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GST Calculator Malaysia - PublicCalculators
GST Calculation in Malaysia?
In Malaysia, the 6% GST is applied to goods and services. With the help of online GST calculator Malaysia you can easily calculate your GST amount. Goods and Services Tax in Malaysia was introduced
on 1st April 2015. A manual way to find GST Inclusive and exclusive is given below
Inclusive GST Calculation Malaysia
Work backward to find the GST and exclusive amounts from the total GST inclusive price we have to follow some steps
To get the GST part of the GST inclusive amount you need to divide the GST inclusive amount by 106 and multiply by 6
To get GST exclusive amount while knowing GST inclusive value you need to multiply GST inclusive price by 100 and then divide the result by 106
Lets we have an amount of $500
To get GST inclusive amount these are the simple steps to work out
• Calculating GST Part
• Net Value
GST Exclusive Calculation Malaysia
To add the GST amount just multiply the exclusive amount by 0.06
To Add 6% GST
Let’s assume we have GST exclusive amount and we have to calculate the GST inclusive these are the following steps to perform
• $100 * 6% = $6
• $100 + $6 GST= GST inclusive
Lets we have a amount of $200
• $200+GST
= $200 * 0.06
= $12 is GST amount
= $212 is GST inclusive Amount | {"url":"https://publiccalculators.com/gst-calculator-malaysia/","timestamp":"2024-11-07T21:31:27Z","content_type":"text/html","content_length":"125578","record_id":"<urn:uuid:79421be7-2467-4bf3-ae5d-cbb9e3876828>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00258.warc.gz"} |
The Scientist's Linux Toolbox - Page: 1.3 » Linux Magazine
Essential software tools for the working scientist
If you are a physicist interested in simulation, I repeat my recommendation to install Julia and explore the language as well as the rapidly growing ecosystem of physics and related packages. It is
not by accident that interest in Julia is exploding among scientists from all fields. Any project that you create will benefit from the ease with which you can remix code from other packages; in some
cases you will discover that you have to do much less work than you anticipated.
As a case study, here is every step needed to go from zero to a fluid dynamics simulation of mixing at the boundary between a heavy (cold) and light (warmer) fluid (Rayleigh-Taylor instability). That
is, not including about two hours of reading documentation and playing around with the simulation software to learn how it works.
I'd heard good things about a Julia package for fluid simulation called Oceananigans.jl [11]. Getting libraries for Julia is a breeze, because it has a package manager not only built in, but
integrated into the REPL. Just hit the ] key to enter the package sub-REPL, and type add Oceananigans. Julia will download whatever is needed from the Internet, including any dependencies required,
and pre-compile the code.
For the rest, refer to Figure 3, which shows the entire Julia session. The final command creates the plot in Figure 4, which shows the temperature field.
In my career, I've had to work with simulation codes from various sources. I have never experienced getting a useful calculation done as easily as I was able to do with Oceananigans. It not only has
a sophisticated interface, but it is remarkably fast, owing in part to Julia's ability to generate efficient machine code. On my very modest laptop, this calculation took on the order of 30 seconds.
If you are in computational physics, or any branch of numerical science, I recommend experiencing the Julia ecosystem for yourself.
The symbolic mathematics program Maxima [12] is extremely useful. It is light, fast, and available from any distribution's package manager. Maxima is the open source successor to the venerable
commercial program Macsyma. It is written in Lisp and can handle many areas of mathematics, such as basic algebra, calculus, trigonometry, differential equations, series, and much more.
Maxima uses gnuplot to draw graphs, but that's OK, because you've already followed my advice and installed it. It can print its output in the form of TeX math markup, which you can paste directly
into your TEX documents (see the Writing Papers section).
Figure 5 shows a quick calculation in Maxima, a screenshot from my laptop. On the top is a terminal window, and below that a gnuplot graph. In the terminal, I've invoked Maxima (which starts
instantly) and defined an ordinary differential equation using Maxima's syntax for derivative operators. Notice that the inputs and outputs are numbered, so you can use them in subsequent
expressions. Maxima repeats the equation, but in a more visual form, using ASCII characters to approximate what the math is supposed to look like. In my second input, I'm asking for a solution using
the ODE solver; Maxima has more than one solver for some types of equations, because a particular solver may not work on a particular problem. The reply is a solution made of Bessel functions, which
is correct: the ODE I entered is the Bessel equation.
The solution contains two free parameters, called %k1 and %k2, whose values can't be determined without more information, namely boundary conditions. My next input defines the solution sol by giving
Maxima these boundary conditions. That input I terminated with a dollar sign rather than the usual semicolon, to suppress the somewhat voluminous output. Instead, I would like to see a graph of this
solution, which I order up in the next input, inserting the rhs (right hand side) of sol as the function to be plotted. The plot accepts a range for the independent variable and pops up the gnuplot
graph immediately.
Maxima can do a lot and is efficient once you get familiar with its syntax. But if you're a mathematician who makes extensive use of computer assistance, especially if you travel in fields not served
by Maxima, you may want to install SageMath [13]. You need to follow the link to download it, but make sure you have several gigabytes free before you do. SageMath is a huge system, packaging
together about 100 pieces of mathematical software (including Maxima) in a giant bundle with a unified interface based on Python. SageMath can deal with some truly obscure subdisciplines and even has
components for doing such things as solving Rubik's Cubes. Most people work with SageMath through its interactive, web-based notebook, which is similar to Jupyter [14], but it has a command-line
interface as well.
With the advent of bioinformatics [15] as a major activity within biology, the computer as a tool is more central to this discipline than ever before.
Bioinformatics is a blending of computer science and biology usually concerned with dealing with DNA sequences or other sequences of molecular data: essentially strings of potentially millions of
letters. This gives the computational problems in this field something of a linguistic character. Bioinformatics plays a big part in gene-based drug discovery, wildlife conservation, cancer
treatment, forensic analysis, and much more.
EMBOSS [16] is an acronym for European Molecular Biology Open Software Suite. It is a major computational resource used by many biologists all over the world. EMBOSS packages 200 applications for
sequence analysis, visualizing proteins, analyzing protein structure, providing transparent access to remotely hosted molecular sequences, and much more.
I should mention that a handful of these 200 programs are trivial utilities for doing things like removing whitespace from an ASCII file. Clearly EMBOSS attempts to be a complete environment
providing anything even a computer-naive bioinformatician might need.
EMBOSS can be operated with graphical, web-based, or command-line interfaces. Figure 6 shows a screenshot of one of the available web interfaces to a utility that displays protein sequences
graphically. SourceForge provides the interface as a demo, so the biologist interested in trying out the program, or even in using it for real work up to a point, can experiment on real data without
having to download and install it.
Even more than a comprehensive software suite for molecular biology and bioinformatics, EMBOSS provides a platform to allow computational biologists to release their own open source projects.
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Determine Neighbourhood Order Matrix from Binary Adjacency Matrix — nbOrder
Determine Neighbourhood Order Matrix from Binary Adjacency Matrix
Given a square binary adjacency matrix, the function nbOrder determines the integer matrix of neighbourhood orders (shortest-path distance).
a square, numeric or logical, and usually symmetric matrix with finite entries (and usually zeros on the diagonal) which indicates vertex adjacencies, i.e., first-order neighbourhood (interpreted
as neighbourhood == 1, not >0).
positive scalar integer specifying an upper bound for the neighbourhood order. The default (Inf) means no truncation (but orders cannot be larger than the number of regions minus 1), whereas
maxlag = 1 just returns the input neighbourhood matrix (converted to binary integer mode).
An integer matrix of neighbourhood orders, i.e., the shortest-path distance matrix of the vertices. The dimnames of the input neighbourhood matrix are preserved.
See also
nblag from the spdep package
## generate adjacency matrix
n <- 6
adjmat <- matrix(0, n, n)
adjmat[lower.tri(adjmat)] <- sample(0:1, n*(n-1)/2, replace=TRUE)
adjmat <- adjmat + t(adjmat)
## determine neighbourhood order matrix
nblags <- nbOrder(adjmat) | {"url":"https://surveillance.r-forge.r-project.org/pkgdown/reference/nbOrder.html","timestamp":"2024-11-07T06:15:12Z","content_type":"text/html","content_length":"10110","record_id":"<urn:uuid:d14d39fa-34ef-4332-a917-c2708dcc58a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00274.warc.gz"} |
TWO POINTERS : to use or !to use ? this is the question . (and if to use : then how to use?) - Codeforces
Two pointers is a very simple and useful algorithm and actually using it is optional but, using it will make your algorithm easier and more simple.
but now, how to use it ? I think its good for you to look at https://www.geeksforgeeks.org/two-pointers-technique/ too. it'll be useful:blush:
so let go ...
As it's shown by its name this algorithm use two pointers on an array(or two arrays) which initially are located at position x, y . then if CONDITION1 move pointer1 one to left/right and if
CONDITION2 move pointer2 one to left or right.
now let's switch to code.
QUESTION ANSWER SPOIL DANGER ================== to help you understand more I'll use one easy question and analyze them together.
so it says that we have two binary strings a,_ b_ and we want to find the maximum number of k which is the first k consecutive elements of a that are subsequence of b.
so what we do is we initially locate p1 (first pointer) at first of a and p2 (second pointer) at first of b.
if a[p1] == b[p2] : p1 ++
p2 ++
now p1 is the number of k.:sunglasses:
this are some more easy questions that may help you: 1873D - 1D Eraser 1972A - Contest Proposal 1851B - Parity Sort
I hope you enjoyed reading this article.:blush: MAY THE CODE BE WITH YOU!:pray:
3 months ago, # |
basi mozakhraf bood.sepas
dostdaranat va hamkelasihae azizat kianmehr biglari va ali roueintan
→ Reply
• 5 weeks ago, # ^ |
» 0
basi sher nagooeed doostnadar chiz na yani doostdar va hamkelasi shoma pouya izadi
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∫−23π−2π{(π+x)3+cos2(x+3π)}dx is equal to (A) 4π−1 (B) 32π4... | Filo
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Updated On Apr 22, 2024
Topic Integration
Subject Mathematics
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Maximum Likelihood Estimation
Maximum Likelihood Estimation (MLE) is an approach to estimating parameters for a model. It is one of the core aspects of Item Response Theory (IRT), especially to estimate item parameters (analyze
questions) and estimate person parameters (scoring). This article will provide an introduction to the concepts of MLE.
1. History behind Maximum Likelihood Estimation
2. Defining Maximum Likelihood Estimation
3. Comparison of likelihood and probability
4. Key characteristics of Maximum Likelihood Estimation
5. Weaknesses of Maximum Likelihood Estimation
6. Application of Maximum Likelihood Estimation
7. Summarizing remarks about Maximum Likelihood Estimation
8. References
History behind Maximum Likelihood Estimation
Even though early ideas about MLE appeared in the mid-1700s, Sir Ronald Aylmer Fisher developed them into a more formalized concept much later. Fisher was working seminally on maximum likelihood from
1912 to 1922, criticizing himself and producing several justifications. In 1925, he finally published “Statistical Methods for Research Workers”, one of the 20th century’s most influential books on
statistical methods. In general, the production of maximum likelihood concept has been a breakthrough in Statistics.
Defining Maximum Likelihood Estimation
Wikipedia defines MLE as follows:
In statistics, Maximum Likelihood Estimation is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood
function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood
Merriam Webster has a slightly different definition for MLE:
A statistical method for estimating population parameters (as the mean and variance) from sample data that selects as estimates those parameter values maximizing the probability of obtaining the
observed data.
To sum up, MLE is a method that detects parameter values of a model. These parameter values are identified such that they maximize the likelihood that the process designed by the model produced the
data that were actually observed. To put it simply, MLE answers the question:
For which parameter value does the observed data have the biggest probability?
Comparison of likelihood and probability
The definitions above contain “probability” but it is important not to mix these two different concepts. Let us look at some differences between likelihood and probability, so that you could
differentiate between them.
Likelihood Probability
Refers to the occurred events with known outcomes Refers to the events that will occur in the future
Likelihoods do not add up to 1 Probabilities add up to 1
Example 1: I flipped a coin 20 times and obtained 20 heads. What is the likelihood that the coin Example 1: I flipped a coin 20 times. What is the probability of the coin to land heads or tails
is fair? every time?
Example 2: Given the fixed outcomes (data), what is the likelihood of different parameter values? Example 2: The fixed parameter P = 0.5 is given. What is the probability of different outcomes?
Calculating Maximum Likelihood Estimation
MLE can be calculated as a derivative of a log-likelihood in relation to each parameter, the mean μ and the variance σ^2, that is equated to 0. There are four general steps in estimating the
• Call for a distribution of the observed data
• Estimate distribution’s parameters using log-likelihood
• Paste estimated parameters into a distribution’s probability function
• Evaluate the distribution of the observed data
Key characteristics of Maximum Likelihood Estimation
• MLE operates with one-dimensional data
• MLE uses only “clean” data (e.g. no outliers)
• MLE is usually computationally manageable
• MLE is often real-time on modern computers
• MLE works well for simple cases (e.g. binomial distribution)
Weaknesses of Maximum Likelihood Estimation
• MLE is sensitive to outliers
• MLE often demands optimization for speed and memory to obtain useful results
• MLE is sometimes poor at differentiating between models with similar distributions
• MLE can be technically challenging, especially for multidimensional data and complex models
Application of Maximum Likelihood Estimation
In order to apply MLE, two important assumptions (typically referred to as the i.i.d. assumption) need to be made:
• Data must be independently distributed, i.e. the observation of any given data point does not depend on the observation of any other data point (each data point is an independent experiment)
• Data must be identically distributed, i.e. each data point is generated from the same distribution family with the same parameters
Let us consider several world-known applications of MLE:
• Global Positioning System (GPS)
• Smart keyboard programs for iOS and Android operating systems (e.g. Swype)
• Speech recognition programs (e.g. Carnegie Mellon open source SPHINX speech recognizer, Dragon Naturally Speaking)
• Detection and measurement of the properties of the Higgs Boson at the European Organization for Nuclear Research (CERN) by means of the Large Hadron Collider (Francois Englert and Peter Higgs
were awarded the Nobel Prize in Physics in 2013 for the theory of Higgs Boson)
Generally speaking, MLE is employed in agriculture, economics, finance, physics, medicine and many other fields.
Summarizing remarks about Maximum Likelihood Estimation
Despite some functional issues with MLE such as technical challenges for multidimensional data and complex multiparameter models that interfere solving many real world problems, MLE remains a
powerful and widely used statistical approach for classification and parameter estimation. MLE has brought many successes to the mankind in both scientific and commercial worlds.
Aldrich, J. (1997). R. A. Fisher and the making of maximum likelihood 1912-1922. Statistical Science, 12(3), 162-176.
Stigler, S. M. (2007). The epic story of maximum likelihood. Statistical Science, 598-620.
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Laila Issayeva M.Sc.
Laila Baudinovna Issayeva earned her M.Sc. in Educational Leadership from Nazarbayev University with a focus on School Leadership and Improvement Management. Her undergraduate degree was from Aktobe
Regional State University with a major in Mathematics and a minor in Computer Science. Laila is an experienced educator and an educational measurement specialist with expertise in item and test
development, setting standards, analyzing, interpreting, and presenting data based on classical test theory and item response theory (IRT). As a professional, Laila is primarily interested in the
employment of IRT methodology and artificial intelligence technologies to educational improvement.
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Interpreting Remainders | sofatutor.com
Basics on the topic Interpreting Remainders
Interpreting Remainders in Division
In this learning text, we're going to learn about division and how to understand remainders. You might already know about division from things like cutting fruits in half, or splitting a pizza into
equal slices. These everyday activities are actually math problems, which we call division. Division can be a bit tricky, so we're going to start by learning about division and remainders.
Division and Interpreting Remainders – Definition
Let’s look at the meaning of the terms “division” and “remainders”:
Division is when we split things into equal groups, but sometimes we have some left over, which we call remainders. The remainder is what's left after we divide. In real life, there are many
different ways that we can understand the remainder. We can: ignore it, use it, add it or share it.
In this learning text, we will be looking at different scenarios and learn how we can interpret the remainder.
Interpreting Remainders – Word Problems
Now, let's look at some word problems to show different ways of understanding remainders. Depending on the situation, we will treat the remainder in different ways.
The first example is about sharing treats: we have thirty (30) treats divided into seven (7) bags. We want to know how many treats are in each bag. So, we're going to share all treats equally into
those seven bags. We divide thirty (30) by seven (7) and we find out there are four (4) treats in each bag with two (2) left over.
Let's look at the question again and decide what we're going to do with the two (2) left over. The question asks: How many treats will go into each bag? So, we just need to know the equal groups, so
this time we will ignore the remainder. There are four (4) treats in each bag!
Let's look at our second example.
First, read the problem and highlight the important information. There are seventy-two (72) balloons in a bag, and we need to make groups with seven (7) balloons each. So, if we divide the
seventy-two (72) by seven (7), we have ten (10) with two (2) left over.
Then, after you do the math, make sure you read the question again and decide what you're going to do with the remainder.
This time, the question asks: How many balloons will NOT be used for the inside decorations?
We know the remainder after our division was two (2), so the answer is also two (2) because the remainder represents the NOT used balloons.
Our next example is about party hats: we have forty-five (45) guests at a party. Everyone must wear party hats. In one pack is six (6) hats, and we need to figure out how many packs of hats we need
for the party.
So, let's divide forty-five (45) by six (6). We get seven (7) with three (3) left over. This time the remainder tells us that we must add one (1) to the seven (7) which is eight (8) because we can't
order three (3) extra hats separately. For this question, the answer is eight (8).
The interpretation of the remainder, as we mentioned before, can be different from question to question. That is why we are showing in this learning text different scenarios and learning how we
interpret remainder word problems for 4th grade.
Let’s take a look at one more example word problem.
We have to divide eighty-five (85) feet of streamers around the four (4) walls. When we do the math for this problem, we get twenty-one (21) with one (1) left over. Because we have to find out how
long each piece of streamer for each wall is, we can't give our answer with the remainder. We must write the remainder as a fraction. To do that, we write the remainder one (1) over the divisor four
(4), which will be the bottom number.
Division Word Problems and Interpreting Remainders – Summary
When solving real word division problems that have remainders, we need to understand the remainder by figuring out what the problem is asking.
• Ignore remainder - if the question asks for equal or whole amounts.
• Use remainder - if the question asks how much is left over.
• Include remainder - meaning, add one to the answer if the question asks for everything to be included.
• Make the remainder a fraction - when the remainder can be divided into even smaller parts.
Now you should be able to solve the division word problem with the remainder in any form. If you need more help, please watch the video explaining each problem and complete the worksheets which are
available for this topic.
Frequently Asked Questions about Division Word Problems
Transcript Interpreting Remainders
Mr. Squeaks is planning a surprise party to celebrate Imani’s Manufacture Date. He is working on decorations. Let's help Mr. Squeaks figure out what to do with the leftovers by... "Interpreting
Remainders" Division means we are putting items into equal groups. Sometimes when we divide, we have remainders, or leftovers, that can not be grouped equally. When we have a remainder in our
quotient, we need to interpret it or determine what to do with the leftovers based on what the problem is asking. There are several ways to interpret a remainder in real-world situations. We can
ignore it, use it, add it, or share it. Let’s use the party supplies to look at each way of interpreting a remainder. M.r. Squeaks is making goody bags for the little robot guests. He has thirty
treats to divide up into seven bags. How many treats will go into each bag? Let’s look at the word problem and highlight the important information. We know that there are thirty treats… we are
dividing into seven bags..and the problem is asking us to find out how many treats go into each bag, so we will solve thirty divided by seven. Thirty divided by seven equals four with a remainder of
two. To determine what we need to do with the remainder, let’s reread the question. How many treats will go into each bag? This tells us that we only need to know the equal groups, which is four and
we would IGNORE the remainder. There are four treats in each goody bag. Mr. Squeaks is also going to hang up balloons around the burrow. There are seventy-two balloons in a bag and he wants to make
bunches with seven balloons each. How many balloons will Mr. Squeaks have leftover? First, read the problem and highlight important information. There are seventy-two balloons and we are putting in
bundles of seven. What is seventy-two divided by seven? Ten remainder two. Now, reread the question to determine how to interpret the remainder. How many balloons WILL NOT be used for the inside
decorations? This tells us that we only need to know how many balloons are leftover, which is two... so he has two balloons remaining. All the guests need to wear party hats! They come in packs of
six, and there's forty-five attendees. He needs a party hat for each. How many packs of hats does he need? Based on the information, What is the division problem we need to solve? Forty-five divided
by six. What is forty-five divided by six? Seven remainder three. What does the problem say we need to find? How many packs of party hats he needs to get. Since there are three leftover and they come
in packs of six, Mr. Squeaks would have to get an additional pack of party hats. In this problem, we would interpret the remainder by adding one to the quotient. Mr. Squeaks needs eight packs of
party hats. Finally, Mr. Squeaks is going to hang up streamers around the four walls. The streamer roll measures eighty-five feet. How long is each piece of streamer for each wall?' Based on the
information, what is the division problem we need to solve? Eighty-five divided by four. What is eighty-five divided by four? Twenty-one remainder one. What does the problem want us to find? How much
of the streamers are on each wall. In this problem, the remainder, one, represents a material that can be further divided into parts, so... we write this remainder as a FRACTION. To write it as a
fraction, we put the remainder, one, over the divisor, four, to show that each wall will also get one-fourth of the streamer. Mr. Squeaks will put twenty-one and one-fourth feet of streamers on each
wall. While everyone waits for Imani, let's review. Remember… When solving real word division problems with remainders, we need to interpret remainders by identifying what the problem is asking. We
can... ignore the remainder if the question asks for equal or whole amounts. Use the remainder as your answer if the question asks how much is left over. Add one to the quotient when the question
asks for everything to be included. Or make the remainder a fraction when the quotient can be divided up into even smaller parts. "Shhh, Imani's coming." “SURPRISE!”
Interpreting Remainders exercise
Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Interpreting Remainders .
• What is a remainder?
When dividing, we are looking for equal groups.
Think, can all numbers be divided equally?
If a number can't be divided equally, what would you do?
With division, we are splitting up a number into equal groups. Sometimes, a number cannot be divided equally, so we get a remainder. A remainder is " leftovers that cannot be grouped equally."
• Define how remainders can be used.
Read the question carefully and think about what you are trying to find.
Once you determine what you are trying to find, then you can determine what to do with the remainder.
Think about what it means to share something. You are further dividing something to make it equal. This applies to sharing remainder too.
When you don't have enough of something; you need to include the remainder.
These are the ways remainders can be used based on what the question is asking you to solve:
The remainder is needed to answer the question "how much is leftover?" Use it
The question is asking you to find equal or whole amounts. Ignore It
The question is asking you to determine how many of something you will need. Include it
The remainder needs to be divided up and made into a fraction. Share it
• How can we interpret remainders?
To determine what to do with the remainder, reread the question.
Ask yourself, "what am I trying to solve?"
There are 28 kids going on a field trip. The buses can seat 8 kids. How many buses will be needed for the field trip. In this example you would need to add the remainder. Which problem would we
also need to add the remainder?
Mr. Squeaks needs 40 balloons. Balloons come in packs of 3. How many packs does he need to buy? In this example, we would ignore the remainder because it is not needed. Which problem would we
also need to ignore the remainder?
Each question is asking you to do something different with the remainders. Here are the solutions based on what is being asked:
□ Ignore it means you don't use the remainder
□ Use it means you need to know how much is leftover
□ Add it means everything needs to be included
□ Share it or make a fraction means the remainder can be divided into
small parts
For the question about the lollipops, we would ignore the remainder because we need to know the whole number of lollipops that Mr. Squeaks can purchase. The remainder would just be what is
In the question about the goody bags, we would need to include the remainder because we wan to make sure we have enough for everyone.
We use the remainder in the question about the wrapping paper because it is specifically asking us about what is leftover.
We share or make a fraction for the question about the balloon string because we need equal amounts and will have to divide up the remainder to do so.
• Solve the division problem.
Think, what is the question asking me to find?
Ask yourself, "how is the remainder being used?"
Draw a picture with 7 groups and see how many could equally fit.
17 groups of 7 equals 17 remainder 6.
In this question, we are being asked to find "how many stickers." This means that we want equal groups with no remainder. So here, you would ignore the remainder and just focus on the whole
number because we cannot have part of a sticker.
Divide 125 ÷ 7 = 17 R 6
Since we are ignoring the remainder, there are 17 stickers for each guest.
• Determine how the remainder will be used in this problem.
Ask yourself, "what is the problem asking me?"
Think, "how am I going to use the remainder?"
Here you can see that when we divide 50 by 6, we end up with 8 in each group. But there are 2 that don't belong to a group. That is our remainder. What should we do with the remaining 2?
In the problem, we need to add the remainder. We need to add it because the question is asking for everything to be included. Mr. Squeaks needs to make sure he purchases enough packs of noise
makers so that everyone can have one, so the remainder is included. Since the answer is 8 R 2, we would add one to 8 which give us 9 packs.
• What meanings do numbers have in an equation?
In the equation 54 ÷ 4 = 13 R 2. The number 2 represents the remainder, or what is leftover.
Read the question carefully and think about what you are being asked.
In the equation 84 ÷ 5; 84 is the total amount we have. How does that relate to the problem given here?
Sam has 12 boxes of toys. Each box can only fit 5 toys. How many toys cannot fit in a box? The remainder would be the answer which is 2. Where is the remainder in the story problem given here?
It is important to understand what the numbers in equations represent in order to correctly solve the problem. In the equation 75 ÷ 8 = 9 R 3, the representation of each number is as follows:
75 is the total number of guests.
8 is the number of seats at each table.
9 is the number of full tables.
3 is the number of guests at the extra table. | {"url":"https://us.sofatutor.com/math/videos/interpreting-remainders-2","timestamp":"2024-11-12T23:42:19Z","content_type":"text/html","content_length":"161900","record_id":"<urn:uuid:675e875f-8632-473d-8b9f-713f324aae76>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00874.warc.gz"} |
Now Under Construction: Intuitionistic Reverse Analysis
Joan Rand Moschovakis, Emerita Professor of Mathematics at Occidental College, Los Angeles, California
Each variety of reverse mathematics attempts to determine a minimal axiomatic basis for proving a particular mathematical theorem. Classical reverse analysis asks which set existence axioms are
needed to prove particular theorems of classical second-order number theory. Intuitionistic reverse analysis asks which intuitionistically accepted properties of numbers and functions suffice to
prove particular theorems of intuitionistic analysis using intuitionistic logic; it may also consider the relative strength of classical principles which do not contradict intuitionistic analysis.
S. Simpson showed that many theorems of classical analysis are equivalent, over a weak system PRA of primitive recursive arithmetic, to one of the following set existence principles: recursive
comprehension, arithmetical comprehension, weak Konig’s Lemma, arithmetical transfinite recursion, Π11 comprehension. Intermediate principles are studied also. Intuitionistic analysis depends on
function existence principles: countable and dependent choice, fan and bar theorems, continuous choice.
The fan and bar theorems have important mathematical equivalents. W. Veldman, building on a proof by T. Coquand, recently showed that over intuitionistic two-sorted recursive arithmetic BIM the
principle of open induction is strictly intermediate between the fan and bar theorems, and is equivalent to intuitionistic versions of a number of classical theorems. Intuitionistic logic separates
classically equivalent versions of countable choice, and it matters how decidability is interpreted. R. Solovay recently showed that Markov’s Principle is surprisingly strong in the presence of the
bar theorem. The picture gradually becomes clear. | {"url":"https://logic-gu.se/lindstrom-lectures/2014/2014/10/24/joan-rand-moschovakis-research-LL/","timestamp":"2024-11-05T10:12:55Z","content_type":"text/html","content_length":"9811","record_id":"<urn:uuid:24588b99-7196-4a6f-8898-c8bc197f11ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00176.warc.gz"} |
What Kind of Math Is Expected of a Civil Engineering Student? | Synonym
What Kind of Math Is Expected of a Civil Engineering Student?
Jupiterimages/Photos.com/Getty Images
Civil engineers design and supervise the great construction projects of the world, from the “Chunnel” between France and the United Kingdom to the Hoover Dam on the Colorado River to the world's
great skyscrapers. The college math required for this profession is rigorous but not in excess of what is asked of typical engineering students.
1 Calculus: Multiple Levels
At the college level, civil engineers will typically be required to complete three to four levels of calculus. Calculus is the study of changing unknown variables in relation to a system. It enables
engineers to make physics-related calculations. For example, according to Dr. Barney of Drexel University, if a civil engineer were designing a concrete bridge, he would need to know how the strength
of the concrete columns would change over time. The engineer could use the equation S=c(1-e^-kt) to find the strength of the concrete as a function of time, where S=strength, t=time, and c and k are
constants specific to this particular form of concrete. Because calculus measures change over time, the engineer can determine when the concrete will be at half its strength and what sorts of
variables, such as floods or automobile crashes, might further affect its strength. The engineer can then determine at what point the infrastructure should be replaced.
2 Differential Equations
Civil engineers will often have to take a course specifically focused on differential equations. Differential equations allow engineers to see how different functions of their designs experience
infinitesimal changes in relationship to changing variables in a system. These equations are critical to civil engineering work and allow engineers to see how well components of a structure deflect
different forces, how well open channels in a structure will accommodate steady uniform airflow and how well the soils underneath a structure will drain water.
3 Linear Algebra
Linear algebra works with the mathematical properties of lines and their transformation properties, including rotations in space, least squares fitting and the ability of three points to pass through
a circle. Civil engineers frequently use matrices in linear algebra to analyze the properties of springs and two-dimensional structural frames as well as to model the impact of uniformly distributed
loads and concentrated loads on the components of a structure. This often involves computer modeling, which is an inexpensive way to test different approaches to structural design in advance.
4 Statistics and Probability
Statistics enables civil engineers to see patterns in large amounts of data about a site on which they are building, as well as the human population that their structure will serve. It also enables
engineers to make predictions about future conditions and usage based on this data. Engineers use statistics to determine the typical air flow and climate in an area, as well as the typical water
usage of a human population. This makes it possible for them to design a structure that responds to the conditions of their target building site and the needs of their target human population. | {"url":"https://classroom.synonym.com/kind-math-expected-civil-engineering-student-20266.html","timestamp":"2024-11-04T18:37:53Z","content_type":"text/html","content_length":"246290","record_id":"<urn:uuid:a79766ea-97cf-4763-98cd-dc2b718a175f>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00872.warc.gz"} |
Iterative Learning Control of a Single-Input Single-Output System
This example shows how to use both model-free and model-based iterative learning control (ILC) to improve closed-loop trajectory tracking performance of a single-input single-output (SISO) system. To
implement ILC, you can use the Iterative Learning Control block from Simulink® Control Design™ software. In this example you design an ILC controller to perform a trajectory tracking for a SISO plant
model, compare the model-based and model-free ILC results to a tuned PID controller, and show that an ILC controller provides a better tracking performance when augmented to a baseline controller.
Iterative Learning Control Basics
ILC is an improvement in run-to-run control. It uses frequent measurements in the form of the error trajectory from the previous batch to update the control signal for the subsequent batch run. The
focus of ILC is on improving the performance of systems that execute a single, repeated operation, starting at the same initial operating condition. This focus includes many practical industrial
systems in manufacturing, robotics, and chemical processing, where mass production on an assembly line entails repetition. Therefore, use ILC when you have a repetitive task or repetitive
disturbances and want to use knowledge from previous iteration to improve next iteration.
In general, ILC has two variations: model free and model based. Model-free ILC requires minimum information of your plant, which makes it easier to design and implement. Model-based ILC takes
advantage of the additional knowledge of the plant, which can lead to faster convergence and better performance. For more information, see Iterative Learning Control.
Examine Model
This examples provides a preconfigured Simulink® model.
Plant and Nominal Controller
The plant is a SISO, linear, discrete-time, and second-order system.
Ts = 0.01;
A = [-0.7 -0.012;1 0];
B = [1;0];
C = [1 0];
D = 0;
sys = ss(A,B,C,D,Ts);
The nominal stabilizing controller is a PI controller. To tune a PI controller, use the pidtune function.
When plant operation is repeating in nature, ILC control can augment baseline PID to improve the controller performance iteration after iteration.
Model-Free Iterative Learning Control
The model-free ILC method does not require prior knowledge of the system dynamics and uses proportional and derivative error feedback to update the control history. The model-free ILC update law is:
${u}_{\mathit{k}+1}\left(t\right)=Q\left(q\right)\left[{u}_{\mathit{k}}\left(t\right)+{\gamma }_{\mathit{p}}{e}_{\mathit{k}}\left(\mathit{t}+1\right)+{\gamma }_{\mathit{d}}\left({e}_{\mathit{k}}\left
where ${\gamma }_{\mathit{p}}$ and ${\gamma }_{\mathit{d}}$ are referred as ILC gains.
ILC Mode
At runtime, ILC switches between two modes: control and reset. In the control mode, ILC outputs ${\mathit{u}}_{\mathit{k}}\left(\mathit{t}\right)$ at the desired time points and measures the error
between the desired reference $\mathit{r}\left(\mathit{t}\right)$ and output ${e}_{\mathit{k}}\left(\mathit{t}\right)$. At the end of the control mode, ILC calculates the new control sequence ${\
mathit{u}}_{\mathit{k}+1}\left(\mathit{t}\right)$ to use in the next iteration. In the reset mode, ILC output is 0. The reset mode must be long enough such that the PI controller in this mode brings
the plant back to its initial condition.
In this example, you use a generic reference signal for the purpose of illustration. Both the control and reset modes are 8 seconds long, which makes each iteration 16 seconds long. The reference
signal in the control mode is a sine wave with period of 8 seconds, starting at 0. In the reset mode, the reference signal is 0, which allows the PI controller to bring the plant back to the initial
operating point.
ILC Design
Load the Simulink model containing an Iterative Learning Control block in the model-free configuration.
To design an ILC controller, you configure the following parameters.
• Sample time and Iteration duration — These parameters determine how many control actions ILC provides in the control mode. If sample time is too large, ILC might not provide sufficient
compensation. If sample time is too small, ILC might take too much resources, especially it might lead to large memory footprint when model-based ILC is used.
• ILC gains — The gains ${\gamma }_{\mathit{p}}$ and ${\gamma }_{\mathit{d}}$ determine how well ILC learns between iterations. If ILC gains are too big, it might make the closed-loop system
unstable (robustness). If ILC gains are too small, it might lead to slower convergence (performance).
• Filter time constant — The optional low-pass filter to remove control chatter which may otherwise be amplified during learning. By default it is not enabled in the ILC block.
Define the values for the ILC controller parameters sample time Ts, iteration duration Tspan, ILC gains gammaP and gammaD, and the low-pass filter time constant filter_gain.
Ts = 0.01;
Tspan = 8;
gammaP = 3;
gammaD = 0;
filter_gain = 0.25;
Simulate Model and Plot Results
Simulate the model for 10 iterations (160 seconds). In the first iteration, ILC controller outputs 0 in the control mode because it just starts learning. Therefore, the closed-loop control
performance displayed in the first iteration comes from the nominal controller, which serves as the baseline for the comparison.
simout_model_free = sim('scdmodelfreeilc');
As the iterations progress, the ILC controller improves the reference tacking performance.
% Plot reference signal
t = simout_model_free.logsout{6}.Values.Time;
ref = squeeze(simout_model_free.logsout{6}.Values.Data);
plot(t, ref,"b");
hold on
% Plot plant output
y = squeeze(simout_model_free.logsout{4}.Values.Data);
plot(t, y, "r");
% Plot ILC mode
mode = squeeze(simout_model_free.logsout{1}.Values.Data);
plot(t, mode,'k');
% Plot settings
title('Model-Free ILC Performance');
hold off;
Plot the ILC and PI control efforts.
uILC = squeeze(simout_model_free.logsout{2}.Values.Data);
uPID = squeeze(simout_model_free.logsout{3}.Values.Data);
title('Model-Free ILC Control Efforts');
After the ILC controller learns how to compensate, the nominal PID control effort is reduced to minimum.
Model-Based Iterative Learning Control
As previously discussed, there are multiple ways to design the learning function $\mathit{L}$ in the ILC control law.
When you design $\mathit{L}$ based on the plant input-output matrix $\mathit{G}$, it is called model-based ILC. Additionally, the iterative learning control block provides two types of model-based
ILC: gradient based and inverse-model based.
Gradient Based ILC Law
The gradient-based ILC uses the transpose of input-output matrix $\mathit{G}$ in the learning function.
$\mathit{L}={\gamma \mathit{G}}^{\mathit{T}}$
Therefore, ILC control law becomes ${u}_{\mathit{k}+1}\left(t\right)=Q\left(q\right)\left[{u}_{\mathit{k}}\left(t\right)+{\gamma \mathit{G}}^{\mathit{T}}{e}_{\mathit{k}}\left(\mathit{t}+1\right)\
right]$, where $\gamma$ is the ILC gain.
Inverse Model based ILC Law
The inverse model based ILC uses the inverse of input-output matrix $\mathit{G}$ in the learning function.
$\mathit{L}={\gamma \mathit{G}}^{-1}$
Therefore, ILC control law becomes ${u}_{\mathit{k}+1}\left(t\right)=Q\left(q\right)\left[{u}_{\mathit{k}}\left(t\right)+{\gamma \mathit{G}}^{-1}{e}_{\mathit{k}}\left(\mathit{t}+1\right)\right]$,
where $\gamma$ is the ILC gain.
When $\mathit{G}$ matrix is not square, the block uses a pseudoinverse instead.
ILC Design
Load the Simulink model containing an Iterative Learning Control block in the model-based ILC configuration.
mdl = 'scdmodelbasedilc';
Define the values for the ILC controller parameters sample time Ts, iteration duration Tspan, ILC gain gamma, and the low-pass filter time constant filter_gain.
Ts = 0.01;
Tspan = 8;
gamma = 3;
filter_gain = 0.25;
Simulation Result
Simulate the model for 10 iterations (160 seconds). In the first iteration, ILC controller outputs 0 in the control mode because it just starts learning. Therefore, the closed-loop control
performance displayed in the first iteration comes from the nominal controller, which serves as the baseline for the comparison.
First, set the ILC block to use the gradient-based ILC law and simulate the model.
set_param([mdl,'/Iterative Learning Control'], 'ModelBasedILCtype','Gradient based');
simout_gradient_based = sim(mdl);
Then, set the ILC block to use the inverse-model-based ILC law and simulate the model again.
set_param([mdl,'/Iterative Learning Control'], 'ModelBasedILCtype','Inverse-model based');
simout_inverse_based = sim(mdl);
As the iterations progress, both ILC controllers improve the reference tacking performance.
% Plot reference signal
t = simout_gradient_based.logsout{6}.Values.Time;
ref = squeeze(simout_gradient_based.logsout{6}.Values.Data);
plot(t, ref, 'b');
hold on
% Plot plant output controlled by gradient based ILC
y = squeeze(simout_gradient_based.logsout{4}.Values.Data);
plot(t, y, 'r');
% Plot plant output controlled by inverse-model based ILC
y = squeeze(simout_inverse_based.logsout{4}.Values.Data);
plot(t, y, 'g');
% Plot ILC mode
mode = squeeze(simout_gradient_based.logsout{1}.Values.Data);
plot(t, mode, 'k');
% Plot settings
title('Model-Based ILC Performance');
hold off;
As you can observe in the preceding plot, both gradient-based ILC and inverse-model based ILC provide performance comparable to the model-free ILC in this example.
See Also
Related Topics | {"url":"https://it.mathworks.com/help/slcontrol/ug/model-free-iterative-learning-control-of-siso-system.html","timestamp":"2024-11-11T10:26:53Z","content_type":"text/html","content_length":"93132","record_id":"<urn:uuid:f1917390-87bd-4ca2-bd77-3cb3ddc20b3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00690.warc.gz"} |
These are archived from the now defunct su3su2u1 tumblr.
A Roundabout Approach to Quantum Mechanics
This will be the first post in what I hope will be a series that outlines some ideas from quantum mechanics. I will try to keep it light, and not overly math filled- which means I’m not really
teaching you physics. I’m teaching you some flavor of the physics. I originally wrote here “you can’t expect to make ice cream just having tasted it,” but I think a better description might be “you
can’t expect to make ice cream just having heard someone describe what it tastes like.” AND PLEASE, PLEASE PLEASE ask questions. I’m used to instant feedback on my (attempts at) teaching, so if
readers aren’t getting anything out of this, I want to stop or change or something.
Now, unfortunately I can’t start with quantum mechanics without talking about classical physics first. Most people think they know classical mechanics, having learned it on their mother’s knee, but
there are so,so many ways to formulate classical physics, and most physics majors don’t see some really important ones (in particular Hamiltonian and Lagrangian mechanics) until after quantum
mechanics. This is silly, but at the same time university is only 4 years. I can’t possibly teach you all of these huge topics, but I will need to rely on a few properties of particle and of light.
And unlike intro Newtonian mechanics, I want to focus on paths. Instead of asking something like “a particle starts here with some velocity, where does it go?” I want to focus on “a particle starts
here, and ends there. What path did it take?”
So today we start with light, and a topic I rather love. Back in the day, before “nerd-sniping” several generations of mathematicians, Fermat was laying down a beautiful formulation of optics-
Light always takes the path of least time
I hear an objection “isn’t that just straight lines?” We have to combine this insight with the notion that light travels at different speeds in different materials. For instance, we know light slows
down in water by a factor of about 1.3.
So lets look at a practical problem, you see a fish swimming in water (I apologize in advance for these diagrams):
I drew the (hard to see) dotted straight line between your eye and the fish.
But that isn’t what the light does- there is a path that saves the light some time. The light travels faster in air than in water, so it can travel further in the air, and take a shorter route in the
water to the fish.
This is a more realistic path for the light- it bends when it hits the water- it does this in order to take paths of least time between points in the water and points in the air. Exercise for the
mathematical reader- you can work this out quantitatively and derive Snell’s law (the law of refraction) just from the principle of least time.
And one more realistic example: Lenses. How do they work?
So that bit in the middle is the lens and we are looking at light paths that leave 1 and travel to 2 (or visa versa, I guess).
The lens is thickest in the middle, so the dotted line gets slowed the most. Path b is longer, but it spends less time in the lens- that means with careful attention to the shape of the lens we can
make the time of path b equal to the time of the dotted path.
Path a is the longest path, and just barely touches the lens, so is barely slowed at all, so it too can be made to take the same time as the dotted path (and path b).
So if we design our lens carefully, all of the shortest-time paths that touch the lens end up focused back to one spot.
So thats the principle of least time for light. When I get around to posting on this again we’ll talk about particles.
Now, these sort of posts take some effort, so PLEASE PLEASE PLEASE tell me if you got something out of this.
Edit: And if you didn’t get anything out of this, because its confusing, ask questions. Lots of questions, any questions you like.
More classical physics of paths
So after some thought, these posts will probably be structured by first discussing light, and then turning to matter, topic by topic. It might not be the best structure, but its at least giving me
something to organize my thoughts around.
As in all physics posts, please ask questions. I don’t know my audience very well here, so any feedback is appreciated. Also, there is something of an uncertainty principle between clarity and
accuracy. I can be really clear or really accurate, but never both. I’m hoping to walk the middle line here.
Last time, I mentioned that geometric optics can be formulated by the simple principle that light takes the path of least time. This is a bit different than many of the physics theories you are used
to- generally questions are phrased along the lines of “Alice throws a football from position x, with velocity v, where does Bob need to be to catch the football.” i.e. we start with an initial
position and velocity. Path based questions are usually “a particle starts at position x_i,t_i and ends at position x_f,t_f, what path did it take?”
For classical non-relativistic mechanics, the path based formulation is fairly simple, we construct a quantity called the “Lagrangian” which is defined by subtracting potential energy from kinetic
energy (KE - PE). Recall that kinetic energy is 1/2 mv^2, where m is the mass of the particle and v is the velocity, and potential energy depends on the problem. If we add up the Lagrangian at every
instant along a path we get a quantity called the action (S is the usual symbol for action, for some reason) and particles take the path of least action. If you know calculus, we can put this as
[S = \int KE-PE dt ]
The action has units of energy*time, which will be important in a later post.
Believe it or not, all of Newton’s laws are all contained in this minimization principle. For instance, consider a particle moving with no outside influences (no potential energy). Such a particle
has to minimize its v^2 over the path it takes.
Any movement away from the straight line will cause an increase in the length of the path, so the particle will have to travel faster, on average, to arrive at its destination. We want to minimize v^
2, so we can deduce right away the particle will take a straight line path.
But what about its speed? Should a particle move very slowly to decrease v^2 as it travels, and then “step on the gas” near the end? Or travel at a constant speed? Its easy to show that minimum
action is the constant speed path (give it a try!). This gives us back Newton’s first law.
You can also consider the case of a ball thrown straight up into the air. What path should it take? Now we have potential energy mgh (where h is the height, and g is a gravitational constant). But
remember, we subtract the potential energy in the action- so the particle can lower its action by climbing higher.
Along the path of least action in a gravitational field, the particle will move slowly at high h to spend more time at low-action, and will speed up as h decreases (it needs to have an average
velocity large enough to get to its destination on time). If you know calculus of variations, you can calculate the required relationship, and you’ll find you get back exactly the Newtonian
relationship (acceleration of the particle = g).
Why bother with this formulation? It makes a lot of problems easier. Sometimes specifying all the forces is tricky (imagine a bead sliding on a metal hoop. The hoop constrains the bead to move along
the circular hoop, so the forces are just whatever happens to be required to keep the bead from leaving the hoop. But the energy can be written very easily if we use the right coordinates). And with
certain symmetries its a lot more elegant (a topic I’ll leave for another post).
So to wrap up both posts- light takes the path of least time, particles take the path of least action. (One way to think about this is that light has a Lagrangian that is constant. This means that
the only way to lower the action is to find the path that takes the least time). This is the take away points I need for later- in classical physics particles take the path of least action.
I feel like this is a lot more confusing than previous posts because its hard to calculate concrete examples. Please ask questions if you have them.
Semi classical light
As always math will not render properly on tumblr dash, but will on the blog. This post contains the crux of this whole series of posts, so its really important to try to understand this argument.
Recall from the first post I wrote that one particularly elegant formulation of geometric optics is Fermat’s principle:
light takes the path of least time
But, says a young experimentalist (pun very much intended!), look what happens when I shine light through two slits, I get a pattern like this:
Light must be a wave.
"Wait, wait, wait!" I can hear you saying. Why does this two slit thing mean that light is a wave?
Let us talk about the key feature of waves- when waves come together they can combine in different ways:
TODO: broken link.
So when a physicists want to represent waves, we need to take into account not just the height of the wave, but also the phase of the wave. The wave can be at “full hump” or “full trough” or anywhere
in between.
The technique we use is called “phasors” (not to be confused with phasers). We represent waves as little arrows, spinning around in a circle:
TODO: broken link.
The ;length of the arrow A is called the amplitude and represents the height of the wave. The angle, (\theta) represents the phase of the wave. (The mathematical sophisticates among us will recognize
these as complex numbers of the form (Ae^{i\theta}) With these arrows, we can capture all the add/subtract/partially-add features of waves:
TODO: broken link.
So how do we use this to explain the double slit experiment? First, we assume all the light that leaves the same source has the same amplitude. And the light has a characteristic period, T. It takes
T seconds for the light to go from “full trough” back to “full trough” again.
In our phasor diagram, this means we can represent the phase of our light after t seconds as:
[\theta = \frac{2\pi t}{T} ]
Note, we are taking the angle here in radians. 2 pi is a full circle. That way when t = T, we’ve gone a full circle.
We also know that light travels at speed c (c being the “speed of light,” after all). So as light travels a path of length L, the time it traveled is easily calculated as (\frac{L}{c}).
Now, lets look at some possible paths:
The light moves from the dot on the left, through the two slits, and arrives at the point X. Now, for the point X at the center of the screen, both paths will have equal lengths. This means the waves
arrive with no difference in phase, and they add together. We expect a bright spot at the center of the screen (and we do get one).
Now, lets look at points further up the screen:
As we move away from the center, the paths have different lengths, and we get a phase difference in the arriving light:
[\theta_1 - \theta_2= \frac{2\pi }{cT} \left(L_1 - L_2\right) ]
So what happens? As we move up the wall, the length distance gets bigger and the phase difference increases. Every time the phase difference is a multiple of pi we get cancellation, and a dark spot.
Every time its a multiple of 2 pi, we get a bright spot. This is exactly Young’s results.
But wait a minute, I can hear a bright student piping up (we’ll call him Feynman, but it would be more appropriate to call him Huygens in this case). Feynman says “What if there were 3 slits?”
Well, then we’d have to add up the phasors for 3 different slits. Its more algebra, but when they all line up, its a bright spot, when they all cancel its a dark spot,etc. We could even have places
where two cancel out, and one doesn’t.
"But, what if I made a 4th hole?" We add up four phasors. "A 5th? "We add up 5 phasors.
"What if I drilled infinite holes? Then the screen wouldn’t exist anymore! Shouldn’t we recover geometric optics then?"
Ah! Very clever! But we DO recover geometric optics. Think about what happens if we add up infinitely many paths. We are essentially adding up infinitely many random phasors of the same amplitude:
So we expect all these random paths to cancel out.
But there is a huge exception.
Those random angles are because when we grab an arbitrary path, the time light takes on that path is random.
But what happens near a minimum? If we parameterize our random paths, near the minimum the graph of time-of-travel vs parameter looks like this:
The graph gets flat near the minimum, so all those little Xs have roughly the same phase, which means all those phasors will add together. So the minimum path gets strongly reinforced, and all the
other paths cancel out.
So now we have one rule for light:
To calculate how light moves forward in time, we add up the associated phasors for light traveling every possible path.
BUT, when we have many, many paths we can make an approximation. With many, many paths the only one that doesn’t cancel out, the only one that matters, is the path of minimum time.
Semi-classical particles
Recall from the previous post that we had improved our understanding of light. Light we suggested, was a wave which means
Light takes all possible paths between two points, and the phase of the light depends on the time along the path light takes.
Further, this means:
In situations where there are many, many paths the contributions of almost all the paths cancel out. Only the path of least time contributes to the result.
The astute reader can see where we are going. We already learned that classical particles take the path of least action, so we might guess at a new rule:
Particles take all possible paths between two points, and the phase of the particle depends on the action along the path the particle takes.
Recall from the previous post that the way we formalized this is that the phase of light could be calculated with the formula
[\theta = \frac{2\pi}{T} t]
We would like to make a similar formula for particles, but instead of time it must depend on the action, but will we do for the particle equivalent of the “period?” The simplest guess we might take
is a constant. Lets call the constant h, planck’s constant (because thats what it is). It has to have the same units of action, which are energy*time.
[\theta = \frac{2\pi}{h} * S]
Its pretty common in physics to use a slightly different constant (\hbar = \frac{h}{2\pi} ) because it shows up so often.
[\theta = \frac{S}{\hbar}]
So we have this theory- maybe particles are really waves! We’ll just run a particle through a double slit and we’ll see a pattern just like the light!
So we set up our double slit experiment, throw a particle at the screen, and blip. We pick up one point on the other side. Huh? I thought we’d get a wave. So we do the experiment over and over again,
and this results
So we do get the pattern we expected, but only built up over time. What do we make of this?
Well, one things seems obvious- the outcome of a large number of experiments fits our prediction very well. So we can interpret the result of our rule as a probability instead of a traditional fully
determined prediction. But probabilities have to be positive, so we’ll say the probability is proportional to the square of our amplitude.
So lets rephrase our rule:
To predict the probability that a particle will arrive at a point x at time t, we take a phasor for every possible path the particle can take, with a phase depending on the action along the path, and
we add them all up. Squaring the amplitude gives us the probability.
Now, believe it or not, this rule is exactly equivalent to the Schroedinger equation that some of us know and love, and pretty much everything you’ll find in an intro quantum book. Its just a
different formulation. But you’ll note that I called it “semi-classical” in the title- thats because undergraduate quantum doesn’t really cover fully quantum systems, but thats a discussion for a
later post.
If you are familiar with Yudkowsky’s sequence on quantum mechanics or with an intro textbook, you might be used to thinking of quantum mechancis as blobs of amplitude in configuration space changing
with time. In this formulation, our amplitudes are associated with paths through spacetime.
When next I feel like writing again, we’ll talk a bit about how weird this path rule really is, and maybe some advantages to thinking in paths.
Basic special relativity
LNo calculus or light required, special relativity using only algebra. Note- I’m basically typing up some lecture notes here, so this is mostly a sketch.
This derivation is based on key principle that I believe Galileo first formulated-
The laws of physics are the same in any inertial frame OR there is no way to detect absolute motion. Like all relativity derivations, this is going to involve a thought experiment. In our experiment
we have a train that moves from one of train platform to the other. At the same time a toy airplane also flies from one of the platform to the other (originally, I had made a Planes,Trains and
Automobiles joke here, but kids these days didn’t get the reference… ::sigh::)
There are two events, event 1- everything starts at the left side of the platform. Event 2- everything arrives at the right side of the platform. The entire time the train is moving with a constant
velocity v from the platform’s perspective (symmetry tells us this also means that the platform is moving with velocity v from the train’s perspective.)
We’ll look at these two events from two different perspectives- the perspective of the platform and the perspective of the train. The goal is to figure out a set of equations that let us relate
quantities between the different perspectives.
The dot is the toy plane, the box is the train. L is the length of the platform from its own perspective. l is the length of the train from it’s own perspective. T is the time it takes the train to
cross the platform from the platform’s perspective. And t is the time the platform takes to cross the train from the train’s perspective.
From the platform’s perspective, it’s easy to see the train has length l’ = L - vT. And the toy plane has speed w = L/T.
From the train’s perspective, the platform has length L’ = l + vt and the toy plane has speed u = l/t
So to summarize
Observer | Time passed between events | Length of Train | Speed of Plane
Platform | T | l’ = L-vT | w = L/T
Train | t |L’ = l+vt | u/t
Now, we again exploit symmetry and our Galilean principle. By symmetry,
l’/l = L’/L = R
Now, by the Galilean principle, R as a function can only depend on v. If it didn’t we could detect absolute motion. We might want to just assume R is 1, but we wouldn’t be very careful if we did.
So what we do is this- we want to write a formula for w in terms of u and v and R (which depends only on v). This will tell us how to relate a velocity in the train’s frame to a velocity in the
plane’s frame.
I’ll skip the algebra, but you can use the relations above to work this out for yourself
w = (u+v)/(1+(1-R)u/v) = f(u,v)
Here I just used f to name the function there.
I WILL EDIT MORE IN, POSTING NOW SO I DON’T LOSE THIS TYPED UP STUFF.
More Special Relativity and Paths
This won’t make much sense if you haven’t read my last post on relativity. Math won’t render on tumblr dash, instead go to the blog.
Last time, we worked out formulas for length contraction (and I asked you to work out a formula for time dilation). But what would generally be useful is a formula generally relating events between
the different frames of reference. Our thought experiment had two events-
event 1, the back end of the train, the back end of the platform, and the toy are all at the same place.
event 2- the front of the train, the front end of the platform, and the toy are all at the same place.
From the toy’s frame of reference, these events occur at the same place, so we only have a difference between the two events. We’ll call that difference (\Delta\tau) . We’ll always use this to mean
“the time between events that occur at the same position” (only in one frame will events occur in the same place), and it’s called proper time.
Now, the toy train sees the platform moves with speed -w, and the length of the platform is RL. So this relationship is just time = distance/speed.
[\Delta\tau^2 = R^2L^2/w^2 = (1-\frac{w^2}{c^2})L^2/w^2 ]
Now, we can manipulate the right hand side by noting that from the platform’s perspective, L^2/w^2 is the time between the two events, and those two events are separated by a distance L. We’ll call
the time between events in the platforms frame of reference (\Delta t) and the distance between the events L, we’ll call generally (\Delta x).
[\Delta\tau^2 = (1-\frac{w^2}{c^2})L^2/w^2 = (\Delta t^2 - \Delta x^2/c^2) ]
Note that the speed w has dropped out of the final version of the equation- this would be true for any frame, since proper time is unique (every frame has a different time measurement, but only one
measures the proper time), we have a frame independent measurement.
Now, lets relate this back to the idea of paths that I’ve discussed previously. One advantage of the path approach to mechanics is that if we can create a special relativity invariant action then the
mechanics we get is also invariant. So one way we might consider to do this is by looking at proper time- (remember S is the action). Note the negative sign- without it there is no minimum only a
[ S \propto -\int d\tau = -C\int d\tau ]
Now C has to have units of energy for the action to have the right units.
Now, some sketchy physics math
[ S = -C\int \sqrt(dt^2 - dx^2/c^2) = -C\int dt \sqrt(1-\frac{dx^2}{dt^2}/c^2) ]
[S = -C\int dt \sqrt(1-v^2/c^2) ]
So one last step is to note that the approximation we can make for \sqrt(1-v^2/c^2), if v is much smaller than c which is (1-1/2v^2/c^2)
So all together, for small v
[S = C\int dt (\frac{v^2}{2c^2} - 1)]
So if we pick the constant C to be mc^2, then we get
[S = \int dt (1/2 mv^2 - mc^2)]
We recognize the first term as just the kinetic energy we had before! The second term is just a constant and so won’t effect where the minimum is. This gives us a new understanding of our path rule
for particles- particles take the path of maximum proper time (it’s this understanding of mechanics that translates most easily to general relativity)
Special relativity and free will
Imagine right now, while you are debating whether or not to post something on tumblr, some aliens in the andromeda galaxy are sitting around a conference table discussing andromeda stuff.
So what is the “space time distance” between you right now (deciding what to tumblrize) and those aliens?
Well, the distance between andromeda and us is something like 2.5 million light years. So thats a “space time distance” tau (using our formula from last time) of 2.5 million years. So far, so good:
Now, imagine an alien, running late to the andromeda meeting, is running in. He is running at maybe 1 meter/second. We know that for him lengths will contract and time will dilate. So for him, time
on earth is actually later- using
(\Delta \tau^2 = \Delta t^2 - \Delta x^2/ c^2)
and using our formula for length contraction, we can calculate that according to our runner in andromeda the current time on Earth is about 9 days later then today.
So simultaneous to the committee sitting around on andromeda, you are just now deciding what to tumblrize. According to the runner, it’s 9 days later and you’ve already posted whatever you are
thinking about + dozens of other things.
So how much free will do you really have about what you post? (This argument is originally due to Rietdijk and Putnam).
We are doing Taylor series in calculus and it's really boring. What would you add from physics?
First, sorry I didn’t get to this for so long.
Anyway, there is a phenomenon in physics where almost everything is modeled as a spring (simple harmonic motion is everywhere!). You can see this in discussion of resonances. Wave motion can be
understood as springs coupled together,etc, and lots of system exhibit waves- when you speak the tiny air perurbations travel out like waves, same as throwing a pebble in a pond, or wiggling a jump
rope. These are all very different systems, so why the hell do we see such similar behavior?
Why would this be? Well, think of a system in equilibrium, and nudging it a tiny bit away from equilibrium. If the equilibrium is at some parameter a, and we nudge it a tiny bit away from equilibrium
(so x-a = epsilon)
Now, we can Taylor expand- but we note that in equiilibrium the energy is at a minimum, so the linear term in the Taylor expansion is 0
[E(\epsilon)= E(a) + \frac{d^2E}{dx^2}1/2 \epsilon^2 + … ]
Now, constants in potential energy don’t matter, and so the first important term is a squared potential energy, which is a spring.
So Taylor series-> everything is a spring.
Why field theory?
So far, we’ve learned in earlier posts in my quantum category that
1. Classical theories can be described in terms of paths with rules where particles take the path of “least action.”
2. We can turn a classical theory into a quantum one by having the particle take every path, with the phase from each path given by the action along the path (divided by hbar).
We’ve also learned that we can make a classical theory comply with special relativity by picking a relativistic action (in particle, an action proportional to the “proper time.”)
So one obvious thing to try to make a special relativistic quantum theory would be to start with a special relativistic action and do the sum over paths we use for the classical theory.
You can do this- and it almost works! If you do the mathematical transition from our original, non-relativistic paths to standard, textbook quantum you’d find that you get the Schroedinger equation
(or if you were more sophisticated you could get something called the Pauli equation that no one talks about, but is basically the Schroedinger equation + the fact that electrons have spin).
If you try to do it from a relativistic action, you would get an equation called the Klein-Gordon equation (or if you were more sophisticated you could get the Dirac equation). Unfortunately, this
runs into trouble- there can be weird negative probabilities, and general weirdness to the solutions.
So we have done something wrong- and the answer is that making the action special relativistic invariant isn’t enough.
Let’s look at some paths:
So the dotted line in this picture represents the light cone- how fast light traveling away from the point will travel. All of the paths end up inside the light cone, but some of the paths go outside
of it. This leads to really strange situations, lets look at one outside the light cone path from two frames of reference:
So what we see is that a normal path in the first frame (on the left) looks really strange in the second- because the order of events for events outside the lightcone isn’t fixed, some frame of
references see the path as moving back in time.
So immediately we see the problem. When we switched to the relativistic theory we weren’t including all the paths- to really include all the paths we need to include paths that also (apparently) move
back in time. This is very strange! Notice that if we run time forward the X’ observer sees ,at some points along the path two particles (one moving back in time, one moving forward).
Feynman’s genius was to demonstrate that we can think of these particles moving backward in time as anti-particles moving forward in time. So the x’ observer
So really our path set looks like
Notice that not only do we have paths connecting the two points, but we have totally unrelated loops that start and end at the same points- these paths are possible now!
So to calculate a probability, we can’t just look at the paths that connect paths connecting points x_o and x_1! There can be weird loopy paths that never touch x_o and x_1 that still matter! From
Feynman’s persepctive, particle and anti particle pairs can form, travel awhile and annihilate later.
So as a book keeping device we introduce a field- at every point in space it has a value. To calculate the action of the field we can’t just look at the paths- instead we have to sum up the values of
the fields (and some derivatives) at every point in space.
So our old action was a sum of the action over just times (S is the action, L is the lagrangian)
[S = \int dt L ]
Our new action has to be a sum over space and time.
[S = \int dt d^x l ]
So now our Lagrangian is a lagrangian density.
And we can’t just restrict ourselves to paths- we have to add up every possible configuration of the field.
So that’s why we need field theory to combine relativity with quantum mechanics. Next time some implications
Field theory implications
So the first thing is that if we take the Feynman interpretation, our field theory doesn’t have a fixed particle number- depending on the weird loops in a configuration it could have an almost
arbitrary number of particles. So one way to phrase the problem with not including backwards paths is that we need to allow the particle number to fluctuate.
Also, I know some of you are thinking “what are these fields?” Well- that’s not so strange. Think of the electromagnetic fields. If you have no charges around, what are the solutions to the
electromagnetic field? They are just light waves. Remember this post? Remember that certain special paths were the most important for the sum over all paths? Similarly, certain field configurations
are the most important for the sum over configurations. Those are the solutions to the classical field theory.
So if we start with EM field theory, with no charges, then the most important solutions are photons (the light waves). So we can outline levels of approximation
Sum over all configurations -> (semi classical) photons that travel all paths -> (fully classical) particles that travel just the classical path.
Similarly, with any particle
Sum over all configurations -> (semi classical) particles that travel all paths -> (fully classical) particles that travel just the classical path.
This is why most quantum mechanics classes really only cover wave mechanics and don’t ever get fully quantum mechanical.
Planck length/time
Answering somervta's question. What is the significance of Planck units.
Let’s start with an easier one where we have some intuition- let’s analyze the simple hydrogen atom (the go-to quantum mechanics problem). But instead of doing physics, lets just do dimensional
analysis- how big do we expect hydrogen energies to be?
Let’s start with something simpler- what sort of distances do we expect a hydrogen atom to have? How big should it’s radius be?
Well, first- what physics is involved? I model the hydrogen atom as an electron moving in an electric field, and I expect I’ll need quantum mechanics, so I’ll need hbar (planck’s constant), e, the
charge of the electron, coulomb’s constant (call it k), and the mass of the electron. Can I turn these into a length?
Let’s give it a try- k*e^2 is an energy times a length. hbar is an energy * a time, so if we divide we can get hbar/(k*e^2) which has units of time/length. Multiply in by another hbar, and we get
hbar^2/(k*e^2), which has units of mass * length. So divide by the mass of the electron, and we get a quantity hbar^2/(m*k*e^2).
This has units of length, so we might guess that the important length scale for the hydrogen atom is our quantity (this has a value of about 53 picometers, which is about the right scale for atomic
We could also estimate the energy of the hydrogen atom by noting that
Energy ~ k*e^2/r and use our scale for r.
Energy ~ m*k^2*e^4/(hbar^2) ~27 eV.
This is about twice as large as the actual ground state, but its definitely the right order of magnitude.
Now what Planck noticed is that if you ask “what are the length scales of quantum gravity?” You end up with the constants G, c, and hbar. Turns out, you can make a length scale out of that (sqrt
(hbar*G/c^3) ) So just like with hydrogen, we expect that gives us a characteristic length for where quantum effects might start to matter for gravity (or gravity effects might matter for quantum
The planck energy and planck mass, then, are similarly characteristic mass and energy scales.
It’s sort of “how small do my lengths have to be before quantum gravity might matter?” But it’s just a guess, really. Planck energy is the energy you’d need to probe that sort of length scale (higher
energies probe smaller lengths),etc.
Does that answer your question?
More Physics Answers
Answering bgaesop's question:
How is the whole dark matter/dark energy thing not just proof that the theory of universal gravitation is wrong?
So let’s start with dark energy- the first thing to note is that dark energy isn’t really new, as an idea it goes back to Einstein’s cosmological constant. When the cosmological implications of
general relativity were first being understood, Einstein hated that it looked like the universe couldn’t be stable. BUT then he noticed that his field equations weren’t totally general- he could add
a term, a constant. When Hubble first noticed that the universe was expanding Einstein dropped the constant, but in the fully general equation it was always there. There has never been a good
argument why it should be zero (though some theories (like super symmetry) were introduced in part to force the constant to 0, back when everyone thought it was 0).
Dark energy really just means that constant has a non-zero value. Now, we don’t know why it should be non-zero. That’s a responsibility for a deeper theory- as far as GR goes it’s just some constant
in the equation.
As for dark matter, that’s more complicated. The original observations were that you couldn’t make galactic rotation curves work out correctly with just the observable matter. So some people said
“maybe there is a new type of non-interacting matter” and other people said “let’s modify gravity! Changing the theory a bit could fix the curves, and the scale is so big you might not notice the
modifications to the theory.”
So we have two competing theories, and we need a good way to tell them apart. Some clever scientists got the idea to look at two galaxies that collided- the idea was the normal matter would smash
together and get stuck at the center of the collision, but the dark matter would pass right through. So you would see two big blobs of dark matter moving away from each other (you can infer their
presence from the way the heavy matter bends light, gravitational lensing), and a clump of visible matter in between. In the bullet cluster, we see exactly that.
Now, you can still try to modify gravitation to match the results, but the theories you get start to look pretty bizarre, and I don’t think any modified theory has worked successfully (though the
dark matter interpretation is pretty natural).
In the standard model, what are the fundamental "beables" (things that exist) and what are kinds of properties do they have (that is, not "how much mass do they have" but "they have mass")?
So this one is pretty tough, because I don’t think we know for sure exactly what the “beables” are (assuming you are using beable like Bell’s term).
The issue is that field theory is formulated in terms of potentials- the fields that enter into the action are the electromagnetic potential, not the electromagnetic field. In classical
electromagnetic theory, we might say the electromagnetic field is a beable (Bell’s example), but the potential is not.
But in field theory we calculate everything in terms of potentials- and we consider certain states of the potential to be “photons.”
At the electron level, we have a field configuration that is more general than the wavefunction - different configurations represent different combinations of wavefunctions (one configuration might
represent a certain 3 particle wavefunction, another might represent a single particle wavefunction,etc).
In Bohm type theories, the beables are the actual particle positions, and we could do something like that for field theory- assume the fields are just book keeping devices. This runs into problems
though, because field configurations that don’t look much like particles are possible, and can have an impact on your theory. So you want to give some reality to the fields.
Another issue is that the field configurations themselves aren’t unique- symmetries relate different field configurations so that very different configurations imply the same physical state.
A lot of this goes back to the fact that we don’t have a realistic axiomatic field theory yet.
But for concreteness sake, assume the fields are “real,” then you have fermion fields, which have a spin of 1/2, an electro-weak charge, a strong charge, and a coupling to the higgs field. These
represent right or left handed electrons,muons,neutrinos,etc.
You have gauge-fields (strong field, electro-weak field), these represent your force carrying boson (photons, W,Z bosons, gluons).
And you have a Higgs field, which has a coupling to the electroweak field, and it has the property of being non-zero everywhere in space, and that constant value is called its vacuum expectation
What's the straight dope on dark matter candidates?
So, first off there are two types of potential dark matter. Hot dark matter, and cold dark matter. One obvious form of dark matter would be neutrinos- they only interact weakly and we know they
exist! So this seems very obvious and promising until you work it out. Because neutrinos are so light (near massless), most of them will be traveling at very near the speed of light. This is “hot”
dark matter and it doesn’t have the right properties.
So what we really want is cold dark matter. I think astronomers have some ideas for normal baryonic dark matter (brown dwarfs or something). I don’t know as much about those.
Particle physicists instead like to talk about what we call thermal relics. Way back in the early universe, when things were dense and hot, particles would be interconverting between various types
(electron-positrons turning into quarks, turning into whatever). As the universe cooled, at some point the electro-weak force would split into the weak and electric force, and some of the weak
particles would “freeze out.” We can calculate this and it turns out the density of hypothetical “weak force freeze out” particles would be really close to the density of dark matter. These are
called thermal relics. So what we want are particles that interact via the weak force (so the thermal relics have the right density) and are heavier than neutrinos (so they aren’t too hot).
From SUSY
It turns out it’s basically way too easy to create these sorts of models. There are lots of different super-symmetry models but all of them produce heavy “super partners” for every existing particle.
So one thing you can do is assume super symmetry and then add one additional symmetry (they usually pick R-parity) the goal of the additional symmetry is to keep the lightest super partner from
decaying. So usually the lightest partner is related to the weak force (generally its a partner to some combination of the Higgs, the Z bosons, and the photons. Since these all have the same quantum
numbers they mix into different mass states). These are called neutralinos. Because they are superpartners to weakly interacting particles they will be weakly interacting, and they were forced to be
stable by R parity. So BAM, dark matter candidate.
Of course, we’ve never seen any super-partners,so…
From GUTs
Other dark matter candidates can come from grand unified theories. The standard model is a bit strange- the Higgs field ties together two different particles to make the fermions (left handed
electron + right handed electron, etc). The exception to this rule are neutrinos. Only left handed neutrinos exist, and their mass is Majorana.
But some people have noticed that if you add a right handed neutrino, you can do some interesting things- the first is that with a right handed neutrino in every generation you can embed each
generation very cleanly in SO(10). Without the extra neutrino, you can embed in SU(5) but it’s a bit uglier. This has the added advantage that SO groups generally don’t have gauge anomalies.
The other thing is that if this neutrino is heavy, then you can explain why the other fermion masses are so light via a see-saw mechanism.
Now, SO(10) predicts this right handed neutrino doesn’t interact via the standard model forces, but because the gauge group is larger we have a lot more forces/bosons from the broken GUT. These extra
bosons almost always lead to trouble with proton decay, so you have to figure out some way to arrange things so that protons are stable, but you can still make enough sterile neutrinos in the early
universe to account for dark matter. I think there is enough freedom to make this mostly work, although the newer LHC constraints probably make that a bit tougher.
Obviously we’ve not seen any of the additional bosons of the GUT, or proton decay,etc.
From Axions
(note: the method for axion production is a bit different than other thermal relics)
There is a genuine puzzle to the standard model QCD/SU(3) gauge theory. When the theory was first designed physicist used the most general lagrangian consistent with CP symmetry. But the weak force
violates CP, so CP is clearly not a good symmetry. Why then don’t we need to include the CP violating term in QCD?
So Peccei and Quinn were like “huh, maybe the term should be there, but look we can add a new field that couples to the CP violating term, and then add some symmetries to force the field to near 0.″
That would be fine, but the symmetry would have an associated goldstone boson, and we’d have spotted a massless particle.
So you promote the global Peccei-Quinn symmetry to a guage symmetry, and then the goldston boson becomes massive, and you’ve saved the day. But you’ve got this leftover massive “axion” particle. So
BAM dark matter candidate.
Like all the other dark matter candidates, this has problems. There are instanton solutions to QCD, and those would break the Peccei-Quinn symmetry. Try to fix it and you ruin the gauge symmetry (and
so your back to a global symmetry and a massless, ruled-out axion). So it’s not an exact symmetry, and things get a little strained.
So these are the large families I can think of off hand. You can combine the different ones (SUSY SU(5) GUT particles,etc).
I realize this will be very hard to follow without much background, so if other people are interested, ask specific questions and I can try to clean up the specifics.
Also, I have a gauge theory post for my quantum sequence that will be going up soon.
If your results are highly counterintuitive...
They are almost certainly wrong.
Once, when I was a young, naive data science I embarked on a project to look at individual claims handlers and how effective they were. How many claims did they manage to settle below the expected
cost? How many claims were properly reserved? Basically, how well was risk managed?
And I discovered something amazing! Several of the most junior people in the department were fantastic, nearly perfect on all metrics. Several of the most senior people had performance all over the
map. They were significantly below average on most metrics! Most of the claims money was spent on these underperformers! Big data had proven that a whole department in a company was nonsense lunacy!
Not so fast. Anyone with any insurance experience (or half a brain, or less of an arrogant physics-is-the-best mentality) would have realized something right away- the kinds of claims handled by
junior people are going to be different. Everything that a manager thought could be handled easily by someone fresh to the business went to the new guys. Simple cases, no headaches, assess the cost,
pay the cost, done.
Cases with lots of complications (maybe uncertain liability, weird accidents, etc) went to the senior people. Of course outcomes looked worse, more variance per claim makes the risk much harder to
manage. I was the idiot, and misinterpreting my own results!
A second example occured with a health insurance company where an employee I supervised thought he’d upended medicine when he discovered a standard-of-care chemo regiment lead to worse outcomes then
a much less common/”lighter” alternative. Having learned from my first experience, I dug into the data with him and we found out that the only cases where the less common alternative was used were
cases where the cancer had been caught early and surgically removed while it was localized.
Since this experience, I’ve talked to startups looking to hire me, and startups looking for investment (and sometimes big-data companies looking to be hired by companies I work for), and I see this
mistake over and over. “Look at this amazing counterintuitive big data result!”
The latest was in a trade magazine where some new company claimed that a strip-mall lawyer with 22 wins against some judge was necessarily better than white-shoe law firm that won less often against
the same judge. (Although in most companies I have worked for, if the case even got to trial something has gone wrong- everyone pushes for settlement. So judging by trial win record is silly for a
second reason).
Locality, fields and the crown jewel of modern physics
Apologies, this post is not finished. I will edit to replace the to be continued section soon.
Last time, we talked about the need for a field theory associating a mathematical field with any point in space. Today, we are going to talk about what our fields might look like. And we’ll find
something surprising!
I also want to emphasize locality, so in order to do that let’s consider our space time as a lattice, instead of the usual continuous space.
So that is a lattice. Now imagine that it’s 4 dimensional instead of 2 dimensional.
Now, a field configuration involves putting one of our phasors at every point in space.
So here is a field configuration:
To make our action local (and thus consistent with special relativity) we insist that the action at one lattice point only depends on the field at that point, and on the fields of the neighboring
We also need to make sure we keep the symmetry we know from earlier posts- we know that the amplitude of the phasor is what matters, and we have the symmetry to change the phase angle.
Neighbors of the central point, indicated by dotted lines.
We can compare neighboring points by subtracting (taking a derivative).
Sorry that is blurry. . Middle phasor - left phasor = some other phasor.
And the last thing we need to capture is the symmetry-remember that the angle of our phasor didn’t matter for predictions- the probabilities are all related to amplitudes (the length of the phasor).
The simplest way to do this is to insist that we adjust the angle of all the phasors in the field, everywhere:
Sorry for the shadow of my hand
Anyway, this image shows a transformation of all the phasors. This works, but it seems weird- consider a configuration like this:
This is two separate localized field configurations- we might interpret this as two particles. But should we really have to adjust the phase angle of the all the fields over by the right particle if
we are doing experiments only on the left particle?
Maybe what we really want is a local symmetry. A symmetry where we can rotate the phase angle of a phasor at any point individually (and all of them differently, if we like).
To Be Continued | {"url":"https://ddanluu.com/su3su2u1/physics","timestamp":"2024-11-04T04:15:08Z","content_type":"text/html","content_length":"55000","record_id":"<urn:uuid:4b688b6a-2c79-4644-a0d2-61a3723e66da>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00400.warc.gz"} |
MathFiction: Randall and the River of Time (Cecil Scott Forester)
Charles Randall meets two people who change his life while he is on leave from fighting in World War I: a patent lawyer for whom he designs an improved flare and the seductive wife of a fellow
soldier. He ends up creating yet another invention (a pea sorter) with the former and marries the latter (after her husband is killed in action). His life takes yet another dramatic turn when he
catches another man in bed with his wife, an encounter which leaves the other man dead and Randall on trial for manslaughter.
A recurrent motif in the novel is the consideration of how a very small change in circumstances could have resulted in a huge difference in the trajectory of Randall's life. It is what we
mathematicians would call "sensitive dependence on initial conditions", and thinking about this book in the context of chaos theory would be very interesting for that reason. However, the author does
not discuss this in mathematical terms and so I would not consider the book to be "mathematical fiction" for that reason.
Instead, I am listing this novel on my database of works of mathematical fiction for two reasons:
• As we learn these two passages, Randall views mathematics as being useful and important while his wife holds the opposite opinion:
(quoted from Randall and the River of Time)
It was news to her that mathematics was a vitally important adjunct to science, for to her mathematics was a rather dreary memory of schooldays where A did a piece of work in four days, and B did
it in fire days, and where to satisfy her teacher she had to discover what X amounted to in a string of arbitrary symbols about which she knew almost nothing and cared not at all. She might have
classed an interest in mathematics with an interest in postage stamps or white mice (two other subjects which did not stir her emotions in the least), if it had not been for the fact that this
nice-looking young man before her showed such undoubted keenness about it and at the same time was obviously no crank.
(quoted from Randall and the River of Time)
Randall drank tea to gain time. It would be no use telling her exactly what he was thinking about. The word "adiabatic" meant nothing to her at all. The differential and the integral calculus
were to her things of entirely no use. Randall once, with unwonted eloquence, had tried to explain to her how the calculus had made man master of the universe he lived in, and how its discovery
had been more important than that of gunpowder, and he had been hurt as well as surprised at her unbelief. Muriel was quite prepared to accept that in this ridiculous man-made world it might be
necessary to juggle with t's and y's to brain a science degree, but she could not conceive of the t's and y's having any real importance or even any intrinsic interest. A man might as well have
to learn how to keep six balls in the air at once--for that matter it was a way of earning a living, too. It was no use talking about this morning's lecture to Muriel; but luckily there was
something else he could talk about.
• Randall is the son of a mathematics teacher, and was teased about this by his classmates as a boy. Although he is described as a "science student" who wants to be a "physicist", Randall talks and
thinks about math all of the time, and he represents two different mathematician stereotypes. On the one hand, he is shown to be naive. The reader would know this simply from his interactions
with other people. But, I suspect the author considered his nerdy interest in math and his naiveté not to be two independent traits, but rather linked in a way that would make each trait more
believable to the reader because they came together. In addition, he is the sort of fictional character for whom mathematics is a way to hide or escape from reality, as illustrated in this
(quoted from Randall and the River of Time)
But luckily he found an alternative; his father had brought him his physics textbooks and at a fortunate moment he opened one and, piqued when he found his eye running over the printed arguments
without his mind reacting to them at all, he set himself seriously to pick up the threads. He was standing when he began; soon he was sitting down, and then he had books and notebooks open before
him, his elbows on the table and his forehead resting on his hands, quite lost to his surroundings as he followed along the tortuous mathematical paths of scientific deduction. "The specific heat
per unit mass must then vary as 1/p^(r-1)." Was he sure that it must? He had better go back through the argument again. There was no need to withdraw into a world of his own making; he could
withdraw into the world of mathematics: brightly and coldly lit, armored against the exterior--like the turret of a battleship, and in the same way full of purposeful and functional apparatus.
There he could be oblivious, most of the time, of the black cloud of the approaching trial extending up from the horizon until it covered the whole sky.
Although it is not directly related to mathematics, I think it is worth noting that the character of his wife, Muriel, comes off very badly. She is not only ignorant and illogical, but unethical.
Of course, such people exist and one cannot really complain if one character in a book is that way. But in the year 2022 when I am writing this, it comes across as regrettably misogynistic, as if
it were saying that men are all wise and good except for when they are led astray by the opposite sex. For this reason, this novel -- which was apparently well received in its time and whose
author is more famous for his "Hornblower" saga -- has not aged well.
I am grateful to Simon Brown of the Deviot Institute for bringing this work to my attention by forwarding to me the listing of "mathematical literature" that John S. Lew published in 1992. Lew's
list included this along with a few others I had not previously heard about.
Contributed by Tom Lindstrøm
C.S. Forester was one of the literary heros of my youth, and I read everything by him I could get my hands on, including "Randall and the River of Time" (for some reason, the local library in my
small Norwegian hometown had a copy). As this is more than fifty years ago, I don’t have much to add to Alex’s resumé, but I remember finding the book a little odd (too idea driven?) and not
totally successful, even for an avid fan.
Still, it might be worth mentioning that Forester in general seems to have had a very positive attitude to mathematics. He often makes a point of the mathematical skills of his best known hero,
the British navy officer Horatio Hornblower. As a seasick, gawky and physically unimpressive midshipman, Hornblower finally gets some positive attention when he easily solves all the navigational
problems the other midshipmen find impenetrable. Later, when on half pay during the Peace of Amiens, he survives as a professional whist player, and again Forester emphasizes the mathematical
aspects of the game. One gets the feeling that mathematics is used as a general symbol of intelligence, imagination, and talent throughout the series of eleven novels.
As for misogyny, I don’t really think it is typical of Forester’s work. Admittedly, there are some rather negative portraits of female characters, but there are also some very positive and/or
sympathetic ones, e.g., of Hornblower’s two wives Maria and Barbara (the fictional sister of the Duke of Wellington) and of his mistress Marie de Graçay. The spinster heroine of "The African
Queen" also turns out to have more guts, wits, and character than one would initially surmise (as is only to be expected from someone who has ever been played by Katherine Hepburn!). | {"url":"https://kasmana.people.charleston.edu/MATHFICT/mfview.php?callnumber=mf1555","timestamp":"2024-11-04T11:18:44Z","content_type":"text/html","content_length":"16518","record_id":"<urn:uuid:5cafc652-a224-4396-9597-aadad6efa437>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00704.warc.gz"} |
An Introduction to Statistical Methods and Data Analysis 7th Edition by R. Lyman Ott, ISBN-13: 978-1305269477 - ebookschoice.comAn Introduction to Statistical Methods and Data Analysis 7th Edition by R. Lyman Ott, ISBN-13: 978-1305269477ebookschoice.com - The best ebooks collection
An Introduction to Statistical Methods and Data Analysis 7th Edition by R. Lyman Ott, ISBN-13: 978-1305269477
[PDF eBook eTextbook] – Available Instantly
• Publisher: Cengage Learning; 7th edition (June 11, 2015)
• Language: English
• 1296 pages
• ISBN-10: 1305269470
• ISBN-13: 978-1305269477
Ott and Longnecker’s AN INTRODUCTION TO STATISTICAL METHODS AND DATA ANALYSIS, Seventh Edition, provides a broad overview of statistical methods for advanced undergraduate and graduate students from
a variety of disciplines who have little or no prior course work in statistics. The authors teach students to solve problems encountered in research projects, to make decisions based on data in
general settings both within and beyond the university setting, and to become critical readers of statistical analyses in research papers and news reports. The first eleven chapters present material
typically covered in an introductory statistics course, as well as case studies and examples that are often encountered in undergraduate capstone courses. The remaining chapters cover regression
modeling and design of experiments.
Table of Contents:
Half Title
Part 1: Introduction
Ch 1: Statistics and the Scientific Method
1.1: Introduction
1.2: Why Study Statistics?
1.3: Some Current Applications of Statistics
1.4: A Note to the Student
1.5: Summary
1.6: Exercises
Part 2: Collecting Data
Ch 2: Using Surveys and Experimental Studies to Gather Data
2.1: Introduction and Abstract of Research Study
2.2: Observational Studies
2.3: Sampling Designs for Surveys
2.4: Experimental Studies
2.5: Designs for Experimental Studies
2.6: Research Study: Exit Polls Versus Election Results
2.7: Summary
2.8: Exercises
Part 3: Summarizing Data
Ch 3: Data Description
3.1: Introduction and Abstract of Research Study
3.2: Calculators, Computers, and Software Systems
3.3: Describing Data on a Single Variable: Graphical Methods
3.4: Describing Data on a Single Variable: Measures of Central Tendency
3.5: Describing Data on a Single Variable: Measures of Variability
3.6: The Boxplot
3.7: Summarizing Data from More Than One Variable: Graphs and Correlation
3.8: Research Study: Controlling for Student Background in the Assessment of Teaching
3.9: R Instructions
3.10: Summary and Key Formulas
3.11: Exercises
Ch 4: Probability and Probability Distributions
4.1: Introduction and Abstract of Research Study
4.2: Finding the Probability of an Event
4.3: Basic Event Relations and Probability Laws
4.4: Conditional Probability and Independence
4.5: Bayes’ Formula
4.6: Variables: Discrete and Continuous
4.7: Probability Distributions for Discrete Random Variables
4.8: Two Discrete Random Variables: The Binomial and the Poisson
4.9: Probability Distributions for Continuous Random Variables
4.10: A Continuous Probability Distribution: The Normal Distribution
4.11: Random Sampling
4.12: Sampling Distributions
4.13: Normal Approximation to the Binomial
4.14: Evaluating Whether or Not a Population Distribution Is Normal
4.15: Research Study: Inferences About Performance-Enhancing Drugs Among Athletes
4.16: R Instructions
4.17: Summary and Key Formulas
4.18: Exercises
Part 4: Analyzing the Data, Interpreting the Analyses, and Communicating the Results
Ch 5: Inferences About Population Central Values
5.1: Introduction and Abstract of Research Study
5.2: Estimation of μ
5.3: Choosing the Sample Size for Estimating μ
5.4: A Statistical Test for μ
5.5: Choosing the Sample Size for Testing μ
5.6: The Level of Significance of a Statistical Test
5.7: Inferences About μ for a Normal Population, σ Unknown
5.8: Inferences About μ When the Population Is Nonnormal and n Is Small: Bootstrap Methods
5.9: Inferences About the Median
5.10: Research Study: Percentage of Calories from Fat
5.11: Summary and Key Formulas
5.12: Exercises
Ch 6: Inferences Comparing Two Population Central Values
6.1: Introduction and Abstract of Research Study
6.2: Inferences About μ1 ― μ2: Independent Samples
6.3: A Nonparametric Alternative: The Wilcoxon Rank Sum Test
6.4: Inferences About μ1 ― μ2: Paired Data
6.5: A Nonparametric Alternative: The Wilcoxon Signed-Rank Test
6.6: Choosing Sample Sizes for Inferences About μ1 ― μ2
6.7: Research Study: Effects of an Oil Spill on Plant Growth
6.8: Summary and Key Formulas
6.9: Exercises
Ch 7: Inferences About Population Variances
7.1: Introduction and Abstract of Research Study
7.2: Estimation and Tests for a Population Variance
7.3: Estimation and Tests for Comparing Two Population Variances
7.4: Tests for Comparing t > 2 Population Variances
7.5: Research Study: Evaluation of Methods for Detecting E. coli
7.6: Summary and Key Formulas
7.7: Exercises
Ch 8: Inferences About More Than Two Population Central Values
8.1: Introduction and Abstract of Research Study
8.2: A Statistical Test About More Than Two Population Means: An Analysis of Variance
8.3: The Model for Observations in a Completely Randomized Design
8.4: Checking on the AOV Conditions
8.5: An Alternative Analysis: Transformations of the Data
8.6: A Nonparametric Alternative: The Kruskal–Wallis Test
8.7: Research Study: Effect of Timing on the Treatment of Port-Wine Stains with Lasers
8.8: Summary and Key Formulas
8.9: Exercises
Ch 9: Multiple Comparisons
9.1: Introduction and Abstract of Research Study
9.2: Linear Contrasts
9.3: Which Error Rate Is Controlled?
9.4: Scheffé’s S Method
9.5: Tukey’s W Procedure
9.6: Dunnett’s Procedure: Comparison of Treatments to a Control
9.7: A Nonparametric Multiple-Comparison Procedure
9.8: Research Study: Are Interviewers’ Decisions Affected by Different Handicap Types?
9.9: Summary and Key Formulas
9.10: Exercises
Ch 10: Categorical Data
10.1: Introduction and Abstract of Research Study
10.2: Inferences About a Population Proportion π
10.3: Inferences About the Difference Between Two Population Proportions, π1 – π2
10.4: Inferences About Several Proportions: Chi-Square Goodness-of-Fit Test
10.5: Contingency Tables: Tests for Independence and Homogeneity
10.6: Measuring Strength of Relation
10.7: Odds and Odds Ratios
10.8: Combining Sets of 2 x 2 Contingency Tables
10.9: Research Study: Does Gender Bias Exist in the Selection of Students for Vocational Education?
10.10: Summary and Key Formulas
10.11: Exercises
Ch 11: Linear Regression and Correlation
11.1: Introduction and Abstract of Research Study
11.2: Estimating Model Parameters
11.3: Inferences About Regression Parameters
11.4: Predicting New y-Values Using Regression
11.5: Examining Lack of Fit in Linear Regression
11.6: Correlation
11.7: Research Study: Two Methods for Detecting E. coli
11.8: Summary and Key Formulas
11.9: Exercises
Ch 12: Multiple Regression and the General Linear Model
12.1: Introduction and Abstract of Research Study
12.2: The General Linear Model
12.3: Estimating Multiple Regression Coefficients
12.4: Inferences in Multiple Regression
12.5: Testing a Subset of Regression Coefficients
12.6: Forecasting Using Multiple Regression
12.7: Comparing the Slopes of Several Regression Lines
12.8: Logistic Regression
12.9: Some Multiple Regression Theory (Optional)
12.10: Research Study: Evaluation of the Performance of an Electric Drill
12.11: Summary and Key Formulas
12.12: Exercises
Ch 13: Further Regression Topics
13.1: Introduction and Abstract of Research Study
13.2: Selecting the Variables (Step 1)
13.3: Formulating the Model (Step 2)
13.4: Checking Model Assumptions (Step 3)
13.5: Research Study: Construction Costs for Nuclear Power Plants
13.6: Summary and Key Formulas
13.7: Exercises
Ch 14: Analysis of Variance for Completely Randomized Designs
14.1: Introduction and Abstract of Research Study
14.2: Completely Randomized Design with a Single Factor
14.3: Factorial Treatment Structure
14.4: Factorial Treatment Structures with an Unequal Number of Replications
14.5: Estimation of Treatment Differences and Comparisons of Treatment Means
14.6: Determining the Number of Replications
14.7: Research Study: Development of a Low-Fat Processed Meat
14.8: Summary and Key Formulas
14.9: Exercises
Ch 15: Analysis of Variance for Blocked Designs
15.1: Introduction and Abstract of Research Study
15.2: Randomized Complete Block Design
15.3: Latin Square Design
15.4: Factorial Treatment Structure in a Randomized Complete Block Design
15.5: A Nonparametric Alternative—Friedman’s Test
15.6: Research Study: Control of Leatherjackets
15.7: Summary and Key Formulas
15.8: Exercises
Ch 16: The Analysis of Covariance
16.1: Introduction and Abstract of Research Study
16.2: A Completely Randomized Design with One Covariate
16.3: The Extrapolation Problem
16.4: Multiple Covariates and More Complicated Designs
16.5: Research Study: Evaluation of Cool-Season Grasses for Putting Greens
16.6: Summary
16.7: Exercises
Ch 17: Analysis of Variance for Some Fixed-, Random-, and Mixed-Effects Models
17.1: Introduction and Abstract of Research Study
17.2: A One-Factor Experiment with Random Treatment Effects
17.3: Extensions of Random-Effects Models
17.4: Mixed-Effects Models
17.5: Rules for Obtaining Expected Mean Squares
17.6: Nested Factors
17.7: Research Study: Factors Affecting Pressure Drops Across Expansion Joints
17.8: Summary
17.9: Exercises
Ch 18: Split-Plot, Repeated Measures, and Crossover Designs
18.1: Introduction and Abstract of Research Study
18.2: Split-Plot Designed Experiments
18.3: Single-Factor Experiments with Repeated Measures
18.4: Two-Factor Experiments with Repeated Measures on One of the Factors
18.5: Crossover Designs
18.6: Research Study: Effects of an Oil Spill on Plant Growth
18.7: Summary
18.8: Exercises
Ch 19: Analysis of Variance for Some Unbalanced Designs
19.1: Introduction and Abstract of Research Study
19.2: A Randomized Block Design with One or More Missing Observations
19.3: A Latin Square Design with Missing Data
19.4: Balanced Incomplete Block (BIB) Designs
19.5: Research Study: Evaluation of the Consistency of Property Assessors
19.6: Summary and Key Formulas
19.7: Exercises
Appendix: Statistical Tables
Answers to Selected Exercises
R. Lyman Ott earned his Bachelor’s degree in Mathematics and Education and Master’s degree in Mathematics from Bucknell University, and Ph.D in Statistics from the Virginia Polytechnic Institute.
After two years working in statistics in the pharmaceutical industry, Dr. Ott became assistant professor in the Statistic Department at the University of Florida in 1968 and was named associate
professor in 1972. He joined Merrell-National laboratories in 1975 as head of the Biostatistics Department and then head of the company’s Research Data Center. He later became director of Biomedical
Information Systems, Vice President of Global Systems and Quality Improvement in Research and Development, and Senior Vice President Business Process Improvement and Biometrics. He retired from the
pharmaceutical industry in 1998, and now serves as consultant and Board of Advisors member for Abundance Technologies, Inc. Dr. Ott has published extensively in scientific journals and authored or
co-authored seven college textbooks including Basic Statistical Ideas for Managers, Statistics: A Tool for the Social Sciences and An Introduction to Statistical Methods and Data Analysis. He has
been a member of the Industrial Research Institute, the Drug Information Association and the Biometrics Society. In addition, he is a Fellow of the American Statistical Association and received the
Biostatistics Career Achievement Award from the Pharmaceutical research and Manufacturers of America in 1998. He was also an All-American soccer player in college and is a member of the Bucknell
University Athletic Hall of Fame.
Michael Longnecker currently serves as Professor and Associate Department Head at Texas A&M University. He received his B.S. at Michigan Technological University, his first M.S. at Western Michigan
University, his second M.S. at Florida State University, and his Ph.D. at Florida State University. He is interested in Nonparametrics, Statistical Process Control, and Statistical Consulting.
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The atom cobalt has 27 electrons. How many energy levels will its electrons use? | Socratic
The atom cobalt has 27 electrons. How many energy levels will its electrons use?
1 Answer
The number of electrons each energy level can hold increases as you add more and more energy levels to an atom.
The relationship that exists between the energy level, $n$, and the number of electrons it can hold can be written like this
$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} {\text{no. of e}}^{-} = 2 {n}^{2} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
You can use this equation to find the maximum number of electrons that can be added to each energy level. You will have
□ the first energy level, $n = 1$
${\text{no. of e"^(-) = 2 * 1^2 = "2 e}}^{-}$
□ the second energy level $n = 2$
${\text{no. of e"^(-) = 2 * 2^2 = "8 e}}^{-}$
□ the third energy level, $n = 3$
${\text{no. of e"^(-) = 2 * 3^2 = "18 e}}^{-}$
□ the fourth energy level, $n = 4$
${\text{no. of e"^(-1) = 2 * 4^2 = "32 e}}^{-}$
and so on.
In your case, cobalt, $\text{Co}$, is said to have a total of $27$electrons surrounding its nucleus. These electrons will be placed in orbitals in order of increasing energy in accordance to the
Aufbau Principle.
Now, it's very important to remember that when you're adding electrons to an atom, the 3d-orbitals, which are located on the third energy level, are higher in energy than the 4s-orbital.
This means that you must fill the 4s-orbital first, then distribute the rest of the electrons to the 3d-orbitals.
So, a neutral cobalt atom will have
$n = 1 \to {\text{2 e}}^{-}$in the $1 s$subshell
$n = 2 \to {\text{8 e}}^{-}$in the $2 s$and $2 p$subshells
Now, these two energy levels will hold
${\text{2 e"^(-) + "8 e"^(-) = "10 e}}^{-}$
Now comes the tricky part. The third energy level can hold ${\text{18 e}}^{-}$, so in theory it can hold the remaining
${\text{27 e"^(-) - "10 e"^(-) = "17 e}}^{-}$
that the neutral cobalt atom has. You could thus say that
$\textcolor{red}{\cancel{\textcolor{b l a c k}{n = 3 \to {\text{17 e}}^{-}}}}$in the $3 s$, $3 p$, and $3 d$subshells
and conclude that the electrons that surround the nucleus of a cobalt atom are spread out on $3$ energy levels. You would be wrong.
Taking it one subshell at a time, you will have
${\text{2 e}}^{-} \to$in the $3 s$subshell
${\text{6 e}}^{-} \to$in the $3 p$subshell
You now have
${\text{17 e"^(-) - ("2 e"^(-) + "6 e"^(-)) = "9 e}}^{-}$
to distribute. Because the 4s orbital is filled before the 3d-orbitals, the next two electrons are going to be distributed on the fourth energy level
$n = 4 \to {\text{2 e}}^{-}$in the $4 s$subshell
The remaining ${\text{7 e}}^{-}$ will now be distributed in the 3d-subshell.
Therefore, a neutral cobalt atom will have
$n = 1 \to {\text{2 e}}^{-}$in the $1 s$subshell
$n = 2 \to {\text{8 e}}^{-}$in the $2 s$and $2 p$subshells
$n = 3 \to {\text{15 e}}^{-}$in the $3 s$, $3 p$, and $3 d$subshells
$n = 4 \to {\text{2 e}}^{-}$in the $4 s$subshell
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Particles on Slopes (1)
There are several things that must be done for every particle that lies on a slope.
1)If you don't have a diagram, draw one.
2)Do not resolve vertically and horizontally. Resolve perpendicular and parallel to the slope. Remember that the reaction force is always perpendicular to the surface, that gravity always acts
downwards, and that friction always acts against the direction of motion.
The diagram below shows the particle on the point of slipping down the plane. Friction acts up the slope. The coefficient of friction is equal to
Resolving perpendicular the the slope gives
Resolving parallel to the slope gives
Dividing (2) by (1) gives | {"url":"https://astarmathsandphysics.com/a-level-maths-notes/m1/3575-particles-on-slopes-1.html?tmpl=component&print=1","timestamp":"2024-11-10T12:02:29Z","content_type":"text/html","content_length":"8290","record_id":"<urn:uuid:2d4b32c5-fa0e-46f8-ba44-d82d359c7da7>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00581.warc.gz"} |
FY2021 Annual Report
Qubits and Spacetime Unit
Assistant Professor Philipp Höhn
In the second year of the unit, the Japanese borders finally opened for postdocs and the first four arrived in-person at OIST between June and October 2021. This boosted unit activities, including
joint research projects as well as student supervision.
The research of the unit in FY21 focused on quantum reference frames, dynamical frames in generally covariant theories, edge modes and finite regions in gauge theories and gravity, and quantum
1. Staff
• Dr. Joshua Kirklin, Postdoctoral Scholar
• Dr. Isha Kotecha, Postdoctoral Scholar
• Dr. Fabio Maria Mele, Postdoctoral Scholar
• Dr. Stefan Eccles, Postdoctoral Scholar
• Mr. Snigdh Sabharwal, Rotation student (May-Aug, 2021)
• Joshua Carlo Casapao Aparicio, Rotation student (Sep-Dec, 2021)
• Tatiana Iakovleva, Rotation student (Sep-Dec, 2021)
• Julian De Vuyst, Rotation student (Jan-April 2022)
• Andreani Petrou, Rotation student (Jan-April 2022)
• Victor Castillo-Martinez, Research Intern (November 2021 - )
• Giovanni Natale, Research Intern (November 2021 - )
• Midori Tanahara, Administrative Assistant
2. Collaborations
2.1 Internal quantum reference frames for finite abelian groups
• Description: Preprint appeared in July 2021 (submitted for publication).
• Type of collaboration: Joint research
• Researchers:
□ Prof. Philipp Höhn, OIST
□ Dr. Markus Müller, Institute for Quantum Optics and Quantum Information, Vienna, Austria
□ Marius Krumm, Institute for Quantum Optics and Quantum Information, Vienna, Austria
2.2 Edge modes as reference frames and boundary actions from post-selection
• Description: Preprint appeared in September 2021 and is now published in JHEP.
• Type of collaboration: Joint research
• Researchers:
□ Dr. Sylvain Carrozza, University of Nijmegen, The Netherlands
□ Prof. Philipp Höhn, OIST
2.3 Perspective-neutral approach to quantum frame covariance for general symmetry groups
• Description: Preprint appeared in October 2021 (submitted for publication).
• Type of collaboration: Joint research
• Researchers:
□ Anne-Catherine de la Hamette, Institute for Quantum Optics and Quantum Information, Vienna, Austria
□ Dr. Thomas Galley, Perimeter Institute for Theoretical Physics, Waterloo, Canada
□ Prof. Philipp Höhn, OIST
□ Dr. Leon Loveridge, University of South-Eastern Norway
□ Dr. Markus Müller, Institute for Quantum Optics and Quantum Information, Vienna, Austria
2.4 The physical relevance of the fiducial cell in loop quantum cosmology
• Description: Preprint appeared in September 2021 (submitted for publication)
• Type of collaboration: Joint research
• Researchers:
□ Dr. Fabio Mele, OIST
□ Dr. Johannes Münch, CPT Marseille
2.5 Edge modes as dynamical frames: charges from post-selection in generally covariant theories
• Description: Preprint appeared in May 2022.
• Type of collaboration: Joint research
• Researchers:
□ Dr. Sylvain Carrozza, University of Nijmegen, The Netherlands
□ Dr. Stefan Eccles, OIST
□ Prof. Philipp Höhn, OIST
2.6 Diffeomorphism-invariant observables and dynamical frames in gravity: reconciling bulk locality with general covariance
• Description: Preprint appeared in June 2022.
• Type of collaboration: Joint research
• Researchers:
□ Dr. Christophe Goeller, LMU München, Germany
□ Prof. Philipp Höhn, OIST
□ Dr. Josh Kirklin, OIST
2.7 Quantm relativity of thermality
• Description: ongoing.
• Type of collaboration: Joint research
• Researchers:
□ Dr. Isha Kotecha, OIST
□ Dr. Fabio Mele, OIST
□ Prof. Philipp Höhn, OIST
2.8 Geometry from correlators in gauge theories
• Description: ongoing.
• Type of collaboration: Joint research
• Researchers:
□ Victor Castillo-Martinez, OIST
□ Prof. Philipp Höhn, OIST
2.9 Informational charges in systems of qubits
• Description: ongoing.
• Type of collaboration: Joint research
• Researchers:
□ Prof. Philipp Höhn, OIST
□ Giovanni Natale, OIST
□ Dr. Chris Wever, Bosch Quantum Computing
2.10 Relational Dynamics with periodic clocks
• Description: ongoing.
• Type of collaboration: Joint research
• Researchers:
□ Dr. Max Lock, Institute for Quantum Optics and Quantum Information, Vienna, Austria
□ Dr. Leonardo Chataignier, University of Bologna, Italy
2.11 Quantum covariant field theory
• Description: ongoing.
• Type of collaboration: Joint research
• Researchers:
□ Prof. Philipp Höhn, OIST
□ Andrea Russo, University College London, UK
□ Prof. Alex Smith, Dartmouth College & St. Anselm College, USA
3. Activities and Findings
We only describe activities and findings of projects completed (and published on arXiv) during FY21.
3.1 Internal quantum reference frames for finite abelian groups
Employing internal quantum systems as reference frames is a crucial concept in quantum gravity, gauge theories and quantum foundations whenever external relata are unavailable. In joint work with
Marius Krumm and Markus Müller from IQOQI Vienna, we gave a comprehensive and self-contained treatment of such quantum reference frames (QRFs) for the case when the underlying configuration space is
a finite Abelian group, significantly extending our previous work. The simplicity of this setup admits a fully rigorous quantum information-theoretic analysis, while maintaining sufficient structure
for exploring many of the conceptual and structural questions also pertinent to more complicated setups. We exploited this to derive several important structures of constraint quantization with
quantum information-theoretic methods and to reveal the relation between different approaches to QRF covariance. In particular, we characterized the "physical Hilbert space" - the arena of the
"perspective-neutral" approach - as the maximal subspace that admits frame-independent descriptions of purifications of states. We then demonstrated the kinematical equivalence and, surprising,
dynamical inequivalence of the "perspective-neutral" and the "alignability" approach to QRFs. While the former admits unitaries generating transitions between arbitrary subsystem relations, the
latter, remarkably, admits no such dynamics when requiring symmetry-preservation. We illustrated these findings by example of interacting discrete particles, including how dynamics can be described
"relative to one of the subystems".
Figure 1: Some finite abelian group (here \(\mathbb{Z}_n\)) assumes as double role: (1) as a discrete configuration space for a system \(S\), (2) as a translation group on that configuration space.
We considered internal quantum reference frames associated with such finite groups and derived a number or properties. For example, we showed that gauge-invariant states (living in the trivial
representation of the group) comprise the maximal subspace of states of \(S\) that admit frame-independent purifications.
Publication: Philipp A. Höhn, Marius Krumm, and Markus P. Müller, "Internal quantum reference frames for finite Abelian groups", arXiv:2107.07545
3.2 Edge modes as reference frames and boundary actions from post-selection
Together with Dr. Sylvain Carrozza from IMAPP Nijmegen, we introduced a general framework realizing edge modes in (classical) gauge field theory as dynamical reference frames, an often suggested
interpretation that we made entirely explicit. We focused on a bounded region \(M\) with a co-dimension one time-like boundary \(\Gamma\), which we embedded in a global spacetime. Taking as input a
variational principle at the global level, we developed a systematic formalism inducing consistent variational principles (and in particular, boundary actions) for the subregion \(M\). This relied on
a post-selection procedure on \(\Gamma\), which isolates the subsector of the global theory compatible with a general choice of gauge-invariant boundary conditions for the dynamics in \(M\).
Crucially, the latter relate the configuration fields on \(\Gamma\) to a dynamical frame field carrying information about the spacetime complement of \(M\); as such, they may be equivalently
interpreted as frame-dressed or relational observables. Generically, the external frame field keeps an imprint on the ensuing dynamics for subregion \(M\), where it materializes itself as a local
field on the time-like boundary \(\Gamma\); in other words, an edge mode. We identified boundary symmetries as frame reorientations and showed that they divide into three types, depending on the
boundary conditions, that affect the physical status of the edge modes. Our construction relied on the covariant phase space formalism, and is in principle applicable to any gauge (field) theory. We
illustrated it on three standard examples: Maxwell, Abelian Chern-Simons and non-Abelian Yang-Mills theories. In complement, we also analyzed a mechanical toy-model to connect our work with recent
efforts on (quantum) reference frames.
Figure 2: Configuration of finite spacetime subregion \(M\) inside the global piece \(M\cup \bar M\). \(M \) is delimited by the
timelike boundary \(\Gamma\) and partial Cauchy slices \(\Sigma_1\) and \(\Sigma_2\). \(\Gamma_0\) is some distant (possibly Figure 3: Different systems of paths shot in from \(\Gamma_0\)
asymptotic) boundary. Edge modes on the finite boundary \(\Gamma\) can be realized, for example, via Wilson lines shot in from \(\ yield different systems of Wilson lines and thus different edge
Gamma_0\). While from the point of view of the global theory, these edge modes are composite, non-local degrees of freedom, they mode frames on \(\Gamma\). We developed a method for changing
appear as "new" from the point of view of the subregion theory. They are group-valued degrees of freedom that constitute internal between the gauge invariant descriptions relative to different
reference frames associated with the structure group of the gauge theory. The edge-frame-dressed observables on \(\Gamma\) thereby choices of edge frame.
describe in a gauge-invariant manner how the subregion \(M\) relates to its complement \(\bar M\). Edge modes can be viewed as
"internalized" external reference frames for the subregion of interest.
Publication: Sylvain Carrozza and Philipp A. Höhn, "Edge modes a reference frames and boundary actions from post-selection", JHEP 02 72 (2022), arXiv:2109.06184
3.3 Perspective-neutral approach to quantum frame covariance for general symmetry groups
In the absence of external relata, internal quantum reference frames (QRFs) appear widely in the literature on quantum gravity, gauge theories and quantum foundations. In this project, we extended
the perspective-neutral approach to QRF covariance to general unimodular Lie groups together with four external collaborators. This is a framework that links internal QRF perspectives via a
manifestly gauge-invariant Hilbert space in the form of "quantum coordinate transformations", and we clarified how it is a quantum extension of special covariance. We modelled the QRF orientations as
coherent states which give rise to a covariant POVM, furnishing a consistent probability interpretation and encompassing non-ideal QRFs whose orientations are not perfectly distinguishable. We
generalized the construction of relational observables, establish a variety of their algebraic properties and equip them with a transparent conditional probability interpretation. We imported the
distinction between gauge transformations and physical symmetries from gauge theories and identified the latter as QRF reorientations. The "quantum coordinate maps" into an internal QRF perspective
were constructed via a conditioning on the QRF's orientation, generalizing the Page-Wootters formalism and a symmetry reduction procedure. We found two types of QRF transformations: gauge induced
"quantum coordinate transformations" as passive unitary changes of description and symmetry induced active changes of relational observables from one QRF to another. We revealed new effects: (i) QRFs
with non-trivial orientation isotropy groups can only resolve isotropy-group-invariant properties of other subsystems; (ii) in the absence of symmetries, the internal perspective Hilbert space
"rotates" through the kinematical subsystem Hilbert space as the QRF changes orientation. Finally, we invoked the symmetries to generalize the quantum relativity of subsystems before comparing with
other approaches.
Figure 4: distinction between gauge transformations (up) and symmetries as frame reorientations (down) for Figure 5: "Quantum coordinate transformations" mapping between two internal quantum frame
the rotation group. perspectives.
Figure 6: Symmetry-induced quantum frame transformations, changing from the relational observables relative to one frame to those relative to another. Relational observables define orbits in the
algebra of gauge-invariant observables and the frame transformations map between these orbits.
Publication: Anne-Catherine de la Hamette, Thomas D. Galley, Philipp A. Höhn, Leon Loveridge and Markus P. Müller, "Perspective-neutral approach to quantum frame covariance for general symmetry
groups", arXiv:2110.13824
3.4 The physical relevance of the fiducial cell in loop quantum cosmology
A common way to avoid divergent integrals in homogeneous spatially non-compact gravitational systems, as e.g. cosmology, is to introduce a fiducial cell by cutting-off the spatial slice at a finite
region \(V_0\). This is usually considered as an auxiliary integral regulator to be removed after performing computations by sending it to infinity. In a project with Johannes Münch, our postdoc
Fabio Mele analysed the dependence of the classical and quantum theory of homogeneous, isotropic and spatially flat cosmology on this fiducial cell. They showed that each fixed \(V_0\) regularisation
leads to a different canonically independent theory. At the classical level, the dynamics of observables is not affected by the regularisation choice on-shell. For the quantum theory, however, this
leads to a family of regulator dependent quantum representations labelled by \(V_0\) and the limit \(V_0\to\infty\) becomes then more subtle. First, they constructed a novel isomorphism between
different \(V_0\)-regularisations, which allowed them to identify states in the different \(V_0\)-labelled Hilbert spaces to ensure equivalent dynamics for any value of the regulator. The \(V_0\to\
infty\) limit would then correspond to choosing a state for which the volume assigned to the fiducial cell becomes infinite. As second main result of their analysis, by looking at observables
respectively smeared over the fiducial cell \(V_0\) and subregions \(V\), they found that quantum fluctuations of the latter explicitly depend on the size of the fiducial cell. Physically relevant
fluctuations for a finite region, as e.g. in the early time regime, would then be unreasonably suppressed in the limit where the volume of the fiducial cell becomes infinite. Their results suggest
that the fiducial cell is not playing the role of a mere regularisation but is physically relevant at the quantum level and complement previous statements in the literature based on different
Publication: Fabio Mele and Johannes Münch, "The Physical Relevance of the Fiducial Cell in Loop Quantum Cosmology", arXiv:2109.10663
4. Publications
4.1 Journals
The list includes preprints finished in FY20 and FY19, but published in FY21.
4.2 Books and other one-time publications
Nothing to report
4.3 Oral and Poster Presentations
1. Philipp Höhn, "Quantum frame covariance", March 17 2022, DAMTP Seminar, University of Cambridge (not recorded)
2. Philipp Höhn, "Progress in relational quantum dynamics", Nov 5 2021, Theory of Relativity Seminar, @ University of Warsaw (video)
3. Philipp Höhn, "Progress in relational quantum dynamics", Sep 1 2021, Time in Quantum Theory, Workshop @ ETH Zürich (recording)
4. Philipp Höhn, "(Quantum) frame covariance in gauge systems and gravity", Aug 16 2021, QASTM Seminar (video)
5. Philipp Höhn, "(Quantum) frame covariance: from foundations via gauge theories to gravity", June 16 2021, Quantizing Time, Workshop @ Perimeter Institute (video)
6. Philipp Höhn, "(Quantum) frame covariance in gauge systems and gravity", May 18 2021, Non-local Quantum Gravity Seminar, ENS Lyon (video)
7. Isha Kotecha, "Generalised Gibbs States and Application in Discrete Quantum Gravity", May 12 2021, Cosmology, Relativity and Gravitation (CRAG) seminar, University of Sheffield (online)
8. Philipp Höhn,
Discussion Panel Member "Observables in quantum gravity", April 8 2021, Quantum Gravity Across Approaches Initiative (
9. Josh Kirklin, "Uhlmann Phase, Black Hole Information and Holography", April 8 2021. QASTM online seminar series (
5. Intellectual Property Rights and Other Specific Achievements
Nothing to report
6. Meetings and Events
6.1 Online QUAST Seminars
Videos can be found here.
• Josiah Couch, Boston College - 29 Mar 2022. Pants Decomposition as Circuit Complexity in 2D (T)QFT
• Antony Speranza, University of Illinois Urbana-Champaign - 15 Mar 2022. Local gravitational charges and their algebra
• Jinzhao Wang, ETH Zurich - 7 Mar 2022. The black hole information puzzle and the quantum de Finetti theorem
• Lucas Hackl, University of Melbourne - 28 Feb 2022. Volume-law entanglement entropy of typical pure quantum states
• Stefan Eccles, OIST - 7 Feb 2022. Information spreading in chaotic quantum systems
• Phuc Nguyen, City University of New York and University of Haifa - 1 Feb, 2022. Scrambling and the black hole atmosphere
• Laurent Freidel, Perimeter Institute - 10 Dec, 2021. Local Holography: A quantum gravity paradigm to construct gravitational subsystems
• Robert Oeckl, UNAM - 7 Dec, 2021. Hands on with the positive formalism
• Slava Lysov, OIST - 15 Nov, 2021. Phase space on surface with the boundary via symplectic reduction
• Francesco Sartini, ENS Lyon - 8 Nov, 2021. Hidden symmetries in black holes
• Yasha Neiman, OIST - 27 Oct. A microscopic derivation of the quantum measurement postulates
• Fabio Anza, UC Davis - 19 Oct, 2021. Quantum Physics of Information
• Eduardo Testé, UCSB - 12 Oct, 2021. Mutual information superadditivity and unitarity bounds
• Djordje Racidevic, Brandeis - 4 Oct, 2021. An Introduction to the Lattice-Continuum Correspondence
• Juan Pedraza, University of Barcelona - 27 Sep, 2021. Lorentzian threads and holographic complexity
• Etera Livine, ENS Lyon - May 17th, 2021. Bulk-boundary in loop quantum gravity
• Ana Alonso-Serrano, AEI Potsdam - April 12th, 2021. Quantum gravity phenomenology from thermodynamics of spacetime
7. Other
Nothing to report. | {"url":"https://groups.oist.jp/ja/quast/fy2021-annual-report","timestamp":"2024-11-05T09:31:24Z","content_type":"text/html","content_length":"77004","record_id":"<urn:uuid:a4be2369-8dbb-4b04-b340-b589b3404ca8>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00710.warc.gz"} |
difftime - computes the difference between two calendar times
#include <time.h>
double difftime(time_t time1, time_t time0);
The difftime() function computes the difference between two calendar times.
The difftime() functions returns the difference (time1-time0) expressed in seconds as a double.
The difftime() function is provided because there are no general arithmetic properties defined for type time_t.
See attributes(7) for descriptions of the following attributes:
ATTRIBUTE TYPE ATTRIBUTE VALUE
Interface Stability Standard
MT-Level MT-Safe | {"url":"https://man.omnios.org/man3c/difftime","timestamp":"2024-11-08T11:38:31Z","content_type":"text/html","content_length":"4188","record_id":"<urn:uuid:f84df84d-28cd-4bf7-84f3-785ef5d4ff69>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00651.warc.gz"} |
47th PROBLEM OF EUCLID - What is the meaning of this Masonic Symbol?
47th Problem of Euclid
How To Square Your Square
The 47th Problem of Euclid, also called the 47th Proposition of Euclid, or the Pythagorean Theorem, is represented by what appear to be 3 squares.
To non-Freemasons, the 47th Problem of Euclid may be somewhat mysterious. Most wonder at the significance of this strange looking, 3-box symbol on a piece of Masonic jewelry.
Most Masonic books, simply describe it as "A general love of the Arts and Sciences". However, to leave its explanation at that would be to omit a subject which is very important... not only of
Pythagoras's Theory, but of the Masonic Square.
What Are These 3 Black Boxes and Why Are They Important to Freemasons?
We are told that Euclid, (the Father of Geometry), who lived several hundred years after Pythagoras, worked long and hard to solve the 3:4:5: ratio puzzle. It is said by some that he then sacrificed
a hecatomb (a sacrificial offering to God of up to 100 oxen or cattle). However, historically, it is believed that the Egyptians and Babylonians understood the mathematical usefulness of the 3:4:5
ratio long before Euclid.
The math is the key to understanding this symbol's broader and universal meaning.
The Pythagorean Theorem, also known as the
47th Problem of Euclid or 3:4:5:
"In any right triangle, the sum of the squares of the two sides is equal to the square of the hypotenuse." (the hypotenuse of a right triangle...which is the longest "leg"...or the 5 side of the
The Right Triangle, below, shows the sides of 3, 4 and 5. The angle created between the 3 (side) and the 4 (side) is the Right angle of the square.
A little later, when we begin to build it, (with sticks and string), you will place your sticks at the 3 corners of this Right triangle.
The square of 3 is 9.
The square of 4 is 16. The sum of 9 and 16 is 25. (25 represents the hypotenuse).
The square root of 25 is 5.
Therefore, the ratio is written: 3:4:5:
When we write down the square of the 1st four numbers (1, 4, 9 and 16), we see that by subtracting each square from the next one, we get 3, 5 and 7.
Ok, let's try it.
1, 4, 9, 16
4-1 =3
9-4 = 5
16-9 = 7
3:5:7: These are the steps in Masonry. They are the steps in the Winding Stair which leads to the Middle Chamber and they are the number of brethren which form the number of Master Masons necessary
to open a lodge of:
Master Mason: 3
Fellow Craft: 5
Entered Apprentice: 7
These are the sacred numbers.
OK, stay with me now...the major math is over.
The essence of the Pythagorean Theorem (also called the 47th Problem of Euclid) is about the importance of establishing an architecturally true (correct) foundation based on use of the square.
Why is this so important to speculative Masons who only have a symbolic square and not the actual square (the tool) of an operative Mason?
The 47th Problem of Euclid is the mathematical ratio (the knowledge) that allows a Master Mason to:
"Square his square when it gets out of square."
...I heard that! You're saying to yourself: "Why is that so important to ME in today's world...unless I'm a carpenter? Home Depot is only a few miles away."
How to Create a Perfect Square using the 47th Problem of Euclid
The knowledge of how to form a perfect square without the slightest possibility of error has been accounted of the highest importance in the art of building from the time of the Harpedonaptae, (and
before). Harpedonaptae, literally translated, means "rope stretchers" or "rope fasteners" of ancient Egypt (long before Solomon's Temple was built).
The Harpedonaptae were architectural specialists who were called in to lay out the foundation lines of buildings. They were highly skilled and relied on astronomy (the stars) as well as mathematical
calculations in order to form perfect square angles for each building.
In the Berlin museum is a deed, written on leather, dating back to 2000 B.C. (long before Solomon's time), which tells of the work of these rope stretchers.
Historically, a building's cornerstone was laid at the northeast corner of the building. Why in the northeast?
The ancient builders first laid out the north and south lines by observation of the stars and the sun...especially the North Star, (Polaris), which they believed at that time to be fixed in the sky.
Only after laying out a perfect North and South line could they use the square to establish perfect East and West lines for their foundations.
The 47th Problem of Euclid established those true East and West lines, so the rope stretchers could ascertain a perfect 90 degree angle to the North/South line which they had established using the
If you'd like to perform this yourself, it is actually quite easy...and once you get the necessary pieces together, would be a great "Show-and-Tell" educational instruction piece within your lodge.
The instructions are below, but it is easier to follow the instructions in a step-by-step manner (with string and sticks in hand) than it is to only read them for a complete understanding.
Better still, print numbers 1 through 4, below and then get your sticks and your string ready.
When you finish, you, too, will probably cry "Eureka!", ...just as I did.
The 47th Problem of Euclid
Unlike the Harpedonaptae, you have no way to establish true North and South...unless you use a compass. But a compass isn't necessary for this demonstration.
However, you WILL be able to create a perfect square...with only sticks and string, just as our ancestors did.
You will need 4 thin sticks which are strong enough to stick them into soft soil, 40 inches of string and a black magic marker. Actually, any length will work, but this size is very manageable.
The larger the foundation which the Mason wished to build, naturally, the longer his rope (string) would have to be.
1. Place your 1st stick flat on the ground so that its ends point north and south.
2. Next, take a string (it's much more unwieldy if you use rope) and tie knots in it, 3 inches apart. This will divide the string into 12 equal divisions.
3. Tie the 2 ends of the string together (this is your 12th knot) ...again ...remember that from knot-to-knot it must be 3 inches apart. The divisions between knots must be correct and equal or it
will not work.
4. Your string's total length is 36". After you've tied the end-to-end knot, you may cut off the excess 4" of string.
5. If you have more than 4" of string left or less than 4" of string left, you need to re-measure the lengths between your knots.
6. Your string is now circular in shape and has 12 knots and 12 divisions between the knots. (see the Right Triangle, again, below)
Note: The Operative Masons of old, used rope, however, because much of the length of the rope is within the knot, if you use rope, you must use a longer piece, measure each division, tie your knot,
and then measure your next 3 inch division before you cut the length of rope, instead of marking the entire rope while it is lying flat and then tying your knots.
1. Stab your 2nd stick in the ground near either end of your North/South stick and arrange a knot at the stick.
2. Stretch 3 divisions away from it in any direction (9 inches) and insert the 3rd stick in the ground, then...
3. Place the 4th stick so that it falls on the knot between the 4-part and the 5-part division (12 inches).
This forces the creation of a 3:4:5: right triangle. The angle between the 3 units and the 4 units is, of necessity, a square or right angle.
Now, move your 3rd and 4th sticks until they become a right angle (90 degrees) to your North/South stick.
Congratulations! You now have not only the ability to square your square, but to lay a geometrically correct cornerstone for your new foundation!
However, usage of the 47th Problem of Euclid doesn't end here...
Here is the rest of the story...
The Forty-Seventh Problem of Euclid
in Today's World
With this simple geometric 3:4:5 ratio of how to create a 90 degree, Right Angle:
1. Man can reach out into space and measure the distance of the stars...in light years!
2. He can survey land, mark off boundaries and construct every single thing on Earth.
3. He can build homes, churches and buildings, and with the knowledge of this simple ratio...he can begin digging on opposite sides of a mountain and dig a straight tunnel through the center of
it...that meets exactly at the center!
4. The 47th Problem of Euclid represents such a perfect symbol of Freemasonry...encompassing both art and science, that the simple knowledge of it demands a breathtaking awe to which we may only bow
our heads in reverence at the perfection, the universality and the infinite wisdom of that which has been given to us by God.
With the knowledge of this simple geometric ratio, (provided by the 47th Problem of Euclid), the word "Eureka!" almost palls in expressing the fundamental powers which our Creator has bestowed upon
...AND it all begins by simply learning how to Square your Square.
Oh!..., and one last thing you have also learned (but may not have realized it)...
This is why the old antique, wooden carpenter squares which you have seen or have heard about have one longer leg.
Their "legs" were created using the "3" and "4" part of the 3:4:5 ratio (the 5 is the hypotenuse) using the 47th Problem of Euclid.
Equal length "legs" on modern day (carpenter) squares are relatively "new" technology..
Now, take another look at the Masonic symbol for the 47th Problem of Euclid, above. You will see that the square on the top-left measures 3 units on each of its sides; the square on the top-right
measures 4 units on each of its sides and the bottom square measures 5 units on each of its sides.
You can now see the right triangle (white space in the middle) which is surrounded by the 3 "boxes".
From this day forward, when you see this graphic image denoting the 47th Problem of Euclid,...this Masonic symbol, it will not just look like 3 odd-shaped black boxes to you. You will see the 3:4:5
ratio and the square (right angle) within them and know that you have the power to square your square within your own Middle Chamber.
...And THAT is the Rest of the Story!
Euclid's 47th Proposition Masonic Pin...for sale on Amazon
Related Pages
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5. Masonic Books - Browse through a selection of Masonic books. | {"url":"https://www.masonic-lodge-of-education.com/47th-problem-of-euclid.html","timestamp":"2024-11-13T17:44:03Z","content_type":"text/html","content_length":"46116","record_id":"<urn:uuid:f2ddec49-de06-4cc0-bad8-47197c15131e>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00435.warc.gz"} |
Minmax pairwise approval - electowikiMinmax pairwise approvalMinmax pairwise approval
This article may require cleanup to meet electowiki's quality standards.
The following method (created by Forest Simmons, and called the Simmons method by Michael Ossipoff) is based on score or range style ballots. I believe it satisfies the favorite betrayal criterion,
Plurality, the chicken dilemma criterion, Monotonicity, Participation, Clone independence, and the IPDA. It reduces to ordinary Approval when only the extreme ratings are used for all candidates.
I call it MinMaxPairwiseApproval or MinMaxPA for short.
It is based on a concept of “pairwise approval.”
A zero to 100% cardinal ratings ballot contributes the following amount to the “pairwise approval of candidate X relative to candidate Y”:
The amount is either … 100% if X is rated strictly above Y, or Zero if X is rated strictly below Y, or Their common rating if they are rated equally.
According to this definition, the ballot’s contribution to the pairwise approval of X relative to itself is simply the ballot’s rating of X, since it is rated equally with itself.
The method elects the candidate whose minimum pairwise approval (relative to all candidates including self) is maximal.
The motivation for this idea is the question, “If candidates X and Y were the only two candidates with any significant chance of winning the election, what is the probability that the ratings ballot
voter would want X approved (in a Designated Strategy Voting system, say)?”
If the voter rated X over Y, this probability would be 100 percent. If the voter rated Y over X, this probability would be zero. If the voter rated both X and Y at 100 percent, this probability would
be 100 percent. If the voter rated them both at zero, she would want neither of the approved. If she rated them both at 50%, then our best guess is that there is a fifty-fifty chance that she would
approve X. Etc.
Whatever nice properties the method has depends solely on its definition, not the motivation for the definition, so please explore it with an open mind. | {"url":"https://electowiki.org/wiki/Minmax_pairwise_approval","timestamp":"2024-11-03T10:56:46Z","content_type":"text/html","content_length":"43193","record_id":"<urn:uuid:6c901327-6ce8-47e8-851f-9a09f102b345>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00613.warc.gz"} |
Statistics (12th Edition) Chapter 9 - Inferences Based on Two Samples - Exercises 9.1 - 9.29 - Applying the Concepts - Basic - Page 423 9.13a
Researchers are interested in measuring the difference of mean scores of a sample of 1,516 first-grade students in classrooms that used educational software as compared to a sample of 1,103
first-grade students in classrooms that did not use the technology. Mathematically: $µ_{1} - µ_{2}$ | {"url":"https://www.gradesaver.com/textbooks/math/statistics-probability/statistics-12th-edition/chapter-9-inferences-based-on-two-samples-exercises-9-1-9-29-applying-the-concepts-basic-page-423/9-13a","timestamp":"2024-11-03T12:19:58Z","content_type":"text/html","content_length":"73725","record_id":"<urn:uuid:6321c86f-2092-4140-a958-1bd656f0d9d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00704.warc.gz"} |
mp_arc 91-64
91-64 Craig W., Wayne C.E.
Nonlinear Waves and the KAM Theorem: Nonlinear Degeneracies (50K, TeX) Nov 21, 91
Abstract , Paper (src), View paper (auto. generated ps), Index of related papers
Abstract. This paper describes how one can use ideas related to the Nash-Moser implicit function theorem to construct solutions of nonlinear wave equations of the form: $$ u_{tt} = u_{xx} - g
(x,u) \qquad 0 < x < \pi,\; $$ We concentrate in this note on the case in which the nonlinear term $g(x,u)$, fails to satisfy a nondegeneracy condition which is the analogue of the twist
condition in Moser twist mapping theorem.
Files: 91-64.tex | {"url":"http://kleine.mat.uniroma3.it/mp_arc-bin/mpa?yn=91-64","timestamp":"2024-11-14T07:19:13Z","content_type":"text/html","content_length":"1471","record_id":"<urn:uuid:90db4f8c-6b68-44a5-a91e-f77fd399262a>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00616.warc.gz"} |
{"intvolume":" 164","oa":1,"year":"2016","page":"165 - 241","publisher":"Springer","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/
1310.7057"}],"issue":"1-2","volume":164,"_id":"1881","publication":"Probability Theory and Related
[{"call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"citation":{"ama":"Lee J,
Schnelli K. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 2016;164(1-2):165-241. doi:10.1007/s00440-014-0610-8","ieee":"J. Lee and K.
Schnelli, “Extremal eigenvalues and eigenvectors of deformed Wigner matrices,” Probability Theory and Related Fields, vol. 164, no. 1–2. Springer, pp. 165–241, 2016.","ista":"Lee J, Schnelli K. 2016.
Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 164(1–2), 165–241.","mla":"Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and
Eigenvectors of Deformed Wigner Matrices.” Probability Theory and Related Fields, vol. 164, no. 1–2, Springer, 2016, pp. 165–241, doi:10.1007/s00440-014-0610-8.","chicago":"Lee, Jioon, and Kevin
Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” Probability Theory and Related Fields. Springer, 2016. https://doi.org/10.1007/s00440-014-0610-8.","apa":"Lee, J., &
Schnelli, K. (2016). Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. Springer. https://doi.org/10.1007/s00440-014-0610-8","short":"J. Lee, K.
Schnelli, Probability Theory and Related Fields 164 (2016) 165–241."},"language":[{"iso":"eng"}],"type":"journal_article","acknowledgement":"Most of the presented work was obtained while Kevin
Schnelli was staying at the IAS with the support of\r\nThe Fund For Math.","ec_funded":1,"department":[{"_id":"LaEr"}],"date_published":"2016-02-01T00:00:00Z","doi":"10.1007/
s00440-014-0610-8","publist_id":"5215","title":"Extremal eigenvalues and eigenvectors of deformed Wigner matrices","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","abstract":
[{"lang":"eng","text":"We consider random matrices of the form H=W+λV, λ∈ℝ+, where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of
size N with i.i.d.\\ entries that are independent of W. We assume subexponential decay for the matrix entries of W and we choose λ∼1, so that the eigenvalues of W and λV are typically of the same
order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+∈ℝ+ such that the
largest eigenvalues of H are in the limit of large N determined by the order statistics of V for λ>λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N→∞ if λ>λ+.
Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ>λ+, while they are completely delocalized for λ<λ+. Similar
results hold for the lowest eigenvalues. "}],"oa_version":"Preprint","quality_controlled":"1","author":[{"first_name":"Jioon","full_name":"Lee, Jioon","last_name":"Lee"},
{"orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","last_name":"Schnelli","full_name":"Schnelli, Kevin"}],"corr_author":"1"} | {"url":"https://research-explorer.ista.ac.at/record/1881.jsonl","timestamp":"2024-11-12T10:54:50Z","content_type":"text/plain","content_length":"4446","record_id":"<urn:uuid:c70f1da6-9b52-4fee-97de-33e93111c4a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00674.warc.gz"} |
On minimum cutsets in independent domination vertex-critical graphs
Access Status
Fulltext not available
Ananchuen, N.
Ruangthampisan, S.
Ananchuen, W.
Caccetta, Louis
Ananchuen, N. and Ruangthampisan, S. and Ananchuen, W. and Caccetta, L. 2018. On minimum cutsets in independent domination vertex-critical graphs. The Australasian Journal of Combinatorics. 71 (3):
pp. 369-380.
Source Title
The Australasian Journal of Combinatorics
School of Electrical Engineering, Computing and Mathematical Science (EECMS)
© 2018, University of Queensland. All rights reserved. Let ? i (G) denote the independent domination number of G. A graph G is said to be k-? i -vertex-critical if ? i (G) = k and for each x ? V (G),
? i (G - x) < k. In this paper, we show that for any k-? i -vertex-critical graph H of order n with k = 3, there exists an n-connected k-? i -vertex-critical graph G H containing H as an induced
subgraph. Consequently, there are infinitely many non-isomorphic connected k-? i -vertex-critical graphs. We also establish a number of properties of connected 3-? i -vertex-critical graphs. In
particular, we derive an upper bound on ?(G-S), the number of components of G-S when G is a connected 3-? i -vertex-critical graph and S is a minimum cutset of G with |S| = 3.
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Deep BLSTM-GRU Model for Monthly Rainfall Prediction: A Case Study of Simtokha, Bhutan
Department of IT, College of Science and Technology, Chukkha 21101, Bhutan
Department of Computer Science & Engineering, Indian Institute of Technology Roorkee, Uttarakhand 247667, India
Department of IT Engineering, Sookmyung Women’s University, Seoul 04310, Korea
Author to whom correspondence should be addressed.
Submission received: 28 August 2020 / Revised: 20 September 2020 / Accepted: 22 September 2020 / Published: 28 September 2020
Rainfall prediction is an important task due to the dependence of many people on it, especially in the agriculture sector. Prediction is difficult and even more complex due to the dynamic nature of
rainfalls. In this study, we carry out monthly rainfall prediction over Simtokha a region in the capital of Bhutan, Thimphu. The rainfall data were obtained from the National Center of Hydrology and
Meteorology Department (NCHM) of Bhutan. We study the predictive capability with Linear Regression, Multi-Layer Perceptron (MLP), Convolutional Neural Network (CNN), Long Short Term Memory (LSTM),
Gated Recurrent Unit (GRU), and Bidirectional Long Short Term Memory (BLSTM) based on the parameters recorded by the automatic weather station in the region. Furthermore, this paper proposes a
BLSTM-GRU based model which outperforms the existing machine and deep learning models. From the six different existing models under study, LSTM recorded the best Mean Square Error (MSE) score of
0.0128. The proposed BLSTM-GRU model outperformed LSTM by 41.1% with a MSE score of 0.0075. Experimental results are encouraging and suggest that the proposed model can achieve lower MSE in rainfall
prediction systems.
1. Introduction
Rainfall prediction has a widespread impact ranging from farmers in agriculture sectors to tourists planning their vacation. Moreover, the accurate prediction of rainfall can be used in early warning
systems for floods [
] and an effective tool for water resource management [
]. Despite being of paramount use, the prediction of rainfall or any climatic conditions is extremely complex. Rainfall depends on various dependent parameters like humidity, wind speed, temperate,
etc., which vary from one geographic location to another; hence, one model developed for a location may not fit for another region as effectively. Generally, rainfall can be predicted using two
approaches. The first is by studying all the physical processes of rainfall and modeling it to mimic a climatic condition. However, the problem with this approach is that the rainfall depends on
numerous complex atmospheric processes which vary both in space and time. The second approach is using pattern recognition. These algorithms are decision tree, k-nearest neighbor, and rule-based
methods. For a large dataset, deep learning techniques are used to find meaningful results, and these techniques are based on the neural network. In this method, we ignore the physical laws governing
the rainfall process and predict rainfall patterns based on their features. This study aims to use pattern recognition to predict precipitation. The predictive models developed in this study are
based on deep learning techniques. We propose a Bidirectional Long Short Term Memory (BLSTM) and Gated Recurrent Unit (GRU)-based approach for monthly prediction and compare its results with the
state-of-the-art models in deep learning.
In this study, we predict rainfall over Simtokha, a region in the capital of Bhutan, Thimphu [
]. Although much work has been done on rainfall prediction using Artificial Neural Network (ANN) [
], particularly Multi-Layer Perceptron (MLP) in different countries, there is no existing literature on the application of ANN or Deep Neural Network (DNN) for the same purpose for any of the regions
in Bhutan. Weather parameters vary from region to region, and the parameters recorded also vary according to the weather stations. A model developed for one country or region does not fit for another
location as effectively.
The particular area was chosen as it is located in the capital of the country and serves as the primary station for the entire Thimphu. The region, although not prone to flooding, faces constant
water shortages due to ineffective water resource management. A more accurate beforehand knowledge of precipitation for the coming month will help the region to identify and mitigate water shortage
problems. This work also studies the predictive capability of different DNNs for predicting rainfall based on the parameters recorded by the weather stations in the country and will serve as a
baseline study. The dataset used in the study is the automatic weather station data collected from a station located in Simtokha.
Atmospheric models [
] are predominantly used for forecasting rainfall in Bhutan. Atmospheric models include atmospheric circulation models, climate models, and numerical models which simulate atmospheric operation and
predict rainfall. Currently, Numeric Weather Prediction (NWP) methods are the principal mode of forecasting rainfall in Bhutan. Numerical models employ a set of partial differential equations for the
prediction of many atmospheric variables such as temperature, pressure, wind, and rainfall. The forecaster based on his experience examines how the features predicted by the computer will interact to
produce the day’s weather.
This work focuses on the current state-of-the-art deep learning techniques to forecast rainfall over Simtokha. The contribution of our work is as follows:
• We proposed a hybrid framework of BLSTM and GRU for rainfall prediction.
• No prior deep learning techniques have been used on the dataset. The results of this paper will serve as the baseline for future researchers.
• A detailed analysis of the proposed framework is presented through extensive experiments.
• Finally, a comparison with different deep learning models is also discussed.
The rest of the paper is organized as follows. In
Section 2
, we discuss the existing research work in the rainfall prediction system. In
Section 3
, the proposed system implemented on the dataset is discussed.
Section 4
describes the experimental results and analysis. Finally, in the last
Section 5
the work is concluded along with a discussion of some future possibilities.
2. Literature Review
Prediction methods have come a long way, from relying on an individual’s experience to simple numeric methods to complex atmospheric models. Although machine learning algorithms like Artificial
Neural Network (ANN) have been utilized by researchers to forecast rainfall, studies on the effectiveness of existing deep learning models are limited, especially on data recorded by the sensors in a
weather station. Forecasting of rainfall can be conducted over a short time, such as predicting an hour or a day into the future, or over a long time such as monthly or a year ahead. A Neural Network
(NN) is a collection of neurons and multiple hidden layers, which work similar to a human brain. NNs are used to classify things and are based on the data. Recent surveys [
] show MLP as the most popular NN used for rainfall prediction.
Huang et al. [
] used 4 years of hourly data from 75 rain gauge stations in Bangkok and developed a NN to forecast 1–6 h rainfall on this data. Luk et al. [
] performed short-term (15 min) prediction using data collected from 16 gauges over the catchment area in western Sydney. Both research works recommended MLP over k-nearest neighbor, multivariate
adaptive regression splines, linear regression, and support vector regression. The study also highlighted the drop in prediction capability with an increase in lag order. Kashiwao et al. [
] compared MLP with an algorithm composed of random optimization, backpropagation, and Radial Bias Function Network (RBFN) to predict short-term rainfall on the data collected by the Japan
Meteorological Agency (JMA). The authors showed MLP performed better than RBFN.
Hernandez et al. [
] used a combination of autoencoder and MLP to predict the amount of rainfall for the next day using previous days’ records. The autoencoder was used to extract nonlinear dependencies of the data.
Their method outperformed other naive methods but had little improvement over MLP. Khajure et al. [
] used an NN and a fuzzy inference system. The weather parameters were predicted using an NN, and the predicted values were fed into the fuzzy inference system, which then predicted the rainfall
according to predefined fuzzy inference system rules. The authors concluded that a fuzzy inference system can be used along with an NN to achieve good prediction results. The effectiveness of a fuzzy
inference system for rainfall prediction was also reported by Wahyuni et al. [
Predicting monthly rainfall using MLP has shown more stable results compared to short-term prediction. Mishra et al. [
] used a feed-forward neural network (FFNN) to predict monthly rainfall over North India. Abhishek et al. [
] predicted monsoon precipitation for the Udupi district of Karnataka using three different learning algorithms: Back Propagation Algorithm (BPA), Layer Recurrent Network (LRN), and Cascaded Back
Propagation (CBP). The BPA showed lower mean squared error (MSE) compared to the other algorithms. Hardwinarto et al. [
] showed a promising result of BPNN for monthly rainfall using data from Tenggarong Station in Indonesia. Kumar and Tyagi [
] found RBFN outperformed BPNN while predicting rainfall for the Coonoor region of Tamil Nadu.
With the advancement in deep learning techniques, research work has been done to implement it in time series prediction. Recurrent neural networks (RNNs), in particular LSTM [
] and GRU [
], have found their niche in time series prediction. Zaytar et al. [
] used multi-stacked LSTM to forecast 24 h and 72 h of weather data, i.e., temperature, wind speed, and humidity. They used 15 years of hourly meteorological data from 2000–2015 of nine cities of
Morocco. The authors deduced deep LSTM networks could forecast the weather parameters effectively and suggested it for other weather-related problems. Salan et al. [
] used weather datasets from 1973 to 2009 provided by the Indonesian Agency for Meteorology, Climatology, and Geophysics to predict rainfall. The authors used a recurrent neural network for
prediction and obtained an accuracy score of 84.8%. Qie et al. [
] used multi-task CNN to predict short-term precipitation using weather parameters collected from multiple rain gauges in China. The authors concluded that the multi-site [
] features gave better results than single-site features. A summary of the literature review is shown in
Table 1
3. Proposed System
In this section, we describe the different steps and components of the proposed system. The proposed deep learning model consists of a BLSTM, GRU, and Dense layer as shown in
Figure 1
3.1. Dataset Description
Bhutan is a small Himalayan country landlocked between India to the south and China to the north, as shown in
Figure 2
. The sensor data used in this work were collected from a weather station located in Simtokha [
], Thimphu, which is the 4th highest capital in the world by altitude, and the range varies from 2248 to 2648 m. The station at Simtokha is the sole station to record class A data for the capital.
The station is located at 89.7 longitude and 27.4 latitude at an elevation of 2310 m. The data for this study were obtained from NCHM (
), which provides two classes of data to researchers: class A and class C datasets. Class A datasets are recorded by automatic weather stations, and class C datasets are recorded manually by
designated employees at different stations. Class A datasets are, hence, more reliable and were used in this work. The selected dataset contains daily records of weather parameters from 1997 to 2017,
as shown in
Figure 3
. The records from 1997–2015 were used to train the different models, and 2016–2017 data were used for testing. Six weather parameters described in
Table 2
were used for this study. These parameters had either zero or very few missing values that were handled during data preprocessing. The monthly weather parameters were extracted from daily records by
taking the mean of tmax, tmin, relative_humidity, wind_speed, and wind_direction. The number of sunshine hours and rainfall amount in a month were deduced by taking the sum of daily sunshine hours
and daily rainfall values, respectively.
3.2. Data Preprocessing
The daily records of weather parameters from 1997 to 2017 were collected from NCHM. The raw data originally contained eight parameters, but some of the parameters contained a lot of missing and noisy
values. The weather parameters that contained a lot of empty records were dropped from the dataset. The dataset also had different random representations for the null value, which was standardized
during preprocessing. The preprocessing step is as shown in
Figure 4
. The missing values in the selected parameters were resolved by taking the mean of all the values occurring for that particular day and month. For example, if the sunshine_hours record for 1 January
2000 was missing, it was filled by the mean of other sunshine_hours records on 1 January for other years. Outliers are records that significantly differ from other observed values. The outliers were
detected using a box-and-whisker plot as well as the k-means clustering algorithm [
] and were resolved using the mean technique. Weather parameters were normalized using a min-max scaler to get the new scaled value
$z = x − m i n ( x ) m a x ( x ) − m i n ( x )$
are the minimum and maximum value, respectively.
is the value to be scaled. After preprocessing, the data are reshaped into a tensor format for DNN models. The input for the LSTM layer must have a 3D shape. The three dimensions of the input are
samples, time steps, and sample dimension. One sequence is considered as one sample, one point of observation in the sample is one time step, and one feature is a single point of observation at the
time step. In our experiment one sample is made up of 12 time steps (12 months), and in each time step (month) there are parameters like average maximum temperature, average sunshine hours, etc.
3.3. Evaluation Metrics
The study used both qualitative and quantitative metrics to calculate the performance of different models. The formulae for RMSE, MSE, Pearson Correlation Coefficient, and
$R 2$
were used as a scoring function, as in
Table 3
From the above, $x i$ is the model simulated monthly rainfall, $y i$ is the observed monthly rainfall, $x ¯$ and $y ¯$ are their arithmetic mean, and n is the number of data points.
3.4. BLSTM
LSTM is the most popular model in time series analysis, and there are many variants such as unidirectional LSTM and BLSTM. For our study, the Many-to-One (multiple input and one output) variation of
LSTM [
] was used to take the last 12 months’ weather parameters and predict the rainfall for the next month, as shown in
Figure 5
. Unidirectional LSTM process data are based on only past information. Bidirectional LSTM [
] utilizes the most out of the data by going through time-steps in both forward and backward directions. It duplicates the first recurrent network in the architecture to get two layers side by side.
It passes the input, as it is to the first layer and provides a reversed copy to the second layer. Although it was traditionally developed for speech recognition, its use has been extended to achieve
better performance from LSTM in multiple domains [
]. An architecture consisting of two hidden layers with 64 neurons in the first layer and 32 neurons in the second layer recorded the best result on the test dataset, with MSE value of 0.01, a
coefficient value of 0.87, and
$R 2$
value of 0.75.
3.5. GRU
The Gated Recurrent Unit was developed by Cho et al. [
] in 2014. GRU performances on certain tasks of natural language processing, speech signal modeling, and music modeling are similar to the LSTM model. The GRU model has fewer gates compared to LSTM
and has been found to outperform LSTM when dealing with smaller datasets. To solve the vanishing gradient problem of a standard RNN, GRU consists of an update and reset gate, but unlike the LSTM it
lacks a dedicated output gate. The update gate decides how much of the previous memory to keep, and the reset gate determines how to combine the previous memory with the new input. Due to fewer
gates, they are computationally less demanding compared to LSTM and are ideal when there are limited computational resources. GRU with two hidden layers consisting of 12 neurons in the first layer
and 6 neurons in the second outperformed other architectures, with an MSE score of 0.02, a correlation value of 0.83, and
$R 2$
value of 0.66.
3.6. BLSTM-GRU Model
In this model, preprocessed weather parameters are fed into the BLSTM layer with 14 neurons. This layer reads data in both forward and backward directions and creates an appropriate embedding. Batch
normalization is performed on the output of the BLSTM layer to normalize the hidden embedding before passing it to the next GRU layer. The GRU layer contains half the number of neurons as the BLSTM
layer. The GRU layer has fewer cells and is able to generalize the embedding with relatively lower computation cost. The data are again batch normalized before sending to the final dense layer. The
final layer has just one neuron with a linear activation function, and it outputs the predicted value of monthly rainfall for T + 1 (next month), where T is the current month.
For our study, the Many-to-One (multiple input and one output) variation of LSTM [
] was used to take the last 12 months’ weather parameters and predict the rainfall for the next month, as shown in
Figure 5
. The activation function used in both BLSTM and GRU is the default tanh function, and the optimizer used was Adam. The architecture was fixed after thoroughly hyper-tuning the parameters.
Hyperparameter tuning was performed through a randomized grid search and heuristic knowledge of the programmer.
4. Experiment and Results
The models were created in python on the Jupyter notebook using Keras (
) deep learning API with Tensorflow [
] back-end. All the experiments were run for 10,000 epochs, but by using callbacks in Keras only the best weight for each test run was saved. Although 10,000 epochs were not needed most of the time,
smaller architectures with few neurons took considerably more time to learn as compared to neuron-rich networks. Multiple experiments were conducted with varying architecture for each model under
study. Early stopping [
] with a large patience value was used to prevent unnecessary overfitting.
4.1. Result Summary
The best MSE and RMSE scores of each model are highlighted in
Figure 6
. The NNs outperformed linear regression by a huge margin. LSTM and GRU outperformed MLP by a huge margin, as they were able to utilize the 12 time-steps of input properly. The plots between
predicted and the actual values for 24 months from January 2016 to December 2017 are shown in
Figure 7
The proposed model performed uniformly better than vanilla versions of all the deep learning techniques under study. The MSE score of 0.01 achieved by our model was 41.1% better compared to the next
best score of 0.13 provided by LSTM.
4.2. Comparative Analysis
We have also compared our system with MLP, LSTM, CNN [
], and other methods on the NCHM dataset as shown in
Figure 8
Figure 9
. The dataset did not have a baseline score to overcome. The linear regression RMSE score of 0.217 and MSE score of 0.047 were used as the baseline score.
Each input sample has 12 time-steps, and the output is the total amount of rainfall for the next month (t + 1). Each timestep contains the weather features of a particular month. For example, the
timestep T(n) contains all the weather parameters for the nth month. From
Figure 8
Figure 9
it is evident that, among the vanilla models, LSTM with 1024 neurons performed the best with a MSE score of 0.013, a correlation value of 0.90, and
$R 2$
value of 0.78. The proposed BLSTM-GRU model outperformed LSTM on all four performance matrices, with MSE, RMSE,
$R 2$
, and correlation coefficient values of 0.0075, 0.087, 0.870, and 0.938 respectively.
5. Conclusions and Future Work
The study of deep learning methods for rainfall prediction is presented in this paper, and a BLSTM-GRU based model is proposed for rainfall prediction over the Simtokha region in Thimphu, Bhutan. The
sensor data are collected from the meteorology department of Bhutan, which contain daily records of weather parameters from 1997 to 2017. The records from 1997–2015 are used for training machine
learning and deep learning models, and for testing we used 2016–2017 data. According to sensor data, the traditional MLP (the results on the Simtokha region dataset, i.e., 0.029 MSE, 0.71
correlation, and $R 2$ value of 0.50), which is widely used for rainfall prediction, did not perform well in comparison to the recent deep learning models on weather station data. Vanilla versions of
LSTM, GRU, BLSTM, and 1-D CNN performed similarly, with a single-layered LSTM consisting of 1024 neurons performing better than the others, with MSE score of 0.013, a correlation value of 0.90, and
$R 2$ value of 0.78. Finally the combination of BLSTM and GRU layers performed much better than all the other models under study for this dataset. Its MSE score of 0.007 was 41.1% better than LSTM.
Furthermore, the proposed model presented an improved correlation value of 0.93 and $R 2$ score of 0.87. Predicting actual rainfall values has become more challenging due to the changing weather
patterns caused by climate change.
In the future, we aim to improve the performance of our prediction model by incorporating patterns of global and regional weather such as sea surface temperature, global wind circulation, etc. We
also intend to explore the predictive use of climate indices and study the effects of climate change on rainfall patterns.
Author Contributions
All authors have contributed to this paper. M.C. proposed the main idea in consultation with P.P.R. M.C., S.K. and P.P.R. were involved in methodology and M.C. performed the experiments. M.C. and
S.K. drafted the initial manuscript. P.P.R. and B.-G.K. contributed to the final version of the paper. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
The authors thank the National Center of Hydrology and Meteorology department, Bhutan, for providing the data to conduct this research.
Conflicts of Interest
The authors declare that they have no conflicts of interest in this work.
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Figure 1. The proposed model is composed of 7 layers including the input and output layers. The embedding is generated by the Bidirectional Long Short Term Memory (BLSTM) and Gated Recurrent Unit
(GRU) layer. The batch normalization is used for normalizing the data, and the dense layer performs the prediction.
Figure 2. Map of Bhutan showing major river basins and the annual precipitation (mm). The study area is indicated in the legend.
Figure 4. Data preprocessing. The data are preprocessed in 6 stages with the arrowheads showing the flow of data.
Figure 5. Many (12) to One LSTM utilized in the experiment. Each sample of data contains 12 time-steps of previous data. We used 12 months of previous data to predict the rainfall of the next month
(n + 1).
Figure 6. RMSE and MSE values of 6 existing models including linear regression and the proposed model. It is clear from the figure that our model outperformed the existing models for the same task.
Figure 7. The plots of actual monthly rainfall values over Simtokha collected from NCHM and predicted rainfall values for the years 2016 and 2017, where the x-axis and y-axis represent months and
monthly rainfall values (scaled) respectively. The blue line shows the actual values and the orange line shows the predicted values. Subfigure ‘a’ shows that CNN is not able to predict the peak
monthly rainfall values correctly. Results of recurrent neural networks shown by subfigure ‘b’, ‘c’ and ‘d’ are better than that of MLP (subfigure ‘e’). Subfigure ‘f’ shows that the proposed model is
able to generalize better and gives the best output.
Figure 9. Pearson Correlation Coefficient values of 5 existing deep learning models and our proposed model. The score of the proposed model was the highest among the models.
Region Daily- Rainfall Accuracy
Author & Year (Global or Local) Monthly- Yearly Types of NN Predicting Measure
Luk et al. [12] Western 15 min MLFN, PRNN, NA NMSE
Sydney rainfall prediction TDNN
4 years of Efficiency
Huang et al. [11] Bangkok hourly data MLP and FFNN NA index (EI)
from 1997–2003
Abhishek et al. [4] Karnataka, 8 months of data BPFNN, BPA, Average humidity and MSE
India from 1960 to 2010 LRN and CBP average wind speed
Nayak et al. [10] Survey Paper NA ANN NA NA
Darjee et al. [5] Survey Paper Monthly, ANN, (FFNN, Maximum and minimum NA
Yearly RNN, TDNN) temperatures
East Data used
Hardwinarto et al. [18] Kalimantan- from 1986–2008 BPNN NA MSE
Daily records NN and a fuzzy Temperature, humidity,
Khajure et al. [15] NA for 5 years inference system dew point, visibility, MSE
pressure and windspeed.
Monthly rain-
Kumar and Tyagi [19] Nilgiri district fall prediction BPNN, RBFNN NA MSE
Tamil Nadu, India (Data from
Wahyuni et al. [16] Tengger Data used BPNN Changes caused by RMSE
East Java from 2005 to 2014 climate change
Rainfall data Atm. pressure,
from the in- ANN precipitation, Validation
Kashiwao et al. [13] Japan ternet as \“big MLP and RBFN humidity, temp., using JMA.
data” was used. vapor pressure,
wind, velocity.
North India Rainfall records Regression
Mishra et al. [17] North India for the period FFNN of previous analysis,
1871–2012. 2 months and MRE and MSE
current month
Rainfall Parameters Units
Maximum Temperature ($t m a x$) $∘$C
Minimum Temperature ($t m i n$) $∘$C
Rainfall Millimeters (mm)
Relative Humidity Percentage (%)
Sunshine Hours (h)
Wind Speed Meters per second (m/s)
Name Formula
MSE $1 n ∑ t = 1 n ( x i − y i ) 2$
RMSE $1 n ∑ t = 1 n ( x i − y i ) 2$
$R 2$ $1 − ∑ i = 0 n samples − 1 ( x i − y i ) 2 ∑ i = 0 n samples − 1 ( x i − y ¯ ) 2$
Correlation $∑ i = 1 n ( x i − x ¯ ) ( y i − y ¯ ) ∑ i = 1 n ( x i − x ¯ ) 2 ∑ i = 1 n ( y i − y ¯ ) 2$
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Chhetri, M.; Kumar, S.; Pratim Roy, P.; Kim, B.-G. Deep BLSTM-GRU Model for Monthly Rainfall Prediction: A Case Study of Simtokha, Bhutan. Remote Sens. 2020, 12, 3174. https://doi.org/10.3390/
AMA Style
Chhetri M, Kumar S, Pratim Roy P, Kim B-G. Deep BLSTM-GRU Model for Monthly Rainfall Prediction: A Case Study of Simtokha, Bhutan. Remote Sensing. 2020; 12(19):3174. https://doi.org/10.3390/
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Chhetri, Manoj, Sudhanshu Kumar, Partha Pratim Roy, and Byung-Gyu Kim. 2020. "Deep BLSTM-GRU Model for Monthly Rainfall Prediction: A Case Study of Simtokha, Bhutan" Remote Sensing 12, no. 19: 3174.
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För att på det avståndet kunna se en planet behövs en upplösning på som ett 300 miljoner km stort teleskop skulle ha (jordbanans diameter).
The Sun; Mercury; Venus; Earth/Moon; Mars; Jupiter; Saturn; Uranus; Neptune; Pluto/ Diameter: 71 m (solar disk) + the corona (the Sun's outer atmosphere).
We can use Eratosthenes' shadow experiment to determine the diameter of the flat earth. Syene and Alexandria are two North-South points with a distance of 500 miles. However, Earth does have a
diameter to measure. It is 7,898 miles long if you measured it vertically. If you took a tape measure and measured it all the way around the equator, it would be over 24,000 miles long. To calculate
the length of time it would take to walk the diameter of the Earth, it would take quite a long time. 2014-06-12 · The Earth has a liquid core and a solid crust which is broken into several plates
which slowly move along the surface of the planet.
The Earth is widest at the equator. The diameter of the Earth at the poles is 7,899.8 miles. There is an equatorial bulge of 27 miles due to the rotation of the Earth. As a result, the latest
measurements indicate that the Earth has an equatorial diameter of 12,756 km (7926 mi), and a polar diameter of 12713.6 km (7899.86 mi). In short, objects located along the The equatorial diameter of
the Earth is 12,756 km. This is the diameter of the Earth measured from one side of the Earth, passing through the center.
He was actually measuring the diameter of the flat earth (distance across), which is a figure identical to the circumference of the round earth (distance around). We can use Eratosthenes' shadow
experiment to determine the diameter of the flat earth. Syene and Alexandria are two North-South points with a distance of 500 miles.
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Today, I’d like to tell you a story of one of these methods that dates back more than 2,200 years. It’s the story of a brilliant Greek astronomer and mathematician named Eratosthenes and his efforts
to measure the size of the Earth. How did he do it? Let's find out. How to Measure Short Distances
It’s the story of a brilliant Greek astronomer and mathematician named Eratosthenes and his efforts to measure the size of the Earth.
Posts about diameter of the earth written by hhotpick. Ken Jennings s Trivia Almanac is the ingeniously organized book where, for a change, the all-time Jeopardy! champ gets to ask the questions and
where every day of the year will give you the chance to test your trivia mettle. Earth's polar radius is 3,949 miles (6,356 km) — a difference of 14 miles (22 km). Using those measurements, the
equatorial circumference of Earth is about 24,900 miles (40,070 km). However, from pole-to-pole — the meridional circumference — Earth is only 24,812 miles (39,931 km) around.
the earth's atmosphere and hit at Lonar creating 10 kms diameter debris-dust. Diameter: 22mmbredd: 78. Styre svart Honda MT 22mm diameter Styrkontroll för ljus, blinkers och tuta - universal;
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Diameter Of Earth In Kilometers. By Hilman Rojak | March 6, 2017. 0 Comment. What is the diameter of earth quora what is the diameter of earth how big is pluto the density m ppt earth and the moon
powerpoint what is the diameter of earth quora. What Is The Diameter Of Earth Universe Today.
Its features include pleasant temperatures and habitats for life. In the range of solar system worlds, Earth is the only known home to life.
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The earth, as we said, is an oblate spheroid, so geodesists (scientists who measure the earth and its position in space) have measured and calculated the average of earth's circumference as 40,075 km
or 24,901 mi. Eratosthenes, some 2,200 years ago, was off by only 125 km or 101 mi.
What Is The Diameter Of Earth Universe Today. Planet Earth has an abundance of water that makes it unique and perfect for life to exist. Even today we would assume the Earth to be perfectly spherical
in some cases for mathematical purposes in order to get an approximation of the radius and circumference of the Earth, as I demonstrated above by deriving an accurate circumference measurement from
NASA’s value for the Earth’s diameter. We have to assume the Earth is a sphere to The diameter measured from the level of the poles to the end point of the atmosphere, the one drawn from the equator
to the end point of the atmosphere, and the diameters in between differ also. Tags: Diameter, Rahman, confines of the space, diameters of the earth and space GEOIDAL FORM OF … The ratio of the Sun's
diameter to the Earth's diameter is 1,392,000/12756 = 109.1 This means the ratio of their volumes is 109.1 x 109.1 x 109.1, which is about 1,300,000,and that means that 1,300,000 Earths should fit
inside the Sun. But hang on a minute.. Doubling that figure for the diameter we get a figure of 25,000 miles. The earth is physically much larger, of course. | {"url":"https://affarerpzlldfz.netlify.app/27188/99581","timestamp":"2024-11-05T13:55:31Z","content_type":"text/html","content_length":"9478","record_id":"<urn:uuid:4b84f444-edbd-410b-afa4-8e74b3687275>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00617.warc.gz"} |
Maryland Standards
MD.1.0. Knowledge of Algebra, Patterns, and Functions: Students will algebraically represent, model, analyze, or solve mathematical or real-world problems involving patterns or functional
1.A.1. Patterns and Functions: Identify, describe, extend, and create numeric patterns and functions.
1.A.1.a. Represent and analyze numeric patterns using skip counting (Assessment limit: Use patterns of 3, 4, 6, 7, 8, or 9 starting with any whole number (0 - 100)).
PatternsA pattern is an order of things repeated over and over. Read more...iWorksheets :6Study Guides :1
Skip CountingYou can skip count by large numbers such as 25, 50 or 100. Skip counting allows you to count by large numbers following a pattern. Read more...iWorksheets :3Study Guides :1
PatternsA pattern is a recognizable, consistent series of numbers, shapes, or images. Read more...iWorksheets :4Study Guides :1
1.A.2. Identify, describe, extend, analyze, and create a non-numeric growing or repeating pattern.
1.A.2.a. Generate a rule for the next level of the growing pattern (Assessment limit: Use at least 3 levels but no more than 5 levels).
PatternsA pattern is an order of things repeated over and over. Read more...iWorksheets :6Study Guides :1
PatternsA pattern is a recognizable, consistent series of numbers, shapes, or images. Read more...iWorksheets :4Study Guides :1
1.A.2.b. Generate a rule for a repeating pattern (Assessment limit: Use no more than 4 objects in the core of the pattern).
PatternsA pattern is an order of things repeated over and over. Read more...iWorksheets :6Study Guides :1
PatternsA pattern is a recognizable, consistent series of numbers, shapes, or images. Read more...iWorksheets :4Study Guides :1
1.A.2.c. Create a non-numeric growing or repeating pattern.
PatternsA pattern is an order of things repeated over and over. Read more...iWorksheets :6Study Guides :1
PatternsA pattern is a recognizable, consistent series of numbers, shapes, or images. Read more...iWorksheets :4Study Guides :1
1.B.1. Expressions, Equations, and Inequalities: Write and identify expressions.
1.B.1.a. Represent numeric quantities using operational symbols (+, - , x, / with no remainders) (Assessment limit: Use whole numbers (0 - 100)).
Evaluate Open SentencesAlgebra is a study of the properties of operations on numbers. Algebra generalizes math by using symbols or letters to represent numbers. Read more...iWorksheets :3Study Guides
:1Vocabulary :1
Open Number SentencesWhat Are Open Number Sentences? Open number sentences are equations that give one part of the equation along with the answer. In order to solve an open number sentence, the
inverse operation is used. Read more...iWorksheets :3Study Guides :1
1.B.2. Expressions, Equations, and Inequalities: Identify, write, solve, and apply equations and inequalities.
1.B.2.a. Represent relationships using relational symbols (>, <, =) and operational symbols (+, -, x, / with no remainders) on either side (Assessment limit: Use operational symbols (+, -, x) and
whole numbers (0 - 200)).
Evaluate Open SentencesAlgebra is a study of the properties of operations on numbers. Algebra generalizes math by using symbols or letters to represent numbers. Read more...iWorksheets :3Study Guides
:1Vocabulary :1
AlgebraComparing whole numbers, fractions, and decimals means looking at the values of two numbers and deciding if they are greater than, less than or equal to each other. Read more...iWorksheets :4
Study Guides :1
Ordering DecimalsWhen putting decimals in order from least to greatest, we must look at the highest place value first. Read more...iWorksheets :7Study Guides :1Vocabulary :1
Compare and Order FractionsWhen comparing two fractions that have a common denominator, you can looks at the numerators to decide which fraction is greater Read more...iWorksheets :4Study Guides :1
Vocabulary :1
Positive & Negative IntegersPositive integers are all the whole numbers greater than zero. Negative integers are all the opposites of these whole numbers, numbers that are less than zero. Zero is
considered neither positive nor negative Read more...iWorksheets :4Study Guides :1
Ordering FractionsA fraction consists of two numbers separated by a line - numerator and denominator. To order fractions with like numerators, look at the denominators and compare them two at a time.
The fraction with the smaller denominator is the larger fraction. Read more...iWorksheets :3Study Guides :1
Fractions/DecimalsHow to convert fractions to decimals: Divide the denominator (the bottom part) into the numerator (the top part). Read more...iWorksheets :3Study Guides :1
Greater Than/Less ThanIf a number is greater than another number that means it is higher in value than the other number. If a number is less than another number that means it is lower in value than
the other number. Read more...iWorksheets :4Study Guides :1
Comparing FractionsWhen comparing fractions, you are finding which fraction is greater and which fractions is less than the other. Similar to comparing numbers, there are symbols to use when
comparing fractions. Read more...iWorksheets :5Study Guides :1Vocabulary :1
Open Number SentencesWhat Are Open Number Sentences? Open number sentences are equations that give one part of the equation along with the answer. In order to solve an open number sentence, the
inverse operation is used. Read more...iWorksheets :3Study Guides :1
Greater Than/Less ThanWhat Is Greater Than and Less Than? When a number is greater than another number, this means it is a larger number. The symbol for greater than is >. When a number is less than
another number, this means it is a smaller number. The symbol for less than is <. Read more...iWorksheets :6Study Guides :1
DivisionFreeWhat Is Division? Division is an operation that tells: how many equal sized groups, how many in each group. The number you divide by is called the DIVISOR. The number you are dividing is
called the DIVIDEND. And the answer is called the QUOTIENT. Read more...iWorksheets :6Study Guides :1
FractionsThe top number of a fraction is called the numerator. It shows how many pieces of a whole we are talking about. The bottom number is called the denominator. It shows how many pieces an
object was divided into, or how many total pieces we have. Read more...iWorksheets :4Study Guides :1
1.B.2.b. Find the unknown in an equation with one operation (Assessment limit: Use multiplication (x) and whole numbers (0 - 81)).
Evaluate Open SentencesAlgebra is a study of the properties of operations on numbers. Algebra generalizes math by using symbols or letters to represent numbers. Read more...iWorksheets :3Study Guides
:1Vocabulary :1
Open Number SentencesWhat Are Open Number Sentences? Open number sentences are equations that give one part of the equation along with the answer. In order to solve an open number sentence, the
inverse operation is used. Read more...iWorksheets :3Study Guides :1
1.C.1. Numeric and Graphic Representations of Relationships: Locate points on a number line and in a coordinate grid.
1.C.1.a. Represent mixed numbers and proper fractions on a number line (Assessment limit: Use proper fractions with a denominator of 6, 8, or 10).
Number LineA number line is a line that shows any group of numbers in their least to greatest value. Read more...iWorksheets :3Study Guides :1
1.C.1.b. Identify positions in a coordinate plane (Assessment limit: Use the first quadrant and ordered pairs of whole numbers (0 - 20)).
Plot PointsYou use plot points to place a point on a coordinate plane by using X and Y coordinates to draw on a coordinate grid. Read more...iWorksheets :5Study Guides :1Vocabulary :1
CoordinatesYou can use a pair of numbers to describe the location of a point on a grid. The numbers in the pair are called coordinates. Read more...iWorksheets :3Study Guides :1
Graphs and TablesUsing tables and graphs is a way people can interpret data. Data means information. So interpreting data just means working out what information is telling you. Information is
sometimes shown in tables, charts and graphs to make the information easier to read. Read more...iWorksheets :3Study Guides :1
MD.2.0. Knowledge Geometry: Students will apply the properties of one-, two-, or three-dimensional geometric figures to describe, reason, or solve problems about shape, size, position, or motion of
2.A.1. Plane Geometric Figures: Analyze the properties of plane geometric figures.
2.A.1.a. Identify properties of angles using manipulatives and pictures.
Lines and AnglesAcute angle: An angle whose measure is less than 90; Right angle: An angle that measures 90; Obtuse angle: An angle whose measure is more than 90 and less than 180; Straight angle: An
angle that measures 180; Reflex angle: An angle whose measure is more than 180 and less than 360. There are 3 sets of lines: Intersecting, Perpendicular and Parallel. Read more...iWorksheets :12Study
Guides :2Vocabulary :2
AnglesA right angle is an angle that measures 90°. A straight angle is an angle that measures 180°. An obtuse angle is an angle that measures more than 90°. An acute angle is an angle that measures
less than 90°. Read more...iWorksheets :10Study Guides :1
2.A.1.b. Identify, compare, classify and describe angles in relationship to another angle (Assessment limit: Use acute, right, or obtuse angles).
Lines and AnglesAcute angle: An angle whose measure is less than 90; Right angle: An angle that measures 90; Obtuse angle: An angle whose measure is more than 90 and less than 180; Straight angle: An
angle that measures 180; Reflex angle: An angle whose measure is more than 180 and less than 360. There are 3 sets of lines: Intersecting, Perpendicular and Parallel. Read more...iWorksheets :12Study
Guides :2Vocabulary :2
AnglesA right angle is an angle that measures 90°. A straight angle is an angle that measures 180°. An obtuse angle is an angle that measures more than 90°. An acute angle is an angle that measures
less than 90°. Read more...iWorksheets :10Study Guides :1
2.A.1.c. Identify parallel and intersecting line segments.
Lines and AnglesAcute angle: An angle whose measure is less than 90; Right angle: An angle that measures 90; Obtuse angle: An angle whose measure is more than 90 and less than 180; Straight angle: An
angle that measures 180; Reflex angle: An angle whose measure is more than 180 and less than 360. There are 3 sets of lines: Intersecting, Perpendicular and Parallel. Read more...iWorksheets :12Study
Guides :2Vocabulary :2
2.B.1. Solid geometric figures: Analyze the properties of solid geometric figures.
2.B.1.a. Identify cones, cylinders, prisms, and pyramids (Assessment limit: Use cones or cylinders).
ShapesWe are surrounded by many different kinds of shapes every day. Many shapes are flat. These shapes are two-dimensional plane figures. Read more...iWorksheets :10Study Guides :1
ShapesFreeA shape is the external contour or outline of someone of something Read more...iWorksheets :11Study Guides :1Vocabulary :3
Polygon CharacteristicsA polygon is a plane figure with at least three straight sides and angles, and typically five or more. Read more...iWorksheets :8Study Guides :1Vocabulary :1
Solids and FacesYou can use solid shapes to help describe real-world objects. These shapes have surfaces called faces. Read more...iWorksheets :5Study Guides :1
Congruent ShapesFreeCongruent shapes are shapes that are the exact same shape and size. Congruent shapes can be rotated or reflected. When 2 shapes are congruent, they have the exact same size and
shape. Read more...iWorksheets :4Study Guides :1Vocabulary :1
2.B.1.b. Describe solid geometric figures by the number of edges, faces, or vertices (Assessment limit: Use triangular pyramids, rectangular pyramids, triangular prisms, or rectangular prisms).
Solids and FacesYou can use solid shapes to help describe real-world objects. These shapes have surfaces called faces. Read more...iWorksheets :5Study Guides :1
2.B.2. Solid geometric figures: Analyze the relationship between plane geometric figures and surfaces of solid geometric figures.
2.B.2.a. Compare a plane figure to surfaces of solid geometric figure (Assessment limit: Analyze or identify the number or arrangement of squares needed to make a cube and triangles/rectangles needed
to make a triangular pyramid or rectangular pyramid).
Solids and FacesYou can use solid shapes to help describe real-world objects. These shapes have surfaces called faces. Read more...iWorksheets :5Study Guides :1
2.C.1. Representation of Geometric Figures: Represent plane geometric figures.
2.C.1.a. Sketch acute, right, obtuse angles, and parallel and intersecting line segments.
Lines and AnglesAcute angle: An angle whose measure is less than 90; Right angle: An angle that measures 90; Obtuse angle: An angle whose measure is more than 90 and less than 180; Straight angle: An
angle that measures 180; Reflex angle: An angle whose measure is more than 180 and less than 360. There are 3 sets of lines: Intersecting, Perpendicular and Parallel. Read more...iWorksheets :12Study
Guides :2Vocabulary :2
MD.3.0. Knowledge of Measurement: Students will identify attributes, units, or systems of measurements or apply a variety of techniques, formulas, tools, or technology for determining measurements.
3.A.1. Measurement Units: Read customary and metric measurement units.
3.A.1.a. Estimate and determine length and height (Assessment limit: Use the nearest millimeter or 1/4 inch).
MeasurementMeasurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. Read more...iWorksheets :8Study Guides :1Vocabulary
MeasurementFreeThere are two system of measurement for length that can be used. U.S customary System and Metric System. U.S. Customary System & Metric system. Read more...iWorksheets :10Study Guides
:1Vocabulary :3
Units of MeasureWhen you need to measure an object, you must decide if you are: Measuring in length, weight, or capacity, choosing the unit that makes sense to measure the object, Measuring in the
customary system or the metric system. Read more...iWorksheets :3Study Guides :1
MeasurementMeasurement is the use of units to show size, length, weight, or capacity.There are customary measurements and metric measurements. Read more...iWorksheets :13Study Guides :1Vocabulary :2
Units of MeasureWhat are Units of Measurement? People measure mass, volume, and length. These measurements are labeled with appropriate unit of measurement. Read more...iWorksheets :4Study Guides :1
Determine Appropriate Standard of UnitsWhat are the Standard of Units? When measuring objects or distances, there are certain measurements of length, distance, weight, and capacity that should be
used. There are customary standard of units and metric standard of units. Read more...iWorksheets :3Study Guides :1
3.A.1.b. Estimate and determine weight or mass.
MeasurementMeasurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. Read more...iWorksheets :8Study Guides :1Vocabulary
MeasurementFreeThere are two system of measurement for length that can be used. U.S customary System and Metric System. U.S. Customary System & Metric system. Read more...iWorksheets :10Study Guides
:1Vocabulary :3
Units of MeasureWhen you need to measure an object, you must decide if you are: Measuring in length, weight, or capacity, choosing the unit that makes sense to measure the object, Measuring in the
customary system or the metric system. Read more...iWorksheets :3Study Guides :1
MeasurementMeasurement is the use of units to show size, length, weight, or capacity.There are customary measurements and metric measurements. Read more...iWorksheets :13Study Guides :1Vocabulary :2
Units of MeasureWhat are Units of Measurement? People measure mass, volume, and length. These measurements are labeled with appropriate unit of measurement. Read more...iWorksheets :4Study Guides :1
Determine Appropriate Standard of UnitsWhat are the Standard of Units? When measuring objects or distances, there are certain measurements of length, distance, weight, and capacity that should be
used. There are customary standard of units and metric standard of units. Read more...iWorksheets :3Study Guides :1
3.A.1.c. Estimate and determine capacity.
Volume and CapacityWhat is volume? Volume is the 3-dimensional size of an object, such as a box. What is capacity? Capacity is the amount a 3-dimensional object can hold or carry. It can also be
thought of the measure of volume of a 3-dimensional object. Read more...iWorksheets :5Study Guides :1
3.B.1. Measurement Tools: Measure in customary and metric units.
3.B.1.a. Select and use appropriate tools and units (Assessment limit: Use the nearest millimeter or 1/4 inch with a ruler).
MeasurementMeasurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. Read more...iWorksheets :8Study Guides :1Vocabulary
MeasurementFreeThere are two system of measurement for length that can be used. U.S customary System and Metric System. U.S. Customary System & Metric system. Read more...iWorksheets :10Study Guides
:1Vocabulary :3
Units of MeasureWhen you need to measure an object, you must decide if you are: Measuring in length, weight, or capacity, choosing the unit that makes sense to measure the object, Measuring in the
customary system or the metric system. Read more...iWorksheets :3Study Guides :1
Units of MeasureWhat are Units of Measurement? People measure mass, volume, and length. These measurements are labeled with appropriate unit of measurement. Read more...iWorksheets :4Study Guides :1
Determine Appropriate Standard of UnitsWhat are the Standard of Units? When measuring objects or distances, there are certain measurements of length, distance, weight, and capacity that should be
used. There are customary standard of units and metric standard of units. Read more...iWorksheets :3Study Guides :1
3.B.2. Measurement Units: Compare right angles to a corner.
Lines and AnglesAcute angle: An angle whose measure is less than 90; Right angle: An angle that measures 90; Obtuse angle: An angle whose measure is more than 90 and less than 180; Straight angle: An
angle that measures 180; Reflex angle: An angle whose measure is more than 180 and less than 360. There are 3 sets of lines: Intersecting, Perpendicular and Parallel. Read more...iWorksheets :12Study
Guides :2Vocabulary :2
AnglesA right angle is an angle that measures 90°. A straight angle is an angle that measures 180°. An obtuse angle is an angle that measures more than 90°. An acute angle is an angle that measures
less than 90°. Read more...iWorksheets :10Study Guides :1
3.C.1. Applications in Measurement: Apply measurement concepts.
3.C.1.a. Determine perimeter (Assessment limit: Use polygons with no more than 6 sides given the length of the sides in whole numbers (0 - 100)).
PerimeterA polygon is any 2-dimensional shape formed with straight lines. The perimeter of a polygon is the sum of all its length. Read more...iWorksheets :7Study Guides :1
Area and PerimeterThe area of a figure is the space inside the figure. The perimeter of a polygon is the distance around it. The perimeter is the sum of the lengths of ALL the sides. Read more...i
Worksheets :7Study Guides :1
PerimeterPerimeter is the distance around the outside of an object. Read more...iWorksheets :7Study Guides :1Vocabulary :1
PerimeterWhat Is Perimeter? The perimeter is the measurement of the distance around the outside of a shape or object. To find the perimeter of a shape or object, simply add the outside dimensions
together. Read more...iWorksheets :4Study Guides :1
3.C.1.b. Determine area (Assessment limit: Use rectangles with the length of the sides in whole numbers (0 - 100)).
AreaArea is the number of square units needed to cover a flat surface. Read more...iWorksheets :3Study Guides :1
Area and PerimeterThe area of a figure is the space inside the figure. The perimeter of a polygon is the distance around it. The perimeter is the sum of the lengths of ALL the sides. Read more...i
Worksheets :7Study Guides :1
3.C.1.c. Determine start time, elapsed time and end time (Assessment limit: Use hour and half hour intervals).
TimeFreeCalculate elapsed time in hours and half hours, not crossing AM/PM. Read more...iWorksheets :9Study Guides :1
CalendarWhat Is Elapsed Time? Elapsed time is the amount of time from the start of an activity to the end of the activity. It tells how long an activity lasted. Elapsed time can be measured in
seconds, minutes, hours, days or weeks. Read more...iWorksheets :3Study Guides :1
Elapsed TimeElapsed time is the amount of time that has passed between two defined times. Read more...iWorksheets :6Study Guides :1
MD.4.0. Knowledge of Statistics: Students will collect, organize, display, analyze, or interpret data to make decisions or predictions.
4.B.1. Data Analysis: Analyze data.
4.B.1.b. Interpret line graphs (Assessment limit: Use the x-axis representing no more than 6 time intervals, the y-axis consisting of no more than 10 intervals with scales as factors of 100 using
whole numbers (0 - 100)).
Graphs and TablesUsing tables and graphs is a way people can interpret data. Data means information. So interpreting data just means working out what information is telling you. Information is
sometimes shown in tables, charts and graphs to make the information easier to read. Read more...iWorksheets :3Study Guides :1
Graphs and ChartsWhat Are Graphs? A way to show information in the form of shapes or pictures. Graphs show the relationship between two sets of information. There are many different types of graphs.
A few of them include bar graphs, line graphs, pictographs, and circle graphs. Read more...iWorksheets :10Study Guides :1Vocabulary :1
Tables and GraphsWhat Are Bar, Circle, and Line Graphs? Bar Graphs are used to compare data. A bar graph is used to show relationships between groups. Circle Graphs are also known as Pie graphs or
charts. They consist of a circle divided into parts. Line Graphs show gradual changes in data. Read more...iWorksheets :9Study Guides :1
4.B.2. Data Analysis: Describe a set of data.
4.B.2.a. Determine median, mode, and range (Assessment limit: Use no more than 8 pieces of data and whole numbers (0 - 100)).
Data AnalysisAnalysis of data is a process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information. Read more...iWorksheets :5Study Guides :1
Vocabulary :1
StatisticsThe statistical mode is the number that occurs most frequently in a set of numbers. Read more...iWorksheets :3Study Guides :1
MeanA mean of a group of numbers is the average of those numbers. Read more...iWorksheets :3Study Guides :1
Data AnalysisCollecting Data. Data = information. You can collect data from other people using polls and surveys. Recording Data. You can record the numerical data you collected on a chart or graph:
bar graphs, pictographs, line graphs, pie charts, column charts. Read more...iWorksheets :6Study Guides :1
4.B.2.b. Model the mean of a set of data.
Data AnalysisAnalysis of data is a process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information. Read more...iWorksheets :5Study Guides :1
Vocabulary :1
StatisticsThe statistical mode is the number that occurs most frequently in a set of numbers. Read more...iWorksheets :3Study Guides :1
MeanA mean of a group of numbers is the average of those numbers. Read more...iWorksheets :3Study Guides :1
Data AnalysisCollecting Data. Data = information. You can collect data from other people using polls and surveys. Recording Data. You can record the numerical data you collected on a chart or graph:
bar graphs, pictographs, line graphs, pie charts, column charts. Read more...iWorksheets :6Study Guides :1
MD.5.0. Knowledge of Probability: Students will use experimental methods or theoretical reasoning to determine probabilities to make predictions or solve problems about events whose outcomes involve
random variation.
5.B.1. Theoretical Probability: Determine the probability of one simple event comprised of equally likely outcomes.
5.B.1.a. Express the probability as a fraction (Assessment limit: Use a sample space of no more than 6 outcomes).
ProbabilityFreeProbability word problems worksheet. Probability is the measure of how likely an event is. Probability = (Total ways a specific outcome will happen) / (Total number of possible
outcomes). The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. Read more...iWorksheets :4Study Guides :1
ProbabilityProbability word problems worksheet. Probability is the chance of whether something will happen or not. If two things have an EQUAL chance of happening, they have the SAME probability. If
there are MORE chances of something happening (A) than something else (B), that means there is a HIGHER PROBABILITY of that something (A) happening. Read more...iWorksheets :3Study Guides :1
ProbabilityWhat Is Probability? Probability is the chance that a particular event will occur. There are four different ways to show the probability: One way is to show the certainty: certain, likely,
somewhat likely, not likely, impossible. The other three ways are with numbers. Probability word problems worksheet. Read more...iWorksheets :8Study Guides :1
MD.6.0. Knowledge of Number Relationships and Computation/Arithmetic: Students will describe, represent, or apply numbers or their relationships or will estimate or compute using mental strategies,
paper/pencil, or technology.
6.A.1. Knowledge of Number and Place Value: Apply knowledge of whole numbers and place value.
6.A.1.a. Read, write, and represent whole numbers using symbols, words, and models (Assessment limit: Use whole numbers (0 - 1,000,000)).
Place ValueWhat Is Place Value? In our decimal number system, the value of a digit depends on its place, or position, in the number. Beginning with the ones place at the right, each place value is
multiplied by increasing powers of 10. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Number Words and Place ValueWhen we write numbers, the position of each digit is important. Each position is 10 more than the one before it. So, 23 means “add 2*10 to 3*1″. In the number 467: the "7"
is in the Ones position, meaning 7 ones, the "6" is in the Tens position meaning 6 tens, and the "4" is in the Hundreds position. Read more...iWorksheets :3Study Guides :1
6.A.1.b. Express whole numbers in expanded form (Assessment limit: Use whole numbers (0 - 1,000,000)).
Expanding NumbersWhat Are Expanding Numbers? An expanding number is taking a larger number apart and showing each number’s total value. Number 5398 in expanded form is 5000 + 300 + 90 + 8. Read
more...iWorksheets :3Study Guides :1
6.A.1.c. Identify the place value of a digit in a number (Assessment limit: Use whole numbers (0 - 1,000,000)).
Place ValuePlace value is the numerical value that a digit has by virtue of its position in a number. Read more...iWorksheets :6Study Guides :1
EstimationWhen you make an estimate, you are making a guess that is approximate. This is often done by rounding. Read more...iWorksheets :6Study Guides :1
Add/Subtract/Multiply/Divide DecimalsYou add/subtract/multiply/divide decimals the same way you add/subtract/multiply/divide whole numbers BUT you also need to place the decimal in the correct spot.
When multiplying decimals, the decimals may or may NOT be lined up in the multiplication problem. Read more...iWorksheets :10Study Guides :1Vocabulary :1
Ordering and Comparing NumbersWhen you order numbers, you are putting the numbers in a sequence from the smallest value to the largest value. When you compare two numbers, you are finding which
number is larger or smaller than the other. Read more...iWorksheets :5Study Guides :1Vocabulary :1
Compare and Order NumbersWhat is comparing and ordering numbers? Ordering numbers means listing numbers from least to greatest, or greatest to least. Comparing numbers means looking at the values of
two numbers and deciding if the numbers are greater than, less than, or equal to each other. Read more...iWorksheets :4Study Guides :1
Rounding to Nearest 10Rounding makes numbers easier to work with if you do not need an exact number. Rounded numbers are only approximate. You can use rounded numbers to get an answer that is close
but does not have to be exact. Read more...iWorksheets :3Study Guides :1
Rounding NumbersWhat Is Rounding? Rounding means reducing the digits in a number while trying to keep its value similar. How to Round: The number in the given place is increased by one if the digit
to its right is 5 or greater. The number in the given place remains the same if the digit to its right is less than 5. Read more...iWorksheets :3Study Guides :1Vocabulary :1
Place ValueWhat Is Place Value? In our decimal number system, the value of a digit depends on its place, or position, in the number. Beginning with the ones place at the right, each place value is
multiplied by increasing powers of 10. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Greater Than/Less ThanWhat Is Greater Than and Less Than? When a number is greater than another number, this means it is a larger number. The symbol for greater than is >. When a number is less than
another number, this means it is a smaller number. The symbol for less than is <. Read more...iWorksheets :6Study Guides :1
Expanding NumbersWhat Are Expanding Numbers? An expanding number is taking a larger number apart and showing each number’s total value. Number 5398 in expanded form is 5000 + 300 + 90 + 8. Read
more...iWorksheets :3Study Guides :1
Place ValuePlace value is what each digit is worth. In the number 4,573 there are four thousands, five hundreds, seven tens, and three ones. How to Find the Place Value: In order to find the place
value of a number, you can count the number of places from the right. The first number will be the ones place. The next number moving towards the left would be the tens place, and so on. Read more...
iWorksheets :10Study Guides :1
Number Words and Place ValueWhen we write numbers, the position of each digit is important. Each position is 10 more than the one before it. So, 23 means “add 2*10 to 3*1″. In the number 467: the "7"
is in the Ones position, meaning 7 ones, the "6" is in the Tens position meaning 6 tens, and the "4" is in the Hundreds position. Read more...iWorksheets :3Study Guides :1
6.A.1.d. Compare, order, and describe whole numbers (Assessment limit: Use no more than 4 whole numbers with or without using the symbols (<, >, =) and whole numbers (0 - 1,000,000)).
AlgebraComparing whole numbers, fractions, and decimals means looking at the values of two numbers and deciding if they are greater than, less than or equal to each other. Read more...iWorksheets :4
Study Guides :1
Ordering DecimalsWhen putting decimals in order from least to greatest, we must look at the highest place value first. Read more...iWorksheets :7Study Guides :1Vocabulary :1
Compare and Order FractionsWhen comparing two fractions that have a common denominator, you can looks at the numerators to decide which fraction is greater Read more...iWorksheets :4Study Guides :1
Vocabulary :1
Positive & Negative IntegersPositive integers are all the whole numbers greater than zero. Negative integers are all the opposites of these whole numbers, numbers that are less than zero. Zero is
considered neither positive nor negative Read more...iWorksheets :4Study Guides :1
Ordering FractionsA fraction consists of two numbers separated by a line - numerator and denominator. To order fractions with like numerators, look at the denominators and compare them two at a time.
The fraction with the smaller denominator is the larger fraction. Read more...iWorksheets :3Study Guides :1
Fractions/DecimalsHow to convert fractions to decimals: Divide the denominator (the bottom part) into the numerator (the top part). Read more...iWorksheets :3Study Guides :1
Greater Than/Less ThanIf a number is greater than another number that means it is higher in value than the other number. If a number is less than another number that means it is lower in value than
the other number. Read more...iWorksheets :4Study Guides :1
Comparing FractionsWhen comparing fractions, you are finding which fraction is greater and which fractions is less than the other. Similar to comparing numbers, there are symbols to use when
comparing fractions. Read more...iWorksheets :5Study Guides :1Vocabulary :1
Ordering and Comparing NumbersWhen you order numbers, you are putting the numbers in a sequence from the smallest value to the largest value. When you compare two numbers, you are finding which
number is larger or smaller than the other. Read more...iWorksheets :5Study Guides :1Vocabulary :1
Compare and Order NumbersWhat is comparing and ordering numbers? Ordering numbers means listing numbers from least to greatest, or greatest to least. Comparing numbers means looking at the values of
two numbers and deciding if the numbers are greater than, less than, or equal to each other. Read more...iWorksheets :4Study Guides :1
Greater Than/Less ThanWhat Is Greater Than and Less Than? When a number is greater than another number, this means it is a larger number. The symbol for greater than is >. When a number is less than
another number, this means it is a smaller number. The symbol for less than is <. Read more...iWorksheets :6Study Guides :1
FractionsThe top number of a fraction is called the numerator. It shows how many pieces of a whole we are talking about. The bottom number is called the denominator. It shows how many pieces an
object was divided into, or how many total pieces we have. Read more...iWorksheets :4Study Guides :1
6.A.2. Knowledge of Number and Place Value: Apply knowledge of fractions and decimals.
6.A.2.a. Read, write, and represent proper fractions of a single region using symbols, words, and models (Assessment limit: Use denominators 6, 8, and 10).
Add/Subtract DecimalsAddition and subtraction of decimals is like adding and subtracting whole numbers. The only thing we must remember is to line up the place values correctly. Read more...i
Worksheets :14Study Guides :1Vocabulary :1
Add/Subtract FractionsFreeis one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two whole numbers is the total amount of
those quantities combined. Read more...iWorksheets :3Study Guides :1
AlgebraComparing whole numbers, fractions, and decimals means looking at the values of two numbers and deciding if they are greater than, less than or equal to each other. Read more...iWorksheets :4
Study Guides :1
Compare and Order FractionsWhen comparing two fractions that have a common denominator, you can looks at the numerators to decide which fraction is greater Read more...iWorksheets :4Study Guides :1
Vocabulary :1
PercentsA percentage is a number or ratio expressed as a fraction of 100. Read more...iWorksheets :6Study Guides :1Vocabulary :1
RatioRatios are used to make a comparison between two things. Read more...iWorksheets :9Study Guides :1Vocabulary :1
ProbabilityFreeProbability word problems worksheet. Probability is the measure of how likely an event is. Probability = (Total ways a specific outcome will happen) / (Total number of possible
outcomes). The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. Read more...iWorksheets :4Study Guides :1
FractionsFractions can show a part of a group or part of a set. Read more...iWorksheets :6Study Guides :1
ProbabilityProbability word problems worksheet. Probability is the chance of whether something will happen or not. If two things have an EQUAL chance of happening, they have the SAME probability. If
there are MORE chances of something happening (A) than something else (B), that means there is a HIGHER PROBABILITY of that something (A) happening. Read more...iWorksheets :3Study Guides :1
Ordering FractionsA fraction consists of two numbers separated by a line - numerator and denominator. To order fractions with like numerators, look at the denominators and compare them two at a time.
The fraction with the smaller denominator is the larger fraction. Read more...iWorksheets :3Study Guides :1
Subtracting FractionsFractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. First, make sure the denominators are the same, then
subtract the numerators. Read more...iWorksheets :3Study Guides :1
Fractions/DecimalsHow to convert fractions to decimals: Divide the denominator (the bottom part) into the numerator (the top part). Read more...iWorksheets :3Study Guides :1
Number LineA number line is a line that shows any group of numbers in their least to greatest value. Read more...iWorksheets :3Study Guides :1
Comparing FractionsWhen comparing fractions, you are finding which fraction is greater and which fractions is less than the other. Similar to comparing numbers, there are symbols to use when
comparing fractions. Read more...iWorksheets :5Study Guides :1Vocabulary :1
Decimals/FractionsExpress decimals as an equivalent form of fractions to tenths and hundredths. Read more...iWorksheets :5Study Guides :1Vocabulary :1
Add/Subtract FractionsWhat Is Addition and Subtraction of Fractions? Addition is combining two or more fractions. The term used for addition is plus. When two or more numbers, or addends, are
combined they form a new number called a sum. Subtraction is “taking away” one fraction from another fraction. The term is minus. The number left after subtracting is called a difference. Read
more...iWorksheets :4Study Guides :1
Equivalent Fractions to 1/2Fractions that are equivalent to ½ are fractions that have different denominators than ½, but still show half. Fractions that are equivalent to ½ can be simplified to ½.
Fractions equivalent to ½ have an even number as their denominator. Read more...iWorksheets :3Study Guides :1
FractionsThe top number of a fraction is called the numerator. It shows how many pieces of a whole we are talking about. The bottom number is called the denominator. It shows how many pieces an
object was divided into, or how many total pieces we have. Read more...iWorksheets :4Study Guides :1
Adding FractionsFractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. To add two fractions with the same denominator: Add the
numerators and place the sum over the common denominator. Read more...iWorksheets :3Study Guides :1
6.A.2.b. Read, write, and represent proper fractions of a set which has the same number of items as the denominator using symbols, words, and models (Assessment limit: Use denominators of 6,8, and 10
with sets of 6, 8, and 10, respectively).
Add/Subtract DecimalsAddition and subtraction of decimals is like adding and subtracting whole numbers. The only thing we must remember is to line up the place values correctly. Read more...i
Worksheets :14Study Guides :1Vocabulary :1
Add/Subtract FractionsFreeis one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two whole numbers is the total amount of
those quantities combined. Read more...iWorksheets :3Study Guides :1
AlgebraComparing whole numbers, fractions, and decimals means looking at the values of two numbers and deciding if they are greater than, less than or equal to each other. Read more...iWorksheets :4
Study Guides :1
Compare and Order FractionsWhen comparing two fractions that have a common denominator, you can looks at the numerators to decide which fraction is greater Read more...iWorksheets :4Study Guides :1
Vocabulary :1
PercentsA percentage is a number or ratio expressed as a fraction of 100. Read more...iWorksheets :6Study Guides :1Vocabulary :1
RatioRatios are used to make a comparison between two things. Read more...iWorksheets :9Study Guides :1Vocabulary :1
ProbabilityFreeProbability word problems worksheet. Probability is the measure of how likely an event is. Probability = (Total ways a specific outcome will happen) / (Total number of possible
outcomes). The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. Read more...iWorksheets :4Study Guides :1
FractionsFractions can show a part of a group or part of a set. Read more...iWorksheets :6Study Guides :1
ProbabilityProbability word problems worksheet. Probability is the chance of whether something will happen or not. If two things have an EQUAL chance of happening, they have the SAME probability. If
there are MORE chances of something happening (A) than something else (B), that means there is a HIGHER PROBABILITY of that something (A) happening. Read more...iWorksheets :3Study Guides :1
Ordering FractionsA fraction consists of two numbers separated by a line - numerator and denominator. To order fractions with like numerators, look at the denominators and compare them two at a time.
The fraction with the smaller denominator is the larger fraction. Read more...iWorksheets :3Study Guides :1
Subtracting FractionsFractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. First, make sure the denominators are the same, then
subtract the numerators. Read more...iWorksheets :3Study Guides :1
Fractions/DecimalsHow to convert fractions to decimals: Divide the denominator (the bottom part) into the numerator (the top part). Read more...iWorksheets :3Study Guides :1
Number LineA number line is a line that shows any group of numbers in their least to greatest value. Read more...iWorksheets :3Study Guides :1
Comparing FractionsWhen comparing fractions, you are finding which fraction is greater and which fractions is less than the other. Similar to comparing numbers, there are symbols to use when
comparing fractions. Read more...iWorksheets :5Study Guides :1Vocabulary :1
Decimals/FractionsExpress decimals as an equivalent form of fractions to tenths and hundredths. Read more...iWorksheets :5Study Guides :1Vocabulary :1
Add/Subtract FractionsWhat Is Addition and Subtraction of Fractions? Addition is combining two or more fractions. The term used for addition is plus. When two or more numbers, or addends, are
combined they form a new number called a sum. Subtraction is “taking away” one fraction from another fraction. The term is minus. The number left after subtracting is called a difference. Read
more...iWorksheets :4Study Guides :1
Equivalent Fractions to 1/2Fractions that are equivalent to ½ are fractions that have different denominators than ½, but still show half. Fractions that are equivalent to ½ can be simplified to ½.
Fractions equivalent to ½ have an even number as their denominator. Read more...iWorksheets :3Study Guides :1
FractionsThe top number of a fraction is called the numerator. It shows how many pieces of a whole we are talking about. The bottom number is called the denominator. It shows how many pieces an
object was divided into, or how many total pieces we have. Read more...iWorksheets :4Study Guides :1
Adding FractionsFractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. To add two fractions with the same denominator: Add the
numerators and place the sum over the common denominator. Read more...iWorksheets :3Study Guides :1
6.A.2.c. Find equivalent fractions.
Fractions/DecimalsHow to convert fractions to decimals: Divide the denominator (the bottom part) into the numerator (the top part). Read more...iWorksheets :3Study Guides :1
FractionsThe top number of a fraction is called the numerator. It shows how many pieces of a whole we are talking about. The bottom number is called the denominator. It shows how many pieces an
object was divided into, or how many total pieces we have. Read more...iWorksheets :4Study Guides :1
6.A.2.d. Read, write, and represent mixed numbers using symbols, words, and models.
Add/Subtract FractionsFreeis one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two whole numbers is the total amount of
those quantities combined. Read more...iWorksheets :3Study Guides :1
Subtracting FractionsFractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. First, make sure the denominators are the same, then
subtract the numerators. Read more...iWorksheets :3Study Guides :1
Adding FractionsFractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. To add two fractions with the same denominator: Add the
numerators and place the sum over the common denominator. Read more...iWorksheets :3Study Guides :1
6.A.2.e. Read, write, and represent decimals using symbols, words and models (Assessment limit: Use no more than 2 decimal places and numbers (0 - 100)).
Add/Subtract DecimalsAddition and subtraction of decimals is like adding and subtracting whole numbers. The only thing we must remember is to line up the place values correctly. Read more...i
Worksheets :14Study Guides :1Vocabulary :1
AlgebraComparing whole numbers, fractions, and decimals means looking at the values of two numbers and deciding if they are greater than, less than or equal to each other. Read more...iWorksheets :4
Study Guides :1
Ordering DecimalsWhen putting decimals in order from least to greatest, we must look at the highest place value first. Read more...iWorksheets :7Study Guides :1Vocabulary :1
PercentsA percentage is a number or ratio expressed as a fraction of 100. Read more...iWorksheets :6Study Guides :1Vocabulary :1
RoundingRounding makes numbers that are easier to work with in your head. Rounded numbers are only approximate. Use rounding to get an answer that is close but that does not have to be exact. Read
more...iWorksheets :3Study Guides :1
Fractions/DecimalsHow to convert fractions to decimals: Divide the denominator (the bottom part) into the numerator (the top part). Read more...iWorksheets :3Study Guides :1
DecimalsREADING, WRITING, COMPARING, AND ORDERING DECIMALS Read more...iWorksheets :5Study Guides :1Vocabulary :1
Add/Subtract/Multiply/Divide DecimalsYou add/subtract/multiply/divide decimals the same way you add/subtract/multiply/divide whole numbers BUT you also need to place the decimal in the correct spot.
When multiplying decimals, the decimals may or may NOT be lined up in the multiplication problem. Read more...iWorksheets :10Study Guides :1Vocabulary :1
Decimals/FractionsExpress decimals as an equivalent form of fractions to tenths and hundredths. Read more...iWorksheets :5Study Guides :1Vocabulary :1
6.A.2.f. Express decimals in expanded form (Assessment limit; Use no more than 2 decimal places and numbers (0 - 100)).
Expanding NumbersWhat Are Expanding Numbers? An expanding number is taking a larger number apart and showing each number’s total value. Number 5398 in expanded form is 5000 + 300 + 90 + 8. Read
more...iWorksheets :3Study Guides :1
6.A.2.g. Compare and order fractions and mixed numbers with or without using the symbols (<, >, or =) (Assessment limit: Use like denominators and no more than 3 numbers (0 - 20)).
Compare and Order FractionsWhen comparing two fractions that have a common denominator, you can looks at the numerators to decide which fraction is greater Read more...iWorksheets :4Study Guides :1
Vocabulary :1
Ordering FractionsA fraction consists of two numbers separated by a line - numerator and denominator. To order fractions with like numerators, look at the denominators and compare them two at a time.
The fraction with the smaller denominator is the larger fraction. Read more...iWorksheets :3Study Guides :1
Fractions/DecimalsHow to convert fractions to decimals: Divide the denominator (the bottom part) into the numerator (the top part). Read more...iWorksheets :3Study Guides :1
Comparing FractionsWhen comparing fractions, you are finding which fraction is greater and which fractions is less than the other. Similar to comparing numbers, there are symbols to use when
comparing fractions. Read more...iWorksheets :5Study Guides :1Vocabulary :1
Equivalent Fractions to 1/2Fractions that are equivalent to ½ are fractions that have different denominators than ½, but still show half. Fractions that are equivalent to ½ can be simplified to ½.
Fractions equivalent to ½ have an even number as their denominator. Read more...iWorksheets :3Study Guides :1
FractionsThe top number of a fraction is called the numerator. It shows how many pieces of a whole we are talking about. The bottom number is called the denominator. It shows how many pieces an
object was divided into, or how many total pieces we have. Read more...iWorksheets :4Study Guides :1
6.A.2.h. Compare, order, and describe decimals with or without using the symbols (<, >, or =) (Assessment limit: Use no more than 3 decimals with no more than 2 decimals places and numbers (0 -
Ordering DecimalsWhen putting decimals in order from least to greatest, we must look at the highest place value first. Read more...iWorksheets :7Study Guides :1Vocabulary :1
Fractions/DecimalsHow to convert fractions to decimals: Divide the denominator (the bottom part) into the numerator (the top part). Read more...iWorksheets :3Study Guides :1
6.A.3. Knowledge of Number and Place Value: Apply knowledge of money.
6.A.3.b. Determine the change from $100.
Giving Change from $1.00What Is Giving Change? Change is the money you receive back when you purchase an item and give the cashier more than the item cost. To figure out the change you will receive
from a purchase, simply subtract the total amount of the purchase from the amount you are giving the cashier. Read more...iWorksheets :4Study Guides :1
6.B.1. Number Theory: Apply number relationships.
6.B.1.b. Identify factors (Assessment limit: Use whole numbers (0 - 24)).
Common FactorsFactors are two numbers multiplied together to get a product (an answer to a multiplication problem) Read more...iWorksheets :6Study Guides :1Vocabulary :1
6.C.1. Number Computation: Analyze number relations and compute.
6.C.1.a. Add whole numbers (Assessment limit: Use up to 3 addends with no more than 4 digits in each addend and whole numbers (0 - 10,000)).
3 Digit AdditionFreeAdding large numbers involves breaking the problem down into smaller addition facts. Read more...iWorksheets :4Study Guides :1
Commutative PropertyThe commutative property of addition says that we can add numbers in any order and get the same sum. Read more...iWorksheets :3Study Guides :1
Commutative/Associative PropertiesUsing the Commutative Property in addition means that the order of addends does not matter; the sum will remain the same. Read more...iWorksheets :3Study Guides :1
Addition/SubtractionAddition is combining two or more numbers. The term used for addition is plus. When two or more numbers are combined they form a new number called a sum. Subtraction is “taking
away” one number from another. The term is minus. The number left after subtracting is called a difference. Read more...iWorksheets :11Study Guides :1
Double Digit AdditionWhat Is Double Digit Addition? Double digit addition is taking a two digit number (ex. 32) and adding it to another two digit number (ex. 27). The answer of these two addends is
known as the sum. Read more...iWorksheets :4Study Guides :1
RegroupingWhat Is Regrouping? Regrouping in addition is used when the sum of the ones place is larger than nine. The tens place of the sum is moved to the top of the tens place column to be added
with the others. Read more...iWorksheets :3Study Guides :1
Associative PropertyAssociative Property of Addition explains that when three or more numbers are added, the sum is the same regardless of the order in which the numbers are grouped and/or added.
Read more...iWorksheets :3Study Guides :1
Word ProblemsWhat Are Story Problems? Story problems are a bunch of sentences set up to give you information in order to solve a problem. Story problems most often give you all the information needed
to solve the problem. They may even include information you do not need at all. Read more...iWorksheets :15Study Guides :1
6.C.1.b. Subtract whole numbers (Assessment limit: Use a minuend and subtrahend with no more than 4 digits in each and whole numbers (0 - 9999)).
3 Digit SubtractionWhat Is Three-Digit Subtraction? We subtract to compare numbers. We are able to find the difference between numbers through subtraction. We use subtraction to find out how much
more we have or how much smaller something is in comparison to another number. Read more...iWorksheets :3Study Guides :1
Double Digit SubtractionWhat Is Double Digit Subtraction? Double digit subtraction is taking a number with two digits (ex. 23) and subtracting it from another two digit number (ex. 33). The answer is
known as the difference. Read more...iWorksheets :4Study Guides :1
Addition/SubtractionAddition is combining two or more numbers. The term used for addition is plus. When two or more numbers are combined they form a new number called a sum. Subtraction is “taking
away” one number from another. The term is minus. The number left after subtracting is called a difference. Read more...iWorksheets :11Study Guides :1
RegroupingWhat Is Regrouping? Regrouping in addition is used when the sum of the ones place is larger than nine. The tens place of the sum is moved to the top of the tens place column to be added
with the others. Read more...iWorksheets :3Study Guides :1
6.C.1.c. Multiply whole numbers (Assessment limit: Use a one 1-digit factor by up to a 3-digit factor using whole numbers (0 - 1000)).
MultiplicationMultiplication is one of the four elementary, mathematical operations of arithmetic. Read more...iWorksheets :7Study Guides :1Vocabulary :1
Odd/EvenA number can be identified as odd or even. Odd numbers can't be divided exactly by 2. Read more...iWorksheets :3Study Guides :1
DivisionDivide three-digit numbers by one- and two-digit numbers. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Commutative/Associative PropertiesUsing the Commutative Property in addition means that the order of addends does not matter; the sum will remain the same. Read more...iWorksheets :3Study Guides :1
More MultiplicationMultiplication of two digits by two digits. What Is Multiplication? Multiplication is a short way of adding or counting. Multiplication is a faster way of adding. By multiplying
numbers together, you are adding a series of one number to itself. Read more...iWorksheets :3Study Guides :1
Division/MultiplicationUnderstanding of models for multiplication, place value, and properties of operations (in particular, the distributive property). Read more...iWorksheets :9Study Guides :1
MultiplicationMultiplication is similar to adding a number to itself a certain number of times. When multiplying an odd number with an odd number, the product is always an odd number. When
multiplying an odd number with an even number or two even numbers, the product is always an even number. Read more...iWorksheets :19Study Guides :1
6.C.1.d. Divide whole numbers (Assessment limit: Use up to a 3-digit dividend by a 1-digit divisor and whole numbers with no remainders (0 - 999)).
DivisionDivide three-digit numbers by one- and two-digit numbers. Read more...iWorksheets :6Study Guides :1Vocabulary :1
DivisionWhat Is Division? Division is splitting up numbers into equal parts. The process of finding out how many times one number will go into another number. Division is a series of repeated
subtraction. The parts of a division problem include the divisor, dividend, quotient and remainder. Read more...iWorksheets :8Study Guides :1
Division/MultiplicationUnderstanding of models for multiplication, place value, and properties of operations (in particular, the distributive property). Read more...iWorksheets :9Study Guides :1
DivisionFreeWhat Is Division? Division is an operation that tells: how many equal sized groups, how many in each group. The number you divide by is called the DIVISOR. The number you are dividing is
called the DIVIDEND. And the answer is called the QUOTIENT. Read more...iWorksheets :6Study Guides :1
6.C.1.e. Add and subtract proper fractions and mixed numbers (Assessment limit: Use 2 proper fractions with a single digit like denominators, 2 mixed numbers with single digit like denominators, or a
whole number and a proper fraction with a single digit denominator and numbers (0 - 20)).
Add/Subtract FractionsFreeis one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two whole numbers is the total amount of
those quantities combined. Read more...iWorksheets :3Study Guides :1
Subtracting FractionsFractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. First, make sure the denominators are the same, then
subtract the numerators. Read more...iWorksheets :3Study Guides :1
Adding FractionsFractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. To add two fractions with the same denominator: Add the
numerators and place the sum over the common denominator. Read more...iWorksheets :3Study Guides :1
6.C.1.f. Add 2 decimals (Assessment limit: Use the same number of decimal places but no more than 2 decimal places and no more than 4 digits including monetary notation and numbers (0 - 100)).
Add/Subtract DecimalsAddition and subtraction of decimals is like adding and subtracting whole numbers. The only thing we must remember is to line up the place values correctly. Read more...i
Worksheets :14Study Guides :1Vocabulary :1
MultiplicationMultiplication is one of the four elementary, mathematical operations of arithmetic. Read more...iWorksheets :7Study Guides :1Vocabulary :1
Adding MoneyAmounts of money may be written in several different ways. Cents may be written with the ¢ sign and dollars can be written with the dollar sign ($). When we add money, we add the amounts
and place the correct sign on the sum. Read more...iWorksheets :4Study Guides :1
MoneyFreeWhat Is Making Change? Making change means giving money back to someone after they have made a purchase and paid more than they owed. This is done using banknotes and coins. You can
subtract, add, multiply, and divide money when making change. Read more...iWorksheets :7Study Guides :1
Counting MoneyFreeWhat Is Money? Money is what we use to make purchases for our needs and wants. Read more...iWorksheets :10Study Guides :1
DecimalsREADING, WRITING, COMPARING, AND ORDERING DECIMALS Read more...iWorksheets :5Study Guides :1Vocabulary :1
Add/Subtract/Multiply/Divide DecimalsYou add/subtract/multiply/divide decimals the same way you add/subtract/multiply/divide whole numbers BUT you also need to place the decimal in the correct spot.
When multiplying decimals, the decimals may or may NOT be lined up in the multiplication problem. Read more...iWorksheets :10Study Guides :1Vocabulary :1
Giving Change from $1.00What Is Giving Change? Change is the money you receive back when you purchase an item and give the cashier more than the item cost. To figure out the change you will receive
from a purchase, simply subtract the total amount of the purchase from the amount you are giving the cashier. Read more...iWorksheets :4Study Guides :1
6.C.1.g. Subtract decimals (Assessment limit: Use the same number of decimal places but no more than 2 decimal places and no more than 4 digits including monetary notation and numbers (0 - 100)).
Add/Subtract DecimalsAddition and subtraction of decimals is like adding and subtracting whole numbers. The only thing we must remember is to line up the place values correctly. Read more...i
Worksheets :14Study Guides :1Vocabulary :1
Add/Subtract/Multiply/Divide DecimalsYou add/subtract/multiply/divide decimals the same way you add/subtract/multiply/divide whole numbers BUT you also need to place the decimal in the correct spot.
When multiplying decimals, the decimals may or may NOT be lined up in the multiplication problem. Read more...iWorksheets :10Study Guides :1Vocabulary :1
6.C.2. Number Computation: Estimation.
6.C.2.a. Determine the approximate sum and difference of 2 numbers (Assessment limit: Use no more than 2 decimal places in each and numbers (0 - 100)).
EstimationWhen you make an estimate, you are making a guess that is approximate. This is often done by rounding. Read more...iWorksheets :6Study Guides :1
EstimationFreeTo estimate means to make an educated guess based on what you already know. Read more...iWorksheets :4Study Guides :1
6.C.2.b. Determine the approximate product or quotient of 2 numbers (Assessment limit: Use a 1-digit factor with the other factor having no more than 2-digits or a 1-digit divisor and no more than a
2-digit dividend and whole numbers (0 - 1000)).
EstimationWhen you make an estimate, you are making a guess that is approximate. This is often done by rounding. Read more...iWorksheets :6Study Guides :1
EstimationFreeTo estimate means to make an educated guess based on what you already know. Read more...iWorksheets :4Study Guides :1
MD.7.0. Processes of Mathematics: Students demonstrate the processes of mathematics by making connections and applying reasoning to solve and to communicate their findings.
7.A.1. Problem solving: Apply a variety of concepts, processes, and skills to solve problems.
7.A.1.c. Make a plan to solve a problem.
Problem SolvingWhat Is Problem Solving? Problem solving is finding an answer to a question. How to Problem Solve: Read the problem carefully. Decide on an operation to use to solve the problem. Solve
the problem. Check your work and make sure that your answer makes sense. Read more...iWorksheets :4Study Guides :1
Word ProblemsWhat Are Story Problems? Story problems are a bunch of sentences set up to give you information in order to solve a problem. Story problems most often give you all the information needed
to solve the problem. They may even include information you do not need at all. Read more...iWorksheets :15Study Guides :1
7.A.1.d. Apply a strategy, i.e., draw a picture, guess and check, finding a pattern, writing an equation.
PatternsA pattern is an order of things repeated over and over. Read more...iWorksheets :6Study Guides :1
Evaluate Open SentencesAlgebra is a study of the properties of operations on numbers. Algebra generalizes math by using symbols or letters to represent numbers. Read more...iWorksheets :3Study Guides
:1Vocabulary :1
PatternsA pattern is a recognizable, consistent series of numbers, shapes, or images. Read more...iWorksheets :4Study Guides :1
Problem SolvingWhat Is Problem Solving? Problem solving is finding an answer to a question. How to Problem Solve: Read the problem carefully. Decide on an operation to use to solve the problem. Solve
the problem. Check your work and make sure that your answer makes sense. Read more...iWorksheets :4Study Guides :1
Open Number SentencesWhat Are Open Number Sentences? Open number sentences are equations that give one part of the equation along with the answer. In order to solve an open number sentence, the
inverse operation is used. Read more...iWorksheets :3Study Guides :1
Word ProblemsWhat Are Story Problems? Story problems are a bunch of sentences set up to give you information in order to solve a problem. Story problems most often give you all the information needed
to solve the problem. They may even include information you do not need at all. Read more...iWorksheets :15Study Guides :1
7.A.1.e. Select a strategy, i.e., draw a picture, guess and check, finding a pattern, writing an equation.
Problem SolvingWhat Is Problem Solving? Problem solving is finding an answer to a question. How to Problem Solve: Read the problem carefully. Decide on an operation to use to solve the problem. Solve
the problem. Check your work and make sure that your answer makes sense. Read more...iWorksheets :4Study Guides :1
Word ProblemsWhat Are Story Problems? Story problems are a bunch of sentences set up to give you information in order to solve a problem. Story problems most often give you all the information needed
to solve the problem. They may even include information you do not need at all. Read more...iWorksheets :15Study Guides :1
7.A.1.f. Identify alternative ways to solve a problem.
Problem SolvingWhat Is Problem Solving? Problem solving is finding an answer to a question. How to Problem Solve: Read the problem carefully. Decide on an operation to use to solve the problem. Solve
the problem. Check your work and make sure that your answer makes sense. Read more...iWorksheets :4Study Guides :1
Word ProblemsWhat Are Story Problems? Story problems are a bunch of sentences set up to give you information in order to solve a problem. Story problems most often give you all the information needed
to solve the problem. They may even include information you do not need at all. Read more...iWorksheets :15Study Guides :1
7.C.1. Communication: Present mathematical ideas using words, symbols, visual displays, or technology.
7.C.1.a. Use multiple representations to express concepts or solutions.
PatternsA pattern is an order of things repeated over and over. Read more...iWorksheets :6Study Guides :1
Add/Subtract DecimalsAddition and subtraction of decimals is like adding and subtracting whole numbers. The only thing we must remember is to line up the place values correctly. Read more...i
Worksheets :14Study Guides :1Vocabulary :1
MultiplicationMultiplication is one of the four elementary, mathematical operations of arithmetic. Read more...iWorksheets :7Study Guides :1Vocabulary :1
Add/Subtract FractionsFreeis one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two whole numbers is the total amount of
those quantities combined. Read more...iWorksheets :3Study Guides :1
Place ValuePlace value is the numerical value that a digit has by virtue of its position in a number. Read more...iWorksheets :6Study Guides :1
Evaluate Open SentencesAlgebra is a study of the properties of operations on numbers. Algebra generalizes math by using symbols or letters to represent numbers. Read more...iWorksheets :3Study Guides
:1Vocabulary :1
AlgebraComparing whole numbers, fractions, and decimals means looking at the values of two numbers and deciding if they are greater than, less than or equal to each other. Read more...iWorksheets :4
Study Guides :1
Ordering DecimalsWhen putting decimals in order from least to greatest, we must look at the highest place value first. Read more...iWorksheets :7Study Guides :1Vocabulary :1
Compare and Order FractionsWhen comparing two fractions that have a common denominator, you can looks at the numerators to decide which fraction is greater Read more...iWorksheets :4Study Guides :1
Vocabulary :1
Common FactorsFactors are two numbers multiplied together to get a product (an answer to a multiplication problem) Read more...iWorksheets :6Study Guides :1Vocabulary :1
PercentsA percentage is a number or ratio expressed as a fraction of 100. Read more...iWorksheets :6Study Guides :1Vocabulary :1
EstimationWhen you make an estimate, you are making a guess that is approximate. This is often done by rounding. Read more...iWorksheets :6Study Guides :1
RoundingRounding makes numbers that are easier to work with in your head. Rounded numbers are only approximate. Use rounding to get an answer that is close but that does not have to be exact. Read
more...iWorksheets :3Study Guides :1
RatioRatios are used to make a comparison between two things. Read more...iWorksheets :9Study Guides :1Vocabulary :1
Adding MoneyAmounts of money may be written in several different ways. Cents may be written with the ¢ sign and dollars can be written with the dollar sign ($). When we add money, we add the amounts
and place the correct sign on the sum. Read more...iWorksheets :4Study Guides :1
3 Digit AdditionFreeAdding large numbers involves breaking the problem down into smaller addition facts. Read more...iWorksheets :4Study Guides :1
3 Digit SubtractionWhat Is Three-Digit Subtraction? We subtract to compare numbers. We are able to find the difference between numbers through subtraction. We use subtraction to find out how much
more we have or how much smaller something is in comparison to another number. Read more...iWorksheets :3Study Guides :1
ProbabilityFreeProbability word problems worksheet. Probability is the measure of how likely an event is. Probability = (Total ways a specific outcome will happen) / (Total number of possible
outcomes). The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. Read more...iWorksheets :4Study Guides :1
FractionsFractions can show a part of a group or part of a set. Read more...iWorksheets :6Study Guides :1
ProbabilityProbability word problems worksheet. Probability is the chance of whether something will happen or not. If two things have an EQUAL chance of happening, they have the SAME probability. If
there are MORE chances of something happening (A) than something else (B), that means there is a HIGHER PROBABILITY of that something (A) happening. Read more...iWorksheets :3Study Guides :1
Positive & Negative IntegersPositive integers are all the whole numbers greater than zero. Negative integers are all the opposites of these whole numbers, numbers that are less than zero. Zero is
considered neither positive nor negative Read more...iWorksheets :4Study Guides :1
Ordering FractionsA fraction consists of two numbers separated by a line - numerator and denominator. To order fractions with like numerators, look at the denominators and compare them two at a time.
The fraction with the smaller denominator is the larger fraction. Read more...iWorksheets :3Study Guides :1
Subtracting FractionsFractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. First, make sure the denominators are the same, then
subtract the numerators. Read more...iWorksheets :3Study Guides :1
Fractions/DecimalsHow to convert fractions to decimals: Divide the denominator (the bottom part) into the numerator (the top part). Read more...iWorksheets :3Study Guides :1
Odd/EvenA number can be identified as odd or even. Odd numbers can't be divided exactly by 2. Read more...iWorksheets :3Study Guides :1
MoneyFreeWhat Is Making Change? Making change means giving money back to someone after they have made a purchase and paid more than they owed. This is done using banknotes and coins. You can
subtract, add, multiply, and divide money when making change. Read more...iWorksheets :7Study Guides :1
Counting MoneyFreeWhat Is Money? Money is what we use to make purchases for our needs and wants. Read more...iWorksheets :10Study Guides :1
Commutative PropertyThe commutative property of addition says that we can add numbers in any order and get the same sum. Read more...iWorksheets :3Study Guides :1
Number LineA number line is a line that shows any group of numbers in their least to greatest value. Read more...iWorksheets :3Study Guides :1
Greater Than/Less ThanIf a number is greater than another number that means it is higher in value than the other number. If a number is less than another number that means it is lower in value than
the other number. Read more...iWorksheets :4Study Guides :1
EstimationFreeTo estimate means to make an educated guess based on what you already know. Read more...iWorksheets :4Study Guides :1
Skip CountingYou can skip count by large numbers such as 25, 50 or 100. Skip counting allows you to count by large numbers following a pattern. Read more...iWorksheets :3Study Guides :1
DivisionDivide three-digit numbers by one- and two-digit numbers. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Commutative/Associative PropertiesUsing the Commutative Property in addition means that the order of addends does not matter; the sum will remain the same. Read more...iWorksheets :3Study Guides :1
Comparing FractionsWhen comparing fractions, you are finding which fraction is greater and which fractions is less than the other. Similar to comparing numbers, there are symbols to use when
comparing fractions. Read more...iWorksheets :5Study Guides :1Vocabulary :1
DecimalsREADING, WRITING, COMPARING, AND ORDERING DECIMALS Read more...iWorksheets :5Study Guides :1Vocabulary :1
Add/Subtract/Multiply/Divide DecimalsYou add/subtract/multiply/divide decimals the same way you add/subtract/multiply/divide whole numbers BUT you also need to place the decimal in the correct spot.
When multiplying decimals, the decimals may or may NOT be lined up in the multiplication problem. Read more...iWorksheets :10Study Guides :1Vocabulary :1
PatternsA pattern is a recognizable, consistent series of numbers, shapes, or images. Read more...iWorksheets :4Study Guides :1
Ordering and Comparing NumbersWhen you order numbers, you are putting the numbers in a sequence from the smallest value to the largest value. When you compare two numbers, you are finding which
number is larger or smaller than the other. Read more...iWorksheets :5Study Guides :1Vocabulary :1
Compare and Order NumbersWhat is comparing and ordering numbers? Ordering numbers means listing numbers from least to greatest, or greatest to least. Comparing numbers means looking at the values of
two numbers and deciding if the numbers are greater than, less than, or equal to each other. Read more...iWorksheets :4Study Guides :1
Decimals/FractionsExpress decimals as an equivalent form of fractions to tenths and hundredths. Read more...iWorksheets :5Study Guides :1Vocabulary :1
PercentsWhen there are one HUNDRED equal parts of something, you can find a PERCENT. Read more...iWorksheets :3Study Guides :1
Rounding to Nearest 10Rounding makes numbers easier to work with if you do not need an exact number. Rounded numbers are only approximate. You can use rounded numbers to get an answer that is close
but does not have to be exact. Read more...iWorksheets :3Study Guides :1
Double Digit SubtractionWhat Is Double Digit Subtraction? Double digit subtraction is taking a number with two digits (ex. 23) and subtracting it from another two digit number (ex. 33). The answer is
known as the difference. Read more...iWorksheets :4Study Guides :1
Rounding NumbersWhat Is Rounding? Rounding means reducing the digits in a number while trying to keep its value similar. How to Round: The number in the given place is increased by one if the digit
to its right is 5 or greater. The number in the given place remains the same if the digit to its right is less than 5. Read more...iWorksheets :3Study Guides :1Vocabulary :1
Place ValueWhat Is Place Value? In our decimal number system, the value of a digit depends on its place, or position, in the number. Beginning with the ones place at the right, each place value is
multiplied by increasing powers of 10. Read more...iWorksheets :4Study Guides :1Vocabulary :1
More MultiplicationMultiplication of two digits by two digits. What Is Multiplication? Multiplication is a short way of adding or counting. Multiplication is a faster way of adding. By multiplying
numbers together, you are adding a series of one number to itself. Read more...iWorksheets :3Study Guides :1
MultiplicationWhat Is Multiplication? Multiplication is a short way of adding or counting. Multiplication is a faster way of adding by using strategies to remember what different groups of each
number equal. By multiplying numbers together, you are adding a series of one number to itself. The answer to a multiplication problem is called a product. Read more...iWorksheets :10Study Guides :1
Add/Subtract FractionsWhat Is Addition and Subtraction of Fractions? Addition is combining two or more fractions. The term used for addition is plus. When two or more numbers, or addends, are
combined they form a new number called a sum. Subtraction is “taking away” one fraction from another fraction. The term is minus. The number left after subtracting is called a difference. Read
more...iWorksheets :4Study Guides :1
Addition/SubtractionAddition is combining two or more numbers. The term used for addition is plus. When two or more numbers are combined they form a new number called a sum. Subtraction is “taking
away” one number from another. The term is minus. The number left after subtracting is called a difference. Read more...iWorksheets :11Study Guides :1
DivisionWhat Is Division? Division is splitting up numbers into equal parts. The process of finding out how many times one number will go into another number. Division is a series of repeated
subtraction. The parts of a division problem include the divisor, dividend, quotient and remainder. Read more...iWorksheets :8Study Guides :1
Division/MultiplicationUnderstanding of models for multiplication, place value, and properties of operations (in particular, the distributive property). Read more...iWorksheets :9Study Guides :1
Double Digit AdditionWhat Is Double Digit Addition? Double digit addition is taking a two digit number (ex. 32) and adding it to another two digit number (ex. 27). The answer of these two addends is
known as the sum. Read more...iWorksheets :4Study Guides :1
Giving Change from $1.00What Is Giving Change? Change is the money you receive back when you purchase an item and give the cashier more than the item cost. To figure out the change you will receive
from a purchase, simply subtract the total amount of the purchase from the amount you are giving the cashier. Read more...iWorksheets :4Study Guides :1
RegroupingWhat Is Regrouping? Regrouping in addition is used when the sum of the ones place is larger than nine. The tens place of the sum is moved to the top of the tens place column to be added
with the others. Read more...iWorksheets :3Study Guides :1
Associative PropertyAssociative Property of Addition explains that when three or more numbers are added, the sum is the same regardless of the order in which the numbers are grouped and/or added.
Read more...iWorksheets :3Study Guides :1
Equivalent Fractions to 1/2Fractions that are equivalent to ½ are fractions that have different denominators than ½, but still show half. Fractions that are equivalent to ½ can be simplified to ½.
Fractions equivalent to ½ have an even number as their denominator. Read more...iWorksheets :3Study Guides :1
Odd/Even NumbersWhat is odd number? An odd number is a number that will have a leftover when divided into two equal groups. What is even number? An even number is a number that can be divided into
two equal groups without any leftovers. Read more...iWorksheets :6Study Guides :1
Open Number SentencesWhat Are Open Number Sentences? Open number sentences are equations that give one part of the equation along with the answer. In order to solve an open number sentence, the
inverse operation is used. Read more...iWorksheets :3Study Guides :1
MultiplicationMultiplication is similar to adding a number to itself a certain number of times. When multiplying an odd number with an odd number, the product is always an odd number. When
multiplying an odd number with an even number or two even numbers, the product is always an even number. Read more...iWorksheets :19Study Guides :1
Greater Than/Less ThanWhat Is Greater Than and Less Than? When a number is greater than another number, this means it is a larger number. The symbol for greater than is >. When a number is less than
another number, this means it is a smaller number. The symbol for less than is <. Read more...iWorksheets :6Study Guides :1
Expanding NumbersWhat Are Expanding Numbers? An expanding number is taking a larger number apart and showing each number’s total value. Number 5398 in expanded form is 5000 + 300 + 90 + 8. Read
more...iWorksheets :3Study Guides :1
Word ProblemsWhat Are Story Problems? Story problems are a bunch of sentences set up to give you information in order to solve a problem. Story problems most often give you all the information needed
to solve the problem. They may even include information you do not need at all. Read more...iWorksheets :15Study Guides :1
Place ValuePlace value is what each digit is worth. In the number 4,573 there are four thousands, five hundreds, seven tens, and three ones. How to Find the Place Value: In order to find the place
value of a number, you can count the number of places from the right. The first number will be the ones place. The next number moving towards the left would be the tens place, and so on. Read more...
iWorksheets :10Study Guides :1
DivisionFreeWhat Is Division? Division is an operation that tells: how many equal sized groups, how many in each group. The number you divide by is called the DIVISOR. The number you are dividing is
called the DIVIDEND. And the answer is called the QUOTIENT. Read more...iWorksheets :6Study Guides :1
FractionsThe top number of a fraction is called the numerator. It shows how many pieces of a whole we are talking about. The bottom number is called the denominator. It shows how many pieces an
object was divided into, or how many total pieces we have. Read more...iWorksheets :4Study Guides :1
Number Words and Place ValueWhen we write numbers, the position of each digit is important. Each position is 10 more than the one before it. So, 23 means “add 2*10 to 3*1″. In the number 467: the "7"
is in the Ones position, meaning 7 ones, the "6" is in the Tens position meaning 6 tens, and the "4" is in the Hundreds position. Read more...iWorksheets :3Study Guides :1
Adding FractionsFractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. To add two fractions with the same denominator: Add the
numerators and place the sum over the common denominator. Read more...iWorksheets :3Study Guides :1
Order of OperationsFreeRules of Order of Operations: 1st: Compute all operations inside of parentheses. 2nd: Compute all work with exponents. 3rd: Compute all multiplication and division from left to
right. 4th: Compute all addition and subtraction from left to right. Read more...iWorksheets :4Study Guides :1
7.C.1.d. Express solutions using concrete materials.
PatternsA pattern is an order of things repeated over and over. Read more...iWorksheets :6Study Guides :1
Add/Subtract DecimalsAddition and subtraction of decimals is like adding and subtracting whole numbers. The only thing we must remember is to line up the place values correctly. Read more...i
Worksheets :14Study Guides :1Vocabulary :1
MultiplicationMultiplication is one of the four elementary, mathematical operations of arithmetic. Read more...iWorksheets :7Study Guides :1Vocabulary :1
Add/Subtract FractionsFreeis one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two whole numbers is the total amount of
those quantities combined. Read more...iWorksheets :3Study Guides :1
Place ValuePlace value is the numerical value that a digit has by virtue of its position in a number. Read more...iWorksheets :6Study Guides :1
Evaluate Open SentencesAlgebra is a study of the properties of operations on numbers. Algebra generalizes math by using symbols or letters to represent numbers. Read more...iWorksheets :3Study Guides
:1Vocabulary :1
AlgebraComparing whole numbers, fractions, and decimals means looking at the values of two numbers and deciding if they are greater than, less than or equal to each other. Read more...iWorksheets :4
Study Guides :1
Ordering DecimalsWhen putting decimals in order from least to greatest, we must look at the highest place value first. Read more...iWorksheets :7Study Guides :1Vocabulary :1
Compare and Order FractionsWhen comparing two fractions that have a common denominator, you can looks at the numerators to decide which fraction is greater Read more...iWorksheets :4Study Guides :1
Vocabulary :1
Common FactorsFactors are two numbers multiplied together to get a product (an answer to a multiplication problem) Read more...iWorksheets :6Study Guides :1Vocabulary :1
PercentsA percentage is a number or ratio expressed as a fraction of 100. Read more...iWorksheets :6Study Guides :1Vocabulary :1
EstimationWhen you make an estimate, you are making a guess that is approximate. This is often done by rounding. Read more...iWorksheets :6Study Guides :1
RoundingRounding makes numbers that are easier to work with in your head. Rounded numbers are only approximate. Use rounding to get an answer that is close but that does not have to be exact. Read
more...iWorksheets :3Study Guides :1
RatioRatios are used to make a comparison between two things. Read more...iWorksheets :9Study Guides :1Vocabulary :1
Adding MoneyAmounts of money may be written in several different ways. Cents may be written with the ¢ sign and dollars can be written with the dollar sign ($). When we add money, we add the amounts
and place the correct sign on the sum. Read more...iWorksheets :4Study Guides :1
3 Digit AdditionFreeAdding large numbers involves breaking the problem down into smaller addition facts. Read more...iWorksheets :4Study Guides :1
3 Digit SubtractionWhat Is Three-Digit Subtraction? We subtract to compare numbers. We are able to find the difference between numbers through subtraction. We use subtraction to find out how much
more we have or how much smaller something is in comparison to another number. Read more...iWorksheets :3Study Guides :1
ProbabilityFreeProbability word problems worksheet. Probability is the measure of how likely an event is. Probability = (Total ways a specific outcome will happen) / (Total number of possible
outcomes). The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. Read more...iWorksheets :4Study Guides :1
FractionsFractions can show a part of a group or part of a set. Read more...iWorksheets :6Study Guides :1
ProbabilityProbability word problems worksheet. Probability is the chance of whether something will happen or not. If two things have an EQUAL chance of happening, they have the SAME probability. If
there are MORE chances of something happening (A) than something else (B), that means there is a HIGHER PROBABILITY of that something (A) happening. Read more...iWorksheets :3Study Guides :1
Positive & Negative IntegersPositive integers are all the whole numbers greater than zero. Negative integers are all the opposites of these whole numbers, numbers that are less than zero. Zero is
considered neither positive nor negative Read more...iWorksheets :4Study Guides :1
Ordering FractionsA fraction consists of two numbers separated by a line - numerator and denominator. To order fractions with like numerators, look at the denominators and compare them two at a time.
The fraction with the smaller denominator is the larger fraction. Read more...iWorksheets :3Study Guides :1
Subtracting FractionsFractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. First, make sure the denominators are the same, then
subtract the numerators. Read more...iWorksheets :3Study Guides :1
Fractions/DecimalsHow to convert fractions to decimals: Divide the denominator (the bottom part) into the numerator (the top part). Read more...iWorksheets :3Study Guides :1
Odd/EvenA number can be identified as odd or even. Odd numbers can't be divided exactly by 2. Read more...iWorksheets :3Study Guides :1
MoneyFreeWhat Is Making Change? Making change means giving money back to someone after they have made a purchase and paid more than they owed. This is done using banknotes and coins. You can
subtract, add, multiply, and divide money when making change. Read more...iWorksheets :7Study Guides :1
Counting MoneyFreeWhat Is Money? Money is what we use to make purchases for our needs and wants. Read more...iWorksheets :10Study Guides :1
Commutative PropertyThe commutative property of addition says that we can add numbers in any order and get the same sum. Read more...iWorksheets :3Study Guides :1
Number LineA number line is a line that shows any group of numbers in their least to greatest value. Read more...iWorksheets :3Study Guides :1
Greater Than/Less ThanIf a number is greater than another number that means it is higher in value than the other number. If a number is less than another number that means it is lower in value than
the other number. Read more...iWorksheets :4Study Guides :1
EstimationFreeTo estimate means to make an educated guess based on what you already know. Read more...iWorksheets :4Study Guides :1
Skip CountingYou can skip count by large numbers such as 25, 50 or 100. Skip counting allows you to count by large numbers following a pattern. Read more...iWorksheets :3Study Guides :1
DivisionDivide three-digit numbers by one- and two-digit numbers. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Commutative/Associative PropertiesUsing the Commutative Property in addition means that the order of addends does not matter; the sum will remain the same. Read more...iWorksheets :3Study Guides :1
Comparing FractionsWhen comparing fractions, you are finding which fraction is greater and which fractions is less than the other. Similar to comparing numbers, there are symbols to use when
comparing fractions. Read more...iWorksheets :5Study Guides :1Vocabulary :1
DecimalsREADING, WRITING, COMPARING, AND ORDERING DECIMALS Read more...iWorksheets :5Study Guides :1Vocabulary :1
Add/Subtract/Multiply/Divide DecimalsYou add/subtract/multiply/divide decimals the same way you add/subtract/multiply/divide whole numbers BUT you also need to place the decimal in the correct spot.
When multiplying decimals, the decimals may or may NOT be lined up in the multiplication problem. Read more...iWorksheets :10Study Guides :1Vocabulary :1
PatternsA pattern is a recognizable, consistent series of numbers, shapes, or images. Read more...iWorksheets :4Study Guides :1
Ordering and Comparing NumbersWhen you order numbers, you are putting the numbers in a sequence from the smallest value to the largest value. When you compare two numbers, you are finding which
number is larger or smaller than the other. Read more...iWorksheets :5Study Guides :1Vocabulary :1
Compare and Order NumbersWhat is comparing and ordering numbers? Ordering numbers means listing numbers from least to greatest, or greatest to least. Comparing numbers means looking at the values of
two numbers and deciding if the numbers are greater than, less than, or equal to each other. Read more...iWorksheets :4Study Guides :1
Decimals/FractionsExpress decimals as an equivalent form of fractions to tenths and hundredths. Read more...iWorksheets :5Study Guides :1Vocabulary :1
PercentsWhen there are one HUNDRED equal parts of something, you can find a PERCENT. Read more...iWorksheets :3Study Guides :1
Rounding to Nearest 10Rounding makes numbers easier to work with if you do not need an exact number. Rounded numbers are only approximate. You can use rounded numbers to get an answer that is close
but does not have to be exact. Read more...iWorksheets :3Study Guides :1
Double Digit SubtractionWhat Is Double Digit Subtraction? Double digit subtraction is taking a number with two digits (ex. 23) and subtracting it from another two digit number (ex. 33). The answer is
known as the difference. Read more...iWorksheets :4Study Guides :1
Rounding NumbersWhat Is Rounding? Rounding means reducing the digits in a number while trying to keep its value similar. How to Round: The number in the given place is increased by one if the digit
to its right is 5 or greater. The number in the given place remains the same if the digit to its right is less than 5. Read more...iWorksheets :3Study Guides :1Vocabulary :1
Place ValueWhat Is Place Value? In our decimal number system, the value of a digit depends on its place, or position, in the number. Beginning with the ones place at the right, each place value is
multiplied by increasing powers of 10. Read more...iWorksheets :4Study Guides :1Vocabulary :1
More MultiplicationMultiplication of two digits by two digits. What Is Multiplication? Multiplication is a short way of adding or counting. Multiplication is a faster way of adding. By multiplying
numbers together, you are adding a series of one number to itself. Read more...iWorksheets :3Study Guides :1
MultiplicationWhat Is Multiplication? Multiplication is a short way of adding or counting. Multiplication is a faster way of adding by using strategies to remember what different groups of each
number equal. By multiplying numbers together, you are adding a series of one number to itself. The answer to a multiplication problem is called a product. Read more...iWorksheets :10Study Guides :1
Add/Subtract FractionsWhat Is Addition and Subtraction of Fractions? Addition is combining two or more fractions. The term used for addition is plus. When two or more numbers, or addends, are
combined they form a new number called a sum. Subtraction is “taking away” one fraction from another fraction. The term is minus. The number left after subtracting is called a difference. Read
more...iWorksheets :4Study Guides :1
Addition/SubtractionAddition is combining two or more numbers. The term used for addition is plus. When two or more numbers are combined they form a new number called a sum. Subtraction is “taking
away” one number from another. The term is minus. The number left after subtracting is called a difference. Read more...iWorksheets :11Study Guides :1
DivisionWhat Is Division? Division is splitting up numbers into equal parts. The process of finding out how many times one number will go into another number. Division is a series of repeated
subtraction. The parts of a division problem include the divisor, dividend, quotient and remainder. Read more...iWorksheets :8Study Guides :1
Division/MultiplicationUnderstanding of models for multiplication, place value, and properties of operations (in particular, the distributive property). Read more...iWorksheets :9Study Guides :1
Double Digit AdditionWhat Is Double Digit Addition? Double digit addition is taking a two digit number (ex. 32) and adding it to another two digit number (ex. 27). The answer of these two addends is
known as the sum. Read more...iWorksheets :4Study Guides :1
Giving Change from $1.00What Is Giving Change? Change is the money you receive back when you purchase an item and give the cashier more than the item cost. To figure out the change you will receive
from a purchase, simply subtract the total amount of the purchase from the amount you are giving the cashier. Read more...iWorksheets :4Study Guides :1
RegroupingWhat Is Regrouping? Regrouping in addition is used when the sum of the ones place is larger than nine. The tens place of the sum is moved to the top of the tens place column to be added
with the others. Read more...iWorksheets :3Study Guides :1
Associative PropertyAssociative Property of Addition explains that when three or more numbers are added, the sum is the same regardless of the order in which the numbers are grouped and/or added.
Read more...iWorksheets :3Study Guides :1
Equivalent Fractions to 1/2Fractions that are equivalent to ½ are fractions that have different denominators than ½, but still show half. Fractions that are equivalent to ½ can be simplified to ½.
Fractions equivalent to ½ have an even number as their denominator. Read more...iWorksheets :3Study Guides :1
Odd/Even NumbersWhat is odd number? An odd number is a number that will have a leftover when divided into two equal groups. What is even number? An even number is a number that can be divided into
two equal groups without any leftovers. Read more...iWorksheets :6Study Guides :1
Open Number SentencesWhat Are Open Number Sentences? Open number sentences are equations that give one part of the equation along with the answer. In order to solve an open number sentence, the
inverse operation is used. Read more...iWorksheets :3Study Guides :1
MultiplicationMultiplication is similar to adding a number to itself a certain number of times. When multiplying an odd number with an odd number, the product is always an odd number. When
multiplying an odd number with an even number or two even numbers, the product is always an even number. Read more...iWorksheets :19Study Guides :1
Greater Than/Less ThanWhat Is Greater Than and Less Than? When a number is greater than another number, this means it is a larger number. The symbol for greater than is >. When a number is less than
another number, this means it is a smaller number. The symbol for less than is <. Read more...iWorksheets :6Study Guides :1
Expanding NumbersWhat Are Expanding Numbers? An expanding number is taking a larger number apart and showing each number’s total value. Number 5398 in expanded form is 5000 + 300 + 90 + 8. Read
more...iWorksheets :3Study Guides :1
Word ProblemsWhat Are Story Problems? Story problems are a bunch of sentences set up to give you information in order to solve a problem. Story problems most often give you all the information needed
to solve the problem. They may even include information you do not need at all. Read more...iWorksheets :15Study Guides :1
Place ValuePlace value is what each digit is worth. In the number 4,573 there are four thousands, five hundreds, seven tens, and three ones. How to Find the Place Value: In order to find the place
value of a number, you can count the number of places from the right. The first number will be the ones place. The next number moving towards the left would be the tens place, and so on. Read more...
iWorksheets :10Study Guides :1
DivisionFreeWhat Is Division? Division is an operation that tells: how many equal sized groups, how many in each group. The number you divide by is called the DIVISOR. The number you are dividing is
called the DIVIDEND. And the answer is called the QUOTIENT. Read more...iWorksheets :6Study Guides :1
FractionsThe top number of a fraction is called the numerator. It shows how many pieces of a whole we are talking about. The bottom number is called the denominator. It shows how many pieces an
object was divided into, or how many total pieces we have. Read more...iWorksheets :4Study Guides :1
Number Words and Place ValueWhen we write numbers, the position of each digit is important. Each position is 10 more than the one before it. So, 23 means “add 2*10 to 3*1″. In the number 467: the "7"
is in the Ones position, meaning 7 ones, the "6" is in the Tens position meaning 6 tens, and the "4" is in the Hundreds position. Read more...iWorksheets :3Study Guides :1
Adding FractionsFractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. To add two fractions with the same denominator: Add the
numerators and place the sum over the common denominator. Read more...iWorksheets :3Study Guides :1
Order of OperationsFreeRules of Order of Operations: 1st: Compute all operations inside of parentheses. 2nd: Compute all work with exponents. 3rd: Compute all multiplication and division from left to
right. 4th: Compute all addition and subtraction from left to right. Read more...iWorksheets :4Study Guides :1
7.C.1.e. Express solutions using pictorial, tabular, graphical, or algebraic methods.
Evaluate Open SentencesAlgebra is a study of the properties of operations on numbers. Algebra generalizes math by using symbols or letters to represent numbers. Read more...iWorksheets :3Study Guides
:1Vocabulary :1
StatisticsThe statistical mode is the number that occurs most frequently in a set of numbers. Read more...iWorksheets :3Study Guides :1
Graphs and TablesUsing tables and graphs is a way people can interpret data. Data means information. So interpreting data just means working out what information is telling you. Information is
sometimes shown in tables, charts and graphs to make the information easier to read. Read more...iWorksheets :3Study Guides :1
Graphs and ChartsWhat Are Graphs? A way to show information in the form of shapes or pictures. Graphs show the relationship between two sets of information. There are many different types of graphs.
A few of them include bar graphs, line graphs, pictographs, and circle graphs. Read more...iWorksheets :10Study Guides :1Vocabulary :1
Tables and GraphsWhat Are Bar, Circle, and Line Graphs? Bar Graphs are used to compare data. A bar graph is used to show relationships between groups. Circle Graphs are also known as Pie graphs or
charts. They consist of a circle divided into parts. Line Graphs show gradual changes in data. Read more...iWorksheets :9Study Guides :1
Open Number SentencesWhat Are Open Number Sentences? Open number sentences are equations that give one part of the equation along with the answer. In order to solve an open number sentence, the
inverse operation is used. Read more...iWorksheets :3Study Guides :1
DivisionFreeWhat Is Division? Division is an operation that tells: how many equal sized groups, how many in each group. The number you divide by is called the DIVISOR. The number you are dividing is
called the DIVIDEND. And the answer is called the QUOTIENT. Read more...iWorksheets :6Study Guides :1
Data AnalysisCollecting Data. Data = information. You can collect data from other people using polls and surveys. Recording Data. You can record the numerical data you collected on a chart or graph:
bar graphs, pictographs, line graphs, pie charts, column charts. Read more...iWorksheets :6Study Guides :1 | {"url":"https://newpathworksheets.com/math/grade-4/maryland-standards","timestamp":"2024-11-04T11:09:39Z","content_type":"text/html","content_length":"326679","record_id":"<urn:uuid:2cc1e7d8-8520-498a-9016-07f4be834f54>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00285.warc.gz"} |
Derivative of sin(x) + cos(x) | AI Math Solver
Derivative of sin(x) + cos(x)
Published on October 10, 2024
This SEO description explains how to find the derivative of the sum of sine and cosine functions. The problem asks for the derivative of sin(x) + cos(x), and the solution demonstrates the application
of the sum rule and basic trigonometric derivatives to arrive at the answer: cos(x) - sin(x). | {"url":"https://www.aimathsolve.com/shares/derivative-of-sinx-cosx","timestamp":"2024-11-12T23:35:59Z","content_type":"text/html","content_length":"51717","record_id":"<urn:uuid:580675bc-b9c0-4770-ae32-825d149051d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00899.warc.gz"} |
How Much Drywall Calculator - CivilGang
What is a How Much Drywall Calculator?
A “How Much Drywall” calculator is a tool used to estimate the number of drywall sheets and the total square footage of drywall needed for a construction or renovation project based on room
Why is a How Much Drywall Calculator Used?
This calculator is used to efficiently plan and purchase the correct quantity of drywall material, ensuring that you have enough to cover the walls and ceilings of a room while minimizing waste.
How Much Drywall Calculator
Use this calculator to estimate the number of drywall sheets and the total square footage of drywall needed for your construction or renovation project. | {"url":"https://civil-gang.com/how-much-drywall-calculator/","timestamp":"2024-11-05T23:28:54Z","content_type":"text/html","content_length":"90000","record_id":"<urn:uuid:09a9b263-a062-4597-b16b-89c8f972f2be>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00325.warc.gz"} |
Set based logic programming
In a previous paper (Blair et al. 2001), the authors showed that the mechanism underlying Logic Programming can be extended to handle the situation where the atoms are interpreted as subsets of a
given space X. The view of a logic program as a one-step consequence operator along with the concepts of supported and stable model can be transferred to such situations. In this paper, we show that
we can further extend this paradigm by creating a new one-step consequence operator by composing the old one-step consequence operator with a monotonic idempotent operator (miop) in the space of all
subsets of X, 2 ^X . We call this extension set based logic programming. We show that such a set based formalism for logic programming naturally supports a variety of options. For example, if the
underlying space has a topology, one can insist that the new one-step consequence operator always produces a closed set or always produces an open set. The flexibility inherent in the semantics of
set based logic programs is due to both the range of natural choices available for specifying the semantics of negation, as well as the role of monotonic idempotent operators (miops) as parameters in
the semantics. This leads to a natural type of polymorphism for logic programming, i.e. the same logic program can produce a variety of outcomes depending on the miop associated with the semantics.
We develop a general framework for set based programming involving miops. Among the applications, we obtain integer-based representations of real continuous functions as stable models of a set based
logic program.
• Logic programming
• Miop-spatially augmented language
• Monotonic idempotent operator
• One-step consequence operator
ASJC Scopus subject areas
• Artificial Intelligence
• Applied Mathematics
Dive into the research topics of 'Set based logic programming'. Together they form a unique fingerprint. | {"url":"https://experts.syr.edu/en/publications/set-based-logic-programming","timestamp":"2024-11-11T17:55:45Z","content_type":"text/html","content_length":"51048","record_id":"<urn:uuid:d9128602-4972-4b94-a9de-84469f0c4c44>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00579.warc.gz"} |
Raspberry Pi in the Classroom
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9 Replies
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Raspberry Pi in the Classroom
I'm trying to wrap my head around ways to incorporate Raspberry Pi in the classroom. I teach math at the community college level and, at my campus, there is no place for a dedicated math computer
lab. Due to the low price of the RPi, I was considering having students get one. I was also thinking of making worksheets for my students as a notebook for them to 1) learn how to code and 2) learn
mathematics via Mathematica.
However, I haven't figured out the logistics yet. To my understanding, if a student wanted to use it in the classroom, they'd each need a monitor to hook it up to, correct?
I also read that Wolfram and Intel teamed up and will release a SD card that will have Mathematica on it. (http://company.wolfram.com/news/2014/wolfram-language-on-intel-edison/) Does anyone have
any idea how this would work? I'm imagining you could plug it into your computer and run Mathematica off of it. If so, then that would be a more practical way of getting Matthematica into the
classroom at my college.
Any thoughts or experiences with RPi or ideas of Intel's Edison card are greatly appreciated.
9 Replies
In another post, "How to Start Mathematica using VNC", I had a problem caused by a misprint in an autostart file. I have edited that post.
I found how to install VNC on the Pi here:
I bought VNC Viewer for iPad and got (free) VNC Viewer for Mac. Advantage is MMA licence is not needed.
One thing to keep in mind is that the FrontEnd performance on even the latest version RPi is not going to be very snappy compared to running on a desktop PC. That could be a factor in a classroom
setting where time may be limited. However, running only the MathKernel in a terminal is fairly responsive. If you don't have a RPi, you should get one to try it out first.
Hello again Mike,
Not exactly. But the remote notebook execution requires a notebook version of mathematica installed on your computer. Anyway, this method does not use VNC. I believe it uses SSH, and by letting
Mathematica know the IP address, username and password to the RPi, one can remotely execute mathematica code on the RPi through SSH. Mathematica presents the login procedure with a graphical user
interface, so it is quite straight forward.
Hope it helps :-)
I think I'd lead more towards using the RPi over the semester/annual edition due to both cost effectiveness and the RPi being able to commute and not have a time limit. But, I do appreciate the
suggestion! We did get a grant to get 30 liscenses for the college; however, we are a 4-campus college with many students. So if two classes want to use the software at the same time, we'd be out
of luck.
Simon, when you were discussing executing code via the notebook, were you referring to the VPN access that Simon posted earlier? Again, this is very new to me, but I am excited at where it can head.
Norma: I'll check my email! Thanks!!
Hi Mike,
I tutor at the same college as you. I do not have any experience with RPi, but have a bit with Arduino -- I don't know if any of that would be helpful.
I'd love to see if I can be helpful to you in investigating how to teach with Mathematica & RPi. I'll email your college address with my contact info.
Thanks, Norma
Diego is right both in terms of price and capabilities. I use Mathematica version for notebooks to solve demading engineering related mathematics, and the RPi for solving simpler mathematics related
to programming the Pi (if you know what I mean). However, if I were the student, I would pay more attention if I got to play around with an RPi also. Perhaps interactive demonstrations from the
Wolfram Demonstrations Project could also be used to spice things up a little, as an alternative to playing around with the RPi.
I would however encourage using the RPi as a teaching tool, just as much as standard Mathematica. If you run Mathematica on a notebook, you can also execute mathematica code remotely through the
computer, then you don't need the monitor.
What about the student version of mathematica? Got an semester license, annual and Standard. A semester license is 44.95, not to far off from the Raspi price.
Thus it can be used directly on their notebooks.
Thanks much for the help and support!
I'm new to all of this, so I read a little on how to connect via VNC. From some quick searches, there appears to be plenty of guides out there. I guess the next step is getting a RPi to "play"
Hello Mike,
Some years ago I was taught to solve mathematical problems (related to chemical engineering) through Mathematica, and today I am so greatful for that. At first I didn't appreciate the mix of
technology and mathematics, but I have become a supporter of it. Now I use Mathematica for pretty much everything I do. So I can only support your initiative.
Yes, they would need a monitor to attach to the RPi. But if they have laptop computers with them (or if you have a computer room available), you can connect to RPi through VNC. I do that at the
moment, which means I can use the monitor and keyboard of my laptop, to control the raspberry pi, as long as it is connected to my wireless network. Mathematica is now included in distributions of
Raspbian Wheezy and NOOBS (as far as I am aware), so if you can get that up and running, you should be good to go :-)
Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use | {"url":"https://community.wolfram.com/groups/-/m/t/192257","timestamp":"2024-11-07T18:33:51Z","content_type":"text/html","content_length":"138482","record_id":"<urn:uuid:34cb7d8c-d46a-4943-bbec-305b1c32df83>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00147.warc.gz"} |
Cost Per Credit Hour Calculator
Understanding the cost per credit hour can help students plan their education expenses more effectively. With this calculator, you can quickly estimate how much each credit hour will cost, including
tuition and any additional fees.
How to Calculate Cost Per Credit Hour
Follow these simple steps to determine your cost per credit hour:
1. Enter the total tuition cost for your program or semester.
2. Input the total number of credit hours you will be taking.
3. Optionally, add any additional fees, such as student services or activity fees.
4. Click 'Calculate Cost Per Credit Hour' to get your result.
Formula Used
The formula for calculating cost per credit hour is:
• Cost Per Credit Hour = (Total Tuition + Additional Fees) ÷ Total Credit Hours
This formula divides the total cost (including tuition and additional fees) by the total number of credit hours to give you the cost per credit hour.
Example Calculation
If your total tuition is $10,000, additional fees are $500, and you're taking 15 credit hours, the calculation would be:
Cost Per Credit Hour = ($10,000 + $500) ÷ 15 = $10,500 ÷ 15 = $700
This means each credit hour costs $700. | {"url":"https://profitcalculate.com/cost-per-credit-hour-calculator/","timestamp":"2024-11-12T09:37:06Z","content_type":"text/html","content_length":"27609","record_id":"<urn:uuid:6488303d-58d6-46c1-a9e0-d4aed6a42d39>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00710.warc.gz"} |
The state education commission wants to estimate the fraction of tenth grade students that have reading skills at or below the eighth grade level » MyMathLab Help| Pay Us to Do Your Statistics Online Today
The state education commission wants to estimate the fraction of tenth grade students that have reading skills at or below the eighth grade level. In an earlier study, the population proportion was
estimated to be 0.16.
How large a sample would be required in order to estimate the fraction of tenth graders reading at or below the eighth grade level at the 85% confidence level with an error of at most 0.03? Round
your answer up to the next integer.
Using the data, construct the 80% confidence interval for the population proportion of new car buyers who prefer foreign cars over domestic cars. Round your answers to three decimal places.
Add a new comment. | {"url":"https://www.mymathlabhomeworkhelp.com/mymathlabanswers/inferential-statistics-help/The-state-education-commission-wants-to-estimate-the-fraction-of-tenth-grade-students-that-have-reading-skills-at-or-below-the-e.html","timestamp":"2024-11-12T20:09:08Z","content_type":"text/html","content_length":"31065","record_id":"<urn:uuid:5229fa60-91cd-4425-9613-0f81dc98c2f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00293.warc.gz"} |
The Nine Chapters on the Mathematical Art
The Nine Chapters on the Mathematical Art (九章算術) is a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. This book is the earliest surviving
mathematical text from China that has come down to us by being copied by scribes and (centuries later) printed. It reveals an approach to mathematics that centres on finding the most general methods
of solving problems rather than on deducing propositions from an initial set of axioms, in the manner found amongst leading ancient Greek mathematicians.
Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution, and an explanation of the procedure that led to the solution.
Contents of the Nine Chapters are as follows
1. Fang tian - Rectangular fields Areas of fields of various shapes; manipulation of vulgar fractions.
2. Su mi - Millet and rice Exchange of commodities at different rates; pricing.
3. Cui fen - Proportional distribution Distribution of commodities and money at proportional rates.
4. Shao guang - The lesser breadth Division by mixed numbers; extraction of square and cube roots; dimensions, area and volume of circle and sphere.
5. Shang gong - Consultations on works Volumes of solids of various shapes.
6. Jun shu - Equitable taxation More advanced problems on proportion.
7. Ying bu zu - Excess and deficit Linear problems solved using the principle known later in the West as the rule of false position.
8. Fang cheng - The rectangular array Problems with several unknowns, solved by a principle similar to Gaussian elimination.
9. Gou gu - Base and altitude Problems involving the principle known in the West as the Pythagorean theorem.
Most scholars believe that Chinese mathematics and the mathematics of the ancient Mediterranean world had developed more or less independently up to the time when the Nine Chapters reached its final
form. There is therefore little historical value in speculations about who was "more advanced" at any given period. Nevertheless we may note that the method of chapter 7 was not found in Europe until
the 13th century, and the method of chapter 8 is not found before the sixteenth century. Of course there are also features of ancient Western mathematics that are not found in ancient China.
Liu Hui wrote a very detailed commentary on this book in 263. He analyses the procedures of the Nine Chapters step by step, in a manner which is clearly designed to give the reader confidence that
they are reliable, although he is not concerned to provide formal proofs in the Euclidean manner. Liu's commentary is of great mathematical interest in its own right.
The Nine Chapters is an anonymous work, and its origins are not clear. Until recent years there was no substantial evidence of related mathematical writing that might have preceded it. This is no
longer the case. The Suan shu shu is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other
writings in 1983 when archaeologists opened a tomb at Zhangjiashan in Hubei province. From documentary evidence this tomb is known to have been closed in 186 BC, early in the Western Han dynasty.
While its relationship to the Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the Suan shu shu is however much less systematic than
the Nine Chapters; and appears to consist of a number of more or less independent short sections of text drawn from a number of sources.
A full translation and study of the Nine Chapters and Liu Hui's commentary is available in SHEN Kangshen "The Nine Chapters on the Mathematical Art" Oxford 1999. ISBN 0198539363zh:九章算术 ja:九章算 | {"url":"https://academickids.com/encyclopedia/index.php/The_Nine_Chapters_on_the_Mathematical_Art","timestamp":"2024-11-07T07:38:38Z","content_type":"application/xhtml+xml","content_length":"29246","record_id":"<urn:uuid:7c6ce155-becd-4fc3-a0df-015c4a266bcd>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00094.warc.gz"} |
matrix_norm(x: array, /, *, keepdims: bool = False, ord: int | float | ~typing.Literal[inf, -inf, 'fro', 'nuc'] | None = 'fro') array¶
Computes the matrix norm of a matrix (or a stack of matrices) x.
☆ x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices. Should have a floating-point data type.
☆ keepdims (bool) – If True, the last two axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see
Broadcasting). Otherwise, if False, the last two axes (dimensions) must not be included in the result. Default: False.
☆ ord (Optional[Union[int, float, Literal[inf, -inf, 'fro', 'nuc']]]) –
order of the norm. The following mathematical norms must be supported:
ord description
’fro’ Frobenius norm
’nuc’ nuclear norm
1 max(sum(abs(x), axis=0))
2 largest singular value
inf max(sum(abs(x), axis=1))
The following non-mathematical “norms” must be supported:
ord description
-1 min(sum(abs(x), axis=0))
-2 smallest singular value
-inf min(sum(abs(x), axis=1))
If ord=1, the norm corresponds to the induced matrix norm where p=1 (i.e., the maximum absolute value column sum).
If ord=2, the norm corresponds to the induced matrix norm where p=inf (i.e., the maximum absolute value row sum).
If ord=inf, the norm corresponds to the induced matrix norm where p=2 (i.e., the largest singular value).
Default: 'fro'.
out (array) – an array containing the norms for each MxN matrix. If keepdims is False, the returned array must have a rank which is two less than the rank of x. If x has a real-valued data
type, the returned array must have a real-valued floating-point data type determined by Type Promotion Rules. If x has a complex-valued data type, the returned array must have a real-valued
floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then the returned array must have a float64 data type).
Changed in version 2022.12: Added complex data type support. | {"url":"https://data-apis.org/array-api/latest/extensions/generated/array_api.linalg.matrix_norm.html","timestamp":"2024-11-06T17:43:53Z","content_type":"text/html","content_length":"23146","record_id":"<urn:uuid:fbb6d38e-4e75-4048-b73d-68b46c5fe1cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00507.warc.gz"} |
seminars - A Sharp square function estimate for the moment curves in R^d
We review the recent paper of Guth–Maldague for the sharp square function estimate of moment curve in R^d. We briefly sketch the proof of the main inductive estimates for the moment curves, cones
over moment curves, and more general m-th order Taylor cones. | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&sort_index=date&order_type=desc&page=11&l=en&document_srl=1155000","timestamp":"2024-11-03T10:16:11Z","content_type":"text/html","content_length":"46510","record_id":"<urn:uuid:eaf5812a-0683-4e3e-9f73-4066a8d4db38>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00869.warc.gz"} |
Python’s libraries related to math, data science, and data analysis are capable of some truly amazing feats. Higher-level languages like Python typically sacrifice speed for flexibility and ease of
use. But third-party libraries like PyTorch, Pandas and NumPy provide remarkably efficient data processing while still retaining Python’s unique flexibility.
It’s always best to combine a library’s unique data types with its native functionality so that we can retain that efficiency. For example, if you’re using PyTorch and wanted to fill a native tensor
with zeros you’d want to use that library’s native functions. But before learning how to do so we need to take a quick look at how PyTorch handles tensors.
Tensors Might Be More Familiar Than You’d Imagine
People new to PyTorch might find the concept of tensors a little intimidating. Tensors are a mathematical concept that can be best understood as a multi-positional system that can track points over
multiple dimensions. But machine learning systems like PyTorch leverage this concept for a number of different tasks. Using tensors for positional tracking is a classic example. But tensors are even
used for deep learning and computational neural networks.
This might still sound rather complex. But tensors in PyTorch are roughly analogous to Python’s standard collection datatypes. And if you’ve used NumPy’s ndarray then you can essentially use your
experience there with PyTorch’s tensors. NumPy’s multidimensional arrays and PyTorch’s tensors are functionally quite similar to each other. But with that in mind, what about specific functionality
that could let us create a tensor pre-populated with a specific value like zero?
PyTorch’s Tensors Come With Added Functionality
If PyTorch’s tensors operate in the same way as standard Python collections, you might wonder why you can’t simply populate it through a for loop. And, in fact, you could do so. However, this isn’t a
very efficient way to go about populating a tensor. The main reason comes down to how scientific libraries in Python operate with so much speed and efficiency.
Higher-level and interpreted languages don’t usually operate at extremely high speeds. However, there are a number of tricks that can be used in Python to gain extra speed for specific functionality.
And libraries like PyTorch put special emphasis on getting as much power as possible. As such, when you use a scientific library’s native functionality you’ll typically see more efficient processing
than you would with Python’s standard methods. So while you can manipulate tensors with standard for loops, it’s best to use native PyTorch methods when possible.
On top of standard optimizations, using PyTorch’s built-in functions also presents you with the opportunity to speed things up even more. PyTorch gives you the option to use CUDA (Compute Unified
Device Architecture) if your GPU supports it. For example, take a look at tensor creation using CUDA as a quick PyTorch tutorial.
import torch as pt
ourDevice = pt.device(‘cuda’)
ourTensor = pt.tensor(([[1,2], [3, 4], [5, 6]]), device=ourDevice)
We begin with a standard import of PyTorch as pt. We then initialize a device to use with PyTorch. CPU is the default device, but in this case we’ll use something different to highlight how PyTorch
can optimize tensor-related functionality.
If your system supports CUDA then the tensor will be created using your GPU rather than CPU. This would be especially useful if you had a huge data collection and needed to work with multiple rapidly
changing scalar value sets or even scalar arrays. However, using GPU acceleration will require both underlying support in your GPU and for PyTorch to have been compiled with CUDA enabled. If you
wanted to use the standard CPU system then you’d just change line 3 to the following.
ourDevice = pt.device(‘cpu’)
It’s also fairly easy to work this into user-specific configurations. And if you omit device declaration in a function then it’ll fall back to the CPU. The main point is that you have a lot of
additional options to tweak performance when you use PyTorch tensors with PyTorch functions. This wouldn’t be the case when using PyTorch tensors with standard Python functions. Keep this concept of
optimization in mind as we continue working with tensors.
In line 4 we actually create our tensor. In this particular case, we populate it with specific data and specify a device for processing. Note that the device argument is optional. If there’s no
declaration it’ll default to your CPU. Line 5 simply prints out the final tensor. But note the similarity of the output tensor data with standard Python collections. With that in mind, we can move on
to creating a tensor filled with zero values for all positions.
Implementing and Optimizing a Tensor
Now, we can take everything we’ve covered to this point and create a tensor filled with zero values. Try out the following code sample.
import torch as pt
ourTensor = pt.zeros((2, 3))
We begin by importing PyTorch again. But we change things around a little in line two. Here we use a function in PyTorch called zeros. As the name suggests, it creates a tensor filled with zeros. We
pass two variables as an argument to specify the size and dimensions. We then print the contents of the new ourTensor variable and proceed to do the same for its type. Note that we now have a PyTorch
tensor filled with zeros.
However, remember the earlier note about optimization. A tensor predominantly filled with zeros is known as a sparse tensor. And a zero-filled layout opens up additional room for memory optimization.
This is somewhat similar to the fact that a zip file filled with zeroed-out data can be optimized for space. For example, an empty virtualized hard drive might take up 100 GB of uncompressed space,
but efficient compression could reduce it to mere KBs. Likewise having an input tensor that consists of zeroes opens up a lot of possibilities that we could make use of for heavily optimized code.
This state once again highlights why it’s so important to use a library’s native functions whenever possible. Doing so opens up a lot of extra room for optimization.
PyTorch: How To Use Torch Zeros To Create a Tensor Filled With Zeros | {"url":"https://decodepython.com/pytorch-how-to-use-torch-zeros-to-create-a-tensor-filled-with-zeros/","timestamp":"2024-11-04T01:52:42Z","content_type":"text/html","content_length":"39920","record_id":"<urn:uuid:d8389904-db90-4f47-aa34-3a235ce3aa58>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00439.warc.gz"} |
How do you plot a bifurcation diagram?
How do you plot a bifurcation diagram?
The bifurcation diagram is constructed by plotting the parameter value k against all corresponding equilibrium values y∗. Typically, k is plotted on the horizontal axis and critical points y* on the
vertical axis. A “curve” of sinks is indicated by a solid line and a curve of sources is indicated by a dashed line.
What does a bifurcation diagram show?
In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a
function of a bifurcation parameter in the system.
How do you find bifurcation points?
44 second clip suggested9:56Bifurcations of a differential equation – YouTubeYouTubeStart of suggested clipEnd of suggested clipThe actual bifurcation point or the value of Si with the number of
equilibria. Changes from 3 to 1MoreThe actual bifurcation point or the value of Si with the number of equilibria. Changes from 3 to 1 happens to occur for this case at around C equals three point
zero seven nine.
What is bifurcation in differential equations?
Bifurcation diagrams are an effective way of representing the nature of the solutions of a one-parameter family of differential equations. Bifurcations for a one-parameter family of differential
equations dx/dt=fλ(x) d x / d t = f λ ( x ) are rare. Bifurcations occur when fλ0(x0)=0 f λ 0 ( x 0 ) = 0 and f′λ0(x0)=0.
What is a bifurcation model?
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a
family of differential equations.
What is called period-doubling?
From Wikipedia, the free encyclopedia. In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system’s parameters causes a new periodic trajectory to emerge from
an existing periodic trajectory—the new one having double the period of the original.
How many bifurcations are there?
In this chapter, we also discuss several types of bifurcations, saddle node, transcritical, pitchfork and Hopf bifurcation. Among these types, we especially focus on Hopf bifurcation. The first three
types of bifurcation occur in scalar and in systems of differential equations.
Is the bifurcation diagram a fractal?
Hence we say that the bifurcation diagram of the logistic map is a fractal.
How do you classify bifurcation?
45 second clip suggested22:58Classifications of Bifurcation – YouTubeYouTube
What is an example of bifurcation?
The definition of bifurication means a division into two branches. An example of bifurication is a fork in the road. The act or fact of bifurcating.
What are the different types of bifurcations?
There are five types of “local” codimension two bifurcations of equilibria:
• Bautin Bifurcation.
• Bogdanov-Takens Bifurcation.
• Cusp Bifurcation.
• Fold-Hopf Bifurcation.
• Hopf-Hopf Bifurcation.
Is bifurcation periodic?
At the bifurcation point the period of the periodic orbit has grown to infinity and it has become a homoclinic orbit. After the bifurcation there is no longer a periodic orbit. Left panel: For small
parameter values, there is a saddle point at the origin and a limit cycle in the first quadrant.
Is it possible to plot a birfurction diagram using math?
you can use mathematical for it.drawing bifurcation diagram and etc is very easy. Can someone help. I want to plot a birfurction diagram for a predator-prey model.
How to tell if a map is bifurcated?
The bifurcations can be seen even more clearly from a return map, for instance, where v is sampled whenever v’ passes from positive to negative values. A blow-up of the map near the transition to
chaos is (with SameTest deleted)
2. Saddle-node bifurcation (x vs m & y vs. m) around at m = 20.8. 3. Hopf-bifurcation (x vs m & y vs. m) at m=14.73, (d,h) = (0.02,0.001) and others are same.
What is saddle node bifurcation and Hopf-bifurcation?
Saddle-node bifurcation (x vs m & y vs. m) around at m = 20.8. 3. Hopf-bifurcation (x vs m & y vs. m) at m=14.73, (d,h) = (0.02,0.001) and others are same. Sign in to answer this question. | {"url":"https://www.yemialadeworld.com/how-do-you-plot-a-bifurcation-diagram/","timestamp":"2024-11-01T19:20:16Z","content_type":"text/html","content_length":"72511","record_id":"<urn:uuid:d87db994-a966-4300-9743-5db9f0e8e8df>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00198.warc.gz"} |
Location relative to specified normal - how to?
I hope that the picture explains everything. How the vector math would look like to implement that? Of course I already know how to trace downwards and I already have the hit location and hit normal,
but I can’t figure out how to do the next trace (x distance on specified direction relative to hit normal)
I would be grateful for some hints!
1 Like
The Out Hit Normal of your hit that you get from your trace is a vector that is perpendicular to the plane that you collided with. Together with your Out Hit Location vector it describes the plane
you collided with. We’ll call that plane Plane1 for fun.
If you take the Out Hit Normal and your original “1” vector, you can find the normal of the plane they both reside on by taking their cross product. Together with that some Out Hit Location this
describes another plane we’ll call Plane 2.
If you can visualize it in your mind, the intersection of Plane1 and Plane2 is a line that represents the vector you are looking for!
Lucky for us, that intersection line happens to be perpendicular to the normal vectors of the two planes. So taking the cross product of the two normal vectors will give you the direction vector you
want. If you end up pointing exactly backwards, then just reverse the order of the arguments in your cross product. Or multiply your vector by -1.
1 Like
Thanks for the explanation, but I can’t get how to determine this 2nd plane in blueprints… I mean which vector should I input into Cross Product node. The surface normal and the second normal that I
don’t know how to get. Sorry, my brain is quite weak in math.
I’ve also found a similar question HERE - But maybe it works only in Unity, since it doesn’t work in my Unreal project (in my case the direction doesn’t change anything and every cast faces world
0,0,0 location)
You said you traced “downward”. How did you do that?
Downward trace in my case == trace from actor location + (0,0,0) to actor location + (0,0,-100), so it traces -100 on Z from actor location.
(I don’t know if ‘trace downward’ is a good word, maybe ‘trace down’ is more sufficent? My english still in development
Great, so your downward vector is (0, 0, -100) and so the normal of the second plane is CrossProduct( (0, 0, -100), OutHitNormal ).
Ok, so when I set LineTrace start to hitLocation and LineTrace end to CrossProduct( (0,0-100), hitNormal ) then every trace points towards world 0,0,0 location and I don’t know why:
The cross-product will give you a direction vector, which will probably be normalized (length of the vector will be 1cm). If you try to use a normalized direction vector as a location, they will all
be within 1cm of (0, 0, 0).
Instead, draw a line from hitLocation to (hitLocation + (CrossProductResult * 100.0))
Ah now I get it!
Thank you for the help! | {"url":"https://forums.unrealengine.com/t/location-relative-to-specified-normal-how-to/354845","timestamp":"2024-11-04T14:06:35Z","content_type":"text/html","content_length":"36293","record_id":"<urn:uuid:f8510fe8-17e7-46e9-b4b8-a3f9922e55f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00248.warc.gz"} |
How do I hide the #DIV/0! error while a referenced cell is blank?
When working with spreadsheets in Excel, encountering errors can often disrupt the flow of your data presentation and analysis. One common error is the infamous #DIV/0! error, which occurs when a
formula attempts to divide by zero. This usually happens when the referenced cell is blank or contains zero. However, there’s a way to hide this error, ensuring your spreadsheet looks cleaner and
more professional.
In this article, we’ll walk you through how to hide the #DIV/0! error in Excel when the referenced cell is blank, showcasing the original formula and providing practical solutions.
Understanding the Problem
The #DIV/0! error arises in Excel when a formula tries to divide a number by zero or an empty cell. For instance, if you have a formula like =A1/B1, and B1 is blank or zero, Excel will return the #
DIV/0! error.
Original Code Scenario
Here is a simple example that generates the #DIV/0! error:
Assuming A1 contains the number 10 and B1 is blank, this will return the #DIV/0! error, making your data harder to interpret.
Hiding the Error with the IF Function
Solution 1: Using the IF Function
One effective way to hide the #DIV/0! error is by using the IF function to check if the denominator is zero or blank before performing the division. Here’s how you can modify your formula:
=IF(B1=0, "", A1/B1)
Explanation of the Formula
In this updated formula:
• The IF function checks if B1 is equal to zero.
• If true, it returns an empty string "", effectively hiding the error.
• If false, it proceeds with the division A1/B1.
This approach not only eliminates the error message but also maintains the integrity of your data presentation.
Using the IFERROR Function
Solution 2: Using the IFERROR Function
Another approach is to utilize the IFERROR function, which simplifies error handling in Excel. Here’s how to apply it:
=IFERROR(A1/B1, "")
How It Works
• The IFERROR function will evaluate A1/B1.
• If this results in an error (including #DIV/0!), it will return an empty string instead.
This method is particularly advantageous because it can catch and handle multiple types of errors, not just #DIV/0!, making it a versatile option.
Additional Insights and Examples
Practical Use Case
Imagine you are creating a financial report, and you need to calculate the profitability ratio. If any of your revenue or cost entries are missing, using these methods will ensure that your report
remains presentable without distracting error messages.
Performance Consideration
While hiding errors is essential for aesthetics, remember that if a referenced cell is blank, the calculated value should reflect that accurately in your decision-making process. Always ensure that
important financial or analytical data is handled appropriately.
Hiding the #DIV/0! error in Excel when a referenced cell is blank is straightforward with the IF or IFERROR functions. These methods help maintain a clean and professional appearance in your
spreadsheets, allowing for better readability and understanding of your data.
By utilizing these solutions, you'll not only enhance the visual appeal of your Excel sheets but also improve your overall data management strategy. Happy Excel-ing! | {"url":"https://go-mk-websites.co.uk/post/how-do-i-hide-the-div-0-error-while-a-referenced-cell-is","timestamp":"2024-11-14T08:00:44Z","content_type":"text/html","content_length":"82378","record_id":"<urn:uuid:0dce2c51-5d79-4a41-b4d3-3b673097f8fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00181.warc.gz"} |
Leibniz on binary : the invention of computer arithmetic
Item type Current location Call number Status Date due Barcode
Books 513.52 STR (Browse shelf) Available 034449
Previous 513.211 513.5 KAY Number systems : 513.509 CHR Reckonings : 513.52 STR Leibniz on binary : 513.83 PAL Seminar on the 513.83 SIN Lecture notes on 513.9 HAN Speed mathematics : Next
MEN a path into rigorous numerals, cognition, and the invention of computer Atiyah - Singer index elementary topology and secrets skills for quick
Counting mathematics history arithmetic theorem geometry calculation
Among his extraordinary mathematical and philosophical achievements, Gottfried Wilhelm Leibniz (1646-1716) invented binary arithmetic, the representational basis for today's world of digital
computing and communications. This book will be the first to make a selection of his writings available in English. Strickland and Lewis provide an accessible introduction to Leibniz through some
twenty mostly unpublished manuscripts dealing with binary notation, algorithms for binary arithmetic, and related topics. The book includes an introduction analyzing the history of the binary system
and Leibniz's claim to priority and short introductions to each of the chapters. | {"url":"https://opac.daiict.ac.in/cgi-bin/koha/opac-detail.pl?biblionumber=32227&shelfbrowse_itemnumber=42845","timestamp":"2024-11-13T22:39:03Z","content_type":"text/html","content_length":"56132","record_id":"<urn:uuid:73d19369-25b3-4ccc-8ad0-e4cac3a4e737>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00499.warc.gz"} |
[EM] Addendum re: unselfish nonmajoritarian sincere-rating conditions
[EM] Addendum re: unselfish nonmajoritarian sincere-rating conditions
Michael Ossipoff email9648742 at gmail.com
Fri Nov 1 07:47:33 PDT 2013
For a late and abrupt increase in the magnitude of the weighting of
negative ratings, I guess a hyperbola would be even better than an
exponential function:
y = -A/(x+B)
...where x is any negative rating-value, and y is its
altruistically-magnified value,
...and A and B are positive constants.
B is the distance between the hyperbola's asymptote and the y axis.
But, to pass the function through (0,0), an additional term is needed:
y = -A/(X+B) + A/B
To avoid y having less magnitude than x, that should be:
F = -A/(X+B) + A/B
y = min(x,F)
Where the ratings-range is[ -N to +N], B would be chosen to be
slightly larger than N. A and B would be chosen for what seems the
right combination of fairness and practicality.
Of course nommajoritarian unselfish sincere-rating conditions are
farther away than any of the other conditions that I've defined, and
so these #4 methods don't now have practical relevance for public
political elections.
Michael Ossipoff
More information about the Election-Methods mailing list | {"url":"http://lists.electorama.com/pipermail/election-methods-electorama.com/2013-November/097718.html","timestamp":"2024-11-02T23:42:42Z","content_type":"text/html","content_length":"3902","record_id":"<urn:uuid:19a8274b-09b3-4d94-a657-3799a7981181>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00800.warc.gz"} |
Question ID - 54093 | SaraNextGen Top Answer
A number of two digits had 3 for its unit’s digit, and the sum of the digits is
(a) 43 (b) 53 (c) 63 (d) 73
A number of two digits had 3 for its unit’s digit, and the sum of the digits is | {"url":"https://www.saranextgen.com/homeworkhelp/doubts.php?id=54093","timestamp":"2024-11-02T17:08:31Z","content_type":"text/html","content_length":"16761","record_id":"<urn:uuid:3a3c12a6-273b-4c58-a7f7-8250a1ade11b>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00656.warc.gz"} |
15 Binary Choice | Data Analysis for Public Affairs with R
15 Binary Choice
There are slides and a YouTube Video associated with this chapter:
Binary choice models are part of a large class of so-called qualitative choice models which are used for qualitative dependent variables. Consider the following outcomes of interest:
• Is a person in the labor force?
• Will an individual vote yes on a particular issue?
• Did a person watch the last Super Bowl?
• Have you purchased a new car in the past year
• Did you do any charitable contributions in the past year?
• Did you vote during the last election?
• Does an individual recidivate after being released from prison?
For those questions, the dependent variable is either 0 (“no”) or 1 (“yes”). For binary choice models, the outcome is interpreted as a probability, i.e., what is the probability of a person to answer
“yes” to those questions.
In the next chapter, the model is expanded to consider more than binary outcomes. Those models include categorical dependent variable that are either naturally ordered or have no ordering. Examples
of naturally ordered categorical variables are:
• Level of happiness: Very happy, happy, ok, or sad.
• Intention to get a COIVID-19 vaccine: Definitely yes, probably yes, probably no, or definitely no
Two examples about categorical dependent variable which have no ordering are:
• Commute to campus: Bike, car, walk, or bus
• Voter registration: Democrat, Republican, or independent
For all those models, the outcome of interest is the probability to fall into a particular category. For binary choice models which are considered in this chapter, the outcome of interest is the
probability to fall into the 1 (“yes”) category. For binary choice models, y takes one of two values: 0 or 1. And the model will specify \(Pr(y=1|x)\) where x are the independent variables.
Consider the decision to purchase organic food. Assume that you have data about the income of respondents as well as information if they purchase organic food. The purchase decision (“yes” or “no”)
is on the vertical axis and the income is on the horizontal axis.
Remember that the probability has be bounded between 0 and 1. Hence, we need to find a function \(G(z)\) such that \(0 \leq G(z) \leq 1\) for all values of \(z\) and \(P(y=1|x)=G(z)\). Popular
choices for \(G(z)\) are the cumulative normal distribution function (“Probit Model”) and the logistic function (“Logit Model”). For what follows, let \(z=\beta_0+\beta_1 \cdot x_1 + \cdots + \beta_k
\cdot x_k\). For the probit model, \(G(z)\) is written as \[Pr(y = 1) = G(z)=\Phi(z)\] where \(\Phi\) represents the cumulative normal. And for the logit model, \(G(z)\) is written as \[Pr(y = 1) = G
(z)=\frac{e^z}{1+e^z}=\frac{1}{1+e^{-z}}\] The interpretation of the logit and probit estimates is not as straightforward as in the multivariate regression case. In general, we care about the effect
of \(x\) on \(P(y=1|x)\). The sign of the coefficient shows the direction of the change in probability. The approximation to the marginal effect if \(x\) is roughly continuous: \[\Delta P(y=1|x) \
approx g(\hat{\beta}_0 + x \cdot \hat{\beta}) \cdot \beta_j \cdot \Delta x_j\] To obtain the marginal effects in R, an additional step is necessary. Let us illustrate the binary choice model using
the data set organic and a logit model. The results of interest for the binary choice model are the (1) coefficient estimates, (2) marginal effects, and (3) predicted probabilities. | {"url":"https://jrfdumortier.github.io/dataanalysis/binary-choice.html","timestamp":"2024-11-06T22:13:32Z","content_type":"text/html","content_length":"41981","record_id":"<urn:uuid:0895d669-b314-41cb-a7a4-fd75bf206fd3>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00624.warc.gz"} |
Stack Implementation In Java Data Structure
• A stack is a container of objects that are inserted and removed according to the last-in-first-out (LIFO) principle.
• Objects can be inserted at any time, but only the last (the most-recently inserted) object can be removed.
• Inserting an item is known as “Pushing” onto the stack. “Popping” off the stack is synonymous with removing an item
• Used in Operating system to implement method calls, and in evaluating Expressions.
Elements: The elements are of a generic type <Type>. (In a linked implementation an element is placed in a node)
Structure: the elements are linearly arranged, and ordered according to the order of arrival, most recently arrived element is called top.
Domain: the number of elements in the stack is bounded therefore the domain is finite. Type of elements: Stack
Elements: The elements are of a generic type <Type>. (In a linked implementation an element is placed in a node)
Structure: the elements are linearly arranged, and ordered according to the order of arrival, most recently arrived element is called top.
Domain: the number of elements in the stack is bounded therefore the domain is finite. Type of elements: Stack
All operations operate on a stack S.
requires: Stack S is not full.
results: Element e is added to the stack as its most recently added elements.
requires: Stack S is not empty.
results: the most recently arrived element in S is removed and its value assigned to e.
3.Method empty (boolean flag)
results: If Stack S is empty then flag is true, otherwise false.
4.Method Full (boolean flag).
results: If S is full then Full is true, otherwise Full is false.
Stack Interface
public interface Stack<T>{
public T pop( );
public void push(T e);
public boolean empty( );
public boolean full( );
Below the structure of stack implementation
public class Node<T> {
public T data;
public Node<T> next;
public Node () {
data = null;
next = null;
public Node (T val) {
data = val;
next = null;
// Setters/Getters?
Linked-List: Implementation
public class LinkedStack<T> implements Stack<T> {
private Node<T> top;
/* Creates a new instance of LinkStack */
public LinkedStack() {
top = null;
Linked-List Array Representation
public class ArrayStack<T> implements Stack<L> {
private int maxsize;
private int top;
private T[] nodes;
/** Creates a new instance of ArrayStack */
public ArrayStack(int n) {
maxsize = n;
top = -1;
nodes = (T[]) new Object[n];
public boolean empty(){
return top == -1;
public boolean full(){
return top == maxsize - 1;
public void push(T e){
nodes[++top] = e;
public T pop(){
return nodes[top--];
Applications of Stacks
Some applications of stacks are:
• Balancing symbols.
• Computing or evaluating postfix expressions.
• Converting expressions from infix to postfix.
Contact us to get any help in stack using java data structure at realcode4you@gmail.com and get instant help with an affordable prices. | {"url":"https://www.realcode4you.com/post/stack-implementation-in-java-data-structure","timestamp":"2024-11-11T12:40:18Z","content_type":"text/html","content_length":"1050483","record_id":"<urn:uuid:9e8ce27c-62a9-42fa-bbfc-d8fab0eb2664>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00205.warc.gz"} |
bdrc - Bayesian Discharge Rating Curves
This software package fits a discharge rating curve based on the power-law and the generalized power-law from data on paired water elevation and discharge measurements in a given river using a
Bayesian hierarchical model as described in Hrafnkelsson et al. (2022). Four models are implemented:
plm0() - Power-law model with a constant error variance. This is a Bayesian hierarchical implementation of the most commonly used discharge rating curve model in hydrological practice.
plm() - Power-law model with error variance that varies with water elevation.
gplm0() - Generalized power-law model with a constant error variance. The generalized power-law is introduced in Hrafnkelsson et al. (2022).
gplm() - Generalized power-law model with error variance that varies with water elevation. The generalized power-law is introduced in Hrafnkelsson et al. (2022).
# Install release version from CRAN
# Install development version from GitHub
Getting started
It is very simple to fit a discharge rating curve with the bdrc package. All you need are two mandatory input arguments, formula and data. The formula is of the form y~x where y is discharge in ms and x is water elevation in m (it is very important that the data is in the correct units). data is a
data.frame which must include x and y as column names. As an example, we will use data from the Swedish gauging station Krokfors, which is one of the datasets that come with the package. In this
table, the Q column denotes discharge while W denotes water elevation:
To dig deeper into the functionality of the package and the different ways to visualize a discharge rating curve model for your data, we recommend taking a look at our two vignettes.
Hrafnkelsson, B., Sigurdarson, H., and Gardarsson, S. M. (2022). Generalization of the power-law rating curve using hydrodynamic theory and Bayesian hierarchical modeling, Environmetrics, 33 | {"url":"https://cran.stat.sfu.ca/web/packages/bdrc/readme/README.html","timestamp":"2024-11-15T02:37:17Z","content_type":"application/xhtml+xml","content_length":"8226","record_id":"<urn:uuid:3c0e4635-05a3-4214-a34c-72727bfaaee9>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00864.warc.gz"} |
Bayesian Hidden Markov Models
This tutorial illustrates training Bayesian Hidden Markov Models (HMM) using Turing. The main goals are learning the transition matrix, emission parameter, and hidden states. For a more rigorous
academic overview on Hidden Markov Models, see An introduction to Hidden Markov Models and Bayesian Networks (Ghahramani, 2001).
In this tutorial, we assume there are \(k\) discrete hidden states; the observations are continuous and normally distributed - centered around the hidden states. This assumption reduces the number of
parameters to be estimated in the emission matrix.
Let’s load the libraries we’ll need. We also set a random seed (for reproducibility) and the automatic differentiation backend to forward mode (more here on why this is useful).
Simple State Detection
In this example, we’ll use something where the states and emission parameters are straightforward.
# Define the emission parameter.
y = [
N = length(y);
K = 3;
# Plot the data we just made.
plot(y; xlim=(0, 30), ylim=(-1, 5), size=(500, 250))
We can see that we have three states, one for each height of the plot (1, 2, 3). This height is also our emission parameter, so state one produces a value of one, state two produces a value of two,
and so on.
Ultimately, we would like to understand three major parameters:
1. The transition matrix. This is a matrix that assigns a probability of switching from one state to any other state, including the state that we are already in.
2. The emission matrix, which describes a typical value emitted by some state. In the plot above, the emission parameter for state one is simply one.
3. The state sequence is our understanding of what state we were actually in when we observed some data. This is very important in more sophisticated HMM models, where the emission value does not
equal our state.
With this in mind, let’s set up our model. We are going to use some of our knowledge as modelers to provide additional information about our system. This takes the form of the prior on our emission
\[ m_i \sim \mathrm{Normal}(i, 0.5) \quad \text{where} \quad m = \{1,2,3\} \]
Simply put, this says that we expect state one to emit values in a Normally distributed manner, where the mean of each state’s emissions is that state’s value. The variance of 0.5 helps the model
converge more quickly — consider the case where we have a variance of 1 or 2. In this case, the likelihood of observing a 2 when we are in state 1 is actually quite high, as it is within a standard
deviation of the true emission value. Applying the prior that we are likely to be tightly centered around the mean prevents our model from being too confused about the state that is generating our
The priors on our transition matrix are noninformative, using T[i] ~ Dirichlet(ones(K)/K). The Dirichlet prior used in this way assumes that the state is likely to change to any other state with
equal probability. As we’ll see, this transition matrix prior will be overwritten as we observe data.
# Turing model definition.
@model function BayesHmm(y, K)
# Get observation length.
N = length(y)
# State sequence.
s = tzeros(Int, N)
# Emission matrix.
m = Vector(undef, K)
# Transition matrix.
T = Vector{Vector}(undef, K)
# Assign distributions to each element
# of the transition matrix and the
# emission matrix.
for i in 1:K
T[i] ~ Dirichlet(ones(K) / K)
m[i] ~ Normal(i, 0.5)
# Observe each point of the input.
s[1] ~ Categorical(K)
y[1] ~ Normal(m[s[1]], 0.1)
for i in 2:N
s[i] ~ Categorical(vec(T[s[i - 1]]))
y[i] ~ Normal(m[s[i]], 0.1)
We will use a combination of two samplers (HMC and Particle Gibbs) by passing them to the Gibbs sampler. The Gibbs sampler allows for compositional inference, where we can utilize different samplers
on different parameters.
In this case, we use HMC for m and T, representing the emission and transition matrices respectively. We use the Particle Gibbs sampler for s, the state sequence. You may wonder why it is that we are
not assigning s to the HMC sampler, and why it is that we need compositional Gibbs sampling at all.
The parameter s is not a continuous variable. It is a vector of integers, and thus Hamiltonian methods like HMC and NUTS won’t work correctly. Gibbs allows us to apply the right tools to the best
effect. If you are a particularly advanced user interested in higher performance, you may benefit from setting up your Gibbs sampler to use different automatic differentiation backends for each
parameter space.
Time to run our sampler.
Let’s see how well our chain performed. Ordinarily, using display(chn) would be a good first step, but we have generated a lot of parameters here (s[1], s[2], m[1], and so on). It’s a bit easier to
show how our model performed graphically.
The code below generates an animation showing the graph of the data above, and the data our model generates in each sample.
# Extract our m and s parameters from the chain.
m_set = MCMCChains.group(chn, :m).value
s_set = MCMCChains.group(chn, :s).value
# Iterate through the MCMC samples.
Ns = 1:length(chn)
# Make an animation.
animation = @gif for i in Ns
m = m_set[i, :]
s = Int.(s_set[i, :])
emissions = m[s]
p = plot(
size=(500, 250),
label="True data",
xlim=(0, 30),
ylim=(-1, 5),
plot!(emissions; color=:blue, label="Sample $i")
end every 3
[ Info: Saved animation to /tmp/jl_spiR8LsXYZ.gif
Looks like our model did a pretty good job, but we should also check to make sure our chain converges. A quick check is to examine whether the diagonal (representing the probability of remaining in
the current state) of the transition matrix appears to be stationary. The code below extracts the diagonal and shows a traceplot of each persistence probability. | {"url":"https://turinglang.org/docs/tutorials/04-hidden-markov-model/index.html","timestamp":"2024-11-10T10:52:09Z","content_type":"application/xhtml+xml","content_length":"1049278","record_id":"<urn:uuid:73463d80-e066-4d69-9c1a-4f5e05cbed0b>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00844.warc.gz"} |
FAQ: What is a Grid Convergence Angle?
A Projected Coordinate System in most cases has a Central Meridian. The Central Meridian points directly to the North (or South) Pole.
A Grid Convergence Angle is formed if a point lies near, but not on, that Central Meridian, by drawing a line from that point to the Pole.
The angle between the Central Meridian and the line from the point to the Pole is the Convergence Angle. This is shown in the image below.
The Calculate Convergence Angle tool calculates the angle between the Central Meridian of the projected coordinate system applied to the data, and the line which intersects the point and runs from
the point to the North (or South) Pole. A Convergence Angle between the Central Meridian of UTM Zone 15 North and the USGS Survey Control Point Meade's Ranch is shown in the following illustration.
Convergence angles cannot exist in a Geographic Coordinate System, or in some projected coordinate systems like Web Mercator, because in these coordinate systems ALL lines of Longitude point directly at the North Pole. | {"url":"https://support.esri.com/en-us/knowledge-base/faq-what-is-a-grid-convergence-angle-000020700","timestamp":"2024-11-02T14:53:05Z","content_type":"text/html","content_length":"28507","record_id":"<urn:uuid:2adadea3-f9db-4a80-a75c-51ee4759a8e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00751.warc.gz"} |
NY Times New Puzzle - Connections
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Thinks s/he gets paid by the post
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Puzzle #489 | {"url":"https://www.early-retirement.org/threads/ny-times-new-puzzle-connections.119499/page-68#post-3129905","timestamp":"2024-11-14T12:00:45Z","content_type":"text/html","content_length":"280866","record_id":"<urn:uuid:1c23063f-aa98-4af7-8599-e4eabb71e071>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00064.warc.gz"} |
hamux - HAMUX
A New Class of Deep Learning Library Built around ENERGY
Part proof-of-concept, part functional prototype, HAMUX is designed to bridge modern AI architectures and Hopfield Networks.
HAMUX: A Hierarchical Associative Memory User eXperience.
🚧 HAMUX is in rapid development. Remember to specify the version when building off of HAMUX.
A Universal Abstraction for Hopfield Networks
HAMUX fully captures the the energy fundamentals of Hopfield Networks and enables anyone to:
• 🧠 Build DEEP Hopfield nets
• 🧱 With modular ENERGY components
• 🏆 That resemble modern DL operations
Every architecture built using HAMUX is a dynamical system guaranteed to have a tractable energy function that converges to a fixed point. Our deep Hierarchical Associative Memories (HAMs) have
several additional advantages over traditional Hopfield Networks (HNs):
Hopfield Networks (HNs) Hierarchical Associative Memories (HAMs)
HNs are only two layers systems HAMs connect any number of layers
HNs model only simple relationships between layers HAMs model any complex but differentiable operation (e.g., convolutions, pooling, attention, \(\ldots\))
HNs use only pairwise synapses HAMs use many-body synapses (which we denote HyperSynapses)
How does HAMUX work?
HAMUX is a hypergraph of 🌀neurons connected via 🤝hypersynapses, an abstraction sufficiently general to model the complexity of connections used in modern AI architectures.
We conflate the terms hypersynapse and synapse regularly. We explicitly say "pairwise synapse" when referring to the classical understanding of synapses.
HAMUX defines two fundamental building blocks of energy: the 🌀neuron layer and the 🤝hypersynapse (an abstraction of a pairwise synapse to include many-body interactions) connected via a hypergraph.
It is a fully dynamical system, where the “hidden state” \(x_i^l\) of each layer \(l\) (blue squares in the figure below) is an independent variable that evolves over time. The update rule of each
layer is entirely local; only signals from a layer’s connected synapses (red circles in the figure below) can tell the hidden state how to change. This is shown in the following equation:
\[\tau \frac{d x_{i}^{l}}{dt} = -\frac{\partial E}{\partial g_i^l}\]
where \(g_i^l\) are the activations (i.e., non-linearities) on each neuron layer, described in the section on Neuron Layers. Concretely, we implement the above differential equation as the following
discretized equation (where the bold \({\mathbf x}_l\) is the collection of all elements in layer \(l\)’s state):
\[ \mathbf{x}_l^{(t+1)} = \mathbf{x}_l^{(t)} - \frac{dt}{\tau} \nabla_{\mathbf{g}_l}E(t)\]
HAMUX handles all the complexity of scaling this fundamental update equation to many layers and hyper synapses. In addition, it provides a framework to:
1. Implement your favorite Deep Learning operations as a HyperSynapse
2. Port over your favorite activation functions as Lagrangians
3. Connect your layers and hypersynapses into a HAM (using a hypergraph as the data structure)
4. Inject your data into the associative memory
5. Automatically calculate and descend the energy given the hidden states at any point in time
Use these features to train any hierarchical associative memory on your own data! All of this made possible by JAX.
The examples/ subdirectory contains a (growing) list of examples on how to apply HAMUX on real data.
Explaining the "energy fundamentals" of HAMUX (Layers and Synapses, left) using a 4-layer, 3-synapse example HAM (middle) that can be built using the code on the right.
🌀Neuron Layers
Neuron layers are the recurrent unit of a HAM; that is, 🌀neurons keep a state that changes over time according to the dynamics of the system. These states always change to minimize the global energy
function of the system.
For those of us familiar with traditional Deep Learning architectures, we are familiar with nonlinear activation functions like the ReLU and SoftMax. A neuron layer in HAMUX is exactly that: a
nonlinear activation function defined on some neuron. However, we need to express the activation function as a convex Lagrangian function \(\mathcal{L}\) that is the integral of the desired
non-linearity such that the derivative of the Lagrangian function \(\nabla \mathcal{L}\) is our desired non-linearity. E.g., consider the ReLU:
\[ \begin{align*} \mathcal{L}(x) &:= \frac{1}{2} (\max(x, 0))^2\\ \nabla \mathcal{L} &= \max(x, 0) = \mathrm{relu}(x)\\ \end{align*} \]
We need to define our activation layer in terms of the Lagrangian of the ReLU instead of the ReLU itself. Extending this constraint to other nonlinearities makes it possible to define the scalar
energy for any neuron in a HAM. It turns out that many activation functions used in today’s Deep Learning landscape are expressible as a Lagrangian. HAMUX is “batteries-included” for many common
activation functions including relus, softmaxes, sigmoids, LayerNorms, etc. See our documentation on Lagrangians for examples on how to implement efficient activation functions from Lagrangians in
JAX. We show how to turn Lagrangians into usable energy building blocks in our documentation on neuron layers.
A 🤝hypersynapse ONLY sees activations of connected 🌀neuron layers. Its one job: report HIGH ⚡️energy if the connected activations are dissimilar and LOW ⚡️energy when they are aligned. Hypersynapses
can resemble convolutions, dense multiplications, even attention… Take a look at our documentation on (hyper)synapses.
🚨 Point of confusion: modern AI frameworks have ConvLayers and NormalizationLayers. In HAMUX, these would be more appropriately called ConvSynapses and NormalizationLagrangians.
From pip:
pip install hamux
If you are using accelerators beyond the CPU you will need to additionally install the corresponding jax and jaxlib versions following their documentation. E.g.,
pip install --upgrade "jax[cuda]" -f https://storage.googleapis.com/jax-releases/jax_cuda_releases.html
From source:
After cloning:
cd hamux
conda env create -f environment.yml
conda activate hamux
pip install --upgrade "jax[cuda]" -f https://storage.googleapis.com/jax-releases/jax_cuda_releases.html # If using GPU accelerator
pip install -e .
pip install -r requirements-dev.txt # To run the examples
How to Use
We can build a simple 4 layer HAM architecture using the following code
layers = [
hmx.TanhLayer((32,32,3)), # e.g., CIFAR Images
hmx.SigmoidLayer((11,11,1000)), # CIFAR patches
hmx.SoftmaxLayer((10,)), # CIFAR Labels
hmx.SoftmaxLayer((1000,)), # Hidden Memory Layer
synapses = [
hmx.ConvSynapse((3,3), strides=3),
connections = [
([0,1], 0),
([1,3], 1),
([2,3], 2),
rng = jax.random.PRNGKey(0)
param_key, state_key, rng = jax.random.split(rng, 3)
states, ham = hmx.HAM(layers, synapses, connections).init_states_and_params(param_key, state_key=state_key);
Notice that we did not specify any output channel shapes in the synapses. The desired output shape is computed from the layers connected to each synapse during hmx.HAM.init_states_and_params.
We have two fundamental objects: states and ham. The ham object contains the connectivity structure of the HAM (e.g., layer+hypersynapse+hypergraph information) alongside the parameters of the
network. The states object is a list of length nlayers where each item is a tensor representing the neuron states of the corresponding layer.
We make it easy to run the dynamics of any HAM. Every forward function is defined external to the memory and can be modified to extract different memories from different layers, as desired. The
general steps for any forward function are:
1. Initialize the dynamic states
2. Inject an initial state into the system
3. Run dynamics, calculating energy gradient at every point in time.
4. Return the layer state/activation of interest
def fwd(model, x, depth=15, dt=0.1):
"""Assuming a trained HAM, run association with the HAM on batched inputs `x`"""
# 1. Initialize model states at t=0. Account for batch size
xs = model.init_states(x.shape[0])
# Inject initial state
xs[0] = x
energies = []
for i in range(depth):
energies.append(model.venergy(xs)) # If desired, observe the energy
dEdg = model.vdEdg(xs) # Calculate the gradients
xs = jtu.tree_map(lambda x, stepsize, grad: x - stepsize * grad, xs, model.alphas(dt), dEdg)
# Return probabilities of our label layer
probs = model.layers[-2].activation(xs[-2])
return jnp.stack(energies), probs
x = jax.random.normal(jax.random.PRNGKey(2), (batch_size, 32,32,3))
energies, probs = fwd(ham, x, depth=20, dt=0.3)
print(probs.shape) # batchsize, nclasses
assert jnp.allclose(probs.sum(-1), 1)
More examples coming soon!
The Energy Function vs the Loss Function
We use JAX’s autograd to descend the energy function of our system AND the loss function of our task. The derivative of the energy is always taken wrt to our states; the derivative of the loss
function is always taken wrt our parameters. During training, we change our parameters to optimize the Loss Function. During inference, we assume that parameters are constant.
Autograd for Descending Energy
Every HAM defines the energy function for our system, which is everything we need to compute memories of the system. Naively, we can calculate \(\nabla_x E\): the derivative of the energy function
wrt the states of each layer:
But it turns out we improve the efficiency of our network if we instead take \(\nabla_g E\): the derivative of the energy wrt the activations instead of the states. They have the same local minima,
even though the trajectory to get there is different. Some nice terms cancel, and we get:
\[\nabla_g E_\text{HAM} = x + \nabla_g E_\text{synapse}\] | {"url":"https://bhoov.com/hamux/index.html","timestamp":"2024-11-10T07:47:51Z","content_type":"application/xhtml+xml","content_length":"42192","record_id":"<urn:uuid:1f1acc15-48cd-4077-8639-c6003f1e3bc7>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00403.warc.gz"} |
Different time solutions for the firing squad synchronization
problem on basic grid networks
Different time solutions for the firing squad synchronization problem on basic grid networks
We present several solutions to the Firing Squad Synchronization Problem on grid networks of different shapes. The nodes are finite state processors that work in unison with other processors and in
synchronized discrete steps. The networks we deal with are: the line, the ring and the square. For all of these models we consider one- and two-way communication modes and we also constrain the
quantity of information that adjacent processors can exchange at each step. We first present synchronization algorithms that work in time n2, nlogn, $n\sqrt{n}$, 2n, where n is a total number of
processors. Synchronization methods are described through so called signals that are then used as building blocks to compose synchronization solutions for the cases that synchronization times are
expressed by polynomials with nonnegative coefficients.
Gruska, Jozef, et al. "Different time solutions for the firing squad synchronization problem on basic grid networks." RAIRO - Theoretical Informatics and Applications 40.2 (2006): 177-206. <http://
abstract = { We present several solutions to the Firing Squad Synchronization Problem on grid networks of different shapes. The nodes are finite state processors that work in unison with other
processors and in synchronized discrete steps. The networks we deal with are: the line, the ring and the square. For all of these models we consider one- and two-way communication modes and we also
constrain the quantity of information that adjacent processors can exchange at each step. We first present synchronization algorithms that work in time n2, nlogn, $n\sqrt n$, 2n, where n is a total
number of processors. Synchronization methods are described through so called signals that are then used as building blocks to compose synchronization solutions for the cases that synchronization
times are expressed by polynomials with nonnegative coefficients. },
author = {Gruska, Jozef, La Torre, Salvatore, Napoli, Margherita, Parente, Mimmo},
journal = {RAIRO - Theoretical Informatics and Applications},
language = {eng},
month = {7},
number = {2},
pages = {177-206},
publisher = {EDP Sciences},
title = {Different time solutions for the firing squad synchronization problem on basic grid networks},
url = {http://eudml.org/doc/249723},
volume = {40},
year = {2006},
TY - JOUR
AU - Gruska, Jozef
AU - La Torre, Salvatore
AU - Napoli, Margherita
AU - Parente, Mimmo
TI - Different time solutions for the firing squad synchronization problem on basic grid networks
JO - RAIRO - Theoretical Informatics and Applications
DA - 2006/7//
PB - EDP Sciences
VL - 40
IS - 2
SP - 177
EP - 206
AB - We present several solutions to the Firing Squad Synchronization Problem on grid networks of different shapes. The nodes are finite state processors that work in unison with other processors and
in synchronized discrete steps. The networks we deal with are: the line, the ring and the square. For all of these models we consider one- and two-way communication modes and we also constrain the
quantity of information that adjacent processors can exchange at each step. We first present synchronization algorithms that work in time n2, nlogn, $n\sqrt n$, 2n, where n is a total number of
processors. Synchronization methods are described through so called signals that are then used as building blocks to compose synchronization solutions for the cases that synchronization times are
expressed by polynomials with nonnegative coefficients.
LA - eng
UR - http://eudml.org/doc/249723
ER -
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Advanced Algorithms Lecture Notes (MIT 6.854J) - DOKUMEN.PUB
Citation preview
6.854 Advanced Algorithms Lecture 1: September 7, 2005 Lecturer: David Karger Scribes: David G. Andersen, Ioana Dumitriu, John Dunagan, Akshay Patil (2003)
Fibonaccci Heaps
Motivation and Background
Priority queues are a classic topic in theoretical computer science. As we shall see, Fibonacci Heaps provide a fast and elegant solution. The search for a fast priority queue implementation is
motivated primarily by two network optimization algorithms: Shortest Path and Minimum Spanning Tree (MST).
Shortest Path and Minimum Spanning Trees
Given a graph G(V, E) with vertices V and edges E and a length function l : E → + . We define the Shortest Path and MST problems to be, respectively: shortest path. For a fixed source s ∈ V , find the
shortest path to all vertices v ∈ V minimum spanning tree (MST). Find the minimum length set of edges F ⊂ E such that F connects all of V . Note that the MST problem is the same as the Shortest Path
problem, except that the source is not fixed. Unsurprisingly, these two problems are solved by very similar algorithms, Prim’s for MST and Djikstra’s for Shortest Path. The algorithm is: 1. Maintain a
priority queue on the vertices 2. Put s in the queue, where s is the start vertex (Shortest Path) or any vertex (MST). Give s a key of 0. 3. Repeatedly delete the minimum-key vertex v from the queue
and mark it “scanned”
For each neighbor w of v:
If w is not in the queue and not scanned, add it with key:
• Shortest Path: key(v) + length(v → w) • MST: length(v → w) If, on the other hand, w is in the queue already, then decrease its key to the minimum of the value calculated above and w’s current key.
Lecture 1: September 7, 2005
The classical answer to the problem of maintaining a priority queue on the vertices is to use a binary heap, often just called a heap. Heaps are commonly used because they have good bounds on the
time required for the following operations: insert delete-min decrease-key
O(log n) O(log n) O(log n)
If a graph has n vertices and m edges, then running either Prim’s or Djikstra’s algorithms will require O(n log n) time for inserts and deletes. However, in the worst case, we will also perform m
decrease-keys, because we may have to perform a key update every time we come across a new edge. This will take O(m log n) time. Since the graph is connected, m ≥ n, and the overall time bound is
given by O(m log n). Since m ≥ n, it would be nice to have cheaper key decreases. A simple way to do this is to use d-heaps.
d-heaps make key reductions cheaper at the expense of more costly deletions. This trade off is accomplished by replacing the binary heap with a d-ary heap—the branching factor (the maximum number of
children for any node) is changed from 2 to d. The depth of the tree then becomes logd (n). However, delete-min operations must now traverse all of the children in a node, so their cost goes up to d
logd (n). Thus, the running time of the algorithm becomes O(nd logd (n) + m logd (n)). Choosing the optimal d = m/n to balance the two terms, we obtain a total running time of O(m logm/n n). When m =
n2 , this is O(m), and when m = n, this is O(n log n). This seems pretty good, but it turns out we can do much better.
Amortized Analysis
Amortized analysis is a technique for bounding the running time of an algorithm. Often we analyse an algorithm by analyzing the individual operations that the algorithm performs and then multiplying
the total number of operations by the time required to perform an operation. However, it is often the case that an algorithm will on occasion perform a very expensive operation, but most of the time
the operations are cheap. Amortized analysis is the name given to the technique of analyzing not just the worst case running time of an operation but the average case running time of an operation.
This will allow us to balance the expensive-but-rare operations against their cheap-and-frequent peers. There are several methods for performing amortized analysis; for a good treatment, see
Introduction to Algorithms by Cormen, Leiserson, and Rivest. The method of amortized analysis used to analyze Fibonacci heaps is the potential method: • Measure some aspect of the data structure
using a potential function. Often this aspect of
Lecture 1: September 7, 2005
the data structure corresponds to what we intuitively think of as the complexity of the data structure or the amount by which it is out of kilter or in a bad arrangement. • If operations are only
expensive when the data structure is complicated, and expensive operations can also clean up (“uncomplexify”) the data structure, and it takes many cheap operations to noticeably increase the
complexity of the data structure, then we can amortize the cost of the expensive operations over the cost of the many cheap operations to obtain a low average cost. Therefore, to design an efficient
algorithm, we want to force the user to perform many operations to make the data structure complicated, so that the work doing the expensive operation and cleaning up the data structure is amortized
over those many operations. We compute the potential of the data structure by using a potential function Φ that maps the data structure (DS) to a real number Φ(DS). Once we have defined Φ, we
calculate the cost of the ith operation by: costamortized (operationi ) = costactual (operationi ) + Φ(DSi ) − Φ(DSi−1 ) where DSi refers to the state of the data structure after the ith operation.
The sum of the amortized costs is then �
costactual (operationi ) + Φfinal − Φinitial
. If we can prove� that Φf inal ≥ Φinitial � , then we’ve shown that the amortized costs bound the real costactual . Then we can just analyze the amortized costs and costs, that is, costamortized ≥
show that this isn’t too much, knowing that our analysis is useful. Most of the time it is obvious that Φf inal ≥ Φinitial and the real work is in coming up with a good potential function.
Fibonacci Heaps
The Fibonacci heap data structure invented by Fredman and Tarjan in 1984 gives a very efficient implementation of the priority queues. Since the goal is to find a way to minimize the number of
operations needed to compute the MST or SP, the kind of operations that we are interested in are insert, decrease-key, merge, and delete-min. (We haven’t covered why merge is a useful operation yet,
but it will become clear.) The method to achieve this minimization goal is laziness – “do work only when you must, and then use it to simplify the structure as much as possible so that your future
work is easy”. This way, the user is forced to do many cheap operations in order to make the data structure complicated. Fibonacci heaps make use of heap-ordered trees. A heap-ordered tree is one
that maintains the heap property, that is, where key(parent) ≤ key(child) for all nodes in the tree. A Fibonacci heap H is a collection of heap-ordered trees that have the following properties:
Lecture 1: September 7, 2005
1. The roots of these trees are kept in a doubly-linked list (the “root list” of H); 2. The root of each tree contains the minimum element in that tree (this follows from being a heap-ordered tree);
3. We access the heap by a pointer to the tree root with the overall minimum key; 4. For each node x, we keep track of the rank (also known as the order or degree) of x, which is just the number of
children x has; we also keep track of the mark of x, which is a Boolean value whose role will be explained later.
For each node, we have at most four pointers that respectively point to the node’s parent, to one of its children, and to two of its siblings. The sibling pointers are arranged in a doubly-linked
list (the “child list” of the parent node). Of course, we haven’t described how the operations on Fibonacci heaps are implemented, and their implementation will add some additional properties to H.
Here are some elementary operations used in maintaining Fibonacci heaps.
Inserting, merging, cutting, and marking.
Inserting a node x. We create a new tree containing only x and insert it into the root list of H; this is clearly an O(1) operation. Merging two trees. Let x and y be the roots of the two trees we
want to merge; then if the key in x is no less than the key in y, we make x the child of y; otherwise, we make y the child of x. We update the appropriate node’s rank and the appropriate child list;
this takes O(1) operations. Cutting a node. If x is a root in H, we are done. If x is not a root in H, we remove x from the child list of its parent, and insert it into the root list of H, updating
the appropriate variables (the rank of the parent of x is decremented, etc.). Again, this takes O(1) operations. (We assume that when we want to find a node, we have a pointer hanging around that
accesses it directly, so actually finding the node takes O(1) time.) Marking. We say that x is marked if its mark is set to “true”, and that it is unmarked if its mark is set to “false”. A root is
always unmarked. We mark x if it is not a root and it loses a child (i.e., one of its children is cut and put into the root-list). We unmark x whenever it becomes a root. We will make sure later that
no marked node loses another child before it itself is cut (and reverted thereby to unmarked status).
Decreasing keys and Deleting mins
At first, decrease-key does not appear to be any different than merge or insert ; just find the node and cut it off from its parent, then insert the node into the root list with a new key. This requires
removing it from its parent’s child list, adding it to the root list, updating the parent’s rank, and (if necessary) the pointer to the root of smallest key. This takes O(1) operations.
Lecture 1: September 7, 2005
The delete-min operation works in the same way as decrease-key: Our pointer into the Fibonacci heap is a pointer to the minimum keyed node, so we can find it in one step. We remove this root of
smallest key, add its children to the root-list, and scan through the linked list of all the root nodes to find the new root of minimum key. Therefore, the cost of a delete-min operation is O(# of
children ) of the root of minimum key plus O(# of root nodes); in order to make this sum as small as possible, we have to add a few bells and whistles to the data structure.
Population Control for Roots
We want to make sure that every node has a small number of children. This can be done by ensuring that the total number of descendants of any node is exponential in the number of its children. In the
absence of any “cutting” operations on the nodes, one way to do this is by only merging trees that have the same number of children (i.e, the same rank). It is relatively easy to see that if we only
merge trees that have the same rank, the total number of descendants (counting onself as a descendant) is always (2# of children ). The resulting structure is called a binomial � � tree because the
number of descendants at distance k from the root in a tree of size n is exactly nk . Binomial heaps preceded Fibonacci heaps and were part of the inspiration for them. We now present Fibonacci heaps
in full detail.
Actual Algorithm for Fibonacci Heaps
• Maintain a list of heap-ordered trees. • insert : add a degree 0 tree to the list. • delete-min: We can find the node we wish to delete immediately since our handle to the entire data structure is a
pointer to the root with minimum key. Remove the smallest root, and add its children to the list of roots. Scan the roots to find the next minimum. Then consolidate all the trees (merging trees of
equal rank) until there is ≤ 1 of each rank. (Assuming that we have achieved the property that the number of descendants is exponential in the number of children for any node, as we did in the
binomial trees, no node has rank > c log n for some constant c. Thus consolidation leaves us with O(log n) roots.) The consolidation is performed by allocating buckets of sizes up to the maximum
possible rank for any root node, which we just showed to be O(log n). We put each node into the appropriate bucket, at cost O(log n) + O(# of roots). Then we march through the buckets, starting at
the smallest one, and consolidate everything possible. This again incures cost O(log n) + O(# of roots). • decrease-key: cut the node, change its key, and insert it into the root list as before,
Additionally, if the parent of the node was unmarked, mark it. If the parent of the node was marked, cut it off also. Recursively do this until we get up to an unmarked node. Mark it.
Actual Analysis for Fibonacci Heaps
Define Φ(DS) = (k· # of roots in DS + 2 · # marked bits in DS). Note that insert and delete-min do not ever cause nodes to be marked - we can analyze their behaviour without reference to marked
Lecture 1: September 7, 2005
and unmarked bits. The parameter k is a constant that we will conveniently specify later. We now analyze the costs of the operations in terms of their amortized costs (defined to be the real costs
plus the changes in the potential function). • insert : the amortized cost is O(1). O(1) actual work plus k * O(1) change in potential for adding a new root. O(1) + kO(1) = O(1) total amortized cost.
• delete-min: for every node that we put into the root list (the children of the node we have deleted), plus every node that is already in the root list, we do constant work putting that node into a
bucket corresponding to its rank and constant work whenever we merge the node. Our real costs are putting the roots into buckets (O(#roots)), walking through the buckets (O(log n)), and doing the
consolidating tree merges (O(#roots)). On the other hand, our change in potential is k∗(log n−#roots) (since there are at most log n roots after consolidation). Thus, total amortized cost is O(#
roots) + O(log n) + k ∗ (log n − #roots) = O(log n). • decrease-key: The real cost is O(1) for the cut, key decrease and re-insertion. This also increases the potential function by O(1) since we are
adding a root to the root list, and maybe by another 2 since we may mark a node. The only problematic issue is the possibility of a “cascading cut” - a cascading cut is the name we give to a cut that
causes the node above it to cut because it was already marked, which causes the ndoe above it be cut since it too was alrady marked, etc. This can increase the actual cost of the operation to (# of
nodes already marked). Luckily, we can pay for this with the potential function! Every cost we incur from having to update pointers due to a marked node that was cut is offset by the decrease in the
potential function when that previously marked node is now left unmarked in the root list. Thus the amortized cost for this operation is just O(1). The only thing left to prove is that for every node
in every tree in our Fibonacci heap, the number of descendants of that node is exponential in the number of children of that node, and that this is true even in the presence of the “weird” cut rule
for marked bits. We must prove this in order to substantiate our earlier assertion that all nodes have degree ≤ log n.
The trees are big
Consider the children of some node x in the order in which they were added to x. Lemma : The ith child to be added to x has rank at least i − 2. P roof : Let y be the ith child to be added to x. When
it was added, y had at least i − 1 children. This is true because we can currently see i − 1 children that were added earlier, so they were there at the time of the y’s addition. This means that y
had at least i − 1 children at the time of it’s merger, because we only merge equal ranked nodes. Since a node could not lose more than one child without being cut itself, it must be that y has at
least i − 2 children (i − 1 from when it was added, and no more than a potential 1 subsequently lost). Note that if we had been working with a binomial tree, the appropriate lemma would have been
rank = i − 1 not ≥ i − 2.
Lecture 1: September 7, 2005
Let Sk be the minimum number of descendants of a node with k children. We have S0 = 1, S1 = 2 and, k−2 � Si Sk ≥ i=0
This recurrence is solved by Sk ≥ Fk+2 , the (k+2) Fibonacci number. Ask anyone on the street and that person will tell you that the Fibonacci numbers grow exponentially; we have proved Sk ≥ 1.5k ,
completing our analysis of Fibonacci heaps. th
Only recently have problem sizes increased to the point where Fibonacci heaps are beginning to appear in practice. Further study of this issue might make an interesting term project; see David Karger
if you’re curious. Fibonacci Heaps allow us to improve the running time in Prim’s and Djikstra’s algorithms. A more thorough analysis of this will be presented in the next class.
6.854 Advanced Algorithms Lecture 2: September 9, 2005
Lecturer: David Karger Scribes: Sommer Gentry, Eddie Kohler
Persistent Data Structures 2.1
Introduction and motivation
So far, we’ve seen only ephemeral data structures. Once changes have been made to an ephemeral data structure, no mechanism exists to revert to previous states. Persistent data structures are really
data structures with archaeology. Partial persistence lets you make modifications only to the present data structure but allows queries of any previous version. These previous versions might be
accessed via a timestamp. Full persistence lets you make queries and modifications to all previous versions of the data structure. With this type of persistence the versions don’t form a simple linear
path — they form a version tree. The obvious way to provide persistence is to make a copy of the data structure each time it is changed. This has the drawback of requiring space and time proportional
to the space occupied by the original data structure. In turns out that we can achieve persistence with O(1) additional space and O(1) slowdown per operation for a broad class of data structures.
In addition to the obvious ‘look-back’applications, we can use persistent data structures to solve problems by representing one of their dimensions as time. Once example is the computational geometry
problem of planar point location. Given a plane with various polygons or lines which break the area up into a number of regions, in which region is a query point is located? In one dimension, the
linear point location problem can be solved with a splay tree or a binary tree that simply searches for the two objects on either side of the query point. To solve the problem in two dimensions,
break the plane into vertical slices at each vertex or point where lines cross. These slices are interesting because crossovers don’t happen inside slices: inside each slice, the dividing lines
between regions appear in a fixed order, so the problem reduces to the
Lecture 2: September 9, 2005
Figure 2.1: Breaking the plane into slices for planar point location linear case and requires a binary search (plus a bit of linear algebra). Figure 2.1 shows an example of how these slices look. To
locate a point, first find the vertical slice it is in with a search on the point’s x coordinate, and then, within that slice, find the region it is in with a search on the point’s y coordinate (plus
algebra). To do two binary searches takes only O(log n) time, so we can locate a point in O(log n) time. However, setting up the trees for searching a figure with n vertices will require n different
trees, taking O(n2 log n) time and O(n2 ) space to do the preprocessing. Notice that between two adjacent slices of the picture there will only be one change. If we treat the horizontal direction as
a timeline and use a persistent data structure, we can find the horizontal location of the point as a ’version’ of the vertical point location data structure. In this way, we can preserve the O(log n)
query time and use only O(n) space and O(n log n) preprocessing time.
Making pointer-based data structures persistent
Now let’s talk about how to make arbitrary pointer-based data structures persistent. Eventually, we’ll reveal a general way to do this with O(1) additional space and O(1) slowdown, first published by
Sleator and Tarjan et al. We’re mainly going to discuss partial persistence to make the explanation simpler, but their paper achieves full persistence as well.
First try: fat nodes
One natural way to make a data structure persistent is to add a modification history to every node. Thus, each node knows what its value was at any previous point in time. (For a fully persistent
structure, each node would hold a version tree, not just a version history.) This simple technique requires O(1) space for every modification: we just need to store the new data. Likewise, each
modification takes O(1) additional time to store the modification at the end of the modification history. (This is an amortized time bound, assuming we store the modification history in a growable array.
A fully persistent data structure would add O(log m) time to every modification, since the version history would have to be kept in a tree of some kind.)
Lecture 2: September 9, 2005
Unfortunately, accesses have bad time behavior. We must find the right version at each node as we traverse the structure, and this takes time. If we’ve made m modifications, then each access operation
has O(log m) slowdown. (In a partially persistent structure, a version is uniquely identified by a timestamp. Since we’ve arranged the modifications by increasing time, you can find the right version by
binary search on the modification history, using the timestamp as key. This takes O(log m) time to find the last modification before an arbitrary timestamp. The time bound is the same for a fully
persistent structure, but a tree lookup is required instead of a binary search.)
Second try: path copying
Another simple idea is to make a copy of any node before changing it. Then you have to cascade the change back through the data structure: all nodes that pointed to the old node must be modified to
point to the new node instead. These modifications cause more cascading changes, and so on, until you reach a node that nobody else points to—namely, the root. (The cascading changes will always reach
the root.) Maintain an array of roots indexed by timestamp; the data structure pointed to by time t’s root is exactly time t’s data structure. (Some care is required if the structure can contain
cycles, but it doesn’t change any time bounds.) Figure 2.2 shows an example of path copying on a binary search tree. Making a modification creates a new root, but we keep the old root around for later
use; it’s shown in dark grey. Note that the old and new trees share some structure (light grey nodes).
Figure 2.2: Path copying on binary search trees Access time does better on this data structure. Accesses are free, except that you must find the correct root. With m modifications, this costs O(log m)
additive lookup time—much better than fat nodes’ multiplicative O(log m) slowdown. Unfortunately, modification time and space is much worse. In fact, it’s bounded by the size of the structure, since a
single modification may cause the entire structure to be copied. That’s O(n). Path copying applies just as well to fully persistent data structures.
Sleator, Tarjan et al.
Sleator, Tarjan et al. came up with a way to combine the advantages of fat nodes and path copying, getting O(1) access slowdown and O(1) modification space and time. Here’s how they did it, in the
special case of trees. In each node, we store one modification box. This box can hold one modification to the node—either a modification to one of the pointers, or to the node’s key, or to some other
piece of node-specific
Lecture 2: September 9, 2005
data—and a timestamp for when that modification was applied. Initially, every node’s modification box is empty. Whenever we access a node, we check the modification box, and compare its timestamp
against the access time. (The access time specifies the version of the data structure that we care about.) If the modification box is empty, or the access time is before the modification time, then we
ignore the modification box and just deal with the normal part of the node. On the other hand, if the access time is after the modification time, then we use the value in the modification box,
overriding that value in the node. (Say the modification box has a new left pointer. Then we’ll use it instead of the normal left pointer, but we’ll still use the normal right pointer.) Modifying a
node works like this. (We assume that each modification touches one pointer or similar field.) If the node’s modification box is empty, then we fill it with the modification. Otherwise, the modification
box is full. We make a copy of the node, but using only the latest values. (That is, we overwrite one of the node’s fields with the value that was stored in the modification box.) Then we perform the
modification directly on the new node, without using the modification box. (We overwrite one of the new node’s fields, and its modification box stays empty.) Finally, we cascade this change to the node’s
parent, just like path copying. (This may involve filling the parent’s modification box, or making a copy of the parent recursively. If the node has no parent—it’s the root—we add the new root to a
sorted array of roots.) Figure 2.3 shows how this works on a persistent search tree. The modification boxes are shown in grey.
Figure 2.3: Modifying a persistent search tree. With this algorithm, given any time t, at most one modification box exists in the data structure with time t. Thus, a modification at time t splits the
tree into three parts: one part contains the data from before time t, one part contains the data from after time t, and one part was unaffected by the modification.
Figure 2.4: How modifications split the tree on time.
Lecture 2: September 9, 2005
How about time bounds? Well, access time gets an O(1) slowdown (plus an additive O(log m) cost for finding the correct root), just as we’d hoped! (We must check the modification box on each node we
access, but that’s it.) Time and space for modifications require amortized analysis. A modification takes O(1) amortized space, and O(1) amortized time. To see why, use a potential function ϕ, where ϕ
(T ) is the number of full live nodes in T . The live nodes of T are just the nodes that are reachable from the current root at the current time (that is, after the last modification). The full live
nodes are the live nodes whose modification boxes are full. So, how much does a modification cost? Each modification involves some number of copies, say k, followed by 1 change to a modification box.
(Well, not quite—you could add a new root—but that doesn’t change the argument.) Consider each of the k copies. Each costs O(1) space and time, but decreases the potential function by one! (Why?
First, the node we copy must be full and live, so it contributes to the potential function. The potential function will only drop, however, if the old node isn’t reachable in the new tree. But we
know it isn’t reachable in the new tree—the next step in the algorithm will be to modify the node’s parent to point at the copy! Finally, we know the copy’s modification box is empty. Thus, we’ve
replaced a full live node with an empty live node, and ϕ goes down by one.) The final step fills a modification box, which costs O(1) time and increases ϕ by one. Putting it all together, the change in
ϕ is ∆ϕ = 1 − k. Thus, we’ve paid O(k + ∆ϕ) = O(1) space and O(k + ∆ϕ + 1) = O(1) time! What about non-tree data structures? Well, they may require more than one modification box. The limiting factor
is the in-degree of a node: how many other nodes can point at it. If the in-degree of a node is k, then we must use k extra modification boxes to get O(1) space and time cost.
The geometric search problem
Let’s return to the geometric search problem discussed in Section 2.1. We now know how to make a persistent tree; but what kind of balanced tree should we use? It turns out that this is one
application where splay trees crash and burn. The reason is splaying. Every rotation while we access a splay tree is a modification, so we do O(log n) modifications (costing an additional O(log n)
space) per access—including reads! A less sexy balanced tree, like a red-black tree, is a better choice. Red-black trees keep themselves balanced with at most one rotation per modification (and a
bunch of fiddling with red/black bits). This looks good—accesses are cheaper, and modifications cost O(1)—almost. The “almost” is because of red/black bit fiddling, which may affect a lot more than one
node on the tree. A fully persistent red-black tree would need to keep the proper values for the red/black bits for every single version of the tree (so that further modifications could be made). This
would mean that changing a red/black bit would count as a modification, and would have a persistence-related cost. Luckily, in the geometric search problem, we won’t need to look at the red/black bits
for old versions of the tree, so we can keep them only for the latest version of the tree and pay O(1) persistence-related cost per modification.
6.854 Advanced Algorithms Lecture 3: 09/12/2005
Lecturer: David Karger Scribes: Xin Zhang
Splay Trees
Splay trees are binary search trees with good balance properties when amortized over a sequence of operations. When a node x is accessed, we perform a sequence of splay steps to move x to the root of
the tree. There are 6 types of splay steps, each consisting of 1 or 2 rotations (see Figures 3.1, 3.2, and 3.3).
y x
Figure 3.1: The rr splay step: This is performed when x and x’s parent are both left children. The splay step consists of first a right rotation on z and then a right rotation on y (hence rr). The ll
splay step, for x and x’s parent being right children, is analogous. We perform splay steps to x (rr, ll, lr, or rl, depending on whether x and x’s parent are left or right children) until x is
either the root or a child of the root. In the latter case, we need to perform either a r or l splay step to make x the root. This completes a splay of x. We will show that splay operations have
amortized cost O(log n), and that consequently all splay tree operations have amortized cost O(log n).
Analysis of Splay Steps
For amortized analysis, we define the following for each node x:
Lecture 3: 09/12/2005
z x 1
D y
A B
Figure 3.2: The lr splay step: This is performed when x is a right child and x’s parent is a left child. The splay step consists of first a left rotation on y and then a right rotation on z. The rl
splay step, for x being a left child and x’s parent being a right child, is analogous.
y x
C B
A B
Figure 3.3: The r splay step: This is performed when x is the left child of the root y. The splay step consists of a right rotation on the root. The l splay step, for x being the right child of the
root, is analogous.
• a constant weight w(x) > 0 (for the analysis, this can be arbitrary) • weight sum s(x) = y∈subtree(x) w(y) (where subtree(x) is the subtree rooted at x, including x) • rank r(x) = log s(x) We use r
(x) as the potential of a node. The potential function after i operations is thus φ(i) = x∈tree r(x). Lemma 1 The amortized cost of a splay step on node x is ≤ 3(r (x) − r(x)) + 1, where r is rank
before the splay step and r is rank after the splay step. Furthermore, the amortized cost of the rr, ll, lr, and rl splay steps is ≤ 3(r (x) − r(x)). Proof:
Lecture 3: 09/12/2005
We will consider only the rr splay step (refer to Figure 3.1). The actual cost of the splay step is 2 (for 2 rotations). The splay step only affects the potentials/ranks of nodes x, y, and z; we
observe that r (x) = r(z), r(y) ≥ r(x), and r (y) ≤ r (x). The amortized cost of the splay step is thus: amortized cost = =
2 + φ(i + 1) − φ(i) 2 + (r (x) + r (y) + r (z)) − (r(x) + r(y) − r(z))
= ≤
2 + (r (x) − r(z)) + r (y) + r (z) − r(x) − r(y) 2 + 0 + r (x) + r (z) − r(x) − r(x)
2 + r (x) + r (z) − 2r(x)
b . Thus we also have (s is weight sum before The log function is concave, i.e., log a+log ≤ log a+b 2 2 the splay step and s is weight sum after the splay step): s(x) + s (z) log(s(x)) + log(s (z))
≤ log 2 2 s(x) + s (z) r(x) + r (z) ≤ log (note that s(x) + s (z) ≤ s (x)) 2 2 s (x) ≤ log 2 = r (x) − 1 r (z) ≤ 2r (x) − r(x) − 2 Thus the amortized cost of the rr splay step is ≤ 3(r (x) − r(x)).
The same inequality must hold for the ll splay step; the inequality also holds for the lr (and rl) splay steps. The +1 in the lemma applies for the r and l cases.
Corollary 1 The amortized cost of a splay operation on node x is O(log n). Proof: The amortized cost of a splay operation on x is the sum of the amortized costs of the splay steps on x involved:
amortized cost = cost(splay stepi ) i
≤ 3(ri+1 (x) − ri (x) + 1 i
= 3(r(root) − r(x)) + 1
Lecture 3: 09/12/2005
The +1 comes from the last r or l splay step (if necessary). If we set w(x) = 1 for all nodes in the tree, then r(root) = log n and we have: amortized cost ≤ 3 log n + 1 = O(log n)
3.3 3.3.1
Analysis of Splay Tree Operations Find
For the find operation, we perform a normal BST find followed by a splay operation on the node found (or the leaf node last encountered, if the key was not found). We can charge the cost of going down
the tree to the splay operation. Thus the amortized cost of find is O(log n).
For the insert operation, we perform a normal BST insert followed by a splay operation on the node inserted. Assume node x is inserted at depth k. Denote the parent of x as y1 , y1 ’s parent as y2 ,
and so on (the root of the tree is yk ). Then the change in potential due to the insertion of x is (r is rank before the insertion and r is rank after the insertion, s is weight sum before the
insertion): ∆φ
(r (yj ) − r(yj ))
(log(s(yj ) + 1) − log(s(yj ))
k j=1
= ≤ = ≤
s(yj ) + 1 s(yj )
⎞ k ) + 1 s(y j ⎠ (note that s(yj ) + 1 ≤ s(yj+1 )) log ⎝ s(y ) j j=1 s(y2 ) s(y3 ) s(yk ) s(yk ) + 1 · ··· · log s(y1 ) s(y2 ) s(yk−1 ) s(yk ) s(yk ) + 1 log s(yk )
log n
The amortized cost of the splay operation is also O(log n), and thus the amortized cost of insert is O(log n).
Lecture 3: 09/12/2005 We have proved the following:
Theorem 1 All splay tree operations have amortized cost O(log n).
6.854 Advanced Algorithms Lecture 4: September 14, 2005
Lecturer: David Karger
Suffix Trees and Fibonacci Heaps
Suffix Trees
Recall that our goal is to find a pattern of length m in a text of length n. Also recall that the trie will contain a size |Σ| array at each node to give O(m) lookups.
Size of the Trie
Previously, we have seen a construction algorithm that was linear in the size of the trie. We would like to show that the size of the trie is linear in the size of the text, so that the construction
algorithm takes O(n) time. We can achieve this size goal by using a compressed suffix tree. In a compressed tree, each node has strictly more than one child. Below, we see the conversion from a
uncompressed suffix tree to a compressed suffix tree.
\ b
o \ b o \ b
How will this change the number of nodes in the trie? Since there are no nodes with only one child, this is a full binary tree (i.e. every internal node has degree ≥ 2). Lemma 1 In any full tree, the
number of nodes is not more than twice the number of leaves. When we use the trailing $, the number of leaves in the trie is the number of suffixes. So does this mean that there are n leaves and the
tree is of size O(n)? Yes; however, the number of nodes isn’t necessarily the full size of the tree – we must store the substrings as well. For a string with distinct characters, storing strings on
the edges could lead to a O(n2 ) size algorithm. Instead, we just store the starting and ending index in the original text on each edge, meaning that the storage at each node is O(1), so the total
size for the tree is in fact O(n).
Lecture 4: September 14, 2005
With the compressed tree, we can still perform lookups in O(m) time using the slowfind algorithm which compares one character at a time from the pattern to the text in the trie. When slowfind
encounters a compressed node, it checks all of the characters in the node, just as if it were traversing the uncompressed series of nodes.
Building the Trie
A simple approach to building the compressed tree would be to build an uncompressed tree and then compress it. However, this approach would require quadratic time and space. The construction
algorithm for compressed tries will still insert S1 ...Sn in order. As we go down the trie to insert a new suffix, we may need to leave in the middle of an edge. For example, consider the trie that
contains just bbb. To insert ba, we must split the edge :
o \
o \ bbb \ o
\ b o a / \ bb o o
Splitting an edge is easy; we will create one new node from the split and then one new node (a leaf) from the insertion. One problem with compressed trie is where to put suffix links from the
compressed edges. Another problem is that we previously described the time to operate on the tree in terms of n (the number of characters in the text); however, n may now be greater than the number
of nodes. fastfind is an algorithm for descending the trie if you know that the pattern is in the trie. fastfind only checks the first character of a compressed edge; all the other characters must match
if the first does because the pattern is in the trie and because there is no branch in the edge (ie, if the pattern is there and there is no branch, it must match the entire edge or stop in the middle
of the edge). If the pattern is shorter than the edge, then fastfind will stop in the middle of the edge. Consequently, the number of operations in a fastfind is linear in the number of checked nodes
in the trie rather than the length of the pattern. Suppose we have just inserted Si = aw and are at the newly created leaf which has parent node pi . We maintain the following invariant:
Lecture 4: September 14, 2005
Invariant: Every internal node except for the current parent has a suffix link (ignore for now the issue of where the suffix links point). /
/ SL
g_i o ==============o
/ \ | w1 w1 / ... | o p_i o alpha | | w2 | o s_i Now we describe the consruction in detail. Let gi be a parent of pi . To insert Si+1 : ascend to gi , traverse the suffix link there, and do a fastfind
of w1 , which takes you to node α (thus maintaining the invariant for next time). Make a suffix link from pi to α. From there, do a slowfind on w2 and do the insertions that you need. Since pi was
previously a leaf node, it has no suffix link yet. gi was previously an internal node, so it has a suffix link. w1 is the part of Si that was already in the trie below gi (i.e., it was pi ), which is why
we can use fastfind on it. w2 is the part of Si that was not previously in the trie. The running time analysis will be in two parts. The first part is the cost from the suffix of gi to the suffix of pi .
The second is the cost from the suffix of pi to the bottom of the search. The cost of going up is constant, since there are only two steps thanks to the compressed edges. Looking at the second cost
(the slowfind part), we see that it is the number of charactersr in the length difference between the suffix of pi and pi+1 , which is |pi+1 | − |pi | + 1. The sum of this term over all i is |pn | − |p0
| + n = O(n). For the first cost, recall that fastfind ’s runtime will be upperbounded by the runtime of slowfind. It takes at most |gi+1 | − |gi | time to reach gi+1 . If gi+1 is below the suffix of pi ,
then there is no cost. If the suffix of pi is below gi+1 , then the suffix of pi is pi+1 and the fastfind only takes one step from gi+1 to pi+1 , so the cost is O(1). The progression of the insert is
• suffix of gi • gi+1 • suffix of pi • pi+1 The total time is linear in the size of the compressed tree, which is linear in the size of the input.
Lecture 4: September 14, 2005
Prim and Dijkstra’s algorithms for shortest paths and minimum spanning trees were covered in 6.046. Both are greedy algorithms that start by setting node distances to infinity and then relaxing the
distances while choosing the shortest. To perform these operations, we use a priority queue (generally implemented as a heap). A heap is a data structure that will support insert, decrease-key, and
delete-min operations (and perhaps others). With a standard binary heap, all operations run in O(log n) time, so both algorithms take O(m log n) time. We’d like to improve the performance of the heap
to get a better running time for these algorithms. We could show that O(log n) is a lower bound on the time for delete-min, so the improvement will have to come from somewhere else. The Fibonacci
Heap performs a decrease-key operation in O(1) time such that Prim and Dijkstra’s algorithms require only O(m + n log n) time, Idea: During insertions, perform the minimal work possible. Rather than
performing the whole insert, we’ll just stick the node onto the end of some list, taking O(1) time. This would require us to do O(n) work to perform delete-min. However, we can put that linear amount
of work to good use to make the next delete-min more efficient. The Fibonacci heap uses “Heap Ordered Trees,” meaning that the children of every node have a key greater than their parent and that the
minimum element is at the root. For Fibonacci heaps, we will have only 1 child pointer, a doubly linked list of children, and parent pointers at every node. The time to merge two HOTs is constant:
compare the two root keys and attach the HOT with the larger root as a child of the smaller root. To insert into a HOT, compare the new element x and the root. If x is smaller, it becomes the new
root and the old root is its child. If x is larger, it is added to the list of children. To decrease a key, you prune the node from the list of children and then perform a merge. The expensive
operation is delete-min. Finding the minimum node is easy; it is the root. However, when we remove the root, we might have a large number of children that need to be processed. Therefore, we wish to
keep the number of children of any node in the tree relatively small. We will see how to do this next lecture
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0, are included in Gf . Note that the feasibility conditions imply that uf (v, w) ≥ 0 and uf (v, w) ≤ u(v, w) + u(w, v). This means all capacities in the residual network will be non-negative.
Definition 11 An augmenting path is a directed path from the node s to node t in the residual network Gf . 1
s 1
Figure 6.5: An example of a residual network. This residual network corresponds to the network depicted in Figure 6.1 and the flow in Figure 6.2. The dashed line corresponds to a possible augmenting
Lecture 6: 9/24/2003
Note that if we have an augmenting path in Gf , then this means we can push more flow along such a path in the original network G. To be more precise, if we have an augmenting path (s, v1 , v2 , . . .
vk , t), the maximum flow we can push along that path is min{uf (s, v1 ), uf (v1 , v2 ), uf (v2 , v3 ), . . . uf (vk−1 , vk ), uf (vk , t)}. Therefore, for a given network G and flow f , if there
exists an augmenting path in Gf , then the flow f is not a maximum flow. More generally, if f is a feasible flow in Gf , then f + f is a feasible flow in G. The flow f + f still satisfies conservation
because flow conservation is linear. The flow f + f is feasible because we can rearrange the inequality f (e) ≤ uf (e) = u(e) − f (e) to get f (e) + f (e) ≤ u(e). Conversely, if f is a feasible flow in
G, then the flow f − f is a feasible in Gf . Using residual networks and augmenting paths, we can state and prove the max-flow min-cut theorem. Theorem 1 (Max-flow min-cut theorem). In a flow network G,
the following conditions are equiv alent: 1. A flow f is a maximum flow. 2. The residual network Gf has no augmenting paths. 3. |f | = u(S) for some cut S. These conditions imply that the value of the
maximum flow is equal to the value of the minimum s-t cut: maxf |f | = minS u(S), where f is a flow and S is as-t cut. Proof: We show that each condition implies the other two. • 1 ⇒ 2: If there is an
augmenting path in Gf , then we previously argued that we can push additional flow along that path, so f was not a maximum flow. 1 ⇒ 2 is the contrapositive of this statement. • 2 ⇒ 3: If the residual
network Gf has no augmenting paths, s and t must be disconnected. Let S = {vertices reachable from s in Gf }. Since t is not reachable, the set S describes a s-t cut.
� s
Figure 6.6: Network Gf is disconnected. The set S contains all the nodes that are reachable from s. By construction, all edges (v, w) straddling the cut have residual capacity 0. This means in the
original network G, these edges have f (v, w) = u(v, w). Therefore, |f | = f (S) = u(S).
Lecture 6: 9/24/2003
• 3 ⇒ 1: If for some cut S, |f | = u(S), we know f must be a maximum flow. Otherwise, we would have a flow g with |g| > u(S), contradicting Lemma 3. From (1) and (3), we know that the maximum flow can
not be less than the value of the minimum cut, because for some S, |f | = u(S) and u(S) is at least as big as the minimum cut value. Lemma 3 tells us that the maximum flow can not be greater than the
minimum cut value. Therefore, the maximum flow value and the minimum cut value are the same.
Ford-Fulkerson Algorithm
The Ford-Fulkerson algorithm solves the problem of finding a maximum flow for a given network. The description of the algorithm is as follows: 1. Start with f (v, w) = 0. 2. Find an augmenting path
from s to t (using, for example, a depth first search or similar algorithms). 3. Use the augmenting path found in the previous step to increase the flow. 4. Repeat until there are no more augmenting
paths. If the capacities are all integers, then the running time is O(m|f |). This is true because finding an augmenting path and updating the flow takes O(m) time, and every augmenting path we find
must increase the flow by an integer that is at least 1. In general, if we have integral capacities, then our solution satisfies an integrality property: there exists an integral maximal flow. This
happens because every augmenting path increases flows by an integer amount. Since the running time is directly proportional to the value of the maximal flow, this particular algorithm is only good for
cases when the value |f | is small. For example, when all capacities are at most 1, the maximum flow |f | is at most n. In general, the algorithm may be as bad as linear in unary representation of the
input. Figure 6.7 illustrates a bad case for this form of the Ford-Fulkerson algorithm. We describe such an algorithm as being pseudo-polynomial, because it is polynomial in terms of variables we
care about (but not necessarily the input). If the capacities are rational, then it can be shown that the algorithm will finish. It might, however, require more than O(m|f |) time. If the capacities
are real, the algorithm might never finish, or even converge to a non-optimal value. If we setup better rules for selecting the augmentation paths however, we might get better results. Before showing
some improvements to the Ford-Fulkerson algorithm, we will introduce some new notions on the running time of algorithms.
Lecture 6: 9/24/2003
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Figure 6.7: An example for which the Ford-Fulkerson, in the stated form, might perform very badly. The algorithm runs slowly if at each step, the augmentation path is either s → 1 → 2 → t or s → 2 →
1 → t (shown with dashed lines). At an augmentation, the flow will increase by at most 2.
Definition 12 An algorithm is psuedo-polynomial if it is polynomial in the unary representation of the input. Definition 13 An algorithm is weakly polynomial if it is polynomial in the binary
representation of the input. Definition 14 An algorithm is strongly polynomial if it is polynomial in combinatorial complexity of input. (For example, in the case of max-flow problem, the algorithm
would have to be polynomial in n and m.)
Improvements to the Ford-Fulkerson Algorithm
The are at least two possible ideas for improving the Ford-Fulkerson algorithm. Both of the improvements rely on a better choice of an augmenting path (rather than a random selection of an augmenting
path). 1. Using breadth-first search, we can choose shortest-length augmenting path. With this pathselection rule, the number of augmentations is bounded by n · m, and thus the running time of the
algorithm goes down to O(nm2 ) time. 2. We can also choose the maximum-capacity augmenting path: the augmenting path among all augmenting paths that increases the flow the most (max-capacity
augmenting path). It is possible to find such a path in O(m log n) time using a modified Dijkstra’s algorithm (ignoring the cycles). The number of augmentations will be at most m ln |f | ≤ m ln(nU ),
where U = max{u(v, w)} (for integral capacities). In this lecture we prove the time bound for the second improvement. Consider the maximum flow f in the current residual network. We apply the
flow-decomposition lemma, Lemma 1 (discarding the cycles because they do not modify |f |). There are at most m paths carrying all the flow, so there must be at least one path carrying at least |f |/m
flow. Therefore, the augmenting path with
Lecture 6: 9/24/2003
maximum capacity increases the flow in the original network by at least |f |/m. This decreases the maximum possible flow in the residual graph from |f | to (1 − 1/m)|f | (remember, the smaller is the
maximum possible flow in the residual graph, the greater is the corresponding flow in the original graph). We need to decrease the flow |f | by a factor of (1 − 1/m) about m ln |f | times before we
decrease the max flow in the residual graph to 1. This is because � �ln |f | �m ln |f | � 1 1 ≈ |f | ≈ 1. |f | 1 − m e In one more step, the residual graph will have a maximum flow of 0, meaning that
the corresponding flow in the original graph is maximal. Thus, we need O(m ln |f |) augmentations. Since one augmentation step takes about O(m log n) time, the total running time is O(m2 ln |f | · ln
n). This algorithm is weakly polynomial, but not strongly polynomial.
Scaling Algorithm
We can also improve the running time of the Ford-Fulkerson algorithm by using a scaling algorithm. The idea is to reduce our max flow problem to the simple case, where all edge capacities are either 0
or 1. The scaling idea, described by Gabow in 1985 and also by Dinic in 1973, is as follows: 1. Scale the problem down somehow by rounding off lower order bits. 2. Solve the rounded problem. 3. Scale
the problem back up, add back the bits we rounded off, and fix any errors in our solution. In the specific case of the maximum flow problem, the algorithm is: 1. Start with all capacities in the graph at
0. 2. Shift in the higher-order bit of each capacity. Each capacity is then either 0 or 1. 3. Solve this maximum flow problem. 4. Repeat this process until we have processed all remaining bits. This
description of the algorithm tells us how to scale down the problem. However, we also need to describe how to scale our algorithm back up and fix the errors. To scale back up: 1. Start with some max
flow for the scaled-down problem. Shift the bit of each capacity by 1, doubling all the capacities. If we then double all our flow values, we still have a maximum flow.
Lecture 6: 9/24/2003
2. Increment some of the capacities. This restores the lower order bits that we truncated. Find augmenting paths in the residual network to re-maximize the flow. We will need to find at most m
augmenting paths. Before we scaled our problem back up, we had solved a maximum flow problem, so some cut in the residual network had 0 capacity. Doubling all the capacities and flows keeps this the
same. When we increment the edges however, we increase the cut capacity by at most m: once for each edge. Each augmenting path we find increases the flow by at least 1, so we need at most m augmenting
paths. Each augmenting path takes at most O(m) time to find, so we spend O(m2 ) time in each iteration of the scaling algorithm. If the capacity of any edge is at most U , which is an O(lg U ) bit
number, we require O(lg U ) iterations of the scaling algorithm. Therefore the total running time of the algorithm is O(m2 lg U ). This algorithm is also a weakly polynomial algorithm.
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modulators in the modulation rack. In MainStage Ctrl A (CC2 Breath): MIDI continuous controller 2 (Breath). Ctrl B (CC4 av M Kodama · 2006 · Citerat av 10 — the continuous intake of erythromycin and
chloramphenicol have been found The bone marrow function of one male patient with interstitial pneumonia was Kontinuerlig funktion - Continuous function. Från Wikipedia, den fria encyklopedin.
Matematisk funktion utan plötsliga värdeförändringar continuous function, zero for all negative values of the independent variable time and increasing proportionally with a slope equal to one for
its positive values, NVRAM 128 kB.
2011-02-07 · A function,, is called continuous at a point in, say, the variable if the restriction of to the set is continuous at, that is, if the function of the single variable is continuous at. A
function,,, can be continuous at in every variable, but need not be continuous at this point jointly in the variables. Continuous Functions. Here's the intuitive idea: a function is continuous if you
can draw its graph without lifting your pencil from the paper.
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difference between the groups. Interval training and Answer to let h to be an odd and a continuous function like h(1) = 32. Questions is to calculate f'(1).
Application of continuous-time random walk to statistical
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Fri frakt. The transfer function is a continuous function with greater variations close to the resonant frequencies. Överföringsfunktionen är en kontinuerlig funktion med The transfer function is a
continuous function with greater variations close to the resonant frequencies. Överföringsfunktionen är en kontinuerlig funktion med for which every continous function f on K is the restriction to K
of a continuous potential Uσfk of an absolutely continuous measure σ f supported in an arbitrarily Continuous Function: Russell Jesse: Amazon.se: Books.
expand_more Det är ett fortgående arbete och det finns alltid utrymme för förbättring, alltid ett behov att göra mer. A mathematical function is called continuous if, roughly said, a small change in
the input only causes a small change in the output.
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Here's the intuitive idea: a function is continuous if you can draw its graph without lifting your pencil from the paper. In other words, continuous functions are the ones without gaps, jumps or
holes. The picture below shows a continuous function: In layman’s terms, a continuous function is one that you can draw without taking your pencil from the paper. If you have holes, jumps, or
vertical asymptotes, you will have to lift your pencil up and so do not have a continuous function. If your function jumps like this, it isn’t continuous.
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Khalimsky-continuous approximation of a real-valued function (IWCIA 2004 (Eds. R. Klette & J. Zunic) Lecture Notes in Computer Science, 3322 Record linkage analysis with death hazard estimated as a
continuous function of INR. Data sources. 46 anticoagulation clinics in Sweden with computerised At which points are the following functions {\mathbb R}\to {\mathbb R} Show that all the functions
are continuous. Problem Is the function f:[0,1]\to{\mathbb R} With continuous light/continuous off function (8 hours), activation via intelligent extension unit input With selectable warm-up
function for fluorescent lamps. Programmeringsspråk enligt IEC 61131-3, Instruktionslista (IL) Ladder Diagram (LD) Function Block Diagram (FBD) (Continuous Function Chart (CFC))i VEGA KSC Continuous
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In mathematics, a property of functions and their graphs. A continuous function is one whose Definition av continuous function. a function whose value at any point in its domain is equal its limit
at the same point. Liknande ord. continuous · continuously Minimally thin sets below a function graph. Artikel i In 1991 Gardiner showed that the same criterion holds for the class of Lipschitz
continuous functions.
The graph of such a function is given in Figure 4, which depicts the first stages of the construction, consisting in the indefinite replacement of the middle third of each line segment by a broken
line made up of two segments: the ratio of the lengths is selected such that in Computes and draws a function as a continuous curve. This makes it easy to superimpose a function on top of an existing
plot. The function is called with a grid of evenly spaced values along the x axis, and the results are drawn (by default) with a line. Continuous Functions In this chapter, we define continuous
functions and study their properties. 3.1. Continuity According to the definition introduced by Cauchy, and developed by Weierstrass, continuous functions are functions that take nearby values at
nearby points. De nition 3.1.
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Artikel i In 1991 Gardiner showed that the same criterion holds for the class of Lipschitz continuous functions. Regression analysis, dose-response modelling of continuous and quantal data, in the
calculation of upper confidence limits on the risk function for continuous High Flow Nasal Cannula Versus Nasal Continuous Positive Airway Pressure. Pulmonary Function Testing in Infants With
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2.1 Integration of piecewice continuous functions. Recall the I, and let g be a function defined on I. Then g is called Riemann-Stieltjes. The Limit Distribution of a Continuous Function of Random
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In a right triangle ABC, right angled at B, a circle is drawn with AB as diameter intersecting the hypotenuse AC at P. How do you prove that the tangent to the circle at P bisects BC? | HIX Tutor
In a right triangle ABC, right angled at B, a circle is drawn with AB as diameter intersecting the hypotenuse AC at P. How do you prove that the tangent to the circle at P bisects BC?
Answer 1
The required proof would be to prove that points B,P and D lie on a semicircle with centre D.
OB= OP , radii of the given circle. Hence angle OBP= angle OPB. angle OPD is a rt angle ( PD is a tangent). Hence angle BPD= 90-OPB. Similarly, angle PBD= 90- angle OBP. Since OBP and OPB ar equal,
angle BPD= angle PBD and therefore DB=DP.
Next, angle OPX is rt. angle and angle APX= angleDPC, hence angle BPC= Angle BPD+angle APX
Since, angle APX +APO =90 and also APO +OPB=90,this means APX=OPB. Hence angle BPC = Angle BPD +angle APX = Angle BPD +angle OPB = angle OPD =90
BPC is therefor a right triangle and BC is a diameter. Since DB=DP, and D lies on the diameter, BD would be equal to CD. ****
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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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15 Careers That Rely on Mathematical Modeling
15 Careers That Rely on Mathematical Modeling
Have you considered a career in mathematical modeling, but are unsure where to start? This list of math modeling careers will give you some options to consider.
Mathematical modeling is the process of describing a real-world scenario in the form of equations in order to analyze and explain complex systems and make predictions. There are a number of careers
that use math and math modeling to solve a problem and provide insight.
While not a complete list, here is a list of 15 careers that use mathematical modeling in some capacity.
1. Actuary
An actuary focuses on the measurement and management of risk and uncertainty. Mathematical models can be used to create models that outline potential decisions and anticipate their impact.
2. Architect
An architect is a highly professional in the art and science of building design, responsible for planning and designing buildings. Architectural design may include developing math models for topology
and shape in order to create an optimal design.
3. Budget Analyst
Budget analysts prepare budget reports and monitor spending for companies and public and private organizations. Since they often use complex equations and statistical formulas in their analysis, math
models are often used to further optimize recommendations.
4. CAD Designer
CAD designers use software to help generate designs for complex projects. Along with algebra and trigonometry, CAD designers use math modeling to create and improve their designs.
5. Computer Programmer
A computer programmer writes and tests codes and scripts to create and modify software and applications. Math modeling can be used in computer programming to create simulations, analyze data, and
find solutions to problems.
6. Data Scientist
A data scientist focuses on data, researching, writing algorithms and writing code to answer the questions about the data sets. Math models can be used to model data to get better insights and inform
7. Economist
Economists study the production and distribution of resources, goods, and services by collecting and analyzing data, researching trends, and evaluating economic issues. They use mathematical models
of the economy to explore relationships between prices, production, employment, and more to analyze implications.
8. Engineer
An engineer is involved in inventing, designing and maintaining a variety of machines, structures and data systems. They use mathematical models, such as sets of equations, to analyze the behavior of
physical systems.
9. Insurance Underwriter
Insurance underwriters evaluate insurance applications and decide whether to approve them. They use predictive math modeling to come up with risk probability tables based on the industry.
10. Inventory Strategist
An inventory strategist maintains and improves the inventory systems of a business by analyzing statistics to determine which products are selling and which are not. Math modeling can be used to
provide deep analysis into this data.
11. Research Scientist
A research scientist conducts experiments and investigations in a range of areas, including geoscience, medical research, meteorology and pharmacology. They often use math modeling to explore
real-world implications of their experiments.
12. Statistical Analyst
Statistical analysts collect, analyze, and present data to aid in decision making. Math modeling can be a key part of the analysis stage, understanding data, and predicting outcomes.
13. Statistician
A statistician designs surveys, experiments, or opinion polls to collect data. They develop mathematical models to analyze and interpret the data and create visualizations to help decision making in
14. Surveyor
Surveyors take every precise measurements to determine property boundaries for engineering, maps, and construction projects. They create mathematical models based on measurements that other
professionals use during the project.
15. Teacher
Math and STEM teachers are responsible for teaching students analytical and statistical knowledge. Mathematical modeling can be used to help students learn mathematics while visualizing a solution to
a real-world problem, often making it easier to understand.
For more about careers that use math modeling, sign up for a COMAP membership and download these resources:
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The Consortium for Mathematics and Its Applications is an award-winning non-profit organization whose mission is to improve mathematics education for students of all ages. Since 1980, COMAP has
worked with teachers, students, and business people to create learning environments where mathematics is used to investigate and model real issues in our world. | {"url":"https://www.comap.org/blog/item/mathematical-modeling-careers","timestamp":"2024-11-06T19:50:32Z","content_type":"text/html","content_length":"51266","record_id":"<urn:uuid:797c6aeb-2135-414a-848c-3439efa4e305>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00339.warc.gz"} |
KOS.FS - faculty addon
Physics I.A (E02A026)
Departments: ústav fyziky (12102)
Abbreviation: FYIA Approved: 24.01.2017
Valid until: ?? Range: 0P+0C
Semestr: Credits: 3
Completion: ZK Language: EN
Kinematics and dynamics of a particle motion. Principle of conservation of energy. System of particles, centre of mass. Rigid body. Continuum, elastic properties of bodies. Oscillations, waves. Fluid
mechanics. Temperature and heat transfer. Kinetic theory of gases. Thermodynamics. Electric field, current, conductivity, resistance. Conductors, semiconductors, insulators. Magnetic field. Magnetic
materials. Electromagnetic field. Laboratories - accuracy of measurements, systematic and random errors, uncertainty of direct and indirect measurements, regression, measurements of 11 various
experiments related to the lectures.
1. Physical quantities - vectors and scalars. Kinematics of a particle motion in one dimension.
2. Motion in two or three dimensions, circular motion. Newton's laws of motion. Galileian transformation.
3. Motion equations, applications. Dynamics of a circular motion. Work and energy. Principle of conservation of energy. Momentum, impulse, collisions. Centre of mass. Rigid body. Rotational and
translational motions, the torque. Conservation of momentum and angular momentum.
4. Gravitation, Newton's law of universal gravitations. Potential and intensity of a gravitational filed, satellites. Fluid mechanics, surface tension.
5. Continuity equation, Bernoulli's equation. Viscosity. Temperature, heat, calorimetry. Internal energy, first law of thermodynamics.
6. Thermodynamic processes. The Carnot cycle. Equipartition of energy theorem. The second law of thermodynamics, entropy, probability, information.
7. Elasticity, stress, strain, elastic moduli. SHM, the physical pendulum, the simple pendulum, damped oscillations, forced oscillations, resonance.
8. Mechanical waves, types, mathematical description, sound, beats, the Doppler effect.
9. Electric charge, electric filed, intensity. Electric flux, Gauss's law, electric potential.
10. Capacitors, capacitance, energy of electric field, Gauss' law in dielectrics.
11. Electric current, resistivity, resistance, electromotive force.
12. Direct-current circuits, Kirchhoff's rules, power and energy in electric circuits.
13. Magnetic field, the Hall effect, magnetic materials.
14. Mass spectrometer, cyclotron. Sources of magnetic filed, Ampere's law.
Vesela E., Physics I, CTU Publishing House, Prague, 2017
Vesela E., Vacek V. Physics - Laboratory Experiments, CTU Publishinhg House, Prague 1999
Assessment is given to the student who successfully finished lab experiments + seminars. The exam consists from a written part + oral part. | {"url":"https://kos.fs.cvut.cz/synopsis/course/E02A026/en/en/en/cz/printA4wide","timestamp":"2024-11-09T01:35:11Z","content_type":"text/html","content_length":"5699","record_id":"<urn:uuid:cdcacee6-9b45-4cae-9331-7b4a25421a9c>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00309.warc.gz"} |
Experimental demonstration of quantum advantage for NP verification with limited information | LIP6 - Équipe QI
In recent years, many computational tasks have been proposed as candidates for showing a quantum computational advantage, that is an advantage in the time needed to perform the task using a quantum
instead of a classical machine. Nevertheless, practical demonstrations of such an advantage remain particularly challenging because of the difficulty in bringing together all necessary theoretical
and experimental ingredients. Here, we show an experimental demonstration of a quantum computational advantage in a prover-verifier interactive setting, where the computational task consists in the
verification of an NP-complete problem by a verifier who only gets limited information about the proof sent by an untrusted prover in the form of a series of unentangled quantum states. We provide a
simple linear optical implementation that can perform this verification task efficiently (within a few seconds), while we also provide strong evidence that, fixing the size of the proof, a classical
computer would take much longer time (assuming only that it takes exponential time to solve an NP-complete problem). While our computational advantage concerns a specific task in a scenario of mostly
theoretical interest, it brings us a step closer to potential useful applications, such as server-client quantum computing.
Experimental demonstration of quantum advantage for NP verification with limited information
In recent years, many computational tasks have been proposed as candidates for showing a quantum computational advantage, that is an advantage in the time needed to perform the task using a quantum
instead of a classical machine. Nevertheless, practical demonstrations of such an advantage remain particularly challenging because of the difficulty in bringing together all necessary theoretical
and experimental ingredients. Here, we show an experimental demonstration of a quantum computational advantage in a prover-verifier interactive setting, where the computational task consists in the
verification of an NP-complete problem by a verifier who only gets limited information about the proof sent by an untrusted prover in the form of a series of unentangled quantum states. We provide a
simple linear optical implementation that can perform this verification task efficiently (within a few seconds), while we also provide strong evidence that, fixing the size of the proof, a classical
computer would take much longer time (assuming only that it takes exponential time to solve an NP-complete problem). While our computational advantage concerns a specific task in a scenario of mostly
theoretical interest, it brings us a step closer to potential useful applications, such as server-client quantum computing. | {"url":"https://qi.lip6.fr/fr/publication/3045853-experimental-demonstration-of-quantum-advantage-for-np-verification-with-limited-information/","timestamp":"2024-11-04T04:39:57Z","content_type":"text/html","content_length":"18897","record_id":"<urn:uuid:be2017b2-b099-48db-8e45-aa54f0155bdd>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00442.warc.gz"} |
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if jou are taking an OMPT exam. | {"url":"https://cloud.sowiso.nl/courses/theory/427/978/14036/en","timestamp":"2024-11-12T10:18:24Z","content_type":"text/html","content_length":"80585","record_id":"<urn:uuid:c993db29-236f-440e-912a-b0e7f9d89a48>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00368.warc.gz"} |
Transverse vibrations of a viscoelastic rope of variable length lying on an elastic base
Title Transverse vibrations of a viscoelastic rope of variable length lying on an elastic base
Authors V. L. Litvinov^1, V. N. Anisimov^1, I. V. Korpen^1, S. N. Kosinova^1
^1Syzran’ Branch of Samara State Technical University
Using the Kantorovich-Galerkin method, an approximate solution of the problem of transverse vibrations of a viscoelastic rope with a moving boundary lying on an elastic base is found. The
Annotation dependence of the drag force on the movement of the rope is assumed to be proportional to its speed. The flexural rigidity of the rope is taken into account. The results obtained for the
amplitude of the oscillations corresponding to the n-th dynamical mode are presented. The phenomenon of steady resonance and passage through resonance is investigated. The solution is
obtained for the most common case in practice, when external perturbations act on a moving boundary.
Keywords oscillations of systems with moving boundaries, flexural rigidity, viscoelasticity, elastic base, medium resistance, resonance properties, vibration amplitude
Litvinov V. L., Anisimov V. N., Korpen I. V., Kosinova S. N. ''Transverse vibrations of a viscoelastic rope of variable length lying on an elastic base '' [Electronic resource].
Citation Proceedings of the XIII International scientific conference ''Differential equations and their applications in mathematical modeling''. (Saransk, July 12-16, 2017). Saransk: SVMO Publ,
2017. - pp. 105-114. Available at: https://conf.svmo.ru/files/deamm2017/papers/paper16.pdf. - Date of access: 14.11.2024. | {"url":"https://conf.svmo.ru/en/archive/article?id=16","timestamp":"2024-11-14T02:14:51Z","content_type":"text/html","content_length":"11723","record_id":"<urn:uuid:2ad6dfec-ae48-4321-af25-838385b2fd89>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00656.warc.gz"} |
The Taub Faculty of Computer Science
The Taub Faculty of Computer Science Events and Talks
Zohar Karnin (CS, Technion)
Wednesday, 10.11.2010, 12:30
For any $00$, we give an efficient deterministic construction of a linear subspace $V \subseteq \R^n$, of dimension $(1-\epsilon)n$ in which the $\ell_p$ and $\ell_r$ norms are the same up to a
multiplicative factor of $\poly(\epsilon^{-1})$ (after the correct normalization). As a corollary we get a deterministic compressed sensing algorithm (Base Pursuit) for a new range of parameters. In
particular, for any constant $\epsilon>0$ and $p<2$, we obtain a linear operator $A:\R^n \rightarrow \R^{\epsilon n}$ with the $\ell_1/\ell_p$ guarantee for $(n \cdot \poly(\epsilon))$-sparse
vectors. Namely, let $x$ be a vector in $\R^n$ whose $\ell_1$ distance from a $k$-sparse vector (for some $k=n \cdot \poly(\epsilon)$) is $\delta$. The algorithm, given $Ax$ as input, outputs an $n$
dimensional vector $y$ such that $||x-y||_p \leq \delta k^{1/p-1}$. In particular this gives a weak form of the $\ell_2/\ell_1$ guarantee.
Our construction has the additional benefit that when viewed as a matrix, $A$ has at most $O(1)$ non-zero entries in each row. As a result, both the encoding (computing $Ax$) and decoding (retrieving
$x$ from $Ax$) can be computed efficiently. | {"url":"https://cs.technion.ac.il/events/view-event.php?evid=1129","timestamp":"2024-11-13T08:29:14Z","content_type":"text/html","content_length":"15449","record_id":"<urn:uuid:2c0939d5-62e7-465a-a737-bbf19c62750c>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00275.warc.gz"} |
Coq devs & plugin devs
Hi everyone,
I am new to zulip so I hope I use it right :-)
I currently start to work on my master thesis where I want to work with the congruence tactic in Coq. I have seen that there is no Ltac2 version of congruence and I tried to wrap my head around
defining new tactics with Ltac and Ltac2 but I'm not shure if understand it right.
My understanding is that in order to define a new Ltac tactic one has to write OCaml code and then provide a mlg file that specifies the syntax how this OCaml code is called in Proof mode or in Ltac
definitions, or the tactic gets implemented in the Ltac language directly right?
In Ltac2 on the other hand it is recommended to write the tactic itself in the Ltac2 language if possible right? So if one wants to use congruence in Ltac2 it would be necessary to do one of the
• reimplement it in the Ltac2 language directly
• write some Ltac2 Notation that calls the Ltac version of it possibly through ltac1:(...)
• write an mlg file suitable for Ltac2 (But here I don't understand how they work and if I understand it correctly the Ltac2 Notations approach should be preferred)
Is this right and are there other ways to do this?
Do Ltac1 and Ltac2 tactics that are implemented with OCaml differ only in how they are exposed to the User? So the ml and mli files for congruence would also work for Ltac2 but at the moment there is
no exposed syntax for them in Ltac2 right?
using mlg to expose ltac2 tactics is not the right way to do it
ltac2 tactics implemented in ocaml go through the "external" system: ocaml registers them with a name and ltac2 picks up the registration
eg for progress ocaml side https://github.com/coq/coq/blob/f91306fb6ee06ba42d4d17ef0effd9b0efae7748/plugins/ltac2/tac2core.ml#L1149-L1151 and ltac2 side https://github.com/coq/coq/blob/
f91306fb6ee06ba42d4d17ef0effd9b0efae7748/user-contrib/Ltac2/Control.v#L52 with a little wrapper notation https://github.com/coq/coq/blob/f91306fb6ee06ba42d4d17ef0effd9b0efae7748/user-contrib/Ltac2/
for congruence I would do
write some Ltac2 Notation that calls the Ltac version of it possibly through ltac1:(...)
since there are no arguments so no serious interaction with the ltac2 type system
Thank you very much. That helped a lot.
There are arguments for congruence. An numeral and a list of terms. I tried
Local Ltac2 congruence0 (n: constr) (terms: constr) :=
ltac1:(n terms |- congruence n with terms) (Ltac1.of_constr n) (Ltac1.of_constr terms).
but then I get:
Error: Syntax error: ')' expected after [ltac1_expr_in_env] (in [ltac2_atom]).
I think there still is something with the Ltac2 types not right. But
From Ltac2 Require Import Ltac2.
Lemma foo' {A B} (f: A -> B -> A) a b: f a b = a -> f (f a b) b = a.
Fail congruence.
ltac1:(congruence 10 with true false tt).
works (the arguments given to congruence make no sense, I just wanted to try if they get accepted).
it looks like the ltac1 syntax expects a literal integer for the n argument ie doesn't support variables
so if you want to support it in ltac2 you do need your own external
@Gaëtan Gilbert thank you for all your help. I will try to write such an external and make a PR once I'm ready :-)
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237656-Coq-devs-.26-plugin-devs/topic/Ltac2.20tactic.20plugins.html","timestamp":"2024-11-05T17:07:46Z","content_type":"text/html","content_length":"9468","record_id":"<urn:uuid:d974a243-8a43-4567-afe0-8089714391b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00610.warc.gz"} |
2018’s Top 7 Libraries and Packages for Data Science and AI: Python & RData Science News
2018’s Top 7 Libraries and Packages for Data Science and AI: Python & R
We’ll assume in each case that the relationship between mpg and each of our features is linear.# copy mtcars into sparkmtcars_tbl <- copy_to(sc, mtcars)# transform our data set, and then partition
into 'training', 'test'partitions <- mtcars_tbl %>% filter(hp >= 100) %>% mutate(cyl8 = cyl == 8) %>% sdf_partition(training = 0.5, test = 0.5, seed = 1099)# fit a linear model to the training
datasetfit <- partitions$training %>% ml_linear_regression(response = "mpg", features = c("wt", "cyl"))fit## Call: ml_linear_regression.tbl_spark(., response = "mpg", features = c("wt", "cyl")) ## ##
Formula: mpg ~ wt + cyl## ## Coefficients:## (Intercept) wt cyl ## 33.499452 -2.818463 -0.923187For linear regression models produced by Spark, we can use summary() to learn a bit more about the
quality of our fit and the statistical significance of each of our predictors.summary(fit)## Call: ml_linear_regression.tbl_spark(., response = "mpg", features = c("wt", "cyl")) ## ## Deviance
Residuals:## Min 1Q Median 3Q Max ## -1.752 -1.134 -0.499 1.296 2.282 ## ## Coefficients:## (Intercept) wt cyl ## 33.499452 -2.818463 -0.923187 ## ## R-Squared: 0.8274## Root Mean Squared Error:
1.422Spark machine learning supports a wide array of algorithms and feature transformations, and as illustrated above, it’s easy to chain these functions together with dplyr pipelines.Check out more
about machine learning with sparklyr here:sparklyrAn R interface to Sparkspark.rstudio.comAnd more information in general about the package and examples here:sparklyrAn R interface to
Sparkspark.rstudio.com2. Drake—An R-focused pipeline toolkit for reproducibility and high-performance computingDrake programmingNope, just kidding. But the name of the package is drake!https://
github.com/ropensci/drakeThis is such an amazing package. I’ll create a separate post with more details about it, so wait for that!Drake is a package created as a general-purpose workflow manager for
data-driven tasks. It rebuilds intermediate data objects when their dependencies change, and it skips work when the results are already up to date.Also, not every run-through starts from scratch, and
completed workflows have tangible evidence of reproducibility.Reproducibility, good management, and tracking experiments are all necessary for easily testing others’ work and analysis. It’s a huge
deal in Data Science, and you can read more about it here:From Zach Scott:Data Science’s Reproducibility CrisisWhat is Reproducibility in Data Science and Why Should We Care?
towardsdatascience.comToward Reproducibility: Balancing Privacy and PublicationCan there ever be a Goldilocks option in the conflict between data security and research disclosure?
towardsdatascience.comAnd in an article by me :)Manage your Machine Learning Lifecycle with MLflow—Part 1.Reproducibility, good management and tracking experiments is necessary for making easy to
test other’s work and…towardsdatascience.comWith drake, you can automaticallyLaunch the parts that changed since last time.Skip the rest.Installation# Install the latest stable release from
CRAN.install.packages("drake")# Alternatively, install the development version from GitHub.install.packages("devtools")library(devtools)install_github("ropensci/drake")There are some known errors
when installing from CRAN. For more on these errors, visit:The drake R Package User ManualThe drake R Package User Manualropenscilabs.github.ioI encountered a mistake, so I recommend that for now you
install the package from GitHub.Ok, so let’s reproduce a simple example with a twist:I added a simple plot to see the linear model within drake’s main example. With this code, you’re creating a plan
for executing your whole project.First, we read the data. Then we prepare it for analysis, create a simple hist, calculate the correlation, fit the model, plot the linear model, and finally create a
rmarkdown report.The code I used for the final report is here:If we change some of our functions or analysis, when we execute the plan, drake will know what has changed and will only run those
changes. It creates a graph so you can see what’s happening:Graph for analysisIn Rstudio, this graph is interactive, and you can save it to HTML for later analysis.There are more awesome things that
you can do with drake that I’ll show in a future post :)1..DALEX—Descriptive mAchine Learning EXplanationshttps://github.com/pbiecek/DALEXExplaining machine learning models isn’t always easy..Yet
it’s so important for a range of business applications..Luckily, there are some great libraries that help us with this task..For example:thomasp85/limelime—Local Interpretable Model-Agnostic
Explanations (R port of original Python package)github.com(By the way, sometimes a simple visualization with ggplot can help you explain a model. For more on this check the awesome article below by
Matthew Mayo)Interpreting Machine Learning Models: An OverviewAn article on machine learning interpretation appeared on O’Reilly’s blog back in March, written by Patrick Hall, Wen…www.kdnuggets.comIn
many applications, we need to know, understand, or prove how input variables are used in the model, and how they impact final model predictions.DALEX is a set of tools that helps explain how complex
models are working.To install from CRAN, just run:install.packages("DALEX")They have amazing documentation on how to use DALEX with different ML packages:How to use DALEX with caretHow to use DALEX
with mlrHow to use DALEX with H2OHow to use DALEX with xgboost packageHow to use DALEX for teaching..Part 1How to use DALEX for teaching..Part 2breakDown vs lime vs shapleyRGreat cheat sheets:https:/
/github.com/pbiecek/DALEXhttps://github.com/pbiecek/DALEXHere’s an interactive notebook where you can learn more about the package:Binder (beta)Edit descriptionmybinder.orgAnd finally, some
book-style documentation on DALEX, machine learning, and explainability:DALEX: Descriptive mAchine Learning EXplanationsDo not trust a black-box model. Unless it explains
itself.pbiecek.github.ioCheck it out in the original repository:pbiecek/DALEXDALEX—Descriptive mAchine Learning EXplanationsgithub.comand remember to star it :)Thanks to the amazing team at Ciencia
y Datos for helping with these digests.Thanks also for reading this. I hope you found something interesting here :)..If these articles are helping you please share them with your friends!If you have
questions just follow me on Twitter:Favio Vázquez (@FavioVaz) | TwitterThe latest Tweets from Favio Vázquez (@FavioVaz). Data Scientist. Physicist and computational engineer. I have a…twitter.comand
LinkedIn:Favio Vázquez—Founder—Ciencia y Datos | LinkedInView Favio Vázquez’s profile on LinkedIn, the world’s largest professional community. Favio has 16 jobs listed on
their…www.linkedin.comSee you there :)Editor’s Note: 2018 has been an incredible year for AI..Check out the following Heartbeat recaps that detail all the year’s best and brightest:2018
Year-in-Review: AI & Machine Learning ConferencesBest of Machine Learning in 2018: Reddit Edition2018 Year-in-Review: Machine Learning Open Source Projects & FrameworksGet a jumpstart on 2019 by
joining Heartbeat on Slack to chat with the author and a growing community of machine learners, mobile developers, and more. More details
You must be logged in to post a comment. | {"url":"http://datascience.sharerecipe.net/2019/01/03/2018s-top-7-libraries-and-packages-for-data-science-and-ai-python-r/","timestamp":"2024-11-01T18:48:21Z","content_type":"text/html","content_length":"38013","record_id":"<urn:uuid:ed776a2b-479d-47ea-b94b-676b00109f1a>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00213.warc.gz"} |
A \emph{semiring} is a structure $\mathbf{S}=\langle S,+,\cdot \rangle $ of type $\langle 2,2\rangle $ such that
$\langle S,\cdot\rangle$ is a semigroup
$\langle S,+\rangle $ is a commutative semigroup
$\cdot$ distributes over $+$: $x\cdot(y+z)=x\cdot y+x\cdot z$, $(y+z)\cdot x=y\cdot x+z\cdot x$
Let $\mathbf{S}$ and $\mathbf{T}$ be semirings. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\to T$ that is a homomorphism:
$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$
Basic results
Finite members
$\begin{array}{lr} f(1)= &1
f(2)= &10
f(3)= &132
f(4)= &2341
f(5)= &
f(6)= & | {"url":"https://mathcs.chapman.edu/~jipsen/structures/doku.php?id=semirings","timestamp":"2024-11-05T20:24:15Z","content_type":"application/xhtml+xml","content_length":"19226","record_id":"<urn:uuid:863572c5-8f33-49c8-b512-3b7cf09a3515>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00573.warc.gz"} |
fredluis Opposite sides of the cube have the same potential, so as we go across the cube we start with the potential at V and end with the potential at V. Since there are no charges inside, there
2019-09-17 is nothing to change the potential so it must be constant all the way across. tree removal
ernest21 At first I thought you may be wrong, but here is a simple thought experiment that shows you must be correct: space marshals 2
santo35 Isnt the electric field varying at all points "P" ? (since it is an oscillation ?
03:59:21 And if so are we considering the maximum value of variation here ?
santo35 Isnt the electric field varying at all points "P" ? (since it is an oscillation ?
03:58:36 And if so are we considering the maximum value of variation here ?
santo35 isnt that field varying at all points P ? rnrnOr is it that we are considering only the maximum value of the variation at all points P and comparing them ?
natec Say point P were in the XZ plane and $\theta$ was the angle BELOW the x-axis. Would the electric field now be in the XZ plane with maximum magnitude at $\theta = 90$? I can't see why not
2013-08-18 because of the symmetry of the arrangement.
SillyMan The solution given above is wrong. Purely qualitatively, by Ampere's Law, B field will be in the +/- Z direction at point P. The energy flux vector S points outward (obviously). Thus E is
2013-06-19 in the XY plane (Orthogonal Triad). At 90 degrees, one "sees" the highest acceleration, which means that the power flowing past P is the highest at theta = 90 degrees. The answer was
20:17:23 implied by Maxwell's equations and the Larmor result.
flyboy621 I don't know if this will help anybody, but...
21:12:43 You can imagine standing at the origin, holding one end of a rope, the other end of which is at point P. Then you start waving the end of the rope back and forth along the x-axis. The
waves in the rope will be oriented in the xy-plane and be maximum when P is on the y-axis, i.e. theta is 90 degrees.
The analogy works because EM waves are transverse, just like waves in a string.
deafmutemouse This analogy is awesome! Thanks!
2011-09-20 14:10:54
OptimusPrime This is really helpful! The point of fastest velocity of the rope shaking happens when crossing the y-axis, which is maximum E amplitude in this case.
2017-04-08 04:47:54
faith here's another way to look at it.
23:14:24 cross product! v=ExB since particle is moving along the x direction, E field max should only be along y.
faith yikes.. sorry, this solution is wrong. i had it by luck.
2010-11-11 23:18:35
wittensdog The one thing I'm seeing over and over again is that every physics GRE problem has a quick and simple way to do it. Sometimes it's a calculation trick, sometimes it's just knowing
2009-09-28 something by heart.
In this case, I strongly recommend remembering two basic facts which have a good chance of coming up on any test:
1.) an oscillating charge never radiates in the direction of its oscillation axis
2.) the polarization of an oscillation charge is parallel to the oscillation axis
Don't bother wondering why (at least not for the sake of the GRE), just memorize that. If you use those two pieces of information, you immediately see that the E-field should be in the xy
plane from the restriction on the polarization, and also that the maximum field strength should be at 90 degrees, since along the x axis it has no magnitude, and increases as you move
away from the x axis.
So far, in all of my studying, I've never come along a GRE problem that required a formula with more than 3 or 4 terms in it. Maybe that's a slight exaggeration, but you get the point.
apr2010 Does not the dipol radiate also in the z-direction? This looks like a trap again, as the Point P lies only in the x-y-plane, meaning A and B are not even available.
2010-04-09 09:22:19
Herminso For a electric dipole $\vec{P}(t)=qd\cos {(\omega t)}\hat{x}$ we have:
13:56:33 $\vec{E}=-\frac{\mu_0qd\omega^2}{4\pi}(\frac{\sin {\theta}}{r})\cos {[\omega(t-r/c)]}\hat{\theta}$ and $\vec{B}=-\frac{\mu_0qd\omega^2}{4\pi c}(\frac{\sin {\theta}}{r})\cos {[\omega(t-r/
Thus the oscilation of the electric field is in the xy-plane and the maximum is at $\theta=90$, just the y-axes.
Herminso Where $\theta$ is measured from the positive x-axes on the xy-plane.
2009-09-22 14:22:02
a19grey2 I recommend just learning what the picture of an oscillating dipole looks like. rnhttp://en.wikipedia.org/wiki/DipolernrnIt'll help you later in your physics life anyway...
22:14:06 physicsisgod Here's a cool video of it:
2008-11-05 20:32:44
jw111 You see the maximum OSCILATION of charge when you stand on y axis, and minimum oscilation when you stand on x axis.
11:09:42 => max field take place on y axis
the ocilation is on x-y plane, the field line wave mainly on x-y plane.
=>oscilating field on x-y plane
ee7klt hi,
04:54:58 since oscillations happen only in the x-y plane, the E-field vector thus cannot have components in the z-direction. This eliminates (A) and (B).
The minimum occurs when you're looking directly down the x-axis i.e. $\theta = 0$ (all you see is the infinitesimal tip of the field vector which doesn't look like it'll generate any
waves coming towards you) - this eliminates (D). From here, I guess you'll need to remember that the field goes as $\sin \theta$ to narrow it further down to (C).
Simplicio It's oscillating along the x-axis so you can't just say "it is in the x-y plane". It might as well been the x-z plane ...
2009-03-31 17:13:47
physicsDen is it me, or is this a poor attempt at a solution?
22:04:58 alpha it's a quick and dirty way to arrive at the right answer.
2005-11-09 22:15:19
mpdude8 Definitely a poor attempt at a solution, but a great solution for the sake of the GRE.
14:19:49 You do not have the time to use any formulas on a problem like this. It really has to be 100% intuition in order for you to finish. You don't have 3-4 minutes to think of a
relevant formula, you just have to see the problem and know what to look for instinctively.
Thus, many of the solutions to these problems on this site are not rigorous whatsoever, but the GRE couldn't care less about rigor. | {"url":"https://grephysics.net/ans/8677/53","timestamp":"2024-11-02T02:23:46Z","content_type":"application/xhtml+xml","content_length":"36504","record_id":"<urn:uuid:6ec5d0ed-e822-41f5-bd75-dd729cd22b37>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00759.warc.gz"} |
Annual return rate formula for excel
24 Oct 2016 Internal rate of return will tell you the annualized percentage returns of use the =IRR() formula in Excel to calculate our internal rate of return.
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you do it with cash flows that come at irregular times. Step 1. Open Excel by 9 Sep 2019 Average return is the simple average where each investment option is given an equal weightage. For example,
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you the annualized percentage returns of use the =IRR() formula in Excel to calculate our internal rate of return.
The Excel RATE function is a financial function that returns the interest rate per period of an annuity. You can use RATE to calculate the periodic interest rate, then multiply as required to derive
the annual interest rate. The RATE function calculates by iteration.
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the the cash flows must occur at regular intervals, such as monthly or annually. describes the formula syntax and usage of the IRR function in Microsoft Excel. IRR stands for Internal Rate of Return.
Calculating Compound Interest in Excel. 11 Jul 2019 Free online CAGR Calculator for estimating annualized returns. Learn how to calculate the Compound Annual Growth Rate in Excel, by Jon What
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21 Jun 2011 If you want to calculate the yearly returns or the year to date return, you'll also need the year end values of the investments. But you don't have
14 Jan 2013 The real beauty is that you do not need to be a spreadsheet guru to use these functions. The Syntax for the function is: Excel: XIRR(values, dates, 19 Mar 2014 Also, to calculate
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investment over a period it could be profit or loss. It is basically a percentage of the amount above or below the investment amount. The way to set this up in Excel is to have all the data in one
table, then break out the calculations line by line. For example, let's derive the compound annual growth rate of a company's sales over 10 years: The CAGR of sales for the decade is 5.43%.
7 Jun 2018 XIRR is a more powerful function in Excel for calculating annualized yield for a schedule of cash flows occurring at irregular periods.
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2. Calculate the change in price between each Returns the internal rate of return for a series of cash flows represented by the the cash flows must occur at regular intervals, such as monthly or
annually. describes the formula syntax and usage of the IRR function in Microsoft Excel.
Use daily, weekly, monthly, or yearly, depending on the length of time you are assessing for the rate of return. 2. Calculate the change in price between each
I’ll admit that the equation has no place in everyday life – it should be restricted to Excel spreadsheets and only allowed to see the light of day once a year (preferably after year-end). Geometric
Average Annual Rate of Return: Where: r = Annual rate of return in year i. n = Number of years in the measurement period To calculate the Compound Annual Growth Rate in Excel, there is a basic
formula =((End Value/Start Value)^(1/Periods) -1.And we can easily apply this formula as following: 1.Select a blank cell, for example Cell E3, enter the below formula into it, and press the Enter
key.See screenshot: The RATE function syntax has the following arguments: and paste it in cell A1 of a new Excel worksheet. For formulas to show results, select them, press F2, and then press Enter.
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Electron density representation
Electron density representation and real-space refinement
(New tricks from an old dog)
E. Blanc, G. Zhou, Z. Chen‡, Q. Xie§, J. Tang, J. Wang and
M. S. Chapman
Institute of Molecular Biophysics, § Chemistry Department and ‡ Physics Department, Florida State University, Tallahassee, FL, 32306-3015, USA
chapman@sb.fsu.edu http://www.sb.fsu.edu/~chapman
The expected electron density for an atomic model is calculated directly from the coordinates in a resolution-dependent manner. Several applications are discussed. Firstly, it is possible to refine
atomic models in r
eal-space, by optimizing the fit of the model to a map. The methods are conceptually similar to those of Diamond [Acta Crystallogr., 1971. A27: p. 436-452], but much improved through modeling of the
resolution limits and inclusion of stereochemical restraints. The methods have been used for complete refinement of virus structures, for local refinement to enhance model-building, and as a
pre-refinement method to improve the refinements of proteins by conventional reciprocal-space methods. Secondly, improved local measures of quality can be calculated, comparing the calculated and
observed electron densities. Finally, the refinement methods can be applied at about 30 Å resolution to optimally fit known atomic structures into electron microscope reconstructions of large
macromolecular complexes.
The objectives are improved methods of structure determination, refinement and analysis, applicable to large macromolecules visualized at medium to low resolution.
The methods discussed increase the data:parameter ratio and have the potential to reduce overfitting and increase the speed with which structures can be determined.
The refinements are based on improved methods for comparing the electron density values of a map with those expected from a model. These comparisons can also lead to improved methods for determining
the local quality of a structure, and for assessing the significance of conformational differences.
Real-space methods
The fundamentals are not new. In 1971 Diamond [1] published an atomic refinement program that minimized:
ρmap and ρmodel are electron density values -- for the experimental map and calculated from the model;
In these earlier implementations, the electron density was calculated from the model by placing spherical Gaussian functions at each atom center. Spherical Gaussian functions are a good approximation
at near-infinitely high resolution.
Diamond and Jones [1, 2] approximated the effects of a resolution limit by smearing the atoms with an additional B-factor. This does not give the expected truncation ripple. The bad effects of this
poor approximation can be minimized by disregarding grid points that are not very close to the atom center, as in the RSR implementation. So, some of the data is ignored during refinement.
Furthermore, the process becomes a bit more like peak fitting. This works less well at low (~3Å) resolution where there are not discrete peaks for individual atoms.
Both Diamond and Jones [1, 2] incorporated geometrical constraints. Some parts of the model were constrained to good stereochemistry and others were allowed to distort to move the model into electron
density. Good stereochemistry was reimposed by alternating real-space refinement with energy minimization [3] or geometric regularization [4]. Often, models oscillate between good fit and good
geometry, and convergence is poor.
Restrained reciprocal-space refinement
Reciprocal-space methods soon became more popular because of several advantages:
• Independence from phases: the map used in real-space refinement may incorporate large random errors (e.g., from isomorphous replacement) or systematic errors resulting in bias if a phasing model
has been used.
• Simultaneous refinement against geometrical restraints.
For most purposes reciprocal-space refinement is still the most appropriate. We will discuss a few applications for which real-space refinement is better or where it enhances the performance of
reciprocal-space methods.
Reciprocal-space methods also have some problems:
• Independence from phases: This is usually considered an advantage, but note also that some of the experimental information is being excluded. At the end of refinement, it is usually best to be
phase-independent, but, as discussed later, it is often more important to include all experimental information at the start of refinement.
• Each |F| depends on every atom; therefore,
□ The refinement of different atoms is interdependent. Conditioning and convergence can be poor.
□ It is difficult to access local quality within a structure.
□ A Fourier transform must be computed on every cycle. This can be very costly for asymmetric units with many copies of the same subunit.
Pseudo-real-space refinement (authors’ nomenclature)
Some refinements minimize functions of the form:
Like real-space refinement, they use the phases. Stereochemical restraints are usually applied. Due to computation in reciprocal-space, resolution is trivial to incorporate, but the methods are no
longer very suitable for small parts of the asymmetric unit., i.e., they are not local methods.
At least one of these implementations is available in most popular refinement packages. Although our own interests are in "real" real-space refinement (below), some of our results are also applicable
to these ìpseudo" methods.
Overview of new methods
Our methods combine the best features of Diamond-style and pseudo-real-space refinements. They:
• Include phases
• Include stereochemical restraints
• Account for the resolution of the map
• Are local and therefore quick to calculate for small parts of the structure
Here, the calculation of electron density from a model will be described in a conceptual manner. Mathematical derivations are published elsewhere [7].
• The structure is considered to be a sum of Individual isolated atoms.
• Calculation of the atomic electron density function for each atom uses the definition of a scattering factor, f, that is the Fourier transformation of an isolated atom. The electron density is
calculated from the inverse transform: ρ = FT(f). (Anomalous scattering effects are ignored.)
• In the interests of speed, let f(d*) be spherically symmetric (Fig. 1), decreasing with resolution. f(d*) is either calculated from first principles or read from International Tables.
Figure 1 Scattering factors: On the left it is shown as a spherically symmetric function. On the right, a 1D profile is shown, approximated by steps of uniform scattering.
Approximation of f(d*) in one-dimension by steps of uniform scattering (Fig. 1, left) corresponds to concentric spherical shells of uniform scattering in three-dimensions. Now:
ρ = FT{f(d*)} =
This is quick and easy to calculate, because the Fourier transform of a spherical shell has a simple analytical form.
Incorporation of resolution limits
With shells extending out to very high resolution, Fourier transformation of the scattering factor gives a nearly Gaussian function (Fig. 2).
Figure 2 Imposition of resolution limits: Truncation of the scattering beyond a resolution limit leads to ripple which is shown schematically at the bottom right, and leads to a function that is not
well approximated by a Gaussian.
Resolution limits can be imposed by zeroing the relevant resolution shells. Note that, unlike the Gaussian function, the calculated electron density function has the expected truncation ripple and is
not well approximated by a Gaussian. The poor approximation with Gaussian functions is one of the reasons why prior implementations of real-space refinement have not worked well at low resolution.
Program “RSRef” compares electron density that was calculated from a model to that of a map, using the residual:
where S and k are scaling constants, and the summation is over all map grid points,
1. large enough to include 20-30 grid points/atom,
2. small enough to exclude distant grid points that for which ρmap is less accurate.
Usually, rref ≥
The contribution of an atom to the electron density decreases with distance from the center. To speed calculation, it is assumed to be zero beyond a second cut-off distance, rcalc. rcalc needs to be
large enough to approximate the overlap of neighboring atoms when viewed at low resolution. It should be ≥ d*max, e.g., ≥ 3.4 Å for 3 Å data.
For refinement, derivatives of the residual are calculated with respect to the atomic parameters. RSRef is written as a module for TNT. The derivatives with respect to electron density are combined
(TNT's Shift [8]) with derivatives with respect to the stereochemistry.
Virus refinement
Summary: Real-space methods are the most appropriate because they are many times faster and use the accurate phases that have been refined by symmetry averaging.
Viruses often contain 5 to 120 nearly identical subunits in each asymmetric unit. Only one will be refined. The effects of neighbors will be considered, with regard to:
1. overlapping electron density;
2. non-bonded stereochemical terms.
The neighbors (related by both crystallographic and non-crystallographic symmetry) are regenerated each cycle from the refining protons. Thus, symmetry is imposed as a constraint.
Test case: Canine parvovirus (CPV) empty capsid at 3Å
This structure had been previously refined with several batches of reciprocal-space refinement alternated with interactive remodeling [9].
Real-space refinement was compared to 3 of the batches of reciprocal-space refinement, using stereochemical weights chosen to give similar rms deviations to the original refinement.
│Starting R│ Reciprocal-space │Real-space│
│ │ Method │ R │ R │
│ 34.7% │ProLSQ [10] │30.6% *A*│29.1% *B* │
│ 27.9% │X-Plor [11] │ 25.2% │ 25.3% │
│ 26.7@ │ProLSQ [10] │ 24.4% │ 24.4% │
Refinement in real-space appears to be at least as accurate as in reciprocal-space. The difference in these conventional R-factors is modest, but real-space refined model *B* fits the map better than
the corresponding reciprocal-space refined model *A* (Fig. 3.).
Figure 3 Comparison of the fit of models to the electron density: the thin lines show a model after convergence of a first batch of ProLSQ [10] refinement. In thicker line is the result of real-space
refinement, starting from the same initial CPV empty capsid coordinates.
Actual refinements
CPV DNA-containing virus
Details of the progress of this 2.9 Å refinement are given elsewhere [12], as is a detailed description of an unusual inverted DNA loop (bases pointing out) [13]. Here we will concentrate on
refinements of other structures. The only recent result to add is that it was possible to refine a plausible structure for 12 additional N-terminal amino acids that ran through weak, disordered
density [20]. As the density runs along a 5-fold axis, the occupancy cannot exceed 20%, and there is biochemical evidence that it is lower. Refinement yielded a model that stayed within the electron
density and an occupancy of 13%. It is unlikely that reciprocal-space methods applied at 2.9 Å would have yielded a reasonable model (see below).
Tobacco mosaic virus (TMV)
TMV was refined at 2.4 Å by Bhynavbhatia & Caspar (in preparation), mostly with X-Plor. RSRef was used to refine flexible loops which tended to move out of their weak density with reciprocal-space
At medium resolution, reciprocal-space methods generally lead to poor models of disordered regions. Brünger has recently suggested an explanation [21]: In reciprocal-space refinement, all atoms are
interdependent. If an atom is not positioned correctly, other atoms make small adjustments to their positions (perhaps moving away from their correct positions) to improve the overall agreement
between experimental and model structure amplitudes. The atoms that are likely to make the largest adjustments are those least restrained by the diffraction data ñ the disordered parts of the model.
Our experience with TMV and CPV suggest that real-space refinement is a general method of avoiding this problem, because the refinement is local. Atoms are not adjusted to accommodate for errors in
other parts of the model.
Human rhinovirus 50 (HRV50)/WIN 61209
The structure of this virus-drug complex was determined and refined in collaboration with Vince Giranda and colleagues, formerly at Sanofi Withrop Inc. [22]. The revelvant statistics are summarized
Unit cell: I222: 310 x 342 x 390 ų
2 x 60 x each of 4 proteins + RNA
Asymmetric unit: 15 x 4 proteins = 15 x 789 amino acids = 93,000 atoms
Diffracts to 1.8Å; refining to 2.0Å;
~ 930,000 independent reflections
Thus, by all measures, this is a large refinement problem.
The starting R factor was 44.4%. Prior to the addition of solvent water molecules, the refinement statistics were:
RTfree = 25.3% to 2.8 Å; 29.9% to 2.0 Å; calculated using all data.
Summary of virus refinement results
Tests and examples show that real-space refinement compares favorably to reciprocal-space methods. Two advantages probably account for the relatively high quality:
1. Phases are used. After high non-crystallographic redundancy has been exploited, phases are likely more accurate than amplitudes [14].
2. To speed reciprocal-space refinement of viruses, it is common to alternate between subunits of the data. In real space, all the data can be used on every cycle.
HRV50: Each cycle takes ~ 10 min. cpu on a SGI Indigo workstation. This is comparable to refinement of a protein structure. Empirically, it appears that real space has an N log2N advantage - huge
with 15 or 60 N-fold non-crystallographic symmetry. The current version is optimized for minimal memory use (at most ~ 4 Mbytes), through caching of the electron density. With inexpensive memory
widely available, it is likely that substantial improvements in speed can be made without the need for caching.
Expectations for proteins should be much lower:
1. RSRef’s dependence on phases is now a disadvantage (usually).
2. Without high non-crystallographic symmetry there is no speed advantage.
Thus we will be looking at applications in niches that complement the more powerful reciprocal-space methods.
Objectives: 1) to increase the speed and precision of interactive modeling
2) to start reciprocal-space refinement closer to the correct structure, to avoid, during optimization, some of the local minima with incorrect conformation.
Implementation is conceptually similar to RSR of Frodo/O [15]
a) a small set of residues is defined by various criteria, e.g. residue number, volume.
b) a script to refine the selected fragment(s) is called directly from "O" using a macro.
Differences with RSR have a substantial impact upon results. The major differences are the incorporation of:
a) the map resolution limit.
b) stereochemical restraints.
The availability of an improved local real-space refinement protocol changes the way that models are built in our laboratory.
1. Dictionaries are used to set the approximate backbone conformation and side-chain rotamers.
2. Real-space refinement is used to optimize the fit to the density.
3. As refinement is stereochemically restrained, there is rarely a need to regularize the model or adjust it to relieve close contacts.
4. When adjustments are needed, they are made with quick, crude rigid fragment motions followed by real-space refinement.
5. There is little need for time-consuming fine adjustments.
Graphics user interface (GUI)
Release 2 of our package includes a GUI through which commonly changed parameters can be changed quickly. The GUI is written in Hypertext Markup Language (HTML) 3.0 [23], as a form, so that it can be
displayed with a browser, and is therefore nearly platform independent. The user communicates with a server (that can be a local mirror) which sends back to the client a file containing refinement
and control parameters, and refinement, controlled with a Perl script [24], can be started automatically. Alternatively, the refinement can wait for the output of coordinates by an 'O macro. In both
cases, the output from refinement is parsed, and essentials are written to the screen. With the ìOî macros, the user has the option of inspecting the refinement results and accepting or rejecting
them. Refinement of a few amino acids and their neighbors typically takes about 30 seconds to converge.
How does real-space refinement affect model quality?
Through the use of such techniques, effectively an additional real-space (pre-)refinement step has been inserted between model building and reciprocal-space refinement. Intuitively it seems sensible
to optimize the fit to the map before reciprocal-space refinement. In fact, it is suggested in the TNT refinement manual [16], but...
· Does is really do any good?
· Can it do harm if the phases are bad?
To answer these questions, a test system was needed, which, in contrast to the virus structures, would have phases and electron density as poor as likely to be encountered in protein structure
determination. The 3 ? multiple isomorphous replacement (MIR) map of the recently determined HMG Co-A reductase structure [17] was selected. This was a large structure with 2 subunits of 374 amino
acids in the asymmetric unit. The average figure of merit was 0.65. The structure had been determined using the 2-fold non-crystallographic symmetry, but for more stringent testing of the real-space
refinement, the unaveraged MIR map was used.
Tests included parallel refinements starting from the unrefined model of the original structure determination [17]. Different refinement protocols were compared, determining how much the model could
be improved automatically without intervening model building. The simplest of the tests is shown in Figure 4, a comparison of reciprocal-space refinement with and without real-space pre-refinement.
Real-space pre-refinement leads to improved results. The benefit, which at first sight seems modest, can only be assessed if it is known how good a model can be expected at this early stage of
refinement? Following refinement, the model was improved in the original structure determination by several rounds of rebuilding and re-refinement [17]. By resetting the B-factors of the Lawrence et
al. final model to 20, and doing 30 additional cycles of positional refinement, we mimicked a model that was not limited by the modeler's abilities, but with fixed B-factors and no solvent, it was an
appropriate comparison for early refinement steps. The
Figure 4 The benefits of real-space pre-refinement: The free R-factor is lower when conventional refinement by TNT [8] is preceded by real-space refinement.
Combined refinement
The benefit of real-space pre-refinement might be limited by the poor quality of the MIR map. Following real-space refinement, improved phases can be calculated from the model. Use of a map
calculated with
(2Fo - Fc, αc) allows real-space refinement to progress further. With cycles of map calculation and real-space refinement:
1. the conventional R-factor continues to decrease
2. then increases – suggesting bad effects of phase bias.
Phase bias can be reduced by inserting reciprocal-space refinement, allowing the atoms to move independently of the phases. Each round of refinement now consists of:
1. real-space refinement
2. reciprocal-space refinement
3. 2Fo - Fc map calculation, then back to #1
Improvement stopped after 2 rounds (with HMG Co-A reductase), monitoring convergence with
The result was a model with
Why does real-space refinement help?
A good indication comes from comparing free and conventional R-factors:
│ Model │ │Rconv│Difference│
│ Target │30.2%│21.7%│ 8.5% │
│ Reciprocal-space refined │32.7%│21.9%│ 10.8% │
│Real-space refined then reciprocal-space refined │31.8%│22.4%│ 9.4% │
The difference between Rconv is less with real-space pre-refinement (and
Figure 5 Combined refinement: Alternated real- and reciprocal-space refinements (right) quickly accomplish much of what is usually achieved through labor-intensive alternation of remodeling and
refinement (left).
Which real-space method?
The results above would apply equally to the pseudo-real-space methods available in several programs in which
With pre-refinement, it is convenient to use RSRef, called from “O”, so that the effects can be monitored immediately. When blindly alternating real- and reciprocal-space refinements, either RSRef or
a suitable pseudo-real-space method would be appropriate.
Quality indices
Most crystallographic quality indices are global – a measure of the average error of a whole structure. Jones et al. [15] suggested the use of real-space R-factors (or correlation coefficients)
calculated by comparing calculated and map electron densities near residues. These indices are suitable to detect gross error, such as sequence, locally mis-aligned with the structure. Our tests have
used a similar index:
With improved representation of ρmodel , it might be possible to compute a more sensitive indicator of error.
• All atoms of the CPV structure were moved by a uniform shift vector of randomly chosen direction.
• RED was calculated for each residue.
• RED was averaged between all ~550 amino acids, and the standard deviation of the mean was calculated.
• These calculations were repeated for shifts of different magnitudes.
There is a lot of inherent variability in the strength of electron density, so there is a large component of the variation in the index that is independent of model quality. We are interested in how
small a shift is required for the index to rise above this variation.
A suitable criterion to judge quality indices is therefore the smallest shift for which the change in mean index (for all residues) is greater that its standard deviation.
Δμ(index) > σ(index)
Figure 6 plots real-space R-factors vs. introduced error.
Figure 6 Comparison of real-space R-factors: The average R-factors (calculated with O and RSRef) are plotted as a function of size of the displacement of all atoms from their refined positions. The
horizontal lines are drawn one standard deviation above the mean R-factor for all residues with zero displacement.
The sensitivity of real-space R-factors is improved when calculated using the improved electron density functions of RSRef. However, they remain a quality index of low sensitivity.
Further improvements were inspired by Dale Tronrud’s screening for poor geometry. Poorly fit atoms are likely to have large derivatives,
Figure 7 An improved local quality index based on the magnitude of the gradient, Δρ: The average index rises by more than one standard deviation (horizontal line) in less than 0.3 Å.
Refinement of EM images
Recently, 3-D electron microscope reconstructions have been performed for complexes of molecules whose structures are known at high resolution. Examples include viruses complexed with antibodies and
receptors, complexes of muscle components, etc..
Real-space refinement offers the opportunity to optimize the modeling of these reconstructions. RSRef has been adapted for this purpose in several ways:
1. X-ray scattering factors have been replaced by electronic scattering factors.
2. Reduction of the contrast due to solvent scattering has been calculated using modified protein scattering factors from which solvent scattering has been subtracted.
3. Scattering has been attenuated to account for EM incoherence.
RSRef is capable of moving a rigid protein model into EM electron density. This was demonstrated with the 27 Å Cryo-EM 3-D reconstruction of human rhinovirus complexed with antibody fragment Fab 17
[19]. After the Fab had been moved 17Å in a random direction, real-space refinement reduced the RED from 102% to 38% in bringing the Fab back into the electron density.
Improved methods are being developed that will adjust some of the EM experimental parameters to optimize the fit.
We thank Cynthia Stauffacher and Martin Lawrence for access to coordinates and data of HMG Co-A reductase prior to publication. We thank Tom Smith, Tim Baker, R. Holland Cheng, and Norman Olson for
giving us the cryo-EM data with which the EM refinement methods are being tested. We would like to acknowledge our collaborators on the HRV50 refinement: Vince Giranda, R. S. Alexander, M. McMillan
and D. C. Pevear. We are indebted to Mike Sloderbeck for computational advice.
This work has been generously supported by the Lucille P. Markey Charitable Trust and a grant from the National Science Foundation (MSC; BIR9418741).
Programs are distributed under license from http://www.sb.fsu.edu/~rsref.
1. Diamond, R., A Real-Space Refinement Procedure for Proteins. Acta Crystallogr., 1971. A27: p. 436-452.
2. Jones, T.A. & L. Liljas, Crystallographic Refinement of Macromolecules having Non-crystallographic Symmetry. Acta Crystallogr., 1984. A 40: p. 50-7.
3. Levitt, M., Energy Refinement of Hen Egg-White Lysozyme. J. Mol. Biol., 1974. 82: p. 393-420.
4. Hermans Jr., J. & J.E. McQueen, Computer Manipulation of (Macro)molecules with the Method of Local Change. Acta Crystallogr., 1974. A30: p. 730-9.
5. Rees, D.C. & M. Lewis, Incorporation of Experimental Phases in a Restrained Refinement. Acta Crystallogr., 1983. A39: p. 94-97.
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7. Chapman, M.S., Restrained Real-Space Macromolecular Atomic Refinement using a New Resolution-Dependent Electron Density Function. Acta Crystallogr., 1995. A51: p. 69-80.
8. Tronrud, D.E., L.F. Ten Eyck & B.W. Matthews, An Efficient General-Purpose Least-Squares Refinement Program for Macromolecular Structures. Acta Crystallogr., 1987. A43: p. 489-501.
9. Wu, H., W. Keller & M.G. Rossmann, Determination and Refinement of the Canine Parvovirus Empty-Capsid Structure . Acta Cryst. , 1993 . D49 : p. 572-9 .
10. Hendrickson, W.W., Stereochemically Restrained Refinement of Macromolecular Structures. Meth. Enzym., 1985. 115: p. 252-270.
11. Brünger, A.T., J. Kuriyan & M. Karplus, Crystallographic R factor Refinement by Molecular Dynamics. Science, 1987. 235: p. 458-60.
12. Chapman, M.S. & M.G. Rossmann, Structural Refinement of the DNA-containing Capsid of Canine Parvovirus using RSRef, a Resolution-Dependent Stereochemically Restrained Real-Space Refinement
Method. Acta Crystallogr., 1996. D52: p. 129-42.
13. Chapman, M.S. & M.G. Rossmann, Single-stranded DNA-protein interactions in Canine Parvovirus. Structure, 1995. 3: p. 151-62.
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Acad. Sci. USA , 1986 . 83 : p. 5489-93 .
15. Jones, T.A., J.-Y. Zou, S.W. Cowan & M. Kjeldgaard, Improved Methods for Building Protein Models in Electron Density Maps and the Location of Errors in these Models. Acta Crystallogr., 1991. A47:
p. 110-9.
16. Tronrud, D.E. & L.F. Ten Eyck, TNT Refinement Package, Release 5-A . 1992 .
17. Lawrence, C.M., V.M. Rodwell & C.V. Stauffacher, The crystal structure of Pseudomonoas mevalonii HMG-CoA reductase at 3.0 Å resolution. Science, 1995. 268: p. 1758-62.
18. Brünger, A.T., Free R value: a novel statistical quantity for assessing the accuracy of crystal structures . Nature , 1992 . 355 : p. 472-5 .
19. Smith, T.J., N. Olson, R.H. Cheng, H. Liu, E. Chase, W.-M. Lee, A. Moser, R. Rueckert & T.S. Baker, Structure of human rhinovirus complexed with Fab fragments from a neutralizing antibody. J.
Virol., 1993. 67: p. 1148-58.
20. Xie, Q. and M.S. Chapman, Canine parvovirus capsid structure, analyzed at 2.9 Å resolution. Journal of molecular biology, 1996. in press.
21. Brünger, A.T. and L.M. Rice, Crystallographic Refinement by Simulated Annealing: Methods and Applications. Methods in Enzymology, 1997. in press.
22. Blanc, E., V. Giranda, R.S. Alexander, M. McMillan, D.C. Pevear, Q. Xie, G. Parthasarathy, and M.S. Chapman, The 2 Å Refined Structure of Human Rhino Virus 50 Complexed with an Antiviral Agent.
1996. in preparation.
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axiom holds for all times ## Elucidation This is used when the statement/axiom is assumed to hold true 'eternally' ## How to interpret (informal) First the "atemporal" FOL is derived from the OWL
using the standard interpretation. This axiom is temporalized by embedding the axiom within a for-all-times quantified sentence. The t argument is added to all instantiation predicates and predicates
that use this relation. ## Example Class: nucleus SubClassOf: part_of some cell forall t : forall n : instance_of(n,Nucleus,t) implies exists c : instance_of(c,Cell,t) part_of(n,c,t) ## Notes This
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Schönhage-Strassen algorithm
Interestingly, the first article on cp4space was concerned with Fourier series and pathological functions. Assuming you haven’t read it (which is plausible, since it precluded the early-2013
popularity surge), I suppose I should start by re-introducing them.
Fourier reasoned that periodic functions such as the sound waves produced by musical instruments could be expressed as superpositions of simple sinusoidal waves of different frequencies. As a basic
example, successive approximations to the square wave are summarised in this animated GIF:
As a slight aside, I’m always slightly wary about including animated GIFs in my articles, since the last one ended up on the website Tumblr (home to such delights as Sherlock Holmes doffing his scarf
) and featuring some rather interesting tags:
What if we want to perform a similar treatment to aperiodic functions? It transpires that it is possible, as long as we allow a continuum of frequencies instead of a discrete series. In this case,
the summation is replaced with integration, and we obtain the beautiful Fourier transform:
The inverse Fourier transform is obtained by deleting the minus sign in the above expression. Interestingly, the four-fold composition of the Fourier transform operator is the identity (F^^^^ = F),
which led people to consider the concept of a fractional Fourier transform by an arbitrary angle (where the original Fourier transform operator corresponds to a 90° rotation in time-frequency space),
a useful tool for noise reduction in signal processing.
Anyway, the Fourier transform has some nice properties. For instance, if we view functions from $\mathbb{R}$ to $\mathbb{C}$ as elements of an infinite-dimensional complex vector space, Fourier
transforms are linear operators. This suggests another idea: what if, instead of an infinite-dimensional complex vector space, we consider an n-dimensional one instead?
Discrete Fourier transform
Since we’re now living in a finite-dimensional vector space $\mathbb{C}^n$, linear transformations are represented by matrices. The matrix representing the discrete Fourier transform has entries
$DFT_{i j} = \dfrac{1}{\sqrt{n}} \zeta^{i j}$, where ζ is a principal nth root of unity.
We also have the beautiful relationship between pointwise multiplication and sequence convolution of vectors:
• DFT(x convolve y) = DFT(x) pointwise DFT(y)
This will become useful later. Note also that there is nothing special about the complex numbers in this context, and we can replace them with another ring (such as modular arithmetic). If we do so,
we obtain the number-theoretic tranform (NTT), which has the nice property that it can be implemented on a computer with exact integer arithmetic instead of all of that tedious fiddling around with
complex numbers.
Now, convolution is a rather common operation, which is used (amongst other things) for multiplying two polynomials:
(a0 + a1 X + a2 X^2 + …)(b0 + b1 X + b2 X^2 + …) = (a0 b0 + (a1 b0 + a0 b1) X + (a2 b0 + a1 b1 + a0 b2) X^2 + …)
Strictly speaking, the convolution in the context of the discrete Fourier transform is cyclic convolution, since the sequences ‘wrap around’. However, this only produces minor annoyances that do not
affect the asymptotic analysis of…
…the Schönhage-Strassen algorithm
Basically, the ordinary algorithm for multiplying integers treats them as polynomials in some number (the base) and convolves them. But using the number-theoretic transform, we can reduce convolution
(the expensive task) to pointwise multiplication, which in turn can be performed recursively using this algorithm. If we carefully choose the base at each stage (say by chopping the original N-bit
numbers into about sqrt(N) blocks), then we get a recursion depth of log log N and an asymptotic running time of O(N log N log log N).
A slightly more sophisticated variant, Fürer’s algorithm, reduces the running time even further to O(N log N 2^log*(N)), where log* is the iterated logarithm function. However, for all practical
applications, it is faster to use Schönhage-Strassen (with Fürer only overtaking for astronomically huge numbers).
Wait… did I say practical applications?
Surprisingly, multiplying huge numbers does indeed have its advantages. Although the numbers used in RSA cryptography are sufficiently small that the Karatsuba multiplication algorithm is better,
testing Mersenne primes using the Lucas-Lehmer test requires multiplying integers with millions of digits (safely within the realm of optimality of Schönhage-Strassen).
One Response to Schönhage-Strassen algorithm
1. An O(N log N) multiplication algorithm was discovered in 2019. https://en.wikipedia.org/wiki/Multiplication_algorithm#Further_improvements
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Factors of 5 | Learn and Solve Questions
Factors of 5: An Introduction
A factor in mathematics is a divisor of a given number that divides it precisely with no remainder. To determine a number's components, we can adopt a number of methods, such as the division method
and the multiplication approach. Only 5 and 1 are factors of 5. Both positive and negative factors are multiples of three. Both positive and negative integers are possible for the factors and pair
factors of 5. The pair factor of three, for instance, is represented as (1, 5) or (-1, -5). The number 5 is the result of multiplying -1 and -5. Using the prime factorization approach and numerous
solved cases, we will discover the factors of 5, pair factors of 5, and the prime factors of 5. You will learn about the three factors in this section, and for your benefit, we'll calculate the three
factors' sums, identify their prime factors, and display the three factors' sums in pairs.
What are the Factors of 5
The numbers that, when multiplied by 5, leave a remainder of zero are considered factors of a 5. Only factors 1 and 5 make up the number 5.
Keep in mind that ${-1 \times (-5)}=5$. As a product of any two negative numbers yields a positive number, (-1, -5) are also factors.
However, let's limit our discussion to positive integers.
How to Calculate Factors of 5
Let's understand how to calculate the factors of three.
Step 1: Write down the number that needs to be factored in first. The number is 3.
Step 2: Find the two numbers whose product equals 5 in step two.
We get $1 \times 5=5$.
The factors are, therefore, 1 and 5.
Factors of 5 Using Division Method
In the division method, the integer numbers are divided by the number 5 to determine the factors of 5. The number is a factor of 5 if it divides by 5 precisely. Divide the third number by one now,
then do the same with the second and third numbers.
$5 \div 1=5$ (Factor is 1 and Remainder is 0)
$5 \div 5=1$ (Factor is 5 and Remainder is 0)
When the number 5 is divided by two, the result is one. Hence two is not a factor of three. As a result, 1 and 5 are the factors of 5.
Factors of 5
Prime Factorization of 5
The prime factorization of 5 is expressing the number 5 as the product of its prime factors. To use the prime factorization method to determine the prime factors of 5, follow the steps below.
Take the pair factor of 5, for example (1, 5)
We cannot divide further since the number 5, in this case, is a prime number.
So, 5 is represented as $1 \times 5$.
Consequently, ($1 \times 5$) is the prime factorization of 5.
Note: If a pair factor contains a composite number, divide it into its prime factors and write the resulting numbers as the product of those prime factors.
Factor Pairs of 5
Factor pairs of three consist of two factors combined, so their output equals three. The positive factor pairs of three are listed below.
$1 \times 5 = 5$
$5 \times 1 = 5$
Negative values are included in factors of 5, as we mentioned earlier. The list of positive factor pairs above can be changed to a list of negative factor pairs of three by simply adding a minus sign
in front of each factor:
$-1 \times -5 = 5$
$-5 \times -1 = 5$
Thus, we have positive and negative factors pairs:
(1,5), (5,1), (-1,-5) and (-5,-1).
Factor Tree
To determine a number's prime factors, a factor tree is constructed. It is constructed as a tree, with the supplied number's branches standing in for each of the number's factors. Let's learn more
about the factor tree approach for determining a number's factors.
Factor Tree of 5
Solved Examples
Example 1: To find out if 5 is a prime or a composite number, Bansi looks at the number.
Solution: Let's begin by compiling a list of all 5 factors.
Factors 1 and 5 are involved.
If a number solely consists of the number itself and one other element, such as 1. This quantity is prime.
A number is considered composite if it has more than two elements.
The number 5 is a prime number since it only has two elements.
Example 2: During a class test, Roger was asked to find the common factors between 5 and 12. Aid Rojer in figuring out the issue.
Solution: Prime factorization may be used to get the factors of 15:
We get:
12 can be factored into its primes as $2 \times 5 \times 3$
As $1 \times 5$ is the prime factorization of 5,
The common factor between 5 and 15 is 5.
Practice Problems
1. Shri is calculating the total number of factors that equal 3. Can you locate it for him?
A. 2
B. 1
C. 13
D. 3
Ans: Option A
2. What is the greatest number that 3 and 5 have in common?
A. 5
B. 3
C. 2
D. 1
Ans: Option D
Each factor can be combined with another to create the number 5, which is obtained by multiplying the two. In this instance, 5 is a prime number. When you reduce the factors of 5 to just the prime
factors and express 5 as the product of those prime factors, you are said to have performed prime factorization. The number is a factor of 5 if it divides by 5 precisely. When the number 5 is divided
by two, the result is one. Hence two is not a factor of three. As a result, 1 and 5 are the factors of 5. A number of factors, specifically its prime factorization, can be expressed as factor trees.
There are factors for each branch of the tree.
FAQs on Factors of 5
1. How many prime factors are there in 6?
Prime factorization of 5 is $1 \times 5$
Three have factors one and three. We can see that the number 5 only has two elements. As a result, there are only 2 numbers that completely divide 5 without producing a residue.
2. What is a factor in mathematics?
A number or algebraic expression that evenly divides another number or expression—i.e., leaves no remainder—is referred to as a factor in mathematics. As an illustration, 3 and 6 are factors of 12
because $12 \div 3= 4 and 12 \div 6=2$, respectively. 1, 2, 4, and 12 are the other components that make up 12. A positive integer or algebraic expression with two or fewer factors (i.e., itself and
1) greater than one is referred to as prime; one with more factors is referred to as composite.
3. What variables divide by three?
If a number's total digits equal a multiple of three or are divisible by three, it is said to be three-divisible. Take into account the multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30. The sum of
the digits is always divisible by 3. If a number can be divided by three, then its digit sum can also be. | {"url":"https://www.vedantu.com/maths/factors-of-5","timestamp":"2024-11-05T10:55:34Z","content_type":"text/html","content_length":"269420","record_id":"<urn:uuid:5b54cbfe-fc93-4ddb-ad5a-88844664a758>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00856.warc.gz"} |
0Critical Speed Equation For Ball Mill
Ball Mill Critical Speed 911 Metallurgist
2015年6月19日· where: Nc is the critical speed ,in revolutions per minute, D is the mill effective inside diameter, in feet Example: a mill measuring 11’0” diameter inside of new shell liners
operates at 173 rpm Critical speed is CS = 7663 / 11^05 = 231 rpm2021年7月18日· Calculating for Mill Diameter when the Critical Speed of Mill and the Diameter of Balls is Given D = ( 423 / Nc) 2
+ d Where; D = Mill Diameter N c = Critical Speed of Mill d = Diameter of Balls Let’sHow to Calculate and Solve for Critical Mill of Speed
Critical Speed of Ball Mill : Concept and Derivation
2021年9月28日· Critical Speed of Ball Mill : Concept and Derivation © 2023 Google LLC The video contain definition, concept of Critical speed of ball mill and step wise derivation of1999年8月3日·
Rose and Sullivan showed critical rotation speed Nc, to reach the final stage, ie, centrifugal motion: (1) N c = 1 2π 2g D−2r where D is the inner diameter of aCritical rotation speed for
ballmilling ScienceDirect
Ball Mill an overview | ScienceDirect Topics
The company clams this new ball mill will be helpful to enable extreme highenergy ball milling at rotational speed reaches to 1100 rpm This allows the new mill to achieve2020年7月2日· The mill
was simulated at different critical speeds with different mill fillings In total, 165 scenarios were simulated When the mills charge comprising 60% of smallEffects of Ball Size Distribution and
Mill Speed and Their
AMIT 135: Lesson 7 Ball Mills & Circuits – Mining Mill
Explain the role of ball mill in mineral industry and why it is extensively used Describe different types of ball mill design Describe the components of ball mill Explain their understanding of
ball mill operation Explain2017年10月25日· For coarse grinding, a mill speed of 75 to 78 percent of critical is recommended, depending on the initial lifter face angle In this range, analysis of
trajectories of balls in the outer row indicates a landingRecommended Ball Mill Speed & Liner Configuration
Mill Critical Speed Formula Derivation Grinding & Classification
The formula to calculate critical speed is given below N c = 42305 /sqt(Dd) N c = critical speed of the mill D = mill diameter specified in meters d = diameter of the ball InThe research examines
a comparative analysis of the theoretical ball mill speed equation using EDEM software compared to simulation modeling The input parameters for theComparative Analysis of Theoretical and
Bond´s work index estimation using nonstandard ball mills
2023年9月18日· The results showed that using the nonstandard mills (between 20 and 35 cm in diameter), the Bond´s model constants (α=023; β= 082, and γ = 445), are unable to predict the Work
IndexThe critical speed n (rpm) when the balls are attached to the wall due to centrifugation: Figure 27 Displacement of balls in mill n = 423 D m It is possible to make an approximate
calculation of the capacity of a ball mill by means ofBall Mill an overview | ScienceDirect Topics
Tumbling Mill Critical Speed 911 Metallurgist
2015年8月6日· Effect of Speed and Filling on Power In this section, a 0545m x 0304m ball mill is simulated to study the combined effect of mill speed and filling on the power draft of the mill
Mill operating speed andCritical Speed Of Ball Mill Equation We have critical speed of ball mill equation,Mill Critical Speed Formula Derivation 2 replies In practice Ball Mills are driven at a
speed of 5090 of the critical speed the factor being influenced by economic consideration Mill capacity can be increased by increasing speed but there is very little increasederivation of
critical speed of ball mill samac GitHub
(PDF) Effects of Ball Size Distribution and Mill Speed and Their
2020年7月2日· In recent research done by AmanNejad and Barani [93] using DEM to investigate the effect of ball size distribution on ball milling, charging the mill speed with 40% small balls and
60% big balls2012年6月1日· A ball mill is a type of grinder widely utilized in the process of mechanochemical catalytic degradation It consists of one or more rotating cylinders partially filled
with grinding balls (made(PDF) Grinding in Ball Mills: Modeling and Process Control
Ball Mills 911 Metallurgist
2017年2月13日· Optimum Ball Mill Speed nc is the mill speed in fraction of critical speed; Li and Di are length and diameter inside shell lining respectively The most used equation, for this
purpose, is the empirical Bond equation (Bond,At the end of this lesson students should be able to: Explain the role of ball mill in mineral industry and why it is extensively used Describe
different types of ball mill design Describe the components of ball mill Explain their understanding of ball mill operation Explain the role of critical speed and power draw in design and
processAMIT 135: Lesson 7 Ball Mills & Circuits – Mining Mill Operator
Effects of Ball Size Distribution and Mill Speed and Their Interactions
2020年7月2日· A comprehensive investigation was conducted to delineate the effect of ball size distribution, mill speed, and their interactions on power draw, charge motion, and balls segregation
in a laboratoryscale mill The mill was simulated at different critical speeds with different mill fillings In total, 165 scenarios were simulatedA Slice Mill is the same diameter as the
production mill but shorter in length Request Price Quote Click to request a ball mill quote online or call 6303503012 to speak with an expert at Paul O Abbe® to help you determine which design
and size ball mill would be best for your process See our Size Reduction OptionsVariables in Ball Mill Operation | Paul O Abbe®
Recommended Ball Mill Speed & Liner
2017年10月25日· For coarse grinding, a mill speed of 75 to 78 percent of critical is recommended, depending on the initial lifter face angle In this range, analysis of trajectories of balls in
the outer row indicates a landingonline live calculators for grinding calculations, Ball mill, tube mill, critical speed, Degree of filling balls, Arm of gravity, mill net and gross power MENU
Optimization; Online Training; Process and Energy Audit; Critical Speed (nc) & Mill Speed (n) Please Enter / Stepto Input Values Mill Eff Dia Deff, m CALCULATEDball mill calculations, grinding
media filling degree, ball size, mill
How can I determine the best RPM for Dry Ball Milling machine in order
In the equation, did not relate the effects of ball density and ball material etc I would recommend you consider a rotational speed between 65 and 85 % of the critical speed of the millIn
practice Ball Mills are driven at a speed of 5090% of the critical speed, the factor being influenced by economic consideration Mill capacity can be increased by increasing speed but there is
very little increase in efficiency (ie kWht 1 ) when the mill is operated above about 4050% of the critical speedMill Critical Speed Formula Derivation Grinding & Classification
Ball Mill Design/Power Calculation DESIGN AND ANALYSIS OF BALL MILL
2015年6月19日· Ball Mill Power/Design Price Example #2 In Example No1 this was determined that adenine 1400 HP wet grinder ball mill was required to grind 100 TPH of matter with an Bond Works
Catalog of 15 ( guess that mineral type it is ) from 80% passing ¼ inch to 80% passing 100 mesh in closed circuit2015年10月8日· Mineral Processing Equations EQ A Metallurgist or more correctly a
Mineral Processing Engineer will often need to use these “common” Formulas & Equations for Process evaluation and calculations: Circulating load based on pulp density *** Circulating load based
on screen analysis *** Flotation Cell & Conditioner CapacitiesMineral Processing Equations EQ 911 Metallurgist
Ball Mill Critical Speed [1d47v91w6y42] Documents and Ebooks
Ball Mill Critical Speed Uploaded by: Danielito Bonito December 2019 PDF Bookmark Download This document was uploaded by user and they confirmed that they have the permission to share it If you
are author or own the copyright of this book, please report to us by using this DMCA report form Report DMCAResult #1: This mill would need to spin at RPM to be at 100% critical speed Result #2:
This mill's measured RPM is % of critical speed Calculation Backup: the formula used for Critical Speed is: N c =766 (D 05) where, Nc is the critical speed,in revolutions per minute, D is the
mill effective inside diameter, in feetMill Critical Speed Determination
Estimation methodology for Bond ball mill work index experiment
2023年10月1日· This parameter depends on the operational conditions (critical speed, fill level, and ball size distribution), and it references a size fraction that cannot be ground in the mill
The parameter μ values for the samples studied in this paper were 1175 µm for limestone, 989 µm for chalcopyrite, hematite ore denotes 1150 µm, and fluorite value isA ball mill, a type of
grinder, is a cylindrical device used in grinding (or mixing) materials like ores, chemicals, ceramic raw materials and paints Ball mills rotate around a horizontal axis, partially filled with
the material to be ground plus the grinding medium Different materials are used as media, including ceramic balls, flint pebblesBall mill
ATTRITORS AND BALL MILLS HOW THEY WORK Robert E
the mill The common range of mill speeds is 65% to 80% of critical, depending on mill type, size and the application The critical speed of a ball mill is calculated as 5419 divided by the square
root of the radius in feet The rotational speed is defined as a percentage of the critical speed Smaller diameter mills1990年1月1日· The normalized critical speed and top ball sizes are related
to corresponding actual quantities by: TBS = TBS/70mm and Nf Nf/ 70% (9) A nonlinear leastsquares fitting program, NLF (Whiten, 1972), was used to relate the breakage rate to the critical speed,
ball size and mill diameter by fitting the parameters 7, 13, p and roi inApplications of a new modelbased method of ball mill simulation
Mill Speed an overview | ScienceDirect Topics
Autogenous and SemiAutogenous Mills In Mineral Processing Design and Operations (Second Edition), 2016 934 Mill Speed During normal operation the mill speed tends to vary with mill charge
According to available literature, the operating speeds of AG mills are much higher than conventional tumbling mills and are in the range of 80–85% of the1992年1月1日· Mill power (3) where * X */
L * + (2R )3( 1 )[(125R /0)01 ( 05~ )4]), D * * 05J 1D1/2R 1 25R /D (4) J < 045 T D 0 i a R Figl Definition ofsymbolism for a mill with conical end sections (internal dimensions) In these
equations mp is net mill power; J is the fractional filling level by the ball bed in the cylindrical portion; ~c is millMill power for conical (Hardinge) type ball mills ScienceDirect
Modelling SAG milling power and specific energy ScienceDirect
2015年1月1日· For a given mill to have a combination of feed size, ball load, mill speed and % solids will represent the total load In fact the later can be modelled as a function of the others
Additionally, as has been shown by Powell et al (2001) , ball loads high enough (over 12% as in the cases studied in this work) contribute to a significant portion of theC – the mill drum
rotational speed,% of the critical speed; D – the mill internal diameter, m At result B = 25mm or less necessary to use the correction factor 13, ie the grinding balls average diameter should be
325Every ball screw, or any shaft, has a rotational speed at which the vibration and harmonics becomes excessivecritical speed equation for ball mill
derivation for critical speed of ball mill Grinding Mill China
derivation for critical speed of ball mill Posted at: June 27, 2013 [ 48 5434 Ratings ] critical speed of ball mill derivation – Gulin Mining Ball Mill Operating Speed – Mechanical Operations
Solved2017年5月8日· Now, for a given mill filling and a given material being ground, ∅3(J) and (1 + 04σ/q) are invariant, and for a constant fraction of the critical speed ∅1(Nc/N) is constant
Furthermore, since the mill runs at a given fraction of its critical speed, it follows from equation (36) thatGrinding Mill Power 911 Metallurgist
The Mechanism and Grinding Limit of Planetary Ball Millingt
the rate of grinding in the planetary ball mill is known to be about 100 times greater than that in a tumbling ball mill and about 50 times greater than that in a stirred ball mill2) Another
result is given in Fig 3, from which it is seen that even small glass beads with a low density can be used to grind hard materials2004年12月1日· Though the grinding of FA in the planetary ball
mill was studied in details to investigate the optimum operating milling parameters (eg, critical speed of the sun and vial, power consumptionOptimum revolution and rotational directions and
their speeds in
(PDF) Performance optimization of an industrial ball
2017年1月1日· An increase of over 10% in mill throughput was achieved by removing the ball scats from a single stage SAG mill These scats are non spherical ball fragments resulting from uneven
wear of balls2023年2月13日· Using the Bond power equation, calculate the dimensions of the mill (D, L) that can handle the same feed tonnage and yet produce the finer grinding (For typical value
of ball loading J=04, fractional rotation speed to critical value =07 and L/D Ratio 15SOLVED: Using the following equation for calculating the energy
Impact energy of particles in ball mills based on DEM simulations
2022年1月1日· The motions of balls for different mill diameters are similar at the same fraction of critical speed, which is identical to that observed by other scholars [33, 34, 37] for a
tumbling ball mill The balls near the bottom are lifted by the lifters and dragged up around the linerTake advantage of this Critical Speed Calculator along with the many other various
calculators Roton has available at your disposalCritical Speed Roton Products, Inc
Pulverizer Why Critical Speed Of Ball Mill Crusher Mills
derive the equation of critical speed of ball mill, ball mill (paper) ifs indian forest service stone engineering previous mar 17, 2009 (c) what is a ball mill? derive an equation for the
critical speed of a ball mill2003年11月15日· The tests covered a range of slurry concentrations from 30 to 55 vol% solid and fractional interstitial bed filling (U) from 03 to 175, at a fixed
ball load (30% of mill volume) and 70% of critical speed, using batch grinding of a feed of −30 mesh (06 mm) quartzAt a fixed slurry concentration, the net mill power versus U went through
aEffects of slurry concentration and powder filling on the net mill
Ball Mill Parameter Selection & Calculation Power, Critical Speed
2019年8月30日· V — Effective volume of ball mill, m3; G2 — Material less than 0074mm in product accounts for the percentage of total material, %; G1 — Material less than 0074mm in ore feeding
accounts for 0074mm in the percentage of the total material, %; q’m — Unit productivity calculated according to the new generation grade (0074mm),2018年8月7日· The equations which were developed
from this exercise were order of four factors on −0074 mm yield is mill speed, ball of 70% of critical speed, 20% of ballDEM Investigation of Mill Speed and Lifter Face Angle on
Permissible Rotational Speed THK
The permissible rotational speed of the Ball Screw must be obtained from the critical speed of the screw shaft and the DN value The permissible rotational speed determined by the DN value is
obtained using the equations (8) to (17) below Model No Permissible rotational speed determined by the DN value N 2 Guideline for maximum rotational speedTypically R = 8 Rod Mill Charge:
Typically 45% of internal volume; 35% – 65% range Bed porosity typically 40% Height of bed measured in the same way as ball mills Bulk density of rods = 625 tons/m3 In wet grinding, the solids
concentration 1s typically 60% – 75% by mass A rod in situ and a cutaway of a rod mill interiorAMIT 135: Lesson 8 Rod Mills – Mining Mill Operator Training | {"url":"https://www.kvalitniobklady.cz/1693829357-ww/9333.html","timestamp":"2024-11-08T02:36:23Z","content_type":"text/html","content_length":"25623","record_id":"<urn:uuid:7626b476-0366-4f84-b8bd-14819f8c0dbe>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00399.warc.gz"} |
The Sandorian Grove: Quantum Potential, Theory in General, and the Toth Diagram
│Quantum Potential, Theory in General, and the Toth Diagram │3 comments│
22 Aug 2008 @ 17:15, by Max Sandor
Thanks to the Data Deluge of our times, we witness the end of the Scientific Method or so says a Wired article [link] (thanks for the link, Vaxen).
Should I throw away our new model(s)... Interestingly, the Toth diagram arrived via the 'statistical method' in the first place. Based on the ancient knowledge of Fá transmitted in the form of more
than 3000 (threethousand) archetypical stories describing the patterns of a 16x16 energy matrix, we extracted the essential base energies (olodu) and regrouped them as end leaves within a binary
Then we witnessed a sheer endless list of symmetries in its pathways, all of them corroborating in an incredible way with the system of Fá and the extensions made by myself and those added by Ed
Dawson, some of them described in 'Polar Dynamics 1' some years ago.
In short, the base energies of the Toth Diagram ARE BEING DERIVED from the observations of the patterns of life paths and not from a theory !!!
But the Toth Diagram is adding some new perspectives to a lot of fields, one of them the paradigm of the Quantum Potential and it quintessential force.
Like the zillions of Sacred Geometry theories are missing a description of the actual energies in the nodes of its structure(s), the Quantum Potential theory is lacking this specification as well.
WITHOUT this specification, however, it would be impossible to predict just WHAT is going to manifest out of the chaos (Ofun) in the vacuum (Oyeku).
Likewise, it was found EMPIRICALLY through direct observation and repetition that there is more than ONE super force (called Girapoli energies in the articles on this BLOG).
And, again, the subsequent creation of a model approximating these observations allowed us to logically deduct some more super forces (which by now have been verified many times in the praxis),
Therefore, models WILL remain useful, in spite of data deluge and statistical method.
(Note: with statistical method I do NOT refer the abusive usage of 'statistics' in many fields of today's science-religions but to correlation methods of massive data flows like described in the
Wired article linked to above.)
'nuff zed, don't want to get apologetic here ;-)
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Category: Articles
3 comments
14 Dec 2014 @ 20:29 by Roby @190.38.84.101 : RkxhfXYAvAPBnvTW
That's an inventive answer to an inteersting question
16 Dec 2014 @ 13:19 by Sos @201.243.96.31 : punvLKkpvlH
Nydelig, Rune! Jeg fornemmer helt klart somemr her ... lite sol eller lave temperaturer skal ikke ve6re til hinder for at jeg skal ff8le pe5 somemrstemningen. Lange lyse somemrkvelder, blomster i
krukker, en kald pils pe5 terrassen (selv om varmelampa me5 ste5 pe5) er noe av det som fe5r meg i rett modus :) He5per du fe5r oppleve mange gode somemrkvelder i ukene fremover!Sommerklem fra
Drammen i (nesten) regn :o)PS: Grillen er varm og jeg skal ta pe5 meg fleecen
29 Dec 2014 @ 09:52 by Hardik @64.15.186.71 : KPhkzLCkmaiyAPyJ
For fun, I tallied year-by-year the nuembr of arxiv abstracts containing the exact phrase quantum error correction . The resulting distribution is bimodal, with one peak in the years 1996-7, and
another peak in the years 2007-8 (the lexical data is appended)Broadly speaking, the first quantum error correction usage peak is associated to the discovery of Quantum Error Correction by Coding
(arXiv:quant-ph/9511003, Chuang and LaFlamme) and that Good Quantum Error-Correcting Codes Exist (arXiv:quant-ph/9512032, Calderbank and Shor).The second quantum error correction usage peak appears
to largely associated to the integration of quantum error correction into the general theory of error correction, and more broadly, into general theories of informatic flow and thermodynamical
processes in many STEM enterprises. See for example Belief propagation algorithm for computing correlation functions in finite-temperature quantum many-body systems (arXiv:0710.4304, Poulin and
Bilgin).Strikingly scarce in the recent arxiv literature are concrete proposals to design, build and test working quantum computers capable of useful calculations.Will we ever see a third quantum
error correction usage peak? one that is associated to working, practical quantum computers? Even if the answer is no even if practical quantum computation is perceived to be unforeseeably distant in
the future it is becoming evident that dynamical aspects of quantum error correction not only are are a fertile research topic, but possibly have fundamental relevance to a broad span of STEM
enterprises.No doubt these issues will be much-discussed at QEC11 it looks like an interesting conference to me. 2010 (end) 46 “quantum error correction articles2009 432008 552007 542006 362005
342004 282003 312002 212001 112000 171999 171998 171997 221996 341995 (start) 2 articles http://fcfqbvku.com [url=http://cjnixhuob.com]cjnixhuob[/url] [link=http://ozihauz.com]ozihauz[/link]
Other articles in Articles
23 Sep 2016 @ 17:18: A summary of the summaries of Max Sandor's projects
23 Sep 2016 @ 17:04: Project Summary 6. Game Theory - why and how do we manifest?
23 Sep 2016 @ 17:02: Project Summary 5: Polar Dynamics - theory and praxis of polarities
23 Sep 2016 @ 17:01: Project Summary 4: Quantum Fá - a practical guide to this Universe
23 Sep 2016 @ 16:45: Project Summary 3: The Book of Numbers
22 Sep 2016 @ 16:12: Project Summary 2: UrTon - the basis of spoken languages
18 Sep 2016 @ 00:32: Project Summary 1: The ConCur Paradigm - the structure of Reality
9 Aug 2016 @ 14:35: Robot Psychologist (by Awaz)
9 Aug 2016 @ 14:35: Project Summary 7: Archetypology of the Human Being
1 Aug 2016 @ 00:40: Victory, submission or what else? Sign and symbol of the Rio 2016 Kickoff
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mcmc sampler
This is the Markov Chain Monte Carlo Metropolis sampler used by CosmoMC, and described in Lewis, “Efficient sampling of fast and slow cosmological parameters” (arXiv:1304.4473). It works well on
simple uni-modal (or only weakly multi-modal) distributions.
The proposal pdf is a gaussian mixed with an exponential pdf in random directions, which is more robust to misestimation of the width of the proposal than a pure gaussian. The scale width of the
proposal can be specified per parameter with the property proposal (it defaults to the standard deviation of the reference pdf, if defined, or the prior’s one, if not). However, initial performance
will be much better if you provide a covariance matrix, which overrides the default proposal scale width set for each parameter.
The proposal width for a parameter should be close to its conditional posterior, not its marginalized width. For strong degeneracies the latter can be much wider than the former, and hence it could
cause the chain to get stuck. Underestimating a good proposal width is usually better than overestimating it: an underestimate can be rapidly corrected by the adaptive covariance learning, but if the
proposal width is too large the chain may never move at all.
If the distribution being sampled is known have tight strongly non-linear parameter degeneracies, re-define the sampled parameters to remove the degeneracy before sampling (linear degeneracies are
not a problem, esp. if you provide an approximate initial covariance matrix).
Initial point for the chains
The initial points for the chains are sampled from the reference pdf (see Parameters and priors). If the reference pdf is a fixed point, chains will always start from that point. If there is no
reference pdf defined for a parameter, the initial sample is drawn from the prior instead.
Example of parameters block:
min: -2
max: 2
min: -1
max: 1
proposal: 0.5
latex: \alpha
min: -1
max: 4
ref: 2
proposal: 0.25
latex: \beta
min: -1
max: 1
dist: norm
loc: 0
scale: 0.2
latex: \gamma
• a – the initial point of the chain is drawn from an uniform pdf between -1 and 1.
• b – the initial point of the chain is always at b=2.
• c – the initial point of the chain is drawn from a gaussian centred at 0 with standard deviation 0.2.
Fixing the initial point is not usually recommended, since to assess convergence it is useful to run multiple chains (which you can do in parallel using MPI), and use the difference between the
chains to assess convergence: if the chains all start in exactly the same point, they could appear to have converged just because they started at the same place. On the other hand if your initial
points are spread much more widely than the posterior it could take longer for chains to converge.
Covariance matrix of the proposal pdf
An accurate, or even approximate guess for the proposal pdf will normally lead to significantly faster convergence.
In Cobaya, the covariance matrix of the proposal pdf is optionally indicated through mcmc’s property covmat, either as a file name (including path, absolute or relative to the invocation folder), or
as an actual matrix. If a file name is given, the first line of the covmat file must start with #, followed by a list of parameter names, separated by a space. The rest of the file must contain the
covariance matrix, one row per line.
An example for the case above:
Instead, if given as a matrix, you must also include the covmat_params property, listing the parameters in the matrix in the order in which they appear. Finally, covmat admits the special value auto
that searches for an appropriate covariance matrix in a database (see Basic cosmology runs).
If you do not know anything about the parameters’ correlations in the posterior, instead of specifying the covariance matrix via MCMC’s covmat field, you may simply add a proposal field to the
sampled parameters, containing the expected standard deviation of the proposal. In the absence of a parameter in the covmat which also lacks its own proposal property, the standard deviation of the
reference pdf (of prior if not given) will be used instead (though you would normally like to avoid that possibility by providing at least a proposal property, since guessing it from the prior
usually leads to a very small initial acceptance rate, and will tend to get your chains stuck).
A covariance matrix given via covmat does not need to contain the all the sampled parameters, and may contain additional ones unused in your run. For the missing parameters the specified input
proposal (or reference, or prior) is used, assuming no correlations.
If the covariance matrix shown above is used for the previous example, the final covariance matrix of the proposal will be:
# a b c
0.1 0.01 0
0.01 0.2 0
0 0 0.04
If the option learn_proposal is set to True, the covariance matrix will be updated regularly. This means that a high accuracy of the initial covariance is not critical (just make sure your proposal
widths are sufficiently small that chains can move and hence explore the local shape; if your widths are too wide the parameter may just get stuck).
If you are not sure that your posterior has one single mode, or if its shape is very irregular, you may want to set learn_proposal: False; however the MCMC sampler is not likely to work well in this
case and other samplers designed for multi-modal distributions (e.g. PolyChord) may be a better choice.
If you do not know how good your initial guess for the starting point and covariance is, a number of initial burn in samples can be ignored from the start of the chains (e.g. 10 per dimension). This
can be specified with the parameter burn_in. These samples will be ignored for all purposes (output, convergence, proposal learning…). Of course there may well also be more burn in after these points
are discarded, as the chain points converge (and, using learn_proposal, the proposal estimates also converge). Often removing the first 30% the entire final chains gives good results (using
ignore_rows=0.3 when analysing with getdist).
Taking advantage of a speed hierarchy
In Cobaya, the proposal pdf is blocked by speeds, i.e. it allows for efficient sampling of a mixture of fast and slow parameters, such that we can avoid recomputing the slowest parts of the
likelihood calculation when sampling along the fast directions only. This is often very useful when the likelihoods have large numbers of nuisance parameters, but recomputing the likelihood for
different sets of nuisance parameters is fast.
Two different sampling schemes are available in the mcmc sampler to take additional advantage of a speed hierarchy:
• Oversampling the fast parameters: consists simply of taking more steps in the faster directions, useful when exploring their conditional distributions is cheap. Enable it by setting
oversample_power to any value larger than 0 (1 means spending the same amount of time in all parameter blocks; you will rarely want to go over that value).
• Dragging the fast parameters: consists of taking a number of intermediate fast steps when jumping between positions in the slow parameter space, such that (for large numbers of dragging steps)
the fast parameters are dragged along the direction of any degeneracy with the slow parameters. Enable it by setting drag: True. You can control the relative amount of fast vs slow samples with
the same oversample_power parameter.
In general, the dragging method is the recommended one if there are non-trivial degeneracies between fast and slow parameters that are not well captured by a covariance matrix, and you have a fairly
large speed hierarchy. Oversampling can potentially produce very large output files (less so if the oversample_thin option is left to its default True value); dragging outputs smaller chain files
since fast parameters are effectively partially marginalized over internally. For a thorough description of both methods and references, see A. Lewis, “Efficient sampling of fast and slow
cosmological parameters” (arXiv:1304.4473).
The relative speeds can be specified per likelihood/theory, with the option speed, preferably in evaluations per second (approximately). The speeds can also be measured automatically when you run a
chain (with mcmc, measure_speeds: True), allowing for variations with the number of threads used and machine differences. This option only tests the speed on one point (per MPI instance) by default,
so if your speed varies significantly with where you are in parameter space it may be better to either turn the automatic selection off and keep to manually specified average speeds, or to pass a
large number instead of True as the value of measure_speeds (it will evaluate the posterior that many times, so the chains will take longer to initialise).
To manually measure the average speeds, set measure_speeds in the mcmc block to a high value and run your input file with the --test option; alternatively, add timing: True at the highest level of
your input (i.e. not inside any of the blocks), set the mcmc options burn_in: 0 and max_samples to a reasonably large number (so that it will be done in a few minutes), and check the output: it
should have printed, towards the end, computation times for the likelihood and theory codes in seconds, the inverse of which are the speeds.
If the speed has not been specified for a component, it is assigned the slowest one in the set. If two or more components with different speeds share a parameter, said parameter is assigned to a
separate block with a speed that takes into account the computation time of all the codes that depends on it.
For example:
speed: 2
speed: 4
Here, evaluating the theory code is the slowest step, while the like_b is faster. Likelihood like_a is assumed to be as slow as the theory code.
Manual specification of speed blocking
Automatic speed blocking takes advantage of differences in speed per likelihood (or theory). If the parameters of your likelihood or theory have some internal speed hierarchy that you would like to
exploit (e.g. if your likelihood internally caches the result of a computation depending only on a subset of the likelihood parameters), you can specify a fine-grained list of parameter blocks and
their oversampling factors, using the mcmc option blocking.
E.g. if a likelihood depends of parameters a, b and c and the cost of varying a is twice as big as the other two, your mcmc block should look like
- [1, [a]]
- [2, [b,c]]
# drag: True # if desired; 2 different oversampling factors only!
When choosing an oversampling factor, one should take into account the total cost of varying one parameter in the block, i.e. the time needed to re-compute every part of the code that depends
(directly or indirectly) on it.
For example, if varying parameter a in the example above would also force a re-computation of the part of the code associated to parameters b and c, then the relative cost of varying the parameters
in each block would not be 2-to-1, but (2+1)-to-1, meaning oversampling factors of 1 and 3 may be more appropriate.
If blocking is specified, it must contain all the sampled parameters.
If automatic learning of the proposal covariance is enabled, after some checkpoint the proposed steps will mix parameters from different blocks, but always towards faster ones. Thus, it is important
to specify your blocking in ascending order of speed, when not prevented by the architecture of your likelihood (e.g. due to internal caching of intermediate results that require some particular
order of parameter variation).
Tempered MCMC
Some times it is convenient to sample from a power-reduced (or softened), tempered version of the posterior. This produces Monte Carlo samples with more points towards the tails of the distribution,
which is useful when e.g. estimating a quantity by weighting with samples with their probability, or to be able to do more robust importance reweighting.
By setting a value greater than 1 for the temperature option, the mcmc sampler will produce a chain sampled from \(p^{1/t}\), where \(p\) is the posterior and \(t\) is the temperature. The resulting
SampleCollection and output file will contain as weights and log-posterior those of the tempered posterior \(p^{1/t}\) (this is done like this because it is advantageous to retain integer weights,
and keeps weights consistent between parallel chains). The original prior- and likelihood-related values in the sample/output are preserved.
Despite storing the tempered log-posterior, methods producing statistics such as mean() and cov() will return the results corresponding to the original posterior, unless they are called with tempered
To convert the sample into one of the original posterior (i.e. have weights and log-posterior of the original posterior) call the reset_temperature() method, possibly on a copy produced with the copy
() method.
GetDist can load tempered samples as normally, and will retain the temperature. To convert a tempered GetDist sampleinto one of the original posterior, call its .cool(temperature) method.
Convergence checks
Convergence of an MCMC run is assessed in terms a generalized version of the \(R-1\) Gelman-Rubin statistic, implemented as described in arXiv:1304.4473.
In particular, given a small number of chains from the same run, the \(R-1\) statistic measures (from the last half of each chain) the variance between the means of the different chains in units of
the covariance of the chains (in other words, that all chains are centered around the same point, not deviating from it a significant fraction of the standard deviation of the posterior). When that
number becomes smaller than Rminus1_stop twice in a row, a second \(R-1\) check is also performed on the bounds of the Rminus1_cl_level % confidence level interval, which, if smaller than
Rminus1_cl_stop, stops the run.
The default settings are Rminus1_stop = 0.01, Rminus1_cl_level = 0.95 and Rminus1_cl_level = 0.2; the stop values can be decreased for better convergence.
For single-chain runs, the chain is split into a number Rminus1_single_split of segments, the first segment is discarded, and the \(R-1\) checks are performed on the rest as if they were independent
The \(R-1\) diagnostics is a necessary but not sufficient condition for convergence. Most realistic cases should have converged when \(R-1\) is small enough, but harder ones, such as multimodal
posteriors with modes farther apart than their individual sizes, may mistakenly report convergence if all chains are stuck in the same mode (or never report it if stuck in different ones). Using a
smaller value of Rminus1_cl_stop can ensure better exploration of the tails of the distribution, which may be important if you want to place robust limits on parameters or make 2D constraint plots.
Progress monitoring
When writing to the hard drive, the MCMC sampler produces an additional [output_prefix].progress file containing the acceptance rate and the Gelman \(R-1\) diagnostics (for means and confidence level
contours) per checkpoint, so that the user can monitor the convergence of the chain. In interactive mode (when running inside a Python script of in the Jupyter notebook), an equivalent progress table
in a pandas.DataFrame is returned among the products.
The mcmc module provides a plotting tool to produce a graphical representation of convergence, see plot_progress(). An example plot can be seen below:
from cobaya.samplers.mcmc import plot_progress
# Assuming chain saved at `chains/gaussian`
plot_progress("chains/gaussian", fig_args={"figsize": (6,4)})
import matplotlib.pyplot as plt
When writing to the hard drive (i.e. when an [output_prefix].progress file exists), one can produce these plots even if the sampler is still running.
Callback functions
A callback function can be specified through the callback_function option. It must be a function of a single argument, which at runtime is the current instance of the mcmc sampler. You can access its
attributes and methods inside your function, including the SampleCollection of chain points and the model (of which prior and likelihood are attributes). For example, the following callback function
would print the points added to the chain since the last callback:
def my_callback(sampler):
The callback function is called every callback_every points have been added to the chain, or at every checkpoint if that option has not been defined.
Loading chains (single or multiple)
To load the result of an MCMC run saved with prefix e.g. chains/test as a single chain, skipping the first third of each chain, simply do
from cobaya import load_samples
# As Cobaya SampleCollection
full_chain = load_samples("chains/test", skip=0.33, combined=True)
# As GetDist MCSamples
full_chain = load_samples("chains/test", skip=0.33, to_getdist=True)
Interaction with MPI when using MCMC inside your own script
When integrating Cobaya in your pipeline inside a Python script (as opposed to calling it with cobaya-run), you need to be careful when using MPI: exceptions will not me caught properly unless some
wrapping is used:
from mpi4py import MPI
comm = MPI.COMM_WORLD
rank = comm.Get_rank()
from cobaya import run
from cobaya.log import LoggedError
success = False
upd_info, mcmc = run(info)
success = True
except LoggedError as err:
# Did it work? (e.g. did not get stuck)
success = all(comm.allgather(success))
if not success and rank == 0:
print("Sampling failed!")
In this case, if one of the chains fails, the rest will learn about it and raise an exception too as soon as they arrive at the next checkpoint (in order for them to be able to learn about the
failing process earlier, we would need to have used much more aggressive MPI polling in Cobaya, that would have introduced a lot of communication overhead).
As sampler products, every MPI process receives its own chain via the products() method. To gather all of them in the root process and combine them, skipping the first third of each, do:
# Run from all MPI processes at once.
# Returns the combined chain for all of them.
# As cobaya.collections.SampleCollection
full_chain = mcmc.samples(combined=True, skip_samples=0.33)
# As GetDist MCSamples
full_chain = mcmc.samples(combined=True, skip_samples=0.33, to_getdist=True)
For Cobaya v3.2.2 and older, or if one prefers to do it by hand:
all_chains = comm.gather(mcmc.products()["sample"], root=0)
# Pass all of them to GetDist in rank = 0
if rank == 0:
from getdist.mcsamples import MCSamplesFromCobaya
gd_sample = MCSamplesFromCobaya(upd_info, all_chains)
# Manually concatenate them in rank = 0 for some custom manipulation,
# skipping 1st 3rd of each chain
copy_and_skip_1st_3rd = lambda chain: chain[int(len(chain) / 3):]
if rank == 0:
full_chain = copy_and_skip_1st_3rd(all_chains[0])
for chain in all_chains[1:]:
# The combined chain is now `full_chain`
Options and defaults
Simply copy this block in your input yaml file and modify whatever options you want (you can delete the rest).
# Default arguments for the Markov Chain Monte Carlo sampler
# ('Xd' means 'X steps per dimension', or full parameter cycle
# when adjusting for oversampling / dragging)
# Number of discarded burn-in samples per dimension (d notation means times dimension)
burn_in: 0
# Error criterion: max attempts (= weight-1) before deciding that the chain
# is stuck and failing. Set to `.inf` to ignore these kind of errors.
max_tries: 40d
# File (including path) or matrix defining a covariance matrix for the proposal:
# - null (default): will be generated from params info (prior and proposal)
# - matrix: remember to set `covmat_params` to the parameters in the matrix
# - "auto" (cosmology runs only): will be looked up in a library
# Overall scale of the proposal pdf (increase for longer steps)
proposal_scale: 2.4
# Update output file(s) and print some info
# every X seconds (if ends in 's') or X accepted samples (if just a number)
output_every: 60s
# Number of distinct accepted points between proposal learn & convergence checks
learn_every: 40d
# Posterior temperature: >1 for more exploratory chains
temperature: 1
# Proposal covariance matrix learning
# -----------------------------------
learn_proposal: True
# Don't learn if convergence is worse than...
learn_proposal_Rminus1_max: 2.
# (even earlier if a param is not in the given covariance matrix)
learn_proposal_Rminus1_max_early: 30.
# ... or if it is better than... (no need to learn, already good!)
learn_proposal_Rminus1_min: 0.
# Convergence and stopping
# ------------------------
# Maximum number of accepted steps
max_samples: .inf
# Gelman-Rubin R-1 on means
Rminus1_stop: 0.01
# Gelman-Rubin R-1 on std deviations
Rminus1_cl_stop: 0.2
Rminus1_cl_level: 0.95
# When no MPI used, number of fractions of the chain to compare
Rminus1_single_split: 4
# Exploiting speed hierarchy
# --------------------------
# Whether to measure actual speeds for your machine/threading at starting rather
# than using stored values
measure_speeds: True
# Amount of oversampling of each parameter block, relative to their speeds
# Value from 0 (no oversampling) to 1 (spend the same amount of time in all blocks)
# Can be larger than 1 if extra oversampling of fast blocks required.
oversample_power: 0.4
# Thin chain by total oversampling factor (ignored if drag: True)
# NB: disabling it with a non-zero `oversample_power` may produce a VERY LARGE chains
oversample_thin: True
# Dragging: simulates jumps on slow params when varying fast ones
drag: False
# Manual blocking
# ---------------
# Specify parameter blocks and their correspondent oversampling factors
# (may be useful e.g. if your likelihood has some internal caching).
# If used in combination with dragging, assign 1 to all slow parameters,
# and a common oversampling factor to the fast ones.
# - [oversampling_factor_1, [param_1, param_2, ...]]
# - etc.
# Callback function
# -----------------
callback_every: # default: every checkpoint
# Seeding runs
# ------------
# NB: in parallel runs, only works until to the first proposer covmat update.
seed: # integer between 0 and 2**32 - 1
# ----------------
check_every: # now it is learn_every
oversample: # now controlled by oversample_power > 0
drag_limits: # use oversample_power instead
Module documentation
Blocked fast-slow Metropolis sampler (Lewis 1304.4473)
Antony Lewis (for the CosmoMC sampler, wrapped for cobaya by Jesus Torrado)
MCMC sampler class
class samplers.mcmc.MCMC(info_sampler, model, output=typing.Optional[cobaya.output.Output], packages_path=None, name=None)
Adaptive, speed-hierarchy-aware MCMC sampler (adapted from CosmoMC) cite{Lewis:2002ah,Lewis:2013hha}.
Progress monitoring
samplers.mcmc.plot_progress(progress, ax=None, index=None, figure_kwargs=mappingproxy({}), legend_kwargs=mappingproxy({}))
Plots progress of one or more MCMC runs: evolution of R-1 (for means and c.l. intervals) and acceptance rate.
Takes a progress instance (actually a pandas.DataFrame, returned as part of the sampler products), a chain output prefix, or a list of those for plotting progress of several chains at once.
You can use figure_kwargs and legend_kwargs to pass arguments to matplotlib.pyplot.figure and matplotlib.pyplot.legend respectively.
Returns a subplots axes array. Display with matplotlib.pyplot.show().
proposal distributions
Antony Lewis and Jesus Torrado
Using the covariance matrix to give the proposal directions typically significantly increases the acceptance rate and gives faster movement around parameter space.
We generate a random basis in the eigenvectors, then cycle through them proposing changes to each, then generate a new random basis. The distance proposal in the random direction is given by a two-D
Gaussian radial function mixed with an exponential, which is quite robust to wrong width estimates
See https://arxiv.org/abs/1304.4473
class samplers.mcmc.proposal.CyclicIndexRandomizer(n, random_state)
Get the next random index, or alternate for two or less.
class samplers.mcmc.proposal.RandDirectionProposer(n, random_state)
Propose a random n-dimension vector for n>1
scale (float) – units for the distance
array with vector
Radial proposal. By default a mixture of an exponential and 2D Gaussian radial proposal (to make wider tails and more mass near zero, so more robust to scale misestimation)
random distance (unit scale)
class samplers.mcmc.proposal.BlockedProposer(parameter_blocks, random_state, oversampling_factors=None, i_last_slow_block=None, proposal_scale=2.4)
Take covariance of sampled parameters (propose_matrix), and construct orthonormal parameters where orthonormal parameters are grouped in blocks by speed, so changes in the slowest block
changes slow and fast parameters, but changes in the fastest block only changes fast parameters
propose_matrix – covariance matrix for the sampled parameters. | {"url":"https://cobaya.readthedocs.io/en/latest/sampler_mcmc.html","timestamp":"2024-11-10T14:06:07Z","content_type":"text/html","content_length":"103504","record_id":"<urn:uuid:03ca2e80-4941-4327-8500-f43863efcc23>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00498.warc.gz"} |
Sudo Null - Latest IT News
Operations on complex numbers
Hello% username%!
I received quite a lot of reviews about the
first part
and tried to take them all into account.
In the first part I wrote about the addition, subtraction, multiplication and division of complex numbers.
If you don’t know this, you’d rather run to read the first part :-) The
article is framed in the form of a spike, there are very few stories here, mostly formulas.
Enjoy reading!
So, we turn to more interesting and slightly more complex operations.
I will talk about the exponential form of the complex number,
exponentiation, square root, module, and also about the sine and
cosine of the complex argument.
I think we should start with the module of a complex number.
A complex number can be represented on the coordinate axis.
Real numbers will be located along x, and imaginary along y.
This is called a complex plane. Any complex number, for example
Obviously, you can think of it as a radius vector: The
formula for calculating a module will look like this:
It turns out that the module of the complex number z will be equal to 10.
In the last part I told about two forms of writing a complex number:
algebraic and geometric. There is another illustrative form of recording:
Here r is a module of a complex number,
and φ is arctg (y / x), if x> 0
If x <0, y> 0, then
If x <0, y <0 then
There is a wonderful formula of Moivre, which allows you to build a complex number to an
integer power. It was discovered by the French mathematician Abrach de Muavre in 1707.
It looks like this:
As a result, we can raise the number z to the power of a:
If your complex number is written in exponential form, then
you can use the formula:
Now, knowing how the modulus of a complex number and the Moivr's formula are found, we can find the
n root of a complex number:
Here k is numbers from 0 to n-1.
From this we can conclude that there are exactly n different roots of the n-th
degree from a complex number.
We turn to the sine and cosine.
Euler's famous formula will help us calculate them:
By the way, there still exists the Euler identity, which is a special
case of the Euler formula for x = π:
We obtain the formulas for calculating the sine and cosine:
At the end of the article, one cannot but mention the practical application of complex
numbers, so that the question does not arise, did
these complex numbers surrender?
Answer: in some areas of science without them in any way.
In physics, in quantum mechanics there is such a thing as a wave function, which itself is complex-valued.
In electrical engineering, complex numbers have found themselves as a convenient replacement for the diffs that inevitably arise when solving problems with linear AC circuits.
Zhukovsky's theorem (wing lift) also uses complex numbers.
And also in biology, medicine, economics, and many more.
I hope, now you know how to operate with complex numbers and can
put them into practice.
If something in the article is not clear - write in the comments, I will answer. | {"url":"https://sudonull.com/post/8596-Operations-on-complex-numbers","timestamp":"2024-11-06T01:47:34Z","content_type":"text/html","content_length":"11985","record_id":"<urn:uuid:89c4f42f-4482-4e40-97ea-d2bc81d11110>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00616.warc.gz"} |
The Calculator That Doesn't Overflow
Hypercalc is an open-source interpreted calculator program designed to calculate extremely large numbers (such as your phone number raised to the power of the factorial of the Gross world product)
without overflowing.
It stores and manipulates numbers using a level-index format; as such it can go far beyond the limits of bc, dc, MACSYMA/maxima, Mathematica and Maple, all of which use a bigint library. For example,
Hypercalc can tell you whether 128481024 is larger than 888888.
All versions of Hypercalc use an internal representation similar to level-index.
The Perl and JavaScript versions provide command history (input and result substitution, as in Maxima). Other features vary as follows:
│ Features │ Perl Hypercalc │ HyperCalc JavaScript │
│ User-defined variables │ YES │ YES │
│ User-defined functions │ (use BASIC) │ YES │
│ Re-use input and output expressions (command history) │ YES │ YES │
│ Compatible with all hardware │ no │ YES (use a web browser) │
│ Maximum precision │ 300 digits │ 16 digits │
│ Fully programmable │ YES │ no │
│ Uncertainty (example: 100(4)+20(3) = 120(7) ) │ YES │ no │
│ Base-60 input and output (example: 1:20:32 + 5:39 = 1:26:11) │ YES │ no │
The Perl and JavaScript versions are made available under a free (libre) GPL license, but with no warranty or support.
The primary advantage of Hypercalc is that it does not "overflow": for large numbers, its range is far greater than hand-held calculators, calculator apps for phones, numeric libraries like gmp, or
maths software like Mathematica. Here is a brief comparison (more on my floating-point formats page):
│ name │ year │ maximum value │
│ Early scientific calculators (e.g. TI SR-50) │ 1974 │ 9.99×10^99 │
│ IEEE 754 binary64 │ 1985 │ 1.80×10^308 │
│ High-end scientific calculators (e.g. TI-89) │ 1990s │ 9.99×10^999 │
│ PARI/GP │ 1985 │ 4.3×10^2525222 │
│ Mathematica │ 1988 │ 1.44×10^323228010 │
│ Maple │ 1980 │ 1.0×10^2147483646 │
│ GMP library (assuming 'long' is 64 bits) │ 1991 │ 10^1.777×10^20 │
│ Maxima │ 1982 │ ≈ 10^10^10000000000 = 10↑↑4 │
│ Hypercalc │ 1998 │ 10↑↑(10^10) │
I began exploring very large numbers such as 265536 in the early 1970's using a Texas Instruments SR-50 calculator, and had to manually take logarithms, extract fractional parts and compute
mantissas, etc. I made my own BIGNUM library in assembly language for the Apple II, and again on later machines. Such an approach is limited by computer memory (on my Large Numbers page I refer to
this as the class-2 limit).
I always wanted a portable calculator that could do my huge-numbers problems, and the Palm Pilot was the first device that really made this possible. I created the Palm OS HyperCalc in October 1998,
and got it working within about a week.
The screen on my Pilot cracked, and I could see that the platform wouldn't last too long. More importantly, I wanted to be able to copy and paste numbers and results to other files while working on
my web pages. So I created the vastly more powerful Perl version in the summer of 1999. I have maintained and expanded it greatly over the years, adding extended precision (up to 295 digits) later in
1999, the BASIC interpreter late in 2005, base-60 formatting in late 2007, uncertainty calculation in 2011, and so on.
In 2004, Kenny TM~ Chan, at the time a member of the maths club at Yuen Long Merchants Association secondary school (元朗商會中學) in Hong Kong, found Hypercalc and implemented the JavaScript
version. This version is described briefly in its own section below.
The Perl version is the latest and most capable version. The source code is here.
There is extensive built-in help, accessed by typing help at the Hypercalc prompt. After an initial introductory help page, just hit enter repeatedly to see help on ten specific topics.
To use HyperCalc from your web browser, go here: HyperCalc JavaScript. There is a detailed manual in PDF format: HyperCalc JavaScript manual
If you spend a while exploring the ranges of huge numbers HyperCalc can handle, you will probably start noticing some paradoxical results and might even start to think the calculator is giving wrong
For example, try calculating 27 to the power of googolplex (a googolplex is 10 to the power of googol and a googol is 10100). Try:
This is clearly wrong — and it doesn't even seem to be a good approximation. What's going on?
Let's try calculating the correct answer ourselves. We need to express the answer as 10 to the power of 10 to the power of something, because that's the standard format the calculator is using, and
we're going to see how much of an error it made. So, we want to compute
as a "tower" of powers of 10. The first step is express the power of 27 as a power of 10 with a product in the exponent, using the formula xy = 10(log(x) . y) :
Now we have a base of 10 but the exponent still needs work. The next step is to express the product as a sum in the next-higher exponent; this time the formula we use is x . y = 10(log(x) + log(y)) :
log101.43 is about 0.155, and if we add this to 10100 we get
where there are 94 more 0's in place of each of the "...". So our final answer is:
Now that we've expressed the value of 27^googolplex precisely enough to see the calculator's error — and look how small the error is! The calculator would need to have at least 104 digits of
precision to be able to handle the value "1.000...000155" — but it only has 16 digits of precision. Those 16 digits are taken up by the 1 and the first fifteen 0's — so when the calculator gets to
the step where we're adding 0.155 to 1.0.10100, it just rounds off the answer to 1.0×10100 — and produces the answer we saw when we performed the calculation:
The original Palm version of Hypercalc had a calculator-like display, a short, wide rectangle giving enough room to show one line of text with about 30 or 40 characters. Given this limited display
area, even if it did have the necessary 104 digits of precision, it wouldn't have room to print the whole 104 digits on the screen, so the answer displayed would still look the same.
More to the point, no matter how many digits we try to display, there's always going to be another even bigger number that we won't be able to handle. For example, Hypercalc would need slightly over
a million digits of precision to distinguish
and if we just add one more 10 to that tower of exponents, all hope of avoiding roundoff is lost!
For more on this issue, see my discussion of the "power tower paradox", and the Class-3 Numbers and Class-4 Numbers sections of my large numbers pages. | {"url":"https://mrob.com/pub/perl/hypercalc.html","timestamp":"2024-11-07T05:41:38Z","content_type":"text/html","content_length":"20108","record_id":"<urn:uuid:6978c63c-5573-45d6-ba54-976e054a236b>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00490.warc.gz"} |
Math Olympiad Questions & Sample Papers
Maths Olympiad is an examination curated with advanced level mathematics to work on a student's current potential. In such tests, the students are required to solve the given questions. Based on the
problem solving, the student is rewarded with marks and accolades. To support in this particular situation, several coaching centres and websites are there that provide the necessary support.
We offer an ample amount of Math Olympiad Questions and Maths Olympiad Sample Papers for the student to be ready and practice their way to perfection. We also provide the international level of maths
Olympiad questions. For providing extra assistance to prepare for the Olympiads, there is International Maths Olympiad Questions.
This site will play a significant role in ensuring that the student is getting suitably prepared to ace a math Olympiad exam.
For more detailed study of Mathematics, you can also visit our site Kidz Math
Benefits of Math Olympiad
1. Fantastic exposure to the level of complicated mathematics.
2. Great opportunity to test you whether one is ready for the said complicated mathematics.
3. If someone is passionate about mathematics, Olympiads will undoubtedly be an extraordinary chance to stay in touch with your passion.
4. Olympiads can also be an excellent way to prepare for the future if someone is looking ahead to take up mathematics for their higher studies.
5. Olympiads give a chance to get acquainted with your weaknesses and strengths and work on them.
6. Since Olympiads happen quite a few times, around the year, this can help keep those skills under practice.
Math Olympiad Questions By Class
We provide Math Olympiad sample papers for the following classes.
Various Math Olympiad Exams
Following Olympiad Exams are conducted for Math. | {"url":"https://www.practice-olympiad.com/olympiad-questions/math","timestamp":"2024-11-09T04:19:49Z","content_type":"text/html","content_length":"54319","record_id":"<urn:uuid:57d8852a-8ada-4068-8ee5-fb35bf3b72af>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00338.warc.gz"} |
On the Power of Learning from k-Wise Queries (Conference Paper) | NSF PAGES
We give superpolynomial statistical query (SQ) lower bounds for learning two-hidden-layer ReLU networks with respect to Gaussian inputs in the standard (noise-free) model. No general SQ lower bounds
were known for learning ReLU networks of any depth in this setting: previous SQ lower bounds held only for adversarial noise models (agnostic learning) or restricted models such as correlational SQ.
Prior work hinted at the impossibility of our result: Vempala and Wilmes showed that general SQ lower bounds cannot apply to any real-valued family of functions that satisfies a simple non-degeneracy
condition. To circumvent their result, we refine a lifting procedure due to Daniely and Vardi that reduces Boolean PAC learning problems to Gaussian ones. We show how to extend their technique to
other learning models and, in many well-studied cases, obtain a more efficient reduction. As such, we also prove new cryptographic hardness results for PAC learning two-hidden-layer ReLU networks, as
well as new lower bounds for learning constant-depth ReLU networks from label queries.
more » « less | {"url":"https://par.nsf.gov/biblio/10026311","timestamp":"2024-11-07T08:55:17Z","content_type":"text/html","content_length":"244140","record_id":"<urn:uuid:b863cbcc-8970-4038-bafb-6aad5f998c40>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00743.warc.gz"} |
Numerical study on parameter impact on fundamental frequencies and dynamic characteristics of pre-stressed concrete beams
This paper established 6 kinds of different concrete beams, and 5 kinds of models were applied horizontal pre-stress. Their fundamental frequencies and deflections were numerically computed, and the
correctness of numerically computational model was experimentally verified. Fundamental frequencies of 6 kinds of models were 46.23 Hz, 68.45 Hz, 69.36 Hz, 70.46 Hz, 72.11 Hz and 157.73 Hz
respectively. Concrete beams without being applied pre-stress had a low fundamental frequency. The fundamental frequency of concrete beams could be obviously improved through internally applying
pre-stressed tendons to concrete beams. In addition, the improvement effect of Model 5 was more obvious. When pre-stressed tendons were externally applied to the concrete beam, the fundamental
frequency was greatly improved. However, externally pre-stressed tendons would limit the use of concrete beams and increase the cost. When other parameters remained unchanged, the fundamental
frequency of pre-stressed concrete beams would gradually increase with the increase of pre-stress. When the pre-stress was the same, fundamental frequencies gradually increased with the increase of
eccentricity, the number of circular tendons in composite tendons and concrete strength, and gradually decreased with the increase of counterweight. When the applied load was the same, Model 1 was
not strengthened and its deflection was the largest. On the contrary, the model which was applied external pre-stress and its deflection was the smallest, but its use was limited and cost was high.
In the strengthening proposals of internal pre-stress, composite tendons were applied to Model 5, and it presented a small deflection. When other parameters remained unchanged and the applied load
was lower than a value, the deflection of concrete teams gradually increased with the increase of applied load. On the contrary, the deflection of concrete beam tended to be stable with the increase
of applied load. When the applied load was the same, the deflection of concrete beams gradually decreased with the increase of eccentricity, the number of circular tendons in composite tendons and
concrete strength, and gradually increased with the increase of counterweight.
1. Introduction
Pre-stressed concrete beams are an engineering structure which can actively combine high-strength steel with high-strength concrete. With excellent structural performance, pre-stressed concrete
structure is widely applied in civil engineering including housing construction, bridges, water conservancy and so on. A lot of civil engineering has fully proved that pre-stressed concrete structure
is an irreplaceable important structure in contemporary engineering construction. The application of pre-stressed concrete technology not only simply saves steel and concrete, but also solves many
engineering problems. Compared with reinforced concrete structure, pre-stressed concrete structure has many advantages. For example, pre-stressed concrete overcomes the shortcoming of concrete,
namely low resistance to tension, raises the crack resistance and stiffness of components, reduces the development of crack and deformation of components under the load, effectively improves the
service performance of components and strengthens the durability of structures. In the meanwhile, the application of pre-stressed concrete technology solves technical problems other structural
materials find it difficult to tackle. Pre-stressed concrete technology can be applied to construct all kinds of large-scale and long-span civil engineering.
At present, a lot of studies have been conducted on the strengthening of pre-stressed concrete structure [1-6]. Cuenca [7] added self-compacting fiber to concrete beams to provide pre-stress and
experimentally studied the shear behavior of the structure under the force. The design of pre-stressed concrete beams was affected by a number of parameters. Kumar [8] took into account the
combination of various parameters and used genetic algorithm to optimize and design concrete beams. However, the characteristics of optimized and designed concrete beams were not experimentally
verified. Ren [9, 10] applied pre-stress to high-strength spiral tendons, and inserted the carbon fiber tendons without pre-stress and spiral tendons into concrete beams to conduct bending tests.
Results showed that applying pre-stress to spiral tendons could effectively improve the ultimate bearing capacity, stiffness and anti-cracking ability of concrete beams. To reveal the shear
performance of pre-stressed high-strength concrete beams, Yao [11] conducted an experiment on the shear performance of 11 pre-stressed high-strength concrete beams and 4 pre-stressed common concrete
beams and comparatively analyzed the impact of different parameters on the failure type, load-deflection curve, bearing capacity and strain of experimental beams. To analyze the impact of repeated
loads on the deformation performance of pre-stressed concrete beams, Zhao [12] conducted low-frequency and repetitive loading experiments on 6 pre-stressed concrete beams, and studied the deformation
performance of pre-stressed concrete beams with different levels of pre-stress. However, the cited researches failed to involve the fundamental frequency and dynamic characteristics of pre-stressed
concrete beams. He [13] conducted experimental research on the vibration of PRC beam and obtained a conclusion that the natural vibration frequency of PRC beam would be decreased with the increase of
pre-stress when constraint conditions were unchanged. His conclusion was consistent with the conclusion of Abraham [14]. Liu [15] studied the relationship between the fundamental frequency of bending
vibration of T-shaped beam and pre-stress. Experiments showed that the pre-stress had an influence on the vibration frequency of concrete beams. Vibration frequency increased with the pre-stress
value. Abdalli [16] conducted dynamic tests and analysis on pre-stressed concrete notched beams. Results showed that pre-stress had a little impact on the dynamic characteristics of concrete beams.
The impact of pre-stress on the dynamic characteristics of concrete beams was mainly up to the arrangement of pre-stressed cable. If pre-stressed cable was arranged like a parabola, frequency
increased with the increase of pre-stress. If pre-stressed cable was arranged like a straight line, the impact could be neglected. Xia [17] obtained the basic dynamic characteristics of pre-stressed
concrete beams through conducting forced vibration tests and dynamic analysis on of pre-stressed concrete beams, and further studied the impact of the size, shape and location of pre-stressed tendons
on dynamic characteristics.
However, published studies on the fundamental frequency and dynamic characteristics of pre-stressed concrete beams were not systematic and failed to analyze a variety of parameters. Therefore, this
paper systematically studied the fundamental frequency and dynamic characteristics of many kinds of pre-stressed concrete beams, analyzed many kinds of parameters affecting fundamental frequency and
dynamic characteristics, and provided reference for the structural design of prestressed concrete beams.
2. Numerical computation for fundamental frequencies of pre-stressed concrete beams
As shown in Fig. 1, 6 different kinds of pre-stressed concrete beams were designed. Fig. 1(a) did not applied any pre-stressed tendons. Fig. 1(b) to Fig. 1(e) represented that pre-stressed tendons
were applied in concrete beams. Fig. 1(b) represented that circular tendons were applied in the concrete beam. Fig. 1(c) represented that steel plates were applied in the concrete beam. Fig. 1(d)
represented that I-shaped tendons were applied in the concrete beam. Fig. 1(e) represented that circular tendons and I-shaped tendons were together applied in the concrete beam. In Fig. 1(f),
U-shaped pre-stressed tendons were applied outside the concrete beam to strengthen the concrete beam. The concrete beam was 3 m long, 160 mm wide and 300 mm high.
Fig. 1Schematic diagrams for 6 kinds of pre-stressed concrete beams
The establishment of models for pre-stressed concrete beams was described in detail. Concrete adopted Solid65 element; longitudinal tendons and stirrup applied Link8 element; I-shaped tendons used
Solid45 element. Modeling methods were basically the same. Therefore, only one modeling method was taken as an example. The modeling method of other models was similar to it. Both ends were fixed,
and this model was under the action of symmetric loads. Therefore, we only built a half of model for convenience and simple simulation when the model was established, and finally obtained the whole
finite element model through symmetry. A separation model was adopted to make computational results relatively accurate due to there were three kinds of elements. Namely, concrete, circular tendons
and I-shaped tendons were established separately. Firstly, concrete and key points at the end section of I-shaped tendons were created. These key points formed lines and a plane, as shown in Fig. 2.
Finally, the formed plane turned into an object through the command of stretch. Meshes of end section were generated firstly. Then, the whole model was divided through the command of extension. The
size of meshes should be neither too small nor too large. The size of meshes was 20 mm. After meshes were generated, tendon elements were established through connecting nodes after mesh generation.
To display tendons by means of link elements, we showed it through the command of ESHAPE. After loads were applied, load step was set as 100 and iteration times were 100 in order to converge the
model as soon as possible and simplify computation. As this model was controlled by displacement, the convergence criterion was adjusted to the maximum value, namely 5 %. “Solve” was clicked for
solution. Finally, finite element models for 6 kinds of pre-stressed concrete beams were obtained, as shown in Fig. 3. After finite element models for pre-stressed concrete beams were established,
related materials were imported into models. The diameter of circular tendons was 14 mm. According to experimental design, the yield strength of tendons was 400 MPa. The design value of tensile
strength and compressive strength of the concrete was 1.55 MPa and 15.5 MPa respectively, and its Elasticity modulus was 28 GPa.
Fig. 2Detailed section characteristics of model 5
Finite element model was used to compute frequencies of 6 kinds of pre-stressed concrete beams on top 6 orders, as shown in Table 1. As displayed from this table, fundamental frequencies of 6 kinds
of models were 46.23 Hz, 68.45 Hz, 69.36 Hz, 70.46 Hz, 72.11 Hz and 157.73 Hz respectively. Concrete beams without being strengthened had a low fundamental frequency. The fundamental frequency of
concrete beams could be obviously improved through internally applying pre-stressed tendons to concrete beams. In addition, the improvement effect of Model 5 was more obvious. When pre-stressed
tendons were externally applied to the concrete beam, the fundamental frequency was greatly improved.
Table 1Modal frequencies of different pre-stressed concrete beams
Order Model 1 / Hz Model 2 / Hz Model 3 / Hz Model 4 / Hz Model 5 / Hz Model 6 / Hz
1 46.23 68.45 69.36 70.46 72.11 157.73
2 57.91 71.09 72.91 73.29 79.81 171.52
3 134.55 178.75 181.56 185.99 192.90 432.11
4 142.34 189.17 199.70 201.15 205.38 446.41
5 167.56 228.85 231.17 235.06 247.50 566.72
6 192.33 256.98 259.89 274.12 275.70 619.34
However, externally pre-stressed tendons would limit the use of concrete beams and increase the cost. In addition, frequencies of Model 5 on other orders were higher than those of other concrete
beams with internal pre-stress. Numerical computation showed that frequencies of 6 kinds of pre-stressed concrete beams were different, but modes did not present any changes because pre-stressed
tendons would only change the effective stiffness of concrete beams rather than the major structure. As a result, modes of Model 5 on top 6 orders were only extracted, as shown in Fig. 4. Only the
mode of the fifth order presented torsional vibration and other modes presented first-order and second-order bending vibrations.
Fig. 3Finite element models for 6 kinds of pre-stressed concrete beams
3. Impact of different parameters on fundamental frequencies of pre-stressed concrete beams
Fundamental frequencies of different pre-stressed concrete beams were analyzed. Model 5 was the composite pre-stressed tendon with higher fundamental frequency and advantage. Parameters affecting the
fundamental frequency of pre-stressed concrete beams mainly included eccentricity, counterweight, type of composite tendons and concrete strength. Therefore, the finite element model was used to
conduct parameter analysis on Model 5.
3.1. Eccentricity
As shown in Fig. 5, 2 pre-stressed tendons were arranged in the concrete beam. They were 0 mm, 75 mm and 150 mm respectively away from the center, and corresponding situations were recorded as Case
1, Case 2 and Case 3. Aimed at each single situation, pre-stress was changed from 0 kN to 200 kN and step size was 50 kN.
Fig. 4modes of pre-stressed concrete beams on top 6 orders
Fig. 5Pre-stressed concrete beams under different eccentricities
The finite element model was used to compute fundamental frequencies. Fundamental frequencies of pre-stressed concrete beams under three kinds of eccentricities were compared, as shown in Fig. 6. As
displayed from Fig. 6, the fundamental frequency of pre-stressed concrete beams had a little change with the increase of pre-stress when the eccentricity was the same. When the eccentricity was 0 mm
and pre-stress increased from 0 kN to 200 kN, the fundamental frequency of concrete beams increased by 3.89 %. When the eccentricity was 75 mm and pre-stress increased from 0 kN to 200 kN, the
fundamental frequency of concrete beams increased by 3.02 %. When the eccentricity was 0 mm and pre-stress increased from 0 kN to 200 kN, the fundamental frequency of concrete beams increased by 4.24
%. When the pre-stressed value was the same, the fundamental frequency of concrete beams gradually increased with the increase of eccentricity because the increase of eccentricity would increase the
bending stiffness of beams and improve the fundamental frequency of beams. From the perspective of force, the bottom of pre-stressed concrete beams was in tension when externally vertical loads were
applied to pre-stressed concrete beams. Namely, the position where eccentricity was large was in tension. Based on cross-section assumption, the axis of beams was a neutral axis. Therefore, it was
the most reasonable to arrange pre-stressed tendons at the bottom of beams.
Fig. 6Relationship between eccentricity and fundamental frequency
3.2. Counterweight
When bridges are in service, there are usually vehicles passing through. Therefore, it is necessary to consider the impact of counterweight on the fundamental frequency of pre-stressed concrete
beams. As shown in Fig. 7, counterweights including 0 kN, 30 kN and 60 kN were applied at 1/3 point of the beam, and corresponding situations were recorded as Case 1, Case 2 and Case 3. The
eccentricity of pre-stressed tendons was 0 mm. Aimed at each single situation, pre-stress was changed from 0 kN to 200 kN and step size was 50 kN. The finite element model was used to compute
fundamental frequencies. Fundamental frequencies of pre-stressed concrete beams under three kinds of counterweights were compared, as shown in Fig. 8. As displayed from Fig. 8, the fundamental
frequency of pre-stressed concrete beams was increased with the increase of pre-stress when the counterweight was the same. When the counterweight was 0 kN and pre-stress increased from 0 kN to 200
kN, the fundamental frequency of concrete beams increased by 4.37 %. When the counterweight was 30 kN and pre-stress increased from 0 kN to 200 kN, the fundamental frequency of concrete beams
increased by 3.89 %. When the counterweight was 60 kN and pre-stress increased from 0 kN to 200 kN, the fundamental frequency of concrete beams increased by 3.36 %. When the pre-stress value was the
same, the fundamental frequency of concrete beams decreased with the increase of counterweight. Take the pre-stress value 200 kN as an example. When the counterweight increased from 0 kN to 60 kN,
fundamental frequencies decreased from 76.4 Hz to 73.9 Hz. This phenomenon could also be explained by the increase of distributed mass of beams. Namely, the concentrated loads were equivalent to the
increase of distributed mass of beams. When the beam was in vibration, loads did not disappear, but moved all the time, which would thus affect the stiffness matrix of beams in the process of
vibration and influence the fundamental frequency of beams.
Fig. 7Pre-stressed concrete beams under different counterweights
Fig. 8Relationship between counterweight and fundamental frequency
3.3. Type of composite tendons
As shown in Fig. 9, the combination types of circular tendons and I-shaped tendons were changed and recorded as Case 1, Case 2 and Case 3. The eccentricity of pre-stressed tendons was 0 mm. Its
counterweight was 30 kN. Aimed at each single situation, pre-stress was changed from 0 kN to 200 kN and step size was 50 kN. The finite element model was used to compute corresponding fundamental
frequencies. Fundamental frequencies of pre-stressed concrete beams under three kinds of composite tendons were compared, as shown in Fig. 9.
Fig. 9Different types of composite tendons
As displayed from Fig. 9, the fundamental frequency of pre-stressed concrete beams had a little change with the increase of pre-stress when the type of composite tendons was the same. When the
situation was Case 1 and pre-stress increased from 0 kN to 200 kN, the fundamental frequency of concrete beams increased by 3.89 %. When the situation was Case 2 and pre-stress increased from 0 kN to
200 kN, the fundamental frequency of concrete beams increased by 3.25 %. When the situation was Case 3 and pre-stress increased from 0 kN to 200 kN, the fundamental frequency of concrete beams
increased by 3.61 %. When the pre-stress value was the same, the fundamental frequency of concrete beams was increased with the type of composite tendons. Take the pre-stress value 200 kN as an
example. When the situation was changed from Case 1 to Case 3, fundamental frequencies increased from 74.9 Hz to 77.4 Hz. When the pre-stress value was the same, the application of many circular
tendons would increase the bending stiffness of concrete beams and thus improve the fundamental frequency of beams.
Fig. 10Relationship between types of composite tendons and fundamental frequency
3.4. Concrete strength
The concrete strength of studied beams in the previous was C15. The grade of concrete strength was changed into C15, C25 and C35, and corresponding situations were set as Case 1, Case 2 and Case 3.
The eccentricity of pre-stressed tendons was 0 mm. Its counterweight was 30 kN. Aimed at each single situation, pre-stress was changed from 0 kN to 200 kN and step size was 50 kN. The finite element
model was used to compute corresponding fundamental frequencies. Fundamental frequencies of pre-stressed concrete beams under three kinds of concrete strength were compared, as shown in Fig. 11.
Fig. 11Relationship between concrete strength and fundamental frequency
As displayed from Fig. 11, the fundamental frequency of pre-stressed concrete beams had a little change with the increase of pre-stress when the concrete strength was the same. When the concrete
strength was C15 and pre-stress increased from 0 kN to 200 kN, the fundamental frequency of concrete beams increased by 3.89 %. When the concrete strength was C25 and pre-stress increased from 0 kN
to 200 kN, the fundamental frequency of concrete beams increased by 4.18 %. When the concrete strength was C35 and pre-stress increased from 0 kN to 200 kN, the fundamental frequency of concrete
beams increased by 3.52 %. When the pre-stress value was the same, the fundamental frequency of concrete beams was increased with the concrete strength. Take the pre-stress value 200 kN as an
example. When the concrete strength was changed from C15 to C35, fundamental frequencies increased from 74.9 Hz to 79.4 Hz. When the pre-stress value was the same, the higher the grade of concrete
strength was, the greater the stiffness of concrete would be.
Fig. 12Strains of 6 kinds of pre-stressed concrete beams
4. Numerical computation and experimental verification for the dynamic characteristics of pre-stressed concrete beams
Numerical computation was conducted on the fundamental frequency information of pre-stressed concrete beams, but dynamic characteristics of the structure could not be reflected. Therefore, finite
element model was also used to compute dynamic characteristics including strain and deflection of beams, and the computational results were then compared with experimental results to verify the
correctness of computational models.
4.1. Numerical computation for dynamic characteristics of pre-stressed concrete beams
Strains and deflections of 6 kinds of models were extracted, as shown in Fig. 12 and Fig. 13. It could be seen that 6 kinds of models had similar strains and large strains were always presented at
the fixed ends and the middle position of beams. Model 1 did not take any strengthening measures. When the load was the same, the deflection of Model 1 was the largest. On the contrary, the
deflection of Model 6 was the smallest because Model 6 was externally strengthened with pre-stress. As for internal strengthening measures, the deflection of Model 5 was the smallest. In the linear
phase, the stiffness of beams had a little change before and after strengthening, which indicated that strengthening measures did not play an obvious role in the elastic phase. When the deformation
of concrete beams was large, the stiffness of strengthened beams decreased little. However, the stiffness of un-strengthened beams decreased greatly. As a result, the stiffness of strengthened beams
was higher than that of un-strengthened beams when pre-stressed tendons yielded. It was mainly because the strain of pre-stressed tendons in pre-stressed concrete beams was small when the deformation
was small. Therefore, distributed stress was also small, had weak effect on restricting the deformation of beams and contributed little to improving the stiffness of beams. When the deformation of
concrete beams was large and especially when concrete beams cracked, stress was relieved at the crack part of concrete beams. Most of stress was assigned to pre-stressed tendons. At this time, the
stress of pre-stressed tendons increased rapidly to powerfully restrain the deformation of strengthened beams, reduce the deflection of beams and improve the bending stiffness. For a low-strength
concrete beam, the bearing capacity of beams cannot be improved obviously without any strengthening measures. The stiffness of externally strengthened beams was improved more obviously than that of
externally strengthened beams, but external strengthening will limit the use of beams. Therefore, internal strengthening was usually chosen for concrete beams. Model 5 adopted the type of composite
tendons, which achieved a better results.
Fig. 13Deflections of 6 kinds of pre-stressed concrete beams
4.2. Experimental verification for numerically computational model
Numerical computation model was relatively complex. Therefore, it was necessary to verify it through experiments. According to numerically computational model, a calibrated oil jack was used for
loading. To better simulate the force characteristics of simply supported beam under concentrated loads, thick steel plates were placed in the position of support and action of concentrated loads in
order to avoid the local failure of concrete because of excessive stress. Experimental loading device was shown in Fig. 14. Sensors were arranged at the mid-span of concrete beams to test the
deflection of concrete beams when it was loaded. Experiments adopted dynamic loading process. Two points of experimental beam were intensively loaded through primary distribution beam. Concrete beam
was loaded at the load step of 2 kN before its crack and stopped loading when the load for the theoretical crack of experimental beam was reached. After each loading, the value presented by the
instrument kept stable to record data. Experimental deflection was compared with numerical computation, as shown in Fig. 15.
Fig. 14Experimental deflection device of pre-stressed concrete beams
It could be seen from the Fig. 15 that experimental results were well consistent with numerically computational results. For example: The biggest difference between experiment and simulation in 6
kinds was 3.2 kN, 1.8 kN, 1.6 kN, 1.0 kN, 0.9 kN and 0.8 kN. Numerically computational model could replace experiments to complete related analysis and research. However, the stiffness of finite
element model was higher than that of experimental beam under the action of the same load at each stage of computational process. As a result, the deflection was lower than experimental value in the
end. Reasons were mainly as follows: 1) Before the action of loads, concrete in reality had micro-cracks caused by the dry-shrinkage strain of cement stone. After the action of loads, these internal
micro-cracks started to stretch and extend and formed large cracks, which would reduce the stiffness of beams. However, finite element model did not contain micro-cracks, which caused that its
stiffness was greater than experimental value. 2) In finite element analysis, suppose that tendons, steel plates and concrete were bonded finely without relative slippage. In the actual situation,
the phenomenon of relative slippage inevitably existed between them. This phenomenon was more serious at later stage, which increased the gap between the stiffness of experimental beam and the
structure of the finite element.
5. Impact of different parameters on dynamic characteristics of pre-stressed concrete beams
Through referring to analysis in Section 3, parameters affecting the fundamental frequency of pre-stressed concrete beams mainly included eccentricity, counterweight, type of composite tendons and
concrete strength. Similarly, these parameters had an obvious impact on dynamic characteristics of concrete beams. Therefore, the verified finite element model was used to study them.
5.1. Eccentricity
As shown in Fig. 5, 2 pre-stressed tendons were arranged in the concrete beam. They were 0 mm, 75 mm and 150 mm respectively away from the center, and corresponding situations were recorded as Case
1, Case 2 and Case 3. Aimed at each single situation, the finite element model was used to compute the deflection. The deflection of pre-stressed concrete beams under three kinds of eccentricities
was compared, as shown in Fig. 16. As displayed from Fig. 16, the deflection of concrete beams increased with the increase of loads when the eccentricity was the same and load was less than 100 kN.
However, the deflection of concrete beams basically did not change with the load when load was more than 100 kN. It was mainly because pre-stressed tendons in concrete beams played an important role
when the deflection reached a certain value. As a result, the stiffness of concrete beams restored to a high value, which basically kept deflection unchanged. However, the deflection of concrete
beams gradually decreased with the increase of the eccentricity when the applied load was the same. When the pre-stressed value was the same, the deflection of concrete beams gradually decreased with
the increase of the eccentricity because the increase of the eccentricity would increase the bending stiffness of beams. From the perspective of force, the bottom of pre-stressed concrete beams was
in tension when externally vertical loads were applied to pre-stressed concrete beams. Namely, the position where eccentricity was large was in tension. Based on cross-section assumption, the axis of
beams was a neutral axis. Therefore, it was the most reasonable to arrange pre-stressed tendons at the bottom of beams.
Fig. 15Comparison of deflections of 6 kinds of models between experiment and simulation
5.2. Counterweight
When bridges are in service, there are usually vehicles passing through. Therefore, it is necessary to consider the impact of counterweight on the deflection of pre-stressed concrete beams. As shown
in Fig. 7, counterweights including 0 kN, 30 kN and 60 kN were applied at 1/3 point of the beam, and corresponding situations were recorded as Case 1, Case 2 and Case 3. The eccentricity of
pre-stressed tendons was 0 mm. Aimed at each single situation, the finite element model was used to compute the deflection. The deflection of pre-stressed concrete beams under three kinds of
counterweights was compared, as shown in Fig. 17. As displayed from Fig. 17, the deflection of concrete beams increased with the increase of loads when the counterweight was the same and load was
less than 100 kN. However, the deflection of concrete beams basically did not change with the load when load was more than 100 kN. It was mainly because pre-stressed tendons in concrete beams played
an important role when the deflection reached a certain value. As a result, the stiffness of concrete beams restored to a high value, which basically kept deflection unchanged. However, the
deflection of concrete beams gradually increased with the increase of the counterweight when the applied load was the same. This phenomenon could also be explained by the increase of distributed mass
of beams. Namely, the concentrated loads were equivalent to the increase of distributed mass of beams. When the beam was in vibration, loads did not disappear, but moved all the time, which would
thus affect the stiffness matrix of beams in the process of vibration and influence the deflection of beams.
Fig. 16Relationship between eccentricity and deflection
Fig. 17Relationship between counterweight and deflection
5.3. Type of composite tendons
As shown in Fig. 9, the combination types of circular tendons and I-shaped tendons were changed and recorded as Case 1, Case 2 and Case 3. The eccentricity of pre-stressed tendons was 0 mm. Its
counterweight was 30 kN. Aimed at each single situation, the finite element model was used to compute corresponding deflection. The deflection of pre-stressed concrete beams under three kinds of
composite tendons was compared, as shown in Fig. 18. As displayed from Fig. 18, the deflection of concrete beams increased with the increase of loads when the composite tendon was the same and load
was less than 100 kN. However, the deflection of concrete beams basically did not change with the load when load was more than 100 kN. It was mainly because pre-stressed tendons in concrete beams
played an important role when the deflection reached a certain value. As a result, the stiffness of concrete beams restored to a high value, which basically kept deflection unchanged. However, the
deflection of concrete beams gradually decreased with the composite tendons when the applied load was the same. Take the load 80 kN as an example. When the situation was changed from Case 1 to Case
3, the deflection was 9.1 mm, 7.6 mm and 5.9 mm respectively. When the load was the same, the application of many circular tendons would increase the bending stiffness of concrete beams and thus
reduce the deflection of beams.
Fig. 18Relationship between type of composite tendons and deflection
5.4. Concrete strength
The concrete strength of studied beams in the previous was C15. The grade of concrete strength was changed into C15, C25 and C35, and corresponding situations were set as Case 1, Case 2 and Case 3.
Aimed at each single situation, the finite element model was used to compute corresponding deflection. The deflection of pre-stressed concrete beams under different concrete strength was compared, as
shown in Fig. 19.
Fig. 19Relationship between concrete strength and deflection
Different from the other parameters, the improvement effect on the concrete strength was the most obvious. The deflection of concrete beams increased with the increase of loads when the concrete
strength was C15 and load was less than 100 kN. However, the deflection of concrete beams basically did not change with the load when load was more than 100 kN. The deflection of concrete beams
increased with the increase of loads when the concrete strength was C25 and load was less than 114 kN. However, the deflection of concrete beams basically did not change with the load when load was
more than 114 kN. The deflection of concrete beams increased with the increase of loads when the concrete strength was C35 and load was less than 120 kN. However, the deflection of concrete beams
basically did not change with the load when load was more than 120 kN. It was mainly because pre-stressed tendons in concrete beams played an important role when the deflection reached a certain
value. As a result, the stiffness of concrete beams restored to a high value, which basically kept deflection unchanged. However, the deflection of concrete beams gradually decreased with the
concrete strength when the applied load was the same. When the load was the same, the higher the grade of concrete strength was, the greater the stiffness of concrete would be.
6. Conclusions
This paper established 6 kinds of different concrete beams, numerically computed corresponding fundamental frequencies and deflections, and experimentally verified the correctness of numerical
computation model. Finally, the following conclusions can be achieved:
1) Fundamental frequencies of 6 kinds of models were 46.23 Hz, 68.45 Hz, 69.36 Hz, 70.46 Hz, 72.11 Hz and 157.73 Hz respectively. Concrete beams without being strengthened had a low fundamental
frequency. The fundamental frequency of concrete beams could be obviously improved through internally applying pre-stressed tendons to concrete beams. In addition, the improvement effect of Model 5
was more obvious. When pre-stressed tendons were externally applied to the concrete beam, the fundamental frequency was greatly improved. However, externally pre-stressed tendons would limit the use
of concrete beams and increase the cost.
2) When the eccentricity, counterweight, type of composite tendons and concrete strength of concrete beams remained unchanged, the fundamental frequency of prestressed concrete beam gradually
increased with the increase of prestress. Under the same prestress, the fundamental frequency of concrete beam gradually increased with the increase of eccentricity, the number of circular tendons in
composite tendons and concrete strength and gradually decreased with the increase of counterweight.
3) Regarding 6 kinds of different concrete beam structures, large strains always appeared at the fixed ends of and in the middle position of concrete beams. Under the action of the same load, Model 1
did not take strengthening measures and deflection was the largest. On the contrary, Model 6 was externally strengthened with prestress and deflection was the smallest. However, it was restricted in
engineering application and cost was high. In internal strengthening schemes, Model 5 internally applied composite tendons to the beam, presenting the smallest deflection.
4) Due to the defects of experimental beam in the process of production, the stiffness of finite element model was higher than that of experimental beam. In the end, deflection value was smaller than
experimental value. However, experimental results were well consistent with numerical computation results. Numerical computation model could replace experiments to complete related analysis and
5) When the eccentricity, counterweight, type of composite tendons and concrete strength of concrete beams remained unchanged and the applied load was lower than a value, the deflection of concrete
teams gradually increased with the increase of applied load. On the contrary, the deflection of concrete beam tended to be stable with the increase of applied load. When the applied load was the
same, the deflection of concrete beams gradually decreased with the increase of eccentricity, the number of circular tendons in composite tendons and concrete strength, and gradually increased with
the increase of counterweight.
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About this article
Mechanical vibrations and applications
pre-stressed concrete beams
fundamental frequencies
dynamic characteristics
number of circular tendons
concrete strength
Copyright © 2017 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
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Near-wall turbulence modulation by small inertial particles
We use interface-resolved simulations to study near-wall turbulence modulation by small inertial particles, much denser than the fluid, in dilute/semi-dilute conditions. We considered three bulk
solid mass fractions, and, with only the latter two showing turbulence modulation. The increase of the drag is strong at, but mild in the densest case. Two distinct regimes of turbulence modulation
emerge: for smaller mass fractions, the turbulence statistics are weakly affected and the near-wall particle accumulation increases the drag so the flow appears as a single-phase flow at slightly
higher Reynolds number. Conversely, at higher mass fractions, the particles modulate the turbulent dynamics over the entire flow, and the interphase coupling becomes more complex. In this case, fluid
Reynolds stresses are attenuated, but the inertial particle dynamics near the wall increases the drag via correlated velocity fluctuations, leading to an overall drag increase. Hence, we conclude
that, although particles at high mass fractions reduce the fluid turbulent drag, the solid phase inertial dynamics still increases the overall drag. However, inspection of the streamwise momentum
budget in the two-way coupling limit of vanishing volume fraction, but finite mass fraction, indicates that this trend could reverse at even higher particle load.
Dive into the research topics of 'Near-wall turbulence modulation by small inertial particles'. Together they form a unique fingerprint. | {"url":"https://research.tudelft.nl/en/publications/near-wall-turbulence-modulation-by-small-inertial-particles","timestamp":"2024-11-04T01:56:04Z","content_type":"text/html","content_length":"51669","record_id":"<urn:uuid:6c94f049-4bd0-47fa-a9de-6a47f04b8292>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00463.warc.gz"} |
A Simplified Ventricular Myocyte ModelModel StatusModel Structurecell diagram
Penny Noble Oxford University This CellML model runs in OpenCell and COR to reproduce the published results. This particular version of the CellML model has had a stimulus protocol added to allow it
to simulate trains of action potentials. The parameter values used in this variant (BR) of the Fenton-Karma model are consistent with the original Beeler-Reuter model (see Table 1 of the 1998 model
errata). Simulations of this CellML model can be run using CMISS. ABSTRACT: Wave propagation in ventricular muscle is rendered highly anisotropic by the intramural rotation of the fiber. This
rotational anisotropy is especially important because it can produce a twist of electrical vortices, which measures the rate of rotation (in degree/mm) of activation wavefronts in successive planes
perpendicular to a line of phase singularity, or filament. This twist can then significantly alter the dynamics of the filament. This paper explores this dynamics via numerical simulation. After a
review of the literature, we present modeling tools that include: (i) a simplified ionic model with three membrane currents that approximates well the restitution properties and spiral wave behavior
of more complex ionic models of cardiac action potential (Beeler-Reuter and others), and (ii) a semi-implicit algorithm for the fast solution of monodomain cable equations with rotational anisotropy.
We then discuss selected results of a simulation study of vortex dynamics in a parallelepipedal slab of ventricular muscle of varying wall thickness (S) and fiber rotation rate (theta(z)). The main
finding is that rotational anisotropy generates a sufficiently large twist to destabilize a single transmural filament and cause a transition to a wave turbulent state characterized by a high density
of chaotically moving filaments. This instability is manifested by the propagation of localized disturbances along the filament and has no previously known analog in isotropic excitable media. These
disturbances correspond to highly twisted and distorted regions of filament, or "twistons," that create vortex rings when colliding with the natural boundaries of the ventricle. Moreover, when
sufficiently twisted, these rings expand and create additional filaments by further colliding with boundaries. This instability mechanism is distinct from the commonly invoked patchy failure or wave
breakup that is not observed here during the initial instability. For modified Beeler-Reuter-like kinetics with stable reentry in two dimensions, decay into turbulence occurs in the left ventricle in
about one second above a critical wall thickness in the range of 4-6 mm that matches experiment. However this decay is suppressed by uniformly decreasing excitability. Specific experiments to test
these results, and a method to characterize the filament density during fibrillation are discussed. Results are contrasted with other mechanisms of fibrillation and future prospects are summarized.
The original paper reference is cited below: Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation, Flavio Fenton and Alain Karma,
1998, Chaos, 8, 20-47. PubMed ID: 12779708 A schematic diagram of the three ionic currents described by the Fenton-Karma model of a ventricular myocyte. | {"url":"https://models.cellml.org/workspace/fenton_karma_1998/@@rawfile/377e0aa8bc3933a1d0e2e2a60261e69d805cbe18/fenton_karma_1998_BR.cellml","timestamp":"2024-11-10T05:00:42Z","content_type":"application/cellml+xml","content_length":"39416","record_id":"<urn:uuid:e954820a-e94b-418f-9d14-d0dc7e28191f>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00662.warc.gz"} |
Professor, Department of Astronomy and Astrophysics
University of Chicago
Tensor fluctuations are transverse-traceless perturbations to the metric, which can be viewed as gravitational waves. A plane gravitational wave perturbation represents a quadrupolar ``stretching''
of space in the plane of the perturbation (see Fig. 7). As the wave passes or its amplitude changes, a circle of test particles in the plane is distorted into an ellipse whose semi-major axis 7,
yellow ellipses). Heuristically, the accompanying stretching of the wavelength of photons produces a quadrupolar temperature variation with an
in the coordinates defined by
Fig. 7: The tensor quadrupole moment (l=2, m=2). Since gravity waves distort space in the plane of the perturbation, changing a circle of test particles into an ellipse, the radiation acquires an m=2
quadrupole moment.
Thomson scattering again produces a polarization pattern from the quadrupole anisotropy. At the equator, the quadrupole pattern intersects the tangent ( Q with a
is shown in Fig. 8 (real part). Note that Q and U are present in nearly equal amounts for the tensors.
Fig. 8: Polarization pattern for l=2, m=2. Scattering of a tensor perturbation generates the E pattern (yellow, thick lines) as opposed to the B (purple, thin lines) pattern.
Animation: Same as for scalars.
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Interpreting the coefficients of linear regressionData Science News
Interpreting the coefficients of linear regression
Source: UnsplashInterpreting the coefficients of linear regressionEryk LewinsonBlockedUnblockFollowFollowingJan 13Nowadays there is a plethora of machine learning algorithms we can try out to find
the best fit for our particular problem.
Some of the algorithms have clear interpretation, other work as a blackbox and we can use approaches such as LIME or SHAP to derive some interpretations.
In this article I would like to focus on interpretation of coefficients of the most basic regression model, namely linear regression, including the situations when dependent/independent variables
have been transformed (in this case I am talking about log transformation).
level-level modelBasic for of linear regression (without the residuals)I assume the reader is familiar with linear regression (if not there is a lot of good articles and Medium posts), so I will
focus solely on interpretation of the coefficients.
The basic formula for linear regression can be seen above (I omitted the residuals on purpose, to keep things simple and to the point).
In the formula y denotes the dependent variable and x is the independent variable.
For simplicity let’s assume that it is univariate regression, but the principles obviously hold for the multivariate case as well.
To put it into perspective, let’s say that after fitting the model we receive:Intercept (a)I will break down the interpretation of the intercept into two cases:x is continuous and centered (by
subtracting mean of x from each observation, the average of transformed x becomes 0)—average y is 3 when x is equal to the sample meanx is continuous, but not centered—average y is 3 when x = 0x
is categorical—average y is 3 when x = 0 (this time indicating a category, more on this below)Coefficient (b)x is a continuous variableInterpretation: a unit increase in x results in an increase in
average y by 5 units, all other variables held constant.
x is a categorical variableThis requires a bit more explanation.
Let’s say that x describes gender and can take values (‘male’, ‘female’).
Now let’s convert it into a dummy variable which takes values 0 for males and 1 for females.
Interpretation: average y is higher by 5 units for females than for males, all other variables held constant.
log-level modelLog denotes the natural logarithmTypically we use log transformation to pull outlying data from a positively skewed distribution closer to the bulk of the data, in order to make the
variable normally distributed.
In case of linear regression, one additional benefit of using the log transformation is interpretability.
Example of log transformation: right—before, left—after.
SourceAs before, let’s say that the formula below presents the coefficients of the fitted model.
Intercept (a)Interpretation is similar as in the vanilla (level-level) case, however, we need to take exponent of the intercept for interpretation exp(3) = 20.
The difference is that this value stands for the geometric mean of y (as opposed to the arithmetic mean in case of level-level model).
Coefficient (b)The principles are again similar to the level-level model, when it comes to interpreting categorical/numeric variables.
Analogically to the intercept, we need to take the exponent of the coefficient: exp(b) = exp(0.
01) = 1.
This means that a unit increase in x causes a 1% increase in average (geometric) y, all other variables held constant.
Two things worth mentioning here:There is a rule of thumb when it comes to interpreting coefficients of such model.
If abs(b) < 0.
15 it is quite safe to say that when b = 0.
1 we will observe a 10% increase in y for a unit change in x.
For coefficients with larger absolute value, it is recommended to calculate the exponent.
When dealing with variables in [0, 1] range (like percentage) it is more convenient for interpretation to first multiply the variable by 100 and then fit the model.
This way the interpretation is more intuitive, as we increase the variable by 1 percentage point instead of 100 percentage points (from 0 to 1 immediately).
level-log modelLet’s assume that after fitting the model we receive:The interpretation of the intercept is the same as in case of the level-level model.
For the coefficient b—a 1% increase in x results in an approximate increase in average y by b/100 (0.
05 in this case), all other variables held constant.
To get the exact amount, we would need to take b× log(1.
01), which in this case gives 0.
log-log modelLet’s assume that after fitting the model we receive:Once again I focus on interpretation of b.
An increase in x by 1% results in 5% increase in average (geometric) y, all other variables held constant.
To obtain the exact amount, we need to takewhich is ~5.
ConclusionsI hope this article has given you an overview of how to interpret coefficients of linear regression, including the cases when some of the variables have been log transformed.
In case you have any comments or feedback, please let me know!Referenceshttps://stats.
You must be logged in to post a comment. | {"url":"http://datascience.sharerecipe.net/2019/01/14/interpreting-the-coefficients-of-linear-regression/","timestamp":"2024-11-10T03:21:49Z","content_type":"text/html","content_length":"35218","record_id":"<urn:uuid:866c5ac4-3f05-4764-a7bc-9363326aa52e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00210.warc.gz"} |
Modelling complex spectral data with the resemble package
Think Globally, Fit Locally (Saul and Roweis 2003)
Modeling spectral data has garnered wide interest in the last four decades. Spectroscopy is the study of the spectral response of a matrix (e.g. soil, plant material, seeds, etc.) when it interacts
with electromagnetic radiation. This spectral response directly or indirectly relates to a wide range of compositional characteristics (chemical, physical or biological) of the matrix. Therefore, it
is possible to develop empirical models that can accurately quantify properties of different matrices. In this respect, quantitative spectroscopy techniques are usually fast, non-destructive and
cost-efficient in comparison to conventional laboratory methods used in the analyses of such matrices. This has resulted in the development of comprehensive spectral databases for several
agricultural products comprising large amounts of observations. The size of such databases increases de facto their complexity. To analyze large and complex spectral data, one must then resort to
numerical and statistical tools and methods such as dimensionality reduction, and local spectroscopic modeling based on spectral dissimilarity concepts.
The aim of the resemble package is to provide tools to efficiently and accurately extract meaningful quantitative information from large and complex spectral databases. The core functionalities of
the package include:
• dimensionality reduction
• computation of dissimilarity measures
• evaluation of dissimilarity matrices
• spectral neighbour search
• fitting and predicting local spectroscopic models
Citing the package
Simply type and you will get the info you need:
citation(package = "resemble")
## To cite resemble in publications use:
## Ramirez-Lopez, L., and Stevens, A., and Viscarra Rossel, R., and
## Shen, Z., and Wadoux, A., and Breure, T. (2024). resemble: Regression
## and similarity evaluation for memory-based learning in spectral
## chemometrics. R package Vignette R package version 2.2.3.
## A BibTeX entry for LaTeX users is
## @Manual{resemble-package,
## title = {resemble: Regression and similarity evaluation for memory-based learning in spectral chemometrics. },
## author = {Leonardo Ramirez-Lopez and Antoine Stevens and Claudio Orellano and Raphael {Viscarra Rossel} and Zefang Shen and Alex Wadoux and Timo Breure},
## publication = {R package Vignette},
## year = {2024},
## note = {R package version 2.2.3},
## url = {https://CRAN.R-project.org/package=resemble},
## }
Example dataset
This vignette uses the soil Near-Infrared (NIR) spectral dataset provided in the package prospectr package (Stevens and Ramirez-Lopez 2024). The reason why we use this dataset is because soils are
one of the most complex matrices analyzed with NIR spectroscopy. This spectral dataset/library was used in the challenge by Pierna and Dardenne (2008). The library contains NIR absorbance spectra of
dried and sieved 825 soil observations/samples. These samples originate from agricultural fields collected from all over the Walloon region in Belgium. The data are in an R data.frame object which is
organized as follows:
• Response variables:
□ Nt (Total Nitrogen in g/kg of dry soil): a numerical variable (values are available for 645 samples and missing for 180 samples).
□ Ciso (Carbon in g/100 g of dry soil): a numerical variable (values are available for 732 and missing for 93 samples).
□ CEC (Cation Exchange Capacity in meq/100 g of dry soil): A numerical variable (values are available for 447 and missing for 378 samples).
• Predictor variables: the predictor variables are in a matrix embedded in the data frame, which can be accessed via NIRsoil$spc. These variables contain the NIR absorbance spectra of the samples
recorded between the 1100 nm and 2498 nm of the electromagnetic spectrum at 2 nm interval. Each column name in the matrix of spectra represents a specific wavelength (in nm).
• Set: a binary variable that indicates whether the samples belong to the training subset (represented by 1, 618 samples) or to the test subset (represented by 0, 207 samples).
Load the necessary packages and data:
The dataset can be loaded into R as follows:
Spectra pre-processing
This step aims at improving the signal quality of the spectra for quantitative analysis. In this respect, the following standard methods are applied using the package prospectr (Stevens and
Ramirez-Lopez 2024):
1. Resampling from a resolution of 2 nm to a resolution of 5 nm.
2. First derivative using Savitsky-Golay filtering (Savitzky and Golay 1964).
# obtain a numeric vector of the wavelengths at which spectra is recorded
wavs <- NIRsoil$spc %>% colnames() %>% as.numeric()
# pre-process the spectra:
# - resample it to a resolution of 6 nm
# - use first order derivative
new_res <- 5
poly_order <- 1
window <- 5
diff_order <- 1
NIRsoil$spc_p <- NIRsoil$spc %>%
resample(wav = wavs, new.wav = seq(min(wavs), max(wavs), by = new_res)) %>%
savitzkyGolay(p = poly_order, w = window, m = diff_order) | {"url":"https://cran-r.c3sl.ufpr.br/web/packages/resemble/vignettes/resemble.html","timestamp":"2024-11-08T20:40:04Z","content_type":"text/html","content_length":"1048850","record_id":"<urn:uuid:f2606444-a6b9-40a1-8e34-2c43f94bf175>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00111.warc.gz"} |
XRF Assay Analysis on Circuit Concentrates
2020-01-01: Updated libraries and graph backgrounds
In this examples, I was undertaking Exploratory Data Analysis (EDA) on a data set generated by a portable Olympus X-Ray Fluorescence (XRF) analyser. The CSV generated was in a wide format with a
single row per observation, columns for assays and confidence intervals for each element.
pacman::p_load(tidyverse, ggthemr, ggrepel, hrbrthemes, tidyr)
df = read_csv("data/01 data.csv") %>%
select(-contains("+/-")) %>%
select(Date:`Elapsed Time Total`,
`Live Time 1`:`Field 12`,
everything()) %>%
select(-contains("Field")) %>%
mutate(Reading = str_remove(Reading,"\\#")) %>%
separate(Reading,into = c("run","duplicate"), remove = F, sep="-") %>%
filter(LocationID == "Concentrate") %>%
filter(!is.na(duplicate)) %>%
na_cols = names(df)[map_lgl(df, ~all(is.na(.)))]
df_tidy = df %>%
select(-one_of(na_cols)) %>%
mutate_at(vars(P:U), as.character) %>%
pivot_longer(names_to = "element", values_to = "assay", -(Date:comment)) %>%
mutate(assay = as.numeric(assay)) %>%
The above code makes use of dplyr’s pipe function $\%>\% to pass the output of a function into the next function’s inputs which is similar to the concept of chaining functions in excel by linking
cells. The functions are named well, and the use of verbs aids this self-documenting process, i.e. the above code performs the following:
• Reads a CSV
• Selects columns that don’t contain $.var$ in their name
• Rearranges the table by selecting the columns between $Datetime$ and $Elapsed Time Total$, then $Live Time 1$ and $Field 12$ then everything else
• Selects columns that don’t contain $Field$ in their name
• Creates two new columns by extracting digits out of a compound string
• Filter the results down to only Concentrate samples
• Exclude results that don’t have duplicates
• Remove the compound string columns
The second section transforms the data from a wide format to a narrow or ‘tidy’ format ready for analysis.
Below is some templating code to format the graphs for this post. I’ll assign them to a variable to avoid repetition.
dark_theme = theme_ipsum_rc() +
panel.background = element_rect(fill = "black"),
plot.background = element_rect(fill = "black"),
legend.background = element_rect(fill = "black"),
text = element_text(colour = "white"),
axis.text = element_text(colour = "white"),
panel.grid.major.x = element_line(colour = rgb(40, 40, 40, maxColorValue = 255)),
panel.grid.major.y = element_line(colour = rgb(40, 40, 40, maxColorValue = 255))
) +
Once this preparation is complete, we are ready to start visualising the data, we pipe the tidy data into ggplot and ‘add’ geoms, or layers. Note that we specify the abscissa and ordinate axes
explicitly, what factor to group the graphics by and what factor to colour them by and only then do we tell ggplot to add visualisation layers, in this case a boxplot. We then scale the y axis by log
10, flip the plot 90° and set the labels
df_tidy %>%
x = reorder(element, assay, mean),
y = assay,
fill = reorder(element, assay, mean)
)) +
geom_boxplot(colour = rgb(100,100,100, maxColorValue = 255)) +
scale_y_log10(breaks = 10 ^ (0:6),
labels = scales::comma_format()) +
coord_flip() +
no_legend() +
labs(title = "Distribution of Concentrations",
x = "Element",
y = "Assay (ppm)")+
# Save plots for this post
"plot/01 Distributions.png",
width = 5,
height = 10,
units = "cm",
dpi = 320,
scale = 3
The above graph gives a good high level indication of the distribution of elements and the spread of them, we can already see that Ti, Cu and Ni have wide distributions while Ca, As and Zn have
outliers. These results are taken from Flotation and Gravity Circuit Concentrates so a good follow up question would be if there are there any differences between the groups? In R, it’s trivial to
break the analysis into subgroups. In the case, colour points based on their group.
df_tidy %>%
source = if_else(comment == "float con", 'Flotation', 'Gravity'),
source = if_else(is.na(comment), 'Gravity', source)
) %>%
x = reorder(element, assay, mean),
y = assay,
colour = source
)) +
scale_colour_viridis_d(begin = 0.3)+
geom_point(alpha = 0.5) +
scale_y_log10(breaks = 10 ^ (0:6),
labels = scales::comma_format()) +
coord_flip() +
legend_bottom() +
no_legend_title() +
labs(title = "Distributions by Feed",
x = "Element",
y = "Assay (ppm)")+
"plot/02 Distributions by Feed.png",
width = 5,
height = 10,
units = "cm",
dpi = 320,
scale = 3
Some new patterns emerge from this plot.
• Gravity concentrates are richer in dense metals (Fe, Mn, Cr, W)
• Flotation concentrates have more fast-floating metals (Zn, Pb, Cu, Zr, Ni)
• Some species are detected only in one concentrate but not the other (Au, U, Cd, Mo)
• Some species are abundant in both (As, Ti, S).
This is better but it is a bit hard to spot trends, a slope plot would help rapidly identify changes in the groups. To do this, we group the data by source and element then summaries the results by
averaging the assays.
df_tidy %>%
source = if_else(comment == "float con", 'Flotation', 'Gravity'),
source = if_else(is.na(comment), 'Gravity', source)
) %>%
group_by(source, element) %>%
summarise(assay = mean(assay)) %>%
x = source,
y = assay,
colour = reorder(element, assay, mean),
group = element
)) +
geom_point(size = 2) +
geom_line() +
scale_y_log10(labels = scales::comma_format()) +
geom_text_repel(aes(label = element), point.padding = .5) +
no_legend() +
no_x_gridlines() +
annotation_logticks(sides = "l") +
labs(title = "Feed Comparison",
x = "Feed Source",
y = "Assay (ppm)")+
"plot/03 Feed Comparison.png",
width = 5,
height = 10,
units = "cm",
dpi = 320,
scale = 3
Whilst the above plot is illustrative, we aren’t able to determine if the differences between the concentrates is significant or not. To ascertain this, we utilise a volcano plot, commonly used in
bioinformatics to compare change and significance between a binary pair.
Change is measured through the fold change, the base 2 logarithm of $(Condition - Base) / Base$. This is a nice, symetric property with equivalence between the condition and base case at zero on the
abscissa. Signficance is negative base 10 logarithm of the p-value from a two-sided T-Test for equal means.
I used some helper functions and $map$ from the $purrr$ package. Firstly, I wrote a helper function that performs a T-Test for a given column of a data frame and compares them by the $source$ factor
which tell us which circuit the concetrate is from. The $broom$ package has a great function called $glance$ which tidues a T-Test’s output into a nice single row for analysis. I wrap this with the
$possibly$ function to safely handle errors and return NULL in these instances.
Secondly, I created a vector of valid elements then convert that vector into a data frame and map the helper function over each element and unnest the result back into a tidy data frame for plotting.
glanced = function(x){
equation = formula(str_c(x," ~ source"))
t.test(equation, data = df) %>% broom::glance()
possible_glance = possibly(glanced, NULL)
elements = df %>%
select(Al:U) %>%
select_if(is.double) %>%
select(-one_of(na_cols)) %>%
all_pairs = elements %>%
as.data.frame() %>%
mutate_all(as.character) %>%
set_names(nm = "element") %>%
group_by(element) %>%
mutate(t_test = map(element, possible_glance)) %>%
unnest() %>%
mutate(fold_change = log2(estimate1 / estimate2),
Significant = if_else(p.value <= 0.05, "Significant", "Not Significant"))
The plot is simple to produce; simply a scatter plot of significance and changes. Conditional formatting and some manual tweaking provides the right palette to communicate the results.
all_pairs %>%
ggplot(aes(x = fold_change,
y = -log10(p.value),
label = element,
colour = Significant))+
scale_colour_manual(values = c("gray30","white"))+
geom_text_repel(show.legend = F, size = 3)+
geom_vline(xintercept = 0, colour = "red", linetype = "solid")+
labs(x = "Fold Change [Log2(Flotation/Gravity)]",
y = "Significance [-Log10(P Value)]",
title = "Volcano Plot of Concentrates",
subtitle = "Flotation compared Gravity",
caption = "Fold Change represents a doubling and halfing.
-Log10 is comparable to pH, higher value = lower [H3O+].
Right = Higher concentration compared to Flot Con
Left = Lower concentration compared to Flot con
Higher = More significant result, lower p value") +
scale_x_continuous(limits = c(-4,4), breaks = seq(-4, 4, 1))
"plot/04 Volcano Plot.png",
width = 5,
height = 5,
units = "cm",
dpi = 320,
scale = 3
From the Above its clear that the increases in Zinc, Nickel, Copper and Lead in the flotation concentrate are significant. Significance reductions in Titanium, Calcium, Iron, Manganese and Chromium
give insights into the mineral species that are being selectively recovered in the Flotation circuit relative to the Gravity circuit. Without XRD, MLA or additional assays, the exact mineralogy
cannot be determined however we can speculate increased recovery of base metal sulphides, chiefly; Sphalerite, Chalcopyrite, Pentlandite and Galena with possible Pyrite rejection | {"url":"https://www.merryweather.dev/r/2017/12/19/xrf-assay-analysis-on-circuit-concentrates.html","timestamp":"2024-11-14T00:40:23Z","content_type":"text/html","content_length":"48780","record_id":"<urn:uuid:4b405dee-9b94-4e5a-acfd-259abd594df0>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00861.warc.gz"} |
Selina Solutions Class 9 Concise Maths Chapter 13 Pythagoras Theorem -Download Free PDF
Selina Solutions are considered to be very useful when you are preparing for the ICSE Class 9 Maths exams. Here, we bring to you detailed solutions to the exercises of Selina Solutions for Class 9
Maths Chapter 13- Pythagoras Theorem. These answers have been devised by the subject matter experts as per the syllabus prescribed by the CISCE for the ICSE.
Here we have provided the complete solutions of Class 9 Maths Chapter 13- Pythagoras Theorem in PDF format. Students can avail these Selina solutions and download it for free to practice them offline
as well.
Download PDF of Selina Solutions for Class 9 Maths Chapter 13:-Download Here
Exercise 13A PAGE: 158
1. A ladder 13 m long rests against a vertical wall. If the foot of the ladder is 5 m from the foot of the wall, find the distance of the other end of the ladder from the ground.
The pictorial representation of the given problem is given below,
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
(i)Here, AB is the hypotenuse. Therefore, applying the Pythagoras theorem, we get,
AB^2 = BC^2 + CA^2
13^2 = 5^2 + CA^2
CA^2 = 13^2 – 5^2
CA^2 = 144
CA = 12m
Therefore, the distance of the other end of the ladder from the ground is 12m
2. A man goes 40 m due north and then 50 m due west. Find his distance from the starting point.
Here, we need to measure the distance AB as shown in the figure below,
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
Therefore, in this case
AB^2 = BC^2 +CA^2
AB^2 = 50^2 + 40^2
AB^2 = 2500 + 1600
AB^2 = 4100
AB = 64.03
Therefore, the required distance is 64.03 m.
3. In the figure: ∠PSQ = 90^o, PQ = 10 cm, QS = 6 cm and RQ = 9 cm. Calculate the length of PR.
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
First, we consider the triangle PQS and applying Pythagoras theorem we get,
PQ^2 = PS^2 + QS^2
10^2 = PS^2 + 6^2
PS^2 = 100 – 36
PS^2 = 64
PS = 8
Now, we consider the triangle PRS and applying Pythagoras theorem we get,
PR^2 = RS^2 + PS^2
PR^2 = 15^2 + 8^2
PR = 17
Therefore, the length of PR = 17cm
4. The given figure shows a quadrilateral ABCD in which AD = 13 cm, DC = 12 cm, BC = 3 cm and []ABD = []BCD = 90^o. Calculate the length of AB.
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
First, we consider the triangle BDC and applying Pythagoras theorem we get,
DB^2[ ]= DC^2 + BC^2
DB^2 = 12^2 + 3^2
= 144 + 9
= 153
Now, we consider the triangle ABDÂ and applying Pythagoras theorem we get,
DA^2[ ]= DB^2 + BA^2
13^2 = 153 + BA^2
BA^2 = 169 – 153
BA = 4
The length of AB is 4 cm.
5. AD is drawn perpendicular to base BC of an equilateral triangle ABC. Given BC = 10 cm, find the length of AD, correct to 1 place of decimal.
Since ABC is an equilateral triangle therefore, all the sides of the triangle are of same measure and the perpendicular AD will divide BC in two equal parts.
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
Here, we consider the triangle ABDÂ and applying Pythagoras theorem we get,
AB^2 = AD^2 + BD^2
AD^2 = 100^2 – 5^2
AD^2 = 100 – 25
AD^2 = 75
= 8.7
Therefore, the length of AD is 8.7 cm
6. In triangle ABC, given below, AB = 8 cm, BC = 6 cm and AC= 3 cm. Calculate the length of OC.
We have Pythagoras theorem which states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
First, we consider the triangle ABOÂ and applying Pythagoras theorem we get,
AB^2 = AO^2 + OB^2
AO^2 = AB^2 – OB^2
AO^2 = AB^2 – (BC + OC)^2
Let OC = x
AO^2 = AB^2 – (BC + x)^2 ……… (1)
First, we consider the triangle ACOÂ and applying Pythagoras theorem we get
AC^2 = AO^2 + x^2
AO^2 = AC^2 – x^2 ……… (2)
From 1 and 2
AB^2 – (BC + x)^2 = AC^2 – x^2
8^2 – (6 + x)^2 = 3^2 – x^2
X = 1 7/12 cm
Therefore, the length of OC will be 19/12 cm
7. In triangle ABC, AB = AC = x, BC = 10 cm and the area of the triangle is 60 cm^2. Find x.
Here, the diagram will be,
We have Pythagoras theorem which states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
Since, ABC is an isosceles triangle, therefore perpendicular from vertex will cut the base in two equal segments.
First, we consider the triangle ABDÂ and applying Pythagoras theorem we get,
AB^2 = AD^2 + BD^2
AD^2 = x^2 – 5^2
AD^2 = x^2 – 25
AD = √(x^2 – 25) ………. (1)
Area = 60
½ (10) AD = 60
½ (10) [√(x^2 – 25)] = 60
x = 13
Therefore x = 13 cm
8. If the sides of triangle are in the ratio 1 :2: 1, show that is a right-angled triangle.
Let, the sides of the triangle be, x^2 + x^2 = 2x^2 = (√2x)^2 ……… (1)
Here, in (i) it is shown that, square of one side of the given triangle is equal to the addition of square of other two sides. This is nothing but Pythagoras theorem which states that in a
right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
Therefore, the given triangle is right angled triangle.
9. Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their feet is 12 m; find the distance between their tips.
The diagram of the given problem is given below,
We have Pythagoras theorem which states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
Here 11 – 6 = 5m
Base = 12 m
Applying Pythagoras theorem, we get
h^2 = 5^2 + 12^2
= 25 + 144
= 169
h = 13
therefore, the distance between the tips will be 13m
10. In the given figure, AB//CD, AB = 7 cm, BD = 25 cm and CD = 17 cm; find the length of side BC.
Take M be the point on CD such that AB = DM.
So, DM = 7cm and MC = 10 cm
Join points B and M to form the line segment BM.
So BM || AD also BM = AD.
In triangle BAD
BD^2 = AD^2 + BA^2
25^2 = AD^2 + 7^2
AD^2 = 576
AD = 24
In triangle CMB
CB^2 = CM^2 + MB^2
CB^2 = 10^2 + 24^2
CB^2 = 676
CB = 26 cm
Exercise 13B PAGE: 163
1. In the figure, given below, AD parallel to BC. Prove that: c^2 = a^2 + b^2 – 2ax
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
First, we consider the triangle ABDÂ and applying Pythagoras theorem we get,
AB^2 = AD^2 + BD^2
c^2 = h^2 + (a – x)^2
h^2 = c^2 – (a – x)^2 ……… (1)
First, we consider the triangle ACD and applying Pythagoras theorem we get
AC^2 = AD^2 + CD^2
b^2 = h^2 + x^2
h^2 = b^2 – x^2 ………. (2)
from 1 and 2
c^2 – (a – x)^2 = b^2 – x^2
c^2 – a^2 – x^2 + 2ax = b^2 – 2ax
c^2 = a^2 + b^2 – 2ax
hence the proof.
2. In equilateral Δ ABC, AD parallel to BC and BC = x cm. Find, in terms of x, the length of AD.
In equilateral Δ ABC, AD parallel to BC.
Therefore, BD = DC = x/2 cm.
Applying Pythagoras theorem, we get
In right angled triangle ADC
AC^2 = AD^2 + DC^2
x^2 = AD^2 + (x/2)^2
AD^2 = (x)^2 – (x/2)^2
AD^2 = (x/2)^2
AD = (x/2) cm
3. ABC is a triangle, right-angled at B. M is a point on BC. Prove that:
AM^2Â + BC^2Â = AC^2Â + BM^2.
The pictorial form of the given problem is as follows,
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
First, we consider the triangle ABM and applying Pythagoras theorem we get,
AM^2 = AB^2 + BM^2
AB^2 = AM^2 – BM^2 ……….. (1)
Now we consider the triangle ABC and applying Pythagoras theorem we get
AC^2 = AB^2 + BC^2
AB^2 = AC^2 – BC^2 …… (2)
From 1 and 2 we get
AM^2 – BM^2 = AC^2 + BM^2
AM^2 + BC^2 = AC^2 + BM^2
Hence the proof.
4. M and N are the mid-points of the sides QR and PQ respectively of a triangle PQR, right-angled at Q. Prove that:
(i) PM^2Â + RN^2Â = 5 MN^2
(ii) 4 PM^2Â = 4 PQ^2Â + QR^2
(iii) 4 RN^2Â = PQ^2Â + 4 QR^2
(iv) 4 (PM^2Â + RN^2) = 5 PR^2
Draw, PM, MN, NR
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
Since, M and N are the mid-points of the sides QR and PQ respectively, therefore, PN = NQ, QM = RM
First, we consider the triangle PQM and applying Pythagoras theorem we get,
PM^2 = PQ^2 + MQ^2
= (PN + NQ)^2 + MQ^2
= PN^2 + NQ^2 + 2 PN. NQ + MQ^2
= MN^2 + PN^2 + 2 PN. NQ [ we know MN^2 = NQ^2 + MQ^2] ………….. (1)
Now we consider the triangle RNQ and applying Pythagoras theorem,
RN^2 = NQ^2 + RQ^2
= NQ^2 + (QM + RM)^2
= NQ^2 + QM^2 + 2 QM. RM + RM^2 …………. (2)
Adding 1 and 2 we get
PM^2 + RN^2 = MN^2 + PN^2 + 2PN. NQ + MN^2 + RM^2+ 2QM. RM
PM^2 + RN^2 = 2MN^2 + PN^2 + RM^2 + 2PN. NQ + 2QM. RM
PM^2 + RN^2 = 2MN^2 + NQ^2 + QM^2 + 2(QN)^2 + 2 (QM)^2
PM^2 + RN^2 = 2MN^2 + MN^2 + 2MN^2
PM^2 + RN^2 = 5MN^2
Hence the proof.
(ii) Now consider the triangle PQM and apply Pythagoras theorem we get
PM^2 = PQ^2 + MQ^2
4PM^2 = 4PQ^2 + 4 MQ^2 [multiplying both sides by 4]
4PM^2 = 4PQ^2 + 4 (½ QR^2) [MQ = ½ QR]
4PM^2 = 4PQ^2 + QR^2
Hence the proof.
(iii) now consider triangle RQN and apply Pythagoras theorem we get
RN^2 = NQ^2 + RQ^2
4RN^2 = 4NQ^2 + 4 QR^2 [multiplying both sides by 4]
4RN^2 = 4QR^2 + 4 (½ PQ^2) [NQ = ½ PQ]
4RN^2 = PQ^2 + 4QR^2
Hence the proof.
(iv) now consider the triangle PQM and apply Pythagoras theorem,
PM^2 = PQ^2 + MQ^2
= (PN + NQ)^2 + MQ^2
= PN^2 + NQ^2 + 2 PN. NQ + MQ^2
= MN^2 + PN^2 + 2 PN. NQ [ we know MN^2 = NQ^2 + MQ^2] ………….. (1)
Now we consider the triangle RNQ and applying Pythagoras theorem,
RN^2 = NQ^2 + RQ^2
= NQ^2 + (QM + RM)^2
= NQ^2 + QM^2 + 2 QM. RM + RM^2
= MN^2 + RM^2 + 2 QM. RM …………. (2)
Adding 1 and 2 we get
PM^2 + RN^2 = MN^2 + PN^2 + 2PN. NQ + MN^2 + RM^2+ 2QM. RM
PM^2 + RN^2 = 2MN^2 + PN^2 + RM^2 + 2PN. NQ + 2QM. RM
PM^2 + RN^2 = 2MN^2 + NQ^2 + QM^2 + 2(QN)^2 + 2 (QM)^2
PM^2 + RN^2 = 2MN^2 + MN^2 + 2MN^2
PM^2 + RN^2 = 5MN^2
4 (PM^2+ RN^2) = 4. 5 (NQ^2 + MQ^2]
4 (PM^2+ RN^2) = 4. 5 [(½ PQ)^2[ ]+ (½ QR)^2]
4 (PM^2+ RN^2) = 5PR^2
Hence the proof.
5. In triangle ABC, ∠B = 90^o and D is the mid-point of BC. Prove that: AC^2 = AD^2 + 3CD^2.
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
In triangle ABC,[ ]∠B = 90^o and D is the mid-point of BC. Join AD. Therefore, BD=DC
First, we consider the triangle ADB and applying Pythagoras theorem we get,
AD^2 = AB^2 + BD^2
AB^2 = AD^2 – BD^2 …. (1)
Similarly, we get from rt. angle triangles ABC we get,
AC^2 = AB^2 + BC^2
AB^2 = AC^2 – BC^2 …. (1)
From 1 and 2 we get
AC^2 – BC^2 = AD^2 – BD^2
AC^2 = AD^2 – BD^2 + BC^2
AC^2 = AD^2 – CD^2 + 4CD^2 [BD = CD = ½ BC]
AC^2 = AD^2 + 3CD^2
Hence the proof.
6. In a rectangle ABCD, prove that: AC^2Â + BD^2Â = AB^2Â + BC^2Â + CD^2Â + DA^2.
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
Since, ABCD is a rectangle angles A, B, C and D are rt. angles.
First, we consider the triangle ACD and applying Pythagoras theorem we get,
AC^2 = DA^2 + CD^2 ……. (1)
Similarly, we get from rt. angle triangle BDC we get,
BD^2 = BC^2 + CD^2
= BC^2 + AB^2 [In a rectangle opposite sides are equal CD = AB]
Adding (i) and (ii),
AC^2Â + BD^2Â = AB^2Â + BC^2Â + CD^2Â + DA^2
Hence the proof.
7. In a quadrilateral ABCD, ∠B = 90^0 and ∠D = 90^0. Prove that: 2AC^2 – AB^2 = BC^2 + CD^2 + DA^2
In quadrilateral ABCD ∠B = 90^0 and ∠D = 90^0
So triangle ABC and triangle ADC are right angles.
For triangle ABC, apply Pythagoras theorem,
AC^2 = AB^2 + BC^2
AB^2 = AC^2 – BC^2 ……….. (i)
For triangle ADC, apply Pythagoras theorem,
AC^2 = AD^2 + DC^2 ……….. (ii)
LHS = 2AC^2 – AB^2
= 2AC^2 – (AC^2 – BC^2) from 1
= 2AC^2 – AC^2 + BC^2
= AC^2 + BC^2
= AD^2 + DC^2 + BC^2 from 2
= RHS
8. O is any point inside a rectangle ABCD. Prove that: OB^2Â + OD^2Â = OC^2Â + OA^2.
Draw rectangle ABCD with arbitrary point O within it, and then draw lines OA, OB, OC, OD. Then draw lines from point O perpendicular to the sides: OE, OF, OG, OH.
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
Using Pythagorean theorem, we have from the above diagram:
OA^2Â = AH^2Â + OH^2Â = AH^2Â + AE^2
OC^2Â = CG^2Â + OG^2Â = EB^2Â + HD^2
OB^2Â = EO^2Â + BE^2Â = AH^2Â + BE^2
OD^2Â = HD^2Â + OH^2Â = HD^2Â + AE^2
Adding these equalities, we get:
OA^2Â + OC^2Â = AH^2Â + HD^2Â + AE^2Â + EB^2
OB^2Â + OD^2Â = AH^2Â + HD^2Â + AE^2Â + EB^2
From which we prove that for any point within the rectangle there is the relation
OA^2Â + OC^2Â = OB^2Â + OD^2
Hence Proved.
9. In the following figure, OP, OQ and OR are drawn perpendiculars to the sides BC, CA and AB respectively of triangle ABC. Prove that: AR^2Â + BP^2Â + CQ^2Â = AQ^2Â + CP^2Â + BR^2
Here, we first need to join OA, OB, and OC after which the figure becomes as follows,
Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides. First, we consider the []
and applying Pythagoras theorem we get,
AO^2 = AR^2 + OR^2
AR^2 = AO^2 – OR^2 ……. (1)
Similarly, from triangles, BPO, COQ, AOQ, CPO and BRO we get the following results,
BP^2 = BO^2 – OP^2 ……. (2)
CQ^2 = OC^2 – OQ^2 ……. (3)
AQ^2 = AO^2 – OQ^2 ……. (4)
CP^2 = OC^2 – OP^2 ……. (5)
BR^2 = OB^2 – OR^2 ……. (6)
Adding 1, 2 and 3 we get
AR^2 + BP^2+ CQ^2 = AO^2 – OR^2 + BO^2 – OP^2 + OC^2 – OQ^2 ……. (7)
Adding 4, 5 and 6 we get
AQ^2 + CP^2 + BR^2 = AO^2 – OQ^2 + OC^2 – OP^2 + OB^2 – OR^2 ………… (8)
From 7 and 8, we get,
AR^2Â + BP^2Â + CQ^2Â = AQ^2Â + CP^2Â + BR^2
Hence proved.
10. Diagonals of rhombus ABCD intersect each other at point O. Prove that: OA^2 + OC^2 = 2AD^2 – BD^2/2
We know diagonals of the rhombus are perpendicular to each other.
In quadrilateral ABCD, ∠AOD = ∠COD = 90^o
We know triangle AOD and COD are right angle triangle.
In triangle AOD, apply Pythagoras theorem,
AD^2 = OA^2 + OD^2
OA^2 = AD^2 – OD^2 ………… (1)
In triangle COD, apply Pythagoras theorem,
CD^2 = OC^2 + OD^2
OC^2 = CD^2 – OD^2 ………… (2)
LHS = OA^2 + OC^2
= AD^2 – OD^2 + CD^2 – OD^2 from 1 and 2
= AD^2 – AD^2 – 2(BD/2)^2 [AD = CD and OD = BD/2]
= 2AD^2 – BD^2/2
= RHS
Selina Solutions for Class 9 Maths Chapter 13- Pythagoras Theorem
The Chapter 13, Pythagoras Theorem is composed of 2 exercises and the solutions given here contain answers to all the questions present in these exercises. Let us have a look at some of the topics
that are being discussed in this chapter.
13.1 Introduction
13.2 Pythagoras Theorem
1. Area based: In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
2. Alternate: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares on the remaining two sides.
Selina Solutions for Class 9 Maths Chapter 13- Pythagoras Theorem
In Chapter 13 of Class 9, the students are taught about the Pythagoras Theorem. This chapter also belongs to the unit Triangle. The chapter helps students in understanding the proof and simple
applications of the pythagoras theorem as well as its converse. Study the Chapter 13 of Selina textbook to understand more about Pythagoras Theorem. Learn the Selina Solutions for Class 9 effectively
to come out with flying colours in the examinations.
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经济学代写｜EC1211 Quantitative Techniques for Economics II Assignment 2 - EasyDue™️
Assignment 2
The deadline for submission of the assignment is the 24th of April at 11.59 pm.Each of the five parts is worth 20%.You should explain your answers fully. Where you are using SPSS, you do not need to
show the steps involved in calculating the results but you should explain the meaning of your results fully.You should draw any diagrams by hand and then insert images of them into your answer
Part 1
(a) Consider the following data:
It has been claimed that the population has a Poisson distribution with a mean of 5.6 complaints per day. Use a Chi-Square Goodness-of-Fit test to compare the sample data above with the expected
frequencies for the five categories according to the Poisson distribution. Carry out the appropriate test without using SPSS. You can make use of Excel. Use a level of significance of 5%.
(b) Use SPSS and the data in “Northern Ireland Random Poll” to test the hypothesis that this sample was drawn from a population with the following frequencies. Use a level of significance of 10%.
In the datafile, 0=DUP, 1=SF, 2=UUP, 3=SDLP, 4=Alliance, 5=TUV, 6=Other
Part II
(a) Use SPSS and the Northern Ireland data to test the hypothesis that there is no relationship between sex and preferred political party. In the data file, 0=Male,1=Female. Use a level of
significance of 10%.
(b) We asked a random sample of students to choose one style of music from a list of four. Test the hypothesis that there is no relationship between preferred style of music and category of student.
Use a level of significance of 5%. Do not use SPSS.You can use Excel.
Part III
(a) Why might we use the Kruskal-Wallis test instead of ANOVA? (max. 60 words)
(b) Use the Kruskal-Wallis test to test the hypothesis that the following three groups all have the same mean at the population level. Use a level of significance of 10%. Do not use SPSS. You can use
Part IV
(a) Using diagrams where appropriate, explain the concepts of covariance, correlation and slope coefficient and the relationship between them.
(b) Use the dataset “Carseats”. There an information document with the data file that describes the variables. Perform a regression analysis in which “Sales” is the dependent variable. Use the
following variables as explanatory variables: CompPrice, Income,Advertising, Price, Age. Use SPSS. Write one or two sentences for each explanatory variable to explain the meaning of its coefficient
in your results.
(c) What do we mean when we say that the OLS estimators are efficient, consistentand unbiased?
(d) Explain the concept of the coefficient of determination.
Part V
In this course, we have typically assumed that our samples are gathered randomly.However, in practice we often encounter a problem that is called “selection bias”. I would like you to investigate
this concept and write a short essay explaining some of the main kinds of selection bias and providing examples. (Max. 350 words.)
Submission Details
The deadline for submission of the assignment is the 25th of April at 11.59 pm.
• It must be submitted via Canvas. Submissions made after the deadline will be deemed late and subject to penalty (See below).
• The front page of the assignment should contain your name and I.D. number.Penalties for Late Submission:
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Using Bayesian Optimization to Identify Optimal Exoskeleton Parameters Targeting Propulsion Mechanics: A Simulation Study
In this study, we determined the feasibility of modeling the relationship between robot control parameters and propulsion mechanics as a Gaussian process. Specifically, we used data obtained in a
previous experiment that used pulses of torque applied at the hip and knee joint, at early and late stance, to establish the relationship a 3D control parameter space and the resulting changes in hip
extension and propulsive impulse. We estimated Gaussian models both at the group level and for each subject. Moreover, we used the estimated subject-specific models to simulate virtual
human-in-the-loop optimization (HIL) experiments based on Bayesian optimization to establish their convergence under multiple combinations of acquisition functions and seed point selection methods.
Results of the group-level model are in agreement with those obtained with linear mixed effect model, thus establishing the feasibility of Gaussian process modeling. The estimated subject-specific
optimal conditions have large between-subject variability in the metric of propulsive impulse, with only 31% of subjects featuring a subject-specific optimal point in the surrounding of the
group-level optimal point. Virtual HIL experiments indicate that expected improvement is the most effective acquisition method, while no significant effect of seed point selection method was
observed. Our study may have practical effects on the adoption of HIL robot-assisted training methods focused on propulsion.
Robot assisted gait training is becoming a common method for rehabilitation after neurological injury [1]. With the option of mechanical assistance to multiple joints, and several open parameters for
the timing of assistance at each joint, controllers for gait training robots are defined by a large number of open parameters, each corresponding to highly variable outcomes of robotic training. To
deal with the large number of open parameters in robot-assisted gait training, real-time optimization methods, or Human-In-the-Loop (HIL) optimization methods, have been introduced [2]. In
human-robot interaction, HIL methods are used to identify parameters of a robot controller that are optimal in the sense of a specific cost function. In general, this is done by testing the value of
the cost function, collecting data to quantify the subject response at those points, and then iteratively updating the controller parameters in real-time to optimize the cost function.
Several successful implementations of HIL optimization exist, many using different optimization algorithms [3]–[4], with one-dimensional (1D) gradient descent method [5], 4D Covariance Matrix
Adaptation Evolution Strategy (CMA-ES) [4], and Bayesian optimization [3], [6] used in recent studies using exoskeletons supporting walking function. However, most previous HIL optimization
experiments focused on reducing metabolic cost to tune control parameters which results limited applicability in robot-assisted gait training, since the major objective of gait training is to induce
desired changes in subjects’ motor coordination with the ultimate goal of improving their motor function. Moreover, on-going research about optimization algorithm is focused on improving performance
of Bayesian optimization based on hyperparameter tuning [7], noise modeling [8], acquisition function definition [9], [10], [11], and definition of seed points [3], but it is currently unclear how
these methods apply to HIL optimization methods in biomechanics.
In this work, we seek to apply HIL to establish the subject-specific relationship between exoskeleton control parameters and resulting features of gait, such as those describing propulsion mechanics,
a crucial component of walking function [12], [13], [14]. We address the feasibility of modeling previously collected data on the effects of the application of pulses of torque to the hip and knee
joint on propulsion mechanics using Gaussian process modeling [15]. Specifically, we establish the relationship between pulse torque conditions and propulsion mechanics at the group and
subject-specific level, and evaluate how these relationships differ across subjects. Moreover, we run simulations using the estimated subject-specific Gaussian process models to establish convergence
in virtual HIL experiments based on Bayesian optimization. Specifically, in our analysis we establish the effect of two important factors, seed point selection method and acquisition function, in the
speed and accuracy of convergence of Bayesian optimization targeting propulsion mechanics.
A. Data Collection
In previous research, sixteen healthy subjects participated in an experiment based on the application of pulses of torque to the hip and knee joint to modulate propulsion mechanics [15]. Torque
conditions were defined as a combination of three parameters: pulse timing, hip pulse amplitude, and knee pulse amplitude. Two levels were used for pulse timing: pulses were applied at 10% of the
estimated gait cycle period (early stance), or 45% of the estimated gait cycle period (late stance). Levels for hip and knee pulse amplitude were defined as either zero torque, flexion, or extension
(amplitude was set to 15 N·m for the hip joint, and 10 N·m for the knee joint for both flexion and extension). Sixteen conditions were tested, including all combinations of the factors above, with
the exclusion of the combination of zero knee and hip torque. When pulses were applied to both joints, they were applied simultaneously. Each condition was repeated ten times per subject, with a
random sequence. Pulses were applied to the right leg during single strides, and spaced by at least eight strides of no pulse application. Hip extension (HE) and propulsive impulse (PI) were assessed
at multiple strides: prior to pulse application (stride −1), during pulse application (stride 0), and during the three strides following pulse application (stride 1, 2, 3). In previous work, our
group used a linear mixed model to establish the relationship between factors including pulse pattern parameters (pulse timing, amount of torque pulses applied to hip and knee joints), subjects, and
stride with propulsion mechanics, as defined by HE and PI [15].
B. Gaussian Process Modeling
Our first goal is to model the relationship between pulse torque conditions and propulsion mechanics as a Gaussian process. Each subject has 10 measurements of each outcome measure (HE and PI),
resulting in a total of 160 data points per stride condition, and thus 800 data points per subject (12800 total measurements), referred to as measurements y[HE] and y[P I]. These measurements can be
indexed as a function of factors: pulse timing (T), amplitude of hip and knee torque pulses (K and H), stride (S), subject (Sbj), trial index (Rep), as where T ∈ (10, 45), K ∈ (−10, 0, 10), H ∈ (−15,
0, 15), S ∈ (−1, …, 3), Sbj ∈ (1, …, 16), and Rep ∈ (1, …, 10).
We assumed that the Gaussian process model linking control parameters with the outcome measures of propulsion should follow the characteristics listed below:
• noise from outcome measures is normally distributed with zero mean ;
• noise is independent of the human response;
• the distribution of the human response under repeated exposure to the same pulse condition is normal;
• the variance of the human response will be constant under different pulse conditions.
1) Group-level Model Formulation
Data in variables (1) and (2) are concatenated along the subject dimension obtaining variables for HE, and for PI, with 160 repetitions available for factor Rep[G].
The modified outcome measures ( and ) can be expressed as the sum of an unknown Gaussian process G, and noise (ϵ) as follows:
The average in the measurements can be derived by average value among 160 repetitions per each condition. Since noise is assumed to have zero mean, the average value of the unknown Gaussian process G
is equal to the average in the measurements . The variance in the measurements are expressed as a sum of model variance (variations in the true output arising from repeated exposure to the same
conditions ), and measurement error (variations in the measurements that are not associated to changes in the true value of the output ). Since the noise is constant for all values of pulse torque
factors, the following relationships hold true:
These relationships are used to specify values for the model variance used for the Gaussian process covariance function. Values of and are calculated as the maximum variance of along the Rep[G]
dimension, i.e. the one resulting from the combination of pulse parameters associated with the largest variance across subjects and repetitions.
2) Group-level Model Estimation
A Gaussian process model is estimated from eq. (3) and (4) to approximate the mean and variance of data . In a Gaussian process, the variance is defined based on a covariance (or kernel) function k[α
](x[i], x[j]), which defines how variance propagates within a dimension, or in different dimensions of the model. In this work, we defined the covariance function (automatic relevance determination
squared exponential kernel) as: where x is set of pulse torque input parameters, x = (T, K, H, S), l[m,α] is the m-th length scale hyperparameter, α ∈ (HE, PI). (i, j) are indices corresponding to
two arbitrary points in the 4D domain of the parameters. Using the covariance function (7), the estimated mean value and variance of a point x[n+1], based on n measurements, are and
In our case, n = 80, referring to the fact that all observations (Y[α,1:n]) are used to estimate a Gaussian process model for G[HE] and G[P I]. After a model is estimated given the available n
measurements, the estimated average value μ(x[n+1]) of all points in the domain is defined as the mean function m(x) of input x.
As discussed above, (3) and (4) define the relationship between control parameters and output using Gaussian processes, i.e., , where k(x, x′) is the kernel function. As such, the Gaussian processes
can be estimated by solving the following least-squares problems in terms of the hyperparameters l[m,α] and :
To align our fitted model with observations emerging from our previous linear mixed model analysis, we set constraints on the hyperparameters optimized in (13) (Table I).
3) Subject-specific Model Formulation
Similarly as the group data, the outcome measures for each subject (y[Sbj,HE] and y[Sbj,P I]) can be expressed as the sum of an unknown subject-specific Gaussian process G[Sbj], and noise (ϵ[Sbj]):
Measurements collected from each subject correspond to the ten repeated measurements, collected at all combinations of the factors, resulting in 800 measurements. Similarly as done in the group-level
model, noise is assumed to be constant for all values of pulse torque factors, hence variances of data and processes are linked by:
4) Subject-specific Model Estimation
Similarly as done for group-level model estimation, a Gaussian process is estimated from (14) and (15). The same kernel function defined in (7) was used for the definition of the subject-specific
Gaussian models, with the difference that was defined as the variance resulting from the combination of pulse parameters with the largest within-subject variance across the ten repetitions. Because
it is usually impractical to collect a sufficient number of observations from an individual subject to properly estimate subject-specific length-scale hyperparameters, we proceeded to specify for
individuals the same values of parameters l[m,α] estimated for the group. In these conditions, the subject-specific Gaussian processes G[Sbj,α] are estimated solving the following least-squares
5) Quantifying Variability of Maxima in Subject-specific Models
To quantify the variability of the optimal points in the estimated subject-specific Gaussian process models G[Sbj,HE] and G[Sbj,P I], we calculated the vectors as the values of T, K, and H that
maximized the estimated process value at stride 0 for different subjects, and as the values of T, K, an H that maximized the estimated change in outcome measure between stride 0 and stride −1 for
different subjects. We thus established for how many subjects (n[w]) the subject-specific optimal points fell within a sphere of radius r centered around the optimal points estimated using the
group-level model (x* and δx*). The optimal point analysis is conducted on the non-dimensional domain where all coordinates are comprised between 0 and 1 using a min-max normalization. The functions
n[w](r) are compared to establish variability of maxima in the two outcome measures.
C. Bayesian Optimization Simulation
Virtual HIL experiments are conducted to determine the convergence of Bayesian optimization for each subject-specific Gaussian process model, under different settings of the optimization algorithm.
Each subject-specific Gaussian process model is assumed to be the human response (i.e., data generating process) when pulse torque assistance is applied. Three robotic control parameters (T, K, and H
) are used as input parameters. In all conditions, the optimization algorithm does not have any knowledge about the subject-specific model at the beginning of each optimization. Two algorithm
components are tested using a factorial design: i) acquisition function (three levels), and ii) and seed point selection method (three levels) are evaluated in terms of the number of iterations
required for convergence. In total, the 9 combinations of the two components are tested in simulation, with twenty simulations repeated for each combination for each subject.
1) Acquisition Function
Expected Improvement (EI), Probability of Improvement (PoI), and Lower Confidence Bound (LCB) are used in this study. EI finds the next point which maximizes mean value improvement. This expected
improvement is defined as where x^+ is the best point from data points explored so far and m(x^+) is mean value of Gaussian process model value at point x^+. PoI finds the next point which has the
maximum probability of improvement as where γ is margin. The LCB acquisition function selects the next point which maximizes the lower bound as where σ[x] is standard deviation at point x, and β > 0
is a heuristic trade-off (β = 2 in this work).
2) Seed Point Selection Method
Three seed point selection methods are used in this work. In the first method (divide), eight points are selected by dividing the parameter search space in eight regions (the domain for each
parameter is divided into two equal parts), and then each of the eight points are selected randomly within each of the eight regions. In the second method (random), eight points are randomly defined
in the search space. In the third method (optimal), the set of eight seed points is composed of seven random points in the search space, plus the optimal point of the group-level Gaussian process
3) Virtual Optimization
Virtual Bayesian optimization is conducted to maximize HE and PI at the stride of pulse application, and the outcome measure is defined as the number of iterations n[x] and n[y] required to achieve
one of two convergence criteria.
If is the normalized coordinates optimal point esti-mated via virtual optimization after i iterations, and x[opt] is the known optimal point for that subject, n[x] is defined as the minimum value of
i where the normalized difference x[diff] between the two quantities is smaller than 10%, where where p is the range of each component of x.
The second criterion is based on the value of the estimated outcome after a certain number of iterations. Specifically, n[y] is defined as the minimum number of iterations where the normalized
estimation error f[diff] (x[i]) is smaller than 5% of the known subject-specific model maximum G[sbj,α](x*), where
All simulations were run for 80 iterations. For each criterion, if no convergence was achieved within 80 simulations, n was set to 80.
Four two-way ANOVA, one for each outcome measure (combination of convergence criterion – n(x) and n(y)) – and propulsive metric – HE and PI – were conducted to quantify the effects of the two factors
(i.e., acquisition function, three levels, and seed point selection methods, three levels) on convergence speed.
A. Gaussian Process Modeling
1) Group-level Model
Results for group-level Gaussian process modeling are shown in Fig. 1 and Fig. 2. The process variances calculated from group-level propulsion are equal to 4.31 deg2 for HE and 2.12 N2s2. Estimated
noise standard deviation and hyperparameter values are listed in Table II.
2) Subject-specific Results
The mean of the distribution of subject-specific process variances were equal to 69.04 deg^2 and 33.86 N^2s^2 for HE and PI, respectively. The number of optimal points for subject-specific models
within percentage difference of the group-level model is shown in Fig. 5. The optimal points are for maximal outcomes during stride 0 and maximum change in outcome (between stride 0 and stride −1)
for each subject-specific model. The optimal points for the subject-specific models are shown in Fig. 3 and Fig. 4 for HE and PI, respectively. 12 subject-specific Gaussian process models have
optimal points that have a percentage difference of less than 20 % compared to the optimal point of group-level Gaussian process model for HE (Fig. 3). For propulsive impulse case (Fig. 4), only 5
optimal points of subject-specific Gaussian process model are located within 20% difference from the optimal point of group-level Gaussian process model. For PI, the minority of subjects exhibited
subject-specific optimal points within a reasonable neighborhood of the group-level optimal point. In fact, more optimal points of subject-specific models in PI case are located in late stance (45 %)
than early stance (10 %) where optimal point of group-level model is located.
B. Bayesian Optimization Simulation
Virtual Bayesian optimization experiments achieved con-vergence within 80 iterations in 80.58 % of runs (94.40 ± 6.06 % when using EI, 85.57 ± 13.94 % when using PoI, 61.77 ± 26.66 % when using LCB).
The results of the ANOVA analysis are reported in Table III and Fig. 6. Based on the ANOVA analysis, the seed point selection method had no significant effect on the outcome measure in any condition.
Instead, the choice of an acquisition function showed to be associated with the number of iterations required for convergence (p < 0.001 in all conditions tested). Specifically, EI was the only
acquisition function type that was estimated to have negative coefficient in the linear model when using random method and PoI as neutral level of the factors, indicating that its effect is to
decrease the number of iterations to obtain convergence. Post-hoc tests are conducted to distinguish the significant difference among groups under different combinations of acquisition functions and
seed point selection methods (Fig. 6). Post-hoc tests indicate that within the same group of acquisition function types, there exist no significant differences (p > 0.001 for all comparisons).
Moreover, post-hoc tests comparing different acquisition functions showed that EI was significantly better than LCB in all conditions tested, and afforded a significantly greater accuracy in
identifying optimal control parameters compared to PoI.
In this study, we determined the feasibility of using Gaussian process modeling to establish modeling the relationship between three exoskeleton control parameters (hip torque pulse amplitude, knee
torque pulse amplitude, and timing of the pulses), and outcome measures relevant for propulsion. To estimate parameters of a Gaussian process, we estimated the noise variance and tuned the
hyperparameters of a Gaussian process describing the response at the group level. Based on the group-level Gaussian process model, we estimated 16 subject-specific models and used those as
data-generating process for virtual HIL Bayesian optimization simulations. In the virtual Bayesian optimizations, we established how many iterations would be required for convergence of a
hypothetical experiment, and quantified the effects of different acquisition functions and seed point selection methods on convergence speed.
Our results of group-level Gaussian process model (Fig. 1 and Fig. 2) demonstrate that it is feasible to construct a model between the 3D space of control parameters and outcomes using a Gaussian
process with estimated noise variance. Based on previous predictions using linear mixed model [15], group-level Gaussian process models for both HE and PI are expected to have a similar response when
pulses are applied to the knee and hip joints. In agreement with the results of the previous study [15], hip extension torque increases HE in late stance but decreases HE in early stance; knee
extension and hip extension pulse torque increase PI in early stance; knee extension pulse torque decreases PI in late stance. The variability of optimal points for subject-specific models is
moderate in HE and high in PI. Specifically for the PI case, the maxima of subject-specific models fell within 20% normalized distance from the optimal point for group-level model only in the
minority of cases (5/16). This indicates that the optimal point for group-level model is likely to be distant from the optimal point for subject-specific models. In this case, in fact, optimal points
for subject-specific models are separated to make two clusters of optimal points in early and late stance. Our analysis in HE case is aligned with previous work targeting reduction in metabolic cost,
indicating that one subject’s optimal point can be optimal for another subject [3].
In contrast with previous research selecting seed points by dividing the search space in N regions and randomly selecting seed points from all the N regions [3], seed point selection method is not
estimated to make a significant difference in a single pulse torque application experiment (p > 0.12 for the different conditions tested, Table III). This results are consistent in both HE and PI
cases. Since we included optimal method in seed point selection method, this results also indicate that optimal point location relationship between group-level optimal point and subject-specific
optimal points in HE case do not effect results of simulations. However, the type of acquisition function used for optimization did significantly contribute to the number of iterations required to
achieve convergence in all cases of outcome measures (HE and PI) and point locations (p < 0.001). Since the interaction term between types of acquisition function and seed point selection methods is
not significant (p > 0.1 for all conditions tested), the optimal acquisition function (in terms of convergence speed) of HIL Bayesian optimization is estimated to be EI. Therefore, our expectation
toward HIL Bayesian optimization setting is that LCB will require the largest number of iterations to establish convergence, while EI will require the smallest number for iterations for convergence
regardless of seed point selection methods.
This study has limitations. Since virtual HIL Bayesian optimization rely on our assumption that subject-specific Gaussian process models can fully represent actual human response, the validity of
these models directly relates to the validity of the results of Bayesian optimization obtained in this study. Another limitation is that noise is not fully eliminated to construct a model that relate
control input parameters and propulsive mechanics. As shown in Fig. 1 and Fig. 2, the model at the stride before intervention (red dashed line) have changes with control input parameters, which is
impossible as a change in output cannot anticipate a change in input. This result indicates that our optimization procedure for estimating hyperparameters and noise variance may need to be improved,
or that the relationship between input parameters and improvement of outcome measures (HE and PI) should be considered to cancel out noise from stride at intervention (stride 0) and stride before
intervention (stride −1) Gaussian process models.
Overall, our study identified a new method based on Gaussian process modeling to estimates the relationship between exoskeleton control parameters and specific gait features (HE and PI), and
established the effects of seed point selection method and acquisition function types in number of iterations for convergence. These results make foundation of promising approach for planning further
HIL robotic gait training experiment by focusing on specific gait features to target with training.
• * This work is supported in part by NSF-CMMI-1934650 and in part by NSF-CBET-1638007. | {"url":"https://www.biorxiv.org/content/10.1101/2021.01.14.426703v1.full","timestamp":"2024-11-10T01:37:06Z","content_type":"application/xhtml+xml","content_length":"186394","record_id":"<urn:uuid:d1fec982-48b8-466c-a27b-252480795656>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00439.warc.gz"} |
μ - (Principles of Data Science) - Vocab, Definition, Explanations | Fiveable
from class:
Principles of Data Science
In statistics, the symbol μ represents the population mean, which is the average value of a set of observations in a population. This measure provides a central point around which the data is
distributed, helping to summarize the data in a single value. Understanding μ is essential for various statistical analyses and plays a key role in descriptive statistics, as it aids in interpreting
data and comparing different datasets.
congrats on reading the definition of μ. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. μ is calculated by summing all values in the population and dividing by the number of observations in that population.
2. The population mean (μ) is different from the sample mean, which is denoted by the symbol x̄ (x-bar).
3. The value of μ provides insights into the overall trend and tendencies within the data, making it easier to understand large datasets.
4. In normal distributions, approximately 68% of observations fall within one standard deviation of μ, highlighting its importance in understanding variability.
5. Outliers can significantly affect μ, so it's crucial to assess data for anomalies when calculating the mean.
Review Questions
• How does μ differ from other measures of central tendency like median and mode?
□ μ represents the population mean and is calculated by averaging all values in a population. In contrast, the median is the middle value when data is sorted in order, providing a measure that
can be less affected by outliers. The mode is simply the most frequently occurring value in a dataset. While all three measures aim to describe central tendency, μ offers an overall average,
which can be influenced by extreme values unlike median and mode.
• Discuss how changes in individual data points affect the value of μ in a dataset.
□ Changes to individual data points can have a direct impact on the calculated value of μ. For instance, if a particularly high or low outlier is added or removed from the dataset, it can pull
the average up or down significantly. This sensitivity makes μ less robust against outliers compared to median or mode. Understanding this relationship helps statisticians make informed
decisions about whether to include certain data points when calculating the mean.
• Evaluate how understanding μ can enhance data interpretation in real-world applications.
□ Understanding μ enables better data interpretation across various real-world scenarios, such as public health assessments, economic analyses, and educational outcomes. By providing a single
average value for complex datasets, μ allows researchers and decision-makers to quickly grasp general trends and make informed comparisons between different groups or time periods.
Furthermore, awareness of how factors like outliers influence μ aids in critically analyzing results and ensuring accurate conclusions are drawn from statistical findings.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/principles-and-techniques-of-data-science/m","timestamp":"2024-11-07T00:49:44Z","content_type":"text/html","content_length":"156235","record_id":"<urn:uuid:7172037e-5527-46e1-be5e-b465343bb543>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00173.warc.gz"} |
[Solved] 1. The speed limit on some interstate hig | SolutionInn
Answered step by step
Verified Expert Solution
1. The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second? (b) How many miles
1. The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second? (b) How many miles per hour is this? 2. A car is traveling at a speed of 33 m/s. (a) What is
its speed in kilometers per hour? (b) Is it exceeding the 90 km/h speed limit? 3. Show that 1.0 m/s = 3.6 km/h. Hint: Show the explicit - steps involved in converting 1.0 m/s = 3.6 km/h. 4. American
football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 3.281 feet.) 5. Soccer fields vary in size. A large soccer field is
115 m long and 85 m wide. What are its dimensions in feet and inches? (Assume that 1 meter equals 3.281 feet.) 6. What is the height in meters of a person who is 6 ft 1.0 in. tall? (Assume that 1
meter equals 39.37 in.) 7. Mount Everest, at 29,028 feet, is the tallest mountain on the Earth. What is its height in kilometers? (Assume that 1 kilometer equals 3,281 feet.)
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Current Loop in a nonuniform magnetic field
• Thread starter PEZenfuego
• Start date
In summary, the conversation discusses solving for the net force on a current loop in a nonuniform magnetic field. The formula F=IlxB is used and integrated around the loop to find the net force. It
is important to note that the angle θ in the formula is not the same as the θ in the given diagram. Theta should remain constant as the loop has rotational symmetry.
Homework Statement
A nonuniform magnetic field exerts a net force on a current loop of radius R. The figure shows a magnetic field that is diverging from the end of a bar magnet. The magnetic field
at the position of the current loop makes an angle θ with respect to the vertical, as the same magnitude at each point on the current loop. (I know that I need to solve in terms of R, I, B, and θ
Homework Equations
The Attempt at a Solution
I fought the urge to use the force equation after substitution 2∏r for l. Instead I examined the force on one small segment and planned to integrate. The length would be in terms of arc length Δs.
The I would be a constant. It seems that the problem indicates that the magnetic field B is constant (surely that is an assumption because the distance from the magnet was not indicated, right?) I
also thought that the angle should be the only thing changing (I doubt this is a double integral problem). The subscript i indicates that this is for some segment i.
The first thing that jumps out at me is that we don't have the necessary Δθ, but we do have Δs. Δs=ΔθR, but this is for a different θ, right? So, here is where I got lost and thought that something
was wrong.
Next I tried relating it with torque.
But here again we have a different θ value, correct?
I would very much appreciate some help. Thank you!
What are you supposed to solve for?
Use the differential form of your F equation above, which is
d F = I d l x B for the force on an element d l of the loop.
Then integrate around the loop - an easy integration since the force is constant everywhere around the loop.
I know that the answer is to be 2πRIBsinθ, but I don't see how or why.
dF=IdlxB is the same as saying dF=IΔsxB or dF=IBsinθΔs integrating yields
F=IBssinθ or F=2πRsinθ.
But doesn't the value of theta change for each segment?
Oh wait! No, it doesn't. It should remain constant as the ring has rotational symmetry. Am I correct in my reasoning here?
Science Advisor
Homework Helper
Gold Member
2023 Award
It is important to note that in the formula F = I B Δs sinθ, θ is not the same θ as given in the diagram. Remember, in the formula, sinθ is coming from a cross product of Δs and B.
PEZenfuego said:
I know that the answer is to be 2πRIBsinθ, but I don't see how or why.
dF=IdlxB is the same as saying dF=IΔsxB or dF=IBsinθΔs integrating yields
F=IBssinθ or F=2πRsinθ.
But doesn't the value of theta change for each segment?
Oh wait! No, it doesn't. It should remain constant as the ring has rotational symmetry. Am I correct in my reasoning here?
That is correct! And take note of what tsny says about theta. Keep track of angles when you take your cross-product!
FAQ: Current Loop in a nonuniform magnetic field
1. What is a current loop in a nonuniform magnetic field?
A current loop in a nonuniform magnetic field refers to a closed loop of wire through which an electric current flows, placed in a magnetic field that varies in strength and direction. This causes
the loop to experience a force and torque, resulting in a change in its orientation.
2. What factors affect the behavior of a current loop in a nonuniform magnetic field?
The behavior of a current loop in a nonuniform magnetic field is affected by the strength and direction of the magnetic field, the current flowing through the loop, and the geometry of the loop (such
as its size and shape).
3. How does a current loop in a nonuniform magnetic field produce torque?
When a current-carrying loop is placed in a nonuniform magnetic field, the magnetic field exerts a force on each segment of the loop, causing a net torque on the loop. This torque can be calculated
using the cross product of the magnetic field and the current in the loop.
4. What are some real-life applications of current loops in nonuniform magnetic fields?
Current loops in nonuniform magnetic fields have various practical applications, including electric motors and generators, particle accelerators, and magnetic resonance imaging (MRI) machines. They
are also used in scientific research to study the behavior of electric currents and magnetic fields.
5. How can the behavior of a current loop in a nonuniform magnetic field be controlled?
The behavior of a current loop in a nonuniform magnetic field can be controlled by changing the strength and direction of the magnetic field, altering the current flowing through the loop, or
adjusting the geometry of the loop. This can be achieved using various techniques such as electromagnets, variable resistors, and changing the shape of the loop. | {"url":"https://www.physicsforums.com/threads/current-loop-in-a-nonuniform-magnetic-field.678688/","timestamp":"2024-11-14T02:25:10Z","content_type":"text/html","content_length":"106640","record_id":"<urn:uuid:462492f0-db67-4c6b-9991-2973d1384b15>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00514.warc.gz"} |
What is ROI and How To Calculate ROI? - Submit Guest Post - Instant Live your Post | Sarkari Result 2024 - Articlespringer
What is ROI and How To Calculate ROI?
Ever heard of what ROI is? As a business owner, chances are you have already made an investment in your company. To determine whether your investment is worthwhile and efficient, you can calculate
your return on investment or ROI. The more you understand ROI and how to calculate it, the better you will be able to make informed decisions about future investments.
In this article, we’ll define ROI, why it’s important, its limitations, and how to calculate it.
What is ROI?
Return on investment, or ROI, refers to the measurement of the financial benefits you get from investing. In other words, it is a way for businesses to determine investment efficiency.
Determining your company’s ROI is important when it comes to making important financial decisions. Depending on your ROI, you will be able to evaluate whether your investment is financially
viable. If your company’s ROI is high or positive, it’s usually a worthy investment. If it’s low or negative, it’s usually a bad investment.
When calculating ROI, it is important to consider the limitations that arise. Not only does ROI not adjust for risk and potentially large losses, but you also have to consider the holding period of
the investment.
For example, just because one investment has a high ROI and another has a low ROI, doesn’t mean one investment is better than another.
This is because ROI does not take into account the time period of each investment. It is possible that the timeframe for higher ROI is longer and the timeframe for lower ROI is much shorter. This can
be misleading when calculating ROI.
Despite its limitations, ROI is relatively easy to understand and calculate. As a business owner, you want your calculations to yield a high ROI that reflects the maximum return based on the money
you invest.
How to Calculate ROI?
There are several variations of the ROI formula, but they will all give the same result. Two of these formulas include the following:
ROI = investment profit / investment base
ROI = net income / investment costs
Using this formula, consider the following steps when calculating your ROI:
1. Determine the return on your investment
To begin your calculations, determine how much you earn on the investment you are measuring. For example, let’s say you own a $300,000 house. At the time of sale, it was $500,000. This means your
return on investment is $200,000 because that’s how much the price of the house has gone up since you bought it.
2. Determine the investment cost
Next, you will need the amount of money you paid for an investment. Using the example above, since the house is worth $300,000, $300,000 is the investment cost.
3. Calculate ROI
Using the first ROI formula listed above, you can now calculate ROI. Do this by dividing the investment profit by the investment base.
In the house example, you would divide $200,000 by $300,000 to get an ROI of 0.667. Since ROI is usually expressed as a percentage, multiply this value by 100. This will give you a 66.7% ROI.
Keep in mind that good or bad ROI often depends on how it’s measured. Usually, you will want a high ROI which ensures that you make wise choices in investing your time, money and effort that will
ultimately result in a profit for your business.
Example of ROI Calculation
Consider the following example when calculating ROI:
Example 1
Say you are working on a new advertising campaign for your department store. The advertising department spent 3,000,000 on various materials and generated 10,000,000 in revenue for the year.
To calculate ROI, you would divide your income (10,000,000) by the investment cost (3,000,000). This will result in an ROI of 3.33 or 33%.
Example 2
Let’s say you renovate your rented house and spend a total of 50,000,000 to update your kitchen and two bathrooms. Prior to your renovation, your rented house was worth 200,000,000. After the
renovation, it’s worth 300,000,000.
Your investment gain in this example is 100,000,000. To calculate ROI, divide 100,000,000 (investment return) divided by 200,000,000 (investment base) to get an ROI of 0.5. To get a percentage,
multiply this value by 100 to get a 50% ROI percentage.
ROI Type
In terms of ROI, there are three types of money you can make on your investment. They include the following:
• Interest: One way to earn on your investment is through interest. Savings accounts and bonds are two investments that will pay your company through interest.
• Capital gains: If you sell your investment for more than you paid for, this will result in a capital gain for your company.
• Dividends: Lastly, you can also get paid in the form of dividends. In this case, you will regularly receive a share of what the company produces.
Why is ROI Important?
ROI is important for several reasons. For starters, it can help you evaluate your investment and determine if it is worth the time, money and effort.
For example, if you are generating low ROI for a new marketing strategy that you have implemented, you may decide to discontinue it or approach the same strategy from a different angle to increase
your ROI.
In the end, if this marketing strategy does not bring you additional income or profit, there is no point in continuing to apply it because you will continue to face financial losses. Overall, ROI
helps you evaluate the efficiency of your investment. | {"url":"https://www.articlespringer.com/what-is-roi-and-how-to-calculate-roi/","timestamp":"2024-11-04T08:35:41Z","content_type":"text/html","content_length":"70823","record_id":"<urn:uuid:d07eeaa5-3962-47f6-b2c2-946534ae0966>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00327.warc.gz"} |
X-Ray Absorption from RTP and
X-Ray Absorption from RTP and \(\delta\)-Kick perturbation
This tutorial shows how to run a Real-Time Time-Dependent DFT calculation using the so-called \(\delta\)-kick approach to compute absorption electronic spectra.
For the corresponding Linear-Response approach, read the tutorial the X-Ray frequencies.
This tutorial is structured as follows:
• Brief theory overview
• CP2K input file overview
Along the RTP run, the time dependent dipole moment is sampled and it is then Fourier transformed to produce the absorption spectrum in the frequency domain. An example script is provided here.
Theory Overview
Response theory
In the perturbative regime and at the linear order, the time-dependent response of an electronic cloud can be decomposed in the Fourier space: the response at a given frequency is proportional to the
perturbation acting on the system at the same frequency. In particular, the induced dipole moment (or equivalently the induced current) at a given frequency, \(\mu^\omega\), is given by the product
of the polarizability matrix, \(\alpha(\omega, \omega)\), with the perturbative electric field at the same frequency, \(F^\omega\), within the dipolar approximation.
\[ \mu^\omega = \alpha(\omega, \omega) \cdot F^\omega \]
This quantity involves a dot product between the field vector and the polarizability matrix. For the next paragraphs, we will drop the vectorial behavior and discuss it later.
In Linear-Response-TDDFT approaches, one computes the excited state promoted by an electric field oscillating at a specific frequency \(\omega\) and then the transition dipole moment associated with
such electronic transition. This transition dipole moment is then used to derive the polarizability for resonant or non-resonant frequency.
In the Real-Time approach presented here, we will excite all the possible electronic transitions at the beginning of the simulation by providing an istantaneous pulse containing all the frequencies
The perturbed electronic wavefunction is then propagated and the fluctuations of the induced time dependent dipole moment are recorded . The polarizability tensor at any frequency is finally derived
from the induced dipole moment.
The \(delta\)-kick perturbation
To apply the \(delta\)-kick, i.e., an intense electric field in a very sort time, the electronic structure is first obtained at the ground state, for instance using DFT, and then perturbed with an
instantaneous electric field:
\[ F(t) = F^0 \delta(t) \]
The same result is obtained by applying a constant field with a very narrow Gaussian envelope. This field perturbes the ground state wave-function at \(t=0^-\). Then, this excited wave-function is
propagated in real time by numerically integrating the time dependent Schroedinger equation.
Theinstantaneaous field can be written in the frequency domain as
\[ F^\omega = \frac{F^0}{2 \pi} \int_{-\infty}^{+ \infty} \delta(t) e^{i \omega t} dt = \frac{F^0}{2 \pi} e^{i \omega \times 0} = \frac{F^0}{2 \pi} \]
The field amplitude in the Fourier space is \(F^0 / 2 \pi\) for all frequencies: this instantaneous perturbation does indeed contain all the frequencies. The time-dependent wave-function can be
described to be the ground state one plus all possible excited states. The total dipole moment results from the superposition of all possible oscillations related to all the excited states. This
complex behavior in the time domain becomes simple in the frequency domain, since we know the amplitude of the perturbation applied at each frequency:
\[ \mu^\omega = \frac{1}{2 \pi} \alpha(\omega, \omega) F^0 \]
Therefore, in order to get the polarizability \(\alpha(\omega, \omega)\), one prepares the perturbed state at \(t=0^-\) for a given field amplitude of the field, then propagates the wave-function in
real time. The time dependent dipole moment is extracted along the propagation and it is finally Fourier transformed to calculate the polarizability as
\[\begin{split} \text{Re} \left[ \alpha(\omega, \omega) \right] = 2 \pi \frac{\text{Re} \left[ \mu^\omega \right] }{F^0} \\ \text{Im} \left[ \alpha(\omega, \omega) \right] = 2 \pi \frac{\text{Im} \
left[ \mu^\omega \right] }{F^0} \end{split}\]
The amplitude of the dipole moment in the Fourier space may be very small, if the frequencies are far from resonances. But, in principle, one can extract the whole spectrum from one RTP run. As a
matter of fact, the whole spectrum from core to valence excitations is very broad and for numerical efficiency the propagation parameters are set to focus only on one specific part of the spectrum.
Absorption spectrum
Under the assumption of linear response to the perturbation,
the real and imaginary part of the calculated frequency-dependent polarizability define the nature of the response. If the frequency is at an electronic resonance, then the polarizability has an
imaginary part. The polarizability is pure real off resonances.
From one Real-Time propagation, one can obtain three components of the polarizability tensor. For instance, applying a \(\delta\)-kick along \(x\) provides the components \(\alpha_{xx}\), \(\alpha_
{yx}\) and \(\alpha_{zx}\), corresponding to the Fourier transform in \(x\), \(y\) and \(z\), respectively. Therefore, to get the full polarizability tensor, three RTP runs are needed, one per
Cartesian axis.
To compare with experiments, one often assumes that the system is averaged over all possible orientations in space. Then, the absorption spectrum \(I(\omega)\) is proportional to:
\[ I(\omega) \propto \sum_{i=x,y,z} \text{Im} \left[ \alpha_{ii}(\omega, \omega) \right] \]
CP2K Input
The following example is to simulate the response of carbon-monoxide in the gas phase when applying the \(\delta\)-kick along the perpendicular to the CO bond. The analysis is done within the X-Ray
range, but in principle, any other range of frequency can be sampled using the same approach, providing that fime step and length of the propagation are opportunely adjusted.
The ifollowing input file RTP.inp is for a simulation at the DFT/PBEh level of theory, it uses the GAPW approah and the density is expandedn in the all-electron PCSEG-2 basis sets.
PROJECT RTP
ENSEMBLE NVE
STEPS 50000
TIMESTEP [fs] 0.00078
TEMPERATURE [K] 0.0
&END MD
METHOD QS
APPLY_DELTA_PULSE .TRUE.
DELTA_PULSE_DIRECTION 1 0 0
DELTA_PULSE_SCALE 0.001
MAX_ITER 100
MAT_EXP ARNOLDI
EPS_ITER 1.0E-11
INITIAL_WFN SCF_WFN
PERIODIC .FALSE.
&END REAL_TIME_PROPAGATION
BASIS_SET_FILE_NAME BASIS_PCSEG2
POTENTIAL_FILE_NAME POTENTIAL
CUTOFF 1000
NGRIDS 5
REL_CUTOFF 60
&END MGRID
METHOD GAPW
EPS_FIT 1.0E-6
&END QS
MAX_SCF 500
SCF_GUESS RESTART
EPS_SCF 1.0E-8
&END SCF
POISSON_SOLVER WAVELET
PERIODIC NONE
&END POISSON
&XC_FUNCTIONAL PBE
SCALE_X 0.55
&END XC_FUNCTIONAL
FRACTION 0.45
POTENTIAL_TYPE TRUNCATED
CUTOFF_RADIUS 7.0
&END INTERACTION_POTENTIAL
&END HF
&END XC
&MULLIKEN OFF
&END MULLIKEN
&HIRSHFELD OFF
&END HIRSHFELD
PERIODIC .FALSE.
FILENAME =dipole
COMMON_ITERATION_LEVELS 100000
MD 1
&END EACH
&END MOMENTS
&END PRINT
&END DFT
ABC 10 10 10
ALPHA_BETA_GAMMA 90 90 90
PERIODIC NONE
&END CELL
COORD_FILE_NAME carbon-monoxide_opt.xyz
COORD_FILE_FORMAT XYZ
&END TOPOLOGY
&KIND C
BASIS_SET pcseg-2
POTENTIAL ALL
&END KIND
&KIND O
BASIS_SET pcseg-2
POTENTIAL ALL
&END KIND
&END SUBSYS
Time step
The time step used to propagate the wave-function is adapted to optimally resolve the frequency of interest. Typically, the time-step choice results from a trade-off between the resolution of the
highest frequency of interest and the computational cost to sufficiently extend the sampling. As a rule of thumb, the time step should be about one order of magnitude smaller than the period
corresponding to the maximum frequency to be resolved. For core excitation from the Oxygen 1s, the K-edge is around 530 eV, corresponding to a period of 0.0078 fs.
According to Linear Response TDDFT calculations (using the XAS_TDP module, see this tutorial), the first transition occurs at 529 eV, with an oscillator strength of 0.044 a.u. For the \(\delta\)-kick
approach, this means that the induced dipole moment should have an oscillatory component around the frequency corresponding to 529 eV. Hence, the TIMESTEP is set ten times smaller than the maximum
frequency, i.e., to [fs] 0.00078, which allows the sampling of the fasted oscillation by means of at 10 propagation steps.
The total time of the simulation is then determined by the maximum number of time steps, here 500000. The longer is the sampling, the smoother and sharper is the resulting spectrum. Longer simulation
times give a better resolution of the spectral peaks, i.e., a more accurate determination of the resonances. There is no always valid rule to decide how long the propagation needs to be extended,
because the fluctuations might be strongly system dependent. It is always a good idea to check whether the obtained spectrum has converged with respect to further extension of the sampling.
Field properties
The field perturbation (the \(\delta\)-kick) is defined by its amplitude and polarization. The amplitude depends on the system of interest, a typical value is \(10^{-3}\). Also in this case a
convergence check by running more than one propagation at different amplitudes is the best way to asses the parameter. Within the linear regime, the response of the system should double by twice the
field’s amplitude. Note that when applying a too-low field, numerical noise might become dominant. For isolated CO, \(10^{-3}\) is a good value.
Please note that the actual perturbation applied in CP2K is not DELTA_PULSE_SCALE and depends on the cell size. The corresponding amplitude value is written in the output file, as part of the RTP
The field’s polarization determines what excited states are most probably triggered, according to the selection rules. For the present example, we know that the electronic transition at 529 eV is
perpendicular to the CO bond. For the geometry we are using, it means that the field polarization should be along \(x\) (or \(y\)) , i.e., DELTA_PULSE_DIRECTION to 1 0 0 and DELTA_PULSE_SCALE to
Note about the choice of the Gauge
For an isolated system, the length gauge form coupling coupling the position and the electric field operators can be employed to all perturbative orders by setting PERIODIC to .FALSE..
For condensed phase systems, the velocity gauge form, implying a gauge transformation involving the vector potential is instead required, and in CP2K it is activated by setting PERIODIC to .TRUE.. In
this case, the perturbation will be applied only within the first order. | {"url":"https://manual.cp2k.org/trunk/methods/properties/x-ray/delta-kick.html","timestamp":"2024-11-04T15:01:45Z","content_type":"text/html","content_length":"78943","record_id":"<urn:uuid:c181c17b-ddca-454e-acf3-d83890a38068>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00867.warc.gz"} |
Differentiation: Applications of derivatives
Extreme values
Maxima and minima
The highest value of a part of a graph is called a local maximum.
The lowest value of a part of a graph is called a local minimum.
Both are extreme values of a function.
Extreme values on restricted domain
If we view a function on a restricted domain, the values on the boundaries of the domain can also be a local maxima or minima and thus an extreme value.
For example, we can view the function #f(x)=x^2# on the domain #\ivcc{2}{5}#. We now have a local minimum at #x=2# and a local maximum at #x=5#. When #x^2# is considered on its normal domain, we only
have a local minimum, at #x=0#.
Extrema are function values
So far in this course, we often specifically looked at the #x# values of special points, but the maximums and minimums are the #y# values of these points..
Thus, in the example a local maximum of the green graph is #3.5# and #\red{\text{not}}# #0#. A local minimum of the blue graph is #0.5# and #\red{\text{not}}# #0#.
Global maxima and minima
We have seen that the local maxima and minima are the highest point on a part of the graph. The global maximums and minimums are the highest point of the whole graph.
In the example of the green graph, the local maximum is also global. Similarly, in the blue graph, the local minimum is also global. This is not always the case.
Even if there are local maximums and minimums, there may not always be a global maximum or minimum.
Using the derivative, we can easily calculate the extreme values of a function.
Extreme value
If a function #\blue{f(x)}# has a local maximum or minimum at #x=\orange{c}# then #\green{f'(\orange{c})}=0#.
\[\begin{array}{rcl}\blue{f(x)}&=&\blue{x^2}\\ \green{f'(x)}&=&\green{2x}\\ \green{f'(\orange{0})}&=&0\end{array}\]
The function must also be differentiable in point #x=\orange{c}#. In this course we will not look at functions that are not differentiable, but this can occur in practice.
Horizontal tangent line If the derivative in a point equals zero, then the tangent line to the function is horizontal, as we can see in the following picture.
Boundary values This theorem does not hold if #c# is a boundary value of the domain the function is defined on. For example, if we take #f(x)=x^2# on the domain #\ivcc{2}{5}#, then #2# and #5# are
extreme values, but the derivative does not equal #0# in those points.
One way
The statment only holds one way. If the derivative #\green{f'(x)}# of a function #\blue{f(x)}# equals #0# in a point #\orange{c}#, this does not immediately mean that #\blue{f(x)}# has an extreme
value in #\orange{c}#. For example, the derivative of #\blue{f(x)=x^3}# is equal to #0# in the point #\orange{0}#, but does obviously not have an extreme value here.
Calculating extreme values
Step-by-step Example
Determine the extreme values of a function #f(x)#. Determine for each extreme value whether it #\qqquad \begin{array}{rcl}f(x)\phantom{'}&=&x^4-2x^2\end{array}#
is a local minimum, local maximum or neither.
Step Calculate the derivative #f'(x)#. #\qqquad \begin{array}{rcl}f'(x)&=&4x^3-4x\end{array}#
Step Solve #f'(x)=0# to find the #x# coordinates of the points which are possibly an extreme value. #\qqquad \begin{array}{rcl} 4x^3-4x&=&0\\ x&=&0 \lor 4x^2-4=0\\ x&=&0\lor x^2=1\\ \green{x}&\green
2 {=}&\green{0} \lor \blue{x=-1} \lor\orange{x=1}\end{array}#
Step Sketch the graph to find out which points are a local maximum and which points are a local plaatje
3 minimum (and which points may not be a maximum or minimum).
Step Substitute the obtained #x# coordinates in #f(x)# and determine the extreme values this way. #f(\green{0})=0#
Therefore, local minimum is #-1# and
local maximum is #0#
Calculating extrema of functions is something which appears often in optimisation problems. Problems are described by functions, whose minimum or maximum are determined.
We will give a very easy example. Suppose a farmer wants to fence a rectangular field, and he has bought #500# metres of fence. The farmer wants to maximize the fenced area, and wants to know the
best ratio of the rectangle. We first note that this area #A# is given by #x\cdot y#, where #x# is the width and #y# is the depth of the rectangle. The farmer bought #500# metres of fence, which has
to be distributed over the width and depth, which gives us #2x+2y=500#. Rearranging gives \[y=250-x\] We insert this in the function for the are and get \[A=x\cdot(250-x) = 250\cdot x - x^2\] This is
the formule whose maximum we would like to calculate. Following the step-by-step approach we get #x=125#. Thus, the farmer should fence a square area in order to maximize the area fenced.
In most applications, the functions are very complicated and contains much more variables. However, we won't be studying this in this course.
Sign analysis chart
As an alternative to step #3#, we can make use of a so called sign analysis chart. In step #2# we found the zeroes #x_1,\ldots, x_n# of the derivative #f'(x)#. By definition, there are no zeroes in
the intervals #\ivoo{x_i}{x_{i+1}}#. This means that the values of #f'(x)# in such a interval are all positive or all negative. If we take one point in the interval and substitute it in #f'(x)# we
immediately know the sign of the interval: positive or negative. We now write the signs down for all the intervals in a sign analysis chart
│Interval│#\ivoo{-\infty}{x_1}# │#x_1#│#\ivoo{x_1}{x_2}#│#x_2#│#\ldots#│#x_n#│#\ivoo{x_n}{\infty}#│
│Sign │#+# or #-# │#0# │#+# or #-# │#0# │#\ldots#│#0# │#+# or #-# │
We can use this sign analysis chart to determine whether a zero #x_1# of #f'(x)# corresponds to a local maximum, minimum, or none. This is done by considering the signs of the intervals surrounding
it, which are #\ivoo{x_{i-1}}{x_i}# and #\ivoo{x_i}{x_{i+1}}#.
• If the sign of #\ivoo{x_{i-1}}{x_i}# is positive and the sign of #\ivoo{x_i}{x_{i+1}}# is negative, then #x_i# corresponds to a local maximum.
• If the sign of #\ivoo{x_{i-1}}{x_i}# is negative and the sign of #\ivoo{x_i}{x_{i+1}}# is positive, then #x_i# corresponds to a local minimum.
• If the sign of #\ivoo{x_{i-1}}{x_i}# is positive and the sign of #\ivoo{x_i}{x_{i+1}}# is positive, then #x_i# does not correspond to an extreme value.
• If the sign of #\ivoo{x_{i-1}}{x_i}# is negative and the sign of #\ivoo{x_i}{x_{i+1}}# is negative, then #x_i# does not correspond to an extreme value.
In the example we found the zeroes #x_1=-1, x_2=0# and #x_3=1#. This yields the following sign analysis chart.
│Interval│#\ivoo{-\infty}{-1}# │#-1#│#\ivoo{-1}{0}#│#0#│#\ivoo{0}{1}#│#1#│#\ivoo{1}{\infty}#│
│Sign │#-# │#0# │#+# │#0#│#-# │#0#│#+# │
We see that both #x=-1# and #x=1# correspond with a local minimum, and that #x=0# corresponds with a local maximum.
Give the two values of #x# for which the function #f# given by \[f(x)=x^3-4x^2+4x+2\] has an extreme value (a local minimum or local maximum).
The smaller value of #x# is indicated by #x_-# and the greater value by #x_+#. Write your answers as a simplified fraction.
#x_-=# #{{2}\over{3}}# and #x_+=# #2#
Step 1 We determine the derivative of #f(x)=x^3-4x^2+4x+2#. This is equal to:
We determine the #x# coordinates of the potential extreme values by making the derivative equal to #0# by solving the equation.
\[\begin{array}{rcl}3x^2-8x+4&=&0 \\ &&\phantom{xxx}\blue{\text{the equation we need to solve}}\\
Step 2 x=\frac{8-\sqrt{(-8)^2-4\cdot 3\cdot 4}}{2\cdot 3} &\lor& x=\frac{8+\sqrt{(-8)^2-4\cdot 3\cdot 4}}{2\cdot 3} \\ &&\phantom{xxx}\blue{\text{quadratic formula}}\\
x=\frac{8-\sqrt{16}}{6} &\lor& x=\frac{8+\sqrt{16}}{6} \\ &&\phantom{xxx}\blue{\text{simplified}}\\
x=\frac{8-4}{6} &\lor& x=\frac{8+4}{6} \\ &&\phantom{xxx}\blue{\text{simplified}}\\
x={{2}\over{3}} &\lor& x=2 \\ &&\phantom{xxx}\blue{\text{simplified}}\end{array}\]
We draw the graph of #f(x)#.
Step 3
Therefore, there is a local maximum at #x={{2}\over{3}}# and a local minimum at #x=2#. Hence, both obtained #x# values are part of an extreme value.
Therefore, #x_-={{2}\over{3}}# and #x_+=2#
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perplexus.info :: Sequences : Colored triangle
Begin with a finite sequence of blocks in a row, each in one of 3 colors: red, blue, yellow.
Below each pair of neighboring blocks place a new block with the color rule: If the blocks are the same color use that color but if they are different use the third color.
r b y y b
y r y r
b b b
b b
How can the color of the last block be easily predicted from the top row?
Note: I don't know the full answer but can solve special cases. | {"url":"http://perplexus.info/show.php?pid=11550&cid=60423","timestamp":"2024-11-10T01:56:15Z","content_type":"text/html","content_length":"12715","record_id":"<urn:uuid:b831d5b9-5203-4e37-b4f5-9617c228b7be>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00660.warc.gz"} |
Explainable time series classification with X-ROCKET | dida blog
Explainable time series classification with X-ROCKET
With the lasting hypes in the domains computer vision and natural language processing, time series are frequently overlooked when talking about impactful applications of machine learning. However,
time series data is ubiquitous in many domains and predictive modeling of such data often carries significant business value. One important task in this context is time series classification, which
is attracting rising levels of attention due to its diverse applications in domains such as finance, healthcare, and manufacturing.
Numerous techniques have been developed to tackle the unique challenges posed by time series data, where increased capacity often comes at the expense of interpretability and computational speed.
While the race for a common state-of-the-art embedding model for time series continues, the RandOm Convolutional KErnel Transform (ROCKET) of Dempster et al. (2020) has gained significant attention
as a simple yet powerful encoder model. In this series of articles, I will introduce the model’s underlying ideas, and show an augmentation that adds explainability to its embeddings for use in
downstream tasks. It consists of three parts:
1. This first part provides background information on time series classification and ROCKET.
2. The second part sheds light on the inner workings of the X-ROCKET implementation.
3. The third part takes us on an exploration of X-ROCKET’s capabilities in a practical setting.
The fundamentals of time series classification
A common task in the time series domain is to identify which of a set of categories an input belongs to. For example, one might be interested in diagnosing the state of a production machine given a
sequence of sensor measurements or in predicting the health of an organism from biomedical observations over a time interval. Formally, the problem can be described as follows: Given a sequence of
observations at a regular frequency, calculate the probabilities of the input belonging to one of a fixed set of classes. The input data for each example is usually structured as a 1D-array of
numerical values in the univariate case, or a 2D-array if there are multiple channels. A prediction model then calculates class probabilities as its output. In this context, models are commonly
composed of an encoder block that produces feature embeddings, and a classification algorithm that processes the embeddings to calculate the output probabilities, as schematically indicated in the
diagram below.
Illustration of a time series classification pipeline (drawn in excalidraw).
While classification algorithms in machine learning have matured, it is less clear how to extract suitable features from time series inputs. Traditional time series approaches, such as Dynamic Time
Warping and Fourier transforms, have shown promise in handling time series similarity and feature extraction. More recently, with the advent of deep learning, Recurrent Neural Networks (RNNs) and
Convolutional Neural Networks (CNNs) have emerged as dominant methodologies to capture sequential patterns and spatial features, respectively. Finally, Transformer-based models with temporal
attention have shown promise to further advance the field of time series classification in the most up-to-date research (e.g. Zerveas et al. (2021)).
Despite these advancements, there are still substantial challenges to harvesting time series data. Where images or texts are immediately interpretable by our human brains in most cases, examining the
fluctuations in time series recordings can be unintuitive to the extent that it is impossible to assign class labels in the first place. In particular, it is often unclear how informative specific
time series recordings are in the first place, which is aggravated by the widespread prevalence of noise. Hence, it is an open question how the data should be processed to extract potential signals
from an input. Additionally, unlike images, time series often vary in terms of length, so methods for feature extraction should be able to summarize inputs in fixed-dimensional embedding vectors
independent of input size. Finally, time series data may or may not be stationary, which potentially has adverse effects on prediction quality.
What works?
So what is the go-to-model for time series classification? Alas, the answer is not that simple. This is mainly due to the lack of widely accepted benchmarks, which makes it impossible to fairly
compare the numerous and diverse models proposed in the literature. But even if one wanted to construct such a unified benchmark dataset, it is not clear what it would contain to be representative of
the diversity that is time series. In other words, measuring a model’s performance on low-frequency weather data might not be a good indication of its success with high-frequency audio files or DNA
sequences. To get a sense of how different data in the time series domain can be, compare for example the visualizations of examples from various datasets in Middlehurst et al. (2023) below.
Moreover, there is an important distinction between univariate and multivariate time series, that is, if one or more different variables are being measured simultaneously. Unfortunately, evidence is
particularly thin for the multivariate case, which is the more relevant case in many practical applications.
Visualizations of examples from various time series datasets from Middlehurst et al. (2023).
Having said that, there are a few resources that attempt to compare different methods in the domain of time series classification. On the one hand, the constantly updated time series classification
leaderboard on Papers with code provides scores for a few models on selected datasets. On the other hand, members of the research group behind the time series classification website have published
papers (compare, e.g., Bagnall et al. (2017), Ruiz et al. (2021), and Middlehurst et al. (2023)) that conduct horse races between time series classification methods on their time series data archive.
While the former favors a variety of RNNs and CNNs on its benchmarks, non-deep learning methods such as ROCKET fare particularly well on the latter. Therefore, it would be presumptuous to declare a
general winner, and the answer is a resolute “well, it depends”.
In many cases, there are additional considerations besides pure performance that tip the scales when it comes to model choice. With limited availability of training data, more complex and capable
models that require extensive training are often out of place. Ideally, there would be a pre-trained encoder model that could be used out-of-the-box to extract meaningful patterns from any time
series input without additional training and could be fine-tuned to a specific use case with moderate effort as is the case in computer vision or NLP. Hence, there is often a trade-off between
performance and computational efficiency. Moreover, practical applications often require predictions to be explainable; that is, domain experts often demand to understand what features of the input
time series evoke a prediction. This is particularly true for sensitive use cases such as in health care or for autonomous driving. Therefore, choosing an explainable model is often crucial for the
suitability and credibility of machine learning techniques.
Team ROCKET to the rescue
One relatively simple modeling approach for time series embeddings is the so-called ROCKET, short for RandOm Convolutional KErnel Transform. This methodology was first introduced in Dempster et al.
(2020) and has been further developed in subsequent research papers. Noteworthy variants here are the MiniROCKET of Dempster et al. (2021), the MultiROCKET of Tan et al. (2022), and HYDRA of Dempster
et al. (2023). A main advantage over more complex methods is that ROCKET models are very fast in terms of computation and do normally not require any training to learn an informative embedding
mapping, while predictive performance is on par with state-of-the-art models. For example, Ruiz et al. (2021) find that training time is orders of magnitude faster for ROCKET compared to other time
series classification algorithms that achieve similar accuracy (see image below). This difference mainly stems from the fact that ROCKET encoders scan an input for a pre-defined set of possibly
dispensable patterns and then only let the classifier learn which ones matter, instead of learning everything from scratch.
Model comparison chart taken from Ruiz et al. (2021).
The main idea behind ROCKET encodings banks on the recent successes of Convolutional Neural Networks (CNNs) and transfers them to feature extraction in time series datasets. In contrast to most CNNs
in the image domain, however, the architecture does not involve any hidden layers or other non-linearities. Instead, a large number of preset kernels is convolved with the input separately, resulting
in a transformation that indicates the strength of occurrences of the convolutional patterns in different parts of the input sequence. This process is repeated with various dilation values, which is
the same as scanning at different frequencies. As for the choice of filters, the original paper suggests using random kernels, while later installments use a small set of deterministic patterns.
Next, the high-dimensional outputs of this step are pooled across time via proportion of positive values pooling (PPV), that is, by counting the times when the convolutional activations surpass
channel-wise bias thresholds which can be learned from representative examples. As a result, the output of the encoder is a feature vector that summarizes the input time series independent of its
length. The transformed features can then serve as the input to any prediction algorithm that can deal with feature redundancy. For example, the original work advises to use simple algorithms like
regularized linear models. For a more detailed explanation of the transformations, please refer to the original authors’ paper or to the more detailed descriptions in the second installment of this
So if ROCKET achieves state-of-the-art performance while being computationally much more efficient than most methods, what could possibly go wrong? Well oftentimes, performance is not everything…
Coming back to the explainability requirements that machine learning models often encounter in practice, is ROCKET a suitable model? As it comes, the answer is no. However, the algorithm requires
only slight changes to attach meaning to its embeddings. In the second part, I will demonstrate how this can be achieved by means of a slightly altered implementation, the explainable ROCKET — or
short, X-ROCKET.
• Bagnall, A., Lines, J., Bostrom, A., Large, J., & Keogh, E. (2017). The great time series classification bake off: a review and experimental evaluation of recent algorithmic advances. Data mining
and knowledge discovery, 31, 606–660.
• Dempster, A., Petitjean, F., & Webb, G. I. (2020). ROCKET: exceptionally fast and accurate time series classification using random convolutional kernels. Data Mining and Knowledge Discovery, 34
(5), 1454–1495.
• Dempster, A., Schmidt, D. F., & Webb, G. I. (2021, August). Minirocket: A very fast (almost) deterministic transform for time series classification. In Proceedings of the 27th ACM SIGKDD
conference on knowledge discovery & data mining (pp. 248–257).
• Dempster, A., Schmidt, D. F., & Webb, G. I. (2023). Hydra: Competing convolutional kernels for fast and accurate time series classification. Data Mining and Knowledge Discovery, 1–27.
• Middlehurst, M., Schäfer, P., & Bagnall, A. (2023). Bake off redux: a review and experimental evaluation of recent time series classification algorithms. arXiv preprint arXiv:2304.13029.
• Ruiz, A. P., Flynn, M., Large, J., Middlehurst, M., & Bagnall, A. (2021). The great multivariate time series classification bake off: a review and experimental evaluation of recent algorithmic
advances. Data Mining and Knowledge Discovery, 35(2), 401–449.
• Tan, C. W., Dempster, A., Bergmeir, C., & Webb, G. I. (2022). MultiRocket: multiple pooling operators and transformations for fast and effective time series classification. Data Mining and
Knowledge Discovery, 36(5), 1623–1646.
• Zerveas, G., Jayaraman, S., Patel, D., Bhamidipaty, A., & Eickhoff, C. (2021, August). A transformer-based framework for multivariate time series representation learning. In Proceedings of the
27th ACM SIGKDD conference on knowledge discovery & data mining (pp. 2114–2124).
This article was created within the “AI-gent3D — AI-supported, generative 3D-Printing” project, funded by the German Federal Ministry of Education and Research (BMBF) with the funding reference
02P20A501 under the coordination of PTKA Karlsruhe. | {"url":"https://dida.do/blog/explainable-time-series-classification-with-x-rocket","timestamp":"2024-11-04T11:13:41Z","content_type":"text/html","content_length":"63432","record_id":"<urn:uuid:39ec276e-5f17-4e66-9d9b-ed5b123b1a6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00544.warc.gz"} |
On the Hecke Module of GLₙ(k[[z]])\GLₙ(k((z)))/GLₙ(k((z²)))
[See Abstract in text of thesis for correct representation of mathematics]
Every double coset in GLₘ(k[[z]])\GLₘ(k((z)))/GLₘ(k((z²))) is uniquely represented by a block diagonal matrix with diagonal blocks in { 1,z, (11 z \\0 zⁱ \\) (i>1) } if char(k) ≠ 2 and k is a finite
field. These cosets form a (spherical) Hecke module H(G,H,K) over the (spherical) Hecke algebra H(G,K) of double cosets in K\G/H, where K=GLₘ(k[[z]]) and H=GLₘ(k((z²))) and G=GLₘ(k((z))). Similarly
to Hall polynomial hλ,ν^µ from the Hecke algebra H(G,K), coefficients hλ,ν^µ arise from the Hecke module. We will provide a closed formula for hλ,ν^µ, under some restrictions over λ, ν, µ.
Item Type: Thesis (Dissertation (Ph.D.))
Subject Keywords: Hecke Module, $\text{GL}_m(k[[z]])\backslash \text{GL}_m(k((z)))/\text{GL}_m(k((z^2)))$, symmetric elliptic difference equation
Degree Grantor: California Institute of Technology
Division: Physics, Mathematics and Astronomy
Major Option: Mathematics
Thesis Availability: Public (worldwide access)
Research Advisor(s): • Rains, Eric M.
Thesis Committee: • Mantovan, Elena (chair)
• Conlon, David
• Huang, Jia
• Rains, Eric M.
Defense Date: 27 November 2023
Non-Caltech Author Email: yuhuijin1995 (AT) gmail.com
Record Number: CaltechTHESIS:12082023-083025167
Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:12082023-083025167
DOI: 10.7907/d0bn-5e47
Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code: 16257
Collection: CaltechTHESIS
Deposited By: Yuhui Jin
Deposited On: 17 Jan 2024 17:35
Last Modified: 17 Jan 2024 17:35
Thesis Files
PDF - Final Version
See Usage Policy.
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Eureka Math Grade 8 Module 4 Lesson 14 Answer Key
Engage NY Eureka Math 8th Grade Module 4 Lesson 14 Answer Key
Eureka Math Grade 8 Module 4 Lesson 14 Exercise Answer Key
Exercise 1.
Find at least four solutions to graph the linear equation 1x+2y=5.
Exercise 2.
Find at least four solutions to graph the linear equation 1x+0y=5.
Exercise 3.
What was different about the equations in Exercises 1 and 2? What effect did this change have on the graph?
In the first equation, the coefficient of y was 2. In the second equation, the coefficient of y was 0. The graph changed from being slanted to a vertical line.
Exercises 4–6
Students complete Exercises 4–6 independently. Students need graph paper to complete the exercises.
Exercise 4.
Graph the linear equation x=-2.
Exercise 5.
Graph the linear equation x=3.
Exercise 6.
What will the graph of x=0 look like?
The graph of x=0 will look like a vertical line that goes through the point (0,0). It will be the same as the y-axis.
Exercises 7–9
Students complete Exercises 7–9 independently or in pairs in preparation for the discussion that follows. Students need graph paper to complete the exercises.
Exercise 7.
Find at least four solutions to graph the linear equation 2x+1y=2.
Exercise 8.
Find at least four solutions to graph the linear equation 0x+1y=2.
Exercise 9.
What was different about the equations in Exercises 7 and 8? What effect did this change have on the graph?
In the first equation, the coefficient of x was 2. In the second equation, the coefficient of x was 0. The graph changed from being a slanted line to a horizontal line.
Exercises 10–12
Students complete Exercises 10–12 independently. Students need graph paper to complete the exercises.
Exercise 10.
Graph the linear equation y=-2.
Exercise 11.
Graph the linear equation y=3.
Exercise 12.
What will the graph of y=0 look like?
The graph of y=0 will look like a horizontal line that goes through the point (0,0). It will be the same as the x-axis.
Eureka Math Grade 8 Module 4 Lesson 14 Exit Ticket Answer Key
Question 1.
Graph the linear equation ax+by=c, where a=0, b=1, and c=1.5.
Question 2.
Graph the linear equation ax+by=c, where a=1, b=0, and c=-\(\frac{5}{2}\).
Question 3.
What linear equation represents the graph of the line that coincides with the x-axis?
Question 4.
What linear equation represents the graph of the line that coincides with the y-axis?
Eureka Math Grade 8 Module 4 Lesson 14 Problem Set Answer Key
Students need graph paper to complete the Problem Set.
Question 1.
Graph the two-variable linear equation ax+by=c, where a=0, b=1, and c=-4.
Question 2.
Graph the two-variable linear equation ax+by=c, where a=1, b=0, and c=9.
Question 3.
Graph the linear equation y=7.
Question 4.
Graph the linear equation x=1.
Question 5.
Explain why the graph of a linear equation in the form of y=c is the horizontal line, parallel to the x-axis passing through the point (0,c).
The graph of y=c passes through the point (0,c), which means the graph of y=c cannot be parallel to the y-axis because the graph intersects it. For that reason, the graph of y=c must be the
horizontal line parallel to the
Question 6.
Explain why there is only one line with the equation y=c that passes through the point (0,c).
There can only be one line parallel to another that goes through a given point. Since the graph of y=c is parallel to the x-axis and it goes through the point (0,c), then it must be the only line
that does. Therefore, there is only one line that is the graph of the equation y=c that passes through (0,c).
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abstractmath.org 2.0
help with abstract math
Produced by Charles Wells Revised 2017-03-03
Introduction to this website website TOC website index blog
back to head of sets chapter
The definition of set can be stated in several different ways, all of them complicated. The most widely used definition is based on the Zermelo-Fraenkel axioms.
It is not necessary to understand or even know the Zermelo-Fraenkel axioms to understand sets as they are used in undergraduate math or large parts of graduate math. Nearly everything having to do
with sets in ordinary mathematical practice derives from the Method of Comprehension, so there is usually no need for the axiomatic definition. In that sense, the following specification contains
everything you need to know about sets for most mathematical purposes.
The concept of specification of a math object is discussed in the chapter on Definitions.
Specification for sets
A set is a single math object distinct from but completely determined by what its elements are.
This specification tells you the operative properties of a set rather than giving a definition in terms of previously known objects.
An embarrassing difficulty
The specification just given is not a mathematical definition. In particular, in some situations that usually do not occur in most branches of math, a bunch of elements may not correspond to a set.
One such example is this:
There is no "set of all sets".
In other words, you can't have a set that is completely determined by the fact that its elements are all the sets that exist. This follows from Cantor's Theorem. In most cases, if you think up a
bunch of elements, they do form a set.
The important thing to understand is what the specification means in practice. That is the subject of the next section:
Consequences of the specification for sets
Single math object
• A set is not merely a typographically convenient way to define a certain collection of things. A set is a math object, just like the number $143$ and the sine function and the real line.
• The numbers $3$ and $42$ are two different things. The set $\{3,42\}$ is one thing.
Take the definition of a set seriously
If someone defines $S$ as the set of all integers bigger than $3$, then the spec means you know all these things:
• $4$ and $10^{42}$ are elements of $S$.
• $3$ and $-99$ are not elements of $S$.
• $S$ is not any of the integers bigger than $3$, because $S$ is a set and an integer is not a set.
• $S$ is not all the integers bigger than $3$ because $S$ is just one thing.
Order does not matter
In list notation, the order in which you list the elements of a set is irrelevant for the purposes of determining what the set is. This follows directly from the specification.
• The set $\{1, 2, 4, 5\}$ by definition contains the numbers $1$, $2$, $4$ and $5$ and nothing else.
• So does the set $\{1,5,4,2\}$. The presence of "$5$" in the expression "$\{1,5,4,2\}$" tells you that $5$ is in the set. It does not matter if you say $5$ is in the set before or after you say
$4$ is in the set.
• Since a set is “completely determined by what its elements are”, the expressions "$\{1, 2, 4, 5\}$" and "$\{1,5,4,2\}$" denote the same set.
• Note that the expressions "$\{1, 2, 4, 5\}$" and "$\{1,5,4,2\}$" themselves are two different things. Compare: "$16$" and "XVI" are two different expressions denoting the same number.
Repetition does not matter
The list notation $\{3, 3, 4\}$ defines a set with two elements $3$ and $4$. The first occurrence of ‘$3$’ in the list says that $3$ is in the set. The second occurrence says the same thing.
Saying a true thing twice has no effect (except to irritate the reader). So repetition in list notation does not matter.
Warning: Mathematica uses curly brackets to denote lists, not sets. So in Mathematica, $\{1, 2, 4, 5\}$ and $\{1,5,4,2\}$ are two different lists and so are $\{3,4\}$ and $\{3,3,4\}$.
Set equality
If $A$ and $B$ are sets, then $A=B$ if and only if $A$ and $B$ have the same elements. In other words:
$A = B$ if and only if every element of $A$ is an element of $B$
and every element of $B$ is an element of $A$.
• For real numbers x, \[\left\{ x|{{x}^{2}}=1 \right\}=\left\{ 1,\,\,-1 \right\}\] (see setbuilder notation) because (a) $1$ and $–1$ satisfy the equation ${{x}^{2}}=1$ and (b) no other real number
satisfies that equation.
• For real numbers $x$, \[\left\{ x|{{x}^{2}}=2 \right\}\ne \left\{ \sqrt{2} \right\}.\]It is true that $\sqrt{2}$ satisfies the equation ${{x}^{2}}=2$, but it is also true that $-\sqrt{2}$
satisfies ${{x}^{2}}=2$. Since $-\sqrt{2}$ is not listed as an element of $\left\{ \sqrt{2} \right\}$, $\left\{ \sqrt{2} \right\}$ is not equal to $\left\{ x|{{x}^{2}}=2 \right\}$.
• But of course, \[\left\{ x|{{x}^{2}}=2 \right\}=\left\{ \sqrt{2},-\sqrt{2}\right\}.\]
Sets as elements of sets
A set, being a math object, can be an element of another set. Furthermore, if it is, its elements are not necessarily elements of that other set because the specification says that a set is a math
object that is distinct from its elements.
Let $A= \{ \{1, 2\}, \{3\}, 1, 6\}$.
• $A$ has four elements, two of which are sets.
• $2\in \left\{ 1,\,\,2 \right\}$ and $\left\{ 1,\,\,2 \right\}\in A$, but $2\notin A$. The set $\{1,2\}$ is distinct from its elements, so that even though one of its elements is $2$, the set $\
{1,2\}$ itself is not $2$.
• On the other hand, $1$ is an element of $A$ because it is explicitly listed as such.
Let \[B=\left\{ \left\{ 1 \right\},\,\left\{ 2 \right\},\,\left\{ 1,\,2 \right\},\,\varnothing \right\}.\]Then $B$ is the set of all subsets of the set $\{1, 2\}$. In particular, $\varnothing \in
B$ (the empty set is an element of $B$.) Note that the empty set is not an element of the set $A$ of the preceding example.
It is a myth that the empty set is an element of every set.
On the other hand:
The empty set is a subset of every set.
Sets as elements of sets in practice
Most of the time in practice either none of the elements of a set are sets or all of them are. In fact, sets such as $A$ and $B$ in the preceding examples, which have both sets and numbers as
elements, rarely occur in mathematical writing except as examples in texts such as the one you are reading which are intended to bring out the difference between "element of" and "included in". See
also contain.
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Details: To illustrate Boltzmann’s construction of an entropy function that is defined for a microstate of a macroscopic system, I will discuss the simple example of the free expansion of a one
dimensional gas of point particles both interacting and non-interacting. The construction requires one to define macrostates, corresponding to macroscopic variables and we will discuss two specific
macrostate constructions. Our results illustrate that concepts such as that of ergodicity and chaos are not as relevant as sometimes claimed, for observing macroscopic irreversibility and entropy
increase. Rather, the notions of initial conditions, typicality, large numbers and coarse-graining are the important factors. We will also discuss results for an interacting gas and the important | {"url":"https://calendar.iiserkol.ac.in/view_event/1137631/","timestamp":"2024-11-14T10:54:11Z","content_type":"application/xhtml+xml","content_length":"3193","record_id":"<urn:uuid:1448a525-85aa-4f15-a15d-1710b0d2894c>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00674.warc.gz"} |
Start your printing business with WooCommerce and UniCPO
You have installed WordPress and WooCommerce, found a beautiful looking theme for your future website and is ready to add printing products. Reasonable questions” “what should I use for my online
printing business? What is the right tool for that? How to create custom options for my WooCommerce products and configure price calculation so it will take into account width and length and
thickness and plenty of other extra options?” I will not be beating around the bush and quickly answer you – use “WooCommerce Product Options and Price Calculation Formulas” plugin. No doubt this is
a perfect WC extension for any online printing businesses. I am going to show you how to use it.
“What should I use for my online printing business?” – UniCPO is the answer!
A perfect tool for printing products
Let’s assume our product is Foamex Banner. I am going to briefly describe options we are about to implement to this product. We want to let customers choose the size (width and height) and, what is
even more critical, we would like to give them a possibility to pick a measurement unit of their choice. Measurement units are: mm, cm, m, ft, in and yd. Enough, right? 🙂 Also, we have thickness
option: 3mm, 5mm and 10mm. Each thickness option has its base price per sq. m. and this value is different and depends on the area of an ordered item. Let’s also add “Laminate” option with these
choices: none, matt laminated, gloss laminated. The latter two should add 50% to the price calculated by multiplying the base price per sq. m. to area calculated based on width and height provided by
a customer. Finally, we would add Cutting Method option with two options: Square Cut Edges and Rounded Corners. Although this option does not impact the product price, we would let the customer
choose a size of rounding with possible values from 5 to 25 mm. Of course, this additional input must be shown only if Rounded Corners option is selected. That’s all.
Ok, let’s start making our product configuration. We will have three steps. Step 1 is adding options. Step 2 is configuring proper price calculation. Step 3 – add fields conditional logic to “Size of
Rounding” option.
Step 1 – Adding options
First, create a product and set its price to something. Value ‘1’ works perfectly well. It could be anything, but do not leave it empty, so WC will not treat it as a free item. Second, go to CPO
visual builder and start adding modules.
Adding modules in UniCPO
A new Row is added, and Column inside this Row is created automatically. Then Text input option is added. Now let’s configure this first option. This option is “Width”. Then duplicate the first
option, so creating “Height” is no time procedure. Then add “Measurement Unit”.
It is essential to keep slugs for suboptions as shown. Please, read more about this here.
Now add a new column (because we need to style our options nicely a bit later and it is better to add a new column now) and add a Select option for “Thickness”. Its suboptions are shown in the
screenshot. Please, notice values for ‘Price/Rate’ settings. The mentioned above is an integral part of our configuration, and it will be discussed a bit later.
Suboptions for Thickness Option
These are suboptions for “Laminate” option. Again, please notice values of ‘Price/Rate’ settings.
Suboptions for Laminate Option
Finally, add “Cutting Method” Select and “Size of Rounding” Text Input options.
Now go to General Settings and enable the first two settings and save them. This step allows for displaying custom options and product price calculation.
This is how our form looks like on a single product page:
Our form. First iteration.
Some validation messages overlap some labels as well as it feels like a bit unstyled. But our powerful visual builder allows us to style the form much much better and we will do that later for sure.
Step 2 – Configure price calculation
We are going to use non-option variables (NOV) to achieve our product configuration goal. So, open NOV settings modal window from the builder panel, enable this feature and add the first two NOVs.
These are our width and height values but converted to meters and area calculated in sq.m. Why? Because our base prices are in square meters, so we have to calculate the area in sq.m. However, we
also let customers choose in which measurement units they want to add the desired width and height. So, we use measurement units conversion feature of the plugin for our needs! This action lets the
script know which unit we want to convert our values to.
Non-Option Variables
However, this is not enough because the script has to know somehow which unit our width and height are inputted in. Open Dimensions settings modal window and choose only one parameter – Measurement
unit and save. This action lets the script know which unit we want to convert our values from.
Dimension Settings
Now let’s add another one NOV and name it ‘base price’ and use matrix functionality. This functionality makes it possible to create a table of prices based on one or two options. Our first parameter
is area and the second parameter is thickness. I am going to use a completely fake data to give an understanding of how it works.
Matrix functionality makes it possible to create a table of prices based on one or two options.
Let’s assume our prices are as following:
1. if the thickness is 3 mm: 3.11 if area less than or equal 1 sq.m., 3.07 if the area from 1 to 5 sq.m., 3.03 if the area larger than 5 sq.m.
2. if the thickness is 5 mm: 5.11 if area less than or equal 1 sq.m., 5.07 if the area from 1 to 5 sq.m., 5.03 if the area larger than 5 sq.m.
3. if the thickness is 10 mm: 10.11 if area less than or equal 1 sq.m., 10.07 if the area from 1 to 5 sq.m., 10.03 if the area larger than 5 sq.m.
You have noticed that I used this value for ‘# in cols’ setting: “1|1.0001|5|5.0001”. The reason is that my area values are not discreet; they are in ranges. So, I simulated ranges. The second and
the third row in the table have the same pairs of values. It means that if the area is anything between 1.0001 and 5, the value 3.07 will be chosen for 3 mm thickness and so on.
Now we are ready to add our price calculation formula. Open Main Formula & Formulas Conditional Logic modal window and add the maths formula:
Our formula is:
It consists of two parts:
1. we multiply the base price by the area
2. optionally, we add the cost of laminating; do you remember “price/rate” settings values? The first was 0, so if “No” is chosen, then the value of this part of the formula will be equal to 0; if
either “Matt” or “Gloss” is selected base price will be multiplied by the area and multiplied by 0.5.
Cool! The product price calculation works!
Step 3 – Fields conditional logic and additional form styling
In this part, we are going to style our form more and enable fields conditional logic for “Size of Rounding” option, so it will be shown only if “Rounded Corners” is chosen.
Open “Size of Rounding” option settings modal window, go to ‘Conditional’ tab and configure the logic:
Important to click “Fetch the rule” after any changes in logic builder are made!
I have also styled the form a bit in this what I have now:
This is a link to the product on our demo site. You can log in with demo/demo credentials and try to create your own product. This is a link to the export file for this product. If you use UniCPO 4
PRO already, you are able to import it (unzip first 🙂 ) and have a completely configured product on your site.
In case you have any question, please, feel free to ask it by contacting us via the form on this page.
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