers,
which may complement the entry localic topos for the newbie. Could of course be expanded a lot…
v1, current
changed page name to singular
v1, current
realized that we did have a brief entry “spatial topos”. Have merged the two. Made “spatial topos” now a redirect to this one.
v1, current | {"url":"https://nforum.ncatlab.org/discussion/8650/sheaf-on-a-topological-space/?Focus=69650","timestamp":"2024-11-13T09:50:13Z","content_type":"application/xhtml+xml","content_length":"40986","record_id":"<urn:uuid:a59b7351-1fad-44f3-8787-c0eac11b9f80>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00393.warc.gz"} |
Straight Line Motion Revisited Homework Answersl New! A new
Straight Line Motion Revisited Homework Answersl
New! A new student section has been written called Calculus from 30,000 feet. Borrowed from Steven Strogatz's book Infinite Powers, this addendum is meant to be used after the AP exam and gives
students a greater insight into how the Infinity Guideline breaks down comples probems into simper ones and then reassembles them. No homework or problems - just an attempy to see the Calculus forest
from the trees.
Straight Line Motion Revisited Homework Answersl
DOWNLOAD: https://www.google.com/url?q=https%3A%2F%2Furlcod.com%2F2u6IXy&sa=D&sntz=1&usg=AOvVaw2JnySWPMsbmBgE7snK_aI7
To find the observable consequences of this curvature we need to take another step. When spacetime is flat freely moving objects obey Newton's first law i.e. they move in a straight line at constant
speed. When spacetime is curved freely moving objects obey a different equation called the geodesic equation:
But all this is a bit abstract, and I suspect you're after a more intuitive feel for how does spacetime curvature cause straight lines not to be straight? Well the most obvious definition of a
straight line is the path of a light ray, because we all learn in school that light travels in straight lines. And the obvious demonstration of this is the bending of light rays by the Sun.
If we shine a light ray so that it just grazes the Sun's surface then that light doesn't travel in a straight line. We can calculate the deflection using the geodesic equation and the values of the
Cristoffel symbols, and we find the light ray is bent by about $1.75$ arcseconds. But this is a tiny, tiny amount. $1.75$ arcseconds is about the angle subtended by a baseball at a distance of $9$
kilometers - you couldn't even see a baseball $9$ km away!
11. Consider two cylinders that start down identical inclines from rest except that one is frictionless. Thus one cylinder rolls without slipping, while the other slides frictionlessly without
rolling. They both travel a short distance at the bottom and then start up another incline. (a) Show that they both reach the same height on the other incline, and that this height is equal to their
original height. (b) Find the ratio of the time the rolling cylinder takes to reach the height on the second incline to the time the sliding cylinder takes to reach the height on the second incline.
(c) Explain why the time for the rolling motion is greater than that for the sliding motion.
An object at rest (not moving) will remain at rest unless acted on by an unbalanced force. Also, an object in motion will continue to move at a constant speed in a straight line unless acted on by an
unbalanced force.
In 2-d projectile motion without air resistance the equation relating the velocity ##\vec v_\!f## at time ##t_\!f## and the initial velocity ##\vec v_0## is $$\vec v_\!f=\vec v_0+\vec gt_\!f.$$The
three vectors participating in the equation form a closed triangle the sides of which are ##OA = v_0##, ##OF = v_\!f## and a third vertical side ##AF = g t_\!f## (see figure 1). Triangle OAF in the
figure corresponds to a projectile that is at positive vertical displacement ##\Delta y## at time ##t_\!f## after reaching maximum height. I call this the velocity triangle; its geometric
construction from three given lengths is outlined in Appendix I. Triangle OAB is another velocity triangle corresponding to the projectile returning to launch level and is drawn for later reference.
Beyond being a curiosity, the position triangle can serve as a useful tool for solving problems that cannot be addressed by a velocity triangle construction. Consider the problem in which a
projectile is fired up a plane inclined at angle ##\alpha## relative to the horizontal. The projection angle is ##\beta## relative to the incline and we are seeking the landing distance ##d## up the
incline. The traditional strategy is to find the trajectory parabola by eliminating ##t_\!f## from the horizontal and vertical position equations, find the horizontal distance ##x_\!f## at which the
parabola intersects the straight line ##y = x \tan\alpha##, find the final height ##y_\!f## and finally use the Pythagorean theorem to find the required distance ##d##. This two-page solution can be
reduced to two lines using the position triangle.
3.4 Velocity & Other Rates of Change\n \n \n \n \n "," \n \n \n \n \n \n AP Physics Monday Standards: 1)a. Students should understand the general relationships between position velocity &
acceleration for a particle.\n \n \n \n \n "," \n \n \n \n \n \n Projectiles Horizontal Projection Horizontally: Vertically: Vertical acceleration g \uf0af 9.8 To investigate the motion of a
projectile, its horizontal and.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 8\/24 Warm up 1. Do in notebook Estimate the instantaneous rate of change at x = 3 for the function by picking values close
to 3.\n \n \n \n \n "," \n \n \n \n \n \n SECTION 4-4 A Second Fundamental Theorem of Calculus.\n \n \n \n \n "," \n \n \n \n \n \n 4-4 THE FUNDAMENTAL THEOREM OF CALCULUS MS. BATTAGLIA \u2013 AP
CALCULUS.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 1. Do in notebook. Be seated before the bell rings DESK homework Warm-up (in your notes) Agenda : warm-up Go over homework homework quiz Notes.\n
\n \n \n \n "," \n \n \n \n \n \n Agenda 9\/23\/13 Hand in Great Graphing homework Quiz: Graphing Motion Rearranging equations practice Discuss homework p. 44, p. 49 Notes\/ Discussion: Kinematic.\n
\n \n \n \n "," \n \n \n \n \n \n Velocity - time graph 1. The velocity \u2013 time graph shows the motion of a particle for one minute. Calculate each of the following. (a) The acceleration.\n \n \n
\n \n "," \n \n \n \n \n \n Warm up 8\/26 Warm up 1. Do in notebook True or False, if false explain why or give example 1. If, then 2. If, then 3. If, then 4. If, then.\n \n \n \n \n "," \n \n \n \n
\n \n Warm-up 8\/31. Finding the derivative and calculating the derivative at a value.\n \n \n \n \n "," \n \n \n \n \n \n \uf09e The derivative of a function f(x), denoted f\u2019(x) is the slope of
a tangent line to a curve at any given point. \uf0a1 Or the slope of a curve at any given.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 8\/25 Warm up 1. Do in notebook Expand the binomials.\n \n \n \n
\n "," \n \n \n \n \n \n Warm up 9\/25 Finalize your letter to the MANufacture. Be ready to shared.\n \n \n \n \n "," \n \n \n \n \n \n 2.1 Position, Velocity, and Speed 2.1 Displacement \uf044 x \
uf0ba x f - x i 2.2 Average velocity 2.3 Average speed \uf0ba \uf0ba\n \n \n \n \n "," \n \n \n \n \n \n 2.2 Differentiation Techniques: The Power and Sum-Difference Rules 1.5.\n \n \n \n \n "," \n \
n \n \n \n \n Warm Up 10\/13 Simplify each expression. 16, (3 2 )\n \n \n \n \n "," \n \n \n \n \n \n Warm up 9\/08 1. Factor 2. Solve by Factor. Be seated before the bell rings DESK homework Warm-up
(in your notes) Ch 5 test tues 9\/15 Agenda: Warmup Go.\n \n \n \n \n "," \n \n \n \n \n \n 3.2 Notes - Acceleration Part A. Objectives \uf070 Describe how acceleration, time and velocity are
related. \uf070 Explain how positive and negative acceleration.\n \n \n \n \n "," \n \n \n \n \n \n Warmup. Be seated before the bell rings DESK homework Warm-up (in your notes) Agenda : go over hw
Hw quiz Notes lesson 3.2.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 10\/16 (glue in). Be seated before the bell rings DESK homework Warm-up (in your notes) Agenda : go over hw Finish Notes lesson
4.5 Start 4.6.\n \n \n \n \n "," \n \n \n \n \n \n Warmup 9\/18 Use 1 st derivative test to determine maximum and minimums.\n \n \n \n \n "," \n \n \n \n \n \n 5.3: Position, Velocity and
Acceleration. Warm-up (Remember Physics) m sec Find the velocity at t=2.\n \n \n \n \n "," \n \n \n \n \n \n Instantaneous Rate of Change The instantaneous rate of change of f with respect to x is.\n
\n \n \n \n "," \n \n \n \n \n \n A car is moving along Highway 50 according to the given equation, where x meters is the directed distance of the car from a given point P at t hours. Find.\n \n \n \
n \n "," \n \n \n \n \n \n 3023 Rectilinear Motion AP Calculus. Position Defn: Rectilinear Motion: Movement of object in either direction along a coordinate line (x-axis, or y-axis)\n \n \n \n \n ","
\n \n \n \n \n \n Warm up 9\/23 Solve the systems of equations by elimination.\n \n \n \n \n "," \n \n \n \n \n \n February 6, 2014 Day 1 Science Starters Sheet 1. Please have these Items on your
desk. I.A Notebook 2- Science Starter: Two Vocabulary Words: Motion Speed.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 8\/19 Warm up 1. Do in notebook Explain why these are incorrect :\n \n \n \n \n
"," \n \n \n \n \n \n Ch. 8 \u2013 Applications of Definite Integrals 8.1 \u2013 Integral as Net Change.\n \n \n \n \n "," \n \n \n \n \n \n Warmup 9\/17 Mr. Xiong\u2019s car passes a stationary
patrol cars at 65 mph. He passes another patrol car 8 min later going 55 mph. He was immediately stop and.\n \n \n \n \n "," \n \n \n \n \n \n 1.A Function and a Point 2.Equation of a line between 2
points 3.A point on the graph of a function 4.Information from a function\u2019s graph 5.Symbols on.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 10\/15. Review: 1)Explain what is the FTC # 2. 2)
Explain how to do each of these three problems a) b)C)\n \n \n \n \n "," \n \n \n \n \n \n Warm-upWarm-up 1.Find all values of c on the interval that satisfy the mean value theorem. 2. Find where
increasing and decreasing.\n \n \n \n \n "," \n \n \n \n \n \n 3-4 VELOCITY & OTHER RATES OF CHANGE. General Rate of Change The (instantaneous) rate of change of f with respect to x at a is the
derivative! Ex 1a)\n \n \n \n \n "," \n \n \n \n \n \n Motion Graphs Learning Target: Be able to relate position, velocity, and acceleration quantitatively and qualitatively.\n \n \n \n \n "," \n \n
\n \n \n \n Speed vs. Velocity.\n \n \n \n \n "," \n \n \n \n \n \n Bell Work 12\/1\/14 Pick up a Topic Reinforcement page from the front table. Take a seat and this week into your agenda. Begin
working quietly on the Topic.\n \n \n \n \n "," \n \n \n \n \n \n Warm up Warm up 1. Do in notebook\n \n \n \n \n "," \n \n \n \n \n \n Ch.5, Sec.1 \u2013 Measuring Motion\n \n \n \n \n "," \n \n \n
\n \n \n What is Motion?.\n \n \n \n \n "," \n \n \n \n \n \n Motion Review Challenge\n \n \n \n \n "," \n \n \n \n \n \n Lecture 2 Chapter ( 2 ).\n \n \n \n \n "," \n \n \n \n \n \n Motion and Force
A. Motion 1. Motion is a change in position\n \n \n \n \n "," \n \n \n \n \n \n Graphing Motion Walk Around\n \n \n \n \n "," \n \n \n \n \n \n Take out your homework from Friday (only front should
be completed)\n \n \n \n \n "," \n \n \n \n \n \n A function F is the Antiderivative of f if\n \n \n \n \n "," \n \n \n \n \n \n 2.3B Higher Derivatives.\n \n \n \n \n "," \n \n \n \n \n \n The
Kinematics Equations\n \n \n \n \n "," \n \n \n \n \n \n Warm-up Enter the two functions into the y = in your\n \n \n \n \n "]; Similar presentations | {"url":"https://www.wiatribe.com/group/mysite-231-group/discussion/e9464ad4-9f59-4708-b3c8-a335f83414c1","timestamp":"2024-11-09T20:23:26Z","content_type":"text/html","content_length":"1050477","record_id":"<urn:uuid:2c0b1284-2e0b-4087-a70f-73b384544d68>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00793.warc.gz"} |
Solver Tutorial - Step by Step Product Mix Example in Visual Basic
VB.NET Product Mix Example
Essential Steps
Follow these steps to define and solve the Product Mix problem in a Visual Basic .NET program (the steps in another Windows programming language would be very similar):
1. Create a variable in your program to hold the value of each decision variable in your model.
2. Create an assignment statement in your program, that calculates the objective function in your model.
3. Similarly, create assignment statements in your program to calculate the left hand sides of your constraints.
4. Use SDK objects in your program to tell the Solver about your decision variables, objective and constraint calculations, and desired bounds on constraints and variables.
5. Call the Optimize() method in your program to find the optimal solution.
Creating a Visual Basic Program
Assuming that you have Visual Studio and Solver Platform SDK installed, and you've opened a new or existing project, the next step is to define a VB.NET function where the formulas for the objective
function and the constraints are calculated.
In the VB.NET code excerpt below, we define a function NLPEvaluator_OnEvaluate() that accesses elements of the model through Solver SDK objects. (Click on the code excerpt for a full-size image.)
In the above code, we've written the formulas to correspond directly to our earlier outline of the problem (click on Writing the Formulas to see this again). But we could also use Visual Basic
arrays and FOR loops to compute these values. (In fact, we can let Solver SDK compute the objective and constraint values for us, using just the coefficients such as 450, 1150, 800 and 400 for the
objective function. Read Using the LP Coefficient Matrix Directly in our Advanced Tutorial for more about this option.)
Using the Solver SDK Objects
Next, we must tell the Solver SDK about elements of the model that aren't included in the NLPEvaluator_OnEvaluate() function above. For example, the left hand sides of the constraints are computed
in NLPEvaluator_OnEvaluate(), but the constant right hand sides (5800 quarts of glue, 730 hours of pressing capacity, etc.), as well as lower bounds of 0 on the variables, must be specified
To do this, we define an SDK Problem object and set its properties as shown in the code excerpt below. In the last line, we call prob.Solver.Optimize() to find the optimal solution. (Click on the
code excerpt for a full-size image.)
In this code excerpt, FcnUB, FcnLB, VarUB, and VarLB are previously defined arrays, where FcnUB contains the right hand sides of the constraints, and VarLB contains lower bounds of 0 for the
variables. Symbolic constants such as Eval_Type.Function, Solver_Type.Maximize, and Problem_Type.OptNLP are predefined in the Solver SDK.
Accessing and Using the Solution
Once the problem is defined and solved, we can use Solver SDK Problem object properties to access the optimal solution. In the code excerpt below, we retrieve the solution status, the objective
function value, and the decision variable values at the optimal solution. (Click on the code excerpt for a full-size image.)
The OptimizeStatus value is an integer code reporting the status of the optimization -- 0 means that an optimal solution was found.
Learning More
If you've gotten to this point, congratulations! You've successfully set up and solved a simple optimization problem using Frontline's Solver Platform SDK. If you'd like, you can see how to set up
and solve the same Product Mix problem using Excel's built-in Solver or using Risk Solver Platform in Excel. If you haven't yet read the other parts of the tutorial, you may want to return to the
Tutorial Start and read the overviews "What are Solvers Good For?", "How Do I Define a Model?", "What Kind of Solution Can I Expect?" and "What Makes a Model Hard to Solve?"
This was an example of a linear programming problem. Other types of optimization problems may involve quadratic programming, mixed-integer programming, constraint programming, smooth nonlinear
optimization, and nonsmooth optimization. To learn more, click Optimization Problem Types. For a more advanced explanation of linearity and sparsity in optimization problems, continue with our
Advanced Tutorial. | {"url":"https://www.solver.com/product-mix-vb","timestamp":"2024-11-09T12:29:30Z","content_type":"text/html","content_length":"61291","record_id":"<urn:uuid:4936791a-651f-4421-8422-f70e316670dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00809.warc.gz"} |
Source code for openquake.hazardlib.mfd.evenly_discretized
# The Hazard Library
# Copyright (C) 2012-2019 GEM Foundation
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU Affero General Public License as
# published by the Free Software Foundation, either version 3 of the
# License, or (at your option) any later version.
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Affero General Public License for more details.
# You should have received a copy of the GNU Affero General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
Module :mod:`openquake.hazardlib.mfd.evenly_discretized` defines an evenly
discretized MFD.
from openquake.hazardlib.mfd.base import BaseMFD
from openquake.baselib.slots import with_slots
class EvenlyDiscretizedMFD(BaseMFD):
Evenly discretized MFD is defined as a precalculated histogram.
:param min_mag:
Positive float value representing the middle point of the first
bin in the histogram.
:param bin_width:
A positive float value -- the width of a single histogram bin.
:param occurrence_rates:
The list of non-negative float values representing the actual
annual occurrence rates. The resulting histogram has as many bins
as this list length.
MODIFICATIONS = set(('set_mfd',))
_slots_ = 'min_mag bin_width occurrence_rates'.split()
def __init__(self, min_mag, bin_width, occurrence_rates):
self.min_mag = min_mag
self.bin_width = bin_width
self.occurrence_rates = occurrence_rates
[docs] def check_constraints(self):
Checks the following constraints:
* Bin width is positive.
* Occurrence rates list is not empty.
* Each number in occurrence rates list is non-negative.
* Minimum magnitude is positive.
if not self.bin_width > 0:
raise ValueError('bin width must be positive')
if len(self.occurrence_rates) == 0:
raise ValueError('at least one bin must be specified')
if not all(value >= 0 for value in self.occurrence_rates):
raise ValueError('all occurrence rates must not be negative')
if not any(value > 0 for value in self.occurrence_rates):
raise ValueError('at least one occurrence rate must be positive')
if not self.min_mag >= 0:
raise ValueError('minimum magnitude must be non-negative')
[docs] def get_annual_occurrence_rates(self):
Returns the predefined annual occurrence rates.
return [(self.min_mag + i * self.bin_width, occurrence_rate)
for i, occurrence_rate in enumerate(self.occurrence_rates)]
[docs] def get_min_max_mag(self):
Returns the minumun and maximum magnitudes
return self.min_mag, self.min_mag + self. bin_width * (
len(self.occurrence_rates) - 1)
[docs] def modify_set_mfd(self, min_mag, bin_width, occurrence_rates):
Applies absolute modification of the MFD from the ``min_mag``,
``bin_width`` and ``occurrence_rates`` modification.
:param min_mag:
Positive float value representing the middle point of the first
bin in the histogram.
:param bin_width:
A positive float value -- the width of a single histogram bin.
:param occurrence_rates:
The list of non-negative float values representing the actual
annual occurrence rates. The resulting histogram has as many bins
as this list length.
self.min_mag = min_mag
self.bin_width = bin_width
self.occurrence_rates = occurrence_rates | {"url":"https://docs.openquake.org/oq-engine/3.7/_modules/openquake/hazardlib/mfd/evenly_discretized.html","timestamp":"2024-11-10T08:32:51Z","content_type":"application/xhtml+xml","content_length":"16232","record_id":"<urn:uuid:724f616a-4008-4b5b-85cf-38faeb1df734>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00600.warc.gz"} |
Swimming training
1. Swimming efficiency, or also called stroke index, is an important concept for improving performance in the water. There are several formulas for calculating swimming efficiency, but let's look at
two common approaches:
Swimming Equation: The swimming equation is a formula that considers variables such as reaction time, time under water, turn time, stroke count, and stroke frequency
Here's how it's written: [ ST = S + (UT + TT) + (CC \cdot SR) ]
• (ST): Swimming time, in seconds..
• (S): Start (Reaction Time + Time in Air), in seconds.
• (UT):Time underwater, i.e. how much time you spend underwater after starting or turning.
• (TT): Time underwater, i.e. how much time you spend underwater after starting or turning.
• (CC): Number of strokes.
• (SR): Stroke rate, in seconds per stroke.
This equation represents two components: time under water and time above water. Time under water is the sum of (UT) and (TT), while time above water is a function of (CC) multiplied by (SR)..
2. Speed Formula: Another approach involves swimming speed.
The formula is: [ v = \frac{{P_m \cdot E_p}}{D} ]
• (v): Swimming speed.
• (P_m): Muscle power.
• (E_p): Propulsive efficiency.
• (D): Water resistance.
• cdot: represents the multiplication operation between the various terms of the formula ( cdot indicates that the metabolic power developed by the swimmer (( P_m )) is multiplied by the propulsive
efficiency of the swimmer (( E_p )), and the product of these two terms is then divided by the drag (( D )), which represents the resistance of the water.
This formula takes into account muscle power, propulsive efficiency and water resistance to calculate swimming speed.
In short, improving swimming efficiency requires good technique, correct stroke frequency, and effective time management under and above the water. | {"url":"https://www.managersport.it/en/swimming-training.html?start=3","timestamp":"2024-11-04T10:12:59Z","content_type":"text/html","content_length":"36388","record_id":"<urn:uuid:6edd05f2-de83-4da9-96df-4014e4f64826>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00798.warc.gz"} |
Cumulative Distribution Functions and Probability Density Functions
Probability Theory Mastering
Probability Theory Mastering
Cumulative Distribution Function (CDF)
The Cumulative Distribution Function (CDF) is a function that describes the cumulative probability of a random variable taking on a value less than or equal to a given value.
Mathematically, the CDF of a random variable X, denoted as F(x), is defined as:
F(x) = Probability that variable X is less or equal to value x.
Using this function, it is easy to describe continuous random variables.
Look at the example below: we will use a normally distributed random variable and look at its CDF using the .cdf() method.
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
# Generate a random variable following a normal distribution
mu = 0 # mean
sigma = 1 # standard deviation
x = np.linspace(-5, 5, 100) # x values
rv = norm(loc=mu, scale=sigma) # create a normal distribution with given mean and standard deviation
# Compute the CDF for the random variable
cdf = rv.cdf(x)
# Plot the CDF
plt.plot(x, cdf, label='CDF')
plt.title('CDF of a Standard Normal Distribution')
Using CDF, we can determine the probability that our random variable belongs to any of the intervals of interest. Assume that X is a random variable, and F(x) is its CDF.
To determine the probability that the variable X belongs to the interval [a, b], we can use the following formula:
P{X є [a,b]} = F(b) - F(a).
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
# Generate a random variable following a normal distribution
mu = 0 # mean
sigma = 1 # standard deviation
rv = norm(loc=mu, scale=sigma)
# Calculate probabilities for different ranges
print('Normally distributed variable belongs to [-1, 1] with probability:', round(rv.cdf(1) - rv.cdf(-1), 3))
print('Normally distributed variable belongs to [-2, 2] with probability:', round(rv.cdf(2) - rv.cdf(-2), 3))
print('Normally distributed variable belongs to [-3, 3] with probability:', round(rv.cdf(3) - rv.cdf(-3), 3))
Percent Point Function (PPF)
Percent Point Function (PPF), also known as the inverse of the cumulative distribution function (CDF). It is used to find the value of a random variable that corresponds to a given probability. In
Python it is implemented using .ppf() method:
from scipy.stats import norm
# Define probabilities
probabilities = [0.1, 0.5, 0.85]
# Iterate over each probability and print the corresponding value of the variable
for i in probabilities:
# Calculate the value of the variable using the percent point function (inverse of the cumulative distribution function)
value = norm.ppf(i)
# Round the value to 3 decimal places for clarity
value = round(value, 3)
# Print the result
print('Normally distributed variable is less than', value, 'with probability', i)
Probability Density Function (PDF)
Probability Density Function (PDF) is a function that provides information about the likelihood of a random variable taking on a particular value at a specific point in the continuous range. Its
interpretation is similar to that of the PMF but is specifically used for describing continuous random variables.
The PDF defines the shape of the probability distribution of a continuous random variable.
Let's consider the following example of PDF calculated using the .pdf() method.
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
# Generate x values for plotting
x = np.linspace(-3, 3, 100)
# Calculate the probability density function (PDF) values for the standard normal distribution
pdf_values = norm.pdf(x, loc=0, scale=1)
# Plot the PDF
plt.plot(x, pdf_values, label='PDF') # Plot PDF values against x values
plt.xlabel('X') # Label for x-axis
plt.ylabel('PDF') # Label for y-axis
plt.title('PDF of a Standard Normal Distribution') # Title of the plot
plt.legend() # Show legend
plt.show() # Display the plot
The PDF provides insight into the likelihood or probability density of a random variable assuming a specific value. Higher PDF values suggest a greater likelihood, while lower values suggest a lesser
To determine the probability of a continuous variable falling within a specific range, similar to using the PMF, we calculate the sum of the PDF for all values within that range. However, since
continuous variables can have an infinite number of values within any range, we calculate the area under the PDF curve within the specified range instead of a simple sum.
Thanks for your feedback! | {"url":"https://codefinity.com/courses/v2/ec6e4978-d3b4-4612-b2be-894504d0f970/d7a3e2ee-c102-44c5-abc4-6f2d0d3f3c15/19db718a-373d-4fda-acda-c4236f1b5bc1","timestamp":"2024-11-06T15:08:38Z","content_type":"text/html","content_length":"446418","record_id":"<urn:uuid:f795d916-81e7-4373-8d38-dae97531cc2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00055.warc.gz"} |
CS4114 Formal Languages and Automata: Spring 2022
6.1. Context-Free Grammars Part 1¶
6.1.1. Context-Free Languages¶
In previous chapters, we saw that some languages are regular languages, which means that we can define a DFA, NFA, regular expression, or regular grammar for them. We also discussed using closure
operators to show that other languages are regular if they can be derived from known regular languages using operators known to be closed for regular languages. Examples of regular languages:
• keywords in a programming language
• names of identifiers
• integers
• a finite list of miscillaneous symbols such as = \ ;
Then we discussed ways to prove that a language is non-regular, such as using the Pumping Lemma or using operators known to be closed for regular language to derive a known non-regular language.
Exxamples of non-regular languages include:
• \(\{a^ncb^n | n > 0\}\)
• expressions: \(((a + b) - c)\)
• block structures (\(\{\}\) in Java/C++ and begin … end in Pascal)
• Balanced parentheses
We know that not all languages are not regular, since we’ve proved that some are not.
(Something to think about: If you were to write a program in your favorite programming language to recognize any of those languages, what is the minimum memory that you need for each?)
Now we will look at a class of languages that is larger than the class of regular languages, context-free languages. And we will discuss ways to represent them.
6.1.2. String Derivation¶
6.1.4. Derivation Trees Example¶
6.1.5. Practice question 1¶
6.1.6. Membership Problem¶
6.1.7. Practice question 2¶ | {"url":"https://opendsa-server.cs.vt.edu/OpenDSA/Books/PIFLAS22/html/CFG1.html","timestamp":"2024-11-02T13:47:35Z","content_type":"text/html","content_length":"21802","record_id":"<urn:uuid:3694f99b-83c6-41d3-a0c9-d64c7f7965a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00009.warc.gz"} |
Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are mathematical expressions which consist of one or more terms, each of which has a variable raised to a power. Dividing polynomials is a crucial operation in algebra that includes
finding the remainder and quotient as soon as one polynomial is divided by another. In this article, we will explore the different approaches of dividing polynomials, including synthetic division and
long division, and provide examples of how to use them.
We will further discuss the importance of dividing polynomials and its uses in different domains of mathematics.
Prominence of Dividing Polynomials
Dividing polynomials is an essential operation in algebra which has several utilizations in many domains of arithmetics, consisting of number theory, calculus, and abstract algebra. It is applied to
solve a broad spectrum of problems, including working out the roots of polynomial equations, figuring out limits of functions, and calculating differential equations.
In calculus, dividing polynomials is applied to work out the derivative of a function, which is the rate of change of the function at any moment. The quotient rule of differentiation consists of
dividing two polynomials, that is applied to work out the derivative of a function which is the quotient of two polynomials.
In number theory, dividing polynomials is used to learn the properties of prime numbers and to factorize large values into their prime factors. It is further applied to study algebraic structures
such as fields and rings, that are fundamental concepts in abstract algebra.
In abstract algebra, dividing polynomials is applied to specify polynomial rings, that are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are used in various
fields of math, comprising of algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is a method of dividing polynomials that is used to divide a polynomial with a linear factor of the form (x - c), where c is a constant. The method is based on the fact that if f
(x) is a polynomial of degree n, subsequently the division of f(x) by (x - c) gives a quotient polynomial of degree n-1 and a remainder of f(c).
The synthetic division algorithm includes writing the coefficients of the polynomial in a row, applying the constant as the divisor, and carrying out a chain of workings to figure out the quotient
and remainder. The answer is a streamlined form of the polynomial that is simpler to function with.
Long Division
Long division is an approach of dividing polynomials that is used to divide a polynomial with another polynomial. The approach is relying on the reality that if f(x) is a polynomial of degree n, and
g(x) is a polynomial of degree m, where m ≤ n, subsequently the division of f(x) by g(x) gives a quotient polynomial of degree n-m and a remainder of degree m-1 or less.
The long division algorithm involves dividing the greatest degree term of the dividend with the highest degree term of the divisor, and subsequently multiplying the answer by the total divisor. The
answer is subtracted from the dividend to obtain the remainder. The method is repeated until the degree of the remainder is less in comparison to the degree of the divisor.
Examples of Dividing Polynomials
Here are a number of examples of dividing polynomial expressions:
Example 1: Synthetic Division
Let's assume we have to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 with the linear factor (x - 1). We can apply synthetic division to simplify the expression:
1 | 3 4 -5 2 | 3 7 2 |---------- 3 7 2 4
The result of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Thus, we can express f(x) as:
f(x) = (x - 1)(3x^2 + 7x + 2) + 4
Example 2: Long Division
Example 2: Long Division
Let's assume we have to divide the polynomial f(x) = 6x^4 - 5x^3 + 2x^2 + 9x + 3 with the polynomial g(x) = x^2 - 2x + 1. We could apply long division to streamline the expression:
First, we divide the largest degree term of the dividend by the largest degree term of the divisor to get:
Subsequently, we multiply the whole divisor with the quotient term, 6x^2, to get:
6x^4 - 12x^3 + 6x^2
We subtract this from the dividend to attain the new dividend:
6x^4 - 5x^3 + 2x^2 + 9x + 3 - (6x^4 - 12x^3 + 6x^2)
that simplifies to:
7x^3 - 4x^2 + 9x + 3
We recur the procedure, dividing the largest degree term of the new dividend, 7x^3, by the highest degree term of the divisor, x^2, to get:
Next, we multiply the whole divisor with the quotient term, 7x, to obtain:
7x^3 - 14x^2 + 7x
We subtract this from the new dividend to achieve the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
that streamline to:
10x^2 + 2x + 3
We repeat the method again, dividing the highest degree term of the new dividend, 10x^2, with the highest degree term of the divisor, x^2, to achieve:
Subsequently, we multiply the whole divisor with the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this from the new dividend to achieve the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that streamlines to:
13x - 10
Thus, the outcome of the long division is the quotient polynomial 6x^2 - 7x + 9 and the remainder 13x - 10. We can express f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
In Summary, dividing polynomials is a crucial operation in algebra which has many utilized in multiple domains of mathematics. Comprehending the different methods of dividing polynomials, for
instance long division and synthetic division, could guide them in figuring out intricate problems efficiently. Whether you're a learner struggling to get a grasp algebra or a professional working in
a domain that consists of polynomial arithmetic, mastering the concept of dividing polynomials is essential.
If you desire support comprehending dividing polynomials or any other algebraic concept, consider calling us at Grade Potential Tutoring. Our expert tutors are accessible online or in-person to offer
personalized and effective tutoring services to help you be successful. Contact us right now to schedule a tutoring session and take your mathematics skills to the next level. | {"url":"https://www.santamonicainhometutors.com/blog/dividing-polynomials-definition-synthetic-division-long-division-and-examples","timestamp":"2024-11-11T14:46:21Z","content_type":"text/html","content_length":"77945","record_id":"<urn:uuid:e99be670-78b2-4231-966f-3259f465ee9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00046.warc.gz"} |
The Open Educator - 4.4.2. Outlier, Leverage, and Influential Points Unusual Observations Check
Unusual Observations
Outlier, Leverage, and Influential Points
An observation could be unusual with respect to its y-value or x-value. However, rather than calling them x- or y-unusual observations, they are categorized as outlier, leverage, and influential
points according to their impact on the regression model.
Outlier – an outlier is defined by an unusual observation with respect to either x-value or y-value. An x-outlier will make the scope of the regression too broad, which is usually considered less
accurate. An x-outlier is uncommon, it may seriously affect the regression outcomes though. However, in an unplanned study, often the data is collected before putting much thought into it. In those
situations, there could be a possibility of having x-outliers. The y-outliers are very common, and it is usually not as severe as the x-outlier. Nevertheless, the effects of the y-outliers must be
investigated further to check whether it is just a simple data entry error, or some severe issue in the process, or just a random phenomenon. Figure 7 shows both x-outlier (left) and y-outlier
(right). Both plots show that a better linear relationship will be possible without these outliers. In this situation, the x-outlier is rotating the line clockwise to change both the slope and the
intercept of the relationship, while the y-outlier is moving the predicted line upward. The solid line shows the predicted relationship without the outliers.
Figure 7. Outlier with Respect to x-Value (Left) and y-Value (Right)
Leverage – a data point whose x-value (independent) is unusual, y-value follows the predicted regression line though (Figure 8). A leverage point may look okay as it sits on the predicted regression
line. However, a leverage point will inflate the strength of the regression relationship by both the statistical significance (reducing the p-value to increase the chance of a significant
relationship) and the practical significance (increasing r-square). Unfortunately, leverage points have no impact on the coefficients because the point follows the predicted regression line.
Practical significance of leverage point – think about a relationship between the muscle mass and the power. In the study, if most individuals weigh around 200 pounds and only one person weighs about
400 lbs. This 400-pound and his extreme y-value (power) will dictate the relationship more than all other individuals weighing near 200 pounds. Therefore, the conclusions for the study could be
misleading. The Leverage points usually make the functional regression relationship too broad. Generally, a wider (too broad) model is conserved less accurate as compared to a shorter one. To improve
the regression model accuracy, shorter models are recommended.
Figure 8. Leverage Point (Right) in a Regression Analysis
Influential – a data point that unduly influences the regression analyses outputs (Figure 9). A point is considered influential if its exclusion causes major changes in the fitted regression
function. Depending on the location of the point, it may affect all statistics, including the p-value, r-square, coefficients, and intercept. Figure 9 shows the impact of an influential point on the
regression statistics, including the r-square, slope, and the intercept.
Figure 9. Influential Point in a Regression Analysis
Statistical Diagnostic Tests for Unusual Observations
Any statistical software, including MS Excel will produce the diagnostic statistics results. Video 2 provides the diagnostic analysis using Minitab software. It also provides an explanation of the
analysis results.
Video 2. How to Explain and Interpret the Linear Regression Diagnostics Analysis Explained Example in Minitab
Diagnostic analysis for each data point is provided in Table 2. An observation is generally considered an outlier if the absolute value of the residual (RESI) is higher. For example, the data point #
6 has a very high residual compared to any other data points of the data set. The absolute values for the other diagnostic statistics such as scaled or adjusted residuals, standardized residuals
(SRES) and deleted residuals (TRES) are also observed to be higher for point # 6. Generally, higher absolute value for any of these diagnostic statistics for a point is considered an outlier.
Table 2
Regression Diagnostic Analysis: Detection of Outliers
An x-outlier is determined from the diagonal element of the hat matrix, HI. The diagonal elements of the hat matrix HI has some interesting properties, including
1. HI measures the weighted distance from the x-mean (mean of the independent variables).
2. The sum of all diagonal elements of the hat matrix, HI is equal to the sum of the total number of parameters and the intercept, p. In this example, there is one parameter and one intercept, which
is equal to 2 = p. Therefore, the sum of HI column in Table 3 is equal to 2.
3. Therefore, any large value for the HI is considered an outlier with respect to the x-values.
4. Generally, any value exceeds twice the mean value of the HI (=2*(p/n)) is considered an x-outlier.
Point #11 produces a value of 0.73 for the diagonal element of the hat matrix, HI, which is larger than the 2p/n (= 0.036). Therefore, this point #11 is considered an outlier with respect the x
Table 3
Regression Diagnostic Analysis: Detection of x-Outlier and Leverage Points
A leverage point is determined by a point whose x-value is an outlier, while the y-value is on the predicted line (y-value is not an outlier). Therefore, this point is undetected by the y-outlier
detection statistics, including the RESI, SRES, and TRES. For example, the RESI, SRES, and TRES values for the point # 11 are NOT considered large at all, rather they are very consistent with other
points. Therefore, the point #11 is not considered an outlier with respect to the y-value. However, the value for the diagonal element of the hat matrix, HI is very large. Any point whose diagonal
element of the hat matrix value exceeds 2p/n (2*2/11=0.36 for this example) is considered a leverage point. Therefore, the point # 11 is considered an x-outlier and it has high leverage on the
regression analysis.
DFIT and COOK distance is used to statistically determine the influential point. If the absolute value of DFIT exceeds 1 for small to medium data sets and for large data set, the point is considered
as influential to the fitted regression. In this small data set example in Table 4, the absolute value of DFIT for point # 11 is observed to be 3.63 which exceeds 1 (one), and therefore, the point #
11 is considered an influential point. While the DFIT measures the influence of the i^th case on the fitted value for this case, Cook’s distance, COOK measures the influence of the i^th case on all n
fitted values. A very large Cook’s distance for a point indicates a potential influence on the fitted regression line. However, to statistically determine the influential point, the probability is
calculated using the Cook’s distance as the value for the F-distribution. If the probability value for the Cook’s distance is 50% or more, the point has a major significant influence on the fitted
regression line. Probability value between 10-20% indicates a very small influence, while 20-50% indicates moderate to high influence on the fitted regression. The probability for Cook’s distance is
calculated using an F-distribution of p and n-p degrees freedom for the numerator and the denominator, respectively. For this example in Table 4, type /write/input = 1-FDIST(1.637,2,9) in MS Excel to
calculate the p-value for the point # 11. The probability value calculated for point #11 is 75.2% indicating a major influence on the regression.
Table 4
Detection of Influential Point | {"url":"https://www.theopeneducator.com/doe/Regression/outlier-leverage-influential-points","timestamp":"2024-11-09T09:59:55Z","content_type":"text/html","content_length":"368735","record_id":"<urn:uuid:6b0d27af-62e3-473d-a72d-70e3edbadfd1>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00096.warc.gz"} |
48/50 Linear System Solver
02-13-2014, 01:30 AM
(This post was last modified: 02-13-2014 01:34 AM by Tim Wessman.)
Post: #1
Tim Wessman Posts: 2,293
Senior Member Joined: Dec 2013
48/50 Linear System Solver
Have someone with a complaint about the linear system solver on the 48 series (50g specifically).
x - 5y + 4z = -3
2x - 7y + 3z = -2
-2x + y + 7z =-1
Now he knows these are dependent, but the concern is that instead of giving "no solutions" it returns
x = -0.04869...
y = 0.25937...
z = -0.221295...
In looking at the old 48 manual, it talks about the linear solver in 18-11 where it talks about 3 different cases.
I am curious as to where exactly these numbers are coming from. Anyone know which case, or what they are supposed to represent? Or is this an artifact of some lower level matrix routines?
The output of the LINSOLVE command is interesting, where it returns a '1=0' as the Z solution. :-)
Although I work for HP, the views and opinions I post here are my own.
02-13-2014, 04:01 PM
Post: #2
Manolo Sobrino Posts: 179
Member Joined: Dec 2013
RE: 48/50 Linear System Solver
Yes, this happens too with emu48 for the 48g. I guess it's the LU decomposition. You can try to decompose the coefficient matrix with the 50g and check by yourself that there is a nasty 1E-14 for the
(3,3) element in the matrix L. It should be zero as it comes from a singular matrix (flag 54 doesn't seem to help BTW). The linear solver should check first that the matrix is invertible IMO, at
least for small dimension matrices and/or integer coefficients, these potential side effects can cause quite a bit of trouble.
02-16-2014, 06:00 PM
(This post was last modified: 02-16-2014 06:01 PM by Dieter.)
Post: #3
Dieter Posts: 2,397
Senior Member Joined: Dec 2013
RE: 48/50 Linear System Solver
(02-13-2014 01:30 AM)Tim Wessman Wrote: Have someone with a complaint about the linear system solver on the 48 series (50g specifically).
The system is exactly determined (three variables in three equations), but its determinant is zero. Which is a quite challenging case for iterative numerical algorithms like a LU-decomposition: due
to accumulated roundoff errors the result may be close to, but not exactly zero. Which makes the calculator think a valid solution can be calculated, and so it continues.. with a meaningless result.
I do not have a 48-series calculator here. What does it return if you try to evaluate the determinant? Something very close to (but not equal to) zero?
BTW: that's why I like simple direct methods like Cramer's rule for small linear systems with 2, 3 or maybe even 4 variables. ;-)
02-16-2014, 06:49 PM
Post: #4
Marcus von Cube Posts: 760
Senior Member Joined: Dec 2013
RE: 48/50 Linear System Solver
(02-16-2014 06:00 PM)Dieter Wrote: The system is exactly determined (three variables in three equations), but its determinant is zero.
If I try to solve the system manually I get 0 = 5 which is obviously false irrespective of the values of x, y or z. So the system has no solution. A determinant of zero can say "no solution" or "a
linear set of solutions" (depending on the right hand side) but it certainly does not say "exactly one solution".
Any calculator should error out on this equation. If I understand the excerpt from the 48G manual correctly, the result of LINSOLV occasionally is a least squares approximation. This is OK if the
coefficients come from real world data and a solution is expected to exist but does not because of limited accuracy of the input.
Marcus von Cube
Wehrheim, Germany
02-16-2014, 06:54 PM
Post: #5
Manolo Sobrino Posts: 179
Member Joined: Dec 2013
RE: 48/50 Linear System Solver
To put it bluntly, the problem is that the HP 48/50 can find easily that the determinant is zero, yet the linear solver just calls the LU decomposition routine without checking the coefficient matrix
02-16-2014, 11:40 PM
Post: #6
Thomas Klemm Posts: 2,271
Senior Member Joined: Dec 2013
RE: 48/50 Linear System Solver
(02-16-2014 06:00 PM)Dieter Wrote: What does it return if you try to evaluate the determinant? Something very close to (but not equal to) zero?
Exactly 0. The RANK of the matrix gives 2.
When flag -22 is cleared (Infinite -> error) I get an error (Infinite Result) when I use / or 1/x to solve the equation. However I get Tim's result when the flag -22 is set or when I use LSQ or the
linear solver. But when you calculate the residual with RSD you will notice that the computed solution isn't close to an actual solution. Furthermore the condition number is huge: 9.8e15. The
approximate number of accurate digits is thus 0. When the flag -22 is cleared I get the same error: Infinite Result.
We get a slightly different result when the last equation is replaced by 4*Ⅰ-3*Ⅱ:
-2x + y + 7z =-6
Now we get a real solution which can be checked with the residual.
02-16-2014, 11:54 PM
Post: #7
Thomas Klemm Posts: 2,271
Senior Member Joined: Dec 2013
RE: 48/50 Linear System Solver
(02-16-2014 06:49 PM)Marcus von Cube Wrote:
(02-16-2014 06:00 PM)Dieter Wrote: The system is exactly determined (three variables in three equations), but its determinant is zero.
A determinant of zero can say "no solution" or "a linear set of solutions" (depending on the right hand side) but it certainly does not say "exactly one solution".
Exactly! Who said so?
02-17-2014, 07:19 AM
Post: #8
gestrickland Posts: 7
Junior Member Joined: Feb 2014
RE: 48/50 Linear System Solver
I don't consider that the calculator is giving a "meaningless result", although it is certainly the case that the offered result does not satisfy the equation with zero or negligible residuals.
To start with a simple illustration, consider the two-dimensional case with the equations x - y = 1 and x - y = -1 . If you put this system into the linear solver you get the "solution" of x = 0, y =
0 . Obviously this does not solve the equations in the usual sense, but it, and any other point midway between the two lines defined by the equations, will give smaller residuals than other points.
Among all these midway points, the calculator chooses the one with the smallest distance from the origin.
In the three dimensional case differing pathologies are possible. In Tim's case it seems that the three planes defined by the three equations intersect in three parallel distinct lines. Since these
never meet in a single point, there is no exact solution. In order to have something a bit easier to think about, let us consider an analogous case where the three equations are y + z = 1, y = 0, z =
0 . Here we have three planes, and their three lines of intersection are parallel. If we put this system into the linear solver, we get the "solution" x = 0, y = 0.3333..., z = 0.3333... . Again,
this is not a solution in the usual sense, but it, and any other point with the same y and z values will give smaller residuals than all other points. Again, the calculator offers the one closest to
the origin.
I agree that it would be nice if the calculator gave warning for this kind of a situation. I don't know whether there might be cases where this type of "solution" might be of practical value, but it
is not meaningless.
02-17-2014, 08:18 AM
Post: #9
Werner Posts: 902
Senior Member Joined: Dec 2013
RE: 48/50 Linear System Solver
There are a few issues here..
for me, my flag settings are as follows:
-20 Clear (underflow -> 0)
-21 Set (Overflow -> error)
-22 Clear (Infinite -> error)
Then there's another flag that influences the outcome: -54 Use tiny element.
With flag -54 Clear (Tiny element ->0), the 'stack solve' (/) returns Infinite Result, and the interactive Linear Solver will return the minimum norm Least Squares solution, because the system is
In this mode, the determinant is returned as 0 exactly, because there's a check to see if the matrix contains integers only, and with flag -54 clear, the calculator knows the determinant must be
integer and will round the result to the nearest integer.
While solving systems, elements less than 1e-14 (relative size) will be set to zero.
With flag -54 Set (use tiny element), both methods will return
[ -5.41666..e14 -2.0833..e14 -1.25..e14]
The determinant now is calculated as -1.2e-13, due to roundoff. So the system is considered NOT singular, and a result can be produced.
In any case, it pays to compute the condition number (COND) that returns 9.8e15.
As a rule of thumb, the exponent signifies the number of wrong digits in your result, indicating here that the matrix is singular to working precision.
In any case
02-17-2014, 02:32 PM
Post: #10
Manolo Sobrino Posts: 179
Member Joined: Dec 2013
RE: 48/50 Linear System Solver
So that was the minimum norm solution! I assumed it was just garbage coming from the LU. I should have calculated it through (don't really do dynamical systems... didn't see that coming.)
I'm used to 20 clear, 21 clear, 22 check. With flag 54 checked I get for the determinant +1.2E-13, and the same solution as Werner. With flag 54 cleared, the determinant is zero and I get Tim's
Looks like the linear solver is a powerful beast indeed, although giving that kind of answer without a warning is kind of mean. If I asked for a Coke, I'm not expecting a Pepsi.
(Please, don't "the user should know better" me... I'm probably using a calculator to solve a 3x3 system because I'm in a hurry, if the system is singular just give it to me straight.)
01-14-2015, 05:35 PM
(This post was last modified: 01-14-2015 06:53 PM by Han.)
Post: #11
Han Posts: 1,882
Senior Member Joined: Dec 2013
RE: 48/50 Linear System Solver
This was a question that I stumbled upon while working on the solver an equation library program. Solve for x in Ax=b if:
Let A = [[12.25, 1.25, 4.5],[1.25,12.25,4.5],[4.5,4.5,3]]
Let b = [[-166.2],[157.2],[-3]]
Note that A is singular. On the HP48, with flag -22 SET (infinite -> MAXREAL), I get:
b A / -> x=[[-25.0333...],[4.3666...],[30]]
b A LSQ -> x=[[-14.972727...],[14.42727...],[-.1818...]]
The b A LSQ method gives a solution with smaller norm than the b A / method. I am able to compute the same solution using SVD decomposition on A and applying the pseudo-inverse.
OBJ-> DROP DROP @ get nonzero eigenvalues
{} + + INV @ turn into a list and compute reciprocals
0 + EVAL ->V3 { 3 3 } DIAG-> @reconstruct inverse 3x3 diagonal matrix
SWAP TRN SWAP * SWAP TRN * @ compute the pseudo inverse
b *
However, I am not sure how the HP48 got the answer for b A / ... Comparing the norms of the residuals, b A / gives a smaller norm for Ax-b, but clearly the norm of the solution is larger than that
from b A LSQ. So what algorithm does b A / apply? (Would really like to not have to break out Jazz and decompile some ROM routines just to figure out the algorithm...)
EDIT: Additionally, RSD according the the advanced user's guide is supposed to compute Ax-b. However, I seem to get different answers when I manually compute Ax-b as compared to using RSD. Is this
the case for anyone else? What am I overlooking?
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01-14-2015, 07:59 PM
(This post was last modified: 01-15-2015 11:17 AM by Gilles.)
Post: #12
Gilles Posts: 244
Member Joined: Oct 2014
RE: 48/50 Linear System Solver
By the way, in exact mode with the 50G :
'x - 5*y + 4*z = -3'
'2*x - 7*y + 3*z = -2'
'-2*x + y + 7*z =-1'
['x' 'y' 'z']
I sometimes used SOLVE instead of LINSOLVE. As far I know It's not documented, but works in most cases even with things like :
'x^2- 5*y - 4*z = 3'
'2*x - 7*y + 3*z = -2'
'-2*x + y + 7*z =-1'
['x' 'y' 'z']
{ [ 'x=-((-37+2*√911)/26)' 'y=-((-513+20*√911)/676)' 'z=-((-105+12*√911)/676)' ] [ 'x=(37+2*√911)/26' 'y=(513+20*√911)/676' 'z=(105+12*√911)/676' ] }
01-15-2015, 12:17 PM
Post: #13
Werner Posts: 902
Senior Member Joined: Dec 2013
RE: 48/50 Linear System Solver
(01-14-2015 05:35 PM)Han Wrote: Let A = [[12.25, 1.25, 4.5],[1.25,12.25,4.5],[4.5,4.5,3]]
Let b = [[-166.2],[157.2],[-3]]
Note that A is singular. On the HP48, with flag -22 SET (infinite -> MAXREAL), I get:
b A / -> x=[[-25.0333...],[4.3666...],[30]]
b A LSQ -> x=[[-14.972727...],[14.42727...],[-.1818...]]
As I said before:
with flag -54 SET, both b A / and the linear solver application return x=[[-25.0333...],[4.3666...],[30]]. In this case, the 48 does not know A is singular - the computed determinant is 1.485e-12,
and a result can be calculated using the normal LU decomposition and subsequent solve. Of course, COND returns 3.e15 meaning the result is meaningless.
with flag -54 CLEAR, b A / returns Infinite Result, and the solver app returns the Least Squares solution.
Cheers, Werner
01-15-2015, 05:35 PM
(This post was last modified: 01-15-2015 06:32 PM by Han.)
Post: #14
Han Posts: 1,882
Senior Member Joined: Dec 2013
RE: 48/50 Linear System Solver
(01-15-2015 12:17 PM)Werner Wrote: As I said before:
with flag -54 SET, both b A / and the linear solver application return x=[[-25.0333...],[4.3666...],[30]]. In this case, the 48 does not know A is singular - the computed determinant is
1.485e-12, and a result can be calculated using the normal LU decomposition and subsequent solve. Of course, COND returns 3.e15 meaning the result is meaningless.
with flag -54 CLEAR, b A / returns Infinite Result, and the solver app returns the Least Squares solution.
Cheers, Werner
Thank you, Werner. I don't know how I glossed over your original post and didn't see your comments -- probably had tunnel vision and too focused on something else. Your explanation is very helpful.
Edit: So I did the LU decomposition of the matrix, and got a different solution: [[84.9666...],[114.3666...],[-300]]. In original example posted by Tim:
Quote:With flag -54 Set (use tiny element), both methods will return
[ -5.41666..e14 -2.0833..e14 -1.25..e14]
Using LU, I got the same result. However, for my particular example, LU doesn't give the same answer as the / operator (under the same flag settings) which suggests that the HP48 doesn't seem to
always use LU.
Just to be clear, I'm not so much after meaningful solutions as I am about understanding how the HP48 solves linear systems using the / operator. So far I've tried QR, LU, and pseudo-inverse (least
squares) and none match. QR gives [ 139.27, -181.42, 5.43 ], LU gives something with large norm (useless answer), and least squares I already listed.
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01-16-2015, 09:52 AM
Post: #15
Werner Posts: 902
Senior Member Joined: Dec 2013
RE: 48/50 Linear System Solver
(01-15-2015 05:35 PM)Han Wrote: Edit: So I did the LU decomposition of the matrix, and got a different solution: [[84.9666...],[114.3666...],[-300]].
How did you arrive at this?
And the '/' operator does LU-decomposition.
01-16-2015, 01:44 PM
Post: #16
Han Posts: 1,882
Senior Member Joined: Dec 2013
RE: 48/50 Linear System Solver
(01-16-2015 09:52 AM)Werner Wrote:
(01-15-2015 05:35 PM)Han Wrote: Edit: So I did the LU decomposition of the matrix, and got a different solution: [[84.9666...],[114.3666...],[-300]].
How did you arrive at this?
And the '/' operator does LU-decomposition.
My stack was:
2: [[-166.2],[157.2],[-3]]
1: [[12.25,1.25,4.5],[1.25,12.25,4.5],[4.5,4.5,3]]
I typed in the following in the command line; remove the comments after the @'s
LU @ b L U P
4 ROLL * @ L U P*b
ROT INV SWAP * @ solve Ly=P * b by y=L^-1 * P *b
SWAP INV SWAP * @ solve Ux = y by x = U^-1 y
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01-16-2015, 02:15 PM
Post: #17
Werner Posts: 902
Senior Member Joined: Dec 2013
RE: 48/50 Linear System Solver
Ah OK.
Two remarks here:
- the condition number of the matrix is 3e15, meaning that very small perturbations in the input lead to huge changes in the calculated solution - exactly what we see here.
- And then, to solve Ly = Pb or Uy=x, simply use / instead of INV SWAP *. Faster and more accurate, but of course not in this case.
(now you get [-1621.7 -1592.3 4820 ], equally meaningless, but that's what the condition number tells you)
Calculating Ax=b with / vs with LU and then manually doing the triangular solves makes a difference: the former uses 15 digits throughout, while the second has the intermediate amounts that make up L
and U rounded to 12 digits. And that small difference is enough to completely change the calculated solution.
01-16-2015, 04:18 PM
Post: #18
Han Posts: 1,882
Senior Member Joined: Dec 2013
RE: 48/50 Linear System Solver
(01-16-2015 02:15 PM)Werner Wrote: Ah OK.
Two remarks here:
- the condition number of the matrix is 3e15, meaning that very small perturbations in the input lead to huge changes in the calculated solution - exactly what we see here.
- And then, to solve Ly = Pb or Uy=x, simply use / instead of INV SWAP *. Faster and more accurate, but of course not in this case.
(now you get [-1621.7 -1592.3 4820 ], equally meaningless, but that's what the condition number tells you)
Calculating Ax=b with / vs with LU and then manually doing the triangular solves makes a difference: the former uses 15 digits throughout, while the second has the intermediate amounts that make
up L and U rounded to 12 digits. And that small difference is enough to completely change the calculated solution.
Aha! Thank you so much for your patience, Werner!
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Automated hypothesis testing
Automated hypothesis testing
A method of automatically applying a hypothesis test to a data set. The method reduces errors made in failing to appreciate predicate assumptions of various statistical tests, and elicits a series of
indications from the user regarding characteristics of interest embodied by the data set to select an appropriate statistical test. The system also reduces errors in constructing competing null and
alternative hypothesis statements by generating a characterization of the data and defining null and alternative hypotheses according to the indications, selected statistical test, and conventions
adopted with respect to the tests. The system also establishes a significance level, calculates the test statistic, and generates an output. The output of the system provides a plain interpretation
of the quantitative results in the terms indicated by the user to reduce errors in interpretation of the conclusion.
The invention relates to methods and systems for using statistical analysis tools, and more particularly to methods and systems for automatically constructing and interpreting hypothesis tests.
Statistical tests provide a mechanism for making quantitative conclusions about characteristics or behavior of a process as represented by a sample of data drawn from the process. Statistical tests
also are used to compare characteristics or behaviors of two or more processes based on respective data sets or samples drawn from the processes.
The term “hypothesis testing” describes a broad topic within the field of statistical analysis. Hypothesis testing entails particular methodologies using statistical tests to calculate the likely
validity of a claim made about a population under study based on observed data. The claim, or theory, to which the statistical testing is applied is called a “hypothesis” or “hypothesis statement”,
and the data set or sample under study usually represents a sampling of data reflecting an input to, or output of, a process. A well-constructed hypothesis statement specifies a certain
characteristic or parameter of the process. Typical process characteristics used in hypothesis testing include statistically meaningful parameters such as the average or mean output of a process
(sometimes also referred to as the “location” of the process) and/or the dispersion/spread or variance of the process.
When constructing a hypothesis test, a hypothesis statement is defined to describe a process condition of interest that, for the purpose of the test, is alleged to be true. This initial statement is
referred to as the “null hypothesis” and is often denoted algebraically by the symbol H[0]. Typically the null hypothesis is a logical statement describing the putative condition of a process in
terms of a statistically meaningful parameter. For example, consider an example of hypothesis testing as applied to the discharge/output of a wastewater treatment process. Assume there are concerns
that the process recently has changed such that the output is averaging a higher level of contaminants than the historical (and acceptable) output of 5 parts of contaminant per million (ppm). A null
hypothesis based on this data could be stated as follows: the level of contaminants in the output of the process has a mean value equal to or greater than 5 ppm. The null hypothesis is stated in
terms of a meaningful statistical parameter, i.e., process mean, and in terms of the process of interest, i.e., the level of contaminants in the process output.
Likewise, hypothesis testing also entails constructing an alternative hypothesis statement regarding the process behavior or condition. For the purpose of the test, the status of the alternative
hypothesis statement is presumed to be uncertain, and is denoted by the symbol H[1]. An alternative hypothesis statement defines an uncertain condition or result in terms of the same statistical
parameter as the null hypothesis, e.g., process mean, in the case of the wastewater treatment example. In that example, an alternative hypothesis statement would be defined along the following lines:
the level of contaminants in the output of the process has a mean value of less than 5 ppm. In constructing null and alternative hypotheses, it is imperative that the statements be stated in terms
that are mutually exclusive and exhaustive, i.e., such that there is neither overlap in possible results nor an unaccounted for or “lurking” hypothesis.
One object in applying hypothesis testing is to see if there is sufficient statistical evidence (data) to reject a presumed null hypothesis H[0 ]in favor of an alternative hypothesis H[1]. Such a
rejection would be appropriate under circumstances wherein the null hypothesis statement is inconsistent with the characteristics of the sampled data. In the alternative, in the event the data are
not inconsistent with the statement made by the null hypothesis, then the test result is a failure to reject the null hypothesis—meaning the data sampling and testing does not provide a reason to
believe any statement other than the null hypothesis. In short, application of a hypothesis test results in a statistical decision based on sampled data, and results either in a rejection of the null
hypothesis H[0], which leaves a conclusion in favor of the alternative H[1], or a failure to reject the null hypothesis H[0], which leaves a conclusion wherein the null hypothesis cannot be found
false based on the sampled data.
Any Hypothesis Test can be conducted by following the four steps outlined below:
Step 1—State the null and alternative hypotheses. This step entails generating a hypothesis of interest that can be tested against an alternative hypothesis. The competing statements must be mutually
exclusive and exhaustive.
Step 2—State the decision criteria. This step entails articulating the factors upon which a decision to reject or fail to reject the null hypothesis will be based. Establishing appropriate decision
criteria depends on the nature of the null and alternative hypotheses and the underlying data. Typical decision criteria include a choice of a test statistic and significance level (denoted
algebraically as “alpha” α) to be applied to the analysis. Many different test statistics can be used in hypothesis testing, including use of a standard or test value associated with the process
data, e.g., the process mean or variance, and/or test values associated with the differences between two processes, e.g., differences between proportions/means/medians, ratios of variances and the
like. The significance level reflects the degree of confidence desired when drawing conclusions based on the comparison of the test statistic to the reference statistic.
Step 3—Collect data relating to the null hypothesis and calculate the test statistic. At this step, data is collected through sampling and the relevant test statistic is calculated using the sampled
Step 4—State a conclusion. At this step, the appropriate test statistic is compared to its corresponding reference statistic (based on the null distribution) which shows how the test statistic would
be distributed if the null hypothesis were true. Generally speaking, a conclusion can be properly drawn from the resultant value of the test statistic in one of several different ways: by comparing
the test statistic to the predetermined cut-off values, which were established in Step 2; by calculating the so-called “p-value” and comparing it to the predetermine significance level a alpha; or by
computing confidence intervals. The p-value is quantitative assessment of the probability of observing a value of the test statistic that is either as extreme as or more extreme than the calculated
value of the test statistic, purely by random chance, under the assumption that the null hypothesis is true.
There are several different forms of statistical tests that are useful in hypothesis testing. Those of skill in the art will understand how tests such as t-tests, Z-tests and F-tests can be used for
hypothesis testing by way of the above methodology, but each may be appropriate only if a variety of predicates are found. In particular, the applicability of a particular test depends on, among
other things, the nature of the hypothesis statements, the nature of the data available, and assumptions relating to the distributions and sampling of the data. For example, sometimes the hypotheses
under consideration entail a comparison of statistical means, a comparison of variances, or a comparison of proportions. Similarly, the data may be either attribute data or variable/continuous data.
With respect to assumptions of the distributions and sampling of data, different statistical tests are appropriate depending upon, for example, whether the sample sizes are large or small, time
ordered or not, paired samples or not, or whether variances of the samples are know or not. The selection of an appropriate test for a particular set of predicates is imperative because application
of an inappropriate test can result in unfounded or erroneous conclusions, which in turn lead to faulty decisions.
The proper construction of the null and alternative hypotheses also requires an understanding of the statistical test and its underlying assumptions. In addition, it is imperative that the null and
alternative hypotheses be constructed so as to be mutually exclusive and exhaustive. Moreover, it is sometimes difficult for the practitioner to construct a meaningful set of competing null and
hypothesis statements in terms of the process and data of interest.
Interpreting the conclusions of a hypothesis test can also be difficult, even when the appropriate test is selected, an appropriate null hypothesis is subjected to the test statistic, and the test
statistic is accurately calculated. This difficulty in interpretation can arise if the results of the test are not expressed in terms that relate the quantitative analysis to the terms used in
describing the process, or if the basis for the conclusion is not clear. Indeed, in some cases, the construction of a null hypothesis and the associated data analysis results in an appropriate (but
counterintuitive) conclusion that the null should not be rejected due to the absence of a data-supported basis that the null hypothesis is false. The logic underlying the conclusion is sound, but
often misunderstood.
Hypothesis testing thus has the potential to bring powerful tools to bear on the understanding of complex process behaviors, particularly processes that behave in a manner that is not intuitive.
Hypothesis testing brings the power and focus of data-driven analysis to decision making, which sometimes can be lead astray by the complexities of the process of interest or biases of the decision
maker. However, despite the power and usefulness of hypothesis testing, it remains a difficult tool to apply. One of the difficulties often encountered in applying hypothesis tests is the fact that
each statistical test depends on multiple predicates or assumptions for validity. Applying a test statistic to a data set that does not embody the predicate assumptions can result in conclusions that
are unsupported by the data, yet are not obviously so. Consequently, it is possible to make unfounded decisions in error.
Another problem with the application of hypothesis testing is the somewhat counter-intuitive requirement that the null hypothesis be stated and then the conclusion be drawn so as to either reject or
fail to reject the null hypothesis (rather than merely accepting the null hypothesis). This difficulty is common for a variety of reasons, among them the requirements that the statements be mutually
exclusive and exhaustive, the statements be posed in terms of a statistically meaningful parameter that is appropriate in view of the process data to be sampled, and the statements should be stated
in terms that will provide meaningful insight to the process, i.e., will be useful in making a decision based on the data.
In this regard, while it is important to state the hypothesis test in terms of the problem, it is equally important (and perhaps more important) to interpret the conclusions of the test in practical
terms. Whether the test statistic supports either rejection or failure to reject the null hypothesis, the result needs to be correctly stated and understood in practical terms so that the results of
the test can guide decisions pertaining to the process or processes.
Accordingly, the invention provides, in one embodiment, a method of automatically applying hypothesis testing to a data set. The method provides a plurality of statistical tests and, through a series
of queries and indications, the method assures that multiple predicates or assumptions for validity of each statistical test are affirmatively considered. By confirming the assumptions and providing
confirmatory notifications relating to the implications of the queries and indications, the method assures application of a statistical test to the data appropriate for the predicate assumptions of
the test.
In another embodiment, the invention provides a method of automatically applying hypothesis testing to a data set including generating definitions of the null and alternative hypotheses in terms of
the a statistical test, its underlying assumptions, so as to be mutually exclusive and exhaustive, and in terms indicated by the user as being descriptive of the processes and data of interest.
In yet another embodiment, the invention provides a method of automatically applying hypothesis testing to a data set including generating test conclusions expressed in terms relating the
quantitative analysis to terms indicated as describing the process, and providing the basis for conclusions in terms describing the process.
Other features and advantages of the invention will become apparent to those skilled in the art upon review of the following detailed description, claims, and drawings.
FIG. 1 is a schematic diagram of a computer system for implementing a software program embodying the invention.
FIG. 2 is a schematic diagram of a logic tree for implementing the hypothesis testing effected by the software program.
FIG. 3 illustrates a user interface generated by the software.
FIG. 4 illustrates a user interface generated by the software.
FIG. 5 illustrates a user interface generated by the software.
FIG. 6 illustrates a user interface generated by the software.
FIG. 7 illustrates a user interface generated by the software.
FIG. 8 illustrates a user interface generated by the software.
FIG. 9 illustrates a user interface generated by the software.
FIG. 10 illustrates a user interface generated by the software.
FIG. 11 illustrates a user interface generated by the software.
FIG. 12 illustrates a user interface generated by the software.
FIG. 13 illustrates a user interface generated by the software.
FIG. 14 illustrates a user interface generated by the software.
FIG. 15 is similar to FIG. 2 and schematically illustrates a second set of statistical tests that can be performed using the software.
Before any aspects of the invention are explained in detail, it is to be understood that the invention is not limited in its application to the details of construction and the arrangement of
components set forth in the following description or illustrated in the following drawings. The invention is capable of other embodiments and of being practiced or of being carried out in various
ways. Also, it is to be understood that the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of “including,” “comprising,” or
“having” and variations thereof herein is meant to encompass the items listed thereafter and equivalents thereof as well as additional items. The terms “connected,” “coupled,” and “mounted” and
variations thereof herein are used broadly and, unless otherwise stated, encompass both direct and indirect connections, couplings, and mountings. In addition, the terms connected and coupled and
variations thereof herein are not restricted to physical and mechanical connections or couplings. As used herein the term “computer” is not limited to a device with a single processor, but may
encompass multiple computers linked in a system, computers with multiple processors, special purpose devices, computers or special purpose devices with various peripherals and input and output
devices, software acting as a computer or server, and combinations of the above. In general, computers accept and process information or data according to instructions (i.e., computer instructions).
The drawings illustrate a system for automatically applying hypothesis testing to one or more data sets having a variety of statistically significant characteristics. Specifically, with reference
initially to FIG. 1, the system includes a general purpose computer 10. The computer 10 provides a platform for operating a software program that applies hypothesis testing to one or more data sets.
In the system identified, data and program files are input to the computer 10, which reads the files and executes the programs therein. Some of the elements of the computer 10 include a processor 12
having an input/output (IO) section 14, a central processing unit (CPU) 16, and a memory module 18. In one form, the software program for applying hypothesis testing is loaded into memory 18 and/or
stored on a configured CD ROM (not shown) or other storage device (not shown). The IO section 14 is connected to keyboard 20 and an optional user input device or mouse 22. The keyboard 20 and mouse
22 enable the user to control the computer 10. IO section 14 is also connected to monitor 24. In operation, computer 10 generates the user interfaces identified in FIGS. 3-14 and displays those user
interfaces on monitor 24. The computer also includes CD ROM drive 26 and data storage unit 28 connected to 10 section 14. In some embodiments, the software program for effecting hypothesis testing
may reside on storage unit 28 or in memory unit 18 rather than being accessed through the CD ROM drive using a CD ROM. Alternatively, CD ROM drive 26 may be replaced or supplemented by a floppy drive
unit, a tape drive unit, or other data storage device. The computer 10 also includes a network interface 30 connected to 10 section 14. The network interface 30 can be used to connect the computer 10
to a local area network (LAN), wide are network (WAN), internet based portal, or other network 32. Any suitable interface can suffice, including both wired and wireless interfaces. Thus, the software
may be accessed and run locally as from CD ROM drive 26, data storage device 28, or memory 18, or may be remotely accessed through network interface 30. In the networked embodiment, the software
would be stored remote from the computer 10 on a server or other appropriate hardware platform or storage device.
The software program provides algorithms relating to a plurality of statistical tests that can be applied under a variety of circumstances to the data sets. For example, the illustrated system
provides the following statistical tests: one proportion Z-test, one proportion binomial test, two proportion Z-test, multi proportion Chi-square test, one mean Z-test, one mean t-test, two means
Z-test, two sample t-test; F-test Anova, Chi square test, and an F-ratio test. FIG. 14 illustrates a second set of statistical tests, known as parametric tests: One Sample Sign Test, Paired Samples
Sign Test, One Sample Wilcoxon Signed Ranks Test, Paired Samples Wilcoxon Signed Ranks Test, Mann Whitney Wilcoxon Test, Kruskal Wallis Test, and The Friedman Test. The system of course could include
other statistical tests useful for hypothesis testing. These statistical tests are useful when applied to data embodying various characteristics. For example, some of the tests are useful when
applied to attribute data while others are not. Similarly, some of the tests are useful when applied to data wherein the mean or location of the process from which the data is drawn is known, and
others are not. Applying a test to a data set without understanding the assumptions underlying the test can generate erroneous or unfounded results.
The system also establishes conventions associated with each test of the plurality of tests. Among the conventions incorporated in the system is the convention of stating the null hypothesis as an
equality for those tests wherein such a logical statement is appropriate. For example, the system avoids stating the null as being “greater than or equal to” a reference value. Another convention
adopted by the system is to state the alternative hypothesis statement as an inequality, which logically follows from the convention of defining the null hypothesis.
The system automatically determines the appropriate statistical test. The determination of the appropriate statistical test is made automatically in response to indications or choices made by the
user in response to queries or prompts generated by the system. The system design follows a logic map that forces the user to confront and affirm choices regarding the data and information available
to the user seeking to apply hypothesis testing to the data. Not only does the system drive the user to application of the correct test, but it also informs the user of the implications of the
choices and consequences of making inappropriate indications.
Initially, the system provides this determination process by seeking an indication as to whether the data set the user seeks to assess is time ordered. In response to this indication, the system
generates a confirmatory notification explaining the importance in hypothesis testing of process stability. More particularly, assuming the data subjected to the hypothesis testing is randomly drawn
from a process of interest, it is imperative that the process be stable. Otherwise, the results drawn from the hypothesis test are not meaningful.
The test determining step also includes seeking an indication of the nature of data as being attribute data or continuous data, again because some of the statistical tests are useful with attribute
data and some are not. In the illustrated system, if the indication is that the data are attribute data, then the system further seeks an indication as to the number of samples from which the data is
drawn, an indication of sample size, and seeks an indication of normality of the data. Likewise, in response to an indication that the data are continuous, the system then seeks an indication as to
the number of samples from which the data are drawn, an indication of sample size and seeks an indication as to whether the data are normal, not normal, or if normalcy is unknown.
The system, in determining the test, responds to the indications of normality. If the indication is that the data are either not normal or the normality of the data is unknown, then the system
provides a confirmatory notification either to use a normality test to determine normality, to use non-parametric tests or to use data transformation functions.
Determining the test also includes identifying a statistical parameter of interest. Identifying a statistical parameter of interest includes selecting a parameter of interest from among the following
commons statistical parameters: proportion, mean, median, and variance of the data.
Determining the test also includes seeking an indication of whether, depending on the number of samples indicated, the data sample includes either paired data or differences between paired data
samples. Likewise, if the parameter of interest is indicated as being the mean, then the system also seeks an indication of whether variance of population is known.
Ultimately, the system automatically selects the appropriate statistical test from among the plurality of tests based on the indications and established conventions, and further provides a
confirmatory notification of the nature of the selected test, the indications and established conventions.
The system also automatically characterizes the data set by establishing test criteria, selecting an appropriate reference test value depending on the test selected; and eliciting an indication of a
description of the data of interest. Specifically, the system prompts the user to identify values for the statistic of interest, e.g., proportion, variance or mean. The system confirms the value and
will prompt the user if inappropriate values are indicated. For example, the system will advise the user that the value of a population proportion must lie between zero and one. Likewise, the system
prompts the user to provide descriptions of the data, e.g., names for the methods or treatments subjected to the hypothesis test. As described below, these indications, provided in the user's own
language or terms, are used in confirmatory notifications, construction of the null and alternative hypotheses, and in explaining and interpreting conclusions drawn from the hypothesis test.
The system also automatically constructs the null and alternative hypothesis statements based in large part on the test selected, the data characterizations as indicated by the user, and according to
the various conventions associated with the tests and data. Defining the null hypothesis includes generating a confirmation of the indications made by the user and the implications of the chosen
fields. The system also provides a confirmatory notification of the null hypothesis statement. In one embodiment, the null hypothesis statement is made in terms of an equality. The system likewise
automatically constructs the alternative hypothesis statement based on the selected test and assumed conventions relating to the selected test and indications of the test criteria and population
description. The system provides a confirmatory notification of the alternative hypothesis statement and the implications of the choices made by the user.
The system also seeks an indication of the desired significance level to be applied to the hypothesis test, and describes the implications of the choice of significance level in hypothesis testing.
The system then automatically conducts the selected test and generates an output. Preferably, the output is in graphical and numeric form, and includes text using the terms provided by the user in
describing the data.
In this regard, the system generates an output including calculations of the values of the test statistic, calculating cut-off values, confidence intervals, and calculating p-values; comparing the
calculated p-value to the indicated significance level, comparing the value of the test statistic to one or more of the reference values, the cut-off values or confidence intervals in view of the
null hypothesis statement. The system formulates and expresses the conclusion in terms of the selected test, the indicated test criteria and population descriptions, in terms indicated by the user,
as to whether to reject the null hypothesis or not to reject the null hypothesis, and also states the basis for the conclusion. By using the terms supplied by the user and explaining the conclusion
using both the indicated terms and the automatically calculated values of the test statistic, the system provides a tool for using hypothesis testing that reduces the likelihood or errors occurring
though misunderstanding predicate assumptions of the tests, flawed null/alternative hypothesis statements and misinterpretation of the test results.
The system is preferably in the form of computer-readable modules capable of providing the functionalities of the system. Those of skill in the art will also readily recognize that the description
and disclosure of the system herein also describes and discloses a method for automatically applying hypothesis testing to a data set. While there are many possible embodiments of the software
program, one commercially available embodiment is the Engine Room® data analysis software provided by Moresteam.com, and which can be purchased online at www.moresteam.com/engineroom.
Various other features and advantages of the invention are set forth in the following claims.
1. A method of automatically applying hypothesis testing to at least one data set, the method comprising:
A. selecting a plurality of statistical tests applicable to the at least one data set having a variety of characteristics and establishing conventions associated with each test of the plurality
of tests; including establishing the convention of stating the null as an equality and the alternative hypothesis statement as an inequality, the plurality of tests including at least one of one
proportion Z-test, one proportion binomial test, two proportion Z-test, multi proportion Chi-square test, one mean Z-test, one mean t-test, two means Z-test, two sample t-test; F-test Anova, Chi
square test, and F-ratio test;
B. determining the test, seeking an indication as to the time ordered nature of the data set, generating a confirmatory notification of process stability, seeking an indication of the nature of
data as being attribute data or continuous data; if the indication is that the data are attribute data, then seeking an indication as to the number of samples from which the data are drawn,
seeking an indication of sample size, seeking an indication of normality of the data (based on large sample theory); if the indication is that the data are continuous, then seeking an indication
as to the number of samples from which the data are drawn, seeking an indication of sample size and seeking an indication as to whether the data are normal, not normal, or if the distribution is
unknown, if the indication is that the data are either not normal or unknown, then prompting confirmatory notifications either to use a normality test to determine normality, to use
non-parametric tests or to use data transformation functions, identifying a statistical parameter of interest, including selecting the parameter of interest from among the proportion, mean, and
variance of the data, seeking an indication of whether, depending on the number of samples indicated, the data sample includes either paired data or differences between paired data samples, if
the parameter of interest is indicated as being the mean, then seeking an indication of whether or not variance of population is known, and selecting a test from among the plurality of tests
based on the indications and established conventions, providing a confirmatory notification of the nature of the selected test, the indications and established conventions;
C. characterizing the data set, including establishing test criteria, selecting an appropriate reference test value depending on the test selected; and eliciting an indication of a description of
the data of interest;
D. constructing null and alternative hypothesis statements, including defining the null hypothesis based on the selected test and assumed conventions relating to the selected test and the
indications of the test criteria and population description, providing a confirmatory notification of the null hypothesis statement, defining an inequality statement for the alternative
hypothesis based on the selected test and assumed conventions relating to the selected test and indications of the test criteria and population description, providing a confirmatory notification
of the alternative hypothesis statement;
E. selecting a significance level,
F. conducting the selected test; and
G. stating a conclusion, including calculating the values of the test statistic, calculating cut-off values, confidence intervals, and calculating p-values; comparing the calculated p-value to
the indicated significance level, comparing the value of the test statistic to one or more reference values, the cut-off values or confidence intervals in view of the null hypothesis statement;
and stating a conclusion in terms of the selected test, indicated test criteria and population descriptions whether to reject the null hypothesis or not to reject the null hypothesis based on the
comparing step, and stating the basis for the conclusion using the results of the comparing step.
2. Software for automatically applying hypothesis testing to at least one data set, the software including computer readable modules for:
A. selecting a plurality of statistical tests applicable to the at least one data set having a variety of characteristics and establishing conventions associated with each test of the plurality
of tests; including establishing the convention of stating the null as an equality and the alternative hypothesis statement as an inequality, the plurality of tests including at least one of one
proportion Z-test, one proportion binomial test, two proportion Z-test, multi proportion Chi-square test, one mean Z-test, one mean t-test, two means Z-test, two sample t-test; F-test Anova, Chi
square test, F-ratio test, One Sample Sign Test, Paired Samples Sign Test, One Sample Wilcoxon Signed Ranks Test, Paired Samples Wilcoxon Signed Ranks Test, Mann Whitney Wilcoxon Test, Kruskal
Wallis Test, and The Friedman Test;
B. determining the test, seeking an indication as to the time ordered nature of the data set, generating a confirmatory notification of process stability, seeking an indication of the nature of
data as being attribute data or continuous data, if the indication is that the data are attribute data, then seeking an indication as to the number of samples from which the data are drawn,
seeking an indication of sample size, seeking an indication of normality of the data; if the indication is that the data are continuous, then seeking an indication as to the number of samples
from which the data are drawn, seeking an indication of sample size and seeking an indication as to whether the data are normal, not normal, or if normalcy is unknown, if the indication is that
the data is either not normal or unknown, then prompting confirmatory notifications either to use a normality test to determine normality, to use non-parametric tests or to use data
transformation functions, identifying a statistical parameter of interest, including selecting the parameter of interest from among the proportion, mean, and variance of the data, seeking an
indication of whether, depending on the number of samples indicated, the data sample includes either paired data or differences between paired data samples, if the parameter of interest is
indicated as being the mean, then seeking an indication of whether variance of population is known, and selecting a test from among the plurality of tests based on the indications and established
conventions, providing a confirmatory notification of the nature of the selected test, the indications and established conventions;
C. characterizing the data set, including establishing test criteria, selecting an appropriate reference test value depending on the test selected; and eliciting an indication of a description of
the data of interest;
D. constructing null and alternative hypothesis statements, including defining the null hypothesis based on the selected test and assumed conventions relating to the selected test and the
indications of the test criteria and population description, providing a confirmatory notification of the null hypothesis statement, defining an inequality statement for the alternative
hypothesis based on the selected test and assumed conventions relating to the selected test and indications of the test criteria and population description, providing a confirmatory notification
of the alternative hypothesis statement;
E. seeking an indication of confidence lever;
F. conducting the selected test; and
G. stating a conclusion, including calculating the values of the test statistic, calculating cut-off values, confidential intervals, and calculating p-values; comparing the calculated p-value to
the indicated significance level, comparing the value of the test statistic to one or more of the reference values, the cut-off values or confidence intervals in view of the null hypothesis
statement; and stating a conclusion in terms of the selected test, indicated test criteria and population descriptions whether to reject the null hypothesis or not to reject the null hypothesis
based on the comparing step, and stating the basis for the conclusion using the results of the comparing step.
3. A method for conducting a hypothesis test to at least one data set, the method comprising:
A. selecting a plurality of statistical tests applicable to the at least one data set having a variety of characteristics and establishing conventions associated with each test of the plurality
of tests;
B. determining the appropriate test including seeking indications regarding the data set and the plurality of statistical tests;
C. characterizing the data set, including establishing test criteria, selecting an appropriate reference test value depending on the test selected; and eliciting an indication of a description of
the data of interest;
D. constructing null and alternative hypothesis statements, including defining the null hypothesis based on the selected test and assumed conventions relating to the selected test and the
indications of the test criteria and population description, providing a confirmatory notification of the null hypothesis statement, defining an inequality statement for the alternative
hypothesis based on the selected test and assumed conventions relating to the selected test and indications of the test criteria and population description, providing a confirmatory notification
of the alternative hypothesis statement;
E. seeking an indication of a significance level;
F. conducting the selected test; and
G. stating a conclusion, including calculating the values of the test statistic, calculating cut-off values, confidential intervals, and calculating p-values; comparing the calculated p-value to
the indicated significance level, comparing the value of the test statistic to one or more of the reference values, the cut-off values or confidence intervals in view of the null hypothesis
statement; and stating a conclusion in terms of the selected test, indicated test criteria and population descriptions whether to reject the null hypothesis or not to reject the null hypothesis
based on the comparing step, and stating the basis for the conclusion using the results of the comparing step and the population descriptions.
4. The method set forth in claim 3 wherein establishing conventions includes stating the null as an equality and the alternative hypothesis statement as an inequality.
5. The method set forth in claim 3 wherein the plurality of tests includes at least one of: one proportion Z-test, one proportion binomial test, two proportion Z-test, multi proportion Chi-square
test, one mean Z-test, one mean t-test, two means Z-test, two sample t-test; F-test Anova, Chi square test, F-ratio test, One Sample Sign Test, Paired Samples Sign Test, One Sample Wilcoxon Signed
Ranks Test, Paired Samples Wilcoxon Signed Ranks Test, Mann Whitney Wilcoxon Test, Kruskal Wallis Test, and The Friedman Test.
6. The method set forth in claim 3 wherein selecting the test includes seeking an indication as to the time ordered nature of the data set.
7. The method set forth in claim 3 wherein selecting the test includes generating a confirmatory notification of process stability.
8. The method set forth in claim 3 wherein selecting the test includes seeking an indication of the nature of data as being attribute data or continuous data.
9. The method set forth in claim 8 wherein selecting the test includes, if the indication is that the data are attribute data, seeking an indication as to the number of samples from which the data
are drawn.
10. The method set forth in claim 9 wherein selecting the test includes seeking an indication of sample size.
11. The method set forth in claim 10 wherein selecting the test includes seeking an indication of normality of the data.
12. The method set forth in claim 8 wherein selecting the test includes, if the indication is that the data is continuous, seeking an indication as to the number of samples from which the data are
13. The method set forth in claim 12 wherein selecting the test includes seeking an indication of sample size.
14. The method set forth in claim 13 wherein selecting the test includes seeking an indication as to whether the data are normal, not normal, or if normalcy is unknown.
15. The method set forth in claim 13 wherein selecting the test includes, if the indication is that the data is either not normal or unknown, then prompting confirmatory notifications either to use a
normality test to determine normality, to use non-parametric tests or to use data transformation functions.
16. The method set forth in claim 3 wherein selecting the test includes identifying a statistical parameter of interest.
17. The method set forth in claim 16 wherein selecting the test includes selecting the parameter of interest from among the proportion, mean, and variance of the data.
18. The method set forth in claim 3 wherein selecting the test includes seeking an indication of samples from which the data are drawn and seeking an indication of whether, depending on the number of
samples indicated, the data sample includes either paired data or differences between paired data samples.
19. The method set forth in claim 16 wherein selecting the test includes, if the parameter of interest is indicated as being the mean, seeking an indication of whether variance of population is
20. The method set forth in claim 3 wherein selecting the test includes selecting a test from among the plurality of tests based on elicited indications and established conventions, and providing a
confirmatory notification of the nature of the selected test, the indications and established conventions.
Patent History
Publication number
: 20070239361
: Apr 11, 2006
Publication Date
: Oct 11, 2007
Patent Grant number
7725291 Inventor
William Hathaway
(Powell, OH)
Application Number
: 11/401,555
Current U.S. Class: 702/19.000
International Classification: G06F 19/00 (20060101); | {"url":"https://patents.justia.com/patent/20070239361","timestamp":"2024-11-10T06:57:53Z","content_type":"text/html","content_length":"100512","record_id":"<urn:uuid:58dfe634-e97e-4d44-b416-df2b0d0cd2bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00827.warc.gz"} |
St Patrick's School
St Patrick's School
Mathematics – The Australian Curriculum
Download a copy of The Australian Curriculum – Mathematics document.
Learning mathematics creates opportunities for and enriches the lives of all Australians. The Australian Curriculum: Mathematics provides students with essential mathematical skills and knowledge in
Number and Algebra, Measurement and Geometry, and Statistics and Probability. It develops the numeracy capabilities that all students need in their personal, work and civic life, and provides the
fundamentals on which mathematical specialties and professional applications of mathematics are built.
At St Patrick’s, we use i-Maths to develop students mathematical understanding, logical reasoning and problem-solving skills. Teachers at St Patrick’s connect the students Mathematical learning to
real-world situations/problems during their Mathematics lesson and as well as in other learning areas, such as in history, students need to be able to imagine timelines and time frames to reconcile
related events.
Teachers use a combination of teaching strategies to teach Mathematical concepts to students, such as concrete materials and manipulatives, digital technologies and interactive applications.
Differentiation is at the forefront of our teachers practice, they know the curriculum content and enable all children to represent their learning, which can be in different ways.
The content strands are Number and Algebra, Measurement and Geometry, and Statistics and Probability. They describe what is to be taught and learnt.
The proficiency strands are Understanding, Fluency, Problem Solving, and Reasoning. They describe how content is explored or developed, that is, the thinking and doing of mathematics. They provide
the language to build in the developmental aspects of the learning of mathematics and have been incorporated into the content descriptions of the three content strands described above. This approach
has been adopted to ensure students’ proficiency in mathematical skills develops throughout the curriculum and becomes increasingly sophisticated over the years of schooling.
Content Strands
Number and Algebra
Number and Algebra are developed together, as each enriches the study of the other. Students apply number sense and strategies for counting and representing numbers. They explore the magnitude and
properties of numbers. They apply a range of strategies for computation and understand the connections between operations. They recognise patterns and understand the concepts of variable and
function. They build on their understanding of the number system to describe relationships and formulate generalisations. They recognise equivalence and solve equations and inequalities. They apply
their number and algebra skills to conduct investigations, solve problems and communicate their reasoning.
Measurement and Geometry
Measurement and Geometry are presented together to emphasise their relationship to each other, enhancing their practical relevance. Students develop an increasingly sophisticated understanding of
size, shape, relative position and movement of two-dimensional figures in the plane and three-dimensional objects in space. They investigate properties and apply their understanding of them to
define, compare and construct figures and objects. They learn to develop geometric arguments. They make meaningful measurements of quantities, choosing appropriate metric units of measurement. They
build an understanding of the connections between units and calculate derived measures such as area, speed and density.
Statistics and Probability
Statistics and Probability initially develop in parallel and the curriculum then progressively builds the links between them. Students recognise and analyse data and draw inferences. They represent,
summarise and interpret data and undertake purposeful investigations involving the collection and interpretation of data. They assess likelihood and assign probabilities using experimental and
theoretical approaches. They develop an increasingly sophisticated ability to critically evaluate chance and data concepts and make reasoned judgments and decisions, as well as building skills to
critically evaluate statistical information and develop intuitions about data.
Proficiency Strands
The proficiency strands describe the actions in which students can engage when learning and using the content. While not all proficiency strands apply to every content description, they indicate the
breadth of mathematical actions that teachers can emphasise.
Students build a robust knowledge of adaptable and transferable mathematical concepts. They make connections between related concepts and progressively apply the familiar to develop new ideas. They
develop an understanding of the relationship between the ‘why’ and the ‘how’ of mathematics. Students build understanding when they connect related ideas, when they represent concepts in different
ways, when they identify commonalities and differences between aspects of content, when they describe their thinking mathematically and when they interpret mathematical information.
Students develop skills in choosing appropriate procedures, carrying out procedures flexibly, accurately, efficiently and appropriately, and recalling factual knowledge and concepts readily. Students
are fluent when they calculate answers efficiently, when they recognise robust ways of answering questions, when they choose appropriate methods and approximations, when they recall definitions and
regularly use facts, and when they can manipulate expressions and equations to find solutions.
Problem Solving
Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. Students formulate and solve problems when they
use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they
verify that their answers are reasonable.
Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. Students are
reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning
from one context to another, when they prove that something is true or false and when they compare and contrast related ideas and explain their choices.
Content descriptions are grouped into sub-strands to illustrate the clarity and sequence of development of concepts through and across the year levels. They support the ability to see the connections
across strands and the sequential development of concepts from Foundation to Year 10.
│Number and Algebra │Measurement and Geometry │Statistics and Probability │
│Number and place value (F-8) │Using units of measurement (F-10) │Chance (1-10) │
│Fractions and decimals (1-6) │Shape (F-7) │Data representation and interpretation (F-10) │
│Real numbers (7-10) │Geometric reasoning (3-10) │ │
│Money and financial mathematics (1-10) │Location and transformation (F-7) │ │
│Patterns and algebra (F-10) │Pythagoras and trigonometry (9-10)│ │
│Linear and non-linear relationships (7-10)│ │ │
Learning mathematics creates opportunities for and enriches the lives of all Australians. | {"url":"https://www.stpatskatanning.wa.edu.au/mathematics/","timestamp":"2024-11-08T01:58:18Z","content_type":"text/html","content_length":"59624","record_id":"<urn:uuid:c0662831-6bb8-459b-8d89-52fb8b9f1e42>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00103.warc.gz"} |
Processing Of Iron Ore With Titanium
Titanium Ore Processing and Beneficiation
May 09, 2016 · Metallurgical ContentTitanium Ore Extracting FlowsheetCrushing of Ti OreGrinding and Coarse ConcentrationHydraulic Classification and TablingFlotation of Titanium FinesFiltering and
DryingMagnetic and Electrostatic SeparationPossibilities For All Flotation Treatment of Titanium Ore To develop a flowsheet for separation of high grade titanium-rutile from ilmenite, that will meet
market ...
titanium processing | Technology, Methods, & Facts ...
Titanium processing, the extraction of titanium from its ores and the preparation of titanium alloys or compounds for use in various products. Titanium (Ti) is a soft, ductile, silvery gray metal
with a melting point of 1,675 °C (3,047 °F).
processing of iron ore with titanium
Titaniferous Iron Ore Beneficiation - godrejseethru. Processing titaniferous ore to titanium dioxide pigment. . The chloride process relies on chlorination of a low iron titanium ore followed by the
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Titanium Iron Ore Processing Equipment
Titanium Iron Ore Processing Equipment. 2019-12-17iron processing, use of a smelting process to turn the ore into a form from which products can be fashionedncluded in this article also is a
discussion of the mining of iron and of its preparation for smeltingron fe is a relatively dense metal with a silvery white appearance and distinctive....
Titanium,Vanadium and Iron Ore Processing
Iron And Titanium From Vanadium Processing. titanium vanadium and iron ore processing. Nov 30, 2017 A process has been developed at the laboratory scale for the recovery of titanium, vanadium and
iron from the vanadium bearing titanomagnetite deposit at Pipestone Lake, Manitoba, by combined pyroand hydrometallurgical processing route.
milling and grinding titanium ore
grinding equipment with low cost for bentonite ore . Basics in Minerals Processing BASICS IN MINERAL PROCESSING . Cost of grinding typical .. Major process equipment components of iron ore pellet
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China Titanium Iron Ore Processing Equipment Flotation ...
China titanium iron ore processing equipment flotation machine - find detail titanium iron ore processing equipment from zhengzhou yufeng heavy machinery co., ltd.
titanium iron ore processing equipment
Apr 12, 2013· titanium mining processing plant for iron ore. Titanium is a mineral of iron and ilmenite oxides, titanium ore refining. Dried ore by the permanent magnet machine, the time on magnetism
of low grade and then electrostatic separation electric rutile, separate the red and gold and zircon.
titanium ore beneficiation process
titanium ore beneficiation process_ball mill in iron ore beneficiation pdf arpainternational.incapacity of ball mill for fe beneficiation pdf. Flow Chart Of Iron Ore Beneficiation Process superior
quality Low Iron Ore Beneficiat ... Titanium Ore Processing and Beneficiation 911 Metallurgist May 9, 2016 ... To develop a flowsheet for separation ...
Iron processing | Britannica
Iron processing, use of a smelting process to turn the ore into a form from which products can be fashioned.Included in this article also is a discussion of the mining of iron and of its preparation
for smelting. Iron (Fe) is a relatively dense metal with a silvery white appearance and distinctive magnetic properties. It constitutes 5 percent by weight of the Earth’s crust, and it is the ...
Mining firm Vale completes first iron ore sale via ...
Sep 04, 2020 · Brazilian mining giant Vale has completed its first sale of iron ore using blockchain technology. The transaction with Nanjing Iron & Steel involved a … | {"url":"http://tanuloszoba.eu/cone/2022-processing-of-iron-ore-with-titanium-6408/","timestamp":"2024-11-09T14:33:02Z","content_type":"application/xhtml+xml","content_length":"10833","record_id":"<urn:uuid:b30a5643-4393-43e5-bb31-6df35d03fcd1>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00675.warc.gz"} |
Announcing the WikiChallenge Winners
Wikipedia Participation Challenge
Over the past couple of months, the Wikimedia Foundation,
organized a data competition. We
data scientists around the world to use Wikipedia editor data and develop an algorithm that predicts the number of future edits, and in particular predicts correctly who will stop editing and who
will continue to edit.
The response has been great! We had 96 teams compete, comprising in total 193 people who jointly submitted 1029 entries. You can have a look for yourself at the
We are very happy to announce that the brothers
and Fridolin Roth (team prognoZit) developed the winning algorithm. It is elegant, fast and accurate. Using
they developed a linear regression algorithm. They used 13 features (2 are based on reverts and 11 are based on past editing behavior) to predict future editing activity. Both the
source code
and the
wiki description
of their algorithm are available. Congratulations to Ben and Fridolin!
Second place goes to
Keith Herring
. Submitting only 3 entries, he developed a highly
accurate model
, using random forests, and utilizing a total of 206 features. His model shows that a randomly selected Wikipedia editor who has been active in the past year has approximately an 85 percent
probability of being inactive (no new edits) in the next 5 months. The most informative features captured both the edit timing and volume of an editor. Asked for his reasons to enter the challenge,
Keith named his fascination for datasets and that
“I have a lot of respect for what Wikipedia has done for the accessibility of information. Any small contribution I can make to that cause is in my opinion time well spent.”
We also have two Honourable Mentions for participants who only used open source software. The first Honorable Mention is for
Dell Zang
(team zeditor) who used a machine learning technique called
gradient boosting
. His
mainly uses recent past editor activity.
The second Honourable Mention is for Roopesh Ranjan and Kalpit Desai (team Aardvarks). Using Python and R, they developed a random forest model as well. Their model used 113 features, mainly based on
the number of reverts and past editor activity, see the
describing their model.
All the documentation and source code has been made available, the main entry page is
WikiChallenge on Meta
What the four winning models have in common is that past activity and how often an editor is reverted are the strongest predictors for future editing behavior. This confirms our intuitions, but the
fact that the three winning models are quite similar in terms of what data they used is a testament to the importance of these factors.
We want to congratulate all winners, as they have showed us in a quantitative way important factors in predicting editor retention. We also hope that people will continue to investigate the training
dataset and keep refining their models so we get an even better understanding of the long-term dynamics of the Wikipedia community.
We are looking forward to use the algorithms of Ben & Fridolin and Keith in a production environment and particularly to see if we can forecast the cumulative number of edits.
Finally, we want to thank the Kaggle people for helping in organizing this competition and our anonymous donor who has generously donated the prizes.
Diederik van Liere
External Consultant, Wikimedia Foundation
Howie Fung
Senior Product Manager, Wikimedia Foundation
2011-10-26: Edited to correct description of the winning algorithm
Can you help us translate this article?
In order for this article to reach as many people as possible we would like your help. Can you translate this article to get the message out?
Start translation | {"url":"https://diff.wikimedia.org/2011/10/26/announcing-the-wikichallenge-winners/","timestamp":"2024-11-02T04:24:17Z","content_type":"text/html","content_length":"255845","record_id":"<urn:uuid:a05e824d-1022-4fc4-ac52-ce3a05abe2c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00490.warc.gz"} |
If you are like me, your first contact with the hyperbolic functions was as "this strange, useless something on the calculator". There were just some weird buttons labeled "sinh" and "cosh". The
school finally explained what "sin" and "cos" are, but there was no mention of those variants with the final "h". What is this about? The names suggest some similarity to the trigonometric functions,
let's see what happens:
(You will get these results if you have the calculator set to radians - if you use degrees, then the cosine results will be different; it has no influence on the hyperbolic functions and we'll see
later why that is.)
Right, these 11 thousand for cosh(10) look very similar to the trigonometric functions. This "h" apparently changes quite a bit, but what exactly...?
If you encountered complex numbers during your later education, you could stumble upon such definitions:
Some similarity is visible here, but... Why such a form? What does this have to do with hyperbolas? If you don't know it yet, you will know after reading this article.
The shape of a black hole's event horizon
Yesterday, while browsing the internet, I stumbled upon a thread which looked like a typical question asked by someone interested in science, and turned out to be a really interesting problem.
The question that has been asked concerned the shape of a black hole. A few people replied that the event horizon (the boundary - or the "surface" in a way - of a black hole) has the shape of a ball
(which should be actually described as a sphere, since the horizon is a 2-dimensional surface, and not a 3-dimensional shape). Someone suggested that it's not exactly true, because black holes
usually spin, which flattens them. I entered the thread then and said that even when a black hole is spinning, its horizon is still spherical - it's described by an equation like r = const. But is
that really so...?
Part 3 - the metric
We already mentioned the notion of the magnitude of a vector, but we said nothing about what it actually is. On a plane it's easy - when we move by $v_x$ in the $x$ axis and by $v_y$ in the $y$ axis,
the distance between the starting and the ending point is $\sqrt{v_x^2 + v_y^2}$ (which can be seen by drawing a right triangle and using the Pythagorean theorem - see the picture). It doesn't have
to be always like that, though, and here is where the metric comes into play.
The metric is a way of generalizing the Pythagorean theorem. The coordinates don't always correspond to distances along perpendicular axes, and it is even not always possible to introduce such
coordinates (but let's not get ahead of ourselves). We want then to have a way of calculating the distance between points $\Delta x^\mu$ apart, where $x^\mu$ are some unspecified coordinates.
Part 2 - coordinates, vectors and the summation convention
The basic object in GR is the spacetime. As a mathematical object, formally it is a differential manifold, but for our purposes it is enough to consider it as a set of points called events, which can
be described by coordinates. In GR, the spacetime is 4-dimensional, which means that we need 4 coordinates - one temporal and three spatial ones.
The coordinates can be denoted by pretty much anything (like $x$, $y$, $z$, $t$), but since we will refer to all four of them at multiple occasions, it will be convenient to denote them by numbers.
It is pretty standard to denote time by 0, and the spatial coordinates by 1, 2 and 3. The coordinate number $\mu$ will be written like this: $x^\mu$ (attention: in this case it is not a power!). $\
mu$ here is called an index (here: an upper one). By convention, if we mean one of the 4 coordinates, we use a greek letter as the index; if only the spatial ones are to be considered, we use a
letter from the latin alphabet.
Part 1 - partial derivatives
As I mentioned in the introduction, I assume that the reader knows what a derivative of a function is. It is a good foundation, but to get our hands wet in relativity, we need to expand that concept
a bit. Let's then get to know the partial derivative. What is it?
Let's remember the ordinary derivatives first. We denote a derivative of a function $f(x)$ as $f'(x)$ or $\frac{df}{dx}$. It means, basically, how fast the value of the function changes while we
change the argument x. For example, when $f(x) = x^2$, then $\frac{df}{dx} = 2x$.
But what if the function depends on more than one variable? Like if we have a function $f(x,y) = x^2 + y^2$ that assigns to each point of the plane the square of its distance from the origin. How do
we even define the derivative of such a function? | {"url":"https://ebvalaim.net/en/category/articles/page/2/","timestamp":"2024-11-01T20:28:30Z","content_type":"text/html","content_length":"61771","record_id":"<urn:uuid:21f4de39-060d-4b49-8bdb-f8f9a1573278>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00177.warc.gz"} |
Help - Oliver Lehmann's 75 questions explantion
Submitted by rebeccafred on Sun, 04/22/2012 - 00:57
I attempted the Oliver Lehmanns's 75 Qs. I am stumped by Q 27 on the Std Deviation Calculation.
Can someone post an explanation please?
Thanks in advance.
Sun, 04/22/2012 - 01:17
You need to calculate variances of all 4 cases and then take sqr root of that to get the std variance.
I hope this would help
Oliver F. Lehmann
Tue, 04/24/2012 - 19:21
the formula is that the path standard deviation is the square root of the summed up squares of the single SDs.
A single SD (= sigma) is defined as a 6th (±3 Sigmas) of Pess. minus Opt.
Single SDs are: Act. A: 2 days, Act. B: 1 day; Act. C: 2 days, Act D: 3 days, Act. E: 3 days.
SD squares are therefore: Act. A: 4; Act. B: 1; Act. C: 4; Act D: 9; Act. E: 9
SD squares, sum: 27
Square root of that: 5.19 days
This means that the distance between a pessimistic and an optimistic estimate for the entire path duration should be 6 x 5.2 = 31.2 days.
It is one of my toughest questions, and I hope that you won't see it in the exam, but who knows for sure?
Tue, 04/24/2012 - 19:48
I have explained and detailed these kinds of question in my thread at pmptrend.com forum with graphs u can visit that thread
Mon, 06/12/2017 - 08:32 | {"url":"https://pmzilla.com/help-oliver-lehmanns-75-questions-explantion","timestamp":"2024-11-10T19:22:18Z","content_type":"text/html","content_length":"36605","record_id":"<urn:uuid:e88102aa-7be9-42b4-9118-7f6b1b4ae960>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00743.warc.gz"} |
Can someone please match the terms to the right definition.1 A rule for generating terms of a sequence that depends on one or more previous terms of the sequence.2 A rule that can be used to find the nth term of a sequence without calculating previous terms of the sequence. Also called the Explicit Rule.3 A value that is held constant in a specific function, or model, while other variables might change.4 The constant d of the sequence.5 A sequence where the difference between consecutive terms is a constant.6 A variable that takes on different values in a specific function, or model, while other values are held constant.7 The constant r of the sequence.8 A list of numbers, finite or infinite, that follow a particular pattern.9 The numbers in a sequence.10 A sequence where the ratio between consecutive terms is a constant.A. SequenceB. Terms of a sequenceC. Recursive ruleD. Iterative ruleE. Arithmetic sequenceF. Geometric sequenceG. Common differenceH. Common ratioI. Model parameterJ. Model variable
1. Home
2. General
3. Can someone please match the terms to the right definition.1 A rule for generating terms of a sequen... | {"url":"http://thibaultlanxade.com/general/can-someone-please-match-the-terms-to-the-right-definition-1-a-rule-for-generating-terms-of-a-sequence-that-depends-on-one-or-more-previous-terms-of-the-sequence-2-a-rule-that-can-be-used-to-find-the-nth-term-of-a-sequence-without-calculating-previou","timestamp":"2024-11-04T14:03:57Z","content_type":"text/html","content_length":"33494","record_id":"<urn:uuid:f6d7d9ba-38c9-4ce9-8f9f-4a8edd0eda6d>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00269.warc.gz"} |
Overlapping Subproblem | Dynamic Programming | PrepInsta
Overlapping Subproblem
Overlapping Subproblem
Overlapping Subproblem is one of the main features of the Dynamic Programming Problem. This is an article to briefly discuss this feature of Dynamic Programming, the cause of it and the optimization
of it.
Overlapping Subproblem | PrepInsta
What is Overlapping Subproblem?
The phrase consists of two parts. Subproblem clearly means when there are sub parts of a comparably bigger problem, and overlapping indicates if the similar kind of data is parsed, used or came
across more than once.
Suppose we have to find the fibonacci Series. So, if we say F(n) is the nth Fibonacci number, we can say F(n) = F(n-1) + F(n-2). So suppose we have to find the 5th FIbonacci number. So, F(5) = F(4) +
F(3). So if we create a function F(n) that will recursively call F(n-1) and return n+F(n-1) until n=0.
So, when we need F(5), we have to call F(4) and F(3). Again when F(4) is needed, we need F(3) and F(2). So here we can see, F(3) is actually needed two times only in this instance. Let us see how
many times we need to find the F(3).
The code to implement this:
#include <bits/stdc++.h>
using namespace std;
map<int,int> m;
int Fibonacci(int n)
cout<<"Fibonacci ("<<n<<") is called "<<m[n]++<<" times."<<endl;
if(n==0|n==1) return n;
int a=Fibonacci(n-1)+Fibonacci(n-2);
return a;
int main()
cout<<"This is Now the 5th Fibonacci Number in Recursive way: \n";
This is Now the 5th Fibonacci Number in Recursive way:
Fibonacci (5) is called 0 times.
Fibonacci (4) is called 0 times.
Fibonacci (3) is called 0 times.
Fibonacci (2) is called 0 times.
Fibonacci (1) is called 0 times.
Fibonacci (0) is called 0 times.
Fibonacci (1) is called 1 times.
Fibonacci (2) is called 1 times.
Fibonacci (1) is called 2 times.
Fibonacci (0) is called 1 times.
Fibonacci (3) is called 1 times.
Fibonacci (2) is called 2 times.
Fibonacci (1) is called 3 times.
Fibonacci (0) is called 2 times.
Fibonacci (1) is called 4 times.
So, here some of the numbers are needed more than 1 time. This is called the Overlapping Subproblems. This increase in the time complexity and also in the stack space. If we want to reduce the time
complexity, we can use memoization and store the already got values in a lookup table for future use. Check this Page: Memoization vs Tabulation for more details.
30+ Companies are Hiring
Get Hiring Updates right in your inbox from PrepInsta | {"url":"https://prepinsta.com/competitive-advanced-coding/overlapping-subproblem/","timestamp":"2024-11-15T03:02:41Z","content_type":"text/html","content_length":"171189","record_id":"<urn:uuid:299a9060-86f9-4435-a4a0-b1407ac0d50f>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00306.warc.gz"} |
Chi-Value Calculat
Chi-Value Calculator
The Chi Value is the point thermal transmission coefficients. You can enter everything manually in the Chi Value Calculator or use the tools that AnTherm already provides.
1. Leitwert 3D: In case of a 3D model, this is automatically calculated by AnTherm. The Leitwert 3D can be manually entered or automatically derived from the Thermal Coupling Coefficients report by
first performing all the calculations and then opening the Chi Calculator.
2. U-Value: It is possible to use the U-Value Calculator included in AnTherm by clicking the button "Layered construct".
3. Area: This must be entered manually by the user.
4. Then you get U * A.
5. For the Psi Calculations you can use the manual Psi Calculator, included in AnTherm. (If you created 2D slices of the model that show the corners, you can use the automatic Psi Calculator to
derive U-Values and lengths.) To access the Psi Calculator, click the respective button. For the Psi computation in the manual Psi Calculator, you also need the Leitwert 2D.
6. It is required to perform a separate calculation for each Psi value. By clicking "OK" you copy the result of the calculation into the Chi Calculator.
7. It is possible to save these calculations and make a screenshot (camera icon).
See also: Coupling Coefficient Report, Material Report Psi-Value Calculator (Tool), U-Value Calculator (Tool), Project types, Thermal transmittance coefficient, Zur Berechnung von Ψ-Werten für
Baukonstruktionen im Bereich bodenberührter Bauteile, EN ISO 14683, EN ISO 13789 | {"url":"https://help.antherm.eu/Forms/ChiValueCalculatorForm/ChiValueCalculatorForm.htm","timestamp":"2024-11-04T22:06:46Z","content_type":"text/html","content_length":"6929","record_id":"<urn:uuid:02adf661-a809-49b1-91a8-659fb2d25345>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00567.warc.gz"} |
Circle calculator
Do not you remember the formula for calculating the area, diameter or perimeter of a circle ? Enter the circle's area, diameter, or circumference and this calculator finds the other two for you!
How to use it?
Enter the value you have beside the specified device. Touch "Solve the others" to get the results. It does not matter which length unit (milimeter, centimeter, meter, kilometer, etc.) has the value.
How to figure this out?
Area, perimeter, radius and diameter of a circle can also be easily calculated manually using some formulas and pi ( π ). Find out how in the table below.
designation Name Formula Explanation
π Pi π = O / D π is the relationship between a circle's circumference and its diameter
D Diameter D = 2 · R The diameter is the right line that goes through the circle
O Circumference D · π = O Circumference is the circle line that goes around the circle
R Radius R = D / 2 Radius is the line that runs from the center to the circle line
A Area A = π · R · R Area is the content of the circle | {"url":"https://www.calculatemate.com/calculators/calculators/circle-calculator","timestamp":"2024-11-14T07:03:41Z","content_type":"text/html","content_length":"16661","record_id":"<urn:uuid:a75dacb4-4ef8-4bbb-a02c-f0a6d6a6b1d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00191.warc.gz"} |
'Wechsler, Risa H.'
Searching for codes credited to 'Wechsler, Risa H.'
➥ Tip! Refine or expand your search. Authors are sometimes listed as 'Smith, J. K.' instead of 'Smith, John' so it is useful to search for last names only. Note this is currently a simple phrase
[ascl:1210.011] Consistent Trees: Gravitationally Consistent Halo Catalogs and Merger Trees for Precision Cosmology
Consistent Trees generates merger trees and halo catalogs which explicitly ensure consistency of halo properties (mass, position, velocity, radius) across timesteps. It has demonstrated the ability
to improve both the completeness (through detecting and inserting otherwise missing halos) and purity (through detecting and removing spurious objects) of both merger trees and halo catalogs.
Consistent Trees is able to robustly measure the self-consistency of halo finders and to directly measure the uncertainties in halo positions, halo velocities, and the halo mass function for a given
halo finder based on consistency between snapshots in cosmological simulations.
[ascl:1708.026] XDGMM: eXtreme Deconvolution Gaussian Mixture Modeling
XDGMM uses Gaussian mixtures to do density estimation of noisy, heterogenous, and incomplete data using extreme deconvolution (XD) algorithms which is compatible with the scikit-learn machine
learning methods. It implements both the astroML and Bovy et al. (2011) algorithms, and extends the BaseEstimator class from scikit-learn so that cross-validation methods work. It allows the user to
produce a conditioned model if values of some parameters are known.
[ascl:1708.027] empiriciSN: Supernova parameter generator
empiriciSN generates realistic supernova parameters given photometric observations of a potential host galaxy, based entirely on empirical correlations measured from supernova datasets. It is
intended to be used to improve supernova simulation for DES and LSST. It is extendable such that additional datasets may be added in the future to improve the fitting algorithm or so that additional
light curve parameters or supernova types may be fit.
[ascl:2210.029] paltas: Simulation-based inference on strong gravitational lensing systems
paltas conducts simulation-based inference on strong gravitational lensing images. It builds on lenstronomy (ascl:1804.012) to create large datasets of strong lensing images with realistic low-mass
halos, Hubble Space Telescope (HST) observational effects, and galaxy light from HST's COSMOS field. paltas also includes the capability to easily train neural posterior estimators of the parameters
of the lensing system and to run hierarchical inference on test populations.
[ascl:2211.009] ovejero: Bayesian neural network inference of strong gravitational lenses
ovejero conducts hierarchical inference of strongly-lensed systems with Bayesian neural networks. It requires lenstronomy (ascl:1804.012) and fastell (ascl:9910.003) to run lens models with
elliptical mass distributions. The code trains Bayesian Neural Networks (BNNs) to predict posteriors on strong gravitational lensing images and can integrate with forward modeling tools in
lenstronomy to allow comparison between BNN outputs and more traditional methods. ovejero also provides hierarchical inference tools to generate population parameter estimates and unbiased posteriors
on independent test sets.
[ascl:2302.011] UniverseMachine: Empirical model for galaxy formation
The UniverseMachine applies simple empirical models of galaxy formation to dark matter halo merger trees. For each model, it generates an entire mock universe, which it then observes in the same way
as the real Universe to calculate a likelihood function. It includes an advanced MCMC algorithm to explore the allowed parameter space of empirical models that are consistent with observations.
[ascl:2303.008] nd-redshift: Number Density Redshift Evolution Code
Comparing galaxies across redshifts via cumulative number densities is a popular way to estimate the evolution of specific galaxy populations. nd-redshift uses abundance matching in the ΛCDM paradigm
to estimate the median change in number density with redshift. It also provides estimates for the 1σ range of number densities corresponding to galaxy progenitors and descendants.
[ascl:2406.006] anzu: Measurements and emulation of Lagrangian bias models for clustering and lensing cross-correlations
The anzu package offers two independent codes for hybrid Lagrangian bias models in large-scale structure. The first code measures the hybrid "basis functions"; the second takes measurements of these
basis functions and constructs an emulator to obtain predictions from them at any cosmology (within the bounds of the training set). anzu is self-contained; given a set of N-body simulations used to
build emulators, it measures the basis functions. Alternatively, given measurements of the basis functions, anzu should in principle be useful for constructing a custom emulator. | {"url":"https://ascl.net/code/cs/Wechsler%2C%20Risa%20H.","timestamp":"2024-11-06T02:21:18Z","content_type":"text/html","content_length":"13799","record_id":"<urn:uuid:2846a9e0-30ae-4030-919b-39473221ed97>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00156.warc.gz"} |
Symmetries in Quantum Field Theory and Quantum Gravity
Total Page：16
File Type：pdf, Size：1020Kb
[email protected]
[email protected]
Abstract: In this paper we use the AdS/CFT correspondence to refine and then es- tablish a set of old conjectures about symmetries in quantum gravity. We first show that any global symmetry, discrete
or continuous, in a bulk quantum gravity theory with a CFT dual would lead to an inconsistency in that CFT, and thus that there are no bulk global symmetries in AdS/CFT. We then argue that any \
long-range" bulk gauge symmetry leads to a global symmetry in the boundary CFT, whose consistency requires the existence of bulk dynamical objects which transform in all finite-dimensional irre-
ducible representations of the bulk gauge group. We mostly assume that all internal symmetry groups are compact, but we also give a general condition on CFTs, which we expect to be true quite
broadly, which implies this. We extend all of these results to the case of higher-form symmetries. Finally we extend a recently proposed new motivation for the weak gravity conjecture to more general
gauge groups, reproducing the \convex hull condition" of Cheung and Remmen. An essential point, which we dwell on at length, is precisely defining what we mean by gauge and global symmetries in the
bulk and boundary. Quantum field theory results we meet while assembling the necessary tools include continuous global symme- arXiv:1810.05338v2 [hep-th] 6 Jun 2019 tries without Noether currents, new
perspectives on spontaneous symmetry-breaking and 't Hooft anomalies, a new order parameter for confinement which works in the presence of fundamental quarks, a Hamiltonian lattice formulation of
gauge theories with arbitrary discrete gauge groups, an extension of the Coleman-Mandula theorem to discrete symmetries, and an improved explanation of the decay π0 ! γγ in the standard model of
particle physics. We also describe new black hole solutions of the Einstein equation in d + 1 dimensions with horizon topology Tp × Sd−p−1. Contents 1 Introduction1 1.1 Notation9 2 Global symmetry 13
2.1 Splittability 19 2.2 Unsplittable theories and continuous symmetries without currents 24 2.3 Background gauge fields 31 2.4 't Hooft anomalies 35 2.5 ABJ anomalies and splittability 41 2.6 Towards
a classification of 't Hooft anomalies 48 3 Gauge symmetry 53 3.1 Definitions 54 3.2 Hamiltonian lattice gauge theory for general compact groups 62 3.3 Phases of gauge theory 70 3.4 Comments on the
topology of the gauge group 73 3.5 Mixing of gauge and global symmetries 76 4 Symmetries in holography 77 4.1 Global symmetries in perturbative quantum gravity 77 4.2 Global symmetries in
non-perturbative quantum gravity 82 4.3 No global symmetries in quantum gravity 88 4.4 Duality of gauge and global symmetries 93 5 Completeness of gauge representations 96 6 Compactness 99 7
Spacetime symmetries 102 8 p-form symmetries 109 8.1 p-form global symmetries 109 8.2 p-form gauge symmetries 114 8.3 p-form symmetries and holography 119 8.4 Relationships between the conjectures?
122 { i { 9 Weak gravity from emergent gauge fields 124 A Group theory 128 A.1 General structure of Lie groups 128 A.2 Representation theory of compact Lie groups 129 B Projective representations 134
C Continuity of symmetry operators 136 D Building symmetry insertions on general closed submanifolds 142 E Lattice splittability theorem 144 F Hamiltonian for lattice gauge theory with discrete gauge
group 146 G Stabilizer formalism for the Z2 gauge theory 148 H Multiboundary wormholes in three spacetime dimensions 153 I Sphere/torus solutions of Einstein's equation 159 1 Introduction It has long
been suspected that the consistency of quantum gravity places constraints on what kinds of symmetries can exist in nature [1]. In this paper we will be primarily interested in three such conjectural
constraints [2,3]: Conjecture 1. No global symmetries can exist in a theory of quantum gravity. Conjecture 2. If a quantum gravity theory at low energies includes a gauge theory with compact gauge
group G, there must be physical states that transform in all finite- dimensional irreducible representations of G. For example if G = U(1), with allowed charges Q = nq with n 2 Z, then there must be
states with all such charges. Conjecture 3. If a quantum gravity theory at low energies includes a gauge theory with gauge group G, then G must be compact. { 1 { These conjectures are quite
nontrivial, since it is easy to write down low-energy effective actions of matter coupled to gravity which violate them. For example Einstein gravity coupled to two U(1) gauge fields has a Z2 global
symmetry exchanging the two gauge fields, and also has no matter fields which are charged under those gauge fields. If we instead use two R gauge fields, then we can violate all three at once.
Conjectures 1-3 say that such effective theories cannot be obtained as the low-energy limit of a consistent theory of quantum gravity: they are in the \swampland" [4{7].1 The \classic" arguments for
conjectures1-3 are based on the consistency of black hole physics. One argument for conjecture1 goes as follows [3]. Assume that a con- tinuous global symmetry exists. There must be some object which
transforms in a nontrivial representation of G. Since G is continuous, by combining many of these objects we can produce a black hole carrying an arbitrarily complicated representation of G.2 We then
allow this black hole to evaporate down to some large but fixed size in Planck units: the complexity of the representation of the black hole will not decrease during this evaporation since the Hawking
process depends only on the geometry and is uncorrelated with the global charge (for example if G = U(1) then positive and nega- tive charges are equally produced). According to Bekenstein and
Hawking the entropy of this black hole is given by [8,9] Area SBH = ; (1.1) 4GN but this is not nearly large enough to keep track of the arbitrarily large representa- tion data we've stored in the
black hole. Thus either (1.1) is wrong, or the resulting object cannot be a black hole, and is instead some kind of remnant whose entropy can arbitrarily exceed (1.1). There are various arguments
that such remnants lead to inconsistencies, see eg [10], but perhaps the most compelling case against either of these possibilities is simply that they would necessarily spoil the
statistical-mechanics interpretation of black hole thermodynamics first advocated in [8]. This interpretation has been confirmed in many examples in string theory [11{16]. The classic argument for
conjecture2 is simply that once a gauge field exists, then so does the appropriate generalization of the Reissner-Nordstrom solution for any representation of the gauge group G. The classic argument
for conjecture3 is that at least if G were R, the non-quantization of charge would imply a continuous infinity in 1Note however that the charged states required by conjecture2 might be heavy, and in
particular they might be black holes. 2More rigorously, given any faithful representation of a compact Lie group G, theorem A.11 below tells us that all irreducible representations of G must
eventually appear in tensor powers of that representation and its conjugate. If G is continuous, meaning that as a manifold it has dimension greater than zero, then there are infinitely many
irreducible representations available. { 2 { the entropy of black holes in a fixed energy band, assuming that black holes of any charge exist, which again contradicts the finite Bekenstein-Hawking
entropy. Moreover non-abelian examples of noncompact continuous gauge groups are ruled out already in low-energy effective field theory since they do not have well-behaved kinetic terms (for noncompact
simple Lie algebras the Lie algebra metric Tr (TaTb) is not positive- definite). These arguments for conjectures1-3 certainly have merit, but they are not com- pletely satisfactory. The argument for
conjecture1 does not apply when the symmetry group is discrete, for example when G = Z2 then there is only one nontrivial irreducible representation, but why should continuous symmetries be special?
In arguing for con- jecture2, does the existence of the Reissner-Nordstrom solution really tell us that a charged object exists? As long as it is non-extremal, this solution really describes a
two-sided wormhole with zero total charge. It therefore does not obviously tell us any- thing about the spectrum of charged states with one asymptotic boundary.3 We could instead consider \one-sided"
charged black holes made from gravitational collapse, but then we must first have charged matter to collapse: conjecture2 would then already be satisfied by this charged matter, so why bother with the
black hole at all? To really make an argument for conjecture2 based on charged solutions of general relativity that do not already have charged matter, we need to somehow satisfy Gauss's law with a
non-trivial electric flux at infinity but no sources. It is not possible to do this with triv- ial spatial topology. One possibility is to consider one-sided charged \geons" created by quotienting some
version of the Reissner-Nordstrom wormhole by a Z2 isometry [18], but this produces a non-orientable spacetime and/or requires that we gauge a discrete Z2 symmetry that flips the sign of the field | {"url":"https://docslib.org/doc/1478343/symmetries-in-quantum-field-theory-and-quantum-gravity","timestamp":"2024-11-14T10:55:41Z","content_type":"text/html","content_length":"63762","record_id":"<urn:uuid:54581ef6-3fac-4557-b150-813429f8a50e>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00151.warc.gz"} |
Project Risk Analysis Software and Project Risk Management Software Forum
Moderator: Intaver Support
Posts: 17
Joined: Sat Nov 26, 2005 11:06 am
Location: London, UK
I assigned the risk to the resource(s) as a relative time increase and then also as a relative cost increase – both at 15% chance with 10% outcome. But the issue I seem to have when I do this is that
the probability of it occurring goes above 15%. Do I potentially need to add the time delay to the task and cost increase to the resources(s) Or am I going about this all the wrong way.
Posts: 1008
Joined: Wed Nov 09, 2005 9:55 am
Calculated probability can vary from the input. But normally not by much. The reason is similar to throwing a dice. At each iteration, the risk has a 15% of occurring, so after 1 iteration, the
probability will be 0 or 100%, after 2 0,30, or 100%. After 100 iterations, it could occur 15x (ie 15%) but could be less or more. The more iterations that are run, the closer the probability will be
to 15%.
Intaver Support Team
Intaver Institute Inc.
Home of Project Risk Management and Project Risk Analysis software RiskyProject | {"url":"http://intaver.com/IntaverFrm/viewtopic.php?p=3485&sid=d069c72ea66f6869321f23faa0e16754","timestamp":"2024-11-01T20:46:58Z","content_type":"text/html","content_length":"23595","record_id":"<urn:uuid:b99bacb4-e474-485c-8c5d-d2f99ef07999>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00039.warc.gz"} |
Majors, Minors & Specializations - UCLA Mathematics
majors, Minors, & Specializations
All pre-major & major courses MUST be taken for letter grades! Please see each the latest undergraduate handbook for further details on the course and letter grade requirements of each program.
Detailed requirements are also listed in the UCLA General Catalog.
For more clarification on our major policies, contact Math Student Services.
Students who are planning to pursue graduate studies in mathematics or related fields are strongly encouraged to major in Mathematics, Applied Mathematics, or Mathematics of Computation.
Designed for students who are interested in the theory of mathematics. Pure mathematicians often pursue master and doctorate degree in mathematics in order to prepare for a career in research or
university level teaching.
Students should review the general catalog for more detailed information.
Ideal for students who are interested in mathematics used in the physical, life and social sciences, and engineering, as well as practical applications in society, including business, finance,
medicine and the internet. Students in this major frequently add the Specialization in Computing.
Students should review the general catalog for more detailed information.
Trains students, through theory and practice, in the mathematical, statistical, and computational principles of data science. Top graduates will be prepared for graduate studies in a field related
to data science or an initial technical position in the field with leadership potential. In collaboration with Statistics, it is a capstone major with a data-based project in the senior year.
Designed for students interested in financial mathematics and its applications. Graduates typically go on to MFE/MBA programs, the actuarial field, banking and/or business.
Freshmen students who entered UCLA in Fall 2019 and forward must follow the new requirements, please click here
Freshmen students who entered UCLA between Fall 2013 and Fall 2018 and Transfers who entered Fall 2019, please click here
Students should review the general catalog for more detailed information.
Designed for individuals who are interested in the mathematical theory and the applications of computing. These students often seek employment in areas similar to the applied mathematicians.
Students should review the general catalog for more detailed information.
Designed for students who have a substantial interest in teaching mathematics at the secondary level. Visit the Curtis Center website for more information about other undergraduate teacher
preparation programs such as the Joint Mathematics Education Program and the Subject Matter Preparation Program.
Students should review the general catalog for more detailed information.
The Mathematics/Applied Science major is intended for students who are interested in applications of mathematics to other areas. Students majoring in Mathematics/Applied Science often pursue careers
in medical professions, professional programs, or graduate programs in business or law. Students who major in Mathematics/Applied Science must pursue one of the following plans:
Designed to give students a solid foundation in both mathematics and economics, stressing those areas of mathematics and statistics that are most relevant to economics and the parts of economics that
emphasize the use of mathematics and statistics. It is ideal for students who may wish to complete a higher degree in economics.
Scope and Objectives
In recent years economics has become increasingly dependent on mathematical methods, and the mathematical tools it employs have become more sophisticated. Mathematically competent economists, with
bachelor’s degrees and with advanced degrees, are needed in industry and government. Graduate programs in economics and finance programs in graduate schools of management require strong undergraduate
preparation in mathematics for admission.
This degree program is designed to give students a solid foundation in both mathematics and economics, stressing those areas of mathematics and statistics that are most relevant to economics and the
parts of economics that emphasize the use of mathematics and statistics.
Undergraduate Study
For students who declared the major from Fall 2015 to Summer 2016.
For students who declared the major in Fall 2016.
Students should review the general catalog for more detailed information.
Designed to provide students who are non-math major the opportunity to widen their background and general comprehension of the role of mathematics in various disciplines. Students in a mathematics
major cannot add the Mathematics minor.
Designed for students majoring in fields other than mathematics who plan to teach secondary mathematics after graduation. Mathematics minors are not available for students in a Mathematics for
Teaching major.
The Department of Mathematics offers a Specialization in Computing. All mathematics majors can add the specialization except for Mathematics of Computation and Data Theory.
• Courses from the Specialization can overlap with its corresponding major, another major, and/or minor with no unit limitations.
• CS 31, 32, and 33 can substitute PIC 10A, 10B, and 10C, respectively.
Students should review the general catalog for more detailed information.
Departmental Scholars and Honors Programs
The Mathematics Departmental Scholars Program is the most advanced program offered to undergraduate mathematics majors. It provides excellent preparation for graduate school for exceptional students.
Scholars complete both Bachelor’s (BS) and Master’s (MA) degrees, within a four-year period.
Admission to the Departmental Scholars Program is by application only. Students must apply before the end of the spring quarter of the junior year.
Applications which satisfy the conditions below, and which are submitted by the end of spring quarter of the junior year, are guaranteed consideration. Applications are reviewed and decided by the
Undergraduate and Graduate program faculty, in consultation with other faculty.
To apply for the Departmental Scholars Programs, students must meet the following requirements:
• Be declared in a mathematics major and have fulfilled all premajor math requirements for that major
• Have completed 96 quarter course units or 24 courses at UCLA or another institution
• Have passed the Basic Exam
• Have completed the WII requirement
• Have at least a 3.6 gpa in all upper division math courses
• Have at least a 3.5 gpa overall in UCLA courses
• Have 2 letters of recommendation from Mathematics permanent faculty who have taught the student
• Have a detailed study plan for completing both the Bachelor’s degree and the Masters degree by the end of Year 4
• Have a Statement of Purpose, about one page but no more than two, which explains the student’s interest in the program
Note that these are the minimal requirements for application. Fulfilling them does not guarantee admission to the program.
To remain in the Departmental Scholars Program, students must meet the following requirements:
• Remain a UCLA mathematics student in good academic standing
• Maintain at least a 3.5 GPA in mathematics courses at all times
The following timeline is recommended for students in the Departmental Scholars Program:
│ First year │ Complete or have credit from another institution/standardized test (AP or IB Exams) all lower-division Calculus-based courses (Math 31A, 31B, 32A, 32B, 33A, 33B). If possible take │
│ at UCLA │ 115AH in spring. │
│ Second year │ Complete pre-major courses, take Math 115AH (Honors Linear Algebra), Math 115B (Linear Algebra), Math 131AH (Honors Analysis) and 131BH (Honors Analysis).Take major honors courses │
│ at UCLA │ in area in which you plan to take first grad courses. Begin preparation for Basic Exam (offered Sept. and March) using online copies of past exams. │
│ Third Year │ Pass the Basic Qualifying Exam. Apply to Scholars program no later than Spring Quarter. Complete remaining undergraduate math major courses, and most general UCLA required │
│ at UCLA │ courses. Begin graduate courses, │
│ Fourth year │ Complete remaining graduate (and undergraduate) level coursework. │
│ at UCLA │ │
The MA requirements include 11 additional courses, of which 8 must be graduate math courses. Typically, Scholars follow at least two of the Core graduate course sequences. These classes start in the
fall quarter The normal course load for beginning graduate students is 3 math courses, with a least 2 in core sequences. In addition, candidates must fulfill all University level requirements. Note,
in particular, that no course may be used to fulfill requirements for both degrees.
An important note on taking graduate courses as an undergraduate. Graduate courses which are taken more than two quarters prior to the quarter of application to the Scholars program may not be used
for credit toward the Masters degree. Specifically, this means that if your application file is submitted to the Undergraduate Office and is complete in:
-Fall Quarter, then you can count any course taken in Winter or Spring Quarter of the previous year.
-Winter Quarter, then you can count any course taken in Fall quarter of the same year or Spring quarter of the previous year
-Spring Quarter, then you can count any course taken in Winter or Fall Quarter of the same year.
Please use these rules in planning your studies and please consult with Connie Jung, or either the Graduate or Undergraduate Vice chair, if you have any questions. Graduate courses may be used to
fulfill undergraduate major electives, even if they were taken too early to be used for the Masters degree.
More information on the Basic Exam, including old exams and dates of upcoming exams: https://ww3.math.ucla.edu/qualifying-exam-dates/
Contact ugrad[at]math.ucla.edu with questions.
Honors Program in Mathematics
Admission to the Program:
To be considered for admission to the Honors Program in Mathematics, a student must:
• be officially enrolled in the Mathematics major;
• have completed at least four courses at UCLA in the Mathematics Department from those required in the “Preparation for the Major” or Major; and
• have at least a 3.6 GPA in such mathematics courses taken at UCLA.
Requirements For Honors At Graduation:
The student must have completed, in addition to usual course requirements:
• Mathematics 115AH, 131AH, 131BH, 132H, 110AH, 110BH and 110C;
• One of the following:
□ Mathematics 191; or
□ take, as an approved active participant, any graduate seminar offered by the Department of Mathematics; or
□ submit an original project, which can be done as part of a regular course, a special course (Mathematics 199), or by special arrangement.
• Earn a 3.6 GPA or higher in approved upper division and graduate mathematics courses
Original Project:
• The project should involve some aspects of mathematical theory.
• The project is to be carried out under the sponsorship of a faculty advisor.
• The project may be done as part of a regular course, a special course (Mathematics 199), Summer REU project or by a special arrangement.
• No later than one quarter prior to graduation, the student must submit a project proposal to the Honors Committee for approval. The project itself must be submitted not later than the fifth week
of the last quarter before graduation.
• The Honors Committee will evaluate the project in consultation with the faculty sponsor, and may at it’s discretion, require a personal presentation by the student.
• Upper division seminars in Mathematics automatically count as mathematics electives for the major.
Requirements For Highest Honors At Graduation:
In addition to the above, the student must:
• complete at least one approved graduate mathematics course; and
• earn a 3.8 GPA or higher in approved upper division and graduate mathematics courses.
This Honors Program is independent of the honors sections of the mathematics courses. Graduation with Honors in Mathematics is also distinct from graduation with College Honors. Applications for
the Honors Program in Mathematics are available in the Student Services Office, 6356 Math Sciences. If you have any questions about the program, or special requests, you are welcome to consult any
members of the Mathematics Honors Committee, or see an Undergraduate Mathematics Counselor in 6356 Math Sciences.
December 2007
Honors Program in Applied Mathematics
Admission to the Program:
To be considered for admission to the Honors Program in Applied Mathematics, a student must:
• be officially enrolled in the Applied Mathematics major;
• have completed at least four courses at UCLA in the Mathematics Department from those required in the “Preparation for the Major” or Major; and
• have at least a 3.6 GPA in such mathematics courses taken at UCLA.
Requirements For Honors At Graduation:
1. The student must have completed:
Mathematics 115AH, 131AH, 131BH, and 132H AND
One of the following three quarter sequences:
• Mathematics 151AB and any course 152-159;
• 170AB and 171;
• Statistics 100ABC; or
• 3 from Mathematics 133, 134, 135, 136, 146.
(Other appropriate courses may be substituted for this requirement upon approval of the Honors Committee.)
2. The student must either:
• submit an original project as described below; or
• take, as an approved active participant, any upper division or graduate seminar offered by the Department of Mathematics. Such participation is described below.
3. The student must have a GPA of at least 3.6 in upper division mathematics and statistics courses taken for the major.
Original Project:
• The project should involve some aspects of mathematical theory.
• The project is to be carried out under the sponsorship of a faculty advisor.
• The project may be done as part of a regular course, a special course (Mathematics 199), Summer REU project or by a special arrangement.
• No later than one quarter prior to graduation, the student must submit a project proposal to the Honors Committee for approval. The project itself must be submitted not later than the fifth week
of the last quarter before graduation.
• The Honors Committee will evaluate the project in consultation with the faculty sponsor, and may at it’s discretion, require a personal presentation by the student.
Approval as an active participant requires all of the following:
• two lectures;
• a written statement, signed by the instructor, describing the nature of the participation. This statement must be submitted to the Honors Committee no later than the end of the quarter in which
the seminar is given or the fifth week of the last quarter before graduation, whichever is sooner;
• approval of the Honors Committee.
• Upper division seminars in Mathematics automatically count as mathematics electives for the major.
Requirements For Highest Honors At Graduation:
In addition to the above:
• Students who demonstrate exceptional achievement will be awarded Highest Honors.
• Decisions regarding projects, seminar participation, and Highest Honors will be made by the Honors Committee.
This Honors Program is independent of the honors sections of the mathematics courses. Graduation with Honors in Mathematics is also distinct from graduation with College Honors. Applications for
the Honors Program in Mathematics are available in the Student Services Office, 6356 Math Sciences. If you have any questions about the program, or special requests, you are welcome to consult any
members of the Mathematics Honors Committee, or see an Undergraduate Mathematics Counselor in 6356 Math Sciences.
December 2007
Honors Program in Financial Actuarial Mathematics
Admission to the Program:
To be considered for admission to the Honors Program in Financial Actuarial Mathematics, a student must:
• be officially enrolled in the Financial Actuarial Mathematics major;
• have completed all of the “Preparation for the Major” courses;
• have at least a 3.5 GPA in the Mathematics “Preparation for the Major” courses;
• have at least a 3.5 GPA in the Economics “Preparation for the Major” courses
Requirements For Honors At Graduation:
1. Complete Mathematics 115AH, 131AH and 131BH;
2. Complete Mathematics 170AB and 171;
3. Complete Mathematics 172ABC and 173AB.
(Other appropriate courses may be substituted for this requirement upon approval of the Honors Committee.)
4. The student must either:
• submit an original project as described below; or
• take, as an approved active participant, any upper division or graduate seminar offered by the Department of Mathematics. Such participation is described below.
5. The student must have a GPA of at least 3.6 in upper division mathematics and economics/statistics courses (calculated separately) taken for the major.
Original Project:
• The project should involve some aspects of mathematical theory.
• The project is to be carried out under the sponsorship of a faculty advisor.
• The project may be done as part of a regular course, a special course (Mathematics 199), Summer REU project or by a special arrangement.
• The project may be done by enrolling in Economics 198A for preparation for Economics 198B (the thesis process requires enrollment in a two-quarter sequence of Economics courses).
□ Present thesis in Economics 198B.
• No later than one quarter prior to graduation, the student must submit a project proposal to the Honors Committee for approval. The project itself must be submitted not later than the fifth week
of the last quarter before graduation.
• The Honors Committee will evaluate the project in consultation with the faculty sponsor, and may at its discretion, require a personal presentation by the student.
Approval as an active participant requires all of the following:
1. two lectures;
2. a written statement, signed by the instructor, describing the nature of the participation. This statement must be submitted to the Honors Committee no later than the end of the quarter in which
the seminar is given or the fifth week of the last quarter before graduation, whichever is sooner;
3. approval of the Honors Committee.
Requirements For Highest Honors At Graduation:
In addition to the above:
• Students who demonstrate exceptional achievement will be awarded Highest Honors.
• Decisions regarding projects, seminar participation, and Highest Honors will be made by the Honors Committee.
This Honors Program is independent of the honors sections of the mathematics courses. Graduation with Honors in Mathematics is also distinct from graduation with College Honors. Applications for
the Honors Program in Mathematics are available in the Student Services Office, 6356 Math Sciences. If you have any questions about the program, or special requests, you are welcome to consult any
members of the Mathematics Honors Committee, or see an Undergraduate Mathematics Counselor in 6356 Math Sciences.
September 2013
Honors Program in Mathematics of Computation
Admission to the Program:
To be considered for admission to the Honors Program in Mathematics of Computation, a student must:
• be officially enrolled in the Mathematics of Computation major;
• have completed at least four courses at UCLA in the Mathematics Department from those required in the “Preparation for the Major” or Major; and
• have at least a 3.6 GPA in such mathematics courses taken at UCLA.
Requirements For Honors At Graduation:
1. The student must have completed:
Mathematics 115AH, 131AH, 131BH and 132H; and
• Mathematics 151AB and any one course from 152-159 and
• Mathematics 134, 135, and any one course from 133, 136 or 146.
2. The student must either:
• submit an original project as described below; or
• take, as an approved active participant, any upper division or graduate seminar offered by the Department of Mathematics. Such participation is described below.
3. The student must have a GPA of at least 3.6 in upper division mathematics and statistics courses taken for the major.
Original Project:
• The project should involve some aspects of mathematical theory.
• The project is to be carried out under the sponsorship of a faculty advisor.
• The project may be done as part of a regular course, a special course (Mathematics 199), Summer REU project or by a special arrangement.
• No later than one quarter prior to graduation, the student must submit a project proposal to the Honors Committee for approval. The project itself must be submitted not later than the fifth week
of the last quarter before graduation.
• The Honors Committee will evaluate the project in consultation with the faculty sponsor, and may at it’s discretion, require a personal presentation by the student.
• Upper division seminars in Mathematics automatically count as mathematics electives for the major.
Requirements For Highest Honors At Graduation:
In addition to the above, the student must:
• earn a 3.8 GPA or higher in approved upper division and graduate mathematics courses.
• Decisions regarding projects, seminar participation, and Highest Honors will be made by the Honors Committee.
This Honors Program is independent of the honors sections of the mathematics courses. Graduation with Honors in Mathematics is also distinct from graduation with College Honors. Applications for
the Honors Program in Mathematics are available in the Student Services Office, 6356 Math Sciences. If you have any questions about the program, or special requests, you are welcome to consult any
members of the Mathematics Honors Committee, or see an Undergraduate Mathematics Counselor in 6356 Math Sciences.
December 2007
Honors Program in Mathematics/Economics
Admission to the Program:
To be considered for admission to the Honors Program in Mathematics/Economics, a student must:
• be officially enrolled in the Mathematics/Economics major;
• have completed all of the “Preparation for the Major” courses;
• have at least a 3.5 GPA in the Mathematics “Preparation for the Major” courses;
• have at least a 3.5 GPA in the Economics “Preparation for the Major” courses; and
• have completed Econ 11, Econ 101 and Econ 102 with an overall GPA of 3.5.
Requirements For Honors At Graduation:
• Complete Mathematics 115AH;
• Complete Mathematics 131AH and 131BH;
• Enroll in Economics 198A for preparation for Economics 198B (the thesis process requires enrollment in a two-quarter sequence of Economics courses).
• Present thesis in Economics 198B.
• Complete major requirements with at least a 3.5 GPA in the upper division Mathematics courses.
• Complete major requirements with at least a 3.5 GPA in the upper division Economics courses.
Requirements For Highest Honors At Graduation:
In addition to the above:
• Highest honors, awarded at the discretion of the Mathematics Departmental Honors Committee and the Mathematics-Economics IDP Committee, in consultation with the Economics Department, and are
based on grade-point average and quality of the senior thesis.
This Honors Program is independent of the honors sections of the mathematics courses. Graduation with Honors in Mathematics is also distinct from graduation with College Honors. Applications for
the Honors Program in Mathematics/Economics are available in the Student Services Office, 6356 Math Sciences. If you have any questions about the program, or special requests, you are welcome to
consult any members of the Mathematics Honors Committee, or see an Undergraduate Mathematics Counselor in 6356 Math Sciences.
January 2011
For more information, contact student services, ugrad[at]math.ucla.edu.
If you are declared into a math program (major, minor, or specialization) and DARS is not recognizing some of your completed coursework for the program, please check below to see if these scenarios
apply to you.
**ALL courses toward a math program be taken for letter grades.**
I took…
1. CS 31, 32, and 33 in lieu of PIC 10A, 10B, and 10C, respectively
2. Math 170E and 170S in lieu of Math 170A and 170B, respectively
3. PIC 16B for the specialization in computing
4. An IB Math exam. I have written approval (via email or Message Center) that I will be waived from Math 31A and/or Math 31B.
5. An equivalent course to one of the Math/PIC courses. My Transfer Credit Petitionwas approved (in writing, via email, or Message Center) for that course.
6. A graduate course in lieu of an upper division math course for the major. My Transfer Credit Petition was approved (in writing, through email or via Message Center) for that course.
7. AP Physics “C” Mechanics, AP Stats, and/or AP Chemistry exam(s) and received a 4 or 5. I contacted the respective departments to get credit for Physics 1A, Stats 10, and/or Chem 20A.
8. An equivalent course to one of the non-Math/PIC courses. My Transfer Credit Petition was approved (in writing, via email, or Message Center) for that course by the department that oversees that
• For Cases 1-6: Math advisors are aware of these course substitutions and account for them when we review your petition to declare a math program or audit your coursework in your last quarter for
• For Case 7 or 8: Please make sure the advisor of that department makes a note on your Record of Interview (ROI) about these approvals so that math advisors can account for them we review your
petition to declare a math program or audit your coursework in your last quarter for graduation.
If any of the above applies to you, we will update your DARS by Friday of Week 7 of your LAST quarter at UCLA. Please run a new audit in Week 8 and visit us if there are still any discrepancies.
I took upper division courses for my non-math minor, but DARS is using these courses for my upper division math major. I need 20.0 exclusive units to the stats minor.
We will update your DARS by Friday of Week 7 of your LAST quarter at UCLA by excluding the courses that should be exclusive to your minor. E.g. If you are a stats minor, we will exclude 20.0 units of
stats courses from your upper division math coursework.
Please run a new audit in Week 8 and visit us if there are still any discrepancies.
I can have 20.0 units of overlapping courses, but DARS is overlapping too many math courses between my majors.
We will update your DARS by Friday of Week 7 of your LAST quarter at UCLA by excluding some of your upper division courses from your math major.
Please run a new audit in Week 8 and visit us if there are still any discrepancies.
1. Model Your DARS
1. Go to DARS.
2. Click Run Audit.
3. Click “Select a Different Program.”
4. Select “College of Letters and Science – LS” and the term in which you plan to declare.
5. Select your prospective program. This is called “modeling your DARS.” You can model three different majors and two minors to compare different major requirements all at once.
6. Click Run Different Program. This will show you your pre-major and major course requirements. You can click on the courses to see its description and prerequisites.
2. Check the Tentative Schedule
• Use our Tentative Schedule to see when we plan to offer each course. We try to keep our offerings as consistent as possible each year. Use the current school year’s tentative schedule as a soft
guide to plan every quarter, but please remember that things may change based on our resources.
3. Fill Out the Degree Plan Contract
• Fill in the Degree Plan Contract (DPC), using the list of courses from DARS and the tentative schedule. Remember that some courses have prerequisites and are offered in specific quarters.
□ List specific course numbers if the course is EXPLICITLY listed for your major.
☆ E.g., Math 115A and 131A are specifically listed for every major. Math students should explicitly list Math 115A and 131A in their plan.
☆ If you are listing a specific class, make sure you know what the prerequisites are. You can find prerequisites by clicking on the class name. A course catalog will pop up in a new window
with details on the prerequisites and course description.
□ Upper division electives just need to be listed as “[Subject] UD.”
☆ E.g., The math of computation major requires “6 upper division math electives” and “3 upper division computer science electives.” Math of Comp students should just list “Math UD” 6 times
and “CS UD” 3 times in their plan.
4. Have an Advisor Review Your Plan
All mathematics preparation (pre-major) courses for mathematics majors (with the exception of the three majors below) must be passed with a “C” or better and an overall pre-major GPA of 2.5. All
pre-major & major courses MUST be taken for letter grades!
Data Theory
• All pre-major courses must be passed with a “C” or better and an overall pre-major GPA of 3.3. Math 115A is included as a pre-major course and must be passed with a “C” or better.
Financial Actuarial Mathematics
• Each pre-major course must be passed with a “C” or better. Pre-major GPAs for mathematics and economics courses are calculated separately, and both pre-major GPAs must meet a minimum 2.5.
• Each pre-major course must be passed with a “C” or better. Pre-major GPAs for mathematics and economics courses are calculated separately, and both pre-major GPAs must meet a minimum 2.7.
Each pre-major has a corresponding major. All students must declare a math major before completing 160.0 units (minus AP credit).
All mathematics majors (with the exception of Data Theory) must pass Math 115A with a “C-” or better. All mathematics majors (except Data Theory) must pass Math 131A with a “C-” or better.
Financial Actuarial Mathematics and Mathematics/Economics majors must pass Econ 101 and 102 with a “C-” or better.
Detailed requirements are listed in the general catalog. For more information, contact Student Services: ugrad[at]math.ucla.edu.
Students who are planning to pursue graduate studies in mathematics or related fields are strongly encouraged to major in Mathematics, Applied Mathematics, or Mathematics of Computation. | {"url":"https://ww3.math.ucla.edu/majors-minors-specializations/","timestamp":"2024-11-08T11:04:41Z","content_type":"text/html","content_length":"178384","record_id":"<urn:uuid:f1527a6e-b996-482c-b808-2cc96514cf3e>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00236.warc.gz"} |
STATEMENT-1 : In the expansion of (5+31/5)10, sum of integral ... | Filo
Question asked by Filo student
STATEMENT-1 : In the expansion of , sum of integral terms is 3134 . and STATEMENT-2 : .
a. Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement
b. Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statemer
c. Statement-1 is True, Statement-2 is False
d. Statement-1 is False, Statement-2 is True
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Question Text STATEMENT-1 : In the expansion of , sum of integral terms is 3134 . and STATEMENT-2 : .
Updated On Nov 26, 2022
Topic Algebra
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 104
Avg. Video Duration 19 min | {"url":"https://askfilo.com/user-question-answers-mathematics/statement-1-in-the-expansion-of-sum-of-integral-terms-is-32383838393038","timestamp":"2024-11-14T08:37:10Z","content_type":"text/html","content_length":"273821","record_id":"<urn:uuid:6bd15e6a-e260-44bb-b92c-5fabd834d1e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00265.warc.gz"} |
What are the signs of mycoplasma?
My 7yo has had a persistent dry cough for over a week, and he is just not feeling well, and his temperament is quite affected. Pandas ds13 is acting out. Could it be that the 7yo has walking
pneumonia? I took him in Saturday and they said his lungs where clear. I know something's up, but I really am afraid if I go there playing the paranoid mother I may loose whatever little credibility
I have left at the peds, and then I'll get no help.
What are the clinical signs of m. pneumonia? Would they hear anything in the lungs? Do they do x-rays? How is it diagnosed?
My son had or still has mycoplasma pne. He did have a cough but we would not have known he had it had we not had a blood test. Just like testing for strep titers they test for mycoplasma pneumonia
igg and igm.
My non-PANDAS son had a cough when he was dx with Myco P. We dismissed it since we thought the cough was due to allergies. Anyway, our doc just listened to his chest and diagnosed it.
My non-PANDAS son had a cough when he was dx with Myco P. We dismissed it since we thought the cough was due to allergies. Anyway, our doc just listened to his chest and diagnosed it.
We have allergies in the picture too, but the lungs checked out clear. I just don't know what to do anymore. I can't watch another child go down the PANDAS hill. If his mood and the cough don't clear
in a couple more days, I'll have to take him back in and insist they run some bloodwork
Common symptoms include the following:
•Chest pain
•Cough, usually dry and not bloody
•Excessive sweating
•Fever (may be high)
•Sore throat
Less common symptoms include:
•Ear pain
•Eye pain or soreness
•Muscle aches and joint stiffness
•Neck lump
•Rapid breathing
•Skin lesions or rash
I agree. If the cough stays then get it rechecked.Cough was my son's only symptom.
My non-PANDAS son had a cough when he was dx with Myco P. We dismissed it since we thought the cough was due to allergies. Anyway, our doc just listened to his chest and diagnosed it.
We have allergies in the picture too, but the lungs checked out clear. I just don't know what to do anymore. I can't watch another child go down the PANDAS hill. If his mood and the cough don't
clear in a couple more days, I'll have to take him back in and insist they run some bloodwork
My non-PANDAS daughter had Myco last year, and a dry cough was her only noticiable symptom. Dr. dx with just an office visit.
My ds13 has had a lingering dry cough for over a month - he's been on zithromax for about a week now - is myco not responsive to zithromax or should we be looking for a different culprit?
Our non PANDAS son had surgery in October and about 5 days post op (on a weekend, of course), he developed a dry cough that was persistent and sounded a bit odd. Since he had been intubated during
surgery for several hours, I suspected mycoplasm. I took him to an urgent pediatric care center close to our home and they said he just had a cold virus. Two weeks later, the cough was a bit worse, ,
and our PANDAS daughter, who had been symptom free for months, walked into the kitchen after being mildly scolded and flipped a chair over on its side (very uncharacteristic for her). I turned to my
husband and said, "Our son has mycoplasm and it's causing our daughter's symptoms to flare". I made an appointment the next morning with our pediatrician and he listened to our son's chest and
thought it sounded suspicious. He sent us for x-rays and sure enough, it was mycoplasm. Within 72 hours of our son starting antibiotics, our PANDAS daughter was back to her normal self. She never
contracted the illness as she was on a prophylactic dose of azith. However, she was obviously making antibodies to it which caused a mild exacerbation of behavior.
Edited by DebC
Myco P is suppose to be responsive to Zithromax but our PANDAS PITAND kids don't always fit the majority rule. My non-PANDAS son was give a 5 day script for Zith, the cough lasted for about a week
after the script ended (doc said that's expected and normal) then he was fine.
I remember reading when Myco P first became a highly discussed subject on here that some were being given Biaxin. Whether or not that finally cleared it, I don't know.
My ds13 has had a lingering dry cough for over a month - he's been on zithromax for about a week now - is myco not responsive to zithromax or should we be looking for a different culprit?
Edited by Vickie
Just learned that our 4 y/o has/had this by chance with a blood test Dr B. ordered. No symptoms upper resp.-wise, but rashy as could be for several weeks and behavioral symptoms in that time period.
Last thing I would have ever guess to come back postitive and it was a really high number-tested with Igg & Igm.
• 1 month later...
My dd9 has chronic mycoplasma and does not have anything ever show in her chest. And she has battled that dry cough for years:( It wasn't until it turned into PITANDS recently that we did the blood
work and was positive.
If you suspect, you should just take him to the immunologist and get tested.
I had persistent cough and asthma that I just couldn't get rid of for 3 years! When symptoms would escalate, the only abx that ever worked was azith (which happens to be one of the two top abx for
mycoP.) Even though I went to 2 asthma specialists and a pulmonologist, no one thought to test for mycoP. When my kids went to see Dr. B. he tested the whole family for strep and mycoP. That's how I
found out that my IgM and IgG were both very positive for it (I had managed to spread it to everyone else.)
If you are even slightly suspicious, ask the dr. to test for it. It's a simple blood test. I'm still trying to get over the symptoms 10 months later, and am seeing a lung specialist at the end of May
to make certain that we are getting it all, and that I don't have permanent lung damage from it.
• 1 year later...
I had persistent cough and asthma that I just couldn't get rid of for 3 years! When symptoms would escalate, the only abx that ever worked was azith (which happens to be one of the two top abx
for mycoP.) Even though I went to 2 asthma specialists and a pulmonologist, no one thought to test for mycoP. When my kids went to see Dr. B. he tested the whole family for strep and mycoP.
That's how I found out that my IgM and IgG were both very positive for it (I had managed to spread it to everyone else.)
If you are even slightly suspicious, ask the dr. to test for it. It's a simple blood test. I'm still trying to get over the symptoms 10 months later, and am seeing a lung specialist at the end of
May to make certain that we are getting it all, and that I don't have permanent lung damage from it.
Good to know! My youngest and myself are both allergic. I'm asthmatic and he was just evaluated for asthma two weeks ago. At the same time, my oldest started to flare up with PANDAS symptoms again.
I'm thinking we should all be tested for mycoplasma? The blood test will be simple for me, but not so simple for the 6 and 7 years olds in question
For my son, then just turned 4,cough, typical kid fever (not terribly high but in the low 100's for a day or 2), chest that got tighter sounding. Took him in and he was dx w/walking pneumonia. Titers
showed Myco was elevated whereas 6 weeks prior to the pneumonia when titers were taken, Myco was normal. | {"url":"https://latitudes.org/forums/topic/13014-what-are-the-signs-of-mycoplasma/","timestamp":"2024-11-03T03:36:56Z","content_type":"text/html","content_length":"233864","record_id":"<urn:uuid:1675cd1a-cc89-4a34-9bfd-b48173d68153>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00785.warc.gz"} |
Quasicrystals - Quantum Gravity Research
The existence of quasicrystals in matter was firmly believed by the scientific community to be absolutely impossible.
Then Paul Steinhardt predicted that they must exist.
Then Dan Schechtman discovered them in matter. Synthetic matter, but matter.
And then they were discovered in nature. In meteor fragments – but nature.
A quasicrystal is an aperiodic, but not random, pattern. A quasicrystal in any given dimension is created by projecting a crystal – a periodic pattern – from a higher dimension to a lower one. For
example, imagine projecting a 3-dimensional checkerboard – or cubic lattice made of equally sized and equally spaced cubes – onto a 2D plane at a certain angle. The 3D cubic lattice is a periodic
pattern that stretches out infinitely in all directions. The 2-dimensional, projected object is not a periodic pattern. Rather, it is distorted due to the angle of projection, and instead of
containing only one shape that repeats infinitely like the 3D crystal does, it contains a finite number of different shapes (called proto-tiles) that combine with one another in specific ways
governed by a set of mathematical/geometrical rules to fill the 2D plane in all directions. It is possible, with the correct mathematical and trigonometrical toolkit to actually recover the mother
object in 3D (the cubic lattice in this example) by analyzing the 2D projection. A famous example of a 2D quasicrystal is the Penrose tiling conceived by Roger Penrose in the 1970’s, in which a 2D
quasicrystal is created by projecting a 5-dimensional cubic lattice to a 2D plane.
Emergence theory focuses on projecting the 8-dimensional E8 crystal to 4D and 3D. When the fundamental 8D cell of the E8 lattice (a shape with 240 vertices known as the “Gosset polytope”) is
projected to 4D, two identical, 4D shapes of different sizes are created. The ratio of their sizes is the golden ratio. Each of these shapes are constructed of 600 3-dimensional tetrahedra rotated
from one another by a golden ratio based angle. We refer to this 4D shape as the “600-Cell”. The 600-cells interact in specific ways (they intersect in 7 golden-ratio related ways and “kiss” in one
particular way) to form a 4D quasicrystal. We then project this 4D quasicrystal to 3D to form a 3D quasicrystal that has one type of proto-tile: a 3D tetrahedron.
To learn more on the fascinating new world of quasicrystals, here are some academic papers on the subject:
Dan Shechtman, Ilan Blech (1984). “Metallic Phase with Long-Range Orientational Order and No Translational Symmetry.”
Alan Mackay (1982). “Crystallography and the Penrose Pattern.”
Veit Elser, N.J.A Sloane. “A Highly Symmetric Four Dimensional Quasicrystal”.
Michael Baake, Franz Gähler. “Symmetry Structure of the Elser-Sloane Quasicrystal”.
Justus A. Kromer, et al. “What Phasons Look Like: Particle Trajectories in a Quasicrystalline Potential.”
Marjolein N. van der Linden, Jonathan P.K. Doye, Ard A. Louis (2012) “Formation of dodecahedral quasicrystals in two-dimensional systems of patchy particles”
Pablo F. Damasceno, et al. (2012) “Predictive Self-Assembly of Polyhedra into Complex Structures.”
John Gardiner (2012) “Fibonacci, quasicrystals and the beauty of flowers.”
Kleman, Maurice (2011). Cosmic Forms.
Amir Haji-Akbari, et al. (2009) “Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra.”
Kleman, Maurice (2002). Phasons and the Plastic Deformation of Quasicrystals.
Dmitrienko, V E.; Kléman, M. (2001). Tetrahedral structures with icosahedral order and their relation to quasicrystals.
Dmitrienko, V E.; Kléman, M. (1999). Icosahedral order and disorder in semiconductors.
Henley, C.L. (1986). Sphere Packings and local environments in Penrose tilings. | {"url":"https://quantumgravityresearch.org/portfolio/quasicrystals/","timestamp":"2024-11-07T16:55:49Z","content_type":"text/html","content_length":"81330","record_id":"<urn:uuid:0dd1b871-519c-449f-a1ea-0c944a1c3776>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00217.warc.gz"} |
Laws of thermodynamics
From the article you will learn what entropy is, how to quantify disorder and how to calculate, will there be a reaction?
Laws of thermodynamics
The first law of thermodynamics states that energy is conserved during any process, but this is not enough, to determine if a reaction will occur with enough energy. For example, if in a room with a
book if there is a lot of heat on the floor, then the book will not rise on the table, even if the amount of heat supplied it will be disproportionately more than the required energy. Reactions that
occur by themselves are often accompanied by loss energy, but this cannot serve as a criterion by which we can predict: will a reaction occur?
Any process occurs in such a way that the more desirable state of the system is less ordered (everything tends to disorder). Mathematically, disorder is more likely, for example: imagine two
connected vessels, put one molecule in them. The probability that the molecule will be in vessel A is 50%, in vessel B is 50%. If we put two molecules, we will get four possible combinations: the
molecules will be in the vessel A - 25%, in the vessel B - 25%, in two different vessels - 50%. If we try to place four molecules, then the probability that the molecules grouped in pairs - 3/8. If
we increase the number of molecules, then we will see a tendency that the probability of finding molecules in one vessel becomes infinitely small, whereas the probability of a uniform distribution
always more.
Thus, any system tends to a more probable state - a less ordered one. A mess it can be expressed in terms of the number of possible combinations of energy distribution, for example, if we take a
crystal containing eight atoms, lower the temperature to absolute zero, then all eight atoms will be to be in the vertices of the crystal and the transfer of energy between them is impossible, so
there is only one possible combination and W=1. If you give the crystal enough energy in order to transfer one of the atoms to an excited state, we will get eight possible states of the system, so W=
The third law of thermodynamics
The third law of thermodynamics can be formulated as follows: the entropy of a pure solid crystal is zero at absolute zero.
Entropy is a measure of disorder, expressed by S = k • ln W, where k is the Boltzmann constant defining the dependence of energy on temperature. In an isolated system (a system in which energy
transfer is impossible or matter beyond its limits) each change of state means that the system is moving into a more probable state, i.e. entropy increases.
In an uninsulated system, energy can be exchanged with the environment. Thus, the entropy change this is the sum of the ΔS[environment] and ΔS[system], i.e. the entropy change is:
ΔS = ΔS[environments] + ΔS[systems] (1)
So we can deduce that the sum of any system and environment is the universe. Rudolf Clausius formulated the first and second laws of thermodynamics are as follows:
• The energy of the universe is constant
• The entropy of the universe is constantly increasing
Gibbs energy
For any process that occurs at a constant temperature, the change in the entropy of the medium depends on the amount of the heat absorbed by the system and the temperature at which the heat was
ΔS = heat/T (2)
The heat absorbed by the medium is -q, the system is - +q, at constant pressure q =ΔH, hence, at constant P and T we get:
ΔS = ΔH[environments]/T= -ΔH[systems]/T (3)
ΔS[Σ] = ΔS[systems] + ΔS[environments] (4)
ΔS[Σ] =ΔS[systems] - ΔH[systems]/T (5)
multiplying by -T, we get:
-TΔS[Σ] = ΔH[systems] - TΔS[systems] (6)
The value -TΔS[Σ] is called Gibbs free energy or simply free energy:
G = H - TS
H and S are state functions, so G will also be a state function, that is, it does not depend on how the system came to this state:
ΔG = G[2] - G[1] = H[2] - H[1] - (T[2]S[2] - T[1]S[1]) (7)
According to the second law of thermodynamics, entropy increases in processes that occur by themselves, thus, at constant temperature and pressure, if ΔG > 0, then the reaction will not occur.
Using the third law, we can calculate ΔS for the temperature change from absolute zero to T: ΔS[0→T] = S[T] - S[0] = S[T] - 0 = S[T]. So, S[T], the absolute entropy of a substance at temperature T is
calculated and tabulated for many substances, entropy values for a given temperature are taken from the reference book. | {"url":"https://en.k-tree.ru/articles/chemistry/general/thermodynamics_laws","timestamp":"2024-11-04T10:36:51Z","content_type":"application/xhtml+xml","content_length":"384858","record_id":"<urn:uuid:b3b8e0af-184e-4c5d-ba4e-673423b9c568>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00768.warc.gz"} |
Subsets – Sets and functions – Mathigon
Sets and functionsSubsets
The idea of set equality can be broken down into two separate relations: two sets are equal if the first set contains all the elements of , and .
Definition (Subset)
Suppose and are sets. If every element of is also an element of , then we say is a subset of , denoted .
If we visualize a set as a potato and its elements as dots in the blob, then the subset relationship looks like this:
Here has elements, and has elements.
Two sets are equal if .
The relationship between "⊂" and "=" has a real-number analogue: we can say that x=y if and only if .
Think of four pairs of real-world sets which satisfy a subset relationship. For example, the set of cars is a subset of the set of vehicles.
Suppose that is the set of even positive integers and that is the set of positive integers which are one more than an odd integer. Then .
Solution. We have , since the statement " is a positive even integer" the statement " is one more than an odd number". In other words, implies that .
Likewise, we have , because " is one more than an positive odd integer" " is a positive even integer".
Finally, we have , since .
Drag the items below to put the sets in order so that each set is a subset of the one below it. | {"url":"https://et.mathigon.org/course/sets-and-functions/subsets","timestamp":"2024-11-11T18:11:39Z","content_type":"text/html","content_length":"68987","record_id":"<urn:uuid:57437915-8b33-4969-bda8-9cb5ea85c97a>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00838.warc.gz"} |
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Keras documentation: GroupNormalization layer
GroupNormalization layer
GroupNormalization class
Group normalization layer.
Group Normalization divides the channels into groups and computes within each group the mean and variance for normalization. Empirically, its accuracy is more stable than batch norm in a wide range
of small batch sizes, if learning rate is adjusted linearly with batch sizes.
Relation to Layer Normalization: If the number of groups is set to 1, then this operation becomes nearly identical to Layer Normalization (see Layer Normalization docs for details).
Relation to Instance Normalization: If the number of groups is set to the input dimension (number of groups is equal to number of channels), then this operation becomes identical to Instance
• groups: Integer, the number of groups for Group Normalization. Can be in the range [1, N] where N is the input dimension. The input dimension must be divisible by the number of groups. Defaults
to 32.
• axis: Integer or List/Tuple. The axis or axes to normalize across. Typically, this is the features axis/axes. The left-out axes are typically the batch axis/axes. -1 is the last dimension in the
input. Defaults to -1.
• epsilon: Small float added to variance to avoid dividing by zero. Defaults to 1e-3
• center: If True, add offset of beta to normalized tensor. If False, beta is ignored. Defaults to True.
• scale: If True, multiply by gamma. If False, gamma is not used. When the next layer is linear (also e.g. nn.relu), this can be disabled since the scaling will be done by the next layer. Defaults
to True.
• beta_initializer: Initializer for the beta weight. Defaults to zeros.
• gamma_initializer: Initializer for the gamma weight. Defaults to ones.
• beta_regularizer: Optional regularizer for the beta weight. None by default.
• gamma_regularizer: Optional regularizer for the gamma weight. None by default.
• beta_constraint: Optional constraint for the beta weight. None by default.
• gamma_constraint: Optional constraint for the gamma weight. None by default. # Input shape Arbitrary. Use the keyword argument input_shape (tuple of integers, does not include the samples axis)
when using this layer as the first layer in a model. # Output shape Same shape as input.
Call arguments
• inputs: Input tensor (of any rank).
• mask: The mask parameter is a tensor that indicates the weight for each position in the input tensor when computing the mean and variance.
Reference: - Yuxin Wu & Kaiming He, 2018 | {"url":"https://keras.io/2.15/api/layers/normalization_layers/group_normalization/","timestamp":"2024-11-07T20:14:55Z","content_type":"text/html","content_length":"17057","record_id":"<urn:uuid:66a03710-72ef-4548-8c67-2f6c667db40b>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00619.warc.gz"} |
The Stacks project
Lemma 67.40.8. Let $S$ be a scheme. Let
\[ \xymatrix{ X \ar[rr]_ h \ar[rd]_ f & & Y \ar[ld]^ g \\ & B } \]
be a commutative diagram of morphism of algebraic spaces over $S$. Assume
1. $X \to B$ is a proper morphism,
2. $Y \to B$ is separated and locally of finite type,
Then the scheme theoretic image $Z \subset Y$ of $h$ is proper over $B$ and $X \to Z$ is surjective.
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GP CGPA to Percentage Calculator | myCBSEguide
GP CGPA to Percentage Calculator
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GP CGPA to Percentage Calculator CBSE Class 10 report card has GP and CGPA only. There is no percentage of marks. You can calculate estimated percentage in individual subjects and overall percentage.
The above formula given by CBSE to convert grades into marks and percentage is only an approximation and not exact.
CBSE CCE CGPA & Overall Percentage Calculator
New Features GP CGPA to Percentage Calculator
• Rename subject name to any alphanumeric value. For example, from “SUBJECT 1” to “Mathematics”, “English” etc.
• Your input will saved temporarily within the browser, which can be “reloaded” on your next visit.
• Grade Chart and Result tables can now be downloaded in a single PDF file.
GP CGPA to Percentage Calculator
Calculation of GP to subject wise percentage:
│GP (Grade Points) │Calculation│Estimated Percentage │
│10 │10 X 9.5 │95 % │
│09 │9 X 9.5 │85.5% │
│08 │8 X 9.5 │76% │
│07 │7 X 9.5 │66.5% │
│06 │6 X 9.5 │57% │
│05 │5 X 9.5 │47.5% │
│04 │4 X 9.5 │38% │
GP CGPA to Percentage Calculator
Calculation of overall percentage :
CGPA X 9.5 = overall percentage
Example :
Your CGPA (Cumulative Grade Point Average) is : 7.6
Your overall percentage will be : 7.6 X 9.5 = 72.2 %
Note : Abbreviations used against Result :
** = Upgraded grade by one level
QUAL – Eligible for Qualifying Certificate,
EIOP – Eligible for Improvement of Performance,
NIOP – Not Eligible for Improvement of Performance,
XXXX – Upgradation of Performance/Additional Subject
N.E. – Not Eligible,
R.W. – Result Withheld,
R.L. – Result Later,
UFM – Unfair means,
SJD – Subjudice?
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21 thoughts on “GP CGPA to Percentage Calculator”
1. i m very thankful to this site for giving every possible news at time .every news was enough and correct accept few
2. to calculate % above 95, I think this formula will work.
percentage= (CGPA x 15) – 50 ;
3. if cgpa X 9.5 = percentage then the highest percentage obtained will be 95%, i.e. 10.0 X 9.5 = 95%. so wat about others who get more than 95%??
4. if i have got 98.79%in summative 1 so what i should do how i calculate my cgpa
5. WHAT ABOUT THE G.P. LESS THAN 04?????????????????????????????? IT SHOULD BE GIVEN NA?????????????? CAUSE WE ARE SILLY CHILDREN—- OUR PRINCIPAL AND TEACHERS TOLD ABOUT IT…………….
6. sir,
Please the me how to calculate CGPA with full details and an example
7. what to do to calculate accurate % ????????
i got cgpa 10 ………
8. i don no wat is cgpa
9. in my school, there are 11 students who got cgpa as 10….. so how to know who is the highest… and my questions also resemble with deblina and vaibhav
10. Yaa this is a bst website for me . It gives me true information. Thanks
11. is cgpa only calculated for the final board exam ? or is it calculated for the whole 10th grade? pls answer fast
12. plz tell me how to calculate cgpa coz i dont know grace point of particular subject and want to convert subject marks into grace point like i f i got 65 out of 90 then what will be gpa of this
13. i got only 7.6 as cgpa is it a good mark
14. My friend is got marks in science FA-A 1,SA-C 1,TOTAL-A 2**,GP-09 and I got same marks in science FA,SA but my total is B 1**and GP is 08 why?
15. CGPA 7 belongs in which category Average or below average or very good or excellent
16. How did you get that formula?
17. 10 CGPA
18. If I got 65 percent in school then how much I required marks in each sub to get 10 cgp
19. what is full form of FT written on 12th std result marks sheet
20. Science paper very low number coming pleas ree chacking my copy
21. Great article on grading systems! It’s fascinating to see the diversity in grading practices across different schools and states. The detailed explanations on letter grades, GPA, and alternative
assessment methods are incredibly insightful. This comprehensive guide is perfect for anyone navigating the complexities of educational evaluation. By the way, the CGPA to Percentage Converter
tool mentioned here is a fantastic resource for students. Keep up the fantastic work!
Leave a Comment | {"url":"https://mycbseguide.com/blog/gp-cgpa-to-percentage-calculator/","timestamp":"2024-11-03T20:30:09Z","content_type":"text/html","content_length":"125480","record_id":"<urn:uuid:8093919f-93d4-47e0-b6da-24bcc4d1d1e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00500.warc.gz"} |
Complexometric titration - equivalence point calculation
At equivalence point we have just a solution of complex, and calculation of concentration of the ion is very similar to the precipitation titration case, we just have to account for the complex
concentration. For that we have to know formation constant and complex stoichiometry. We could derive a formula similar to that derived in precipitation titration section, but as it was already
signalled its use will be very limited. Most chelating reagents are weak acids and/or bases, so you quite often have to account for pH and speciation of ligand forms.
What is pAg at the equivalence point if the 0.01 M solution of cyanide is titrated with AgNO[3] solution? Assume pH is so high CN^- hydrolysis can be neglected, ignore dilution effects. log K[f] =
First of all - please note, that we are asked to calculate not concentration of titrated substance, but concentration of titrant (Ag^+). Second, in the Liebig-Dénigès method of cyanides determination
(which the question asks about), titrated solution is alkalized with ammonia, so the pH is high and assumption about lack of the hydrolysis is close to the reality.
Equation of the complexation reaction is
Ag^+ + 2CN^- → Ag(CN)[2]^-
and the formation constant is
At the equivalence point there are stoichiometric amounts of metal and ligans mixed so solution doesn't differ from the one prepared just by dissolving complex - and that means that
so formation constant can be written as
This can be easily solved for [Ag^+]. However, to calculate concentration of Ag^+ we need one additional assumption - that Ag(CN)[2]^- concentration can be treated as constant. Formation constant is
relatively high, so this assumption is probably correct - but we will check it once we calculate Ag^+ concentration.
As is obvious from the stoichiometry of the reaction, concentration of the complex is half that of initial cyanide concentration (in the real experiment it would be even lower, but we were asked to
ignore dilution effects).
and finally
It is time to check if our assumption was correct. Seems it was. Concentration of silver is 2×10^-6 that of the complex concentration, which means we can safely ignore changes of the complex
dissociation due to its dissociation.
What is pZn at equivalence point if 0.005 M ZnCl[2] solution is titrated with 0.01 M solution of EDTA at pH 10? Complex formation constant log K[f] = 16.8, pK[a1]=2.07, pK[a2]=2.75, pK[a3]=6.24, pK
At the equivalence point we have just a solution of complex ZnEDTA^2-. Complex dissociates according to the reaction
ZnEDTA^2- → Zn^2+ + EDTA^4-
and the formation constant is
Problem is, EDTA^4- is a weak acid. Once the complex dissociates, ligand reacts with water, its concentration goes down and complex dissociates further. To be able to calculate concentration of Zn^2+
we have to account for EDTA^4- hydrolyzis.
How to do it? Let's start taking a look at given pK[a] values. We know that when pH=pK[a] concentrations of acid and conjugate base are identical. When pH changes by one unit, ratio of concentrations
changes tenfold. That means that we can safely ignore pK[a1], pK[a2] and pK[a3] - pH is so high concentrations of these acids will be neglectfully low. But what about pK[a4]?
This one can be not ignored, we are just 0.34 pH unit below. Let's take a look at acid dissociation constant:
After some rearranging (very similar to that used when deriving Henderson-Hasselbalch equation) this equation takes form
which allows easy calculation of ratio of concentrations of acid and conjugate base. Note, that the same equation can be used for any acid.
From the stoichiometry and mass balance we also know that concentration of Zn^2+ present in the solution must be identical to sum of concentrations of both EDTA^4- and HEDTA^3-:
We can easily get rid of [HEDTA^3-] - it can be expressed in terms of known pH, pK[a4] and yet unknown [EDTA^4-]:
(notice sign change in exponent) and substituted into mass balance:
Combined with complex formation constant:
In general we should also account for the fact, that some amount of ZnEDTA^2- complex dissociated, but we won't - formation constant is high enough so that we can treat [ZnEDTA^2-] as constant. To be
sure we will check correctness of the assumption in the final step of calculations.
Solving for [Zn^2+]:
The only thing we need now is the equivalence point concentration of ZnEDTA^2- complex. We started with 0.005 M solution of Zn^2+ and 0.01 M solution of EDTA. As these substances react 1:1 to each V
[Zn] we had to add V[Zn]/2 of EDTA solution (compare concentrations if you don't know why), final volume is 1.5V[Zn] and complex concentration is 0.005V[Zn]/1.5V[Zn]. Combining this concentration
with pH and equation derived above, we get
Finally, we should check if our assumption that concentration of the complex can be treated as constant is valid - and it obviously is, as concentration of dissociation products is about 8 orders of
magnitude lower. | {"url":"http://www.titrations.info/complexometric-titration-equivalence-point-calculation","timestamp":"2024-11-06T20:27:09Z","content_type":"text/html","content_length":"50250","record_id":"<urn:uuid:a30d0b14-b72a-4a2e-a811-c7c6b2f79352>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00750.warc.gz"} |
08 Span
If $S=\{\vec{v}_{1}, \vec{v}_{2}, \dots, \vec{v}_{k}\}$ is a set of vectors, then the span of $S$, denoted by $\text{span}\{S\}$. is the set of all possible linear combinations of the vectors in $S$.
$$ \text{span}\{\vec{u}, \vec{v}\} = \{ c\vec{u}+d\vec{v}\; |\; c,d \in R \} $$
• Asking whether a vector is in the span of some other vectors is the same as asking if it is in the span of those other vectors. | {"url":"https://samienr.com/notebook/linalg/08-span/","timestamp":"2024-11-06T04:06:21Z","content_type":"text/html","content_length":"9275","record_id":"<urn:uuid:144093f9-88e7-47ca-8fd2-b02222c7c636>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00017.warc.gz"} |
The Wars of the Matrix GANGs
The Wars of the Matrix GANGs is a Rounding the Earth article series written by Mathew Crawford.
GANG is an acronym for Governance by Aggressive Nonsensical Gusuism.
See also The Information Wars.
Glitches in the Matrix
Magic of the Matrix
The Science of the Matrix
Technocracy and the Matrix | {"url":"https://www.campfire.wiki/doku.php?id=rounding_the_earth:the_wars_of_the_matrix_gangs","timestamp":"2024-11-09T10:37:54Z","content_type":"application/xhtml+xml","content_length":"22693","record_id":"<urn:uuid:e08023d3-6c10-4084-8032-4a4a0c705e1c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00434.warc.gz"} |
Mathematica/3DContours - Wikibooks, open books for an open world
Mathematica supports 3D contours on surfaces using the option MeshFunctions.
MeshFunctions draws ISO lines of an arbitary function of the x,y,z values, and for some functions additional parameters.
For a conventional set of altitude contours we just use the identity function of the z value (the third parameter) using MeshFunctions->{#3&}
Plot3D[Sin[x y], {x, 0, 3}, {y, 0, 3}, MeshFunctions -> {#3 &}]
In addition the region between the contours can be colored using the option MeshShading. The following example divides the image into 10 altitude contours and colors them in gray scale.
Plot3D[Sin[x y], {x, 0, 3}, {y, 0, 3}, Mesh -> 10, MeshFunctions -> {#3 &}, MeshShading -> GrayLevel /@ Range[0, 1, 0.1], Lighting -> "Neutral"] | {"url":"https://en.m.wikibooks.org/wiki/Mathematica/3DContours","timestamp":"2024-11-13T02:14:20Z","content_type":"text/html","content_length":"21277","record_id":"<urn:uuid:4bf58147-e53d-4218-81ae-6e5f26b19aa6>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00471.warc.gz"} |
Square root of a complex number
1. Square root
Square root of a complex number
This online tool calculates the square root of a complex number.
Square root of a complex number
Let z be a complex number written in its algebraic form,
`z = a + i * b`, a and b are two real numbers.
Then, the square root of z is the complex number R such as,
`R = x + i * y`, x and y are real numbers and,
`R^2 = z`
`(x + i * y)^2 = a + i * b`
We are searching for real numbers x and y satisfying,
`(x + i * y)^2 = a + i * b`
`x^2 - y^2 + 2*x*i*y = a + i * b`
We then get a system of two equations and two unknown x and y.
`{(x^2 - y^2 = a),(2*x*y = b):}`
We notice that it will be easier to calculate `x^2` and `y^2` first. To do this, we use the modulus as follows,
`|R^2| = |z|`
`x^2+y^2 = sqrt(a^2+b^2)`
We rewrite our system of equations,
`{(x^2 - y^2 = a),(2*x*y = b),(x^2+y^2 = sqrt(a^2+b^2)):}`
Using Equations (1) and (3), we deduce,
`x^2 = (sqrt(a^2+b^2)+a)/2`
`y^2 = (sqrt(a^2+b^2)-a)/2`
`x = +-sqrt((sqrt(a^2+b^2)+a)/2)`
`y = +-sqrt((sqrt(a^2+b^2)-a)/2)`
To determine the signs of x and y, just use equation (2).
- if b > 0 then x and y have the same sign, the 2 solutions of the equation system are
First solution: `x = sqrt((sqrt(a^2+b^2)+a)/2)` and `y = sqrt((sqrt(a^2+b^2)-a)/2)`
x + i* y is a first root of z
Second solution: `x = -sqrt((sqrt(a^2+b^2)+a)/2)` and `y = -sqrt((sqrt(a^2+b^2)-a)/2)`
x + i * y is the second root of z.
- if b < 0 then x and y have opposite signs, the solutions of the system of equations are
First solution: `x = sqrt((sqrt(a^2+b^2)+a)/2)` and `y = -sqrt((sqrt(a^2+b^2)-a)/2)`
x + i* y is a first root of z
Second solution: `x = -sqrt((sqrt(a^2+b^2)+a)/2)` and `y = sqrt((sqrt(a^2+b^2)-a)/2)`
x + i * y is the second root of z.
- if b = 0 then y=0 and z is a real number (z = a), we find the trivial roots of a real number,
`x = sqrt(a)` or
`x = -sqrt(a)`
Calculate the root of z = 1-i
The root of z is denoted by R, so we have `R^2 = z = 1 - i`
`R = x+i*y`
`(x+i*y)^2 = 1-i`
(1) `x^2 - y^2 = 1`
(2) `2*x*y = -1`
By the way, `|R^2| = |z|` therefore,
(3) `x^2+y^2 = sqrt(2)`
By combining (1) and (3) we obtain,
`x^2 = (sqrt(2)+1)/2`
`y^2 = (sqrt(2)-1)/2`
`x = +-sqrt((sqrt(2)+1)/2)`
`y = +-sqrt((sqrt(2)-1)/2)`
Now according to (2) `x*y = -1/2` therefore x and y have opposite signs,
The solutions of the system are,
First solution: `x = sqrt((sqrt(2)+1)/2)` and `y = -sqrt((sqrt(2)-1)/2)`
x + i* y is a first root of z
Second solution: `x = -sqrt((sqrt(2)+1)/2)` and `y = sqrt((sqrt(2)-1)/2)`
x + i * y is the second root of z.
See also
Algebraic form of a complex number
Modulus of a complex number | {"url":"https://www.123calculus.com/en/complex-number-root-page-1-45-170.html","timestamp":"2024-11-03T09:11:33Z","content_type":"text/html","content_length":"18770","record_id":"<urn:uuid:c81d9ea3-7c68-488a-b3da-40559c7ea8a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00020.warc.gz"} |
2.7 – A Geometric Problem
A Geometric Problem
Consider the square in the xy-plane bounded by the lines x = 0, x = 1, y = 0 and y = 1. Now consider a vertical line with equation x = c, where 0 ≤ c ≤ 1 is fixed. Note that this line will intersect
the unit square defined previously.
Suppose we select a point inside this square, uniformly at random. If we let X be the x-coordinate of this random point, what is the probability that X is in the interval [0 , c]?
An illustration of our problem is given in the figure below. Graphically, we are trying to find the probability that a randomly selected point inside the square lies to the left of the red line.
Let's begin by considering two cases:
• Case 1: Find Pr(0 ≤ X ≤ c) if c = 1
• Case 2: Find Pr(0 ≤ X ≤ c) if c is in the interval [0, 1)
Case 1
If c = 1, then the random point must certainly have x-coordinate between 0 and c. The area to the left of the red line is the entire area of the square, and our random point has to lie inside the
square. Thus, we are certain to find the x-coordinate in the interval [0,1]; i.e. Pr(0 ≤ X ≤ c) = Pr(0 ≤ X ≤ 1) = 1.
Case 2
The region to the left of the red line is a rectangle with area equal to c. The probability that our random point lies inside this rectangle is proportional to the area of that rectangle, since the
larger the area of the rectangle, the larger the probability that the point is inside of it.
Thinking of probabilities in terms of areas, consider the following cases:
• if the probability that the point is between 0 and c were equal to 0.50, then the red line would have to divide the square into two equal halves: c = 0.5
• if the probability that the point is between 0 and c were equal to 0.25, then the red line would have to divide the square at 1/4: c = 0.25
• if the probability that the point is between 0 and c were equal to 0.10, then the red line would have to divide the square at 1/10: c = 0.1
In general, it is clear that we should have Pr(0 ≤ X ≤ c) = c.
Notice that this result matches with the definition of our random variable X. Recall that we wanted to select a random point uniformly at random from the unit square. Thus the random variable X
giving the x-coordinate of this random point should be a continuous uniform random variable on the interval [0,1]. We have already seen that the PDF of a general uniform random variable defined on a
generic interval [a,b] is
Thus, the CDF of a generic uniform random variable is
For the example under examination, we have a = 0 and b = 1. Therefore, Pr(0 ≤ X ≤ c) = Pr(X ≤ c) - Pr(X ≤ 0) = c, which agrees with the answer we derived using purely geometric considerations.
source: http://wiki.ubc.ca/Science:MATH105_Probability/Lesson_2_CRV/2.09_Additional_PDF_Example | {"url":"https://blogs.ubc.ca/math105/continuous-random-variables/additional-pdf-example-2/","timestamp":"2024-11-08T19:25:29Z","content_type":"text/html","content_length":"46171","record_id":"<urn:uuid:823acd7b-cb92-448c-a4a8-6f157fd454f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00248.warc.gz"} |
Santa Routing and Heuristics in Operations Research
As a child, I always wondered how Santa Claus could visit all children of the world overnight. I now know that he must have a very optimized route plan. And this tells me one thing: Santa Claus is a
math genius!
For simplicity, I only put some of the cities Santa needs to visit… And by no way do I mean that English kids have been too bad for Santa’s visit! Also, I can’t resist giving you this link to
CGPGrey’s awesome video on Santa
, even though it has nothing to do with vehicle routing.
What does math have to do with Santa?
In the last decades, similar problems called vehicle routing problems have become crucial for many companies, like Fedex and UPS. And these are so difficult, that companies asked help from applied
mathematicians! Since then, applied mathematicians have enjoyed the beautifully combinatorial structure of these problems, as well as the millions of dollars companies have been willing to pay for
answers! Let’s see what these mathematicians have come up with! This will draw a global picture of the world of operations research, which, more generally, aims at solving complex combinatorial
industrial problems through mathematics.
I’m very thankful for Marilène Cherkesly for explaining me all the different technics in this article and helping me design the outline of this article!
Santa’s Vehicle Routing Problem
If there were no constraint on Santa Claus’ journey, the visiting-all-children problem would be a Traveling Salesman Problem (TSP): It would consist in searching for the shortest path visiting all
cities and coming back to the original point. This class of problem has become iconic in the theory of computation, as it was proved to be very hard to solved!
So what’s a vehicle routing problem?
To make Santa Claus’ problem a vehicle routing problem, we merely have to assume that Santa Claus’ reindeers cannot carry too heavy quantities. So, for instance, they can only carry presents for 5
cities. This will force Santa to come back to his toy factory several times to reload his sleigh. Typically, a Santa’s journey on the night of December 24th could look as follows:
To have a chance to make it on time, it’s crucial for Santa to minimize the total distance of his journey. This corresponds to searching for a shortest overall journey. And that’s precisely what the
Vehicle Routing Problem (VRP) is about.
Note that the assumption that reindeers can visit 5 cities only greatly simplifies the problem, as we now know that Santa will need 4 and only 4 tours. In the more common capacitated vehicle routing
problem (CVRP), reindeers can carry a certain weight and different cities need to be delivered different amounts of toys, depending for instance on their sizes. In CVRP, the priority is to minimize
the number of tours, which is even harder than our example where tours have a strong structure.
So how can it be solved?
The VRP is actually known to as hard as the TSP. This means that, when the number of cities isn’t small, there is virtually no hope to solve the VRP exactly in a reasonable amount of time, even with
tomorrow’s greatest computer power. In fact, so far, no best solution is known for problems with more than a few hundreds of cities to visit!
What I’m saying here assumes that $P\neq NP$, which is the dominant belief in the complexity theorist community. In fact, I’m even assuming $BQP \neq NP$, where $BQP$ corresponds to the class of
problems solvable efficiently with quantum computers. Find out more about $BQP$ in
my article on probabilistic algorithms
So, rather than solving the VRP exactly, operational researchers have been designing clever technics to find nearly optimal solutions in a reasonable amount of time. These technics are called
Greedy Algorithms
The most straightforward heuristics is the greedy algorithm.
What’s a greedy algorithm?
It consists in choosing the next best step without much thinking, like a greedy Santa who is given a bunch of candies to choose from. He’d repeatedly eat his favorite one among the remaining candies,
until he’s full… or until Mother Christmas tells him to stop!
It sounds like a great way to choose the best candies!
It is! In many cases, like for Santa’s candies, the greedy algorithm turns out to be optimal!
But how would that work for VRP?
A simple greedy algorithm would be to get Santa to the closest unvisited city.
Can you illustrate?
Sure! Starting in Santa’s workshop in Rovaniemi, and using this link, you can determine that the closest city is Moscow, at nearly 1,300 km. Then, the closest city to Moscow is Tehran. Then, the next
closest cities to visit are Cairo, Paris and Marrakesh. Now that Santa has visited 5 cities, he must get back to Rovaniemi to pick up presents for other cities. Then, he goes on with his visits.
Eventually, here’s the Santa’s journey we get:
Such a whole journey is called a solution. This is not to be confused with the optimal solution which would be the best Santa’s journey. Here, the solution found by the greedy algorithm is 124,800
kilometers long. Can we do better… And how?
Local Search
One way to do better is to improve what we have.
Isn’t it the only way?
Hummm… Yes. But what I mean by that is that we can try to make small modifications to the best solution we’ve found, and see if these modifications lead to an improvement. This technic is called
local search. It’s pretty much what people use when they play the colder-warmer game, where each step is warmer when it improves one’s location.
So what kind of modifications are we talking about here?
It’s up to you to invent them! Experiments characterized a simple and efficient kind of modification called 2-opt. It consists in removing two arcs and reconnecting cities of these arcs in a
consistent way.
Can you give an example?
Sure! Let’s remove the arcs New York – Mexico and Vancouver – Lima. Then, we have two possible reconstructions. The New York – Lima and Vancouver – Mexico reconstruction breaks Santa’s tour so it’s
not acceptable. Conversely, the New York – Vancouver and Mexico – Lima reconstruction is consistent with Santa’s tour:
Note that the paths we removed add up to 11,600 km, while these we’ve added in the consistent reconstruction add up to 8,200km. Thus, by modifying Santa’s 2nd tour, we’ve reduced the total distance
by 3,400km!
You chose to remove 2 arcs that crossed… Was that on purpose?
Great observation! By removing 2 arcs that crossed in the 2-opt modification, I was sure to greatly improve the current solution! Let me apply that to all crossing arcs. This is what we get:
Note that 2-opt on all crossing arcs may not work as we consider constraints such as pick-up and delivery or time windows.
Network of Solutions
Cool! But why is this method called this a local search?
To fully understand what is meant by local search, let me introduce the network of solutions. This network connects two solutions if a 2-opt modification enables to move from one to the other.
Typically, the following figure displays a few of the solutions of the larger network of solutions:
Any solution is connected to a bunch of others, which form what its neighborhood. And since we only looked at 2-opt modifications, our local search corresponds to exploring the neighborhood of best
solutions found. So, the search is dubbed local, because we’re looking for better solutions only in the neighborhood of the current best solution.
In fact, let’s pause a bit on this network of solutions. Let’s forget about the details of the network and, rather, let’s ponder upon the global structure of the network of solutions…
What do you mean?
Let’s look at the network as a cloud of solutions with connections between some of these solutions. When we apply a local search to this cloud, links can only be taken in certain direction: From the
worse to the better solution. Thus, links actually have an orientation. In graph theoretical terms, this means that we actually have a directed (acyclic) graph of Santa’s journeys. And a local search
will simply consist in taking a path in this graph until we face a dead-end.
This network of solution is an image I always try to keep in mind while doing optimization. It’s applicable to a huge class of optimization technics, including the state-of-the-art and fundamental
linear programming
, and to a lesser extent, the gradient methods. Indeed, in gradient methods, the graph is sort of dynamics as it depends on the step we take which usually decrease in time, and a good understanding
of how we move from one solution to another is better reached with
But I guess it’s not sure that we’ll get to the actual optimal solution with a local search…
I’d even add that it’s highly unlikely we will. One thing I can guarantee though is that we will reach a local optimum.
What’s a local optimum?
A local optimum is a solution which is better than any of its neighborhood. In other words, no arrow points out of a local optimum. So, crucially, it’s impossible with local search to get out of a
local optimum. An that may be quite a problem, since a local optimum can be very bad compared to the global optimum.
Let me guess… The global optimum is a solution better than all others, right?
Exactly! And there’s worse than the fact that our algorithm may not lead us to the global optimum: Depending on the initial solution, it may simply be impossible for local searches to get to the
global optimum, as there may be no path from the initial solution to the global optimum! This remark has led to new ideas for optimization.
Adaptive Large Neighborhood Search
Building upon this remark, applied mathematicians have then invented the Adaptive Large Neighborhood Search (ALNS). The idea of ALNS is to make radical modifications of the current solution to move
away from local optima. Sort of like you sometimes need a revolution to really change the world…
What kind of radical modifications can we do in VRP?
Typically, we’d only keep half of the arrows of the current solution, and try to fill the gaps using a basic method. It’s common to include randomness both in construction and destruction to make
sure that we will rebuild a very different solution.
Can you illustrate?
Sure. Let’s be clever and keep the arrows which seem relevant. Then let’s assign randomly colors to these arrows, and reconstruct afterwards how we can. Here’s what I got:
Note that, by now, we’ve improved the greedy solution (which was already not too bad) by over 10%! As turnovers of companies who require such models add up to billions of dollars, just think about
how much we may save them… and how much they would owe us! It’s no surprise that some companies specialized in optimization have millions of turnover!
For good performance of ALNS, the level of destruction needs to be managed accordingly to the success of previous local search attempts. Note that there are several variants, like Variable
Neighborhood Search (VNS) and Iterated Local Search (ILS).
Simulated Annealing and Tabu
But I guess that it’s quite unlikely for a radical modifications to improve the current solution, right?
Indeed. This led to simulated annealing. The idea is to still carry out radical modifications even though they are worse than the current best solution, and as long as there is hope they will lead to
good solutions after local searches. However, after more and more iterations, we may become more and more demanding and eventually require a strict improvement from the current best solution.
Why is it called simulated annealing?
By analogy with metal annealing! In a hot metal, particles still jiggle around at all kinds of energy levels. Importantly, as long as the metal is hot, particles can leave low energy state and get to
a higher energy one. But as the metal cools down, particles reach lower energy states and have more trouble leaving them. Eventually, they rest at a low energy state which is often lower than the one
they’d reach had the metal been frozen instantly.
So, eventually, in simulated annealing, no worse solution would be accepted, right?
Exactly! Here’s how to visualize simulated annealing used combined to local search and ALNS:
Given the arrows of simulated annealing, a path may walk back on its footsteps. To avoid that, the Tabu search has been introduced. It simply consists in remembering the last footsteps made to makes
sure we never come back to a solution we already explored.
Genetic Algorithms
In our example, combining all the heuristics discussed so far would yield nearly unbeatable solutions. But, as VRP models are made more realistic, they become much harder to solve, even
heuristically! Examples of realistic constraints we may want to add include differences in quantity of toys required for cities, time windows to visit cities or reindeer required rests. For
constraints like these, even slight modifications may greatly modify the quality of a solution. This leads to an explosion of the number of local optima. For this reason, the methods introduced so
far can turn out quite limited.
So what can be done then?
One great idea came from evolutionary biology: Genetic Algorithms. The core idea lies in the fact that a mixture of two parents can make a kid much better than the average of the parents. An example
of that is how crossing English Father Christmas with Dutch Sinterklaas has given birth to the world-renowned Santa Claus.
How would we do that in VRP?
The key to powerful genetic algorithms lies in the way we cross the parents’ genes. Researchers have come up with plenty of ways to do so, as listed in this article. Let me consider one to
How does that work?
First, we represent solutions as the ordered sequence of all visited cities. Then, the father solution yields half of the cities to the kid, all placed exactly as they appear in the father’s ordering
of cities. The gaps are then filled by the other half of the cities, in the order in which they appear in the mother solution.
Hummm… Can you give an example?
Sure. Let’s mix the greedy solution and the ALNS solution we have found earlier:
It actually looks unlikely that the kid is any good…
He needs education! Funnily, this term has been used to applying local search to the kid to make it an interesting solution!
Even with education, it doesn’t seem likely that the kid becomes the global optimum…
But, at least, he’ll contribute to a greater diversity of solutions. Indeed, the problem with local search is that the intensification leads to solutions which all look the same. It’d be a bit like
having all physicists working on string theory alone. Sure, string theory is promising and it probably deserves intensification… But if all physicists looked in a single direction, they’d be missing
out on a lot of other less promising but eventually more fruitful leads! Conversely, many breakthroughs in History have resulted from a clever mix of good ideas, like how Newton merged Galileo’s laws
of motions on Earth with Kepler’s laws of planetary motions!
So a balance needs to be met between diversification and intensification, right?
Exactly! A classical approach consists in maintaining a pool of diverse not-too-bad solutions, called population. This population can be initially generated by a greedy randomized adaptive search
procedure (GRASP). Then, two solutions of the population are picked and crossed over. Their kid is inserted in the population and replaces an unfit or non-original solution of the population. The
choice of the solution to remove is called selection, and it regulates the level of diversification. This is represented below:
This sounds and looks nice, but do genetic algorithms really work?
Amazingly, since a paper by Christian Prins in 2004, genetic approaches have surpassed local searches in many problems.
Why did this success only occurred recently? Is it only now that researchers have thought of that?
Actually, genetic algorithms have been introduced as soon as the 1950s! But, crucially, computing power has greatly increased since, and you actually need to have this much greater computing capacity
to get genetic algorithms running at their full potential!
So, do you think genetic algorithms will be ruling the future of optimization?
I’m sure that their ideas will be useful. But I rather believe in even more advanced optimization technics.
Integer Programming and Cuts
Like most combinatorial optimization problem, the VRP can be written as an integer linear program.
What’s an integer linear program?
It’s an optimization problems whose variables are all integers and which have a linear structure. More precisely, if you forget the fact that variables must take integer values, then you should be
left with a linear program. This linear program has a linear objective function and linear constraints.
Hummm… What would an integer program for VRP look like?
Let’s encode whether Santa goes from city $i$ to city $j$ in tour number $k$ by binary variables $x_{ijk}$. More, precisely $x_{ijk}$ equals 1 if Santa does, and 0 otherwise.
Now, to have an integer program, we also need an objective function and constraints.
What’s the objective function?
It’s what we want to minimize (or maximize). So, in our case, it’s the total distance of Santa’s journey. Crucially, it can be deduced from the variables. Indeed, denoting $d_{ij}$ between cities $i$
and $j$, the term $d_{ij} x_{ijk}$ equals $d_{ij}$ if Santa goes from $i$ to $j$ in tour $k$, and 0 otherwise. Thus, by summing over all tours $k$ and over all pairs of cities $i$ and $j$, we add up
all the distances of intercity trips of Santa: That’s also the total distance of Santa’s journey! So, VRP consists in finding the variables $x_{ijk}$ which minimize the sum of terms $d_{ij}x_{ijk}$
over all values of $i$, $j$ and $k$.
Wait… Now it looks like always having $x_{ijk} = 0$ is the optimal solution!
You’re right! And that’s because our integer program so far includes solution which are inconsistent with the actual VRP. We now need to add constraints to make the integer program consistent with
the problem it’s supposed to solve.
We need to add constraints? Like what?
Santa must go through all cities. In particular, if $i$ is a city which is not Santa’s, he must go through it exactly once. This implies that he must also leave it once. So, there must be one and
only one variable $x_{ijk}$ which equals 1. Thus, all variables $x_{ijk}$ must add up to 1. So, for all cities $i$ different from Santa’s, we’ll add the constraint that the sum of $x_{ijk}$ equals 1.
But that’s still not enough, right?
Indeed. Let’s add the following constraints:
• Santa goes and leaves its home at most once in each tour (0 if he makes no visit in the tour).
• In each tour, through each city, there must be as many incoming arcs as outgoing ones.
• A tour cannot be made of more than 5 cities other than Santa’s (and thus 6 arcs).
Now, given these constraints, it’s still possible to have inconsistent loops like below:
Such a solution is said to have a subtour. To avoid these, a common method is to add subtour elimination constraints. They are based on the two following remarks. First, for any subset $S$ of cities
not including Santa’s, there must be at least one arc going out of the subset. Thus, this outgoing arc comes from a city which has no outgoing arc which remains in the subset $S$. Second, the number
of arcs within the subset $S$ equals the number of cities of the subset with an outgoing arc remaining in the subset $S$. Thus, by combining the two remarks, we deduce that the number of arcs within
$S$ cannot equal or exceed the number of cities of the subset!
Is that it? Are there still more constraints to add?
Nope! We’re done! Here’s the complete integer programming formulation, with the mathematical translation of our constraints added on the right:
Note that index 0 of cities is Santa’s toy factory city Rovaniemi, and the subset $S$ must not include 0.
This formulation is nearly perfect to put it directly into an integer program optimizer and quickly get a great solution!
What do you mean by “nearly”?
There’s one trouble though… The integer program above has an exponential number of constraints! Indeed, there are as many subtour elimination constraints as the number of subsets of the set of
cities. And that’s bad, very bad…
Is it that bad?
If there are 100 cities, then there are $2^{100} \approx 10^{30}$ subtour elimination constraints! That’s more than the size of the web, which is only about $10^{18}$ bits! So, you can’t even write
down the subtour elimination constraints for a VRP of 100 cities on the whole web!
This kind of make the formulation above useless, doesn’t it?
Fortunately, applied mathematicians have more than a few tricks up their sleeves! They have proposed to start the optimization without subtour elimination constraints. Then, for each solution found,
we add the subtour elimination constraints the solution does not verify. If there is no such subtour elimination constraint, then this means that our solution is consistent… And is the global optimum
too! This technic is called cutting planes, and has turned out to be today’s state-of-the-art technique for many problems. In fact, other constraints than subtour eliminations can be used in cutting
plane methods to efficiently reach quality solutions.
Let’s Conclude
So, would you now agree with Santa’s being a math genius?
I guess… Unless his elves are the geniuses who plan the routing for him!
Haha… Now, I should say that there’s one last more state-of-the-art optimization technic I have not mentioned here. And yet, it’s the specialty of my research lab! It’s called column generation, and
exploits the structure of the integer program to write it differently, using the Dantzig-Wolfe decomposition. Find out more with this article!
Do these more advanced technics mean that local searches and genetic algorithms are pointless?
Not at all. First, because other problems don’t have a structure as rich as the VRP does, which means that integer programming and column generation may not work for them, or would involve too much
work and tests to get right. Second, and more importantly, these other methods can actually be used in combination with more advanced technics. This is particularly the case with column generation
which decomposes the problem in smaller subproblems which may require local search or genetic algorithms! Overall, a clever balance of all these technics seems to be key for powerful optimization.
Cool! Also, you mentioned something about making these models more realistic…
Yes! By now, there have been a huge number of variants of the VRP. Among others are VRPPD (where cities correspond to pick-up or deliveries, and capacities must be modeled accordingly), LIFO-VRPPD
(where the order in which you load and unload goods in trucks matter), VRPTW (where each city must be visited within a certain time window), SDVRP (where deliveries for a customer can be split in
several trucks), HVRP (where the fleet of vehicles is heterogeneous), SVRP and RVRP (with uncertainties where we minimize the average case or the worst case). And, these can be combined, yielding
problems like LIFO-SDVRPPDTW… As you can imagine, I like to tease my VRP colleagues about how they create new problems by just adding letters to the acronym!
No idea of gift? Send this article to your nerdy friends! Merry Christmas and Happy New Year! With a special dedication to all my vehicle routing researcher friends: Marilène, Charles, Claudio,
Heiko, Remy, Sana, Vincent, Guillaume, François, Guy and those I forget!
1. Great post! As a PhD student working in the same field, I find it difficult sometimes to explain to others what I’m doing. This is a great way to get people interested in the field. Good job!
2. Good day! This is my first visit to your blog! We are a
team of volunteers and starting a new initiative in a community
in the same niche. Your blog provided us beneficial information to
work on. You have done a extraordinary job! | {"url":"https://www.science4all.org/article/vehicle-routing-problem/","timestamp":"2024-11-06T14:37:57Z","content_type":"text/html","content_length":"94134","record_id":"<urn:uuid:1a8243a7-99d7-4b52-b3f3-5dc342b83939>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00832.warc.gz"} |
Area Principle in Statistics
Descriptive Statistics > Area Principle
What is the Area Principle?
The area principle states that the area of a graph should equal the magnitude of the data it is representing. For a simple example, let’s say you took three height measurements of 4 feet, 5 feet, and
6 feet. The top part of the image below shows data that agrees with the area principle:
• The 4 feet measurement has an area of 4 (1 unit up * 4 units across);
• The 5 feet measurement has an area of 5 (1 unit up * 5 units across);
• The 6 feet measurement has an area of 6 (1 unit up * 5 units across);
The bottom part shows data that violates the area principle. The eye is drawn towards the 4 foot measurement because of the increased area that is out of proportion to the measurement. Even though 6
feet is clearly the larger measurement, the eye is drawn to the larger area.
• The 4 feet measurement has an area of 8 (2 units up * 4 units across);
• The 5 feet measurement has an area of 5 (1 unit up * 5 units across);
• The 6 feet measurement has an area of 6 (1 unit up * 5 units across);
The Area Principle and 3D Pie Charts
This 3D graph makes it look like the yellow wedge is about the same size (or even larger) as the green wedge. If you look at the key, and at the surface area of the wedges, you’ll notice the green
wedge is actually 10% larger. Not all 3D software obeys the principle rule, so check your graph after you make it.
The Area Principle and Bar Charts
Bar charts can violate the area principle if the vertical axis starts at a number other than zero. This tactic is one commonly employed by the media to make numbers look scarier than they actually
are. This graph comes from Fox News, which clearly showed the disaster that would happen if Bush’s tax cuts expired. Notice the vertical axis starting on the right at 34.
Figure 1. Source: https://twitter.com/DanaDanger/status/230851016344600576/photo/1/large
Vertical axes usually start on the left, at zero. The placement of the axis on the right is probably deliberate, as the casual viewer would be looking on the left for the axis, and not seeing it,
would only see the “huge” jump in tax rates. The tax rate jump would be, in fact, only about 4.6%.
Tabor, J. & Franklin, C. (2011). Statistical Reasoning in Sports. W. H. Freeman. | {"url":"https://www.statisticshowto.com/area-principle-in-statistics/","timestamp":"2024-11-13T15:43:10Z","content_type":"text/html","content_length":"70055","record_id":"<urn:uuid:07131995-2f26-4026-addb-850f3b8e5466>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00262.warc.gz"} |
Drake Concepts | Drake Tutorial
Drake's core library has 3 big parts:
I. Dynamical Systems Modeling
Drake provides tools to model the physics of a dynamic system, which can be used for analysis and simulation.
Drake's system modeling works like Matlab Simulink. Drake constructs complex systems from blocks called system. system has input/output ports that could be connected with other systems. A system
block can be a diagram or a leafsystem. leafsystem is the minimum build block, and a diagram is composed of multiple leafsystems or diagrams.
leafsystem functions as basic components in robotics systems, such as signals, sensors, controllers, planners, etc.
Drake uses diagram to represent compound systems. diagram internally contain several connected subsystems. diagram itself is a system and can be nested. The root diagram contains all the subsystems
and subdiagrams.
The main function in Drake usually starts from a blank root diagram. systems are added to the rootdiagram and connected through their input/output ports.
context contains the data of system states and parameters cached in a separate place. Each diagram and system has its own context. The context and the diagram are the only information a simulator
requires to simulate. Given the context, all methods called on a system should be completely deterministic and repeatable (Ref. Underactuated Robotics textbook).
Drake has a method diagram->CreateDefaultContext() to create the context with default values for all the subsystems. Values in the context, such as the initial state and the initial time, can be
independently set before the simulation begins.
A context can have continuous state, discrete state, and abstract variable. Based on the variable type, the simulator would update the context data by either numerically integrating the continuous
derivative or updating the states using state-space dynamics.
Drake is a simulation software. The Drake simulator takes in the system diagram together with its context to simulate by updating parameters such as integral continuous state derivatives, compute
discrete state updates, allocates the various outputs of a system, etc.
II. Mathematical Programs Solving
Drake incorporates famous and useful optimization tools, for example, Gurobi, SNOPT, IPOPT, SCS, MOSEK. These tools help to solve mathematical problems in robotics, such as motion planning and
To use Mathematical Programming, there is a very good starting point written in python. The same idea applies to C++.
III. Multibody Kinematics and Dynamics
Multibody means multiple rigid bodies connected in an articulated structure. The root diagram that contains a unique leafsystem namedMultibodyPlant is considered as a robotic system. MultibodyPlant
internally uses rigid body tree algorithms to compute the robot's kinematics, dynamics, jacobian, etc.
MultibodyPlant is a system. So it has input/output ports that could be connected to other systems like controllers and visualizers.
Tools that Drake uses
1. Eigen
Eigen is a C++ library with linear algebra operations and algorithms.
A convenient technique to compute Derivative. Computing Integral is trivial and computing Derivative is non-trivial. AutoDiff is a good solution that Eigen provides to solve the derivative.
2. Lightweight Communications and Marshalling (LCM)
LCM is a multi-process communication tool. LCM is everywhere in Drake. It serves as the bridge between system ports, so all the communications between systems are transported using LCM, which can be
inspected by LCM spy tool.
Visualize data in LCM
3. Tinyxml2
A handy tool that parses XML files, enables Drake to parse URDF and SDF, thus creating MultibodyPlant for simulation.
4. The Visualization Toolkit (VTK)
Drake uses VTK as a geometry rendering tool. The Drake visualizer communicates with the simulator through LCM. | {"url":"https://drake.guzhaoyuan.com/introduction/drake-concept","timestamp":"2024-11-09T00:49:28Z","content_type":"text/html","content_length":"282693","record_id":"<urn:uuid:609b05ef-fd12-44ac-9bb3-8a6ea8772307>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00459.warc.gz"} |
Statistics Sunday: Null and Alternative Hypotheses
In my writing about statistics, there is one topic - considered basic - that I haven't covered. This is the issue of null and alternative hypotheses, which are key components in any
inferential statistical
analysis you conduct. The thing is, I've seen it cause nothing but confusion in both new and experienced researchers, who don't seem to understand the difference between these statistical hypotheses
and the research hypotheses you are testing in your study. I've rolled my eyes through doctoral research presentations and wielded my red pen in drafts of grant applications as researchers have
informed me of the null and alternative hypotheses (which are implied when you state which statistical analysis you're using) alongside their research hypotheses (which require stating).
Frankly, I've been so frustrated by the lack of understanding that I questioned whether to even address these topics. When I teach, I downplay the significance (pun fully intended) of null and
alternative hypotheses. (And in fact, many in the field are trying to move us away from the so-called Null Hypothesis Significance Testing, or NHST, approach, but that's another post for another
day.) In any case, I treat this topic as an annoying subject to get through before getting to the fun stuff: analyzing data. Not that I think this is a boring topic, or that I have a problem with
boring topics - when you love a field, you have to love the boring stuff too.
Rather, I questioned whether the topic was even necessary. You can conduct statistical analysis without thinking about null and alternative hypotheses - I often do.
I realize now that the topic
important, but it's not really explained why. So statistics professors and textbook authors continue to address the topic without addressing the purpose. Today, I'd like to do both.
First, we need to think about what it means to be a science, or for a line of inquiry to be scientific. Science is about generating knowledge in a specific way - through systematic empirical methods.
We don't want just any knowledge. It has to meet very high and specific standards. It has to be testable, and, more importantly,
. If a hypothesis is wrong, we need to be able to show that. In fact, we set up our studies with specific controls and methods so that if a hypothesis is wrong, it can show us it's wrong.
If, after all that, we find support for a hypothesis, we accept that... for now. But we keep watching that little supported hypothesis out of the corner of our eyes, just in case it shows us its true
(or rather, false) colors. See, if we conduct a study to test our research hypothesis, we will use the results of the study to reject (if it's false) or support (if it doesn't appear to be false). We
don't prove anything, nor do we call hypotheses true. We're still looking for evidence to falsify it. That is the purpose of science. To study something again and again, not to see if we can prove it
true, but to see if we can falsify it. It's as though every time we do a study of a hypothesis that's been supported, we're saying, "Oh yeah? Well, what about if I do
This is the nature of scientific skepticism. There could be evidence out there that shows a hypothesis is false; we just haven't found it yet. Karl Popper addressed this facet of science directly in
the black swan problem. You can do study after study to support the hypothesis that all swans are white, but it takes only one case - one black swan - to refute that hypothesis.
So essential is this concept to science that we build it into our statistical analysis. The specifics of the null and alternative hypotheses vary depending on which statistic you're using, but the
basic premise is this:
Null hypothesis
: There's nothing going on here. There is no difference or relationship in your data. These are not the droids you're looking for.
Alternative hypothesis
: There's something here. Whatever difference or relationship you're looking for exists in your data. These are, in fact, the droids you're looking for. Go you.
(Come to think of it, that Jedi mind trick is the perfect demonstration of a
Type II error
. But pretend for a moment that they really
the droids they were looking for.)
This is your basic reminder that we first look for evidence to
before we look for evidence to support your research hypothesis. We then run our statistical analysis and look at the results. If we find something - the difference or relationship we expect - we
reject the null, because it doesn't apply in this situation (although because of the possibility of
Type I error
, we never lose the null completely). And we have support for our alternative hypothesis. If, on the other hand, we don't find a significant difference or relationship, we
fail to reject
the null. (Yes, that is the exact language you would use. You don't "accept" or "support" the null, because nonsignificant results could simply mean low
You also use the null and alternative hypotheses to state if there is an expected direction of the effect. For example, to go back to the
caffeine study example
, we expect caffeine will improve test performance (this is our research hypothesis). So we would write our null and alternative hypotheses to demonstrate that direction:
: The mean test score of the caffeine group will be less than or equal to the mean test score of the non-caffeine group or M
≤ M
[Decaf] Alternative
: The mean test score of the caffeine group will be greater than the mean test score of the non-caffeine group or M
> M
Notice how the null and alternative hypotheses are both mutually exclusive and exhaustive (together they cover all possible directions). If we conducted our statistical analysis in this way, we would
only support our research hypothesis if the caffeine group had a significantly
test score. If their test score was lower - even
lower - we would still fail to reject the null. (In fact, if we follow this strict, directional hypothesis, finding a significantly lower score when we expected a significantly higher score would
simply be considered Type I error.)
If we didn't specify the direction, we would simply state the scores will be equal or unequal:
: The mean test score of the caffeine group will be equal to the mean test score of the non-caffeine group or M
= M
[Decaf] Alternative
: The mean test score of the caffeine group will be different from the mean test score of the non-caffeine group or M
≠ M
These hypotheses are implicit when doing statistical analysis - they're for your benefit, but you wouldn't spend time in your dissertation defense, journal article, or grant application stating the
null and alternative. (Maybe if you were writing an article on a new statistical analysis.) Readers who know about statistics will understand they're implied. And readers who don't know about
statistics will prefer concrete differences - what you hypothesize will happen in your study, and what specific differences you found and what they mean.
As you continue learning and practicing statistics skills, you may find that you don't really think about the null and alternative hypothesis. And that's okay. In fact, I wrote two posts that tie
directly into null and alternative hypotheses without once referencing these concepts. Remember
? And
? I said in these posts that these refer to probabilities of finding an effect of a certain size by chance alone. Specifically, they refer to probabilities of finding an effect of a certain size
if the null hypothesis is actually true -
if we could somehow pull back the curtain of the universe and discover once and for all the truth
We can't do that, of course, but we can build that uncertainty into our analysis. That is what is meant by Null Hypothesis Significance Testing.
But, as you saw, I could still describe these topics without even using the phrase "null or alternative hypotheses." As long as you stay a skeptical scientist, who remembers all findings are
tentative, pending new evidence, you're doing it right. | {"url":"http://www.deeplytrivial.com/2017/07/statistics-sunday-null-and-alternative.html","timestamp":"2024-11-04T12:00:41Z","content_type":"text/html","content_length":"96729","record_id":"<urn:uuid:9c012dbf-cee0-4592-b0c2-215106fa8446>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00184.warc.gz"} |
Single-Source Shortest Paths
Motivating Problem : Given a weighted graph G and a starting source vertex s, what are the shortest paths from s to every other vertices of G ?
This problem is well known as SSSP (Single-Source shortest path) problem on a weighted graph. There are efficient algorithm to solve this SSSP problem.
If graph is unweighted (or all edges have equal or constant weight), we can use the efficient $O(V+E)$ BFS algorithm. For general weighted graph, BFS doesn’t work and we should use algorithms like
$ O((V+E) \log V)$ Dijkstra’s algorithm or the $O(VE)$ Bellman Ford’s algorithm.
SSSP on Unweighted Graph
Since graph is unweighted graph, distance between two vertices is 1 unit and BFS explores graph layer by layer.
Some problems may require us to reconstruct the actual shortest path, not just the shortest path length and can be achieved easily by vi p.
void printPath(int u) { // extract info from `vi p`
if(u == a)
printf("%d",s); return; // base case, at source s
printf(" %d",u); }
// inside int main()
vi dist(V, INF); dist[s] = 0; // dist s to s is zero
queue<int> q; q.push(s);
vi p;
while(!q.empty()) {
int u = q.front(); q.pop();
for(auto v : adj[u]) { // integer pair
if(dist[v] == INF) {
dist[v] = dist[u] + 1;
p[v] = u;
} } }
printPath(t), printf("\n"); // prints path from vertex p
SSSP on Weighted Graph
For this problem BFS won’t work, we use greedy Edsger Wyde Dijkstra’s algorithm. There are many ways to implement the algorithm. We will use C++ STL’s priority_queue implementation for sake of
This Dijkstra’s variant maintains a priority queue called as pq that stores pairs of vertex information. The first and the second item of the pair is distance from vertex from the source and vertex
number, respectively.
NOTE : code below assumes graph is implemented as (v,d) format in adjacency list while pq flips this format for the sake of utilizing the priority queue.
This pq is sorted based on increasing distance from the source and tie, by vertex number. This is different from another Dijkstra’s implementation that uses binary heap feature that is not
supported in this built-in library.
So pq contains (0,s) base case initially and greedily picks a vertex (d,u) from the front of pq. If the distace to u from source recorded in d greater than dist[u], it ignores u; otherwise, it
process u. The real reason for this check is explained below.
When this algorithm process u, it tries to relax all neighbors v of u. Every time it relaxes an edge u -> v, it will en-queue a pair(newer/shorter distance to v from source) into pq and leave the
inferior pair (older/longer distance to v from source) inside pq. This is called as Lazy Deletion.
This approach leaves more than one copy of the same vertex in pq with different distances from source. That is why we have the check earlier to process only first dequeued vertex information pair
which has correct/shorter distance (other copies are longer).
vi dist(V, INF); dist[s] = 0;
priority_queue< ii, vector<ii>, greater<ii>> pq;
while(!pq.empty()) {
ii front pq.top(); pq.pop();
int d = front.first, u = front.second;
if(d > dist[u]) continue;
for(auto v : adj[u]) {
if(dist[u] + v.second < dist[v.first]) {
dist[v.first] = dist[u] + v.second; // relaxation
Sample Application : UVa 11367 - Full Tank ?
Abridged Problem : Given a connected weighted graph length that stores the road lengths between E pairs of cities i and j, the prices p[i] of fuel at each city i, and the fuel tank capacity c of a
car. Determine the cheapest trip cost from starting city s to ending city e using the fuel capacity c.
This problem actually shows the importance of graph modeling. The explicitly given graph in this problem is weighted graph of road networks. However we cannot solve this problem with just this graph.
This is because the state of this problem requires not just the current location but also the fuel level at that location. Otherwise we cannot determine whether car is able to complete the trip.
Therefore we require a pair of information to represent the state : (location, fuel) and by doing so, the total number of vertices of the modified graph explodes from just 1000 vertices to 100000
vertices. We call the modified graph ‘state-space graph’
Solution : In the state-space graph, the source vertex is state (s,0) - at starting city s with empty fuel tank and target vertices are states (e,any) - at ending city with any level of fuel between
[0….c]. There are two types of edges in State-Space Graph : 0-weighted edge that goes from vertex (x, fuel_x) to vertex(y,fuel_x - len(x,y)) if the car has sufficient fuel to travel from vertex x
to vertex y, and the p[x] weighted edge that goes from vertex (x, fuel_x) to vertex (x, fuel_x + 1) if the car can be refuel at vertex x by one unit of fuel (fuel capacity < c). Now running dijkstra
on this State-Space graph gives us the solution for this problem.
SSSP on Graph with Negative Weight Cycle
typical implementation of dijkstra’s algorithm on negative-graph can produce wrong answer. However, Dijkstra’s implementation variant shown before will work just fine, only lit bit of slower.
Because Dijkstra’s implementation we saw before will keep inserting new vertex information pair into the priority queue every time it does a relax operation. So if graph has no negative weight
cycle, the variant will keep propagating the shortest path distance information until there is no more possible relaxation (which implies shortest path is found). However, when given a graph with
negative weight cycle, the variant - if implemented as shown before will be trapped in an infinite loop.
To solve SSSP prooblem in potential presence of negative cycle, the more generic (slower) Bellman Ford’s algorithm must be used. Idea is simple :
Relax all E edges (in arbitrary order) V-1 times!
Main part of the code is more simpler than BFS and Dijkstra’s code.
vi dist(V,INF); dist[s] = 0;
for(int i = 0; i < V-1; i++) {// relax V-1 times
for(int u = 0; u < V; u++) {
for(auto v: adj[u])
dist[v.first] = min(dist[v.first], dist[u]+v.second);
Here Complexity becomes $O(V^3)$ if graph is Adjacency Matrix or $O(VE)$ if graph is stored as adjacency list. Both are slow as compared to Dijkstra’s.
Corollary: we proved that after relaxing all E edges V-1 times, the SSSP should have been solved, i.e. we cannot relax any more edge. If we can still relax an edge, there must be negative cycle in
our graph.
bool hasNegativeCycle = false;
// after running above bellman ford
// one more pass to check for possible relaxation
// relax V-1 times
for(int u = 0; u < V; u++) {
for(auto v: adj[u])
if(dist[v.first] > dist[u] + v.second)
hasNegativeCycle = true;
printf("Negative Cycle Exists ? %s\n", hasNegativeCycle ? "YES" : "NO");
Given a connected, weighted graph G with V $\le$ 100 and two vertices s and d. Find the maximum possible value of dist[s][i] + dist[i][d] over all possible $i \in [0 … V-1]$ .
This problem requires shortest path information from all possible sources (all possible vertices) of G. We can make V calls of Dijkstra’s algorithm that we have learned earlier. However we can
solve the problem in shorter way in terms of code. If weighted graph has V $\le$ 400, then there is another algorithm that is simple to code.
Load the small graph into a Adjacency Matrix and run the following 4-liner. When it terminates AdjMat [i][j] will contain the shortest path distance beween two vertices i and j in G.
// inside main
// adj[i][j] contains the weight of edge i to j
// or INF (1B) if there is no such edge
for(int k = 0; k < V; k++)
for(int i = 0; i < V; i++)
for(int j = 0; j < V; j++)
adj[i][j] = min(adj[i][j], adj[i][k] + adj[k][j])
Above algorithm is called as Floyd Warshall’s algorithm, invented by Robert W Floyd and Stephen Warshall. This is a DP algorithm that clearly runs in O(V^3) due to 3 nested loops.
In general above code solves the APSP problem rather can calling SSSP algorithm multiple times.
1. V calls of O((V+E) log V) Dijkstr’s = $O(V^3 \log V)$ if $E = O(V^2)$.
2. V calls of O(VE) Bellman Ford’s = $O(V^4)$ if $E = O(V^2)$.
Other Applications
Solving SSSP Problem on Small Weighted Graph
If we have the APSP information we also know the SSSP information.
Printing Shortest Path
A common issue encountered by programmers who use four-liner Floyd Warshall’s without understanding how it works is when they are asked to print shortest paths too. In BFS/Dijkstra’s/ Bellman
Ford’s algorithm we just need to remember the shortest paths spanning tree by using a 1D vector to store parent information for each vertex. In Floyd Warshall’s we need to store a 2D parent
matrix. The modified code is as
for(int i = 0; i < V; i++)
for(int j = 0; j < V; j++)
p[i][j] = i; // initialize the parent matrix
for(int k = 0; k < V; k++)
for(int i = 0; i < V; i++)
for(int j = 0; j < V; j++)
if(adj[i][k] + adj[k][j] < adj[i][j) {
adj[i][j] = adj[i][k] + adj[k][j];
p[i][j] = p[k][j];
void printPath(int i, int j) {
if(i!=j) printPath(i,p[i][j]);
printf(" %d", j);
Transitive Closure (Warshall’s Algorithm)
Given a graph, determine if vertex i is connected to j, directly or indirectly. This variant uses bitwise operators which are much faster than arithmatic operators.
Initially set entire adj to 1 if i connected to j otherwise 0.
After running Floyd’s Algorithm we can check if any two vertices are connected by check adj matrix
for(int k = 0; k < V; k++)
for(int i = 0; i < V; i++)
for(int j = 0; j < V; j++)
adj[i][j] != (adj[i][k] & adj[k][j]);
Minimax and Maximin
We have seen the minimax(and maximin) path problem earlier. The solution using Floyd Warshall’s is shown below. First intialize adj[i][j] to be the weight of edge (i,j). This is default minimax
cost for two vertices that are directly connected. For pair i-j without any direct edge, set adj[i][j] = INF. Then we try all possible intermediate vertex k. The minimax cost adj[i][j] is minimum of
either (itself) or (the maximum between adj[i][k]] or adj[k][j]) However this approach works only for V $\le$ 400.
for (int k = 0; k < V; k++)
for(int i = 0; i < V; i++)
for(int j = 0; j < V; j++)
adj[i][j] = min(adj[i][j], max(adj[i][k],adj[k][j]))
Finding the Cheapest/Negative Cycle
We know Bellman Ford will terminate after O(VE) steps regardless of the type of input graph, same is the case with Floyd Warshall it terminates in $O(V^3)$.
To solve this problem, we intially set the main diagonal of the adj matrix to a very large value, i.e. adj[i][i] = INF, which now mean shortest cyclic paht weight strating from vertex i goes thru up
to V-1 other intermediate vertices and return back to i. If adj[i][i] is no longer INF for any $i \in [0…V-1]$, then we have a cycle. The smallest non-negative adj[i][j] $\forall i \in [0…V-1]$
is the cheapest cycle.
If adj[i][j] < 0 for any $i \in [0…V-1]$, then we have a negative cycle because if we take this cyclic path one more time, we will get even shorter shortest path.
Finding the Diameter of a Graph
The diameter is maximum shortest path distance between any pair, just find the i and j s.t. its maximum of the matrix adj.
Finding the SCCs of a Directed Graph
Tarjan’s algorithm can be used to identify SCCs of a digraph. However code is bit long. If input graph is small, we can also identify the SCCs of graph in $O(V^3)$ using Warshall’s transitive
closure algorithm and then use the following check.
To find all the members of SCC that contains vertex i check all other vertices $V\in [0…V-1]$. If adj[i][j] && adj[j][i] is true, then vertex i and j belong to same SCC. | {"url":"https://algo.minetest.in/Data_Structures_library/Graphs/shortest_paths/","timestamp":"2024-11-10T01:11:37Z","content_type":"text/html","content_length":"209199","record_id":"<urn:uuid:0c9ce27d-0c05-47dc-9c95-9ec2b1ba42ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00742.warc.gz"} |
Pure Math Phd for Computer Science student
I'm an international student currently pursuing my masters in computer science.My bachelor's was also in computer science.I'm interested in doing my Phd in pure mathematics.
I had taken many math courses in during my undergraduate years and my present research is also very math oriented (algorithmic algebra and computational algebraic geometry).Also, I've independently
studying many other topics.
My first question s that will I be eligible for Phd in Pure mathematics ?
If yes, do I have any realistic chance of getting admission in any good university (say top 50)?
What else can I do to improve my chances of being accepted?
Re: Pure Math Phd for Computer Science student
You will need to check the websites - generally it should be fine as long as you have a degree in a field related to mathematics.
However, you may be at a disadvantage in the admissions process if most of your coursework is computational in nature. Plus, you would need to write a convincing statement about wanting to pursue a
pure math advanced degree.
Computational algebraic geometry can arguably be considered pure math, depending on what you mean by the word "computational"; if this means using extremely elementary theory to generate computations
using a machine, that's probably less helpful. More helpful would be if you perform difficult computations in commutative algebra using the insights of algebraic geometry, etc, and use a computer as
an aid. There are lots of schools with strong algebraic geometry doing things like this.
Your admission will depend on your strength of *pure* math program, letters of recommendation (very important in your case, since your background may be unique, and admissions must be especially sure
of your having strong direction in pure mathematics), and if relevant, standardized testing (of course, consult the school websites for what is needed). If you come from a not so famous school,
standardized testing is more important than it would otherwise have been, given you do not have the same widely recognized marker of competence relative to other backgrounds.
Re: Pure Math Phd for Computer Science student
What math did you take - you will obviously need the big two, Algebra and Analysis (full years worth). However, as someone said, you will be disadvantaged against those who have gone the pure route
and simply have better "training" than you.
I would perhaps suggest doing the master in math, and then transitioning to PhD, instead of jumping right into a math doctoral program.
Re: Pure Math Phd for Computer Science student
ANDS wrote:What math did you take - you will obviously need the big two, Algebra and Analysis (full years worth). However, as someone said, you will be disadvantaged against those who have gone
the pure route and simply have better "training" than you.
I would perhaps suggest doing the master in math, and then transitioning to PhD, instead of jumping right into a math doctoral program.
Well,I have taken Algebra , Real Analysis, Galois Theory and Commutative Algebra.
This sem I'll be taking Algebraic geometry.
Actually, I'm taking only math courses in my masters as my work revolves much more around pure math than computer science. I can safely say it's computer science only for namesake. | {"url":"https://mathematicsgre.com/viewtopic.php?f=1&t=586","timestamp":"2024-11-11T01:40:58Z","content_type":"text/html","content_length":"25225","record_id":"<urn:uuid:21cfc4a9-813e-483d-9eed-81fec3967987>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00739.warc.gz"} |
• Introduction：
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Priority support | {"url":"https://toolful.ai/t/math-solver-scanner","timestamp":"2024-11-09T03:15:52Z","content_type":"text/html","content_length":"91765","record_id":"<urn:uuid:17872a7a-24f2-4da3-963a-660c01c9f45c>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00100.warc.gz"} |
Modified theories of gravity
The recent observational data in cosmology seem to indicate that the universe is currently expanding in an accelerated way. This unexpected conclusion can be explained assuming the presence of a
non-vanishing yet extremely fine tuned cosmological constant, or invoking the existence of an exotic source of energy, dark energy, which is not observed in laboratory experiments yet seems to
dominate the energy budget of the Universe. On the other hand, it may be that these observations are just signalling the fact that Einstein's General Relativity is not the correct description of
gravity when we consider distances of the order of the present horizon of the universe.
In order to study if the latter explanation is correct, we have to formulate new theories of the gravitational interaction, and see if they admit cosmological solutions which fit the observational
data in a satisfactory way. Quite generally, modifying General Relativity introduces new degrees of freedom, which are responsible for the different large distance behaviour. On one hand, often these
new degrees of freedom have negative kinetic energy, which implies that the theory is plagued by ghost instabilities. On the other hand, for a modified gravity theory to be phenomenologically viable
it is necessary that the extra degrees of freedom are efficiently screened on terrestrial and astrophysical scales. One of the known mechanisms which can screen the extra degrees of freedom is the
Vainshtein mechanism, which involves derivative self-interaction terms for these degrees of freedom.
In this thesis, we consider two different models, the Cascading DGP and the dRGT massive gravity, which are candidates for viable models to modify gravity at very large distances. Regarding the
Cascading DGP model, we consider the minimal (6D) set-up and we perform a perturbative analysis at first order of the behaviour of the gravitational field and of the branes position around background
solutions where pure tension is localized on the 4D brane. We consider a specific realization of this set-up where the 5D brane can be considered thin with respect to the 4D one.
We show that the thin limit of the 4D brane inside the (already thin) 5D brane is well defined, at least for the configurations that we consider, and confirm that the gravitational field on the 4D
brane is finite for a general choice of the energy momentum tensor. We also confirm that there exists a critical tension which separates background configurations which possess a ghost among the
perturbation modes, and background configurations which are ghost-free. We find a value for the critical tension which is different from the value which has been obtained in the literature; we
comment on the difference between these two results, and perform a numeric calculation in a particular case where the exact solution is known to support the validity of our analysis.
Regarding the dRGT massive gravity, we consider the static and spherically symmetric solutions of these theories, and we investigate the effectiveness of the Vainshtein screening mechanism. We focus
on the branch of solutions in which the Vainshtein mechanism can occur, and we truncate the analysis to scales below the gravitational Compton wavelength, and consider the weak field limit for the
gravitational potentials, while keeping all non-linearities of the mode which is involved in the screening.
We determine analytically the number and properties of local solutions which exist asymptotically on large scales, and of local (inner) solutions which exist on small scales. Moreover, we analyze in
detail in which cases the solutions match in an intermediate region. We show that asymptotically flat solutions connect only to inner configurations displaying the Vainshtein mechanism, while non
asymptotically flat solutions can connect both with inner solutions which display the Vainshtein mechanism, or with solutions which display a self-shielding behaviour of the gravitational field. We
show furthermore that there are some regions in the parameter space of the theory where global solutions do not exist, and characterize precisely in which regions the Vainshtein mechanism takes
Date of Award Dec 2013
Original language English
Awarding Institution • University of Portsmouth
Supervisor Kazuya Koyama (Supervisor) & Marco Bruni (Supervisor) | {"url":"https://researchportal.port.ac.uk/en/studentTheses/modified-theories-of-gravity","timestamp":"2024-11-06T04:39:17Z","content_type":"text/html","content_length":"29621","record_id":"<urn:uuid:014f957f-5d5b-48bf-876e-176868a497b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00821.warc.gz"} |
Solid State Physics
1. Fundamental questions in the theory of solids. Phenomenological and microscopic approaches. Types of binding forces and structure of solids. Description of symmetry of crystalline solids.
Adiabatic approximation in the study of motion of electrons and nuclei.
2. Vibrations of crystal lattice and its thermal properties. Harmonic approximation and normal vibrations of crystalline solids. Acoustic and optical branches of vibrations of ions in crystals,
vibration spectrum of real crystals. Phonons as quasiparticles in the systém of collectively vibrating ions in crystals. Specific heat of solids.
3. Electron theory of ideal crystalline solids. The Hartree - Fock approximation of self - consistent field. Bloch theory of motion of electrons in a periodic electric field in the crystal.
Properties of wave functions and of energy spectrum, quasimomentum of itinerant electrons, the approximation of effective mass. Positive holes in an almost completely filled electron bands.
4. Methods of calculation of band electron structure of solids. The approximation of nearly free electrons, the method of tightly bound electrons, the augmented plane wave and orthogonalized plane
wave methods, the pseudopotential method. Band structure of various types of solids.Fermi surfaces of energy of itinerant electrons in metals. Properties of electrons in valence and conduction bands
in semiconductors.
5. Electron theory of real solids. Wannier theory of motion of electron in perturbed periodic electric field. Localized states of itinerant electrons in crystals with imperfections. States of
electrons near of surface of solids and in thin films. Donor and acceptor energy levels of impurities in semiconductors. Excitons. States of electrons in disordered solids.
6. Electric, magnetic and optical properties of solids. Properties of ensemble of itinerant electrons in statistical equilibrium. Dynamics of itinerant electrons in external electric and magnetic
fields. Paramagnetism and diamagnetism of itinerant electrons. Interband and intraband optical transitions.
7. Transport phenomena in solids. Boltzmann kinetic equation. Scattering of itinerant electrons by phonons and impurities. Relaxation time of conduction electrons in metals and semiconductors.
Galvanomagnetic, thermoelectric and photoelectric phenomena. | {"url":"https://explorer.cuni.cz/class/NFPL181?query=nested","timestamp":"2024-11-10T16:19:43Z","content_type":"text/html","content_length":"37491","record_id":"<urn:uuid:b42a3e50-1760-40fc-a58c-96a7a9fe95b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00234.warc.gz"} |
Dart Scoring
Problem B
Dart Scoring
You have invented a new form of the game of darts, and it is based on the idea that the tightest cluster of darts wins — but they don’t necessarily have to be close to a target. In fact, the game is
easy to play because it doesn’t require a target; players can just throw darts against a wall (preferably one that you don’t care about). When a player is done throwing their $n$ darts, they wrap a
string tightly around the outside of all the outermost darts, so that the string encompasses all of the thrown darts. If $s$ is the length of the string needed to enclose the darts, then the score
for that player’s darts is $100 \cdot n / (1 + s)$.
Now you need a program to determine the score of different dart configurations.
Input consists of up to $200$ player turns, one turn per line. Each player’s turn has a list of $1$ to $100$ pairs of real numbers with at most $3$ digits past the decimal, which are the $(x, y)$
locations where the dart has landed. The bounds are $-30 \le x, y \le 30$. Input ends at the end of file.
For each turn, print the score obtained, with at most $0.0001$ error.
Sample Input 1 Sample Output 1
10.8 -13.7 100.0000000000
0.278 2.555 2.815 3.800 3.920 1.510 29.5344250128
2.358 1.731 0.663 3.485 4.276 6.242 3.858 0.089 0.409 1.460 2.578 4.539 34.3557840779
0.117 5.881 4.655 2.766 0.941 0.213 6.180 6.550 4.215 6.723 5.822 4.367 5.464 1.001 30.3425374145 | {"url":"https://liu.kattis.com/courses/AAPS/AAPS24/assignments/jjtpsu/problems/dartscoring","timestamp":"2024-11-13T14:34:11Z","content_type":"text/html","content_length":"26243","record_id":"<urn:uuid:9e5dda9d-e2fc-455e-a101-b570af8d820b>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00658.warc.gz"} |
ANSTO :: Browsing by Author "Soda, M"
Browsing by Author "Soda, M"
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• Magnetic order in frustrated Kagome-Triangular lattice antiferromagnet NaBa2Mn3F11
(American Physical Society, 2017-03) Hayashida, S; Ishikawa, H; Okamoto, Y; Okubo, T; Hiroi, Z; Avdeev, M; Manuel, P; Hagihala, M; Soda, M; Masuda, T
We performed powder neutron diffraction experiments on NaBa2Mn3F11 [1], a model compound of \textit{Kagome-Triangular} lattice where three of six next-nearest neighbor interactions are
non-negligible. More than 10 magnetic Bragg peaks are clearly observed below T= 2 K, meaning that the ground state is a magnetically ordered state. From indexing the magnetic Bragg peaks,
magnetic propagation vector of \textbf{\textit{q}}0= (0, 0, 0) and two incommensurate vectors which are close to (1/3, 1/3, 0) are identified. Combination of representation analysis and Rietveld
refinement reveals that the propagation vector of \textbf{\textit{q}}0 exhibits the 120º structure in the \textit{ab}-plane. Our calculation of the ground state suggests that the non-negligible
magnetic dipolar interaction is responsible for the determined 120º structure in NaBa2Mn3F11. © 2021 American Physical Society
• Magnetic ordering of the buckled honeycomb lattice antiferromagnet Ba2NiTeO6
(American Physical Society, 2016-01-19) Asai, S; Soda, M; Kasatani, K; Ono, T; Avdeev, M; Masuda, T
We investigate the magnetic order of the buckled honeycomb lattice antiferromagnet Ba2NiTeO6 and its related antiferromagnet Ba3NiTa2O9 by neutron diffraction measurements. We observe magnetic
Bragg peaks below the transition temperatures, and identify propagation vectors for these oxides. A combination of representation analysis and Rietveld refinement leads to a collinear magnetic
order for Ba2NiTeO6 and a 120∘ structure for Ba3NiTa2O9. We find that the spin model of the bilayer triangular lattice is equivalent to that of the two-dimensional buckled honeycomb lattice
having magnetic frustration. We discuss the magnetic interactions and single-ion anisotropy of Ni2+ ions for Ba2NiTeO6 in order to clarify the origin of the collinear magnetic structures. Our
calculation suggests that the collinear magnetic order of Ba2NiTeO6 is induced by the magnetic frustration and easy-axis anisotropy. ©2016 American Physical Society
• Magnetic state selected by magnetic dipole interaction in the kagome antiferromagnet NaBa2Mn3F11
(American Physical Society, 2018-02-12) Hayashida, S; Ishikawa, H; Okamoto, Y; Okubo, T; Hiroi, Z; Avdeev, M; Manuel, P; Hagihala, M; Soda, M; Masuda, T
We haved studied the ground state of the classical kagome antiferromagnet NaBa2Mn3F11. Strong magnetic Bragg peaks observed for d spacings shorter than 6.0 Å were indexed by the propagation
vector of k0=(0,0,0). Additional peaks with weak intensities in the d-spacing range above 8.0 Å were indexed by the incommensurate vector of k1=[0.3209(2),0.3209(2),0] and k2=[0.3338(4),0.3338
(4),0]. Magnetic structure analysis unveils a 120∘ structure with the tail-chase geometry having k0 modulated by the incommensurate vector. A classical calculation of the Heisenberg kagome
antiferromagnet with antiferromagnetic second-neighbor interaction, for which the ground state a k0120∘ degenerated structure, reveals that the magnetic dipole-dipole (MDD) interaction including
up to the fourth neighbor terms selects the tail-chase structure. The observed modulation of the tail-chase structure is attributed to a small perturbation such as the long-range MDD interaction
or the interlayer interaction. ©2018 American Physical Society
• Magnetic structure and dielectric state in the multiferroic Ca2CoSi2O7
(The Physical Society of Japan, 2017-05-10) Soda, M; Hayashida, S; Yoshida, T; Akaki, M.; Hagiwara, M; Avdeev, M; Zaharko, O; Masuda, T
The magnetic structure of the multiferroic Ca2CoSi2O7 was determined by neutron diffraction techniques. A combination of experiments on polycrystalline and single-crystal samples revealed a
collinear antiferromagnetic structure with the easy axis along the 〈100〉 directions. The dielectric state is discussed in the framework of the spin-dependent d–p hybridization mechanism,
leading to the realization of the antiferroelectric structure. The origin of the magnetic anisotropy is discussed in comparison with that of the isostructural Ba2CoGe2O7. ©2017 The Physical
Society of Japan
• Neutron scattering study in breathing pyrochlore antiferromagnet Ba3Yb2Zn5O11
(International Conference on Neutron Scattering, 2017-07-12) Masuda, T; Haku, T; Soda, M; Sera, M; Kimura, K; Taylor, J; Itoh, S; Yokoo, T; Matsumoto, Y; Yu, DH; Mole, RA; Takeuchi, T; Nakatsuji,
S; Kohno, Y; Sakakibara, T; Chang, LJ
Comprehensive study on breathing pyrochlore antiferromagnet Ba3Yb2Zn5O11 [1] is presented. To identify the energy scheme of crystalline electric field (CEF), we performed inelastic neutron
scattering (INS) measurement in high energy range. The observed dispersionless excitations are explained by a CEF Hamiltonian of Kramers ion Yb3+ of which the local symmetry exhibits C3v point
group symmetry. The magnetic susceptibility is consistently reproduced by the energy scheme of the CEF excitations. To identify the spin Hamiltonian we performed INS experiment in low energy
range and thermodynamic property measurements at low temperatures. The INS spectra are quantitatively explained by spin-1/2 single-tetrahedron model having XXZ anisotropy and
Dzyaloshinskii-Moriya interaction. This model has a two-fold degeneracy of the lowest-energy state per tetrahedron and well reproduces the magnetization curveat 0.5 K and heat capacity above 1.5
K. At lower temperatures, however, we observe a broad maximum in the heat capacity around 63 mK, demonstrating that a unique quantum ground state is selected due to extra perturbations with
energy scale smaller than the instrumental resolution of INS. Possible mechanisms for the ground state selection are discussed [2].
• Neutron scattering study of the quasi-one-dimensional antiferromagnet Ba2CoSi2O7
(American Physical Society, 2019-10-07) Soda, M; Hong, T; Avdeev, M; Yoshizawa, H; Masuda, T; Kawano-Furukawa, H
Magnetization and neutron scattering measurements have been carried out on an antiferromagnet Ba2CoSi2O7. The observed magnetic excitation is almost dispersionless, and the neutron intensity is
only modulated along the [101] direction. The dispersionless magnetic excitation suggest that the Ba2CoSi2O7 system is a quasi-one-dimensional antiferromagnet. Classical spin-wave theory for a
one-dimensional antiferromagnet can explain the dispersionless spin excitation. The magnetic structure determined by the measurement of the neutron powder diffraction is consistent with no
observation of the multiferroic property in this system. ©2019 American Physical Society
• Stripy order in buckled honeycomb lattice antiferromagnet Ba2NiTeO6
(International Conference on Neutron Scattering, 2017-07-12) Asai, S; Soda, M; Kasatani, K; Ono, T; Avdeev, M; Garlea, VO; Winn, B; Masuda, T
Ba NiTeO is a rare experimental realization of a buckled honeycomb lattice antiferromagnet. The nearest-neighbor and next-nearest-neighbor interactions in the honeycomb lattice are comparative
due to the buckled geometry, leading to magnetic frustration. A magnetic transition is observed at 8.6 K in the susceptibility and heat capacity measurements [1]. The frustration parameter /T is
18.6, where is Weiss temperature and is the magnetic transition temperature. In order to investigate the low temperature state we performed neutron scattering experiments. In the diffraction
profile magnetic Bragg peaks are observed at < , and the propagation vector is identified as (0, 1/2,1). Combination of the representation analysis and Rietveld refinement reveals that a
collinear stripy structure [2] is realized [3]. Our calculation suggests that the stabilization of the stripy structure instead of spiral structure is ascribed to the competition between magnetic
frustration and easy-axis type anisotropy. In the inelastic neutron spectrum at 2 K a magnetic excitation with an energy gap of 2 meV is observed. Spin-wave calculation based on two-dimensional
frustrated honeycomb lattice antiferromagnet having easy-axis anisotropy reproduces the experimental data. The obtained parameters are consistent with Weiss temperature estimated from the bulk
magnetic susceptibility measurement. | {"url":"https://apo.ansto.gov.au/browse/author?value=Soda,%20M","timestamp":"2024-11-12T18:40:10Z","content_type":"text/html","content_length":"482627","record_id":"<urn:uuid:ea696c09-c5f9-4abe-91ed-2fd9733aab58>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00646.warc.gz"} |
Cauchy sequence
Define a Cauchy sequence. Prove: a) In any metric space \(\mathrm{X}\), every convergent sequence is a Cauchy sequence. b) Suppose \(\left\\{p_{n}\right\\}\) is a Cauchy sequence in a compact metric
space \(\mathrm{X}\), then \(\left\\{\mathrm{p}_{\mathrm{n}}\right\\}\) converges to some point of \(\mathrm{X}\)
Short Answer
Expert verified
A Cauchy sequence is a sequence \(\left\\{p_n\right\\}\) in a metric space \(X\) with distance function \(d\) such that, for any given \(\epsilon > 0\), there exists a natural number \(N\) with \(d\
left(p_m, p_n\right) < \epsilon\) for all \(m,n \ge N\). (a) Every convergent sequence in a metric space is a Cauchy sequence, as proven using the triangle inequality and taking \(N = \max\{N_1, N_2
\}\) for appropriate choices of \(N_1\) and \(N_2\). (b) If \(\left\\{p_n\right\\}\) is a Cauchy sequence in a compact metric space \(X\), then there exists a point \(p \in X\) such that \(\lim_{n \
to \infty} p_n = p\), as proven by showing the existence of a convergent subsequence and applying the triangle inequality again.
Step by step solution
Definition of a Cauchy Sequence
A sequence \(\left\\{p_n\right\\}\) in a metric space \(X\) with distance function \(d\) is called a Cauchy sequence if, for any given \(\epsilon > 0\), there exists a natural number \(N\) such that
for any \(m,n \ge N\), we have \(d\left(p_m, p_n\right) < \epsilon\).
Proof of (a): Every convergent sequence is a Cauchy sequence
Let \(\left\\{p_n\right\\}\) be a convergent sequence in a metric space \(X\) with distance function \(d\). Let \(p\) be the limit of this convergent sequence. That is, \(\lim_{n \to \infty} p_n = p
\). Given \(\epsilon > 0\), we need to show that there exists an \(N\) such that if \(m, n \ge N\), then \(d\left(p_m, p_n\right) < \epsilon\). Since \(\lim_{n \to \infty} p_n = p\), for \(\frac{\
epsilon}{2} > 0\), there exists an \(N_1\) such that for all \(n \ge N_1\), \(d\left(p_n, p\right) < \frac{\epsilon}{2}\). Similarly, there exists an \(N_2\) such that for all \(m \ge N_2\), \(d\left
(p_m, p\right) < \frac{\epsilon}{2}\). Now, let \(N = \max\{N_1, N_2\}\). If \(m, n \ge N\), then by the triangle inequality, we have \[d\left(p_m, p_n\right) \le d\left(p_m, p\right) + d\left(p, p_n
\right) < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.\] Therefore, the sequence \(\left\\{p_n\right\\}\) is a Cauchy sequence.
Proof of (b): A Cauchy sequence in a compact metric space converges
Suppose \(\left\\{p_n\right\\}\) is a Cauchy sequence in a compact metric space \(X\) with distance function \(d\). We need to prove that there exists a point \(p \in X\) such that \(\lim_{n \to \
infty} p_n = p\). Since \(X\) is compact, every sequence in \(X\) has a convergent subsequence. So, there exists a subsequence \(\left\\{p_{n_k}\right\\}\) of \(\left\\{p_n\right\\}\) which converges
to a point \(p \in X\). Now, given \(\epsilon > 0\), since \(\left\\{p_n\right\\}\) is a Cauchy sequence, there exists an \(N_1\) such that for any \(m, n \ge N_1\), \(d\left(p_m, p_n\right) < \frac
{\epsilon}{2}\). Also, since \(\lim_{k \to \infty} p_{n_k} = p\), there exists an \(N_2\) such that for any \(k \ge N_2\), \(d\left(p_{n_k}, p\right) < \frac{\epsilon}{2}\). Let \(N = \max\{N_1, N_2
\}\). For any \(n \ge N\), we have: 1. By the definition of a Cauchy sequence, \(d(p_n, p_{n_N}) < \frac{\epsilon}{2}\). 2. Since \(n_N \ge N \ge N_2\), by the convergence of the subsequence, \(d(p_
{n_N}, p) < \frac{\epsilon}{2}\). Now, using the triangle inequality, we have: \[d(p_n, p) \le d(p_n, p_{n_N}) + d(p_{n_N}, p) < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.\] Thus, \(\lim_{n
\to \infty} p_n = p\), and the Cauchy sequence \(\left\\{p_n\right\\}\) converges to a point of \(X\).
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding a Metric Space
When studying advanced mathematics, you will likely encounter the concept of a
metric space
. It's a key idea in analysis and topology that helps us understand the distance between points in a more abstract sense beyond just the familiar Euclidean space.
A metric space is a set, let's call it
, paired with a function known as a metric or distance function, denoted as
. This function measures the distance between any two points within the set. The metric space must adhere to specific properties:
• The distance between two points is always a non-negative number.
• The distance from a point to itself is zero.
• The distance from point A to point B is the same as from point B to point A (symmetry).
• The metric also must satisfy the triangle inequality. Which means the direct distance between two points is always less than or equal to the sum of the distances when taking a detour through a
third point.
Grasping the notion of a metric space is vital, as it lays the foundation for understanding sequences within that space, such as Cauchy sequences and their behavior with respect to convergence.
Convergent Sequences and Cauchy Sequences
A convergent sequence is a series of points in a metric space that come increasingly closer to a specific point, called the limit. Formally, a sequence {p_n} converges to a point p if, as n gets
larger and larger, the distance d(p_n, p) becomes arbitrarily small.
A Cauchy sequence, while similar to a convergent sequence, does not require knowledge of the limit. Instead, it focuses on the distances between points within the sequence itself. A sequence is
Cauchy if for any small number ε you choose, you can find a point in the sequence after which all subsequent points are within ε distance of each other.
Relationship between Cauchy and Convergent Sequences
In metric spaces, every convergent sequence is a Cauchy sequence. This is because, as the points get close to the limit, they also get closer to each other. However, the reverse isn't always true;
not all Cauchy sequences converge in every metric space. This leads us to explore special types of metric spaces where Cauchy sequences do indeed always converge - which brings us to compact metric
Compact Metric Spaces: Guaranteeing Convergence
A compact metric space is a special type of metric space that holds the property wherein every sequence of points has a subsequence that converges to a point within the space
Compactness is a form of 'limitedness' ensuring that a space is not too large or spread out, making it possible to control sequences and their limits effectively. An important property of compact
metric spaces is that every Cauchy sequence in the space will converge to a point within the space. This is a significant result because it guarantees the completeness of the space in terms of
sequence convergence.
This is particularly useful when working with Cauchy sequences since, in compact spaces, you don't need to know ahead of time what the limit is. You have the assurance that a limit exists and that
the sequence will get arbitrarily close to it, which is essential for many areas of analysis and is key to understanding continuity, integration, and other fundamental concepts in mathematics. | {"url":"https://www.vaia.com/en-us/textbooks/math/advanced-calculus-1-edition/chapter-11/problem-326-define-a-cauchy-sequence-prove-a-in-any-metric-s/","timestamp":"2024-11-12T00:32:33Z","content_type":"text/html","content_length":"259337","record_id":"<urn:uuid:2727c2b3-7cf2-4a98-a984-34a0de89dbf7>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00083.warc.gz"} |
Ethan T. Parcell
The first thing to keep in mind is
I'm left handed, by the way
Don't be alarmed
We're going to start with the head
It is rounded
In a particular way
Imagine a pretty narrow letter “U”
turned upside down
Here we go
Now we're moving on to the beak.
This is unrealistic
Forgive me
It's a similar shape to the head
Perhaps a little less narrow
Make sure to overlap the right edge of your head shape
with the start of the beak
creating a nice little 90 degree angle on both sides
of the intersection
here we go
Onto the duck's body
Have you ever seen an oval?
Imagine an oval but leave out the top right corner
Ovals don't have corners
Imagine instead
If the oval were to blossom
into a circle
the section of it
you could maybe call
“1 through 3 o'clock”
do you follow?
Remove that part, because that's where our head and beak lives
maybe I'll just show you
This part is fun
It's the duck's feet
Don't worry about them
They are easy
Now that the body is there
Imagine back
to our
“oval becomes a circle which is a clock” scenario
the feet are probably at 7 and 8 o'clock respectively
and they are just lines
The eyes of the duck are next
Eyes are what helps us to see
That's even true about real ducks
But with a real duck-
if you were looking at it in profile like we are
with ours-
you would probably only see one eye
but that's the case the case with this duck
We want to see both
so as to understand its expression more clearly
above the beak
inside the head
and now for our finishing touches
the wing
and the smile
both of these components
are false for distinct reasons
I don't think ducks smile
Our scenario in which we see both of our duck's eyes
Despite being in profile view
might not align with only seeing one wing
maybe it does
the wing is probably not just a straight line
but pretty close to one
the smile is nearly identical in shape
but inside the beak
here's one
and here's the other
This is a duck
It is what we can draw
we can draw it many times
we can tie knots
and eat breakfast most days
and attempt
to fill the planet with the highest quantity
of harmless things
and rid the planet of the highest quantity
of harmful things
or at least not fill the planet
with very many harmful things
can you hear my voice?
Listen to my voice
isn't this nice?
go ahead and add another zero or two
remove punctuation
for now
Day is good and long
Short is a smaller word than little
a horizon is horizontal
don't ask if me if anything is vertical
there are places that change
electricity into something else
whatever electricity isn't already
which isn't much but is probably plenty | {"url":"http://www.ethantparcell.info/p/whyducks.html","timestamp":"2024-11-08T02:03:46Z","content_type":"application/xhtml+xml","content_length":"35470","record_id":"<urn:uuid:c340bbca-bf29-43dd-a510-493fd3685e1e>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00180.warc.gz"} |
How to Calculate a Rank in Excel - EasyClick AcademyHow to Calculate a Rank in Excel
In this tutorial, we’re going to have a look at how to calculate a rank in Excel. Based on the calculation, you’ll be able to compare data of any size and determine their position within the data
Would you rather watch this tutorial? Click the play button below!
To rank data, first we need to click into the cell where we want to display the result.
We start typing the equal sign and the first letters of the word ‘RANK’, and Excel will come up with the right function suggestion immediately.
Here we need to note that the Rank function is available for compatibility with Excel 2007 and earlier.
Let’s click on the suggested function and fill in the missing bits. First, we need to enter the number we want to calculate the rank for. This value is stored in the cell C3, which contains the
information on Tommy’s sales.
We can add a comma and now we need to include all the numbers of the data set which we want to compare and evaluate, and against which we want to determine Tommy’s position within the ranking. So,
here we’ll select the sales by the rest of the salesmen.
However, we need to be careful not to forget to lock the reference to this set of data. Click on C3 and press and hold the Function key along with F4. Some of you don’t have to use the Function key –
this depends on the keyboard type you’re using.
The dollar sign appeared next to the cell reference, which means that the value for calculating the rank will stay the same once the function is copied in the rows below.
We use the same keys to fix the coordinates C7 to make sure that the reference to the whole area will remain unchanged.
If we copy the Rank function to the rest of the rows below, the reference to the relevant cells won’t change, which is important as we want to find out the rank for each of the salesmen, so we need
to keep the set of data the same in each calculation. If you’re interested to know more on how to use the absolute cell reference in Excel, check out a separate tutorial the link to which is in the
list below.
But let’s move on now and add another comma. Now it’s time to specify whether we want the rank to be descending or ascending.
If we go for descending, the highest number on the list will become number one and the lowest sales count will be the last in the rank. This option is set as default in Excel, so if we close the
bracket at this point, without adding any extra information, Excel will go on and calculate a descending rank.
To calculate the ascending rank, we need to choose the option ‘ascending’. In that case, the lowest number will become number one and the highest number will be marked as the last in the ranking.
Let’s go for ‘descending’ now – click on it, close the brackets, hit Enter and here we are! You can see the position of Tommy’s sales compared to the sales by the rest of the salesmen.
Copy the function simply by dragging the bottom right-hand corner of the cell containing the complete formula and the ranking for other salesmen appears right away.
As you can see, the first one in the ranking is the salesman who got the highest number of sales and the last one is the one with the lowest sales count, just as we wanted.
There’s one important thing to be aware of – if there are two values which are the same within the selected data area, Excel will do the evaluation in a specific way.
Here you can see that the third position belongs to two salesmen with the same sales counts. They’ve been both put in the same rank, which is three, but since there are two people with the same
numbers, Excel skipped position four and the ranking continues with position five.
If there were three identical values on the same position, the following two places would get skipped, so make sure to keep this rule in mind when calculating a rank in Excel.
If you found this tutorial helpful, give us a like and watch other tutorials by EasyClick Academy. Learn how to use Excel in a quick and easy way!
Is this your first time on EasyClick? We’ll be more than happy to welcome you in our online community. Hit that Subscribe button and join the EasyClickers!
Thanks for watching and I’ll see you in the next tutorial! | {"url":"https://www.easyclickacademy.com/how-to-calculate-a-rank-in-excel/","timestamp":"2024-11-12T19:49:46Z","content_type":"text/html","content_length":"117161","record_id":"<urn:uuid:5cd3c5fe-94e3-4b40-a650-7de76d96639f>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00572.warc.gz"} |
Calculate proportions by category and create histogram in ggplot
Visualizing Proportions with Histograms in ggplot2: A Step-by-Step Guide
Data visualization is crucial for understanding complex datasets, and histograms offer a powerful way to represent the distribution of continuous variables. When working with categorical data,
however, we often need to visualize the proportions within each category. This article will guide you through calculating proportions by category and creating informative histograms in ggplot2, a
popular R package for data visualization.
Let's consider a hypothetical dataset of customer purchases, where we want to analyze the distribution of purchase amounts across different customer segments. Here's a simplified example:
# Sample data frame
purchases <- data.frame(
customer_segment = c(rep("Gold", 100), rep("Silver", 150), rep("Bronze", 50)),
purchase_amount = c(runif(100, 10, 100), runif(150, 5, 50), runif(50, 1, 20))
# Original code - calculating proportions by category and creating histogram
# Incorrect code
ggplot(purchases, aes(x = purchase_amount, fill = customer_segment)) +
geom_histogram(position = "fill")
This original code aims to create a histogram where the bars represent the proportion of each customer segment within each purchase amount bin. However, the position = "fill" argument in
geom_histogram is not designed to calculate proportions across categories. Instead, it stacks bars on top of each other, making it difficult to visualize the proportions correctly.
Correcting the Approach
To achieve our desired visualization, we need to first calculate the proportions of each customer segment within each purchase amount bin. We can achieve this using the dplyr package:
# Calculate proportions by category and bin
proportions <- purchases %>%
mutate(bin = cut(purchase_amount, breaks = seq(0, 100, by = 10))) %>%
group_by(customer_segment, bin) %>%
summarize(count = n()) %>%
mutate(proportion = count / sum(count))
# Create the histogram
ggplot(proportions, aes(x = bin, y = proportion, fill = customer_segment)) +
geom_bar(stat = "identity", position = "dodge") +
labs(x = "Purchase Amount (Bins)", y = "Proportion", fill = "Customer Segment")
1. We use cut() to divide the purchase amounts into bins of size 10.
2. We group the data by customer segment and bin using group_by().
3. We calculate the count of observations within each group using summarize().
4. We calculate the proportion of each customer segment within each bin using mutate().
5. Finally, we create the histogram using ggplot(), where we map the proportion to the y-axis and the bin to the x-axis. We use geom_bar(stat = "identity", position = "dodge") to create separate
bars for each customer segment within each bin.
Advantages of this approach:
• Clear Visualization: The histogram now accurately represents the proportion of each customer segment within each purchase amount bin, allowing for easy comparison between categories.
• Flexibility: You can easily adjust the bin width and number of bins using the breaks argument in cut().
• Customization: You can further customize the histogram using various ggplot2 options, such as adding titles, changing colors, and adding annotations.
Practical Applications
This approach is widely applicable in various scenarios where you need to visualize proportions across categories. For example:
• Marketing Analysis: Analyzing the proportion of customers who respond to different marketing campaigns by demographic group.
• Sales Analysis: Visualizing the proportion of sales by product category in different regions.
• Financial Analysis: Comparing the proportion of investments in different asset classes across different portfolio types.
By calculating proportions and creating histograms in ggplot2, you gain valuable insights into the distribution of data within different categories. This approach allows you to create clear,
informative visualizations that help you understand complex datasets and make informed decisions. | {"url":"https://laganvalleydup.co.uk/post/calculate-proportions-by-category-and-create-histogram-in","timestamp":"2024-11-10T16:00:59Z","content_type":"text/html","content_length":"83745","record_id":"<urn:uuid:2a62b147-ca98-4172-8402-7c62a2e51fcf>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00326.warc.gz"} |
The Homogeneous Broadcast Problem in Narrow and Wide Strips I: Algorithms
Let P be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let s∈ P be a given source node. Each node p can transmit information to all other nodes within
unit distance, provided p is activated. The (homogeneous) broadcast problem ... read more | {"url":"https://dspace.library.uu.nl/handle/1874/382144","timestamp":"2024-11-13T08:28:55Z","content_type":"application/xhtml+xml","content_length":"18902","record_id":"<urn:uuid:fdc1f164-da14-4298-a377-4e6b9667b2b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00850.warc.gz"} |
COUNT AND SUMIFS ... TO NOT INCLUDE CERTAIN CRITERIAS
Hi, I am wanting to =countif({Postcode 2}, does not contain any of the listed postcodes ....
and then the same for a sumif however my brain is not working :(
help please :)
• Hi @Stacy Meadows,
You have a couple of choices on this one:
You can make a long formula to do the OR & NOT:
=COUNTIF({Postcode 2}, NOT(OR(@cell = Postcode1, @cell = Postcode2, @cell = Postcode3)))
^ This is just doing the first 3, but you should get the idea - the numbers will need changing to the relevant row numbers to accommodate the "title" rows.
Alternatively, if your data source only has rows with information (no totals etc.) then you could use:
=COUNT({Postcode 2})-SUM(Count1:Count 14)
This would find the total and then subtract the ones which have already been accounted for.
The SUMIF/SUM formulas would be much the same, just substituting the relevant function in.
Hope this helps, if you've any questions etc. then just post! ☺️
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/105405/count-and-sumifs-to-not-include-certain-criterias","timestamp":"2024-11-11T07:25:03Z","content_type":"text/html","content_length":"392423","record_id":"<urn:uuid:2aa969ed-eba9-45b7-8879-6b2121bcd7f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00872.warc.gz"} |
Drawing Graphs as Spanners
We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph G, the goal is to construct a straight-line drawing Γ of G in the plane such that, for any two
vertices u and v of G, the ratio between the minimum length of any path from u to v and the Euclidean distance between u and v is small. The maximum such ratio, over all pairs of vertices of G, is
the spanning ratio of Γ. First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio 1, a proper straight-line drawing with spanning ratio 1, and a planar
straight-line drawing with spanning ratio 1 are NP-complete, ∃ R-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a
vertex. Second, we show that moving from spanning ratio 1 to spanning ratio 1 + ϵ allows us to draw every graph. Namely, we prove that, for every ϵ> 0 , every (planar) graph admits a proper (resp.
planar) straight-line drawing with spanning ratio smaller than 1 + ϵ. Third, our drawings with spanning ratio smaller than 1 + ϵ have large edge-length ratio, that is, the ratio between the length of
the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that
distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line
drawing with constant spanning ratio.
ASJC Scopus subject areas
• Theoretische Informatik
• Geometrie und Topologie
• Diskrete Mathematik und Kombinatorik
• Theoretische Informatik und Mathematik
Fields of Expertise
• Information, Communication & Computing | {"url":"https://graz.elsevierpure.com/de/publications/drawing-graphs-as-spanners-2","timestamp":"2024-11-09T19:22:31Z","content_type":"text/html","content_length":"43925","record_id":"<urn:uuid:86e64c8e-182f-4faf-9f6d-e7cf09ddac6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00232.warc.gz"} |
Assignment 3: Introduction to Data Science and AI
import pandas as pd import numpy as np import matplotlib.pyplot as plt import matplotlib.cm as cm import math from scipy.ndimage.filters import gaussian_filter import seaborn as sns from matplotlib
import mlab as ml from sklearn.datasets import make_blobs from sklearn import metrics from sklearn.cluster import KMeans from sklearn.preprocessing import StandardScaler
# Importing the data set to a data frame file_name = "data_all.csv" df = pd.read_csv(file_name) # Checking for any NaN values in the data set df.isnull().values.any()
# Extracting the variables X_phi = df['phi'] X_psi = df['psi'] # Making a scatterplot plt.figure(figsize=(14,9)) plt.scatter(X_phi, X_psi, s = 10, c = 'darkcyan') plt.grid(True) plt.title(
'Distribution of Phi and Psi combinations for protein molecules') plt.xlabel('Phi, in degrees') plt.ylabel('Psi, in degrees') plt.show()
# Generating some test data plt.figure(figsize=(14,9)) heatmap, xedges, yedges = np.histogram2d(X_phi, X_psi, bins = 220) heatmap = gaussian_filter(heatmap, sigma = 32) extent = [xedges[0], xedges[-1
], yedges[0], yedges[-1]] plt.clf() plt.axis([X_phi.min(), X_phi.max(), X_psi.min(), X_psi.max()]) plt.imshow(heatmap.T, extent=extent, origin='lower', cmap=cm.jet) cb = plt.colorbar() cb.set_label(
'Number of samples per bin') plt.title("Distribution of Phi and Psi combinations for protein molecules") plt.xlabel('Phi, in degrees') plt.ylabel('Psi, in degrees') plt.grid(True) plt.show()
# Function for conducting K-means Clustering and plotting the results def kmeans_clustering(X, n_clusters): plt.figure(figsize=(6,4)) # Perform K-means clustering kmeans = KMeans(n_clusters =
n_clusters, random_state = 42) y_pred = kmeans.fit_predict(X) plt.scatter(X[:,0], X[:,1], c = y_pred, cmap='gist_rainbow', edgecolor='black', s = 20) plt.scatter(kmeans.cluster_centers_[:, 0], kmeans
.cluster_centers_[:, 1], marker='x', c='black') plt.title(f'Data points with K-means algorithm when k = {n_clusters}') plt.show()
# Scaling the data as the euclidean distance is used for K-means algorithm X = df[['phi', 'psi']] scaler = StandardScaler() X = scaler.fit_transform(X) # Setting the range of k that we want to test
between 2 - 8 k_values = range(2, 8) # For each value in our lenght, do the K-means clustering algorithm and display the results for i in k_values: kmeans_clustering(X, i)
distortions = [] n_clusters = range(1,10) # We try out different k:s and get the inertia for cluster in n_clusters: kmean_model = KMeans(n_clusters = cluster) kmean_model.fit(X) distortions.append(
plt.figure(figsize=(6,4)) plt.plot(n_clusters, distortions, 'bx-') plt.xlabel('Number of Clusters') plt.ylabel('Sum of Squared Distance') plt.title('Elbow Method Showing The Optimal # Clusters') plt.
grid(True) plt.show()
# Selecting k as 4 k = 4 percentages = [1,0.75,0.5,0.25,0.10] plots = len(percentages) fig, axs = plt.subplots(1, plots, figsize=(20,4)) for i in range(0, plots): # choosing different sample lengths
for the plots to show difference in centroids being selected n_sample_size = math.floor(len(X) * percentages[i]) # Creating blobs of the data in order to change the sample sizes X, y = make_blobs(
n_samples = n_sample_size, centers = df[['phi', 'psi']]) kmeans = KMeans(n_clusters = k, random_state = 0) y_pred = kmeans.fit_predict(X) axs[i].scatter(X[:,0], X[:,1], c = y_pred, cmap =
'gist_rainbow', edgecolor='black', s = 20) # Plot kmeans cluster centers axs[i].scatter(kmeans.cluster_centers_[:, 0], kmeans.cluster_centers_[:, 1], marker='x', c= 'black') axs[i].set_title(
f'Displaying {percentages[i]*10}% of the total data points') plt.tight_layout() plt.show()
This plot illustrates how the clustering is affected by the removal of random data points. The results show a strong consistency in the clustering, using k=4. The groups remain consistent through out
the iterations and keep the overarching structure, where you can clearly see that points which were together in a previous iteration remain clustered together.
# Selecting k as 4 X = df[['phi', 'psi']] scaler = StandardScaler() X = scaler.fit_transform(X) random_init = [0, 1, 2, 3, 4, 5] plots = len(random_init) fig, axs = plt.subplots(1, plots, figsize=(20
,4)) for i in range(0, plots): kmeans = KMeans(n_clusters = k, random_state = i) y_pred = kmeans.fit_predict(X) axs[i].scatter(X[:,0], X[:,1], c = y_pred, cmap= 'gist_rainbow', edgecolor='black', s =
20) axs[i].scatter(kmeans.cluster_centers_[:, 0], kmeans.cluster_centers_[:, 1], marker = 'x',c = 'black') axs[i].set_title(f'Random state {i}') plt.tight_layout() plt.show()
The plot displays an incredible consistency in the clustering, when using 4 groups, regardless of initial position for the centroids in K-means algorithm. This speaks very highly of the consistency
of k=4 and the validity of the clusters.
X = df[['phi', 'psi']] k_values = range(2, 8) k_opt = 0 high_score = 0 for k in k_values: kmeans = KMeans(n_clusters = k, random_state = 0).fit(X) labels = kmeans.labels_ score = metrics.
silhouette_score(X, labels, metric = 'euclidean') print(f"Silhoutte score for k = {k} is: {score}") if (score > high_score): k_opt = k high_score = score print(f"The optimal silhoutte score is for k
= {k_opt} and is: {high_score}")
The silhoutte score ranges from -1 to 1. Values closer to 1 indicate a good clustering method. Looking for the different suggestions for k (ranging from 2 to 8) we find that k=3 and k=4 provides the
best sihoutte scores, as they are so similar it is hard to make a judgement. Using the findings from the 'elbow method' and visual inspection of plots, we choose k=4 as the best fit.
df_mod = df.copy() df_mod['phi']=(df['phi']+360)%360 df_mod['psi']=(df['psi']+360)%360 # Extracting the variables X_phi_mod = df_mod['phi'] X_psi_mod = df_mod['psi'] # Making a scatterplot plt.figure
(figsize=(14,9)) plt.scatter(X_phi_mod, X_psi_mod, s = 10, c = 'darkcyan', label = 'Distribution of phi and psi') plt.grid(True) plt.title('Distribution of Phi and Psi combinations for protein
molecules (shifted by 360 degrees)') plt.xlabel('Phi, in degrees') plt.ylabel('Psi, in degrees') plt.legend(loc ='upper left') plt.show()
X = df_mod[['phi', 'psi']] X = StandardScaler().fit_transform(X) k_values = range(2, 8) for value in k_values: kmeans_clustering(X, value)
It appears as though the most intuitive fit is now instead 3 clusters. Let's see how this looks for the silhoutte score. Let's compare k=3 and k=4 (which was the most effective before shifting the
data set).
X = df_mod[['phi', 'psi']] k_values = [3, 4] k_opt=0 high_score = 0 for k in k_values: kmeans = KMeans(n_clusters=k, random_state=0).fit(X) labels = kmeans.labels_ score = metrics.silhouette_score(X,
labels, metric='euclidean') print(f"Silhoutte score for k = {k} is: {score}") if (score > high_score): k_opt = k high_score = score print(f"The optimal silhoutte score is for k = {k_opt} and is: {
The silhoutte score is now clearly optimal for k=3 instead. Which is consistent with the graphical displays above.
from sklearn.cluster import DBSCAN from sklearn import metrics from sklearn.preprocessing import StandardScaler from sklearn.neighbors import NearestNeighbors import collections
X = df[['phi', 'psi']] scaler = StandardScaler() X = scaler.fit_transform(X)
# Function for creating and plotting a DBSCAN for different values of eps and min_samples def createDBSCAN(X, eps = 0.5, min_samples = 100, add_bar_plot = False): # Fitting and predicting given
values provided for eps and min_samples dbscan = DBSCAN(eps = eps, min_samples = min_samples) y = dbscan.fit_predict(X) labels = dbscan.labels_ # Number of clusters in labels, ignoring noise if
present. n_clusters_ = len(set(labels)) - (1 if -1 in labels else 0) n_noise_ = list(labels).count(-1) print('Estimated number of clusters: %d' % n_clusters_) print('Estimated number of noise points:
%d' % n_noise_) outliers_df = df[labels == -1] clusters_df = df[labels != -1] color_clusters = labels[labels != -1] color_outliers = 'black' plt.figure(figsize=(7,4)) plt.scatter(clusters_df['phi'],
clusters_df['psi'], c = color_clusters, edgecolors = 'black', cmap='gist_rainbow', s = 30) plt.scatter(outliers_df['phi'], outliers_df['psi'],c = color_outliers, edgecolors = 'black', label =
'Outliers', s = 30) plt.title(f"Datapoints with DBSCAN, minimum samples:{min_samples}, eps: {eps}") plt.xlabel('Phi, in degrees') plt.ylabel('Psi, in degrees') plt.legend(loc ='upper left') plt.show(
) if add_bar_plot == True: bar = outliers_df['residue name'].value_counts(sort=True).plot.bar() bar.set_title('Amino acid residue types that are most frequently outliers') # Tryng the function with
eps = 0.5 and different values for min_samples createDBSCAN(X, min_samples = 10) createDBSCAN(X, min_samples = 100) createDBSCAN(X, min_samples = 500)
# Minimum samples to test min_samples = [200, 250, 300]
color_list = ['orchid', 'darkcyan', 'darkviolet'] i = 0 for value in min_samples: neigh = NearestNeighbors(n_neighbors = value) # Fitting NearestNeighbors to the data nbrs = neigh.fit(X) # Retrieving
the distances and indices from Kneigbors distances, indices = nbrs.kneighbors(X) distances = np.sort(distances, axis=0) distances = distances[:,1] plt.plot(distances, c = color_list[i], linewidth = 3
) plt.xlabel('Number of points') plt.ylabel('Average Distance') plt.title(f'Finding optimal eps, # of nearest neighbours:{value}') plt.grid(True) plt.show() i = i + 1
eps = [0.3, 0.4, 0.5] for i in eps: print(f'DBSCAN with eps = {i} and various values for min_samples') for j in min_samples: createDBSCAN(X, eps = i, min_samples = j)
# Plotting the cluster found with DBSCAN with epsilon = 0.4 and min_samples = 150 createDBSCAN(X, 0.4, 200, True)
print('For non-translated data, k = 4 is optimal') kmeans_clustering(X, 4) createDBSCAN(X, 0.4, 200)
pro_df = df[(df['residue name'] == 'PRO')].copy() X = pro_df[['phi', 'psi']] X = StandardScaler().fit_transform(X) min_samples = 200 eps = 0.5 # Fitting and predicting given values provided for eps
and min_samples dbscan = DBSCAN(eps = eps, min_samples = min_samples) y = dbscan.fit_predict(X) labels = dbscan.labels_ # Number of clusters in labels, ignoring noise if present. n_clusters_ = len(
set(labels)) - (1 if -1 in labels else 0) n_noise_ = list(labels).count(-1) print('Estimated number of clusters: %d' % n_clusters_) print('Estimated number of noise points: %d' % n_noise_)
outliers_df = pro_df[labels == -1] clusters_df = pro_df[labels != -1] color_clusters = labels[labels != -1] color_outliers = 'black' plt.figure(figsize=(7,4)) plt.scatter(clusters_df['phi'],
clusters_df['psi'], c = color_clusters, edgecolors = 'black', cmap='gist_rainbow', s = 30) plt.scatter(outliers_df['phi'], outliers_df['psi'],c = color_outliers, edgecolors = 'black', label =
'Outliers', s = 30) plt.title(f"Datapoints with DBSCAN for PRO, minimum samples:{min_samples}, eps: {eps}") plt.xlabel('Phi, in degrees') plt.ylabel('Psi, in degrees') plt.legend(loc ='upper left')
The initial parameters seem to produce consistent results, even varying them slightly does not impact the solution.
The clustering using only the residue type PRO differs from the general DBSCAN clustering by not having any clusters with positive Phi values. Furthermore, it produces two well defined clusters, and
does not find any values in the top left corner, which was very prevalent in previous DBSCAN clusters. This is interesting, as DBSCAN never seems to cluster these exact spots, however, for large k,
the k means algorithm seems to find these clusters (found in residue type PRO) more accurately.
pro_df = df[(df['residue name'] == 'GLY')].copy() X = pro_df[['phi', 'psi']] X = StandardScaler().fit_transform(X) min_samples = 200 eps = 0.5 # Fitting and predicting given values provided for eps
and min_samples dbscan = DBSCAN(eps = eps, min_samples = min_samples) y = dbscan.fit_predict(X) labels = dbscan.labels_ # Number of clusters in labels, ignoring noise if present. n_clusters_ = len(
set(labels)) - (1 if -1 in labels else 0) n_noise_ = list(labels).count(-1) print('Estimated number of clusters: %d' % n_clusters_) print('Estimated number of noise points: %d' % n_noise_)
outliers_df = pro_df[labels == -1] clusters_df = pro_df[labels != -1] color_clusters = labels[labels != -1] color_outliers = 'black' plt.figure(figsize=(7,4)) plt.scatter(clusters_df['phi'],
clusters_df['psi'], c = color_clusters, edgecolors = 'black', cmap='gist_rainbow', s = 30) plt.scatter(outliers_df['phi'], outliers_df['psi'],c = color_outliers, edgecolors = 'black', label =
'Outliers', s = 30) plt.title(f"Datapoints with DBSCAN for GLY, minimum samples:{min_samples}, eps: {eps}") plt.xlabel('Phi, in degrees') plt.ylabel('Psi, in degrees') plt.legend(loc ='upper left')
The initial parameters seem to produce consistent results, even varying them slightly does not impact the solution.
The residue type GLY seems to represent somewhat more of the clusters found in the general case. We see one cluster with phi>0, we find clusters both in the upper left and the middle left.
However, some data points fall in the remaining clusters found in the general case, however, these are deemed outliers by the DBSCAN method.
It is important to consider that in previous tasks we found that the GLY residue had the highest number of outliers, by multiple factors. This can be displayed in the clustering of only GLY residues
also, as we can see there are no clear clusters, there seems to be data points in each quadrant of the graph, and some almost randomly scattered. | {"url":"https://deepnote.com/app/sara/Assignment-3-Introduction-to-Data-Science-and-AI-9241b934-f86b-48b9-8336-9a9583ab85b6","timestamp":"2024-11-08T11:37:47Z","content_type":"text/html","content_length":"271112","record_id":"<urn:uuid:dc7a40ac-256d-4abf-9b62-5acddd374d51>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00788.warc.gz"} |
Transformers for Natural Language Reasoning and Automated Symbolic Reasoning Tasks
Natural language reasoning is an application of deductive reasoning that draws a conclusion from given premises and rules stated in natural language. The goal of neural network architecture research
is to figure out how to use these premises and rules to draw new conclusions.
In the past, a comparable task would have been performed by systems that were pre-programmed with the knowledge already stored in a formal way, as well as the rules to follow to infer new
information. However, the use of formal representations has proven to be a significant obstacle for this branch of research (Mark A. Musen, 1988). Now, thanks to the development of transformers and
the exceptional performance they have shown in a wide variety of NLP tasks, it is possible to avoid the need for formal representations and to have transformers participate directly in reasoning
through the use of natural language. In this article, we will highlight some of the most important transformers for natural language reasoning tasks.
In the research conducted in 2020 by Clark et al. (2020b), the transformer was given a binary classification task to perform. The goal of this work was to determine whether or not a given proposition
could be derived from a given collection of premises and rules expressed in natural language.
It was pre-trained on a dataset consisting of questions from standardized high school tests that required the use of reasoning skills. Because of this pre-training, the Transformer was able to
achieve a very high level of accuracy on the test dataset, which was 98%. The data set contained hypotheses that were randomly selected and generated using predefined sets of names and
The task asked the transformer to determine whether the given premises and rules (the context) led to the conclusion that the given claim (the assertion) followed from them (Clark et al., 2020b).
In the study by Richardson and Sabharwal (2022), the authors attempted to address a shortcoming identified in the paper by Clark et al. (2020b) regarding the technique used to generate the dataset.
They pointed out that the uniform random selection of theories used in (Clark et al., 2020b) did not always result in challenging cases.
They provided an innovative way to generate challenging data sets for algorithmic reasoning to get around this problem. The most important component of their technique is an approach that takes hard
examples of standard SAT propositional formulas and translates them into natural language using a predetermined collection of English rule languages. By following this technique, they were able to
build a more challenging dataset, which is important for training robust models and for trustworthy evaluation.
The authors conducted experiments in which they evaluated the models trained on the dataset from (Clark et al., 2020b) on their newly built dataset. The goal of these tests was to illustrate the
effectiveness of the technique they had developed. According to the results, the models achieved an accuracy of 57.7% and 59.6% for T5 and RoBERTa, respectively. These results highlight the fact that
models trained on simple datasets may not be able to solve challenging cases.
In a separate but similar paper, Saha et al. (2020) developed a model they call PRover. This model is an interpretable joint transformer that has the potential to provide a matching proof that is
accurate 87% of the time. The problem that PRover attempts to solve is identical to the one studied by Clark et al. (2020b) and Richardson & Sabharwal (2022), which is to determine whether or not a
particular conclusion follows logically from the premises and rules presented.
The proof generated by PRover is given in the form of a directed graph, where the propositions and rules are represented by nodes, and the edges of the graph indicate which new propositions result
from the application of the rules to the previous propositions. In general, the strategy proposed by Saha et al. (2020) is a promising way to achieve interpretable and correct reasoning models.
As described in (Picco et al., 2021), improved generalization performance on the RuleTaker dataset was achieved by using a BERT-based architecture (called a "neural unifier"). Even when trained only
on shallow queries, the model should be able to answer complex questions, so the authors tried to mimic certain aspects of the backward chaining reasoning technique. This should improve the model's
ability to handle questions that require more than one step to answer.
The neural unifier has two components: a fact-checking unit and a unification unit, both of which are typical BERT transformers.
• The fact-checking unit is designed to determine whether an embedding vector C represents a knowledge base from which a query of depth 0 (represented by the embedding vector q-0) follows.
• The unification unit performs backward chaining by taking the embedding vector q-n of a depth-n query and the embedding vector of the knowledge base, vector C, as input and trying to predict an
embedding vector q0.
In contrast to the above work, Sinha et al. (2019) created a dataset called CLUTRR in which the rules are not provided in the input. Instead, the BERT transformer model must extract the links between
entities and infer the rules that govern them. For example, if the network is provided with a knowledge base containing assertions such as "Alice is Bob's mother" and "Jim is Alice's father", it can
infer that "Jim is Bob's grandfather".
Automated symbolic reasoning
The branch of computer science known as automated symbolic reasoning focuses on using computers to solve logical problems such as SAT solving and theorem proving. Heuristic search techniques have
long been the standard method for solving such problems. However, recent research has explored the possibility of improving the effectiveness and efficiency of such approaches by using learning-based
One method is to study how classical algorithms choose among multiple heuristics to find the most effective one. Another is to use a solution based on learning data from start to the end. Results
from both methods have been promising, and both hold promise for future developments in automated symbolic reasoning (Kurin et al., 2020; Selsam et al., 2019). Some transformer-based models have
performed very well in automated symbolic reasoning tasks.
To solve the SAT problem for boolean formulae, Shi et al. introduced SATformer in 2022 (Shi et al., 2021)—a hierarchical transformer design that provides a comprehensive, learning-based solution to
the issue. It is common practice to convert a boolean formula into its conjunctive normal form (CNF) before feeding it into a SAT solver. Each sentence in the CNF formula is a disjunction of the
literals that make up the formula, hence the formula itself is a conjunction of boolean variables and their negations. Using boolean variables in a CNF formula would result in the notation (A OR B)
AND (NOT A OR C), where the clauses (A OR B) and (NOT A OR C) are composed entirely of literals.
The authors employ a graph neural network (GNN) to obtain the embeddings of the clauses in the CNF formula. SATformer then operates on these clause embeddings to capture the interdependencies
among clauses, with the self-attention weight being trained to be high when groups of clauses that could potentially lead to an unsatisfiable formula are attended together, and low otherwise.
Through this approach, SATformer efficiently learns the correlations between clauses, resulting in improved SAT prediction capabilities (Shi et al., 2022b)
In 2021, Shi et al. performed research on MaxSAT, a special case of the boolean SAT issue, and presented a transformer model called TRSAT, which functions as an end-to-end learning-based SAT solver
(Shi et al., 2021). The satisfiability of a linear temporal formula (Pnueli, 1977) is analogous to the boolean SAT in which the objective is to find a satisfactory symbolic trace to the formula given
the linear temporal formulation.
Hahn et al. (2021) handled challenging problems than the binary classification tasks previously studied by addressing the boolean SAT issue and the temporal satisfiability problem. The objective of
these issues is not to merely categorize whether or not a given formula is met, but rather to construct a suitable sequence assignment for that formula. Using classical solvers, the authors created
linear temporal formulae with satisfying symbolic traces and boolean formulas with corresponding satisfying partial assignments to create their datasets. To perform this operation, they used a
conventional transformer design. Some of the satisfying traces generated by the Transformer were not seen during training, demonstrating its capacity to solve the issue and not only copy the behavior
of the traditional solvers.
Polu and Sutskever (2020) introduced GPT-F, an automated prover and proof assistant that, like GPT-2 and GPT-3, is built on a decoder-only transformers architecture. The set.mm dataset, which
includes over 38,000 proofs, was used to train GPT-F. The authors' largest model has 774 million trainable parameters over 36 layers. Many of the innovative proofs generated by this deep learning
network have been adopted by groups and libraries devoted to the study of mathematics.
In this article we have talked about some powerfull transformers that we can use for natural language reasoning. There has been some encouraging research on the use of transformers for theorem
proving and propositional reasoning, but their performance on natural language reasoning tasks is still far from ideal. Further research is needed to improve the reasoning abilities of transformers
and to find new, challenging tasks for them. Despite these caveats, transformer-based models have undoubtedly advanced the state of the art in natural language processing and opened up exciting new
avenues for research in language understanding and reasoning.
A Comprehensive Survey on Applications of Transformers for Deep Learning Tasks
Transformer is a deep neural network that employs a self-attention mechanism to comprehend the contextual relationships within sequential data. Unlike conventional neural networks or updated versions
of Recurrent Neural Networks (RNNs) such as Long Short-Term Memory (LSTM), transformer models excel…
Reasoning with Transformer-based Models: Deep Learning, but Shallow...
Recent years have seen impressive performance of transformer-based models on different natural language processing tasks. However, it is not clear to what degree the transformers can reason on… | {"url":"https://blog.paperspace.com/telling-creative-stories-with-generative-visual-aid/","timestamp":"2024-11-10T01:28:13Z","content_type":"text/html","content_length":"93859","record_id":"<urn:uuid:8e4f23bb-5635-4bc0-beec-d47fb3815885>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00784.warc.gz"} |
Reverse-engineering the Globus INK, a Soviet spaceflight navigation computer
One of the most interesting navigation instruments onboard Soyuz spacecraft was the Globus INK,1 which used a rotating globe to indicate the spacecraft's position above the Earth. This
electromechanical analog computer used an elaborate system of gears, cams, and differentials to compute the spacecraft's position. The globe rotates in two dimensions: it spins end-over-end to
indicate the spacecraft's orbit, while the globe's hemispheres rotate according to the Earth's daily rotation around its axis.2 The spacecraft's position above the Earth was represented by the fixed
crosshairs on the plastic dome. The Globus also has latitude and longitude dials next to the globe to show the position numerically, while the light/shadow dial below the globe indicated when the
spacecraft would enter or leave the Earth's shadow.
The INK-2S "Globus" space navigation indicator.
Opening up the Globus reveals that it is packed with complicated gears and mechanisms. It's amazing that this mechanical technology was used from the 1960s into the 21st century. But what are all
those gears doing? How can orbital functions be implemented with gears? To answer these questions, I reverse-engineered the Globus and traced out its system of gears.
The Globus with the case removed, showing the complex gearing inside.
The diagram below summarizes my analysis. The Globus is an analog computer that represents values by rotating shafts by particular amounts. These rotations control the globe and the indicator dials.
The flow of these rotational signals is shown by the lines on the diagram. The computation is based around addition, performed by ten differential gear assemblies. On the diagram, each "⨁" symbol
indicates one of these differential gear assemblies. Other gears connect the components while scaling the signals through various gear ratios. Complicated functions are implemented with three
specially-shaped cams. In the remainder of this blog post, I will break this diagram down into functional blocks and explain how the Globus operates.
This diagram shows the interconnections of the gear network in the Globus.
For all its complexity, though, the functionality of the Globus is pretty limited. It only handles a fixed orbit at a specific angle, and treats the orbit as circular. The Globus does not have any
navigation input such as an inertial measurement unit (IMU). Instead, the cosmonauts configured the Globus by turning knobs to set the spacecraft's initial position and orbital period. From there,
the Globus simply projected the current position of the spacecraft forward, essentially dead reckoning.
A closeup of the gears inside the Globus.
The globe
On seeing the Globus, one might wonder how the globe is rotated. It may seem that the globe must be free-floating so it can rotate in two axes. Instead, a clever mechanism attaches the globe to the
unit. The key is that the globe's equator is a solid piece of metal that rotates around the horizontal axis of the unit. A second gear mechanism inside the globe rotates the globe around the
North-South axis. The two rotations are controlled by concentric shafts that are fixed to the unit. Thus, the globe has two rotational degrees of freedom, even though it is attached at both ends.
The photo below shows the frame that holds and controls the globe. The dotted axis is fixed horizontally in the unit and rotations are fed through the two gears at the left. One gear rotates the
globe and frame around the dotted axis, while the gear train causes the globe to rotate around the vertical polar axis (while the equator remains fixed).
The axis of the globe is at 51.8° to support that orbital inclination.
The angle above is 51.8° which is very important: this is the inclination of the standard Soyuz orbit. As a result, simply rotating the globe around the dotted line causes the crosshair to trace the
orbit.3 Rotating the two halves of the globe around the poles yields the different paths over the Earth's surface as the Earth rotates. An important consequence of this design is that the Globus only
supports a circular orbit at a fixed angle.
Differential gear mechanism
The primary mathematical element of the Globus is the differential gear mechanism, which can perform addition or subtraction. A differential gear takes two rotations as inputs and produces the
(scaled) sum of the rotations as the output. The photo below shows one of the differential mechanisms. In the middle, the spider gear assembly (red box) consists of two bevel gears that can spin
freely on a vertical shaft. The spider gear assembly as a whole is attached to a horizontal shaft, called the spider shaft. At the right, the spider shaft is attached to a spur gear (a gear with
straight-cut teeth). The spider gear assembly, the spider shaft, and the spider's spur gear rotate together as a unit.
Diagram showing the components of a differential gear mechanism.
At the left and right are two end gear assemblies (yellow). The end gear is a bevel gear with angled teeth to mesh with the spider gears. Each end gear is locked to a spur gear and these gears spin
freely on the horizontal spider shaft. In total, there are three spur gears: two connected to the end gears and one connected to the spider assembly. In the diagrams, I'll use the symbol below to
represent the differential gear assembly: the end gears are symmetric on the top and bottom, with the spider shaft on the side. Any of the three spur gears can be used as an output, with the other
two serving as inputs.
The symbol for the differential gear assembly.
To understand the behavior of the differential, suppose the two end gears are driven in the same direction at the same rate, say upwards.4 These gears will push on the spider gears and rotate the
spider gear assembly, with the entire differential rotating as a fixed unit. On the other hand, suppose the two end gears are driven in opposite directions. In this case, the spider gears will spin
on their shaft, but the spider gear assembly will remain stationary. In either case, the spider gear assembly motion is the average of the two end gear rotations, that is, the sum of the two
rotations divided by 2. (I'll ignore the factor of 2 since I'm ignoring all the gear ratios.) If the operation of the differential is still confusing, this vintage Navy video has a detailed
The controls and displays
The diagram below shows the controls and displays of the Globus. The rotating globe is the centerpiece of the unit. Its plastic cover has a crosshair that represents the spacecraft's position above
the Earth's surface. Surrounding the globe itself are dials that show the longitude, latitude, and the time before entering light and shadow. The cosmonauts manually initialize the globe position
with the concentric globe rotation knobs: one rotates the globe along the orbital path while the other rotates the hemispheres. The mode switch at the top selects between the landing position mode,
the standard Earth orbit mode, and turning off the unit. The orbit time adjustment configures the orbital time period in minutes while the orbit counter below it counts the number of orbits. Finally,
the landing point angle sets the distance to the landing point in degrees of orbit.
The Globus with the controls labeled.
Computing the orbit time
The primary motion of the Globus is the end-over-end rotation of the globe showing the movement of the spacecraft in orbit. The orbital motion is powered by a solenoid at the top of the Globus that
receives pulses once a second and advances a ratchet wheel (video).5 This wheel is connected to a complicated cam and differential system to provide the orbital motion.
The orbit solenoid (green) has a ratchet that rotates the gear to the right. The shaft connects it to differential gear assembly 1 at the bottom right.
Each orbit takes about 92 minutes, but the orbital time can be adjusted by a few minutes in steps of 0.01 minutes6 to account for changes in altitude. The Globus is surprisingly inflexible and this
is the only orbital parameter that can be adjusted.7 The orbital period is adjusted by the three-position orbit time switch, which points to the minutes, tenths, or hundredths. Turning the central
knob adjusts the indicated period dial.
The problem is how to generate the variable orbital rotation speed from the fixed speed of the solenoid. The solution is a special cam, shaped like a cone with a spiral cross-section. Three followers
ride on the cam, so as the cam rotates, the follower is pushed outward and rotates on its shaft. If the follower is near the narrow part of the cam, it moves over a small distance and has a small
rotation. But if the follower is near the wide part of the cam, it moves a larger distance and has a larger rotation. Thus, by moving the follower to a particular point on the cam, the rotational
speed of the follower is selected. One follower adjusts the speed based on the minutes setting with others for the tenths and hundredths of minutes.
A diagram showing the orbital speed control mechanism. The cone has three followers, but only two are visible from this angle. The "transmission" gears are moved in and out by the outer knob to
select which follower is adjusted by the inner knob.
Of course, the cam can't spiral out forever. Instead, at the end of one revolution, its cross-section drops back sharply to the starting diameter. This causes the follower to snap back to its
original position. To prevent this from jerking the globe backward, the follower is connected to the differential gearing via a slip clutch and ratchet. Thus, when the follower snaps back, the
ratchet holds the drive shaft stationary. The drive shaft then continues its rotation as the follower starts cycling out again. Each shaft output is accordingly a (mostly) smooth rotation at a speed
that depends on the position of the follower.
A cam-based system adjusts the orbital speed using three differential gear assemblies.
The three adjustment signals are scaled by gear ratios to provide the appropriate contribution to the rotation. As shown above, the adjustments are added to the solenoid output by three differentials
to generate the orbit rotation signal, output from differential 3.8 This signal also drives the odometer-like orbit counter on the front of the Globus. The diagram below shows how the components are
arranged, as viewed from the back.
A back view of the Globus showing the orbit components.
Displaying the orbit rotation
Since the Globus doesn't have any external position input such as inertial guidance, it must be initialized by the cosmonauts. A knob on the front of the Globus provides manual adjustment of the
orbital position. Differential 4 adds the knob signal to the orbit output discussed above.
The orbit controls drive the globe's motion.
The Globus has a "landing point" mode where the globe is rapidly rotated through a fraction of an orbit to indicate where the spacecraft would land if the retro-rockets were fired. Turning the mode
switch caused the globe to rotate until the landing position was under the crosshairs and the cosmonauts could evaluate the suitability of this landing site. This mode is implemented with a landing
position motor that provides the rapid rotation. This motor also rotates the globe back to the orbital position. The motor is driven through an electronics board with relays and a transistor,
controlled by limit switches. I discussed the electronics in a previous post so I won't go into more details here. The landing position motor feeds into the orbit signal through differential 5,
producing the final orbit signal.
The landing position motor and its associated gearing. The motor speed is geared down and then fed through a worm gear (upper center).
The orbit signal from differential 5 is used in several ways. Most importantly, the orbit signal provides the end-over-end rotation of the globe to indicate the spacecraft's travel in orbit. As
discussed earlier, this is accomplished by rotating the globe's metal frame around the horizontal axis. The orbital signal also rotates a potentiometer to provide an electrical indication of the
orbital position to other spacecraft systems.
The light/shadow indicator
Docking a spacecraft is a tricky endeavor, best performed in daylight, so it is useful to know how much time remains until the spacecraft enters the Earth's shadow. The light/shadow dial under the
globe provides this information. This display consists of two nested wheels. The outer wheel is white and has two quarters removed. Through these gaps, the partially-black inner wheel is exposed,
which can be adjusted to show 0% to 50% dark. This display is rotated by the orbital signal, turning half a revolution per orbit. As the spacecraft orbits, this dial shows the light/shadow transition
and the time to the transistion.9
The light/shadow indicator, viewed from the underside of the Globus. The shadow indicator has been set to 35% shadow. Near the hub, a pin restricts motion of the inner wheel relative to the outer
You might expect the orbit to be in the dark 50% of the time, but because the spacecraft is about 200 km above the Earth's surface, it will sometimes be illuminated when the surface of the Earth
underneath is dark.10 In the ground track below, the dotted part of the track is where the spacecraft is in the Earth's shadow; this is considerably less than 50%. Also note that the end of the orbit
doesn't match up with the beginning, due to the Earth's rotation during the orbit.
Ground track of an Apollo-Soyuz Test Project orbit, corresponding to this Globus. Image courtesy of
The latitude indicator
The latitude indicator to the left of the globe shows the spacecraft's latitude. The map above shows how the latitude oscillates between 51.8°N and 51.8°S, corresponding to the launch inclination
angle. Even though the path around the globe is a straight (circular) line, the orbit appears roughly sinusoidal when projected onto the map.11 The exact latitude is a surprisingly complicated
function of the orbital position.12 This function is implemented by a cam that is attached to the globe. The varying radius of the cam corresponds to the function. A follower tracks the profile of
the cam and rotates the latitude display wheel accordingly, providing the non-linear motion.
A cam is attached to the globe and rotates with the globe.
The Earth's rotation
The second motion of the globe is the Earth's daily rotation around its axis, which I'll call the Earth rotation. The Earth rotation is fed into the globe through the outer part of a concentric
shaft, while the orbital rotation is provided through the inner shaft. The Earth rotation is transferred through three gears to the equatorial frame, where an internal mechanism rotates the
hemispheres. There's a complication, though: if the globe's orbital shaft turns while the Earth rotation shaft remains stationary, the frame will rotate, causing the gears to turn and the hemispheres
to rotate. In other words, keeping the hemispheres stationary requires the Earth shaft to rotate with the orbit shaft.
A closeup of the gear mechanisms that drive the Globus, showing the concentric shafts that control the two rotations.
The Globus solves this problem by adding the orbit rotation to the Earth rotation, as shown in the diagram below, using differentials 7 and 8. Differential 8 adds the normal orbit rotation, while
differential 7 adds the orbit rotation due to the landing motor.14
The mechanism to compute the Earth's rotation around its axis.
The Earth motion is generated by a second solenoid (below) that is driven with one pulse per second.13 This motion is simpler than the orbit motion because it has a fixed rate. The "Earth" knob on
the front of the Globus permits manual rotation around the Earth's axis. This signal is combined with the solenoid signal by differential 6. The sum from the three differentials is fed into the
globe, rotating the hemispheres around their axis.
This solenoid, ratchet, and gear on the underside of the Globus drive the Earth rotation.
The solenoid and differentials are visible from the underside of the Globus. The diagram below labels these components as well as other important components.
The underside of the Globus.
The longitude display
The longitude cam and the followers that track its radius.
The longitude display is more complicated than the latitude display because it depends on both the Earth rotation and the orbit rotation. Unlike the latitude, the longitude doesn't oscillate but
increases. The longitude increases by 360° every orbit according to a complicated formula describing the projection of the orbit onto the globe. Most of the time, the increase is small, but when
crossing near the poles, the longitude changes rapidly. The Earth's rotation provides a smaller but steady negative change to the longitude.
The computation of the longitude.
The diagram above shows how the longitude is computed by combining the Earth rotation with the orbit rotation. Differential 9 adds the linear effect of the orbit on longitude (360° per orbit) and
subtracts the effect of the Earth's rotation (360° per day). The nonlinear effect of the orbit is computed by a cam that is rotated by the orbit signal. The shape of the cam is picked up and fed into
differential 10, computing the longitude that is displayed on the dial. The differentials, cam, and dial are visible from the back of the Globus (below).
A closeup of the differentials from the back of the Globus.
The time-lapse video below demonstrates the behavior of the rotating displays. The latitude display on the left oscillates between 51.8°N and 51.8°S. The longitude display at the top advances at a
changing rate. Near the equator, it advances slowly, while it accelerates near the poles. The light/shadow display at the bottom rotates at a constant speed, completing half a revolution (one light/
shadow cycle) per orbit.
The Globus INK is a remarkable piece of machinery, an analog computer that calculates orbits through an intricate system of gears, cams, and differentials. It provided astronauts with a
high-resolution, full-color display of the spacecraft's position, way beyond what an electronic space computer could provide in the 1960s.
The drawback of the Globus is that its functionality is limited. Its parameters must be manually configured: the spacecraft's starting position, the orbital speed, the light/shadow regions, and the
landing angle. It doesn't take any external guidance inputs, such as an IMU (inertial measurement unit), so it's not particularly accurate. Finally, it only supports a circular orbit at a fixed
angle. While a more modern digital display lacks the physical charm of a rotating globe, the digital solution provides much more capability.
I recently wrote blog posts providing a Globus overview and the Globus electronics. Follow me on Twitter @kenshirriff or RSS for updates. I've also started experimenting with Mastodon recently as @
[email protected]. Many thanks to Marcel for providing the Globus. I worked on this with CuriousMarc, so check out his Globus videos.
Notes and references
1. In Russian, the name for the device is "Индикатор Навигационный Космический" abbreviated as ИНК (INK). This translates to "space navigation indicator." but I'll use the more descriptive nickname
"Globus" (i.e. globe). The Globus has a long history, back to the beginnings of Soviet crewed spaceflight. The first version was simpler and had the Russian acronym ИМП (IMP). Development of the
IMP started in 1960 for the Vostok (1961) and Voshod (1964) spaceflights. The more complex INK model (described in this blog post) was created for the Soyuz flights, starting in 1967. The landing
position feature is the main improvement of the INK model. The Soyuz-TMA (2002) upgraded to the Neptun-ME system which used digital display screens and abandoned the Globus. ↩
2. According to this document, one revolution of the globe relative to the axis of daily rotation occurs in a time equal to a sidereal day, taking into account the precession of the orbit relative
to the earth's axis, caused by the asymmetry of the Earth's gravitational field. (A sidereal day is approximately 4 minutes shorter than a regular 24-hour day. The difference is that the sidereal
day is relative to the fixed stars, rather than relative to the Sun.) ↩
3. To see how the angle between the poles and the globe's rotation results in the desired orbital inclination, consider two limit cases. First, suppose the angle between is 90°. In this case, the
globe is "straight" with the equator horizontal. Rotating the globe along the horizontal axis, flipping the poles end-over-end, will cause the crosshair to trace a polar orbit, giving the
expected inclination of 90°. On the other hand, suppose the angle is 0°. In this case, the globe is "sideways" with the equator vertical. Rotating the globe will cause the crosshair to remain
over the equator, corresponding to an equatorial orbit with 0° inclination. ↩
4. There is a bit of ambiguity when describing the gear motions. If the end gears are rotating upwards when viewed from the front, the gears are both rotating clockwise when viewed from the right,
so I'm referring to them as rotating in the same direction. But if you view each gear from its own side, the gear on the left is turning counterclockwise, so from that perspective they are
turning in opposite directions. ↩
5. The solenoids are important since they provide all the energy to drive the globe. One of the problems with gear-driven analog computers is that each gear and shaft has a bit of friction and loses
a bit of torque, and there is nothing to amplify the signal along the way. Thus, the 27-volt solenoids need to provide enough force to run the entire system. ↩
6. The orbital time can be adjusted between 86.85 minutes and 96.85 minutes according to this detailed page that describes the Globus in Russian. ↩
7. The Globus is manufactured for a particular orbital inclination, in this case 51.8°. The Globus assumes a circular orbit and does not account for any variations. The Globus does not account for
any maneuvering in orbit. ↩
8. The outputs from the orbit cam are fed into the overall orbit rotation, which drives the orbit cam. This may seem like an "infinite loop" since the outputs from the cam turn the cam itself.
However, the outputs from the cam are a small part of the overall orbit rotation, so the feedback dies off. ↩
9. The scales on the light/shadow display are a bit confusing. The inner scale (blue) is measured in percentage of an orbit, up to 100%. The fixed outer scale (red) measures minutes, indicating how
many minutes until the spacecraft enters or leaves shadow. The spacecraft completes 100% of an orbit in about 90 minutes, so the scales almost, but not quite, line up. The wheel is driven by the
orbit mechanism and turns half a revolution per orbit.
The light and shadow indicator is controlled by two knobs.
10. The Internation Space Station illustrates how an orbiting spacecraft is illuminated more than 50% of the time due to its height. You can often see the ISS illuminated in the nighttime sky close
to sunset and sunrise (link). ↩
11. The ground track on the map is roughly, but not exactly, sinusoidal. As the orbit swings further from the equator, the track deviates more from a pure sinusoid. The shape will depend, of course,
on the rectangular map projection. For more information, see this StackExcahnge post. ↩
12. To get an idea of how the latitude and longitude behave, consider a polar orbit with 90° angle of inclination, one that goes up a line of longitude, crosses the North Pole, and goes down the
opposite line of latitude. Now, shift the orbit away from the poles a bit, but keeping a great circle. The spacecraft will go up, nearly along a constant line of longitude, with the latitude
increasing steadily. As the spacecraft reaches the peak of its orbit near the North Pole, it will fall a bit short of the Pole but will still rapidly cross over to the other side. During this
phase, the spacecraft rapidly crosses many lines of longitude (which are close together near the Pole) until it reaches the opposite line of longitude. Meanwhile, the latitude stops increasing
short of 90° and then starts dropping. On the other side, the process repeats, with the longitude nearly constant while the latitude drops relatively constantly.
The latitude and longitude are generated by complicated trigonometric functions. The latitude is given by arcsin(sin i * sin (2πt/T)), while the longitude is given by λ = arctan (cos i * tan(2πt/
T)) + Ωt + λ[0], where t is the spaceship's flight time starting at the equator, i is the angle of inclination (51.8°), T is the orbital period, Ω is the angular velocity of the Earth's rotation,
and λ[0] is the longitude of the ascending node. ↩
13. An important function of the gears is to scale the rotations as needed by using different gear ratios. For the most part, I'm ignoring the gear ratios, but the Earth rotation gearing is
interesting. The gear driven by the solenoid has 60 teeth, so it rotates exactly once per minute. This gear drives a shaft with a very small gear on the other end with 15 teeth. This gear meshes
with a much larger gear with approximately 75 teeth, which will thus rotate once every 5 minutes. The other end of that shaft has a gear with approximately 15 teeth, meshed with a large gear with
approximately 90 teeth. This divides the rate by 6, yielding a rotation every 30 minutes. The sequence of gears and shafts continues, until the rotation is reduced to once per day. (The tooth
counts are approximate because the gears are partially obstructed inside the Globus, making counting difficult.) ↩
14. There's a potential simplification when canceling out the orbital shaft rotation from the Earth rotation. If the orbit motion was taken from differential 5 instead of differential 4, the landing
motor effect would get added automatically, eliminating the need for differential 7. I think the landing motor motion was added separately so the mechanism could account for the Earth's rotation
during the landing descent. ↩
2 comments:
jan swenker said...
The Rolleiflex 2.8F from 1960 uses a differential gear to multiply shutter speed and diaphragma for the lightmeter dial. Both have a logarithmic scale as in a conventional light meter.
Salty said...
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Paint the edges of the tree¶
This is a fairly common task. Given a tree $G$ with $N$ vertices. There are two types of queries: the first one is to paint an edge, the second one is to query the number of colored edges between two
Here we will describe a fairly simple solution (using a segment tree) that will answer each query in $O(\log N)$ time. The preprocessing step will take $O(N)$ time.
First, we need to find the LCA to reduce each query of the second kind $(i,j)$ into two queries $(l,i)$ and $(l,j)$, where $l$ is the LCA of $i$ and $j$. The answer of the query $(i,j)$ will be the
sum of both subqueries. Both these queries have a special structure, the first vertex is an ancestor of the second one. For the rest of the article we will only talk about these special kind of
We will start by describing the preprocessing step. Run a depth-first search from the root of the tree and record the Euler tour of this depth-first search (each vertex is added to the list when the
search visits it first and every time we return from one of its children). The same technique can be used in the LCA preprocessing.
This list will contain each edge (in the sense that if $i$ and $j$ are the ends of the edge, then there will be a place in the list where $i$ and $j$ are neighbors in the list), and it appear exactly
two times: in the forward direction (from $i$ to $j$, where vertex $i$ is closer to the root than vertex $j$) and in the opposite direction (from $j$ to $i$).
We will build two lists for these edges. The first one will store the color of all edges in the forward direction, and the second one the color of all edges in the opposite direction. We will use $1$
if the edge is colored, and $0$ otherwise. Over these two lists we will build each a segment tree (for sum with a single modification), let's call them $T1$ and $T2$.
Let us answer a query of the form $(i,j)$, where $i$ is the ancestor of $j$. We need to determine how many edges are painted on the path between $i$ and $j$. Let's find $i$ and $j$ in the Euler tour
for the first time, let it be the positions $p$ and $q$ (this can be done in $O(1)$ if we calculate these positions in advance during preprocessing). Then the answer to the query is the sum $T1
[p..q-1]$ minus the sum $T2[p..q-1]$.
Why? Consider the segment $[p;q]$ in the Euler tour. It contains all edges of the path we need from $i$ to $j$ but also contains a set of edges that lie on other paths from $i$. However there is one
big difference between the edges we need and the rest of the edges: the edges we need will be listed only once in the forward direction, and all the other edges appear twice: once in the forward and
once in the opposite direction. Hence, the difference $T1[p..q-1] - T2[p..q-1]$ will give us the correct answer (minus one is necessary because otherwise, we will capture an extra edge going out from
vertex $j$). The sum query in the segment tree is executed in $O(\log N)$.
Answering the first type of query (painting an edge) is even easier - we just need to update $T1$ and $T2$, namely to perform a single update of the element that corresponds to our edge (finding the
edge in the list, again, is possible in $O(1)$, if you perform this search during preprocessing). A single modification in the segment tree is performed in $O(\log N)$.
Here is the full implementation of the solution, including LCA computation:
const int INF = 1000 * 1000 * 1000;
typedef vector<vector<int>> graph;
vector<int> dfs_list;
vector<int> edges_list;
vector<int> h;
void dfs(int v, const graph& g, const graph& edge_ids, int cur_h = 1) {
h[v] = cur_h;
for (size_t i = 0; i < g[v].size(); ++i) {
if (h[g[v][i]] == -1) {
dfs(g[v][i], g, edge_ids, cur_h + 1);
vector<int> lca_tree;
vector<int> first;
void lca_tree_build(int i, int l, int r) {
if (l == r) {
lca_tree[i] = dfs_list[l];
} else {
int m = (l + r) >> 1;
lca_tree_build(i + i, l, m);
lca_tree_build(i + i + 1, m + 1, r);
int lt = lca_tree[i + i], rt = lca_tree[i + i + 1];
lca_tree[i] = h[lt] < h[rt] ? lt : rt;
void lca_prepare(int n) {
lca_tree.assign(dfs_list.size() * 8, -1);
lca_tree_build(1, 0, (int)dfs_list.size() - 1);
first.assign(n, -1);
for (int i = 0; i < (int)dfs_list.size(); ++i) {
int v = dfs_list[i];
if (first[v] == -1)
first[v] = i;
int lca_tree_query(int i, int tl, int tr, int l, int r) {
if (tl == l && tr == r)
return lca_tree[i];
int m = (tl + tr) >> 1;
if (r <= m)
return lca_tree_query(i + i, tl, m, l, r);
if (l > m)
return lca_tree_query(i + i + 1, m + 1, tr, l, r);
int lt = lca_tree_query(i + i, tl, m, l, m);
int rt = lca_tree_query(i + i + 1, m + 1, tr, m + 1, r);
return h[lt] < h[rt] ? lt : rt;
int lca(int a, int b) {
if (first[a] > first[b])
swap(a, b);
return lca_tree_query(1, 0, (int)dfs_list.size() - 1, first[a], first[b]);
vector<int> first1, first2;
vector<char> edge_used;
vector<int> tree1, tree2;
void query_prepare(int n) {
first1.resize(n - 1, -1);
first2.resize(n - 1, -1);
for (int i = 0; i < (int)edges_list.size(); ++i) {
int j = edges_list[i];
if (first1[j] == -1)
first1[j] = i;
first2[j] = i;
edge_used.resize(n - 1);
tree1.resize(edges_list.size() * 8);
tree2.resize(edges_list.size() * 8);
void sum_tree_update(vector<int>& tree, int i, int l, int r, int j, int delta) {
tree[i] += delta;
if (l < r) {
int m = (l + r) >> 1;
if (j <= m)
sum_tree_update(tree, i + i, l, m, j, delta);
sum_tree_update(tree, i + i + 1, m + 1, r, j, delta);
int sum_tree_query(const vector<int>& tree, int i, int tl, int tr, int l, int r) {
if (l > r || tl > tr)
return 0;
if (tl == l && tr == r)
return tree[i];
int m = (tl + tr) >> 1;
if (r <= m)
return sum_tree_query(tree, i + i, tl, m, l, r);
if (l > m)
return sum_tree_query(tree, i + i + 1, m + 1, tr, l, r);
return sum_tree_query(tree, i + i, tl, m, l, m) +
sum_tree_query(tree, i + i + 1, m + 1, tr, m + 1, r);
int query(int v1, int v2) {
return sum_tree_query(tree1, 1, 0, (int)edges_list.size() - 1, first[v1], first[v2] - 1) -
sum_tree_query(tree2, 1, 0, (int)edges_list.size() - 1, first[v1], first[v2] - 1);
int main() {
// reading the graph
int n;
scanf("%d", &n);
graph g(n), edge_ids(n);
for (int i = 0; i < n - 1; ++i) {
int v1, v2;
scanf("%d%d", &v1, &v2);
--v1, --v2;
h.assign(n, -1);
dfs(0, g, edge_ids);
for (;;) {
if () {
// request for painting edge x;
// if start = true, then the edge is painted, otherwise the painting
// is removed
edge_used[x] = start;
sum_tree_update(tree1, 1, 0, (int)edges_list.size() - 1, first1[x],
start ? 1 : -1);
sum_tree_update(tree2, 1, 0, (int)edges_list.size() - 1, first2[x],
start ? 1 : -1);
} else {
// query the number of colored edges on the path between v1 and v2
int l = lca(v1, v2);
int result = query(l, v1) + query(l, v2);
// result - the answer to the request | {"url":"https://cp-algorithms.com/graph/tree_painting.html","timestamp":"2024-11-08T06:12:42Z","content_type":"text/html","content_length":"172034","record_id":"<urn:uuid:4703021a-10e1-4352-a237-32955e218135>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00489.warc.gz"} |
Mathema Foundation
Gallery Guides
Visitors can learn more about mathematical ideas and mathematicians through the Gallery Guide worksheets. These worksheets are designed to be fun and challenging exercises that further explain some
of the displays and interactive items of the Mathema Gallery.
Archimedes lived in Syracuse on the island of Sicily which was a Greek territory during that time. He is universally considered one of the most creative individuals that has ever lived; often ranked
with Newton as a supreme mathematical genius (Hollingdale, 1989). He is known best for his novel discovery of the law of buoyancy which supposedly caused him to streak naked from the public baths.
Archimedes is credited with inventions such as the compound pulley, Archimedean screw used to raise and move water, and powerful levers that were often used to build war machines.
Archimedes achievements were amazing give that mathematical notation was very limited during this era. Like all of the Greek mathematicians, Archimedes solved problems using geometry. Algebra had not
been invented and the number system was antiquated. So, problem solving was slow, challenging and required immense powers of concentration.
Archimedes used geometrical methods to solve his problem, without algebra. It was another 1800 years before Newton and Leibniz would be give dual credit for creating the calculus. The sequence of
pictures below shows how Archimedes calculated the circumference of a circle. He squeezed the circumference between regular polygons with an increasing number of sides. The lengths of the sides were
added (to find the perimeter). A greater number of sides got closer and closer to the circumference of the circle.
The activities in this Gallery Guide relate to some interesting contributions Archimedes made that are less well known but very novel.
Hindu-Arabic Number System
When we think of numbers, we tend to do so quite naturally and unthinkingly. Almost everything we do – from learning to tell the time to knowing how old we are – is dictated by numbers. The counting
system we use today-comprising the familiar set of digits from ‘0’ through ‘9’-has become so set in our minds, cultures and educational systems that we rarely pause to consider its historical
origins. Where did these symbols come from? Who invented them? Where would we be without them?
Archaeological evidence suggests that humans could have been counting as far back as 50,000 years ago. Early human societies had unique ways for counting and representing value, such as matching the
number of pebbles or sticks with whatever they were trying to count. At Mathema, learn about where today’s counting system first emerged and how it ultimately made its way to the Western world. Find
out about its long road to universal acceptance, and how the system clashed with other popular methods of counting such as the Roman abacus system.
Ada Lovelace
The advent of computer technology is one of the most remarkable milestones in human history. Nowadays, computers are used in almost all professional spaces and nearly all school curriculums require
students to learn essential skills in computer use. Without them, we would still be grappling with difficult equations by hand and sifting through thick books and manuscripts to find information we
can now access with the click of a button. Having computers allows us to do these same tasks at half the speed, and with double the accuracy.
At Mathema, learn about the brilliant mind of Ada Lovelace – one of the earliest pioneers behind computer technology. Building on Charles Babbage’s idea of a difference engine, Ada was instrumental
in designing the first general-purpose computer with the same basic mechanical and logical structure as the modern-day PC. It was the first time that the idea was floated for a mechanical device
which could compute algorithms and perform a range of simple arithmetic calculations. See how Ada Lovelace helped usher in the technology age and even had one of the earliest programming languages
named in her honour.
Albrecht Durer
Albrecht Durer is one of the most famous artists of the German Renaissance period. One of his most well-known works is this print which he created in the year 1514. It continues to captivate viewers
with its visually intriguing and conceptually challenging themes.
The name of the piece is Melencolia I, which carries a connotation of sadness. One interpretation of Melencolia I is that it reflects the perpetually unsatisfied mind of the mathematician, who can
never fully succeed at perfecting his craft. Even with its melancholic sentiment, the artwork is paradoxically a celebration of mathematical genius. This is evidenced (among other things) by the
iconic magic square displayed in the top right corner of the frame. At Mathema, learn what gives this 4×4 grid its magical reputation.
Alan Turing
While the Second World War is undeniably one of the greatest tragedies in human history, not all stories that emerged from it were ones of devastation and loss. There were also the fantastic stories
of success, bravery and strategic victory. One such success story was that of Alan Turing, a British genius who was instrumental in decoding secret ‘Enigma’ communications which were being used by
the Germans to guide their military plans. The intelligence uncovered by Turing was central to Britain’s war effort, with some suggesting it could have shortened the conflict by up to two years!
Turing, however, was more than just a codebreaker. He was also a mathematician, philosopher and computing pioneer who grappled with the fundamental problems of life itself. His ideas have helped
shape the modern world, including early computer programming and even the seeds of artificial intelligence. This exhibition offers an absorbing retrospective view of one of Britain’s greatest
20th-century thinkers.
Ken Ono
Some mathematical propositions throughout history involve such technically rigorous methods of proof that they have never officially been solved. One such proposition is the Riemann hypothesis which
was first postulated by Bernhard Riemann in 1859. The Riemann hypothesis is of great interest in number theory because of what it tells us about prime numbers and their distribution.
As a concept, prime numbers are relatively straightforward to grasp. They simply refer to those numbers >1 whose only positive divisors are 1 and itself (e.g. 2, 3, 5, 7, 11…). Although the concept
of prime numbers is one that most of us grasp in elementary mathematics class, the pattern by which primes occur from 1 through to infinity is much less understood. The Riemann hypothesis is one of
the seven ‘Millennium Problems’ laid out in 2000, with a $1 million bounty for anyone capable of solving it.
At Mathema, learn about the awe-inspiring career of Ken Ono and his role in devising a criterion which implies the Riemann hypothesis in most cases. See if you can build on Ono’s legendary theorems
to find a conclusive solution to this 160-year-old mathematical mystery once and for all.
Platonic Solids
The universe is made up of shapes. If you look at your immediate surroundings, chances are you could describe most of the things you see in terms of their structure or shape. The mobile device you
own; the plates that you use to eat breakfast; even the computer monitor you are now reading from – these are just a handful of examples of the daily contact we have with the shapes of our universe.
Some of the shapes we encounter come in simple two-dimensional forms (like squares and circles), while others are slightly more complex three- dimensional structures called polyhedra. Polyhedra come
in many shapes and sizes (no pun intended), and can include anything from pyramids to prisms of all sorts. One rather special group of polyhedra are the Platonic Solids.
At Mathema, learn about how the Platonic Solids were used in early philosophical accounts to describe the essential elements of the universe. You will also find out what makes the Platonic Solids so
special, and why there are only five such structures known to exist. The answer is simple: it is all about geometry!
Leonhard Euler
Born on April 15, 1707, in Basel, Switzerland, Leonhard Euler was one of mathematics’ most pioneering thinkers. He established a career as an academy scholar and contributed greatly to the fields of
geometry, trigonometry and calculus, among many others.
One of Euler’s greatest discoveries was a formula to represent the number of edges, faces and vertices on any polyhedron. Instead of going through the arduous exercise of counting each individual
edge, face and vertex on a given polyhedron, we can make matters simpler for ourselves by applying a simple yet ingenious mathematical rule: the number of edges will always be 2 less than the number
of faces and vertices combined.
Euler’s name is also often credited with the famous Konigsberg Bridge problem. Besides laying the foundations for modern-day graph and network theory, this problem was presented to Euler as a
mathematical puzzle set in the old Prussian city of Konigsberg (now Kaliningrad, Russia). The task for Euler was to see if a single line could be drawn to connect the various parts of the town
without crossing any bridge more than once.
Though a historical recreational puzzle, the applications of the Konigsberg problem are still relevant today, as seen in transportation routes, communication channels and even the flow of electrical
currents. At Mathema, delve into the secrets of Euler’s mind and see whether he was ultimately successful at solving this historical mathematical problem.
Pascal's Triangle
Although Pascal’s triangle owes its name to 17th century French mathematician Blaise Pascal, the history books suggest that the triangle originated in China as early as the 11th century. Pascal’s
triangle is one of the most compelling number patterns in all of mathematics. Some patterns are easy to find and apply, while others are more hidden and complex. Even though some of these patterns
were detected many centuries ago, they continue to be applied by mathematicians and students to solve problems from algebra through to probability.
At Mathema, see what fascinating patterns emerge from Pascal’s triangle. Immerse yourself in a world of discovery that will take you from figurate numbers to the Sierpinski fractal, and everything
in- between. You can even ‘try your luck’ on the ‘stepping stones’ game to see how the overlap between probability and Pascal’s triangle plays out in real life!
Srinivasa Ramanujan
The story of Srinivasa Ramanujan is one of true ‘rags to mathematical riches’. Ramanujan was a prodigy who came from a small town outside Madras in South India. He came from a very poor family and
had virtually no formal training in pure mathematics. The rise of this mathematical genius is a story of great success and the triumph of perseverance and intellect in the face of all the cultural,
economic and racial resistance that stood in his way.
Not many people in 20th century India who came from such backgrounds would go on to make a lasting impact in the world of mathematics. Ramanujan was an exception to that general rule, his aptitude
for numbers catching the eye of notable Cambridge professors and eventually winning him a prestigious collaboration with English mathematician Geoffrey Hardy.
While Ramanujan understood numbers in highly complex ways, his methods for working with them were unorthodox and cryptic to most. He had the ability to come up with ground-breaking new theorems
without necessarily knowing how he got there. Many of Ramanujan’s theoretical propositions have since been proven to be correct.
At Mathema, you will have the chance to see mathematics through Ramanujan’s eyes. See if you have what it takes to solve Goldbach’s conjecture, and put Ramanujan’s theory of partitions to the test.
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