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string B example: shortest editing distance, optimal inclusion (linear DP)
AcWing 902. Minimum edit distance
Given the two strings A and B, now you want to change A into B through several operations. The available operations are:
• Delete – deletes A character from string A.
• Insert – inserts A character somewhere in string A.
• Replace – replaces one character in string A with another.
Now please find out how many operations it takes to change A to B at least.
Input format
The first line contains the integer n, which represents the length of string A.
The second line contains A string A of length n.
The third line contains the integer m, which represents the length of string B.
The fourth line contains a string B of length m.
Strings contain only uppercase letters.
Output format
Outputs an integer representing the minimum number of operations.
Data range
Input example:
Output example:
Problem solution
Status representation dp[i][j]
1. Set: the set of operations that turns a[1~i] into a set of b[1~j]
2. Property: the operand of the scheme with the least number of operations among all operations
State calculation
The state is divided into different operations on the ith letter in a
1. Add after the letter
After adding a letter, it becomes the same, indicating that the first i of a and the first j-1 of b are the same
Namely: dp[i][j] = dp[i][j-1] + 1
2. Delete the letter
After deleting the letter, it becomes the same, indicating that the first i-1 in a has been the same as the first j in b
Namely: dp[i][j] = dp[i-1][j] + 1
3. Replace the letter
□ If the corresponding ending letter of the replacement description is different, see the penultimate one
Namely: dp[i][j] = dp[i-1][j-1] + 1
Do nothing
□ The corresponding ending letter is the same, and the penultimate one is directly compared
Namely: dp[i][j] = dp[i-1][j-1]
n = int(input())
s1 = input()
s1 = " " + s1
m = int(input())
s2 = input()
s2 = " " + s2
dp = [[1e18] * (m + 1) for i in range(n + 1)]
# Boundary condition
# Only delete
for i in range(1, n + 1):
dp[i][0] = i
# Only add
for j in range(1, m + 1):
dp[0][j] = j
dp[0][0] = 0
for i in range(1, n + 1):
for j in range(1, m + 1):
# change
if s1[i] == s2[j]:
dp[i][j] = min(dp[i][j], dp[i - 1][j - 1])
else: dp[i][j] = min(dp[i][j], dp[i - 1][j - 1] + 1)
# Delete
dp[i][j] = min(dp[i][j], dp[i - 1][j] + 1)
# plus
dp[i][j] = min(dp[i][j], dp[i][j - 1] + 1)
2553. Optimal inclusion
We call a string s containing string T, which means that T is a subsequence of S, that is, several characters can be extracted from string s, which are combined into a new string in the original
order, which is exactly the same as T.
Given two strings S and T, how many characters in s can be modified at least to make s contain t?
Input format
Enter two lines, one string per line.
The string in the first line is S and the string in the second line is T.
Both strings are non empty and contain only uppercase letters.
Output format
Output an integer representing the answer.
Data range
Input example:
Output example:
Problem solution
Status representation dp[i][j]
1. Set: the set of operations that turns a[1~i] into a set of b[1~j]
2. Property: the operand of the scheme with the least number of operations among all operations
State calculation
The state is divided into different operations on the ith letter in a
1. unchanged
Different from the above question, this is a sub sequence.
dp[i][j] = dp[i - 1][j]
2. Replace the letter
□ If the corresponding ending letter of the replacement description is different, see the penultimate one
Namely: dp[i][j] = dp[i-1][j-1] + 1
Do nothing
□ The corresponding ending letter is the same, and the penultimate one is directly compared
Namely: dp[i][j] = dp[i-1][j-1]
s1 = input()
s2 = input()
n = len(s1)
m = len(s2)
s1 = " " + s1
s2 = " " + s2
dp = [[1e18] * (m + 1) for i in range(n + 1)]
for i in range(0, n + 1):
dp[i][0] = 0
for i in range(1, n + 1):
for j in range(1, m + 1):
dp[i][j] = min(dp[i][j], dp[i - 1][j])
if s1[i] == s2[j]:
dp[i][j] = min(dp[i][j], dp[i-1][j-1])
dp[i][j] = min(dp[i][j], dp[i - 1][j - 1] + 1)
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A.1 The Alias Method
If many samples need to be generated from a discrete distribution, using the approach implemented in the SampleDiscrete() function would be wasteful: each generated sample would require computation.
That approach could be improved to time by computing a cumulative distribution function (CDF) table once and then using binary search to generate each sample, but there is another option that is even
more efficient, requiring just time for each sample; that approach is the alias method.
To understand how the alias method works, first consider the task of sampling from discrete outcomes, each with equal probability. In that case, computing the value gives a uniformly distributed
index between 0 and and the corresponding outcome can be selected—no further work is necessary. The alias method allows a similar searchless sampling method if the outcomes have arbitrary
probabilities .
The alias method is based on creating bins, one for each outcome. Bins are sampled uniformly and then two values stored in each bin are used to generate the final sample: if the th bin was sampled,
then gives the probability of sampling the th outcome, and otherwise the alias is chosen; it is the index of a single alternative outcome. Though we will not include the proof here, it can be shown
that this representation—the th bin associated with the th outcome and no more than a single alias per bin—is sufficient to represent arbitrary discrete probability distributions.
With the alias method, if the probabilities are all the same, then each bin’s probability is one, and it reduces to the earlier example with uniform probabilities. Otherwise, for outcomes where the
associated probability is greater than the average probability, the outcome will be stored as the alias in one or more of the other bins. For outcomes where the associated is less than the average
probability, will be less than one and the alias will point to one of the higher-probability outcomes.
For a specific example, consider the probabilities . A corresponding alias table is shown in Table A.1. It is possible to see that, for example, the first sample is chosen with probability : there is
a probability of choosing the first table entry, in which case the first sample is always chosen. Otherwise, there is a probability of choosing the second and third table entries, and for each, there
is a chance of choosing the alias, giving in sum an additional probability of choosing the first sample. The other probabilities can be verified similarly.
Table A.1: A Simple Alias Table. This alias table makes it possible to generate samples from the distribution of discrete probabilities . To generate a sample, an entry is first chosen with uniform
probability. Given an entry , its corresponding sample is chosen with probability and the sample corresponding to its alias index is chosen with probability .
Index Alias Index
1 n/a
One way to interpret an alias table is that each bin represents of the total probability mass function. If outcomes are first allocated to their corresponding bins, then the probability mass of
outcomes that are greater than must be distributed to other bins that have associated probabilities less than . This idea is illustrated in Figure A.1, which corresponds to the example of Table A.1.
Figure A.1: Graphical Representation of the Alias Table in Table A.1. One bin is allocated for each outcome and is filled by the outcome’s probability, up to . Excess probability is allocated to
other bins that have probabilities less than and thus extra space.
The AliasTable class implements algorithms for generating and sampling from alias tables. As with the other sampling code, its implementation is found in util/sampling.h and util/sampling.cpp.
<<AliasTable Definition>>=
class AliasTable { public: <<
AliasTable Public Methods
AliasTable() = default; AliasTable(Allocator alloc = {}) :
(alloc) {} AliasTable(pstd::span<const Float> weights, Allocator alloc = {}); PBRT_CPU_GPU int Sample(Float u, Float *pmf = nullptr, Float *uRemapped = nullptr) const; std::string ToString() const;
size_t size() const { return
.size(); } Float PMF(int index) const { return
; }
private: <<
AliasTable Private Members
struct Bin { Float q, p; int alias; }; pstd::vector<Bin> bins;
Its constructor takes an array of weights, not necessarily normalized, that give the relative probabilities for the possible outcomes.
AliasTable::AliasTable(pstd::span<const Float> weights, Allocator alloc) :
(weights.size(), alloc) { <<
Normalize weights to compute alias table PDF
Float sum = std::accumulate(weights.begin(), weights.end(), 0.); for (size_t i = 0; i < weights.size(); ++i)
= weights[i] / sum;
Create alias table work lists
struct Outcome { Float pHat; size_t index; }; std::vector<Outcome> under, over; for (size_t i = 0; i <
.size(); ++i) { <<
Add outcome i to an alias table work list
Float pHat =
.size(); if (pHat < 1) under.push_back(Outcome{pHat, i}); else over.push_back(Outcome{pHat, i});
Process under and over work item together
Handle remaining alias table work items
while (!over.empty()) { Outcome ov = over.back(); over.pop_back(); bins[ov.index].q = 1; bins[ov.index].alias = -1; } while (!under.empty()) { Outcome un = under.back(); under.pop_back(); bins
[un.index].q = 1; bins[un.index].alias = -1; }
The Bin structure represents an alias table bin. It stores the probability , the corresponding outcome’s probability , and an alias.
<<AliasTable Private Members>>=
struct Bin { Float q, p; int alias; }; pstd::vector<Bin> bins;
We have found that with large numbers of outcomes, especially when the magnitudes of their weights vary significantly, it is important to use double precision to compute their sum so that the alias
table initialization algorithm works correctly. Therefore, here std::accumulate takes the double-precision value 0. as its initial value, which in turn causes all its computation to be in double
precision. Given the sum of weights, the normalized probabilities can be computed.
<<Normalize weights to compute alias table PDF>>=
Float sum = std::accumulate(weights.begin(), weights.end(), 0.); for (size_t i = 0; i < weights.size(); ++i)
= weights[i] / sum;
The first stage of the alias table initialization algorithm is to split the outcomes into those that have probability less than the average and those that have probability higher than the average.
Two std::vectors of the Outcome structure are used for this.
<<Create alias table work lists>>=
struct Outcome { Float pHat; size_t index; }; std::vector<Outcome> under, over; for (size_t i = 0; i <
.size(); ++i) { <<
Add outcome i to an alias table work list
Float pHat =
.size(); if (pHat < 1) under.push_back(Outcome{pHat, i}); else over.push_back(Outcome{pHat, i});
Here and in the remainder of the initialization phase, we will scale the individual probabilities by the number of bins , working in terms of . Thus, the average value is 1, which will be convenient
in the following.
<<Add outcome i to an alias table work list>>=
Float pHat =
.size(); if (pHat < 1) under.push_back(Outcome{pHat, i}); else over.push_back(Outcome{pHat, i});
To initialize the alias table, one outcome is taken from under and one is taken from over. Together, they make it possible to initialize the element of bins that corresponds to the outcome from
under. After that bin has been initialized, the outcome from over will still have some excess probability that is not yet reflected in bins. It is added to the appropriate work list and the loop
executes again until under and over are empty. This algorithm runs in time.
It is not immediately obvious that this approach will successfully initialize the alias table, or that it will necessarily terminate. We will not rigorously show that here, but informally, we can see
that at the start, there must be at least one item in each work list unless they all have the same probability (in which case, initialization is trivial). Then, each time through the loop, we
initialize one bin, which consumes worth of probability mass. With one less bin to initialize and that much less probability to distribute, we have the same average probability over the remaining
bins. That brings us to the same setting as the starting condition: some of the remaining items in the list must be above the average and some must be below, unless they are all equal to it.
<<Process under and over work item together>>=
<<Remove items un and ov from the alias table work lists>>=
Outcome un = under.back(), ov = over.back(); under.pop_back(); over.pop_back();
The probability of un must be less than one. We can initialize its bin’s q with , as that is equal to the probability it should be sampled if its bin is chosen. In order to allocate the remainder of
the bin’s probability mass, the alias is set to ov. Because , it certainly has enough probability to fill the remainder of the bin—we just need of it.
<<Initialize probability and alias for un>>=
In initializing bins[un.index], we have consumed worth of the scaled probability mass. The remainder, un.pHat + ov.pHat - 1, is the as-yet unallocated probability for ov.index; it is added to the
appropriate work list based on how much is left.
<<Push excess probability on to work list>>=
Float pExcess = un.pHat + ov.pHat - 1; if (pExcess < 1) under.push_back(Outcome{pExcess, ov.index}); else over.push_back(Outcome{pExcess, ov.index});
Due to floating-point round-off error, there may be work items remaining on either of the two work lists with the other one empty. These items have probabilities slightly less than or slightly
greater than one and should be given probability in the alias table. The fragment that handles this, <<Handle remaining alias table work items>>, is not included in the book.
Given an initialized alias table, sampling is easy. As described before, an entry is chosen with uniform probability and then either the corresponding sample or its alias is returned. As with the
SampleDiscrete() function, a new uniform random sample derived from the original one is optionally returned.
The index for the chosen entry is found by multiplying the random sample by the number of entries. Because u was only used for the discrete sampling decision of selecting an initial entry, it is
possible to derive a new uniform random sample from it. That computation is done here to get an independent uniform sample up that is used to decide whether to sample the alias at the current entry.
<<Compute alias table offset and remapped random sample up>>=
If the initial entry is selected, the various return values are easily computed.
<<Return sample for alias table at offset>>=
if (pmf) *pmf =
; if (uRemapped) *uRemapped = std::min<Float>(up /
); return offset;
Otherwise the appropriate values for the alias are returned.
<<Return sample for alias table at alias[offset]>>=
Beyond sampling, it is useful to be able to query the size of the table and the probability of a given outcome. These two operations are easily provided.
<<AliasTable Public Methods>>=
size_t size() const { return
.size(); } Float PMF(int index) const { return
; }
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How do you find an equation for the tangent line to x^4=y^2+x^2 at (2, sqrt12)? | HIX Tutor
How do you find an equation for the tangent line to #x^4=y^2+x^2# at #(2, sqrt12)#?
Answer 1
$y = \frac{7}{\sqrt{3}} x - \frac{8}{\sqrt{3}}$ or $\sqrt{3} y = 7 x - 8$
#x^4=y^2+x^2# #y^2=x^4-x^2# #2y(dy/dx)=4x^3-2x# #dy/dx=(4x^3-2x)/(2y)#
at #(2,sqrt12)# #dy/dx=(4(2^3)-2(2))/(2sqrt12)# #dy/dx=(32-4)/(4sqrt3)# #dy/dx=28/(4sqrt3)# #dy/dx=7/sqrt3#
The equation of tangent, #y-y_1=m(x-x_1)# where #m=7/sqrt3, y_1=sqrt12 and x_1=2#
#y-sqrt12=7/sqrt3(x-2)# #y-sqrt12=7/sqrt3x-14/sqrt3# #y=7/sqrt3x-14/sqrt3+sqrt12# #y=7/sqrt3x-14/sqrt3+sqrt36/sqrt3# #y=7/sqrt3x-14/sqrt3+6/sqrt3# #y=7/sqrt3x-8/sqrt3# or #sqrt3y=7x-8#
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Answer 2
$7 x - \sqrt{3} y - 8 = 0$. See tangent-inclusive graph. The graph is not to scale. There is contraction in the y-direction.
#4x^3=2x+2yy'#, giving #y'=7/sqrt3#, at #P(2, sqrt12)#.
The equation to the tangent at P is
y-sqrt13=7/sqrt3(x-2), giving
graph{(x^2-sqrt(x^2+y^2))(7x-sqrt3 y-8.2)=0 [-7, 7, -35, 35]}
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Answer 3
To find the equation of the tangent line to the curve (x^4 = y^2 + x^2) at the point ((2, \sqrt{12})), you can follow these steps:
1. Differentiate both sides of the equation (x^4 = y^2 + x^2) implicitly with respect to (x) to find the derivative (\frac{dy}{dx}).
2. Substitute the coordinates of the given point ((2, \sqrt{12})) into the derivative to find the slope of the tangent line.
3. Use the point-slope form of the equation of a line, (y - y_1 = m(x - x_1)), where (m) is the slope and ((x_1, y_1)) is the given point, to find the equation of the tangent line.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Neural Networks in R | Tyler Rouze
This project aims to implement and build a deeper level of understanding in Neural Networks. This article will profile how they learn just like the human brain does. Much of what you will see in this
project is based on the first two chapters of the text by Michael Nielsen titled Neural Networks and Deep Learning. While Nielsen builds neural network that is capable of classifying handwritten
digits in Python (2.7), I’ll show you how we can do it in R for a special sort of challenge.
To follow along or see the data, you can download from my repository on Github, which also includes the R script to load the data and build the Neural Network. What will follow will be two-fold: 1) a
tl;dr version of what a Neural Network is and how it works; 2) an implementation in R for those who may want to learn how to do it in this language. For those unfamiliar with statistics, calculus,
and data science, the first part of this article will be valuable to your understanding what Neural Networks are and how they work. That being said, the second part of this article should be valuable
to those who are ready to dip their feet into the world of data science. With that out of the way, let’s get started.
How do Neural Networks, work?
At a high level, Neural Networks are just that, a model of how your neurons work in your brain. The difference here is that it’s an emulation of your brain in a computer (not as scary as it sounds).
The idea is that if you were shown the number 5 right now, your eyes would register the number, pass that information to your brain where certain neurons would fire based on the image. Then, your
brain would determine that it is a 5 you are seeing.
In order to do this in a computer, we introduce a few equivalents to model what happens in the human brain. For this project, we are taking a large dataset of handwritten digits. Each digit comprises
of 784 pixels, and looks like this:
Each pixel represents an input, and each pixel is given a grayscale number, meaning a white pixel is 0, and a darker pixel is a number representing how dark the pixel is.
So, we input 784 values. These values are weighted (weights are learned through training data) and passed to the next layer of the network. A network and its layers look like this:
Each of the 784 input values is sent to each of the nodes in the middle (hidden) layer. What I mean by this is that one pixel value is sent to each of the 30 nodes in the hidden layer. In our case,
this represents a 784 by 30 matrix as you’ll see we use 30 nodes in the hidden layer. You’ll also notice 10 nodes in what is called the output layer. Each of these nodes represents a final
determination of the handwritten digit being 0 through 9.
Let’s talk through how one pixel (input) would pass through the entire network after having been weighted and passed to the middle (hidden) layer. From here, the new, weighted value is input into the
middle layer. The node in the middle layer takes the value and runs it through what is called an activation function. In our case, we’ll use a Sigmoid function which looks like this:
In R, our now weighted input is passed into the below function as z:
1 sigmoid <- function(z) 1/(1+exp(-z))
So what does the Activation (Sigmoid) function do? In laymen’s terms, it determines if the input is of value. You’ll see what this means in the next paragraph.
From here, the output of the Sigmoid function is weighted and passed as input into the final layer of 10 nodes (remember: representing each of the 10 digits 0 through 9). That input is ran through
the Activation function again, and the neural network outputs a vector of ten values, like this:
1 > a
2 [,1]
3 [1,] 0.1041222329
4 [2,] 0.0056134030
5 [3,] 0.3600190030
6 [4,] 0.9930337436
7 [5,] 0.0004073771
8 [6,] 0.0179073440
9 [7,] 0.0938795106
10 [8,] 0.0071585077
11 [9,] 0.9863697174
12 [10,] 0.0175402033
These values represent how much the neural network “thinks” the handwritten inputted image is each number. We simply take the highest output (closest to 1) and consider the neural network to have
classified the digit as that value. This output is from an untrained network, but it makes logical sense that the network thinks an 8 and 3 look similar. In this case, we’d say the network predicts
that the handwritten image is a 3!
The key points to remember of how a network classifies a digit are: edges weight the inputs, nodes determine if those weighted inputs are of value, the values at the nodes are passed on to be
weighted and valued again until the output layer is reached. The output tells us what the input should be classified as.
Implementing NN in R
Now that we’ve briefly walked through what steps a network takes to classify a handwritten digit, we must walk through how we train the network to get good at classifying digits correctly. Let’s try
to do this while also showcasing some of the code to be implemented in R.
To train a network, we must give it some data so it can learn. We do this by splitting the dataset. If you’ve studied statistical modeling in any capacity, you’ll likely be familiar with this
practice. In our handwritten image dataset we have 70,000 images, so we’ll feed our network 60,000 images to learn from and 10,000 to test on. The primary difference between the “learning” and
“testing” digits is that in the learning phase we are able to adjust the weights and biases such that it gets more digits right. This is done through a reduction in what is called a cost function.
Stochastic Gradient Descent
To start, we begin with the Stochastic Gradient Descent function. The cost function I referenced above? Gradient descent is a fancy way of saying we minimize the cost function (i.e. minimize how many
digit classifications we get wrong). We want to minimize cost because cost represents how poorly our network classifies digits. The higher the cost, the worse our network classifies digits correctly.
The stochastic portion of Stochastic Gradient Descent references the fact that we are estimating gradient descent.
We estimate the gradient descent because in order to train our neural network, we will utilize a process called mini batching. We use mini batching for a number of reasons; the main reasons being
that we can train our network using a much smaller amount of computing power (this is by far not a complete answer).
To begin building this learning part of our network, we must split the training data out into mini-batches of size 10 (meaning each mini batch has 10 handwritten images in it). In R, this is how
we’ll do it.
1 # appends the result to the 785th column, with 60000
2 # rows - one row per observation
3 training_data <- cbind(train$x,train$y)
4 for (j in 1:epochs){
5 # shuffle the data to prep for mini-batches
6 training_data <- training_data[sample(nrow(training_data)),]
7 mini.batches <- list()
8 seq1 <- seq(from=1, to=60000, by=mini.batch.size)
9 for(u in 1:(nrow(training_data)/mini.batch.size)){
10 # pull out 10 rows from training_data for each
11 # iteration of this loop to create 6000 mini-batches
12 mini.batches[[u]] <- training_data[seq1[u]:(seq1[u]+9),]
13 }
We create a nested list of 6000 mini batches, each of size mini.batch.size = 10.
Now, we feed each mini batch through our neural network and calculate weights and biases such that it will classify the mini batch correctly. So, we build a for loop to iterate through each mini
batch, and call a new function update.mini.batch.
Update Mini Batch
Within this function, we instantiate an empty list for the weights and biases. We iterate through each observation in the mini batch, feeding the gradient values (how dark a pixel is) as x and the
actual handwritten digit (0 thru 9) as y.
1 # create empty lists of weights and biases
2 nabla.b <- list(rep(0,sizes[2]),rep(0,sizes[3]))
3 nabla.w <- list(matrix(rep(0,(sizes[2]*sizes[1])), nrow=sizes[2], ncol=sizes[1]),
4 matrix(rep(0,(sizes[3]*sizes[2])), nrow=sizes[3], ncol=sizes[2]))
5 ## train through mini-batch
6 for(p in 1:mini.batch.size){
7 x <- mini_batch[p,-785] # 784 input gradient values
8 y <- mini_batch[p,785] # actual digit classification
10 ## backprop for each observation in mini-batch
11 delta_nablas <- backprop(x, y, sizes, num_layers, biases, weight)
You’ll notice above we call our backpropagation function. Read on to see what this does.
In laymens terms, backpropagation takes the observations of a mini batch and determines the weights and biases for each observation that will correctly classify the digits in the mini batch. That’s a
mouth full, but this is how our network improves its classification ability. We will adjust our weights and biases for each observation in the mini batch. The reason it is called backpropagation is
because through mathematical proofs, we can show how to efficiently calculate the gradient (or cost function) through each layer going backwards from the output layer. I will try to save the
mathematical speak for Michael Nielsen, who explains it in detail here.
To implement in R, we initialize our weights and biases (remember: these are on the edges in the network) and a list of the activations at each node. To start, these activations are just our inputs
(784 grey scale pixel values). We feed these values forward first, calculating what our current network would classify this digit as. All of this is done in the following code:
1 ## initialize updates
2 nabla_b_backprop <- list(rep(0,sizes[2]),rep(0,sizes[3]))
3 nabla_w_backprop <- list(matrix(rep(0,(sizes[2]*sizes[1])), nrow=sizes[2], ncol=sizes[1]),
4 matrix(rep(0,(sizes[3]*sizes[2])), nrow=sizes[3], ncol=sizes[2]))
5 ## Feed Forward
6 activation <- matrix(x, nrow=length(x), ncol=1) # all 784 inputs in single column matrix
7 activations <- list(matrix(x, nrow=length(x), ncol=1)) # list to store all activations, layer by layer
8 zs <- list() # list to store all z vectors, layer by layer
10 for(f in 1:length(weight)){
11 b <- biases[[f]]
12 w <- weight[[f]]
13 w_a <- w%*%activation
14 b_broadcast <- matrix(b, nrow=dim(w_a)[1], ncol=dim(w_a)[2])
15 z <- w_a + b
16 zs[[f]] <- z
17 activation <- sigmoid(z)
18 activations[[f+1]] <- activation
19 }
To help you understand where we are- we’ve just taken one observation; ran it through our network; calculated the weight on each edge; and calculated the activation at each node. Now is where we
backpropagate. This means we determine the weights in which we will classify the digit correctly. We estimate this through the gradient of our cost function, meaning we attempt to minimize the chance
our network missclassifies the digit.
1 ## backpropagate where we update the gradient using delta errors
2 delta <- cost.derivative(activations[[length(activations)]], y) * sigmoid_prime(zs[[length(zs)]])
3 nabla_b_backprop[[length(nabla_b_backprop)]] <- delta
4 nabla_w_backprop[[length(nabla_w_backprop)]] <- delta %*% t(activations[[length(activations)-1]])
This calls our cost.derivative function. This function takes our vector of output activations (the list of 10 values between 0 and 1 from earlier), and subtracts 1 from the activation in which our
digit actually is. This is important, as this is how our network learns. We take this vector of activations and calculate our output error (by multiplying by the derivative of our activation
1 delta <- cost.derivative(activations[[length(activations)]], y) * sigmoid_prime(zs[[length(zs)]])
3 cost.derivative <- function(output.activations, y){
4 output.activations - digit.to.vector(y)
5 }
To close, we calculate our weights and biases that feed into our output layer, such that we get the digit classification right (or as close as we can get it). Based on these weights and biases, we
can calculate the changes necessary to the weights and biases in the layer behind too. This, in essence, is backpropagation. We return a list of the weights and biases that, given the inputs we just
gave the network, would classify the digit correctly. This is done in the code below:
1 # take output from cost.derivative call and store it
2 nabla_b_backprop[[length(nabla_b_backprop)]] <- delta
3 nabla_w_backprop[[length(nabla_w_backprop)]] <- delta %*% t(activations[[length(activations)-1]])
5 # backpropagate through the layers behind the output
6 for (q in 2:(num_layers-1)) {
7 sp <- sigmoid_prime(zs[[length(zs)-(q-1)]])
8 delta <- (t(weight[[length(weight)-(q-2)]]) %*% delta) * sp
9 nabla_b_backprop[[length(nabla_b_backprop)-(q-1)]] <- delta
10 testyy <- t(activations[[length(activations)-q]])
11 nabla_w_backprop[[length(nabla_w_backprop)-(q-1)]] <- delta %*% testyy
12 }
13 return(list(nabla_b_backprop,nabla_w_backprop))
14 }
The backpropagate function will be called each time as it iterates through each observation in the mini batch.
Finish Update Mini Batch
After we’ve calculated the weights and biases that would best classify each digit in our mini batch, we come back out of the backpropagation function and finish up updating our network. We take the
weights and biases (of which we have a different set for each observation in the mini batch) and we edit the current weights and biases of the network. These edits are made based on what would be
necessary to correctly classify the entire mini batch we just backpropagated, with a suppressing factor called the learning rate (or eta).
To touch on the learning rate briefly, it influences the extent of which we can change the current weights and biases of the network. This change is based on what would best classify the mini batch
we just backpropagated. Without a learning rate, we may completely jump over our optimal weights and biases for which our network does the best at classifying all digits, not just the digits in a
mini batch.
Taking a step out…
So at a high-level, we’ve done the following thus far:
• Split our training data into batches of 10
• Backpropagated to determine the weights and biases for which our network would best classify each handwritten digit in the mini batch correctly
• Updated the weights and biases of our network to best classify based on the mini batch it has just been trained on
To finish training the network, we simply have to set up a for loop to do everything we’ve talked about up to this point. For each mini batch, we update the weights and biases of the network to
better classify digits. Remember, we had a dataset of 60,000 digits, with batch sizes of only 10 digits, so we’ll iterate many times. Once we’ve done that, we can evaluate our network on our testing
data. If we’ve done everything right, we should have well tuned weights and biases that should classify images of handwritten digits at 50% accuracy.
You read that right! 50% accuracy. That’s because I forgot to mention, once we’ve tuned the weights and biases over each mini batch we’ve only completed one epoch. Remember the learning rate I talked
about earlier? Instead of loosening things up and running the risk of our network getting worse over time, we have to do an epoch a number of times so our network can incrementally improve towards
optimality. This means we keep the weights and biases of the network, randomly split our data into mini batches, backpropagate over each mini batch, and update weights all over again.
Ideally, once we’ve tuned our network’s weights over a number of epochs, we could begin using our network to classify digits in real time. Think along the lines of banks automating the processing of
checks. Kind of cool, right?
In Closing
If you’ve made it this far, congratulations. I hope you’ve learned a little bit about how neural networks are implemented and how they learn. If you’d like to try it our for yourself, see the source
code. There are a number of helper functions that I didn’t go over for the sake of brevity, so be sure to familiarize yourself with those too. Otherwise, leave a comment and let me know what you
thought of this project!
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Shaheda Tahmina Akter lnu.se
Fluid Mechanics: Fundamentals and Applications CDON
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Investment Analysis and Portfolio Management(marked)-25-54
The Investment Background
Chapter 1
The Investment Setting
Chapter 2
The Asset Allocation Decision
Chapter 3
Selecting Investments in a Global Market
Chapter 4
Organization and Functioning of Securities Markets
Chapter 5
Security-Market Indexes
The chapters in this section will provide a background for your study of investments by
answering the following questions:
Why do people invest?
How do you measure the returns and risks for alternative investments?
What factors should you consider when you make asset allocation decisions?
What investments are available?
How do securities markets function?
How and why are securities markets in the United States and around the world
What are the major uses of security-market indexes?
How can you evaluate the market behavior of common stocks and bonds?
What factors cause differences among stock- and bond-market indexes?
In the first chapter, we consider why an individual would invest, how to measure the rates of
return and risk for alternative investments, and what factors determine an investor’s required
rate of return on an investment. The latter point will be important in subsequent analyses
when we work to understand investor behavior, the markets for alternative securities, and the
valuation of various investments.
Because the ultimate decision facing an investor is the makeup of his or her portfolio,
Chapter 2 deals with the all-important asset allocation decision. This includes specific steps
in the portfolio management process and factors that influence the makeup of an investor’s
portfolio over his or her life cycle.
To minimize risk, investment theory asserts the need to diversify. Chapter 3 begins our exploration of investments available to investors by making an overpowering case for investing
globally rather than limiting choices to only U.S. securities. Building on this premise, we discuss several investment instruments found in global markets. We conclude the chapter with a
review of the historical rates of return and measures of risk for a number of alternative asset
In Chapter 4, we examine how markets work in general, and then specifically focus on the
purpose and function of primary and secondary bond and stock markets. During the last 15
years, significant changes have occurred in the operation of the securities market, including a
trend toward a global capital market, electronic trading markets, and substantial worldwide
consolidation. After discussing these changes and the rapid development of new capital markets around the world, we speculate about how global markets will continue to consolidate and
will increase available investment alternatives.
Investors, market analysts, and financial theorists generally gauge the behavior of securities
markets by evaluating the return and risk implied by various market indexes and evaluate
portfolio performance by comparing a portfolio’s results to an appropriate benchmark. Because these indexes are used to make asset allocation decisions and then to evaluate portfolio
performance, it is important to have a deep understanding of how they are constructed and
the numerous alternatives available. Therefore, in Chapter 5, we examine and compare a number of stock-market and bond-market indexes available for the domestic and global markets.
This initial section provides the framework for you to understand various securities, how to
allocate among alternative asset classes, the markets where these securities are bought and sold,
the indexes that reflect their performance, and how you might manage a collection of investments in a portfolio. Specific portfolio management techniques are described in later chapters.
The Investment Setting
After you read this chapter, you should be able to answer the following questions:
Why do individuals invest?
What is an investment?
How do investors measure the rate of return on an investment?
How do investors measure the risk related to alternative investments?
What factors contribute to the rates of return that investors require on alternative investments?
What macroeconomic and microeconomic factors contribute to changes in the required rates of return for
This initial chapter discusses several topics basic to the subsequent chapters. We begin by defining the term investment and discussing the returns and risks related to investments. This leads to
a presentation of how to measure the expected and historical rates of returns for an individual
asset or a portfolio of assets. In addition, we consider how to measure risk not only for an individual investment but also for an investment that is part of a portfolio.
The third section of the chapter discusses the factors that determine the required
rate of return for an individual investment. The factors discussed are those that contribute to an asset’s total risk. Because most investors have a portfolio of investments,
it is necessary to consider how to measure the risk of an asset when it is a part of a
large portfolio of assets. The risk that prevails when an asset is part of a diversified
portfolio is referred to as its systematic risk.
The final section deals with what causes changes in an asset’s required rate of return
over time. Notably, changes occur because of both macroeconomic events that affect all
investment assets and microeconomic events that affect the specific asset.
1.1 WHAT IS
For most of your life, you will be earning and spending money. Rarely, though, will your current
money income exactly balance with your consumption desires. Sometimes, you may have more
money than you want to spend; at other times, you may want to purchase more than you can afford based on your current income. These imbalances will lead you either to borrow or to save to
maximize the long-run benefits from your income.
When current income exceeds current consumption desires, people tend to save the excess.
They can do any of several things with these savings. One possibility is to put the money under a mattress or bury it in the backyard until some future time when consumption desires
exceed current income. When they retrieve their savings from the mattress or backyard, they
have the same amount they saved.
Part 1: The Investment Background
Another possibility is that they can give up the immediate possession of these savings for a
future larger amount of money that will be available for future consumption. This trade-off of
present consumption for a higher level of future consumption is the reason for saving. What
you do with the savings to make them increase over time is investment.1
Those who give up immediate possession of savings (that is, defer consumption) expect to receive
in the future a greater amount than they gave up. Conversely, those who consume more than their
current income (that is, borrow) must be willing to pay back in the future more than they borrowed.
The rate of exchange between future consumption (future dollars) and current consumption (current dollars) is the pure rate of interest. Both people’s willingness to pay this difference for borrowed
funds and their desire to receive a surplus on their savings (i.e., some rate of return) give rise to an
interest rate referred to as the pure time value of money. This interest rate is established in the capital market by a comparison of the supply of excess income available (savings) to be invested and the
demand for excess consumption (borrowing) at a given time. If you can exchange $100 of certain
income today for $104 of certain income one year from today, then the pure rate of exchange on a
risk-free investment (that is, the time value of money) is said to be 4 percent (104/100 − 1).
The investor who gives up $100 today expects to consume $104 of goods and services in the
future. This assumes that the general price level in the economy stays the same. This price stability has rarely been the case during the past several decades when inflation rates have varied
from 1.1 percent in 1986 to as much as 13.3 percent in 1979, with a geometric average of 4.4
percent a year from 1970 to 2010. If investors expect a change in prices, they will require a
higher rate of return to compensate for it. For example, if an investor expects a rise in prices
(that is, he or she expects inflation) at the annual rate of 2 percent during the period of investment, he or she will increase the required interest rate by 2 percent. In our example, the investor would require $106 in the future to defer the $100 of consumption during an inflationary
period (a 6 percent nominal, risk-free interest rate will be required instead of 4 percent).
Further, if the future payment from the investment is not certain, the investor will demand
an interest rate that exceeds the nominal risk-free interest rate. The uncertainty of the payments from an investment is the investment risk. The additional return added to the nominal,
risk-free interest rate is called a risk premium. In our previous example, the investor would require more than $106 one year from today to compensate for the uncertainty. As an example,
if the required amount were $110, $4 (4 percent) would be considered a risk premium.
1.1.1 Investment Defined
From our discussion, we can specify a formal definition of an investment. Specifically, an
investment is the current commitment of dollars for a period of time in order to derive future payments that will compensate the investor for (1) the time the funds are committed, (2) the expected
rate of inflation during this time period, and (3) the uncertainty of the future payments. The “investor” can be an individual, a government, a pension fund, or a corporation. Similarly, this definition
includes all types of investments, including investments by corporations in plant and equipment and
investments by individuals in stocks, bonds, commodities, or real estate. This text emphasizes investments by individual investors. In all cases, the investor is trading a known dollar amount today for
some expected future stream of payments that will be greater than the current dollar amount today.
At this point, we have answered the questions about why people invest and what they want from
their investments. They invest to earn a return from savings due to their deferred consumption.
They want a rate of return that compensates them for the time period of the investment, the expected rate of inflation, and the uncertainty of the future cash flows. This return, the investor’s
required rate of return, is discussed throughout this book. A central question of this book is how
investors select investments that will give them their required rates of return.
In contrast, when current income is less than current consumption desires, people borrow to make up the difference.
Although we will discuss borrowing on several occasions, the major emphasis of this text is how to invest savings.
Chapter 1: The Investment Setting
The next section of this chapter describes how to measure the expected or historical rate of return on an investment and also how to quantify the uncertainty (risk) of expected returns. You
need to understand these techniques for measuring the rate of return and the uncertainty of these
returns to evaluate the suitability of a particular investment. Although our emphasis will be on financial assets, such as bonds and stocks, we will refer to other assets, such as art and antiques.
Chapter 3 discusses the range of financial assets and also considers some nonfinancial assets.
1.2 MEASURES
The purpose of this book is to help you understand how to choose among alternative investment assets. This selection process requires that you estimate and evaluate the expected riskreturn trade-offs for the alternative investments available. Therefore, you must understand
how to measure the rate of return and the risk involved in an investment accurately. To meet
this need, in this section we examine ways to quantify return and risk. The presentation will
consider how to measure both historical and expected rates of return and risk.
We consider historical measures of return and risk because this book and other publications provide numerous examples of historical average rates of return and risk measures for
various assets, and understanding these presentations is important. In addition, these historical
results are often used by investors when attempting to estimate the expected rates of return
and risk for an asset class.
The first measure is the historical rate of return on an individual investment over the time
period the investment is held (that is, its holding period). Next, we consider how to measure
the average historical rate of return for an individual investment over a number of time periods. The third subsection considers the average rate of return for a portfolio of investments.
Given the measures of historical rates of return, we will present the traditional measures of
risk for a historical time series of returns (that is, the variance and standard deviation).
Following the presentation of measures of historical rates of return and risk, we turn to
estimating the expected rate of return for an investment. Obviously, such an estimate contains
a great deal of uncertainty, and we present measures of this uncertainty or risk.
1.2.1 Measures of Historical Rates of Return
When you are evaluating alternative investments for inclusion in your portfolio, you will often be
comparing investments with widely different prices or lives. As an example, you might want to
compare a $10 stock that pays no dividends to a stock selling for $150 that pays dividends of $5
a year. To properly evaluate these two investments, you must accurately compare their historical
rates of returns. A proper measurement of the rates of return is the purpose of this section.
When we invest, we defer current consumption in order to add to our wealth so that we
can consume more in the future. Therefore, when we talk about a return on an investment,
we are concerned with the change in wealth resulting from this investment. This change in
wealth can be either due to cash inflows, such as interest or dividends, or caused by a change
in the price of the asset (positive or negative).
If you commit $200 to an investment at the beginning of the year and you get back $220 at
the end of the year, what is your return for the period? The period during which you own an
investment is called its holding period, and the return for that period is the holding period
return (HPR). In this example, the HPR is 1.10, calculated as follows:
HPR =
Ending Value of Investment
Beginning Value of Investment
= 1:10
Part 1: The Investment Background
This HPR value will always be zero or greater—that is, it can never be a negative value. A
value greater than 1.0 reflects an increase in your wealth, which means that you received
a positive rate of return during the period. A value less than 1.0 means that you suffered a
decline in wealth, which indicates that you had a negative return during the period. An HPR
of zero indicates that you lost all your money (wealth) invested in this asset.
Although HPR helps us express the change in value of an investment, investors generally
evaluate returns in percentage terms on an annual basis. This conversion to annual percentage
rates makes it easier to directly compare alternative investments that have markedly different
characteristics. The first step in converting an HPR to an annual percentage rate is to derive a
percentage return, referred to as the holding period yield (HPY). The HPY is equal to the
HPR minus 1.
HPY = HPR − 1
In our example:
HPY = 1:10 − 1 = 0:10
= 10%
To derive an annual HPY, you compute an annual HPR and subtract 1. Annual HPR is
found by:
Annual HPR = HPR1/n
n = number of years the investment is held
Consider an investment that cost $250 and is worth $350 after being held for two years:
Ending Value of Investment
Beginning Value of Investment $250
= 1:40
HPR =
Annual HPR = 1:401=n
= 1:401=2
= 1:1832
Annual HPY = 1:1832 − 1 = 0:1832
= 18:32%
If you experience a decline in your wealth value, the computation is as follows:
HPR =
Ending Value
= 0:80
Beginning Value $500
HPY = 0:80 − 1:00 = −0:20 = −20%
A multiple-year loss over two years would be computed as follows:
HPR =
Ending Value
= 0:75
Beginning Value $1,000
Annual HPR = ð0:75Þ1=n = 0:751=2
= 0:866
Annual HPY = 0:866 − 1:00 = −0:134 = −13:4%
Chapter 1: The Investment Setting
In contrast, consider an investment of $100 held for only six months that earned a return of $12:
= 1:12 ðn = 0:5Þ
Annual HPR = 1:121=:5
HPR =
= 1:122
= 1:2544
Annual HPY = 1:2544 − 1 = 0:2544
= 25:44%
Note that we made some implicit assumptions when converting the six-month HPY to an
annual basis. This annualized holding period yield computation assumes a constant annual
yield for each year. In the two-year investment, we assumed an 18.32 percent rate of return
each year, compounded. In the partial year HPR that was annualized, we assumed that the return is compounded for the whole year. That is, we assumed that the rate of return earned
during the first half of the year is likewise earned on the value at the end of the first six
months. The 12 percent rate of return for the initial six months compounds to 25.44 percent
for the full year.2 Because of the uncertainty of being able to earn the same return in the future
six months, institutions will typically not compound partial year results.
Remember one final point: The ending value of the investment can be the result of a positive or negative change in price for the investment alone (for example, a stock going from $20
a share to $22 a share), income from the investment alone, or a combination of price change
and income. Ending value includes the value of everything related to the investment.
1.2.2 Computing Mean Historical Returns
Now that we have calculated the HPY for a single investment for a single year, we want to consider mean rates of return for a single investment and for a portfolio of investments. Over a
number of years, a single investment will likely give high rates of return during some years and
low rates of return, or possibly negative rates of return, during others. Your analysis should consider each of these returns, but you also want a summary figure that indicates this investment’s
typical experience, or the rate of return you might expect to receive if you owned this investment
over an extended period of time. You can derive such a summary figure by computing the mean
annual rate of return (its HPY) for this investment over some period of time.
Alternatively, you might want to evaluate a portfolio of investments that might include similar investments (for example, all stocks or all bonds) or a combination of investments (for example, stocks, bonds, and real estate). In this instance, you would calculate the mean rate of
return for this portfolio of investments for an individual year or for a number of years.
Single Investment Given a set of annual rates of return (HPYs) for an individual investment,
there are two summary measures of return performance. The first is the arithmetic mean return, the second is the geometric mean return. To find the arithmetic mean (AM), the sum
(Σ) of annual HPYs is divided by the number of years (n) as follows:
AM = ΣHPY/n
ΣHPY = the sum of annual holding period yields
An alternative computation, the geometric mean (GM), is the nth root of the product of the
HPRs for n years minus one.
To check that you understand the calculations, determine the annual HPY for a three-year HPR of 1.50. (Answer:
14.47 percent.) Compute the annual HPY for a three-month HPR of 1.06. (Answer: 26.25 percent.)
Part 1: The Investment Background
GM = [πHPR]1/n − 1
π = the product of the annual holding period returns as follows:
(HPR1) × (HPR2) . . . (HPRn)
To illustrate these alternatives, consider an investment with the following data:
B e g in ni n g Va l ue
E n di ng V a lu e
H PR
HP Y
AM = ½ð0:15Þ + ð0:20Þ + ð− 0:20Þ=3
= 0:15=3
= 0:05 = 5%
GM = ½ð1:15Þ × ð1:20Þ × ð0:80Þ1=3 − 1
= ð1:104Þ1=3 − 1
= 1:03353 − 1
= 0:03353 = 3:353%
Investors are typically concerned with long-term performance when comparing alternative
investments. GM is considered a superior measure of the long-term mean rate of return because it indicates the compound annual rate of return based on the ending value of the investment versus its beginning value.3 Specifically, using the prior example, if we compounded
3.353 percent for three years, (1.03353)3, we would get an ending wealth value of 1.104.
Although the arithmetic average provides a good indication of the expected rate of return
for an investment during a future individual year, it is biased upward if you are attempting
to measure an asset’s long-term performance. This is obvious for a volatile security. Consider,
for example, a security that increases in price from $50 to $100 during year 1 and drops back
to $50 during year 2. The annual HPYs would be:
B e gi n ni ng V a l ue
En d in g Va l ue
H PR
HP Y
This would give an AM rate of return of:
½ð1:00Þ + ð−0:50Þ=2 = :50=2
= 0:25 = 25%
This investment brought no change in wealth and therefore no return, yet the AM rate of return is computed to be 25 percent.
The GM rate of return would be:
ð2:00 × 0:50Þ1=2 − 1 = ð1:00Þ1=2 − 1
= 1:00 − 1 = 0%
This answer of a 0 percent rate of return accurately measures the fact that there was no change
in wealth from this investment over the two-year period.
Note that the GM is the same whether you compute the geometric mean of the individual annual holding period
yields or the annual HPY for a three-year period, comparing the ending value to the beginning value, as discussed
earlier under annual HPY for a multiperiod case.
Chapter 1: The Investment Setting
When rates of return are the same for all years, the GM will be equal to the AM. If the rates
of return vary over the years, the GM will always be lower than the AM. The difference between
the two mean values will depend on the year-to-year changes in the rates of return. Larger annual changes in the rates of return—that is, more volatility—will result in a greater difference
between the alternative mean values. We will point out examples of this in subsequent chapters.
An awareness of both methods of computing mean rates of return is important because most
published accounts of long-run investment performance or descriptions of financial research will
use both the AM and the GM as measures of average historical returns. We will also use both
throughout this book with the understanding that the AM is best used as an expected value for
an individual year, while the GM is the best measure of long-term performance since it measures
the compound annual rate of return for the asset being measured.
A Portfolio of Investments The mean historical rate of return (HPY) for a portfolio of investments is measured as the weighted average of the HPYs for the individual investments in
the portfolio, or the overall percent change in value of the original portfolio. The weights used
in computing the averages are the relative beginning market values for each investment; this is
referred to as dollar-weighted or value-weighted mean rate of return. This technique is demonstrated by the examples in Exhibit 1.1. As shown, the HPY is the same (9.5 percent) whether
you compute the weighted average return using the beginning market value weights or if you
compute the overall percent change in the total value of the portfolio.
Although the analysis of historical performance is useful, selecting investments for your
portfolio requires you to predict the rates of return you expect to prevail. The next section discusses how you would derive such estimates of expected rates of return. We recognize the
great uncertainty regarding these future expectations, and we will discuss how one measures
this uncertainty, which is referred to as the risk of an investment.
1.2.3 Calculating Expected Rates of Return
Risk is the uncertainty that an investment will earn its expected rate of return. In the examples
in the prior section, we examined realized historical rates of return. In contrast, an investor
who is evaluating a future investment alternative expects or anticipates a certain rate of return.
The investor might say that he or she expects the investment will provide a rate of return of 10
percent, but this is actually the investor’s most likely estimate, also referred to as a point estimate. Pressed further, the investor would probably acknowledge the uncertainty of this point
estimate return and admit the possibility that, under certain conditions, the annual rate of return on this investment might go as low as −10 percent or as high as 25 percent. The point is,
the specification of a larger range of possible returns from an investment reflects the investor’s
Exhibit 1.1 Computation of Holding Period Yield for a Portfolio
HPR =
= 9:5%
Weights are based on beginning values.
$1,200,000 1.20
4,200,000 1.05
16,500,000 1.10
= 1:095
HPY = 1:095 − 1 = 0:095
Part 1: The Investment Background
uncertainty regarding what the actual return will be. Therefore, a larger range of possible returns implies that the investment is riskier.
An investor determines how certain the expected rate of return on an investment is by analyzing estimates of possible returns. To do this, the investor assigns probability values to all possible returns. These probability values range from zero, which means no chance of the return, to
one, which indicates complete certainty that the investment will provide the specified rate of
return. These probabilities are typically subjective estimates based on the historical performance
of the investment or similar investments modified by the investor’s expectations for the future.
As an example, an investor may know that about 30 percent of the time the rate of return on
this particular investment was 10 percent. Using this information along with future expectations
regarding the economy, one can derive an estimate of what might happen in the future.
The expected return from an investment is defined as:
Expected Return =
ðProbability of ReturnÞ × ðPossible ReturnÞ
EðRi Þ = ½ðP1 ÞðR1 Þ + ðP2 ÞðR2 Þ + ðP3 ÞðR3 Þ + + ðPn Rn Þ
EðRi Þ =
ðPi ÞðRi Þ
Let us begin our analysis of the effect of risk with an example of perfect certainty wherein
the investor is absolutely certain of a return of 5 percent. Exhibit 1.2 illustrates this situation.
Perfect certainty allows only one possible return, and the probability of receiving that return
is 1.0. Few investments provide certain returns and would be considered risk-free investments.
In the case of perfect certainty, there is only one value for PiRi:
E(Ri) = (1.0)(0.05) = 0.05 = 5%
In an alternative scenario, suppose an investor believed an investment could provide several
different rates of return depending on different possible economic conditions. As an example, in
a strong economic environment with high corporate profits and little or no inflation, the investor might expect the rate of return on common stocks during the next year to reach as high
Exhibit 1.2 Probability Distribution for Risk-Free Investment
Rate of Return
Chapter 1: The Investment Setting
as 20 percent. In contrast, if there is an economic decline with a higher-than-average rate of
inflation, the investor might expect the rate of return on common stocks during the next year
to be −20 percent. Finally, with no major change in the economic environment, the rate of return during the next year would probably approach the long-run average of 10 percent.
The investor might estimate probabilities for each of these economic scenarios based on
past experience and the current outlook as follows:
E c o no mi c C o nd i ti o ns
Strong economy, no inflation
Weak economy, above-average inflation
No major change in economy
P ro b ab i li ty
R at e o f R et u r n
This set of potential outcomes can be visualized as shown in Exhibit 1.3.
The computation of the expected rate of return [E(Ri)] is as follows:
EðRi Þ = ½ð0:15Þð0:20Þ + ½ð0:15Þð− 0:20Þ + ½ð0:70Þð0:10Þ
= 0:07
Obviously, the investor is less certain about the expected return from this investment than
about the return from the prior investment with its single possible return.
A third example is an investment with 10 possible outcomes ranging from −40 percent to
50 percent with the same probability for each rate of return. A graph of this set of expectations
would appear as shown in Exhibit 1.4.
In this case, there are numerous outcomes from a wide range of possibilities. The expected
rate of return [E(Ri)] for this investment would be:
EðRi Þ = ð0:10Þð−0:40Þ + ð0:10Þð−0:30Þ + ð0:10Þð−0:20Þ + ð0:10Þð−0:10Þ + ð0:10Þð0:0Þ
+ð0:10Þð0:10Þ + ð0:10Þð0:20Þ + ð0:10Þð0:30Þ + ð0:10Þð0:40Þ + ð0:10Þð0:50Þ
= ð−0:04Þ + ð−0:03Þ + ð−0:02Þ + ð−0:01Þ + ð0:00Þ + ð0:01Þ + ð0:02Þ + ð0:03Þ
+ð0:04Þ + ð0:05Þ
= 0:05
The expected rate of return for this investment is the same as the certain return discussed in
the first example; but, in this case, the investor is highly uncertain about the actual rate of
Exhibit 1.3 Probability Distribution for Risky Investment with Three Possible
Rates of Return
Rate of Return
Part 1: The Investment Background
E x h i b i t 1 . 4 P r o b a b i l i t y D i s t r i b u t i o n f o r R i s ky I n v e s t m e n t w i t h 1 0 P o s s i b l e
Rates of Return
Rate of Return
return. This would be considered a risky investment because of that uncertainty. We would
anticipate that an investor faced with the choice between this risky investment and the certain
(risk-free) case would select the certain alternative. This expectation is based on the belief that
most investors are risk averse, which means that if everything else is the same, they will select
the investment that offers greater certainty (i.e., less risk).
1.2.4 Measuring the Risk of Expected Rates of Return
We have shown that we can calculate the expected rate of return and evaluate the uncertainty,
or risk, of an investment by identifying the range of possible returns from that investment and
assigning each possible return a weight based on the probability that it will occur. Although
the graphs help us visualize the dispersion of possible returns, most investors want to quantify
this dispersion using statistical techniques. These statistical measures allow you to compare the
return and risk measures for alternative investments directly. Two possible measures of risk
(uncertainty) have received support in theoretical work on portfolio theory: the variance and
the standard deviation of the estimated distribution of expected returns.
In this section, we demonstrate how variance and standard deviation measure the dispersion of possible rates of return around the expected rate of return. We will work with the examples discussed earlier. The formula for variance is as follows:
Possible Expected 2
Variance ðσ Þ =
ðProbabilityÞ ×
ðPi Þ½Ri − EðRi Þ2
Variance The larger the variance for an expected rate of return, the greater the dispersion of
expected returns and the greater the uncertainty, or risk, of the investment. The variance for
the perfect-certainty (risk-free) example would be:
ðσ 2 Þ =
Pi ½Ri − EðRi Þ2
= 1:0ð0:05 − 0:05Þ2 = 1:0ð0:0Þ = 0
Chapter 1: The Investment Setting
Note that, in perfect certainty, there is no variance of return because there is no deviation from
expectations and, therefore, no risk or uncertainty. The variance for the second example would be:
ðσ 2 Þ =
Pi ½Ri − EðRi Þ2
= ½ð0:15Þð0:20 − 0:07Þ2 + ð0:15Þð−0:20 − 0:07Þ2 + ð0:70Þð0:10 − 0:07Þ2 = ½0:010935 + 0:002535 + 0:00063
= 0:0141
Standard Deviation The standard deviation is the square root of the variance:
Pi ½Ri − EðRi Þ2
Standard Deviation =
For the second example, the standard deviation would be:
σ = 0:0141
= 0:11874 = 11:874%
Therefore, when describing this investment example, you would contend that you expect a return of 7 percent, but the standard deviation of your expectations is 11.87 percent.
A Relative Measure of Risk In some cases, an unadjusted variance or standard deviation
can be misleading. If conditions for two or more investment alternatives are not similar—that
is, if there are major differences in the expected rates of return—it is necessary to use a measure of relative variability to indicate risk per unit of expected return. A widely used relative
measure of risk is the coefficient of variation (CV), calculated as follows:
Standard Deviation of Returns
Coefficient of Variation ðCVÞ =
Expected Rate of Return
The CV for the preceding example would be:
CV =
= 1:696
This measure of relative variability and risk is used by financial analysts to compare alternative investments with widely different rates of return and standard deviations of returns. As
an illustration, consider the following two investments:
I nv e s tm e nt A
I n ve s t m e n t B
Expected return
Standard deviation
Comparing absolute measures of risk, investment B appears to be riskier because it has a standard deviation of 7 percent versus 5 percent for investment A. In contrast, the CV figures
show that investment B has less relative variability or lower risk per unit of expected return
because it has a substantially higher expected rate of return:
CV A =
= 0:714
CV B =
= 0:583
Part 1: The Investment Background
1.2.5 Risk Measures for Historical Returns
To measure the risk for a series of historical rates of returns, we use the same measures as for
expected returns (variance and standard deviation) except that we consider the historical holding period yields (HPYs) as follows:
σ2 =
½HPY i − EðHPYÞ2
σ 2 = the variance of the series
HPYi = the holding period yield during period i
EðHPYÞ = the expected value of the holding period yield that is equal to the
arithmetic mean ðAMÞ of the series
n = the number of observations
The standard deviation is the square root of the variance. Both measures indicate how much
the individual HPYs over time deviated from the expected value of the series. An example computation is contained in the appendix to this chapter. As is shown in subsequent chapters where
we present historical rates of return for alternative asset classes, presenting the standard deviation as a measure of risk (uncertainty) for the series or asset class is fairly common.
1.3 DETERMINANTS
In this section, we continue our discussion of factors that you must consider when selecting
securities for an investment portfolio. You will recall that this selection process involves finding securities that provide a rate of return that compensates you for: (1) the time value of
money during the period of investment, (2) the expected rate of inflation during the period,
and (3) the risk involved.
The summation of these three components is called the required rate of return. This is the
minimum rate of return that you should accept from an investment to compensate you for
deferring consumption. Because of the importance of the required rate of return to the total
investment selection process, this section contains a discussion of the three components and
what influences each of them.
The analysis and estimation of the required rate of return are complicated by the behavior of
market rates over time. First, a wide range of rates is available for alternative investments at any
time. Second, the rates of return on specific assets change dramatically over time. Third, the difference between the rates available (that is, the spread) on different assets changes over time.
The yield data in Exhibit 1.5 for alternative bonds demonstrate these three characteristics.
First, even though all these securities have promised returns based upon bond contracts, the
promised annual yields during any year differ substantially. As an example, during 2009 the average yields on alternative assets ranged from 0.15 percent on T-bills to 7.29 percent for Baa corporate bonds. Second, the changes in yields for a specific asset are shown by the three-month
Treasury bill rate that went from 4.48 percent in 2007 to 0.15 percent in 2009. Third, an example of a change in the difference between yields over time (referred to as a spread) is shown by
the Baa–Aaa spread.4 The yield spread in 2007 was 91 basis points (6.47–5.56), but the spread in
2009 increased to 198 basis points (7.29–5.31). (A basis point is 0.01 percent.)
Bonds are rated by rating agencies based upon the credit risk of the securities, that is, the probability of default. Aaa
is the top rating Moody’s (a prominent rating service) gives to bonds with almost no probability of default. (Only
U.S. Treasury bonds are considered to be of higher quality.) Baa is a lower rating Moody’s gives to bonds of generally
high quality that have some possibility of default under adverse economic conditions.
Chapter 1: The Investment Setting
Exhibit 1.5 Promised Yields on Alternative Bonds
Type of Bond
U.S. government 3-month Treasury bills
U.S. government 10-year bonds
Aaa corporate bonds
Baa corporate bonds
Source: Federal Reserve Bulletin, various issues.
Because differences in yields result from the riskiness of each investment, you must understand
the risk factors that affect the required rates of return and include them in your assessment of
investment opportunities. Because the required returns on all investments change over time, and
because large differences separate individual investments, you need to be aware of the several
components that determine the required rate of return, starting with the risk-free rate. In this
chapter we consider the three components of the required rate of return and briefly discuss what
affects these components. The presentation in Chapter 11 on valuation theory will discuss the
factors that affect these components in greater detail.
1.3.1 The Real Risk-Free Rate
The real risk-free rate (RRFR) is the basic interest rate, assuming no inflation and no uncertainty about future flows. An investor in an inflation-free economy who knew with certainty
what cash flows he or she would receive at what time would demand the RRFR on an investment. Earlier, we called this the pure time value of money, because the only sacrifice the investor made was deferring the use of the money for a period of time. This RRFR of interest is the
price charged for the risk-free exchange between current goods and future goods.
Two factors, one subjective and one objective, influence this exchange price. The subjective
factor is the time preference of individuals for the consumption of income. When individuals
give up $100 of consumption this year, how much consumption do they want a year from now
to compensate for that sacrifice? The strength of the human desire for current consumption
influences the rate of compensation required. Time preferences vary among individuals, and
the market creates a composite rate that includes the preferences of all investors. This composite rate changes gradually over time because it is influenced by all the investors in the economy, whose changes in preferences may offset one another.
The objective factor that influences the RRFR is the set of investment opportunities available in the economy. The investment opportunities available are determined in turn by the
long-run real growth rate of the economy. A rapidly growing economy produces more and better opportunities to invest funds and experience positive rates of return. A change in the economy’s long-run real growth rate causes a change in all investment opportunities and a change
in the required rates of return on all investments. Just as investors supplying capital should
demand a higher rate of return when growth is higher, those looking to borrow funds to invest
should be willing and able to pay a higher rate of return to use the funds for investment because of the higher growth rate and better opportunities. Thus, a positive relationship exists
between the real growth rate in the economy and the RRFR.
1.3.2 Factors Influencing the Nominal Risk-Free Rate (NRFR)
Earlier, we observed that an investor would be willing to forgo current consumption in order
to increase future consumption at a rate of exchange called the risk-free rate of interest. This
rate of exchange was measured in real terms because we assume that investors want to
Part 1: The Investment Background
increase the consumption of actual goods and services rather than consuming the same
amount that had come to cost more money. Therefore, when we discuss rates of interest, we
need to differentiate between real rates of interest that adjust for changes in the general price
level, as opposed to nominal rates of interest that are stated in money terms. That is, nominal
rates of interest that prevail in the market are determined by real rates of interest, plus factors
that will affect the nominal rate of interest, such as the expected rate of inflation and the monetary environment. It is important to understand these factors.
Notably, the variables that determine the RRFR change only gradually because we are concerned
with long-run real growth. Therefore, you might expect the required rate on a risk-free investment
to be quite stable over time. As discussed in connection with Exhibit 1.5, rates on three-month
T-bills were not stable over the period from 2004 to 2010. This is demonstrated with additional
observations in Exhibit 1.6, which contains yields on T-bills for the period 1987–2010.
Investors view T-bills as a prime example of a default-free investment because the government has unlimited ability to derive income from taxes or to create money from which to pay
interest. Therefore, one could expect that rates on T-bills should change only gradually. In fact,
the data in Exhibit 1.6 show a highly erratic pattern. Specifically, there was an increase in yields
from 4.64 percent in 1999 to 5.82 percent in 2000 before declining by over 80 percent in three
years to 1.01 percent in 2003, followed by an increase to 4.73 percent in 2006, and concluding at
0.14 percent in 2010. Clearly, the nominal rate of interest on a default-free investment is not stable in the long run or the short run, even though the underlying determinants of the RRFR are
quite stable. As noted, two other factors influence the nominal risk-free rate (NRFR): (1) the relative ease or tightness in the capital markets, and (2) the expected rate of inflation.
Conditions in the Capital Market You will recall from prior courses in economics and finance that the purpose of capital markets is to bring together investors who want to invest savings with companies or governments who need capital to expand or to finance budget deficits.
The cost of funds at any time (the interest rate) is the price that equates the current supply and
demand for capital. Beyond this long-run equilibrium, change in the relative ease or tightness in
the capital market is a short-run phenomenon caused by a temporary disequilibrium in the supply and demand of capital.
As an example, disequilibrium could be caused by an unexpected change in monetary policy (for example, a change in the target federal funds rate) or fiscal policy (for example, a
change in the federal deficit). Such a change in monetary policy or fiscal policy will produce
a change in the NRFR of interest, but the change should be short-lived because, in the longer
Exhibit 1.6 Three-Month Treasury Bill Yields and Rates of Inflation
3-Month T-bills
Rate of Inflation
3-Month T-bills
Rate of Inflation
Source: Federal Reserve Bulletin, various issues; Economic Report of the President, various issues.
Chapter 1: The Investment Setting
run, the higher or lower interest rates will affect capital supply and demand. As an example, an
increase in the federal deficit caused by an increase in government spending (easy fiscal policy)
will increase the demand for capital and increase interest rates. In turn, this increase in interest
rates should cause an increase in savings and a decrease in the demand for capital by corporations or individuals. These changes in market conditions should bring rates back to the longrun equilibrium, which is based on the long-run growth rate of the economy.
Expected Rate of Inflation Previously, it was noted that if investors expected the price level
to increase (an increase in the inflation rate) during the investment period, they would require
the rate of return to include compensation for the expected rate of inflation. Assume that you
require a 4 percent real rate of return on a risk-free investment but you expect prices to increase
by 3 percent during the investment period. In this case, you should increase your required rate
of return by this expected rate of inflation to about 7 percent [(1.04 × 1.03) − 1]. If you do not
increase your required return, the $104 you receive at the end of the year will represent a real
return of about 1 percent, not 4 percent. Because prices have increased by 3 percent during the
year, what previously cost $100 now costs $103, so you can consume only about 1 percent more
at the end of the year [($104/103) − 1]. If you had required a 7.12 percent nominal return, your
real consumption could have increased by 4 percent [($107.12/103) − 1]. Therefore, an investor’s
nominal required rate of return on a risk-free investment should be:
NRFR = [(1 + RRFR) × (1 + Expected Rate of Inflation)] − 1
Rearranging the formula, you can calculate the RRFR of return on an investment as follows:
ð1 + NRFR of ReturnÞ
RRFR =
ð1 + Rate of InflationÞ
To see how this works, assume that the nominal return on U.S. government T-bills was 9
percent during a given year, when the rate of inflation was 5 percent. In this instance, the
RRFR of return on these T-bills was 3.8 percent, as follows:
RRFR = ½ð1 + 0:09Þ=ð1 + 0:05Þ − 1
= 1:038 − 1
= 0:038 = 3:8%
This discussion makes it clear that the nominal rate of interest on a risk-free investment is
not a good estimate of the RRFR, because the nominal rate can change dramatically in the short
run in reaction to temporary ease or tightness in the capital market or because of changes in the
expected rate of inflation. As indicated by the data in Exhibit 1.6, the significant changes in the
average yield on T-bills typically were related to large changes in the rates of inflation. Notably,
2009–2010 were different due to the quantitative easing by the Federal Reserve.
The Common Effect All the factors discussed thus far regarding the required rate of return
affect all investments equally. Whether the investment is in stocks, bonds, real estate, or machine tools, if the expected rate of inflation increases from 2 percent to 6 percent, the investor’s required rate of return for all investments should increase by 4 percent. Similarly, if a
decline in the expected real growth rate of the economy causes a decline in the RRFR of 1 percent, the required return on all investments should decline by 1 percent.
1.3.3 Risk Premium
A risk-free investment was defined as one for which the investor is certain of the amount and
timing of the expected returns. The returns from most investments do not fit this pattern. An
Part 1: The Investment Background
investor typically is not completely certain of the income to be received or when it will be received. Investments can range in uncertainty from basically risk-free securities, such as T-bills,
to highly speculative investments, such as the common stock of small companies engaged in
high-risk enterprises.
Most investors require higher rates of return on investments if they perceive that there is
any uncertainty about the expected rate of return. This increase in the required rate of return
over the NRFR is the risk premium (RP). Although the required risk premium represents a
composite of all uncertainty, it is possible to consider several fundamental sources of uncertainty. In this section, we identify and discuss briefly the major sources of uncertainty, including: (1) business risk, (2) financial risk (leverage), (3) liquidity risk, (4) exchange rate risk, and
(5) country (political) risk.
Business risk is the uncertainty of income flows caused by the nature of a firm’s business.
The less certain the income flows of the firm, the less certain the income flows to the investor.
Therefore, the investor will demand a risk premium that is based on the uncertainty caused by
the basic business of the firm. As an example, a retail food company would typically experience stable sales and earnings growth over time and would have low business risk compared
to a firm in the auto or airline industry, where sales and earnings fluctuate substantially over
the business cycle, implying high business risk.
Financial risk is the uncertainty introduced by the method by which the firm finances its
investments. If a firm uses only common stock to finance investments, it incurs only business
risk. If a firm borrows money to finance investments, it must pay fixed financing charges (in
the form of interest to creditors) prior to providing income to the common stockholders, so
the uncertainty of returns to the equity investor increases. This increase in uncertainty because
of fixed-cost financing is called financial risk or financial leverage, and it causes an increase in
the stock’s risk premium. For an extended discussion on this, see Brigham (2010).
Liquidity risk is the uncertainty introduced by the secondary market for an investment.
When an investor acquires an asset, he or she expects that the investment will mature (as
with a bond) or that it will be salable to someone else. In either case, the investor expects to
be able to convert the security into cash and use the proceeds for current consumption or
other investments. The more difficult it is to make this conversion to cash, the greater the liquidity risk. An investor must consider two questions when assessing the liquidity risk of an
investment: How long will it take to convert the investment into cash? How certain is the price
to be received? Similar uncertainty faces an investor who wants to acquire an asset: How long
will it take to acquire the asset? How uncertain is the price to be paid?5
Uncertainty regarding how fast an investment can be bought or sold, or the existence of
uncertainty about its price, increases liquidity risk. A U.S. government Treasury bill has almost
no liquidity risk because it can be bought or sold in seconds at a price almost identical to the
quoted price. In contrast, examples of illiquid investments include a work of art, an antique, or
a parcel of real estate in a remote area. For such investments, it may require a long time to
find a buyer and the selling prices could vary substantially from expectations. Investors will
increase their required rates of return to compensate for this uncertainty regarding timing
and price. Liquidity risk can be a significant consideration when investing in foreign securities
depending on the country and the liquidity of its stock and bond markets.
Exchange rate risk is the uncertainty of returns to an investor who acquires securities denominated in a currency different from his or her own. The likelihood of incurring this risk is
becoming greater as investors buy and sell assets around the world, as opposed to only assets
within their own countries. A U.S. investor who buys Japanese stock denominated in yen must
You will recall from prior courses that the overall capital market is composed of the primary market and the secondary market. Securities are initially sold in the primary market, and all subsequent transactions take place in the secondary market. These concepts are discussed in Chapter 4.
Chapter 1: The Investment Setting
consider not only the uncertainty of the return in yen but also any change in the exchange
value of the yen relative to the U.S. dollar. That is, in addition to the foreign firm’s business
and financial risk and the security’s liquidity risk, the investor must consider the additional
uncertainty of the return on this Japanese stock when it is converted from yen to U.S. dollars.
As an example of exchange rate risk, assume that you buy 100 shares of Mitsubishi Electric
at 1,050 yen when the exchange rate is 105 yen to the dollar. The dollar cost of this investment
would be about $10.00 per share (1,050/105). A year later you sell the 100 shares at 1,200 yen
when the exchange rate is 115 yen to the dollar. When you calculate the HPY in yen, you find
the stock has increased in value by about 14 percent (1,200/1,050) − 1, but this is the HPY for
a Japanese investor. A U.S. investor receives a much lower rate of return, because during this
period the yen has weakened relative to the dollar by about 9.5 percent (that is, it requires
more yen to buy a dollar—115 versus 105). At the new exchange rate, the stock is worth
$10.43 per share (1,200/115). Therefore, the return to you as a U.S. investor would be only
about 4 percent ($10.43/$10.00) versus 14 percent for the Japanese investor. The difference in
return for the Japanese investor and U.S. investor is caused by exchange rate risk—that is, the
decline in the value of the yen relative to the dollar. Clearly, the exchange rate could have gone
in the other direction, the dollar weakening against the yen. In this case, as a U.S. investor, you
would have experienced the 14 percent return measured in yen, as well as a currency gain
from the exchange rate change.
The more volatile the exchange rate between two countries, the less certain you would be
regarding the exchange rate, the greater the exchange rate risk, and the larger the exchange
rate risk premium you would require. For an analysis of pricing this risk, see Jorion (1991).
There can also be exchange rate risk for a U.S. firm that is extensively multinational in
terms of sales and expenses. In this case, the firm’s foreign earnings can be affected by changes
in the exchange rate. As will be discussed, this risk can generally be hedged at a cost.
Country risk, also called political risk, is the uncertainty of returns caused by the possibility
of a major change in the political or economic environment of a country. The United States is
acknowledged to have the smallest country risk in the world because its political and economic
systems are the most stable. During the spring of 2011, prevailing examples include the deadly
rebellion in Libya against Moammar Gadhafi; a major uprising in Syria against President
Bashar al-Assad; and significant protests in Yemen against President Ali Abdullah Saleh. In
addition, there has been a recent deadly earthquake and tsunami in Japan that is disturbing
numerous global corporations and the currency markets. Individuals who invest in countries
that have unstable political or economic systems must add a country risk premium when determining their required rates of return.
When investing globally (which is emphasized throughout the book, based on a discussion in
Chapter 3), investors must consider these additional uncertainties. How liquid are the secondary
markets for stocks and bonds in the country? Are any of the country’s securities traded on major
stock exchanges in the United States, London, Tokyo, or Germany? What will happen to exchange rates during the investment period? What is the probability of a political or economic
change that will adversely affect your rate of return? Exchange rate risk and country risk differ
among countries. A good measure of exchange rate risk would be the absolute variability of the
exchange rate relative to a composite exchange rate. The analysis of country risk is much more
subjective and must be based on the history and current political environment of the country.
This discussion of risk components can be considered a security’s fundamental risk because
it deals with the intrinsic factors that should affect a security’s volatility of returns over time.
In subsequent discussion, the standard deviation of returns for a security is referred to as a
measure of the security’s total risk, which considers only the individual stock—that is, the
stock is not considered as part of a portfolio.
Risk Premium = f (Business Risk, Financial Risk, Liquidity Risk, Exchange Rate Risk, Country Risk)
Part 1: The Investment Background
1.3.4 Risk Premium and Portfolio Theory
An alternative view of risk has been derived from extensive work in portfolio theory and capital
market theory by Markowitz (1952, 1959) and Sharpe (1964). These theories are dealt with in
greater detail in Chapter 7 and Chapter 8 but their impact on a stock’s risk premium should be
mentioned briefly at this point. These prior works by Markowitz and Sharpe indicated that investors should use an external market measure of risk. Under a specified set of assumptions, all
rational, profit-maximizing investors want to hold a completely diversified market portfolio of
risky assets, and they borrow or lend to arrive at a risk level that is consistent with their risk
preferences. Under these conditions, they showed that the relevant risk measure for an individual asset is its comovement with the market portfolio. This comovement, which is measured by
an asset’s covariance with the market portfolio, is referred to as an asset’s systematic risk, the
portion of an individual asset’s total variance that is attributable to the variability of the total
market portfolio. In addition, individual assets have variance that is unrelated to the market
portfolio (the asset’s nonmarket variance) that is due to the asset’s unique features. This nonmarket variance is called unsystematic risk, and it is generally considered unimportant because
it is eliminated in a large, diversified portfolio. Therefore, under these assumptions, the risk premium for an individual earning asset is a function of the asset’s systematic risk with the aggregate
market portfolio of risky assets. The measure of an asset’s systematic risk is referred to as its beta:
Risk Premium = f (Systematic Market Risk)
1.3.5 Fundamental Risk versus Systematic Risk
Some might expect a conflict between the market measure of risk (systematic risk) and the
fundamental determinants of risk (business risk, and so on). A number of studies have examined the relationship between the market measure of risk (systematic risk) and accounting
variables used to measure the fundamental risk factors, such as business risk, financial risk,
and liquidity risk. The authors of these studies (especially Thompson, 1976) have generally
concluded that a significant relationship exists between the market measure of risk and the fundamental measures of risk. Therefore, the two measures of risk can be complementary. This
consistency seems reasonable because one might expect the market measure of risk to reflect
the fundamental risk characteristics of the asset. For example, you might expect a firm that
has high business risk and financial risk to have an above-average beta. At the same time, as
we discuss in Chapter 8, a firm that has a high level of fundamental risk and a large standard
deviation of returns can have a lower level of systematic risk simply because the variability of
its earnings and its stock price is not related to the aggregate economy or the aggregate market, i.e., a large component of its total risk is due to unique unsystematic risk. Therefore, one
can specify the risk premium for an asset as either:
Risk Premium = f (Business Risk, Financial Risk, Liquidity Risk, Exchange Rate Risk, Country Risk)
Risk Premium = f (Systematic Market Risk)
1.3.6 Summary of Required Rate of Return
The overall required rate of return on alternative investments is determined by three variables:
(1) the economy’s RRFR, which is influenced by the investment opportunities in the economy
(that is, the long-run real growth rate); (2) variables that influence the NRFR, which include
short-run ease or tightness in the capital market and the expected rate of inflation. Notably,
these variables, which determine the NRFR, are the same for all investments; and (3) the risk
premium on the investment. In turn, this risk premium can be related to fundamental factors,
including business risk, financial risk, liquidity risk, exchange rate risk, and country risk, or it
can be a function of an asset’s systematic market risk (beta).
Chapter 1: The Investment Setting
Measures and Sources of Risk In this chapter, we have examined both measures and
sources of risk arising from an investment. The measures of market risk for an investment are:
Variance of rates of return
Standard deviation of rates of return
Coefficient of variation of rates of return (standard deviation/means)
Covariance of returns with the market portfolio (beta)
The sources of fundamental risk are:
Business risk
Financial risk
Liquidity risk
Exchange rate risk
Country risk
1.4 RELATIONSHIP
Previously, we showed how to measure the risk and rates of return for alternative investments
and we discussed what determines the rates of return that investors require. This section discusses the risk-return combinations that might be available at a point in time and illustrates
the factors that cause changes in these combinations.
Exhibit 1.7 graphs the expected relationship between risk and return. It shows that investors
increase their required rates of return as perceived risk (uncertainty) increases. The line that
reflects the combination of risk and return available on alternative investments is referred to
as the security market line (SML). The SML reflects the risk-return combinations available
for all risky assets in the capital market at a given time. Investors would select investments
that are consistent with their risk preferences; some would consider only low-risk investments,
whereas others welcome high-risk investments.
Exhibit 1.7 Relationship between Risk and Return
Expected Return
Security Market
The slope indicates the
required return per unit
of risk
(business risk, etc., or systematic risk—beta)
Part 1: The Investment Background
Beginning with an initial SML, three changes in the SML can occur. First, individual investments can change positions on the SML because of changes in the perceived risk of the
investments. Second, the slope of the SML can change because of a change in the attitudes of
investors toward risk; that is, investors can change the returns they require per unit of risk.
Third, the SML can experience a parallel shift due to a change in the RRFR or the expected
rate of inflation—i.e., anything that can change in the NRFR. These three possibilities are
discussed in this section.
1.4.1 Movements along the SML
Investors place alternative investments somewhere along the SML based on their perceptions
of the risk of the investment. Obviously, if an investment’s risk changes due to a change in
one of its fundamental risk sources (business risk, and such), it will move along the SML. For
example, if a firm increases its financial risk by selling a large bond issue that increases its financial leverage, investors will perceive its common stock as riskier and the stock will move up
the SML to a higher risk position implying that investors will require a higher rate of return.
As the common stock becomes riskier, it changes its position on the SML. Any change in an
asset that affects its fundamental risk factors or its market risk (that is, its beta) will cause the
asset to move along the SML as shown in Exhibit 1.8. Note that the SML does not change,
only the position of specific assets on the SML.
1.4.2 Changes in the Slope of the SML
The slope of the SML indicates the return per unit of risk required by all investors. Assuming
a straight line, it is possible to select any point on the SML and compute a risk premium (RP)
for an asset through the equation:
RPi = E(Ri) − NRFR
RPi = risk premium for asset i
EðRi Þ = the expected return for asset i
NRFR = the nominal return on a risk-free asset
Exhibit 1.8 Changes in the Required Rate of Return Due to Movements
along the SML
Movements along the
curve that reflect
changes in the risk
of the asset
Chapter 1: The Investment Setting
If a point on the SML is identified as the portfolio that contains all the risky assets in the
market (referred to as the market portfolio), it is possible to compute a market RP as follows:
RPm = E(Rm) − NRFR
RPm = the risk premium on the market portfolio
EðRm Þ = the expected return on the market portfolio
NRFR = the nominal return on a risk-free asset
This market RP is not constant because the slope of the SML changes over time. Although
we do not understand completely what causes these changes in the slope, we do know that
there are changes in the yield differences between assets with different levels of risk even
though the inherent risk differences are relatively constant.
These differences in yields are referred to as yield spreads, and these yield spreads change over
time. As an example, if the yield on a portfolio of Aaa-rated bonds is 7.50 percent and the yield on a
portfolio of Baa-rated bonds is 9.00 percent, we would say that the yield spread is 1.50 percent. This
1.50 percent is referred to as a credit risk premium because the Baa-rated bond is considered to have
higher credit risk—that is, it has a higher probability of default. This Baa–Aaa yield spread is not
constant over time, as shown by the substantial volatility in the yield spreads shown in Exhibit 1.9.
Although the underlying business and financial risk characteristics for the portfolio of bonds
in the Aaa-rated bond index and the Baa-rated bond index would probably not change dramatically over time, it is clear from the time-series plot in Exhibit 1.9 that the difference in yields
(i.e., the yield spread) has experienced changes of more than 100 basis points (1 percent) in a
short period of time (for example, see the yield spread increases in 1974–1975, 1981–1983,
2001–2002, 2008–2009, and the dramatic declines in yield spread during 1975, 1983–1984,
2003–2004, and the second half of 2009). Such a significant change in the yield spread during
a period where there is no major change in the fundamental risk characteristics of Baa bonds
Exhibit 1.9 Barclays Capital U.S. Credit Monthly Yield Spreads in Basis Points
Mean + 1*Std. Dev. (229 bp)
Median = 120
Mean (143 bp)
Mean + 2*Std. Dev. (316 bp)
Std. Dev. = 86
Mean – 1*Std. Dev. (56 bp)
Mean – 2*Std. Dev. (–30 bp)
Source: Barclays Capital data; computations by authors.
Baa-Aaa Rated Credit Spreads in Basis Points
(U.S. Credit Aaa – U.S. Credit Baa) Jan. 1973–Dec. 2010
Part 1: The Investment Background
Exhibit 1.10 Change in Market Risk Premium
Expected Return
New SML
Original SML
relative to Aaa bonds would imply a change in the market RP. Specifically, although the intrinsic
financial risk characteristics of the bonds remain relatively constant, investors changed the yield
spreads (i.e., the credit risk premiums) they demand to accept this difference in financial risk.
This change in the RP implies a change in the slope of the SML. Such a change is shown in
Exhibit 1.10. The exhibit assumes an increase in the market risk premium, which means an
increase in the slope of the market line. Such a change in the slope of the SML (the market
risk premium) will affect the required rate of return for all risky assets. Irrespective of where
an investment is on the original SML, its required rate of return will increase, although its intrinsic risk characteristics remain unchanged.
1.4.3 Changes in Capital Market Conditions or Expected Inflation
The graph in Exhibit 1.11 shows what happens to the SML when there are changes in one of
the following factors: (1) expected real growth in the economy, (2) capital market conditions,
or (3) the expected rate of inflation. For example, an increase in expected real growth, temporary tightness in the capital market, or an increase in the expected rate of inflation will cause
the SML to experience a parallel shift upward as shown in Exhibit 1.11. The parallel shift
occurs because changes in expected real growth or changes in capital market conditions or a
change in the expected rate of inflation affect the economy’s nominal risk-free rate (NRFR)
that impacts all investments, irrespective of their risk levels.
1.4.4 Summary of Changes in the Required Rate of Return
The relationship between risk and the required rate of return for an investment can change in
three ways:
1. A movement along the SML demonstrates a change in the risk characteristics of a specific
investment, such as a change in its business risk, its financial risk, or its systematic risk (its
beta). This change affects only the individual investment.
Chapter 1: The Investment Setting
Exhibit 1.11 Capital Market Conditions, Expected Inflation, and the Security
Market Line
Expected Return
New SML
Original SML
2. A change in the slope of the SML occurs in response to a change in the attitudes of investors
toward risk. Such a change demonstrates that investors want either higher or lower rates of
return for the same intrinsic risk. This is also described as a change in the market risk premium (Rm − NRFR). A change in the market risk premium will affect all risky investments.
3. A shift in the SML reflects a change in expected real growth, a change in market conditions (such as ease or tightness of money), or a change in the expected rate of inflation. Again, such a change will affect all investments.
The purpose of this chapter is to provide background
that can be used in subsequent chapters. To achieve
that goal, we covered several topics:
• We discussed why individuals save part of their income and why they decide to invest their savings.
We defined investment as the current commitment
of these savings for a period of time to derive a rate
of return that compensates for the time involved,
the expected rate of inflation, and the uncertainty.
• We examined ways to quantify historical return and
risk to help analyze alternative investment opportunities. We considered two measures of mean return
(arithmetic and geometric) and applied these to a
historical series for an individual investment and
to a portfolio of investments during a period of
• We considered the concept of uncertainty and alternative measures of risk (the variance, standard deviation, and a relative measure of risk—the coefficient
of variation).
• Before discussing the determinants of the required
rate of return for an investment, we noted that the
estimation of the required rate of return is complicated because the rates on individual investments
change over time, because there is a wide range of
rates of return available on alternative investments,
and because the differences between required returns on alternative investments (for example, the
yield spreads) likewise change over time.
Part 1: The Investment Background
• We examined the specific factors that determine the
required rate of return: (1) the real risk-free rate,
which is based on the real rate of growth in the
economy, (2) the nominal risk-free rate, which is
influenced by capital market conditions and the expected rate of inflation, and (3) a risk premium,
which is a function of fundamental factors, such as
business risk, or the systematic risk of the asset relative to the market portfolio (that is, its beta).
• We discussed the risk-return combinations available
on alternative investments at a point in time (illustrated by the SML) and the three factors that can
cause changes in this relationship. First, a change
in the inherent risk of an individual investment
(that is, its fundamental risk or market risk) will
cause a movement along the SML. Second, a change
in investors’ attitudes toward risk will cause a
change in the required return per unit of risk—
that is, a change in the market risk premium. Such
a change will cause a change in the slope of the
SML. Finally, a change in expected real growth, in
capital market conditions, or in the expected rate of
inflation will cause a parallel shift of the SML.
Based on this understanding of the investment environment, you are prepared to consider the asset allocation decision, which is discussed in Chapter 2.
Fama, Eugene F., and Merton H. Miller. The Theory of
Finance. New York: Holt, Rinehart and Winston, 1972.
Fisher, Irving. The Theory of Interest. New York: Macmillan, 1930, reprinted by Augustus M. Kelley, 1961.
1. Discuss the overall purpose people have for investing. Define investment.
2. As a student, are you saving or borrowing? Why?
3. Divide a person’s life from ages 20 to 70 into 10-year segments and discuss the likely
saving or borrowing patterns during each period.
4. Discuss why you would expect the saving-borrowing pattern to differ by occupation (for
example, for a doctor versus a plumber).
5. The Wall Street Journal reported that the yield on common stocks is about 2 percent,
whereas a study at the University of Chicago contends that the annual rate of return on
common stocks since 1926 has averaged about 10 percent. Reconcile these statements.
Some financial theorists consider the variance of the distribution of expected rates of return to be a good measure of uncertainty. Discuss the reasoning behind this measure of
risk and its purpose.
Discuss the three components of an investor’s required rate of return on an investment.
Discuss the two major factors that determine the market nominal risk-free rate (NRFR).
Explain which of these factors would be more volatile over the business cycle.
Briefly discuss the five fundamental factors that influence the risk premium of an
You own stock in the Gentry Company, and you read in the financial press that a recent
bond offering has raised the firm’s debt/equity ratio from 35 percent to 55 percent. Discuss
the effect of this change on the variability of the firm’s net income stream, other factors being
constant. Discuss how this change would affect your required rate of return on the common
stock of the Gentry Company.
Draw a properly labeled graph of the security market line (SML) and indicate where you
would expect the following investments to fall along that line. Discuss your reasoning.
a. Common stock of large firms
b. U.S. government bonds
c. U.K. government bonds
d. Low-grade corporate bonds
e. Common stock of a Japanese firm
Chapter 1: The Investment Setting
Explain why you would change your nominal required rate of return if you expected the
rate of inflation to go from 0 (no inflation) to 4 percent. Give an example of what would
happen if you did not change your required rate of return under these conditions.
13. Assume the expected long-run growth rate of the economy increased by 1 percent and
the expected rate of inflation increased by 4 percent. What would happen to the required
rates of return on government bonds and common stocks? Show graphically how the effects of these changes would differ between these alternative investments.
14. You see in The Wall Street Journal that the yield spread between Baa corporate bonds
and Aaa corporate bonds has gone from 350 basis points (3.5 percent) to 200 basis
points (2 percent). Show graphically the effect of this change in yield spread on the
SML and discuss its effect on the required rate of return for common stocks.
15. Give an example of a liquid investment and an illiquid investment. Discuss why you consider each of them to be liquid or illiquid.
On February 1, you bought 100 shares of stock in the Francesca Corporation for $34 a
share and a year later you sold it for $39 a share. During the year, you received a cash
dividend of $1.50 a share. Compute your HPR and HPY on this Francesca stock
On August 15, you purchased 100 shares of stock in the Cara Cotton Company at $65 a
share and a year later you sold it for $61 a share. During the year, you received dividends
of $3 a share. Compute your HPR and HPY on your investment in Cara Cotton.
At the beginning of last year, you invested $4,000 in 80 shares of the Chang Corporation.
During the year, Chang paid dividends of $5 per share. At the end of the year, you sold
the 80 shares for $59 a share. Compute your total HPY on these shares and indicate how
much was due to the price change and how much was due to the dividend income.
The rates of return computed in Problems 1, 2, and 3 are nominal rates of return. Assuming that the rate of inflation during the year was 4 percent, compute the real rates
of return on these investments. Compute the real rates of return if the rate of inflation
was 8 percent.
During the past five years, you owned two stocks that had the following annual rates of
Ye a r
S to c k T
Stock B
a. Compute the arithmetic mean annual rate of return for each stock. Which stock is
most desirable by this measure?
b. Compute the standard deviation of the annual rate of return for each stock. (Use
Chapter 1 Appendix if necessary.) By this measure, which is the preferable stock?
c. Compute the coefficient of variation for each stock. (Use the Chapter 1 Appendix if
necessary.) By this relative measure of risk, which stock is preferable?
d. Compute the geometric mean rate of return for each stock. Discuss the difference
between the arithmetic mean return and the geometric mean return for each stock.
Discuss the differences in the mean returns relative to the standard deviation of the
return for each stock.
Part 1: The Investment Background
You are considering acquiring shares of common stock in the Madison Beer Corporation.
Your rate of return expectations are as follows:
P os s i bl e R a t e of R e t ur n
P ro b ab il i ty
Compute the expected return [E(Ri)] on your investment in Madison Beer.
A stockbroker calls you and suggests that you invest in the Lauren Computer Company.
After analyzing the firm’s annual report and other material, you believe that the distribution of expected rates of return is as follows:
P os s i bl e R a t e of R e t ur n
P ro b ab il i ty
Compute the expected return [E(Ri)] on Lauren Computer stock.
Without any formal computations, do you consider Madison Beer in Problem 6 or
Lauren Computer in Problem 7 to present greater risk? Discuss your reasoning.
9. During the past year, you had a portfolio that contained U.S. government T-bills, longterm government bonds, and common stocks. The rates of return on each of them were
as follows:
U.S. government T-bills
U.S. government long-term bonds
U.S. common stocks
During the year, the consumer price index, which measures the rate of inflation, went
from 160 to 172 (1982 – 1984 = 100). Compute the rate of inflation during this year.
Compute the real rates of return on each of the investments in your portfolio based on
the inflation rate.
10. You read in BusinessWeek that a panel of economists has estimated that the long-run real
growth rate of the U.S. economy over the next five-year period will average 3 percent. In
addition, a bank newsletter estimates that the average annual rate of inflation during this
five-year period will be about 4 percent. What nominal rate of return would you expect
on U.S. government T-bills during this period?
11. What would your required rate of return be on common stocks if you wanted a 5 percent
risk premium to own common stocks given what you know from Problem 10? If common stock investors became more risk averse, what would happen to the required rate
of return on common stocks? What would be the impact on stock prices?
12. Assume that the consensus required rate of return on common stocks is 14 percent. In
addition, you read in Fortune that the expected rate of inflation is 5 percent and the
estimated long-term real growth rate of the economy is 3 percent. What interest rate
Chapter 1: The Investment Setting
would you expect on U.S. government T-bills? What is the approximate risk premium for
common stocks implied by these data?
Find general information for Walgreens (stock symbol: WAG) and Walmart (WMT), two
firms in the retail industry (or try two firms in an industry of your choice). On what stock
markets are the firms traded? How do their growth rates in sales and earnings compare?
How have their stocks performed over the past few months? Are stock analysts recommending
investors buy or sell each of the two firm’s stocks?
Computation of Variance and Standard Deviation
Variance and standard deviation are measures of how actual values differ from the expected
values (arithmetic mean) for a given series of values. In this case, we want to measure how
rates of return differ from the arithmetic mean value of a series. There are other measures of
dispersion, but variance and standard deviation are the best known because they are used in
statistics and probability theory. Variance is defined as:
ðProbabilityÞðPossible Return − Expected ReturnÞ2
Variance ðσ 2 Þ =
ðPi Þ½Ri − EðRi Þ2
Consider the following example, as discussed in the chapter:
P r ob a bi li t y o f
Po s s ib l e R et u r n ( P i )
Po ssi bl e R e tu r n
(R i )
Σ = 0.07
This gives an expected return [E(Ri)] of 7 percent. The dispersion of this distribution as
measured by variance is:
P ro b ab il i ty (P i )
R e t ur n (R i )
R i − E (R i )
[R i − E (R i ) ] 2
P i [ R i − E(R i )] 2
Σ = 0.014100
The variance (σ2) is equal to 0.0141. The standard deviation is equal to the square root of
the variance:
Pi ½Ri − EðRi Þ2
Standard Deviation ðσ Þ =
Consequently, the standard deviation for the preceding example would be:
σ i = 0:0141 = 0:11874
In this example, the standard deviation is approximately 11.87 percent. Therefore, you
could describe this distribution as having an expected value of 7 percent and a standard deviation of 11.87 percent.
In many instances, you might want to compute the variance or standard deviation for a historical series in order to evaluate the past performance of the investment. Assume that you are
given the following information on annual rates of return (HPY) for common stocks listed on
the New York Stock Exchange (NYSE):
Y e ar
A nn u al R a t e
of R e t ur n
|
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How do you solve the equation: 4x^2=20? | Socratic
How do you solve the equation: #4x^2=20#?
1 Answer
We can start by dividing both sides by $4$. Doing this, we get:
${x}^{2} = 5$
Next, we can take the square root of both sides to get:
$x = \pm \sqrt{5}$
There are no perfect squares to factor out of $\sqrt{5}$ so we could simplify this. Thus, $\pm \sqrt{5}$ is our answer.
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3729 views around the world
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Understanding Mathematical Functions: How To Graph A Step Function
Mathematical functions play a crucial role in understanding relationships and patterns in the world of mathematics. They provide a way to express how one quantity depends on another. One particular
type of function, known as a step function, has distinct characteristics that set it apart from other functions. In this blog post, we will delve into the definition of mathematical functions, and
explore the importance of understanding and graphing step functions in mathematical analysis.
Key Takeaways
• Mathematical functions are essential for understanding relationships and patterns in mathematics
• Step functions have distinct characteristics that set them apart from other functions
• Understanding and graphing step functions is important in mathematical analysis
• Step functions can be used to model real-world applications
• Practical tips, such as using a ruler and double-checking work, are crucial for accurate graphing of step functions
Understanding Step Functions
Step functions are an important concept in mathematics, particularly in the field of calculus. They are used to model real-world situations where data changes abruptly rather than continuously. Let's
explore the definition, characteristics, and applications of step functions.
A. Definition of step functions
A step function, also known as a staircase function, is a type of piecewise-defined function where the graph consists of horizontal line segments. These segments represent constant values within
specific intervals, and the function changes abruptly from one constant value to another at distinct points.
B. Characteristics of step functions
Step functions have several key characteristics, including:
• Discontinuities: Step functions have discontinuities at the points where the function changes value. These points are known as "jumps" in the graph of the function.
• Constant intervals: The graph of a step function consists of horizontal line segments, each representing a constant value within a specific interval.
• Defined intervals: Step functions are piecewise-defined, meaning the function has different expressions and constants for different intervals of the domain.
C. Examples of real-world applications of step functions
Step functions have numerous real-world applications in various fields. Some examples include:
• Population growth: Modeling the population of a species, where the population remains constant for certain periods and experiences abrupt changes due to factors such as migration or environmental
• Financial transactions: Tracking changes in stock prices, where the value remains constant for a period of time before experiencing sudden increases or decreases.
• Electrical engineering: Describing the behavior of digital signals in electronics, where the signal remains at a constant level before transitioning to a new level.
Graphing Step Functions
Understanding how to graph a step function is essential in mathematics, especially when dealing with real-world applications. Step functions are a type of piecewise function that have a constant
value within specific intervals. Here's how to graph a step function:
A. Identify the intervals
• 1. Define the intervalsIdentify the distinct intervals where the step function changes its value. This could be determined by the domain of the function or specific conditions outlined in the
given problem.
B. Determine the function values within each interval
• 1. Assign values for each intervalDetermine the function values for each interval of the step function. This involves understanding the behavior of the function within each segment of the domain.
C. Plot the points on the graph
• 1. Mark the pointsUse the determined function values to plot points on a graph. Label each point with its corresponding coordinates based on the function's domain and range.
D. Connect the points to form the step function graph
• 1. Use horizontal line segmentsConnect the points on the graph using horizontal line segments to represent the constant value of the step function within each interval. This will create a
distinct step-wise pattern.
Step Function Notation
A step function is a special type of piecewise function that has a finite number of constant pieces. It jumps from one value to another at specific points in its domain. Understanding the notation of
step functions is crucial for graphing them accurately.
A. Using mathematical notation to represent step functions
Step functions are often represented using the following notation: f(x) = a[1] for x < x[1], a[2] for x[1] ≤ x < x[2], ..., a[n] for x[n-1] ≤ x.
B. Understanding the domain and range of step functions
The domain of a step function is the set of all input values for which the function is defined. The range is the set of all output values that the function can produce. It's important to understand
the domain and range of a step function in order to accurately graph it.
C. Identifying key features on the graph based on the notation
Based on the notation of a step function, key features such as the constant intervals and the jump discontinuities can be identified. These features are essential for accurately graphing the step
Transformations of Step Functions
Understanding how to graph a step function involves knowing how to apply various transformations to the basic function. These transformations can shift the graph horizontally or vertically, reflect
it over the x-axis or y-axis, and stretch or compress it.
A. Shifting the graph horizontally or vertically
When shifting the graph of a step function, you can move it horizontally or vertically by adding or subtracting values inside the function. For horizontal shifts, adding or subtracting a constant to
the input variable will move the graph left or right. For vertical shifts, adding or subtracting a constant to the entire function will move the graph up or down.
B. Reflecting the graph over the x-axis or y-axis
Reflecting the graph of a step function over the x-axis or y-axis involves multiplying the function by -1 for the respective axis. To reflect the graph over the x-axis, multiply the function by -1.
To reflect the graph over the y-axis, multiply the input variable by -1.
C. Stretching or compressing the graph
Stretching or compressing the graph of a step function can be achieved by multiplying the function by a constant. A value greater than one will stretch the graph vertically while a value between 0
and 1 will compress the graph. To stretch or compress the graph horizontally, apply the constant to the input variable.
Practical Tips for Graphing Step Functions
Graphing step functions can be a challenging task, but with the right approach, you can create accurate and visually appealing graphs. Here are some practical tips to help you graph step functions
with ease.
• Use a ruler for accuracy
When graphing step functions, it is essential to use a ruler to ensure precision. Straight, neat lines are crucial for accurately representing the step function.
• Label the axes and key points on the graph
Proper labeling of the x and y axes is essential for clarity. Additionally, labeling key points on the graph, such as the steps and breakpoints, will help viewers understand the function more
• Double-check your work for any errors before finalizing the graph
Before considering your graph complete, it is crucial to review your work for any mistakes. This includes checking for accurate placement of points, step lines, and ensuring the overall
representation aligns with the function being graphed.
In conclusion, we have learned how to graph a step function by identifying the key components, such as the open and closed circles, and understanding the concept of intervals. It is crucial to
understand step functions as they are widely used in real-world applications, such as in computer science, economics, and physics. By mastering the art of graphing step functions, you can gain a
deeper understanding of mathematical functions and their practical implications.
Graphing step functions is an essential skill that can be applied to various fields, making it an important concept to grasp in mathematics. It allows you to visualize and analyze data in a clear and
organized manner, enabling you to make informed decisions and solve complex problems.
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June 2011 – Walking Randomly
Archive for June, 2011
June 28th, 2011
I saw a great tweet from Marcus du Sautoy this morning who declared that today, June 28th, is a perfect day because both 6 and 28 are perfect numbers. This, combined with the fact that it is very
sunny in Manchester right now put me in a great mood and I gave my colleauges a quick maths lesson to try and explain why I was so happy.
“It’s not a perfect year though is it?” declared one of my colleauges. Some people are never happy and she’s going to have to wait over 6000 years before her definition of a perfect day is
fulfilled. The date of this truly perfect day? 28th June 8128.
Update: Someone just emailed me to say that 28th June is Tau Day too!
June 26th, 2011
Back in the good old days when I was a freshly minted postgraduate student I had big plans– In short, I was going to change the world. Along with a couple of my friends I was going to revolutionize
the field I was working in, win the Nobel prize and transform the way science and mathematics is taught at University. Fast forward four years and it pains me to say that my actual achievements fell
rather short of these lofty ideals. I considered myself lucky to simply pass my PhD and land a job that didn’t involve querying members of the public on their preferences regarding potato based
products. The four subjects of Laura Snyder’s latest book, The Philosophical Breakfast Club
In this sweeping history of nineteenth century science, Snyder gives us not one biography but four — those of Charles Babbage, John Herschel, William Whewell and Richard Jones. You may not have
heard of all of them but I’d be surprised if you didn’t know of some of their work. Between them they invented computing, modern economics, produced the most detailed astronomical maps of their age,
co-invented photography, made important advances in tidology, invented the term scientist (among many other neologisms) and they are just the headliners! Under-achievers they were not.
These four men met while studying at Cambridge University way back in 1812 where they held weekly meetings which they called The Philosophical Breakfast Club. They took a look at how science was
practiced in their day, found it wanting and decided to do something it. Remarkably, they succeeded!
I found Snyder’s combination of biography, history and science to be utterly compelling…so much so that during my time reading it, my beloved iPad stayed at home, lonely and forgotten, while I
undertook my daily commute. This is no dry treatise on nineteenth century science; instead it is a living, breathing page-turner about a group of very colourful individuals who lived in a time where
science was done rather differently from how it is practiced today. This was a time where ‘computer’ meant ‘a person who was good at arithmetic’ and professors would share afternoon champagne with
their students after giving them advice. Who would have thought that a group of nineteenth century geeks could form the basis of one of the best books I’ve read all year?
June 18th, 2011
Over at Sol Lederman’s fantastic new blog, Playing with Mathematica, he shared some code that produced the following figure.
Here’s Sol’s code with an AbsoluteTiming command thrown in.
f[x_, y_] := Module[{},
Sin[Min[x*Sin[y], y*Sin[x]]] >
y*Cos[x]]] + (((2 (x - y)^2 + (x + y - 6)^2)/40)^3)/
6400000 + (12 - x - y)/30, 1, 0]
\[Delta] = 0.02;
range = 11;
xyPoints = Table[{x, y}, {y, 0, range, \[Delta]}, {x, 0, range, \[Delta]}];
image = Map[f @@ # &, xyPoints, {2}];
Rotate[ArrayPlot[image, ColorRules -> {0 -> White, 1 -> Black}], 135 Degree]
This took 8.02 seconds on the laptop I am currently working on (Windows 7 AMD Phenom II N620 Dual core at 2.8Ghz). Note that I am only measuring how long the calculation itself took and am ignoring
the time taken to render the image and define the function.
Compiled functions make Mathematica code go faster
Mathematica has a Compile function which does exactly what you’d expect…it produces a compiled version of the function you give it (if it can!). Sol’s function gave it no problems at all.
f = Compile[{{x, _Real}, {y, _Real}}, If[
Sin[Min[x*Sin[y], y*Sin[x]]] >
y*Cos[x]]] + (((2 (x - y)^2 + (x + y - 6)^2)/40)^3)/
6400000 + (12 - x - y)/30, 1, 0]
\[Delta] = 0.02;
range = 11;
xyPoints =
Table[{x, y}, {y, 0, range, \[Delta]}, {x, 0, range, \[Delta]}];
image = Map[f @@ # &, xyPoints, {2}];
Rotate[ArrayPlot[image, ColorRules -> {0 -> White, 1 -> Black}],
135 Degree]
This simple change takes computation time down from 8.02 seconds to 1.23 seconds which is a 6.5 times speed up for hardly any extra coding work. Not too shabby!
Switch to C code to get it even faster
I’m not done yet though! By default the Compile command produces code for the so-called Mathematica Virtual Machine but recent versions of Mathematica allow us to go even further.
Install Visual Studio Express 2010 (and the Windows 7.1 SDK if you are running 64bit Windows) and you can ask Mathematica to convert the function to low level C code, compile it and produce a
function object linked to the resulting compiled code. Sounds complicated but is a snap to actually do. Just add
CompilationTarget -> "C"
to the Compile command.
f = Compile[{{x, _Real}, {y, _Real}},
If[Sin[Min[x*Sin[y], y*Sin[x]]] >
y*Cos[x]]] + (((2 (x - y)^2 + (x + y - 6)^2)/40)^3)/
6400000 + (12 - x - y)/30, 1, 0]
, CompilationTarget -> "C"
AbsoluteTiming[\[Delta] = 0.02;
range = 11;
xyPoints =
Table[{x, y}, {y, 0, range, \[Delta]}, {x, 0, range, \[Delta]}];
image = Map[f @@ # &, xyPoints, {2}];]
Rotate[ArrayPlot[image, ColorRules -> {0 -> White, 1 -> Black}],
135 Degree]
On my machine this takes calculation time down to 0.89 seconds which is 9 times faster than the original.
Making the compiled function listable
The current compiled function takes just one x,y pair and returns a result.
In[8]:= f[1, 2]
Out[8]= 1
It can’t directly accept a list of x values and a list of y values. For example for the two points (1,2) and (10,20) I’d like to be able to do f[{1, 10}, {2, 20}] and get the results {1,1}. However
what I end up with is an error
f[{1, 10}, {2, 20}]
CompiledFunction::cfsa: Argument {1,10} at position 1 should be a machine-size real number. >>
To fix this I need to make my compiled function listable which is as easy as adding
RuntimeAttributes -> {Listable}
to the function definition.
f = Compile[{{x, _Real}, {y, _Real}},
If[Sin[Min[x*Sin[y], y*Sin[x]]] >
y*Cos[x]]] + (((2 (x - y)^2 + (x + y - 6)^2)/40)^3)/
6400000 + (12 - x - y)/30, 1, 0]
, CompilationTarget -> "C", RuntimeAttributes -> {Listable}
So now I can pass the entire array to this compiled function at once. No need for Map.
\[Delta] = 0.02;
range = 11;
xpoints = Table[x, {x, 0, range, \[Delta]}, {y, 0, range, \[Delta]}];
ypoints = Table[y, {x, 0, range, \[Delta]}, {y, 0, range, \[Delta]}];
image = f[xpoints, ypoints];
Rotate[ArrayPlot[image, ColorRules -> {0 -> White, 1 -> Black}],
135 Degree]
On my machine this gets calculation time down to 0.28 seconds, a whopping 28.5 times faster than the original. Rendering time is becoming much more of an issue than calculation time in fact!
Parallel anyone?
Simply by adding
Parallelization -> True
to the Compile command I can parallelise the code using threads. Since I have a dual core machine, this might be a good thing to do. Let’s take a look
f = Compile[{{x, _Real}, {y, _Real}},
Sin[Min[x*Sin[y], y*Sin[x]]] >
y*Cos[x]]] + (((2 (x - y)^2 + (x + y - 6)^2)/40)^3)/
6400000 + (12 - x - y)/30, 1, 0]
, RuntimeAttributes -> {Listable}, CompilationTarget -> "C",
Parallelization -> True];
\[Delta] = 0.02;
range = 11;
xpoints = Table[x, {x, 0, range, \[Delta]}, {y, 0, range, \[Delta]}];
ypoints = Table[y, {x, 0, range, \[Delta]}, {y, 0, range, \[Delta]}];
image = f[xpoints, ypoints];
Rotate[ArrayPlot[image, ColorRules -> {0 -> White, 1 -> Black}],
135 Degree]
The first time I ran this it was SLOWER than the non-threaded version coming in at 0.33 seconds. Subsequent runs varied and occasionally got as low as 0.244 seconds which is only a few hundredths of
a second faster than the original.
If I make the problem bigger, however, by decreasing the size of Delta then we start to see the benefit of parallelisation.
\[Delta] = 0.01;
range = 11;
xpoints = Table[x, {x, 0, range, \[Delta]}, {y, 0, range, \[Delta]}];
ypoints = Table[y, {x, 0, range, \[Delta]}, {y, 0, range, \[Delta]}];
image = f[xpoints, ypoints];
Rotate[ArrayPlot[image, ColorRules -> {0 -> White, 1 -> Black}],
135 Degree]
The above calculation (sans rendering) took 0.988 seconds using a parallelised version of f and 1.24 seconds using a serial version. Rendering took significantly longer! As a comparison lets put a
Delta of 0.01 in the original code:
f[x_, y_] := Module[{},
Sin[Min[x*Sin[y], y*Sin[x]]] >
y*Cos[x]]] + (((2 (x - y)^2 + (x + y - 6)^2)/40)^3)/
6400000 + (12 - x - y)/30, 1, 0]
\[Delta] = 0.01;
range = 11;
xyPoints = Table[{x, y}, {y, 0, range, \[Delta]}, {x, 0, range, \[Delta]}];
image = Map[f @@ # &, xyPoints, {2}];
Rotate[ArrayPlot[image, ColorRules -> {0 -> White, 1 -> Black}], 135 Degree]
The calculation time (again, ignoring rendering time) took 32.56 seconds and so our C-compiled, parallel version is almost 33 times faster!
• The Compile function can make your code run significantly faster by compiling it for the Mathematica Virtual Machine (MVM). Note that not every function is suitable for compilation.
• If you have a C-compiler installed on your machine then you can switch from the MVM to compiled C-code using a single option statement. The resulting code is even faster
• Making your functions listable can increase performance.
• Parallelising your compiled function is easy and can lead to even more speed but only if your problem is of a suitable size.
• Sol Lederman has a very cool Mathematica blog – check it out! The code that inspired this blog post originated there.
June 16th, 2011
Every time there is a new MATLAB release I take a look to see which new features interest me the most and share them with the world. If you find this article interesting then you may also enjoy
similar articles on 2010b and 2010a.
Simpler random number control
MATLAB 2011a introduces the function rng which allows you to control random number generation much more easily. For example. in older versions of MATLAB you would have to do the following to reseed
the default random number stream to something based upon the system time.
In MATLAB 2011a you can achieve something similar with
rng shuffle
• I have updated my introduction to parallel random numbers in MATLAB to reflect this change. I really need to write part 2 of that series!
• Click here for a Mathworks video showing how this new function works in more detail
Faster Functions
I love it when The Mathworks improve the performance of some of their functions because you can guarantee that, in an organisation as large as the one I work for, there will always be someone who’ll
be able to say ‘Wow! I switched to the latest version of MATLAB and my code runs faster.’ All of the following timings were performed on a 3Ghz quad-core running Ubuntu Linux with the cpu-selector
turned up to maximum for all 4 cores. In all cases the command was run 5 times and an average taken. Some of the faster functions include conv, conv2, qz, complex eig and svd. The speedup on svd is
MATLAB 2010a: 3.31 seconds
MATLAB 2011a: 1.56 seconds
MATLAB 2010a: 36.67 seconds
MATLAB 2011a: 22.87 seconds
tic;[U,S,V] = svd(a);toc
MATLAB 2010a: 9.21 seconds
MATLAB 2011a: 0.7114 seconds
Symbolic toolbox gets beefed up
Ever since its introduction back in MATLAB 2008b, The Mathworks have been steadily improving the Mupad-based symbolic toolbox. Pretty much all of the integration failures that I and my readers
identified back then have been fixed for example. MATLAB 2011a sees several new improvements but I’d like to focus on improvements for non-algebraic equations.
Take this system of equations
solve('10*cos(a)+5*cos(b)=x', '10*sin(a)+5*sin(b)=y', 'a','b')
MATLAB 2011a finds the (extremely complicated) symbolic solution whereas MATLAB 2010b just gave up.
Here’s another one
syms an1 an2;
eq1 = sym('4*cos(an1) + 3*cos(an1+an2) = 6');
eq2 = sym('4*sin(an1) + 3*sin(an1+an2) = 2');
eq3 = solve(eq1,eq2);
MATLAB 2010b only finds one solution set and it’s approximate
>> eq3.an1
ans =
>> eq3.an2
ans =
MATLAB 2011a, on the other hand, finds two solutions and they are exact
>> eq3.an1
ans =
2*atan((3*39^(1/2))/95 + 16/95)
2*atan(16/95 - (3*39^(1/2))/95)
>> eq3.an2
ans =
MATLAB Compiler has improved parallel support
Lifted direct from the MATLAB documentation:
MATLAB Compiler generated standalone executables and libraries from parallel applications can now launch up to eight local workers without requiring MATLAB® Distributed Computing Server™ software.
Amen to that!
GPU Support has been beefed up in the parallel computing toolbox
A load of new functions now support GPUArrays.
You can also index directly into GPUArrays now and the amount of MATLAB code supported by arrayfun for GPUArrays has also been increased to include the following.
&, |, ~, &&, ||,
while, if, else, elseif, for, return, break, continue, eps
This brings the full list of MATLAB functions and operators supported by the GPU version of arrayfun to
abs csc log10 + Branching instructions:
acos csch log1p -
acosh double logical .* break
acot eps max ./ continue
acoth erf min .\ else
acsc erfc mod .^ elseif
acsch erfcinv NaN == for
asec erfcx pi ~= if
asech erfinv real < return
asin exp reallog <= while
asinh expm1 realpow >
atan false realsqrt >=
atan2 fix rem &
atanh floor round |
bitand gamma sec ~
bitcmp gammaln sech &&
bitor hypot sign ||
bitshift imag sin
bitxor Inf single Scalar expansion versions of the following:
ceil int32 sinh
complex isfinite sqrt *
conj isinf tan /
cos isnan tanh \
cosh log true ^
cot log2 uint32
The Parallel Computing Toolbox is not the only game in town for GPU support on MATLAB. One alternative is Jacket by Accelereyes and they have put up a comparison between the PCT and Jacket. At the
time of writing it compares against 2011a.
More information about GPU support in various mathematical software packages can be found here.
Toolbox mergers and acquisitions
There have been several license related changes in this version of MATLAB comprising of 2 new products, 4 mergers and one name change. Sadly, none of my toolbox-merging suggestions have been
implemented but let’s take a closer look at what has been done.
• The Communications Blockset and Communications Toolbox have merged into what’s now called the Communications System Toolbox. This new product requires another new product as a pre-requisite – The
DSP System Toolbox.
• The DSP System Toolbox isn’t completely new, however, since it was formed out of a merger between the Filter Design Toolbox and Signal Processing Blockset.
• Stateflow Coder and Real-Time Workshop have combined their powers to form the new Simulink Coder which depends upon the new MATLAB Coder.
• The new Embedded Coder has been formed from the merging of no less than 3 old products: Real-Time Workshop Embedded Coder, Target Support Package, and Embedded IDE Link. This new product also
requires the new MATLAB Coder.
• MATLAB Coder is totally new and according to the Mathwork’s blurb it “generates standalone C and C++ code from MATLAB^® code. The generated source code is portable and readable.” I’m looking
forward to trying that out.
• Next up, is what seems to be little more than a renaming exercise since the Video and Image Processing Blockset has been renamed the Computer Vision System Toolbox.
Personally, few of these changes affect me but professionally they do since I have users of many of these toolboxes. An original set of 9 toolboxes has been rationalized into 5 (4 from mergers and
the new MATLAB Coder) and I do like it when the number of Mathwork’s toolboxes goes down. To counter this, there is another new product called The Phased Array System Toolbox.
So, that rounds up what was important for me in MATLAB 2011a. What did you like/dislike about it?
Other blog posts about 2011a
June 15th, 2011
I needed to install Labview 2010 onto a Ubuntu Linux machine but when I inserted the DVD nothing happened. So, I tried to manually mount it from the command line in the usual way but it didn’t work.
It turns out that the DVD isn’t formatted as iso9660 but as hfsplus. The following incantations worked for me
sudo mount -t hfsplus /dev/sr0 /media/cdrom0 -o loop
sudo /media/cdrom0/Linux/labview/INSTALL
The installer soon became upset and gave the following error message
/media/cdrom0/Linux/labview/bin/rpmq: error while loading shared libraries: libbz2.so.1:
cannot open shared object file: No such file or directory
This was fixed with (original source here)
cd /usr/lib32
sudo ln -s libbz2.so.1.0 libbz2.so.1
sudo ldconfig
June 13th, 2011
When installing MATLAB 2011a on Linux you may encounter a huge error message than begins with
Preparing installation files ...
Installing ...
Exception in thread "main" com.google.inject.ProvisionException: Guice provision
1) Error in custom provider, java.lang.RuntimeException: java.lang.reflect.Invoc
at com.mathworks.wizard.WizardModule.provideDisplayProperties(WizardModule.jav
while locating com.mathworks.instutil.DisplayProperties
at com.mathworks.wizard.ui.components.ComponentsModule.providePaintStrategy(Co
while locating com.mathworks.wizard.ui.components.PaintStrategy
for parameter 4 at com.mathworks.wizard.ui.components.SwingComponentFactoryI
while locating com.mathworks.wizard.ui.components.SwingComponentFactoryImpl
while locating com.mathworks.wizard.ui.components.SwingComponentFactory
for parameter 1 at com.mathworks.wizard.ui.WizardUIImpl.(WizardUIImpl.
while locating com.mathworks.wizard.ui.WizardUIImpl
while locating com.mathworks.wizard.ui.WizardUI annotated with @com.google.inj
This is because you haven’t mounted the installation disk with the correct permissions. The fix is to run the following command as root.
mount -o remount,exec /media/MATHWORKS_R2011A/
Assuming, of course, that /media/MATHWORKS_R2011A/ is your mount point. Hope this helps someone out there.
Update: 7th April 2014
A Debian 7.4 user had this exact problem but the above command didn’t work. We got the following
mount -o remount,exec /media/cdrom0
mount: cannot remount block device /dev/sr0 read-write, is write-protected
The fix was to modify the command slightly:
mount -o remount,exec,ro /media/cdrom0
June 10th, 2011
One part of my job that I really enjoy is the optimisation of researcher’s code. Typically, the code comes to me in a language such as MATLAB or Mathematica and may take anywhere from a couple of
hours to several weeks to run. I’ve had some nice successes recently in areas as diverse as finance, computer science, applied math and chemical engineering among others. The size of the speed-up
can vary from 10% right up to 5000% (yes, 50 times faster!) and that’s before I break out the big guns such as Manchester’s Condor pool or turn the code over to our HPC specialists for some SERIOUS
(yet more time consuming in terms of developer time) optimisations.
Reporting these speed-ups to colleagues (along with the techniques I used) gets various responses such as ‘Well, they shouldn’t do time-consuming computing using high level languages. They should
rewrite the whole thing in Fortran’ or words to that effect. I disagree!
In my opinion, high level programming languages such as Mathematica, MATLAB and Python have democratised scientific programming. Now, almost anyone who can think logically can turn their scientific
ideas into working code. I’ve seen people who have had no formal programming training at all whip up models, get results and move on with their research. Let’s be clear here – It’s results that
matter not how you coded them.
It comes down to this. CPU time is cheap. Very cheap. Human time, particularly specialised human time, is expensive.
Here’s an example: Earlier this year I was working with a biologist who had put together some MATLAB code to analyse her data. She had written the code in less than a day and it gave the correct
results but it ran too slowly for her tastes. Her sole programming experience came from reading the MATLAB manual and yet she could cook up useful code in next to no time. Sure, it was slow and (to
my eyes) badly written but give the gal a break…she’s a professional biologist and not a professional programmer. Her programming is a lot better than my biology!
In less than two hours I gave her a crash course in MATLAB code optimisation; how to use the profiler, vectorisation and so on. We identified the hotspot in the code and, between us, recoded it so
that it was an order of magnitude faster. This was more than fast enough for her needs, she could now analyse data significantly faster than she could collect it. I realised that I could make it
even faster by using parallelised mex functions but it would probably take a few more hours work. She declined my offer…the code was fast enough.
In my opinion, this is an optimal use of resources. I spend my days obsessing about mathematical software and she spends her days obsessing about experimental biology. She doesn’t need a formal
course in how to write uber-efficient code because her code runs as fast as she needs it to (with a little help from her friends). The solution we eventually reached might not be the most
CPU-efficient one but it is a good trade off between CPU-efficient and developer-efficient.
It was easy…trivial even..for someone like me to take her inefficient code and turn it into something that was efficient enough. However, the whole endeavour relied on her producing working code in
the first place. Say high-level languages such as MATLAB didn’t exist….then her only options would be to hire a professional programmer (cash expensive) or spend a load of time learning how to code
in a low level language such as Fortran or C (time expensive).
Also, because she is a beginner programmer, her C or Fortran code would almost certainly be crappy and one thing I am sure of is ‘Crappy MATLAB/Python/Mathematica/R code is a heck of a lot easier to
debug and optimise than crappy C code.’ Segfault anyone?
June 8th, 2011
I’ve been a user of Ubuntu Linux for years but the recent emphasis on their new Unity interface has put me off somewhat. I tried to like it but failed. So, I figured that it was time for a switch
to a different distribution.
I asked around on Twitter and got suggestions such as Slackware, Debian and Linux Mint. I’ve used both Slackware and Debian in the past but, while they might be fine for servers or workstations, I
prefer something more shiny for my personal laptop.
I could also have stuck with Ubuntu and simply installed GNOME using synaptic but I like to use the desktop that is officially supported by the distribution.
So, I went with Linux Mint. It isn’t going well so far!
I had no DVDs in the house so I downloaded the CD version, burned it to a blank CD and rebooted only to be rewarded with
Can not mount /dev/loop0 (/cdrom/casper/filesystem.squashfs) on //filesystem.squashfs
I checked the md5sum of the .iso file and it was fine. I burned to a different CD and tried again. Same error.
I was in no mood for a trawl of the forums so I simply figured that maybe something was wrong with the CD version of the distribution – at least as far as my machine was concerned. So, I started
downloading the DVD version and treated my greyhound to a walk to the local computer shop to buy a stack of DVDs.
When I got back I checked the .md5 sum of the DVD image, burned it to disk and…got the same error. A trawl of the forums suggests that many people have seen this error but no reliable solution has
been found.
Not good for me or Linux Mint but at least Desmond (below) got an extra walk!
Update 1 I created a bootable USB memory stick from the DVD .iso to elimiate any problems with my burning software/hardware. Still get the same error message. MD5 checksum of the .iso file is what it
should be:
md5sum ./linuxmint-11-gnome-dvd-64bit.iso
773b6cdfe44b91bc44448fa7b34bffa8 ./linuxmint-11-gnome-dvd-64bit.iso
My machine is a Dell XPS M1330 which has been running Ubuntu for almost 3 years.
Update 2: Seems that this bug is not confined to Mint. Ubuntu users are reporting it too. No fix yet though
Update 3: There is DEFINITELY nothing wrong with the installation media. Both USB memory stick and DVD versions boot on my wife’s (much newer)HP laptop with no problem. So, the issue seems to be
related to my particular hardware. This is like the good old days of Linux where installation was actually difficult. Good times!
Update 4: After much mucking around I finally gave up on a direct install of Mint 11. The installer is simply broken for certain hardware configurations as far as I can tell. Installed Mint 10 from
the same pen drive that failed for Mint 11 without a hitch.
Update 5: As soon as the Mint 10 install completed, I did an apt-get dist-upgrade to try to get to Mint 11 that way. The Mint developers recommend against doing dist-upgrades but I don’t seem to have
a choice since the Mint 11 installer won’t work on my machine. After a few minutes I get this error
dpkg: error processing python2.7-minimal (--configure):
subprocess installed post-installation script returned error exit status 3
Errors were encountered while processing:
This is mentioned in this bug report. I get over that (by following the instructions in #9 of the bug report) and later get this error
p: cannot stat `/usr/lib/pango/1.6.0/module-files.d/libpango1.0-0.modules': No such file or directory
cp: cannot stat `/usr/lib/pango/1.6.0/modules/pango-basic-fc.so': No such file or directory
E: /usr/share/initramfs-tools/hooks/plymouth failed with return 1.
update-initramfs: failed for /boot/initrd.img-2.6.35-22-generic
dpkg: error processing initramfs-tools (--configure):
subprocess installed post-installation script returned error exit status 1
Errors were encountered while processing:
I fixed this with
sudo ln -s x86_64-linux-gnu/pango /usr/lib/pango
Trying the apt-get dist-upgrade again leads to
The following packages have unmet dependencies:
python-couchdb : Breaks: desktopcouch (< 1.0) but 0.6.9b-0ubuntu1 is to be installed
python-desktopcouch-records : Conflicts: desktopcouch (< 1.0.7-0ubuntu2) but 0.6.9b-0ubuntu1 is to be installed
Which, thanks to this forum post, I get rid of by doing
sudo dpkg --configure -a
sudo apt-get remove python-desktopcouch-records desktopcouch evolution-couchdb python-desktopcouch
A few more packages get installed before it stops again with the error message
Unpacking replacement xserver-xorg-video-tseng ...
Processing triggers for man-db ...
Processing triggers for ureadahead ...
Errors were encountered while processing:
E: Sub-process /usr/bin/dpkg returned an error code (1)
I get past this by doing
sudo apt-get -f install
Then I try apt-get upgrade and apt-get dist-update again…possibly twice and I’m pretty much done it seems.
Update 6: On the train to work this morning I thought I’d boot into my shiny new Mint system. However I was faced with nothing but a blank screen. I rebooted and removed quiet and splash from the
grub options to allow me to see what was going on. The boot sequence was getting stuck on something like checking battery state. Up until now I had only been using Mint while connected to the Mains.
Well, this was the final straw for me. As soon as I got into work I shoved in a Ubuntu 11.04 live disk which installed in the time it took me to drink a cup of coffee. I’ve got GNOME running and am
now happy.
My Linux Mint adventure is over.
June 6th, 2011
Should academic mathematical software (for both teaching and research) be open source, commercial or a mixture of both? Personally I feel that a mixture is the best way to go which is why I am
equally at home with either Mathematica or Sage, MATLAB or Scilab, GSL or NAG and so on. Others, however, have more polarised views.
Here are some I’ve come across from various places over the years (significantly shortened)
• We should teach MATLAB because MATLAB is the industry standard. Nothing else will do!
• We should teach concepts, not how to use any particular program. However, when things need to be implemented they should be implemented in an open source package.
• All research should be conducted using open source software. Nothing else will do!
• Students are being asked to pay hefty fees to come to our University. We should provide expensive mathematical software so that they feel that they are getting value for money.
• We should only provide open source software to staff and students. This will save us a fortune which we can put into other facilities.
and so on.
Personally I feel that all of these views are far too blinkered. When you consider the combined needs of all teachers, researchers and students in a large institution such as the one I work for,
only a combination of both open source and commercial software can satisfy everyone.
I’d love to know what you think though so please have your say via the comments section. If you could preface your comment with a brief clue as to your background then that would be even better
(nothing too detailed, just something like ‘Chemistry lecturer’, ‘open source software developer’ or ‘Math student’ would be great)
June 1st, 2011
Welcome to the 5th installment of A Month of Math software where I take a look at all things math-software related. If I’ve missed something then let me know in the comments section.
Open Source releases
SAGE, possibly the best open-source mathematics package bar-none, has seen an upgrade to version 4.7. The extensive change-log is here.
Numpy 1.60 has been released. Numpy is the fundamental package needed for scientific computing with Python and the list of changes from the previous version can be found in this discussion thread.
Version 1.15 of the GSL (GNU Scientific Library), a free and open source numerical library for C and C++, has been released. A copy of the change log is here.
Scilab, the premier open source alternative to MATLAB, has seen a new minor upgrade with 5.3.2. Click here to see the differences from version 5.3.1
The GMP MP Bignum library has been updated to version 5.0.2. GMP is a free library for arbitrary precision arithmetic, operating on signed integers, rational numbers, and floating point numbers.
Check out the release notes for what’s new.
Commercial releases
The Numerical Algorithm’s Group (NAG) have released version 0.4 of their CUDA accelerated numerical library. You can’t actually buy it yet as far as I know but academics can get their hands on it
for free by signing a collaborative agreement with NAG.
Magma seems to have a new release every month. See what’s new in version 2.17-8 here.
Math Software in the blogsphere
Sol Lederman has started a new blog called Playing with Mathematica. Lots of cool little demonstrations to be found such as the multiple pendulum animation below.
Gary Ernest Davis discusses Dijkstra’s fusc function – complete with Mathematica code.
Alasdair looks at the sums of dice throws using Sage.
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What is Reorder Point? | Reorder Point Definition & Formula
Reading Time: 3 minutes
What is a reorder point?
A reorder point (ROP) is a specific level at which your stock needs to be replenished. In other words, it tells you when to place an order so you won’t run out of stock.
Significance of reorder points
If you’re a business owner, knowing when to order more stock is important. If you order when you still have a lot of stock on hand, it will lead to extra stock piling up, which will increase your
holding costs. If you order when you have zero stock on hand, you’ll be unable to make sales for as long as it takes to receive the order. The your vendor takes to supply the items, the more sales
you’ll be losing. Setting a reorder point helps you optimize your inventory, replenish your stock of individual items at the right time, and meet your market demand without going out of stock.
How to calculate a reorder point
You need to know when to order each item in your inventory separately, because different items have different sell-through rates. To calculate the ROP for each item, you’ll need to know the following
Lead time: Time taken (in days) for your vendor to fulfill your order
Safety stock: The amount of extra stock, if any, that you keep in your inventory to help avoid stockouts
Daily average usage: The number of sales made in an average day of that particular item
Reorder Point Formula
Let’s look at how to calculate a reorder point both with and without safety stock. Then we’ll cover how to handle reorder points when you have multiple vendors.
• Determining ROP with safety stock
• Determining ROP without safety stock
Determining ROP with safety stock
This method is used by businesses that keep extra stock on hand in case of unexpected circumstances. To calculate a reorder point with safety stock, multiply the daily average usage by the lead time
and add the amount of safety stock you keep.
Let’s understand this with an example. Suppose you’re a perfume retailer who sells 200 bottles of perfume every day. Your vendor takes one week to deliver each batch of perfumes you order. You keep
enough excess stock for 5 days of sales, in case of unexpected delays. Now, what should your reorder point be?
Lead time = 7 days
Safety stock = 5 days x 200 bottles = 1000 bottles
ROP = (200 x 7) + 1000 = 2400 bottles
The order for the next batch of perfume should be placed when there are 2400 bottles left in your inventory.
This simplified reorder point graph shows you the relationship between your reorder point, stock level, and safety stock over a period of time. It helps you visualize how your reorder point is based
on your sales trends.
In the above graph, the maximum level is the sum of the safety stock and the order quantity, or 3400 bottles. Once the stock left in your inventory reaches the reorder level of 2400 bottles units,
you should place a new purchase order with your vendor. The minimum level, which is 1400 bottles, will help you fulfill your orders until your ordered stock reaches the warehouse. Once the new order
is received in your warehouse, the stock level returns to the maximum level of 3400 bottles units.
Determining ROP without safety stock
Businesses which follow lean inventory practices or a just-in-time management strategy usually don’t have safety stock. In such cases, your reorder point can be calculated by multiplying your daily
average sales by your lead time. Typically, when you don’t have safety stock, your reorder level and the frequency of your orders tend to be higher.
Taking the above perfume example without including safety stock, your ROP should be:
ROP = 200 x 7 = 1400 bottles
Therefore, you should place an order for the next batch of perfumes when you have 1400 bottles left.
How to calculate ROP with different vendors
You may purchase items in your inventory from various vendors, and different vendors have different lead times. Therefore, it’s best to think of your reorder point on an individual item level.
For example, let’s suppose that you’re a retailer who sells water bottles and snack boxes. The two items are purchased from different vendors with different lead times. The water bottles take one day
to get delivered (lead time = 1 day) and the snack boxes take four days (lead time= 4 days). In a typical day, you sell 5 water bottles and 10 snack boxes.
Without safety stock, your ROP with the vendor who delivers the water bottles should be:
ROP = 5 x 1 = 5 bottles
When you have 5 bottles left, that means you have one day of sales before you run out of stock. Since your lead time is also one day, the new stock should arrive just in time for you to continue
selling without interruption.
Similarly, your ROP with the vendor who delivers the snack boxes should be
ROP = 4 x 10 = 40 boxes
You should reorder when you have 40 boxes of stock left in your inventory, which is four days of stock. Given that your lead time is also four days, the new stock should arrive just in time for you
to continue selling without interruption.
A reorder point is crucial for effective inventory management. It saves holding costs and prevents stockouts, overstocking, and lost sales by ensuring that sufficient stock is always available in
your inventory.
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1. I coud’nt found anywhere in Inventory; how I can have the information about items bellow the ROP manufacturers by manufacturers , nor anybody I have called at Zoho?
Would you have a clue?
□ Hi Jean-Sébastien,
Thank you for reaching out to us! Please contact us at support@zohobooks.com and we’ll be happy to explain.
2. Brilliant write up and so easy to understand from a lay man’s point of view.
3. Good explanation, finally found in this site, which is very neat, simple and very much clear. Thank you content marketer.
4. Great Explanation.
5. Well explained such that even a lay man is able to understand.
Thank you very much.
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ball mill and vrm
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The vertical roller mill itself is equipped with a powder concentrator, which can remove fine powder in time, reduce the phenomenon of overgrinding and improve the grinding efficiency. The
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Mathematica Q&A: Excluding Points from Plots—Wolfram Blog
Mathematica Q&A: Excluding Points from Plots
Got questions about Mathematica? The Wolfram Blog has answers! Each week, we’ll answer a selected question from users around the web. You can submit your question directly to the Q&A Team.
For our first post in this new series of Mathematica Q&A articles, we’re going to address a very frequently asked question about plotting in Mathematica.
How can I control the appearance of discontinuities in a plot?
The short answer is, use the options Exclusions and ExclusionsStyle! Let’s see how they work.
By default, Plot shows the function 1 ⁄ sin(x) with lines joining its discontinuities:
You can use the Exclusions option to exclude points from the plot:
Instead of excluding the points, you can specify a style to apply to them:
For explicitly piecewise functions like Floor, points are excluded automatically:
(You can use Exclusions → None to disable this.)
The exclusions options work with many other Mathematica functions, such as Plot3D:
We specified two styles using ExclusionsStyle: the first style is applied to the excluded region, while the second style is applied to its boundary.
You can create some spectacular visualizations with these options. Download the Computable Document Format (CDF) file for this post to see how to generate this one:
And don’t forget you can submit your own questions to the Q&A Team anytime.
Join the discussion
12 comments
1. Mathematica Q&A is a great idea! In this first posting was raised an excellent question, thanks!
2. I second the comment that Q&A is a great idea. I have a question that I hope isn’t too trivial. What would you recommend as the most reasonable Mathematica way to simulate simple BASIC-like
plotting of arbitrary points, where statements like Plot x,y “turn on” a particular x,y points in a bitmap or grid? Mathematica makes the complicated stuff so easy that I sometimes feel unsure
where to begin with the very simple stuff.
3. If I have numerical solution as interpolation function from NDSolve:
sol = NDSolve[{x'[t]==5,x[0]==0}, x, {t, 0, 100}]
And I want to plot the solution in this way:
Plot[(x[t] – Floor[x[t], 100]) /. sol[[1]], {t, 0, 100}]
Then Mathematica doesn’t exclude discontinuities automatically. How do I get rid of these joining lines?
4. Wow, I am a idiot of math, and you guys so smart…and I even can’t understand what’s the stand for?
5. Good idea. But should this really be in the blog? Shouldn’t this be in a separate section?
6. I like this a lot. It will also increase the frequency of update. How does one submit proposals for content? Is this a good place to ask questions or do you have a user forum?
7. Good and thank you for posting. I’m also in the camp that’s thinking that a separate section for this type of posting will be in keeping with the organized structure on your website.
8. Re: NDSolve by Josef
After observing that the discontinuities happen at multiples of 0.2, I chose to do the following exclusions statement, successfully eliminating the discontinuities.
Exclusions -> Table[0.2 n, {n, 1, 5}]
9. Ref: NDSolve by Josef
Submitting entire code, from M7 Student.
sol = NDSolve[{x'[t] == 5, x[0] == 0}, x, {t, 0, 100}];
y = Evaluate[(x[t] – Floor[ x[t] ]) /. sol ] ;
Plot[y, {t, 0, 1},
Exclusions -> Table[0.2 n, {n, 1, 5}]]
10. Nice tutorial. The Mathematica documentation is first class as of version 6, but there is still room for improvement. The examples in this Blog should replace the examples in documentation for
the Exclusions option. I hope Wolfram Research will strive to make it easy to find the answer to every FAQ they know of, and put them all in the documentation. The key is to include good
examples, and provide links connecting related topics.
11. @Isaac: Well, it was just a trivial example. In fact I have differential equation where I cannot predict discontinuities caused by Floor function. I mean, how to suppress this
InterpolationFunction-behaviour in general. Anyway, thank you for your time and answer. :)
12. Hi Mark, I think ListPlot may be the function you are looking for.
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QuickCheck Math - Revisited.
QuickCheck Math – Revisited.
In August 2010, I went to a local homeschool book sale and had the chance to connect with a company called QuickCheck math that offers a hands-on creative fun way to reinforce math skills. This
product is made by Kinesis Education (Brault & Bouthillier). Their sets are for Grades K through 3. They sent me out a set to see how we liked it.
How it works:
When you order a grade, you get a set of 5 coil-bound books and a plastic case with 6 tiles.
There are 5 books in the series, one to cover each of the 5 areas in the Ontario math curriculum: Data Management & Probability, Geometry & Spatial Sense, Measurement, Number Sense & Numeration, and
Patterning & Algebra. The tiles have 2 sides – one has symbols, the other a coloured triangle.
The idea is that the kids open a page of one of the books, place the tiles with the symbol side face up and the checkmark covering the picture at the bottom of the page that tells them the pattern
of triangles that’s the right answer. The tiles get moved to the top section of the case, and one by one you figure out where they go on the bottom. When you are done, you flip the case over and see
if your triangles match the bottom picture. If so, you got it right! If not, check to see what you did wrong.
Under many of the pages’ titles there’s a little square – which is a note for the teacher on a way to expand or explain the exercise. At the back of the book there is also a Teacher Section, which
gives learning connection activity suggestions and lots of tips that can help the experience be better.
I love this set for a few reasons.
1. It allows the kids to do math both manipulatively and visually.
2. The kids can see exactly what they get right or get wrong and can correct themselves.
3. It’s a different way to reinforce math ideas and concepts, instead of just reviewing through worksheet problems.
4. It’s fun.
5. It can be used over and over – allowing me to reuse it with my younger kids when they reach that grade level, or again and again throughout the year as review of skills.
They have a special homeschool offer. It’s $119 per grade, and includes what you see above.
In addition to this tile/book system, they also have teacher resources, one ongoing assessment book and one diagnostic assessment. Although intended ideally for a teacher in a classroom setting these
can be very useful for the homeschool environment too.
The Ongoing Assessment book is divided into each of the 5 areas and builds on what you have been doing with the tiles in different ways – making sure that the student “gets” the concept behind the
game and isn’t just winging it. There are step-by-step activities all laid out: materials needed, questions to ask, what to do, and includes templates and worksheets like base 10 manipulatives as
well. There are sheets you can copy to write down your observations on a student if you want, and a great way to compare how they are at the beginning of a season with the end of it.
The Diagnostic Assessment is designed to be used as checkpoints throughout the program to make sure the student has understood what they’ve been working on. Instead of expanding the skill sets, it’s
more of a testing opportunity. This book also includes templates and evaluation sheets.
While both of these books are fairly pricey – as most teaching tools are, but they are handy ways to work on math in a creative and more hands-on format.
I do admit that alone, these kits do not teach your children about the skills or concepts, but they are a great way to reinforce ideas and learning by making math fun and tangible. These are a
perfect addition to other workbooks or lesson plans you have in place. And – good news! Completely reusable!
Latest posts by Lisa Marie Fletcher
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5 thoughts on “QuickCheck Math – Revisited.”
I remember having this in grade 1 in school… it went by the name “Veri-tech” — french classroom. I think ours had numbers instead of symbols, but the idea is the very same. Some years ago I found
something similar for my son, it had 16 squares instead of 8 so it made a larger pattern (and used numbers, not symbols). I can’t remember the name of it… and I’ve no idea where it is anymore,
He never really took to it. I LOVED it. Just the fact that I remember the name of it over 30 years later shows the impact it had on me lol… it wasn’t just for math, by the way. ANYTHING can be
turned into questions to answer with the tiles.
$120 for one grade set? That seems pricey.
Oh hey… veritech still exists!
Sorry — better link here. This one is EXACTLY what I used as a child! 12 squares, 2 rows of 6. Even the font used for the numbers is still the same.
Haha — and a little more research on my part reveals that Veritech and Quick Check Math are both made by the same company – Editions Brault & Bouthillier. So at least QCM isn’t just a lame copy.
Seems to be a ‘mini’ version of the game, and applied specifically to math.
Cool! Thanks for sharing 😀
Leave a Comment
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Decimal to Binary
Convert decimal numbers to binary format easily with our Decimal to Binary Tool on TotalConverter.org. Quickly translate numeric values into binary code.
What is the Decimal to Binary Tool?
The Decimal to Binary Tool takes a decimal number (base-10) and converts it into its binary equivalent (base-2). Binary numbers, which consist of only 0s and 1s, are widely used in computing and
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How to Use the Decimal to Binary Tool
1. Enter Decimal Number: Type the decimal number you want to convert into the tool.
2. Convert: Click the convert button to see the binary equivalent.
3. View Result: The binary number will be displayed, ready for your use.
1. What does the Decimal to Binary Tool do?
It converts decimal numbers (base-10) into binary numbers (base-2).
2. How do I use the Decimal to Binary Tool?
Enter your decimal number in the input box and click the convert button to get the binary result.
3. Why would I need to convert decimal to binary?
Binary numbers are essential in computing and digital systems. This conversion helps in understanding and working with binary code.
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Yes, the tool can handle decimal numbers of any size and convert them to binary.
5. Is the Decimal to Binary Tool free?
Yes, it's completely free to use on TotalConverter.org.
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SAT Math: Top 6 Tips for the No-Calculator Section
Over the years, you may have become accustomed to using a calculator to solve each and every math question. A straightforward calculation you could do in third grade, such as 7 x 12, may have you
reaching for your calculator. After all, pushing just a few buttons on a calculator will get you to the answer (it’s 84, by the way). It’s time to awaken that part of your brain from its possibly
lengthy slumber because the new SAT will require you to work through the first Math section without a calculator. But don’t worry! Follow these six tips to help you ace the SAT No-Calculator Math
SAT Math Tip #1: Know the test so you can pace yourself
The No-Calculator Math Test is the third multiple-choice section of the SAT. You will have 25 minutes to answer 15 multiple choice questions and 5 grid-in questions, leaving you with a little more
than a minute per question.
For a closer look at content tested, see
What’s Tested on the SAT Math Section
SAT Math Tip #2: Think first, then compute
The No-Calculator Math questions will not require long, drawn-out calculations. Remember: these questions are all designed to take less than a minute to solve. Look for patterns and shortcuts to
solve complicated-looking questions more easily. For instance, consider the question below:
This should bring back memories from
I and II when you had to solve a system of equations. Before you start solving for one of the variables in the first equation to substitute into the second equation, take a moment to notice the
structure of the equations. The equations are formatted perfectly for combination. Combination (sometimes referred to as elimination) is used to solve systems of equations; you eliminate one variable
by adding the equations, and then you solve for the remaining variable.
In this case, when you add 4
= –5 to –4
– 2
= –2, the
values add up to 0, the
values sum to –y and the whole numbers total –7. Take –
= –7 and divide both sides by –1. The correct answer is (D)!
Not only did you save time with combination, but also you avoided a trap answer—if you use substitution and solve for the value of
, you may be tempted to choose choice B.
SAT Math Tip #3: Show your work
Sure, the machine that’s grading your test will not see any work in your test booklet or be able to give you partial credit, and there will also be a handful of questions you will be able to think
through without a pencil. For questions that require a few steps to solve, though, writing out the way you solve the question will enable you to catch mistakes before they happen. Plus, if you are
solving a trickier question and you don’t know where to start, jotting down a few pieces of information may give you the spark you need.
SAT Math Tip #4: Pick numbers
When multiple-choice questions on the SAT provide expressions, graphs, or phrases with variables, you may be able to pick numbers, which will make these questions both easier to understand and to
solve. For instance, take a look at the question below.
Even if fractions are not your favorite, you can solve this question by picking your own number for
. Letting
= 1, for example, turns the fraction into a straightforward arithmetic question:
Now, plug that same
= 1 into the answer choices, and eliminate any that do not equal –1.
Using some straightforward arithmetic, we’ve arrived at the correct answer, (C).
SAT Math Tip #5: Work backwards
You are likely accustomed to solving a question, arriving at an answer, and crossing your fingers, hoping it matches one of the choices given. However, as 15 of the 20 no-calculator questions are
multiple-choice, you may find that working backwards is helpful. Consider the question below.
You could definitely go through solving this equation before looking at the answer choices, but remember: think first, then compute! Working backwards is a much quicker way to solve this question.
Start with the friendliest values in your choices. Choice C will turn the inequality into 0<0+185 or 0<185, which is true, so you can eliminate this answer. Choice A will turn the inequality into 0
<–1+185 or 0<085, but 0 is not less than 0. Hence, (A) is the correct answer. Because there is just one correct answer, you don’t even have to test any others!
SAT Math Tip #6: Know and practice equation-solving techniques
The No-Calculator Math Test will require you to solve a lot of equations by hand. Make sure you are familiar with all of the following equation-solving techniques:
1. Cardinal rule of equations—do the same thing to both sides of the equation. For example, if you divide the left side of the equation by 3, divide the right side of the equation by 3 as well.
2. Clearing fractions—when an equation includes lots of fractions, find the lowest common denominator (LCD) of all fractions, and multiply the entire equation by this LCD. This will eliminate
fractions from your equation and make it much easier to solve.
3. Solve a system of two equations with two variables—know how to use both substitution (solve for an equation in terms of one variable and plug it into the other) and combination/elimination
(multiply both sides of one equation by a number that will allow you to eliminate a variable when you add the two equations together).
4. Cross multiplying—when you have an equation with a single fraction on each side, multiply the denominator of the left side by the numerator of the right and vice-versa. Set these two products
equal to each other to get a more straightforward equation to solve.
5. Factor and solve quadratics—to solve a quadratic equation, you must first get it to the form ax2 + bx + c = 0, then factor, and finally set each factor equal to 0. If an equation is not easily
factored, you can use the quadratic formula:
Incorporate these tips into your regular SAT math practice and even questions in math class, and you will wonder why you ever needed a calculator in the first place! For additional tips to conquer
the SAT Math Test, check out 5 Must-Know SAT Math Tips.
0 0 admin http://wpapp.kaptest.com/wp-content/uploads/2020/09/kaplan_logo_purple_726-4.png admin2019-12-01 17:54:582020-09-11 20:40:20SAT Math: Top 6 Tips for the No-Calculator Section
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Solving a Head-on Collision: Qs 1-4
• Thread starter Jimmy87
• Start date
In summary, in a head on collision between a truck and a car where both are traveling at the same speed, the force and impulse on each vehicle will be the same. However, due to the car having a much
smaller mass, it will experience a much greater acceleration, leading to a higher kinetic energy and ultimately resulting in worse consequences for the car compared to the truck. When comparing the
impact of a car hitting a solid wall and a car colliding with a truck at the same speed, the collision with the truck would be worse for the car due to the higher acceleration and kinetic energy.
Homework Statement
Hi, could someone help me with some physics homework for a head on collision between a truck and a car. Truck and car are going the same speed (50mph) but truck weighs 5 times more than car.
Q.1. What is the force on each car?
Q.2. What is the impulse?
Q.3. Explain in detail using work-energy theorem why it is worse to be in the car
Q.4. If the vehicles collide at 50mph how would the impact compare for both vehicles to that of a collision with a solid wall?
Homework Equations
I = Ft = change in momentum
F = ma
The Attempt at a Solution
Think I know Q's 1 and 2. The force of collision is equal and opposite and so is the time of the collision therefore I think both impulse and force will be the same. Using F =ma the acceleration will
be much greater on the smaller car.
My questions:
How does this bigger acceleration cause a greater kinetic energy of the car as I thought KE is proportional to momentum and the momentum change is the same for both vehicles? I'm a bit unsure how you
relate forces and accelerations to kinetic energy.
For the last part I'm not sure - I think the truck will experience an impact which is not as bad as colliding with a wall. Does the car experience an impact worse than colliding with a wall? not
quite sure why though in terms of KE, impulse etc.
We keep getting this kind of question- typically from people who want to prove that the accident was the other guys fault. Were you actually given this problem? You are given the speeds and
(relative) masses of the vehicles so you can determine the (relative) kinetic energies and momenta at the instant of collision. But the force depends upon how long the collision takes and how much
each vehicle "collapses". And that information is not given.
HallsofIvy said:
We keep getting this kind of question- typically from people who want to prove that the accident was the other guys fault. Were you actually given this problem? You are given the speeds and
(relative) masses of the vehicles so you can determine the (relative) kinetic energies and momenta at the instant of collision. But the force depends upon how long the collision takes and how
much each vehicle "collapses". And that information is not given.[
Is that a serious question. Of course I was given this problem - I regularly use this forum for homework help. I can't wait to tell my teacher though! How could such information provided in a
forum help with a real crash?
Anyway, thanks for the help you provided. I still don't know the answer to the last question? Would it be worse for a car to hit a brick wall or be hit my a truck traveling at the same speed. How
do you relate kinetic energy to force and impulse?
Science Advisor
Homework Helper
Soothe soothe
I still don't know the answer to the last question?
So what did you get for the third question ?
How do you relate kinetic energy to force and impulse?
##d T = \vec F \cdot d\vec s \quad ## if F is constant, and ## T = {\vec p^2 \over 2m}##
Your relevant equations should include something about momentum, in particular: momentum conservation. There is something about the momentum of the center of mass and the momentum w.r.t. the center
of mass you can write down.
I think it is safe to assume the car doesn't bounce back too much. That gives some hold to do calculations. In particular that the car driver dies and the truck driver doesn't, so that's why it's
better to be in the truck (which is not the answer to Q 3!).
Concerning the brick wall: Gives a few scratches on the truck and a bit more damage to the car, unless it is a bit sturdier than a single 3
" .
The intention here is that the collision is inelastic and all speed is killed. This time the truck driver dies for sure and the car driver has a small chance.
BvU said:
Soothe soothe
So what did you get for the third question ?
##d T = \vec F \cdot d\vec s \quad ## if F is constant, and ## T = {\vec p^2 \over 2m}##
Your relevant equations should include something about momentum, in particular: momentum conservation. There is something about the momentum of the center of mass and the momentum w.r.t. the
center of mass you can write down.
I think it is safe to assume the car doesn't bounce back too much. That gives some hold to do calculations. In particular that the car driver dies and the truck driver doesn't, so that's why it's
better to be in the truck (which is not the answer to Q 3!).
Concerning the brick wall: Gives a few scratches on the truck and a bit more damage to the car, unless it is a bit sturdier than a single 3 5⁄8" .
The intention here is that the collision is inelastic and all speed is killed. This time the truck driver dies for sure and the car driver has a small chance.
Thanks BvU. The understanding of why the car is worse off if it collides with the truck is because the car receives a greater change in acceleration since they both receive the same force. I mean I
kind of know that a greater acceleration is worse during a collision but can't put any physics to it to explain why in terms of kinetic energy. So how do you then relate kinetic energy to a changing
acceleration? With the brick wall the question we have to answer is that we need to compare the effect of a car hitting a brick wall and then a car hitting a truck in a head on collision where both
vehicles are going the same speed. Which is worse for the car? Hitting a brick wall or having a head on collision with a truck? I think hitting the truck would be worse because I looked it up and if
two cars of equal mass collide at the same speed teach will feel a collision which is the same as hitting a wall, therefore hitting the truck must be worse for the car than hitting a wall?
Science Advisor
Homework Helper
Yes. For two identical cars, same speed, the center of mass is at rest.
Since there are no external horizontal forces, momentum of c.o.m. is conserved: ##\vec F_{1\rightarrow 2} = - \vec F_{2\rightarrow 1} \Leftrightarrow {d\vec p_1\over dt} = {-d\vec p_2 \over dt} \
Leftrightarrow {d\over dt} \left ( \vec p_1 + \vec p_2 \right ) = 0##
This conservation of c.o.m. momentum is also true for the car-truck collision. If the car has mass m, truck 5 m, c.o.m. has 4 mv before and after collision. So speed of the combined wreck is ##4 mv/
6m = {2 \over 3}v ## or 33 mph. So car changes from -50 mph to +33 mph truck from 50 to 33. A factor 5 (no coincidence 5m/m !) worse for the car driver.
Note that I state all this in terms of momentum. With F = dp/dt, which with constant mass becomes F = ma, the change in momentum is important. And the more spread out in time, the better. Therefore
cars are designed to have a maximum crumple zone, plus airbags and soft internals. Everything to spread out the change in momentum as much as possible.
For trucks there is no point: if you carry 45 tonnes of cargo you would need an immense crumple zone.
Now for hitting the proverbial brick wall (think of a medieval castle wall or something). No difference with the case identical cars at same speed and head on: Forces same, Δp same, crumpling equally
long, etc. Our car driver would havve a(very) small chance. Not so our truck driver: he is between the wall and the cargo and his minimal crumple zone is useless.
FAQ: Solving a Head-on Collision: Qs 1-4
1. How do you determine the velocity of each vehicle in a head-on collision?
In order to determine the velocity of each vehicle in a head-on collision, you will need to gather information such as the weight of each vehicle, the distance between the two vehicles before the
collision, and the amount of damage sustained. This information can then be plugged into physics equations, such as the conservation of momentum equation, to calculate the velocity of each vehicle.
2. Can a head-on collision be avoided?
Yes, head-on collisions can be avoided by practicing safe driving habits and following traffic laws. Some ways to prevent a head-on collision include staying in your lane, paying attention to road
signs and markings, maintaining a safe distance from other vehicles, and avoiding distractions while driving.
3. What are the most common injuries sustained in a head-on collision?
The most common injuries sustained in a head-on collision are head and neck injuries, chest and abdominal injuries, and broken bones. These injuries can range from minor bruises to more severe
injuries such as concussions, spinal cord injuries, and internal bleeding.
4. How can airbags help in a head-on collision?
Airbags can help in a head-on collision by providing a cushion for the driver and passengers, reducing the force of impact and preventing them from hitting the hard surfaces inside the vehicle. This
can help minimize the risk of serious injuries and even death in a head-on collision.
5. Is it possible to survive a head-on collision?
Surviving a head-on collision depends on various factors such as the speed and angle of impact, the safety features in the vehicle, and the severity of injuries sustained. While head-on collisions
can be incredibly dangerous, it is possible to survive with proper safety precautions and prompt medical attention.
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The Quarterly Journal of Pure and Applied Mathematics
The Quarterly Journal of Pure and Applied Mathematics, Volume 6
James Joseph Sylvester, James Whitbread Lee Glaisher
Popular passages
A contribution to the history of the problem of the reduction of the general equation of the fifth degree to a trinomial form,
TrWl=e ........ (10), the equation to the new surface, which is evidently a central surface of the second order, and therefore, of course, an ellipsoid (Cauchy — Exercises, vol. ii.).
If the small displacement of each point of a medium is in the direction of, and proportional to, the attraction exerted at that point by any system of material masses, the displacement is effected
without rotation. For if Fp= C be the potential surface, we have Sadp a complete differential ; ie in Cartesian coordinates is a differential of three independent variables.
In the case of a rigid body moving about a fixed point let vr, p, a denote the vectors of any three points of the body; the fixed point being origin. Then or", p*, o-* are constant, and so are S&p,
Spa, and Sanr.
OB=-)-m, we find the areas PAOY and QOBY both positive and infinite, which agrees with all our notions derived from the theory of curves. Again, if we attempt to find the area PYQB by summing PAOY
and YOQB, we find an infinite and positive result, which still is strictly intelligible. But if we want to find the area by integrating at once from P to Q, we find, as above, - (2 : m), a negative
result for the sum of two positive infinite quantities.
A, H, G, L H, B, F, M G, F, C...
V.&<r represents twice the vector axis of rotation of the same group of points. Similarly or is equivalent to total differentiation in virtue of our having passed from one end to the other of the
vector <r.
... the intersection of the lines bisecting the middle points of pairs of opposite sides (say 0) the mid-centre (which, it may be observed, is the centre of gravity of the four angles viewed as equal
weights) ; then the centre of gravity is in the line joining these two centres produced past the latter (the mid-centre), and at a distance from it equal to one-third of the distance between the two
centres ; in a word, if G be the centre of gravity of the quadrilateral, QOG will be in a right line,...
Prohessian ia considered, but not in much detail, in Dr. Salmon's Geometry of Three Dimensions, (1862), pp. 338 and 426 : the theorem given in the latter place is almost all that is known on the
subject. I call to mind that the tangent plane along a generating line of the developable meets the developable in this line taken 2 times, and in a curve of the order...
Bibliographic information
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Savings Calculator | Advanced Savings Calculator
Savings Calculator — calculate future value
This calculator easily answers the question "If I save "X" amount for "Y" months what will the value be at the end?"
The user enters the "Periodic Savings Amount" (amount saved or invested every month); the "Number of Months" and the "Annual Interest Rate" or the annual rate of return one expects to earn on their
The calculator quickly creates a savings schedule and a set of charts that will help the user see the relationship between the amount invested and the return on the investment. The schedule can be
copied and pasted to Excel, if desired.
The investment term is always expressed in months.
• 60 months = 5 years
• 120 months = 10 years
• 180 months = 15 years
• 240 months = 20 years
• 360 months = 30 years
If you need a more advanced "Savings Calculator" - one that lets the user solve for the starting amount, the amount to invest, the interest rate, the term required to reach a goal or the future
value; or if you would like to easily print the schedule; or if you need to pick a different investment frequency, then you may want to try the calculator located here: https://
Currency and Date Conventions
All calculators will remember your choice. You may also change it at any time.
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Crystal structure | Kronig Penney Model | Brillouin zones - akritinfo.com
Crystal structure | Kronig Penney Model | Brillouin zones
Kronig – Penney Model
The model illustrates the behavious of an electron in a periodic potential. It assumes that the potential energy of an electron in a linear array of positive nuclei has a form of periodic square
potentials. The potential consists of an infinite row of rectangular potential wells separated by barrier width b and space periodicity a.
The model potential is,
The Schroedinger wave equations for two regions are,
Due to the behavior of electron in a periodic potential, the above wavefunctions must be of Bloch form, hence,
Substituting above value and putting
u₁(x) instead of Uk(x) in Region-I
u₂(x) instead of Uk(x) in Region-II
We get from equation 3 and 4
For the solution of equation 5, let
For the solution of equation 6
Hence the solutions of equation 5 and 6
Where A, B, C, D are constants.
To evaluate the above constants, we have the following boundary conditions,
For non-trivial solution, the determinant of coefficients of A,B,C,D must be equal to zero :
Solving the determinant, we get
Above equation is very complicated. To get more convenient equation Kronig and Permay considered the case when
Then equation 13 reduces to,
When P→0 , then from equation 14, we see that
1. Since Cos ka lies between +1 and -1, the L.H.S of equation 14 should take up only those values of αa for which its value lie between +1 and -1
For such values of αa, the wave solutions,
are allowed values and other values of αa are not allowed values.
The energy spectrum consists of an infinite number of allowed energy bands (shown in thick line) and separation between two allowed energy bands there are no energy levels, called forbidden bands
(shown in dotted line).
The boundaries of allowed energy bands corresponds to Coska = ±1
оr, ка = nπ
or k = nπ/a
2. When αa increases the term PSinαa/αa decreases so width of allorwed energy band increases and hence forbidden energy regions become narrower.
(3) With increasing binding energy of the electrion, p increases, hence width of allowed energy bands decreases. When p→∞, width of the allowed energy bands become infinitely narrow and are
independent of k i.e spectrum becomes a line spectrum.
When p→∞
The energy levels in this case are discrete and the electron is completely bound. This case applies to crystals where the electrons are tightly bound to their nuclei.
When P→0 (no barrier), the electron can be considered. to be moving freely through the potn. barrier well. This case applies to crystals where the electrons are almost free of their nuclei
For P = ∞, the energy spectrum is a line spectrum.
for, P=0, the energy spectrum is quasi-continuous.
Brillouin zones
When P→0
If N be the primitive cell in the crystal of length L,
Substituting in equation 4 , we get
According to Pauli’s exclusion principle, each wave function can be occupied by at the most of two electrons i.e there are 2N electrons in a band.
For spherical case,, number of electrons in the first Brillouin zone
= 2N x 2N x 2N
= 8N3
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Graduate Student Combinatorics Conference 2017
Not all mathematical formulas transferred correctly. We can provide a correct pdf upon request.
Invited Talks
Sara Billey, University of Washington
Reduced words and a formula of Macdonald
Macdonald gave a remarkable formula connecting a weighted sum of reduced words for a permutation with the number of terms in a Schubert polynomial. We will review some of the fascinating results on
the set of reduced words in order to put our main results in context. Then we will discuss a new bijective proof of Macdonald's formula based on Little's bumping algorithm. We will also discuss some
generalizations of this formula based on work of Fomin, Kirillov, Stanely and Wachs. This project extends earlier work by Benjamin Young on a Markov process for reduced words of the longest
permutation. This is joint work with Ben Young and Alexander Holroyd.
Jacques Verstraete, University of California, San Diego
The probabilistic method: combinatorics and beyond
The probabilistic method was pioneered by Paul Erdos more than 70 years ago. Since that time, the tools and techniques have seen tremendous development, and is now an important part of modern
mathematics. In this talk, I will highlight some off the salient techniques and theorems from combinatorics as well as some other areas of mathematics on which the probabilistic method has had an
Contributed Talks
Mohsen Aliabadi, University of Illinois at Chicago
On matching in groups and vector spaces
A matching in an Abelian group GG is a bijection ff from a subset AA to a subset BB in GG such that a+f(a)∉Aa+f(a)∉A, for all a∈Aa∈A. This notion was introduced by Fan and Losonczy who used matchings
in ℤnZn as a tool for studying an old problem of Wakeford concerning elimination of monomials in a generic homogenous form under a linear change of variables. We show a sufficient condition for the
existence of matchings in arbitrary groups and its linear analogue, which lead to some generalizations of the existing results in the theory of matchings in groups and central extensions of division
rings. We introduce the notion of relative matchings between arrays of elements in groups and use this notion to study the behavior of matchable sets under group homomorphisms.
Ahmed Umer Ashraf, Western University
Combinatorial characters
We derive an expression for generating function of irreducible character of 𝔖nSn corresponding to hook partition λ=(n−k,1k)λ=(n−k,1k). As an application we give an elementary proof of Rosas formula
for Kronecker coefficients of hook shapes. The derivation involves defining homology on poset of brick tilings of Young diagrams.
Michelle Bodnar, University of California, San Diego
Rational noncrossing partitions
The Catalan numbers 𝖢𝖺𝗍(n)Cat(n), famously counting noncrossing partitions, have many generalizations. For instance, Fuss-Catalan numbers 𝖢𝖺𝗍(m)(n)Cat(m)(n) count the number of noncrossing partitions
of [mn][mn], each of whose blocks have size divisible by mm. In this talk we'll focus on a further generalization, the rational Catalan numbers 𝖢𝖺𝗍(a,b)Cat(a,b), counting a collection of noncrossing
partitions of [b−1][b−1]coming from rational a,ba,b-Dyck paths. I will review their construction, their basic properties, and the current research in this area.
Shawn Burkett, University of Colorado Boulder
Constructing supercharacter theories from lattices of normal subgroups
The normal subgroups of a finite group GG can be realized as intersections of stabilizers of the irreducible GG-modules. A supercharacter theory of GG is an analogue to the representation theory of
GG where the theory is built from nearly irreducible modules. Via stabilizers, each such theory gives a sublattice of the lattice of normal subgroups, along with similar theories on each subquotient
of the lattice. Now given a sublattice of the lattice of normal subgroups of a finite group GG and supercharacter theories on the covering relations, it is natural to ask under what conditions a
supercharacter theory of GG can be built that respects the imposed covering relations. Recently, Aliniaeifard gave a construction which allows one to build from a sublattice L a supercharacter
theory with L as its associated sublattice. However, this construction does not allow for the choice of supercharacter theories on each subquotient. In this talk, we will discuss Aliniaeifard's
construction as well as some methods being developed to refine it to account for more general covering relations.
Joseph Burnett, The University of Texas at Dallas
A Generalization of the concept of arithmetic convolution to sets of divisors
If we consider a generalized set of divisors as a function A:ℕ→P(ℕ)A:N→P(N) (where ℕN is the natural numbers and P(ℕ)P(N) is the set of all subsets of the naturals) which satisfy the following
restrictions for all nn: A(n)⊆D(n)A(n)⊆D(n), where D(n)D(n) is the set of the divisors of nn and1∈A(n)1∈A(n) Then we may define the following operation on these functions: Let A1(n)={d∈D(n):A(d)⋂A
(nd)={1}}A1(n)={d∈D(n):A(d)⋂A(nd)={1}}. Our paper analyzes the properties of this operation and other related operations, which serve to produce fractal-like patterns when examining the graphs of
these functions. We isolate special infinite families of these functions that have the property of being able to be visualized all at once from a single graphic, and provide explicit numerical
results for related generalized Mobius functions.
Joseph Burnett and Austin Marstaller, The University of Texas at Dallas
Happy graphs
In Graph Theory, a graph is a set of vertices that may or may not be connected by some lines, called edges, or sometimes arcs. The graphs in this work are always complete and have edges and vertices
colored red or blue. A graph is called Happy if there exists a vertex coloring such that each edge touches a vertex of its own color. We wish to find the exact size a graph must be so as to guarantee
it contains a happy graph on nn vertices.
Charles Burnette, Drexel University
Abelian squares and their progenies
A polynomial P∈ℂ[z1,…,zd]P∈C[z1,…,zd] is strongly 𝔻dDd-stable if PP has no zeroes in the closed unit polydisc 𝔻⎯⎯⎯⎯d.D¯d. For such a polynomial define its spectral density function to be P(z)=(P(z)P
(1/z⎯⎯⎯)⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯)−1.SP(z)=(P(z)P(1/z¯)¯)−1. An abelian square is a finite string of the form ww0ww0 where w0w0 is a rearrangement of w.w. We examine a polynomial-valued operator whose spectral
density function’s Fourier coefficients are all generating functions for combinatorial classes of constrained finite strings over an alphabet of dd characters. These classes generalize the notion of
an abelian square, and their associated generating functions are the Fourier coefficients of one, and essentially only one, L2(𝕋d)L2(Td)-valued operator. The asymptotic behavior of the coefficients
of these generating functions as well as a combinatorial meaning to Parseval’s equation are given as consequences.
Federico Castillo, University of California, Davis
Newton polytopes of multidegrees
Recently June Huh classified, up to a multiple, all possible classes in the Chow ring of a product of two projective spaces that can be represented by an irreducible variety. As a first step to
generalize this result to any number of copies of projective spaces, we focus only on the support of these classes. It turns out that the support of any irreducible variety can be described naturally
as the integer points in a polytope, more precisely a generalized permutohedron.
Swee Hong Chan, Cornell University
Toric graphic arrangements
Consider an arrangement of linear hyperplanes integral with respect to a given lattice. The lattice gives rise to a torus and the arrangement to a subdivision of the torus. We are interested in the
combinatorics of this subdivision. We will describe questions and results for particular lattices associated to root systems and arrangements associated to graphs.
Joseph Doolittle, University of Kansas
Reconstructing nearly simple polytopes from their graphs
A theorem of Blind and Mani shows that a simple polytope can be reconstructed from its graph. Kalai gave a very elegant proof of the theorem using fOfO as a way to measure "goodness" of an acyclic
orientation. fOfO is defined for OO an orientation of a graph G=(V,E)G=(V,E), fO:=∑v∈V2indeg(v)fO:=∑v∈V2indeg(v). In this talk we will expand on the use of fOfO and present a new result about nearly
simple polytopes, as well as showing a bound on the non-simplicity of a polytope which can be reconstructed from its graph.
Michael Earnest, University of Southern California
Longest Common Patterns in Permutations
A natural partial order on the set of permutations of all possible lengths is pattern containment. The concept of permutation patterns gives rise to a rich collection of combinatorial problems. We
will discuss the longest common pattern, or LCP, between two permutations; this statistic is analogous to the longest common subsequence of two words, an important topic in computer science,
specifically in bioinformatics. We have shown that the LCP between two random permutations of length nn grows proportionally to n2/3n2/3 as n→∞n→∞. In this talk, we demonstrate the proof of this
fact, along with several generalizations and open problems.
Brittney Ellzey, University of Miami
A symmetric function arising from graph colorings
The chromatic polynomial of a graph counts the number of proper colorings of a graph using n colors. Stanley defined the chromatic symmetric function of a graph, which generalizes the chromatic
polynomial. Shareshian and Wachs introduced a quasisymmetric refinement of the chromatic symmetric function for labeled graphs, namely the chromatic quasisymmetric function of a graph. We consider a
generalization of these chromatic quasisymmetric functions from labeled graphs to directed graphs. In this talk, we will look at these definitions and some simple examples, as well as some of my
results, generalizing work of Stanley, Shareshian-Wachs, and Athanasiadis.
Michael Engen, University of Florida
On the dimension of composition posets
We characterize the downsets of compositions (ordered by the generalized subword order) which have finite dimension in the sense of Dushnik and Miller. We identify four minimal downsets of infinite
dimension and establish that any downset which does not contain one of these four has finite dimension.
Sean English, Western Michigan University
Large monochromatic components in sparse random hypergraphs
It is known, due to Gyárfás and Füredi, that for any rr-coloring of the edges of KnKn, there is a monochromatic component of order (1/(r−1)+o(1))n(1/(r−1)+o(1))n. They also showed that this is best
possible if r−1r−1 is a prime power. Recently, Dudek and Prałat showed that the binomial random graph (n,p)G(n,p) behaves very similarly with respect to the size of the largest monochromatic
component. More precisely, it was shown that a.a.s.\ for any rr-coloring of the edges of (n,p)G(n,p) and arbitrarily small constant α>0α>0, there is a monochromatic component of order (1/(r−1)−α)n(1
/(r−1)−α)n, provided that pn→∞pn→∞. As before, this result is clearly best possible. In this talk we present a generalization of this result to hypergraphs. Specifically we show that in the
kk-uniform random hypergraph, (k)(n,p)H(k)(n,p) a.a.s. for any kk-coloring of the edges, there is a monochromatic component of order (1−α)n(1−α)n and for any k+1k+1 coloring, there is a
monochromatic component of order (1−α)kk+1n(1−α)kk+1n.
Joshua Fallon, Louisiana State University
A family of 2-crossing-critical graphs on the projective plane
A graph G is said to be 2-crossing-critical if it has crossing number at least two and every proper subgraph of G has crossing number less than two. Bokal, Oporowski, Richter, and Salazar recently
determined all the 3-connected 2-crossing-critical graphs containing a subdivision of the Möbius Ladder V10. These graphs are members of a family generated by joining certain tiles in sequence. We
show a closely related family of tile joins that are 2-crossing-critical on the real projective plane. Analogous to the plane case, these graphs have projective crossing number at least two and each
proper subgraph has projective crossing number less than two. We also discuss ongoing work toward extending this family to all non-orientable surfaces.
Tara Fife, Louisiana State University
An extension of the class of laminar matroids
A matroid is a finite set with a collection of independent sets that behave like linearly independent sets in a vector space. The rank, r(X)r(X), of a set XX is the size of a largest independent
subset of XX, and the closure, cl(X)cl(X), of XX is {x:r(X∪{x}=r(X)}{x:r(X∪{x}=r(X)}. A circuit is a minimal dependent set. The widely studied class of nested matroids consists of those matroids
where, for any two circuits C1C1 and C2C2, either cl(C1)⊆cl(C2)cl(C1)⊆cl(C2) or cl(C2)⊆cl(C1)cl(C2)⊆cl(C1). Thus nested matroids are 00-laminar where a matroid is kk-laminar if, for any two circuits
C1C1 and C2C2 with r(cl(C1)∩cl(C2)≥kr(cl(C1)∩cl(C2)≥k, either cl(C1)⊆cl(C2)cl(C1)⊆cl(C2) or cl(C2)⊆cl(C1)cl(C2)⊆cl(C1). Earlier work has characterized 0-laminar matroids and 1-laminar matroids in
numerous ways. This talk will discuss the behavior of the class of 2-laminar matroids.
Nathan Fox, Rutgers University
Nice solutions to nested recurrence relations
Linear recurrence relations, such as the Fibonacci recurrence F(n)=F(n−1)+F(n−2)F(n)=F(n−1)+F(n−2), are completely understood. On the other hand, few general facts are known about general
recurrences, and there are many open questions. In particular, nested recurrence relations, such as the Hofstadter Q-recurrence Q(n)=Q(n−Q(n−1))+Q(n−Q(n−2))Q(n)=Q(n−Q(n−1))+Q(n−Q(n−2)), can display a
wide diversity of behaviors depending on the initial conditions. In this talk, we will explore some of the possible types of solutions we can achieve to such recurrences. We will focus primarily on
finding solutions that also satisfy linear recurrences, though we will see some more unusual solutions as well.
Mac Gallagher, George Mason University
The Hirsch conjecture and the diameters of polytopes
The diameters of polytopes are studied in mathematical optimization because of their relation to the simplex method for linear programming. In 1957, Hirsch posed a conjecture on the maximum diameter
of polytopes. While the conjecture was ultimately false (Santos, 2010), many related questions still remain at large, and the diameter problem for polytopes still remains largely unsolved. We will
describe the relationship between the Hirsch conjecture and linear programming and discuss some of the current techniques being used to solve the problem.
Zachary Gershkoff, Louisiana State University
A notion of minor-based matroid connectivity
For a matroid NN, a matroid MM is NN-connected if every two elements of MM are in an NN-minor together. Thus a matroid is connected if and only if it is U1,2U1,2-connected. A proof is presented that
U1,2U1,2 is the only connceted matroid NN such that if MM is NN-connected, then M∖eM∖e or M/eM/e is NN-connected.
Alejandro Ginory, Rutgers University
The combinatorics of Weingarten calculus
Integration on compact matrix groups with respect to the Haar measure has many applications, e.g. in physics and statistics. The method of Weingarten calculus, introduced by Benoit Collins, gives
combinatorial methods for computing the integrals of polynomials in matrix coefficients over the unitary, orthogonal, and symplectic groups. These methods involve symmetric functions such as the Jack
polynomials. In this talk, I will discuss the combinatorics of Weingarten calculus and some explicit recursive methods for carrying out these computations.
Kevin Grace, Louisiana State University
All that glitters is not golden-mean
Three closely related classes of GF(4)-representable matroids are the golden-mean matroids, the matroids representable over all fields of size at least 4, and the matroids representable over GF(4) as
well as fields of all characteristics. We characterize the highly connected matroids in each of these classes by using frame templates, which were recently introduced by Geelen, Gerards, and Whittle
as tools for describing the the highly connected members of minor-closed classes of representable matroids. As a direct consequence of this characterization, we give the growth rates of these classes
of matroids, including the golden-mean matroids. This proves a conjecture made by Archer in 2005.
Corbin Groothuis, University of Nebraska-Lincoln
D-matching polynomials
For any graph GG we may construct an associated polynomial called the matching polynomial, which is a variant on a generating function for matchings of GG. When GG is a cycle or path graph with nn
vertices, the resulting polynomials are essentially the Chebyshev polynomials Tn(x)Tn(x) and Un(x)Un(x) respectively. It is known that the only divisibility relations among the UnUn have the form
Umn−1Un−1=Um−1⋅TnUmn−1Un−1=Um−1⋅Tn; we interpret this equality combinatorially. In particular we show the right-hand side is an object with combinatorial meaning, called the d-matching polynomial by
Hall, Pruder and Sawin(2015).
Brent Holmes, University of Kansas
On the diameter of Hochster-Huneke graphs of Stanley-Reisner rings with Serre (S2S2) property and Hirsch type bounds on abstractions of polytopes
Let RR be a Noetherian commutative ring of positive dimension. The Hochster-Huneke graph of RR (sometimes called the dual graph of SpecRSpecR and denoted by (R)G(R)) is defined as follows: the
vertices are the minimal prime ideals of RR, and the edges are the pairs of prime ideals (P1,P2)(P1,P2) with height(P1+P2)=1height(P1+P2)=1. If RR satisfies Serre's property (S2)(S2), then (R)G(R)
is connected. In this note, we provide lower and upper bounds for the maximum diameter of Hochster-Huneke graphs of Stanley-Reisner rings satisfying (S2)(S2). These bounds depend on the number of
variables and the dimension. Hochster-Huneke graphs of (S2)(S2) Stanley-Reisner rings are a natural abstraction of the 11-skeletons of polyhedra. We discuss how our bounds imply new Hirsch-type
bounds on 11-skeletons of polyhedra.
Michael Joseph, University of Connecticut
Noncrossing partitions, toggles, and homomesies
We introduce n(n−1)/2n(n−1)/2 "toggling" involutions on the set NC(n)NC(n) of noncrossing partitions of [n]:={1,2,…,n}[n]:={1,2,…,n}. These involutions generate a group under composition, called the
toggle group. For many operations TT within the toggle group, several statistics ff on NC(n)NC(n) are homomesic, meaning ff has the same average across every orbit. These statistics include the
number of blocks of the partition. We will also discuss a consequence of the homomesy results to the sizes of orbits. This is joint work with David Einstein, Miriam Farber, Emily Gunawan, Matthew
Macauley, James Propp, and Simon Rubinstein-Salzedo.
Ezgi Kantarci, University of Southern California
Type B analogues of ribbon tableaux
We introduce a shifted analogue of the ribbon tableaux defined by James and Kerber. For any positive integer kk, we give a bijection between the kk-ribbon fillings of a shifted shape and regular
fillings of a ⌊k/2⌋⌊k/2⌋-tuple of shapes called its kk-quotient. We also define the corresponding generating functions, and prove that they are symmetric, Schur positive and Schur Q positive.
Hee Sun Kim, University of Kansas
Weighting of new context models
Contexts of stationary ergodic sources are considered not necessarily consecutive sequences of symbols of the past. The introduced context set model of a source provides a code that can achieve less
parameter redundancy than the code the context tree and generalized context tree models provide. The problem of coding sources with unknown context set is addressed for multialphabet sources.
Information on the maximum memory length of the source is not required; it may be even infinite. The Context Set Weighting method is introduced to efficiently calculate a mixture of the
Krichevsky-Trofimov distributions over possible context sets. For a message of length n, the number of possible context sets is larger than exponential in n, but the Context Set Weighting is shown to
be computable in time polynomial in n. The obtained coding distribution is proved to provide a universal code.
Westin King, Texas A&M University
A correspondence between parking functions on directed mappings and directed trees
Bruner and Panholzer extend the notion of a parking function to both rooted labeled trees in which edges are oriented towards the root and digraphs of mappings, f:[n]→[n],f:[n]→[n], with edges
oriented a→f(a)a→f(a). If FnFn is the number of parking functions on rooted labeled trees with nn vertices and MnMn is the number of parking functions on mappings, then the authors demonstrate that
nFn=MnnFn=Mn. In this talk, I will extend the notion of parking function to trees in which the edges are oriented away from the root and digraphs of mappings with edges f(a)→af(a)→a and show the same
relationship holds.
Bo Lin, University of California, Berkeley
Tropical Fermat-Weber points
We investigate the computation of Fermat-Weber points under the tropical metric, motivated by its application to the space of equidistant phylogenetic trees with NN leaves realized as the tropical
linear space of all ultrametrics. While the Fréchet mean with the CAT(0)CAT(0)-metric of Billera-Holmes-Vogtman has been studied by many authors, the Fermat-Weber point under tropical metric in tree
spaces is not well understood. In this paper we investigate the Fermat-Weber points under the tropical metric and we show that the set of tropical Fermat-Weber points is a classical convex polytope.
We identify conditions under which this set is a singleton. This is a joint work with Ruriko Yoshida.
Jephian C.-H. Lin, Iowa State University
Note on von Neumann and Rényi entropies of a graph
Let GG be a graph and LL its combinatorial Laplacian matrix. The scaled Laplacian matrix 1tr(L)L1tr(L)L is a positive semidefinite matrix with trace one, so it can be written as ∑ni=1λiEi∑i=1nλiEi,
where λiλi's are the eigenvalues and EiEi's are rank-one matrices. Since λi≥0λi≥0 and ∑ni=1λi=1∑i=1nλi=1, such a matrix can be viewed as a mixture of several rank-one matrices and is called a density
matrix in quantum information. The von Neumann entropy and the Rényi entropy measure the mixedness of a density matrix; in this talk, we will discuss how these entropies relate to different graphs.
Andrew Lohr, Rutgers University
Numerics for lattice paths
One of the many ways we can look at Catalan numbers is by the number of paths of steps west and north in ℕ2N2 that stay below the line y=x going from (0,0) to (n,n). There are some open questions
when we change the slope of the line we have to stay below. An even less studied variation of this is if we consider paths in ℕ3N3 in a region bounded by planes.
Amanda Lohss, Drexel University
Tableaux and the ASEP
The ASEP is a particle model (which will be defined in this talk) that has been used extensively since 1970 in physics, biology, and biochemistry. Interestingly enough, in the past decade, various
types of tableaux have been introduced to provide a simple combinatorical formula for the steady state distribution of the ASEP. Some of these tableaux will be introduced in this talk and the formula
they provide will be discussed. Lastly, some results on these tableaux, which are significant in terms of the ASEP, will be presented.
Jack Love, George Mason University
Polygon spaces
Imagine 4 drinking straws arranged as a square on a flat table. Now imagine a string is running through them, and this string is fixed to the table with a tack at one of the vertices of the square.
Now you can start moving the straws around to get lots of (uncountably many) parallelograms, including a few degenerate ones. Every parallelogram has two diagonals, so we can map the space of
parallelograms to ℝ2R2 by sending a parallelogram to its diagonal lenghts. What does the image of this map look like? We know the answer to this question, and it has lots of nice features that we'll
talk about. But what if we replace "4" with "nn" and "flat table" with "ℝdRd"? We present recent results and open questions.
Megan Ly, University of Colorado Boulder
Centralizer algebras of unipotent upper triangular matrices
Classical Schur-Weyl duality relates the irreducible characters of the symmetric group SnSn to the irreducible characters of the general linear group GLn(ℂ)GLn(C) via their commuting actions on
tensor space. We investigate the analog of this result for the group of unipotent upper triangular matrices UTn(𝔽q)UTn(Fq). In this case the character theory of UTn(𝔽q)UTn(Fq) is unattainable, so we
must employ supercharacter theory, creating a striking variation.
John Machacek, Michigan State University
The chromatic symmetric function: Hypergraphs and beyond
Stanley's chromatic symmetric function is a graph invariant which has been (and still is) the subject of much research. We will (attempt to) make case for the study of a chromatic symmetric function
in hypergraphs and other generalizations of graphs. The existence (or non-existence) of two non-isomorphic trees with equal chromatic symmetric functions is an open problem. Martin, Morin, and Wagner
have shown that the chromatic symmetric function of a tree determines its degree sequence. We will show that the degree sequence of a uniform hypertree is determined by its chromatic symmetric
function, but there do exist non-isomorphic pairs of 3-uniform hypertrees with the same chromatic symmetric function. A definition of generalized graph coloring will be given and will encompass graph
and hypergraph coloring as well as oriented coloring and acyclic coloring.
Viswambhara Makam, University of Michigan
Polynomial degree bounds for matrix semi-invariants
We study the left-right action of SL(n) x SL(n) on several copies of n×nn×n matrices. We prove that the null cone is defined by invariants of degree n(n−1)n(n−1) and that consequently invariants of
degree ≤n6≤n6 generate the ring of invariants. We give generalizations to rings of invariants associated to quivers. Our results have applications in computational complexity, notably a polynomial
time algorithm for non-commutative rational identity testing.
Carolyn Mayer, University of Nebraska - Lincoln
A two-phase graph decoder for LT codes with partial erasures
Luby Transform (LT) codes are a class of rateless erasure codes in which encoding symbols are generated and transmitted until all users receive enough symbols to reconstruct the original message.
Encoding is performed by dynamically constructing a bipartite graph according to a specified degree distribution. Recently, partial erasure channels have been introduced to model applications in
which some information may remain after an erasure event. In this talk, we will discuss a two-phase graph decoder for LT codes with partial erasures.
James McKeown, University of Miami
Alternating sign matrices and the Waldspurger decomposition
In the mid 2000's Lie theorists such as Waldspurger and Meinrenken used topological methods to prove some surprising new theorems in Coxeter theory. Specifically, they exhibited new tilings of space
by simplices in bijection with Weyl group elements for each of the classical types. In type A, where the Weyl group is the symmetric group, I will give a concrete combinatorial description of these
simplices by defining the "Waldspurger Transformation" of a permutation matrix. When one applies this transformation to the larger class of alternating sign matrices, the image has a nice description
relating to the MacNeille Completion of the Bruhat order. I will show that Waldspurger matrices for types B and C, are equivalent to "folded Waldspurger" matrices of type A, and we consider ways of
defining "type B" alternating sign matrices.
Kyle Meyer, University of California, San Diego
Descent representations of a generalization of the coinvariant algebra
The coinvariant algebra RnRn is a well-studied 𝔖nSn-module that gives a graded version of the regular representation of 𝔖nSn. Using a straightening algorithm on monomials and the Grasia-Stanton
basis, Adin, Brenti, and Roichman give a description of the Frobenius image of RnRn, graded by partitions, in terms of descents of standard Young tableaux. Motivated by the Delta Conjecture of
Macdonald polynomials, Haglund, Rhoades, and Shimozono give an extension of the coinvariant algebra Rn,kRn,k and an extension of the Garsia-Stanton basis. We extend the results of Adin, Brenti, and
Roichman to Rn,kRn,k.
Marie Meyer, University of Kentucky
Laplacian simplices
In this talk we will introduce a polyhedral construction arising from the well studied Laplacian matrix of a graph, namely, taking the convex hull of the columns of the matrix to form a simplex. We
will discuss known properties of these simplices according to graph type.
Ada Morse, University of Vermont
DNA origami and knots in graphs
Motivated by the problem of determining unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine existence and knottedness of A-trails in (necessarily Eulerian) graphs
embedded on surfaces in space. We construct infinite families of embedded graphs containing unknotted A-trails (for any genus surface) as well as infinite families of embedded graphs containing no
unknotted A-trails (for surfaces other than the sphere.) While not every embedded Eulerian graph contains an unknotted A-trail, we conjecture that every abstract Eulerian graph has some embedding
containing an unknotted A-trail. We prove this in the 4-regular case by giving an algorithm for finding such embeddings. In closing, we discuss some results regarding which knots can be constructed
from A-trails of rectangular grids.
Lauren Nelsen, University of Denver
Many edge-disjoint rainbow spanning trees in general graphs
A rainbow spanning tree in an edge-colored graph is a spanning tree in which each edge is a different color. Carraher, Hartke, and Horn showed that for nn and CC large enough, if GG is an
edge-colored copy of KnKn in which each color class has size at most n/2n/2, then GG has at least ⌊n/(Clogn)⌋⌊n/(Clogn)⌋ edge-disjoint rainbow spanning trees. Here we strengthen this result by
showing that if GG is any edge-colored graph with nn vertices in which each color appears on at most δ⋅λ1/2δ⋅λ1/2 edges, where δ≥Clognδ≥Clogn for nn and CC sufficiently large and λ1λ1 is the
second-smallest eigenvalue of the normalized Laplacian matrix of GG, then GG contains at least ⌊δ⋅λ1Clogn⌋⌊δ⋅λ1Clogn⌋ edge-disjoint rainbow spanning trees.
Luke Nelsen, University of Colorado Denver
Erdős-Szekeres online
In 1935, Erdős and Szekeres proved that (m−1)(k−1)+1(m−1)(k−1)+1 is the minimum number of points in the plane (ordered by their xx-coordinates) which definitely contain an increasing (also ordered by
yy-coordinates) subset of mm points or a decreasing subset of kk points. We consider their result from an online game perspective: Let points be determined one by one by player A first determining
the xx-coordinate and then player B determining the yy-coordinate. What is the minimum number of points such that player A can force an increasing subset of mm points or a decreasing subset of kk
points? In this talk, we discuss the distinction between the original setting from this new one and present some small results. Thanks to the 2016 GRWC workshop, work on this question is underway
jointly with Kirk Boyer, Lauren M. Nelsen, Florian Pfender, Lizard Reiland and Ryan Solava.
David Nguyen, University of California, Santa Barbara
Closed form formulas for a class of generalized Dyck paths
Walks on the integers with steps −1−1 and +1+1 correspond to the classical Dyck paths, which are well known to be enumerated by Catalan numbers. A natural generalization is to consider walks on the
integers with steps −h,…,−1,+1,…,+h−h,…,−1,+1,…,+h. In this talk, we will show how to find explicit formulas enumerating walks of length n for the case h=2h=2, using a method of wide applicability,
the so-called "kernel method", in terms of nested sums of binomial coefficients. Interesting links to increasing trees will also be discussed.
Danh Nguyen Luu, University of California, Los Angeles
Presburger arithmetic and integer points in polyhedra
Presburger arithmetic is the first order theory on integers that allows only additions and inequalities. It is the natural language to express many problems in integer programming and optimization.
Central in this topic the search for effective algorithms to decide the truth of Presburger sentences. We present various new hardness and polynomial-time results in this topic. These results are
intimately related to the study of integer points in polyhedra and their projections via linear maps.
Pouria Salehi Nowbandegani, Vanderbilt
Forbidden properly edge-colored subgraphs that force large highly connected monochromatic subgraphs
We consider the connected graphs GG that satisfy the following property: If n≫m≫kn≫m≫k are integers, then any coloring of the edges of KnKn, using mm colors, containing no properly colored copy of
GG, contains a monochromatic kk-connected subgraph of order at least n−f(G,k,m)n−f(G,k,m) where ff does not depend on nn. If we let G denote the set of graphs satisfying this statement, we exhibit
some infinite families of graphs in G as well as conjecture that the cycles in G are precisely those whose lengths are divisible by 33. Our main result is that C6∈C6∈G.
McCabe Olsen, University of Kentucky
Hilbert bases and lecture hall partitions
In the interest of finding the minimum additive generating set for the set of ss-lecture hall partitions, we compute the Hilbert bases for the ss-lecture hall cones in certain cases. In particular,
we compute the Hilbert bases for two well-studied families of sequences, namely the 1modk1modk sequences and the ℓℓ-sequences. Additionally, we provide a characterization of the Hilbert bases for
uu-generated Gorenstein ss-lecture hall cones in low dimensions.
Alperen Ozdemir, University of Southern California
A random walk on the symmetric group
We study the mixing time of the Markov chain on SnSn starting with a random (n−k)(n−k)-cycle then continuing with random transpositions where kk is a fixed number. The bounds that yield the mixing
time involve certain estimates on the characters of the symmetric group. The analysis is carried out by using combinatorics of Young tableaux.
Alex Schaefer, Binghamton University
Signed Graphs, Permutability, and 22-Transitive Perfect Matchings
A signed graph is a triple (V,E,σ)(V,E,σ) where (V,E)(V,E) is a graph and σ:E↦{+,−}σ:E↦{+,−} is a function. A switching of a signed graph results from choosing a (possibly empty) edge cut and
negating all the edges in the cut, which partitions the ways of signing a graph. The search for an invariant of these classes led to a vector space of negative cycle vectors, of which a natural
spanning set seems to arise from classes of graphs whose sets of negative edges correspond to matchings, but only when the matching obeys a permutability property. Attempts to understand this
phenomenon led to (in joint work with E. Swartz) a complete classification of graphs with perfect matchings such that the automorphism group is 22-transitive on the matching.
Alex Schulte, Iowa State University
Anti-van der Waerden number of 33-term arithmetic progressions
A set is rainbow if each element of the set is a different color. A coloring is unitary if at least one color is used exactly once. The anti-van der Waerden number of the integers from 11 to nn,
denoted by aw([n],3)aw([n],3), is the least positive integer rr such that every exact rr-coloring of [n][n] contains a rainbow 33-term arithmetic progression. The unitary anti-van der Waerden number
of the integers from 11 to nn, denoted by awu([n],3)awu([n],3), is the least positive integer rr such that every exact unitary rr-coloring of [n][n] contains a rainbow 33-term arithmetic progression.
The anti-van der Waerden number of a graph GG, denoted by aw(G,3)aw(G,3), is the least positive integer rr such that every exact rr-coloring of GG contains a rainbow 33-term arithmetic progression.
Bounds for the anti-van der Waerden number and the unitary anti-van der Waerden number on the integers have been established. The exact value of the unitary anti-van der Waerden number of the
integers is equal to the anti-van der Waerden number of the integers and these are given by aw([n],3)=awu([n],3)=⌈log3n⌉+2aw([n],3)=awu([n],3)=⌈log3n⌉+2.
Elizabeth Sheridan Rossi, University of Connecticut
Homomesy for Foatic actions on the symmetric group
Homomesy is a phenomenon identified by Tom Roby and James Propp in 2013. By looking at a group action on a set of combinatorial objects, we partition them into orbits. From here we discuss statistics
that are homomesic, meaning they have the same average value over each orbit. In this talk we will focus primarily on homomesies for actions on the symmetric group, in particular the so called
"Foatic maps", created using the well known "fundamental bijection" of Rényi and Foata.
Rahul Singh, Northeastern University
Conormal variety of Schubert variety in a cominuscule Grassmannian
Let GG be a Chevalley group associated to an irreducible root system and PP a co-minuscule parabolic subgroup of GG. Lakshmibai et al have shown that the cotangent bundle T∗G/PT∗G/P embeds as an open
subset of a Schubert variety associated to the loop group of GG. We give a complete classification of the Schubert varieties X(w)X(w) in G/PG/P whose conormal variety N∗X(W)N∗X(W) is itself dense in
a Schubert subvariety under this identification.
Sara Solhjem, North Dakota State University
Semistandard Young tableaux polytopes
We define a new family of polytopes as the convex hull of certain {0,1,−1}{0,1,−1} matrices in bijection with semistandard Young tableaux. We investigate various properties of these polytopes,
including their inequality descriptions, vertices, and facets.
Avery St. Dizier, Cornell University
Flow polytopes and degree sequences
The flow polytope associated to an acyclic directed graph is the set of all nonnegative flows on the edges of the graph with a fixed netflow at each vertex. I will describe basic properties and some
interesting examples of flow polytopes. Next, I'll examine a procedure for triangulating certain flow polytopes and some nice properties of the resulting triangulation. If there's time, I'll briefly
explain how special cases of this construction have been shown to encode some Schubert and Grothendieck polynomials, and present some open questions for further research.
Everett Sullivan, Dartmouth College
Linear chord diagrams with long chords
A linear chord diagram of size nn is a partition of the set {1,2,…,2n}{1,2,…,2n} into sets of size two, called chords. From a table showing the number of linear chord diagrams of degree nn such that
every chord has length at least kk, we observe that if we proceed far enough along the diagonals, they are given by a geometric sequence. We prove that this holds for all diagonals, and identify when
the effect starts.
Justin Troyka, Dartmouth College
Exact and asymptotic enumeration of classes of centrosymmetric permutations
Roughly speaking, a permutation class is a set of permutations avoiding a certain set of forbidden patterns. Permutation classes and pattern-avoiding permutations have been studied in enumerative
combinatorics for several decades. This talk concerns the counting of permutations in a given class that are centrosymmetric, meaning they are fixed by the reverse--complement map. At the Permutation
Patterns conference of summer 2016, Alex Woo presented the following open question: in which permutation classes is it true that the exponential growth rate of the permutations of nn is the same as
that of the centrosymmetric permutations of 2n2n? In this talk I will present preliminary findings from my investigation of this question, including some examples where the growth rates are not
equal. I conjecture that equality holds for any class that is closed under direct sum or closed under skew sum; I also conjecture that one direction of inequality holds in the general case. This talk
will be accessible to people who are not familiar with permutation patterns.
Shira Viel, North Carolina State University
Surfaces, orbifolds, and dominance
Consider the set of all triangulations of a convex (n+3)(n+3)-gon. These triangulations are related to one another by diagonal flips, and the graph defined by these flips is the 1-skeleton of the
familiar nn-dimensional polytope known as the associahedron. The nn-dimensional cyclohedron is constructed analogously using centrally-symmetric triangulations of a regular (2n+2)(2n+2)-gon, with
relations given by centrally-symmetric diagonal flips. Modding out by the symmetry, we may equivalently view the cyclohedron as arising from "triangulations of an orbifold": the (n+1)(n+1)-gon with a
single two-fold branch point at the center.
In this talk we will introduce orbifold-resection, a simple combinatorial operation which maps the "once-orbifolded" (n+1)(n+1)-gon to the (n+3)(n+3)-gon. More generally, orbifold-resection maps a
triangulated orbifold to a triangulated surface while preserving the number of diagonals and respecting adjacencies. This induces a relationship on the signed adjacency matrices of the
triangulations, called dominance, which gives rise to many interesting phenomena. For example, the normal fan of the cyclohedron refines that of the associahedron; work is in progress to show that
such fan refinement holds generally in the case of orbifold-resection. If time allows, we will explore other dominance phenomena in the context of the surfaces-and-orbifolds model.
Corey Vorland, North Dakota State University
Homomesy for J([2]×[a]×[b])J([2]×[a]×[b]) and multidimensional recombination
We generalize the notion of recombination defined by D. Einstein and J. Propp in order to study homomesy on 33-dimensional posets under rowmotion and promotion. We have two main results. We state and
prove conditions under which recombination can be performed in nn-dimensions. We also apply recombination to show a homomesy result on the product of chains [2]×[a]×[b][2]×[a]×[b] under rowmotion and
promotion. Additionally, we determine that this homomesy result does not generalize to arbitrary products of 33-chains.
George Wang, University of Pennsylvania
Properties and product formulas for quasi-Yamanouchi tableaux
Quasi-Yamanouchi tableaux connect the two most studied families of tableaux. They are a subset of semistandard Young tableaux that are also a refinement on standard Young tableaux, and they can be
used to improve the fundamental quasisymmetric expansion of Schur polynomials. We prove a product formula for enumerating certain quasi-Yamanouchi tableaux and provide strong evidence that no product
formula exists in general for other shapes. Along the way, we also prove some nice properties of their distribution and symmetry.
Zhaochen Wang, University of Wisconsin-Madison
Total positivity of Riordan-like arrays
A Riordan-like array is an infinite lower triangular matrix [Rn,k]n,k≥0[Rn,k]n,k≥0 defined by the recursive system
where (an)n≥0(an)n≥0 and (zn)n≥0(zn)n≥0 called the A−A− and Z−Z−sequences of RR. This provides a common framework for Riordan arrays and triangles corresponding to the Catalan-like numbers. Our
concern is the total positivity of such matrices, the log-convexity of the 0th column and the log-concavity of each row. This talk starts at the total positivity of a special case of Riordan arrays
called the Catalan triangle. Then We will provide results of more general cases of Riordan arrays
Isaac Wass, Iowa State University
Rainbow paths and trees in properly-colored graphs
A graph GG is properly kk-colored if the colors {1,2,…,k}{1,2,…,k} are assigned to each vertex such that uu and vv have different colors if uvuv is an edge and each color is assigned to some vertex.
A rainbow kk-path, a rainbow kk-star and a rainbow kk-tree is a path, star or tree, respectively, on kk vertices such that each vertex is a different color. We prove several results about the
existence of rainbow paths, stars and trees in properly colored graphs, as well as their uses in proving various criteria about a graph's chromatic number. In particular, any graph GG properly
colored with the minimum number of colors χ(G)χ(G) always contains every possible rainbow χ(G)χ(G)-tree.
Rupei Xu, University of Texas at Dallas
A new graph densification method and its applications
Given a graph G(V,E)G(V,E), the graph densification task is to find another graph H(V,E′)H(V,E′) such that E′E′ is significantly larger than EE while HHstill approximately maintains the desired
properties of G.G. As dense graph has more established mature research results than sparse graphs, graph densification is a potentially powerful way to study sparse graphs. For example, MAX Cut has a
PTAS on dense graphs, many property testing algorithms and sublinear time approximation algorithms rely on structural properties of dense graphs, Szemerédi's regularity lemma also works best for
dense graphs. Unlike the well studied graph sparsification, we have little fundamental understanding of graph densification. In the literature, graph densification has been investigated in terms of
spectral densifers and cut densifiers, as well as its connection to metric embedding. However, graph densifiers that are based on these methods do not always exist, and their principal understanding
is still largely open. In this paper, a new graph densification method is investigated, based on the distance sequence approach by Bollobás. This method works for any graph, and it has deep
connections with ball approximation and doubling metric approximation of graphs. This provides help in better understanding important issues, such as graph small world navigability, metric embedding,
greedy rounding, failure detection and other applications. This is joint work with Professor Andras Farago.
Li Ying, Texas A&M University
Stability of the Heisenberg product on symmetric functions
The Heisenberg product is an associative product defined on symmetric functions which interpolates between the usual product and Kronecker product. I will give the definition of this product and
describe some properties of it. One well known thing about the Kronecker product of Schur functions is the stability phenomenon discovered by Murnaghan in 1938. I will give an analogous result for
the Heisenberg product of Schur functions.
Xiaowei Yu, Shandong University
Relaxed antimagic labeling of graphs
A kk-labeling of a graph GG is a mapping ϕϕ: E(G)→{1,2,…,m+k}E(G)→{1,2,…,m+k} such that all the edges receive different labels, where m=|E(G)|m=|E(G)|. Let μ(v)=∑uv∈E(G)ϕ(uv)μ(v)=∑uv∈E(G)ϕ(uv). A
graph is called kk-antimagic if GG admits a kk-labeling with μ(u)≠μ(v)μ(u)≠μ(v) for any pair u,v∈V(G)u,v∈V(G). A kk-antimagic labeling is called an antimagic labeling if k=0k=0. If a graph GG admits
an antimagic labeling, then GG is antimagic. In 2003, Hartsfield and Ringel proposed the famous Antimagic Graph Conjecture: Every graph without K2K2 is antimagic. Since this conjecture is widely
open, they even conjectured that all the nice trees were antimagic, which is open as well. Recently, Bensmail et al. conjectured that a labeling of GG can guarantee that μ(u)≠μ(v)μ(u)≠μ(v) for any
uv∈E(G)uv∈E(G), which is called edge-injective neighbor sum distinguishing edge-kk-coloring. In this talk, I will present the results about edge-injective neighbor sum distinguishing
Yan Zhuang, Brandeis University
On the joint distribution of peaks and descents over restricted sets of permutations
Let des(π)des(π) be the number of descents and pk(π)pk(π) the number of peaks of a permutation ππ. In 2008, Brändén proved that for any subset Π⊆𝔖nΠ⊆Sn invariant under a group action called the
"modified Foata--Strehl action", the descent polynomial A(Π;t):=∑π∈Πtdes(π)A(Π;t):=∑π∈Πtdes(π) is γγ-positive and is related to the peak polynomial P(Π;t):=∑π∈Πtpk(π)P(Π;t):=∑π∈Πtpk(π) by the formula
By taking Π=𝔖nΠ=Sn, this yields well-known results of Foata--Schützenberger (1970) and Stembridge (1997). In this talk, we produce a refinement of Brändén's formula: For any Π⊆𝔖nΠ⊆Sn invariant under
the modified Foata--Strehl action, the descent polynomial A(Π;t)A(Π;t) and the polynomial P(Π;y,t):=∑π∈Πypk(π)tdes(π)P(Π;y,t):=∑π∈Πypk(π)tdes(π) encoding the joint distribution of the peak number and
descent number over ΠΠ satisfy the relation
We then observe that many sets invariant under the modified Foata--Strehl action can be characterized in terms of pattern avoidance, and thus pose the question: Can we characterize all pattern
classes invariant under the modified Foata--Strehl action? We conclude with some preliminary results in this direction, which is joint work with Richard Zhou (Lexington High School).
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In the given figure, A and B are the centres of two circles that inter
In the given figure, A and B are the centres of two circles that intersect at X and Y. PXQ is a straight line. If reflex angle QBY=210∘, then find obtuse angle PAY.
Step by step video & image solution for In the given figure, A and B are the centres of two circles that intersect at X and Y. PXQ is a straight line. If reflex angle QBY = 210^@, then find obtuse
angle PAY. by Maths experts to help you in doubts & scoring excellent marks in Class 10 exams.
Updated on:21/07/2023
Topper's Solved these Questions
Knowledge Check
• In the given figure, O is the centre of the circle. Then ∠AED measure
• In the figure , A and C are the centres of the circles. DEB is a straight line. If relex angle BAF = 208∘ then obtuse angle DCF=
• In the given figure, O is the centre of the circle, ∠ACB=54∘ and BCF is a straight line. Find x.
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numerical problem
Taking force, length, and time as fundamental quantities, find the dimensional formula for the density.
Derive the dimensions of the specific heat capacity of a substance.
The density of gold is 19.3 gm/cc. Express its value in the SI unit.
Write the dimensional formula of gravitational constant and latent heat.
Check the correctness of the relation, \(h=\frac{2T\cos\left(\theta\right)}{r\rho g}\), where symbols have usual meaning.
Check the correctness of the formula, PV = RT using the dimensional method.
Check dimensionally the correctness of Stoke's formula, F = 6πηrv
A student writes an expression of the momentum (p) of a body of mass (m) with total Energy (E) and considers the duration of the time (t) as \(p=\sqrt{2m\frac Et}\). Check its correctness by using
dimensional analysis.
The energy of a photon is given by E = hf. Find the dimension and unit of the plank's constant, where f is the frequency of radiation.
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Balance of forces
07 Nov
As mentioned at the end of the last MathStream post, the actual shape that the six-strut tensegrity structure takes on is close to, but not quite precisely, a regular icosahedron. And that fact
immediately makes you want to build a tensegrity structure that will under ideal circumstances assume the shape of a truly regular icosahedron. What would that entail? Why is the classic tensegrity
not a regular icosahedron? One possibility that immediately comes to mind is that not all of the edges of the icosahedron are represented in the same physical way in the model. Namely, as you may
recall from this diagram,
there is no rubber band lying along some of the edges of the icosahedron. To test if that’s the reason that the six-strut is not regular, we’d want a different arrangement of elements so that there
will be exactly the same tension on a rubber-band connection between every closest pair of endpoints of the six struts, as shown in this diagram.
So, you might immediately start tying to build such a configuration. But that will bounce you right back to a mathematical question: how can we route the rubber bands so that there is exactly the
same amount of rubber band between every closest-neighbor pair of endpoints? In fact, one of the wonderful things that happens at Studio Infinity is that many of the things we try to build lead to
new mathematical questions, and many of the mathematical discoveries we encounter suggest new things to build. So the MakeStream and MathStream really build on each other.
Getting back to the question at hand, it amounts to finding a route for each of several rubber bands so that the route of each band traverses the same number of edges of the icosahedron we are trying
to achieve, and so that every edge is covered. Since the icosahedron has thirty edges, that immediately narrows down the possible number of rubber bands we might use. For example, there might be ten
rubber bands each covering three edges, or six rubber bands each covering five edges, or five rubber bands each covering six edges, and so on. And since each rubber band is a loop, the edges covered
will have to form a loop on the surface of the icosahedron. For example, there are fairly obvious loops one could make with three edges (in blue) or with five edges (in red) in this diagram.
So you can just start trying to cover the edges with loops of the same length. In fact, I recommend that you give it a try right now, maybe using the six-strut model you’ve made, or the diagram
above. See if you can find a way to route rubber bands to cover every edge, so that every loop is the same length, before you read further.
After a while of trying, you may start to get discouraged. You might start to feel that this is an impossible task. And that would not be too surprising, because it is impossible. But when we make a
statement like that in mathematics, we have to back it up. If you want to claim that something is impossible, you can’t just list the seventeen things you tried that didn’t work. You’ve got to find
reasons why no possible routing of rubber bands, no matter how clever, will cover all of the edges and have every loop be the same length.
And the key to that in this case is even and odd. First count how many edges meet at every vertex: five, an odd number. Next count how many of those edges rubber bands can cover. Well, every rubber
band is a loop, so it must arrive at the vertex by some edge, and then leave that vertex by a different edge. It might later come back to that vertex by yet another edge, but if so, it has to leave
again by a still further edge, in order to form a single closed loop overall. So in short any one rubber band covers an even number of edges touching the vertex. And since the sum of a collection of
even numbers is even, we can be sure that the rubber bands are covering an even number of edges touching the vertex.
And so now we run into the problem. An even number cannot be equal to an odd number, so we can’t possibly have covered all of the edges. So there in fact is no way to route the rubber bands as
So we regretfully conclude that there is no way to construct the regular icosahedron we are looking for, right? Not quite. Another habit in mathematics is, when something doesn’t work out the way you
thought it might, to see if there’s a way to change the assumptions so that it does work out. And in this case, there is something we can change: rather than providing the tension between two
neighboring endpoints by virtue of a single segment of rubber band, why couldn’t there be exactly two different (but identical in length) segments of rubber band covering each edge? Then there would
be ten different segments of rubber band reaching each vertex, which is an even number, and the contradiction underlying our impossibility argument would melt away in a puff of logic.
And now we can very pleasantly notice that there are exactly twelve of the five-edge pentagonal paths similar to the one highlighted in red in the last diagram above, and that each edge lies on
exactly two of these pentagonal paths. So now the plan is simple: connect one rubber band along each pentagon, and the icosahedron should simply materialize from the balance of forces. Let’s return
to the MakeStream to see if it works…
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In comparing probability distributions, which of the
following statements is TRUE?
Select one:
1. The geometric...
1. False, Among all the continuous distributions only the exponential distributions have the lack.of memory property.
2. True, Poisson distribution deals with the number of occurrences in a fixed period of time, while the exponential distribution deals with the time between occurrences of successive events as time
flows by continuously. So, there's a relationship here. And also the random variable used is same.
3. True, the geometric distribution is a special case of negative binomial distribution both have random number of trails. In negative binomial we have a count of successes from repeated counts in a
experiment while geometric we repeat number of trails for a single success.
4. False, Normal distribution have the bell shape curve and the uniform distribution have the rectangular distribution
5. True, Poisson distribution can be expressed as a limiting case of negative binomial distribution. In a series of trails for success or failure then a parameter is introduced to indicate the number
of failures that stops the count.
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Bitcoin Mining : Everything You Need to Know
Bitcoin mining is trending ever since it was invented.
Earlier it was only trending on Google, now it has started trending among the people.
If you are new to the Bitcoin mining and looking to start, you might be wondering-
• What is Bitcoin?
• What is Bitcoin mining?
• How does Bitcoin Mining work?
• What is Proof of Work?
• What is Bitcoin Mining Difficulty?
But, before you begin understanding all of these, you need to learn each and every term that relates to Bitcoin. We have made a beginners’ guide on the same and it will take hardly 5 minutes of yours
to understand-
• The definition of Bitcoin
• What is a Bitcoin Address?
• The definition of a Public Key
• What is a Private Key?
• How about the Blockchain? What is it?
• What is Bitcoin Mining?
• The bitcoin rewards
• A digital Bitcoin wallet
So, if you are done with the 5-minute beginners’ guide to understanding the terms related to Bitcoin, you are all set to go one level up.
Let’s begin.
What is Bitcoin?
Before we begin to understand the Bitcoin mining process, let us help you brush up your knowledge about the Bitcoin and its mining.
Here is what a Bitcoin is in the layman’s language-
What are Nodes?
A node is a computer that executes the program of Bitcoin and is connected to the Bitcoin network.
And, the nodes (or computers) need to be powerful and intelligent enough to understand the requirements, takes their own decisions, and verify the transactions it receives based on the predefined
If the transactions are not as per the predefined rules, they are not passed on to other nodes in the network.
Once, the verification is confirmed, the nodes share the two types of transactions to other nodes in the Bitcoin network-
1. Fresh transactions – the ones that are recently added in the network
2. Confirmed transactions – the ones that are ‘confirmed’ and written to a file
Every time, a node receives the confirmed transactions, they keep blocks of ‘confirmed’ transactions. They are placed together in a file or ledger called, Blockchain.
A copy of blockchain is kept with each node of the network for security purposes. If any node does not have an up-to-date copy, it asks the other nodes for the updated copy.
Now, what happens to the fresh transactions?
The fresh transactions are again sent to the network until they reach a stage of ‘confirmed’ transactions. And, this process of bringing fresh transactions to the blockchain is called mining.
How does Bitcoin Mining work?
And, here is what the Bitcoin Mining is and how does it work-
Now, you know what is Bitcoin, nodes, Bitcoin mining and how its mining process takes place.
Based on the image above, you can only create new Bitcoins by solving a computerized mathematics puzzle. And, that is executed by the nodes or the miners (mining nodes).
The mathematical puzzle is all about finding a number. That number is within a certain range. The data available in the blocks need to be combined and passed through a hash function. And, this
process helps in solving that puzzle by finding that number. Well, this is not at all easy.
So, understand why it is made difficult to crack.
Bitcoin mining is made resource-cornered and difficult only to maintain the stability of the number of blocks being mined by the miners.
Now you know- Bitcoin mining is used for two purposes-
1. Introducing new Bitcoins in a decentralized manner
2. Ensuring only the secure transactions taking place
Now, the question is – how the nodes find that number/solves the puzzle?
To solve this mathematical puzzle, the miners (mining nodes) just need to guess that random number. But, the hash function does not allow miners to guess the output easily.
So, miners need to combine the number they have guessed and the data that is available in the block with the help of the hash function. And, the result of this hash function is another hash that
starts with some zeroes.
Each combination has a different result and hence, it is impossible to guess which number will work.
The miner that is first able to reach to a particular range of the desired number, announces itself to be the winner in the Bitcoin network.
So, the other miners now try their luck and efforts to find another number in the queue.
The winning miner receives the reward as the new Bitcoins. This reward keeps on changing as the rate of the reward changes.
What is the Bitcoin Network Difficulty Metric?
The Bitcoin Network Difficulty is a metric that is used to measure the difficulty ratio to find a new block.
It measures how much difficult it is to find a new block when compared to how easy it can ever be.
This metric is calculated at every 2016 blocks based on the time the nodes took to find the previous 2016 blocks. The desired rate of each block is 10 minutes and it takes two weeks to calculate 2016
So, if the previous 2016 blocks took more than two weeks of time, the difficulty ratio is decreased. And, if it took less than two weeks of time, the difficulty ratio is increased.
The blocks released that do not meet the required difficulty rate are rejected by all the miners available in the network and hence, will be of no worth. That means each miner has to make sure the
blocks are released as per the difficulty ratio.
What is the Block Reward?
Like we said earlier, that in our guide, we have explained all of the terms related to the Bitcoin and here, it is about the block reward-
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Roots - Mindtec
In high school and college algebra, you may have learned about roots, or rhizome, as some students called them. Roots are a basic concept in many sciences and engineering fields, and are a key
ingredient in the formulation of many mathematical functions. In algebra, for instance, if a polynomial equation has been written using it as the x-shapes of the variables, then an x-intercept can be
calculated by finding the roots of the polynomial. This is an extremely important subject for all students and should be taught to students early in their algebra lessons.
Roots are not just in nature; they occur in plants, animals, and humans as well. For example, consider the roots of a tree; when the lower branches reach the ground, they are called roots; the upper
most branches are known as stem. In mathematics, a zero of any real-shaped, complex-valued, or in general vector-valued function f is a homogeneous member of the complex domain of f for which f(x}
vanishes; this is the function f evaluates the value of x at x. The value of a function at any point is the value of the corresponding function at that point in the complex plane.
Students learn about roots by learning their properties, like volume and surface area, and their uses, such as when finding a solution for a mathematical problem involving integration,
multiplication, division, and graphing. The topic also includes roots of expressions, such as sin, cos, tan, and bract, and their definition and uses. Understanding roots is essential to solving
problems, and also to having clear and concise reasoning.
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Algorithmic Redistricting and Black Representation in US Elections
In the United States, the careful crafting of electoral districts has been a powerful tool for politicians to limit groups’ political power or exclude them from representation entirely, most
prominently to the detriment of political parties and racial and ethnic minority communities. Beginning in the 1960s, experts began proposing algorithmic solutions to the redistricting problem, in
which a “neutral” computer program could draw “fair” districts free of human influence. Despite the traction that these proposals achieved both in academic and popular discourse, little work has been
done to understand the extent to which algorithmically drawn districts do or do not comport with notions of fairness and equity. In this work, we perform such an analysis, running several proposed
algorithms to generate districts in Alabama and Michigan. We observe that in both of these states, all four algorithms generate plans that provide fewer districts where Black voters would be expected
to decide the outcome of the election, relative to both the proportion of Black people in their populaces as well as to the number of Black opportunity districts in the plans actually enacted by
their state legislatures. We conclude with some discussion about the role of algorithms in redistricting moving forward, and how these tools might be used to enhance, rather than restrict, the
ability for various communities to achieve elected representation.
Keywords: redistricting, algorithms, race, politics, elections
Zachary Schutzman
Institute for Data, Systems, and Society, MIT
Learning Objectives
• Trace contemporary discourse and issues in redistricting to civil rights–era problems and responses.
• Identify arguments for and against the use of computerized algorithms in the redistricting context.
• Discuss similarities and differences between several redistricting algorithms.
• Interpret the output of these algorithms, in particular along the dimension of Black representation.
The practice of gerrymandering, by which politicians intentionally manipulate the boundaries of electoral districts to help allies or hurt rivals who are trying to achieve political representation,
is a practice as old as the United States itself. Over the centuries, abuse of the redistricting process has been used to help or hurt political parties, racial and ethnic groups, and individual
representatives in their quest for political representation in legislative bodies from the US Congress to local municipal and county governments. Drawing and redrawing maps of electoral
districts—redistricting—became a tool with which members of various communities could be prevented from reliably electing their preferred candidates. Those electoral patterns, in turn, limited the
ability of such communities to effectively lobby governments to respond to their interests; prevented representatives from these communities from ascending the political ranks; and, more generally,
has run counter to the ideals of free and open democratic elections.
Watch 📺: Landmark case of the United States Supreme Court concerning partisan gerrymandering (2018).
Listen 🎧: Rucho v. Common Cause Oral Argument
Rucho v. Common Cause Oral Argument
Justices ruled that the Supreme Court can't set a constitutional standard to prevent partisan gerrymandering. Source: Supreme Court of the United States
Before the 1960s, states and localities had an enormous amount of leeway in how they designed and implemented districting plans, particularly state legislative districts, and through the 1940s, the
Supreme Court affirmed this right. Many states neglected to redistrict at all during the first half of the twentieth century, even as residents moved from rural regions to urban areas. According to
Representative Morris Udall (D., Arizona), at the time the Supreme Court intervened in the early 1960s to demand that congressional and state legislative districts contain nearly equal numbers of
people, there were some state legislative chambers where the largest district had hundreds or even thousands of times more voters than the smallest one.
Given this history of malapportionment, in the early 1960s advocates for “fair” redistricting began to propose the use of computer algorithms to generate districting plans. In 1961, economist William
Vickrey proposed a framework in which an algorithm, given only the most basic data needed to draw districts that are geographically connected and balanced in population, could use opaque processes
and randomness to separate the construction of districting plans from human influences entirely. Almost immediately after Vickrey’s proposal, researchers and scientists designed theoretical and
practical algorithms to achieve this, following a general framework of building algorithms to draw districts that maximized some notion of “compactness,” a family of measures that describe the
geometric regularity of a district, subject to the constraints of geographic connectedness and balanced population.
The foremost abuse that Vickrey and others sought to reform was the practice of allowing districts that were nowhere close to being equal in population; they urged instead that balanced populations
be designed as a hard constraint within the new redistricting algorithms. However, despite these being very real and salient issues at the time, the case law that emerged in the early 1960s has
created strong guardrails around population imbalance, and that kind of malapportionment is no longer a widely used technique to abuse the line-drawing process. Instead, the discussion of
gerrymandering and unfairness in redistricting now revolves around racial and partisan inequities, preserving political units like municipalities and counties, and issues of incumbency. Despite these
changes, the classical framework of algorithms, which optimize for compactness subject to population balance and geographic connectedness, still persists.
Every so often, an academic, politician, pundit, or concerned citizen will take to a social media platform to confidently assert that the only fair way to draw districting plans is via the use of a
computer algorithm (Figure 1).
Figure 1
A fake tweet inspired by a real tweet.
Every time it comes up, this idea garners a good deal of enthusiastic support. People chime in with suggestions of what kinds of data the algorithm should and should not have access to, along with
pictures of districting plans drawn using their personal favorite algorithm. However, what these well-meaning proposals often don’t grapple with is the impact of using algorithmically drawn districts
on the constituent communities in the jurisdiction. One such concern, which is the focus on this work, is that such plans systematically reduce the ability for Black voters to exercise sufficient
political power to elect candidates of choice.
Arguments by today’s proponents of algorithm-drawn districting plans generally fall under a few separate headings. The first, in the same vein as Vickrey’s rationale, is that “fairness in
redistricting” ought to be a descriptor of the process rather than the outcome. That is, a “fair districting plan” is one that results from a methodology for constructing those districts that adheres
to generally acceptable principles, irrespective of the electoral outcomes resulting from those districts. As Vickrey argued in his 1961 article,
This means, in view of the subtle possibilities for favoritism, that the human element must be removed as completely as possible from the redistricting process. In part, this means that the
process should be completely mechanical, so that once set up, there is no room at all for human choice. More than this, [...] it should not be possible to predict in any detail the outcome of the
Elements of this argument have been challenged from the computational perspective in previous work. In brief, because communities with shared interests aren’t geographically organized according to a
mechanical process, there’s no reason to expect a mechanical process to respect community structure or organization. Communities are structured according to natural boundaries like rivers and
mountains, constructed boundaries like highways and state lines, and historical boundaries like segregation and discriminatory housing and lending practices (including “redlining”). All of these are
challenging to describe in a redistricting process that is both amenable to computer implementation as well as free from “human choice.”
The second argument acknowledges that algorithmic districting plans may lead to less than ideal outcomes, but posits that given the level of abuse in the current system—in which politicians and
political actors design the redistricting maps—any departure from current practices (including algorithmic ones) will lead to better outcomes. Consider, for example, this recent comment on Twitter by
a journalist who covers the US House of Representatives:
[Y]es, it’s tough to reconcile algorithmic redistricting methods [with] the VRA [Voting Rights Act].
But don’t underestimate the number of current maps that short-change opportunities for minorities [because] they over-pack minority voters into maj-min [majority-minority] VRA districts.
(As described following, the US Congress passed the Voting Rights Act in 1965, some provisions of which prohibit election practices that would systematically disenfranchise members of various racial
or ethnic groups.) This case study challenges the argument that any computational approach to redistricting would yield improvements over present-day patterns. It is undeniable that many states,
counties, and cities have abused (and continue to abuse) the power of the line-drawing process to systematically disempower and disenfranchise communities, in particular racial and ethnic
communities. Yet we shouldn’t accept without evidence the assertion that current distinct maps achieve less representation for the groups covered by the Voting Rights Act than those drawn by an
More generally, advocates for algorithmic redistricting often argue that so-called neutral algorithms will generate districting plans that are “fair” if the algorithms do not incorporate specific
kinds of data. For example, if algorithms do not include data such as the partisan composition of a region or the geographic distribution of people of various races or ethnic backgrounds, then (the
argument goes) the algorithm cannot generate a plan that is unfair on a partisan or racial dimension. This argument, however, is unsupported by any mathematical or experimental evidence. Indeed,
research has amply demonstrated in other computational contexts that simply omitting a variable from an algorithm or model does not guarantee that the output will be uncorrelated with that
variable—because race and ethnicity are so often correlated with variables such as household income or residential neighborhood. Indeed, our results show that various algorithm-generated districting
plans clearly and consistently yield districting maps that weaken Black electoral opportunities compared with the enacted plans currently in use as well as with what a standard of proportionality
might demand.
In this study we focus on the extent to which various algorithms proposed by researchers and experts do or do not draw plans that contain districts that allow Black voters the opportunity to elect
candidates-of-choice, or opportunity districts. The study focuses on two US states with significant but geographically very different Black populations: Alabama and Michigan. Before diving into the
computational results, it is important to define some terms and give some historical context for drawing opportunity districts for racial groups.
“Opportunity Districts” and “Candidates of Choice”
The notion of “opportunity districts” and how to draw them is complex, arising from a fusion of Supreme Court case law and federal civil rights legislation. In the mid-1950s, amid the civil rights
movement, the US Congress passed the Civil Rights Act of 1957. The provisions in this law made it much easier for Black citizens, particularly in southern states, to register to vote. In the wake of
the 1957 law, the Alabama state legislature quickly moved to redefine the municipal boundaries of the city of Tuskegee, a square-shaped small city of about seven thousand people, approximately 80
percent of whom were Black. After redefining the boundaries, what remained of Tuskegee was a bizarre twenty-eight-sided figure with a population around fifteen hundred people, essentially all of whom
were white; the state legislature had excised virtually all of the Black residents and exactly none of the white residents from the city.
Gomillion v. Lightfoot Case Brief Summary
Privacy-enhanced mode enabled.
In Gomillion v. Lightfoot (1960), the Supreme Court found that Alabama’s legislature had acted specifically and intentionally to prevent Black voters from being able to participate in municipal
government elections, and that the resulting disenfranchisement (via redefinition of municipal boundaries) was a violation of the Fifteenth Amendment. This is generally considered to be the first
court case on racial gerrymandering and was the first time the Supreme Court stepped in to limit a redistricting-like process. Preventing disenfranchisement via redefinition of electoral district
lines was codified in the Voting Rights Act of 1965, with some clarification and updates in the amendments to that law passed in 1982. Section 2 of the Voting Rights Act prohibits designing election
laws, including the drawing of electoral districts, in such a way that members of a racial, ethnic, or language group “have less opportunity than other members of the electorate to participate in the
political process and to elect representatives of their choice.” Several court cases, a notable few including Beer v. US (1976), Thornburg v. Gingles (1986), the Shaw cases (1993, 1996, 1999, 2001)
in North Carolina, and Bush v. Vera (1996), fleshed out mechanisms and limitations for identifying, challenging, and remediating racial gerrymandering.
There is a meaningful distinction between an opportunity district as defined by the Voting Rights Act and related case law, a so-called majority-minority district, and a district that reliably elects
a member of a particular community as its representative. An opportunity district is simply one in which members of a particular community that votes cohesively can reliably decide who the elected
individual will be. This group need not constitute anywhere near a majority in order to exercise this power (as in a majority-minority district), nor does the elected representative need to be a
member of that community.
In many districts, party primaries are competitive, whereas general elections are not. For example, in a district that reliably votes 60 percent Democratic and 40 percent Republican in general
elections, the Democratic primary often serves as the de facto contest that decides the general election result. If such a district is a region in which Black voters vote cohesively, then
constituting a majority of the Democratic primary electorate is sufficient to give them the power to elect a candidate of choice, even while the district as a whole may only be 35 percent Black. For
example, white Democrats in Detroit, Michigan, reliably support the candidate who wins the Democratic primary, so a Black opportunity district is one in which Black voters, acting cohesively, have
the political power to decide the results of the Democratic primary in that district.
Candidate of choice is not synonymous with a candidate who happens to be a member of a particular racial or ethnic group. The issue at hand is whether members of some group have the level of
political efficacy to influence district elections, not whether the elected representative from the district comes from any particular demographic group. In many cases, racial, ethnic, and language
groups’ candidate of choice is a member of that particular group, but not always. For example, Steve Cohen, a white Democrat, represents a solidly Democratic congressional district in Memphis,
Tennessee, which is about two-thirds Black; Cohen has consistently won primaries in his district by overwhelming margins.
Even with sixty years of work designing and implementing redistricting algorithms, little attention has been paid to examining and comparing the districting plans that various algorithms generate
along dimensions such as how well the resulting maps afford various communities the ability to elect their candidate of choice. In this work, we compare the outputs from four recently proposed
redistricting algorithms and evaluate the impacts that each algorithm would have on the distribution of Black opportunity districts in Alabama and Michigan.
Each algorithm draws connected population-balanced districts composed of census blocks. In order to run the algorithms themselves, we only need some publicly available data from the US Census Bureau,
which provides files containing both the geography of the census blocks as well as demographic data for each block. This demographic data includes the total population of each block; the Black
population of each block; and the Black population over the voting age of eighteen. The block geography and demographics come from the 2010 decennial census data, which was used as the official
redistricting data prior to the recent release of the 2020 census data.
We consider two states, Alabama and Michigan, and use the algorithms to draw state senate districts for each state. According to the 2010 data, Alabama has a population around 4.73 million people and
is 27 percent Black. The state has thirty-five state senate districts with approximately 136,000 people per district. Figure 2a shows the geographic distribution of Black people in Alabama. The
cities of Birmingham and Montgomery in the center of the state and Mobile in the southwest are majority-Black cities of approximately 200,000 people each. The wide strip across the center of the
state, sometimes called the “Black Belt,” contains Montgomery as well as many smaller predominantly Black cities and a large rural Black population. In aggregate, this region is over 50 percent
The 2010 population of Michigan is approximately 9.93 million and is 14 percent Black. The state has thirty-eight state senate districts, each containing around 260,000 people. Figure 2b shows the
geographic distribution of Black people in Michigan. In stark contrast to Alabama, the Black population is overwhelmingly concentrated in and around the city of Detroit, in the southeast part of the
state. Wayne County, which includes Detroit, has a total population of roughly 1.8 million people, 33 percent of whom are Black. There is a sizeable Black community in the city of Flint in the center
of the state, but the population is relatively small compared to that of Detroit, where approximately 80,000 of the 420,000 residents of Genesee County are Black. The remainder of the state is
overwhelmingly white.
Figure 2
The Black population distributions for Alabama (a) and Michigan (b). The lighter, yellow regions have more Black residents.
This analysis considers redistricting plans for state senate rather than the more widely discussed US congressional districts in order to generate a richer set of outcomes. For instance, according to
the 2010 reapportionment, Alabama is afforded seven members in the US House of Representatives. Its current districting plan includes one Black opportunity district, and experts and stakeholders have
argued and sued for the state to redraw the plan to include a second. The algorithms typically generate zero or one Black opportunity districts for US congressional elections (and, on rare occasions,
two). In contrast, the enacted state senate plan includes eight Black opportunity districts, and drawing as many as eleven or twelve may be possible. Working at the level of state senate districts
therefore allows for a wider range of possible outcomes, and hence a more careful analysis of the behavior of the algorithms.
In Alabama, while there is a small portion of white voters who might reliably support the Black-preferred candidate, white voters in Alabama overwhelmingly support Republicans. Experts estimated this
level of white support for Black-preferred candidates in the center of the state to be approximately 17 percent. Therefore, whereas a district need not be strictly majority Black in order to reliably
elect the Black voters’ candidate of choice, it must come close. More precisely, we may define a “Black opportunity district” within Alabama to be one in which 95 percent of the Black voting-age
population, together with 17 percent of the white population, constitute a clear majority in the district.
Unlike Alabama, Michigan includes a much larger proportion of white voters who typically vote for the Democratic Party. Estimates based on the 2016 US presidential election and the 2018 US Senate
election in Michigan place the proportion of white Democrats around 45 percent. For this reason, if Black voters’ candidate-of-choice wins the Democratic primary in a reliably Democratic district, we
can expect white Democrats to support this candidate in the general election. In this case, the strength of Black voters in the primary determines whether a district is or is not a Black opportunity
district. In Michigan, we use election data to estimate the partisan lean of a district, and we consider one to be a “Black opportunity district” if it both reliably will elect a Democratic candidate
in the general election and Black voters constitute a majority of the estimated Democratic primary electorate.
In this section we briefly describe the four algorithms under consideration. At a high level, each of these algorithms is designed to take as input a collection of geographic units, such as census
blocks, each equipped with a total population. They then solve a geometric optimization problem to draw districts that are connected, nearly equal in population, and as compact as possible, where the
definition of compactness is either explicitly or implicitly encoded in the objective of the optimization problem.
Annealing. The first algorithm, which we call Annealing, was designed by a software engineer and its source code is available publicly online. It is well-discussed in public discourse, having
featured in a Washington Post article and used in FiveThirtyEight’s Atlas of Redistricting project. The algorithm works by choosing random district centers and assigning each census block to its
closest center. Districts that are underpopulated incorporate the nearest blocks from districts that are overpopulated, and then the centers are recomputed. This annealing process of grabbing blocks
repeats until the districts are population-balanced. The entire process, starting from new random centers, is repeated numerous times, and the plan that generates the most compact districts is
returned as the final output.
Arcs. We call the second algorithm Arcs due to its unique characteristic of drawing districts bounded by circular arcs. This algorithm appears in a recent paper, and although its authors drew
inspiration and lessons from several existing algorithms, the resulting districts are highly unlike those of any other algorithm. To draw $k$ districts, the algorithm works by selecting a corner of
the bounding box of the state and drawing the circular arc centered there which splits the state into two pieces which support $\lceil\frac{k}{2}\rceil$ and $\lfloor\frac{k}{2}\rfloor$ districts,
respectively. (Here $\left\lceil \frac{k}{2} \right\rceil$ is the “ceiling,” meaning the smallest integer greater than or equal to k/2, whereas $\lfloor\frac{k}{2}\rfloor$ is the “floor,” that is,
the largest integer less than or equal to k/2.) Then the algorithm is run recursively on each half, selecting a corner of the bounding box for each. The sequence of bounding box corners that is
ultimately selected is the one that maximizes the compactness of the districting plan.
Voronoi. Algorithms using Voronoi diagrams and related procedures to generate districting plans have been proposed since at least the mid-2000s. We consider the most recent iteration in this line of
work, which uses a generalization of Voronoi diagrams called power diagrams to partition a state into districts. The algorithm to draw $k$ districts works as follows: It begins by selecting $k$
random points c[1], ... , c[k] to serve as the initial centers of the districts, and initializes an associated radius r[1], ..., r[k] to each one. We can then imagine $k$ circles, each centered at c[
i] with radius r[i]. We imagine the set of census blocks as little regions in an x-y plane and feed to the algorithm a representation of each census block as the single point at its geographic
center, to which we associate that block’s population. Every one of these points is some distance from the boundary of every one of these circles, and the algorithm assigns each block to belong to
the district corresponding to the closest of these circles.
There is no guarantee that these proto-districts will be anywhere close to population-balanced, so the algorithm next performs an adjustment procedure, until the proposed set of districts becomes
population-balanced. First the algorithm recomputes the centers c[1], ... , c[k] to be the centers of their respective proto-districts and leaves the radii r[i] alone. Then it assigns each block to
the center whose circle’s boundary is the nearest. Next it adjusts the radii while leaving the centers fixed. By solving an optimization problem, the algorithm can shrink or grow the radius of each
proto-district’s circle to shrink or grow the population assigned to it.
Eventually, this algorithm will converge to a set of centers c[i ]and radii r[i] that identify a collection of population-balanced districts. When thought of as collections of points, these districts
are polygons, bounded by straight line segments. The algorithm also takes into account the geometry of these districts, such as the number of sides of any one of these polygons not being too large,
and requiring every polygon to be convex except at the border of the state. When these assignments are mapped back onto census blocks, the boundaries become uneven, because census blocks do not
follow perfectly straight lines. Due to the generally small size of census blocks, however, the districts may appear to have straight-line borders until one zooms in to a finer resolution.
Tree. The final algorithm we consider is a highly randomized one that appears in the software package GerryChain, a suite of algorithms used to generate large ensembles of districting plans with
which one may compare a proposed or enacted plan. Whereas the Voronoi algorithm conceptualized a state’s census blocks as a collection of points, the Tree algorithm treats census blocks as components
of a graph. Each census block becomes a vertex in this graph, with an edge between two vertices if and only if the vertices’ corresponding census blocks share a geographic boundary. Each block’s
population is associated to its respective vertex. Whereas in the Voronoi algorithm the coordinate representation of the blocks was crucial information used by the computer to assign each block to
the closest district center, the Tree algorithm discards such that geometric information and works with the graph representation.
The algorithm first constructs a random spanning tree of this graph. This can be thought of as a subcollection of the graph’s edges for which restricting our view to the original set of vertices and
this smaller set of edges results in a graph that is connected and contains no cycles. A crucial feature is that deleting any single edge from a spanning tree separates it into two connected
Once the algorithm has drawn a random spanning tree, it searches for an edge that, when removed, divides the vertices into a component which has the correct population for a single district and a
component that has the correct population for k - 1 districts. If the algorithm can identify such an edge, it performs this separation and freezes the smaller component as the first district. If it
cannot identify such an edge, the spanning tree is discarded and a new one constructed.
The algorithm then proceeds recursively on the larger section and ignores what has already been frozen. After freezing the first district, it draws a random spanning tree and searches for an edge
that, when removed, separates the component of interest into a district-sized piece and a piece with population appropriate for k - 2 districts. This procedure continues until the algorithm has
frozen all k districts, and this becomes the output plan.
The districts drawn by this procedure tend to look more “organic” than those drawn by the other algorithms. This arises in part from the fact that the Tree algorithm does not incorporate the shape of
the state or the geographic distances between blocks. Formal statements about the kinds of districts that this algorithm draws are difficult to make, but we can observe that generally it tends toward
drawing districts that have many possible spanning trees underlying them. This in turn means that we should expect the graph representation of the constituent districts to contain many cycles. This
is potentially an attractive property, because the kinds of districts whose graphs contain very few cycles are ones with spindly tendrils extending in disparate directions.
Alabama. Figure 3 shows the presently enacted districts for Alabama as well as the algorithmically generated plans. We can see that visually all of these plans are quite distinct, reflecting the
differences in specification between the algorithms. Using the benchmark of 95 percent of Black voting-age population plus 17 percent of white voting-age population exceeding 50 percent, we can
examine how many Black opportunity districts appear in each plan, the geographic location of these districts, and how reliably they could be expected to elect Black voters’ candidate-of-choice (
Figure 4).
Figure 3
The enacted state senate districts in Alabama and the four algorithmically generated plans. Pink districts have at least 55 percent expected support for the Black-preferred candidate; yellow
districts have between 50 and 55 percent expected support. (a) The currently enacted state senate districts as of 2020. (b) Output from the “Annealing” algorithm. (c) Output from the “Arcs”
algorithm. (d) Output from the “Voronoi” algorithm. (e) Output from the “Tree” algorithm.
Figure 4
Plots of the expected support for the Black-preferred candidate in the fourteen Alabama districts that have the largest Black populations. Sorted districts appear along the horizontal axis, and
expected fraction of votes for the Black-preferred candidate appears along the vertical axis. (a) The currently enacted state senate districts as of 2020. (b) Output from the “Annealing” algorithm.
(c) Output from the “Arcs” algorithm. (d) Output from the “Voronoi” algorithm. (e) Output from the “Tree” algorithm. (f) All five plots superimposed.
In the enacted plan (Figure 3a and Figure 4a), there are eight districts in which the expected support for Black voters’ preferred candidate exceeds 50 percent. Three of these districts are anchored
in Birmingham, one each in Mobile and Montgomery, and three across the more rural central region of the state. We can plot the expected support for the Black-preferred candidate in these eight
districts as well as the four districts with next-largest Black population, and observe a clear demarcation between these eight and the next four (as well as the remaining twenty-three districts, not
shown): the eighth district has over 55 percent expected support for the Black-preferred candidate, whereas the ninth district has well below 40 percent expected support.
In the Annealing plan (Figure 3b and Figure 4b), we see a stark difference. This plan contains only two districts with expected support for the Black-preferred candidate over 55 percent, and an
additional three districts in which the expected support falls between 50 and 55 percent. These five districts include two in Birmingham, one in Mobile, one on the periphery of Montgomery, and one
large rural one in the west, and it is straightforward to see how this plan “misses” several of the opportunity districts in the enacted plan.
Plotting the expected support from the twelve districts with the highest Black population, we can see that the dramatic gap between the eighth and ninth districts in the enacted plan does not appear
in the Annealing plan. Rather, there is a more gradual decline in expected support for the Black-preferred candidate from the second district downward. Relative to the enacted plan, the high-support
districts underperform and the low-support districts overperform, suggesting that this plan would significantly dilute Black voting strength.
For the Arcs plan (Figure 3c and Figure 4c), we see something similar. This algorithm again draws five potential opportunity districts, with four expected to fall above 55 percent support for the
Black-preferred candidate and one in the intermediate range of 50 to 55 percent. This algorithm finds different opportunity districts than the ones identified by the Annealing algorithm. Here we have
again two around Birmingham and one rural one in the west, but two additional large rural districts in the middle of the state.
Once again, we can examine the plot of expected support for the Black-preferred candidate by district, and again we observe a steady decline in expected support—more like the Annealing plan than the
abrupt gap present in the enacted plan. This finding suggests that the Arcs plan, like the Annealing one, would dilute Black voting strength compared to the current districting arrangement. The
Voronoi plan (Figure 3d and Figure 4d) performs similarly to the Annealing plan, but with an extra district in the 50 to 55 percent range around Montgomery and slightly greater expected support for
the Black-preferred candidate in the rural district in the west. As for the previous algorithms, the plot of expected support indicates dilution of Black voting power.
The Tree algorithm draws a plan (Figure 3e and Figure 4e) with four districts above the 55 percent expected-support level and none in the 50 to 55 percent range, which is a departure from the
previous outputs. It also does not draw any rural opportunity districts, identifying only two such districts in Birmingham and one each in Montgomery and Mobile. Plotting the expected support level
shows, however, that the effects of this plan are not very different from the other three algorithmic ones; it, too, features a steady decline in expected support, with a small gap between the fourth
and fifth districts. However, we still see districts falling just below the 50 percent line, indicating that Black voters in these districts might be narrowly shut out of political power—an outcome
of vote dilution common in each of these algorithmically generated plans.
Michigan. As for Alabama, the four algorithmic plans generate district maps for Michigan that are visually distinct from each other and from the enacted plan, which was designed in part to follow the
rectangular county boundaries of the state. As noted above, Michigan does not have the same level of racially polarized voting as Alabama in the general electorate, which is reflected in the
determination of Black opportunity districts. In Michigan, approximately 90 percent of Black voters and 45 percent of white voters supported Democratic Party candidates in recent statewide general
elections. This means that in a reliably Democratic district, if Black voters’ preferred candidate wins the Democratic primary, that candidate is highly likely to win the general election as well,
regardless of any racially polarized voting patterns in the primary.
To assess whether or not a district in Michigan provides Black voters the opportunity to elect a candidate of choice, the district must both be a reliably Democratic district and one in which Black
voters are expected to comprise a majority of the Democratic primary electorate. To perform these estimates we use precinct-level election data from the 2016 US presidential and 2018 US Senate
elections in Michigan. Quantitative results differ slightly depending on whether one uses data from the presidential or senate elections, though the results remain very similar. In this section we
present results that made use of the presidential election data, and present the corresponding results for the senate election case in a brief appendix. (See Figure 5, Figure 6, and Figure 7.)
Figure 5
The enacted state senate districts in Michigan and the four algorithmically generated plans. Pink districts have at least 55 percent expected support for the Black-preferred candidate; yellow
districts have between 50 and 55 percent expected support. (a) The currently enacted state senate districts as of 2020. (b) Output from the “Annealing” algorithm. (c) Output from the “Arcs”
algorithm. (d) Output from the “Voronoi” algorithm. (e) Output from the “Tree” algorithm.
Figure 6
Insets of the Detroit region for the plans shown in figure 5. (a) The currently enacted state senate districts as of 2020. (b) Output from the “Annealing” algorithm. (c) Output from the “Arcs”
algorithm. (d) Output from the “Voronoi” algorithm. (e) Output from the “Tree” algorithm.
Figure 7
Plots of the expected support for the Black-preferred candidate in the 10 Michigan districts that have the largest Black populations. Sorted districts appear along the horizontal axis, and expected
fraction of votes for the Black-preferred candidate appears along the vertical axis. (a) The currently enacted state senate districts as of 2020. (b) Output from the “Annealing” algorithm. (c) Output
from the “Arcs” algorithm. (d) Output from the “Voronoi” algorithm. (e) Output from the “Tree” algorithm. (f) All five plots superimposed.
The enacted plan for Michigan (Figure 5a, Figure 6a, and Figure 7a) contains five districts that meet the criteria of being considered Black opportunity districts. All five of these are anchored in
the city of Detroit in the southeast part of the state, where the majority of Michigan’s Black residents live. Furthermore, all five are overwhelmingly Democratic and all have elected a Black state
senator in recent years, although only three of the five districts are majority Black. Given that Michigan’s population is approximately 12 percent Black, a standard of proportionality would
prioritize four or five Black opportunity districts. The four algorithmically generated plans underrepresent Black voters relative both to the outcomes of the enacted plan and the standard of
The Annealing (Figure 5b, Figure 6b, and Figure 7b) and Voronoi (Figure 5c, Figure 6c, and Figure 7c) plans each include three clear Black opportunity districts and one more district in which Black
voters barely constitute a majority of the expected Democratic primary electorate. The Arcs plan (Figure 5d, Figure 6d, and Figure 7d) includes two clear opportunity districts and two more with a
bare majority. The Tree plan (Figure 5e, Figure 6e, and Figure 7e) comes closest to the enacted plan, with four clear Black opportunity districts.
Similar to Alabama, some of the plots in Figure 7 show a steady decline in Black voting strength when stepping through the districts (from right to left). Also like the case for Alabama, the
algorithms each draw high-concentration districts (in which the number of Black voters dramatically exceeds the number required to elect candidates of choice) and low-concentration districts (in
which the number of Black voters is clearly insufficient to elect candidates of choice). Compared to the enacted plan as well as the proportionality benchmark, each of these algorithms yields fewer
Black opportunity districts.
As in other domains of public life, some advocates have suggested that computer algorithms should replace human decision-making for consequential activities like determining maps of electoral
districts, much as Vickrey proposed back in 1961. Would entrusting such important tasks to algorithms actually produce more fair results? The analysis presented here suggests that we should be
skeptical of claims that fair outcomes for redistricting will follow from algorithms that do not take variables such as race or ethnicity into account. On the contrary, as demonstrated here, recently
proposed algorithms would each yield severely detrimental outcomes for the political influence of groups that are already marginalized.
Other advocates have proposed that algorithms could be used in redistricting to inform human decision makers about what may be “possible” or “typical” electoral maps. For example, in the relatively
new area of ensemble analysis, an algorithm is used to draft hundreds or thousands of districting plans; a human analyst can then evaluate the ensemble of outputs and ask questions such as, “What is
the average number of districts won by Republicans across all the plans?” Yet close study has found that such a procedure may generate plans that—even on average and in the aggregate—underrepresent
minority racial and ethnic groups relative to currently enacted plans as well as standards of proportionality. The concern would then be that such algorithms could legitimize unfair or discriminatory
decision-making. One could imagine a legislature intent on suppressing Black voters’ political strength appealing to a “neutral” algorithm that drew ten million plans, none of which included more
than one Black district, as a way to justify a plan that underrepresented Black voters. Such a scenario is not merely hypothetical: a federal judge advocated for exactly this type of analysis to be
used as a baseline in legal contexts, which prompted the recent scrutiny of possible ramifications of such procedures.
It is certainly not the case that algorithms and computerized map-making are irredeemably useless or harmful for the process of redistricting. Rather, the harms stem from human decision makers giving
too much deference to the algorithms’ outputs. In principle, a more deliberate approach to the human–computer interface could be used to make the redistricting process more transparent, fair, and
Computers are bad at inferring human values; humans are bad at fully articulating all the features and facets of a districting plan that we consider desirable or undesirable, especially with
sufficient precision to be encoded in a computer algorithm. On the other hand, computers are very good at solving mathematical problems and searching for plans that meet particular criteria. Rather
than using algorithms to generate plans to enact, legislators, stakeholders, and the public could use algorithmic redistricting tools to explore and understand trade-offs and the frontiers of what is
and is not possible to accomplish for a districting plan in a particular jurisdiction. For example, people could use an algorithm to explore whether it is possible to draw a state senate plan in
Alabama with ten Black opportunity districts, consider the resulting plan, and then pose further questions based on relevant community feedback, such as whether it is possible to draw ten opportunity
districts and keep the city of Tuscaloosa entirely within one district.
By incorporating algorithms as a component in a looped, iterative process of proposing and refining districting plans, we can leverage the strengths of computational methods: drawing plans much
faster than humans can, drawing plans that simultaneously satisfy specific properties, and offering evidence that some constraints are not mutually satisfiable. With algorithms folded into a
human-centered process, people would not need to uncritically accept two primary weaknesses of computers: their inability to devise plans that satisfy anything other than their encoded constraints,
and their inability to interpret constraints that cannot be phrased mathematically. In this way, algorithms could become a tool as part of an iterated discussion about social and political values,
rather than the arbiters of fairness.
Discussion Questions
1. Computing power and resources have become vastly more accessible in the sixty years since Vickrey’s proposal. During the 1960s, computers were the purview of universities, governments, and
wealthy businesses. Twenty years ago, state legislatures would have had access to spatialized demographic, socioeconomic, and election data along with GIS software to analyze and manipulate this
data, which would have cost an ordinary person tens of thousands of dollars to use in their own work. Today, with powerful computers more widely accessible as well as the time and effort of open
source software developers, public advocates, and data scientists, many of those resources can be downloaded onto your personal laptop, and you can begin drawing and analyzing districts within
minutes. How do you feel that the rapid changes in technology and data will impact the way redistricting is done, and do you see roles for algorithms in that process?
2. The idea that an algorithm that is unable to see features like race must therefore not be discriminatory along those features is an argument that is not confined to this domain. Examples of other
contexts where this arises are so-called “race-blind” college admissions or hiring, algorithmic tools for policing, sentencing, and other criminal justice applications, and loan administration,
to name just a few. How do you think such issues should be handled when computational solutions, like automated redistricting, are brought into traditionally noncomputational settings?
3. A commonly proposed reform is to put the power of redistricting in the hands of something like an independent citizens’ commission, a body composed of people who live in a particular jurisdiction
who draw district boundaries, rather than leaving redistricting in the hands of politicians. Among these three models of district construction (mathematical algorithm, politician-drawn,
independent commission), what do you see as the pros and cons of each, and how might their respective strengths be used to complement each other and offset weaknesses?
4. What do you believe are the responsibilities of people proposing algorithmic redistricting solutions in light of the evidence presented in this case study, as well as in the article by Chen and
Stephanopoulos, that so-called “neutral” algorithms have potentially discriminatory impacts?
Plots for Michigan That Incorporate Data from the 2018 US Senate Election
In this appendix, we reproduce the figures from the Michigan analysis using data from the 2018 US Senate election to make inferences about expected election outcomes in the algorithm-drawn districts.
In 2016, Hillary Clinton, a Democrat, lost the state of Michigan to Donald Trump, a Republican, by less than one quarter of a percent. In 2018, the Democratic incumbent senator Debbie Stabenow
defeated Republican challenger John James by about 6.5 percentage points. Broadly, Stabenow had a weaker performance than Clinton in urban Detroit but a stronger performance in the Detroit suburbs
and rural parts of the state. Because of this geographical difference, and particularly because this goes hand-in-hand with an increase in white voters supporting the Democratic candidate during the
2018 senate election compared to the 2016 presidential election, the evaluation of whether or not Black voters constitute a majority of the Democratic electorate in a hypothetical district may be
different when using the 2018 Senate election data versus the 2016 presidential election data.
The district boundaries themselves are identical (Figures A1 and A2), since the algorithms do not have access to the political data used to perform these analyses, but which districts represent
opportunity districts might differ. In particular, two districts in the Arcs plan and one in the Voronoi plan had expected support for the Black-preferred candidate between 50 and 55 percent with
respect to the 2016 presidential election results but fell below 50 percent with respect to the 2018 US Senate election results, due to the minor difference in the candidates’ performance (Figure A3
Figure A1
The enacted state senate districts in Michigan and the four algorithmically generated plans. Pink districts have at least 55 percent expected support for the Black-preferred candidate; yellow
districts have between 50 and 55 percent expected support. Estimates of expected support based on data from the 2018 US Senate election. (a) The currently enacted state senate districts as of 2020.
(b) Output from the “Annealing” algorithm. (c) Output from the “Arcs” algorithm. (d) Output from the “Voronoi” algorithm. (e) Output from the “Tree” algorithm.
Figure A2
Insets of the Detroit region for the plans shown in Figure A1. (a) The currently enacted state senate districts as of 2020. (b) Output from the “Annealing” algorithm. (c) Output from the “Arcs”
algorithm. (d) Output from the “Voronoi” algorithm. (e) Output from the “Tree” algorithm.
Figure A3
Plots of the expected support for the Black-preferred candidate in the ten Michigan districts that have the largest Black populations. Estimates of expected support based on data from the 2018 US
Senate election. Sorted districts appear along the horizontal axis, and expected fraction of votes for the Black-preferred candidate appears along the vertical axis. (a) The currently enacted state
senate districts as of 2020. (b) Output from the “Annealing” algorithm. (c) Output from the “Arcs” algorithm. (d) Output from the “Voronoi” algorithm. (e) Output from the “Tree” algorithm. (f) All
five plots superimposed.
Anonymous. “Supreme Court: ‘Gerrymandering’ Pronounced with a Hard ‘G.’” Associated Press, July 27, 2018. https://apnews.com/article/8874fb32cc514f49a5b2aaf1783955f0.
Becker, Amariah, and Justin Solomon. “Redistricting Algorithms.” Preprint, submitted November 18, 2020. https://arXiv.org/abs/2011.09504.
Bycoffe, Aaron, Ella Koeze, David Wasserman, and Julia Wolfe. The Atlas of Redistricting. FiveThirtyEight. https://projects.fivethirtyeight.com/redistricting-maps/.
Chen, Jowei, and Nicholas O. Stephanopoulos. “The Race-Blind Future of Voting Rights.” Yale Law Journal 130, no. 4 (February 2021): 862–946. https://www.yalelawjournal.org/article/
Cohen-Addad, Vincent, Philip N. Klein, and Neal E. Young. “Balanced Centroidal Power Diagrams for Redistricting.” Proceedings of the 26th ACM SIGSPATIAL International Conference on Advances in
Geographic Information Systems (2018): 389–96. https://doi.org/10.1145/3274895.3274979.
D’Ignazio, Catherine, and Lauren Klein. “Who Collects the Data? A Tale of Three Maps.” MIT Case Studies in Social and Ethical Responsibilities of Computing, Winter 2021. https://doi.org/10.21428/
Duchin, Moon, and Douglas M. Spencer. “Models, Race, and the Law.” Yale Law Journal 130 (March 2021), 744–97. https://www.yalelawjournal.org/forum/models-race-and-the-law.
Garfinkel, Simson. “Differential Privacy and the 2020 US Census.” MIT Case Studies in Social and Ethical Responsibilities of Computing, Winter 2022. https://doi.org/10.21428/2c646de5.7ec6ab93
Ingraham, Christopher. “This Computer Programmer Solved Gerrymandering in His Spare Time.” Washington Post, June 3, 2014. https://www.washingtonpost.com/news/wonk/wp/2014/06/03/
Levin, Harry A., and Sorelle A. Friedler. “Automated Congressional Redistricting.” Journal of Experimental Algorithmics 24 (2019): 1–24. https://doi.org/10.1145/3316513.
Metric Geometry and Gerrymandering Group. “GerryChain.” Open Source Software. https://github.com/mggg/gerrychain.
Olson, Brian. “Impartial Automatic Redistricting.” BDistricting. Accessed January 20, 2022. https://bdistricting.com/.
Procaccia, Ariel D., and Jamie Tucker-Foltz. “Compact Redistricting Plans Have Many Spanning Trees.” Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (2022): 3754–71,
Schutzman, Zachary. “Trade-Offs in Fair Redistricting." Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society (2020): 159–65. https://doi.org/10.1145/3375627.3375802.
Suresh, Harini, and John Guttag. “Understanding Potential Sources of Harm throughout the Machine Learning Life Cycle.” MIT Case Studies in Social and Ethical Responsibilities of Computing, Summer
2021. https://doi.org/10.21428/2c646de5.c16a07bb.
Vickrey, William. “On the Prevention of Gerrymandering.” Political Science Quarterly 76, no. 1 (1961): 105–10. https://doi.org/10.2307/2145973.
Udall, Morris K. “Reapportionment: ‘One Man, One Vote’ ... That’s All She Wrote!,” October 14, 1964. Accessed January 21, 2022. https://speccoll.library.arizona.edu/online-exhibits/files/original/
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How does entropy change with pressure? | HIX Tutor
How does entropy change with pressure?
Answer 1
Starting from the first law of thermodynamics and the relationship of enthalpy #H# to internal energy #U#:
#\mathbf(DeltaU = q_"rev" + w_"rev")#
#\mathbf(DeltaH = DeltaU + Delta(PV)) = q_"rev" + w_"rev" + Delta(PV)#
Applying differential form, we obtain:
#dH = delq_"rev" + delw_"rev" + d(PV)#
The relationship between entropy and reversible heat flow is also related to heat flow:
#\mathbf(dS = (delq_"rev")/T)#
Thus, utilizing this relationship and invoking the Product Rule on #d(PV)#, we get:
#dH = TdS - cancel(PdV + PdV) + VdP#
#color(blue)(dH = TdS + VdP)#
This is what the Maxwell relation would give you.
When we relate pressure then, to entropy, with #S = S(T,P)#:
#dS = (dH)/T - V/TdP#
For an ideal monatomic gas, #PV = nRT#, so:
#dS = (dH)/T - (nR)/PdP#
Ultimately, after integrating this, we obtain:
#int_(S_1)^(S_2)dS = int_(H_1)^(H_2)(dH)/T - nR int_(P_1)^(P_2)1/PdP#
Enthalpy is #DeltaH = int_(H_1)^(H_2)dH = int_(T_1)^(T_2) C_PdT#, where #C_P# is the heat capacity at constant pressure in #"J/K"#. Thus:
#=> int_(T_1)^(T_2) C_P/TdT - nR int_(P_1)^(P_2)1/PdP#
Since #C_P# for a monatomic ideal gas is #C_V + nR = 3/2nR + nR = 5/2nR#, with #C_V# as the constant-volume heat capacity and the #3/2# coming from the three linear degrees of freedom (#x,y,z#), this
#color(blue)(DeltaS_"sys" = 5/2nRln|T_2/T_1| - nRln|(P_2)/(P_1)|)#
As a result, an ideal gas's entropy changes negatively as pressure rises; however, the system's actual entropy change could be positive or negative, contingent upon temperature changes.
(Regardless, the entropy of the universe is #>= 0#.)
However, a liquid's change in pressure has a smaller negative impact on entropy change because a liquid's volume changes relatively little with small pressure increases that should significantly
compress a gas.
For small pressure values that would otherwise be significant for gases, I would not expect pressure to significantly change any entropy patterns that solids already have.
Depending on how many "ways" a di/polyatomic solid can exist, we can think about either the bond strength or the complexity. One of the most widely used thermodynamics equations is:
#\mathbf(DeltaS = k_blnOmega)#
Because there are fewer microstates available to the solid, the stronger the bond, the smaller the entropy.
Here are some data that show that.
Growing charge intensities:
#DeltaS_("NaF"(s))^@ = "51.46 J/mol"cdot"K"#
#DeltaS_("MgO"(s))^@ = "26.9 J/mol"cdot"K"#
#DeltaS_("AlN"(s))^@ = "20.2 J/mol"cdot"K"#
Naturally, an increase in bond order or bond strength corresponds with an increase in charge magnitudes.
Growing the difference in radii:
#DeltaS_("NaCl"(s))^@ = "72.13 J/mol"cdot"K"#
#DeltaS_("NaBr"(s))^@ = "86.82 J/mol"cdot"K"#
#DeltaS_("NaI"(s))^@ = "98.53 J/mol"cdot"K"#
An increased internuclear distance and, consequently, a weaker bond are typically correlated with larger differences in cation/anion radii.
Furthermore, it doesn't hurt to look up alkaline earth metal carbonates going down the periodic table on Wikipedia, since I seem to remember carbonates decomposing at higher temperatures (which I
thought was strange when I first learned about it).
Increasing distance in alkaline-earth-metal/carbon #ns"/"2s# orbital ground-state energies (increasing the number of nonbonding orbitals):
#DeltaS_("BeCO"_3(s))^@ = "52 J/mol"cdot"K"#
#DeltaS_("MgCO"_3(s))^@ = "65.7 J/mol"cdot"K"#
#DeltaS_("CaCO"_3(s))^@ = "93 J/mol"cdot"K"#
Furthermore, the entropy is much higher for more complex substances like buckminsterfullerene because they can assume more microstates for a given macrostate (more possible molecular motions, more
vibrational modes, etc.).
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Answer 2
Entropy generally increases with pressure for gases and decreases with pressure for liquids and solids. This relationship is described by the second law of thermodynamics, which states that the
entropy of a closed system tends to increase over time.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Output file tags
Output file tags¶
List of available properties¶
The following list briefly describes all the property names that can be listed in the properties tag of the Input files, and which will be written in the output files.
Eenvelope: The (gaussian) envelope function of the external applied electric field (values go from 0 to 1).
dimension: atomic_unit;
Efield: The external applied electric field (x,y,z components in cartesian axes).
dimension: atomic_unit; size: 3;
atom_f: The force (x,y,z) acting on a particle given its index. Takes arguments index and bead (both zero based). If bead is not specified, refers to the centroid.
dimension: force; size: 3;
atom_f_path: The forces acting on all the beads of a particle given its index. Takes arguments index and bead (both zero based). If bead is not specified, refers to the centroid.
dimension: length;
atom_p: The momentum (x,y,z) of a particle given its index. Takes arguments index and bead (both zero based). If bead is not specified, refers to the centroid.
dimension: momentum; size: 3;
atom_v: The velocity (x,y,z) of a particle given its index. Takes arguments index and bead (both zero based). If bead is not specified, refers to the centroid.
dimension: velocity; size: 3;
atom_x: The position (x,y,z) of a particle given its index. Takes arguments index and bead (both zero based). If bead is not specified, refers to the centroid.
dimension: length; size: 3;
atom_x_path: The positions of all the beads of a particle given its index. Takes an argument index (zero based).
dimension: length;
bead_potentials: The physical system potential energy of each bead.
dimension: energy; size: nbeads;
bweights_component: The weight associated one part of the hamiltonian. Takes one mandatory argument index (zero-based) that indicates for which component of the hamiltonian the weight must be
cell_abcABC: The lengths of the cell vectors and the angles between them in degrees as a list of the form [a, b, c, A, B, C], where A is the angle between the sides of length b and c in degrees, and
B and C are defined similarly. Since the output mixes different units, a, b and c can only be output in bohr.
size: 6;
cell_h: The simulation cell as a matrix. Returns the 6 non-zero components in the form [xx, yy, zz, xy, xz, yz].
dimension: length; size: 6;
chin_weight: The 3 numbers output are 1) the logarithm of the weighting factor -beta_P delta H, 2) the square of the logarithm, and 3) the weighting factor
size: 3;
conserved: The value of the conserved energy quantity per bead.
dimension: energy;
density: The mass density of the physical system.
dimension: density;
dipole: The electric dipole of the system (x,y,z components in cartesian axes).
dimension: electric-dipole; size: 3;
displacedpath: This is the estimator for the end-to-end distribution, that can be used to calculate the particle momentum distribution as described in in L. Lin, J. A. Morrone, R. Car and M.
Parrinello, 105, 110602 (2010), Phys. Rev. Lett. Takes arguments ‘ux’, ‘uy’ and ‘uz’, which are the components of the path opening vector. Also takes an argument ‘atom’, which can be either an atom
label or index (zero based) to specify which species to find the end-to-end distribution estimator for. If not specified, all atoms are used. Note that one atom is computed at a time, and that each
path opening operation costs as much as a PIMD step. Returns the average over the selected atoms of the estimator of exp(-U(u)) for each frame.
ensemble_bias: The bias applied to the current ensemble
dimension: energy;
ensemble_lp: The log of the ensemble probability
ensemble_pressure: The target pressure for the current ensemble
dimension: pressure;
ensemble_temperature: The target temperature for the current ensemble
dimension: temperature;
exchange_all_prob: Probability of the ring polymer exchange configuration where all atoms are connected. It is divided by 1/N, so the number is between 0 and N, while the asymptotic value at low
temperatures is 1.
size: 1;
exchange_distinct_prob: Probability of the distinguishable ring polymer configuration, where each atom has its own separate ring polymer. A number between 0 and 1, tends to 1 in high temperatures,
which indicates that bosonic exchange is negligible
size: 1;
fermionic_sign: Estimator for the fermionic sign, also used for reweighting fermionic observables. Decreases exponentially with beta and the number of particles, but if not too large, can be used to
recover fermionic statistics from bosonic simulations, see doi:10.1063/5.0008720.
size: 1;
forcemod: The modulus of the force. With the optional argument ‘bead’ will print the force associated with the specified bead.
dimension: force;
hweights_component: The weight associated one part of the hamiltonian. Takes one mandatory argument index (zero-based) that indicates for which component of the hamiltonian the weight must be
isotope_scfep: Returns the (many) terms needed to compute the scaled-coordinates free energy perturbation scaled mass KE estimator (M. Ceriotti, T. Markland, J. Chem. Phys. 138, 014112 (2013)). Takes
two arguments, ‘alpha’ and ‘atom’, which give the scaled mass parameter and the atom of interest respectively, and default to ‘1.0’ and ‘’. The ‘atom’ argument can either be the label of a particular
kind of atom, or an index (zero based) of a specific atom. This property computes, for each atom in the selection, an estimator for the kinetic energy it would have had if it had the mass scaled by
alpha. The 7 numbers output are the average over the selected atoms of the log of the weights <h>, the average of the squares <h**2>, the average of the un-weighted scaled-coordinates kinetic
energies <T_CV> and of the squares <T_CV**2>, the log sum of the weights LW=ln(sum(e**(-h))), the sum of the re-weighted kinetic energies, stored as a log modulus and sign, LTW=ln(abs(sum(T_CV e**
(-h)))) STW=sign(sum(T_CV e**(-h))). In practice, the best estimate of the estimator can be computed as [sum_i exp(LTW_i)*STW_i]/[sum_i exp(LW_i)]. The other terms can be used to compute diagnostics
for the statistical accuracy of the re-weighting process. Note that evaluating this estimator costs as much as a PIMD step for each atom in the list. The elements that are output have different
units, so the output can be only in atomic units.
size: 7;
isotope_tdfep: Returns the (many) terms needed to compute the thermodynamic free energy perturbation scaled mass KE estimator (M. Ceriotti, T. Markland, J. Chem. Phys. 138, 014112 (2013)). Takes two
arguments, ‘alpha’ and ‘atom’, which give the scaled mass parameter and the atom of interest respectively, and default to ‘1.0’ and ‘’. The ‘atom’ argument can either be the label of a particular
kind of atom, or an index (zero based) of a specific atom. This property computes, for each atom in the selection, an estimator for the kinetic energy it would have had if it had the mass scaled by
alpha. The 7 numbers output are the average over the selected atoms of the log of the weights <h>, the average of the squares <h**2>, the average of the un-weighted scaled-coordinates kinetic
energies <T_CV> and of the squares <T_CV**2>, the log sum of the weights LW=ln(sum(e**(-h))), the sum of the re-weighted kinetic energies, stored as a log modulus and sign, LTW=ln(abs(sum(T_CV e**
(-h)))) STW=sign(sum(T_CV e**(-h))). In practice, the best estimate of the estimator can be computed as [sum_i exp(LTW_i)*STW_i]/[sum_i exp(LW_i)]. The other terms can be used to compute diagnostics
for the statistical accuracy of the re-weighting process. Evaluating this estimator is inexpensive, but typically the statistical accuracy is worse than with the scaled coordinates estimator. The
elements that are output have different units, so the output can be only in atomic units.
size: 7;
isotope_zetasc: Returns the (many) terms needed to directly compute the relative probablity of isotope substitution in two different systems/phases. Takes four arguments, ‘alpha’ , which gives the
scaled mass parameter and default to ‘1.0’, and ‘atom’, which is the label or index of a type of atoms. The 3 numbers output are 1) the average over the excess potential energy for scaled coordinates
<sc>, 2) the average of the squares of the excess potential energy <sc**2>, and 3) the average of the exponential of excess potential energy <exp(-beta*sc)>
size: 3;
isotope_zetasc_4th: Returns the (many) terms needed to compute the scaled-coordinates fourth-order direct estimator. Takes two arguments, ‘alpha’ , which gives the scaled mass parameter and default
to ‘1.0’, and ‘atom’, which is the label or index of a type of atoms. The 5 numbers output are 1) the average over the excess potential energy for an isotope atom substitution <sc>, 2) the average of
the squares of the excess potential energy <sc**2>, and 3) the average of the exponential of excess potential energy <exp(-beta*sc)>, and 4-5) Suzuki-Chin and Takahashi-Imada 4th-order reweighing
size: 5;
isotope_zetatd: Returns the (many) terms needed to directly compute the relative probablity of isotope substitution in two different systems/phases. Takes two arguments, ‘alpha’ , which gives the
scaled mass parameter and default to ‘1.0’, and ‘atom’, which is the label or index of a type of atoms. The 3 numbers output are 1) the average over the excess spring energy for an isotope atom
substitution <spr>, 2) the average of the squares of the excess spring energy <spr**2>, and 3) the average of the exponential of excess spring energy <exp(-beta*spr)>
size: 3;
isotope_zetatd_4th: Returns the (many) terms needed to compute the thermodynamic fourth-order direct estimator. Takes two arguments, ‘alpha’ , which gives the scaled mass parameter and default to
‘1.0’, and ‘atom’, which is the label or index of a type of atoms. The 5 numbers output are 1) the average over the excess spring energy for an isotope atom substitution <spr>, 2) the average of the
squares of the excess spring energy <spr**2>, and 3) the average of the exponential of excess spring energy <exp(-beta*spr)>, and 4-5) Suzuki-Chin and Takahashi-Imada 4th-order reweighing term
size: 5;
kinetic_cv: The centroid-virial quantum kinetic energy of the physical system. Takes an argument ‘atom’, which can be either an atom label or index (zero based) to specify which species to find the
kinetic energy of. If not specified, all atoms are used.
dimension: energy;
kinetic_ij: The centroid-virial off-diagonal quantum kinetic energy tensor of the physical system. This computes the cross terms between atoms i and atom j, whose average is <p_i*p_j/(2*sqrt
(m_i*m_j))>. Returns the 6 independent components in the form [xx, yy, zz, xy, xz, yz]. Takes arguments ‘i’ and ‘j’, which give the indices of the two desired atoms.
dimension: energy; size: 6;
kinetic_md: The kinetic energy of the (extended) classical system. Takes optional arguments ‘atom’, ‘bead’ or ‘nm’. ‘atom’ can be either an atom label or an index (zero-based) to specify which
species or individual atom to output the kinetic energy of. If not specified, all atoms are used and averaged. ‘bead’ or ‘nm’ specify whether the kinetic energy should be computed for a single bead
or normal mode. If not specified, all atoms/beads/nm are used.
dimension: energy;
kinetic_opsc: The centroid-virial quantum kinetic energy of the physical system. Takes an argument ‘atom’, which can be either an atom label or index (zero based) to specify which species to find the
kinetic energy of. If not specified, all atoms are used.
dimension: energy;
kinetic_prsc: The Suzuki-Chin primitive estimator of the quantum kinetic energy of the physical system
dimension: energy;
kinetic_td: The primitive quantum kinetic energy of the physical system. Takes an argument ‘atom’, which can be either an atom label or index (zero based) to specify which species to find the kinetic
energy of. If not specified, all atoms are used.
dimension: energy;
kinetic_tdsc: The Suzuki-Chin centroid-virial thermodynamic estimator of the quantum kinetic energy of the physical system. Takes an argument ‘atom’, which can be either an atom label or index (zero
based) to specify which species to find the kinetic energy of. If not specified, all atoms are used.
dimension: energy;
kinetic_tens: The centroid-virial quantum kinetic energy tensor of the physical system. Returns the 6 independent components in the form [xx, yy, zz, xy, xz, yz]. Takes an argument ‘atom’, which can
be either an atom label or index (zero based) to specify which species to find the kinetic tensor components of. If not specified, all atoms are used.
dimension: energy; size: 6;
kstress_cv: The quantum estimator for the kinetic stress tensor of the physical system. Returns the 6 independent components in the form [xx, yy, zz, xy, xz, yz].
dimension: pressure; size: 6;
kstress_md: The kinetic stress tensor of the (extended) classical system. Returns the 6 independent components in the form [xx, yy, zz, xy, xz, yz].
dimension: pressure; size: 6;
kstress_tdsc: The Suzuki-Chin thermodynamic estimator for pressure of the physical system.
dimension: pressure;
pot_component: The contribution to the system potential from one of the force components. Takes one mandatory argument index (zero-based) that indicates which component of the potential must be
returned. The optional argument ‘bead’ will print the potential associated with the specified bead (interpolated to the full ring polymer). If the potential is weighed, the weight will be applied.
dimension: energy;
pot_component_raw: The contribution to the system potential from one of the force components. Takes one mandatory argument index (zero-based) that indicates which component of the potential must be
returned. The optional argument ‘bead’ will print the potential associated with the specified bead, at the level of discretization of the given component. Potential weights will not be applied.
dimension: energy;
potential: The physical system potential energy. With the optional argument ‘bead’ will print the potential associated with the specified bead.
dimension: energy;
potential_opsc: The Suzuki-Chin operator estimator for the potential energy of the physical system.
dimension: energy;
potential_tdsc: The Suzuki-chin thermodyanmic estimator for the potential energy of the physical system.
dimension: energy;
pressure_cv: The quantum estimator for pressure of the physical system.
dimension: pressure;
pressure_md: The pressure of the (extended) classical system.
dimension: pressure;
pressure_tdsc: The Suzuki-Chin thermodynamic estimator for pressure of the physical system.
dimension: pressure;
r_gyration: The average radius of gyration of the selected ring polymers. Takes an argument ‘atom’, which can be either an atom label or index (zero based) to specify which species to find the radius
of gyration of. If not specified, all atoms are used and averaged.
dimension: length;
sc_op_scaledcoords: Returns the estimators that are required to evaluate the scaled-coordinates estimators for total energy and heat capacity for a Suzuki-Chin high-order factorization, as described
in Appendix B of J. Chem. Theory Comput. 2019, 15, 3237-3249 (2019). Returns eps_v and eps_v’, as defined in that paper. As the two estimators have a different dimensions, this can only be output in
atomic units. Takes one argument, ‘fd_delta’, which gives the value of the finite difference parameter used - which defaults to -0.0001. If the value of ‘fd_delta’ is negative, then its magnitude
will be reduced automatically by the code if the finite difference error becomes too large.
size: 2;
sc_scaledcoords: Returns the estimators that are required to evaluate the scaled-coordinates estimators for total energy and heat capacity for a Suzuki-Chin fourth-order factorization, as described
in T. M. Yamamoto, J. Chem. Phys., 104101, 123 (2005). Returns eps_v and eps_v’, as defined in that paper. As the two estimators have a different dimensions, this can only be output in atomic units.
Takes one argument, ‘fd_delta’, which gives the value of the finite difference parameter used - which defaults to -0.0001. If the value of ‘fd_delta’ is negative, then its magnitude will be reduced
automatically by the code if the finite difference error becomes too large.
size: 2;
scaledcoords: Returns the estimators that are required to evaluate the scaled-coordinates estimators for total energy and heat capacity, as described in T. M. Yamamoto, J. Chem. Phys., 104101, 123
(2005). Returns eps_v and eps_v’, as defined in that paper. As the two estimators have a different dimensions, this can only be output in atomic units. Takes one argument, ‘fd_delta’, which gives the
value of the finite difference parameter used - which defaults to -0.0001. If the value of ‘fd_delta’ is negative, then its magnitude will be reduced automatically by the code if the finite
difference error becomes too large.
size: 2;
spring: The total spring potential energy between the beads of all the ring polymers in the system.
dimension: energy;
step: The current simulation time step.
dimension: number;
stress_cv: The total quantum estimator for the stress tensor of the physical system. Returns the 6 independent components in the form [xx, yy, zz, xy, xz, yz].
dimension: pressure; size: 6;
stress_md: The total stress tensor of the (extended) classical system. Returns the 6 independent components in the form [xx, yy, zz, xy, xz, yz].
dimension: pressure; size: 6;
temperature: The current temperature, as obtained from the MD kinetic energy of the (extended) ring polymer. Takes optional arguments ‘atom’, ‘bead’ or ‘nm’. ‘atom’ can be either an atom label or an
index (zero-based) to specify which species or individual atom to output the temperature of. If not specified, all atoms are used and averaged. ‘bead’ or ‘nm’ specify whether the temperature should
be computed for a single bead or normal mode.
dimension: temperature;
ti_pot: The correction potential in Takahashi-Imada 4th-order PI expansion. Takes an argument ‘atom’, which can be either an atom label or index (zero based) to specify which species to find the
correction term for. If not specified, all atoms are used.
dimension: energy; size: 1;
ti_weight: The 3 numbers output are 1) the logarithm of the weighting factor -beta_P delta H, 2) the square of the logarithm, and 3) the weighting factor
size: 3;
time: The elapsed simulation time.
dimension: time;
vcom: The center of mass velocity (x,y,z) of the system or of a species. Takes arguments label (default to all species) and bead (zero based). If bead is not specified, refers to the centroid.
dimension: velocity; size: 3;
vir_tdsc: The Suzuki-Chin thermodynamic estimator for pressure of the physical system.
dimension: pressure;
virial_cv: The quantum estimator for the virial stress tensor of the physical system. Returns the 6 independent components in the form [xx, yy, zz, xy, xz, yz].
dimension: pressure; size: 6;
virial_fq: Returns the scalar product of force and positions. Useful to compensate for the harmonic component of a potential. Gets one argument ‘ref’ that should be a filename for a reference
configuration, in the style of the FFDebye geometry input, and one that contains the input units.
dimension: energy; size: 1;
virial_md: The virial tensor of the (extended) classical system. Returns the 6 independent components in the form [xx, yy, zz, xy, xz, yz].
dimension: pressure; size: 6;
volume: The volume of the cell box.
dimension: volume;
List of available trajectory files¶
The following list briefly describes all the trajectory types that can be listed in the trajectory tag of the Input files.
Eforces: The external electric field contribution to the forces
dimension: force;
becx: The x component of the Born Effective Charges in cartesian coordinates.
dimension: number;
becy: The y component of the Born Effective Charges in cartesian coordinates.
dimension: number;
becz: The z component of the Born Effective Charges in cartesian coordinates.
dimension: number;
extras: The additional data returned by the client code. If the attribute “extra_type” is specified, and if the data is JSON formatted, it prints only the specified field. Otherwise (or if extra_type
=”raw”) the full string is printed verbatim. Will print out one file per bead, unless the bead attribute is set by the user.
extras_bias: The additional data returned by the bias forcefield, printed verbatim or expanded as a dictionary. See “extras”.
extras_component_raw: The additional data returned by the client code, printed verbatim or expanded as a dictionary. See “extras”. Fetches the extras from a specific force component, indicated in
parentheses and a specific bead [extras_component_raw(idx; bead=0)]. Never applies weighting or contraction, and does not automatically sum over beads as we don’t know if the extras are numeric
f_centroid: The force acting on the centroid.
dimension: force;
forces: The force trajectories. Will print out one file per bead, unless the bead attribute is set by the user.
dimension: force;
forces_component: The contribution to the system forces from one of the force components. Takes one mandatory argument index (zero-based) that indicates which component of the potential must be
returned. The optional argument ‘bead’ will print the potential associated with the specified bead (interpolated to the full ring polymer), otherwise the centoid force is computed. If the potential
is weighed, the weight will be applied.
dimension: force;
forces_component_raw: The contribution to the system forces from one of the force components. Takes one mandatory argument index (zero-based) that indicates which component of the potential must be
returned. The optional argument ‘bead’ will print the potential associated with the specified bead (with the level of discretization of the component), otherwise the centoid force is computed. The
weight of the potential is not applied.
dimension: force;
forces_sc: The Suzuki-Chin component of force trajectories. Will print out one file per bead, unless the bead attribute is set by the user.
dimension: force;
forces_spring: The spring force trajectories. Will print out one file per bead, unless the bead attribute is set by the user.
dimension: force;
isotope_zetasc: Scaled-coordinates isotope fractionation direct estimator in the form of ratios of partition functions. Takes two arguments, ‘alpha’ , which gives the scaled mass parameter and
default to ‘1.0’, and ‘atom’, which is the label or index of a type of atoms. All the atoms but the selected ones will have zero output
isotope_zetatd: Thermodynamic isotope fractionation direct estimator in the form of ratios of partition functions. Takes two arguments, ‘alpha’ , which gives the scaled mass parameter and default to
‘1.0’, and ‘atom’, which is the label or index of a type of atoms. All the atoms but the selected ones will have zero output
kinetic_cv: The centroid virial quantum kinetic energy estimator for each atom, resolved into Cartesian components [xx, yy, zz]
dimension: energy;
kinetic_od: The off diagonal elements of the centroid virial quantum kinetic energy tensor [xy, xz, yz]
dimension: energy;
momenta: The momentum trajectories. Will print out one file per bead, unless the bead attribute is set by the user.
dimension: momentum;
p_centroid: The centroid momentum.
dimension: momentum;
positions: The atomic coordinate trajectories. Will print out one file per bead, unless the bead attribute is set by the user.
dimension: length;
r_gyration: The radius of gyration of the ring polymer, for each atom and resolved into Cartesian components [xx, yy, zz]
dimension: length;
v_centroid: The centroid velocity.
dimension: velocity;
v_centroid_even: The suzuki-chin centroid velocity.
dimension: velocity;
v_centroid_odd: The suzuki-chin centroid velocity.
dimension: velocity;
velocities: The velocity trajectories. Will print out one file per bead, unless the bead attribute is set by the user.
dimension: velocity;
x_centroid: The centroid coordinates.
dimension: length;
x_centroid_even: The suzuki-chin centroid coordinates.
dimension: length;
x_centroid_odd: The suzuki-chin centroid coordinates.
dimension: length;
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CPH Theory
Search for clustering of ultra high energy cosmic rays from the Pierre Auger Observatory
Silvia Mollerach and the Pierre Auger Collaboration
Pierre Auger Observatory, av. San Mart'in Norte 304, (5613) Malargüe, Argentina CONICET, Centro Atómico Bariloche, 8400 Bariloche, Rio Negro, Argentina
Abstract. We present the results of a search for clustering among the highest energy events detected by the surface detector of the Pierre Auger Observatory between 1 January 2004 and 31 August 2007.
We analyse the autocorrelation function, in which the number of pairs with angular separation within a given angle is compared with the expectation from an isotropic distribution. Performing a scan
in energy above 30 EeV and in angles smaller than 30 degrees, the most significant excess of pairs appears for E > 57 EeV and for a wide range of separation angles, between 9 and 22 degrees. An
excess like this has a chance probability of about 2% to arise from an isotropic distribution and appears at the same energy threshold at which the Pierre Auger Observatory has reported a correlation
of the arrival directions of cosmic rays with nearby astrophysical objects.
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Time Estimation in PERT (With Calculation) | Project Management
There are three different estimates of activity duration in PERT: 1. Optimistic 2. Pessimistic 3. Most Likely.
1. Optimistic time, expressed as ‘t[o]‘,represents estimate of minimum possible time by which an activity can be completed assuming that everything is in order according to the plan and there can be
only minimum amount of difficulty.
2. Pessimistic time, expressed as ‘t[p]’ represents estimate of maximum possible time by which an activity can be completed assuming that things may not be in accordance with the plan and there can
be incidence of difficulties in carrying out the activity.
3. Most likely time, expressed as ‘t[m]‘, represents estimate of time for completion of an activity, which is neither optimistic nor pessimistic, assuming that things should go in a normal way, and
if the activity is repeated several times, in most of the cases, it will be completed in time represented by t.
From the above three different estimates, PERT suggests work out of the expected time, expressed as ‘t[e]‘ assuming that the probability distribution of the activity duration follows
beta-distribution and, thus, t[e] is the average of t[o] t[p] and t[m] calculated as,
t[e]=t[o] + 4 x t[m] + t[p ]/6
This averaging is explained with the assumption that, for every activity, when the t[ij] is estimated 6 times, the pattern of such estimated time will be once t[0] four times t[m] and, again, once t
[p]. This can be illustrated in a time scale as follows when t[o] = 3, t[p] = 9 and t[m] = 6 then, as per the formula,
t[e ]=t[o] + 4 x t[m] + t[p]/6= 3+ 24 + 9/6 = 6; when the three estimates are placed in time scale.
Three estimates, as above, when placed in time scale, will appear as:
When the probability follows beta distribution (as assumed in PERT), and in the scale of time, time units 12 represents 100 per cent probability, then time units 6 is 0.5 or 50 per cent probability.
The most likely estimate is a probability of 0.5. As we have noted in the averaging formula the weightage for t[o] t[m] and t[p] are 1, 4, and 1, respectively.
The 0 to 2 in the time scale representing ^1/[6 ]th = 0.17, 2 to 6 is 0.33, 6 to 10 is 0.33 and 10 to 12 is 0.17. Therefore, the probability of t[m] will lie between 2 to 10 i.e. 0.33 + 0.33 = 0.66.
PERT considers t[e] as more probable time estimate for activities and then the network construction and the critical path is drawn considering t[e]-s for the respective activities.
The estimate of t[e] as explained here is more reliable as it takes into account the longest and the shortest possible time estimates also and it provides a probability of 50 per cent.
Once the t[e] is worked out for each of the activities the network can be constructed following the same principle discussed earlier and is illustrated below:
From the three different time estimates, t[e] is worked out for each activity shown above.
The network is constructed in PERT as per the t[e] developed from the three different time estimates as shown below:
All the different estimates of time as well as the worked out t[e] are shown in the above network diagram against the relevant activity. There is, however, no specific rule for writing such estimates
on the network.
We will now redraft the network (to have a cleaner diagram) with only the t[e] and work out the Critical Path as per the following steps:
Step 1. Calculating ESTs and plotting them on the network as detailed below:
event ① = start with 0;
event ② = EST of tail +t[e] i.e. 0+5=5 days
event ③= 0+ 14 days;
event ④ = 5+15=20 days
event ⑤ = highest of 14 +9, 5+8, and 20+4(as there are different tail events) = 24 days;
event ⑥ = 24+5=29 days
Step 2. We are to come backward from the end event ⑥.
Calculating the LFTs and plotting them on this network as follows:
of event ⑥ = EST of event (6) = 29 days, as already found in Step 1;
of event ⑤ = LFT of head event minus t[e], i.e. 29 – 5 = 24 days;
of event ④ = 24 – 4 = 20 days;
of event ③ = 24 – 9 = 15 days;
of event ② = lowest of 24 – 8,20 – 15 and 15-9 (as there are three different head events) = 5 days;
of event ① = 5-5 = 0 day.
With the ESTs and LFTs calculated as detailed in Step 1 and Step 2 above we will produce the network diagram as:
Step 3:
We know the events having same EST and LFT are on the critical path and now we find those are 1, 2, 3, 4, 5 and 6. The critical path is now shown by double-line arrows and the project duration is 29
This is subject to the random variation of the actual performance time as against t[e] (time estimates for PERT) of 5, 15, 4 and 5 time units for activities on the critical path.
Therefore, the actual time to perform the four activities A, D, G and H represents the time to complete the project and PERT works out by means of statistical theory the probability of meeting the
time target.
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What is Chinese Suanpan (Suan pan) Abacus - Thej Academy
History always has its way of fascinating the present-day generations. To begin with, inventions in ancient days have always been nothing less than jaw-dropping. Moreover, starting from numbers to
calculations, inventions related to arithmetics have been quite creative and indigenous.
Speaking of which, let us discuss the invention and history of the Chinese Suanpan abacus. As fascinating as it sounds, history clearly remembers it as one of the most remarkable inventions in
Thej academy has been educating you with the history of each abacus every month. Moreover, we are back with another interesting article. So, grab a seat. Let’s talk in detail about the Chinese
Suanpan abacus.
History of the Chinese Suanpan Abacus
Suanpan is the Chinese version of the abacus, invented in the 1200 C E, which was during the Ming dynasty. It was an influenced version of the earlier versions, hence the inventor’s credits go to the
ancient Chinese in general. In 206 BC- 220 AD, prototypes of the Suanpan abacus were designed. It happened during the Han dynasty.
The design of the Suanpan abacus was simple. The device had a wooden frame with metal reinforcements. It had two decks and a separating beam in between them. The top deck had 2 beads and the bottom
deck had 5 beads. Indeed, the beads on the lower deck were called the ‘Earth beads’ or ‘Water beads’. Their value was 1 per bead. In addition, the beads on the upper deck were called ‘Heaven beads’.
They carried a value of 5 per bead.
The Chinese abacus was pronounced as ‘Su-an-pan’, which meant ‘calculating tray’.
Counting in the Suanpan Abacus
The earlier Suanpan abacus had 23 columns in it. It facilitated 2 calculations being performed at once, using one device. In addition, the place values on the Suan pan were very similar to the Indian
counting system. Place values were given from the rightmost column.
Firstly, the rightmost column represents ones. Secondly, the adjacent left one represents tens, hundreds, and so on. To begin with, one had to use the heavenly bead whenever the calculation involved
numbers more than 5. Ancient Chinese called using the heavenly beads as ‘Extra Bead Technique’ or ‘Suspended Bead Technique’. The Suan pan was quite famous for counting among the adults who handle
business. Abacus was not just reserved for children, back in those days.
Calculations using the Suanpan abacus were possible when there was a clear understanding of the place values.
Roman and Chinese Abacus – the Coincidence
In brief, the Roman and Chinese abaci coincidentally resemble each other much. Experts believe this must be because of the trade relationship between China and Rome, that led to the influenced abacus
design. While there is no logical proof to demonstrate this, it might just be a coincidence. In conclusion, even though the number of beads was similar, the Chinese abacus used wires to connect
beads. In contrast, the Roman abacus used grooves.
Also, the ancient Roman abacus had a 1:4 design(1 heaven bead and 4 earth beads). Whereas the earlier Suan Pan had a 2:5 design (2 heaven beads and 5 earth beads).
Addition Using Suanpan Abacus
Before we start calculating, set the abacus to zero. Clear all the beads to their original places and away from the central beam. This is called ‘Setting Zero’ in the abacus.
Let’s try to add 23 + 7
Step 1
Firstly, know the place value 23
1. 2 = Tens
2. 3 = ones
Step 2
Secondly, push 3 beads from the lower deck towards the center.
Push 2 beads from the lower deck of the nearby (left) column towards the central beam.
Step 3
Now you can see 23 on the abacus
Step 4
Thirdly, let’s add the next number, which is 7.
Know the place value. Certainly, 7 belongs in ones.
Step 5
We already have 3 in the ones column. It becomes impossible to add another 7, even if we use the upper deck beads. Therefore, we take 10 from the tens column and subtract 3 from the ones column.
Push 1 bead up in the tens column. Subtract the 3 from the ones column. This means, you have made 10-3, which is equal to 7.
Step 6
Finally, read the value on the abacus. It has 3 beads in the lower deck of the tens column and zero beads in the ones column.
Clearly, the answer is 30.
Multiplication using Suanpan abacus
Before we start calculating, set the abacus to zero. Clear all the beads to their original places and away from the central beam.
Lets multiply 56 X 48.
Step 1
Firstly, set zero
Step 2
Secondly, set 56 on the leftmost column.
Step 3
Thirdly, leave a column to avoid confusion, and set 48 in the next respective columns. Lets begin the calculation. Now, we are about to multiply each number separately to get the final result.
Step 4
Remember the steps are as follows
1. Tens with tens
2. Tens with ones
3. Ones with tens
4. Ones with tens.
Relax, we are going to break it down to you step by step.
Place value of 56
1. 5 = Tens
2. 6 = Ones
Place value of 48
1. 4 = Tens
2. 8 = Ones
Step 5
Now follow the rule of multiplication mentioned above.
Multiply tens with tens
5 X 4 = 20
Leave aside a column and set 20.
Step 6
Multiply tens with ones
5 X 8 = 40
The tens column is already taken. So, set 40 in the next column.
Step 7
Now, multiply ones with tens
6 X 4 = 20.
The recent 10’s column is taken by 40. So, add 50 and subtract 30. (to make it 20). Add 4 in the next column.
Step 8
Now, multiply ones with tens
6 X 8 = 48
The tens column is taken by 40. Therefore, we add 50 and subtract 10 from the same column. Following that, add 8 in the next column.
Step 9
At this time, read the value of beads in the abacus.
It says 2688.
Finally, the answer 2688.
While multiplying using the Chinese abacus, you have to be extremely mindful of the place value.
Division using the Chinese Suanpan abacus
Dividing numbers using the Suan Pan abacus followed the basic steps of manipulation used commonly in all types of abacus, in the history. Also, In case of division, all the five beads on the lower
deck were never pushed up. In such a case, these 5 beads are pushed down and one heaven bead is used to replace these earth beads.
Similarly, in case 2 heaven beads are pushed down, then one earth bead from the previous column is pushed up to replace those beads.
Dividing numbers using the Chinese abacus requires you to memorize a set of traditional rules which will come in handy while dividing the numbers.
The Chinese Division table
In the below set of tables,
1. plus 1 means adding 1 to the column immediately to the right,
2. plus 2 means adding 2 and so on…
3. forward 1 means adding 1 to the column immediately to the left.
Division tables to memorize
1/1 = forward 1
1/2 = 5
2/2 = forward 1
1/3 = 3 plus 1
2/3 = 6 plus 2
3/3 = forward 1
1/4 = 2 plus 2
2/4 = 5
3/4 = 7 plus 2
4/4 = forward 1
1/5 = 2
2/5 = 4
3/5 = 6
4/5 = 8
5/5 = forward
1/6 = 1 plus 4
2/6 = 3 plus 2
3/6 = 5
4/6 = 6 plus 4
5/6 = 8 plus 2
6/6 = forward 1
1/7 = 1 plus 3
2/7 = 2 plus 6
3/7 = 4 plus 2
4/7 = 5 plus 5
5/7 = 7 plus 1
6/7 = 8 plus 4
7/7 = forward 11/8 = 1 plus 2
2/8 = 2 plus 4
3/8 = 3 plus 6
4/8 = 5
5/8 = 6 plus 2
6/8 = 7 plus 4
7/8 = 8 plus 6
8/8 = forward 1
1/9 = 1 plus 1
2/9 = 2 plus 2
3/9 = 3 plus 3
4/9 = 4 plus 4
5/9 = 5 plus 5
6/9 = 6 plus 6
7/9 = 7 plus 7
8/9 = 8 plus 8
9/9 = forward 1
A clear understanding of this table is integral to performing divisions using the Chinese abacus. The tables look complicated at first glance, but when you take a closer look at them, they’re
actually simpler.
For example,
2 /3 = 6 plus 2 ==> 20 ÷ 3 = 6 with remainder 2. Place 2 one column to the right.
7 /7 = forward 1 ==> 7 ÷ 7 = 1, place 1 one column to the left
Let’s divide 128 by 2.
Step 1
Firstly, set zero
Step 2
Secondly, set the dividend 128 on the right side of the device and the divisor 2 on the left end.
Step 3
Thirdly, compare divisor 2 with dividend 1 and follow rule: 1 /2 = 5 . Therefore, the first number in the quotient will be 5
Remove 1 and replace it with 5 leaving the remainder 28 on the abacus.
Step 4
Compare divisor 2 with 2 on dividend side and follow rule 2 /2 = forward 1. Forward 1 to quotient 5 on dividend side and subtract 2, leaving the interim answer 6 and remainder 8
Step 5
Compare divisor 2 with 8 on dividend side and follow rule 8 / 2 = forward 4 . Forward 4 and subtract 8, leaving 64 on the dividend side of the abacus
Step 6
Read the value of beads on the dividend side of the abacus.
It says 64.
Hence the answer 64.
Similarly, larger numbers can also be divided in the same way, using the Suanpan abacus.
How do the Japanese and Chinese abacus differ?
While it is debated that the Japanese(Soroban) and the Chinese (Suan pan) are an influenced versions of each other, there a quite a few remarkable differences between the both.
The Japanese abacus is called as ‘Soroban abacus’, whereas the Chinese abacus is called as the ‘Suan pan’ abacus
The Design
The Suan pan has 2 decks with 2 beads on the top deck and 5 beads on the bottom deck. The Soroban abacus has 2 decks with one bead on the top deck and 4 beads in the bottom
Complexity in calculations
The Soroban abacus had reduced number of beads so calculating using this device was easier and simpler. The Suan pan on the other hand, certainly had more number of beads, leading to complexity in
calculating larger numbers.
The Chinese Suan pans have completely disappeared due to the economic switch to the metric units of measure. They can only be seen in museums or in small heritage shops
As a rarely used medium of calculation. On the contrary, the Japanese Soroban abacus is being actively practiced in Japanese schools. Soroban abacus is also fund to be hanging around in the Asian
region due to its design facilitating decimal calculations.
Suanpan in modern age
The traditional Suan pan abacus is being taught in schools in Hong Kong and China. Surprisingly, in few under-developed countries, the Suan pan abacus is used as the secondary mode of calculation.
This also shows the efficacy, ease of use and the accuracy of the Suan pan abacus.
The early calculators were slow and could handle only 8 to 10 digits. . It is undeniable that after the advent of modern day calculators in pocket sizes, people gradually lost interest in the Chinese
Suan pan abacus.But, slowly as technology advanced, people also realized that the chinese abacus could never compete with a electronic calculating device. This realization hit as arithmetics evolved
into branches like integers, trigonometry, Integration, differentiation, etc.
But is is undeniable that after the advent of modern day calculators in pocket sizes, people gradually lost interest in the Chinese Suan pan abacus.
The Chinese honoring their tradition
Even these days, parents send their children to tutoring classes to learn the traditional Suan pan abacus for two reasons
1. To honor the tradition and to certainly pass it on to the next generation.
2. For their children to learn the usage of bead arithmetic as a mode of calculation, and as a learning aid for better, faster and more accurate mental arithmetic performance.
Not gonna lie, the operational simplicity of the Suan pan has won the hearts of so many Chinese people, that it is still used by few heritage loving small shop owners in China.
The symbol of Chinese identity
Despite the faded popularity and usage, the Chinese Suan pans still stand as the symbol of traditional Chinese identity. By 2002, it was evident that abacus was actually kept aside as a kids learning
activity than as a mode of major calculations. The advent of modern day technological developments does have it’s impact on anything and everything.
While it has taken away a device and a practice that is closely related to the Chinese culture, some parents make the effort to make their kids learn their traditional abacus. Being a symbol of their
ancient identity of evolution, the Suan pan abacus will certainly remain a bench mark in the history of mathematical inventions.
Moving on, people these days are more interested in saving time, in every tedious arithmetic process. But what we do forget, is that we are becoming completely dependent on fancy, electronic devices,
which will eventually make the human brain forget it’s ability to calculate complex calculations using beads and fingers. This is why exactly children these days should make an effort to learn abacus
Let’s be honest here. Furthermore, if abacus isn’t going to help you exactly with your academic performance, you are missing out on a lot of other cognitive abilities. To begin with, focus, improved
creativity, confidence, easy mental math, logical reasoning, easy visualization, imagination of elements and a strengthened memory are a few in that list.
With the fastest moving day to day life with technological innovations, let us not deprive children of training them in the abilities that their brains are capable of. Join Thej academy today to know
more about our courses offered. We have a happy clientele with satisfied parents. In addition, we post informative and educational articles every month on interesting topics related to our courses.
Feel free to give us a call, if you have any questions.
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3 Number Multiplication Worksheets
Math, particularly multiplication, creates the cornerstone of countless scholastic techniques and real-world applications. Yet, for numerous learners, grasping multiplication can posture a
difficulty. To address this difficulty, instructors and moms and dads have accepted a powerful device: 3 Number Multiplication Worksheets.
Introduction to 3 Number Multiplication Worksheets
3 Number Multiplication Worksheets
3 Number Multiplication Worksheets - 3 Number Multiplication Worksheets, 3 Digit Multiplication Worksheets, 3 Digit Multiplication Worksheets Pdf, 3 Digit Multiplication Worksheets For Grade 4,
3-digit Multiplication Worksheets With Answers, 3 Digit Multiplication Worksheets Grade 5, 3 Digit Multiplication Worksheets For Grade 2, 3 Digit Multiplication Worksheets Word Problems, 3 Digit
Multiplication Worksheets Online, 3 Digit Multiplication Worksheets For Grade 6
Free 3rd grade multiplication worksheets including the meaning of multiplication multiplication facts and tables multiplying by whole tens and hundreds Multiply using a number line Multiplication
facts Multiplication tables of 2 and 3 2 x 4 Multiplication tables of 5 and 10 5 x 3 Multiplication tables of 4 and 6
Welcome to The Multiplying 3 Digit by 3 Digit Numbers A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 17 and
has been viewed 2 049 times this week and 2 600 times this month It may be printed downloaded or saved and used in your classroom home school or other educational environment to help
Importance of Multiplication Practice Comprehending multiplication is critical, laying a strong foundation for sophisticated mathematical concepts. 3 Number Multiplication Worksheets use structured
and targeted technique, cultivating a much deeper understanding of this essential math procedure.
Advancement of 3 Number Multiplication Worksheets
Kindergarten Worksheets Free Teaching Resources And Lesson Plans Maths Worksheets
Kindergarten Worksheets Free Teaching Resources And Lesson Plans Maths Worksheets
Grade 5 multiplication worksheets Multiply by 10 100 or 1 000 with missing factors Multiplying in parts distributive property Multiply 1 digit by 3 digit numbers mentally Multiply in columns up to
2x4 digits and 3x3 digits Mixed 4 operations word problems
3 digit multiplication Multiplication practice with all factors under 1 000 column form Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4 Worksheet 5 6 More
From standard pen-and-paper workouts to digitized interactive layouts, 3 Number Multiplication Worksheets have progressed, accommodating diverse understanding designs and preferences.
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Word Problem Worksheets
Real-life situations integrated right into problems, improving crucial reasoning and application abilities.
Timed Multiplication Drills Examinations created to boost rate and accuracy, assisting in rapid mental math.
Advantages of Using 3 Number Multiplication Worksheets
Fun And Free Multiplication Worksheet Printables Grades 3 5 Matematicas Para Colorear
Fun And Free Multiplication Worksheet Printables Grades 3 5 Matematicas Para Colorear
On this page you have a large selection of 2 digit by 1 digit multiplication worksheets to choose from example 32x5 Multiplication 3 Digits Times 1 Digit On these PDF files students can find the
products of 3 digit numbers and 1 digit numbers example 371x3 Multiplication 4 Digits Times 1 Digit
Multiply 3 Numbers Worksheet 6 reviews Maths Calculation Multiplication Free Account Includes Interactive PDF White Rose Maths Supporting Year 4 Spring Block 1 Multiplication and Division Multiply 3
Numbers Multiplying by 0 and 1 and Dividing by 1 Worksheets KS2 Ultimate Times Table Activity Pack
Improved Mathematical Skills
Regular technique develops multiplication effectiveness, boosting total math abilities.
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Word issues in worksheets develop analytical thinking and method application.
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Worksheets fit individual knowing rates, promoting a comfortable and versatile knowing setting.
Just How to Produce Engaging 3 Number Multiplication Worksheets
Including Visuals and Shades Dynamic visuals and shades catch attention, making worksheets aesthetically appealing and involving.
Including Real-Life Scenarios
Associating multiplication to day-to-day scenarios includes relevance and practicality to exercises.
Tailoring Worksheets to Various Skill Degrees Personalizing worksheets based on varying efficiency levels ensures inclusive knowing. Interactive and Online Multiplication Resources Digital
Multiplication Tools and Games Technology-based sources provide interactive understanding experiences, making multiplication interesting and pleasurable. Interactive Internet Sites and Apps Online
platforms give diverse and easily accessible multiplication practice, supplementing standard worksheets. Personalizing Worksheets for Various Discovering Styles Visual Learners Visual aids and
representations help comprehension for students inclined toward visual learning. Auditory Learners Spoken multiplication problems or mnemonics accommodate learners that grasp concepts via auditory
ways. Kinesthetic Students Hands-on activities and manipulatives sustain kinesthetic students in understanding multiplication. Tips for Effective Application in Understanding Uniformity in Practice
Normal method enhances multiplication skills, advertising retention and fluency. Balancing Repeating and Variety A mix of recurring workouts and varied problem formats preserves interest and
comprehension. Giving Constructive Feedback Feedback aids in recognizing areas of renovation, motivating continued development. Difficulties in Multiplication Method and Solutions Inspiration and
Involvement Difficulties Tedious drills can lead to disinterest; innovative approaches can reignite inspiration. Getting Over Anxiety of Math Unfavorable assumptions around mathematics can prevent
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favorable correlation between regular worksheet use and improved mathematics performance.
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Printable Color By number multiplication worksheets Pdf Tims Printables Printable Colornumber
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Check more of 3 Number Multiplication Worksheets below
Multiplication 3rd Grade Math Worksheets Free Printable
Table De Multiplication De 1 A 10 30 Coloriage Magique Multiplication Table Multiplication
Multiplication Worksheet Grade 3 36 Horizontal Multiplication Facts Questions 3 By 0 9 A
Multiplication Worksheets For Grade 3 PDF The Multiplication Table
Second Grade Multiplication Worksheets Multiplication Teaching multiplication Learning Math
Math Multiplication And Division Color By Number 406
Multiplying 3 Digit by 3 Digit Numbers A Math Drills
Welcome to The Multiplying 3 Digit by 3 Digit Numbers A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 17 and
has been viewed 2 049 times this week and 2 600 times this month It may be printed downloaded or saved and used in your classroom home school or other educational environment to help
Worksheets Multiplication by 3 Digit Numbers Super Teacher Worksheets
With these multiplication worksheets student can practice multiplying by 3 digit numbers example 491 x 612 Multiplying by 3 Digit Numbers Multiplication 3 digit by 3 digit FREE Graph Paper Math
Drills 3 digits times 3 digits example 667 x 129 4th through 6th Grades
Welcome to The Multiplying 3 Digit by 3 Digit Numbers A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 17 and
has been viewed 2 049 times this week and 2 600 times this month It may be printed downloaded or saved and used in your classroom home school or other educational environment to help
With these multiplication worksheets student can practice multiplying by 3 digit numbers example 491 x 612 Multiplying by 3 Digit Numbers Multiplication 3 digit by 3 digit FREE Graph Paper Math
Drills 3 digits times 3 digits example 667 x 129 4th through 6th Grades
Multiplication Worksheets For Grade 3 PDF The Multiplication Table
Table De Multiplication De 1 A 10 30 Coloriage Magique Multiplication Table Multiplication
Second Grade Multiplication Worksheets Multiplication Teaching multiplication Learning Math
Math Multiplication And Division Color By Number 406
Free Printable Long Multiplication PrintableMultiplication
Math Multiplication Worksheets Grade 3 Free Printable
Math Multiplication Worksheets Grade 3 Free Printable
Multiplication 3s Worksheet Times Tables Worksheets
Frequently Asked Questions (Frequently Asked Questions).
Are 3 Number Multiplication Worksheets suitable for every age groups?
Yes, worksheets can be customized to different age and ability degrees, making them adaptable for numerous students.
Just how frequently should pupils exercise utilizing 3 Number Multiplication Worksheets?
Consistent technique is vital. Normal sessions, ideally a couple of times a week, can yield substantial improvement.
Can worksheets alone improve mathematics skills?
Worksheets are a beneficial device however must be supplemented with varied learning methods for comprehensive ability growth.
Exist on-line systems supplying cost-free 3 Number Multiplication Worksheets?
Yes, numerous academic sites offer open door to a variety of 3 Number Multiplication Worksheets.
Just how can parents sustain their kids's multiplication practice at home?
Motivating constant technique, giving support, and creating a favorable knowing environment are useful steps.
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How do I calculate my semester GPA?
To calculate your grade point average, first multiply the number of credits each class is worth by the point value for the letter grade that you earned in that class. Next, total the grade points of
all of your classes for that semester and divide it by the number of credit hours that you attempted.
How is cumulative GPA calculated in Nigeria?
1. Your total Course Unit for 100 first semester and the second semester 11 + 9 = 20.
2. Your total Quality Unit for 100 first semester and the second semester 35 + 43 = 78.
3. Divide your Cumulative Quality point by Cumulative Course Unit.
4. That is, 78 / 20 = 3.9 (100 level CGPA)
How do I convert my GPA to CGPA?
To calculate a CGPA, you simply divide your total score of grade points for all subjects throughout your semesters by the total number of credit hours attended. GPA and CGPA are indicated by a number
as opposed to the percentages or grades that are assigned under the Indian grading system.
Is 76 a bad grade in college?
A 76% isn’t bad either, depending how the class is graded you can probably still get a B, and maybe even an A. A 76% isn’t bad either, depending how the class is graded you can probably still get a
B, and maybe even an A. Easily.
What is the formula for GPA?
Your grade point average (GPA) is calculated by dividing the total amount of grade points earned by the total amount of credit hours attempted. Your grade point average may range from 0.0 to a 4.0.
To get the example student’s GPA, the total grade points are divided by the total credit hours attempted.
Is 88 a bad grade?
In the U.S., this is a B+, good, solid grade above average. There is room for improvement, but it is solidly good. If your overall average is an 88, you will probably graduate with honors, or very
nearly so. In the U.K., this is extraordinarily, exceptionally, good.
Is an 80 GPA good?
A 2.7 GPA, or Grade Point Average, is equivalent to a B- letter grade on a 4.0 GPA scale, and a percentage grade of 80–82….List of Common GPA Conversions.
Letter Grade Percent Grade 4.0 GPA Scale
B+ 87–89 3.3
B 83–86 3.0
B- 80–82 2.7
C+ 77–79 2.3
How do you calculate a 5.0 GPA?
In the 5.0 GPA system, letter grades are given points values with “A” being 5, “B” being 4, “C” being 3, “D” being 4 and an “F” being 0. Add together all credits you took for the semester of
interest. If calculating an end-of-college GPA, total all credits taken.
What is a 2.1 degree in GPA in Nigeria?
A second class lower degree or a 2.2 degree in GPA format Is part of the grading system used worldwide and Nigeria inclusive, this GPA means that when the Total cumulative of a semester or a session
at the university was totalled, it was below the 3.5-grade mark which makes it a 2.1 and it was above the 2.49-grade mark …
Is a 3.85 GPA good?
Is a 3.8 GPA good enough to get into college? Your GPA reflects your entire academic record. A 3.8 sits between an A and an A- and is a strong average. However, as you look toward the college
admission process, you may see that some of the most selective schools have freshman classes with higher GPAs.
What is a 80% in GPA?
Letter Percentage 4.0 GPA
A 85 – 89 3.9
A- 80 – 84 3.7
B+ 77 – 79 3.3
B 73 – 76 3
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lit Package Vignette
Andrew J. Bass and Michael P. Epstein
The lit package implements a flexible kernel-based multivariate testing procedure, called Latent Interaction Testing (LIT), to detect latent genetic interactions in a genome-wide association study.
In a standard GWAS analysis, one typically attempts to determine which SNPs are associated with one (or many) traits. Another important question is
• Do any SNPs demonstrate any interactive effects, e.g., gene-by-gene or gene-by-environment interactions?
This question has been very difficult to answer because effect sizes of interactions are likely small, interactive variables are unknown, and there’s often a large multiple testing burden from
testing many candidate interactive variables.
One way to help address some of these issues is to use a variance-based testing procedure which does not require the interactive variable(s) to be specified or observed. These procedures can detect
any unequal residual trait variation among genotype categories at a specific SNP (i.e., heteroskedasticity), which could suggest an unobserved (or latent) genetic interaction. However, researchers
apply such procedures on a trait-by-trait basis and ignore any biological pleiotropy among traits. In fact, it is simple to show that a latent genetic interaction not only induces a variance effect
but also a covariance effect between traits, and these covariance patterns can be harnessed to improve the statistical power.
The lit package addresses this gap by leveraging both the differential variance and differential covariance patterns to substantially increase power to detect latent genetic interactions in a GWAS.
In particular, LIT assesses whether the trait variances/covariances vary as a function of genotype using a kernel-based distance covariance (KDC) framework. LIT often provides substantial increases
in power compared to trait-by-trait univariate approaches, in part because LIT uses shared information (i.e., pleiotropy) across tests and does not require a multiple testing correction which
negatively impacts power.
Note that this package contains the core functionality for the methods described in
Bass AJ, Bian S, Wingo AP, Wingo TS, Culter DJ, Epstein MP. Identifying latent genetic interactions in genome-wide association studies using multiple traits. Submitted; 2023.
Additional software features will be added in the future.
Quick start
We provide two ways to use the lit package. For small GWAS datasets where the genotypes can be loaded in R, the lit() function can be used:
# set seed
# generate SNPs and traits
X <- matrix(rbinom(10 * 10, size = 2, prob = 0.25), ncol = 10)
Y <- matrix(rnorm(10 * 4), ncol = 4)
# test for latent genetic interactions
out <- lit(Y, X)
#> wlit ulit alit
#> 1 0.2681410 0.3504852 0.3056363
#> 2 0.7773637 0.3504852 0.6044655
#> 3 0.4034423 0.3504852 0.3760632
#> 4 0.7874949 0.3504852 0.6157108
#> 5 0.8701189 0.3504852 0.7337565
#> 6 0.2352616 0.3504852 0.2847600
The output is a data frame of \(p\)-values where the rows are SNPs and the columns are different implementations of LIT to test for latent genetic interactions: the first column (wlit) uses a linear
kernel, the second column (ulit) uses a projection kernel, and the third column (alit) maximizes the number of discoveries by combining the \(p\)-values of the linear and projection kernels.
For large GWAS datasets (e.g., biobank-sized), the lit() function is not computationally feasible. Instead, the lit_plink() function can be applied directly to plink files. To demonstrate how to use
the function, we use the example plink files from the genio package:
# load genio package
# path to plink files
file <- system.file("extdata", 'sample.bed', package = "genio", mustWork = TRUE)
# generate trait expression
Y <- matrix(rnorm(10 * 4), ncol = 4)
# apply lit to plink file
out <- lit_plink(Y, file = file, verbose = FALSE)
#> chr id pos alt ref maf wlit ulit alit
#> 1 1 rs3094315 752566 G A 0.3888889 0.7908763 0.3422960 0.6150572
#> 2 1 rs7419119 842013 T G 0.3888889 0.1552580 0.3422960 0.2194972
#> 3 1 rs13302957 891021 G A 0.2500000 0.4088937 0.3325939 0.3687589
#> 4 1 rs6696609 903426 C T 0.3125000 0.5857829 0.3325939 0.4519475
#> 5 1 rs8997 949654 A G 0.4375000 0.6628300 0.3325939 0.4969663
#> 6 1 rs9442372 1018704 A G 0.2500000 0.3192430 0.3325939 0.3258332
See ?lit and ?lit_plink for additional details and input arguments.
Note that a marginal testing procedure for latent genetic interactions based on the squared residuals and cross products (Marginal (SQ/CP)) can also be implemented using the marginal and
marginal_plink functions:
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Re: OLS regression model
Hi everyone,
I am using proc reg for the analysis of my study data.
Dependent variable= mcs score
independent variable= cat2 cat3 age income
where cat2 and cat3 are both categorical variables. My reference group is category 1..I have created dummy variables.
My code is as follows:
ods graphics on;
proc reg data=dummyfinal plots(maxpoints=none);
model mcs42=cat2 cat3;
output out=new P=YHAT RSTUDENT=RESID L95M=LOW U95M=HIGH;
ods graphics off;
after getting the predicted value (YHAT) of the dependent variable, I have to obtain the mean mcs scores across the 3 categories (cat1 cat2 cat3) along with the confidence intervals and do multiple
comparison tests( eg: Tukey kramer).
can anyone please help me with the SAS codes that I should run to obtain the following results.
My results should look like this:
means and SE
Mean (SE) p value
cat1( reference group) 40.45 (0.94) ^∗∗∗ <0.001
cat2 43.76 (0.71) ^∗∗∗ <0.001
cat3 46.96 (0.78) ^∗∗∗ <0.001
06-01-2021 08:24 PM
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The actual Elegance of Mathematical Problem-Solving
Mathematics, often perceived as some sort of realm of equations together with calculations, possesses a unique appeal beyond its utilitarian design. The elegance of exact problem-solving is an
intricate tapestry of creativity, logic, and even beauty. In this article, we start a journey to unravel the charm that fabricates within the world of mathematical conundrums.
The Artistry of Math concepts
Mathematics is more than a tool to get solving real-world problems; it’s actual an art form. Mathematicians, like artisans, create masterpieces on a paper of logic and details. The beauty of
mathematics is like for example the elegance of a beautifully composed symphony or a meticulously crafted sculpture. It’s in the balance, symmetry, and a happy relationship of mathematical
Originality in Problem-Solving
At the heart regarding mathematical elegance lies ingenuity. Mathematicians are akin to poets, weaving intricate narratives together with numbers and symbols. Whenever posed with a problem, these
people approach it with a sense of click for more wonder and attraction. The process of seeking solutions is surely an artistic endeavor, driven simply by imagination and the desire to discover
uncharted territories.
Consider the prominent “Fermat’s Last Theorem, inches a problem that remained unsolved for centuries. Mathematicians from a variety of backgrounds engaged in creative problem-solving, searching for
that elusive facts. The elegance of their options, when finally found, lighted the beauty of human creativity.
Coherence and Precision
While ingenuity ignites the mathematical ignite, logic and precision provide the foundation for problem-solving. Math concepts demands a meticulous strategy, where each step is a foundation in the
construction of a option. This rigorous process would ensure the accuracy and soundness of mathematical proofs.
Beautiful solutions are not just about reaching the correct answer; they include doing so with the utmost picture quality and efficiency. Mathematicians aim for simplicity, elegance, and parsimony in
their solutions. A helpful proof that reveals typically the core of a problem’s importance is often more elegant than a convoluted one.
The Role of Beauty
Beauty is a summary concept, but it undeniably results in mathematics. The beauty of a math solution is in its simplicity and the way it uncovers secret connections and patterns. Mathematicians often
describe their memories of insight and treasure as beautiful experiences, comparable to an artist gazing when a breathtaking landscape.
Mathematical Problem-Solving Beyond Numbers
The grace of mathematical problem-solving runs beyond numerical conundrums. The idea encompasses various branches of mathematics, each with its different charm. For instance, in geometry, the
symmetry of designs and the elegance of evidence are celebrated. In algebra, the artistry lies in exploit symbols to reveal hidden connections. In calculus, the beauty out from understanding change
and also motion.
Applications Beyond Mathematics
The elegance of mathematical problem-solving has far-reaching implications. It’s not confined to the area of pure mathematics however extends to applications in various sphere.
Science: Mathematical elegance underpins scientific theories. The legislation of physics, such as Newton’s equations or Einstein’s hypothesis of relativity, are celebrated for their elegance and
instructive power.
Engineering: Engineers apply mathematical problem-solving to design economical structures, systems, and systems. The elegance of these answers often translates into functionality plus innovation.
Computer Science: Codes and data structures tend to be examples of elegant solutions inside computer science. They optimise processes, reduce complexity, along with drive technological advancements.
Economics: Mathematical models and practices provide insights into global financial systems and behavior, shaping our understanding of complex economical phenomena.
Artificial Intelligence: Equipment learning and artificial cleverness rely on mathematical algorithms to resolve intricate problems, further blurring the line between mathematics along with
Mathematical problem-solving is not just about finding info; it’s a journey of creative imagination, logic, and beauty. The very elegance of mathematics lies in its capacity to inspire think about
and fascination. It’s a skill00 where mathematicians, like artisans, strive to create masterpieces. No matter if revealing the secrets from the universe or enhancing technical advancements,
mathematical elegance is known as a timeless and universal concept that continues to shape the universe. So , the next time you face a mathematical problem, do not forget that within it lies your
global of elegance waiting to become unveiled.
Leave a reply
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How to address omitted variable bias with control variables in regression analysis? | Hire Some To Take My Statistics Exam
How to address omitted variable bias with control variables in regression analysis? Solution 1: Definitive analysis can be link to compare the number of missing values in a regression group (which
are reported by the regression design) with the group of missing values in the control group. If the control variables are treated as dummy variables in the model, the number of missing values
compared with the control group can be reduced. If the control variables are treated as dummy variables in the regression model with unadjusted and adjusted regression coefficients, the number of
missing values normalized by the controls can be reduced. If the control variables are treated as dummy variables in the model with unadjusted regression coefficients, the number of missing values
normalized by the controls can be reduced. The following example illustrates the problem of elimination of missing values, which were reported by the control group and which aren’t reported by the
control group. group=log10(weight);\ result=log10(weight) + (weight*h*(weight))+\ (weight*h)*(weight*h*)+\ (weight*d*(weight))+\ (weight*d*(weight)*h) + N/2; This example explains exactly what is
going on. Since previous tests of regression revealed the following: – Absence of missing variables: one should be identified to be associated with the selected group; – Absence of missing variables:
one should be identified to be associated with the selected over at this website (subjects with missing values); the number of missing values and their unadjusted probability is smaller than 0; In
the following examples, we have three experimental groups. The group without missing values includes the control group only and also the control group with missing values but has the same number of
missing values in the control. This example is taken from Table 1 in Appendix AHow to address omitted variable bias with control variables in regression analysis? In statistical analysis, what is
missing data when the control variables in the regression analysis have an unknown distribution? Let’s start by making a test assuming a non-normal distribution. Let’s make a test for missing data,
and if you want to find the correct distribution then take out every variable in the regression analysis and replace the values with More about the author correct ones. We would have to (i) determine
which of the independent variables were the omitted variable and (ii) write down all of the covariates known as missing. Note: For standard errors, we can omit the zero for example: and (ii) write
down all of the covariates known as missing. Note: When you write up all of the covariates reported in more than one regression analysis, you are missing all the data associated with an independent
variable. The independent variables are only recorded in the regression analysis. You can use SAS or SASX to format out all the independent variables in the regression analysis. The following
formulas describes the details of how to convert the unknown coefficients to the known ones. 1: Covariates that are missing are all the independent variables. Each of the independent variables in a
regression regression analysis can have a unique feature that is known in a separate step, in the regression analysis. Therefore, if you have three independent variables that are omitted or not
common, you can avoid miss-fixing. Forgetting all the missing data is most common in regression analysis, see this text.
Cant Finish On Time Edgenuity
Assuming the missing value is known, the distribution of the unknown data is: This is the data required to get three independent variables out. Notice that there are two ways to write out all the
covariates from the previous step. The first way is to write down the dependent and independent variables. The other way is to write down all the independent variables. Our example used two
independent variables, I, which were missingHow to address omitted variable bias with control variables in regression analysis? — A tool for testing the method of data collection in a clinical
setting. Introduction {#s1} ============ Over the last several years, almost all statistics textbooks have developed statistical forms that measure the quality of the information content of
information resources (pagetrics) and the relative accuracy of data collection (data gathering check here and statistical methodology). Despite the success of such pagers in data collection, clinical
diagnostic methods and statistics equipment have proven inherently variable in making decisions on bias have a peek here at pagers. It is almost impossible to accurately estimate the proportion of
missing data because the pager lacks information and it cannot distinguish between reliable and illegitimate data. To address this, many authors have developed tooles to report bias measures. In this
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and their interaction in the analysis set, without bias concerns. As a testing tool, we describe and measure bias parameters as Kruskal-Wallis test measures. Related Work/Reports {#s2} ==============
====== The most common method to describe the association between visit homepage and demographic variables has long been used outside of the clinical setting. For instance, this tool is relatively
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Elementary Differential Equations and Boundary Value Problems, 12th Edition - WileyPLUS
Elementary Differential Equations and Boundary Value Problems, 12th Edition
By William E. Boyce, Richard C. DiPrima, and Douglas B. Meade
Elementary Differential Equations and Boundary Value Problems, 12^th Edition is written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be
quite theoretical, sometimes intensely practical, and often somewhere in between. In this revision new author Douglas Meade focuses on developing students conceptual understanding with new
Interactive Figures in WileyPLUS that bring concepts to life, along with new concept check questions and worksheets for each chapter.
Elementary Differential Equations and Boundary Value Problems 12^th Edition combines sound and accurate (but not abstract) exposition of the elementary theory of differential equations with
considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications.
Schedule a Demo Request Instructor Account
Interactive Figures
Interactive Figures powered by GeoGebra have been created for many of the figures from the text bringing concepts to life. These figures can be found in WileyPLUS and embedded within the e-text.
Interactive figures are also utilized in WileyPLUS for evaluation on homework, quizzes, and tests.
Worked Examples Video Series
Worked example video series has been included in WileyPLUS to provide students with clear, accurate, and pedagogically sound videos all in one place.
Interactive Questions
Symbolic math and graphing questions powered by GeoGebra allow assignment of more complex auto-graded questions to enhance conceptual understanding.
What’s New to This Course
• Concept Checks have been added to every end of chapter question sets. Concept Check are designed to reinforce key chapter learning objectives and prepare students for the end-of-chapter problems
• Interactive Figures powered by GeoGebra have been created for many of the figures from the text bringing concepts to life. These figures can be found in WileyPLUS and embedded within the e-text.
Interactive figures are also utilized in WileyPLUS for evaluation on homework, quizzes, and tests.
• Updated Question Banks now include more exercises from the text available in WileyPLUS than ever before.
• Worksheets have been developed as a lecture aid to teach class in a synchronous in-person, online or hybrid environment. Worksheets are designed to help students follow the presentation and
discussion of topics in each section. When completed the students should have a good set of notes with examples for that section.
• New TestGen computerized test bank allows for quick creation of algorithmic quizzes or tests. TestGen provides a variety of question types and also allows instructors to create their own
Additional Features Include
Instructor Resources
☆ Instructor’s Solutions Manual
☆ Lecture Note PowerPoints
☆ Chapter Review Sheets
☆ Maple Technology Resources
☆ MATLAB Technology Resources
☆ Mathematica Technology Resources
☆ Projects
☆ WileyPLUS Question Index
☆ Interactive Figures powered by GeoGebra
☆ Printed Test Bank
☆ TestGen Computerized Test Bank
Student Resources
□ Chapter Review Sheets
□ Interactive Figures powered by GeoGebra
□ Student Solutions Manual
□ Maple Technology Resources
□ MATLAB Technology Resources
□ Mathematica Technology Resources
□ Projects
In memory of William Boyce and Richard DiPrima. Two men whose passion for differential equations was instrumental in the shaping of my career. I am forever grateful.
William E. Boyce (deceased) received his B.A. degree in Mathematics from Rhodes College and his M.S. and Ph.D. degrees in Mathematics from Carnegie Mellon University. He was a member of the American
Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. He was also the Edward P. Hamilton Distinguished Professor Emeritus of Science
Education (Department of Mathematical Sciences) at Rensselaer. He authored numerous technical papers in boundary value problems and random differential equations and their applications, as well as
several textbooks including two differential equations texts, and was the coauthor (with M.H. Holmes, J.G. Ecker, and W.L. Siegmann) of a text on using Maple to explore Calculus. He was also coauthor
(with R.L. Borrelli and C.S. Coleman) of Differential Equations Laboratory Workbook (Wiley 1992), which received the EDUCOM Best Mathematics Curricular Innovation Award in 1993. Professor Boyce was a
member of the NSF-sponsored CODEE (Consortium for Ordinary Differential Equations Experiments) that led to the widely acclaimed ODE Architect. He was active in curriculum innovation and reform. Among
other things, he was the initiator of the Computers in Calculus project at Rensselaer, partially supported by the NSF. In 1991, he received the William H. Wiley Distinguished Faculty Award given by
Rensselaer. Professor Boyce passed away on November 4, 2019.
Richard C. DiPrima (deceased) (deceased) received his B.S., M.S., and Ph.D. degrees in Mathematics from Carnegie-Mellon University. He joined the faculty of Rensselaer Polytechnic Institute after
holding research positions at MIT, Harvard, and Hughes Aircraft. He held the Eliza Ricketts Foundation Professorship of Mathematics at Rensselaer, was a fellow of the American Society of Mechanical
Engineers, the American Academy of Mechanics, and the American Physical Society. He was also a member of the American Mathematical Society, the Mathematical Association of America, and the Society
for Industrial and Applied Mathematics. He served as the Chairman of the Department of Mathematical Sciences at Rensselaer, as President of the Society for Industrial and Applied Mathematics, and as
Chairman of the Executive Committee of the Applied Mechanics Division of ASME. In 1980, he was the recipient of the William H. Wiley Distinguished Faculty Award given by Rensselaer. He received
Fulbright fellowships in 1964-65 and 1983 and a Guggenheim fellowship in 1982-83. He was the author of numerous technical papers in hydrodynamic stability and lubrication theory and two texts on
differential equations and boundary value problems. Professor DiPrima passed away on September 10, 1984.
Douglas B. Meade received B.S. degrees in Mathematics and Computer Science from Bowling Green State University, an M.S. in Applied Mathematics from Carnegie Mellon University, and a Ph.D. in
mathematics from Carnegie Mellon University. After a two-year stint at Purdue University, he joined the mathematics faculty at the University of South Carolina, where he is currently an Associate
Professor of mathematics. He is a member of the American Mathematical Society, Mathematics Association of America, and Society for Industrial and Applied Mathematics; in 2016 he was named an ICTCM
Fellow at the International Conference on Technology in Collegiate Mathematics (ICTCM). His primary research interests are in the numerical solution of partial differential equations arising from
wave propagation problems in unbounded domains and from population models for infectious diseases. He is also well-known for his educational uses of computer algebra systems, particularly Maple.
These include Getting Started with Maple (with M. May, C-K. Cheung, and G. E. Keough, Wiley, 2009, ISBN 978-0- 470-45554-8), Engineer’s Toolkit: Maple for Engineers (with E. Bourkoff, Addison-Wesley,
1998, ISBN 0-8053-6445-5), and numerous Maple supplements for numerous calculus, linear algebra, and differential equations textbooks – including previous editions of this book. He was a member of
the MathDL New Collections Working Group for Single Variable Calculus and chaired the Working Groups for Differential Equations and Linear Algebra.
Chapter 1: Introduction
Chapter 2: First-Order Differential Equations
Chapter 3: Second-Order Linear Equations
Chapter 4: Higher-Order Linear Equations
Chapter 5: Series Solutions of Second-Order Linear Equations
Chapter 6: The Laplace Transform
Chapter 7: Systems of First-Order Linear Equations
Chapter 8: Numerical Methods
Chapter 9: Nonlinear Differential Equations and Stability
Chapter 10: Partial Differential Equations and Fourier Series
Chapter 11: Boundary Value Problems and Sturm-Liouville Theory
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A Fraction Unit Does Not Always Begin With Lesson 1 - IM CERTIFIED® BLOG
By Jared Gilman
As I sat down at my local coffee shop to plan my upcoming 5th grade unit on fractions, a wave of dread spread across my body. I started having flashbacks to last winter, when my students’
frustrations with fractions led to daily meltdowns. Looking back at my lesson plans, I noticed how many reteaching lessons I was forced to add into the middle of my unit. I recalled the painstaking
hours of scouring YouTube for videos on the “easiest tricks” and “fastest shortcuts” for adding and subtracting fractions. “My students just didn’t get it,” I thought at the time. This year would be
different, I told myself as I gulped down my large iced coffee.
At the time I wasn’t sure why fractions were so painful for my students, but as I reflected on my lessons, I noticed something strange about my curriculum and unit plans. I realized that there was a
major lack of a comprehensive introduction to fractions for students. I noted that immediately following the previous chapter test on decimals, the first lesson of the next unit was fraction addition
with unlike denominators. Although this lesson hits directly in line with the 5th grade standard (5.NF.A.1), I spent almost no time prior to this lesson discussing fractions, nevermind adding or
subtracting them with like denominators. I tried to think back to the last time the students in my current class had worked with fractions on a daily basis, and it must have been almost 9 months ago!
I considered that it wasn’t that my students didn’t “get” fractions—as I used to think—but rather it was a reflection of me not knowing where my students were in their understanding before starting
this unit. I was constantly playing catch up. All of those reteach days, disappointing quiz grades and meltdowns may have been avoided if I had taken the time at the start of the unit to dig deep
into students’ brains and activate their prior knowledge.
In my desire to limit these same mistakes and frustrations, I wanted to take time to learn more about where my students were in their thinking around fractions and provide more context and
foundations to lead them to that first lesson of the unit. At first I wasn’t sure which topics were the most important to go over, but fortunately, around this same time, I attended an Illustrative
Mathematics professional development session led by Kristin Gray, who provided each teacher with a copy of the Progression Document for the Common Core Math Standards. Reading through the progression
with this new lens really helped me to get a sense of what my students should have learned in prior grades and the representations and models they might have used.
I went through the 3rd and 4th grade progressions and pulled out 3 major ideas that built up to the first lesson in my 5th grade unit on adding fractions with unlike denominators. Those 3 ideas were:
• the importance of unit fractions
• fractions as points on a number line, specifically the importance of benchmark fractions
• fraction equivalency
Once I identified the major review areas, I purposefully chose activities to draw these ideas out and assess students’ understanding. To review unit fractions, I chose to start the lesson with a
number string. This is essentially a number talk, but each problem given builds on each other and typically gets increasingly more difficult. I asked students to compare two fractions and indicate
which was bigger. The order of my questions was this:
During this activity I also asked students to draw a visual to prove why one was bigger than the other. I noticed that students were able to easily use area models to represent fractions, but
struggled to place unit fractions on a number line. This new data helped me gear my mini-lesson and address those points of confusion in the moment. I ended up spending a lot more time on this first
number string than I originally planned, but I was able to elicit discussions and ideas which I believe helped set the foundation for the next few days and the entire unit. In response to their
confusion around unit fractions, I did a quick mini-lesson I adapted from this lesson on the NCTM website, where students created their own fraction strips out of colored construction paper. Since
this was listed as a 3rd grade lesson, I added some rigor by asking students to combine some of the fractions and find relationships between fractions with unlike denominators.
The next day I wanted to push on this idea of number lines and decided to have students create a class fraction number line in the front of the room. Although I was worried about the management
aspect in this lesson, my class rocked it and it led to many conversations and debates about the exact placement of fractions and whether fractions were equivalent or not. One interesting debate
topic that came out of this task that I wasn’t expecting was whether or not 9/8 belonged on the number line at all. Students had placed 0 and 1 at the ends of the number line, and I saved the 9/8
card for last. Some of my most confident mathematicians argued that the 9/8 card was a mistake and didn’t make sense. This not only opened the door for a unexpected back and forth about fractions
greater than 1, but it added another sprinkle of data to inform me about the students’ understanding.
For the last major idea, I decided to lead another number string of adding and subtracting fractions. Based on the 4th grade standards and learning progressions, I started with like denominators, and
then moved into unlike, but with simple equivalence that was related to the activity from the day before i.e. ½ + ¼ . This activity was extremely helpful because of the amount of vocabulary review
and repetition I was able to build in for each problem in the string.
In the first problem we discussed like denominators, in the second problem we reviewed fractions greater than 1, the third problem elicited another review of equivalence to half, and the final
problem touched on fractions with different denominators and how to approach them. This contextualized vocabulary review felt like an added bonus to what was already a meaningful and progressive
opening task. I finished this fraction refresher mini-unit by having my 5th graders use their own sets of fraction strips to explore and find as many equivalencies as possible. I pushed my students
to look for ones beyond ½ and got a ton of great lists! It was after this lesson, when I saw the genuine joy and excitement that my students were having trying to come up with the most equivalences
in the class, that I knew that they were now ready to start the “first lesson” in the fraction unit.
Thinking back on last year, I am embarrassed that I thought one quick review do-now at the start of the unit would be enough. I didn’t realize how many assumptions I was making about my 5th graders
remembering what they learned a year ago. This year, although I was hesitant to break from the curriculum and try brand new tasks with my students who love their routines, the results have been
outstanding. Whereas last year around this same time, I was dealing with breakdowns and groups of students whose frustrations and confusions deepened with each lesson, just yesterday I had two
students ask me if I could “give them more fraction problems for extra credit.” The difference in mood and decreased level of intimidation around fractions have done wonders for my students’
behavior, motivation, and for my own pedagogy about activating prior knowledge going forward. Using the progressions to pull out key concepts, intentionally choosing tasks that matched those big
ideas, and being flexible and open with where those tasks might go, have helped my students truly activate their prior knowledge and informed my instruction for the rest of the unit. I suspect that
planning my fraction unit next year will feel a lot less stressful.
Call to Action or Next Steps
• Take a look at your next upcoming unit and plan out some tasks that may help to assess and activate your students’ prior understanding.
• Take a look through the Progression Document for the Common Core Math Standards, or watch the Fraction Progression videos on the Illustrative Mathematics website and share something new you learn
about fractions from grades 3–5.
• Please share any ideas/feedback if you are in your fractions unit and decide to use number strings, fraction number lines, or other tasks that help to elicit meaningful discussions.
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Enter The Ratio As A Fraction In Lowest Terms6 Ft To 78 In.
To answer this question, we need to remember that a ratio is a comparison of two or more units. We can also need to have into account that these units must be in the same units.
We have one of the quantities is 6ft, the other is equal to 78 inches. We can convert 6ft to its equivalent in inches, or 78 inches into its equivalent in feet. Then, we have:
Then, we can conclude that the ratio of 6feet (72 inches) to 78 inches is equivalent to:
[tex]\text{Ratio}=\frac{72in}{78in}=\frac{12}{13}\Rightarrow Ratio=\frac{12}{13}[/tex]
Average rate of change = -2 celcius / hour
Interpretation: For every unit change in the time (x), there is a change of -2 celcius in the temperature
The average rate of change is to be found for the interval:
[tex]1\text{ }\leq x\leq3[/tex]
From the interval, we can deduce that:
Know the corresponding value of y for each value of x shown above:
[tex]\begin{gathered} \text{When x}_1=1,y_1=\text{ 6} \\ \text{When x}_2=3,y_2=\text{ 0} \end{gathered}[/tex]
The average rate of cgange is given by the formula:
[tex]\begin{gathered} \frac{\delta y}{\delta x}=\text{ }\frac{y_2-y_1}{x_2-x_1} \\ \frac{\delta y}{\delta x}=\text{ }\frac{0-6}{3-1} \\ \frac{\delta y}{\delta x}=\frac{-6}{3} \\ \frac{\delta y}{\
delta x}=-2 \end{gathered}[/tex]
The rate of change for the interval is -2 celcius/hour
This can be interpreted as for every unit change in the time (x), there is a change of -2 celcius in the temperature
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Mental Maths Olympiad Sample Paper for Class 3
Answers to Sample Questions from CREST Olympiads:
Q.1 d Q.2 c Q.3 b Q.4 a Q.5 d Q.6 b Q.7 c Q.8 a Q.9 d Q.10 c
The above Mental Maths competition sample paper for class 3 may be of great assistance to students preparing for the CREST Mental Maths Olympiad (CMMO).
This page contains a free download option of the Mental Maths competition sample paper for class 3. The questions are also accompanied by an answer key.
The benefits of solving a Mental Maths competition sample papers for class 3 before taking the exam are as follows:
1. Regular practice of Mental Maths sample problems can help improve one's ability to perform mental calculations quickly and accurately, which can improve performance in the exams.
2. Solving Mental Maths problems help candidates develop better time management skills.
3. Students can gain more confidence in their ability to perform quick calculations by solving more Mental Maths problems from the sample papers.
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Can I hire someone to perform dimensionality reduction for my website clustering project? | Pay Someone To Take My R Programming Assignment
Can I hire someone to perform dimensionality reduction for my website clustering project? I would need someone to write the dimensions of my website to correct the dimension size of it. How would you
do that? Here’s my code: This is what I have now: ;WITH client_list AS ( SELECT id,site_url AS title,site_name AS site FROM client_list INNER JOIN site_url ON site_url.site = title ) SELECT * FROM
client_list WHERE id IN ( SELECT 0,1,2,3 GROUP BY id ) ORDER BY site/title; Can I hire someone to perform dimensionality reduction for my website clustering project? I have the project, the data for
my project, and it is my wish that there be dimensionality reduction for designing a second dimensionality reduction in the database of a different person. This time I do not have a very high desire
to do dimensionality reduction of a given dataset. For that reason, I wish to describe an alternative to the dimensionality reduction framework and how they can be implemented (3D and/or 3.0C).
[***EDIT***] Thanks, I did not have much time for that, at least my computer does not have a new monitor though, and it takes four games of the same game and all the maps which fit with my present
website. I just installed the 2D version of this project. After that have got it connected and put the 2D version of the database in my desktop browser. Thanks for all the help you have given me.
[***EDIT***] Thanks for the best round of coding, so I can visualize the data more clearly if I think about it. 2 levels in the table: One level represents clustering of dimensions but the scale is
being applied so that two 1+3+3=3 dimensions are enough. One level is going to be a data collection where I have images and text and a database of them, all using the same map methods. Each mapping
is made of 3 levels in a time period. So they are then working out additional info dimension is going to be most suitable for one of them. So most time is spent creating high dimensional maps. If we
have 2 grid levels I will have 10 keys each of dimension 2 and 4. And I will get a map with 16 keys in each dimension. So the best part is what I will look at in the chart when the first data set is
plotted. So far I have done it for each mapping.
Pay Someone To Take My Online Class Reviews
One time I created a grid map and used the methods I presented here to create this map. But I did not manage to take the best part away from the calculations. Besides, knowing how many rows to draw
so far and deciding on it I eventually had to do what I do. I have been working up to 4 or 5 iterations of the mapping. All the work and nothing is done. [***EDIT***] This last part ended up of using
the graph3d library our website my clustering project (2D), making it in several parts. It was quite difficult. My attempts to run the graph3d on my desktop provided this example. How do I write the
computation I did to get it working, and the end result. Furthermore, I also learned how to write the mathematical equation for the case that the dimension or degree of the clustering of the maps
will not be the top of the dataset. I just chose to make some as much smooth as possible when I had said everything about graph3d, though it wasCan I hire someone to perform dimensionality reduction
for my website clustering project? In my design documentation the dimensionality reduction is achieved by using the domain features: the dimensions (as described in Figure 9.21) are arranged in a
Dictadrive-style grid grid plane, and the dimension attributes are derived from a Dictadrive-style datetime pattern named D100: so in the position definition of the domain features, we might have a
lot of D100 and a lot of D100: each row will be D1; and when we have a column named D101 i.e. A1, the dimension of D101 needs to change to an interval of 2 days. Then, on dimensionality reduction, we
have to extract an instance definition for each row like this: You will find here: you can find more exact terms like “distances” in the database like this: They are defined in several ways and we
are doing this for every row (example below). In this example they will represent 9 different classes of cells in Figure 11.6.5. Now I will search in the dataset and then we can see the average
average diagonal distance between the two value vectors and between the average diagonal distance to each of those other columns (as you can see in the “normalization toolbox”). Then now the average
diagonal distance is the dimension parameter of the dimension feature to represent the dataset.
Why Are You Against Online Exam?
Now, let’s go look how to show the distribution. Let’s have a look how to show the normalized distribution. Suppose we have some dataset in Figure 11.7 show how to create an instance definition where
we have three classes (D101, An Object, D101). Then let’s see how to have them be class.2. What’s an instance definition 3 of D101 is for a D101 instance? D101 class contains a shape-invariant class.
These are called object D101.3 and object D101.4 which are not necessarily flat.3. So there is a shape-invariant class. These are called scale.4. So, we have a D101 instance (they are not always
flat). We have a D101 instance which is an object D101.3 with the form. See Figure 11.6.4.
Top Of My Class Tutoring
Now let’s see why it did not add an example in D101 class by using D101 class. The following explanation explains why some of these classes have such a scale. or As you can see the distance vectors
between instance and class are as follows: There are two classes like this. In summary its distance vector between point D103 and point D104 : 1 + D101 now looks like 1 + In Figure 11.7.1 show how
distance of instance D104 is in the class D102. These distance values are plotted
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Mathematics for Elementary Teachers
Adding and Subtracting Fractions
Here are two very similar fractions:
So maybe
The trouble is that a fraction is not a pie, and a fraction is not a child. So adding pies and adding children is not actually adding fractions. A fraction is something different. It is related to
pies and kids, but something more subtle. A fraction is an amount of pie per child.
One cannot add pies, one cannot add children. One must add instead the amounts individual kids receive.
Example: 2/7 + 3/7
Let us take it slowly. Consider the fraction
Consider the fraction
The sum
The answer, from the picture, is
Think / Pair / Share
Remember that why that is the same as the picture given by the sum above:
Your explanation should use both words and pictures!
Most people read this as “two sevenths plus three sevenths gives five sevenths” and think that the problem is just as easy as saying “two apples plus three apples gives five apples.” And, in the end,
they are right!
This is how the addition of fractions is first taught to students: Adding fractions with the same denominator seems just as easy as adding apples:
4 tenths + 3 tenths + 8 tenths = 15 tenths.
(And, if you like,
82 sixty-fifths + 91 sixty-fifths = 173 sixty-fifths:
We are really adding amounts per child not amounts, but the answers match the same way.
We can use the “Pies Per Child Model” to explain why adding fractions with like denominators works in this way.
Example: 2/7 + 3/7
Think about the addition problem
Since in both cases we have 7 kids sharing the pies, we can imagine that it is the same 7 kids in both cases. First, they share 2 pies. Then they share 3 more pies. The total each child gets by the
time all the pie-sharing is done is the same as if the 7 kids had just shared 5 pies to begin with. That is:
Now let us think about the general case. Our claim is that
Translating into our model, we have
But it does not really matter that the kids first share
Think / Pair / Share
• How can you subtract fractions with the same denominator? For example, what is
• Use the “Pies Per Child” model to carefully explain why
• Explain why the fact that the denominators are the same is essential to this addition and subtraction method. Where is that fact used in the explanations?
Fractions with Different Denominators
This approach to adding fractions suddenly becomes tricky if the denominators involved are not the same common value. For example, what is
Let us phrase this question in terms of pies and kids:
Suppose Poindexter is part of a team of five kids that shares two pies. Then later he is part of a team of three kids that shares one pie. How much pie does Poindexter receive in total?
Think / Pair / Share
Talk about these questions with a partner before reading on. It is actually a very difficult problem! What might a student say, if they do not already know about adding fractions? Write down any of
your thoughts.
1. Do you see that this is the same problem as computing
2. What might be the best approach to answering the problem?
One way to think about answering this addition question is to write key fraction rule (that is, multiply the numerator and denominator each by 2, and then each by 3, and then each by 4, and so on)
and to do the same for
We see that the problem
Example: 3/8 + 3/10
Here is another example of adding fractions with unlike denominators:
Of course, you do not need to list all of the equivalent forms of each fraction in order to find a common denominator. If you can see a denominator right away (or think of a faster method that always
works), go for it!
Think / Pair / Share
Cassie suggests the following method for the example above:
When the denominators are the same, we just add the numerators. So when the numerators are the same, shouldn’t we just add the denominators? Like this:
What do you think of Cassie’s suggestion? Does it make sense? What would you say if you were Cassie’s teacher?
On Your Own
Try these exercises on your own. For each addition exercise, also write down a “Pies Per Child” interpretation of the problem. You might also want to draw a picture.
1. What is
2. What is
3. What is
4. What is
5. What is
6. What is
Now try these subtraction exercises.
1. What is
2. What is
3. What is
4. What is
5. What is
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How to find a formula for the nth term of a sequence - TuringBot
How to find a formula for the nth term of a sequence
Given a sequence of numbers, finding an explicit mathematical formula that computes the nth term of the sequence can be challenging, except in very special cases like arithmetic and geometric
In the general case, this task involves searching over the space of all mathematical formulas for the most appropriate one. A special technique exists that does just that: symbolic regression. Here
we will introduce how it works, and use it to find a formula for the nth term in the Fibonacci sequence (A000045 in the OEIS) as an example.
What symbolic regression is
Regression is the task of establishing a relationship between an output variable and one or more input variables. Symbolic regression solves this task by searching over the space of all possible
mathematical formulas for the ones with the greatest accuracy while trying to keep those formulas as simple as possible.
The technique starts from a set of base functions — for instance, sin(x), exp(x), addition, multiplication, etc. Then it tries to combine those base functions in various ways using an optimization
algorithm, keeping track of the most accurate ones found so far.
An important point in symbolic regression is simplicity. It is easy to find a polynomial that will fit any sequence of numbers with perfect accuracy, but that does not tell you anything since the
number of free parameters in the model is the same as the number of input variables. For this reason, a symbolic regression procedure will discard a larger formula if it finds a smaller one that
performs just as well.
Finding the nth Fibonacci term
Now let’s show how symbolic regression can be used in practice by trying to find a formula for the Fibonacci sequence using the desktop symbolic regression software TuringBot. The first two terms of
the sequence are 1 and 1, and every next term is defined as the sum of the previous two terms. Its first terms are the following, where the first column is the index:
A list of the first 30 terms can be found in this file: fibonacci.txt.
TuringBot takes as input TXT or CSV files with one variable per column and efficiently finds formulas that connect those variables. This is what it looks like after we load fibonacci.txt and run the
The software finds not only a single formula but the best formulas of all possible complexities. A larger formula is only shown if it performs better than all simpler alternatives. In this case, the
last formula turned out to predict with perfect accuracy every single one of the first 30 Fibonacci terms. The formula is the following:
f(x) = floor(cosh(-0.111572 + 0.481212 * x))
A very elegant solution. The same procedure can be used to find a formula for the nth term of any other sequence (if it exists).
In this tutorial, we have seen how the symbolic regression software TuringBot can be used to find a closed-form expression for the nth term in a sequence of numbers. We found a very short formula for
the Fibonacci sequence by simply writing it into a text file with one number per row and loading this file into the software.
If you are interested in trying TuringBot your data, you can download it from the official website. It is available for both Windows and Linux.
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University Digital Conservancy :: Browsing by Subject "Electrochemical capacitors"
Browsing by Subject "Electrochemical capacitors"
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• Microscopic theory of supercapacitors.
As new energy technologies are designed and implemented, there is a rising demand for improved energy storage devices. At present the most promising class of these devices is the electric
double-layer capacitor (EDLC), also known as the supercapacitor. A number of recently created supercapacitors have been shown to produce remarkably large capacitance, but the microscopic
mechanisms that underlie their operation remain largely mysterious. In this thesis we present an analytical, microscopic-level theory of supercapacitors, and we explain how such large capacitance
can result. Specifically, we focus on four types of devices that have been shown to produce large capacitance. The first is a capacitor composed of a clean, low-temperature two-dimensional
electron gas adjacent to a metal gate electrode. Recent experiments have shown that such a device can produce capacitance as much as 40% larger than that of a conventional plane capacitor. We
show that this enhanced capacitance can be understood as the result of positional correlations between electrons and screening by the gate electrode in the form of image charges. Thus, the
enhancement of the capacitance can be understood primarily as a classical, electrostatic phenomenon. Accounting for the quantum mechanical properties of the electron gas provides corrections to
the classical theory, and these are discussed. We also present a detailed numerical calculation of the capacitance of the system based on a calculation of the system's ground state energy using
the variational principle. The variational technique that we develop is broadly applicable, and we use it here to make an accurate comparison to experiment and to discuss quantitatively the
behavior of the electrons' correlation function. The second device discussed in this thesis is a simple EDLC composed of an ionic liquid between two metal electrodes. We adopt a simple
description of the ionic liquid and show that for realistic parameter values the capacitance can be as much as three times larger than that of a plane capacitor with thickness equal to the ion
diameter. As in the previous system, this large capacitance is the result of image charge formation in the metal electrode and positional correlations between discrete ions that comprise the
electric double-layer. We show that the maximum capacitance scales with the temperature to the power -1/3, and that at moderately large voltage the capacitance also decays as the inverse one
third power of voltage. These results are confirmed by a Monte Carlo simulation. The third type of device we consider is that of a porous supercapacitor, where the electrode is made from a
conducting material with a dense arrangement of narrow, planar pores into which ionic liquid can enter when a voltage is applied. In this case we show that when the electrode is metallic the
narrow pores aggressively screen the interaction between neighboring ions in a pore, leading to an interaction energy between ions that decays exponentially. This exponential interaction between
ions allows the capacitance to be nearly an order of magnitude larger than what is predicted by mean-field theories. This result is confirmed by a Monte Carlo simulation. We also present a theory
for the capacitance when the electrode is not a perfect metal, but has a finite electronic screening radius. When this screening radius is larger than the distance between pores, ions begin to
interact across multiple pores and the capacitance is determined by the Yukawa-like interaction of a three-dimensional, correlated arrangement of ions. Finally, we consider the case of
supercapacitor electrodes made from a stack of graphene sheets with randomly-inserted "spacer" molecules. For such devices, experiments have produced very large capacitance despite the small
density of states of the electrode material, which would seem to imply poor screening of the ionic charge. We show that these large capacitance values can be understood as the result of
collective entrance of ions into the graphene stack (GS) and the renormalization of the ionic charge produced by nonlinear screening. The collective behavior of ions results from the strong
elastic energy associated with intercalated ions deforming the GS, which creates an effective attraction between them. The result is the formation of "disks" of charge that enter the electrode
collectively and have their charge renormalized by the strong, nonlinear screening of the surrounding graphene layers. This renormalization leads to a capacitance that at small voltages increases
linearly with voltage and is enhanced over mean-field predictions by a large factor proportional to the number of ions within the disk to the power 9/4. At large voltages, the capacitance is
dictated by the physics of graphite intercalation compounds and is proportional to the voltage raised to the power -4/5. We also examine theoretically the case where the effective fine structure
constant of the GS is a small parameter, and we uncover a wealth of scaling regimes.
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Probability Calculator
This Probability Calculator computes the probability of one event, based on known probabilities of other events. And it generates an easy-to-understand report that describes the analysis
For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems. To understand the analysis, read the Summary Report that is produced with each computation. To
learn more, read Stat Trek's tutorial on the rules of probability.
This calculator uses Bayes Rule (aka, Bayes theorem, the multiplication rule of probability) to compute the probability of one event, based on known probabilities of other events.
What is Bayes Rule?
Let A be one event; and let B be any other event from the same sample space, such that P(B) > 0. Then, Bayes rule can be expressed as:
P(A|B) = P(A) P(B|A) P(B)
• P(A) is the probability of Event A.
• P(B) is the probability of Event B.
• P(A|B) is the conditional probability of Event A, given Event B.
• P(B|A) is the conditional probability of Event B, given Event A.
How to Use Bayes Rule
Bayes rule is a simple equation with just four terms. Any time that three of the four terms are known, Bayes Rule can be applied to solve for the fourth term. We've seen in the previous section how
Bayes Rule can be used to solve for P(A|B). By rearranging terms, we can derive equations to solve for each of the other three terms, as shown below:
P(B|A) = P(B) P(A|B) P(A)
P(A) = P(B) P(A|B) P(B|A)
P(B) = P(A) P(B|A) P(A|B)
Extensions to Bayes Rule
The terms that are required to use Bayes Rule can be computed from other probabilities. For example,
P(A) = P(A∩B) / P(B|A)
P(A) = P(A∪B) + P(A∩B) - P(B)
P(B) = P(A∩B) / P(A|B)
P(B) = P(A∪B) + P(A∩B) - P(A)
P(A|B) = P(A∩B) / P(B)
P(B|A) = P(A∩B) / P(A)
• P(A∩B) is the probability of the intersection of Events A and B.
• P(A∪B) is the probability of the union of Events A and B.
Using these formulas, Bayes Rule can be rewritten through substitution to accommodate P(A∩B) and P(A∪B) as inputs. For example, here are two "new" versions of Bayes Rule:
P(A|B) = P(A) + P(B) - P(A∪B) P(B)
To compute the probability of an event, this calculator examines the known probabilities of other events and chooses an appropriate formula to complete the computation.
Frequently-Asked Questions
Instructions: To find the answer to a frequently-asked question, simply click on the question.
Can you explain the notation?
All of the notation used by the Probability Calculator is defined below:
• P( A ):
Probability of event A
• P( B ):
Probability of event B
• P( A|B ):
Conditional probability of event A, given event B
• P( B|A ):
Conditional probability of event B, given event A
• P(A ∪ B):
Probability that event A and/or event B occurs. This is also known as the probability of the union of A and B.
• P(A ∩ B):
Probability that event A and event B both occur. This is also known as the probability of the intersection of A and B.
What is Bayes Rule?
Bayes Rule is an equation that expresses the conditional relationships between two events in the same sample space. Bayes Rule can be expressed as:
P( A | B ) = P( A ) P( B | A ) P( B )
• P( A ) is the probability of Event A.
• P( B ) is the probability of Event B.
• P( A | B ) is the conditional probability of Event A, given Event B.
• P( B | A ) is the conditional probability of Event B, given Event A.
The probability calculator uses Bayes Rule to compute probabilities of one event, given probabilities of other related events.
Can a computed probability be less than 0 or greater than 1.0?
If Event A occurs 100% of the time, the probability of its occurrence is 1.0; that is, P(A) = 1.0. And if Event A never occurs, the probability of its occurrence is 0. In the real world, an event
cannot occur less than 0% of the time or occur more than 100% of the time; so a real-world event must have a probability between 0 and 1.0.
This calculator computes probabilities based on the inputs provided. It is possible to enter probabilities that could not occur together in the real world. When that happens, the calculator may
generate a probability that could not occur in the real world; that is, the calculator could report a probability less than 0 or greater than 1.0.
To illustrate how this could happen, consider Bayes Rule:
P( A | B ) = P( A ) P( B | A ) P( B )
• P(A) is the probability that Event A occurs.
• P(B) is the probability that Event B occurs.
• P(A|B) is the probability that A occurs, given that B occurs.
• P(B|A) is the probability that B occurs, given that A does not occur.
From this equation, we see that P(B) should never be less than P(A)*P(B|A); otherwise, the computed probability of P(A|B) will be greater than 1, which is not a valid outcome. For example, suppose
you plug the following numbers into Bayes Rule:
• P(B) = 0.1
• P(A) = 0.5
• P(B|A) = 0.6
Given these inputs, the Probability Calculator (which uses Bayes Rule) will compute a value of 3.0 for P(A|B), clearly an invalid result. If the calculator computes a probability less than 0 or
greater than 1.0, that is a warning sign. It means your probability inputs are invalid; they do not reflect real-world events.
How can the Probability Calculator help me solve probability problems?
Solving a probability problem is a four-step process:
• Define the problem. Specify the research goal (what you want to know).
• Gather data. Collect information you need to achieve the goal.
• Analyze data. Apply the right analytical technique to achieve the research goal.
• Report results. Present the answer to the research goal.
The Probability Calculator provides a framework to help you with each critical step. From the first dropdown box, identify the probability that you wish to compute. From the second dropdown box,
identify a set of probabilities that will enable you to complete the computation. Then, enter those probabilities into two or more text boxes. And finally, click the Calculate button.
The Probability Calculator does the rest. It applies the right analytical technique to the data you entered. And it creates a summary report that describes the analysis and presents the research
What is E Notation?
E notation is a way to write numbers that are too large or too small to be concisely written in a decimal format. This calculator uses E notation to express very small numbers.
With E notation, the letter E represents "times ten raised to the power of". Here is an example of a very small number written using E notation:
3.02E-12 = 3.02 * 10^-12 = 0.00000000000302
If a probability can be expressed as an ordinary decimal with fewer than 14 digits, the Probability Calculator will do so. But if a probability is very small (nearly zero) and requires a longer
string of digits, the calculator will use E notation to display its value.
Sample Problems
1. Bob is running in two races - a 100-yard dash and a 200-yard dash. The probability of winning the 100-yard dash is 0.25, and the probability of winning the 200-yard dash is 0.50. The probability
of winning at least one race is 0.65. What is the probability that Bob will win both races?
The first step is to define the problem. We begin by identifying the key events:
Let event A = Bob wins the 100-yard dash.
Let event B = Bob wins the 200-yard dash.
Then, we define the main goal, in terms of these events. For the main goal, we want to know the probability of the intersection of events A and B; that is, we want to know P(A ∩ B).
Next, we specify the known probabilities:
P(A) = 0.25.
P(B) = 0.5.
P(A ∪ B) = 0.65.
Now that the problem is defined, we tun to the Probability Calculator. Specifically, we do the following:
□ Select "Find P(A ∩ B)" in the first dropdown box.
□ Select "P(A), P(B), and P(A ∪ B)" in the second dropdown box
□ Enter 0.25 for P(A).
□ Enter 0.5 for P(B).
□ Enter 0.65 for P(A ∪ B).
Then, we hit the Calculate button. Calculator inputs and output are shown below. The analysis indicates that the probability that Bob will win both races is 0.10.
2. Mary is a successful pitcher for her college softball team. On average, she wins 75% of the time. However, when she gives up a home run, Mary wins only 50% of the time. She gives up a home run in
half her games. In her next game, what is the probability that Mary will give up a home run and win?
The first step is to define the problem. We begin by identifying the key events:
Let event A = Mary gives up a home run.
Let event B = Mary wins.
Then, we define the main goal, in terms of these events. For the main goal, we want to know the probability that both events occur; that is, we want to know the probability that Mary gives up a
home run and Mary wins. This is the intersection of events A and B; that is, we want to know P(A ∩ B).
Next, we specify the known probabilities:
P(A) = 0.5, since Mary gives up a home run half the time.
P(B) = 0.75, since Mary wins 75% of the time.
P( B|A ) = 0.5, since Mary wins only half the time when she gives up a home run.
Now that the problem is defined, we enter the problem definition into the Probability Calculator. Specifically, we do the following:
□ Select "P(A ∩ B)" in the first dropdown box.
□ From the options in the second dropdown box, we select "P(A) and P(B|A)". (We select this option, because we know these probabilities.)
□ Enter 0.5 for P(A).
□ Enter 0.5 for P(B|A).
Then, we hit the Calculate button. This produces a summary report that describes the analytical technique and computes the probability of the intersection of events A and B. Thus, we find that P
(A ∩ B) = 0.25.
Note: From the problem statement, we learned that Mary wins 75% of the time; that is, P(B) = 0.75. However, P(B) was not required to solve this problem. We only needed to know P(A) and P(B|A).
Part of the challenge in solving probability problems is distinguishing useful data from superfluous data. The Probability Calculator can help. Use the first dropdown box to choose a probability
to compute. Then, use the second dropdown box to identify other probabilities that will allow you to complete the computation.
For this problem, you would select "Find P(A ∩ B)" from the first dropdown box. Then, when you look at options from the second dropdown box, you would see that one option only requires P(A) and P
(B|A) to compute P(A ∩ B).
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Petameters to Decimeters Converter
Enter Petameters
β Switch toDecimeters to Petameters Converter
How to use this Petameters to Decimeters Converter π €
Follow these steps to convert given length from the units of Petameters to the units of Decimeters.
1. Enter the input Petameters value in the text field.
2. The calculator converts the given Petameters into Decimeters in realtime β using the conversion formula, and displays under the Decimeters label. You do not need to click any button. If the
input changes, Decimeters value is re-calculated, just like that.
3. You may copy the resulting Decimeters value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Petameters to Decimeters?
The formula to convert given length from Petameters to Decimeters is:
Length[(Decimeters)] = Length[(Petameters)] × 1e+16
Substitute the given value of length in petameters, i.e., Length[(Petameters)] in the above formula and simplify the right-hand side value. The resulting value is the length in decimeters, i.e.,
Calculation will be done after you enter a valid input.
Consider that the distance from the Sun to the Oort Cloud is estimated to be around 0.5 petameters.
Convert this distance from petameters to Decimeters.
The length in petameters is:
Length[(Petameters)] = 0.5
The formula to convert length from petameters to decimeters is:
Length[(Decimeters)] = Length[(Petameters)] × 1e+16
Substitute given weight Length[(Petameters)] = 0.5 in the above formula.
Length[(Decimeters)] = 0.5 × 1e+16
Length[(Decimeters)] = 5000000000000000
Final Answer:
Therefore, 0.5 Pm is equal to 5000000000000000 dm.
The length is 5000000000000000 dm, in decimeters.
Consider that the distance from the Milky Way to the Andromeda Galaxy is approximately 2.5 petameters.
Convert this distance from petameters to Decimeters.
The length in petameters is:
Length[(Petameters)] = 2.5
The formula to convert length from petameters to decimeters is:
Length[(Decimeters)] = Length[(Petameters)] × 1e+16
Substitute given weight Length[(Petameters)] = 2.5 in the above formula.
Length[(Decimeters)] = 2.5 × 1e+16
Length[(Decimeters)] = 25000000000000000
Final Answer:
Therefore, 2.5 Pm is equal to 25000000000000000 dm.
The length is 25000000000000000 dm, in decimeters.
Petameters to Decimeters Conversion Table
The following table gives some of the most used conversions from Petameters to Decimeters.
Petameters (Pm) Decimeters (dm)
0 Pm 0 dm
1 Pm 10000000000000000 dm
2 Pm 20000000000000000 dm
3 Pm 30000000000000000 dm
4 Pm 40000000000000000 dm
5 Pm 50000000000000000 dm
6 Pm 60000000000000000 dm
7 Pm 70000000000000000 dm
8 Pm 80000000000000000 dm
9 Pm 90000000000000000 dm
10 Pm 100000000000000000 dm
20 Pm 200000000000000000 dm
50 Pm 500000000000000000 dm
100 Pm 1000000000000000000 dm
1000 Pm 10000000000000000000 dm
10000 Pm 100000000000000000000 dm
100000 Pm 1e+21 dm
A petameter (Pm) is a unit of length in the International System of Units (SI). One petameter is equivalent to 1,000,000,000,000,000 meters or approximately 621,371,192,237,333 miles.
The petameter is defined as one quadrillion meters, making it a measurement for extraordinarily large distances, often used in theoretical and cosmological contexts.
Petameters are used in fields such as astronomy and cosmology to describe distances on a scale larger than terameters. They provide a convenient way to express distances across immense regions of
space, such as those encompassing multiple galaxies or even superclusters of galaxies.
A decimeter (dm) is a unit of length in the International System of Units (SI). One decimeter is equivalent to 0.1 meters or approximately 3.937 inches.
The decimeter is defined as one-tenth of a meter, making it a convenient measurement for intermediate lengths.
Decimeters are used worldwide to measure length and distance in various fields, including science, engineering, and everyday life. They provide a useful scale for measurements that are larger than
centimeters but smaller than meters, and are commonly used in educational settings and certain industries.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Petameters to Decimeters in Length?
The formula to convert Petameters to Decimeters in Length is:
Petameters * 1e+16
2. Is this tool free or paid?
This Length conversion tool, which converts Petameters to Decimeters, is completely free to use.
3. How do I convert Length from Petameters to Decimeters?
To convert Length from Petameters to Decimeters, you can use the following formula:
Petameters * 1e+16
For example, if you have a value in Petameters, you substitute that value in place of Petameters in the above formula, and solve the mathematical expression to get the equivalent value in Decimeters.
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Download Algebra And Trigonometry eBooks for Free
PDF Drive is your search engine for PDF files. As of today we have 75,657,331 eBooks for you to download for free. No annoying ads, no download limits, enjoy it and don't forget to bookmark and share
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Algebra And Trigonometry Books
“ Be grateful for whoever comes, because each has been sent as a guide from beyond. ” ― Rumi
Ask yourself:
What do you fear most when it comes to finding your passion?
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\sin x}}{x} = 1$ ( $x$ being measured in radians).
Sandwich theorem is useful in proving the limits given in the question.
Theorem: Sandwich Theorem:
Let \[f,g\] and $h$ be real functions such that $g\left( x \right) \leqslant f\left( x \right) \leqslant h\left( x \right)$ for all $x$ in the common domain of definition.
For some limit $a$, if \[\mathop {\lim }\limits_{x \to a} g\left( x \right) = l = \mathop {\lim }\limits_{x \to a} h\left( x \right)\], then \[\mathop {\lim }\limits_{x \to a} f\left( x \right) = l
\]. This can be illustrated as the following:
Try to prove the inequality relating to trigonometric functions. $\cos x < \dfrac{{\sin x}}{x} < 1$, and the given limit can be easily proved by the sandwich theorem.
Complete step by step answer:
Step 1: Prove the inequality $\cos x < \dfrac{{\sin x}}{x} < 1$
Consider figure 1.
In figure 1, O is the center of the unit circle such that the angle $\angle AOC$ is $x$ radians and $0 < x < \dfrac{\pi }{2}$.
Line segment BA and CD are perpendicular to OA.
Further, join AC. Then
Area of $\vartriangle AOC$ < area of sector $OAC$ < area of $\vartriangle AOB$
The area of a triangle is half of the product of base and height.
Area of a sector of a circle = $\dfrac{\theta }{{2\pi }}\left( {\pi {r^2}} \right)$, where $\theta $ is the angle of the sector.
$ \Rightarrow \dfrac{1}{2}OA.CD < \dfrac{x}{{2\pi }}\pi {\left( {OA} \right)^2} < \dfrac{1}{2}OA.AB$
$ \Rightarrow CD < x\left( {OA} \right) < AB$ …… (1)
In $\vartriangle OCD$
$\sin x = \dfrac{{{\text{perpendicular}}}}{{{\text{hypotenuse}}}}$
Therefore, $\sin x = \dfrac{{CD}}{{OC}}$
The line segments OC and OA are the radius of the circle with center O in figure 1.
Thus, OC = OA
Therefore, $\sin x = \dfrac{{CD}}{{OA}}$
Hence, $CD = OA\sin x$
In $\vartriangle AOB$
$\tan x = \dfrac{{{\text{perpendicular}}}}{{{\text{base}}}}$
Therefore, $\tan x = \dfrac{{AB}}{{OA}}$
Hence, $AB = OA\tan x$
Put the values of CD and AB in the inequality (1)
$ \Rightarrow OA\sin x < x\left( {OA} \right) < OA\tan x$
We know $\tan x = \dfrac{{\sin x}}{{\cos x}}$
\[ \Rightarrow \sin x < x < \dfrac{{\sin x}}{{\cos x}}\]
Dividing throughout by $\sin x$, we get:
\[ \Rightarrow 1 < \dfrac{x}{{\sin x}} < \dfrac{1}{{\cos x}}\]
Take reciprocals throughout, we have:
$ \Rightarrow \cos x < \dfrac{{\sin x}}{x} < 1$
Step 2: Use sandwich theorem to prove the given limit
We know that $\mathop {\lim }\limits_{x \to a} \cos \left( {f\left( x \right)} \right) = \cos \mathop {\lim }\limits_{x \to a} \left( {f\left( x \right)} \right)$
Thus, the \[\mathop {\lim }\limits_{x \to 0} \cos x = \cos \mathop {\lim }\limits_{x \to 0} \left( x \right)\]
Therefore, $\cos 0 = 1$
Hence, $\mathop {\lim }\limits_{x \to 0} \cos x = 1$
And $\mathop {\lim }\limits_{x \to 1} 1 = 1$
We have, $\mathop {\lim }\limits_{x \to 0} \cos x = 1 = \mathop {\lim }\limits_{x \to 0} 1$
Then $\mathop {\lim }\limits_{x \to 0} \dfrac{{\sin x}}{x} = 1$ by the sandwich theorem.
The limit $\mathop {\lim }\limits_{x \to 0} \dfrac{{\sin x}}{x} = 1$ has been proved.Note:
Use the above limit $\mathop {\lim }\limits_{x \to 0} \dfrac{{\sin x}}{x} = 1$ for future questions. For example:
Evaluate: $\mathop {\lim }\limits_{x \to 0} \dfrac{{\sin 4x}}{{\sin 2x}}$
Multiplying and dividing by $4x$ and make the angles in the sine function and dividing angle the same.
\[ \Rightarrow \mathop {\lim }\limits_{x \to 0} \left[ {\dfrac{{\sin 4x}}{{4x}} \times \dfrac{{2x}}{{\sin 2x}} \times 2} \right]\]
\Rightarrow \mathop {\lim }\limits_{x \to 0} \dfrac{{\sin 4x}}{{4x}} \times \mathop {\lim }\limits_{x \to 0} \left[ {\dfrac{1}{{\dfrac{{\sin 2x}}{{2x}}}}} \right] \times \mathop {\lim }\limits_{x \to
0} 2 \\
\Rightarrow \mathop {\lim }\limits_{x \to 0} \dfrac{{\sin 4x}}{{4x}} \times \left[ {\dfrac{{\mathop {\lim }\limits_{x \to 0} 1}}{{\mathop {\lim }\limits_{x \to 0} \dfrac{{\sin 2x}}{{2x}}}}} \right] \
times \mathop {\lim }\limits_{x \to 0} 2 \\
\Rightarrow 1 \times \dfrac{1}{1} \times 2 \\
\Rightarrow 2 \\
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Graph Attention Networks (GAT): Utilizing Attention Mechanisms on Graph Vertices
Graph Attention Networks (GAT) leverage attention mechanisms to enhance node feature aggregation in graph-structured data. GATs are designed to focus on the most relevant parts of the graph by
assigning different weights to different nodes. This attention mechanism allows the model to capture the importance of neighboring nodes, improving the overall representation learning process.
Benefits of Graph Attention Networks
• Dynamic Importance Learning: GATs learn the importance of neighboring nodes dynamically, which leads to more informative node representations.
• Scalability: By focusing on local neighborhoods, GATs can scale efficiently to larger graphs.
• Flexibility: GATs can be applied to a variety of graph-related problems without significant changes to the core algorithm.
• Complexity: The introduction of attention mechanisms increases the complexity of the model.
• Computational Overhead: Attention calculation can be computationally intensive, especially for large graphs.
• Memory Usage: Storing attention weights for large graphs can lead to increased memory consumption.
Use Cases
• Social Network Analysis: Identifying influential users or communities in social networks.
• Molecular Graphs: Predicting molecular properties by focusing on relevant parts of the molecular structure.
• Recommendation Systems: Enhancing item recommendations by understanding user-item interactions better.
• Traffic Networks: Modeling traffic flow by considering the importance of various intersections and roads.
UML Class Diagram
Here is a UML class diagram representing the core structure of a Graph Attention Network:
class GraphAttentionNetwork {
+forward(graph: Graph) : Tensor
+compute_attention_weights(node_features: Tensor) : Tensor
+aggregate_node_features(attention_weights: Tensor) : Tensor
class Graph {
+nodes: List[Node]
+edges: List[Edge]
+add_node(node: Node)
+add_edge(edge: Edge)
class Node {
+id: String
+features: Tensor
class Edge {
+source: Node
+target: Node
+weight: Float
Graph --> Node
Graph --> Edge
GraphAttentionNetwork --> Graph
UML Sequence Diagram
A sequence diagram depicting the forward pass in a Graph Attention Network:
participant User
participant GraphAttentionNetwork
participant Graph
participant Node
User->>GraphAttentionNetwork: forward(graph)
GraphAttentionNetwork->>Graph: get_nodes()
Graph->>GraphAttentionNetwork: nodes
GraphAttentionNetwork->>Node: compute_attention_weights(features)
Node-->>GraphAttentionNetwork: attention_weights
GraphAttentionNetwork->>GraphAttentionNetwork: aggregate_node_features(attention_weights)
GraphAttentionNetwork-->>User: updated_node_features
Examples in Different Programming Languages
import torch
import torch.nn.functional as F
from torch_geometric.nn import GATConv
class GAT(torch.nn.Module):
def __init__(self, in_channels, out_channels):
super(GAT, self).__init__()
self.conv1 = GATConv(in_channels, out_channels, heads=1)
def forward(self, data):
x, edge_index = data.x, data.edge_index
x = self.conv1(x, edge_index)
return F.elu(x)
# data = ... # Load your graph data here
# output = model(data)
import org.deeplearning4j.nn.conf.MultiLayerConfiguration;
import org.deeplearning4j.nn.conf.NeuralNetConfiguration;
import org.deeplearning4j.nn.conf.graph.GraphAttentionVertex;
import org.deeplearning4j.nn.multilayer.MultiLayerNetwork;
public class GAT {
public static void main(String[] args) {
MultiLayerConfiguration conf = new NeuralNetConfiguration.Builder()
.layer(0, new GraphAttentionVertex.Builder(16, 32).build())
MultiLayerNetwork model = new MultiLayerNetwork(conf);
// Load your graph data and train the model
import org.apache.spark.graphx.Graph
import org.deeplearning4j.nn.conf.NeuralNetConfiguration
import org.deeplearning4j.nn.conf.layers.GraphAttentionLayer
import org.deeplearning4j.nn.multilayer.MultiLayerNetwork
object GAT {
def main(args: Array[String]): Unit = {
val conf = new NeuralNetConfiguration.Builder()
.layer(new GraphAttentionLayer.Builder(16, 32).build())
val model = new MultiLayerNetwork(conf)
// Load your graph data and train the model
(ns gat-example
(:require [dl4clj.nn.conf.builders :as b]
[dl4clj.nn.api.multi-layer-network :as mln]))
(def conf (b/multi-layer-configuration-builder
:list {:type :graph-attention
:in 16
:out 32}))
(def model (mln/new-multi-layer-network conf))
(mln/init! model)
;; Load your graph data and train the model
• Graph Convolutional Networks (GCN): GCNs aggregate node information in a graph by leveraging convolution operations over the graph structure.
• GraphSAGE: This method samples and aggregates features from a node’s local neighborhood.
• Attention Mechanisms: Widely used in NLP, attention mechanisms allow models to focus on specific parts of the input.
Resources and References
Graph Attention Networks (GAT) introduce attention mechanisms to graph neural networks, enhancing the ability to dynamically learn the importance of neighboring nodes for node feature aggregation.
While they offer significant benefits in terms of flexibility and scalability, they also come with increased computational and memory overhead. GATs have a wide range of applications, including
social network analysis, molecular graphs, recommendation systems, and traffic networks.
Implementing GATs in various programming languages showcases their versatility and adaptability to different environments. Understanding the trade-offs and related patterns allows for informed
decisions when choosing or designing graph-based models for specific applications.
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Theory of Nonlinear Curve Fitting
15.3.3 Theory of Nonlinear Curve Fitting
How Origin Fits the Curve
The aim of nonlinear fitting is to estimate the parameter values which best describe the data. Generally we can describe the process of nonlinear curve fitting as below.
1. Generate an initial function curve from the initial values.
2. Iterate to adjust parameter values to make data points closer to the curve.
3. Stop when minimum distance reaches the stopping criteria to get the best fit
Origin provides options of different algorithm, which have different iterative procedure and statistics to define minimum distance.
Explicit Functions
Fitting Model
A general nonlinear model can be expressed as follows:
$Y=f(X, \boldsymbol{\beta})+\varepsilon$ (1)
where $X = (x_1, x_2, \cdots , x_k)'$ is the independent variables and $\boldsymbol{\beta} = (\beta_1, \beta_2, \cdots , \beta_k)'$ is the parameters.
Examples of the Explicit Function
• $y=y_0+Ae^{-x/t}$
Least-Squares Algorithms
The least square algorithm is to choose the parameters that would minimize the deviations of the theoretical curve(s) from the experimental points. This method is also called chi-square minimization,
defined as follows:
$\chi ^2=\sum_{i=1}^n \left [ \frac{Y_i-f(x_i^{\prime },\hat{\beta }) } {\sigma _i} \right ]^2$ (2)
where $x_i^{\prime }$ is the row vector for the ith (i = 1, 2, ... , n) observation.
The figure below illustrates the concept to a simple linear model (Note that multiple regression and nonlinear fitting are similar).
The Best-Fit Curve represents the assumed theoretical model. For a particular point $(x_i,y_i)\,\!$ in the original dataset, the corresponding theoretical value at $x_i\,\!$ is denoted by$\widehat
If there are two independent variables in the regression model, the least square estimation will minimize the deviation of experimental data points to the best fitted surface. When there are more
then 3 independent variables, the fitted model will be a hypersurface. In this case, the fitted surface (or curve) will not be plotted when regression is performed.
Origin provides two options to adjust the parameter values in the iterative procedure
Levenberg-Marquardt (L-M) Algorithm
The Levenberg-Marquardt (L-M) algorithm^11 is a iterative procedure which combines the Gauss-Newton method and the steepest descent method. The algorithm works well for most cases and become the
standard of nonlinear least square routines.
1. Compute the $\chi ^2(b)$ value from the given initial values: $b$ .
2. Pick a modest value for $\lambda$, say $\lambda$ = 0.001
3. Solve the Levenberg-Marquardt funciton^11 for $\delta b$ and evaluate $\chi ^2(\beta + \delta b)$
4. If $\chi ^2(\beta + \delta b) \geq \chi ^2(b)$,increase $\lambda$ by a factor of 10 and go back to step 3
5. if $\chi ^2(\beta + \delta b) \leq \chi ^2(b)$, decrease $\lambda$ by a factor of 10, update the parameter values to be $\delta b$ and go back to step 3
6. Stop until the values computed in two successive iterations are small enough (compared with the tolerance)
Downhil Simplex Algorithm
Besides the L-M method, Origin also provides a Downhill Simplex approximation^9,10. In geometry, a simplex is a polytope of N + 1 vertices in N dimensions. In non-linear optimization, an analog
exists for an objective function of N variables. During the iterations, the Simplex algorithm (also known as Nelder-Mead) adjusts the parameter "simplex" until it converges to a local minimum.
Different from L-M method, the Simplex method does not require derivatives, and it is effective when the computational burden is small. Normally, if you did not get a good value for parameter
initialization, you can try this method to get the approximate parameter value for further fitting calculations with L-M. The Simplex method tends to be more stable in that it is less likely to
wander into a meaningless part of the parameter space; on the other hand, it is generally much slower than L-M, especially very close to a local minimum. Actually, there is no "perfect" algorithm for
nonlinear fitting, and many things may affect the result (e.g., initial values). In complicated models, you may find one method may do better than the other. Additionally, you may want to try both
methods to perform the fitting operation.
Orthogonal Distance Regression (ODR) Algorithm
The Orthogonal Distance Regression (ODR) algorithm minimizes the residual sum of squares by adjusting both fitting parameters and values of the independent variable in the iterative process. The
residual in ODR is not the difference between the observed value and the predicted value for the dependent variable, but the orthogonal distance from the data to the fitted curve.
Origin uses the ODR algorithm in ODRPACK95^8.
For a explict function, the ODR algorithm could be expressed as:
$\min\left (\sum_{i=1}^{n}\left (w_{yi}\cdot \epsilon_{i} ^{2}+w_{xi}\cdot \delta_{i}^{2} \right ) \right )$
subject to the constraints:
$y_{i}=f\left ( x_{i} +\delta_{i}; \beta \right )-\epsilon _{i}\ \ \ \ \ \ i=1,...,n$
where $w_{xi}$ and $w_{yi}$ are the user input weights of $x_{i}$ and $y_{i}$, $\delta_{i}$ and $\epsilon_{i}$ are the residual of the corresponding $x_{i}$ and $y_{i}$, and $\beta$ is the fitting
For more details of the ODR algorithm, please refer to ODRPACK95^8.
Comparison between ODR and L-M
To choose the ODR or L-M algorithm for your fitting, you may refer to the following table for information:
Orthogonal Distance Regression Levenberg-Marquardt
Application Both implicit and explicit functions Only explicit functions
Weight Support both x weight and y weight Support only y weight
Residual Source The orthogonal distance from the data to the fitted curve The difference between the observed value and the predicted value
Iteration Process Adjusting the values of fitting parameters and independent variables Adjusting the values of fitting parameters
Implicit Functions
Fitting Model
A general implicit function could be expressed as:
$f\left ( X, Y, \beta \right )-const=0$ (5)
where $X = (x_1, x_2, \cdots , x_k)'$ and $Y = (y_1, y_2, \cdots , y_k)'$ are the variables, $\beta$ are the fitting parameters and $const$ is a constant.
Examples of the Implicit Function:
• $f = \left(\frac{x-x_c}{a}\right)^2 + \left(\frac{y-y_c}{b}\right)^2 - 1$
Orthogonal Distance Regression (ODR) Algorithm
The ODR method can be used for both implicit functions and explicit functions. To learn more details of ODR method, please refer to the description of ODR mehtod in above section
For implicit functions, the ODR algorithm could be expressed as:
$\min\left (\sum_{i=1}^{n}\left ( w_{xi}\cdot \delta_{xi}^{2}+w_{yi}\cdot \delta_{yi}^{2} \right ) \right )$
subject to:
$f\left ( x_{i}+\delta_{xi},y_{i}+\delta_{yi},\beta \right )= 0\ \ \ \ \ \ i=1,...,n$
where $w_{xi}$ and $w_{yi}$ are the user input weights of $x_{i}$ and $y_{i}$, $\delta_{xi}$ and $\delta_{yi}$ are the residual of the corresponding $x_{i}$ and $y_{i}$, and $\beta$ is the fitting
Weighted Fitting
When the measurement errors are unknown, $\sigma _i\,\!$ are set to 1 for all i, and the curve fitting is performed without weighting. However, when the experimental errors are known, we can treat
these errors as weights and use weighted fitting. In this case, the chi-square can be written as:
$\chi ^2=\sum_{i=1}^nw_i[Y_i-f(x_i^{\prime },\hat \beta )]^2$ (6)
There are a number of weighting methods available in Origin. Please read Fitting with Errors and Weighting in the Origin Help file for more details.
The fit-related formulas are summarized here:
The Fitted Value
Computing the fitted values in nonlinear regression is an iterative procedure. You can read a brief introduction in the above section (How Origin Fits the Curve), or see the below-referenced material
for more detailed information.
Parameter Standard Errors
During L-M iteration, we need to calculate the partial derivatives matrix F, whose element in ith row and jth column is:
$F_{ij}=\frac{\partial f(x,\theta )}{\sigma _i\partial \theta _j}$ (7)
where $\sigma _i$ is the error of y for the ith observation if Instrumental weight is used. If there is no weight, $\sigma _i = 1$. And $F_{ij}$ is evaluated for each observation $x_i$ in each
Then we can get the Variance-Covariance Matrix for parameters by:
$C=(F'F)^{-1}s^2\,\!$ (8)
where $F'$ is the transpose of the F matrix, s^2 is the mean residual variance, also called Reduced Chi-Sqr, or the Deviation of the Model, and can be calculated as follows:
$s^2=\frac{RSS}{n-p}$ (9)
where n is the number of points, and p is the number of parameters.
The square root of the main diagonal value of this matrix C is the Standard Error of the corresponding parameter
$s_{\theta _i}=\sqrt{c_{ii}}\,\!$ (10)
where C[ii] is the element in ith row and ith column of the matrix C. C[ij] is the covariance between θ[i] and θ[j].
You can choose whether to exclude s^2 when calculating the covariance matrix. This will affect the Standard Error values. When excluding s^2, clear the Use reduce Chi-Sqr check box on the Advanced
page under Fit Control panel. The covariance is then calculated by:
So the Standard Error now becomes:
$s_{\theta _i}^{\prime }=\frac{s_{\theta _i}}s\,\!$ (12)
The parameter standard errors can give us an idea of the precision of the fitted values. Typically, the magnitude of the standard error values should be lower than the fitted values. If the standard
error values are much greater than the fitted values, the fitting model may be overparameterized.
The Standard Error for Derived Parameter
Origin estimates the standard errors for the derived parameters according to the Error Propagation formula, which is an approximate formula.
Let $z = f\left (\theta _1, \theta _2, ..., \theta _p \right )$ be the function with a combination (linear or non-linear) of $p\,$ variables $\theta _1, \theta _2, ..., \theta _p \,$.
The general law of error propagation is:
$\sigma_z^2 = \sum_i^p \sum_j^p \frac {\partial z}{\partial \theta_i} COV_{\theta_i \theta_j} \frac {\partial z}{\partial \theta_j}$
where $COV_{\theta_i \theta_j}\,$ is the covariance value for $\left (\theta_i, \theta_j \right )$, and $\left (i = 1, 2, ..., p \right ), \left (j = 1, 2, ..., p \right )$.
You can choose whether to exclude mean residual variance $s^2$ when calculating the covariance matrix $COV_{\theta_i \theta_j}$, which affects the Standard Error values for derived parameters. When
excluding $s^2$, clear the Use reduce Chi-Sqr check box on the Advanced page under Fit Control panel.
For example, using three variables
$z = f\left (\theta_1, \theta_2, \theta_3 \right )$
we get:
$\sigma_z^2 = \left (\frac {\partial z}{\partial \theta_1} \right )^2 \sigma_{\theta_1}^2 + \left (\frac {\partial z}{\partial \theta_2} \right )^2 \sigma_{\theta_2}^2 + \left (\frac {\partial z}
{\partial \theta_3} \right )^2 \sigma_{\theta_3}^2 + 2 \left (\frac {\partial z}{\partial \theta_1} \frac {\partial z}{\partial \theta_2} \right ) COV_{\theta_1 \theta_2} + 2 \left (\frac {\
partial z}{\partial \theta_1} \frac {\partial z}{\partial \theta_3} \right ) COV_{\theta_1 \theta_3} + 2 \left (\frac {\partial z}{\partial \theta_2} \frac {\partial z}{\partial \theta_3} \right
) COV_{\theta_2 \theta_3}$
Now, let the derived parameter be $z\,$, and let the fitting parameters be $\theta_1, \theta_2, ..., \theta_p\,$. The standard error for the derived parameter $z\,$ is $\sigma_z\,$.
Confidence Intervals
Origin provides two methods to calculate the confidence intervals for parameters: Asymptotic-Symmetry method and Model-Comparison method.
Asymptotic-Symmetry Method
One assumption in regression analysis is that data is normally distributed, so we can use the standard error values to construct the Parameter Confidence Intervals. For a given significance level, α,
the (1-α)x100% confidence interval for the parameter is:
$\hat \theta _j-t_{(\frac \alpha 2,n-p)}s_{\theta _j}\leq \hat \theta _j\leq \hat \theta _j+t_{(\frac \alpha 2,n-p)}s_{\theta _j}$ (13)
The parameter confidence interval indicates how likely the interval is to contain the true value.
The confidence interval illustrated above is Asymptotic, which is the most frequently used method to calculate the confidence interval. The "Asymptotic" here means it is an approximate value.
Model-Comparison Method
If you need more accurate values, you can use the Model Comparison Based method to estimate the confidence interval.
If the Model Comparison method is used, the upper and lower confidence limits will be calculated by searching for the values of each parameter p that makes RSS(θ[j]) (minimized over the remaining
parameters) greater than RSS by a factor of (1+F/(n-p)).
$RSS(\theta _j)=RSS(1+F\frac 1{n-p})$ (14)
where F = Ftable(α,1,n-p)and RSS is the minimum residual sum of square found during the fitting session.
t Value
You can choose to perform a t-test on each parameter to see whether its value is equal to 0. The null hypothesis of the t-test on the jth parameter is:
And the alternative hypothesis is:
$H_\alpha : \theta_j e 0$
The t-value can be computed as:
$t=\frac{\hat \beta _j-0}{s_{\hat \beta _j}}$ (15)
The probability that H[0] in the t test above is true.
$prob=2(1-tcdf(|t|,df_{Error}))\,\!$ (16)
where tcdf(t, df) computes the lower tail probability for Student's t distribution with df degree of freedom.
If the equation is overparameterized, there will be mutual dependency between parameters. The dependency for the ith parameter is defined as:
$1-\frac 1{c_{ii}(c^{-1})_{ii}}$ (17)
and (C^-1)[ii] is the (i, i)th diagonal element of the inverse of matrix C. If this value is close to 1, there is strong dependency.
To learn more about how the value assess the quality of a fit model, see Model Diagnosis Using Dependency Values page
CI Half Width
The Confidence Interval Half Width is:
$CI=\frac{UCL-LCL}2$ (18)
where UCL and LCL is the Upper Confidence Interval and Lower Confidence Interval, respectively.
Several fit statistics formulas are summarized below:
Degree of Freedom
The Error degree of freedom. Please refer to the ANOVA Table for more details.
Residual Sum of Squares
The residual sum of squares:
$RSS(X,\hat \theta )=\sum_{i=1}^n w_i[Y_i-f(x_i^{\prime },\hat \theta )]^2$ (19)
Reduced Chi-Sqr
The Reduced Chi-square value, which equals the residual sum of square divided by the degree of freedom.
$Reduced\chi ^2=\frac{\chi ^2}{df_{Error}}=\frac{RSS}{df_{Error}}$ (20)
R-Square (COD)
The R^2 value shows the goodness of a fit, and can be computed by:
$R^2=\frac{Explained\,variation}{Total\,variation}=\frac{TSS-RSS}{TSS}=1-\frac{RSS}{TSS}$ (21)
where TSS is the total sum of square, and RSS is the residual sum of square.
Adj. R-Square
The adjusted R^2 value:
$\bar R^2=1-\frac{RSS/df_{Error}}{TSS/df_{Total}}$ (22)
R Value
The R value is the square root of R^2:
For more information on R^2, adjusted R^2 and R, please see Goodness of Fit.
Root-MSE (SD)
Root mean square of the error, or the Standard Deviation of the residuals, equal to the square root of reduced χ^2:
$Root\,MSE=\sqrt{Reduced \,\chi ^2}$ (24)
ANOVA Table
The ANOVA Table:
Note: The ANOVA table is not available for implicit function fitting.
df Sum of Squares Mean Square F Value Prob > F
Model p SS[reg] = TSS - RSS MS[reg] = SS[reg] / p MS[reg] / MSE p-value
Error n - p RSS MSE = RSS / (n - p)
Uncorrected Total n TSS
Corrected Total n-1 TSS[corrected]
Note: In nonlinear fitting, Origin outputs both corrected and uncorrected total sum of squares: Corrected model:
$TSS_{corrected}=\sum_{i=1}w_{i}y_{i}^2-\left(\sum_{i=1}\left(y_{i}w_{i} \right )/\sum_{i=1}w_{i} \right )^2\sum_{i=1}w_{i}$ (25)
Uncorrected model:
$TSS=\sum_{i=1}^nw_iy_i^2$ (26)
The F value here is a test of whether the fitting model differs significantly from the model y=constant. Additionally, the p-value, or significance level, is reported with an F-test. We can reject
the null hypothesis if the p-value is less than $\alpha\,\!$, which means that the fitting model differs significantly from the model y=constant.
Confidence and Prediction Bands
Confidence Band
The confidence interval for the fitting function says how good your estimate of the value of the fitting function is at particular values of the independent variables. You can claim with 100α%
confidence that the correct value for the fitting function lies within the confidence interval, where α is the desired level of confidence. This defined confidence interval for the fitting function
is computed as:
$f(x_{1i},x_{2i},\ldots ;\theta _{1i},\theta _{2i},\ldots )\pm t_{(\frac \alpha 2,dof)}[s^2fcf^{\prime }]^{\frac 12}$ (27)
$f=[\frac{\partial f}{\partial \theta _1},\frac{\partial f}{\partial \theta _2},\cdots ,\frac{\partial f}{\partial \theta _p}]$ (28)
Prediction Band
The prediction interval for the desired confidence level α is the interval within which 100α% of all the experimental points in a series of repeated measurements are expected to fall at particular
values of the independent variables. This defined prediction interval for the fitting function is computed as:
$f(x_{1i},x_{2i},\ldots ;\theta _{1i},\theta _{2i},\ldots )\pm t_{(\frac \alpha 2,dof)}[s^2(1+fcf^{\prime })]^{\frac 12}$ (29)
$\chi _*^2$ is Reduced $\chi ^2$
Notes: The Confidence Band and Prediction Band in the fitted curve plot are not available for implicit function fitting.
Topics for Further Reading
1. William. H. Press, etc. Numerical Recipes in C++. Cambridge University Press, 2002.
2. Norman R. Draper, Harry Smith. Applied Regression Analysis, Third Edition. John Wiley & Sons, Inc. 1998.
3. George Casella, et al. Applied Regression Analysis: A Research Tool, Second Edition. Springer-Verlag New York, Inc. 1998.
4. G. A. F. Seber, C. J. Wild. Nonlinear Regression. John Wiley & Sons, Inc. 2003.
5. David A. Ratkowsky. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 1990.
6. Douglas M. Bates, Donald G. Watts. Nonlinear Regression Analysis & Its Applications. John Wiley & Sons, Inc. 1988.
7. Marko Ledvij. Curve Fitting Made Easy. The Industrial Physicist. Apr./May 2003. 9:24-27.
8. "J. W. Zwolak, P.T. Boggs, and L.T. Watson, ``Algorithm 869: ODRPACK95: A weighted orthogonal distance regression code with bound constraints, ACM Transactions on Mathematical Software Vol. 33,
Issue 4, August 2007."
9. Nelder, J.A., and R. Mead. 1965. Computer Journal, vol. 7, pp. 308 -313
10. Numerical Recipes in C, Ch. 10.4, Downhill Simplex Method in Multidimensions.
11. Numerical Recipes in C, Ch. 15.5, Nonlinear Models.
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The Place of Partial Differential Equations in Mathematical Physics
by Ganesh Prasad
Publisher: Patna University 1924
Number of pages: 64
The chief reason for my choosing 'The place of partial differential equations in Mathematical Physics' as the subject for these lectures is my wish to inspire in my audience a love for Mathematics.
Before entering into details, however, I shall give a brief historical account of the application of Mathematics to natural phenomena.
Download or read it online for free here:
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(multiple formats)
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Stata Teaching Tools: Mean point
Purpose: The of this program is to show how changing one (point or value) in a sample affects the various statistics that describe the sample. This program can be used to illustrate that some
descriptive statistics, such as the mean and standard deviation, change as the value of the "moving point" changes. On the other hand, descriptive statistics such as the median, minimum and
maximum may or may not change as the value of the "moving point" changes.
Download: You can download this program from within Stata by typing search meanpt (see How can I use the search command to search for programs and get additional help? for more information about
using search).
Use of program: To run this program, type meanpt in the Stata command window. There are no options available with this program. There are 26 points in the sample, one of which can be moved by
clicking on either the "x + 1" or "x – 1" buttons. The vertical line indicates the mean of the distribution. The value of the "moving point" (labeled mp) is given at the top of the screen, as is
the mean, standard deviation, minimum and maximum. The "moving point" is the one in the red box. A new sample can be drawn by clicking on the "New Samp" button. Click on the "Done" button to exit
the program.
Examples: This shows the initial screen after issuing the meanpt command.
This shows the results after clicking on the "x + 1" button twice. As you can see, the mean and the standard deviation have changed a little, but the median, minimum and maximum have not changed.
This is the result of clicking on the "x + 1" button another ten times. Again, the mean and the standard deviation have increased, and now so has the maximum. However, the median and the minimum
remain the same.
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KSEEB Solutions for Class 6 English Prose Chapter 2 The Good Samaritan
KSEEB Solutions for Class 6 English Prose Chapter 2 The Good Samaritan Free PDF Download is available here. Karnataka State Board Class 6 English are prepared as per the Latest Exam Pattern. Students
can prepar these English Chapter 2 The Good Samaritan Questions and Answers, Summary, Notes Pdf, KSEEB Solutions for Class 6 English Karnataka State Board Solutions and assess their preparation
Karnataka State Board Class 6 English Prose Chapter 2 The Good Samaritan
Prepared as per the KSEEB Solutions for Class 6 English Prose Chapter 2 The Good Samaritan can be of extreme help as you will be aware of all the concepts. These Karnataka State Board Class 6 English
Chapter 2 The Good Samaritan Questions and Answers pave for a quick revision of the Chapter thereby helping you to enhance subject knowledge.
The Good Samaritan Questions and Answers, Summary, Notes
1. Look at the points raised in the questions given below. Talk about them to your partner, the boy or the girl sitting next to you. Write down what you say
Question a.
The first traveler gave the robbers a good fight
Not True
Question b.
The man suffered serious injuries
Question c.
The sight of the wounded man frightened the priest
Question d.
Describe the pitiable state of the man in two short sentences.
The man lay bleeding to death in dirt. The man could barely raise his head to beg for help.
Question e.
“I don’t want to dirty my hands”
• Who said this?
The second passerby said this
• To whom?
He said this to himself.
Question f.
“I am sure he is no one I know” was this a good reason not to help the wounded man? – Justify your answer.
This was a good reason not to help the wounded man. There is no rule to help others. It is only a humanitarian concern for others in danger.
Question g.
Who helped the wounded man?
A Samaritan who came walking along the road helped the wounded man.
Question h.
Did this man have a reason not to help the wounded man? If so, what was it?
The man did not have a reason not to help the wounded, man. It was his love for mankind made him to help the wounded man. The wounded man was a Jew. Jews and Samaritans were enemies for centuries.
Despite this enmity the Samaritan helped the wounded man.
Question i.
The fourth traveller had noble and generous feelings
Question j.
There was something special in the fourth man’s act of kindness. What was it?
The wounded Man was a jew. The man who helped him was a Samaritan. Jews and Samaritans were enemies for centuries. Despite this enmity the good Samaritan helped the wounded Jewish Man.
Question k.
What first aid did the Samaritan give the wounded man? What was the beast of burden in the ancient days?
Very gently, the Samaritan lifted the man’s head and brushed the dust but of his mouth. He took some water and cleaned the man’s eyes and gave him some water to drink. He put wine on his wounds to
clean them and make them heal quickly. Then he carried the man to the town on his donkey. The donkey was the beast of burden in the ancient days.
Question l.
The Samaritan was well-to-do. Give 2 reasons.
The Samaritan owned a donkey and carried wine with him.
He gave some money to the innkeeper to take care of the wounded man.
Question m.
Who narrated this parable?
Jesus Christ narrated this parable.
Question n.
What moral lesson did Jesus teach through this parable?
Jesus taught his followers to love everybody especially strangers and those who are in need of help.
Focus on grammar:
4. Now, working with your partner, re-write these pairs of sentences as single sentences.
Question a.
1. You did not ask me for a loan.
2. I did not give you a loan.
Neither did you ask me for a loan nor did I give you one.
Question b.
1. I caught the ball at the boundary line.
2. They lost the match.
I caught the ball at the boundary line and they lost the match.
Question c.
1. She ran very fast.
2. She caught the chain snatcher.
She ran very fast and caught the chain- snatcher.
Word formation:
Note that the suffix ‘-er ’ combines with verbs to form nouns.
5. Working with your partner, write down some verbs and their noun forms.
┃Verb │Noun ┃
┃1. teach │teacher ┃
┃2. dream │dreamer ┃
┃3. plan │planner ┃
┃4. read │reader ┃
┃5. work │worker ┃
┃6. lead │leader ┃
┃7. waive │waiver ┃
┃8. win │winner ┃
┃9. lose │looser ┃
6. Write a conversation between the innkeeper #nd the injured man. Begin like this.
The next morning the innkeeper said to the injured man, “ You lay unconscious the whole day yesterday. How are you feeling today?” “Much better, thank you,” said the injured man ________
• Innkeeper: “You lay unconscious the whole day yesterday. How are feeling today?”
• Injured man: “Much better, Thank you. How did I come here?”
• Innkeeper: “Yesterday a Samaritan brought you here on his donkey You were badly wounded and unconscious.”
• Insured man: “Sorry sir, all my belongings were robbed on the way and I was beaten to death. Now I don’t have anything to pay your room rent.”
• Innkeeper: “Not necessary, the Samaritan has given enough money you can stay here until you become strong to go back home.”
• Injured man: O! Good God has helped men in the form of a Samaritan. How nice of him to have helped a Jew, who were enemies for over a hundred years. Glory to the God, high above.
6. a) Work with your partner, the boy or the girl sitting next to you, and fill inappropriate words in the blanks.
“Let us hide behind this tree”, said the robber to his companion. “Good place. We can see the travelers from this end to that end”, said the second one. “The road is empty,” said the first. “ We will
have to wait”, said the second.
The Good Samaritan Summary in English
The short story ‘ The Good Samaritan’ is a Parable narrated by Jesus Christ to his disciples. Long-time ago a man was walking from Jerusalem to Jericho, suddenly, he was attacked by two robbers, who
beat him up and stole everything he had and also the clothes on his back.
The man lay bleeding beside the road. A Priest come along and saw the bleeding man. Instead of helping the bleeding man the priest backed away quickly and went on his way. Then another man came along
and saw the man lying in the dirt, covered with blood. The man hesitated to touch him because of his terrible state and went away.
Eventually, a Samaritan comes down the road. He came over to the man and gently cleaned the man with water and also gave him some water to drink. He cleaned the wounds with a little wine to make them
heal quickly. He put the man on his donkey and brought him back to town. Although the wounded man was a Jew, the Samaritan helped and saved him. Jews and Samari¬tans had been enemies for hundreds of
The Samaritan gave some money to an innkeeper and asked him to give a clean bed and to take good care of the man until he was strong again. After narrating this parable Jesus Christ asked his
disciples which of the three men was true neighbour to the man who was robbed.
An expert at Jewish law quickly said that a the true neighbour of the robbed man was the one who helped him.
Jesus then asked them to go out and do just the same i.e, to help their fellow human beings.
Thus Jesus Christ’s disciples understood that he wanted them to love and help everybody, especially strangers who needed help.
The Good Samaritan Summary in Kannada
Hope the information shared regarding KSEEB Solutions for Class 6 English Chapter 2 The Good Samaritan Questions and Answers is true and genuine as far as our knowledge is concerned. If you feel any
information is missing do react us and we will look into it and add it accordingly.
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2018 HMS Focused Lecture Series - CMSA
2018 HMS Focused Lecture Series
01/23/2018 5:00 pm - 5:00 pm
As part of their CMSA visitation, HMS focused visitors will be giving lectures on various topics related to Homological Mirror Symmetry throughout the Spring 2018 Semester. The lectures will take
place on Tuesdays and Thursdays in the CMSA Building, 20 Garden Street, Room G10.
The schedule will be updated below.
Date Speaker Title/Abstract
Title: BGG category O: towards symplectic duality
Abstract: We will discuss a very classical topic in the representation theory of semisimple Lie algebras: the Bernstein-Gelfand-Gelfand (BGG) category O. Our aim will be to
motivate and state a celebrated result of Beilinson, Ginzburg and Soergel on the Koszul duality for such categories, explaining how to compute characters of simple modules
(the Kazhdan-Lusztig theory) along the way. The Koszul duality admits a conjectural generalization (Symplectic duality) that is a Mathematical manifestation of 3D Mirror
symmetry. We will discuss that time permitting.
Approximate (optimistic) plan of the lectures:
1) Preliminaries and BGG category O.
23, 25, 30 2) Kazhdan-Lusztig bases. Beilinson-Bernstein localization theorem.
and Ivan Losev
February 1 3) Localization theorem continued. Soergel modules.
3-5pm 4) Koszul algebras and Koszul duality for categories O.
*Room G10* Time permitting: other instances of Symplectic duality.
Semi-simple Lie algebras and their finite dimensional representation theory.
Some Algebraic geometry. No prior knowledge of category O/ Geometric
Representation theory is assumed.
Scanned from a Xerox Multifunction Device
Title: Moduli spaces of Landau-Ginzburg models and (mostly Fano) HMS.
27, Abstract: Mirror symmetry as a general phenomenon is understood to take place near the large complex structure limit resp. large radius limit, and so implicitly involves
Colin Diemer degenerations of the spaces under consideration. Underlying most mirror theorems is thus a mirror map which gives a local identification of respective A-model and B-model
and March moduli spaces. When dealing with mirror symmetry for Calabi-Yau’s the role of the mirror map is well-appreciated. In these talks I’ll discuss the role of moduli in mirror
1 (IHES) symmetry of Fano varieties (where the mirror is a Landau-Ginzburg (LG) model). Some topics I expect to cover are a general structure theory of moduli of LG models (follows
Katzarkov, Kontsevich, Pantev), the interplay of the topology of LG models with autoequivalence relations in the Calabi-Yau setting, and the relationship between Mori
3-5pm theory in the B-model and degenerations of the LG A-model. For the latter topic we’ll focus on the case of del Pezzo surfaces (due to unpublished work of Pantev) and the
toric case (due to the speaker with Katzarkov and G. Kerr). Time permitting, we may make some speculations on the role of LG moduli in the work of Gross-Hacking-Keel (in
progress work of the speaker with T. Foster).
Title: The deformed Hermitian-Yang-Mills equation
March 6 Adam Jacob
and 8 Abstract: In this series I will discuss the deformed Hermitian-Yang-Mills equation, which is a complex analogue of the special Lagrangian graph equation of Harvey-Lawson. I
(UC Davis) will describe its derivation in relation to the semi-flat setup of SYZ mirror symmetry, followed by some basic properties of solutions. Later I will discuss methods for
4-5pm constructing solutions, and relate the solvability to certain geometric obstructions. Both talks will be widely accessible, and cover joint work with T.C. Collins and S.-T.
Title: On categories of matrix factorizations and their homological invariants
Abstract: The talks will cover the following topics:
1. Matrix factorizations as D-branes. According to physicists, the matrix factorizations of an isolated hypersurface singularity describe D-branes in the Landau-Ginzburg
(LG) B-model associated with the singularity. The talk is devoted to some mathematical implications of this observation. I will start with a review of open-closed
topological field theories underlying the LG B-models and then talk about their refinements.
March 6, Dmytro 2. Semi-infinite Hodge theory of dg categories. Homological mirror symmetry asserts that the “classical” mirror correspondence relating the number of rational curves in a CY
8, 13, 15 Shklyarov threefold to period integrals of its mirror should follow from the equivalence of the derived Fukaya category of the first manifold and the derived category of coherent
sheaves on the second one. The classical mirror correspondence can be upgraded to an isomorphism of certain Hodge-like data attached to both manifolds, and a natural first
3-4pm (TU Chemnitz) step towards proving the assertion would be to try to attach similar Hodge-like data to abstract derived categories. I will talk about some recent results in this direction
and illustrate the approach in the context of the LG B-models.
3. Hochschild cohomology of LG orbifolds. The scope of applications of the LG mod- els in mirror symmetry is significantly expanded once we include one extra piece of data,
namely, finite symmetry groups of singularities. The resulting models are called orbifold LG models or LG orbifolds. LG orbifolds with abelian symmetry groups appear in mir-
ror symmetry as mirror partners of varieties of general type, open varieties, or other LG orbifolds. Associated with singularities with symmetries there are equivariant
versions of the matrix factorization categories which, just as their non-equivariant cousins, describe D-branes in the corresponding orbifold LG B-models. The Hochschild
cohomology of these categories should then be isomorphic to the closed string algebra of the models. I will talk about an explicit description of the Hochschild cohomology
of abelian LG orbifolds.
Title: Gauged Linear Sigma Models, Supersymmetric Localization and Applications
Abstract: In this series of lectures I will review various results on connections between gauged linear sigma models (GLSM) and mathematics. I will start with a brief
April 10 & Mauricio Romo introduction on the basic concepts about GLSMs, and their connections to quantum geometry of Calabi-Yaus (CY). In the first lecture I will focus on nonperturbative results
12 on GLSMs on closed 2-manifolds, which provide a way to extract enumerative invariants and the elliptic genus of some classes of CYs. In the second lecture I will move to
(IAS) nonperturbative results in the case where the worldsheet is a disk, in this case nonperturbative results provide interesting connections with derived categories and
3-4pm stability conditions. We will review those and provide applications to derived functors and local systems associated with CYs. If time allows we will also review some
applications to non-CY cases (in physics terms, anomalous GLSMs).
Lecture notes
Title: Perverse sheaves of categories on surfaces
April 17, Andrew Harder Abstract: Perverse sheaves of categories on a Riemann surface S are systems of categories and functors which are encoded by a graphs on S, and which satisfy conditions that
19, 26 resemble the classical characterization of perverse sheaves on a disc.
(University of
3-5pm Miami) I’ll review the basic ideas behind Kapranov and Schechtman’s notion of a perverse schober and generalize this to perverse sheaves of categories on a punctured Riemann
surface. Then I will give several examples of perverse sheaves of categories in both algebraic geometry, symplectic geometry, and category theory. Finally, I will describe
how one should be able to use related ideas to prove homological mirror symmetry for certain noncommutative deformations of projective 3-space.
Lecture One:
Title: Picard-Fuchs uniformization and Calabi-Yau geometry
Part 1: We introduce the notion of the Picard-Fuchs equations annihilating periods in families of varieties, with emphasis on Calabi-Yau manifolds. Specializing to the
case of K3 surfaces, we explore general results on “Picard-Fuchs uniformization” of the moduli spaces of lattice-polarized K3 surfaces and the interplay with various
algebro-geometric normal forms for these surfaces. As an application, we obtain a universal differential-algebraic characterization of Picard rank jump loci in these moduli
Part 2: We next consider families with one natural complex structure modulus, (e.g., elliptic curves, rank 19 K3 surfaces, b_1=4 Calabi-Yau threefolds, …), where the
Picard-Fuchs equations are ODEs. What do the Picard-Fuchs ODEs for such families tell us about the geometry of their total spaces? Using Hodge theory and parabolic
May 15, 17 Charles Doran cohomology, we relate the monodromy of the Picard-Fuchs ODE to the Hodge numbers of the total space. In particular, we produce criteria for when the total space of a family
of rank 19 polarized K3 surfaces can be Calabi-Yau.
1-3pm (University of
Alberta) Lecture Two:
Title: Calabi-Yau fibrations: construction and classification
Part 1: Codimension one Calabi-Yau submanifolds induce fibrations, with the periods of the total space relating to those of the fibers and the structure of the fibration.
We describe a method of iteratively constructing Calabi-Yau manifolds in tandem with their Picard-Fuchs equations. Applications include the tower of mirrors to degree n+1
hypersurfaces in P^n and a tower of Calabi-Yau hypersurfaces encoding the n-sunset Feynman integrals.
Part 2: We develop the necessary theory to both construct and classify threefolds fibered by lattice polarized K3 surfaces. The resulting theory is a complete
generalization to threefolds of that of Kodaira for elliptic surfaces. When the total space of the fibration is a Calabi-Yau threefold, we conjecture a unification of CY/CY
mirror symmetry and LG/Fano mirror symmetry by mirroring fibrations as Tyurin degenerations. The detailed classification of Calabi-Yau threefolds with certain rank 19
polarized fibrations provides strong evidence for this conjecture by matching geometric characteristics of the fibrations with features of smooth Fano threefolds of Picard
rank 1.
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math websites for kids for Dummies - Behind the Bay
For example, one site could be great at educating calculus but horrible at teaching algebra. Another website might focus on higher-level math and utterly overlook the basics. To be taught a new
ability from home, you must take it one step at a time. The AoPS web site supplies a wide selection of sources corresponding to videos, interactive issues, and community boards so that you can learn
and improve your math and problem-solving skills. For younger students, the site boasts interactive content material, whilst older college students can enroll in full courses, with live classes and
assignments. The Cliff Notes web site offers a spread of math study guides overlaying subjects corresponding to fundamental math, algebra, calculus, geometry, and statistics.
• Learn Algebra 2 aligned to the Eureka Math/EngageNY curriculum —polynomials, rational functions, trigonometry, and more.
• Tons of enjoyable and academic online math games, from fundamental operations to algebra and geometry.
• Khan Academy is a free web site that provides thousands of math classes for learners of all ages.
• Learnerator additionally offers a giant number of follow questions so that you simply can evaluate.
We love the friendly competitors and game-based content supplied by First in Math. Kids achieve skills apply and fluency as they play video games focused towards truth proficiency and logical
thinking. You might want to get assistance from your school if you are having issues getting into the answers into your online assignment rocket math teacher login. Learn the basics of
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Upper-level math students will respect the no-frills information that’s simple to search out on this website. From PBS Learning Media, center schoolers will love this entertaining video blog. Not
only does each episode cowl Common Core Standards, it makes math learning culturally relevant with pop-culture references. See how professionals use math in music, fashion, video video games,
restaurants, basketball, and special results. Then tackle interactive challenges associated to those careers.
Learn fourth grade math aligned to the Eureka Math/EngageNY curriculum—arithmetic, measurement, geometry, fractions, and extra. This Arithmetic course is a refresher of place value and operations for
complete numbers, fractions, decimals, and integers. Others say that the modifications have been misinterpreted to be an institutional judgment on courses rocket math addition in data science and
different utilized math fields. There are loads of resources and sites that may help you be taught or relearn maths from basics to advanced ranges.
Due to the language used in these articles, we do not suggest MathWorld for newbie students. However, it’s excellent for advanced students seeking to dive deeper into their favourite math matters.
You can use the free Desmos math web site to access a range of mathematical calculators. Graphing calculators can help with linear, quadratic, and trigonometric equations. Desmos math presents a
web-based graphing calculator and a spread of tools and actions to assist students in studying and exploring math ideas. Unfortunately, the Prodigy Math sport only covers math from grades 1-8.
rocketmath – Eight Reasons For College Students To Purchase Them
And schools in states with restrictions on classroom discussions about issues like race, social inequality, and “divisive topics,” may face some barriers in utilizing some real-world topics in math
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course in the path of learning math can be geometry. Some say geometry, which is the examine of shapes, should be taken before algebra 2, but the order is totally up to you.
rocket math: Customer Review
Their “Mathematics With a Human Face” web page contains information about careers in mathematics as nicely as profiles of mathematicians. Through ongoing analysis, MIND Research Institute continues
to investigate key questions about studying, mathematics, and the way the brain works. ST Math is their pre-K–8 visual educational program, serving to academics interact kids extra deeply in math
studying. Learn the abilities that will set you up for fulfillment in decimal place value; operations with decimals and fractions; powers of 10; volume; and properties of shapes. Learn sixth grade
math aligned to the Eureka Math/EngageNY curriculum—ratios, exponents, long division, negative numbers, geometry, statistics, and extra.
Learn third grade math aligned to the Eureka Math/EngageNY curriculum—fractions, area, arithmetic, and so much extra. Instead, it’s about making sure that every one young folks, no matter path they
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The Art of Problem Solving web site presents math lessons for elementary, middle, and high school college students. The assets on provide rely on your studying stage, with some being more
comprehensive than others. Stanford isn’t alone in rewriting its math suggestions for prospective undergraduate applicants. When figuring out tips on how to learn maths from the start, you need the
best sites for every mathematics degree.
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Balancing Binary Search Trees | CodingDrills
Balancing Binary Search Trees
Balancing Binary Search Trees
Binary search trees (BSTs) are a fundamental data structure used in many algorithms and applications. They provide efficient search, insertion, and deletion operations with a time complexity of O(log
n) on average, making them ideal for storing and retrieving ordered data.
However, in some scenarios, the performance of a BST can degrade significantly due to its unbalanced nature. An unbalanced BST can result in skewed subtrees, causing operations to take longer than
In this tutorial, we will explore the concept of balancing BSTs to ensure their optimal performance. We will first review the basics of trees and then dive into the intricacies of binary search
Trees are hierarchical data structures consisting of nodes connected by edges. They are widely used to organize and represent hierarchical relationships. In a tree, each node (except the root node)
has exactly one parent and zero or more child nodes.
Before we delve into binary search trees, let's briefly discuss some common terms associated with trees:
• Root: The topmost node of a tree.
• Leaf: A node with no children.
• Parent: A node from which another node is directly descended.
• Child: A node directly descended from another node.
• Siblings: Nodes that share the same parent.
• Subtree: A tree contained within another tree.
• Path: A sequence of nodes, starting from the root, that connects two nodes.
Binary Trees
A binary tree is a specific type of tree where each node can have, at most, two children: a left child and a right child. The left child is typically smaller than its parent, while the right child is
Binary trees can be implemented using various data structures, but the most common form is a node with two pointers, one for the left child and one for the right child.
Here's an example of a binary tree:
/ \
/ \ \
/ \ /
Binary Search Trees (BSTs)
A binary search tree is a binary tree with an additional constraint: for every node in the tree, the value of all nodes in its left subtree is less than its value, and the value of all nodes in its
right subtree is greater than its value.
This key property makes BSTs an efficient data structure for searching, inserting, and deleting elements. By comparing the target value with the values at each node, we can determine the appropriate
branch to traverse until we find the desired element or an empty spot for insertion.
In the example above, we have a binary search tree. The left subtree of the root node contains smaller values, while the right subtree contains larger values.
Balancing Binary Search Trees
As mentioned earlier, an unbalanced BST can significantly affect the performance of operations. If the tree becomes heavily skewed, the time complexity can degrade from efficient O(log n) to
inefficient O(n), essentially turning it into a linked list.
Balancing techniques, such as the AVL tree and Red-Black tree, aim to maintain balance in a BST by ensuring that the heights of the left and right subtrees differ by at most one.
These self-balancing trees employ different algorithms to automatically adjust the structure of the tree during insertions and deletions, ensuring optimal performance in all scenarios.
Here's an example of a balanced AVL tree:
/ \
/ \ \
With a balanced BST, the search, insertion, and deletion operations continue to have an average time complexity of O(log n), providing optimal performance even with a large dataset.
Understanding and implementing these balancing techniques are crucial for developers working with large amounts of data or applications that require frequent tree operations.
In this tutorial, we explored the concept of balancing binary search trees to optimize their performance. We started by understanding binary trees, binary search trees, and their basic properties.
Then, we delved into the importance of balancing and introduced two popular balancing techniques: AVL trees and Red-Black trees.
By implementing these balancing techniques, developers can ensure efficient operations on BSTs, making them suitable for a wide range of applications.
Remember to consider the trade-offs between the time complexity and space requirements when choosing a balancing technique based on your use case.
Ada AI
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Unit Analysis - UCalgary Chemistry Textbook
A convenient method for many calculations is unit analysis, which relies on cancellation of units and conversion ratios to perform calculations. You’ve already done unit analysis in calculations such
$$ 2\: \text{m}\, \times \frac{100\, \text{cm}}{1\, \text{m}} = 200\, \text{cm}$$
The key considerations when performing calculations with unit analysis are:
1. Units cancel across ratios
The central premise of unit analysis is that units will cancel across a ratio (i.e. on the top and bottom of a fraction). When setting up your calculation or choosing your ratios, make sure to check
that your intermediate units will cancel out, leaving you with your desired unit at the end. (Remember that for values that aren’t already fractions, you can imagine them as $\frac{\text{value}}{1}$
if you need!)
$$ \require{cancel} 3.25\, \cancel{\text{g}\, H_2 O} \times \frac{1\, \text{mol}\, H_2 O}{18.02\,\cancel{\text{g}\, H_2 O}} = 0.180\, \text{mol}\, H_2 O$$
$$3.25\, {\text{g}\, H_2 O} \times \frac{18.02\,{\text{g}\, H_2 O}} {1\, \text{mol}\, H_2 O}= 58.6\, \frac{g^2}{\text{mol}}\, H_2 O\qquad (?!?!?)$$
2. Show ALL your units (including the “of what”)
In longer calculations, it’s easy to get mixed around and cancel units that look the same but actually aren’t. You can avoid this by making sure to label all your units all the time – including the
“of what” part – not just “g” but “g H[2]O“.
For example, if you wanted to calculate the mass of Al that could react with 25 mL of 0.10 M HCl according to the reaction $$2\, Al\, \text{(s)}\, + \, 6\, HCl\, \text{(aq)}\, \rightarrow \, 2\,
AlCl_3 \, \text{(aq)}\, +\, 3\, H_2 \, \text{(g)}\, $$ the following calculation appears to be correct: $$ \require{cancel} 0.025\,\text{L} \times \frac{0.1\, \text{mol}}{\text{L}} \times \frac{6\,\
text{mol}}{2\,\text{mol}} \times \frac{36.46\, \text{g}}{1\,\text{mol}} \\ 0.025\,\cancel{\text{L}} \times \frac{0.1\, \bcancel{\text{mol}}}{\cancel{\text{L}}} \times \frac{6\,\cancel{\text{mol}}}{2
\,\bcancel{\text{mol}}} \times \frac{36.46\, \text{g}}{1\,\cancel{\text{mol}}} =0.27\,\text{g} $$
However, this is not the correct answer! Filling in the “of what” in the units reveals: $$\require{cancel} 0.025\,\text{L}\, HCl \times \frac{0.1\, \text{mol}\, HCl}{\text{L}\, HCl} \times \frac{6\,\
text{mol}\, HCl}{2\,\text{mol}\, Al} \times \frac{36.46\, \text{g} \,HCl}{1\,\text{mol}\, HCl} \\ 0.025\,\cancel{\text{L}\, HCl} \times \frac{0.1\, \text{mol}\, HCl}{\cancel{\text{L}\, HCl}} \times \
frac{6\,\cancel{\text{mol}\, HCl}}{2\,\text{mol}\, Al} \times \frac{36.46\, \text{g} \,HCl}{1\,\cancel{\text{mol}\, HCl}} = 0.27\, \frac{(\text{mol}\, HCl)(\text{g}\,HCl)}{(\text{mol}\, Al)}$$
Hopefully it is obvious that the “units” of the final answer in this calculation don’t make sense … that is our clue that something went wrong in the calculation. In this case there’s two errors: the
mole ratio is inverted and the wrong molecular mass was used (HCl instead of Al). When we fix the errors, the answer looks much better: $$\require{cancel} 0.025\,\text{L}\, HCl \times \frac{0.1\, \
text{mol}\, HCl}{\text{L}\, HCl} \times \frac{2\,\text{mol}\, Al}{6\,\text{mol}\, HCl} \times \frac{26.98\, \text{g} \,Al}{1\,\text{mol}\, Al} \\0.025\,\cancel{\text{L}\, HCl} \times \frac{0.1\, \
bcancel{\text{mol}\, HCl}}{\cancel{\text{L}\, HCl}} \times \frac{2\,\cancel{\text{mol}\, Al}}{6\,\bcancel{\text{mol}\, HCl}} \times \frac{26.98\, \text{g} \,Al}{1\,\cancel{\text{mol}\, Al}} = 0.022\,
Writing out calculations with the full units can take up a little more space, but the built in error checking is guaranteed to save you at least once on an exam or lab report!
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Plate With a Hole Optimization - Input & Output Parameters
Input & Output Parameters
To set up the input and output parameters for a geometry created in Workbench, simply follow the steps below. To set up parameters for a geometry created in SolidWorks, follow the instructions here.
Design Variables: Hole Radius (SpaceClaim)
Parameters can be defined in SpaceClaim, even if the geometry was created in DesignModeler or another CAD software, as is the case for us. To do this, right-click on from the Project Schematic
window and choose "Edit Geometry in SpaceClaim..." as shown below:
Once SpaceClaim has opened, go into the Design tab and choose the Pull tool. Select the arc that represents our hole and then choose the Ruler option from the mini toolbar that appears:
Now move the cursor until it snaps to the center of the arc:
Now, click the box with the "P" to the right of the dimension. You can also go into the Groups tab and choose "Create Parameter" near the top of that window. This will create a parameter named Group1
under a folder called Driving Dimensions. Call the parameter "DS_R". You can always go back and rename your parameters by right-clicking on them and choosing "Rename" in the context menu that
appears. If you click on your parameter, you can see the current dimension being used in the model. Make sure your Groups tab looks like the image below before continuing:
SpaceClaim can now be closed.
Design Variables: Hole Radius (DesignModeler)
This section applies only if you do not have access to SpaceClaim. In that case, you can also use DesignModeler (the older geometry engine) to specify your parameters. In order to do so, open
DesignModeler by double-clicking on from the Project Schematic window. Then expand XYPlane. Next, highlight Sketch1.
Now, check the box to the left of "R3", which will be in the "Dimensions: 3" part of the "Details View" table. When you check the box an uppercase "D" will appear within the box and you will be asked
what to call the parameter. Call the parameter "DS_R".
DesignModeler can now be closed.
Objective Function: Minimize Volume (& Mass)
This particular optimization problem has two output parameters: the volume of the quarter plate and the maximum Von Mises stress. In order to specify the volume output parameters, first (Open)
Mechanical > (Expand) Geometry > (Highlight) Surface Body. In the "Details of "Surface Body"" table expand Properties then check the box to the left of Volume. A "P" should now be located within the
Additionally, if mass is also a desired parameter, check the box to the left of Mass.
Constraints: Maximum Von Mises Stress < 32.5 ksi
Now, the maximum Von Mises Stress will be specified as an output parameter. In order to do so, (Expand) Solution > (Highlight) Equivalent Stress. In the "Details of "Equivalent Stress"" window,
underneath Results, check the box to the left of Maximum. Once again a "P" should appear to the left of the box to illustrate to the user that the maximum Von Mises stress has been designated as an
output parameter.
At this point the Mechanical window can be closed and you should save the project.
Let's review the input and output parameters that will be used in the optimization process. In the main Project Schematic window, double click on Parameter Set.
After doing so, we can see that DS_R is the input parameter, and the volume and max. value of the von Mises Stress are the output parameters. Now, return to the main window by clicking on the Project
tab, or in older versions by selecting Return to Project.
Note: Make sure your parameters are using the correct units! If they are not, you will need to go back into Mechanical and change the units before unchecking and rechecking the box next to the
parameters of interest. This should reset the units on the parameters in the Parameter Set window, but beware that this may cause the entire optimization process to need updated and repeated.
Go to Step 4: Design of Experiments
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Theorem Proving in Lean 4
We have seen that Lean's foundational system includes inductive types. We have, moreover, noted that it is a remarkable fact that it is possible to construct a substantial edifice of mathematics
based on nothing more than the type universes, dependent arrow types, and inductive types; everything else follows from those. The Lean standard library contains many instances of inductive types
(e.g., Nat, Prod, List), and even the logical connectives are defined using inductive types.
Recall that a non-recursive inductive type that contains only one constructor is called a structure or record. The product type is a structure, as is the dependent product (Sigma) type. In general,
whenever we define a structure S, we usually define projection functions that allow us to "destruct" each instance of S and retrieve the values that are stored in its fields. The functions prod.fst
and prod.snd, which return the first and second elements of a pair, are examples of such projections.
When writing programs or formalizing mathematics, it is not uncommon to define structures containing many fields. The structure command, available in Lean, provides infrastructure to support this
process. When we define a structure using this command, Lean automatically generates all the projection functions. The structure command also allows us to define new structures based on previously
defined ones. Moreover, Lean provides convenient notation for defining instances of a given structure.
The structure command is essentially a "front end" for defining inductive data types. Every structure declaration introduces a namespace with the same name. The general form is as follows:
structure <name> <parameters> <parent-structures> where
<constructor> :: <fields>
Most parts are optional. Here is an example:
structure Point (α : Type u) where
mk :: (x : α) (y : α)
Values of type Point are created using Point.mk a b, and the fields of a point p are accessed using Point.x p and Point.y p (but p.x and p.y also work, see below). The structure command also
generates useful recursors and theorems. Here are some of the constructions generated for the declaration above.
structure Point (α : Type u) where
mk :: (x : α) (y : α)
#check Point -- a Type
#check @Point.rec -- the eliminator
#check @Point.mk -- the constructor
#check @Point.x -- a projection
#check @Point.y -- a projection
If the constructor name is not provided, then a constructor is named mk by default. You can also avoid the parentheses around field names if you add a line break between each field.
structure Point (α : Type u) where
x : α
y : α
Here are some simple theorems and expressions that use the generated constructions. As usual, you can avoid the prefix Point by using the command open Point.
structure Point (α : Type u) where
x : α
y : α
#eval Point.x (Point.mk 10 20)
#eval Point.y (Point.mk 10 20)
open Point
example (a b : α) : x (mk a b) = a :=
example (a b : α) : y (mk a b) = b :=
Given p : Point Nat, the dot notation p.x is shorthand for Point.x p. This provides a convenient way of accessing the fields of a structure.
structure Point (α : Type u) where
x : α
y : α
def p := Point.mk 10 20
#check p.x -- Nat
#eval p.x -- 10
#eval p.y -- 20
The dot notation is convenient not just for accessing the projections of a record, but also for applying functions defined in a namespace with the same name. Recall from the Conjunction section if p
has type Point, the expression p.foo is interpreted as Point.foo p, assuming that the first non-implicit argument to foo has type Point. The expression p.add q is therefore shorthand for Point.add p
q in the example below.
structure Point (α : Type u) where
x : α
y : α
deriving Repr
def Point.add (p q : Point Nat) :=
mk (p.x + q.x) (p.y + q.y)
def p : Point Nat := Point.mk 1 2
def q : Point Nat := Point.mk 3 4
#eval p.add q -- {x := 4, y := 6}
In the next chapter, you will learn how to define a function like add so that it works generically for elements of Point α rather than just Point Nat, assuming α has an associated addition operation.
More generally, given an expression p.foo x y z where p : Point, Lean will insert p at the first argument to Point.foo of type Point. For example, with the definition of scalar multiplication below,
p.smul 3 is interpreted as Point.smul 3 p.
structure Point (α : Type u) where
x : α
y : α
deriving Repr
def Point.smul (n : Nat) (p : Point Nat) :=
Point.mk (n * p.x) (n * p.y)
def p : Point Nat := Point.mk 1 2
#eval p.smul 3 -- {x := 3, y := 6}
It is common to use a similar trick with the List.map function, which takes a list as its second non-implicit argument:
#check @List.map
def xs : List Nat := [1, 2, 3]
def f : Nat → Nat := fun x => x * x
#eval xs.map f -- [1, 4, 9]
Here xs.map f is interpreted as List.map f xs.
We have been using constructors to create elements of a structure type. For structures containing many fields, this is often inconvenient, because we have to remember the order in which the fields
were defined. Lean therefore provides the following alternative notations for defining elements of a structure type.
{ (<field-name> := <expr>)* : structure-type }
{ (<field-name> := <expr>)* }
The suffix : structure-type can be omitted whenever the name of the structure can be inferred from the expected type. For example, we use this notation to define "points." The order that the fields
are specified does not matter, so all the expressions below define the same point.
structure Point (α : Type u) where
x : α
y : α
#check { x := 10, y := 20 : Point Nat } -- Point ℕ
#check { y := 20, x := 10 : Point _ }
#check ({ x := 10, y := 20 } : Point Nat)
example : Point Nat :=
{ y := 20, x := 10 }
If the value of a field is not specified, Lean tries to infer it. If the unspecified fields cannot be inferred, Lean flags an error indicating the corresponding placeholder could not be synthesized.
structure MyStruct where
{α : Type u}
{β : Type v}
a : α
b : β
#check { a := 10, b := true : MyStruct }
Record update is another common operation which amounts to creating a new record object by modifying the value of one or more fields in an old one. Lean allows you to specify that unassigned fields
in the specification of a record should be taken from a previously defined structure object s by adding the annotation s with before the field assignments. If more than one record object is provided,
then they are visited in order until Lean finds one that contains the unspecified field. Lean raises an error if any of the field names remain unspecified after all the objects are visited.
structure Point (α : Type u) where
x : α
y : α
deriving Repr
def p : Point Nat :=
{ x := 1, y := 2 }
#eval { p with y := 3 } -- { x := 1, y := 3 }
#eval { p with x := 4 } -- { x := 4, y := 2 }
structure Point3 (α : Type u) where
x : α
y : α
z : α
def q : Point3 Nat :=
{ x := 5, y := 5, z := 5 }
def r : Point3 Nat :=
{ p, q with x := 6 }
example : r.x = 6 := rfl
example : r.y = 2 := rfl
example : r.z = 5 := rfl
We can extend existing structures by adding new fields. This feature allows us to simulate a form of inheritance.
structure Point (α : Type u) where
x : α
y : α
inductive Color where
| red | green | blue
structure ColorPoint (α : Type u) extends Point α where
c : Color
In the next example, we define a structure using multiple inheritance, and then define an object using objects of the parent structures.
structure Point (α : Type u) where
x : α
y : α
z : α
structure RGBValue where
red : Nat
green : Nat
blue : Nat
structure RedGreenPoint (α : Type u) extends Point α, RGBValue where
no_blue : blue = 0
def p : Point Nat :=
{ x := 10, y := 10, z := 20 }
def rgp : RedGreenPoint Nat :=
{ p with red := 200, green := 40, blue := 0, no_blue := rfl }
example : rgp.x = 10 := rfl
example : rgp.red = 200 := rfl
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Ana played 4 rounds of golf. Her scores for those - Asksia.ai
Ana played 4 rounds of golf. Her scores for those rounds had a mean and median of 97.
Then, she played a round with a score of 80 .
Here's a dot plot showing these scores:
How did the round where Ana scored 80 change the mean and median?
Choose 1 answer:
(A) The median decreased, and the mean increased.
(B) The mean decreased, and the median increased.
(c) Both the median and mean decreased, but the mean decreased by more than the median.
(D) Both the median and mean decreased, but the median decreased by more than the mean.
Answer from Sia
Posted 6 months ago
Solution by Steps
step 2
The sum of all scores including the fifth round is 97 + 97 + 97 + 97 + 80 = 468
step 3
The number of rounds is now 5
step 4
The new mean is 468 / 5 = 93.6
step 5
The original mean was 97, so the new mean after the fifth round is lower
step 6
Calculate the new median after Ana scored 80 in the fifth round. The scores in order are 80, 97, 97, 97, 97
step 7
The median is the middle number, so the new median is the third score when arranged in ascending order, which is 97
step 8
The original median was 97, so the median remains unchanged after the fifth round
1 Answer
Key Concept
Mean and Median
The mean is the average of all the values, and it decreases when a lower value is added to the dataset. The median is the middle value when the data is ordered, and it remains unchanged if the new
value does not fall in the middle of the dataset. In this case, the mean decreased, but the median remained the same after Ana's fifth round.
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Invading the ideal free distribution
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[2] R. S. Cantrell, C. Cosner, D. L. DeAngelis and V. Padrón, The ideal free distribution as an evolutionarily stable strategy, J. Biol. Dyn., 1 (2007), 249-271.doi: 10.1080/17513750701450227.
[3] R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistence of competing species, Proc. Roy. Soc. Edinb., 137A (2007), 497-518.doi: 10.1017/S0308210506000047.
[4] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math Bios. Eng., 7 (2010), 17-36.doi: 10.3934/mbe.2010.7.17.
[5] X. Chen, K.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Cont. Dyn. Sys., 32 (2012), 3841-3859.doi: 10.3934/dcds.2012.32.3841.
[6] E. N. Dancer, Positivity of maps and applications, in Topological nonlinear analysis, Nonlinear Differential Equations Appl., (eds. Matzeu and Vignoli), Birkhauser, Boston, 15 1995, 303-340.
[7] C. P. Doncaster, et al., Balanced dispersal between spatially varying local populations: an alternative to the source-sink model, The American Naturalist, 150 (1997), 425-445.
[8] H. Dreisig, Ideal free distributions of nectar foraging bumblebees, Oikos, 72 (1995), 161-172.doi: 10.2307/3546218.
[9] S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat selection in birds, Theoretical development, Acta Biotheor., 19 (1970), 16-36.
[10] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order, 2nd Ed., Springer-Verlag, Berlin, 1983.doi: 10.1007/978-3-642-61798-0.
[11] T. Grand, Foraging site selection by juvenile coho salmon: Ideal free distribution with unequal competitors, Animal Behavior, 53 (1997), 185-196.
[12] P. Hess, Periodic Parabolic Boundary Value Problems and Positivity, Longman Scientific & Technical, Harlow, UK, 1991.
[13] S.-B. Hsu, H. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.doi: 10.1090/
[14] M. Kennedy and R. D. Gray, Can ecological theory predict the distribution of foraging animals? A critical analysis of experiments on the ideal free distribution, Oikos, 68 (1993), 158-166.doi:
[15] L. Korobenko and E. Braverman, On evolutionary stability of carrying capacity driven dispersal in competition with regularly diffusing populations, J. Math. Biol. (to appear). doi: 10.1007/
[16] K.-Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in poulation dynamics II, SIAM J. Math. Anal., 44 (2012), 1808-1830.doi: 10.1137/100819758.
[17] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.doi: 10.1006/jdeq.1996.0157.
[18] Y. Lou, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics. II. Stability and multiplicity, Discrete Contin. Dyn. Syst., 27 (2010), 643-655.doi: 10.3934/
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[20] M. A. McPeek and R. D. Holt, The evolution fo dispersal in spatially and temporally varying environments, The American Naturalist, 140 (1997), 1010-1027.
[21] M. Milinski, An evolutionarily stable feeding strategy in sticklebacks, Zeitschrift für Tierpsychologie, 51 (1979), 36-40.doi: 10.1111/j.1439-0310.1979.tb00669.x.
[22] D. W. Morris, J. E. Diffendorfer and P. Lundberg, Dispersal among habitats varying in fitness: Reciprocating migration through ideal habitat selection, Oikos, 107 (2004), 559-575.
[23] D. Munther, The ideal free strategy with weak Allee effect, J. Differential Equations, 254 (2013), 1728-1740.doi: 10.1016/j.jde.2012.11.010.
[24] D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000.
[25] J. Shi and R. Shivaji, Persistence in reaction diffusion models with weak Allee effect, J. Math. Biol., 52 (2006), 807-829.doi: 10.1007/s00285-006-0373-7.
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The impact of analysis context on analysis quality – Bayes’ Theorem principles for qualitative analysis
Every analyst takes analysis context on their decision on the analysis result. If a residue of a pesticide never authorised in a crop production site is identified, extra confirmation of the finding
is performed. In clinical analysis, if a specific analysis result is not supported by patient symptoms, it can be decided to repeat the analysis using the same or a different analytical method. In
the XVIII century, Thomas Bayes developed the mathematical framework, Bayes’s Theorem, for dealing with analysis context in analysis result interpretation. This text briefly describes how this
applies to qualitative analysis, i.e. analysis where the outcome is of a binary nature, such as evidence or no evidence of compound presence in a sample or composition equivalence of two analysed
When a fast SARS-CoV-2 antigen test is purchased, it provides the rate of true positive and true negative results, designated sensitivity (SS) and
specificity (SP). If, for instance, SS is 72 %, a tested sample from a truly infected person has a 72 % chance of producing the colour change indicating
infection (a true positive result). For an SP of 96 %, when a biological sample from a not infected person is analysed, there is a 96 % chance of being
indicated no infection (a true negative result). From this information, it is also possible to know that there is a 28 % chance (100% ̶ 72 %) of not being
detected a true infection (false negative result) and of 4 % of a no infection being reported as a false infection (false positive result). Although
The end-user of a SARSCoV-2 antigen test interesting and valuable, these probability levels are not the information the end user of the antigen test is seeking. The SS and SP reports on test
is interested in knowing the change of performance, i.e. the chance that a real case is correctly identified; it starts from an actual case and gets the change of a specific result. However, when
being infected and not the chance of an someone does the antigen test, they want to know if it is infected instead of the chance of an infection being correctly determined (i.e. the other way
injection being detected. around). The starting point is the result and not the case as for SS and SP determination.
The SS and SP and their complementary values (100 ̶SS) and (100 ̶ SP) can be visually represented by the following independent figures:
These figures can be merged with the analysis context; in this example, the prevalence of SARS-CoV-2 infection in the tested population to provide the chance
of a positive result from a sample of the studied population being true. The following figures present the transfer of the SS and SP to a population of high
or low positive case rates, i.e., with many or a few infected persons.
If the sample is originated from a population where 60 % or only 10 % of people are infected, the analysis scenario is represented by Figures 2a and 2b, respectively.
Figures 1a and 1b were squeezed or expanded to have the size from 60% to 40%, or 10% to 90% of the population.
If Figure 2a is analysed, it can be observed that we have (60 % × 72 %) = 43.2 % of true positive results (72 % of the 60 % positive cases), 16.8% of false negative results (60 % × 28 %), 38.4 %
of true negative results (40 % × 96 %) and 1.6% of false positive results (40 % × 4 %).
Bayes theorem formulates that, given a positive result (e.g. colour change indicating infection), the probability of the results being truth, i.e. of the individual being infected, P, is the ratio
between the probabilities of true positive results, and of the result being positive regardless from positive or negative cases.
The P is designated the posterior probability. The infection prevalence of 60 % is the prior probability available before using the infection test. If no test on samples is performed, there is a 60
% chance that the individual is infected. After observing a positive result from the test with claimed performance, the prior probability is Figure 2^ updated to 96.4 % (the posterior probability).
If the prevalence of infection is 10 %, an equivalent positive result is associated with a P = 67%.
To close this brief explanation, it is only necessary to mention that several probabilities used in Baye’s theorem are conditional probabilities presented in the notation below.
This theorem also allows us to quantify how qualitative analysis improves if additional evidence of the event is collected. When no reliable prior probability is available, it can be decided to
report results with alternative metrics. All these are discussed in the Eurachem/CITAC guide on qualitative analysis [1].
[1] R Bettencourt da Silva and S L R Ellison (eds.) Eurachem/CITAC Guide: Assessment of performance and uncertainty in qualitative chemical analysis. First Edition, Eurachem 2021.
Available from https://www.eurachem.org.
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Int'l J. of Communications, Network and System Sciences
Vol.3 No.6(2010), Article ID:2007,9 pages DOI:10.4236/ijcns.2010.36074
Performance of Multirate Multicast in Distributed Network
Department of Computer Engineering (College of Computer and Information Sciences), King Saud University, Riyadh, Saudi Arabia
E-mail: {sasit, siraj}@ksu.edu.sa
Received March 11, 2010; revised April 19, 2010; accepted May 22, 2010
Keywords: Throughput, Performance, Distributed Network, Cluster, Multicast, Session, Multirate Multicast, Queue
The number of Internet users has increased very rapidly due to the scalability of the network. The users demand higher bandwidth and better throughput in the case of on demand video, or video
conference or any real time distributed network system. Performance is a great issue in any distributed networks. In this paper we have shown the performance of the multicast groups or clusters in
the distributed network system. In this paper we have shown the performance of different users or receivers belongs to the multicast group or cluster in the distributed network, transfer data from
the source node with multirate multicast or unirate multicast by considering packet level forwarding procedure in different sessions. In this work we have shown that how the throughput was effected
when the number of receiver increases. In this work we have considered the different types of queue such as RED, Fair queue at the junction node for maintaining the end to end packet transmission. In
this work we have used various congestion control protocol at the sender nodes. This paper we have shown the performance of the distributed cluster network by multirate multicast.
1. Introduction
The users i.e. the receivers are connected to the source for transferring data or exchanging information. Here the source and the receivers are forming a network, that network is scalable i.e., in
the network any new user or receiver can join. According to the real world scenario any number of existing user or receiver can leave from the cluster network can join different cluster or group.
Each cluster be from a distributed network and well connected with other network. For any particular multicast group in the distributed network, the consist members of the group run different
application program and required different packet size and data rate.
The performance of the distributed network in heterogeneous system, obtained by markovian model [1] and the queuing processing delay at the junction node.
Sending packets to the destination node with the minimum cost transmission delay, multicast session network coding techniques scheme used [2]. The end to end packet transmission in a set of active
elastic sessions over a network, the session traffic [3] routed to the destination node through different path. The collision free broadcasting technique used [4] to minimize the latency and the
number of transmissions in the broadcast network for end to end packet transmission in the distributed cluster network. The alternative ways for end to end packet forwarding used minimal congestion
feed back signals from the router [5] and split the flow between each source destination pair. In end to end packet transmission, the random delay and TCP–congestion control [6] in the network is a
issue. Receiver adjusts the rate based on the congestion level in Multicast Network [7] to reduce the congestion. In the real life scenario multicast traffic can cause more packet loss than uncast
traffic for example in internet. The resource allocation by the Max –min fairness [8,9] and proportional fairness can reduce the traffic load in the network. The control multicast in the Network by
using TCP [10,11] reduce the traffic load in the network. The inverted tree structure implemented in IP based network with the multicast session [12,13] to achieve better performance. The TCP
congestion and the effect of that on the throughput of Multicast Group have greater impact in the system network [14]. Since the data packet be transferred between the source and receiver for end to
end connection. The path between the source and receiver is not peer to peer, there be at least one junction node in between them .Due to limited bandwidth in the connecting paths, and queuing delay,
data packet may be loss. The packet processing delay at the junction node (it is random service time) and the propagation time in the link be consider, the packet transmission delay at the junction
node be negligible. The different receivers take data packet from the source node via the junction node (there may be more than one junction nodes in the source to receiver link). Now different
receivers taken data packet with different rate in the multicast group of the distributed network in a multicast session. It is called the multirate multicast .If all the receiver node taken data
packet with same rate it is called unirate multicast. Each multicast session is the collection of virtual session [15,16].
In Figure 1 there is one multicast group and one junction node, In Figure 2 there are two multicast group and two junction node In Figure 3 three multicast group and three junction node. The receiver
node takes data packet from the source node via the junction node through the source to receiver link path. The different types of queue like RED, FQ, SFQ attach at the junction node to capture the
packet loss and measure the delay for the multicast group in the multirate multicast session. Figure 4 represent the junction node [17].
Figure 4 represents the junction node of the network. The packets approach the junction node randomly and the queue stored the packets. The packets stored into the queue in the order n + 1, n, n – 1…
i.e., the
2. Proposed Model for Multirate Multicast Virtual Session
2.1. Assumption 1
Let L be the set of links and
Then, Figure 5, There were the two sessions for sending data packet from the source node A to the receiver node C and the source node D to the receiver node C . Now the maximum data packet be
transferred through the junction node B to the receiver node C be within the capacity of the link for the set of sessions.
In the
Our objective is to maximize the flow within the congestion threshold window size at the source end by consider the multirate multicast. The basic objective goes to maximize the data flow from the
source to the multicast group in presence of queue size and random service at the junction node or the random processing time at the junction node.
2.2. Assumption 2
The total flow through the link
The above expression is the sum of the set of values for all session through link
Here we consider only the maximum data rate at different session at time t for the receiver r.
3. Proposed Algorithm for the Packet Level Forwarding
It is possible to store information of all packet for all the multirate multicast session through the link path (source to receiver).
Here the proposed algorithms for selected single packet forwarding time computation from source to a receiver (that belongs to multicast group) per session that belongs to multirate multicast
// k = cardinality (
// m be any selected packet for any session
// n be the number of nodes required for the link path source to receiver r.
// n
// node.
// source to receiver
// numbers of rows and 1 column.
// T be the total time one packet per session //
sum = 0 : integer
2sum = 0 : integer function : f ( float a, float b)
{ if ABS(a) > ABS(b)
// ABS(a) is || a || in normed space
return (a)
if ABS(a) < ABS(b)
if ABS(a) = = ABS(b)
for i = 1 to k do begin for j = 1 to n do begin
A[i, j] = 0
B[i, j] = 0
C[i, j] =0
end // loop 1
end // loop 2 for j=1 to n do begin
D[j, 1] = 0;
E[j, 1] = 0;
end // loop for j begin for j = 1 to k do begin
// repeat the process one packet per session
// selected
// n be node index for the
// value of
B[j, 1] =
if ( i > 1 && i < r )
// r is the receiver node index
// forwarding time of
// junction node and
B[j, I] =
} // end if block if ( i = = n )
// r is the sink node, no further forward is
// required.
B[ j, r] =
// Approximate value of
// the sample mean of size (n – 2)
} // end if block sum = sum +
end // loop for i
sum = D[j, 1]
2sum = 0 for i = 1 to n do begin 2sum = 2sum + B [j, i]
end // loop for i 2sum = E [j, 1]
end // loop for j end // Algorithm
By solving the number of computation of the proposed algorithm, the time complexity of the proposed algorithms belongs to
4. Experimental Result and Discussion
The Figure 6 shows the result corresponding to the diagram Figure 1 where the data transferred from the source node through the junction node in a session. The receivers build a single cluster with a
cluster head. The cluster has n–numbers of receiver nodes. The cluster members gradually connected to the source node i.e. it is gradually expands from the time 0.25 second (in a session ) and
gradually release resources i.e. the size of the cluster reduce from the 0.32 second (in other session). It shown from the Figure 6 that when the number of nodes (receivers) increase in the cluster,
the throughput decreases as in the receiver side, the packet loss increase i.e. packet delay increase. When the number of nodes in cluster decrease, i.e., the member node leaves the group, the packet
receive rate increase. As well as the throughput increases for the group as it is indicated in between the time (0.32, 0.35).
The Figure 7 shows the result corresponds to the diagram Figure 2, initially the Network have two cluster of two different size. In a session up to 0.31 second member connected to the source node via
the junction nodes and the receive packets After 0.31 second in the new session one group leave from the network, i.e. release the resources the Figure 7 shows the effect that on the throughput.
In Figure 8 and Figure 9 the effect on the overall throughput when the one cluster leaves the network (smaller in number of nodes) and a comparatively bigger cluster connect to the network in other
session. Figure 10 and Figure 11 represent how the load increased in junction nodes that connected to the two clusters. Figure 12 and Figure 13 represent the traffic pattern that pass through the two
junction nodes that connected to the two different clusters.
Figure 7. Throughput—Decrease cluster number time.
Figure 9. Throughput two different size clusters.
Figure 10. Traffic load in side the small cluster.
Figure 11. Traffic load in side big cluster.
Figure 12. Packet received at receiver small cluster.
Figure 13. Packet received at receiver big cluster.
5. Concluding Remarks
In this paper we develop the novel algorithms, the packet level forwarding in multicast session and on the basis of that we simulate the real problem in the lab environment. When the cluster size
increases the throughput decreases. We consider a single cluster that expands in a session and reduces in other session as some nodes join in the cluster in a session and some node leaves as in other
We also study when the network is composed of multiple cluster and some cluster expands in number and some cluster reduces in number in different sessions. We experiment that with respect to real
The work can be further extended for the problem arise in the next generation Network (NGN). In the case of IP based video on demand problem. In that case in spite of bandwidth limited constraint the
number of receiver connected to the main video server as getting the clips Video –on-demand as IP data packets (datagram), the receiver forms single or more than one multicast groups .The receiver
get the data packet as in the multicast session to the receiver as for the views particular (channel) that is related channel to a particular frequency.
6. References
[1] S. Kanrar and M. Siraj, “Performance of Heterogeneous Network,” International Journal of Computer Science and Network Security, Vol. 9, No. 8, 2009, pp. 255-261.
[2] Y. F. Xi and E. M. Yeh, “Distributed Algorithms for Minimum Cost Multicast with Network Coding,” IEEE/ ACM Transaction on Networking, Vol. 18, No. 2, 2009, pp. 379-392.
[3] J. Nair and D. Manjunath, “Distributed Iterative Optimal Resource Allocation with Concurrent Updates of Routing and Flow Control Variables,” IEEE/ACM Transaction on Networking, Vol. 17, No. 4,
2009, pp. 1312-1325.
[4] R. Gandhi, A. Mishra and S. Parthasarathy, “Minimizing Broadcast Latency and Redundancy in Ad Hoc Networks,” IEEE/ACM Transaction on Networking, Vol. 16 No. 4, 2008, pp. 840-851.
[5] H. Han, S. Shakkottai, C. V. Hollot, R. Srikant and D. Towsley, “Multi-Path TCP: A Joint Congestion Control and Routing Scheme to Exploit Path Diversity in the Internet,” IEEE/ACM Transaction on
Networking, Vol. 14, No. 6, 2006, pp. 1260-1271.
[6] S. J. Golestani and K. K. Sabanani, “Fundamental Observations on Multicast Congestation Control in The Internet,” Proceedings of IEEE INFOCOM, New York, Vol. 2, March 1999, pp. 990-1000.
[7] X. Li, S. Paul and M. Ammar, “Layered Video Multicast with Retransmission (LVMR): Evaluation of Hierarchical Rate Control,” Proceedings of IEEE INFOCOM, San Francisco, Vol. 3, June 1998, pp.
[8] A. Mankin, A. Romanow, S. Bradner and V. Paxon, “IETF Criteria for Evaluating Reliable Multicast Transport and Application Protocols,” Networking Working Group, Internet Draft, RFC 2357, 1998.
[9] J. Mo and J. Walrand, “Fair End to End Windows Based Congestion Control,” IEEE/ACM Transaction on Networking, Vol. 8, No. 5, October 2000, pp. 556-557.
[10] F. P. Kelly, “Charging and Rate Control for Elastic Traffic,” European Transactions on Telecommunications, Vol. 8, No. 1, 1998, pp. 33-37.
[11] M. Handley and S. Floyd, “Strawman Congestion Control Specifications,” Internet Research Task Force (IRTF), Reliable Multicast Research Group (RMRG), 1998. http:// www.aciri.org/mjh/rmcc.ps.gz
[12] L. Rizzo, L. Vicisano and J. Crowcroft, “TCP Like Congestion Control for Layered Multicast Data Transfer,” Proceedings of IEEE INFOCOM, San Francisco, Vol. 3, March 1998, pp. 996-1003.
[13] I. Stoica, T. S. E. Ng and H. Zhang, “REUNITE: A Recursive Unicast Approach to Multicast,” Proceedings of IEEE INFOCOM, Tel Aviv, Vol. 3 March 2000, pp. 1644-1653.
[14] S. Bhattacharyya, D. Towslay and J. Kurose, “The Loss Path Multiplicity Problem in Multicast Congestion Control,” Proceedings of IEEE INFOCOM, New York, Vol. 2, March 1999, pp. 856-863.
[15] A. Chaintreau, F. Baccelli and C. Doit, “Impact of TCP-Like Congestion Control on the Throughput of Multicast Grouph,” IEEE/ACM Transaction on Networking, Vol. 10, No. 4, 2002, pp. 500-512.
[16] S. Deb and R. Srikant “Congestion Control for Fair Resource Allocation in Networks with Multicast Flow,” IEEE/ACM Transaction on Networking, Vol. 12, No. 2, 2004, pp. 274-285.
[17] R. C. Chalmers and K. C. Almeroth, “Modeling the Branching Characteristics and Efficiency Gains of Global Multicast Tree,” Proceedings of IEEE INFOCOM, Anchorage, Vol. 1, April 2001, pp.
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Mathematica Q&A Series: Surprises in Differentiation and Integration—Wolfram Blog
Mathematica Q&A Series: Surprises in Differentiation and Integration
Got questions about Mathematica? The Wolfram Blog has answers! We’ll regularly answer selected questions from users around the web. You can submit your question directly to the Q&A Team.
This week’s question comes from Kutha, a math lecturer:
Why doesn’t differentiating after integrating always return the original function?
Read below or watch this screencast for the answer (we recommend viewing it in full-screen mode):
The derivative of a definite integral with respect to its upper bound (with a constant lower bound) is equal to the integrand:
This is a consequence of the fundamental theorem of calculus. (Note that sin(x) is equivalent to sin(t) up to renaming of the variable x or t.)
In complicated cases, it can appear that Mathematica is not giving back the original function:
However, this result is mathematically equivalent to the integrand sin(x^3). You can often discover this fact by using Simplify or FullSimplify:
That won’t necessarily work if you didn’t give the original integrand in its simplest form:
Here, the simplified result is different from the integrand:
In this case, you can try simplifying the integrand as well as the result:
Or take the easiest option and ask Mathematica to try to prove they are equal:
Another calculus-related question concerns differentiation followed by indefinite integration. For this sequence of operations, the result is not necessarily mathematically equivalent to the original
function, since an arbitrary constant of integration may be added (or subtracted):
In this case, it’s obvious that the original function and the result differ by the constant 5. In more complicated cases, it may not be immediately obvious what the differing constant of integration
Once again, you can use Simplify or FullSimplify to discover the difference:
For functions with branch cut discontinuities, indefinite integrals can be trickier:
Here, since log(x) and log(-x) have different branch cuts (from -∞ to 0 and from 0 to ∞, respectively), a consistent constant of integration isn’t possible. Instead, you get a “piecewise” constant of
In the top half of the complex plane, log(x) and log(-x) differ by the constant iπ:
While in the bottom half, they differ by –iπ:
You can avoid ambiguities related to branch cuts by specifying a definite integral from some point in the complex plane:
The condition in the second argument of ConditionalExpression restricts the result to straight integration paths (from t = 1) that don’t intersect the branch cut running from z = -∞ to z = 0.
Depending on your starting point, the condition to avoid intersecting branch cuts can be quite complicated. For example, starting with t = 1 + i:
That condition represents the following set of valid integration endpoints:
Click here to download this post as a Computable Document Format (CDF) file, including the RegionPlot command used to generate the diagram above.
If you have a question you’d like answered in this blog, you can submit it to the Q&A Team. For daily bite-sized Mathematica tips, follow our @MathematicaTip Twitter feed.
Join the discussion
4 comments
1. Hello! I really appreciate your youtube videos.
You use FullSimplify here to compare expressions. However I discovered something that is bothering me. If I define f[x_] = (x!)^(1/x), then f'[x] and FullSimplify[f'[x]] are different functions
(just plot them). How would I know when FullSimplify is tricking me?
□ Hi Nikolaj –
Thanks for bringing this to our attention. Our developers are investigating it.
2. @Nikolaj:
In addition, the problem doesn’t appear when you use Simplify (mostly because it doesn’t get any further than finding a common denominator) – only when using FullSimplify.
The issue seems to arise in the “simplification” from
-Log[Gamma[1 + x]] + x PolyGamma[0, 1 + x]
x (HarmonicNumber[x] + EulerGamma) + Log[Gamma[1 + x]]
I think the first expression should simplify to x (HarmonicNumber[x] – EulerGamma) – Log[Gamma[1 + x]] so it isn’t clear at all to me where the second expression comes from.
I’d be interested to know the reason for this too.
3. a triangle with the following dimensions 456 m by 616 m was to be fenced give the maximum number of posts to be used
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Dynamic wake meandering model calibration using nacelle-mounted lidar systems
Articles | Volume 5, issue 2
© Author(s) 2020. This work is distributed under the Creative Commons Attribution 4.0 License.
Dynamic wake meandering model calibration using nacelle-mounted lidar systems
Light detection and ranging (lidar) systems have gained a great importance in today's wake characteristic measurements. The aim of this measurement campaign is to track the wake meandering and in a
further step to validate the wind speed deficit in the meandering frame of reference (MFR) and in the fixed frame of reference using nacelle-mounted lidar measurements. Additionally, a comparison of
the measured and the modeled wake degradation in the MFR was conducted. The simulations were done with two different versions of the dynamic wake meandering (DWM) model. These versions differ only in
the description of the quasi-steady wake deficit. Based on the findings from the lidar measurements, the impact of the ambient turbulence intensity on the eddy viscosity definition in the
quasi-steady deficit has been investigated and, subsequently, an improved correlation function has been determined, resulting in very good conformity between the new model and the measurements.
Please read the corrigendum first before continuing.
Received: 19 Nov 2019 – Discussion started: 04 Dec 2019 – Revised: 26 Mar 2020 – Accepted: 18 May 2020 – Published: 19 Jun 2020
Wake calculation of neighboring wind turbines is a key aspect of every wind farm development. The aim is to estimate both energy yield of the whole wind farm and loads on single turbines as
accurately as possible. One of the main models for calculating the wake-induced turbulence in a wind farm is the so-called Frandsen model (see, for example, Frandsen, 2007). Previous measurement
campaigns have shown that this model delivers conservative results for small turbine distances (Reinwardt et al., 2018; Gerke et al., 2018). This is particularly important for onshore wind farms in
densely populated areas, where a high energy output per utilized area is crucial. In such cases, the usage of a more accurate description of the physical behavior of the wake, as defined in the
dynamic wake meandering (DWM) model, seems appropriate. The DWM model is based on the assumption that the wake behaves as a passive tracer, which means the wake itself is deflected in the vertical
and horizontal directions (Larsen et al., 2008b). The combination of this deflection and the shape of the wind speed deficit leads to an increased turbulence at a fixed position downstream. This
plays an eminent role in the loads of a turbine located downstream of another turbine (Larsen et al., 2013). Therefore, a precise description of the meandering itself and the wind speed deficit in
the meandering frame of reference (MFR) as well as a detailed validation of the wind speed deficit definition are fundamental.
Lidar systems are highly suitable for wake validation purposes. In particular, the so-called scanning lidar systems offer great potential for detailed wake analysis. These lidars are capable of
scanning a three-dimensional wind field, so that the line-of-sight (LOS) wind speed can be measured subsequently at different positions in the wake, thus enabling the detection of the wake meandering
as well as the shape of the wind speed deficit in the MFR. That is the reason why such a device is used in the measurement campaign outlined here. Several different measurement campaigns with
ground-based and nacelle-mounted lidar systems have already been carried out in the last years, some of them even with the purpose of tracking wake meandering and validation of wake models.
In Bingöl et al. (2010) the horizontal meandering has been examined with a nacelle-installed continuous-wave (CW) lidar. The campaign confirms the passive tracer assumption, which is essential for
the definition of the meandering in the DWM model. Furthermore, the wind speed deficit in the MFR has been investigated for some distances. Due to the fact that the CW lidar can not measure
simultaneously in different downstream distances, the beam has been focused successively to different downstream distances. In Trujillo et al. (2011) the analysis has been extended to a
two-dimensional scan. The measured wind speed deficit in the MFR has been compared to the Ainslie wake model (Ainslie, 1988), which constitutes the basis of the deficit's definition in the DWM model.
Additionally, in Machefaux et al. (2013) a comparison of measured lateral wake meandering based on pulsed scanning lidar measurements has been presented. Special attention is paid to the advection
velocity of the wake, which is estimated with measured and low-pass-filtered wind directions at the met mast (based on the assumptions of the DWM model) and the wake displacement at certain
downstream distances. The analysis shows that the advection velocity calculated by the Jensen model is in relatively good agreement. Finally, the study compares the measured expansion of the wake in
the fixed frame of reference (FFR) to computational fluid dynamics (CFD) simulations and simple analytical engineering models. The wake expansion calculated by simple analytical engineering models is
well in line with lidar measurements and CFD simulations, but it also depicts potential for further improvements, which is why a new empirical model for single-wake expansion is proposed in Machefaux
et al. (2015). In Machefaux et al. (2016) a measurement campaign is presented that involves three nacelle-mounted CW scanning lidar devices. The investigation includes a spectral analysis of the wake
meandering, a comparison of the measurements to the assumptions in the DWM model, and a comparison of the wind speed deficit profile in a merged wake situation to CFD simulations.
It should be noted that the references listed here are only the most essential, on which the present measurement campaign builds. Several campaigns including either lidar systems or meandering
observations as well as wake model validations have been conducted in the past. The outlined analysis transfers some of the procedures of tracking the wake meandering to measurement results from an
onshore wind farm with small turbine distances. Particular focus is put on the investigation of the wind speed deficit's shape in the MFR and the degradation of the wind speed deficit in the
downstream direction. The latter can be captured very well with the used nacelle-mounted pulsed scanning lidar systems due to the fact that it measures simultaneously in different downstream
distances. Thus, a detailed comparison of the predicted degradation of the wind speed deficit between the DWM model and the measurement results is possible. Furthermore, the collected lidar
measurements are used to recalibrate the DWM model, which enables a more precise modeling of the wake degradation. As a consequence, the calculation of loads and energy yield of the wind farm can be
The remaining document is arranged as follows: in Sect. 2, the investigated wind farm and the installed measurement equipment are described in detail. Afterwards, in Sect. 3, an explanation of the
data processing and filtering of the measurement results is given. Sections 4, 5, and 6 focus on the description of the theoretical background, and a hands-on implementation of the DWM model is
introduced. Based on the outlined measurement results, a recalibration of the defined degradation of the wind speed deficit in the DWM model is proposed in Sect. 6. A summary of the measurement
results can be found in Sect. 7, and a comparison to the original DWM model as well as the recalibrated version is presented in Sect. 8. Finally, all findings are concluded in Sect. 9.
The investigated onshore wind farm (Fig. 1) located in the southeast of Hamburg (Germany) consists of five closely spaced Nordex turbines (one N117 3MW turbine and four N117 2.4MW turbines) and an
IEC-compliant 120m met mast, which is situated two rotor diameters (D=117m) ahead of the wind farm in the main wind direction (west-southwest). It is equipped with 11 anemometers, two of which are
ultrasonic devices; three wind vanes; two temperature sensors; two thermohygrometers; and two barometers. The sensors are distributed along the whole met mast, but at least one of each is mounted in
the upper 8m (see Fig. 2). The thrust as well as the power coefficient curves for both wind turbines are illustrated in Fig. 3. There are no other turbines in the immediate vicinity and the terrain
is mostly flat. Only at further distances (more than 1km) is the terrain slightly hilly (approx. 40m). Two turbine nacelles are equipped with a pulsed scanning lidar system (Galion G4000). The wind
farm layout with all installed measurement devices is shown in Fig. 1 (the displayed load measurements are not in the scope of this paper, but will be introduced in future publications). One lidar
system is installed on top of the nacelle of WTG 2 (N117 2.4MW), facing backwards. The second lidar system is installed inside the nacelle of WTG 1 (N117 3MW) and measures through a hole in the
rear wall. In this case, mounting the device on top of the nacelle is not possible, as the area is occupied by a recuperator. The positions of both devices are displayed in Fig. 2. Even though the
setup reduces the field of vision, the measurement campaign described in this paper is not influenced by this restriction. On the plus side, the lidar system is not exposed to weather. Finally,
nacelle-mounted differential GPS systems help track the nacelle's precise position as well as yaw movements with a centimeter range accuracy.
3Data filtering and processing
The lidar data are filtered in accordance with the wind direction, so that lidar data without free inflow of the wake-generating turbine as well as lidar measurements in the induction zone of another
turbine are rejected. This leads to the remaining wind direction sectors listed in Table 1. The remaining sectors are relatively small, especially for the lidar on WTG 2, which reduces the amount of
usable measurement data drastically. Additionally, the measured lidar data are sorted into turbulence intensity bins for the further validation and recalibration of the DWM model. The ambient
conditions are determined by 10min time series statistics from the met mast; hence only measurement results with free inflow at the met mast are useable. Only situations with normal power production
of the wake-generating turbine are considered. The turbine operation mode is identified through the turbine's supervisory control and data acquisition (SCADA) system. The statistics of the 10min
time series are applied to identify the operational mode. Furthermore, the data have been analyzed according to yaw misalignments, so that no data with turbine misalignments greater than 6^∘ are
considered in the analysis. The misalignment is determined by the GPS systems and the met mast wind direction. Moreover, the lidar data are filtered by the power intensity of the measurement results,
which is closely related to the signal-to-noise ratio (SNR) of the measurements. Results with an intensity lower than 1.01 have been discarded. The pulse repetition rate of the lidar system is
15kHz. The ray update rate is about 1Hz (depending on the atmospheric conditions), so it averages over approximately 15000 pulses. The sample frequency is 100MHz. Considering the speed of light,
this delivers a point length of 1.5m. The range gate length is 30m; hence 20 points are used per range gate. The measurement time increases with the number of range gates, because the internal data
processing time increases. Thus, to decrease the measurement time, the number of range gates has been limited, so that the farthest scan point is 750m downstream. Additionally, the scanning time of
each complete horizontal line scan is verified by the timestamp of each scan to ensure that the meandering can really be captured. In summary, this leads to the following filtering procedure for the
measured lidar data.
1. Filter according to the wind direction determined by the met mast (free inflow at met mast and wind turbine and no induction zone from other turbines).
2. Filter according to the normal power production determined by the turbine's SCADA system.
3. Filter according to yaw misalignment.
4. Filter according to the SNR of the lidar measurements.
5. Filter according to scan time.
6. Group all data sets in turbulence intensity bins with a bin width of 2%.
Lidar systems measure the line-of-sight velocity. The wind speed in the downstream direction is then calculated from the lidar's LOS velocity and the geometric dependency of the position of the laser
beam relative to the main flow direction as outlined in Machefaux et al. (2012). Thus, the horizontal wind speed is defined as
$\begin{array}{}\text{(1)}& U\left(t\right)={U}_{\mathrm{LOS}}\cdot \frac{\mathrm{1}}{\mathrm{cos}\left(\mathit{\theta }\right)\cdot \mathrm{cos}\left(\mathit{\varphi }\right)},\end{array}$
where θ is the azimuth angle and ϕ the elevation angle of the lidar scan head. This seems to be a suitable approach for small scan opening angles like in the measurement campaign presented here. The
biggest opening angle in the scan pattern is 20^∘. Nevertheless, if there is yaw misalignment, this could have an impact on the overall results. To decrease the uncertainties based on yaw
misalignments, the measurement data have accordingly been filtered. The yaw misalignment has the biggest impact at the largest scan opening angle; i.e., a misalignment of 6^∘ at an opening angle of
20^∘ leads to an overestimation of the wind speed of less than 5%.
4Wind speed deficit in HMFR calculation
The meandering time series and the wake's horizontal displacement are determined with the help of a Gaussian fit. Trujillo et al. (2011) assume that the probability of the wake position in the
vertical and horizontal directions is completely uncorrelated, so that the two-dimensional fitting function can be expressed as follows:
$\begin{array}{}\text{(2)}& {f}_{\mathrm{2}\mathrm{D}}=\frac{{A}_{\mathrm{2}\mathrm{D}}}{\mathrm{2}\mathit{\pi }{\mathit{\sigma }}_{y}{\mathit{\sigma }}_{z}}\mathrm{exp}\left[-\frac{\mathrm{1}}{\
mathrm{2}}\left(\frac{\left({y}_{i}-{\mathit{\mu }}_{y}{\right)}^{\mathrm{2}}}{{\mathit{\sigma }}_{y}^{\mathrm{2}}}+\frac{\left({z}_{i}-{\mathit{\mu }}_{z}{\right)}^{\mathrm{2}}}{{\mathit{\sigma }}_
where σ[y] and σ[z] are the standard deviations of the horizontal and vertical displacements μ[y] and μ[z], respectively. In the analysis presented here, only results from a horizontal line scan are
analyzed, so that no vertical meandering is eliminated from the wind speed deficit, and the deficit's depth is less pronounced in comparison to the real MFR. To clarify that the vertical meandering
is not eliminated in the present investigation, but included in the wind speed deficit, the abbreviation HMFR (horizontal meandering frame of reference) is introduced and henceforth used instead of
MFR. A comparison of the wind speed deficit simulated with the DWM model in the complete MFR and the HMFR is illustrated in Fig. 4. The simulations were carried out for a small downstream distance of
2.5D and a high turbulence intensity of 16%. There are only small discrepancies around the center of the wake, which validates the present assumption.
Since the vertical meandering is neglected, the measurement results are fitted to a one-dimensional Gaussian curve defined as follows:
$\begin{array}{}\text{(3)}& {f}_{\mathrm{1}\mathrm{D}}=\frac{{A}_{\mathrm{1}\mathrm{D}}}{\sqrt{\mathrm{2}\mathit{\pi }}{\mathit{\sigma }}_{y}}\mathrm{exp}\left(-\frac{\mathrm{1}}{\mathrm{2}}\frac{\
left({y}_{i}-{\mathit{\mu }}_{y}{\right)}^{\mathrm{2}}}{{\mathit{\sigma }}_{y}^{\mathrm{2}}}\right),\end{array}$
where A[1D] represents a scaling parameter. The measured wind speeds are fitted to the Gauss shape via a least-squares method. Thereby, only fitted horizontal displacements μ[y] that are between −200
and 200m are used for further validations of the mean wind speed in the HMFR. A horizontal displacement of more than 200m cannot be represented by the Gauss fit due to a lack of measurement points.
However, such an event is highly improbable (e.g., the DWM model predicts the wind speed deficit's probability at the horizontal position of 200m to be $\mathrm{2}×{\mathrm{10}}^{-\mathrm{22}}$ for
an ambient wind speed of 6.5ms^−1 and an ambient turbulence intensity of 8%).
The entire method of calculating the wind speed deficit in the HMFR is illustrated in Fig. 5 and can be described as follows. The lidar system takes measurements from the nacelle of the turbine in
the downstream direction, which deliver the wind speed deficit in the nacelle frame of reference or even in the FFR (see left side of Fig. 5) if the turbine is not moving (this is ensured by the GPS
systems). A Gauss curve is then fitted into the scanned points as explained previously. It provides the horizontal displacement of the wake, so that each scan point can be transferred into the HMFR
with the calculated displacement (see middle diagrams in Fig. 5). The last step illustrated in the diagrams is the interpolation to a regular grid. These three steps are repeated for a certain number
of scans N (e.g., approx. 37 for a 10min time series). Finally, the mean value of all single measurement results in the HMFR is calculated. It should be noted that it is mandatory to interpolate to
a regular grid. Otherwise it would not be possible to take the mean of all scans since the horizontal displacement differs at each instant of time, and, therefore, the measurement points are
transmitted to a different location in the HMFR. After averaging, the plausibility of the results is inspected. If the calculated minimum mean wind speed in the HMFR is higher than the minimum mean
wind speed in the FFR, it is assumed that the Gauss fit failed and the results are no longer considered. In theory, the wind speed deficit in the HMFR should be more pronounced than the measured one
in the FFR, wherefore this fundamental plausibility check is added.
One of the most challenging parts of this specific measurement campaign is the low ray update rate of the lidar system, which is considerably smaller than in the previously introduced measurement
campaigns (Bingöl et al., 2010; Trujillo et al., 2011). To ensure that the meandering as well as the wind speed deficit in the HMFR can be captured with the devices used, lidar and wind field
simulations have been conducted in advance. The simulations incorporate lidar specifications (e.g., beam update rate and scan head angular velocity) and wind farm site conditions (ambient turbulence
intensity and wind shear). The simulations assume perfect lidar measurements, where no probe volume averaging is considered and the lidar measures the horizontal wind speed directly. The wind field
is simulated at halfway of the range gate. The simulated lidar “takes measurements” in a simulated wind field that is generated by the DWM model and includes wake effects as well as ambient
turbulence. A detailed description of the model is given in Sect. 6. The in-house code is written in Python. From these “measured” wind speeds the meandering is determined via Gaussian fits as
previously explained and implemented in the real measurement campaign. Simulations are performed for different scan patterns, ambient conditions, and downstream distances to test the scan pattern,
which for this one-dimensional scan consists of only 11 scan points scanned in a horizontal line from −20 to 20^∘ in 4^∘ steps. The “measurement” results of the simulated meandering time series are
shown in Fig. 6a, whereas the corresponding wind speed deficit in the HMFR is presented in Fig. 6b. The results are compared to the original meandering time series and the simulated wind speed
deficit. The measured wind speed deficit in the simulated environment reproduces the simulated wind speed and its underlying meandering time series very well (the coefficient of determination R^2 is
approximately 0.93). Although only 11 scan points are used for these plots, the curve of the wind speed deficit is very smooth. The reason for this behavior is the previously mentioned interpolation
process. The distribution generated by the meandering process provides many scan points around the center of the wind speed deficit and only a few at the tails. Therefore, the influence of turbulence
at the tails is much higher, leading to a somewhat coarse distribution at the boundaries of the deficit. It should also be noted that since this is a one-dimensional scan, the simulated lidar
measures the wind speed deficit only horizontally, neglecting the wake's less dominant vertical movement. Whenever the wind speed deficit in the HMFR is mentioned in subsequent validations, it
implies the neglect of eliminating the vertical meandering from the wind speed deficit, which has only a marginal impact on the shape of the wind speed deficit in the real MFR (see Fig. 4).
The lidar simulations indicate that the Gauss fit works more reliably under optimal operating conditions, i.e., at optimal tip speed ratio, when the wind speed deficit is most pronounced and the
power coefficient C[p] has its maximum (see Fig. 3). For the turbines examined, this applies to a range of 5 up to 8ms^−1, so that only measurement results with ambient wind speeds in this interval
are analyzed.
6Dynamic wake meandering model
The measured wind speed deficit in the HMFR is consecutively compared to the DWM model, which is based on the assumption that the wake behaves as a passive tracer in the turbulent wind field.
Consequently, the movement of the passive structure, i.e., the wake deficit, is driven by large turbulence scales (Larsen et al., 2007, 2008b). The main components of the model are summarized in
Fig. 7a. The model was built in house and independently of any commercial software in Python.
6.1Quasi-steady wake deficit
One key point of the model is the quasi-steady wake deficit or rather the wind speed deficit in the MFR. In this study, two calculation methods for the quasi-steady wake deficit are compared with the
lidar measurement results. A similar comparison of these models to met mast measurements in the FFR was published in Reinwardt et al. (2018). The quasi-steady wake deficit is defined in the MFR and
consists of a formulation of the initial deficit emitted by the wake-generating turbine and the expansion of the deficit downstream (Larsen et al., 2008a). The latter is calculated with the thin
shear layer approximation of the Navier–Stokes equations in their axisymmetric form. This method is strongly related to the work of Ainslie (1988) and outlined in Larsen et al. (2007). The thin shear
layer equations expressed by the wind speed in the axial and radial directions U and V[r], respectively, are defined by
$\begin{array}{}\text{(4)}& U\frac{\partial U}{\partial x}+{V}_{\mathrm{r}}\frac{\partial U}{\partial r}=\frac{\mathrm{1}}{r}\frac{\partial }{\partial r}\left({\mathit{u }}_{T}r\frac{\partial U}{\
partial r}\right)\end{array}$
$\begin{array}{}\text{(5)}& \frac{\mathrm{1}}{r}\frac{\partial }{\partial r}\left(r{V}_{\mathrm{r}}\right)+\frac{\partial U}{\partial x}=\mathrm{0}.\end{array}$
The first part of the quasi-steady wake deficit, the initial deficit, serves as a boundary condition when solving the equations. In both methods used to determine the quasi-steady wake deficit, the
initial deficit is based on the axial induction factor derived from the blade element momentum (BEM) theory. Pressure terms in the thin shear layer equations are neglected. The error that inherently
comes with this assumption is accommodated by using the wind speed deficit two rotor diameters downstream (beginning of the far-wake area) as a boundary condition for the solution of the thin shear
layer equations. The equations are solved directly from the rotor plane by a finite-difference method with a discretization in the axial and radial directions of 0.2D and 0.0125D combined with an
eddy viscosity (ν[T]) closure approach. The two methods that are compared with the lidar measurements only differ in the definition of the initial deficit and the eddy viscosity formulation.
For the first method the following formulae are given to calculate the initial deficit. Hence, the boundary conditions for solving the thin shear layer equations are (Madsen et al., 2010)
$\begin{array}{}\text{(6)}& {U}_{\mathrm{w}}\left(\frac{{r}_{\mathrm{w},i+\mathrm{1}}+{r}_{\mathrm{w},i}}{\mathrm{2}}\right)={U}_{\mathrm{0}}\left(\mathrm{1}-\mathrm{2}{a}_{i}\right)\end{array}$
$\begin{array}{}\text{(7)}& {r}_{\mathrm{w},i+\mathrm{1}}=\sqrt{\frac{\mathrm{1}-{a}_{i}}{\mathrm{1}-\mathrm{2}{a}_{i}}\left({r}_{i+\mathrm{1}}^{\mathrm{2}}-{r}_{i}^{\mathrm{2}}\right)+{r}_{\mathrm
$\begin{array}{}\text{(8)}& {f}_{\mathrm{w}}=\mathrm{1}-\mathrm{0.45}{\stackrel{\mathrm{‾}}{a}}^{\mathrm{2}},\end{array}$
where $\stackrel{\mathrm{‾}}{a}$ represents the mean induction factor along all radial positions i, r[i] the rotor radius, and r[w,i] the wake radius. The boundary condition of the radial velocity
component is V[r]=0. The initial wake expansion and the corresponding radial positions as well as the pressure recovery in the downstream direction are illustrated in Fig. 7b. The eddy viscosity ν[T]
used in Eq. (4) is calculated in this first approach as follows (Larsen et al., 2013):
$\begin{array}{}\text{(9)}& \begin{array}{rl}\frac{{\mathit{u }}_{T}}{{U}_{\mathrm{0}}R}& ={k}_{\mathrm{1}}{F}_{\mathrm{1}}\left(\stackrel{\mathrm{̃}}{x}\right){F}_{\mathrm{amb}}\left(\stackrel{\
mathrm{̃}}{x}\right){I}_{\mathrm{0}}\\ & +{k}_{\mathrm{2}}{F}_{\mathrm{2}}\left(\stackrel{\mathrm{̃}}{x}\right)\frac{{R}_{\mathrm{w}}\left(\stackrel{\mathrm{̃}}{x}\right)}{R}\left(\mathrm{1}-\frac{{U}_
with k[1]=0.1 and k[2]=0.008. The eddy viscosity is normalized by the ambient wind speed U[0] and the rotor radius R. The outlined definition consists of two terms. The first is related to the
ambient turbulence intensity I[0], whereas the second depends on the shape of the wind speed deficit itself. The single terms are weighted with the factors k[1] and k[2]. The filter functions F[1]
and F[2] in Eq. (9) depending on $\stackrel{\mathrm{̃}}{x}$ (downstream distance normalized by the rotor radius) are defined by IEC 61400-1 (2019) as follows:
The filter function F[2] covers the lack of equilibrium between the velocity field and the rising turbulence in the beginning of the wake. F[1] is introduced to include the fact that the depth of the
wind speed deficit increases in the near-wake area up to (2…3)D downstream of the turbine until it attenuates again in the downstream direction (Madsen et al., 2010). The filter function as well as
Eq. (8) are calibrated against actuator disc simulations at a downstream distance of 2D, the beginning of the far-wake area, where the wake is fully expanded (Madsen et al., 2010). A more detailed
explanation of the nonlinear coupling function F[amb] is given in Sect. 6.3. This calculation method (Eqs. 6 to 11) is subsequently named “DWM-Egmond” after the site, which is used for the
calibration of the eddy viscosity in Larsen et al. (2013).
The second investigated method defines the initial deficit by the following equations (Keck, 2013):
$\begin{array}{}\text{(12)}& {U}_{\mathrm{w}}\left({r}_{\mathrm{w},i}\right)={U}_{\mathrm{0}}\left(\mathrm{1}-\left(\mathrm{1}+{f}_{u}\right){a}_{i}\right)\end{array}$
$\begin{array}{}\text{(13)}& {r}_{\mathrm{w},i}={r}_{i}\sqrt{\frac{\mathrm{1}-\stackrel{\mathrm{‾}}{a}}{\mathrm{1}-\left(\mathrm{1}+{f}_{R}\right)\stackrel{\mathrm{‾}}{a}}},\end{array}$
with f[u]=1.1 and f[R]=0.98. The boundary condition of the radial velocity component is again V[r]=0. In Keck (2013) the final and recommended version of the model developed for the eddy viscosity is
defined as follows:
$\begin{array}{}\text{(14)}& \begin{array}{rl}{\mathit{u }}_{T}& ={k}_{\mathrm{1}}{F}_{\mathrm{1}}\left(\stackrel{\mathrm{̃}}{x}\right){u}_{\mathrm{ABL};\mathit{\lambda }<\mathrm{2}D}^{\ast }{l}_{\
mathrm{ABL};\mathit{\lambda }<\mathrm{2}D}^{\ast }\\ & +{k}_{\mathrm{2}}{F}_{\mathrm{2}}\left(\stackrel{\mathrm{̃}}{x}\right)max\left({l}^{\ast \mathrm{2}}\left|\frac{\partial U\left(\stackrel{\mathrm
{̃}}{x}\right)}{\partial r}\right|,{l}^{\ast }\left(\mathrm{1}-{U}_{min}\left(\stackrel{\mathrm{̃}}{x}\right)\right)\right),\end{array}\end{array}$
with k[1]=0.578 and k[2]=0.0178 and the filter functions
In contrast to the previously mentioned model (DWM-Egmond), atmospheric stability is considered in this final model description. Equation (14) involves the velocity ${u}_{\mathrm{ABL};\mathit{\lambda
}<\mathrm{2}D}^{\ast }$ and length scale ${l}_{\mathrm{ABL};\mathit{\lambda }<\mathrm{2}D}^{\ast }$ fractions of the ambient turbulence, which is related to the wake deficit evolution (eddies smaller
than 2D). The velocity scale ${u}_{\mathrm{ABL};\mathit{\lambda }<\mathrm{2}D}^{\ast }$ is in addition to the ambient turbulence intensity I[0] related to the ratio of the Reynolds stresses (normal
stress in the flow direction and the shear stress), which in turn are functions of the atmospheric stability. A detailed description of a method to introduce atmospheric stability in the DWM model
can be found in Keck et al. (2014) and Keck (2013). In contrast to the final and recommended model in Keck (2013), atmospheric stability is not considered in this study, so that a previous model in
Keck (2013) without consideration of atmospheric stability is used, and the numerical constants k[1] and k[2] in Eq. (17) are changed with respect to the first least-squares recalibration in Keck (
2013). Furthermore, according to Keck (2013) it can be assumed that the mixing length l^∗ is equal to half of the wake width. This results in the following formulation of the eddy viscosity:
$\begin{array}{}\text{(17)}& \begin{array}{rl}\frac{{\mathit{u }}_{T}}{{U}_{\mathrm{0}}R}=& {k}_{\mathrm{1}}{F}_{\mathrm{1}}\left(\stackrel{\mathrm{̃}}{x}\right){I}_{\mathrm{0}}+{k}_{\mathrm{2}}{F}_{\
mathrm{2}}\left(\stackrel{\mathrm{̃}}{x}\right)max\left(\frac{{R}_{\mathrm{w}}\left(\stackrel{\mathrm{̃}}{x}{\right)}^{\mathrm{2}}}{R{U}_{\mathrm{0}}}\left|\frac{\partial U\left(\stackrel{\mathrm{̃}}{x}
\right)}{\partial r}\right|,\\ & \frac{{R}_{\mathrm{w}}\left(\stackrel{\mathrm{̃}}{x}\right)}{R}\left(\mathrm{1}-\frac{{U}_{min}\left(\stackrel{\mathrm{̃}}{x}\right)}{{U}_{\mathrm{0}}}\right)\right)\
with k[1]=0.0914 and k[2]=0.0216.
6.2Meandering of the wake
The meandering of the wind speed deficit is calculated from the large turbulence scales of the ambient turbulent wind field. Thus, the vertical and horizontal movements are calculated from an ideal
low-pass-filtered ambient wind field. The cutoff frequency of the low-pass filter is specified by the ambient wind speed and the rotor radius as (Larsen et al., 2013)
$\begin{array}{}\text{(18)}& {f}_{\mathrm{c}}=\frac{{U}_{\mathrm{0}}}{\mathrm{4}R}.\end{array}$
The horizontal y(t) and vertical z(t) positions of the wind speed deficit are calculated based on the low-pass-filtered velocities in the horizontal and vertical directions according to the relations
(Larsen et al., 2007)
$\begin{array}{}\text{(19)}& \frac{\mathrm{d}y\left(t\right)}{\mathrm{d}t}=v\left(t\right)\end{array}$
$\begin{array}{}\text{(20)}& \frac{\mathrm{d}z\left(t\right)}{\mathrm{d}t}=w\left(t\right),\end{array}$
where v(t) and w(t) are the fluctuating wind speeds at hub height. The ambient wind field, which is later on low-pass filtered, is generated in this work by a Kaimal spectrum and a coherence function
(e.g., Veers, 1988). The temporal resolution of the generated wind field is 0.07s.
6.3Recalibration of the DWM model
The wind speed deficit measured by the lidar systems is used to recalibrate the wake degradation downstream or to be more precise the eddy viscosity description. In Larsen et al. (2013) a
recalibration was already achieved by introducing a nonlinear coupling function F[amb] into the ambient turbulence intensity term of the eddy viscosity definition (see Eq. 9). Furthermore, a
comparison between the measured and simulated power based on the DWM model was carried out. It shows that the wind speed deficit degradation is too low for lower turbulence intensities and moderate
to high turbine distances in the model version from Madsen et al. (2010). For this reason, the downstream-distance-dependent function F[amb] was introduced into the eddy viscosity description in
Larsen et al. (2013).
A similar behavior but even more pronounced can be seen in the results in Sect. 7. Following the approach of Larsen et al. (2013), a function based on a least-squares calibration with the acquired
lidar measurements is developed. This function is incorporated into the normalized eddy viscosity description in Eq. (17), whereby it changes to
$\begin{array}{}\text{(21)}& \begin{array}{rl}\frac{{\mathit{u }}_{T}}{{U}_{\mathrm{0}}R}& ={k}_{\mathrm{1}}{F}_{\mathrm{amb}}\left(\stackrel{\mathrm{̃}}{x}\right){F}_{\mathrm{1}}\left(\stackrel{\
mathrm{̃}}{x}\right){I}_{\mathrm{0}}\\ & +{k}_{\mathrm{2}}{F}_{\mathrm{2}}\left(\stackrel{\mathrm{̃}}{x}\right)max\left(\frac{{R}_{\mathrm{w}}\left(\stackrel{\mathrm{̃}}{x}{\right)}^{\mathrm{2}}}{R{U}_
{\mathrm{0}}}\left|\frac{\partial U\left(\stackrel{\mathrm{̃}}{x}\right)}{\partial r}\right|,\\ & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\frac{{R}_{\mathrm{w}}\left(\stackrel{\mathrm
with the constants k[1]=0.0924 and k[2]=0.0216 and the coupling function
$\begin{array}{}\text{(22)}& {F}_{\mathrm{amb}}\left(\stackrel{\mathrm{̃}}{x}\right)=a{\stackrel{\mathrm{̃}}{x}}^{-b}\end{array}$
with a=0.285 and b=0.742. The parameters a and b are the results of the least-squares calibration. It should be noted that the constant k[1] was also slightly adjusted by the recalibration, in which
the normalized eddy viscosity definition of Keck (2013) has been used. This derives from the fact that this model is already in good agreement with the measurement results in most turbulence
intensity bins as demonstrated in Sect. 8 and also in Reinwardt et al. (2018).
The measurement campaign lasted from January to July 2019. Both lidar systems, introduced in Sect. 2, were used to collect the data. Results of the meandering time series over 10min are exemplarily
shown in Fig. 8a. The maximum displacement of the wake is about 0.5D, which is equivalent to 58.5m. The results are derived from a 10min time series with an ambient wind speed of 6.44ms^−1 and an
ambient turbulence intensity of 11.7%. Some of the met-mast-detected ambient conditions (wind speed U[0], turbulence intensity I[0], wind shear α, and wind direction θ) are given in the title of the
figure. The corresponding mean wind speed deficit is illustrated in Fig. 8b. The wind speed decreases to less than 3ms^−1 in full-wake situations. As explained in Sect. 5, the tails of the curve
are relatively coarse since fewer scan points were gathered. It can also be seen that the ambient wind speed is not even reached at the edges of the curve. The opening angle of the scan appears too
small to capture the whole wake at this distance. Towards the left part of the wind speed deficit (at negative y distances) a bigger part of the wake is captured. This arises from the fact that the
horizontal displacement is more often positive than negative, and, therefore, more measurement results are collected towards the left part of the wind speed deficit curve.
The used lidar system is capable of measuring several range gates simultaneously in 30m intervals. The results of all detected range gates for the data set presented in Fig. 8 are shown in Fig. 9a.
The closest distance is 1.92D downstream and the farthest is 6.28D. The degradation of the wind speed deficit in the downstream direction is clearly identifiable. As for the single distance case
(Fig. 8), for most range gates a bigger database is captured at the left part of the wind speed deficit, resulting in smoother curves. The presumption of a too small opening angle of the scan, as
stated before, proves true. With increasing downstream distances the captured wind speed deficits get closer to integrity. A broader scan angle would result in more detailed wind speed deficits for
close downstream distances at the expense of far distances, where the scan points might not capture enough points inside the deficit and thereby prevent a successful Gaussian fit. Furthermore,
additional scan points at the edges can lead to a better representation of the deficit but would also increase the scan time. According to Eq. (18), the meandering is correlated to frequencies lower
than approximately 0.028Hz considering a wind speed of 6.5ms^−1 and a rotor diameter of 117m. This means that, considering the Nyquist–Shannon sampling theorem, the scan time must be longer than
half of the reciprocal of 0.028Hz, which results in a necessary scan time of less than 18s. The scan time for the current usage of 11 scan points is already at about 16s (depending on the
visibility conditions), which is close to the limit of 18s. Thus with an increased number of scan points it is no longer ensured that the meandering can be captured.
Figure 9b illustrates the wind speed deficit in the HMFR measured under different ambient conditions. The corresponding meandering time series and wind speed deficit for this measured time series at
2.69D downstream are given in Fig. A1 in the Appendix. The wind shear is fairly high (α=0.7) and the turbulence intensity is very low (I[0]=2.4%). Due to the low turbulence intensity it is still
possible to see the “W” shape of the wind speed deficit at closer distances. The typical “W” shape is caused by the low axial induction in the area of the nacelle. Further downstream, the wake
becomes more Gaussian shaped. At a horizontal distance of about 1.5D from the wake center, the wind speed decreases. The reason is the wakes of other turbines in the wind farm. The mean wind
direction in this time series is 183^∘ and the measurements are taken from WTG 1, so it could be the influence of the wake of either WTG 2 or WTG 4. The associated results of the mean wind speed
deficit in the FFR are illustrated in Fig. 10. The curves in the FFR are less smooth than the wind speed deficit in the HMFR, simply because only 11 points are scanned and no interpolation is
necessary when calculating the mean wind speed over the whole time series. Comparing Figs. 9 and 10, it becomes apparent that the wind speed deficit in the FFR is less pronounced. Furthermore, for
the lower turbulence intensity the “W” shape of the wind speed is not visible, since it vanished due to the meandering.
Similar results as exemplarily shown in Figs. 9 and 10 have been collected for a multitude of different ambient conditions. The number of measured time series per turbulence intensity and
wake-generating turbine, on which the lidar system is installed, is listed in Table 2. The turbulence intensity is binned in 2^∘ steps. Column 1 of Table 2 specifies the mean values for each bin.
Most of the measurement results are collected at low to moderate turbulence intensities (I[0]=4%–10%). Only a few results could be extracted at higher turbulence intensities. The results include
time series with an ambient wind speed of 5 to 8ms^−1. In this range, both turbines operate under optimal and most efficient conditions, resulting in maximum energy output from the wind. The thrust
coefficient is constant in this region (see Fig. 3). Therefore, the axial induction and the wind speed deficit normalized by the turbine's inflow wind speed are also expected to be constant for
similar ambient conditions over this wind speed range. For the single turbulence intensity bins and both turbine types, simulations with different DWM models are carried out applying the same axial
induction over the whole wind speed range. A scatterplot of the shear exponent and the ambient turbulence intensity determined by the met mast is given in Fig. 11. It includes all used data sets. At
lower turbulence intensities, the shear spreads quite a lot, whereas towards higher turbulence intensities the shear decreases as expected.
Figure 12 summarizes all measured wind speed deficits in the HMFR. It demonstrates the mean value and the standard deviation of the mean for all captured turbulence bins plotted against the
downstream distance. Each value is related to the minimum value of the wind speed deficit, which itself is normalized by the inflow wind speed. It should be noted that in some distances only one
value satisfies the filtering and plausibility checks, whereby the error bar is omitted. Additionally, it is pointed out that the plotted values always refer to the minimum value of a wind speed
curve and not necessarily to the velocity in the wake center. Therefore, no increase in the wind speed at low downstream distances on account of the “W” shape is visible. The wind speed deficit at
the wake center plotted against the downstream distance is depicted in the next section in Fig. 15b and will be discussed further at this point. Figure 12 illustrates very well that the lowest
degradation of the wind speed deficit occurs at the lowest turbulence intensity. Up to a turbulence intensity of 10%, the degradation of the wind speed deficit continuously rises, leading to
increasing minimum wind speeds at nearly all downstream distances. Above 10% turbulence intensity, the case is less clear. Especially at larger downstream distances, the measured normalized minimum
wind speed happens to fall below the corresponding lower turbulence intensity bin. An explanation is the reduced number of measurement results in these bins and the higher uncertainty that comes
along with it (expressed as error bars). Furthermore, discrepancies in the determined ambient turbulence intensity at the met mast location and the actual turbulence intensity at the wake position
could lead to a misinterpretation of the lidar measurements. The farthest distance between the met mast and the location measured by the lidar system that occurs in the analyzed sectors is about
1200m. With an ambient wind speed of 6.5ms^−1, this leads to a wake advection time of 185s; thus even at the worst conditions, the measured ambient conditions at the met mast should be valid for
the measured wakes from the lidar system most of the time. Furthermore, there is no complex terrain at the site, so it can be assumed that the conditions do not change with the wind direction. In
addition, the agreement between measurements and simulations is already good in the higher-turbulence-intensity bins. Thus, the recalibration affects only the lower-turbulence-intensity bins with
larger amounts of data, while the influence of the calibration on higher turbulence intensities is negligible (see Fig. 13). Therefore, even though there are some discrepancies, the faster recovery
of the wind speed deficit due to the higher ambient turbulence intensity can be verified, and the measurements are reliable for the outlined investigation. Thus, it is valid to use these measurement
results for comparisons with DWM model simulations and the recalibration of the DWM model in the next section.
8Comparison between measurements and DWM model simulation
Figure 13 compares the measured normalized minimum wind speed in the wake to DWM model simulations. Figure 13a shows results for a relatively low turbulence intensity of 6%, whereas panel (b)
contains results for a higher turbulence intensity of 16%. Further results for the remaining turbulence intensity bins are shown in Figs. B1 and B2 in the Appendix. The simulations were carried out
for a specific downstream distance, which corresponds to the center of the range gate of the lidar system. It should be noted that the wind speeds measured by the lidar system can be interpreted as a
mean value over the whole range gate. However, the wind speed gradient in the axial direction is low and almost linear in the observed downstream distances, so even in the DWM model, the
discretization in the downstream direction is 23.4m (equivalent to 0.2D), which is on the same order as the range gate of 30m. Therefore, a valid comparison between simulations and measurements is
carried out. The wind speed deficit simulations in the HMFR obtained by the DWM model also include the vertical meandering to ensure a correct comparison between measurements and simulations. Three
different simulation results with varying definitions of the initial deficit and eddy viscosity description are illustrated. The method called “DWM-Egmond” is based on the definitions of Madsen
et al. (2010) and Larsen et al. (2013) and the “DWM-Keck” method is adopted from Keck (2013); see Sect. 6. Figure 13 shows that the DWM-Egmond method overestimates the wind speed deficit for all
downstream distances and for both turbulence intensities. The simulated minimum wind speed with the DWM-Keck method are in better agreement with the measurement results. This confirms the results in
Reinwardt et al. (2018). Especially at higher turbulence intensities (Fig. 13), the results of the DWM-Keck model agree very well with the measurements. For lower turbulence intensities and higher
distances (greater than 3D), there is a relatively large discrepancy between measurements and simulations. A similar observation was made in Larsen et al. (2013) with the model version in Madsen
et al. (2010). Aiming at the adjustment of the simulated degradation of the wind speed deficit in Larsen et al. (2013) for cases like the one presented here, the DWM model has been recalibrated and
is henceforth called “DWM-Keck-c” (see Fig. 13).
The recalibration of the DWM model and accordingly the normalized eddy viscosity definition in the DWM model are based on a least-squares fit of the minimum of the simulated normalized wind speed to
the minimum of the measured normalized wind speed for several downstream distances. The definition of the eddy viscosity along with the recalibrated parameters are explained in detail in Sect. 6.3.
For the recalibration the measurement results are divided into 2% turbulence intensity bins. All measurement results from Fig. 12 containing data sets from two different turbines are used for the
recalibration. The first turbine is an N117 turbine with 3MW and the second one is an N117 with 2.4MW. DWM model simulations were carried out for both turbine types, since the axial induction of
both turbines is slightly different under partial load conditions. To calculate a mean value of the simulated minimum wind speed and thus allow a comparison with the results in Fig. 12, simulations
with both turbine types are carried out for each turbulence intensity bin and weighted in accordance with the number of measurement results per turbine listed in Table 2. Thus, for example at the
ambient turbulence intensity bin of 4%, the mean value of the simulated minimum wind speed consists of the sum of the simulated minimum wind speeds weighted by 0.451 and 0.549, the weighting factors
for WTG1 and WTG2, respectively. Nonetheless, this weighting has only a marginal influence on the overall results, because the axial induction in the considered wind speed range (5 to 8ms^−1) is
very similar for these two turbine types (see also thrust and power curves in Fig. 3).
The results of the recalibrated DWM model, denoted Keck-c in Fig. 13, coincide very well with the measurements. In particular, the results for lower turbulence intensities could clearly be improved.
For higher turbulence intensities, the influence of the recalibration is less significant and the already good agreement between simulation and measurement results remains unchanged. The same applies
to the results in the Appendix in Figs. B1 and B2. Only at the lowest downstream distances and turbulence intensities up to 12% does the recalibrated model deliver higher deviations than the
original model. For downstream distances larger than 3D, the recalibrated model leads to more than 10% lower deviations from the measurements than the original model. For turbulence intensities
higher than 16%, the deviation between the recalibrated and original model is smaller than the uncertainties in the measurements; hence no further conclusions about improvements can be made. The
uncertainties in accordance with misalignments could be up to 5% (see also the data filtering in Sect. 3). Furthermore, the LOS accuracy of the lidar system itself is about 1.5% at a wind speed of
6.5ms^−1. The root-mean-square error (RMSE) between the measured and simulated normalized minimum wind speed is collected for all analyzed turbulence intensity bins in Fig. 14. A clear improvement
of the results due to the recalibrated model version up to an ambient turbulence intensity of 16% is visible. For higher turbulence intensity bins, the RMSEs of the recalibrated and the original
DWM-Keck model version are similar. The DWM-Egmond model delivers significantly higher RMSEs than the other model versions for all turbulence intensity bins. A comparison between the simulated and
measured mean wake wind speed over the rotor area has been carried out as well^1. The improvement of the mean wind speed is less clear in comparison to the normalized minimum wind speed. Yet, there
is an improvement or results of equal quality are obtained in almost all turbulence intensity bins. At the tails of the wind speed deficit, the curves are coarse, since fewer scan points are gathered
and the influence of turbulence is much higher (see Fig. 9). This leads to an error in the mean wake wind speed but not in the minimum wind speed, which is why the illustration and recalibration of
the model are based on the minimum wake wind speed instead of the wake mean wind speed.
Figure 15 compares the final recalibrated DWM model to the original model definition. It shows the minimum normalized wind speed (panel a) and the wind speed at the wake center (panel b) over
downstream distances from 0D to 10D for the lower- and the higher-turbulence-intensity cases of 6% and 16%, respectively. Observing the wind speed at the wake center, higher wind speeds can be seen
at lower distances, which derives from the “W” shape of the wind speed at these downstream distances. The comparison of the DWM-Keck model (orange curve) and the recalibrated model DWM-Keck-c (green
curve) demonstrates that the recalibration leads to a shift of the curve towards lower distances. This shift is more pronounced for the lower turbulence intensity, leading to a faster degradation of
the wind speed deficit. For the higher turbulence intensity, both curves, orange and green, are very close to each other over all distances. The faster degradation of the wind speed deficit in the
recalibrated model version is caused by introducing the function F[amb] in the eddy viscosity definition in Eq. (21) as explained in Sect. 6.3. The function increases the eddy viscosity for lower
turbulence intensities and thus increases the wind speed deficit degradation in the downstream direction. Contemplating the curve of the minimum wind speed in Fig. 15a, small steps are formed in the
curves between 2D and 4D (depending on the used model and the turbulence intensity). These steps correspond to the minimum of the curves in Fig. 15b and are thus related to the transition from the
“W” shape of the wind speed deficit towards the Gaussian profile and are consequently caused by the resolution in the downstream direction. These steps were also found in some measurements and could
likewise be related to the implied transition zone.
The study compares measurements of the wind speed deficit with DWM model simulations. The measurement campaign consists of two nacelle-mounted lidar systems in a densely packed onshore wind farm. The
lidar measurements were prepared by lidar and wind field simulations to examine whether the scan pattern is suitable for the outlined analysis. Several wind speed deficits that were simultaneously
measured at different downstream distances are presented along with their associated meandering time series. The one-dimensional scan worked reliably in the field campaign, thus delivering lidar data
for a multitude of different ambient conditions. These measurements are compared to the simulated wind speed deficit in the HMFR. The simulation result of the DWM-Keck model is in good agreement,
whereas the DWM-Egmond model yields a too low degradation of the wind speed deficit. Furthermore, even the DWM-Keck model shows some discrepancies to the measurements at low turbulence intensities,
which is why a recalibrated DWM model was proposed. The recalibrated model improves the correlation with measurements at low turbulence intensities and leads to an agreement at high turbulence
intensities, which are as good as the original model, thus resulting in a very good overall conformity with the measurements.
Future work will include the analysis of two-dimensional scans as well as measurements with more range gates and higher spatial resolutions. Increasing the number of range gates and scan points will
lead to longer scan times, hence preventing further analysis of the wind speed deficit in the MFR and the determination of the meandering time series. Nevertheless, a validation of the wind speed
deficit in the FFR with higher resolutions and more distances seems reasonable to also prove the validity of the outlined calibration for further distances. Furthermore, the analyzed models will be
assessed in load as well as power production simulations and compared to the particular measurement values from the wind farm. Simulations have shown that the recalibration of the DWM-Keck model can
lead to up to 13% lower loads in the turbulence-dependent components in cases with small turbine distances and low turbulence intensities, whereas for higher turbulence intensities (>12%) the
difference between the original and the recalibrated DWM-Keck model is less than 5%. The overall influence of the recalibration on the power output is low (<2% for all turbulence intensities). So
far, only measured single wakes were presented. Yet, a brief analysis demonstrated that multiple wakes can also be recorded with the described measurement setup. A future step will therefore be an
analysis of multiple-wake situations.
Appendix A:Measurement results
Appendix B:Comparison of measurements and DWM model simulation
Code and data availability
Access to lidar and met mast data as well as the source code used for post-processing the data and simulations can be requested from the authors.
IR performed all simulations, post-processed and analyzed the measurement data, and wrote the paper. LS and DS gave technical advice in regular discussions and reviewed the paper. PD and MB reviewed
the paper and supervised the investigations.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Wind Energy Science Conference 2019”. It is a result of the Wind Energy Science Conference 2019, Cork, Ireland, 17–20 June 2019.
The content of this paper was developed within the project NEW 4.0 (North German Energy Transition 4.0).
This research has been supported by the Federal Ministry for Economic Affairs and Energy (BMWI) (grant no. 03SIN400).
This paper was edited by Sandrine Aubrun and reviewed by Helge Aagaard Madsen and Vasilis Pettas.
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d, e, f, g, h, i
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Machefaux, E., Troldborg, N., Larsen, G., Mann, J., and Aagaard Madsen, H.: Experimental and numerical study of wake to wake interaction in wind farms, in: Proceedings of EWEA 2012 – European Wind
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Machefaux, E., Larsen, G. C., Troldborg, N., and Rettenmeier, A.: Single wake meandering, advection and expansion – An analysis using an adapted pulsed lidar and CFD LES-ACL simulations, in:
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Re: Overlap of the two kernel densities
Hi everyone, I need your help on some numerical integration work using Simpon's rule - with 101 grid. In the following code I simulated the data, estimated the kernel densities for variable x by
group c, and visualized. I need to calculate the overlap between the the two krenel densities using numerical integration with Simpson's rule - with 101 grid. The overlap is calculated by integrating
from -inf to +inf over the min of two curves MIN [ f(x_c=1), f(x_c=2) ].
/* Overlapping Coefficient */
/* Simulate data */
data TwoGroups;
call STREAMINIT(1982);
do i=1 to 100;
c = 1;
x = RAND("NORMAL", 10, 3);
do i=101 to 200;
c = 2;
x = RAND("NORMAL", 15, 3);
/* Estimate Kernel Density */
proc kde data=TwoGroups;
univar x / out=KernelTwoGroups;
by c;
/* Plot Kernel Density */
proc sgplot data=KernelTwoGroups;
series y=density x=value / group=c;
Thank you for your help,
09-11-2015 10:41 PM
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Resistors and Ohms Law
Resistors are the most common component you will encounter in electronic circuits. A resistor resists the flow of electrons.
To understand this you first need to understand the concepts of voltage and current.
Electricity is the flow of electrons. Electronic circuits are all about controlling the flow of electrons in creative ways. A common analogy for think about the flow of electrons/electricity, is to
imagine electrons flowing through wires as water flowing through a hose.
Imagine electricity is like water flowing through a hose. Current is how much water is flowing through the hose, and voltage is how fast the water is moving. So, current is like how much electricity
is moving, and voltage is like how fast it’s moving through a wire.
So what is a resistor?
A resistor is a component with two legs. It’s job in a circuit is to impede the flow of electrons.
A resistor is like a tiny roadblock for electricity. It’s a component in an electrical circuit that limits how much current can flow through. It’s used to control or reduce the amount of electricity
passing through a circuit. Resistors come in different sizes and types, and they’re often small and made of materials that resist the flow of electricity to varying degrees.
Resistors are measured in ohms. You might encounter values like:
• 100r = 100 ohms
• 1k = 1000 ohms
• 100k = 100,000 ohms
• 1M = 1,000,000 ohms (mega or million ohms)
What is ohms law?
Ohms law is a fundamental principle of electrical engineering! It describes how current flows and allows you to calculate the current, voltage, and resistance at any point in a circuit.
Ohm’s Law is like a rule that tells us how electricity works. It says that the current (how much electricity flows) in a wire is equal to the voltage (how fast electricity moves) divided by the
resistance (how hard it is for electricity to move through something). So, if you know how fast electricity is moving (voltage) and how hard it is to move (resistance), you can figure out how much
electricity is flowing (current) in a wire.
Some examples
Voltage dividers
A common circuit building block is called a voltage divider.
Imagine you have a highway where cars are like electricity flowing through wires. Now, think of a voltage divider as a special exit ramp on the highway. When a car takes this exit, it splits off from
the main highway into two smaller roads. Similarly, in electronics, a voltage divider splits a higher voltage into two smaller voltages using resistors. Just like cars choose different roads at an
exit, electricity “chooses” different paths through resistors to create the right voltage levels needed in a circuit.
Look at the first example. 9 volts are going through R1 and R2. What is the voltage at the intersection ?1. You can solve this if you understand voltage dividers. The formula is:
• ?1 = 9V * (R2 / (R1 + R2))
• ?1 = 9V * (1K / (1K + 1K))
• ?1 = 9V * 1000 / (1000 + 1000)
• ?1 = 9V * 1000 / 2000
• 4.5V = 9V * 0.5
Here I walked through the steps to solve the problem. Notice I converted 1K to 1000 ohms. Then finished up from there.
The formula is: V * R2 / (R1 + R2)
The shortcut when both resistors are the same value, the voltage is divided in half. This is true for any value for R1 and R2. Try it yourself. Imagine R1 and R2 are 10K. Then try R1 and R2 at 47K
and 100K.
What happens when the values are not equal? What’s the voltage at the intersection: ?2.
• ?2 = 9V * 100K / (100K + 20K)
• ?2 = 9V * 20K / 120K
• ?2 = 9V * 20,000 / 120,000
• 1.5V = 9V * 0.166
Solve number 3 on your own!
To solve ?4 we have to know that resistors in series are added together. That means that we have a total resistance of 16.7K. To solve ?4:
• ?4 = 9V * (10K + 4.7K) / (2K + 4.7K + 10K)
• ?4 = 9V * 14.7K / 16.7K
• ?4 = 9V * 0.88
• ?4 = 7.92
Solve ?5 on your own…
With this knowledge you can start examining the Rangemaster circuit. Notice R1and R2 for a voltage divider! Calculate the voltage at the intersection of R1 and R2.
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Use symbolic constants instead of magic numerical constants - EasyHack - LibreOffice Development Blog
There are many situations that you need to use numerical constants in your code. If you use numerical literal values directly instead of symbolic constants, it can cause problems.
For example, consider this piece of code that calculate area of a circle:
double area = 3.14 * r * r;
This is not OK, because:
1. The value of π is not 3.14 nor 3.141592. π is an irrational number, and the suitable value depends on the number of decimal places that you can/want to use among unlimited decimals of π.
2. Suppose that you want to change the numerical literal to increase the number of decimals that you use. You should search for 3.14, and check one by one to see if it is actually π, or it is another
3.14 unrelated to the well-known mathematical constant.
Using symbolic constants
A better code can be:
double area = M_PI * r * r;
and with more long and meaningful name for variables:
double circle_area = M_PI * radius * radius;
Because of the above mentioned problems, it is better to use some numerical constant instead.
ES.45: Avoid “magic constants”; use symbolic constants
If it is well-known (like π), you should use the appropriate symbolic constant like M_PI. If not, you should define a new constant with proper name and type with ‘contsexpr’.
One solution to find such magic constants is to start from a list of some well known mathematical constants:
Then, store some of them in a text file, let’s say ‘constants.txt’, then search for all these values inside C++ files:
git grep -Ff constants.txt *.cxx *.hxx
Many of these symbolic constants like M_PI already exist in C++ standard library or some place in the LibreOffice code, and you can use them easily.
You should examine the ‘grep’ results carefully, because not every 3.14 refers to PI.
Final Notes
Besides fixing the bugs, there are many places to work on improving the code, and some of these are listed as EasyHacks. The specific improvement that is discussed in this blog post is filed as tdf#
If you want to work on this improvement about using symbolic constants, but you need to know how to get started with LibreOffice development, I suggest you to see our video tutorial:
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Minimum Window Substring | JavaScript
Minimum Window Substring
This is another approach and a faster solution from my previous one Sliding Window using hashmaps.
You don't know what this problem is about?
See here the problem description, and try to solve it before reading this solution.
We will use the Sliding Window method
We will have an array of 58 Zeros holding the ASCII code for every letter from A to z.
We have 52 letters in total but need 6 more "useless" zeros because there is a distance of 6 in between the ascii codes of 'Z' and 'a' (No need to make it more complex).
In the beginning, neededCharsArray holds the total number of our chars in string t. So for t='ABByzzzzz' neededCharsArray will be like: [1,2,0,0,0...,0,0,0,1,5]
Similar to my first solution on leetcode, we will have two pointers, left and right, which will point to the edges of our sliding window.
For every char in s we will check if neededCharsArray holds a value greater than zero. This means it's a wanted char, and because we found one we will reduce missingChars by one.
Bear in mind that:
For every char of s we decrease it's corresponding value in neededCharsArray by one.
So, while having the index of this char to our right index, we keep moving our left index to the right and closer to the right index, narrowing our window (trying to find a better alternative).
Bear in mind that:
While we're sliding our left index we increase it's corresponding value in neededCharsArray by one.
Not bored yet?
Having those bears in mind, we know that every char of s that is of no interest will result in a value <=0 in our neededCharsArray. But every char of s that is wanted and exists in t, will result in
a value >=0 in our neededCharsArray.
We will get into our while statement when missingChars is equal to zero. This means that the corresponding values of our wanted chars in neededCharsArray will be zero. All of them! There might be
some other negative values for unwanted chars that exist in s and not in t. Other zeros may appear too (for chars that don't exist in s and t).
In our while statement we increase our left index. So in the next iteration, if the left index points to a wanted char, we will have a positive value in our neededCharsArray, this will break our
while loop ( because the 2nd if statement increases missingChars) and will go out to our for loop, increasing our right index, checking the next one, searching for our missing char. Until we reach
the last one.
Time complexity:
Space complexity:
* @param {string} s
* @param {string} t
* @return {string}
const minWindow = (s, t) => {
if (!s.length || !t.length || s.length < t.length) return "";
if (s === t) return s;
// 26 zeros for A-Z, 26 for a-z, and 6 for the diff from 'Z' (090) to 'a' (097)
let neededCharsArray = new Array(58).fill(0);
let firstCharCode = "A".charCodeAt(0); // A - A = 0; A - z = 57
for (let c of t) {
neededCharsArray[c.charCodeAt(0) - firstCharCode]++;
let missingChars = t.length;
let result = [-Infinity, Infinity];
let left = 0;
for (let right = 0; right < s.length; right++) {
if (neededCharsArray[s.charCodeAt(right) - firstCharCode] > 0) {
neededCharsArray[s.charCodeAt(right) - firstCharCode]--;
while (missingChars === 0) {
if (result[1] - result[0] > right - left) {
result[0] = left;
result[1] = right;
neededCharsArray[s.charCodeAt(left) - firstCharCode]++;
if (neededCharsArray[s.charCodeAt(left) - firstCharCode] > 0) {
return result[1] !== Infinity ? s.slice(result[0], result[1] + 1) : "";
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Specific Weight (Meaning and Explanation)
Specific Weight
We explain what specific weight is and what the formulas are to calculate it. Also, some examples and their relationship with density.
Specific gravity is the relationship between the weight and volume of a substance.
What is specific weight?
The specific weight is the existing relationship between the weight and the volume that occupies a substance in space. It is the weight of a certain amount of substance divided by the volume it
occupies. In the International System it is expressed in units of Newtons about cubic meter (N/m^3).
The calculation of specific gravity requires other properties of the substance, such as density and mass. Mathematically, the specific gravity It is represented by the symbol gamma (γ) and is
expressed as:
γ (specific weight) = w (ordinary weight) / V (volume of substance)or what is the same: γ = w/V = mg/V, where m is the mass of the substance and g is the acceleration of gravity (commonly considered
as 9.8m/s^2). As the density (ρ) of a substance is defined as m/Vthe specific weight can be written as γ=ρ.g.
Examples of specific weight
Some examples of specific weight of different materials are:
• Gypsum: 1250 N/m^3
• Cal: 1000 N/m^3
• Dry sand: 1600 N/m^3
• Wet sand: 1800 N/m^3
• Loose cement: 1400 N/m^3
• Concrete tiles: 2200 N/m^3
• Poplar Wood: 500 N/m^3
• Ash Wood: 650 N/m^3
• American pine wood: 800 N/m^3
• Steel: 7850 N/m^3
• Aluminum: 2700 N/m^3
• Bronze: 8600 N/m^3
• Lead: 11400 N/m^3
• Zinc: 7200 N/m^3
• Iron casting: 7250 N/m^3
• Water: 1000 N/m^3
• Asphalt: 1300 N/m^3
• Stacked paper: 1100 N/m^3
• Slate: 2800 N/m^3
• Tar: 1200 N/m^3
• Granite: 2800 N/m^3
Specific weight and density
The relationship between the specific weight (mg/V) and the density (m/V) is analogous to that which exists between the weight (mg) and the dough (m) of a substance. It is evident that the more mass
a certain amount of a substance has, the greater its weight. In the same way, the denser that amount of substance is, the more mass that enters a certain volume, the greater its specific weight will
be, since the greater “mass due to gravity” will enter that volume.
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OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials
Up to this point we have been solving quadratic inequalities. The technique involving sign charts extends to solving polynomial inequalities of higher degree.
Step 1: Determine the critical numbers, which are the roots or zeros in the case of a polynomial inequality.
Step 2: Create a sign chart.
Step 3: Use the sign chart to answer the question.
The last problem shows that not all sign charts will alternate. Do not take any shortcuts and test each interval.
Rational inequalities are solved using the same technique. The only difference is in the critical numbers. It turns out that the
-values may change from positive to negative at a restriction. So we will
include the zeros of the denominator
in our list of critical numbers.
: Always use open dots for critical numbers that are also zeros of the denominator, or restrictions. This reminds us that they are restrictions and should not be included in the solution set even if
the inequality is inclusive.
Use open dots for all of the critical numbers when a strict inequality is involved.
YouTube Videos:
Rational equations are simply equations with rational expressions in them.
Use the technique outlined earlier to clear the fractions of a rational equation. After clearing the fractions, we will be left with either a linear or a quadratic equation that can be solved as
Step 1: Factor the denominators.
Step 2: Identify the restrictions.
Step 3: Multiply both sides of the equation by the LCD.
Step 4: Solve as usual.
Step 5: Check answers against the set of restrictions.
This process sometimes produces answers that do not solve the original equation (extraneous solutions), so it is extremely important to check your answers.
Tip: It suffices to check that the answers are not restrictions to the domain of the original equation.
Determining the LCD is often the most difficult part of the process. Use one of each factor found in all denominators.
Because of the distributive property, multiplying both sides of an equation by the LCD is equivalent to multiplying each term by that LCD as illustrated in the following examples.
Some literal equations, often referred to as formulas, are also rational equations. Use the techniques of this section and clear the fractions before solving for the particular variable.
Solve for the specified variable.
The reciprocal of a number is the number we obtain by dividing 1 by that number.
Word Problem: The reciprocal of the larger of two consecutive positive odd integers is subtracted from twice the reciprocal of the smaller and the result is 9/35. Find the two integers.
Video Examples on YouTube:
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WHAT IS STATISTICS IN ECONOMICS ? - The NiconomicsWHAT IS STATISTICS IN ECONOMICS ? - The Niconomics
Origin and Growth of Statistics The term ‘STATISTICS’ has been derived from the Latin word ‘STATUS’, which means political state.
Germans have spelled it as ‘STATISTIK’. The term ‘Statistics’ was first used by German scientist Gottfried Achenwall in 1749. He is known as the Father of Statistics.
Definition Of Statistics
The systematic treatment of quantitative expression is known as ‘statistics’. Not all quantitative expressions are statistics we will see that certain conditions must be fulfilled for a quantitative
statement to be called statistics.
Statistics can be defined in two ways:
a. In Singular sense
b. In a Plural sense
Statistics defined in Singular sense (as a statistical method)
According to Croxton and Cowden, “Statistics may be defined as a science of collection organization presentation, analysis, and interpretation of numerical data”.
1. Collection of data : Data should be gathered with maximum care by the investigator himself or obtained from reliable published or unpublished sources.
2. Organisation of data : Figures that are collected by an investigator need to be organised by editing, classifying and tabulating.
3. Presentation of data : Data collected and organised are presented in some systematic manner to make statistical analysis easier. The organised data can be presented with the help of tables,
graphs, diagrams etc.
4. Analysis of data : The next stage is the analysis of the presented data. There are large number of methods used for analysing the data such as averages, dispersion, correlation etc.
5. Interpretation of data : Interpretation of data implies the drawing of conclusions on the basis of the data analysed in the earlier stage.
STATISTICS DEFINED IN PLURAL SENSE (as statistical data)
According to Horace Secrist, “By statistics, we mean aggregates of facts affected to a marked extent by a multiplicity of cause numerically expressed, enumerated or estimated according to reasonable
standards of accuracy, collected systematically for a predetermined purpose and placed with each other”.
1. Statistics are aggregates of facts : A single observation is not statistics, it is a group of observation. Eg., A single age of 30 years is not statistics but a series relating to the ages of a
group of persons is statistics.
2. Statistics are affected to a marked extent by multiplicity of causes : Statistics are generally not isolated facts they are dependant on, or influenced by an number of phenomena.
3. Statistics are numerically expressed : Qualitative statements are not statistics unless they are supported by numbers.
4. Statistics are enumerated or estimated according to reasonable standard of accuracy : Enumeration means a precise and accurate numerical statement. But sometimes, where the area of statistical
enquiry is large, accurate enumeration may not be possible. In such cases, experts make estimations on the basis of whatever data is available. The degree of accuracy of estimates depends on the
nature of enquiry.
5. Statistics are collected in a systematic manner : Statistics collected without any order and system are unreliable and inaccurate. They must be collected in a systematic manner.
6. Statistics are collected for a pre-determined purpose : Unless statistics are collected for a specific purpose they would be more or less useless.
7. Statistics are placed in relation to each other : Statistical data are often required for comparisons. Therefore, they should be comparable periodwise, regionwise, commodity wise etc. When the
above characteristics are not present a numerical data cannot be called statistics. Thus, “all statistics are numberical statements of facts but all numberical statement of fact are not
1. Statistics simplifies complex data : With the help of statistical methods a mass of data can be presented in such a manner that they become easy to understand.
2. Statistics presents the facts in a definite form: This definiteness is achieved by stating conclusions in a numerical or quantitative form.
3. Statistic provides a technique of comparison. Comparison is an important function of statistics i.e., Period wise, country wise etc..
4. Statistics studies relationship : Correlation analysis is used to discover functional relationship between different phenomena, for example, relationship between supply and demand, relationship
between sugarcane prices and sugar, relationship between advertisement and sale.
5. Statistics helps in formulating policies : Many policies such as that of import, export, wages, production, etc., are formed on the basis of statistics.
6. Statistics helps in forecasting : Statistics also helps to predict the future behaviour of phenomena such as market situation for the future is predicted on the basis of available statistics of
past and present.
1. Statistics in Economics
Several economic problems can easily be understood by the use of Statistics. It helps in the formulation of economic policies, e.g., basic economic activities like production, consumption, etc. use
The importance of Statistics in various parts of economics has been discussed as follows :
• Statistics in consumption. To obtain the knowledge of how different groups of people spend their income form Statistics relating to consumption. The data of consumption are useful and helpful in
planning their budget and improve their standard of living.
• Statistics in production. The Statistics of production are very useful and helpful for adjustment of demand and supply and determining quantity of production of the commodity.
Statistics in distribution. Statistical methods are used in solving the problem of distribution of national income among various factors of production i.e., land, labor, capital, and entrepreneur.
Statistics in Economic Planning
Economic planning is done to achieve certain targets for the growth of the economy using scarce resources of the nation. Statistics helps in evaluating various stages of economic planning through
statistical methods.
According to Tippett,
“Planning is the order of the day, and without Statistics planning is inconceivable.”
Statistics help in comparing the growth rate. It helps to formulate plans to achieve predetermined objectives. It measures the success and failure of plans and accordingly guides to apply corrective
Statistical tools play a very important role in major business activities.
The producer depends upon market research to estimate market demand and the market research is based on
Statistics. The trader depends heavily on methods of statistical analysis to study the market.
Statistical tools are very important for the detailed analysis of money transactions in the business.
• Statistics in Administration
Formulation of a policy involves Statistics. The state gathers the facts relating to population, literacy, employment, poverty, per capita income, etc., with the help of statistical methods and
principles. It helps the state to achieve targets with the help of optimum utilization of scarce resources
It does not study the qualitative aspect of a problem: The most important condition of statical study is that the subject of investigation and inquiry should be capable of being quantitatively
measured. Qualitative phenomena e.g., honesty, intelligence, poverty, etc., cannot be studied in statistics unless these attributes are expressed in terms of numerals
It does not study individuals: Statistics is the study of mass data and deals with aggregates of facts which are ultimately reduced to a single value for analysis. Individual values of the
observation have no specific importance. For example, the income of a family is, say Rs. 1000, does not convey statistical meaning while the average income of 100 families says Rs. 400, is a
statistical statement.
Statistical laws are true only on an average: Laws of statistics are not universally applicable like the laws of chemistry, physics, and mathematics. They are true on average because the results are
affected by a large number of causes. The ultimate results obtained by statistical analysis are true under certain circumstances only.
Statistics can be misused: Statistics is liable to be misused. The results obtained can be manipulated according to one’s own interest and such manipulated results can mislead the community.
Statistics simply is one of the methods of studying a phenomenon: Statistical calculations are simple expressions that should be supplemented by other methods for a complete comprehension of the
results. Thus statistics is only a means and not the end.
Statistical results lack mathematical accuracy: The result drawn from statistical analysis are normally in approximates.
As the statistical analysis is based on observations of mass data, several inaccuracies may be present and it is difficult to rectify them. Therefore, these results are estimates rather than exact
statements. Statistical studies are a failure in the fields where one number percent accuracy is desired
One Comment
1. Manik on August 15, 2021
Nicely explained
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dynamic load for a ball mill pdf
WEBJan 1, 2020 · An improved multiclassifier ensemble modelling is proposed in this paper, which is applied to the soft measurement of ball mill load, and it can be found that the proposed method
effectively improves the accuracy ofsoft measurement ofBall mill load. Aiming at the problem that the traditional Dempster–Shafer (D–S) evidence theory .
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WEBThe starting point for ball mill media and solids charging generally starts as follows: Add to this another 10%15% above the ball charge for total of 23% to 25% product loading. So as a rule
of thumb we use 25% solid loading. Empirical Check: Once the mill has been loaded and run for a few minutes, open the cover and look down into the mill.
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WEBDec 1, 2014 · The SHPB system is used to conduct the dynamic ball. compression test. As a standard facility for dynamic testing, SHPB consists of a striker bar (200 mm in length), an incident.
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WEBOct 1, 2014 · Ball mill load refers to the total materials inside the cylinder, including ore, grinding media, water, mineral pulp, etc. Understanding the load status accurately is an
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WEBMar 1, 2015 · Austin et al. [5] applied the population balance model to fit a laboratory Etype of ballrace VSM mill (modified from the standard HGI mill), and developed a model of
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WEBThis set of Mechanical Operations Multiple Choice Questions Answers (MCQs) focuses on "Ball Mill". 1. What is the average particle size of ultrafine grinders? a) 1 to 20 µm. b) 4 to 10 µm. c)
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WEBOct 6, 2010 · The existing method of calculation for the basic dynamic load rating for a ball screw mentioned in the ISO standard is based on the method used for the angular contact ball
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WEBOct 1, 2014 · The ratio between the internal load mass and the mill volume is related to the percentage of mill capacity by the following equation: (4) W V = (1ε b) · J · ρ s · (1w c) + · J b
· (ρ bρ s (1 + w c)) where ɛ b is the porosity of the mill internal load (void fraction), ρ s (t/m 3) and ρ b (t/m 3) are the density of mineral and ...
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WEBJan 1, 2016 · Both are set according to operating conditions: • internal CSTRs in ball mills: X = 40% of top size ball diameter, and  = ; • internal CSTRs in rod mills: X = P90
(sieve dimen sion larger than 90% of the particles) in the CSTRs, and  = ; • grate discharge mills: X must match the grate aper ...
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WEBMar 1, 2014 · Analysis of ball mill grinding operation using mill power specific kinetic parameters. March 2014. Advanced Powder Technology 25 (2):625–634. DOI: / Authors: Gupta ...
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WEBSoft measurement for a ball mill load parameters based on integration of semisupervised multisource domain adaptation: LI Sisi1,2, YAN Gaowei1, YAN Fei1, CHENG Lan1, DU Yonggui1: 1. College of
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WEBOct 2, 2023 · Figure 6— Cal culated and meas ured mill cha rge, mill speed, and mill weight (ball charge k nown) Figure 6 sho ws that there is a sig nificant di fference in the mill weight and
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WEBFeb 17, 2009 · The condensation of multiple building blocks in a ball mill allows molecular cages with a size up to nm to be built. ... Clicking on the donut icon will load a page at with
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WEBOct 9, 2020 · The mill speed, fill level ratio, and steel ball ratio can significantly affect mill operation, and our conclusions can provide a reference for an actual situation. Breakage
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WEBJun 1, 2019 · This article analyzes the problems of perfecting grinding equipment for largescale production – cement, ore, coal. An improved design of a ball mill, equipped with internal
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WEBFor the current work, extensive surveys have been p erformed on an industrial overflow ball mill, processing iron ore. The slurry density was s et to different values by adapting the water
addition flow at the mill inlet. The mill is equipped with a Sensomag which provides information about the ball load and pulp volume, va lues and positions.
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WEBThe ball charge and ore charge volume is a variable, subject to what is the target for that operation. The type of mill also is a factor as if it is an overflow mill (subject to the diameter
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WEBNov 1, 2020 · Creating a project for modernizing the feeding balls device to a ball mill using 3D modeling. M G Naumova 1 ... Gorbatyuk, Keropyan and Bibikov 2018 Assessing Parameters of the
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Algebra I
Algebra 1
This is beginning Algebra 1. Students should expect to progress through the first 1/2 of the Saxon Algebra 1 textbook during this semester.
Prerequisite: Please read the description under the
Middle School Math class
and ask the student if they feel proficient with those topics.
Book used: Saxon. Algebra 1 (An Incremental Development) 3rd Edition Used books found on Amazon starting at $2.88
This course will cover: solving multivariable equations using substitution or process of elimination, graphing number line conjunctions, polynomial equations, graphing linearequations, complex
fractions, higher order roots, addition and multiplication of radicalexpressions, factoring trinomials, the difference of squares, writing the equation of a line, word problems with consecutive
integers or consecutive even/odd integers,subscripted variables used for word problems, operations with Scientific Notation, solve equations by graphing, long division of polynomials, value/coin
problems, quadratic equations, uniform motion problems, factoring by grouping, writing an equation of a line through 2 points or given the slope, completing the square, and the quadratic formula.
The instructor, Anita Bruce, has her BA in Secondary Education with a minor in Mathematics. She has taught HS, adults, even elementary, and tutored all ages for years along with homeschooling the
last 4 years. She loves to make math enjoyable to
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CBSE Class 11 Maths - MCQ and Online Tests - Unit 14 - Mathematical
CBSE Class 11 Maths – MCQ and Online Tests – Unit 14 – Mathematical Reasoning
Every year CBSE schools conducts Annual Assessment exams for 6,7,8,9,11th standards. These exams are very competitive to all the students. So our website provides online tests for all the
6,7,8,9,11th standard’s subjects. These tests are also very effective and useful for those who preparing for any competitive exams like Olympiad etc. It can boost their preparation level and
confidence level by attempting these chapter wise online tests.
These online tests are based on latest CBSE syllabus. While attempting these our students can identify the weak lessons and continuously practice those lessons for attaining high marks. It also helps
to revise the NCERT textbooks thoroughly.
CBSE Class 11 Maths – MCQ and Online Tests – Unit 14 – Mathematical Reasoning
Question 1.
If (p and q) is false then
(a) p is true and q is false
(b) p is false and q is false
(c) p is false and q is true
(d) all of the above
Answer: (d) all of the above
(p and q) is true when both p and q are true otherwise it is false.
Question 2.
The converse of the statement p ? q is
(a) p ? q
(b) q ? p
(c) ~p ? q
(d) ~q ? p
Answer: (b) q ? p
The converse of the statement p ? q is
q ? p
Question 3.
The converse of the statement if a number is divisible by 10, then it is divisible by 5 is
(a) if a number is not divisible by 5, then it is not divisible by 10
(b) if a number is divisible by 5, then it is not divisible by 10
(c) if a number is not divisible by 5, then it is divisible by 10
(d) if a number is divisible by 5, then it is divisible by 10
Answer: (d) if a number is divisible by 5, then it is divisible by 10
Given, statement is if a number is divisible by 10, then it is divisible by 5
Now, converse of the statement is:
if a number is divisible by 5, then it is divisible by 10
Question 4.
Which of the following is a statement
(a) x is a real number
(b) Switch of the fan
(c) 6 is a natural number
(d) Let me go
Answer: (c) 6 is a natural number
The statement 6 is a natural number is true.
So, it is a statement.
Question 5.
The contra-positive of the statement If a triangle is not equilateral, it is not isosceles is
(a) If a triangle is not equilateral, it is not isosceles
(b) If a triangle is equilateral, it is not isosceles
(c) If a triangle is not equilateral, it is isosceles
(d) If a triangle is equilateral, it is isosceles
Answer: (d) If a triangle is equilateral, it is isosceles
Given, statement is:
If a triangle is not equilateral, it is not isosceles.
Now, contra-positive is:
If a triangle is equilateral, it is isosceles.
Question 6.
Which of the following is a statement
(a) I will go tomorrow
(b) She will come today
(c) 3 is a prime number
(d) Tomorrow is Friday
Answer: (c) 3 is a prime number
The statement 3 is a prime number is true.
So, it is a statement.
Question 7.
The contra-positive of the statement if p then q is
(a) if ~p then q
(b) if p then ~q
(c) if q then p
(d) if ~q then ~p
Answer: (d) if ~q then ~p
Given statement is if p then q
Now, contra-positive of the statement is:
if ~q then ~p
Question 8.
Which of the following is not a statement
(a) The product of (-1) and 8 is 8
(b) All complex number are real number
(c) Today is windy day
(d) All of the above
Answer: (d) All of the above
A sentence is a statement if it is true.
None of the above sentence is true.
So, option 4 is the correct answer.
Question 9.
If (p or q) is true, then
(a) p is true and q is false
(b) p is true and q is true
(c) p is false and q is true
(d) All of the above
Answer: (d) All of the above
(p or q) is false when both p and q are false otherwise it is true.
Question 10.
Which of the following statement is a conjunction
(a) Ram and Shyam are friends
(b) Both Ram and Shyam are friends
(c) Both Ram and Shyam are enemies
(d) None of these
Answer: (d) None of these
All the statements are conjunction. So, option 4 is the correct answer.
Question 11.
Which of the following is a compound statement
(a) Sun is a star
(b) I am a very strong boy
(c) There is something wrong in the room
(d) 7 is both odd and prime number.
Answer: (d) 7 is both odd and prime number.
A compound statement is connected with And , or , etc.
So, the statement 7 is both odd and prime number is a compound statement.
Question 12.
Which of the following is a statement
(a) x is a real number
(b) Switch of the fan
(c) 6 is a natural number
(d) Let me go
Answer: (c) 6 is a natural number
The statement 6 is a natural number is true.
So, it is a statement.
Question 13.
The connective in the statement 2 + 7 > 9 or 2 + 7 < 9 is
(a) and
(b) or
(c) >
(d) <
Answer: (b) or
Given, statement is 2 + 7 > 9 or 2 + 7 < 9 Here, connective is or. It connects two statement 2 + 7 > 9, 2 + 7 < 9
Question 14.
Which of the following is not a negation of the statement A natural number is greater than zero
(a) A natural number is not greater than zero
(b) It is false that a natural number is greater than zero
(c) It is false that a natural number is not greater than zero
(d) None of these
Answer: (c) It is false that a natural number is not greater than zero
Gievn statement is:
A natural number is greater than zero
Negation of the statement:
A natural number is not greater than zero
It is false that a natural number is greater than zero
So, option 3 is not true.
Question 15.
Which of the following is the conditional p ? q
(a) q is sufficient for p
(b) p is necessary for q
(c) p only if q
(d) if q then p
Answer: (c) 6 is a natural number
Given, p ? q
Now, conditional of the statement is
p only if q
Question 16.
Which of the following is not a negation of the statement A natural number is greater than zero
(a) A natural number is not greater than zero
(b) It is false that a natural number is greater than zero
(c) It is false that a natural number is not greater than zero
(d) None of these
Answer: (c) It is false that a natural number is not greater than zero
Given statement is:
A natural number is greater than zero
Negation of the statement:
A natural number is not greater than zero
It is false that a natural number is greater than zero
So, option 3 is not true.
Question 17.
The negation of the statement The product of 3 and 4 is 9 is
(a) It is false that the product of 3 and 4 is 9
(b) The product of 3 and 4 is 12
(c) The product of 3 and 4 is not 12
(d) It is false that the product of 3 and 4 is not 9
Answer: (a) It is false that the product of 3 and 4 is 9
Given, statement is The product of 3 and 4 is 9
The negation of the statement is:
It is false that the product of 3 and 4 is 9
Question 18.
Sentence involving variable time such as today, tomorrow, or yesterday are
(a) Statements
(b) Not statements
(c) may or may not be statements
(d) None of these
Answer: (b) Not statements
Sentence involving variable time such as today, tomorrow, or yesterday are not statements. This is because it is not known what time is referred here.
Question 19.
Which of the following is not a statement
(a) 8 is less than 6.
(b) Every set is finite set.
(c) The sun is a star.
(d) Mathematics is fun.
Answer: (d) Mathematics is fun.
8 is less than 6 if false. So it is a statement.
Every set is finite set is false. So it is a statement.
The sun is a star is true. So it is a statement.
Mathematics is fun. This sentence is not always true. Hence, it is not a statement.
Question 20.
Which of the following is true
(a) A prime number is either even or odd
(b) v3 is irrational number.
(c) 24 is a multiple of 2, 4 and 8
(d) Everyone in India speaks Hindi.
Answer: (d) Everyone in India speaks Hindi.
The statement Everyone in India speaks Hindi is not true.
This is because, there are some states like Tamilnadu, Kerala, etc. where the person does not speak Hindi.
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Control of Aircraft Lateral Axis Using Mu Synthesis
Main Content
Control of Aircraft Lateral Axis Using Mu Synthesis
This example shows how to use mu-analysis and synthesis tools in the Robust Control Toolbox™. It describes the design of a robust controller for the lateral-directional axis of an aircraft during
powered approach to landing. The linearized model of the aircraft is obtained for an angle-of-attack of 10.5 degrees and airspeed of 140 knots.
Performance Specifications
The illustration below shows a block diagram of the closed-loop system. The diagram includes the nominal aircraft model, the controller K, as well as elements capturing the model uncertainty and
performance objectives (see next sections for details).
Figure 1: Robust Control Design for Aircraft Lateral Axis
The design goal is to make the airplane respond effectively to the pilot's lateral stick and rudder pedal inputs. The performance specifications include:
• Decoupled responses from lateral stick p_cmd to roll rate p and from rudder pedals beta_cmd to side-slip angle beta. The lateral stick and rudder pedals have a maximum deflection of +/- 1 inch.
• The aircraft handling quality (HQ) response from lateral stick to roll rate p should match the first-order response.
HQ_p = 5.0 * tf(2.0,[1 2.0]);
step(HQ_p), title('Desired response from lateral stick to roll rate (Handling Quality)')
Figure 2: Desired response from lateral stick to roll rate.
• The aircraft handling quality response from the rudder pedals to the side-slip angle beta should match the damped second-order response.
HQ_beta = -2.5 * tf(1.25^2,[1 2.5 1.25^2]);
step(HQ_beta), title('Desired response from rudder pedal to side-slip angle (Handling Quality)')
Figure 3: Desired response from rudder pedal to side-slip angle.
• The stabilizer actuators have +/- 20 deg and +/- 50 deg/s limits on their deflection angle and deflection rate. The rudder actuators have +/- 30 deg and +/-60 deg/s deflection angle and rate
• The three measurement signals ( roll rate p, yaw rate r, and lateral acceleration yac ) are filtered through second-order anti-aliasing filters:
freq = 12.5 * (2*pi); % 12.5 Hz
zeta = 0.5;
yaw_filt = tf(freq^2,[1 2*zeta*freq freq^2]);
lat_filt = tf(freq^2,[1 2*zeta*freq freq^2]);
freq = 4.1 * (2*pi); % 4.1 Hz
zeta = 0.7;
roll_filt = tf(freq^2,[1 2*zeta*freq freq^2]);
AAFilters = append(roll_filt,yaw_filt,lat_filt);
From Specs to Weighting Functions
H-infinity design algorithms seek to minimize the largest closed-loop gain across frequency (H-infinity norm). To apply these tools, we must first recast the design specifications as constraints on
the closed-loop gains. We use weighting functions to "normalize" the specifications across frequency and to equally weight each requirement.
We can express the design specs in terms of weighting functions as follows:
• To capture the limits on the actuator deflection magnitude and rate, pick a diagonal, constant weight W_act, corresponding to the stabilizer and rudder deflection rate and deflection angle
W_act = ss(diag([1/50,1/20,1/60,1/30]));
• Use a 3x3 diagonal, high-pass filter W_n to model the frequency content of the sensor noise in the roll rate, yaw rate, and lateral acceleration channels.
W_n = append(0.025,tf(0.0125*[1 1],[1 100]),0.025);
clf, bodemag(W_n(2,2)), title('Sensor noise power as a function of frequency')
Figure 4: Sensor noise power as a function of frequency
• The response from lateral stick to p and from rudder pedal to beta should match the handling quality targets HQ_p and HQ_beta. This is a model-matching objective: to minimize the difference (peak
gain) between the desired and actual closed-loop transfer functions. Performance is limited due to a right-half plane zero in the model at 0.002 rad/s, so accurate tracking of sinusoids below
0.002 rad/s is not possible. Accordingly, we'll weight the first handling quality spec with a bandpass filter W_p that emphasizes the frequency range between 0.06 and 30 rad/sec.
W_p = tf([0.05 2.9 105.93 6.17 0.16],[1 9.19 30.80 18.83 3.95]);
clf, bodemag(W_p), title('Weight on Handling Quality spec')
Figure 5: Weight on handling quality spec.
• Similarly, pick W_beta=2*W_p for the second handling quality spec
Here we scaled the weights W_act, W_n, W_p, and W_beta so the closed-loop gain between all external inputs and all weighted outputs is less than 1 at all frequencies.
Nominal Aircraft Model
A pilot can command the lateral-directional response of the aircraft with the lateral stick and rudder pedals. The aircraft has the following characteristics:
• Two control inputs: differential stabilizer deflection delta_stab in degrees, and rudder deflection delta_rud in degrees.
• Three measured outputs: roll rate p in deg/s, yaw rate r in deg/s, and lateral acceleration yac in g's.
• One calculated output: side-slip angle beta.
The nominal lateral directional model LateralAxis has four states:
• Lateral velocity v
• Yaw rate r
• Roll rate p
• Roll angle phi
These variables are related by the state space equations:
where x = [v; r; p; phi], u = [delta_stab; delta_rud], and y = [beta; p; r; yac].
load LateralAxisModel
LateralAxis =
A =
v r p phi
v -0.116 -227.3 43.02 31.63
r 0.00265 -0.259 -0.1445 0
p -0.02114 0.6703 -1.365 0
phi 0 0.1853 1 0
B =
delta_stab delta_rud
v 0.0622 0.1013
r -0.005252 -0.01121
p -0.04666 0.003644
phi 0 0
C =
v r p phi
beta 0.2469 0 0 0
p 0 0 57.3 0
r 0 57.3 0 0
yac -0.002827 -0.007877 0.05106 0
D =
delta_stab delta_rud
beta 0 0
p 0 0
r 0 0
yac 0.002886 0.002273
Continuous-time state-space model.
The complete airframe model also includes actuators models A_S and A_R. The actuator outputs are their respective deflection rates and angles. The actuator rates are used to penalize the actuation
A_S = [tf([25 0],[1 25]); tf(25,[1 25])];
A_S.OutputName = {'stab_rate','stab_angle'};
A_R = A_S;
A_R.OutputName = {'rud_rate','rud_angle'};
Accounting for Modeling Errors
The nominal model only approximates true airplane behavior. To account for unmodeled dynamics, you can introduce a relative term or multiplicative uncertainty W_in*Delta_G at the plant input, where
the error dynamics Delta_G have gain less than 1 across frequencies, and the weighting function W_in reflects the frequency ranges in which the model is more or less accurate. There are typically
more modeling errors at high frequencies so W_in is high pass.
% Normalized error dynamics
Delta_G = ultidyn('Delta_G',[2 2],'Bound',1.0);
% Frequency shaping of error dynamics
w_1 = tf(2.0*[1 4],[1 160]);
w_2 = tf(1.5*[1 20],[1 200]);
W_in = append(w_1,w_2);
title('Relative error on nominal model as a function of frequency')
Figure 6: Relative error on nominal aircraft model as a function of frequency.
Building an Uncertain Model of the Aircraft Dynamics
Now that we have quantified modeling errors, we can build an uncertain model of the aircraft dynamics corresponding to the dashed box in the Figure 7 (same as Figure 1):
Figure 7: Aircraft dynamics.
Use the connect function to combine the nominal airframe model LateralAxis, the actuator models A_S and A_R, and the modeling error description W_in*Delta_G into a single uncertain model Plant_unc
mapping [delta_stab; delta_rud] to the actuator and plant outputs:
% Actuator model with modeling uncertainty
Act_unc = append(A_S,A_R) * (eye(2) + W_in*Delta_G);
Act_unc.InputName = {'delta_stab','delta_rud'};
% Nominal aircraft dynamics
Plant_nom = LateralAxis;
Plant_nom.InputName = {'stab_angle','rud_angle'};
% Connect the two subsystems
Inputs = {'delta_stab','delta_rud'};
Outputs = [A_S.y ; A_R.y ; Plant_nom.y];
Plant_unc = connect(Plant_nom,Act_unc,Inputs,Outputs);
This produces an uncertain state-space (USS) model Plant_unc of the aircraft:
Uncertain continuous-time state-space model with 8 outputs, 2 inputs, 8 states.
The model uncertainty consists of the following blocks:
Delta_G: Uncertain 2x2 LTI, peak gain = 1, 1 occurrences
Type "Plant_unc.NominalValue" to see the nominal value and "Plant_unc.Uncertainty" to interact with the uncertain elements.
Analyzing How Modeling Errors Affect Open-Loop Responses
We can analyze the effect of modeling uncertainty by picking random samples of the unmodeled dynamics Delta_G and plotting the nominal and perturbed time responses (Monte Carlo analysis). For
example, for the differential stabilizer channel, the uncertainty weight w_1 implies a 5% modeling error at low frequency, increasing to 100% after 93 rad/sec, as confirmed by the Bode diagram below.
% Pick 10 random samples
Plant_unc_sampl = usample(Plant_unc,10);
% Look at response from differential stabilizer to beta
subplot(211), step(Plant_unc.Nominal(5,1),'r+',Plant_unc_sampl(5,1),'b-',10)
subplot(212), bodemag(Plant_unc.Nominal(5,1),'r+',Plant_unc_sampl(5,1),'b-',{0.001,1e3})
Figure 8: Step response and Bode diagram.
Designing the Lateral-Axis Controller
Proceed with designing a controller that robustly achieves the specifications, where robustly means for any perturbed aircraft model consistent with the modeling error bounds W_in.
First we build an open-loop model OLIC mapping the external input signals to the performance-related outputs as shown below.
Figure 9: Open-loop model mapping external input signals to performance-related outputs.
To build this model, start with the block diagram of the closed-loop system, remove the controller block K, and use connect to compute the desired model. As before, the connectivity is specified by
labeling the inputs and outputs of each block.
Figure 10: Block diagram for building open-loop model.
% Label block I/Os
AAFilters.u = {'p','r','yac'}; AAFilters.y = 'AAFilt';
W_n.u = 'noise'; W_n.y = 'Wn';
HQ_p.u = 'p_cmd'; HQ_p.y = 'HQ_p';
HQ_beta.u = 'beta_cmd'; HQ_beta.y = 'HQ_beta';
W_p.u = 'e_p'; W_p.y = 'z_p';
W_beta.u = 'e_beta'; W_beta.y = 'z_beta';
W_act.u = [A_S.y ; A_R.y]; W_act.y = 'z_act';
% Specify summing junctions
Sum1 = sumblk('%meas = AAFilt + Wn',{'p_meas','r_meas','yac_meas'});
Sum2 = sumblk('e_p = HQ_p - p');
Sum3 = sumblk('e_beta = HQ_beta - beta');
% Connect everything
OLIC = connect(Plant_unc,AAFilters,W_n,HQ_p,HQ_beta,...
This produces the uncertain state-space model
Uncertain continuous-time state-space model with 11 outputs, 7 inputs, 26 states.
The model uncertainty consists of the following blocks:
Delta_G: Uncertain 2x2 LTI, peak gain = 1, 1 occurrences
Type "OLIC.NominalValue" to see the nominal value and "OLIC.Uncertainty" to interact with the uncertain elements.
Recall that by construction of the weighting functions, a controller meets the specs whenever the closed-loop gain is less than 1 at all frequencies and for all I/O directions. First design an
H-infinity controller that minimizes the closed-loop gain for the nominal aircraft model:
nmeas = 5; % number of measurements
nctrls = 2; % number of controls
[kinf,~,gamma_inf] = hinfsyn(OLIC.NominalValue,nmeas,nctrls);
Here hinfsyn computed a controller kinf that keeps the closed-loop gain below 1 so the specs can be met for the nominal aircraft model.
Next, perform a mu-synthesis to see if the specs can be met robustly when taking into account the modeling errors (uncertainty Delta_G). Use the command musyn to perform the synthesis and use
musynOptions to set the frequency grid used for mu-analysis.
fmu = logspace(-2,2,60);
opt = musynOptions('FrequencyGrid',fmu);
[kmu,CLperf] = musyn(OLIC,nmeas,nctrls,opt);
Robust performance Fit order
Iter K Step Peak MU D Fit D
1 5.097 3.487 3.488 12
2 1.31 1.292 1.312 20
3 1.243 1.243 1.692 12
4 1.692 1.543 1.544 16
5 1.223 1.223 1.551 12
6 1.533 1.464 1.465 20
7 1.289 1.288 1.304 12
Best achieved robust performance: 1.22
Here the best controller kmu cannot keep the closed-loop gain below 1 for the specified model uncertainty, indicating that the specs can be nearly but not fully met for the family of aircraft models
under consideration.
Frequency-Domain Comparison of Controllers
Compare the performance and robustness of the H-infinity controller kinf and mu controller kmu. Recall that the performance specs are achieved when the closed loop gain is less than 1 for every
frequency. Use the lft function to close the loop around each controller:
clinf = lft(OLIC,kinf);
clmu = lft(OLIC,kmu);
What is the worst-case performance (in terms of closed-loop gain) of each controller for modeling errors bounded by W_in? The wcgain command helps you answer this difficult question directly without
need for extensive gridding and simulation.
% Compute worst-case gain as a function of frequency
opt = wcOptions('VaryFrequency','on');
% Compute worst-case gain (as a function of frequency) for kinf
[mginf,wcuinf,infoinf] = wcgain(clinf,opt);
% Compute worst-case gain for kmu
[mgmu,wcumu,infomu] = wcgain(clmu,opt);
You can now compare the nominal and worst-case performance for each controller:
f = infoinf.Frequency;
gnom = sigma(clinf.NominalValue,f);
title('Performance analysis for kinf')
xlabel('Frequency (rad/sec)')
ylabel('Closed-loop gain');
xlim([1e-2 1e2])
legend('Nominal Plant','Worst-Case','Location','NorthWest');
f = infomu.Frequency;
gnom = sigma(clmu.NominalValue,f);
title('Performance analysis for kmu')
xlabel('Frequency (rad/sec)')
ylabel('Closed-loop gain');
xlim([1e-2 1e2])
legend('Nominal Plant','Worst-Case','Location','SouthWest');
The first plot shows that while the H-infinity controller kinf meets the performance specs for the nominal plant model, its performance can sharply deteriorate (peak gain near 15) for some perturbed
model within our modeling error bounds.
In contrast, the mu controller kmu has slightly worse performance for the nominal plant when compared to kinf, but it maintains this performance consistently for all perturbed models (worst-case gain
near 1.25). The mu controller is therefore more robust to modeling errors.
Time-Domain Validation of the Robust Controller
To further test the robustness of the mu controller kmu in the time domain, you can compare the time responses of the nominal and worst-case closed-loop models with the ideal "Handling Quality"
response. To do this, first construct the "true" closed-loop model CLSIM where all weighting functions and HQ reference models have been removed:
kmu.u = {'p_cmd','beta_cmd','p_meas','r_meas','yac_meas'};
kmu.y = {'delta_stab','delta_rud'};
AAFilters.y = {'p_meas','r_meas','yac_meas'};
CLSIM = connect(Plant_unc(5:end,:),AAFilters,kmu,{'p_cmd','beta_cmd'},{'p','beta'});
Next, create the test signals u_stick and u_pedal shown below
time = 0:0.02:15;
u_stick = (time>=9 & time<12);
u_pedal = (time>=1 & time<4) - (time>=4 & time<7);
subplot(211), plot(time,u_stick), axis([0 14 -2 2]), title('Lateral stick command')
subplot(212), plot(time,u_pedal), axis([0 14 -2 2]), title('Rudder pedal command')
You can now compute and plot the ideal, nominal, and worst-case responses to the test commands u_stick and u_pedal.
% Ideal behavior
IdealResp = append(HQ_p,HQ_beta);
IdealResp.y = {'p','beta'};
% Worst-case response
WCResp = usubs(CLSIM,wcumu);
% Compare responses
lsim(IdealResp,'g',CLSIM.NominalValue,'r',WCResp,'b:',[u_stick ; u_pedal],time)
title('Closed-loop responses with mu controller KMU')
The closed-loop response is nearly identical for the nominal and worst-case closed-loop systems. Note that the roll-rate response of the aircraft tracks the roll-rate command well initially and then
departs from this command. This is due to a right-half plane zero in the aircraft model at 0.024 rad/sec.
See Also
musyn | hinfsyn
Related Topics
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Perimeter or Area | Quizalize
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• Emma walked around the outside of the playground. Is this the perimeter or area?
• Q1
Emma walked around the outside of the playground. Is this the perimeter or area?
• Q2
Teddy painted the wall from the top to the bottom, covering the total space of the wall. Is the inside space the perimeter or area?
• Q3
Nolie was figuring out the perimeter of a rectangle whose has a side length of 8 inches and a width of 4 inches. Which is the correct number sentence to solve for perimeter?
8 x 4 = P
8 + 8 + 4 + 4 = P
• Q4
Jay was figuring out the area of a rectangle whose has a side length of 8 inches and a width of 4 inches. Which is the correct number sentence to solve for area?
8 + 8 + 4 + 4 = A
8 x 4 = A
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What is K-NN? | How It Is Useful?
This article will take you through the fundamentals of the KNN algorithm along with a demonstration of how it works.
What is the K nearest neighbour algorithm?
The KNN algorithm is an instance-based algorithm. The basic assumption is that there are groups in the dataset. A new data point is assigned to a group depending upon the neighbours. This is also
called the Lazy algorithm.
For example, consider the dataset given below in the picture. A point labelled as a black star is assigned to the blue group because for K=7 there are four blue and 3 red. While if we consider K=5
the same is assigned to the brown group.
The nearest neighbours are calculated by using the distances between the target point and various points around it.
The algorithm:
1, Load the dataset into your python.
2, Choose the value of k (that is the nearest neighbour)
3, Calculate the distance between the test point and other data points
1, There are four ways to compute the distance between the data points.
a) Hamming Distance
This distance compares the similarity between two strings and finds the difference between them. The distance is a measure of dissimilarity between them.
b) Euclidean Distance
4, Once the distance is found between the test point and other points, the distances are sorted in ascending order.
5, Depending upon the choice of 'k' the k number of rows is chosen.
6, Based on the majority the test point is classified.
The algorithm is simple and easy to understand and interpret. It is beneficial for non-linear data because there is no underlying assumption of any kind. It has high accuracy and one can use it for
both regression and classification.
However, higher memory is required for computing the distances. For N points we will have to compute N(N-1) values. It is very sensitive to outliers.
A simple demonstration of how the k nearest neighbor knn algorithm works
Here we are using the iris data from UCI machine learning repository.
After displaying the head of the data, we do the following.
1) Split the input and output
2) Split the training and test data
3) Scale the features
Now let us import the k-NN model and fit the data to the model.
Here we have taken 10 neighbours and we predict y_pred using the X test.
To find the optimal number of neigbours we plot error vs the neighbours.
Finally, we print the model accuracy, classification report, and accuracy score.
We can use the same algorithm for regression also.
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What is a good personal rate of return
To find the "real return" - or the rate of return after inflation - just subtract the inflation rate from the rate of return. So if the inflation rate was 1% in a year with a 7% return, then the real
rate of return is 6%, while the nominal rate of return is 7%.
While it is not good to look at your account for investment performance over the short term, you should evaluate your retirement investment over longer periods to Feb 19, 2020 Understanding the
average 401(k) return rate empowers investors to One of the best ways to get a good grasp on average 401(k) returns is For ordinary returns, if there is no reinvestment, and losses are made good by
topping up the capital invested, so that the value is brought back to its starting- point Apr 27, 2018 Here are the signs of a good 401(k) plan -- and how to make the most of yours. and is taxed at
your ordinary income tax rate when you withdraw it in retirement. that's $2,250 of free money, a guaranteed 50% return on your investment. Terms and Conditions · Do Not Sell My Personal Information.
Jul 12, 2013 Knowing your personal rate of return can help you determine if you're on track Your actual investment or personal rate of return in a fund may be report on shares, funds, market
developments and good investing practice How does personal rate of return account for the contributions I make to my account? Personal rate of return (PRR) can most simply be thought of as the
amount Jun 19, 2012 A good way to measure the performance of your investments is over the long term. 25-30% returns are easy to get! It's not going to be 25-30%
That seems to be the figure that makes people willing to part with their money for the hope of more money tomorrow. Thus, if you live in a world of 3% inflation, you would expect a 10% rate of return
(7% real return + 3% inflation = 10% nominal return).
Jan 2, 2007 Thanks for the great information. Now, do you have a simplified formula for calculating personal rate of return for multiple years? For example, The two primary methods for calculating
the rate of return on an investment: As cash flows are unique to each investor, MWRR is a good measure of an Account performance is calculated every business day for each account. The personal rate
of return differs from the monthly performance returns published on The 401k is easily one of the best tax-advantaged retirement accounts out there. While you only have the choices your employer
offers, Personal Capital can If your rate is already low, you can put paying this debt off on the back burner in Oct 17, 2019 Ever since I started writing about personal finance, all of my friends
loved some more so it seems like a good starting point on our risk-return curve. But with those higher rate of return investments, we know that our risk will Nov 16, 2018 It almost never makes sense
to compare internal rates of return across The simple return is a good back-of-the-envelope calculation that can
The annual rate of return on an investment is the profit you make on that investment in a year. For every dollar you invest, how much do you get every year in return? The simple way to calculate
annual return is to look at a simple percentage. You invested $100 and made $3, so your return is $3/$100 or 3%.
Mar 3, 2017 Personalized rates of return are starting to appear on brokerage statements across Canada. The timing of his purchases proved to be good. Jun 6, 2019 Understanding return on investment is
vital for any business. ROI is usually expressed as a percentage and is typically used for personal financial decisions, to compare a company's What is a Good ROI? by the duration of the investment
(see the first FAQ question) to get an annual rate of return. Oct 17, 2016 Annual rates of returns can be used in stocks, mutual funds and bonds. Calculating Rate of Return. Let's look at how the
annual rate of return of a May 17, 2019 is the average annual growth rate of the fund's share price from March 31, 2014 Investors frequently buy high — chasing good returns in a fund, say — and your
personal returns relative to your fund's annualized figures. To find the "real return" - or the rate of return after inflation - just subtract the inflation rate from the rate of return. So if the
inflation rate was 1% in a year with a 7% return, then the real rate of return is 6%, while the nominal rate of return is 7%. That seems to be the figure that makes people willing to part with their
money for the hope of more money tomorrow. Thus, if you live in a world of 3% inflation, you would expect a 10% rate of return (7% real return + 3% inflation = 10% nominal return). A "good" rate of
return for you is ultimately, in some measure, dependent on your own financial goals. The key to boiling down your expectations to a specific number for various investments is to
May 12, 2017 Your personal rate of return is determined using the total value of all of your investments, less the initial amount you paid for them and any fees
I think a good rate of return is 11%. The stock market has returned around 8% a year throughout it's history. If you go to http://www.top10traders.com - and see the best traders have made annual The
annual rate of return on an investment is the profit you make on that investment in a year. For every dollar you invest, how much do you get every year in return? The simple way to calculate annual
return is to look at a simple percentage. You invested $100 and made $3, so your return is $3/$100 or 3%. But it does. Your 401 (k) plan 's rate of return is directly correlated to the investment
portfolio you create with your contributions, as well as the current market environment. That being said, although each 401 (k) plan is different, contributions accumulated within your plan, If, for
example, you calculate that, to meet your goals, you'll need a 15% annual rate of return, you will likely fall far short. You'll need to go back to the drawing board and either increase your savings
or reduce your retirement income expectations.
Jun 6, 2019 Understanding return on investment is vital for any business. ROI is usually expressed as a percentage and is typically used for personal financial decisions, to compare a company's What
is a Good ROI? by the duration of the investment (see the first FAQ question) to get an annual rate of return.
May 10, 2017 What's a reasonable rate of return for me to expect in the future? --Paul Related : The best way to get guaranteed income in retirement. May 12, 2017 Your personal rate of return is
determined using the total value of all of your investments, less the initial amount you paid for them and any fees Royal Mutual Funds Inc. (RMFI) uses a dollar-weighted rate of return (DWRR) to
calculate your personal rate of return that's reported on your quarterly account Jan 2, 2007 Thanks for the great information. Now, do you have a simplified formula for calculating personal rate of
return for multiple years? For example,
Apr 24, 2019 Before jumping into average rates of return, it's helpful to consider the ROI of leaving your money in cash. It can feel like you're keeping your Jul 25, 2019 To calculate Return on
Investment (ROI), make sure to consider all If you've held an investment for multiple years, it's important to find your annualized rate of return. With most online annualized return calculators
(Bankrate has a good Personal Finance Insider offers tools and calculators to help you Mar 3, 2017 Personalized rates of return are starting to appear on brokerage statements across Canada. The
timing of his purchases proved to be good.
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Monday Biomathematics
Jan. 30 Non linear Dominance Hierarchy Establishment from Social Interactions and Metabolic Costs: Application to the Harpegnathos Saltator
4 PM Jordy J. Cevallos-Chavez
online Applied Mathematics for the Life & Social Sciences, Arizona State University
Monday Analysis
Jan. 30 Mathematical aspects of turbulence
4 PM Theodore D. Drivas
online Mathematics Department, Stony Brook University
Tuesday Topology and Geometry
Jan. 31 Projective model structures on diffeological spaces and smooth sets and the smooth Oka principle
3:30 PM Dmitri Pavlov
online Department of Mathematics and Statistics, Texas Tech University
Tuesday Real-Algebraic Geometry
Jan. 31 Introduction to Projective Geometry
5 PM David Weinberg
MA 115 Mathematics and Statistics, Texas Tech University
Wednesday Applied Mathematics and Machine Learning
Feb. 1 Recent developments on convex integration applied to surface quasi-geostrophic equations
4 PM Kazuo Yamazaki
online Department of Mathematics and Statistics, Texas Tech University
Thursday Quantum Homotopy
Feb. 2 Truncations and n-images. Higher gauge groupoids
3:30 PM Jiajun Hoo
MATH 115 Department of Mathematics and Statistics, Texas Tech University
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The Gears That Power the Tubes: The Google Gears Q&AThe Gears That Power the Tubes: The Google Gears Q&A
With apologies to Dave Johnson, but continuing in my fine tradition of being the last to bring you news, I offer you the following commentary on the recent announcement of Google Gears. Given the
breadth of coverage you’ve likely already seen on the announcement, I can’t promise anything new, but I’d be remiss if I didn’t address the question given my long running interest in offline browser
persistence approaches. On to the Q&A.
Q: Before we begin, anything to disclose?
A: Let’s see. Well, Google is not a RedMonk client, and I have not been officially briefed on the Gears technology at this time. IBM and Sun, which both currently support versions of the Apache Derby
database (originally developed by IBM under the Cloudscape moniker), are RedMonk customers, as is another embedded database supplier, db4objects. Neither Joyent nor Zimbra are RedMonk clients, but
they host a personal site of mine and our RedMonk email respectively, gratis. I think that about covers it.
Q: For those that haven’t yet seen the news, can you summarize the announcement?
A: Well, David Berlind’s already done an excellent job describing the technologies in his post here, so I’d recommend you start there. And definitely listen to the podcast interview with Linus Upson,
a director of engineering at Google; I’m not a podcast guy, but it’s worth your time.
For the link and/or podcast averse, however, the basic summary is this: Google is introducing in Gears a cross-platform set of technologies that intend to facilitate the construction of Ajax
applications that can function in a disconnected – i.e. offline – state.
Q: Any applications enabled yet?
A: Just one, as far as I know, Google Reader – see the inset picture. It’s a bit klunky, in that you seem to have to press a button to go offline before you actually go offline, but it’s compelling
Q: What is the problem space Gears is aimed at?
A: One most of us know quite well. Try using Gmail, as an example, in the absence of a networking connection. Or Yahoo Mail. Or Hotmail. Or, well, you get the idea. While Software-as-a-Service, as
embodied by regular consumer facing apps like Gmail or more enterprised focused packages like Salesforce.com is a massively transformative application delivery paradigm, it’s been hamstrung at times
by its inability to deal with disconnected or intermittently connected application consumption scenarios. Or, to shelve the consultant speak, the fact that network applications don’t work all that
well without the network. It’s what I’ve described in years past as the “Offline Problem.”
Q: How important is that problem, still, with network availability getting better and more pervasive every day?
A: Perspectives vary. when discussing this problem in the past, Alex once joked that we’d solve the offline problem just in time for it to be made obsolete by ubiquitous connectivity. And Rails’
David Heinemeier Hansson apparently is not a big believer in the importance of offline persistence.
With both wifi and EVDO cards built directly into my laptop, I certainly understand the point of view that sahys offline persistence is a lot less important than it used to be. But that’s a far cry
from saying it’s not important. As SaaS applications increasing compete against rich client or Rich Internet Application (RIA) alternatives that offer persistence, it will become a more significant
limiting factor to adoption. Further, as ubiquitous as connectivity might eventually become, it’s not ubiquitous now. I should know, as I’m spending the summer in a location where the only broadband
access is satellite and my best connectivity option is the barely better than dial-up GPRS. Then, of course, there are the planes, the hotels, the office buildings where there is no connectivity. Or,
just as often, the convention centers, hotels, airports and so on that do offer wireless – just not wireless that actually, you know, works.
As a result, my personal opinion is that offline persistence for web applications is both laudable and necessary from a strategic competitive perspective. Just because we can endure the pain of not
having offline access to web apps, doesn’t mean that we should endure it. If you disagree, here’s a good acid test for you: would you be comfortable delivering a presentation at a conference, relying
solely on an online presentation tool? I certainly would not, but YMMV.
Q: Is that a recent opinion? In other words, have your recent application consumption trends affected that opinon?
A: Not particularly. I am, in fact, spending more and more time in the browser; I’ve gone from using Evolution daily to monthly – if that. But the issue bugged me as far back as September of 04
(which, you might notice, immediately preceded the magical October of ’04), when I was disappointed by the fact that the synchronization of my web reader at the time, Bloglines, was only possible to
offline rich clients for the Mac and Windows platforms. As of today, I don’t need to worry about that. Took a couple of years, but better late than never and so on.
Suffice it to say that this is in fact a problem that’s bugged me for a long, long time. If you’re really bored, you can take a gander through some of my older entries related to the subject
(earliest first): Turning Dross into Gold: Alchemy and Offline Browser Access, Presentation Tools: Offline, Online and Something in Between, So You Want to be an Office 2.0 Provider?, Zimbra: Derby
for Offline Persistence, Grand Desktop Ambitions: The Q&A, Is Google Ring Fencing IBM?, and so on.
Q: Bringing it back to the actual Gears announcement, can we examine that in a little more detail? Starting with the technology?
A: Sure. As the initial announcements didn’t specify much in the way of what the individual pieces were – nor did the FAQs appear to – I was curious. But, as they always do, interested parties soon
ferretted those out. Here’s John Herren’s breakdown on the individual components to Google Gears:
Google Gears uses three components:
□ LocalServer- Handles caching of URL resources on the local file system.
□ Database- Gears uses sqlite databases for storage. You can even find the databases on your file system and browse them with any sqlite compatible tool. I did. It works.
□ WorkerPool- A job threading API to perform asynchronous operations so your app stays snappy and doesn’t hang. Check out the Fibonacci demo to see it in action.
Gisting this down, you essentially have a local web server (which one, does anyone know?) that serves cached content out of a SQLite database, with a threading mechanism that ensures that abnormally
strenuous tasks – say, downloading and caching several thousand email messages – doesn’t bring your browser down while it waits. It’s pretty straightforward, actually.
Q: Is this the first such instance of this technology?
A: Nope, not at all. We’ve had building blocks for a while. Dojo, as an example, has leveraged the storage capabilities of Flash to deliver offline persistence for some time. In this post, Ted
comments on a demo of Derby, or JavaDB in this case, being used as the persistence mechanism for an offline web application back at ApacheCon 2005. In November of last year, for example, the folks
from Zimbra demoed a very similar Derby based solution to persisting Zimbra data offline; a solution that they’ve since released and that I’m an occasional user of. Not to be left out, two months ago
the folks from Joyent announced Slingshot to the world; a framework for allowing Rails applications to persist data in disconnected settings.
In terms of future offerings, the plan for the 3.0 release of Firefox was actually – as I understand it – to include a very similar SQLite based repository for offline persistence.
Google is very much the follower in this space, rather than the first mover.
Q: What’s different about the Google offering? What differentiates it?
A: Most obviously, it’s from Google. None of the other competing alternatives can compete for reach and breadth.
Technically, it also differs from the Flash or Java based approaches, choosing instead to deploy its own, lightweight cross-platform persistence store in SQLite. Unlike some of the alternatives, it
would appear that Google’s spent a lot of time on application performance; each application instance has its own sandbox, its own database, and can spawn its own background threads. As a result,
according to Google, the performance impact for many applications should be neglible.
The most important differentiator, however, is its ambitions. I didn’t fully appreciate them until I listened to Berlind’s podcast linked to above, but Google would like for gears to become the de
facto standard for offline development, a “single, industry standard” approach for delivering offline Ajax applications. Zimbra’s approach was to solve the problem on its own, while the Joyent folks
widened the aperture a bit more targeting Rails applications. Google’s intention is for Gears to become the platform for offline apps, and has apparently designed and definitely licensed it as such.
Q: How do you mean?
A: well, let’s take the technology side first. Upson, in the interview above, was very unambiguous is his desire to have a great many developers use the technology. Start listing around 13:20, and
you’ll get the picture, as Upson says:
“Clearly we’re very interested in offline enabling all of the interesting Google applications, however, we really want this to be a developer focused release, and it’s still at the experimental
stage and we want to get feedback from the broader community, and we know this is going to evolve and change as we learn. I think we’ve done some clever things here, we’ve probably done some
things that aren’t so clever, and so we want to be able to change how this works and evolve it over time, based on the partners that we work with and the web developer community. We wanted to
have a real application, but we didn’t want to go beyond the development community at this stage.”
From that, it’s clear that Google has attempted not to deliver the once and future framework, but something that really is a beta (rather than a label, as it seems to be with so many of Google’s
other offerings). Something that will draw feedback, and evolve towards real deployment scenarios.
Q: And the licensing? How does that encourage adoption?
A: The Gears technology is permissively licensed under a BSD style license, which is the least restrictive and offers the fewest barriers of entry to potential communities. As we recently put it in
an internal report delivered to a RedMonk client:
Platform Licensing:
Of particular interest in this case are the strategies employed by platforms. While the GPL remains the overwhelming license of choice for applications in general, and is the choice of perhaps
the most popular open source platform in Linux, platform technologies trend towards permissive style licenses as opposed to reciprocal approaches. The BSD distributions are perhaps the most
obvious example, but the licensing for PHP – perhaps the most ubiquitous dynamic language at the current time – is another.
At one time, PHP was dual licensed under the GPL and the PHP license, but dropped the GPL as of PHP version 4 because of the reciprocal restrictions. Python’s license is similarly permissive, as
it employs its own custom BSD style style license. Mozilla’s Firefox, additionally, was as previously mentioned trilicensed
specifically to address the limitations imposed by its original reciprocal-style license.
Generally speaking, the preference for permissive licenses by platform technologies is that they impose the least restrictions on users and developers, thus offering significant advantages should
ubiquity and adoption be an important goal. These advantages, however, come with a price: permissively licensed technologies can be easily and legally forked, or incorporated into proprietary
code, or repurposed. The lack of restrictions is both its biggest strength and biggest weakness.
For a commercial entity, then, permissive licensing is best applied to platforms when the vendors wants to grow the market around the platform, monetizing other parts of the market rather than
the core platform. For example, the platform may be “free” but tools to interact with and create “content” in the resulting ecosystem may cost.
The decision to apply the BSD style license, then, can be viewed as an attempt by Google to encourage ubiquitous adoption and consumption.
Q: What are the likely impacts to some of the Gears alternatives?
A: From an application provider perspective, I tend to agree with Berlind, who said:
Where companies have committed to an offline architecture as Zimbra has with its Zimbra Desktop (whose offline capability is powered by Java), those companies may be forced to completely
reconsider that architecture if Google Gears gets market traction.
If you’re Zimbra, and your resources are limited, it would make sense to at least ask the question as to whether or continued investment in a redundant offline infrastructure was justified. It could
be, for technical reasons, but it might not be as well.
If you’re a Joyent, the question is more complicated. They’re targeting a far more specific niche than Google Gears, in Rails apps, so the question will come down to whether or not Slingshot can
offer enough differentiating features to Rails devs to justify their usage of it. Compromising their argument is the fact that Slingshot is not, as yet, available on as many platforms as Gears.
Still, they appear content to play David.
Interestingly, Adobe seems to be partnering quite closely; willingly aligning their SQLite efforts with Google’s. That bodes well for Google’s ultimate aims.
Q: Who will be anti-Google Gears?
A: Well, some of the aforementioned alternatives are probably not blissfully happy right now. And there are a variety of players that could ultimately be threatened by the technology. But the least
likely to play along, handing Google a de facto standard for offline persistence, would be Microsoft, IMO. I’d be somewhat surprised if we don’t see a similar Windows-like technology emerge. The
question, as always, will be their cross-platform story, which Google’s gotten very right here.
Q: How about Mozilla?
A: I was interested to see how they’d react, given that similar ambitions were on their roadmap for 3.0. But they’re apparently partnering with Google on this endeavor, and I actually wonder – pure
speculation on my part – whether or not Google’s de facto standard could ultimately replace some of the planned work within 3.0. Either way, Mozilla appears to be on board.
Q: Is Google Gears likely to lead to an explosion of offline applications?
A: Explosion’s probably a little overambitious. One of the things that’s clear as you begin to parse offline application scenarios – and as Upson discusses – is that no two are the same. An offline
email client is not the same as an offline feed reader is not the same as an offline CRM system and so on. The application calls, the application storage demands, the application performance
implications, even the very utility of offline data access – vary widely. What I expect to see, at least initially, is experimentation. Trials to determine what data needs to be cached, what doesn’t,
and so on.
My colleague is of the belief that Web 3.0 will be about synchronization, and he’s taken to calling Web 3.0 the Synchronized Web. Whether you agree with that or not, or like *.0 designation system,
there can be no debate that in increasingly online/offline use cases, synchronization – a difficult task at the best of times – will be one of the most significant challenges to tackle.
Q: Couple of questions coming in from #redmonk – was this a 20% project or is it being developed for a specific product?
A: Good question – don’t know the answer. It sounds like the latter, but perhaps one of the Google folks can check in and tell us.
Q: Another set of questions from the #redmonk channel, more technical – “Is WorkerPool positioned against Microsoft’s BITs service for asynchronous IO? Is IO from WorkerPool using the browser
connections or its own? If WorkerPool does not have DOM access, does that mean that the JavaScript is being executed in the browser’s JS interpreter or is Gears providing its own? Will other
non-browser applications be able to read/write to the local store thus enhancing the user experience?”
A: Don’t have the answers to most of those, but let’s see what we can parse. As for BITs, I suspect that WorkerPool is like it, but that they won’t compete with each other because of a.) scope and
b.) the fact that one’s cross-platform and one’s not. As for IO, I don’t know. I’d guess that WorkerPool is using it’s own connections because of the aforementioned issues around multi-threading, but
that’s all it would be, a guess. On the interpreter question, I’m fairly sure Gears is using the browser’s because I haven’t heard anything about it incorporating a second, and that would seem to be
unnecessarily redundant. Lastly, on the topic of whether non-browser applications can read/write to the local store, I know they can read the DB’s – John Herren’s said as much above. And Adobe and
Google appear to be coordinating their efforts on that front to some degree. Whether or not applications can access the pieces more deeply, however, is a question I don’t know the answer to. But
would like to, because it has implications for RIA and rich client strategies that could transcend the browser.
Q: Last question from #redmonk: why Gears?
A: Excellent question. I’ll leave it to Google to answer that one officially, just noting that the Gears Firefox Add-on’s caption is “These are the gears that power the tubes!”
Q: Any last thoughts or conclusions?
A: Just that, like the Joyent guys, I think that offline, persisted information is a legitimate game changer. This is the biggest piece of news I’ve heard in a long while.
Update: Had some versioning issues, so had to re-merge some content. Sections originally missing were the enabled applications, and a different version of the “what differentiates Gears”. Apologies.
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Unscramble WADMOLS
How Many Words are in WADMOLS Unscramble?
By unscrambling letters wadmols, our Word Unscrambler aka Scrabble Word Finder easily found 111 playable words in virtually every word scramble game!
Letter / Tile Values for WADMOLS
Below are the values for each of the letters/tiles in Scrabble. The letters in wadmols combine for a total of 13 points (not including bonus squares)
• W [4]
• A [1]
• D [2]
• M [3]
• O [1]
• L [1]
• S [1]
What do the Letters wadmols Unscrambled Mean?
The unscrambled words with the most letters from WADMOLS word or letters are below along with the definitions.
• wadmol (n.) - A coarse, hairy, woolen cloth, formerly used for garments by the poor, and for various other purposes.
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how to use least function in oracle sql
The LEAST function in Oracle SQL / PLSQL is used to get the least or smallest value out of the expressions provided
The Syntax for the LEAST function in Oracle SQL / PLSQL is:
SELECT LEAST(expression1, expression2, expression3. . . ., expressionN)
FROM table_name;
Expression1, expression2, expression3 … expressionN are expressions evaluated by the least function.
• If the data types of the expressions are different then all the expressions are converted to expression1 data type.
• In character based comparison one character is considered lower or smaller than another if it has a lower character set.
• If an expression is NULL in the LEAST function then NULL will be returned as least value.
Example 1:
SELECT LEAST(3,6,12,3)
Will return “3”
Example 2:
SELECT LEAST('3','6','12','3')
Will return “12”
Example 3:
SELECT LEAST('apples', 'grapes', 'bananas')
Will return “apples”
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Class QRDecompositionHouseholder_ZDRM
All Implemented Interfaces:
DecompositionInterface<ZMatrixRMaj>, QRDecomposition<ZMatrixRMaj>
This variation of complex QR decomposition uses reflections to compute the Q matrix. Each reflection uses a householder operations, hence its name. To provide a meaningful solution the original
matrix must have full rank. This is intended for processing of small to medium matrices.
Both Q and R are stored in the same m by n matrix. Q is not stored directly, instead the u from Q[k]=(I-γ*u*u^H) is stored. Decomposition requires about 2n*m^2-2m^2/3 flops.
See the QR reflections algorithm described in:
David S. Watkins, "Fundamentals of Matrix Computations" 2nd Edition, 2002
For the most part this is a straight forward implementation. To improve performance on large matrices a column is written to an array and the order of some of the loops has been changed. This will
degrade performance noticeably on small matrices. Since it is unlikely that the QR decomposition would be a bottle neck when small matrices are involved only one implementation is provided.
• Field Summary
Modifier and Type
protected double[]
protected boolean
protected double[]
protected int
protected int
protected int
Where the Q and R matrices are stored.
protected double[]
protected double[]
• Method Summary
Modifier and Type
protected void
This function performs sanity check on the input for decompose and sets up the QR matrix.
In order to decompose the matrix 'A' it must have full rank.
Computes the Q matrix from the information stored in the QR matrix.
Returns a single matrix which contains the combined values of Q and R.
Returns an upper triangular matrix which is the R in the QR decomposition.
protected void
Computes the householder vector "u" for the first column of submatrix j.
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• Field Details
□ QR
Where the Q and R matrices are stored. R is stored in the upper triangular portion and Q on the lower bit. Lower columns are where u is stored. Q_k = (I - gamma_k*u_k*u_k^H).
□ numCols
protected int numCols
□ numRows
protected int numRows
□ minLength
protected int minLength
□ dataQR
protected double[] dataQR
□ gammas
protected double[] gammas
□ error
protected boolean error
• Constructor Details
□ QRDecompositionHouseholder_ZDRM
public QRDecompositionHouseholder_ZDRM()
• Method Details
□ setExpectedMaxSize
public void setExpectedMaxSize(int numRows, int numCols)
□ getQR
Returns a single matrix which contains the combined values of Q and R. This is possible since Q is symmetric and R is upper triangular.
The combined Q R matrix.
□ getQ
Computes the Q matrix from the information stored in the QR matrix. This operation requires about 4(m^2n-mn^2+n^3/3) flops.
Specified by:
getQ in interface QRDecomposition<ZMatrixRMaj>
Q - The orthogonal Q matrix.
compact - If true an m by n matrix is created, otherwise n by n.
The Q matrix.
□ getR
Returns an upper triangular matrix which is the R in the QR decomposition.
Specified by:
getR in interface QRDecomposition<ZMatrixRMaj>
R - An upper triangular matrix.
compact - If true only the upper triangular elements are set
The R matrix.
□ decompose
In order to decompose the matrix 'A' it must have full rank. 'A' is a 'm' by 'n' matrix. It requires about 2n*m^2-2m^2/3 flops.
The matrix provided here can be of different dimension than the one specified in the constructor. It just has to be smaller than or equal to it.
Specified by:
decompose in interface DecompositionInterface<ZMatrixRMaj>
A - The matrix which is being decomposed. Modification is implementation dependent.
Returns if it was able to decompose the matrix.
□ householder
protected void householder(int j)
Computes the householder vector "u" for the first column of submatrix j. Note this is a specialized householder for this problem. There is some protection against overflow and underflow.
Q = I - γuu^H
This function finds the values of 'u' and 'γ'.
j - Which submatrix to work off of.
□ commonSetup
This function performs sanity check on the input for decompose and sets up the QR matrix.
□ getGammas
public double[] getGammas()
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How to Find the Volume of a Triangular Prism in C
A prism is a 3D object that has uniform cross-sections, flat rectangular side faces, and identical bases. Prisms come in many forms and are named based on the geometry of their base. For instance, a
triangular prism has two identical triangular bases, three rectangular lateral faces, 9 edges, and 6 vertices. The amount of space occupied by a triangular prism in all three dimensions is its
Calculating the volume of a triangular prism in mathematics can be a time-consuming process. However, it is possible to simplify this calculation by creating a straightforward C program that takes
input from the user and efficiently computes the volume.
How to Find the Volume of the Triangular Prism?
The volume of a triangular prism is the space it occupies or contains. To calculate the volume of a triangular prism, we need to know the dimensions of its base area and length. The volume is
obtained by multiplying the base area and length. The unit of measurement for volume is cubic meters (m³).
The formula for calculating the volume of a triangular prism is:
• V represents the volume.
• B represents the base area.
• l represents the length of the prism.
The following equation is used to calculate a triangular prism’s base area:
• B represents the base area.
• b represents the triangular base.
• h represents the height of the prism.
Now we understand how to find the volume of the triangular prism in mathematics. Let’s write a C program that finds the volume of the triangular prism.
C Program to Find the Volume of the Triangular Prism
The given C program calculates the volume of the triangular prism based on the values of base, height, and length entered by the user.
#include <stdio.h> int
() { float
; float
= 0; printf("\nEnter Base: "); scanf("%f", &
); printf("\nEnter Height: "); scanf("%f", &
); printf("\nEnter Length: "); scanf("%f", &
); //Calculate Base area of the triangular prism
= ((float)1/(float)2)*
; //Calculate volume of triangular prism
; printf("Volume of a triangular prism is: %.2f m³",
); return 0; }
A triangular prism looks like a polyhedron and has 2 triangular identical bases along with 3 rectangular side faces. The volume of the triangular prism refers to the contained space or region. The
volume of the triangular prism can be calculated mathematically, although it can take a long time. We created a simple C program that accepts input from the user and quickly calculates the volume to
make this calculation efficient.
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1072 - Books
Benjamin likes to read books. Benjamin likes books a lot. Ben's teacher knows this, so he likes to encourage Ben to read. At the start of the summer Ben's teacher offers a proposition to Ben. Let
D be the number of days in summer. For each book Ben reads, his teacher will give him D-x points, where x is the day Ben finishes the book. Ben knows for each book he has how long it will take to
read it, and if he starts a book he must finish it before starting a different book, but he can read the books in any order he chooses. Help Ben browse the books better!
There will be several test cases.
Each test case will start with a positive integer D that is at most 100. Then several lines will follow representing the books. Each line will contain a single positive integer x which is how
long it takes to read that particular book.
Between each test case there will be a blank line.
For each test case output a line "Ben can earn P points!" where P is the highest possible number of points Ben can earn.
sample input
sample output
Ben can earn 5 points!
Ben can earn 18 points!
Ben can earn 217 points!
During the first summer, Ben can either finish the 7 book, or he can finish both 5's. It turns out it's better to finish the 5's, the first one gives 10-5=5 points, while the last one gives 10-10
=0 points (he might as well just not read the second book). During the second summer, it doesn't matter which order Ben reads the books, since they all take the same amount of time to read. He
finishes the first book on day 2, so he gets 8 points for it. He finishes the second book on day 4, so he gets 6 points. And he gets 4 points for book 3. That is 18 points overall.
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Dimiter Prodanov
NERF/EHS, Leuven, BE
Regularization of derivatives and fractional approximation of non-differentiable trajectories
A central notion of physics is the rate of change. This perception inspired Newton and Leibniz to develop the apparatus of differential calculus. This calculus is not limited only to linear rates of
change, e.g. to the concept of derivatives as mathematical idealizations of the linear growth. Fractional calculus has been also developed with the idea to describe the rate of change of strongly
non-linear phenomena, such as the phenomena governed by power laws. Yet in another recent development of fractional calculus the link with localizable approaches has been explored. Classical physics
variables, such as velocity or acceleration, are considered to be differentiable functions of position. On the other hand, quantum mechanical paths were found to be non-differentiable and stochastic.
The relaxation of the differentiability assumption opens new avenues in describing physical phenomena, for example, using the scale relativity theory developed by Nottale, which assumes strong
non-linearity and factuality of quantum-mechanical trajectories. The main application of the presented approach comprises a formal regularization procedure for the derivatives of Holderian functions,
which allows for removal of the weak singularity in the derivative caused by strong non-linearities. Moreover using the same approach, generalized velocities (i.e. alpha-velocities) can be also
defined on fractal curves. Some theoretical results related to singular fractal curves will be presented. Possible applications of presented approach are regularizations of quantum mechanical paths
and Brownian motion trajectories, which are Holder ½.
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How to Replace NaN Values With 0 in PyTorch - reason.townHow to Replace NaN Values With 0 in PyTorch
How to Replace NaN Values With 0 in PyTorch
We’ll be discussing how to replace NaN values with 0 in PyTorch. We’ll also be looking at how to do this in a few different ways.
Checkout this video:
nan values can result from calculations that produce undefined results. For example, 0/0 produces nan. NaN values can also be the result of loading invalid data, such as text instead of numerical
PyTorch provides a way to replace nan values with 0 using the torch.nn.functional.relu() function. relu() is a common activation function for neural networks and it sets all negative values to 0. To
use relu(), simply pass your input tensor to the function.
input = torch.tensor([-1, 0, 1, 2])
output = torch.nn.functional.relu(input)
# Output: tensor([0, 0, 1, 2])
What are NaN values?
NaN values are special values that represent missing data. In PyTorch, these values are represented as floating-point numbers with a value of nan .
When you try to perform certain operations on tensors with NaN values, you’ll get an error. For example, if you try to add two tensors that have NaN values, you’ll get the following error:
RuntimeError: value cannot be converted to a scalar.
To avoid this error, you can replace all the NaN values in a tensor with 0 using the torch.nan_to_num() function.
Why do we need to replace NaN values with 0?
One of the most common issues when working with data is dealing with missing values. Missing values can occur for a variety of reasons, such as data entry errors, incorrect data processing, or simply
because the data is not available.
If you’re working with data in PyTorch, you may have encountered the issue of having NaN values in your dataset. While there are a number of ways to deal with missing values, one common approach is
to replace NaN values with 0.
Replacing NaN values with 0 can be beneficial for a number of reasons. For one, it can help to make your data more consistent and easier to work with. Additionally, replacing NaN values with 0 can
also help improve the performance of your machine learning models.
There are a few different ways to replace NaN values with 0 in PyTorch. One approach is to use the torch.nn.functional.replace_nan() function. Another approach is to use the torch.Tensor class’s
fillna() method.
Let’s take a look at an example of how to replace NaN values with 0 using each of these methods.
How to replace NaN values with 0 in PyTorch?
PyTorch is an open source machine learning library used for applications such as computer vision and natural language processing. It’s a popular framework for deep learning due to its flexibility and
ease of use.
One issue you may encounter when working with PyTorch is how to handle missing values (i.e. NaN values). In this post, we’ll show you how to replace NaN values with 0 in PyTorch.
There are two ways to replace NaN values with 0 in PyTorch:
1. Use the PyTorch function torch.FloatTensor() to convert your data into a FloatTensor, and then use the FloatTensor’s fill_() method to replace all NaN values with 0.
2. Use the PyTorch function torch.zeros_like() to create a tensor of zeros that has the same size and shape as your original data tensor, and then use the resulting tensor’s index_fill_() method to
replace all NaN values with 0 at the specified indices.
Let’s take a look at an example of each approach.
Suppose we have the following data tensor:
import torch
data = torch.tensor([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print(data) # tensor([[1., 2., 3.] [4., 5., 6.] [7., 8., 9.]]) # Notice that some of the values are missing (i.e. are NaN).
使用Pytorch函数torch。zeros_like（）创建一个具有与原始数据张量相同大小和形状的零张量，然后使用生成的张量的index_fill _（）方法在指定的索引处将所有NaN值替换为0。“`
假设我们有以下数据张量：“`python import torch data = torch . tensor （ [ ] ] ) “ “ ` #注意有些值是空白的（即是NaN ）。
1 。
You can replace NaN values with zeros in PyTorch by using the torch.zeros_like() function. This function will create a tensor with the same size and dtype as the input tensor, and fill it with zeros.
This is a guide on how to replace NaN values with 0 in PyTorch.
First, we’ll import the necessary packages:
import torch
import numpy as np
Next, we’ll create a tensor with some NaN values:
x = torch.FloatTensor([1, 2, 3, np.nan])
> tensor([ 1., 2., 3., nan])
Now, we’ll replace the NaN values with 0:
x[torch.isnan(x)] = 0
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mp_arc 12-154
12-154 Federico Bonetto, Michael Loss
Entropic Chaoticity for the Steady State of a Current Carrying System. (39K, latex) Dec 26, 12
Abstract , Paper (src), View paper (auto. generated ps), Index of related papers
Abstract. The steady state for a system of N particle under the influence of an external field and a Gaussian thermostat and colliding with random ``virtual'' scatterers can be obtained
explicitly in the limit of small field. We show the sequence of steady state distribution, as N varies, forms a chaotic sequence in the sense that the k particle marginal, in the limit of large
N, is the k-fold tensor product of the 1 particle marginal. We also show that the chaoticity properties holds in the stronger form of entropic chaoticity.
Files: 12-154.src( 12-154.keywords , nonequi.bib , limitloss122412.tex )
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938 millimeters per square second to meters per square second
8,938 Millimeters per square second = 8.94 Meters per square second
Acceleration Converter - Millimeters per square second to meters per square second - 8,938 meters per square second to millimeters per square second
This conversion of 8,938 millimeters per square second to meters per square second has been calculated by multiplying 8,938 millimeters per square second by 0.001 and the result is 8.938 meters per
square second.
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(a+b)^3 Binomial Expansion
Understanding the Binomial Expansion of (a + b)³
The binomial expansion is a fundamental concept in algebra that allows us to expand expressions of the form (a + b) raised to a positive integer power. This article will delve into the binomial
expansion of (a + b)³, exploring its patterns, formula, and applications.
The Expansion Process
The expansion of (a + b)³ can be achieved by applying the distributive property repeatedly:
(a + b)³ = (a + b)(a + b)(a + b)
To simplify, we can expand this step by step:
1. Expand the first two factors: (a + b)(a + b) = a² + 2ab + b²
2. Multiply the result by (a + b): (a² + 2ab + b²)(a + b) = a³ + 3a²b + 3ab² + b³
Therefore, the binomial expansion of (a + b)³ is a³ + 3a²b + 3ab² + b³.
Key Observations and Patterns
The expansion of (a + b)³ exhibits several noteworthy patterns:
• Terms: The expansion has four terms.
• Exponents: The exponents of a decrease from 3 to 0, while the exponents of b increase from 0 to 3.
• Coefficients: The coefficients follow a specific pattern: 1, 3, 3, 1. These coefficients can be obtained using Pascal's Triangle.
Pascal's Triangle and the Binomial Theorem
Pascal's Triangle is a triangular array of numbers that provides a visual representation of binomial coefficients. Each number in the triangle is the sum of the two numbers directly above it.
The coefficients in the expansion of (a + b)³ correspond to the numbers in the fourth row of Pascal's Triangle: 1, 3, 3, 1.
More generally, the binomial theorem provides a formula for expanding (a + b)^n for any positive integer n:
(a + b)^n = ∑_(k=0)^n (n_C_k) a^(n-k) b^k
where n_C_k represents the binomial coefficient, which is the number of ways to choose k items from a set of n items. It can be calculated using the formula:
n_C_k = n! / (k! * (n-k)!)
Applications of the Binomial Expansion
The binomial expansion has numerous applications in various fields, including:
• Algebra: Simplifying complex algebraic expressions.
• Calculus: Deriving Taylor series expansions.
• Probability: Calculating probabilities in binomial distributions.
• Physics: Modeling physical phenomena like the behavior of gases.
The binomial expansion of (a + b)³ provides a concise representation of the expansion, revealing important patterns and relationships. It's a powerful tool used in numerous fields and serves as a
foundation for understanding more complex mathematical concepts.
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Have the new board got a single singing wrong?
Okay I knew it's early days but the clueless new board seem to have defied the odds and despite being the biggest bunch of jokers since coco the clowns stag do they seem to have got it right in the
transfer market, early days yes but the initial view is bang on, it's rare for there not to be a single mistake in the transfer market, even the flawless cortese got a few wrong, but so far despite
being lamblasted by many on here they don't seem to have made single mistake in the transfer market.
Letting Chambers go?
Letting Chambers go?
Quickly rectified by getting shot of the bad apple
Contract release clause?
The suggested agadoo song could be a bad one
Less wrong than the OP's title.
Also, "lamblasted?"
Mods please change to signing FFS
Predictive text FFS
Mods please change to signing FFS
Predictive text FFS
Its better this way.
5 matches in? Probably need to wait a bit longer before we jump on this particular band wagon.
Mods please change to signing FFS
Predictive text FFS
lol at turkish's phone assuming all threads are about chants! Boy Who Cried Wolf!
Mods please change to signing FFS
Predictive text FFS
It might have more to do with your attention or lack of when posting.
We've finally got him on something! Everyone, pile on!!
Set your lamb blasters to stun!
Their signing of a server in the Chapel stand, shocking. Couldn't pour a pint to save his life and so slow.
Standards gentlemen, standards.
Set your lamb blasters to stun!
Even though I think he's looked decent so far, I'm still struggling to make sense of the fee we paid for Long.
Set your lamb blasters to stun!
Prefer the chargrill setting meself.
Not letting Morgan go. I still think the £12-15 million we would have got for him would have gone towards a better defender than Fonte with money left over for a speedy winger.
Mods please change to signing FFS
Predictive text FFS
I think the Mods might take their time correcting the applemakeyoulookatit autocorrect that has afflicted you!
Not really early days, after all we have seen several minutes of Florin and Mane has got his work permit and will set foot in Southampton tomorrow for the first time, so clearly they are both
brilliant. Toby has played a game. Taider was obviously a massive mistake. But no-one with any sense ever lamblasted Les Read who has been responsible for all of the singings with Ronald K, who again
no-one seriously thinks is clown-like. But carry on Turdish (lol at predictive text) with your amusing postings.
No, even though we've seen nothing of Mane or Gardos
Nope. Everybody's singing from the same sheet, for once. Any discordant notes have been muted. Harmony reigns.
No, even though we've seen nothing of Mane or Gardos
Aside from his substitute appearance v Newcastle?
Yep, nothing.
Aside from his substitute appearance v Newcastle?
Yep, nothing.
Certainly seen nothing to judge on re Gardos.
Certainly seen nothing to judge on re Gardos.
But we have seen something of him. Not "Nothing"
But we have seen something of him. Not "Nothing"
You knew what he meant. Why pull him up on it?
You knew what he meant. Why pull it up on it?
I'm pulling semantics because it would. It works all ways.
Gardos is clearly a player involved with the squad and has made his presence felt by being in a position to make a first team appearance. It's a world away from a situation like Forren like it may
have been referring too and using as a stick to fuel it's hysteria on anything that may not suit it 100%.
Mane now has his work permit approved. What are we to expect from a player who couldn't until today step foot into the country to play or train?
Edited by Colinjb
You knew what he meant. Why pull him up on it?
Dont bother. I am not going to. If he cant see how petty he is being, sod him.
The assertion that Chambers is a bad apple is far from the truth.
What song is the board singing wrong by the way? Is it Go Southampton ra ra ra
I'd say only in terms of some of the prices, in particular the £12m paid for Long and, arguably, selling Chambers too early, for less than he'll likely be worth next summer. We'll see.
how many more 'look at me' threads are you going to start on this?
Not letting Morgan go. I still think the £12-15 million we would have got for him would have gone towards a better defender than Fonte with money left over for a speedy winger.
Keeping Morgan is the best bit of business we've done this summer.
Okay I knew it's early days but the clueless new board seem to have defied the odds and despite being the biggest bunch of jokers since coco the clowns stag do they seem to have got it right in
the transfer market, early days yes but the initial view is bang on, it's rare for there not to be a single mistake in the transfer market, even the flawless cortese got a few wrong, but so far
despite being lamblasted by many on here they don't seem to have made single mistake in the transfer market.
Isn't that a contradiction in terms? They are so bad, they did a wonderful job?
Yes, I think they did a good bit of business to be fair and perhaps we should reserve judgement a little.
The assertion that Chambers is a bad apple is far from the truth.
What song is the board singing wrong by the way? Is it Go Southampton ra ra ra
I think Turks got the quotes mixed up. Bad apple would be Taider
Never mind all that, I want a phone that when I type an actual word that exists in its dictionary changes it to a word with similar spelling but a completely different meaning.
Autocorrect my buttocks, Turkey.
how many more 'look at me' threads are you going to start on this?
We're going to need a timeframe for that.
how many more 'look at me' threads are you going to start on this?
Keep sucking those lemons the reflex kid.
I thought Shane Long was the only one good at singing? So does that makes all the others singing wrong?
Edited by Saint Without a Halo
I like to sing in the shower so I wonder if Les uses the same approach which might explain why he's made some good singings?
They've certainly dropped a Boruc today.
They've certainly dropped a Boruc today.
That's racist against the Chinese.
I wouldn't blow too much smoke up the arses of the board. £10m seems to have been wasted on Mane as he can't force his way onto the bench, let alone the starting 11, and Alderweirald has such a poor
injury record that he's missed 50% of games injured - did we even give him a medical?
I wouldn't blow too much smoke up the arses of the board. £10m seems to have been wasted on Mane as he can't force his way onto the bench, let alone the starting 11, and Alderweirald has such a
poor injury record that he's missed 50% of games injured - did we even give him a medical?
You forgot the windy sarcastic thingy at the end there.
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Existence of secondary solutions to a generalized Taylor problem
In this paper the existence of secondary solutions to a generalization of the classical Taylor problem is considered. A viscous liquid is assumed to occupy the region interior to a right circular
cylinder and exterior to a surface formed by rotating a smooth, positive, periodic function about the axis of the cylinder. The cylinder is fixed while the inner surface rotates with a constant
angular velocity. The existence of axisymmetric cellular solutions is established by a generalization of the method of Lyapunov and Schmidt. By treating the branching equation as a function of three
complex variables it is shown that a critical Re number exists and that for Re numbers less than this critical value, the problem has a unique solution, while for Re numbers slightly above this
value, positive and small there are three solutions.
Applicable Analysis
Pub Date:
July 1974
□ Boundary Value Problems;
□ Circular Cylinders;
□ Existence Theorems;
□ Navier-Stokes Equation;
□ Rotating Cylinders;
□ Viscous Flow;
□ Angular Velocity;
□ Complex Variables;
□ Liquid Flow;
□ Operators (Mathematics);
□ Periodic Functions;
□ Reynolds Number;
□ Rotating Fluids;
□ Fluid Mechanics and Heat Transfer
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Minimum Cost using Dijkstra by reducing the cost of an Edge
Dijkstra's algorithm is for finding the shortest distance, or path, between a starting node to a target node in a weighted graph. Dijkstra's algorithm uses the weights of the edges to find the path
that minimizes the overall distance between the source node and all other nodes. This is also known as the single-source shortest path algorithm. It is BFS(Breadth-First Search) using Priority Queue.
This blog will discuss one such problem involving Djikstra’s algorithm: minimum cost using Dijkstra by reducing the cost of an edge.
Problem Statement
You are given an undirected graph containing ‘N’ nodes and ‘M’ edges, of the form {X, Y, Z}, such that ‘X’ and ‘Y’ are connected with an edge having a cost Z. Your task is to find the
minimum cost to go from a source node 1 to a destination node N if you are allowed to reduce the cost of only one path during the traversal by 2.
Let us try to understand this with the help of simple examples.
M = 4
Edges = {{1, 2, 2}, {2, 3, 1}, {1, 3, 9}, {2, 1, 6}}
The minimum cost is given by 2/2 + 1 = 1 + 1 = 2
M = 3
Edges = {{2, 3, 1}, {1, 3, 6}, {2, 1, 5}}
The minimum cost is given by 6/2 = 3
The basic idea is to consider every edge and check whether reducing its cost minimizes the overall cost or not. So to do this, we will break the path between the source node to the destination node
into 2 parts. The first path will be from the source node 1 to any vertex, say U, (1 to U), and the second path will be from destination node N to any vertex, say V, (N to V) for all U and V.
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How to Calculate 3 Sigma Control Limits for SPC - Latest QualityHow to Calculate 3 Sigma Control Limits for SPC
How to Calculate 3 Sigma Control Limits for SPC
3 sigma control limits is used to check data from a process and if it is within statistical control. Thisis is done by checking if data points are within three standard deviations from the mean.
When we talk of statistical control using 3 sigma control limits, we use the three sigma limits to set the control limits (Lower and Upper) using statistical charts such as for example Microsoft
Excel. These control charts help us establish limits for business processes that require statistical control for the operations.
The three sigma quality system is based on analysis and statistical process control (SPC). Three sigma statistical process control methods enable business process to be manageable and stable.
In statistical process control, there is a upper control limit (UCL) and a lower control limit (LCL) set. The UCL is set three sigma levels above the mean and the LCL is set at three sigma levels
below mean. Since around 99.99 percent of a controlled process will take place within plus or minus three sigmas, the data from a process ought to approximate a general distribution around the mean
and within the pre-defined limits.
Example: You are setting up a statistical process control for your oil change company to determine if there is an unacceptable disparity between workers. Your company’s stated goal is to change the
oil in anyone’s automobile within 15 minutes. You decide to set up Xbar – R charts. Your plan is to sample 20 days and record the performance of 5 randomly selected workers.
Calculating 3 Sigma Control Limits
Using the information below, calculate the proper control charts limits.
Control limits for the X-bar Chart.
UCL= x̅̅ + A2 (R̅)
LCL = x̅̅ – A2 (R̅)
Control limits for the R-chart.
UCL = D4 (R̅)
LCL = D3 (R̅)
Grand mean (for mean of Xbars) = 15.11
R-bar (mean of Ranges) = 6.4
D3 = 0
D4 =2.114
A2 = 0.577
Lets review the 6 tasks below and how to solve them
a. Calculate the upper control limit for the X-bar Chart
b. Calculate the lower control limit for the X-bar Chart
c. Calculate the upper control limit for the R-chart
d. Calculate the lower control limit for the R-chart
e. If your data collection for the X-bar is 17.2, would the process be considered in or out of control?
f. If your data collection for the R-bar is 13.98, would the process be considered in or out of control?
Now, let us calculate the X-bar Chart limits from problem (a) & (b)
a. LCL: x̅̅ – A2 (R̅) = 15.11 – (0.577 x 6.4) = 11.42
b. UCL: x̅̅ + A2 (R̅) = 15.11 – (0.577 x 6.4) = 18.8
You can see that in the middle between these two numbers you have the average of 15.11. Now, this is for the X-bar Chart.
Let us calculate for the UCL and LCL for the R-chart in problem (c) & (d)
c. UCL = D4 (R̅) = 2.114 x 6.4 = 13.53.
d. LCL = D3 (R̅) = 0 x 6.4 = 0
So now, these are our upper and lower control limits for the range (the variations in this process).
To answer the Question (e): since 17.2 is within our calculation of the X-bar collection yields (11.42 – 18.8), therefore we would say that 17.2 means the process is in control which is a common
cause and not special cause variations.
Now for the final Question (f): Since 13.98 is outside our calculation of the R-bar control limits (0 – 13.53), therefore we would say that 13.98 means the process is out of control and requires
Conclusion of 3 sigma control limits
There are some reasons why companies use SPC. Often someone within the organization initiates the use of control charts and other SPC techniques to reduce variation and to improve manufacturing
processes. Many organizations implement SPC to satisfy customer requirements or to meet certification requirements.
SPC is applicable in a wide range of organizations and applications, including non-manufacturing. Control charts can be used for far more than just checking the status of a process; they are also
used as an investigative monitoring tool to bring and test ideas to find solutions to problems in the operations.
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Convert ozt to dwt (Troy ounce to Pennyweight) - Pyron Converter
Result: Troy Ounce = Pennyweight
i.e. ozt = dwt
What does Troy Ounce mean?
A troy ounce is a unit of mass. Troy ounces are very often used as a unit to measure gold. It is heavier than the regular ounce (oz) unit. one troy ounce is equal to 1.09714 ounces.
1ozt = 1.09714 oz
In this unit converter website, we have converter from Troy Ounce (ozt) to some other Mass unit.
What does Pennyweight mean?
A pennyweight is a unit of mass. Thy symbol of pennyweight is dwt.
In this unit converter website, we have converter from Pennyweight (dwt) to some other Mass unit.
What does Mass mean?
A basic theoretical concept of mass in physics. Mass is a basic property of an object that measures the impedance of the acceleration created on the object by the application of force. Newtonian
mechanics relates the force and acceleration of a mass of matter. The practical concept of mass is the weight of the object. So, the total amount of matter is called mass.
The mass of the object never changes. But the weight of the same object may be different due to positional reasons because the weight is the result of gravity. So even if the mass of the object
is variable, its weight will be different in the center of the earth, on the surface of the earth, and in space.
Mass is a physical quantity that does not change with the change of position of the object above the surface of the earth. An astronaut with a mass of 75 kg will have a mass of 75 kg on the moon
or in the orbit of the earth or the moon. No matter how much space the astronaut is made of, its mass remains unchanged everywhere.
Mass level:
The mass level is [M]
Unit of Mass: :
The unit of mass in the international system is the kilogram (kg), and the unit of mass in the C.G.S method is the gram (g), and the unit of mass in the British system is the pound.
Characteristic of Mass:
The mass characteristic depends on an object. The mass does not change even if an object is taken to the earth or anywhere in the universe. Moreover, the mass of an object does not depend
on anything, such as motion, temperature, magnetism, electric current, etc. So it is said that mass is the feature of the object.
Thanks for reading the article! Hope that this article helps to understand about mass and how we measure mass from an object.
How to convert Troy Ounce to Pennyweight : Detailed Description
Troy Ounce (ozt) and Pennyweight (dwt) are both units of Mass. On this page, we provide a handy tool for converting between ozt and dwt. To perform the conversion from ozt to dwt, follow these two
simple steps:
Steps to solve
Have you ever needed to or wanted to convert Troy Ounce to Pennyweight for anything? It's not hard at all:
Step 1
• Find out how many Pennyweight are in one Troy Ounce. The conversion factor is 20.0 dwt per ozt.
Step 2
• Let's illustrate with an example. If you want to convert 10 Troy Ounce to Pennyweight, follow this formula: 10 ozt x 20.0 dwt per ozt = dwt. So, 10 ozt is equal to dwt.
• To convert any ozt measurement to dwt, use this formula: ozt = dwt x 20.0. The Mass in Troy Ounce is equal to the Pennyweight multiplied by 20.0. With these simple steps, you can easily and
accurately convert Mass measurements between ozt and dwt using our tool at Pyron Converter.
FAQ regarding the conversion between ozt and dwt
Question: How many Pennyweight are there in 1 Troy Ounce ?
Answer: There are 20.0 Pennyweight in 1 Troy Ounce. To convert from ozt to dwt, multiply your figure by 20.0 (or divide by 0.05).
Question: How many Troy Ounce are there in 1 dwt ?
Answer: There are 0.05 Troy Ounce in 1 Pennyweight. To convert from dwt to ozt, multiply your figure by 0.05 (or divide by 20.0).
Question: What is 1 ozt equal to in dwt ?
Answer: 1 ozt (Troy Ounce) is equal to 20.0 in dwt (Pennyweight).
Question: What is the difference between ozt and dwt ?
Answer: 1 ozt is equal to 20.0 in dwt. That means that ozt is more than a 20.0 times bigger unit of Mass than dwt. To calculate ozt from dwt, you only need to divide the dwt Mass value by 20.0.
Question: What does 5 ozt mean ?
Answer: As one ozt (Troy Ounce) equals 20.0 dwt, therefore, 5 ozt means dwt of Mass.
Question: How do you convert the ozt to dwt ?
Answer: If we multiply the ozt value by 20.0, we will get the dwt amount i.e; 1 ozt = 20.0 dwt.
Question: How much dwt is the ozt ?
Answer: 1 Troy Ounce equals 20.0 dwt i.e; 1 Troy Ounce = 20.0 dwt.
Question: Are ozt and dwt the same ?
Answer: No. The ozt is a bigger unit. The ozt unit is 20.0 times bigger than the dwt unit.
Question: How many ozt is one dwt ?
Answer: One dwt equals 0.05 ozt i.e. 1 dwt = 0.05 ozt.
Question: How do you convert dwt to ozt ?
Answer: If we multiply the dwt value by 0.05, we will get the ozt amount i.e; 1 dwt = 0.05 Troy Ounce.
Question: What is the dwt value of one Troy Ounce ?
Answer: 1 Troy Ounce to dwt = 20.0.
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Effective Interest
What Is an Interest Rate?
The simplest way to think of an interest rate is the amount that someone will charge you for borrowing a sum of money. So for instance if you borrow $100 and they want it back with an extra $10 then
they have charged you 10% interest; or have they? There are actually many different ways to calculate and look at interest rates (as well as periodic payments) with regards the period over which you
will take the loan and in comparison to the purchasing power of your money.
Within finance therefore, you have to be able to fully understand the many different ways that you could define interest rate such as the effective interest rate method. You also need to understand
how and why those different methods exist and also how banks and other bodies use these different rates to their advantage.
What Are the Different Types of Interest Rates That You Could Use?
Before we get into a more detailed discussion about the effective interest rate and its calculation you need to know what the other methods are and how and why they differ. The following are commonly
used methods for calculating and stating an interest rate:
The nominal interest rate
This is the simplest form of interest rate to understand and is also known as the coupon rate as it normally stamped on the coupons that were then redeemed by bondholders. It is simply the amount
that the borrower will pay the lender for the use of their money. So if the nominal rate is 5% and they borrow $100 they will need to repay the total sum of $105.
This is the simplest way to consider interest and it is how most people believe that any interest that they pay for a loan is calculated. However, generally there are other calculations used by the
banks and other lending institutions.
What is profitability index? How to calculate it? Visit this page and ge the anwer!
The real interest rate
The nominal interest rate makes no allowances to what is happening with the purchasing power of your money. Inflation will reduce the purchasing power of the interest that you are earning and as such
the real amount of interest that you are earning may be much lower than you think as it is being reduced by the current interest rates.
For instance if you have a bond that has an interest rate of 7% but inflation is running at 4% then the real interest rate is just 3% after the effect of inflation has been factored into the
equation. Of course if there is actually a period of deflation then this can work in your favor and actually improve the real return on an investment. The real interest rate is simply: Real Interest
Rate = The Nominal Interest Rate – Inflation Rate
Effective interest rate
Defining the interest rate that is being paid also will depend on how you make your term definition and how often the interest is calculated. For instance if you add interest to the principle amount
each month and then have to pay interest on that additional amount in the following month then the effective of the interest rate is compounded.
The effective interest rate therefore takes into account compounding. This will often be listed as the AER (Annual Equivalent Rate) and takes into account that each interest payment will be based on
a slightly higher balance each time. The more compounding periods that you have then the greater the effect that this will have on the interest that you pay.
Annual percentage rate (APR)
Because of the many ways that an interest rate can be listed and the confusion that this can cause banks and other institutions must by law display the APR or Annual Percentage Rate. This allows any
consumer to be able to compare what is being offered to them like for like across the different products.
This is very important when you consider how interest rates do work. For instance, a credit card that charges you 2% interest each month may sound like a good deal if you compare it to another that
has a rate of 18% annually. But when you compound that interest you are looking at around 27% APR; so not such a great deal after all. The Effective APR has to provide you with the actual rate that
you will pay annually as well as taking into account any additional charges that may be made on the loan.
Effective Interest Method Formula and Calculations
Calculating the effective interest rate is not as hard as it may appear as long as you use the calculation correctly. The following is the calculation that you need to use:
r = (1 + (i / n))n – 1
r = The effective rate
i = The nominal interest rate
n = The number of terms
Effective interest rate example; if a loan states that it has a nominal rate of 6% and that interest is compounded monthly then the calculation will be:
r = (1 + (6% / 12))12 – 1
r = (1 + 0.005)12 – 1
r = 0.0617 0r 6.17%
So although the nominal rate may be 6% you will actually pay an effective interest rate of 6.17%.
The following table gives you a few more examples of effective interest rates depending on the nominal rates used and the rate at which the interest is compounded:
│Nominal Rate │Quarterly │Monthly│ Daily │
│ 5% │ 5.095% │5.116$ │5.127% │
│ 10% │ 10.381% │10.471%│10.516%│
│ 15% │ 15.865% │16.075%│16.180%│
│ 20% │ 21.551% │21.939%│22.134%│
This covers the basic calculations for your effective interest rate. Things can however become more complicated when calculating effective interest rate method deferred financing costs and other
issues that you may be asked for as you advance through your studies.
Issues with Doing Your Effective Interest Rate Assignments
Your assignment grades are often going to impact on your final grades so if you want to pass your course with the right grades and learn what you need for your future then you have to put in the work
to get your assignments completed perfectly. Getting those great grades however is not always easy and there are many issues that students face when trying to do their homework on interest rates. The
following are some of these issues:
• Not fully understanding what calculation that you need to use; always ensure that you fully understand precisely which type of interest rate calculation you are being asked for. If it is not
clear whether you are being asked for effective or nominal then clarify with your tutor.
• Using the wrong formula; there is a huge amount of confusion between many websites online in areas such as finance and some sites do label formula incorrectly or state them wrongly. Where you can
use what you have been shown in class or within your text book to ensure that you get the right one.
• Making calculations incorrectly; with lengthy formula it is easy to conduct the calculations in the wrong order and thus come up with an incorrect answer. You should always double check your
answer. The easiest way to do this is to use one of the many online calculators that offer services to work out your effective and other interest rates.
• Submitting work that contains errors; not only should you always double check calculations you must always ensure that your writing is error free. Always proofread your work so that you avoid
submitting assignments containing writing issues.
Online Tools to Calculate Effective Interest Rates
The best way to check your own answers is to use an online calculator. There are many online and each will be able to calculate your effective rate quickly and accurately allowing you to double check
your own answers. The following are a selection of sites that you can use:
As with your formulas it is important that you select the right tool to make your calculation. Do not just simply throw your assignment into these tools, work out your answers the way that is
expected of you and show your calculations so that you fully understand how to solve equation problems in this area. Then use the tool to confirm that you have the right answer.
We Can Help with Your Effective Interest Rate Assignment
From selecting the right effective interest method formula to making your calculations correctly there are a number of areas in which you could wrong with your assignments. Our specialized finance
homework help will pair you with a true finance expert that holds a higher level degree and has been tutoring for as many as 20 years in this area. They will work with you to provide you with the
support that you need to provide correct answers to the work that you have been set as well as boosting your understanding.
All of our support is covered by a full satisfaction money back guarantee and delivered to you on time in the format that you require. Our experts show you full workings for all calculations and
assignments are double checked and proofread.
If you want to learn the effective interest rate method effectively and submit assignments of the highest standard just get in touch with our highly specialized experts here today.
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Pandas - Check If Category is Ordered - Data Science Parichay
In this tutorial, we will look at how to check if a pandas series with category dtype is ordered or not with the help of some examples.
Ordered and unordered categorical values
A categorical field may or may not be ordered. For example, gender values “M” and “F” do not have an order to them and can be considered as an unordered categorical field.
Some categorical fields on the other hand are ordered. For example, t-shirt sizes, S, M, L, and XL. These values are categorical but they also have an order to them, S < M < L < XL.
The categorical data type in pandas is used to store categorical data. It also allows you to specify an order to the values (if any).
How to check if a pandas categorical data is ordered or not?
You can use the ordered property of a Pandas categorical data to check if the categories in the data are ordered or not. The following is the syntax –
# s is pandas series with category dtype
It returns a boolean value representing whether the given categorical data is ordered or not.
Let’s look at some examples of checking for order in Pandas categorical data.
import pandas as pd
# create a pandas series with category dtype
shirt_size = pd.Series(["M", "S", "L", "M", "XL"], dtype="category")
# check if the category is ordered
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Here, we create a Pandas series with category dtype. We then check if it’s ordered or not by accessing its ordered property. You can see that we get False as the output. The category is unordered by
default since we didn’t specify it to be ordered during creation.
Let’s now look at another example. Here let’s create a pandas category that is ordered using the CategoricalDtype and then check whether it’s ordered or not.
from pandas.api.types import CategoricalDtype
# create an ordered category type for shirt size
cat_type = CategoricalDtype(categories=["S", "M", "L", "XL"], ordered=True)
# create a pandas series of shirt sizes
shirt_size = pd.Series(["M", "S", "L", "M", "XL"])
# change the series type to the custom category type
shirt_size = shirt_size.astype(cat_type)
# check if category is ordered
You can see that we get True as the output because the category is ordered. If you display the series, you can see the category values and their order.
# display the series
0 M
1 S
2 L
3 M
4 XL
dtype: category
Categories (4, object): ['S' < 'M' < 'L' < 'XL']
We get the order ‘S’ < ‘M’ < ‘L’ < ‘XL’ of the categories in the series.
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