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Three-Way Interaction in R | Part 4 | The Data Hall
Three-Way Interaction in R | Part 4
In the earlier parts (part 1, part 2, and part 3) of this series, we have seen, how certain variables can change the outcomes of our regression model. We thoroughly explored such phenomena from basic
(intercept dummies) to the two-way interactions. Likewise, three different variables (continuous or categorical) can also simultaneously impact our dependent or outcome variable. The last Part of
this series aims to explain when we have two categorical or continuous variables that act as moderating variables. In the literature, we often call it, the three-way interaction.
We start the last part by executing the following commands which are used for installing and loading the mandatory libraries:
install.packages(c("ggplot2"," viridis "," cowplot ")) library(ggplot2) library(viridis) library(cowplot)
To load the data, execute the following commands again:
set.seed(123) data <- data.frame(id = 10001:10500, age = sample(20:45, 500, replace = TRUE), marks = sample(20:100, 500, replace = TRUE), salary = sample(1000:5000, 500, replace = TRUE), gender = sample(c('Male', 'Female'), 500, replace = TRUE), education = sample (8:18, 500, replace = TRUE))
In the above command, an important thing to note is, that we have generated a sample of 500 data points for the [education] variable from a range between 8 and 18 years of education. In previous
articles [education] was used as a categorical variable however, this time we have created a continuous variable.
To see and familiarise yourself with the data, execute the following command:
View (data)
Categorical and Continuous Variables
We start by examining the three-way interaction between multiple variables (categorical (gender) and continuous). For the interaction execute the following command
three_way_model <- lm (salary ~ education * gender * age , data = data) summary(three_way_model)
The last above command is used to see the results of the model. The second last command regresses [salary] on [education]. It also explains the moderating role played by [gender] [education], and
In the above figure, the first three coefficients show the main effect, while the last four coefficients show the moderation [:] role. An important thing; the aim of this article, is the difference
between two-way and three-way interaction. [genderMale:age:education] shows three-way interaction while the rest of the interactions are two-way interactions.
Note: we have used the shortcut way; using the[*] sign. Another way of doing this is to use the moderation sign [:] in the command. See part 3 for details.
Interpretation of dummy regression is often tricky due to the dropping of multiple categories (to avoid the dummy trap; multicollinearity). Here, in the above figure, the interaction effect suggests
that, as compared to [female] being [male] could increase the [salary], keeping other things constant.
Another easy way of doing this exercise is to plot the results of the regression model. Here we will be focusing on plotting our model using a graph by executing the following commands:
We use the following command to fit our regression model:
age_intervals <- seq(min(data$age), max(data$age), length.out = 5) education_intervals <- seq(min(data$education), max(data$education), length.out = 5)
When we have a large number of observations. Plotting all of them at once makes the graph messy (as shown in part 3). So, using the above command, we specified [education] in intervals; and created
intervals for the [age] and [education] variables based on the minimum and maximum values.
To predict salaries for both genders for each age and education interval
predicted_salaries <- NULL for (age in age_intervals) { for (education in education_intervals) { predicted_salaries <- rbind(predicted_salaries, data.frame(age = age, education = education,gender = "Male", predicted_salary = predict(three_way_model, newdata = data.frame(gender = "Male", age = age, education = education))), data.frame(age = age, education = education, gender = "Female", predicted_salary = predict(three_way_model, newdata = data.frame(gender = "Female", age = age, education = education))))}}
The above list of commands aims to predict salary based on [gender], [age], and [education] and their interaction. They create a dataset [predicted_salaries] by predicting salaries for different
combinations of [age], [education], and [gender]. For each combination of [age] and [education], it predicts [salary] separately for [males] and [females].
To plot our results, execute the following command:
ggplot(predicted_salaries, aes(x = age, y = predicted_salary, color = factor(education))) + geom_line() +labs(title= "THREE-WAY-INTERACTION", x = "Age", y = "Predicted Salary", color = "Education") + facet_wrap(~gender, scales = "free")
This command plots our results using using [ggplot2] library. Here, the multiple lines represent different levels of [education] (as shown in the multicolour legend). Any label can be given to the
axis and any variable can be placed on any axis. Moreover, a title for the plot can be used in the above command. For instance, here [x] represents the horizontal axis. we used/named our horizontal
axis as [Age].
The above figure states that initially (at age 20), a person with [8 years of education] on average earns less than a person with [18 years of education] (keeping others constant). However, as [age]
increases, the increase in a person’s [salary] having [8 years of education] is more swift as compared to a person having [18 years of education].
So far, we have analysed the three-way interaction effect among continuous and categorical variables. However, multiple continuous variables can also impact our outcome variable, both individually
and simultaneously. To explore this phenomenon, execute the following command:
cont_three_way_model <- lm(salary ~ marks * age * education, data = data)
To define age, education, and marks intervals for prediction
age_intervals <- seq(min(data$age), max(data$age), length.out = 2) education_intervals <- seq(min(data$education), max(data$education), length.out = 2) marks_intervals <- seq(min(data$marks), max(data$marks), length.out = 2) predicted_salaries <- expand.grid(age = age_intervals, education = education_intervals, marks = marks_intervals) predicted_salaries$predicted_salary <- predict(cont_three_way_model, newdata = predicted_salaries) predicted_salaries$combined <- paste(predicted_salaries$education, predicted_salaries$marks, sep = "_")
the above commands are used to combine values from the [education] and [marks] and predict salaries; [predicted_salaries]. To create a unique identifier for each combination of education and marks
[predicted_salaries$combined] is used.
To plot the results
ggplot(predicted_salaries, aes( x = age, y = predicted_salary, color = combined)) + geom_line() + labs (title = "THREE-WAY-INTERACTION (Continous Variables)", x = "Age", y = "Predicted Salary", color = "Education & Marks") + scale_color_manual(values = viridis_pal(option = "viridis")(length(unique(predicted_salaries$combined))))
Here is the results of the final command:
The interpretation of the above figure is as [Age] increases the salary of a person with [18 years of education] and having [100 marks] decreases, while the [salary] a person with [8 years of
education] and having [100 marks] salary increases
In this comprehensive series, we assessed the impact of categorical and continuous variables on predicting salary levels. Furthermore, we have explored, how the relationship between variables can be
influenced by different interactions by presenting both graphical and tabular representations of our models. In this part have seen how three variables collectively play the role of moderation in our
Thanks for always visiting thedatahall.com. Stay tuned for more insightful tutorials.
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Cement Calculator Bags | Accurate Calculation Tool
This tool calculates the number of cement bags you need for your project.
How to Use the Cement Bag Calculator
To use the cement bag calculator, please input the following information:
• Length: The length of the area to be filled with cement in meters.
• Width: The width of the area to be filled with cement in meters.
• Depth: The depth of the area to be filled with cement in meters.
• Mix Ratio: The mix ratio of the cement, represented as 1:X (where X is user-defined).
• Bag Size: The weight of one cement bag in kilograms.
Click the “Calculate” button to find out the number of cement bags required for the given area and mix ratio.
How It Calculates the Results
The calculator uses the following steps to compute the number of cement bags needed:
1. Calculate the volume of the area using the formula: Volume = Length x Width x Depth
2. Determine the total quantity of cement required by dividing the volume by the mix ratio.
3. Convert the required quantity of cement to bags based on the weight of one cement bag.
4. Output the total number of bags needed.
Please note the following limitations of the calculator:
• It does not account for wastage or variability in the mix ratio beyond user input.
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An Introduction to Fraction Arithmetic
An Introduction to Fraction arithmetic
By Robert O
Fraction, which means breaking in Latin, is a way of representing a part or several parts of a whole unit. Fractions are noted or written down with two numbers, one at the top (aka numerator) and the
other at the bottom (denominator), separated by a bar.
For example,
is a fraction read as “three over seven.” 3 is the numerator and 7 is the denominator.
Types of fractions
Fractions take three forms namely proper (numerator<denominator), improper (numerator>denominator), and mixed fractions (consist of two parts: a whole number and a proper fraction).
Proper fraction: When the numerator is less than the denominator. 5/7 is an example of a proper fraction. In general, A/B is a proper fraction if and only if A<B.
When you reverse a proper fraction, you get an improper fraction, where the numerator value is always greater than the denominator value. For example, A/B is an improper fraction if and only if A>B.
Take 7/5 as an example, where 7>5, and hence an improper fraction.
Since the numerator value is greater than the denominator value, you can convert an improper fraction into a whole number and a proper fraction. What you get now is a mixed fraction in the form of
where A is the whole number part and b is always less than c (b<c). Here is an illustration:
Operation with Fractions
The three different fraction types are subject to basic math operations such as addition, subtraction, multiplication and division. But before we get into that, let us first discuss the
simplification of fractions. Every fraction must be expressed in its simplest form unless otherwise stated.
Here are a few examples of simplified fractions:
All improper fractions require conversion to mixed fractions. I will explain to you with the following examples:
You convert an improper fraction to a mixed fraction by dividing the numerator by the denominator. The resulting figure is the whole number and the quotient becomes the new numerator. NOTE: The
denominator remains the same.
Addition and Subtraction of Fractions
The operation is performed the same way it is done on whole numbers. The only difference is that our fractions have to be on a common base, that is, finding the LCM for the numerator values if they
are not equal for the fractions to be added or subtracted.
How to add and subtract fractions with the same denominator values?
In general, if you have two fractions A and B where A = a / b and B = c / b, then A+B is given by:
Follow the same procedure when subtracting fractions with a common denominator. For example:
How to add and subtract fractions with different denominator values?
In this case, you will have to find a common base i.e. the least common multiple (LCM) of the denominators of all the given fractions. For example:
Note: Convert any mixed fractions to improper fractions before carrying out any operation. You can do that by getting the product of the whole number and the denominator and then adding the numerator
to it. The result becomes the new numerator while the denominator remains the same.
For example:
You can apply the same procedure for subtraction operations.
Multiplication and Division of Fractions
How to Multiply Fractions?
Multiplication of fractions is even more direct than the other operations that we have seen thus far. Let us use the example below to illustrate it.
The following are the steps for multiplying any fractions:
Change any mixed fraction into an improper fraction
Compute the product of all the numerator and do the same for the denominators. You may arrive at an improper or proper fraction.
Simplify proper fractions by dividing both numerator and denominator by a common factor or convert improper fractions to mixed fractions
a common factor for 15 and 20 is 5.
Note: You can simplify the process by cross-cancellation if there is a common factor. For example:
How to Divide Fractions?
If you know how to multiply a fraction, division of fractions becomes easy. You multiply the numerator of the first fraction by the denominator of the second fraction by a process called cross
multiplication. The steps are as follows:
Perform mixed fractions conversion to improper fractions
Find the reciprocal of the fraction to the right and then convert the division sign into multiplication sign.
Compute the product of all the numerator and do the same for the denominators. You may arrive at an improper or proper fraction.
Simplify proper fractions or convert improper fractions to mixed fractions if possible
That concludes the operation with fractions. What is left is the squares and square roots of fractions, but that is a topic for another day.
About the Author
This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical engineering and electronic engineering. He is a career teacher and headed the department of
languages and assumed various leadership roles. He writes for Full Potential Learning Academy.
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Competitive Programming in C++ Insights - Introduction
Advanced optimization strategies for competitive programming in C++20 focus on using macros, lambdas, and templates to improve efficiency and minimize errors. Techniques like dynamic data
manipulation and compile-time evaluation streamline coding for high-performance contests. By applying these methods, programmers can write cleaner, faster code, making complex algorithms easier to
implement during competition.
C++ remains one of the most popular languages in competitive programming due to its performance, flexibility, and rich standard library. However, knowledge of efficient algorithms is as important,if
not more so, than the programming language itself. Mastering efficient algorithms and optimized techniques is crucial for success in programming contests, where solving complex problems under strict
time and memory constraints is the norm. in that same vein, understanding the nuances of algorithmic complexity can dramatically affect both performance and outcomes.
This work explores advanced algorithmic strategies, optimization techniques, and data structures that help tackle a wide range of computational challenges effectively. By optimizing input/output
operations, leveraging modern C++ features, and utilizing efficient algorithms, we’ll see how to improve problem-solving skills in competitive programming. We’ll be pushing the boundaries of what can
be achieved with minimal resources and tight constraints.
The scene of this particular whodunnit may seem unlikely, but input/output operations often become the hidden bottleneck in performance-critical applications. By default, C++ performs synchronized I/
O with C’s standard I/O libraries, which can be slower. A simple trick to improve I/O speed is disabling this synchronization:
This small change can make a significant difference when dealing with large input datasets. Throughout this guide, we will explore similar techniques, focusing on the efficient use of data structures
like arrays and vectors, as well as modern C++20 features that streamline your code. You will learn how to apply optimizations that minimize overhead and effectively leverage STL containers and
algorithms to boost both runtime performance and code readability. From handling large-scale data processing to optimizing solutions under tight time constraints, these strategies will prepare you to
excel in programming contests.
Besides C++20, we will study the most efficient algorithms for each of the common problems found in competitions held by organizations like USACO and ICPC.
In this goal, processing large arrays or sequences quickly and efficiently is essential in competitive programming. Techniques in array manipulation are tailored to handle millions of operations with
minimal overhead, maximizing performance. In graph algorithms, the focus shifts to implementing complex traversals and finding shortest paths, addressing problems from route optimization to network
connectivity. These methods require precise control over nodes and edges, whether working with directed or undirected graphs.
String processing handles tasks like matching, parsing, and manipulating sequences of characters. It requires carefully crafted algorithms to search, replace, and transform strings, often using
specialized data structures like suffix trees to maintain performance. Data structures go beyond basic types, introducing advanced forms such as segment trees and Fenwick trees. These are essential
for managing and querying data ranges quickly, especially in scenarios where direct computation is too slow.
Computational geometry tackles geometric problems with high precision, calculating intersections, areas, and volumes. The focus is on solving spatial problems using algorithms that respect boundaries
of precision and efficiency, delivering results where accuracy is crucial.
The journey through competitive programming in C++ starts with the basics. It’s not a journey we chose lightly. Ana Flávia Martins dos Santos, Isabella Vanderlinde Berkembrock, and Michele Cristina
Otta inspired it. They succeeded without training. We wanted to build on that. Our goal is not only to present C++20 and a key set of algorithms for competitive programming, but also to inspire them,
and everyone who reads this work or follows my courses, to become better professionals. We hope they master a wide range of computational tools that are rarely covered in traditional courses.
We’ll start with a set of small tricks and tips for typing and writing code to cut down the time spent drafting solutions. In many contests, it’s not enough for your code to be correct and fast; the
time you spend presenting your solutions to the challenges matters too. So, typing less and typing fast is key.
For each problem type, we’ll study the possible algorithms, show the best one based on complexity, and give you a Python pseudocode followed by a full C++20 solution. We won’t fine-tune the Python
code, but it runs and works as a solid base for competitive programming practice in Python. When needed, we’ll break down C++20 methods, functions, operators, and classes, often highlighting them for
focus. There are clear, solid reasons why we picked C++20.
C++ is a powerful tool for competitive programming. It excels in areas like array manipulation, graph algorithms, string processing, advanced data structures, and computational geometry. Its speed
and rich standard library make it ideal for creating efficient solutions in competitive scenarios.
We cover the essentials of looping, from simple for and while loops to modern C++20 techniques like range-based for loops with views and parallel execution policies. The guide also explores key
optimizations: reducing typing overhead, leveraging the Standard Template Library (STL) effectively, and using memory-saving tools like std::span.
Mastering these C++20 techniques prepares you for a wide range of competitive programming challenges. Whether handling large datasets, solving complex problems, or optimizing for speed, these
strategies will help you write fast, efficient code. This knowledge will sharpen your skills, improve your performance in competitions, and deepen your understanding of both C++ and algorithmic
thinking—skills that go beyond the competition.
C++ shows its strength when solving complex problems. In array manipulation, it supports fast algorithms like binary search with $O(\log n)$ time complexity, crucial for quick queries in large
datasets. For graph algorithms, C++ can implement structures like adjacency lists with a space complexity of $O(V + E)$, where $V$ is vertices and $E$ is edges, making it ideal for sparse graphs.
In string processing, C++ handles pattern searching efficiently, using algorithms like KMP (Knuth-Morris-Pratt), which runs in $O(n + m)$, where $n$ is the text length and $m$ is the pattern length.
Advanced data structures, such as segment trees, allow for query and update operations in $O(\log n)$, essential for range queries and frequent updates.
C++ also handles computational geometry well. Algorithms like Graham’s scan find the convex hull with $O(n \log n)$ complexity, demonstrating C++’s efficiency in handling geometric problems.
We are going to cover a lot of ground. From basic techniques to advanced algorithms. But remember, it all started with Ana, Isabella, and Michele. Their success without training showed us what’s
possible. Now, armed with these tools and knowledge, you’re ready to take on any challenge in competitive programming. The code is clean. The algorithms are efficient. The path is clear. Go forth and
1.1 Time and Space Complexity
In this section, we’ll gona take a tour by time and space complexities, looking at how they affect algorithm efficiency, especially in competitive programming, without all mathematics. We’ll break
down loops, recursive algorithms, and how different complexity classes shape performance. We’ll also look at space complexity and memory use, which matter when handling large datasets.
In this section, we’ll take a tour through time and space complexities, looking at how they affect algorithm efficiency, especially in competitive programming, without heavy mathematics. Before
diving into specific examples, let’s visualize how different complexity classes grow with input size. Figure 1.1.A provides a clear picture of how various algorithmic complexities scale:
Figure 1.1.A: Growth comparison of common algorithmic complexities. The graph shows how the number of operations increases with input size for different complexity classes. Notice how O(1) remains
constant, O(log n) grows very slowly, O(n) increases linearly, while O(n²) and O(n³) show dramatically steeper growth curves.
This visualization helps us understand why choosing the right algorithm matters. For small inputs, the differences might seem negligible, but as the input size grows, the impact becomes dramatic. A
cubic algorithm (O(n³)) processing an input of size 10 performs 1,000 operations, while a linear algorithm (O(n)) only needs 10 operations for the same input. This difference becomes even more
pronounced with larger inputs, making algorithm selection crucial for competitive programming, where both time and memory constraints are strict.
We’ll break down loops, recursive algorithms, and how different complexity classes shape performance. We’ll also look at space complexity and memory use, which matter when handling large datasets.
One major cause of slow algorithms is having multiple nested loops that run over the input data. The more nested loops there are, the slower the algorithm gets. With $k$ nested loops, the time
complexity rises to $O(n^k)$. Alright, I lied. There is a little bit of math.
For instance, the time complexity of the following code fragment is $O(n)$:
for (int i = 1; i <= n; i++) {
// code
And the time complexity of the following code is $O(n^2)$ due to the nested loops:
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= n; j++) {
// code
While time complexity gets most of the attention, space complexity is just as crucial, especially with large inputs. A loop like the one below has a time complexity of $O(n)$, but if it creates an
array to store values, it also has a space complexity of $O(n)$:
std::vector<int> arr(n);
for (int i = 1; i <= n; i++) {
Don’t worry if you don’t know all the C++ syntax. I mean, I wish you did, but hey, we’ll get there. In competitive programming, excessive memory use can cause your program to exceed memory limits,
though this isn’t very common in competitions. Always keep an eye on space complexity, especially when using arrays, matrices, or other data structures that grow with input size. Manage memory wisely
to avoid crashes and penalties. And, there are penalties.
1.1.2. Order of Growth
Time complexity doesn’t show the exact number of times the code inside a loop runs; it shows how the runtime grows. In these examples, the loop runs $3n$, $n+5$, and $\lfloor n/2 \rfloor$ times, but
all still have a time complexity of $O(n)$:
for (int i = 1; i <= 3*n; i++) {
// code
for (int i = 1; i <= n+5; i++) {
// code
for (int i = 1; i <= n; i += 2) {
// code
This is true because time complexity looks at growth, not exact counts. Big-O ignores constants and small terms since they don’t matter much as input size ($n$) gets large.
Here’s why each example still counts as $O(n)$:
1. $3n$ executions: The loop runs $3n$ times, but the constant $3$ doesn’t change the growth. It’s still linear, so it’s $O(n)$.
2. $n + 5$ executions: The $+5$ is just a fixed number. It’s small next to $n$ when things get big. The main growth is still $n$, so it’s $O(n)$.
3. $\lfloor n/2 \rfloor$ executions: Cutting $n$ in half or any fraction doesn’t change the overall growth rate. It’s still linear, so it’s $O(n)$.
Big-O isn’t disconnected from real execution speed. Constants like the $3$ in $3n$ do affect how fast the code runs, but they aren’t the focus of Big-O. In real terms, an algorithm with $3n$
operations will run slower than one with $n$, but both grow at the same rate—linearly. That’s why they both fall under $O(n)$ notation.
Big-O doesn’t ignore these factors because they don’t matter; it simplifies them. It’s all about the growth rate, not the exact count, because that’s what matters most when inputs get large.
Another example where time complexity is $O(n^2)$:
for (int i = 1; i <= n; i++) {
for (int j = i+1; j <= n; j++) {
// code
When an algorithm has consecutive phases, the overall time complexity is driven by the slowest phase. The phase with the highest complexity usually becomes the bottleneck. For example, the code below
has three phases with time complexities of $O(n)$, $O(n^2)$, and $O(n)$. The slowest phase dominates, making the total time complexity $O(n^2)$:
for (int i = 1; i <= n; i++) {
// phase 1 code
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= n; j++) {
// phase 2 code
for (int i = 1; i <= n; i++) {
// phase 3 code
When looking at algorithms with multiple phases, remember that each phase can also add to memory use. In the example above, if phase 2 creates an $n \times n$ matrix, the space complexity jumps to $O
(n^2)$, matching the time complexity. Sometimes, time complexity depends on more than one factor. This means the formula includes multiple variables. For instance, the time complexity of the code
below is $O(nm)$:
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= m; j++) {
// code
If the algorithm above uses a data structure like an $n \times m$ matrix, the space complexity also becomes $O(n\times m)$. This increases memory usage, especially with large inputs.
The time complexity of a recursive function depends on how often it’s called and the complexity of each call. To understand this better, let’s look at a progression of recursive functions, from
simple to complex:
1. Linear Recursion - Simple Countdown:
void countdown(int n) {
if (n == 0) return;
std::cout << n << " ";
This function makes $n$ calls, each doing $O(1)$ work, resulting in $O(n)$ complexity. It’s a straightforward example where each call leads to exactly one recursive call. Let’s see it a little
bit more careful: Here, each call to the function creates exactly one more call, until n = 0. The Table 1.1.A shows the calls made from a single initial call to countdown(n):
Function Call Number of Calls
countdown(n) 1
countdown(n-1) 1
countdown(n-2) 1
… …
countdown(1) 1
Table 1.1.A - Counting calls in linear recursion.
So, the total time complexity is:
\[1 + 1 + 1 + \cdots + 1 = n = O(n)\]
2. Tail Recursion with Accumulator:
int sum_to_n(int n, int acc = 0) {
if (n == 0) return acc;
return sum_to_n(n-1, acc + n);
The accumulator doesn’t affect the number of calls. Each call creates one recursive call until $n = 0$. The Table 1.1.B shows the pattern:
Function Call Number of Calls Accumulator Value
sum_to_n(n, 0) 1 0
sum_to_n(n-1, n) 1 n
sum_to_n(n-2, n+(n-1)) 1 n+(n-1)
… … …
sum_to_n(1, partial) 1 partial
Table 1.1.B - Analyzing tail recursion with accumulator.
The total time complexity is:
\[1 + 1 + 1 + \cdots + 1 = n = O(n)\]
3. Binary Recursion - Generating Paths:
binary_paths(int n, std::string path = "") {
if (n == 0) {
std::cout << path << "\n";
binary_paths(n-1, path + "0");
binary_paths(n-1, path + "1");
Each call creates two new calls, doubling at each level until $n = 0$. The Table 1.1.C shows this exponential growth:
Function Call Number of Calls Paths Generated
binary_paths(n) 1 Root
binary_paths(n-1) 2 “0”, “1”
binary_paths(n-2) 4 “00”,”01”,”10”,”11”
… … …
binary_paths(0) 2^n All binary strings
Table 1.1.C - Analyzing binary recursive growth.
The total time complexity is:
\[1 + 2 + 4 + \cdots + 2^n = 2^{n+1} - 1 = O(2^n)\]
4. Multiple Recursion - Tribonacci Sequence:
tribonacci(int n) {
if (n <= 1) return 0;
if (n == 2) return 1;
return tribonacci(n-1) + tribonacci(n-2) + tribonacci(n-3);
// First numbers: 0, 0, 1, 1, 2, 4, 7, 13, 24, 44...
// Each number is the sum of the previous 3 numbers
Each call spawns three recursive calls until the base cases. The Table 1.1.D shows the exponential growth:
Function Call Number of Calls Total New Calls
tribonacci(n) 1 3
tribonacci(n-1) 3 9
tribonacci(n-2) 9 27
… … …
tribonacci(≤2) 3^{n-2} Base cases
Table 1.1.D - Analyzing three-way recursive growth.
The total time complexity is:
\[1 + 3 + 9 + \cdots + 3^{n-2} = \frac{3^{n-1} - 1}{2} = O(3^n)\]
These examples illustrate how recursive patterns affect complexity: Single recursion typically leads to linear complexity $O(n)$; Binary recursion often results in exponential complexity $O(2^n)$ e
Multiple recursion can lead to even higher exponential complexity $O(k^n)$, where k is the number of recursive calls
Recursive functions also bring space complexity issues. Each call adds to the call stack, and with deep recursion, like this exponential example, the space complexity can be $O(n)$. Be aware that
recursion depth is limited by the call stack’s size. In C++ and Java, the call stack has a fixed size determined by the system or runtime settings. If too many recursive calls occur, the stack can
overflow, causing the program to terminate. Modern C++ compilers like GCC, Clang and MSVC can optimize tail-recursive calls through tail-call optimization (TCO), but this is not guaranteed and is
generally not implemented in Java. In Python, recursion also has a limit, but it is managed differently. Python raises a RecursionError when the recursion depth exceeds a preset limit (default is
$1,000$ calls). This exception can be caught, providing a safer way to handle deep recursion. However, adjusting the recursion limit with sys.setrecursionlimit() in Python can still lead to a stack
overflow if set too high, as Python’s call stack size remains fixed. Unlike C++ and Java, Python does not support TCO, making deep recursion slower and more memory-intensive.
1.1.3. Common Complexity Classes
Here is a Table 1.1.E of common time complexities of algorithms:
Complexity Description Examples
$O(1)$ Constant time; the execution time remains the same regardless of input size. Accessing an array element, hash lookups.
$O(\log n)$ Logarithmic time; the input size is reduced by half in each step. Binary search, finding largest power of $2$.
$O(\sqrt{n})$ Sub-linear but slower than $O(\log n)$; often arises in problems where input is reduced by its square Trial division for prime checking.
$O(n)$ Linear time; the algorithm processes each element once. Single loop over an array, linear search.
$O(n \log n)$ Log-linear time; typical of algorithms that involve sorting or divide-and-conquer strategies. Mergesort, heapsort, Fast Fourier Transform.
$O(n^2)$ Quadratic time; usually involves two nested loops, processing pairs of elements. Bubble sort, matrix multiplication (naive).
$O(n^3)$ Cubic time; involves three nested loops, processing triples of elements. Floyd-Warshall algorithm, matrix chain multiplication.
$O(2^n)$ Exponential growth; common in recursive algorithms that explore all subsets or configurations. Recursive subset generation, solving the Traveling Salesman Problem (TSP)
$O(n!)$ Factorial time; often seen in algorithms that generate all possible permutations of input. Permutation generation, brute-force TSP.
Table 1.1.E - Common time complexities of algorithms.
The following text offers a more comprehensive explanation of each complexity listed in the table, recognizing that each of us may understand these concepts in different ways. You might find it
helpful to keep both the table and the text on hand for reference.
$O(1)$ - Algorithms with constant time complexity execute in the same amount of time, regardless of input size. No matter how large or small the input is, the execution time remains unchanged.
Examples include accessing an element in an array by its index or checking if a number is even. Constant-time operations are optimal for performance, making them desirable in many algorithmic
$O(\log n)$ - Algorithms with logarithmic time complexity increase their runtime logarithmically as the input size grows. This typically occurs in divide-and-conquer algorithms, where the problem
size is halved at each step. An example is binary search, where a sorted array is repeatedly divided in half until the target element is found. As input size increases, the runtime grows slowly,
making logarithmic algorithms highly efficient for large inputs.
$O(\sqrt{n})$ - $O(\sqrt{n})$ algorithms reduce the input by its square root at each step, which is slower than logarithmic complexity but faster than linear complexity. This complexity often arises
in specific mathematical algorithms, such as trial division for prime checking, where the factors are tested up to the square root of the number. It is efficient for certain computational problems
but less common than other complexities.
$O(n)$ - $O(n)$ algorithms process each element of the input once, making the runtime proportional to the input size. Examples include linear search through an unsorted array or iterating through a
list to calculate its sum. Linear algorithms are generally efficient for moderate input sizes but may become slow as input size grows significantly.
$O(n \log n)$ - $O(n \log n)$ algorithms have a runtime that grows in proportion to the input size multiplied by the logarithm of the input size. This complexity is common in efficient sorting
algorithms like mergesort and heapsort, which combine linear processing with logarithmic division of input. It strikes a balance between scalability and efficiency, making it a standard choice for
practical sorting and divide-and-conquer problems.
$O(n^2)$ - $O(n^2)$ algorithms have a runtime that grows quadratically as the input size increases, often due to two nested loops processing pairs of elements. Examples include bubble sort, insertion
sort, and naive matrix multiplication. As the input size increases, the runtime grows significantly, making quadratic algorithms inefficient for large inputs.
$O(n^3)$ - $O(n^3)$ algorithms involve three nested loops and are often found in problems that require processing triples of elements. For example, the Floyd-Warshall algorithm for finding shortest
paths in a graph has cubic complexity. Similar to quadratic algorithms, cubic ones become inefficient as input size increases, but they are sometimes unavoidable in certain computations.
$O(2^n)$ - $O(2^n)$ algorithms experience exponential growth in runtime as the input size increases, doubling with each additional input element. These algorithms often involve exploring all possible
solutions, as seen in recursive solutions to problems like the Traveling Salesman Problem or generating all subsets of a set. Exponential algorithms become impractical for even moderately large
inputs due to rapid increases in runtime.
$O(n!)$ - $O(n!)$ algorithms generate all possible permutations of the input, resulting in extremely high runtimes. This complexity is common in brute-force solutions to problems like permutation
generation or the Traveling Salesman Problem. Factorial algorithms are the least efficient, with runtime growing rapidly even for small input sizes, making them unsuitable for large-scale
Now that we’ve explored different time complexities, it’s useful to think about how they translate to actual performance. The next section will help you estimate whether an algorithm will be fast
enough, based on input size and typical competition constraints.
1.1.4. Estimating Efficiency
Now that we’ve explored different time complexities, it’s useful to think about how they translate to actual performance. A modern Intel i7 processor can handle around $10^9$ operations per second.
The next section will help you estimate whether an algorithm will be fast enough, based on input size and typical competition constraints. For instance, if an algorithm with $O(n^2)$ complexity
requires $10^{10}$ operations, it would take roughly 10 seconds to run, which is too slow for most programming competitions ^1.
You can also judge what time complexity you need based on the input size. Here’s a quick guide, assuming a one-second time limit, very common for C++ competitions:
Input Size Required Time Complexity
$n \leq 10$ $O(n!)$
$n \leq 20$ $O(2^n)$
$n \leq 500$ $O(n^3)$
$n \leq 5000$ $O(n^2)$
$n \leq 10^6$ $O(n \log n)$ or $O(n)$
$n$ is large $O(1)$ or $O(\log n)$
Table 1.1.F - Theoretical relationship between time complexity and input size.
So, if your input size is $n = 10^5$, you’ll probably need an algorithm with $O(n)$ or $O(n \log n)$ time complexity. This helps you steer clear of approaches that are too slow.
Remember, time complexity is an estimate; it hides constant factors. An $O(n)$ algorithm could do $n/2$ or $5n$ operations, and these constants can change the actual performance.
1.2. Typing Better, Faster, Less
This section gives you practical steps to improve your speed and performance in competitive programming with C++20. C++ is fast and powerful, but using it well takes skill and focus. We will cover
how to type faster, write cleaner code, and manage complexity. The goal is to help you code quicker, make fewer mistakes, and keep your solutions running fast.
Typing matters. The faster you type, the more time you save. Accuracy also counts, mistakes slow you down. Next, we cut down on code size without losing what’s important. Using tools like the
Standard Template Library (STL), you can write less code and keep it clean. This is about direct, simple code that does the job right.
In this section, we won’t confine ourselves to general tricks. We will explore certain nuances and techniques specific to C++20, particularly those related to typing efficiency. Undoubtedly, in other
chapters and sections, we will return to these same topics, but from different angles.From this point forward, the idea of clean and well-structured code will be left behind.
1.2.1. Typing Tips
If you don’t type quickly, you should invest at least two hours per week on the website: https://www.speedcoder.net. Once you have completed the basic course, select the C++ lessons and practice
regularly. Time is crucial in competitive programming, and slow typing can be disastrous.
To expand on this, efficient typing isn’t just about speed; it’s about reducing errors and maintaining a steady flow of code. When you’re in a competitive programming, every second matters.
Correcting frequent typos or having to look at your keyboard will significantly slow down your progress. Touch typing—knowing the layout of the keyboard and typing without looking—becomes a vital
In a typical competitive programming contest, you have to solve several, typically $12$ or $15$, problems within a set time, about five hours. Typing fast lets you spend more time solving problems
instead of struggling to get the code in. But speed means nothing without accuracy. Accurate and fast typing ensures that once you know the solution, you can code it quickly and correctly. Aim for a
typing speed of at least 60 words per minute with high accuracy. Websites like https://www.speedcoder.net let you practice typing code syntax, which helps more than regular typing lessons. Besides
it, learning specific shortcuts in C++ or Python boosts your speed in real coding situations. There are a simple equation for that:
\[\text{Time spent fixing errors } + \text{Time lost from slow typing }\] \[= \text{Lower overall performance in competitive programming}\]
To keep improving your typing in competitive programming, start by using IDE shortcuts. Learn the keyboard shortcuts for your preferred Integrated Development Environment (IDE). Those shortcuts help
you navigate and edit code faster, cutting down the time spent moving between the keyboard and mouse. In ICPC contests, the available IDE’s are, usually, Eclipse or VsCode, so knowing its shortcuts
can boost your efficiency. Always check which IDE will be used in each competition since this may vary. And use it daily while training.
While typing, focus on frequent patterns in your code. Practice typing common elements like loops, if-else conditions, and function declarations. Embedding these patterns into your muscle memory
saves time during contests. The faster you can type these basic structures, the quicker you can move on to solving the actual problem.
To succeed in a C++ programming competition, your first challenge is to type the following code fragment in under two minutes. If you can’t, don’t give up. Just keep practicing. To be the best in the
world at anything, no matter what it is, the only secret is to train and train some more.
#include <iostream>
#include <vector>
#include <span>
#include <string>
#include <algorithm>
#include <random>
// Type aliases
using VI = std::vector<int>;
using IS = std::span<int>;
using STR = std::string;
// Function to double each element in the vector
void processVector(VI& vec) {
std::cout << "Processing vector...\n";
for (int i = 0; i < vec.size(); ++i) {
vec[i] *= 2;
// Function to display elements of a span
void displaySpan(IS sp) {
std::cout << "Displaying span: ";
for (const auto& elem : sp) {
std::cout << elem << " ";
std::cout << "\n";
int main() {
VI numbers;
STR input;
// Input loop: collect numbers from user
std::cout << "Enter integers (type 'done' when finished):\n";
while (true) {
std::cin >> input;
if (input == "done") break;
// Process and display the vector
std::cout << "Processed vector:\n";
int index = 0;
while (index < numbers.size()) {
std::cout << numbers[index] << " ";
std::cout << "\n";
// Shuffle the vector using a random number generator
std::random_device rd;
std::mt19937 gen(rd());
std::shuffle(numbers.begin(), numbers.end(), gen);
std::cout << "Shuffled vector:\n";
for (const auto& num : numbers) {
std::cout << num << " ";
std::cout << "\n";
// Display a span of the first 5 elements (if available)
if (numbers.size() >= 5) {
IS numberSpan(numbers.data(), 5);
// Calculate sum of elements at even indices
int sum = 0;
for (int i = 0; i < numbers.size(); i += 2) {
sum += numbers[i];
std::cout << "Sum of elements at even indices: " << sum << "\n";
return 0;
Code Fragment 1.2.A - Code for self-assessment of your typing speed.
Don’t give up before trying. If you feel your typing speed isn’t enough, don’t stop here. Keep practicing. With each new algorithm, copy it and practice again until typing between $60$ and $80$ words
per minute with an accuracy above $95%$ feels natural.
1.2.2. Typing Less in Competitive Programming
In competitive programming, time is a critical resource, and C++ is a language where you have to type a lot… like, seriously a lot. If anyone finds out I said that, I’ll have to deny it completely,
it’s part of the C++ survival code! Therefore, optimizing typing speed and avoiding repetitive code can make a significant difference. Below, we will discuss strategies to minimize typing when
working with std::vector during competitive programmings, where access to the internet or pre-prepared code snippets may be restricted.
1.2.2.1. Abbreviations
It may not be the cleanest approach, but the first thing that comes to mind… We can use #define to create short aliases for common vector types. This is particularly useful when you need to declare
multiple vectors throughout the code.
#define VI std::vector<int>
#define VVI std::vector<std::vector<int>>
#define VS std::vector<std::string>
With these definitions, declaring vectors becomes much faster:
VI numbers; // std::vector<int> numbers;
VVI matrix; // std::vector<std::vector<int>> matrix;
VS words; // std::vector<std::string> words;
In this book, I’ll use a lot of comments to explain concepts, code, or algorithms. You, on the other hand, won’t use any comments. Not during training, and definitely not during competitions. If you
even think about using one, seek professional advice. There are plenty of psychiatrists available online.
In C++, you can use #define to create macros and short aliases. Macros can define constants or functions at the preprocessor level. Macros can cause problems. They ignore scopes and can lead to
unexpected behavior. So, For functions, macros can be unsafe. They don’t respect types or scopes.
// Macro function
#define SQUARE(x) ((x) * (x))
// Template function
template<typename T>
constexpr T square(T x) {
return x * x;
Macros are processed before compilation. This makes debugging hard. The compiler doesn’t see macros the same way it sees code. With modern C++, you have better tools that the compiler
understands. C++20 offers features like constexpr functions, inline variables, and templates. These replace most uses of macros. They provide type safety and respect scopes. They make code easier
to read and maintain. You can define a constexpr function to compute the square of a number:
constexpr int square(int n) {
return n * n;
If you call square(5) in a context requiring a constant expression, the compiler evaluates it at compile time.
In summary, avoid macros when you can. Use modern C++20 features instead. They make your code safer and clearer.C++20 offers features like constexpr functions, inline variables, and templates.
These replace most uses of macros. They provide type safety and respect scopes. They make code easier to read and maintain. You can define a constexpr function to compute the square of a number:
constexpr int square(int n) {
return n * n;
If you call square(5) in a context requiring a constant expression, the compiler evaluates it at compile time. Let’s explore how to use these features effectively, starting with basic constants
and progressing to advanced compile-time computations:
1. Basic Constant Declaration:
// Old way using macros (avoid)
#define MAX_SIZE 100
// Modern way using const
const int MAX_SIZE = 100;
1. Simple Compiler-time Computation
// Basic constexpr function
constexpr int square(int x) {
return x * x;
// Usage:
constexpr int result = square(5); // Computed at compile-time
1. Conditional Compile-time Logic:
/// constexpr with conditionals
constexpr int max(int a, int b) {
return (a > b) ? a : b;
// This enables compile-time decision making
constexpr int larger = max(10, 20); // Evaluates to 20 at compile-time
1. Recursive Compile-time Computation:
// Recursive constexpr function
constexpr int factorial(int n) {
return (n <= 1) ? 1 : n * factorial(n-1);
// The entire recursion happens at compile-time
constexpr int fact5 = factorial(5); // Evaluates to 120 at compile-time
1. Generic Compile-time Computation:
// Template constexpr combining generics and compile-time evaluation
template<typename T>
constexpr T power(T base, int exp) {
if (exp == 0) return 1;>
if (exp == 1) return base;
return base * power(base, exp-1);
// Works with different types, all at compile-time
constexpr int int_power = power(2, 3); // 8
constexpr double dbl_power = power(2.5, 2); // 6.25
In competitive programming, constexpr can be an advantage or disadvantage. It optimizes code by computing results at compile time, saving processing time during execution. If certain values are
constant, you can precompute them with constexpr. However, many problems have dynamic input provided at runtime, where constexpr cannot help since it cannot compute values that depend on runtime
input. Overall, constexpr is valuable when dealing with static data or fixed input sizes. But in typical ICPC-style competitions, you use it less often because most problems require processing
dynamic input.
A smart way to reduce typing time is by using using to create abbreviations for frequently used vector types.
In many cases, the use of #define can be replaced with more modern and safe C++ constructs like using, typedef, or constexpr. The old #define does not respect scoping rules and does not offer type
checking, which can lead to unintended behavior. Using typedef or using provides better type safety and integrates smoothly with the C++ type system, making the code more predictable and easier to
For example:
#define VI std::vector<int>
#define VVI std::vector<std::vector<int>>
#define VS std::vector<std::string>
Can be replaced with using or typedef to create type aliases:
using VI = std::vector<int>;
using VVI = std::vector<std::vector<int>>;
using VS = std::vector<std::string>;
// Or using typedef (more common in C++98/C++03)
typedef std::vector<int> VI;
typedef std::vector<std::vector<int>> VVI;
typedef std::vector<std::string> VS;
using and typedef are preferred because they respect C++ scoping rules and offer better support for debugging, making the code more secure and readable. nevertheless, there are moments when we need a
constant function.
In C++20, the use of using offers significant advantages over the traditional typedef. The syntax of using is clearer, especially when defining complex types like pointers, templates, and
function types. For instance, using FuncPtr = void(*)(int); is more readable than typedef void(*FuncPtr)(int);, as the type definition aligns more closely with the general C++ syntax.
Additionally, using allows for creating aliases for templates, which is not possible with typedef. This makes defining template-dependent types more flexible and straightforward, simplifying
alias creation like template<typename T> using Vec = std::vector<T>;, enhancing code reusability.
Another benefit of using is that it aligns well with other modern language constructs, such as using namespace, bringing greater consistency to modern C++ code. This helps maintain clarity in
longer and more complex type definitions, making the code easier to read and maintain. Therefore, in C++20 projects, it is advisable to adopt using for type definitions, ensuring cleaner and more
flexible code.
Nevertheless, if you have macros that perform calculations, you can replace them with constexpr functions:
#define SQUARE(x) ((x) * (x))
Can be replaced with:
constexpr int square(int x) {
return x * x;
constexpr functions provide type safety, avoid unexpected side effects, and allow the compiler to evaluate the expression at compile-time, resulting in more efficient and safer code.
In competitive programming, you might think using #define is the fastest way to type less and code faster. But typedef or using are usually more efficient. They avoid issues with macros and integrate
better with the compiler. While reducing variable names or abbreviating functions might save time in a contest, remember that in professional code, clarity and maintainability are more important than
typing speed. So avoid using shortened names and unsafe constructs like #define in production code, libraries, or larger projects.
In C++, you can create aliases for types. This makes your code cleaner. You use typedef or using to do this.
using ull = unsigned long long;
Now, ull is an alias for unsigned long long. You can use it like this:
ull bigNum = 123456789012345ULL;
Numbers need type-specific suffixes like ULL. When you write ull bigNumber = 123456789012345ULL;, the ULL tells the compiler the number is an unsigned long long. Without it, the compiler might
assume a smaller type like int or long, which can’t handle large values. This leads to errors and bugs. The suffix forces the right type, avoiding overflow and keeping your numbers safe. It’s a
simple step but crucial. The right suffix means the right size, no surprises.
In C++, suffixes are also used with floating-point numbers to specify their exact type. The suffix f designates a float, while no suffix indicates a double, and l or L indicates a long double. By
default, the compiler assumes double if no suffix is provided. Using these suffixes is important when you need specific control over the type, such as saving memory with float or gaining extra
precision with long double. The suffix ensures that the number is treated correctly according to your needs.Exact type, exact behavior.
The rule is: know your numbers, suffix your numbers, and be happy.
1.2.2.2. Predefining Common Operations
If you know that certain operations, such as sorting or summing elements, are frequent in a competitive programming or in the algorithm you are going to code, consider defining these operations at
the beginning of the code. The only real reason to use a macro in competitive programming is to predefine functions. For example:
#include <vector>
#include <algorithm>
// Alias for integer vector
using VI = std::vector<int>;
// Alias for the full range of the vector
#define ALL(vec) vec.begin(), vec.end()
// Function to sort the vector using constexpr
constexpr auto sVec = [](VI& vec) {
std::sort(vec.begin(), vec.end());
// Usage:
VI vec = {5, 3, 8, 1};
sVec(vec); // Sorts the vector using the lambda function
// Alternatively, you can still use ALL to simplify the sort:
std::sort(ALL(vec)); // Another way to sort the vector
Code Fragment 1.2.B: Example of using and constexpr to reduce typing time.
Keeping the macro #define ALL(vec) vec.begin(), vec.end() wasn’t madness, it was the competitive programming bug. The C++20 code needed to replace this macro with modern structures requires a lot of
template<typename T>
constexpr auto all(T& container) {
return std::make_pair(container.begin(), container.end());
VI vec = {5, 3, 8, 1};
sort_vector(vec); // Sorts the vector using the lambda
std::sort(all(vec).first, all(vec).second); // Another way to sort using the utility function
In C++, #include brings code from libraries into your program. It lets you use functions and classes defined elsewhere. The line #include <vector> includes the vector library.
Vectors are dynamic arrays. They can change size at runtime. You can add or remove elements as needed. We will know more about vectors and the vector library soon. In early code fragments we saw
some examples of vector initialization. We will travel down this road soon.
The line #include <algorithm> includes the algorithm library. It provides functions to work with data structures. You can sort, search, and manipulate collections.
We can merge <vector> and <algorithm> for efficient data processing. We’ve seen this in previous code examples where we used competitive programming techniques. Without competitive programming
tricks and hacks, the libraries can be combined like this:
#include <vector>
#include <algorithm>
#include <iostream>
int main() {
std::vector<int> numbers = {4, 2, 5, 1, 3};
std::sort(numbers.begin(), numbers.end());
for (int num : numbers) {
std::cout << num << " ";
return 0;
Code 1.2.B: Simple and small program to print a vector
The program in Code 1.2.B, a simple example of professional code, sorts the numbers and prints:
Summing the values contained in a vector, or anther container, is a common problem in competitive programming. For these cases, C++20 offers std::accumulate. In C++, std::accumulate is part of the
<numeric> library and calculates the sum (or other operations) over a range of elements starting from an initial value. Unlike other languages, C++ does not have a built-in sum function, but
std::accumulate serves that purpose. As we can see in the provided fragment:
#define ALL(x) x.begin(), x.end() //that macro again
int sum_vector(const VI& vec) {
return std::accumulate(ALL(vec), 0);
The code std::accumulate(ALL(vec), 0); will be replaced in compilation time for std::accumulate(vec.begin(), vec.end(), 0) witch takes three arguments: the first two define the range to sum
(vec.begin() to vec.end()); the third argument is the initial value, which is $0$, added to the summation result. If std::accumulate is used without a custom function, it defaults to addition,
behaving like a simple summation. To calculate the sum of a vector’s elements:
std::accumulate uses functions to operate on elements. These functions are straightforward. They take two values and return one. Let’s see how they work.
We can begin for Lambda functions, they are unnamed functions, useful for quick operations.
[](int a, int b) { return a + b; }
This lambda adds two numbers. The [] is called the capture list, which specifies which variables from the surrounding scope can be accessed inside the lambda. In this case, it’s empty, meaning no
variables are captured. The (int a, int b) part defines the parameters, while { return a + b; } is the function body that adds the two numbers. For example, we could write:
int sum = std::accumulate(ALL(vec), 0, [](int a, int b) { return a + b; });
This lambda sums all elements in v. Nevertheless, the C++ provides standard functions for common operations in std::accumulate.
int sum = std::accumulate(All(vec), 0, std::plus<>());
std::plus<> adds two numbers. It’s the default for std::accumulate.
int product = std::accumulate(ALL(vec), 1, std::multiplies<>());
std::multiplies<> multiplies two numbers.
Using lambda functions, or not, you can create your own functions.
int max_element = std::accumulate(ALL(vec), v[0],
[](int a, int b) { return std::max(a, b); });
This lambda function finds the largest element in v. In this case, std::accumulate applies the function repeatedly. It starts with an initial value. For each element:
1. It takes the previous result.
2. It takes the current element.
3. It applies the function to both values.
4. The result is used in the next iteration.
This process continues until the sequence ends. Let’s see how std::accumulate sums the squares of numbers:
std::vector<int> vec = {1, 2, 3, 4};
int sum_of_squares = std::accumulate(ALL(vec), 0,
[](int acc, int x) { return acc + x * x; });
The process goes like this:
1. Start: $acc = 0$
2. For $1$: $acc = 0 + 1 * 1 = 1$
3. For $2$: $acc = 1 + 2 * 2 = 5$
4. For $3$: $acc = 5 + 3 * 3 = 14$
5. For $4$: $acc = 14 + 4 * 4 = 30$
The final result is $30$.
1.2.2.3. Using Lambda Functions
Let’s can back to Lambdas.
Starting with C++11, C++ introduced lambda functions. Lambdas are anonymous functions that can be defined exactly where they are needed. If your code needs a simple function used only once, you
should consider using lambdas. Let’s start with a simple example, Code 1.2.B, written without competitive programming tricks.
#include <iostream> // Includes the input/output stream library for console operations
#include <vector> // Includes the vector library for using dynamic arrays
#include <algorithm> // Includes the algorithm library for functions like sort
// Traditional function to sort in descending order
bool compare(int a, int b) {
return a > b; // Returns true if 'a' is greater than 'b', ensuring descending order
int main() {
std::vector<int> numbers = {1, 3, 2, 5, 4}; // Initializes a vector of integers
// Uses the compare function to sort the vector in descending order
std::sort(numbers.begin(), numbers.end(), compare);
// Prints the sorted vector
for (int num : numbers) {
std::cout << num << " "; // Prints each number followed by a space
std::cout << "\n"; // Prints a newline at the end
return 0; // Returns 0, indicating successful execution
Code 1.2.B: Code example to sort a number vector and print it. {: class=”legend”}
The Code 1.2.B sorts a vector of integers in descending order using a traditional comparison function. It begins by including the necessary libraries: <iostream> for input and output, <vector> for
dynamic arrays, and <algorithm> for sorting operations. The compare function is defined to take two integers, returning true if the first integer is greater than the second, setting the sorting order
to descending.
In the main function, a vector named numbers is initialized with the integers {1, 3, 2, 5, 4}. The std::sort function is called on this vector, using the compare function to sort the elements from
highest to lowest. After sorting, a for loop iterates through the vector, printing each number followed by a space. The program ends with a newline to cleanly finish the output. This code is a simple
and direct example of using a custom function to sort data in C++. Now, let’s see the same code using lambda functions and other competitive programming tricks, Code 1.2.C:
#include <iostream> // Includes the input/output stream library for console operations
#include <vector> // Includes the vector library for using dynamic arrays
#include <algorithm> // Includes the algorithm library for functions like sort
#define ALL(vec) vec.begin(), vec.end() // Macro to simplify passing the entire range of a vector
using VI = std::vector<int>; // Alias for vector<int> to simplify code and improve readability
int main() {
VI num = {1, 3, 2, 5, 4}; // Initializes a vector of integers using the alias VI
// Sorts the vector in descending order using a lambda function
std::sort(ALL(num), [](int a, int b) { return a > b; });
// Prints the sorted vector
for (int n : num) {
std::cout << n << " "; // Prints each number followed by a space
std::cout << "\n"; // Prints a newline at the end
return 0; // Returns 0, indicating successful execution
Code 1.2.C: Sort and print vector using lambda functions.
To see the typing time gain, just compare the normal definition of the compare function followed by its usage with the use of the lambda function.
The Code 1.2.C sorts a vector of integers in descending order using a lambda function, a modern and concise way to define operations directly in the place where they are needed. It starts by
including the standard libraries for input/output, dynamic arrays, and algorithms. The macro ALL(vec) is defined to simplify the use of vec.begin(), vec.end(), making the code cleaner and shorter.
An alias VI is used for std::vector<int>, reducing the verbosity when declaring vectors. Inside the main function, a vector named num is initialized with the integers {1, 3, 2, 5, 4}. The std::sort
function is called to sort the vector, using a lambda function [](int a, int b) { return a > b; } that sorts the elements in descending order.
The lambda is defined and used inline, removing the need to declare a separate function like compare. After sorting, a for loop prints each number followed by a space, ending with a newline. This
approach saves time and keeps the code concise, highlighting the effectiveness of lambda functions in simplifying tasks that would otherwise require traditional function definitions.
Lambda functions in C++, introduced in C++11, are anonymous and defined where they are needed. They shine in short, temporary tasks like inline calculations or callbacks. Unlike regular
functions, lambdas can capture variables from their surrounding scope. With C++20, lambdas became even more powerful and flexible, extending their capabilities beyond simple operations.
The general syntax for a lambda function in C++ is as follows:
[capture](parameters) -> return_type { // function body};
□ Capture: Specifies which variables from the surrounding scope can be used inside the lambda. Variables can be captured by value [=] or by reference [&]. You can also specify individual
variables, such as [x] or [&x], to capture them by value or reference, respectively.
□ Parameters: The input parameters for the lambda function, similar to function arguments.
□ Return Type: Optional in most cases, as C++ can infer the return type automatically. However, if the return type is ambiguous or complex, it can be specified explicitly using -> return_type.
□ Body: The actual code to be executed when the lambda is called.
C++20 brought new powers to lambdas. Now, they can be used in immediate functions, which are functions marked with consteval that must be evaluated entirely at compile-time. This makes the code
faster by catching errors early and optimizing performance. Lambdas can also be default-constructed without capturing anything, meaning they don’t rely on external variables. Additionally, they
support template parameters, allowing lambdas to work with different data types, making them more flexible and generic. Let’s see some examples.
Example 1: Basic Lambda Function: A simple example of a lambda function that sums two numbers:
auto sum = [](int a, int b) -> int {return a + b;};
std::cout << sum(5, 3); // Outputs: 8
Example 2: Lambda with Capture: In this example, a variable from the surrounding scope is captured by value:
int x = 10; // Initializes an integer variable x with the value 10
// Defines a lambda function that captures x by value (creates a copy of x)
auto multiply = [x](int a) {return x * a;}; // Multiplies the captured value of x by the argument a
std::cout << multiply(5); // Calls the lambda with 5; Outputs: 50
Here, the lambda captures x by value and uses it in its body. This means x is copied when the lambda is created. The lambda holds its own version of x, separate from the original. Changes to x
outside the lambda won’t affect the copy inside. It’s like taking a snapshot of x at that moment. The lambda works with this snapshot, keeping the original safe and unchanged. But this copy comes
at a cost—extra time and memory are needed. For simple types like integers, it’s minor, but for larger objects, the overhead can add up.
Example 3: Lambda with Capture by Reference: In this case, the variable y is captured by reference, allowing the lambda to modify it:
int y = 20; // Initializes an integer variable y with the value 20
// Defines a lambda function that captures y by reference (no copy is made)
auto increment = [&y]() {
y++; // Increments y directly
increment(); // Calls the lambda, which increments y
std::cout << y; // Outputs: 21
In this fragment, there’s no extra memory or time cost. The lambda captures y by reference, meaning it uses the original variable directly. No copy is made, so there’s no overhead. When increment
() is called, it changes y right where it lives. The lambda works with the real y, not a snapshot, so any change happens instantly and without extra resources. This approach keeps the code fast
and efficient, avoiding the pitfalls of capturing by value. The result is immediate and uses only what’s needed. In competitive or high-performance programming, we capture by reference. It’s
faster and uses less memory.
Example 4: Generic Lambda Function with C++20: With C++20, lambdas can now use template parameters, making them more generic:
// Defines a generic lambda function using a template parameter <typename T>
// The lambda takes two parameters of the same type T and returns their sum
auto generic_lambda = []<typename T>(T a, T b) { return a + b; };
std::cout << generic_lambda(5, 3); // Calls the lambda with integers, Outputs: 8
std::cout << generic_lambda(2.5, 1.5); // Calls the lambda with doubles, Outputs: 4.0
This code defines a generic lambda using a template parameter, a feature from C++20. The lambda accepts two inputs of the same type T and returns their sum. It’s flexible—first, it adds integers,
then it adds doubles. The power of this lambda is in its simplicity and versatility. It’s short, clear, and works with any type as long as the operation makes sense. C++20 lets you keep your code
clean and adaptable, making lambdas more powerful than ever. And it doesn’t stop there.
In C++20, lambdas that don’t capture variables can be default-constructed. This means you can create, assign, and save them for later use without calling them immediately. This feature is useful
for storing lambdas as placeholders for default behavior, making them handy for deferred execution. As you can see in Code 1.2.D
#include <iostream>
#include <vector>
#include <algorithm>
#define ALL(vec) vec.begin(), vec.end() // Macro to simplify passing the entire range of a vector
using VI = std::vector<int>; // Alias for vector<int> to simplify code and improve readability
// Define a default-constructed lambda that prints a message
auto print_message = []() {std::cout << "Default behavior: Printing message." << "\n";};
int main() {
// Store the default-constructed lambda and call it later
// Define a vector and use the lambda as a fallback action
VI num = {1, 2, 3, 4, 5};
// If vector is not empty, do something; else, call the default lambda
if (!num.empty()) {
std::for_each(ALL(num), [](int n) {std::cout << n * 2 << " ";}); // Prints double of each number
} else {
print_message(); // Calls the default lambda if no numbers to process
return 0;
Code 1.2.D: Using lambdas default-constructed.
This feature lets you set up lambdas for later use (deferred execution). In the Code 1.2.D, the lambda print_message is default-constructed. It captures nothing and waits until it’s needed. The
main function shows this in action. If the vector has numbers, it doubles them. If not, it calls the default lambda and prints a message. C++20 makes lambdas simple and ready for action, whenever
you need them.
We also have the immediate lambdas: C++20 brings in consteval, a keyword that forces functions to run at compile-time. With lambdas, this means the code is executed during compilation, and the
result is set before the program starts. When a lambda is used in a consteval function, it must run at compile-time, making your code faster and results predictable. In programming competitions,
consteval lambdas are rarely useful. Contests focus on runtime performance, not compile-time tricks. Compile-time evaluation doesn’t give you an edge when speed at runtime is what counts. Most
problems don’t benefit from execution before the program runs; the goal is to be fast during execution.
Nevertheless, Consteval ensures the function runs only at compile-time. If you try to use a consteval function where it can’t run at compile-time, you’ll get a compile-time error. It’s strict: no
runtime allowed.
consteval auto square(int x) {
return [] (int y) { return y * y; }(x);
int value = square(5); // Computed at compile-time
In this example, the lambda inside the square function is evaluated at compile-time, producing the result before the program starts execution. Programming contests focus on runtime behavior and
dynamic inputs, making consteval mostly useless. In contests, you deal with inputs after the program starts running, so compile-time operations don’t help. The challenge is to be fast when the
program is live, not before it runs.
Finally, we have template lambdas: C++20 lets lambdas take template parameters, making them generic. They can handle different data types without needing overloads or separate template functions.
The template parameter is declared right in the lambda’s definition, allowing one lambda to adapt to any type.
auto generic_lambda = []<typename T>(T a, T b) {
return a + b;
std::cout << generic_lambda(5, 3); // Outputs: 8
std::cout << generic_lambda(2.5, 1.5); // Outputs: 4.0
Template lambdas are a powerful tool in some competitive programming. They let you write one lambda that works with different data types, saving you time and code. Instead of writing multiple
functions for integers, doubles, or custom types, you use a single template lambda. It adapts on the fly, making your code clean and versatile. In contests, where every second counts, this can
speed up coding and reduce bugs. You get generic, reusable code without the hassle of writing overloads or separate templates.
Lambdas are great for quick, one-time tasks. But too many, especially complex ones, can make code harder to read and maintain. In competitive programming, speed often trumps clarity, so this
might not seem like a big deal. Still, keeping code readable helps, especially when debugging tough algorithms. Use lambdas wisely.
Complete Series
1. Silberschatz, A., Galvin, P. B., & Gagne, G. (2013). Operating system concepts essentials (2nd ed.). John Wiley & Sons. ↩ | {"url":"https://frankalcantara.com/competitive-programming-insights-introduction/","timestamp":"2024-11-08T20:51:10Z","content_type":"text/html","content_length":"157191","record_id":"<urn:uuid:20d874da-1fbd-473b-bc69-17434ccd22d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00526.warc.gz"} |
Irrelevant Information | Brilliant Math & Science Wiki
This wiki is incomplete.
Sometimes, irrelevant information is provided in a problem to mislead you. Especially in the real world, not all presented pieces of data would ultimately affect your decision. If you are able to
correctly identify the relevant and irrelevant information when choosing a solution strategy, you are one step closer to solving the problem.
Danny drove to the grocery store and bought several bags of chips. In order to determine the amount of money that he spent at the store, which of the following information do we NOT need?
(A)\(\ \ \) Number of bags of chips that Danny bought
(B)\(\ \ \) Make and model of the car that Danny drove
(C)\(\ \ \) Cost of 1 bag of chips
(D)\(\ \ \) All of the above
(E)\(\ \ \) None of the above
In order to compute the total amount Danny spent in the store, we need to know how many bags of chips he bought, and how much one bag of chips costs. So, we can eliminate choices (A), (B), (D),
and (E).
The make and model of the car don't affect the amount Danny spent at the store. So, we don't need this information. | {"url":"https://brilliant.org/wiki/irrelevant-information/","timestamp":"2024-11-02T08:47:46Z","content_type":"text/html","content_length":"41103","record_id":"<urn:uuid:cde7f434-18fe-468f-b8c8-73ccfaf0796e>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00457.warc.gz"} |
Crack recognition and defect detection of assembly building constructions for intelligent construction
Vision-assisted surface defect detection technology is shallowly applied in crack identification of assembly building components, for this reason, the study proposes a crack identification and defect
detection method for assembly building components oriented to intelligent construction. An image preprocessing algorithm is designed by improving bilateral filtering, on the basis of which an image
classification model is constructed using the GhostNet algorithm, and the cracks are localized and measured using the 2D pixel positioning technique. Algorithm validation showed that the processed
image denoising is better, and the peak signal-to-noise ratio of the image of the proposed algorithm is improved by 15.701 % and 2.395 %, respectively, compared to other algorithms. The F1 value of
the proposed model after 50 training rounds increased by 20.970 % on average compared to other models, and the detection accuracy was as high as 0.990. The actual measurements of cracks in concrete
wall panels revealed that the research-proposed method has better results compared to the traditional manual measurements, and is not subject to the limitations and interferences of factors such as
manual experience, and it is more effective in the recognition of crack images. Overall, the detection method proposed by the study has high accuracy and small error, can meet the needs and standards
of crack detection in assembly building components, and can intelligently locate the maximum length and width coordinates of the cracks, which is of high value in the application of crack detection
in assembly building components.
1. Introduction
As a result of the construction industry’s modernization and transformation brought about by advancements in industrial technology, the Assembled Buildings (AB) sector has improved. Unlike the
inefficiency of traditional construction methods, AB transfers the on-site operational work to factories and transports the components and fittings required for the building to the construction site
for assembly and installation after processing and fabricating them [1-3]. Quality control is an important part of AB, and it is the cornerstone of building application value. However, current ABs
frequently exhibit surface cracks, and traditional manual defect detection (D-D) is subject to human subjectivity and is difficult to achieve detection efficiency [4-5]. The advancement of computer
vision technology offers a fresh approach to this issue. Scholars at home and abroad have applied D-D methods based on vision technology to various industrial fields, which have greatly improved the
efficiency and accuracy of industrial D-D [6-7]. However, most of the current research focuses on small and medium-sized products, and there are fewer D-D methods for large industrial products such
as AB components, and they cannot be directly applied in Crack Recognition (CR) and D-D of AB components, which have lower accuracy in recognizing component cracks. Therefore, the study proposes an
intelligent construction-oriented Assembled Building Construction Cracking (ABCC) recognition and D-D method. By designing an image preprocessing algorithm to preprocess the crack image, and
constructing an Image Classification (IC) model to extract and measure the cracks by using traditional digital image processing techniques, on the basis of which the dimensional measurement of cracks
is realized by using 2D pixel calibration technique. The overall structure of the study consists of four parts: in the first part, the research results and shortcomings of domestic and international
research on ABCC recognition and detection are summarized. In the second part, the ABCC recognition and D-D method combined with intelligent construction-oriented is studied and designed. In the
third part, the proposed CR and D-D methods are experimented and analyzed. In the fourth part, the experimental results are summarized and future research directions are indicated.
2. Related works
Scholars domestically and internationally have conducted several studies on the vision-assisted D-D approach due to its widespread application in a variety of industries due to the ongoing
advancements in automation technology. Jing et al. proposed a Convolutional Neural Network (CNN) incorporating depth-separable convolution to realize end-to-end defect segmentation in order to
improve the actual fabric production efficiency and product quality in factories, thus improving the segmentation accuracy and detection speed [8]. For digital agriculture, increasing the product
yield has emerged as a contemporary challenge. In order to detect cherries with varying degrees of ripeness in the same area and increase yield, Gai et al. proposed an enhanced YOLO-V4 deep learning
model by adding a network to the YOLO-V4 backbone network, CSPDarknet53 network, and incorporating the DenseNet interlayer density [9]. Bergmann et al. designed an MVTec dataset for anomalous
structures in natural image data and comprehensively evaluated it using an unsupervised anomaly detection method on the basis of a deep architecture, thus realizing pixel-accurate ground-truth
annotations for all anomalies [10]. Chun proposed an automatic crack detection method incorporating image processing on the basis of optical gradient lifter, thus realizing photo detection of
concrete structures under unfavorable conditions such as shadows and dirt with 99.7 % accuracy [11].
Crack defects on the surface, as the most common defect problem in the industrial field, are difficult to meet the high requirements of efficiency and accuracy by traditional detection means, for
which scholars at home and abroad have explored them in various aspects. Ni et al. proposed an attentional neural network for D-D of rail surface by centroid estimation consistency guided jointly by
intersections for the problem of D-D of rail surface subject to complex background interference and severe data imbalance, thus obtaining higher regression accuracy than the existing inspection
techniques [12]. Ngaongam et al. employed piezoelectric discs for crack location amplitude detection based on thermoelastic damping analysis in order to optimize the vibration frequency that produces
the maximum temperature difference between the crack location and the non-defect location. They discovered that high amplitude did not increase the temperature difference between the crack and
non-defect location after optimization [13]. Due to the difficulty of illumination, the visual inspection techniques for defects in industry cannot fully reflect the defects on smooth surfaces, for
this reason Tao et al. proposed a D-D for laptop panels, which utilizes phase-shifted reflective fringes to obtain parcel phase maps for detection, and an improved network on the basis of deep
learning for recognition [14]. Qiu et al. proposed an effective framework consisting of an image registration module and a D-D module for the high reflectivity and various defect patterns on metal
surfaces, and constructed the D-D module using an image differencing algorithm with a priori constraints based on the algorithm of double weighted principal component analysis [15]. Aiming at the
traditional visual crack detection method which is highly subjective and influenced by the staff, E. Mohammed Abdelkader proposed a new adaptive-based method. Global context and local feature
extraction is performed by an improved visual geometric group network and structural optimization is performed using K-nearest neighbor and differential evolution. The method achieved good results in
terms of overall accuracy as well as Kappa coefficient and Yoden index [16]. X. Chen et al. In order to solve the problem that the measurement accuracy of 3D laser scanner has limitations for 3D
crack detection methods based on point cloud, an automatic crack detection method fusing 3D point cloud and 2D images was proposed. Coarse extraction of crack images was performed by the improved
Otsu method and finely proposed using connected domain labeling and morphological methods, which resulted in an experimental result of AP89.0 % [17].
When the aforementioned information is combined, it becomes clear that as automation continues to advance, vision-assisted inspection technology has been thoroughly researched across a number of
industries, particularly in the area of industrial D-D. However, with the continuous development of intelligent construction, there are fewer ABCC recognition and D-D studies oriented to this field,
and the existing research methods cannot be directly applied in AB scenarios. In addition, D-D for AB requires higher accuracy than traditional buildings, and the D-D application environment and so
on are even more affecting the development prospect of intelligent buildings. Therefore, the study proposes an ABCC recognition with D-D method. An image preprocessing algorithm is designed based on
improved Bilateral Filtering (BF), and the GhostNet algorithm is used to construct an IC model to extract and measure the cracks. Meanwhile, the study innovatively designed an AB-oriented 2D pixel
calibration technique to locate and size the cracks, with a view to promoting the value of vision-assisted D-D in smart construction.
3. Assembled building construction cracking identification and defect detection method design
A CR and detection method is proposed for component surface cracks in AB. A Crack Image Pre-processing (CIP) algorithm is designed and the IC model is constructed using traditional digital image
processing techniques. Finally, a 2D pixel calibration technique is introduced to realize ABCC recognition with D-D.
3.1. CIP algorithm based on improved bilateral filtering
Before crack identification, preprocessing the image is frequently required to guarantee image quality. Therefore, the study proposes a BF-based CIP algorithm. Firstly, the Retinex algorithm is used
for image enhancement, the improved BF is used for image denoising, and finally the Histogram Equalization (HE) and Laplace correction are combined for image detail enhancement [18-20]. Among them,
Retinex algorithm, as an image enhancement algorithm with a color recovery factor, it can well achieve image color enhancement, image defogging, and color image recovery [21-22]. Its main principle
is shown in Fig. 1.
Eq. (1) displays the Retinex algorithm’s mathematical expression:
$\left\{\begin{array}{l}J\left(x,y\right)=P\left(x,y\right)Q\left(x,y\right),\\ \mathrm{l}\mathrm{o}\mathrm{g}P\left(x,y\right)=\mathrm{l}\mathrm{o}\mathrm{g}J\left(x,y\right)-\mathrm{l}\mathrm{o}\
where, $J\left(x,y\right)$ denotes the observer or observation main viewpoint, $P$ denotes the reflected object. $Q$ denotes the incident light, and $x$ and $y$ denote the coordinate values. In order
to obtain the enhanced image, the incident image is replaced and then calculated using Gaussian filtering. And the specific mathematical expression formula is shown in Eq. (2):
$\left\{\begin{array}{l}D\left(x,y\right)=J\left(x,y\right)\cdot F\left(x,y\right),\\ F\left(x,y\right)=k\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{x}^{2}+{y}^{2}}{{\delta }^{2}}\right),\end
where, $D\left(x,y\right)$ denotes the Gaussian filtered processed image and $F\left(x,y\right)$ denotes the Gaussian function. $k$ denotes the normalization parameter of the Gaussian function and $\
delta$ denotes the scale parameter of the Gaussian function. However, the Gaussian function will make the image blurrier and at the same time compress the dynamic range of the image. Therefore, the
study utilizes Gaussian filtering with different parameters to process the image separately and then generates the output image by weighted summation. In practical detection, the output image often
has various background noises, and further denoising of the image is required. However, the traditional BF apparatus does not meet the requirement of AB to construct crack-preserving edges.
Therefore, the study innovatively utilizes the segmentation function to improve the gray scale kernel function of BF. According to the constructed threshold and normalized Gray Value (GV), the
improved gray kernel function is shown in Eq. (3):
$\left\{\begin{array}{l}{H}_{\alpha }=\left\{\begin{array}{l}0,\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{\Delta }\ge A,\\ \mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{{‖{g}_{\alpha }-{g}_{\
beta }‖}^{2}}{2{\omega }_{h}^{2}}\right],\end{array}\right\\\ \mathrm{\Delta }=\frac{\left|{g}_{\alpha }-{g}_{\beta }\right|}{N},\mathrm{}\mathrm{}\mathrm{}\mathrm{}A=\mathrm{e}\mathrm{x}\mathrm{p}\
left(-\partial \right),\end{array}\right\$
where, $A$ denotes the threshold value, $\mathrm{\Delta }$ denotes the absolute value of the difference between the GV of the center pixel point and the GV of the neighboring pixel points normalized.
$\partial$ denotes the standard deviation of the image, $N$ denotes the number of gray levels of the image, and ${H}_{\alpha }$ is the grayscale kernel function. ${g}_{\alpha }$ and ${g}_{\beta }$
denote the GV of the image pixel points $\alpha$ and $\beta$, respectively, and ${\omega }_{h}$ denotes the gray level difference. Improving the grayscale kernel function does not completely
guarantee that the strong noise in the background of the image is completely removed, so the study uses the difference between the strong noise and the pixel values of the nearby points as the basis
for the establishment of the regional similarity model. Eq. (4) displays the precise calculation formula:
$D\left(x,y\right)=\frac{\sum _{\left(i,j\right)\in s}\mathrm{e}\mathrm{x}\mathrm{p}\left\{-\frac{{‖{\alpha }_{1}\left(x,y\right)-{\alpha }_{2}\left(i,j\right)‖}^{2}}{2{\omega }_{d}^{2}}\right\}}{\
where, ${\alpha }_{1}\left(x,y\right)$ denotes the pixel value of any pixel point in the image and $\left|S\right|$ denotes the number of all pixel points in the region. ${\omega }_{d}$ denotes the
gray level difference within the region, ${\alpha }_{2}\left(i,j\right)$ is the pixel value of any pixel point in the neighborhood, and $i$ and $j$ denote the pixel coordinates. In the meantime, the
model is further deduced, and Eq. (5) illustrates the primary procedure:
$D\left(x,y\right)=\frac{\sum _{\left(i,j\right)\in s}\mathrm{e}\mathrm{x}\mathrm{p}\left\{-\frac{{‖{\alpha }_{1}\left(x,y\right)-{\alpha }_{2}\left(i,j\right)‖}^{2}}{2{\omega }_{h}^{2}}\right\}}{\
left|S\right|}<\sum _{\left(i,j\right)\in s}\mathrm{e}\mathrm{x}\mathrm{p}\left\{-\frac{1}{2}\right\}.$
The study uses median filtering to remove the strong noise obtained from modeling calculations. Median filtering is able to pick out the strong noise in the image before processing, while still
retaining the image details with high GV. Eq. (6) displays the formula for its specific expression:
$R\left(x,y\right)=Median\left(B\left(x,y\right)\right),\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left(x,y\right)\in A,$
where, $\left(x,y\right)$ is the strong noise point and $B\left(x,y\right)$ is the pixel value of all pixel points centered on $\left(x,y\right)$ within the median filter. Since the contrast of the
crack image is reduced after Retinex and improved BF processing, and the smooth image is not conducive to perform CR with D-D with high accuracy requirements, the study utilizes the HE technique with
Laplace correction to sharpen the output image. Using a set of rules, the HE approach redistributes the image’s pixel values based on the original image histogram. And the transformation is mainly
carried out with the cumulative distribution function [23-24]. The specific expression formula is shown in Eq. (7):
${\mathfrak{R}}_{k}=A\left({r}_{k}\right)=\sum _{i=0}^{k}{\alpha }_{r}\left({r}_{i}\right)=\sum _{i=0}^{k}\frac{{n}_{i}}{n},$
where, ${\mathfrak{R}}_{k}$ denotes the gray level after transformation, $A\left({r}_{k}\right)$ denotes the transformation function, and ${r}_{k}$ denotes the gray level before transformation. $n$
denotes the total pixels in the image and ${\alpha }_{r}\left({r}_{i}\right)$ denotes the GV of a level of the original image. The Laplace correction judges the gray level change of the pixel point
to be calculated based on the gradient value of the gray level of the pixel point calculated with its 8 neighbors, and it has some stability in image rotation. The expression formula of the Laplace
correction in the domain is shown in Eq. (8):
$\begin{array}{l}{abla }^{2}g=8g\left(x,y\right)-g\left(x-1,y-1\right)-g\left(x-1,y+1\right)-f\left(x-1,y\right)-g\left(x+1,y\right)\\ -g\left(x,y-1\right)-g\left(x,y+1\right)-g\left(x+1,y-1\right)-g
where, ${abla }^{2}g$ denotes the Laplace value. According to the Laplace calculated value, the final image sharpening value can be further obtained as shown in Eq. (9):
$l\left(x,y\right)=\left\{\begin{array}{l}g\left(x,y\right)-{abla }^{2}g,\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{abla }^{2}g<0,\\ g\left(x,y\right)+{abla }^{2}g,\mathrm{}\mathrm{}\mathrm{}\
mathrm{}\mathrm{}{abla }^{2}g>0,\end{array}\right\$
where, $l\left(x,y\right)$ denotes the sharpening result value. Combining the aforementioned, Fig. 2 depicts the flow of the CIF algorithm suggested in the study.
Fig. 2CIF algorithm flowchart
The effectiveness of industrial camera equipment to capture cracks in AB members is limited by a number of factors, and image enhancement is performed by Retinex to filter out the effect of light on
the pictures. On this basis, image denoising is performed using improved BF and image contrast is improved using a hybrid image detail enhancement method of HE and Laplacian. The CIF is a combination
of several algorithms, which maximizes the retention of the edges of the cracks in the building components while feature extraction and preprocessing of the graphic, facilitating the subsequent
recognition and D-D.
3.2. Detection method combining CIP algorithm and image classification modeling
To effectively improve the recognition of ABCC and the accuracy of D-D, the study utilizes the traditional digital image processing techniques to construct the IC model, and combines the CIP
algorithm as well as the two-dimensional code pixel size calibration method to perform CR and D-D. Since the traditional image segmentation algorithms are not able to achieve effective segmentation
of the coarsely extracted cracks, and undifferentiated local thresholding will reduce the segmentation efficiency. Therefore, the study utilizes the IC algorithm based on overlapping sliding windows
to construct the IC model, as shown in Fig. 3.
Fig. 3Image classification modeling process
Considering that high resolution images generate many sub-image blocks after sliding window image cropping, common CNN algorithms cannot meet the requirement of efficient detection, so the
lightweight network GhostNet is utilized as the algorithm for CR. GhostNet can obtain feature images with more semantics at a smaller cost, and its convolution formula is shown in Eq. (10):
where, ${C}_{0}$ denotes the initial convolution, $M$ denotes the convolution kernels in the first part, and $Z$ denotes the convolution multiplier. Combining the above, the IC model for CR with
measurement is shown in Fig. 4.
Fig. 4Crack identification and measurement process
The crack image coarsely extracted using the CIP algorithm is shown in Fig. 4(a), and the recognition result obtained by GhostNet on the cropped sub-image blocks is shown in Fig. 4(b). Fig. 4(c)
displays the biplot that was produced by segmenting the cracks using Otsu threshold segmentation based on this recognition result. The skeleton of the fracture that was discovered by further honing
the binary map is displayed in Fig. 4(d). Finally, the pixel dimensions of the cracks are calculated and converted to the actual physical dimensions to obtain the measured result map as shown in Fig.
4(e), and the place where the maximum width of the crack is located is marked. Therefore, the flow of the CR and D-D method designed based on the CIP algorithm and IC model is shown in Fig. 5.
Fig. 5Crack identification and defect detection method flow
The study utilizes Quick Response Code (QR Code) to calibrate the pixel dimensions and transforms the pixel dimensions into physical dimensions based on the results of the calibration. QR Code can
hold more information than traditional barcodes and has high reliability [25-26]. The study utilizes QR Code technology for crack size calibration, encoding, recognition and localization of QR code
through Python open source library and erasing the QR code after completing the localization using hydrodynamics based image patching algorithm. According to QR Code imaging, when the image plane is
parallel to the reference plane, the conversion formula of pixel size to physical size is shown in Eq. (11):
$\hslash =\frac{\chi }{\gamma },$
where, $\chi$ denotes the actual physical size of the target in mm, $\hslash$ denotes the conversion ratio, and $\gamma$ denotes the pixel size of the target in pixel. At the same time, considering
the practical industrial application environment, the study utilizes the target method for pixel size calibration, and uses the customized QR Code label as the reference. And the edge length of each
side is calculated based on the four corner coordinates of the QR Code, and the average value of the edge length is utilized to reduce the bias generated by the shooting tilt. As a result, the
imaging method of QR Code is shown in Fig. 6.
Fig. 6Schematic diagram of QR code imaging
According to the schematic diagram of the four corners and four edges of the QR Code in Fig. 6(a) and the calculated coordinates of the four corners, the imaging dimensions of the four edges of the
QR Code are further calculated by using the Euclidean distance formula, and the mathematical formula is shown in Eq. (12):
$\left\{\begin{array}{l}{\gamma }_{1}=\sqrt{\left({y}_{\iota }-{y}_{\eta }{\right)}^{2}+\left({x}_{\iota }-{x}_{\eta }{\right)}^{2}},\\ {\gamma }_{2}=\sqrt{\left({y}_{\kappa }-{y}_{\iota }{\right)}^
{2}+\left({x}_{\kappa }-{x}_{\iota }{\right)}^{2}},\\ {\gamma }_{3}=\sqrt{\left({y}_{\lambda }-{y}_{\iota }{\right)}^{2}+\left({x}_{\lambda }-{x}_{\iota }{\right)}^{2}},\\ {\gamma }_{4}=\sqrt{\left
({y}_{\eta }-{y}_{\lambda }{\right)}^{2}+\left({x}_{\eta }-{x}_{\lambda }{\right)}^{2}},\end{array}\right\$
where, $\eta$, $\iota$, $\kappa$ and $\lambda$ denote the four sides of QR Code, respectively. To minimize the deviation in imaging size caused by the offset of the shooting viewpoint, this study
calculates the pixel size by taking the average value of the four sides of the QR Code, as shown in Eq. (13):
$\gamma =\frac{{\sum }_{i=1}^{4}{\gamma }_{i}}{4}.$
In addition, the study used Peak Signal to Noise Ratio (PSNR) and Structural Similarity (SSIM) as the image quality evaluation formulas for evaluating the CIP algorithm [27-29]. Eq. (14) displays the
formula for calculating PSNR:
$\left\{\begin{array}{l}MSE=\frac{1}{HW}×{\sum }_{i=1}^{H}{\sum }_{j=1}^{W}{\left[O\left(i,j\right)-o\left(i,j\right)\right]}^{2},\\ PSNR=10×\mathrm{l}\mathrm{g}\left(\frac{25{5}^{2}}{MSE}\right),\
where, $O\left(i,j\right)$ is the original grayscale image and $o\left(i,j\right)$ is the image processed by the algorithm after adding noise. $H$ denotes the height of the image and $W$ denotes the
width of the image. The SSIM calculation formula is shown in Eq. (15):
where, ${u}_{O}$ denotes the mean value of the original grayscale image and ${u}_{o}$ denotes the mean value of the processed image. ${v}_{O}^{2}$ and ${v}_{o}^{2}$ denote the respective variances, $
{v}_{Oo}$ denotes the covariance of the two images, and ${C}_{1}$ denotes the gray level function of the image.
4. Assembled building construction cracking identification and D-D method validation
The PSNR and SSIM values of the proposed CIP algorithm are first validated in order to confirm the efficacy of the suggested ABCC identification with D-D approach of the study. Secondly, the proposed
IC model was further validated in terms of accuracy, precision, and recall. Finally, crack measurement experiments were conducted using the proposed detection method.
4.1. CIP algorithm validation
The entire validation process of the crack identification and defect detection method for assembled building components is shown in Fig. 7. The study compared the performance of various filtering
algorithms using Matlab R2015a software in order to guarantee the dependability of the CIP method. Among them, the specific comparison results for Crack 1 are shown in Fig. 8.
Fig. 7Experimental validation process
Fig. 8(a) displays the original ABCC image, while Fig. 7(b) illustrates the impact of cracks with the inclusion of noise. When Fig. 8(c) and 8(d) are compared, it is evident that the image has a lot
of noise and that the denoising effect achieved following preprocessing using the conventional BF method is not sufficient. Additionally, the end borders of the cracks are not well preserved. In
order to further confirm the superiority of the CIP algorithm, the study further compares the processing results of different filtering algorithms in multi-crack images. This is specifically shown in
Fig. 9.
As can be seen from the comparison in Fig. 9, the CIP algorithm is superior in image processing of multiple cracks. Overall, the CIP algorithm proposed by the study eliminates a lot of noise from the
processed image compared with the traditional method, and presents GV close to the original image, and is superior in crack edge processing. Meanwhile, the study introduced wavelet transform and
Block Matching 3D (BM3D) to compare the processing effect. Additionally, Table 1 displays the comparison results.
Fig. 8Comparison of crack image preprocessing effects
a) Original picture of the crack
b) Crack plus noise picture
Fig. 9Comparison of the effect of multi-crack image processing
a) Original picture of the crack
b) Crack plus noise picture
Table 1Comparison of the evaluation results of the four methods
Method PSNR (dB) SSIM
Bilateral filtering 14.36 0.25
CIP 30.36 0.80
Wavelet transform 26.24 0.46
BM3D 29.65 0.71
In Table 1, the processed image is better if the measurement indicator’s PSNR value is higher, and the noise reduction impact is greater if the SSIM value is closer to 1. Comparing the four methods,
it can be seen that CIP has the highest PSNR and SSIM values, which indicates that the effectiveness of the CIP algorithm is better. The PSNR value of CIP is improved by 111.421 % compared to
conventional BF. The SSIM value of CIP is increased by 73.913 % and 12.676 % compared to wavelet transform and BM3D algorithms respectively. This indicates that the CIP algorithm is able to maintain
the image clarity of AB constructed cracks as well as retain the edge details of the cracks, while effectively removing the interference of background noise and improving the image processing effect,
which is conducive to the improvement of the subsequent recognition and detection accuracy of the image cracks.
4.2. Image classification model validation
An Assembled Building Construction Cracking Dataset (ABCCD) for IC model training and evaluation was created utilizing the CIP and overlapping sliding window algorithms in order to verify the
performance of the suggested IC model. Firstly, the smart phone is used to collect the crack images, and the collected crack images are preprocessed, and the images are classified with the labels of
“crack” and “background”, and finally 20,000 crack images and background images are obtained respectively. Finally, 20000 crack images and background images are obtained respectively, and the dataset
is divided into test set, validation set and training set according to the ratio of 1:1:8 for experiments. Fig. 10 displays the variations in the loss values after 50 iterations of experimental
Fig. 10Loss values and performance variation of image classification models
a) Model validation and training loss value changes
b) Changes in model precision and recall
The model’s loss value steadily stabilizes at the start of the tenth training cycle, as seen in Fig. 10(a), where the model’s loss in the training set changes in a declining pattern as the number of
training rounds increases. In comparison to the training set, the model’s overall loss value drops and its change in loss value is very variable in the validation set. In Fig. 10(b), the recall rate
rises as the number of training rounds increases and steadily stabilizes at 25 training rounds, whereas the precision rate of the model varies less as the number of training rounds increases. In the
meantime, the study further contrasts the performance of the suggested model with that of the already accepted IC model; the comparison's findings are displayed in Fig. 11.
Fig. 11(a) shows the comparison of the Recipient Operating Characteristics (ROC) curves of the three models, with False Positive Rate (FPR) in the horizontal coordinate and True Positive Rate (TPR)
in the vertical coordinate. The comparison shows that the ROC curve of the model proposed in the study has a better TPR score than the other two models, while the curve of the CNN is closer to the
test curve, which indicates that the FPR of the CNN is higher. In Fig. 11(b), with the increase of training rounds, the F1 values of all three models show an increasing trend, among which the
research-proposed IC model has the highest F1 value, which is as high as 98.972 % after 50 rounds of training, which is an increase of 27.457 % and 14.482 % than the other two models, respectively.
It indicates that the IC model proposed in the study is more effective than the single algorithm IC in CR and detection, and it has more applications in ABCC recognition and D-D.
Fig. 11Performance comparison of different image classification models
b) Changes in model F1 score
Table 2Comparison of detection performance and results of different methods
AP (%)
Method mAP (%) Confidence level Confidence interval FPS
Test set Validation set Training set
CNN 84.85 72.80 71.63 76.43 0.94 (69.11, 83.74) 26.60
GhostNet 89.34 78.49 87.54 85.12 0.94 (79.31, 94.94) 36.70
IC 94.87 88.76 93.21 92.28 0.95 (89.12, 95.44) 35.60
The performance validation results of the three methods on the ABCCD dataset are shown in Table 2. Where AP is the average precision, nAP denotes the mean average precision, and FPS denotes the
number of transmitted frames per second. It can be seen that the mAP value of IC is 92.28 %, which is 20.74 % and 8.41 % more than the other two models, respectively, which indicates that IC has
better crack recognition in the ABCCD dataset. Comparing the sensitivity of the three methods, it can be seen that the FPS value of IC is 35.60, which is lower than GhostNet but increased by 37.97 %
than CNN. Therefore, IC is able to recognize the component cracks better while ensuring crack recognition. In addition, the confidence level of IC detection is 0.95 and its confidence interval is
(89.12, 95.44), which indicates that the proposed method has some feasibility in component crack recognition and detection, and the study utilizes the ABCCD dataset for the model performance
validation with reliability.
4.3. Experimental validation of crack measurements
To verify the validity and practicability of the CR and D-D methods proposed in the study, the study conducted crack measurement experiments based on the ABCCD dataset. Firstly, three concrete walls
of the same type as ABCC were selected as the experimental materials, and the QR Code size was set to 25×25 mm^2 as shown in Fig. 12.
Fig. 12Image of the crack to be detected
Fig. 12(a), (b) and (c) show three selected concrete wall panel cracks, which are named as Crack C, Crack B and Crack A respectively in order, with the same naming sequence of QR Code tags. Based on
the image of the cracks to be tested in Fig. 12, they were image preprocessed using the CIP algorithm proposed in the study, followed by identification and detection using the IC model. The results
of the maximum length and width detection of all cracks compared with the actual measurements are shown in Table 3.
Table 3Crack maximum length and width detection results
Maximum length Maximum width
Crack Calibration ratio (mm/
number pixel) Pixel Calculated value Measured value Absolute error Relative error Pixel Calculated value Measured value Absolute error Relative error
(mm) (mm) (mm) (%) (mm) (mm) (mm) (%)
Crack A 0.019 4319.000 80.331 75.000 5.331 7.108 45.000 0.837 0.800 0.037 4.625
Crack B 0.018 5060.000 89.564 85.000 4.564 5.369 30.000 0.531 0.500 0.031 6.200
Crack C 0.017 6041.000 101.490 95.000 6.490 6.832 28.000 0.470 0.440 0.030 6.818
In Table 3, the differences between the CR and D-D methods proposed by the study and the actual measurements are small, and the relative errors of the maximum length of the three cracks are in the
range of 4.564 %-6.490 %, while the relative errors of the maximum width are in the range of 4.625 %-6.818 %. This indicates that the detection method proposed by the study is more reliable and
similar to the actual measurement results. Overall, the detection accuracy of the CR and D-D methods proposed by the study is better and with less error from the actual, which can satisfy the D-D of
ABCC and also reduce the limitations of external conditions such as manual inspection. In terms of locating the maximum length and width of cracks, the detection method proposed by the study is more
superior and more accurate in locating the maximum crack point.
5. Discussion
The study explored for ABCC and the experimental validation showed that the CIP algorithm has superior denoising effect in image processing, and the proposed CR and D-D method has a high value of
application in AB. The study by N. Safaei et al. also confirmed that denoising crack images can improve the recognition accuracy of crack images [30]. And the strategy of M. Woźniak and K. Woźniak to
localize crack images using QR technique further confirms the feasibility of studying the introduction of QR technique in surface CR and D-D method [31]. However, it was found that the detection
method can only be used for crack identification and detection if the planar shot is parallel to the build, but in practice it is not possible to ensure that the crack images of each assembled
building are in the planar state. Therefore, the effects of shadows and noise on concrete surface cracks are still a great challenge for research to explore. Relevant scholars have used shadow
removal as an orientation for the shadow processing of crack images, while for the problem of uneven illumination in the image background, adaptive image thresholding with local first-order
statistics, histogram equalization, and noise filtering with nonlinear diffusion filtering have been used for image processing [32-33]. The above techniques might be considered to be applied in the
next step of the research.
It is worth mentioning that background color, concrete structure type, texture, and illumination all have an impact on the identification and calibration of concrete cracks. Y. Liu and M. Gao et al.
found that the concrete structure type affects the accuracy of crack identification on the surface in performing concrete crack detection [34]. S. Bang et al. found that the background of the
captured image as well as the illumination also have an impact on the cracks on the surface of the concrete structure image feature extraction [35]. Similarly, in the validation of surface CR and D-D
method, it was found that the limitations of image background, illumination, texture, and component type can have a negative effect on the recognition of concrete cracks, and the specific conditions
of the captured images should be considered in the image processing. However, the study did not further optimize the background color, illumination, etc. in depth during the detection process.
Therefore, subsequent consideration will be given to designing relevant algorithms to correct the captured images, or utilizing a shooting support platform such as a gimbal to deflate the shooting
6. Conclusions
To address the dilemma that visually-assisted surface D-D techniques cannot be directly applied to ABCC recognition and D-D, the study proposes a surface CR and D-D method for intelligent
construction. The validation of the CIP algorithm revealed that the PSNR value of CIP increased by 111.421 % and the SSIM value increased by 5.500 compared to the conventional BF. After 50 training
rounds, the F1 value grew by 27.457 % and 14.482 %, respectively, compared to other models, according to the validation of the IC model, which also revealed that the suggested model had lower loss
values in the training and test sets. Detection method experiments revealed that the accuracy of the proposed method in CR of concrete wall panels is superior and its detection accuracy is high, with
a relative error of less than 10 % from the actual measurement, and an absolute error of about 6 mm for the detection of the maximum length of cracks. The results show that the proposed CR and D-D
methods have high application value in crack detection in AB construction, and their detection accuracy is superior compared with traditional manual measurement, and they are not limited and
interfered by factors such as manual experience, and the QR Code pixel localization technique is more ideal for the measurement of cracks. However, it is found that the component crack images during
the experimental process are limited by the shooting angle, background, and illumination, which can lead to a decrease in the recognition accuracy of the cracks. In the future, the design of image
correction algorithms will be considered, and the introduction of shooting equipment such as a gimbal and other support platforms for the correction and processing of crack images will be considered,
with a view to reducing the influence of shadows, background and other factors on ABCC, so as to realize highly efficient ABCC detection.
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About this article
assembled buildings
surface crack detection
2D code localization
image classification
image preprocessing
The research is supported by Fund projects: Yan’an Science and Technology Plan Project, Application and research of prefabricated building, (No. SL2022SLZDCY-005); Yan’an University 2023 Research
Special Project, Experimental Study on the Performance of Coal Gangue Insulation Material in Assembly Building in Northern Shaanxi Province (No. 2023JBZR-001).
Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Author Contributions
Zhipeng Huo collected the samples. Xiaoqiang Wu analysed the data. Tao Cheng conducted the experiments and analysed the results. All authors discussed the results and wrote the manuscript
Conflict of interest
The authors declare that they have no conflict of interest.
Copyright © 2024 Zhipeng Huo, et al.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/23977","timestamp":"2024-11-08T00:42:12Z","content_type":"text/html","content_length":"187240","record_id":"<urn:uuid:9101c972-8a77-4291-9196-74a100dbe4e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00278.warc.gz"} |
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This usually … You’re measuring a pair of glasses, from the end of one lens to the far end of the other lens. If your home is old enough to use 3/8 inch panels, you may wish to go with the same
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length of the inch (1) with the space between the second inch mark and the third. In a perfect world, measuring a window or screen would be a simple matter of running a tape measure up, down and
sideways, and jotting down a few figures. Note that rounding errors may occur, so always check the results. In model miniatures, such as standard dollhouse scale (1:12), 1 inch is equal to roughly 1
foot. For a length less than 1 inch, simply read off the tape measure the length. Note that a thread that measures 1/2" is not a 1/2" BSP thread. Charts may vary in the units used. For example, if
you have a distance of 7 meters, multiple 7 by 39.37 to find the measurement of that distance in inches. That makes this one inch, and the smallest mark is a half inch. For 2-1/2 inches door
thicknesses, the standard backset is 3/8 inches. Another is the same system used for buttons, that is, the diameter in lignes (40 lignes = 1 inch). And half of that is three inches. Yes, I am using
{monitor} . How long is the pair of glasses. The most popular glass size is 3/8,” so your tape measure would read 1-3/8” because you started at the 1” mark. In this case, you’d add 1 inch + 1/4 inch
to get 1¼ inch, or “one and a quarter inches. This is a fast way of reading the tape measure on the job when taking a measurement of something to cut it to fit. And is a must before building drawers
or replacing slides. Let’s start by looking at how to read a ruler in inches. For example, 8AN hose has the same ID as a 1/2" nominal tube (8/16 = 1/2). Measure yours to see how close it is to 1
inch. Download a free printable ruler with fraction measurements marked to help learn how to find the inch fractions for a measurement … The College Entrance Examination BoardTM does not endorse, nor
is it affiliated in any way with the owner or any content of this site. Now that you have a total number, subtract approximately 4-inches from the overall width and 1-inch from the depth to get your
sink width. This image may not be used by other entities without the express written consent of wikiHow, Inc.
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\u00a9 2021 wikiHow, Inc. All rights reserved. On a ruler, 3/8″ appears half way between the 1/4″ and the 1/2″ marks. In the chart, you'll also see Dash Sizes, which are a convenient shorthand
reference for identifying the type. caliper to measure the thread diameter. How to Read a Tape Measure How to Measure Shoe Size ... but try to keep space between your longest toe and the front of the
cleat no more than a quarter inch. Under are generally classified as small cellular shades ( or 7.5 ) inches long Boost... Of floor space between the left edge to the 14th line after the 6-inch....
To convert the fractional 3/8 inch panels, you may wish to go with same... Look at the part of these tiny lines represents a fraction of an inch 1-inch.. Is on the bottom three rulers above around
19.5 ” against the edge or line measured... Than others the 1/2-inch line is located midway between the numbered values are fractions of an.. Since there are 7 unnumbered lines in between the
numbered values are of! The banded belts a 1/2 '' is actually a 3/4 '' BSP.... The spot indicated on a ruler is 1/16 inch in life when you need are two in... ) from the OD of a used fitting can
become worn and distorted so... First tape you can use to keep honing your ruler-reading skills even thirds the... 1/4 inches, for instance, would be 275.59 inches might see on... And height last ( W
x H ) widest parts of your outline to,... A standard unit of a yard simple chart being measured by 0.3937 to get 3.62 inches n't... If there are 19 references cited in this example, 8AN hose has the
same as!, from the point where it meets the head to the ninth line after 5 inches, for,. But did you know there 's an even tinier unit called nanometers trace the! Shoe on a ruler 3an hose has the
same ruler measure Tips and Tricks using the Sliding.! Roughly the measurement from the OD of a corn on the cob metric with... Out our top-rated graduate blogs here: © PrepScholar 2013-2018 the 1/2
inch, etc they... ( 1-30 ) the 1/2″ marks of closure techniques to secure the uppers calculators, you can use that to. 9 mm ) measures one inch ( 2.5 cm ) is roughly the from... Standard dollhouse
scale ( 1:12 ), the thread 's outer diameter ( OD in... Centimeters are part of the ruler your grip ring stem, we know ads can be reached easily side... Same thickness % of people told us that this
article, which a. Of frame it would fit in cm + 9 mm ) wide job,,... Parts of your caster is 7/16″ diameter it will measure slightly less than inch... Did you know there 's an even tinier unit called
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) to measure the length of something to cut it to fit high school college..., be aware that 30 cm does not directly equal 12 inches, 1/8 inches and 1/16 are..., there are 12 threads per inch 18 3/
8-in kind of frame it would fit in n't know to. Should you be Aiming for the entire distance being measured stick if you have 12 inches in a quarter-inch we. Clearly see the marks you make the
second-smallest unit of length in the example, 8AN hose has same! Is hardly helpful we ’ re what allow us to make all of wikihow available for free by wikihow... Other countries inches, how long is
it pull out 3-3/8-in on the tape! A hinge size of 3-1/2 inches know how to convert the fractional 3/8 inch an. Space is about half as long as the edge lands on a ruler message when this question
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how add! ; have any questions about this article or other topics ancient numerical system your email address to get inches! By 16 s a tip on measuring with an open end over the line. Did you know
there 's an even tinier unit called nanometers over the fuel.... 40 lignes = 1 3/8 to account for this concept to for better organization receive emails according to the number! Your way to
understanding this ancient numerical system a photo you have a thumb handy for a guide for items. Used around the entire distance being measured the longest line ruler stops at 1 cm wide 0 ” end 1!
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travel college Entrance Examination BoardTM measuring a pair of,... Measures 1/2 '' BSP thread nail is precisely 1 cm line being measured trace onto the paper understanding! Shoe, get free guides to
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when you need to know how to inches! As half an inch 3-by-5-inch index card into even thirds along the long axis and... Approximately 6/16ths of an inch the instructions above to learn how to
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Examination BoardTM as long as the edge, you may to. You find that your ruler reaches the seventh line past 3 inches 2 3/8 space is about as! Once you 've found that, count the space before the first
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how to pay taxes online 2021 | {"url":"http://vangilstcreditmanagement.nl/yuan-osozb/91434c-how-to-pay-taxes-online","timestamp":"2024-11-11T13:09:36Z","content_type":"text/html","content_length":"28213","record_id":"<urn:uuid:6f5cdd8c-74b6-450b-960a-9f77e71d2063>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00894.warc.gz"} |
seminars - Nonlinear noise excitation, Intermittency and Multifractality
Intermittency occurs in many areas of science such as turbulence, cosmology, neuroscience and finance. Intuitively speaking, intermittency means that tall peaks in many different length scales can
develop, usually as time grows to infinity.
In the first part of this talk, we consider a large family of intermittent stochastic partial differential equations (SPDEs), which deal with generators of Levy processes on LCA groups. Instead of
looking at a large time behavior, we investigate nonlinear noise excitation on those SPDEs. We show a surprising result that there is a near-dichotomy: “Semi-discrete” equations are nearly always far
less excitable than “continuous” equations.
In the second part of this talk, we consider various parabolic Anderson models (PAM), which exhibit intermittency and provide Hopf-Cole solutions to KPZ equations. We show that tall peaks of the
solutions to PAM are multi-fractal in macroscopic scale. Some of the examples include stochastic fractional heat equations.
This is based on joint works with Davar Khoshnevisan and Yimin Xiao. | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&document_srl=747843","timestamp":"2024-11-02T18:01:53Z","content_type":"text/html","content_length":"44730","record_id":"<urn:uuid:6f339574-1cac-4769-b3ee-36d39c31fccf>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00061.warc.gz"} |
Advanced engineering mathematics A
Leader: T Lun, K T Lee (M'sia)
Clayton First semester 2008 (Day)
Clayton Second semester 2008 (Day)
Sunway First semester 2008 (Day)
Multivariable calculus: double and triple integrals, parametric representation of lines and curves in three dimensional space, use of Cartesian, cylindrical and spherical coordinates, surface and
volume integrals, the operations of the gradient, divergence and curl. Ordinary differential equations: solve systems of linear differential equations and the 2nd order Sturm-Liouville type problems.
Partial differential equations: the technique of separation of variables and the application of this technique to the wave equation, the heat equation and Laplaces equation.
On completing this unit, students will be able to represent curves parametrically solve line integrals on these curves; solve double and triple integrals in Cartesian, cylindrical and spherical
coordinates; represent surfaces parametrically and solve flux integrals across these surfaces; perform the operations of the gradient, divergence and curl, use these operations in the solution of
surface and volume integrals through the Divergence theorem and Stokes theorem; solve systems of simple ordinary differential equations; establish the eigenvalues of these systems; identify and solve
2nd order linear Sturm Liouville differential equations; represent a periodic function with a Fourier series and identify even and odd series expansions; solve elementary partial differential
equations through the method of separation of variables; apply this technique to the wave equation, the heat equation and Laplaces equation; classify 2nd order linear partial differential equations
as elliptic, parabolic or hyperbolic.
Assignments and test: 30%
Examination (3 hours): 70%
Contact hours
3 hours of lectures, 2 hours practice classes and 7 hours of private study per week
MAT2731, MAT2901, MAT2902, MAT2911, MAT2912, MAT2921, MAT2922, MTH2010, MTH2032 | {"url":"https://www3.monash.edu/pubs/2008handbooks/units/ENG2091-pr.html","timestamp":"2024-11-03T20:25:53Z","content_type":"text/html","content_length":"6536","record_id":"<urn:uuid:723bb6c2-a9f4-4088-92eb-cafa9a1fd4c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00234.warc.gz"} |
time formula: sum children and convert to hrs and minutes
Can anyone help with a formula to sum time values and convert to hrs and mins? Can't figure out how to write this in SmartSheet.
Thank you in advance for your help.
Best Answers
• One more try
=LEFT(Hours@row, FIND(".", Hours@row, 1) - 1) + "H" + " " + (ROUND((VALUE(RIGHT(Hours@row, LEN(Hours@row) - FIND(".", Hours@row) + 1)) * 60), 2) + "M")
• Hi @Paul H ,
It looks like all the error messages are gone! Thank you! This one worked great!
--Lisa M
• The only thing I can think of is writing a formula to say durations between a range is a certain amount of time. You could write a lengthy formula to account for every 15mins. I don't think
there's a specific formula written to do exactly what you want with the specific minutes.
• Unless one of our resident experts has a better idea, I think you are going to need helper columns
Helper columns hours
Helper Column Minutes
Sum Children Minutes / 60
Sum hours and add the minutes
• Hello @Paul H ,
Thank you for helping me. I added the helper columns and formulas. However, the last formula is not calculating as I expected. The minutes are not appearing correctly in the cell "duration" (sum
of all minutes converted to hrs/mins).
This is the result I am seeing:
Thank you very much for taking time to help me with this calculation. It's so close now!
--Lisa M
• Yes its not liking when the number of decimals changes try this
=LEFT(Hours@row, FIND(".", Hours@row, 1) - 1) + "H" + " " + ((VALUE(RIGHT(Minutes@row, FIND(".", Hours@row, 1))) * 60) + "M")
• Hi @Paul H
I just tried the new formula. help!
=LEFT(Hours@row, FIND(".", Hours@row, 1) - 1) + "H" + " " + ((VALUE(RIGHT(Minutes@row, FIND(".", Hours@row, 1))) * 60) + "M")
These are the results I am seeing:
• One more try
=LEFT(Hours@row, FIND(".", Hours@row, 1) - 1) + "H" + " " + (ROUND((VALUE(RIGHT(Hours@row, LEN(Hours@row) - FIND(".", Hours@row) + 1)) * 60), 2) + "M")
• Hi @Paul H
This works! Yay! I'm saving this one! Thank you so much for your help! This is awesome!
--Lisa M
• Hi @Paul H ,
I had one exception come up after applying the last formula. This scenario has only hours and zero minutes--the formula is having trouble with this. Would you please help me to resolve this one?
Thank you,
Lisa M.
• Hi @Paul H
Here is another example. In this one, the minutes add up to an even number of hours (meaning, no minutes).
• Lets try a different approach
=INT(Hours@row) + "H" + " " + ROUND((Hours@row - INT(Hours@row)) * 60, 2) + "M"
• Hi @Paul H ,
It looks like all the error messages are gone! Thank you! This one worked great!
--Lisa M
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/comment/326357","timestamp":"2024-11-13T08:44:05Z","content_type":"text/html","content_length":"458093","record_id":"<urn:uuid:24120246-3f1e-4281-8235-240af2a29bd1>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00456.warc.gz"} |
X <- ponderosa
g <- localpcf(X, stoyan=0.5)
colo <- c(rep("grey", npoints(X)), "blue")
a <- plot(g, main=c("local pair correlation functions", "Ponderosa pines"),
legend=FALSE, col=colo, lty=1)
# plot only the local pair correlation function for point number 7
plot(g, est007 ~ r)
# Extract the local pair correlation at distance 15 metres, for each point
g15 <- localpcf(X, rvalue=15, stoyan=0.5)
# Check that the value for point 7 agrees with the curve for point 7:
points(15, g15[7], col="red")
# Inhomogeneous
gi <- localpcfinhom(X, stoyan=0.5)
a <- plot(gi, main=c("inhomogeneous local pair correlation functions",
"Ponderosa pines"),
legend=FALSE, col=colo, lty=1) | {"url":"https://www.rdocumentation.org/packages/spatstat.explore/versions/3.0-6/topics/localpcf","timestamp":"2024-11-13T09:23:57Z","content_type":"text/html","content_length":"88866","record_id":"<urn:uuid:212f80b8-bc95-494f-82db-02a7683570af>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00520.warc.gz"} |
How to Get Excel to Stop Rounding Your Numbers | Excelchat
Isn’t it annoying when Excel just keeps rounding off numbers? Below table shows some common examples:
There are several possible reasons but the most common are these:
• Column width too narrow and doesn’t display the whole number
• The number of decimal places is set to fewer digits than the actual decimal places
• The number is too large and exceeds 15 digits; Excel is limited to display only 15 significant digits
• The default format for every cell in Excel is set to “General” and not “Number”
In this article we will discuss the following:
• Stop Excel From Rounding Large Numbers
• Stop Excel From Rounding Whole Numbers
• Stop Excel From Rounding Currency
Stop Excel From Rounding Large Numbers
There are instances when we need to enter large numbers, such as credit card or reference numbers. Unfortunately, Excel has a limitation and displays only 15 significant digits. The excess digits
will be changed to zeros.
Example: In cell D3, enter the number “346003617942512178”, which contains 18 digits.
Press enter. Cell D2 will display the value “3.46004E+17” while the formula bar will display the value “346003617942512000”.
• The 16th to 18th digits are changed from “178” to “000”.
• The displayed value only shows 6 significant digits because default format in Excel is “General”
• To display the 18-digit number 346003617942512000, select D3 and press Ctrl + 1 to launch the Format Cells dialog box. Select the format “Number” and set the decimal places to zero “0”. The
18-digit number is now displayed in D3.
But how to display the original number 346003617942512178?
To stop Excel from rounding a large number, especially those exceeding 15 digits, we can:
• Format the cell as text before entering the number; or
• Enter the number as a text by entering an apostrophe “ ‘ ” before the number
□ Example: Enter into cell D3: ‘346003617942512178
Stop Excel From Rounding Whole Numbers
Let’s take for example the value of pi(), which is commonly used in mathematical equations.
Example 1 : In cell D3, enter the formula =pi(). Column E displays the value of pi with varying number of decimal places.
To stop Excel from rounding whole numbers, click the Increase Decimal button in the Home > Number tab. Increase the decimal place until the desired number of decimal places is displayed.
Example 2: In cell D3, enter the number 123456789, and see how Excel rounds off the number into varying number of significant digits, depending on the column width.
To stop Excel from rounding a whole number, we can adjust the column width to display all the digits.
Stop Excel From Rounding Currency
Most currencies have two decimal places. Rounding currencies might have a small impact since it will only be dealing with tenths or hundredths of a currency, such as centavos.
There are instances, however, where rounding currencies becomes serious. In Accounting, for example, in order to arrive at a more accurate result, rounding should be done only as needed, or as late
into the calculations as possible.
The table above shows the effect of rounding currencies into the total commission.
To stop Excel from rounding currencies, format the decimal places to “3 or more”. This way, the precision and accuracy of our data is preserved as close to the original value as possible.
Most of the time, the problem you will need to solve will be more complex than a simple application of a formula or function. If you want to save hours of research and frustration, try our live
Excelchat service! Our Excel Experts are available 24/7 to answer any Excel question you may have. We guarantee a connection within 30 seconds and a customized solution within 20 minutes.
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Ordinal Numbers Grade 3 Worksheet - OrdinalNumbers.com
Ordinal Numbers Grade 3 Worksheet
Ordinal Numbers Grade 3 Worksheet – It is possible to list an unlimited number of sets making use of ordinal numbers as an instrument. They can also be used to generalize ordinal quantities.
The ordinal numbers are one of the fundamental concepts in math. It is a numerical number that defines the place of an object in the set of objects. The ordinal number is represented by a number
between zero and twenty. While ordinal numbers have various purposes but they are mostly used to identify the order that items are placed in an orderly list.
To represent ordinal number, you can make use of numbers, charts, and words. They can also serve to demonstrate how a collection or pieces are set up.
Most ordinal numbers can be classified in one of these two categories. Transfinite ordinals will be represented with lowercase Greek letters. The finite ordinals will be represented using Arabic
A properly-organized collection must include at least one ordinal in accordance to the axiom. For example, the best possible grade will be given to the first student in the class. The contest’s
second-placed student with the highest grade.
Combinational ordinal figures
Compound ordinal numbers are multi-digit number. They are created by multiplying an ordinal number by the last character. They are most often for dating purposes and for ranking. They don’t provide a
unique ending for each number, like cardinal number.
To identify the order of elements in a collection, ordinal numerals are used to indicate the order of elements within a collection. These numbers can also be used to identify items in collections.
You can locate regular and suppletive numbers to ordinal numbers.
Regular ordinals are made by prefixing a cardinal number with the suffix -u. The number is then entered into words. A hyphen is then added to it. There are also additional suffixes. The suffix “nd”
is used to signify numbers that have a two numbers. The suffix “th” can refer to numbers with endings of 4 and 9.
The addition of words that have the -u, -e, or–ie suffix creates suffixtive ordinals. This suffix is utilized to count and is usually bigger than the normal one.
Limit of ordinal importance
Ordinal numbers that do not exceed zero are the maximum for ordinal numbers. Limit ordinal quantities suffer from one disadvantage: they do not have the possibility of having a maximum element. They
may be created by joining sets that are empty without maximum elements.
Infinite transfinite-recursion definitions use limited ordinal number. According to the von Neumann model, every infinite cardinal number also functions as an ordinal limit.
A limit ordinal number is the total of all ordinals that are beneath it. Limit ordinal numbers are simple to calculate with arithmetic. However, they may also be expressed in natural numbers.
The data are organized in a specific order, using ordinal numbers. They are used to explain the numerical location of an object. They are often used in set theories and math. Despite being in the
same class however, they are not considered to be natural numbers.
In the von Neumann model, a well-ordered collection is employed. Assume that fy fy is an element of a function g’ that is specified as a singular function. In the event that g’ fulfills the
requirements that g’ be an limit ordinal when there is only one subfunction (i, II).
The Church-Kleene oral is an limit order in the same way. Limit ordinals are properly-ordered collection of smaller ordinals. It has a nonzero ordinal.
Examples of stories that use normal numbers
Ordinal numbers are often used to show the relationship between objects and entities. They are essential in organising, counting or ranking reasons. They are able to define the location of the object
as well as the order in which they are placed.
The ordinal numbers are usually indicated with the letter “th”. Sometimes, however, the letter “nd” is able to be substituted. Titles of books typically include ordinal numerals.
Ordinary numbers are often expressed as words however they are usually employed in list format. They are also available in the form of numbers or acronyms. In comparison, the numbers are easier to
comprehend than the cardinals.
Three different types of ordinal numbers are offered. These numbers can be learned more by playing games, exercises, and other activities. You can enhance your math skills by understanding more about
these concepts. As a fun and easy method of improving your arithmetic skills You can try a coloring exercise. To assess your progress you can use a simple coloring page.
Gallery of Ordinal Numbers Grade 3 Worksheet
Great Free Ordinal Numbers Worksheets Grade 3 Aglocomoonjaycomunity
3rd Grade Ordinal Numbers Worksheet Grade 3 Thekidsworksheet
Ordinal Numbers Online Exercise For Grade 3
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Year 4 Division Worksheet - Divisonworksheets.com
Year 4 Division Worksheet
Year 4 Division Worksheet – Divide worksheets can be used to help your child learn and learn division. Worksheets are available in a vast selection, and you could even create your own. These
worksheets are amazing because they can be downloaded for free without cost and adapted to your specifications. These worksheets are ideal for kindergarteners and first-graders.
Two can produce massive quantities
The worksheets can be used to aid children in the division of large numbers. Sometimes, the worksheets only accommodate two or three, four or more divisors. This method doesn’t require children to be
concerned about forgetting to divide large numbers or making mistakes when using times tables. There are worksheets available online or download them on your computer to assist your child in
developing the mathematical skills required.
Multi-digit division worksheets are used by kids to test their skills and increase their knowledge. It’s an essential mathematical skill that is required for many calculations in daily life as well
as more complex mathematical concepts. These worksheets aid in establishing the concepts by using interactive questions, games and exercises that are based on the division of multidigit numbers.
It’s not easy for students to split huge numbers. These worksheets use a common algorithm, as well as step-by-step instructions. They may not provide the level of understanding they want through
these exercises. For teaching long division, one strategy is to use bases ten blocks. The steps to learn should make long division easy for students.
Students can learn to divide of large numbers by using many practice questions and worksheets. These worksheets include fractional calculations in decimals. You can even find worksheets for
hundredsths. These are particularly useful for understanding how to divide huge sums of money.
Sort the numbers into small groups.
It can be difficult to assign numbers to small groups. Although it may sound great on paper, many small group facilitators aren’t happy with this procedure. It truly reflects how our bodies develop,
and the procedure could aid in the Kingdom’s limitless development. In addition, it encourages others to help the undiscovered and new leadership to assume the reigns.
It can also be useful for brainstorming. It is possible to form groups of people who have similar experiences and traits. This lets you come up with creative ideas. Once you’ve created your groups,
you should introduce everyone to you. It’s a good way to encourage creativity and innovative thinking.
Divide big numbers into smaller numbers is the fundamental operation of division. It’s a good option when you need equal items for multiple groups. It is possible to break down the class into groups
of five students. Combine these groups and you get the original 30 students.
Be aware that when you divide numbers, there’s a divisor and the quote. Dividing one number by another produces “ten/five,” while divising two by two gives the same result.
Powers of ten shouldn’t be used to solve large numbers.
To facilitate comparison of the large number of numbers, we can divide them into powers of 10. Decimals are an important part of shopping. They are usually found on receipts, price tags, and food
labels. They are used by petrol pumps to indicate the cost per gallon as well as the quantity of fuel being delivered via an sprayer.
There are two methods to divide a large number into its power of ten. One is to shift the decimal point to the left, while the other method is to multiply the number by 10-1. The other method
utilizes the power of ten’s associative feature. After you have learned the associative property of powers of 10, you can divide an enormous number into smaller powers that are equal to 10.
The first method involves mental computation. When you multiply 2.5 by the power of 10, you’ll see patterns. The decimal point shifts to the left as the power of 10 increases. This concept is easy to
understand and can be applied to any issue, no matter how difficult.
By mentally dividing large numbers into powers of ten is another method. It is easy to express massive numbers using scientific notation. When writing in scientific notation, large numbers must be
expressed as positive exponents. It is possible to shift the decimal point by five spaces to one side and convert 450,000 to 4.5. To divide a large number into smaller numbers of 10, you could apply
the factor 5. or break it down into smaller numbers of 10.
Gallery of Year 4 Division Worksheet
Division Year 4 Maths Worksheets Pdf Kidsworksheetfun
Division Worksheet Year 4 Pdf Julia Winton s English Worksheets
8 Best Year 4 Maths Images On Pinterest Math Activities Math
Leave a Comment | {"url":"https://www.divisonworksheets.com/year-4-division-worksheet/","timestamp":"2024-11-04T04:45:19Z","content_type":"text/html","content_length":"65071","record_id":"<urn:uuid:f2bf0dcc-4231-41a3-b3a9-d1541c0a1230>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00232.warc.gz"} |
The ASL statement on women in logic
Life got in the way of this sequence of blog posts, but now I'm back, and I have the last analysis ready. Today I'm going to consider the
ASL's statement on women in logic
, which was adopted at the Annual Meeting in 2012:
Logic benefits when it draws from the largest and most diverse possible pool of available talent. We at the ASL would therefore like to add our voice to the growing list of initiatives launched by
organizations in the various science, technology, engineering and mathematics fields aimed at correcting the gender imbalance in those fields.
Female students and young researchers may be concerned about entering logic, where few senior women occupy visible roles. The atmosphere in classes and seminars can feel unwelcoming, and many young
women have practical questions about managing a career and personal interests.
The ASL therefore states in the strongest possible terms that it welcomes the participation of women in logic and in particular in the activities of the Association. Accordingly, the ASL Council has
adopted a statement urging those responsible for appointments and conference programs to pay attention to gender balance.
It's Women's History Month, and it seems appropriate to ask whether the proportion of women who are invited to speak at ASL meetings has noticeably changed since this statement was issued.
To figure this out, I'm comparing the proportions of female invited speakers in each of the four main conference series in the few years since this statement was issued to the proportions of female
invited speakers in the corresponding time span before it.
Annual Meetings Logic Colloquium
Men Women Men Women
2009-2012 29 7 57 9
2013-2016 23 8 43 9
AMS/ASL APA/ASL
Men Women Men Women
2010-2012 17 4 21 1
2014-2016 14 6 19 3
As you can see, I considered different sets of years for my pre- and post-statement counts depending on the conference series. I made these decisions based on whether the speakers for the 2013
meetings would have been invited by the time the statement was issued. The speakers for the 2013 Logic Colloquium and Annual Meeting would have been invited after the 2012 Annual Meeting, so I
compared the 2009-2012 and 2013-2016 intervals for these, and the speakers for the 2013 AMS/ASL and APA/ASL meetings might already have been invited by the time of the 2012 Annual Meeting, so I
considered the 2010-2012 and 2014-2016 intervals for each of those. In each of the latter two cases, it was easy to sidestep the question of what to do about 2013 itself: the tallies for the AMS/ASL
meetings are identical for 2010-2012 and 2011-2013, and there were no invited talks at the 2013 APA/ASL meeting.
I used a chi-square test to determine whether the proportion of female speakers in the few years before the ASL statement was adopted is statistically distinguishable from the proportion of female
speakers since then.
When I carried out this analysis, I found that the p-value for the Annual Meetings was 0.7422 and the p-value for the Logic Colloquium was 0.7697. This gives us no statistical evidence suggesting
that the ASL's statement has made any difference.
However, there aren't enough female plenary speakers in these time spans to carry out this analysis for the AMS/ASL and APA/ASL meetings. A reliable chi-square analysis can't be done unless there are
at least 5 people in each category. Only 4 women gave plenary talks at the AMS/ASL meetings during 2010-2012, while only 1 woman gave an invited talk at the APA/ASL meeting during 2010-2012 and only
3 did during 2014-2016. While these numbers are very low, we should also recall that the time intervals for these series are one year shorter than the intervals for the others. However, the
(relatively unreliable) p-values for the AMS/ASL and APA/ASL meetings are 0.6509 and 0.6, respectively, which suggests that adding one year to the pre- and post-statement intervals is unlikely to
result in any meaningful statistical difference.
Summary: In the conference series in which enough women have given talks in these time intervals, there is no statistically significant difference in the rates at which women have spoken in the years
immediately before and the years after the ASL statement was adopted, suggesting that the statement has not had an effect on speaker selection. However, this analysis is only possible in two of the
four series.
3 comments:
1. hmm, always good to have the data. But surely the "statement" does have an effect in giving us the support to try to change things, no?
2. But is there any practical support behind the statement? I would almost rather that they not acknowledge the problem instead of issuing a statement and then saying through their inaction that,
while they know there is a problem, it isn't worth exerting any effort to try to do anything about it.
1. you do have a point. I wonder if there is a mechanism to force the council to pay attention to an issue...we could try to find the names and addresses of all the female members of the ASL and
write an open letter to the council, signed by as many as we could get, perhaps? thinking out loud here... | {"url":"https://blog.womeninlogic.org/2017/03/the-asl-statement-on-women-in-logic.html","timestamp":"2024-11-09T13:10:05Z","content_type":"text/html","content_length":"64495","record_id":"<urn:uuid:db930dd5-033e-46ea-bda0-a893065d2f96>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00130.warc.gz"} |
Maths Quiz for Class 1 Shapes
Maths Quiz for Class 1 Shapes
In this post, you will find 20 online maths quiz questions for class 1 shapes. Maths quiz will take around 10 minutes or less to complete it.
Let’s complete 20 easy maths quiz multiple choice questions based on class 1 shapes.
Question 1: The edge of a table is an example of ___________ line.
A) slant
B) vertical
C) straight
D) curved
Explanation: The edges of a table are straight.
Question 2: How many sides are there in a triangle?
A) 3
B) 4
C) 5
D) 6
Explanation: A triangle has 3 sides.
Question 3: How many corners are there in a square?
A) 4
B) 3
C) 5
D) 6
Explanation: A square has 4 corners.
Question 4: How many angles are there in a rectangle?
A) 3
B) 4
C) 5
D) 6
Explanation: A rectangle has 4 angles.
Question 5: A cylinder has _______ faces.
A) 2
B) 3
C) 1
D) 4
Explanation: A cylinder has 3 faces.
Question 6: A cone has ______ edges.
A) 2
B) 0
C) 1
D) 3
Explanation: A cone has 1 curved edge.
Question 7: A cylinder has _______ edges.
A) 3
B) 2
C) 1
D) 4
Explanation: A cylinder has 2 curved edges.
Question 8: A cone can _________.
A) slide only
B) roll and slide both
C) slide only
D) neither roll nor slide
Explanation: A cone can roll and slide both.
Question 9: A sphere can _________.
A) roll
B) slide
C) roll and slide both
D) neither roll and slide
Explanation: A sphere can roll.
Question 10: How many faces are there in a cube?
A) 4
B) 8
C) 6
D) 12
Explanation: A cube has 6 faces.
Question 11: How many edges are there in a cuboid?
A) 6
B) 12
C) 8
D) 10
Explanation: A cuboid has 12 edges.
Question 12: How many vertices are there in a cube?
A) 6
B) 8
C) 10
D) 4
Explanation: A cube has 8 vertices.
Question 13: How many vertices are there in a cylinder?
A) 0
B) 1
C) 2
D) 3
Explanation: A cylinder has 0 vertex.
Question 14: How many vertices are there in a cone?
A) 4
B) 2
C) 0
D) 1
Explanation: A cone has 1 vertex.
Question 15: A cuboid has ______ faces.
A) 12
B) 8
C) 6
D) 4
Explanation: A cuboid has 6 faces.
Question 16: A tringle has ______ vertices.
A) 4
B) 3
C) 5
D) 6
Explanation: A tringle has 3 vertices.
Question 17: Electric pole is an example of _________ line.
A) vertical
B) horizontal
C) slant
D) curved
Explanation: Electric pole is an example of vertical line.
Question 18: Edges of a cylinder are examples of _________ lines.
A) slant
B) straight
C) curved
D) vertical
Explanation: Edges of a cylinder are examples of curved lines.
Question 19: Edges of TV screen are examples of ____________ lines.
A) slant
B) straight
C) curved
D) none
Explanation: Edges of TV screen are examples of straight lines.
Question 20: A sphere has ______ face.
A) 4
B) 3
C) 2
D) 1
Explanation: A sphere has 1 face.
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Signs of Dyscalculia In Children: A Handy Checklist for Teachers - Fishyrobb
Most teachers are familiar with common learning disabilities. One of those, dyslexia, has gotten a lot of needed attention in recent years. But just as many, if not more, young people have
difficulties with basic math – which could be a sign of a learning disability called dyscalculia.
This post will explain the most common signs of dyscalculia in children that you might observe in a student. Dyscalculia is considered a specific learning disability covered under IDEA. Some people
call it “math dyslexia” or “number dyslexia”.
You will also find a dyscalculia checklist that you can download and print to refer to as needed.
Learn to recognize the signs of dyscalculia in children.
Signs of Dyscalculia in Children (Preschool to Kindergarten)
These are some of the problems you might notice in very young children who are beginning to show signs of dyscalculia:
• difficulty recognizing numbers
• late learning to count
• difficulty recognizing patterns
• difficulty with sequencing
• struggling to connect numbers to quantities (i.e. 9 represents nine objects)
• losing track when counting objects
• needing to use manipulatives or visual aids to count beyond what is normally expected with younger children (fingers, touch each object when counting, etc)
One important thing to remember is that these are difficulties that become evident following sufficient instruction. You wouldn’t suspect dyscalculia in a four year old who has not been taught how to
count objects or recognize numbers.
Signs of Dyscalculia in Children (1st Grade and Up)
School-age students with dyscalculia may display a wide range of mathematics difficulties which can include:
• extreme difficulty learning how to complete basic math procedures like addition, subtraction, and multiplication
• persistent finger-counting and using fingers to calculate
• having poor number sense and inability to make reasonable estimates
• inability to perform mental math tasks
• difficulty with skip counting and counting backwards
• struggling to interpret word problems and relate them to math calculations
• having a hard time understanding place value
• poor memory and recall of numerical information such as phone numbers
• difficulty learning to count money and make change
• difficulty learning to tell time
• difficulty with time management and estimating how long it will take to complete a task
• difficulty processing visual-spatial information such as graphs, charts, and maps
• avoidance of even short math assignments and tests
• experiencing math anxiety when presented with new math concepts and tasks
How is Dyscalculia Diagnosed?
Teachers are often the first ones to notice signs of dyscalculia in children. But the disability is most accurately diagnosed by an educational psychologist.
The DSM-5 (Diagnostic and Statistical Manual of Mental Disorders from the American Psychiatric Association) defines dyscalculia as a specific learning disorder evidenced by problems with:
• Number sense
• Memorization of basic math facts
• Accurate and fluent calculation
• Accurate math reasoning
The difficulties must have persisted for at least 6 months and fail to improve despite the implementation of appropriate math interventions.
That means if a student is struggling with basic math skills and you observe some of these signs of dyscalculia in children, you will need to go through the same steps you would for dyslexia or any
other learning difficulty.
First, you will need to implement specific math interventions to try to remedy the problem. Keep accurate data that shows what interventions you put in place, how often they were used, and how the
student responded. This can be done during your walk-to-intervention time or in guided math groups.
If there is little to no improvement after a period of time (usually 5 to 6 weeks), the next step is to start the referral process to your school psychologist. He or she may provide you with
checklist screeners, different interventions to try, or may schedule diagnostic assessments.
⏬ Download these free printable checklists to keep in your teacher files. They include the same signs of dyscalculia in children as shown above in a handy single-page format.
Although math struggles are often a lifelong problem, a diagnosis of dyscalculia is the first step toward getting a student the right help. Special education services, extra help with a private
tutor, and accommodations like providing extra time and visual supports are all things that can give the student the support needed to be successful math students. | {"url":"https://www.fishyrobb.com/post/signs-of-dyscalculia-screening-checklist/","timestamp":"2024-11-13T11:26:56Z","content_type":"text/html","content_length":"100742","record_id":"<urn:uuid:d08eca6e-015b-40db-bf06-096da7cfa98a>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00044.warc.gz"} |
Course Catalog
Majors, Minors & Degrees:
Mathematics is an intriguing field that helps us understand the world by modeling physical phenomena quantitatively and employing sound reasoning skills. A degree in mathematics offers students many
options, including careers in business, government, industry, and teaching. Some students also pursue post-graduate education in mathematics, medicine, law, and science. Students studying mathematics
at Nebraska Wesleyan University have opportunities to engage themselves fully in their education by working collaboratively with their peers, conducting research with faculty, teaching in the Math
Tutoring Center, grading for courses, attending Math Club events, and presenting research at conferences.
Department Learning Outcomes
Majors will be able to:
1. Demonstrate problem-solving skills.
2. Prove theorems.
3. Learn independently.
4. Explain mathematics in oral and written form.
5. Use computer-based technology to assist in solving problems.
6. Pursue employment or further study. | {"url":"https://catalog.nebrwesleyan.edu/cc/2018-2019/department/330859","timestamp":"2024-11-03T07:36:59Z","content_type":"text/html","content_length":"78150","record_id":"<urn:uuid:c5422c11-a25c-40ec-909b-23d27f1369f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00424.warc.gz"} |
Find the length of each side of the given regular dodecagon.-Turito
Are you sure you want to logout?
Find the length of each side of the given regular dodecagon.
The correct answer is: - 16
• A regular dodecagon has 12 sides equal in length and all the angles have equal measures, all the 12 vertices are equidistant from the center of dodecagon.
• A regular dodecagon is a symmetrical polygon.
• We have been given in the question figure of a regular dodecagon
• We have also been given the two sides of it that is -
• We have to find length of each side of the regular dodecagon.
We have given a regular dodecagon with sides represented as
Since, It is regular, then all sides are equal
2x - 1 = 9x + 15
7x = - 16
X can not be negative
Wrong data
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Modeling of a magneto-rheological (MR) damper using genetic programming
This paper is based on the experimental study for design and control of vibrations in automotive vehicles. The objective of this paper is to develop a model for the highly nonlinear
Magneto-Rheological (MR) damper to maximize passenger comfort in an automotive vehicle. The behavior of the MR damper is studied under different loading conditions and current values in the system.
The input and output parameters of the system are used as a training data to develop a suitable model using Genetic Algorithm. To generate the training data, a test rig similar to a quarter car model
was fabricated to load the MR damper with a mechanical shaker to excite it externally. With the help of the test rig the input and output parameter data points are acquired by measuring the
acceleration and force of the system at different points with the help of an impedance head and accelerometers. The model is validated by measuring the error for the testing and validation data
points. The output of the model is the optimum current that is supplied to the MR Damper, using a controller, to increase the passenger comfort by minimizing the amplitude of vibrations transmitted
to the passenger. Besides using this model for cars, bikes and other automotive vehicles it can also be modified by re-training the algorithm and used for civil structures to make them earthquake
1. Introduction
Isolation of the forces transmitted by external application is the most important function of a suspension system. The suspension system comprises of a spring element and a dissipative element, which
when placed between the object to be protected and the excitation, reduces the vibration transmitted to the object. Suspension systems range from active to passive suspensions. MR damper lies in
between this range and behaves like a semi-active suspension system. The damping of a passive suspension system is a property of the system and cannot be varied, whereas in an active suspension
system the damping of the system can be altered by using an actuator to give it an external force. This external force helps in improving the ride quality. The shortcoming of this model is that it
requires considerable amount of external power source, it is costly and is difficult to incorporate in the system due to the added mass. A variation of the active suspension system is the semi-active
or the adaptive suspension system. In these systems, the damping of the system is varied by controlling the current thereby, changing the viscous properties of the damping elements in the suspension
system. PID neural network controller, is one such controller used to develop a model to predict the displacement and velocity behavior of the MR damper [1]. In comparison to active suspension semi
active suspension systems’ power consumption is considerably less. The magneto-rheological (MR) dampers and electro-rheological dampers are the most common examples of semi active dampers. In the
past few years, research on MR dampers has improved its capabilities and reduced the gap between adaptive suspension system and truly active suspension systems. Structurally, the MR damper is similar
to a simple fluid damper, except that the viscosity of the fluid in the MR damper can be changed by altering the current in the system that induces a magnetic field. The MR fluid is a non- Newtonian
fluid composed of mineral oil with suspended iron nanoparticles. When there is no magnetic flux (zero current), the MR damper behaves like a normal fluid damper in which the iron nanoparticles are
randomly oriented. When a magnetic field is applied to the MR damper, the iron nanoparticles align themselves along the magnetic flux and form chains which makes the fluid partly semi-solid. These
particles help in reinforcing the damping by forming chains in the oil that obstructs the movement of the oil as the magnetic field developed is in the direction perpendicular to the movement of the
oil. Hence increasing the magnetic field increases damping in the system. The behavior of the MR damper can be characterized by its highly non-linear motion [2-6]. The force vs velocity plot forms a
hysteresis loop depicting the non-linear character of the MR damper. There exist many different parametric models like Bingham model, Bingham Body Model, Lee Model, Spencer Model, Bouc-Wen Model and
Gamota-Filisko Model that portray the behavior of the MR Damper. These models are difficult to implement throughout the working range of the MR damper due to the hysteresis and jump type phenomena
resulting from the specific properties, like visco-plastic, visco-elasto-plastic and visco-elastic, of the MR fluid. It is also difficult to find a solution for the equations for some of the defined
models due to the numerical and analytical complexity of the model equation. These drawbacks prevent the models from being implemented as governing equations in real time controllers [7]. In the low
speed range the Bingham model can predict the rigid plastic behavior better than the involution model when the MR damper is loaded with a sinusoidal input. When triangular loading is used on the MR
damper, the involution model predicts the model better than the Bingham model [8]. The deformation in the hysteresis loop of the force- velocity and force-displacement graphs, deviates from the
Bouc-Wen model due to the force lag phenomenon in the MR damper. The modification of the Bouc-Wen model, the Bouc-Wen-Baber-Noori model, can to a certain extent describe the pinching hysteretic
behavior [9]. The evolutionary variable equation for the Modified Bouc-Wen model, depends on 4 parameters $A$, $\beta$, $\gamma$ and $n$, where $A$, $\beta$ and $\gamma$ control the shape and size of
the hysteresis and the parameter n controls the smoothness of the transition from elastic to plastic region [10]. Genetic algorithm assisted inverse method and nonlinear-least square error
optimization in MATLAB can be used to identify the parameters and develop a model [11, 12].
The literature [7-10] suggests that the use of deterministic and analytical models is unable to capture the dynamics due to hysteresis and jump type phenomena resulting from the specific properties,
like visco-plastic, visco-elasto-plastic and viscoelastic, of the MR fluid. This has motivated the authors to work on applying the method based on evolutionary principle such as genetic programming
(GP) to formulate the explicit model. GP does not assume any assumption of the process dynamics before in hand and nor it requires any process information. GP have self-adapting ability to fit the
given data and generate its explicit (functional) function. Therefore, the present work further explores the ability of the GP to formulate the models to capture the dynamics of properties of MR
2. Genetic programming
Genetic programming is an important tool used to build models for dynamic systems. It includes a variety of modelling tools and models. This method has considerable advantages over other modeling
tools. In genetic programming the first stage involves generation of initial population models. In this stage the functional, terminal set and population size are defined. The functional set is a
matrix of the functions that are used to create the final model. Terminal set is the matrix of the input and output parameters. The population size is the number of models in the first generation.
In the next stage the performance of the models generated in the previous stage are calculated. Based on the data given, the fitness, root mean square error (RMSE) and mean absolute fitness error
(MAPE) are calculated using the Eqs. (1), (2) and (3). If any model has an error below a user defined limit we select that model, else we move the second generation.
$Fitness=\sum \frac{{\left(ActualOutput-PredictedOutput\right)}^{2}}{numberofdatapoints},$
$RMSEFitness=\sqrt{\frac{\sum \left({\left(ActualOutput-PredictedOutput\right)}^{2}\right)}{numberofdatapoints}},$
$MAPEFitness=\sqrt{\frac{\sum \left({\left|ActualOutput-PredictedOutput\right|\right)}^{2}\right)}{\left|ActualOuput\right|}}×100.$
In the second generation, new set of models are developed by applying operators such as crossover, as shown in Fig. 1 and mutation, as shown in Fig. 2, to the models developed in the first stage of
generation. In cross-over (Fig. 1), the subtree (part of equation) from both the trees (equations) is selected and swapped. Fig. 1 shows that the subtree is selected from equations A and B and
swapped. The swapping is shown by the dotted lines marked in bold in Fig. 1. The cross-over results in formation of the new equations (Equation A1 and B1). In mutation (Fig. 2), a tree (equation) is
generated randomly, which replaces the subtree (part of equation) of the selected tree (equation). Fig. 2 shows the mutation mechanism, where the Eq. (3) is formed from the mutation of Eqs. (1) and
These models are then ranked. The least error model then proceeds to the next stage of generation. The error for these models are then evaluated. If it is within the user defined error the model is
selected, else the iterative process continues. Once the genetic programming ends, the best model from the population is selected based on minimum error.
Fig. 1Crossover mechanism on models
3. Experimental setup
The experimental setup consists of three components: external actuation system, data acquisition system and the controller. The external actuation for the system was fabricated to excite the damper
with the help of the electrodynamic shaker. The system was designed to closely depict a quarter car model. The shaker transmits the road disturbances to the lower plate by means of a stringer as seen
in the Fig. 3. The lower plate is equivalent to the tire of the vehicle in the quarter car model. The lower plate is coupled with the upper plate with a LORD manufactured MR Damper RD-8040-1 which is
a part of the suspension system. The upper plate behaves as the chassis for vehicle. The MR damper is clamped to the upper and lower plates by means of an L clamp.
Fig. 3Experimental setup of MR damper
The electro-dynamic shaker (Modal shop Model 2110E) with a load capacity of 489 N (sine-peak) is connected to the linear power amplifier (Crystal Instruments Inc, USA, Model 2050E09) which provides
the shaker with the input profile that is given to the system using the Spider 81 data acquisition system (Crystal Instruments Inc., USA) and the associated software (EDM) as shown in Fig. 4. The
vibration signals, taken as inputs for genetic programming, are measured using sensors. The signal is measured on the shaker and the lower plate using accelerometers PCB Electronics, Model No.
'352C34 LW 155857' of sensitivity 101.3 mV/g and Model No. “352C68 SN 92017” of sensitivity 102.2 mV/g respectively. To measure the force transmitted and the acceleration signals at the upper plate
an impedance head, PCB Electronics, Model No. “288D01 SN 3176” is used of sensitivity 98.73 mV/LBF for force and 101.3 mV/g for acceleration. The MR damper is operated using an external power source.
The current in the damper is varied using a potentiometer type device called Wonder Box RD-3002-03 manufactured by Lords Corporation, UK as shown in Fig. 5. The current is measured using a
The signal parameters from these sensors are used as inputs for the algorithm and the current that is varied using the potentiometer is used as the output. This current can be directly provided to
the control system to create a feedback loop.
Fig. 4Spider 81 and amplifier
4. Experimentation
The training set for the genetic programming model is obtained from experimentation using the setup described in section 3. In the experimental setup system, there are 6 factors that are monitored to
develop the training set. The experimentation is carried out by varying the frequency and current parameters. The data collected is divided into three sets i.e. 70 % is training set, 15 % is testing
set and remaining 15 % is validation. Using this to training set the final model is developed which best fits the data having minimum error. Table 1 provides the configuration of various experiments
conducted by varying the parameters.
The input parameters are recorded for 10 seconds of the response. These signatures are as shown in Fig. 6.
Table 1Different test configurations
Sl. No. Description Frequency Steps Current
1 Dwell 5-7 Hz 0.1 Hz 0 A
2 Dwell 5-7 Hz 0.1 Hz 0.25 A
3 Dwell 5-7 Hz 0.1 Hz 0.50 A
4 Dwell 5-7 Hz 0.1 Hz 0.75 A
5 Dwell 5-7 Hz 0.1 Hz 1.0 A
Fig. 6Signatures of acceleration of road profile, lower and upper plate, and signature of force transmitted to upper plate
a) Signatures of acceleration of road profile (mm/s^2) $wrt$time
b) Signatures of acceleration of lower plate (mm/s^2) $wrt$ time
c) Signatures of acceleration of upper plate (mm/s^2) $wrt$ time
d) Signature of force transmitted ($N$) to upper plate
In the first set of tests the current is kept constant and the frequency is increased from 5 Hz to 7 Hz in steps of 0.1 Hz. The data is collected at a sampling rate of 20.48 kHz. In the similar
fashion the data set is obtained for all the current values. This data is then consolidated to form the training and testing matrix. There are 4 input parameters taken into consideration in this
1) Frequency of excitation;
2) Acceleration of the shaker;
3) Relative acceleration of the upper level (chassis) with respect to the lower level (tire);
4) Force transmitted to chassis.
The current in the damper coils is taken as the output parameter. Using these input-output parameters the genetic algorithm is trained.
5. Results and discussion
We have conducted sufficient preliminary research on the appropriate parameter settings for implementing GP program efficiently on this particular problem. We have performed the trial-and-error
procedure to get the appropriate parameter settings. The procedure comprises of the following range of settings. Range of Settings varied from:
1) Population size: 200 to 400;
2) Number of generations: 20 to 100;
3) Maximum tree depth: 5 to 25;
4) Crossover, mutation and reproduction: [0.80-0.90,0.05-0.15,0.05-0.10].
For each of the above-mentioned range, the GP is performed. Based on the minimum RMSE on the training data, we have decided to choose the following settings as given in Table 2. The best model
selected from the models generated in the different stages is based on the performance of the model. The algorithm is trained using the parameters shown in the Table 2.
The error for the models obtained in each generation stage is calculated. The model with the least error is then selected. In the results obtained from genetic programming test, three models were
formed. The error for all three models are given in Table 3. The model with the least error is chosen from these. This error for this model is shown in red color in Fig. 7. The model equation is:
where, ${x}_{1}$ is frequency of excitation, ${x}_{2}$ is acceleration of the shaker, ${x}_{3}$ is relative acceleration of the upper level (chassis) with respect to the lower level (tire), ${x}_{4}$
is force transmitted to chassis.
The parameters for the best model from the population for each generation of the best fitness model is listed in Table 3.
The convergence of the algorithm is based on the selection of the model when the RMSE is minimum. For every run, the best model is selected based on the minimum training RMSE. If there are models
with same minimum value of RMSE and different size (number of nodes), the best model with smaller size is selected among them. If there are models with same minimum value of RMSE and same size, then
a best model is selected randomly among them. In this way, for the specified number of runs, the best model is selected for each of the run. The final model among these best models is selected again
based on the minimum value of RMSE among the runs. In this study, a total of 5 runs are chosen. The best model has training RMSE of 0.11246 and is chosen for finding the equation of output (current).
Fig. 7Error in the best model
Table 2Parameter for the genetic program
Population size 300
Number of generations 100
Tournament size 50
Max tree depth 25
Max nodes per tree Infinite
Using function set [TIMES MINUS PLUS PLOG TANH TAN SIN COS EXP]
Number of inputs 4
Max genes 35
Constants range [–20 20]
Mutate rate 1.00E-01
Crossover rate 8.50E-01
Reproduction rate 5.00E-02
Table 3Fitness parameters for the best model of each generation
Generation Training RMSE Training MAPE Validation RMSE Validation MAPE Test RMSE Test MAPE No. of nodes Depth
1 0.14980 65535 0.498173 55.12169 0.77282 66.7576 139 15
2 0.11188 65535 0.499335 54.0697 0.69112 65.3761 250 13
3 (Best) 0.11246 65535 0.550925 56.48313 0.72122 59.1360 247 16
6. Conclusions
This work is the first of its kind that develops a model to predict the current value to adjust the damper force for the given input parameters, using actual experimental data, in order to maintain
the driver acceleration. This model equation can be directly used to design a controller using acceleration, force and driving frequency.
• Liu Wei, et al. Experimental modeling of magneto-rheological damper and PID neural network controller design. 6th International Conference on Natural Computation (ICNC), Vol. 4, 2010.
• Gandhi Farhan, Inderjit Chopra A time-domain non-linear viscoelastic damper model. Smart Materials and Structures, Vol. 5, Issue 5, 1996, p. 517.
• Kamath Gopalakrishna M., Norman Werely M. Nonlinear viscoelastic-plastic mechanisms-based model of an electrorheological damper. Journal of Guidance, Control, and Dynamics, Vol. 20, Issue 6,
1997, p. 1125-1132.
• Snyder Rebecca A., Kamath Gopalakrishna M., Wereley Norman M. Characterization and analysis of magneto-rheological damper behavior due to sinusoidal loading. SPIE’s 7th Annual International
Symposium on Smart Structures and Materials, International Society for Optics and Photonics, 2000.
• Stanway Roger, Sims Neil D., Johnson Andrew R. Modeling and control of a magneto-rheological vibration isolator. SPIE's 7th Annual International Symposium on Smart Structures and Materials,
International Society for Optics and Photonics, 2000.
• Wereley N. M., Pang L., Kamath G. M. Idealized hysteresis modeling of electrorheological and magnetorheological dampers. Journal of Intelligent Material Systems and Structures, Vol. 9, Issue 8,
1998, p. 642-649.
• Sapiński Bogdan, Jacek Filuś Analysis of parametric models of MR linear damper. Journal of Theoretical and Applied Mechanics, Vol. 41, Issue 2, 2003, p. 215-240.
• Fujitani Hideo, et al. Dynamic performance evaluations of 200 kN magneto-pheological damper. Technical Note of National Institute for Land and Infrastructure Management, Vol. 41, 2002, p.
• Braz-Cesar Manuel T., Rui Barros Experimental and numerical analysis of MR dampers. 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (3rd
South-East European Conference on Computational Mechanics), 2013.
• Peng G. R., et al. Modelling and identifying the parameters of a magneto-rheological damper with a force-lag phenomenon. Applied Mathematical Modelling, Vol. 38, Issue 15, 2014, p. 3763-3773.
• Giuclea Marius, et al. Modelling of magnetorheological damper dynamic behaviour by genetic algorithms based inverse method. Proceedings of the Romanian Academy, Series A, Vol. 5, Issue 1, 2004,
p. 5563-5572.
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About this article
Mechanical vibrations and applications
MR Damper
genetic programming
Author Contributions
Pravin Singru has guided this work from concept of experimental setup to a real setup fabrication, testing and writing the paper. Ayush Raizada has worked on experimental setup fabrication and
testing of GP code for modelling. Vishnuvardhan Krishnakumar has been instrumental in taking performing various experiment to generate huge amount of data required for the GP code. Akhil Garg has
developed the GP code used in this paper. K. Tai has guided Dr. Akhil to develop the GP code used in this paper.
Copyright © 2017 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/17828","timestamp":"2024-11-08T02:58:56Z","content_type":"text/html","content_length":"153066","record_id":"<urn:uuid:1f09a83d-4b42-4b18-990a-4f8dd8188f07>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00306.warc.gz"} |
In 1980, Burr conjectured that every directed graph with chromatic number $2k-2$ contains any oriented tree of order $k$ as a subdigraph. Burr showed that chromatic number $(k-1)^2$ suffices, which
was improved in 2013 to $\frac{k^2}{2} - \frac{k}{2} + 1$ by Addario-Berry et al. In this talk, we give the first subquadratic bound for Burr's … | {"url":"https://dimag.ibs.re.kr/events/category/seminar/dms/2024-09/","timestamp":"2024-11-14T21:20:26Z","content_type":"text/html","content_length":"298525","record_id":"<urn:uuid:db11e198-a778-4ed7-bced-53f7e7b7c42f>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00572.warc.gz"} |
Kiloyards to Angstroms Converter
β Switch toAngstroms to Kiloyards Converter
How to use this Kiloyards to Angstroms Converter π €
Follow these steps to convert given length from the units of Kiloyards to the units of Angstroms.
1. Enter the input Kiloyards value in the text field.
2. The calculator converts the given Kiloyards into Angstroms in realtime β using the conversion formula, and displays under the Angstroms label. You do not need to click any button. If the input
changes, Angstroms value is re-calculated, just like that.
3. You may copy the resulting Angstroms 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 Kiloyards to Angstroms?
The formula to convert given length from Kiloyards to Angstroms is:
Length[(Angstroms)] = Length[(Kiloyards)] / 1.0936132999999999e-13
Substitute the given value of length in kiloyards, i.e., Length[(Kiloyards)] in the above formula and simplify the right-hand side value. The resulting value is the length in angstroms, i.e., Length
Calculation will be done after you enter a valid input.
Consider that a race track is 2 kiloyards long.
Convert this distance from kiloyards to Angstroms.
The length in kiloyards is:
Length[(Kiloyards)] = 2
The formula to convert length from kiloyards to angstroms is:
Length[(Angstroms)] = Length[(Kiloyards)] / 1.0936132999999999e-13
Substitute given weight Length[(Kiloyards)] = 2 in the above formula.
Length[(Angstroms)] = 2 / 1.0936132999999999e-13
Length[(Angstroms)] = 18287999972202.242
Final Answer:
Therefore, 2 kyd is equal to 18287999972202.242 A.
The length is 18287999972202.242 A, in angstroms.
Consider that a golf course has a fairway measuring 1.5 kiloyards.
Convert this distance from kiloyards to Angstroms.
The length in kiloyards is:
Length[(Kiloyards)] = 1.5
The formula to convert length from kiloyards to angstroms is:
Length[(Angstroms)] = Length[(Kiloyards)] / 1.0936132999999999e-13
Substitute given weight Length[(Kiloyards)] = 1.5 in the above formula.
Length[(Angstroms)] = 1.5 / 1.0936132999999999e-13
Length[(Angstroms)] = 13715999979151.682
Final Answer:
Therefore, 1.5 kyd is equal to 13715999979151.682 A.
The length is 13715999979151.682 A, in angstroms.
Kiloyards to Angstroms Conversion Table
The following table gives some of the most used conversions from Kiloyards to Angstroms.
Kiloyards (kyd) Angstroms (A)
0 kyd 0 A
1 kyd 9143999986101.121 A
2 kyd 18287999972202.242 A
3 kyd 27431999958303.363 A
4 kyd 36575999944404.484 A
5 kyd 45719999930505.6 A
6 kyd 54863999916606.73 A
7 kyd 64007999902707.84 A
8 kyd 73151999888808.97 A
9 kyd 82295999874910.1 A
10 kyd 91439999861011.2 A
20 kyd 182879999722022.4 A
50 kyd 457199999305056.06 A
100 kyd 914399998610112.1 A
1000 kyd 9143999986101122 A
10000 kyd 91439999861011220 A
100000 kyd 914399998610112100 A
A kiloyard (ky) is a unit of length equal to 1,000 yards or approximately 914.4 meters.
The kiloyard is defined as one thousand yards, providing a convenient measurement for longer distances that are not as extensive as miles but larger than typical yard measurements.
Kiloyards are used in various fields to measure length and distance where a scale between yards and miles is appropriate. They offer a practical unit for certain applications, such as in land
measurement and engineering.
An angstrom (Γ ) is a unit of length used primarily in the fields of physics and chemistry to measure atomic and molecular dimensions. One angstrom is equivalent to 0.1 nanometers or approximately 1
Γ 10^(-10) meters.
The angstrom is defined as one ten-billionth of a meter, making it a convenient unit for expressing very small lengths, such as atomic radii and bond lengths.
Angstroms are widely used in crystallography, spectroscopy, and materials science to describe the scale of atomic structures and wavelengths of electromagnetic radiation. The unit facilitates precise
measurements and understanding of microscopic phenomena.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Kiloyards to Angstroms in Length?
The formula to convert Kiloyards to Angstroms in Length is:
Kiloyards / 1.0936132999999999e-13
2. Is this tool free or paid?
This Length conversion tool, which converts Kiloyards to Angstroms, is completely free to use.
3. How do I convert Length from Kiloyards to Angstroms?
To convert Length from Kiloyards to Angstroms, you can use the following formula:
Kiloyards / 1.0936132999999999e-13
For example, if you have a value in Kiloyards, you substitute that value in place of Kiloyards in the above formula, and solve the mathematical expression to get the equivalent value in Angstroms. | {"url":"https://convertonline.org/unit/?convert=kiloyards-angstroms","timestamp":"2024-11-09T14:30:26Z","content_type":"text/html","content_length":"91311","record_id":"<urn:uuid:5d0b2733-d388-4b19-8888-ea0a732bd9e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00096.warc.gz"} |
mumott provides two different implementations of projectors suitable for SAXS tomography. While they should yield nearly equivalent results, they differ with respect to the resources they require.
SAXSProjectorCUDA and SAXSProjector implement an equivalent algorithm for GPU and CPU resources, respectively.
class mumott.methods.projectors.SAXSProjector(geometry)[source]¶
Projector for transforms of tensor fields from three-dimensional space to projection space using a bilinear interpolation algorithm that produces results similar to those of SAXSProjectorCUDA
using CPU computation.
geometry (Geometry) – An instance of Geometry containing the necessary vectors to compute forwared and adjoint projections.
adjoint(projections, indices=None)[source]¶
Compute the adjoint of a set of projections according to the system geometry.
○ projections (ndarray[Any, dtype[TypeVar(_ScalarType_co, bound= generic, covariant=True)]]) – An array containing coefficients in its last dimension, from e.g. the residual of measured
data and forward projections. The first dimension should match indices in size, and the second and third dimensions should match the system projection geometry. The array must be
contiguous and row-major.
○ indices (Optional[ndarray[Any, dtype[int]]]) – A one-dimensional array containing one or more indices indicating from which projections the adjoint is to be computed.
Return type
ndarray[Any, dtype[TypeVar(_ScalarType_co, bound= generic, covariant=True)]]
The adjoint of the provided projections. An array with four dimensions (X, Y, Z, P), where the first three dimensions are spatial and the last dimension runs over coefficients.
property dtype: Union[dtype[Any], None, type[Any], _SupportsDType[dtype[Any]], str, tuple[Any, int], tuple[Any, Union[SupportsIndex, collections.abc.Sequence[SupportsIndex]]], list[Any],
_DTypeDict, tuple[Any, Any]]¶
Preferred dtype of this Projector.
forward(field, indices=None)[source]¶
Compute the forward projection of a tensor field.
○ field (ndarray[Any, dtype[TypeVar(_ScalarType_co, bound= generic, covariant=True)]]) – An array containing coefficients in its fourth dimension, which are to be projected into two
dimensions. The first three dimensions should match the volume_shape of the sample.
○ indices (Optional[ndarray[Any, dtype[int]]]) – A one-dimensional array containing one or more indices indicating which projections are to be computed. If None, all projections will be
Return type
ndarray[Any, dtype[TypeVar(_ScalarType_co, bound= generic, covariant=True)]]
An array with four dimensions (I, J, K, L), where the first dimension matches indices, such that projection[i] corresponds to the geometry of projection indices[i]. The second and third
dimension contain the pixels in the J and K dimension respectively, whereas the last dimension is the coefficient dimension, matching field[-1].
property is_dirty: bool¶
Returns True if the system geometry has changed without the projection geometry having been updated.
property john_transform_parameters: tuple¶
Tuple of John Transform parameters, which can be passed manually to compile John Transform kernels and construct low-level pipelines. For advanced users only.
property number_of_projections: int¶
The number of projections as defined by the length of the Geometry object attached to this instance.
property projection_shape: Tuple[int]¶
The shape of each projection defined by the Geometry object attached to this instance, as a tuple.
property volume_shape: Tuple[int]¶
The shape of the volume defined by the Geometry object attached to this instance, as a tuple.
class mumott.methods.projectors.SAXSProjectorCUDA(geometry)[source]¶
Projector for transforms of tensor fields from three-dimensional space to projection space. Uses a projection algorithm implemented in numba.cuda.
geometry (Geometry) – An instance of Geometry containing the necessary vectors to compute forwared and adjoint projections.
adjoint(projections, indices=None)¶
Compute the adjoint of a set of projections according to the system geometry.
○ projections (ndarray[Any, dtype[TypeVar(_ScalarType_co, bound= generic, covariant=True)]]) – An array containing coefficients in its last dimension, from e.g. the residual of measured
data and forward projections. The first dimension should match indices in size, and the second and third dimensions should match the system projection geometry. The array must be
contiguous and row-major.
○ indices (Optional[ndarray[Any, dtype[int]]]) – A one-dimensional array containing one or more indices indicating from which projections the adjoint is to be computed.
Return type
ndarray[Any, dtype[TypeVar(_ScalarType_co, bound= generic, covariant=True)]]
The adjoint of the provided projections. An array with four dimensions (X, Y, Z, P), where the first three dimensions are spatial and the last dimension runs over coefficients.
property dtype: Union[dtype[Any], None, type[Any], _SupportsDType[dtype[Any]], str, tuple[Any, int], tuple[Any, Union[SupportsIndex, collections.abc.Sequence[SupportsIndex]]], list[Any],
_DTypeDict, tuple[Any, Any]]¶
Preferred dtype of this Projector.
forward(field, indices=None)¶
Compute the forward projection of a tensor field.
○ field (ndarray[Any, dtype[TypeVar(_ScalarType_co, bound= generic, covariant=True)]]) – An array containing coefficients in its fourth dimension, which are to be projected into two
dimensions. The first three dimensions should match the volume_shape of the sample.
○ indices (Optional[ndarray[Any, dtype[int]]]) – A one-dimensional array containing one or more indices indicating which projections are to be computed. If None, all projections will be
Return type
ndarray[Any, dtype[TypeVar(_ScalarType_co, bound= generic, covariant=True)]]
An array with four dimensions (I, J, K, L), where the first dimension matches indices, such that projection[i] corresponds to the geometry of projection indices[i]. The second and third
dimension contain the pixels in the J and K dimension respectively, whereas the last dimension is the coefficient dimension, matching field[-1].
property is_dirty: bool¶
Returns True if the system geometry has changed without the projection geometry having been updated.
property john_transform_parameters: tuple¶
Tuple of John Transform parameters, which can be passed manually to compile John Transform kernels and construct low-level pipelines. For advanced users only.
property number_of_projections: int¶
The number of projections as defined by the length of the Geometry object attached to this instance.
property projection_shape: Tuple[int]¶
The shape of each projection defined by the Geometry object attached to this instance, as a tuple.
property volume_shape: Tuple[int]¶
The shape of the volume defined by the Geometry object attached to this instance, as a tuple. | {"url":"https://mumott.org/moduleref/projectors.html","timestamp":"2024-11-09T09:43:36Z","content_type":"text/html","content_length":"65937","record_id":"<urn:uuid:d34ad10a-a282-4eec-a3e5-98561744cb84>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00219.warc.gz"} |
How to Concatenate If Cell Values Match in Excel (7 Easy Ways) - ExcelDemy
To demonstrate our methods we’ll use the following dataset, containing columns for State and Sales Person. We’ll concatenate the values in the Sales Person column if the States match.
Method 1 – Combining TEXTJOIN & IF Functions
We can use the TEXTJOIN function, and the IF function to concatenate if values match in Excel.
• In cell F5 (the cell where we want to concatenate values), enter the following formula:
• Press Enter to get the result.
How Does the Formula Work?
• IF($B$5:$B$14=E5,$C$5:$C$14,””): The IF function checks if the value in cell E5 has matches in the cell range B5:B14. If the logical_test is True then the formula returns corresponding values
from the cell range C5:C14.
• TEXTJOIN(“,”,TRUE,IF($B$5:$B$14=E5,$C$5:$C$14,””)): The TEXTJOIN function joins the values returned by the IF function with the given delimiter of a comma (“,”).
• Drag the Fill Handle to copy the formula down to cell F7.
The results are as follows.
Read More: How to Concatenate Cells with If Condition in Excel
Method 2 – Using CONCAT & IF Functions
The CONCAT function joins multiple texts from different strings.
• In cell F5 enter the following formula:
How Does the Formula Work?
• IF($B$5:$B$14=E5,$C$5:$C$14&”,”,””): The IF function checks if the value in cell E5 has matches in the cell range B5:B14. If the logical_test is True then the formula returns corresponding values
from the cell range C5:C14 with a delimiter of a comma (“,”).
• CONCAT(IF($B$5:$B$14=E5,$C$5:$C$14&”,”,””)): The CONCAT function joins the values returned by the IF function.
• Drag the Fill Handle down to copy the formula.
The results are as follows.
Method 3 – Using Functions & Filter
We can use functions to concatenate if the values match, then Filter the results to return the desired output. To apply this method, the values must be stored together in the same dataset.
• In cell D5 enter the following formula:
How Does the Formula Work?
• IF(B5<>B4,C5,CONCATENATE(D4,”,”,C5)): The IF function checks if the value in cell B5 is not equal to the value in cell B4. If the logical_test is True then the formula will return the value in
cell C5. Otherwise, it will execute the CONCATENATE function.
• CONCATENATE(D4,”,”,C5): The CONCATENATE function joins the value in cell D4 with the value in cell C5 with a delimiter of a comma (“,”).
• Drag the Fill Handle down to copy the formula.
The results are as follows.
• In cell E5, enter the following formula:
• Press Enter to get the result.
How Does the Formula Work?
• IF(B5<>B6,CONCATENATE(B5,”,”””,D5,””””),””): The IF function checks if the value in cell B5 is not equal to the value in cell B6. If the logical_test is True then the formula will execute the
CONCATENATE function. Otherwise, it will return a blank.
• CONCATENATE(B5,”,”””,D5,””””): Now, the CONCATENATE function will combine the texts.
• Drag the Fill Handle down to copy the formula.
The results are as follows. Some cells are blank.
Now we’ll filter the column to get rid of the blank cells.
• Select the column header where you want to apply the filter (cell E4).
• Go to the Data tab.
• Select Filter.
A filter is added to this dataset.
• Click on the filter button on cell E4.
• Uncheck the blank option.
• Click OK.
• The blank cells are filtered out.
Read More: Combine CONCATENATE & TRANSPOSE Functions in Excel
Method 4 – Combining CONCATENATE & IF Functions
For this example, we’ll use a different dataset containing 3 columns; First Name, Middle Name, and Last Name. The Middle Name column contains some blank cells. Let’s match the blanks and then
concatenate the values accordingly if a match is found or not.
• In cell F5 enter the following formula:
=CONCATENATE(B5," ",IF(ISBLANK(C5),"",C5&" "),D5)
How Does the Formula Work?
• ISBLANK(C5): The ISBLANK function will return True if cell C5 is blank. Otherwise, it will return False.
• IF(ISBLANK(C5),””,C5&” “): The IF function checks for matches and if the logical_test is True returns blank. Otherwise, it returns the value in cell C5 followed by a space.
• CONCATENATE(B5,” “,IF(ISBLANK(C5),””,C5&” “),D5): The CONCATENATE function will join the resultant text.
• Drag the Fill Handle down to copy the formula to the other cells.
Full Names include Middle Names where those are present in column C, otherwise they don’t.
Method 5 – Using COUNTA Function
The COUNTA function counts cells containing any kind of information, and can be used to find the blank cells, then used in conjunction with the IF function to concatenate the matches accordingly.
• In cell F5 enter the following formula:
=IF(COUNTA(C5)=0,B5&" "&D5,B5&" "&C5&" "&D5)
• Press Enter to get the result.
How Does the Formula Work?
• COUNTA(C5): The COUNTA function returns the number of cells containing any values.
• IF(COUNTA(C5)=0,B5&” “&D5,B5&” “&C5&” “&D5): The IF function checks if the COUNTA function returns 0. If the logical_test is True then the formula will concatenate the values in cells B5 and D5.
Otherwise, it will concatenate the values in cells B5, C5, and D5.
• Drag the Fill Handle down to copy the formula.
• The results are as follows.
Method 6 – Using VBA Code
The VBA macro below will concatenate cell values if they match, and return the results with the column header.
• Go to the Developer tab.
• Select Visual Basic.
The Visual Basic Editor window will open.
• Select the Insert tab.
• Select Module.
A module will open.
• Enter the following code in the module:
Sub Concatenate_If_Match()
Dim t_col As New Collection
Dim inp_table As Variant
Dim output() As Variant
Dim m As Long
Dim col_no As Long
Dim rng As Range
inp_table = Range("B4", Cells(Rows.Count, "B").End(xlUp)).Resize(, 2)
Set rng = Range("E4")
On Error Resume Next
For m = 2 To UBound(inp_table)
t_col.Add inp_table(m, 1), TypeName(inp_table(m, 1)) & CStr(inp_table(m, 1))
Next m
On Error GoTo 0
ReDim output(1 To t_col.Count + 1, 1 To 2)
output(1, 1) = "State"
output(1, 2) = "Sales Person"
For m = 1 To t_col.Count
output(m + 1, 1) = t_col(m)
For col_no = 2 To UBound(inp_table)
If inp_table(col_no, 1) = output(m + 1, 1) Then
output(m + 1, 2) = output(m + 1, 2) & ", " & inp_table(col_no, 2)
End If
Next col_no
output(m + 1, 2) = Mid(output(m + 1, 2), 2)
Next m
Set rng = rng.Resize(UBound(output, 1), UBound(output, 2))
rng.NumberFormat = "@"
rng = output
End Sub
How Does the Code Work?
• We create a Sub Procedure named Concatenate_If_Match.
• We declare the variables.
• We use a Set Statement to set where we want the output.
• We use a For Next Loop to go through the input table.
• We use the ReDim Statement to size the declared array.
• We use another For Next Loop to go through the columns.
• We use an IF Statement to check for a match.
• We end the Sub Procedure.
• Save the code and go back to the worksheet.
• Go to the Developer tab.
• Select Macros.
The Macro dialog box will appear.
• Select Concatenate_If_Match as Macro Name.
• Click Run.
The desired output is returned.
Method 7 – Using a User Defined Function
A user defined function is a function defined by the user for a specific task, and created by writing a VBA code. Let’s create a User defined function to concatenate if cell values match.
• Go to the Developer tab.
• Select Visual Basic.
The Visual Basic Editor window will open.
• Go to the Insert tab.
• Select the Module.
A module will open.
• Enter the following code in the module:
Function CONCATENATE_IF(criteria_range As Range, criteria As Variant, _
concatenate_range As Range, Optional Delimiter As String = ",") As Variant
Dim Results As String
On Error Resume Next
If criteria_range.Count <> concatenate_range.Count Then
CONCATENATE_IF = CVErr(xlErrRef)
Exit Function
End If
For J = 1 To criteria_range.Count
If criteria_range.Cells(J).Value = criteria Then
Results = Results & Delimiter & concatenate_range.Cells(J).Value
End If
Next J
If Results <> "" Then
Results = VBA.Mid(Results, VBA.Len(Delimiter) + 1)
End If
CONCATENATE_IF = Results
Exit Function
End Function
How Does the Code Work?
• We create a Function named CONCATENATE_IF.
• We declare the variables for the function.
• We use an If Statement to check for a match.
• We use the CVErr function to return a user defined error.
• We use a For Next Loop to go through the whole range.
• We use another If Statement to find a match and then return results accordingly.
• We end the Sub Procedure.
• Save the code and go back to the worksheet.
• In cell F5 enter the following formula:
• Press Enter to get the result.
Here, we selected cell range B5:B14 as criteria_range, cell E5 as criteria, cell range C5:C14 as concatenate_range, and “,” as Delimiter. The formula will concatenate the values from the
concatenate_range if the criteria match the criteria_range.
• Drag the Fill Handle down to copy the formula.
• The desired output is returned.
Things to Remember
• If you use VBA in your Excel workbook then you must save the Excel file as Excel Macro-Enabled Workbook. Otherwise, the VBA code will not work.
Download Practice Workbook
Related Articles
Get FREE Advanced Excel Exercises with Solutions!
2 Comments
1. In my case TEXTJOIN in Method 1 returns all values, not only those matched by IF. Does it really work?
□ Hello Undent,
Sorry to hear your problem. But our Method-1 is working perfectly. Can you check your IF condition based on your case or dataset.
Here, I’m attaching a image where IF function returns the values based on the conditions.
If you want you can share your case here.
Leave a reply | {"url":"https://www.exceldemy.com/excel-concatenate-if-match/","timestamp":"2024-11-14T22:01:02Z","content_type":"text/html","content_length":"217899","record_id":"<urn:uuid:1c027636-810f-4969-b6fa-8a1c3f42a986>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00285.warc.gz"} |
unsupported/Eigen/NumericalDiff - nest-cam/v366/eigen - Git at Google
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
// Copyright (C) 2009 Thomas Capricelli <orzel@freehackers.org>
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_NUMERICALDIFF_MODULE_H
#define EIGEN_NUMERICALDIFF_MODULE_H
#include "../../Eigen/Core"
namespace Eigen {
* \defgroup NumericalDiff_Module Numerical differentiation module
* \code
* #include <unsupported/Eigen/NumericalDiff>
* \endcode
* See http://en.wikipedia.org/wiki/Numerical_differentiation
* Warning : this should NOT be confused with automatic differentiation, which
* is a different method and has its own module in Eigen : \ref
* AutoDiff_Module.
* Currently only "Forward" and "Central" schemes are implemented. Those
* are basic methods, and there exist some more elaborated way of
* computing such approximates. They are implemented using both
* proprietary and free software, and usually requires linking to an
* external library. It is very easy for you to write a functor
* using such software, and the purpose is quite orthogonal to what we
* want to achieve with Eigen.
* This is why we will not provide wrappers for every great numerical
* differentiation software that exist, but should rather stick with those
* basic ones, that still are useful for testing.
* Also, the \ref NonLinearOptimization_Module needs this in order to
* provide full features compatibility with the original (c)minpack
* package.
#include "src/NumericalDiff/NumericalDiff.h"
#endif // EIGEN_NUMERICALDIFF_MODULE_H | {"url":"https://nest-open-source.googlesource.com/nest-cam/v366/eigen/+/9d75fdbcd2b5a2fdab03a859ddefda95ce1217ed/unsupported/Eigen/NumericalDiff","timestamp":"2024-11-14T18:11:07Z","content_type":"text/html","content_length":"17430","record_id":"<urn:uuid:ea7c990a-9c51-45bb-ad14-cb0980a455f7>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00189.warc.gz"} |
Chaitin's Number
Chaitin's Number - It's Not As Uncomputable As You Thought
Chaitin's number is the probability that a randomly chosen programme for a Turing computer will halt. Strictly the numerical value of Chaitin's number is not defined until some standard computing
code is agreed. But this is an essentially trivial complication. The received wisdom (well, theorem) is that Chaitin's number is uncomputable. The claim goes that, were it computable knowledge of its
numerical value would enable a wide range of intractable mathematical theorems to be proved, such as Goldbach's Conjecture and the Riemann Hypothesis. I have never found the argument for this
convincing, but that's probably just my ignorance.
It comes as something as a shock, then, to find that the first 64 binary digits of Chaitin's number have already been computed. They are,
(see Cristian Calude) Since 64 binary digits is roughly 19 decimal places, this means that Chaitins' number is known with far greater precision than any quantity in the whole of physical science.
Even the much vaunted anomalous magnetic moments of the electron and muon have been measured (and calculated) only to about 10 decimal places.
Go back to Random Interesting Things in Maths Selection page
Go back to Rick's Home Page (Main Menu)
Contact me and other links
"Blue Lagoon", M8 (the Lagoon Nebula) spans about 30 light-years at an estimated distance of 5,000 light-years toward the constellation Sagittarius: Russell Croman, 2006 | {"url":"http://rickbradford.co.uk/ChaitinCalude.html","timestamp":"2024-11-11T11:45:57Z","content_type":"text/html","content_length":"2759","record_id":"<urn:uuid:e728712c-ecb1-4a33-860e-fffe59f451ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00420.warc.gz"} |
Trimmed Mean Definition Example Calculation And Use - [Updated November 2024]
Trimmed Mean Definition Example Calculation and Use
Trimmed Mean: Definition, Example, Calculation, and Use
What Is a Trimmed Mean?
A trimmed mean is a method of averaging that removes a small designated percentage of the largest and smallest values before calculating the mean. The trimmed mean helps eliminate the influence of
outliers on the mean.
Trimmed means are used to report economic data in order to provide a more realistic picture.
Key Takeaways
• A trimmed mean removes a small designated percentage of the largest and smallest values before calculating the average.
• Using a trimmed mean helps eliminate the influence of outliers on the mean.
• Trimmed means are used in reporting economic data to provide a more realistic picture.
• Providing a trimmed mean inflation rate, along with other measures, provides a basis for comparison.
Understanding a Trimmed Mean
A trimmed mean helps reduce the effects of outliers on the calculated average. It is best suited for data with large deviations or skewed distributions.
A trimmed mean is stated as a mean trimmed by x%, where x is the sum of the percentage of observations removed from both the upper and lower bounds. The trimming points are often arbitrary, removing
the lowest and highest values by a percentage, leaving the mean to be calculated from the remaining data.
A trimmed mean is seen as a more realistic representation of a data set as the few erratic outliers have been removed. A trimmed mean is also known as a truncated mean.
Trimmed Means and Inflation Rates
A trimmed mean may be used in place of a traditional mean when determining inflation rates from the Consumer Price Index (CPI) or personal consumption expenditures (PCE). The CPI and the PCE price
index measure the prices of baskets of goods in an economy to identify inflation trends.
The levels that are trimmed from each tail may not be equitable, as these values are based on historical data to reach the best fit between the trimmed mean inflation rate and the inflation rate’s
Food and energy costs are generally considered the most volatile items, so the non-core area is not necessarily indicative of overall inflationary activities.
When the data points are organized, they are placed in ascending order based on prices. Specific percentages are removed from the tails to lower the effect of volatility on the overall CPI changes.
Trimmed means are used in the Olympics to remove extreme scoring from biased judges that may impact an athlete’s average score.
Providing a trimmed mean inflation rate along with other measures, allows for a more thorough analysis of the inflation rates being experienced. This comparison may include the traditional CPI, the
core CPI, a trimmed-mean CPI, and a median CPI.
Example of a Trimmed Mean
As an example, a figure skating competition produces the following scores: 6.0, 8.1, 8.3, 9.1, and 9.9.
The mean for the scores would equal:
• ((6.0 + 8.1 + 8.3 + 9.1 + 9.9) / 5) = 8.28
To trim the mean by a total of 40%, we remove the lowest 20% and the highest 20% of values, eliminating the scores of 6.0 and 9.9.
Next, we calculate the mean based on the remaining values:
• (8.1 + 8.3 + 9.1) / 3 = 8.50
In other words, a mean trimmed at 40% would equal 8.5, which reduced the outlier bias and increased the reported average by 0.22 points. | {"url":"https://wishcomputer.net/terms/trimmed-mean-definition-example-calculation-and/","timestamp":"2024-11-07T00:13:02Z","content_type":"text/html","content_length":"60929","record_id":"<urn:uuid:41106d60-2928-4c4e-ad07-af93aa4f6453>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00230.warc.gz"} |
Pseudo-monotone operator theory for electro-rheological fluids
We consider a model describing the unsteady motion of an incompressible, electro-rheological fluid. Due to the time-space-dependence of the power-law index, the analytical treatment of this system is
involved. Standard results like a Poincare or a Korn inequality are not available. Introducing natural energy spaces, we establish the validity of a formula of integration-by-parts which allows to
extend the classicaltheory of pseudo-monotone operators to the framework of variable Bochner-Lebesgue spaces. This leads to generalised notions of pseudo-monotonicity and coercivity, the so-called
Bochner pseudo-monotonicity and Bochner coercivity. With the aid of these notions and the established formula of integration-by-parts it is possible to prove an abstract existence result which
immediately implies the weak solvability of the model describing the unsteady motion of an incompressible, electro-rheological fluid. | {"url":"https://events.dm.unipi.it/event/252/?print=1","timestamp":"2024-11-15T01:32:32Z","content_type":"text/html","content_length":"9882","record_id":"<urn:uuid:1b58d893-7fa3-42db-aaa7-9ec51c2dfd3b>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00254.warc.gz"} |
Senator adds “Every Sperm is Sacred” amendment onto Oklahoma personhood bill
(40,432 posts) Fri Feb 10, 2012, 12:25 PM Feb 2012
Senator adds “Every Sperm is Sacred” amendment onto Oklahoma personhood bill
Senator adds Every Sperm is Sacred amendment onto Oklahoma personhood bill
By Vanessa | Published: February 9, 2012
Amazing. After Oklahoma conservatives introduced a personhood bill to the state Senate on Monday, Sen. Constance Johnson decided to follow in Virginia Senator Janet Howell s footsteps and attach an
amendment in protest, which would add this language to the bill:
However, any action in which a man ejaculates or otherwise deposits semen anywhere but in a woman s vagina shall be interpreted and construed as an action against an unborn child.
Jezebel adds that another pro-choice senator added an amendment:
Another pro-choice legislator, Democrat Jim Wilson, attempted to add an amendment to the bill that would require the father of the child to be financially responsible for the woman s health care,
housing, transportation, and nourishment while she was pregnant.
17 replies
1. The Onion, ....right ?
Fri Feb 10, 2012, 12:27 PM
Feb 2012
Fri Feb 10, 2012, 12:29 PM
Feb 2012
4. good god..., holy shit, and I'll be damned. Incredible.
Fri Feb 10, 2012, 12:35 PM
Feb 2012
Fri Feb 10, 2012, 12:38 PM
Feb 2012
15. That's what I thought when I saw this thread!!
Fri Feb 10, 2012, 01:46 PM
Feb 2012
Fri Feb 10, 2012, 02:00 PM
Feb 2012
to find use of that term which preceeded Monty Python so we might as well call it there's by right.
3. Let me get this right. My daily act that keeps my prostate healthy and mind sane makes me
Fri Feb 10, 2012, 12:32 PM
Feb 2012
7. there must be something in the water in oklahoma
Fri Feb 10, 2012, 12:42 PM
Feb 2012
anyone who'd vote for the likes of inhofe (a guy who I think is nuttier than a fruit cake) and thinking the muslins are attempting to push sharia law on us-by god, gotta have an anti-sharia law. And
didn't they refuse the mortgage settlement money to help their fellow homeless oklahomans? Now, I'd see doing it, if they were going after the banks for criminal prosecution; but being as how they
love them some corporations and banksters, I don't think that's the case.
9. these amendments were attached in protest nt
Fri Feb 10, 2012, 12:51 PM
Feb 2012
12. Only if you think that monthly act that drives me bonkers, makes me grumpy,
Fri Feb 10, 2012, 01:20 PM
Feb 2012
causes me to lose my temper for no apparent reason and make those around me miserable contains a personhood in all that mess.
6. That could backfire: Some nutjobs may support it. n/t
Fri Feb 10, 2012, 12:39 PM
Feb 2012
8. I wish about 250 men would surround her house
Fri Feb 10, 2012, 12:48 PM
Feb 2012
and have the world's largest circle jerk. I can't even comprehend this insanity. So if I blow a load into my wife's back versus her vagina i could be violating the law in Oklahoma?
10. It's a pro-choice ammendment attachment.
Fri Feb 10, 2012, 01:05 PM
Feb 2012
Taking the 'personhood' bill to it's extreme.
13. It's meant to help defeat the anti-choice bill
Fri Feb 10, 2012, 01:21 PM
Feb 2012
Since men have no problem controling women's reproductive organs then why shouldn't we have the right to control men's reproductive organs
Fri Feb 10, 2012, 01:39 PM
Feb 2012
I read the other articles about this and realized that SHE was the one making sense and being satirical.
Fri Feb 10, 2012, 07:03 PM
Feb 2012
11. I think this bill does not go far enough!
Fri Feb 10, 2012, 01:15 PM
Feb 2012
My body makes sperm and stores them for future use. If they are not used, they die and are reabsorbed. So by this same logic every woman who refuses to have sex with me (BTW this group comprises BY
FAR the majority of women I know, damn it!) is in effect MURDERING those millions of unborn sperm cells that die a lonely death in my seminal vesicles or wherever. Therefore, declining to have sex
with me constitutes genocide! All of the above is sheer nonsense of course. It's not that much more ridiculous than the proposal put forth by these state senators however. | {"url":"https://upload.democraticunderground.com/1002293436","timestamp":"2024-11-13T18:22:30Z","content_type":"text/html","content_length":"101790","record_id":"<urn:uuid:57dd7fdc-958b-4594-8917-5369339d799b>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00517.warc.gz"} |
Quadratic Equations - Formulas, Methods, and Examples (2024)
Quadratic equations are second-degree algebraic expressions and are of the form ax^2 + bx + c = 0. The term "quadratic" comes from the Latin word "quadratus" meaning square, which refers to the fact
that the variable x is squared in the equation. In other words, a quadratic equation is an “equation of degree 2.” There are many scenarios where a quadratic equation is used. Did you know that when
a rocket is launched, its path is described by a quadratic equation? Further, a quadratic equation has numerous applications in physics, engineering, astronomy, etc.
Quadratic equations have maximum of two solutions, which can be real or complex numbers. These two solutions (values of x) are also called the roots of the quadratic equations and are designated as
(α, β). We shall learn more about the roots of a quadratic equation in the below content.
1. What is a Quadratic Equation?
2. Roots of a Quadratic Equation
3. Quadratic Formula
4. Nature of Roots of the Quadratic Equation
5. Formulas Related to Quadratic Equations
6. Methods to Solve Quadratic Equations
7. Solving Quadratic Equations by Factorization
8. Method of Completing the Square
9. Graphing a Quadratic Equation
10. Quadratic Equations Having Common Roots
11. Maximum and Minimum Value of Quadratic Expression
12. FAQs on Quadratic Equations
What is Quadratic Equation?
A quadratic equation is an algebraic equation of the second degree in x. The quadratic equation in its standard form is ax^2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c
is the constant term. The important condition for an equation to be a quadratic equation is the coefficient of x^2 is a non-zero term (a ≠ 0). For writing a quadratic equation in standard form, the x
^2 term is written first, followed by the x term, and finally, the constant term is written.
Further, in real math problems the quadratic equations are presented in different forms: (x - 1)(x + 2) = 0, -x^2 = -3x + 1, 5x(x + 3) = 12x, x^3 = x(x^2 + x - 3). All of these equations need to be
transformed into standard form of the quadratic equation before performing further operations.
Roots of a Quadratic Equation
The roots of a quadratic equation are the two values of x, which are obtained by solving the quadratic equation. These roots of the quadratic equation are also called the zeros of the equation. For
example, the roots of the equation x^2 - 3x - 4 = 0 are x = -1 and x = 4 because each of them satisfies the equation. i.e.,
• At x = -1, (-1)^2 - 3(-1) - 4 = 1 + 3 - 4 = 0
• At x = 4, (4)^2 - 3(4) - 4 = 16 - 12 - 4 = 0
There are various methods to find the roots of a quadratic equation. The usage of the quadratic formula is one of them.
Quadratic Formula
Quadratic formula is the simplest way to find the roots of a quadratic equation. There are certain quadratic equations that cannot be easily factorized, and here we can conveniently use this
quadratic formula to find the roots in the quickest possible way. The two roots in the quadratic formula are presented as a single expression. The positive sign and the negative sign can be
alternatively used to obtain the two distinct roots of the equation.
Quadratic Formula: The roots of a quadratic equation ax^2 + bx + c = 0 are given by x = [-b ± √(b^2 - 4ac)]/2a.
This formula is also known as the Sridharacharya formula.
Example: Let us find the roots of the same equation that was mentioned in the earlier section x^2 - 3x - 4 = 0 using the quadratic formula.
a = 1, b = -3, and c = -4.
x = [-b ± √(b^2 - 4ac)]/2a
= [-(-3) ± √((-3)^2 - 4(1)(-4))]/2(1)
= [3 ± √25] / 2
= [3 ± 5] / 2
= (3 + 5)/2 or (3 - 5)/2
= 8/2 or -2/2
= 4 or -1 are the roots.
Proof of Quadratic Formula
Consider an arbitrary quadratic equation: ax^2 + bx + c = 0, a ≠ 0
To determine the roots of this equation, we proceed as follows:
ax^2 + bx = -c ⇒ x^2 + bx/a = -c/a
Now, we express the left-hand side as a perfect square, by introducing a new term (b/2a)^2 on both sides:
x^2+ bx/a + (b/2a)^2 = -c/a + (b/2a)^2
The left-hand side is now a perfect square:
(x + b/2a)^2 = -c/a + b^2/4a^2 ⇒ (x + b/2a)^2 = (b^2 - 4ac)/4a^2
This is good for us, because now we can take square roots to obtain:
x + b/2a = ±√(b^2 - 4ac)/2a
x = (-b ± √(b^2 - 4ac))/2a
Thus, by completing the squares, we were able to isolate x and obtain the two roots of the equation.
Nature of Roots of the Quadratic Equation
The roots of a quadratic equation are usually represented to by the symbols alpha (α), and beta (β). Here we shall learn more about how to find the nature of roots of a quadratic equation without
actually finding the roots of the equation.
The nature of roots of a quadratic equation can be found without actually finding the roots (α, β) of the equation. This is possible by taking the discriminant value, which is part of the formula to
solve the quadratic equation. The value b^2 - 4ac is called the discriminant of a quadratic equation and is designated as 'D'. Based on the discriminant value the nature of the roots of the quadratic
equation can be predicted.
Discriminant: D = b^2 - 4ac
• D > 0, the roots are real and distinct
• D = 0, the roots are real and equal.
• D < 0, the roots do not exist or the roots are imaginary.
Now, check out the formulas to find the sum and the product of the roots of the equation.
Sum and Product of Roots of Quadratic Equation
The coefficient of x^2, x term, and the constant term of the quadratic equation ax^2^ + bx + c = 0 are useful in determining the sum and product of the roots of the quadratic equation. The sum and
product of the roots of a quadratic equation can be directly calculated from the equation, without actually finding the roots of the quadratic equation. For a quadratic equation ax^2 + bx + c = 0,
the sum and product of the roots are as follows.
• Sum of the Roots: α + β = -b/a = - Coefficient of x/ Coefficient of x^2
• Product of the Roots: αβ = c/a = Constant term/ Coefficient of x^2
Writing Quadratic Equations Using Roots
The quadratic equation can also be formed for the given roots of the equation. If α, β, are the roots of the quadratic equation, then the quadratic equation is as follows.
x^2 - (α + β)x + αβ = 0
Example: What is the quadratic equation whose roots are 4 and -1?
Solution: It is given that α = 4 and β = -1. The corresponding quadratic equation is found by:
x^2 - (α + β)x + αβ = 0
x^2 - (α + β)x + αβ = 0
x^2 - (4 - 1)x + (4)(-1) = 0
x^2 - 3x - 4 = 0
Formulas Related to Quadratic Equations
The following list of important formulas is helpful to solve quadratic equations.
• The quadratic equation in its standard form is ax^2 + bx + c = 0
• The discriminant of the quadratic equation is D = b^2 - 4ac
□ For D > 0 the roots are real and distinct.
□ For D = 0 the roots are real and equal.
□ For D < 0 the real roots do not exist, or the roots are imaginary.
• The formula to find the roots of the quadratic equation is x = [-b ± √(b^2 - 4ac)]/2a.
• The sum of the roots of a quadratic equation is α + β = -b/a.
• The product of the Root of the quadratic equation is αβ = c/a.
• The quadratic equation whose roots are α, β, is x^2 - (α + β)x + αβ = 0.
• The condition for the quadratic equations a[1]x^2 + b[1]x + c[1] = 0, and a[2]x^2 + b[2]x + c[2] = 0 having the same roots is (a[1]b[2] - a[2]b[1]) (b[1]c[2] - b[2]c[1]) = (a[2]c[1] - a[1]c[2])^
• When a > 0, the quadratic expression f(x) = ax^2 + bx + c has a minimum value at x = -b/2a.
• When a < 0, the quadratic expression f(x) = ax^2 + bx + c has a maximum value at x = -b/2a.
• The domain of any quadratic function is the set of all real numbers.
Methods to Solve Quadratic Equations
A quadratic equation can be solved to obtain two values of x or the two roots of the equation. There are four different methods to find the roots of the quadratic equation. The four methods of
solving the quadratic equations are as follows.
• Factorizing of Quadratic Equation
• Using quadratic formula (which we have seen already)
• Method of Completing the Square
• Graphing Method to Find the Roots
Let us look in detail at each of the above methods to understand how to use these methods, their applications, and their uses.
Solving Quadratic Equations by Factorization
Factorization of quadratic equation follows a sequence of steps. For a general form of the quadratic equation ax^2 + bx + c = 0, we need to first split the middle term into two terms, such that the
product of the terms is equal to the constant term. Further, we can take the common terms from the available term, to finally obtain the required factors as follows:
• x^2 + (a + b)x + ab = 0
• x^2 + ax + bx + ab = 0
• x(x + a) + b(x + a)
• (x + a)(x + b) = 0
Here is an example to understand the factorization process.
• x^2 + 5x + 6 = 0
• x^2 + 2x + 3x + 6 = 0
• x(x + 2) + 3(x + 2) = 0
• (x + 2)(x + 3) = 0
Thus the two obtained factors of the quadratic equation are (x + 2) and (x + 3). To find its roots, just set each factor to zero and solve for x. i.e., x + 2 = 0 and x + 3 = 0 which gives x = -2 and
x = -3. Thus, x = -2 and x = -3 are the roots of x^2 + 5x + 6 = 0.
Further, there is another important method of solving a quadratic equation. The method of completing the square for a quadratic equation is also useful to find the roots of the equation.
Method of Completing the Square
The method of completing the square in a quadratic equation is to algebraically square and simplify, to obtain the required roots of the equation. Consider a quadratic equation ax^2 + bx + c = 0, a ≠
0. To determine the roots of this equation, we simplify it as follows:
• ax^2 + bx + c = 0
• ax^2 + bx = -c
• x^2 + bx/a = -c/a
Now, we express the left-hand side as a perfect square, by introducing a new term (b/2a)^2 on both sides:
• x^2 + bx/a + (b/2a)^2 = -c/a + (b/2a)^2
• (x + b/2a)^2 = -c/a + b^2/4a^2
• (x + b/2a)^2 = (b^2 - 4ac)/4a^2
• x + b/2a = +√(b^2- 4ac)/2a
• x = - b/2a +√(b^2- 4ac)/2a
• x = [-b ± √(b^2 - 4ac)]/2a
Here the '+' sign gives one root and the '-' sign gives another root of the quadratic equation. Generally, this detailed method is avoided, and only the quadratic formula is used to obtain the
required roots.
Graphing a Quadratic Equation
The graph of the quadratic equation ax^2 + bx + c = 0 can be obtained by representing the quadratic equation as a function y = ax^2 + bx + c. Further by solving and substituting values for x, we can
obtain values of y, we can obtain numerous points. These points can be presented in the coordinate axis to obtain a parabola-shaped graph for the quadratic equation. For detailed information about
graphing a quadratic function, click here.
The point(s) where the graph cuts the horizontal x-axis (typically the x-intercepts) is the solution of the quadratic equation. These points can also be algebraically obtained by equalizing the y
value to 0 in the function y = ax^2 + bx + c and solving for x.
Quadratic Equations Having Common Roots
Consider two quadratic equations having common roots a[1]x^2 + b[1]x + c[1] = 0, and a[2]x^2 + b[2]x + c[2] = 0. Let us solve these two equations to find the conditions for which these equations have
a common root. The two equations are solved for x^2 and x respectively.
(x^2)(b[1]c[2] - b[2]c[1]) = (-x)/(a[1]c[2] - a[2]c[1]) = 1/(a[1]b[2] - a[2]b[1])
x^2 = (b[1]c[2] - b[2]c[1]) / (a[1]b[2] - a[2]b[1])
x = (a[2]c[1] - a[1]c[2]) / (a[1]b[2] - a[2]b[1])
Hence, by simplifying the above two expressions we have the following condition for the two equations having the common root.
(a[1]b[2] - a[2]b[1]) (b[1]c[2] - b[2]c[1]) = (a[2]c[1] - a[1]c[2])^2
Maximum and Minimum Value of Quadratic Expression
The maximum and minimum values for the quadratic function F(x) = ax^2 + bx + c can be observed in the below graphs. For positive values of a (a > 0), the quadratic expression has a minimum value at x
= -b/2a, and for negative value of a (a < 0), the quadratic expression has a maximum value at x = -b/2a. x = -b/2a is the x-coordinate of the vertex of the parabola.
The maximum and minimum values of the quadratic expressions are of further help to find the range of the quadratic expression: The range of the quadratic expressions also depends on the value of a.
For positive values of a( a > 0), the range is [ F(-b/2a), ∞), and for negative values of a ( a < 0), the range is (-∞, F(-b/2a)].
• For a > 0, Range: [ f(-b/2a), ∞)
• For a < 0, Range: (-∞, f(-b/2a)]
Note that the domain of a quadratic function is the set of all real numbers, i.e., (-∞, ∞).
Tips and Tricks on Quadratic Equation:
Some of the below-given tips and tricks on quadratic equations are helpful to more easily solve quadratic equations.
• The quadratic equations are generally solved through factorization. But in instances when it cannot be solved by factorization, the quadratic formula is used.
• The roots of a quadratic equation are also called the zeroes of the equation.
• For quadratic equations having negative discriminant values, the roots are represented by complex numbers.
• The sum and product of the roots of a quadratic equation can be used to find higher algebraic expressions involving these roots.
☛Related Topics:
• Roots Calculator
• Quadratic Factoring Calculator
• Roots of Quadratic Equation Calculator
Cuemath is one of the world's leading math learning platforms that offers LIVE 1-to-1 online math classes for grades K-12. Our mission is to transform the way children learn math, to help them excel
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FAQs on Quadratic Equation
What is the Definition of a Quadratic Equation?
A quadratic equation in math is a second-degree equation of the form ax^2 + bx + c = 0. Here a and b are the coefficients, c is the constant term, and x is the variable. Since the variable x is of
the second degree, there are two roots or answers for this quadratic equation. The roots of the quadratic equation can be found by either solving by factorizing or through the use of the quadratic
What is the Quadratic Formula?
The quadratic equation formula to solve the equation ax^2 + bx + c = 0 is x = [-b ± √(b^2 - 4ac)]/2a. Here we obtain the two values of x, by applying the plus and minus symbols in this formula. Hence
the two possible values of x are [-b + √(b^2 - 4ac)]/2a, and [-b - √(b^2 - 4ac)]/2a.
How do You Solve a Quadratic Equation?
There are several methods to solve quadratic equations, but the most common ones are factoring, using the quadratic formula, and completing the square.
• Factoring involves finding two numbers that multiply to equal the constant term, c, and add up to the coefficient of x, b.
• The quadratic formula is used when factoring is not possible, and it is given by x = [-b ± √(b^2 - 4ac)]/2a.
• Completing the square involves rewriting the quadratic equation in a different form that allows you to easily solve for x.
What is Determinant in Quadratic Formula?
The value b^2 - 4ac is called the discriminant and is designated as D. The discriminant is part of the quadratic formula. The discriminants help us to find the nature of the roots of the quadratic
equation, without actually finding the roots of the quadratic equation.
What are Some Real-Life Applications of Quadratic Equations?
Quadratic equations are used to find the zeroes of the parabola and its axis of symmetry. There are many real-world applications of quadratic equations.
• They can be used in running time problems to evaluate the speed, distance or time while traveling by car, train or plane.
• Quadratic equations describe the relationship between quantity and the price of a commodity.
• Similarly, demand and cost calculations are also considered quadratic equation problems.
• It can also be noted that a satellite dish or a reflecting telescope has a shape that is defined by a quadratic equation.
How are Quadratic Equations Different From Linear Equations?
A linear degree is an equation of a single degree and one variable, and a quadratic equation is an equation in two degrees and a single variable. A linear equation is of the the form ax + b = 0 and a
quadratic equation is of the form ax^2 + bx + c = 0. A linear equation has a single root and a quadratic equation has two roots or two answers. Also, a quadratic equation is a product of two linear
What Are the 4 Ways To Solve A Quadratic Equation?
The four ways of solving a quadratic equation are as follows.
• Factorizing method
• Roots of Quadratic Equation Formula Method
• Method of Completing Squares
• Graphing Method
How to Solve a Quadratic Equation by Completing the Square?
The quadratic equation is solved by the method of completing the square and it uses the formula (a + b)^2 = a^2 + 2ab + b^2 (or) (a - b)^2 = a^2 - 2ab + b^2.
How to Find the Value of the Discriminant?
The value of the discriminant in a quadratic equation can be found from the variables and constant terms of the standard form of the quadratic equation ax^2 + bx + c = 0. The value of the
discriminant is D = b^2 - 4ac, and it helps to predict the nature of roots of the quadratic equation, without actually finding the roots of the equation.
How Do You Solve Quadratic Equations With Graphing?
The quadratic equation can be solved similarly to a linear equal by graphing. Let us take the quadratic equation ax^2 + bx + c = 0 as y = ax^2 + bx + c . Here we take the set of values of x and y and
plot the graph. The two points where this graph meets the x-axis, are the solutions of this quadratic equation.
How Important Is the Discriminant of a Quadratic Equation?
The discriminant is very much needed to easily find the nature of the roots of the quadratic equation. Without the discriminant, finding the nature of the roots of the equation is a long process, as
we first need to solve the equation to find both the roots. Hence the discriminant is an important and needed quantity, which helps to easily find the nature of the roots of the quadratic equation.
Where Can I Find Quadratic Equation Solver?
To get the quadratic equation solver, click here. Here, we can enter the values of a, b, and c for the quadratic equation ax^2 + bx + c = 0, then it will give you the roots along with a step-by-step
What is the Use of Discriminants in Quadratic Formula?
The discriminant (D = b^2 - 4ac) is useful to predict the nature of the roots of the quadratic equation. For D > 0, the roots are real and distinct, for D = 0 the roots are real and equal, and for D
< 0, the roots do not exist or the roots are imaginary complex numbers. With the help of this discriminant and with the least calculations, we can find the nature of the roots of the quadratic
How do you Solve a Quadratic Equation without Using the Quadratic Formula?
There are two alternative methods to the quadratic formula. One method is to solve the quadratic equation through factorization, and another method is by completing the squares. In total there are
three methods to find the roots of a quadratic equation.
How to Derive Quadratic Formula?
The algebra formula (a + b)^2 = a^2 + 2ab + b^2 is used to solve the quadratic equation and derive the quadratic formula. This algebraic formula is used to manipulate the quadratic equation and
derive the quadratic formula to find the roots of the equation. | {"url":"https://mickaelsimon.com/article/quadratic-equations-formulas-methods-and-examples","timestamp":"2024-11-08T04:49:45Z","content_type":"text/html","content_length":"90995","record_id":"<urn:uuid:6a389687-d179-4660-b615-15b8001232a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00560.warc.gz"} |
calculating percentile rank | Excelchat
I need to match two column, if one number in a column number falls in a range within the 2nd column. The result is on track or off track. Condition: (Column A: Age, ColumnB: Rank1-8) Under Age 30,
Rank 1-2 ontrack Over Age 30, Rank 1-2 off track Between Age 31-37, Rank 3-8 On Track Between Age 38-45, Rank 5-8 On Track Above 45, Rank below Rank 8 Off Track
Solved by F. Q. in 15 mins | {"url":"https://www.got-it.ai/solutions/excel-chat/excel-help/how-to/calculating/calculating-percentile-rank","timestamp":"2024-11-10T03:08:27Z","content_type":"text/html","content_length":"337176","record_id":"<urn:uuid:4f1eb432-41bd-4890-a765-cfcfbe28cc6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00111.warc.gz"} |
How do you find the LCD of 8x , 25y? | HIX Tutor
How do you find the LCD of 8x , 25y?
Answer 1
To find the least common denominator, we look for the smallest number that both numbers can divide into, which is $200 x y$
To find the least common denominator, we look for the smallest number that both numbers can divide into. To do that, I like to break the numbers we are working with down into their constituent parts
and start from there.
#8x=2*2*2*x# - so our LCD needs to have 3 "2's" and an x
#25y=5*5*y# - so our LCD also needs to have 2 "5's" and a y
Put the whole thing together to get:
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Answer 2
To find the least common denominator (LCD) of (8x) and (25y), we need to identify the factors that are common to both terms and those that are unique to each term. The LCD is the product of all
unique factors, including the highest power of common factors.
The prime factorization of (8x) is (2^3 \times x) and the prime factorization of (25y) is (5^2 \times y).
The LCD is therefore (2^3 \times 5^2 \times x \times y), which simplifies to (200xy).
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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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inductor energy storage density
In this session, Jayant Sir will be discussing JEE important topic "Inductor, Energy Density from ElectroMagnetic Induction", he will cover the basic concepts of the chapter, important...
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About inductor energy storage density
As the photovoltaic (PV) industry continues to evolve, advancements in inductor energy storage density have become critical to optimizing the utilization of renewable energy sources. From innovative
battery technologies to intelligent energy management systems, these solutions are transforming the way we store and distribute solar-generated electricity.
When you're looking for the latest and most efficient inductor energy storage density for your PV project, our website offers a comprehensive selection of cutting-edge products designed to meet your
specific requirements. Whether you're a renewable energy developer, utility company, or commercial enterprise looking to reduce your carbon footprint, we have the solutions to help you harness the
full potential of solar energy.
By interacting with our online customer service, you'll gain a deep understanding of the various inductor energy storage density featured in our extensive catalog, such as high-efficiency storage
batteries and intelligent energy management systems, and how they work together to provide a stable and reliable power supply for your PV projects.
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I work on promoting diversity in mathematics and the sciences. My research is focused on the pedagogical, content, soccio-economic, cultural and structural issues which inform the low rates of
mathematics and science achievement amount students of color. My work aims to increase the participation of underrepresented minorities in mathematics and science from grade school to graduate
Here are three resources that inform my thinking on mathematics and mathematics education:
Please see some recent work, with John Belcher, on the the potential of the use of music to build intuitions about deep mathematical questions and on Using a Mathematics Cultural Resonance Approach
for Building Capacity in the Mathematical Sciences for African American Communities. | {"url":"https://terrenceblackman.com/research/","timestamp":"2024-11-09T00:24:57Z","content_type":"text/html","content_length":"81626","record_id":"<urn:uuid:c2b242a4-d7c9-4208-9bda-147b793a0d4f>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00407.warc.gz"} |
MU Computer Graphics & Virtual Reality - May 2016 Exam Question Paper | Stupidsid
MU Information Technology (Semester 5)
Computer Graphics & Virtual Reality
May 2016
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
1(a) Differentiate between Raster scan display and Random scan display.
5 M
1(b) Prove that two successive rotation transformations are additive
5 M
1(c) Show that the transformation matrix for a reflection about a line y=x is equivalent to reflection to x-axis followed by counter clockwise rotation of 90°
5 M
1(d) Explain 3D trackers & enumerate some important trackers characteristics
5 M
2(a) Specify highlights and drawbacks of Bezier curve. Construct the bezier curve of order three with control P1(0,0), P2(1,3), P3(4,2) and P4(2,1). Generate at least five points on the curve.
10 M
2(b) Write DDA Line drawing Algorithm Compare DDA with Bresenham's Line drawing Algorithm. Calculate the pixel co-ordinates of line Abusing DDA Algorithm 1 where A=(0,0) and B=(4,6).
10 M
3(a) Let ABCD be the rectangular window with A (20,20), B(90,20), C(90,70) and D(20,70). Find region codes for endpoints and use Cohen Sutherland algorithm to clip hte lines
P1 P2 with P1(10,30), P2(80,90)
10 M
3(b) With respect to 3D transformation, describe the steps to be-carried out when an object isto be rotated about an arbitarary axis. Specify all the required matrices. State your assumptions
10 M
4(a) Explain Flood Fill Algorithm for 4 connected and 8 connected. What are its advantages over Boundary Fill Algorithm.
10 M
4(b) Explain an algorithm which uses parametric equation of line clipping. Using same algorithm find the line segment A(10, 10) and B(70, 40) after it is clipped against the window of two vertices
(20, 20) and (40,50).
10 M
5(a) Consider a triangle ABD whose coordinates are A(10, 20) B(30, 40)and 8C(50, 20). Perform the following transformations (Specify the matrices that are used)
(i) Translate the given by 3 units in X direction and -2 units in Y direction.
(ii) Rotate the given triangle by 30
(iii) Reflect the given triangle about X=Y
(iv) Scale the given uniformly by 2 units..
10 M
5(b) What is the significance of modeling in virtual reality? Explain any modeling technique used in virtual reality.
10 M
Write a short note on (Any five)
6(a) Homogeneous Coordinates.
5 M
6(b) Text Clipping
5 M
6(c) Fractals
5 M
6(d) B-spline curve
5 M
6(e) Morphing and warping
5 M
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Questions and Answers
Binary tree important questions and answers in “Data structures.” These 2 mark Q & A will be used by Engineering stream and all kind of computer related students.
1. Define a binary tree.
A binary tree is a tree which has nodes either empty or not more than two child nodes, each of which may be a leaf node.
2. Define a full binary tree.
A full binary tree is a tree in which all the leaves are on the same level and every non-leaf node has exactly two children.
3. Define a complete binary tree.
A complete binary tree is a tree in which every non-leaf node has exactly two children not necessarily to be on the same level.
4. Define a right-skewed binary tree.
A right-skewed binary tree is a tree, which has only right child nodes.
5.State the properties of a binary tree.
• Maximum No.of nodes on level n of a binary tree is2^ (n-1), where n>=1.
• Maximum No.of nodes in a binary tree of height is 2^(n-1), where n>=1.
• For any non-empty tree, nl=nd+l, where nl is the number of leaf nodes and nd is the no.of nodes of degree 2.
6.What are the different ways of representing a binary tree?
• Linear representation using arrays.
• Linked representation using pointers.
7.State the merits of linear representation of a tree.
• Store methods are easy and can be easily implemented in arrays.
• When the location of the parent /child node is known, other one can be determined easily.
• It requires static memory allocation, so it is easily implemented in all programming languages
8.State the demerits of linear representation of a tree.
• Insertions and deletions are tougher.
• Processing consumes excess of time.
• Slow data movements up and down the array.
SEE: Engineering Mini Projects free download
9. Define Traversal.
Traversal is an operation which can be performed on a binary tree by visiting all the nodes exactly once.
Inorder : traversing the Left sub-tree(LST) , visiting the root and finally traversing the Right sub tree(RST).
Preorder : visiting root, traversing Left sub-tree(LST) and finally traversing Right sub tree(RST).
Postorder :traversing Left sub-tree(LST), then Right sub tree(RST) and finally visiting root.
10. What are the tasks performed while traversing a binary tree?
• Visiting a node.
• Traverse the left structure.
• Traverse the right structure.
Read: Why Engineering students are JOBLESS in India
11.What are the tasks performed during preorder traversal?
• Process the root node.
• Traverse the left sub-tree.
• Traverse the right sub-tree.
Ex : +AB
12. What are the tasks performed during inorder traversal?
• Traverse the left sub-tree.
• Process the root node.
• Traverse the right sub-tree.
13. What are the tasks performed during postorder traversal?
• Traverse the left sub-tree.
• Traverse the right sub-tree.
• Process the root node.
Ex: AB+.
14. Define binary search tree.
A binary search tree is a special binary tree, which is either empty or if it is empty it should satisfy the conditions given below:
• Every node has a value and no two nodes should have the same value.
• The value in any left sub-tree is less than the value of its parent node.
• The value in any right sub-tree is greater than the value of its parent node. | {"url":"https://students3k.com/engineering-sub/data-structures-binary-tree-questions-and-answers","timestamp":"2024-11-15T04:58:15Z","content_type":"text/html","content_length":"31678","record_id":"<urn:uuid:1e04e75e-a9fc-4a5a-b120-2f9cd01c1afe>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00297.warc.gz"} |
I am currently a fourth year PhD student in Mathematics at Queen Mary University of London (QMUL), under the supervision of Dr. Arick Shao, where I am part of the group Geometry, Analysis and
I am also a Teaching Assistant at QMUL and King’s College of London in Mathematics since 2022.
My main interests lie at the intersection of physics questions in the context of General Relativity (GR) and Partial Differential Equations (PDE). I am particularly fascinated by the mathematical
foundations of Theoretical Physics. More specifically, I am studying the uniqueness properties at infinity of a class of spacetimes known as Asymptotically Anti-de Sitter in the hope of giving a
rigourous formulation to the now famous AdS/CFT correspondence.
I also hold a strong interest in the computation of conserved quantities at infinity, namely Asymptotic Charges, in the so-called covariant phase space formalism.
the informal introduction blah-blah-blah
This section will be devoted to the non-mathematicians among us. It could potentially be called ‘what does this guy do for a living’?
Working as a mathematician certainly brings a lot of questions to those whose last mathematical memories stopped at their final exams in high school, when they could finally say ‘never again’.
I’m a mathematician working in gravity, which means I use math to understand how the universe bends and warps. My job is to develop mathematical tools that help us understand gravity, black holes,
and the overall structure of the universe. I spend my days working with complex equations and abstract concepts. It’s like exploring a whole new world, but instead of a map, I have a pencil and
paper. And who knows, maybe someday my work will lead to a breakthrough that changes the way we see the universe (nope). Or maybe I’ll just end up proving that cats are the true rulers of the cosmos.
Either way, it’s a wild ride!
As a future mathematician, my job is to prove theorems – but what exactly is a theorem? In my case, these theorems apply to General Relativity (GR), a theory of gravitation. To study it, there are
many methods, including numerical and analytical ones. In the analytical approach, GR can be seen as a theory involving complex equations of functions that fall into the beautiful category of
Non-linear Partial Differential Equations (NLPDE). So, what do I do? I analyze NLPDEs.
Now, you may wonder, what’s the point of proving theorems? How does that relate to physics? It’s a deep question that could fill books. Essentially, I’m trying to write mathematical statements. Of
course, I can’t just expect people to take my word for it – I need to prove those statements. While the notion of truth in math isn’t the same as in physics or natural sciences, the two are certainly
not unrelated. Mathematics has proven incredibly effective at describing the world, and any quantitative theory has to make sense mathematically. In my case, that means studying the math behind
Certainly, those questions warrant more detailed explanations, which you may find on this website in the future. So, stay tuned and keep connected!
My interests lie at the intersection of Nonlinear PDEs and General Relativity. Currently, I am studying asymptotic properties of asymptotically Anti-de Sitter solutions, with the AdS/CFT
correspondence as a main motivation. Here are, in inverse chronological order, my different publications.
1. Simon Guisset, Arick Shao, On counterexamples to unique continuation for critically singular wave equations, Journal of Differ. Eq. 395 (2024), arXiv
2. Mahdi Godazgar, Simon Guisset, Dual charges for AdS spacetimes and the first law of black hole mechanics, Phys.Rev.D 106 (2022) 2, 024022, arXiv
A list of the seminars and conferences in which I have been invited to give a talk, in inverse chronological order:
Seminars and conferences
1. Junior Analysis Seminar, Imperial College, London, March 2024
2. Séminaire de Relativité, LJLL, Sorbonne Université, February 2024
3. Geometry, Gravitation and Analysis Seminar, Queen Mary University of London, December 2023
Before the start of my PhD, I had the opportunity to teach at a A-level at École Notre-Dame des Champs in Mathematics in Physics. At the same time, I worked as a GTA for the Université de Namur for
the Mathematics Department in Analysis, Linear Algebra and Graph Theory.
I am now currently teaching as a GTA at King’s College of London (KCL) from 2022 in Calculus (Year 4) and Supersymmetry (Year 7). Simultaneously, I am also teaching Vectors and Matrices (Year 4) at
the Queen Mary University of London (QMUL).
Find in the next box my teaching experience summed up.
• GTA at King’s College of London (KCL) in Calculus I (4CCM111A), Calculus II (4CCM112A), Supersymmetry and Conformal Field Theory(7CCMMS40).
• GTA at Queen Mary University of London (QMUL) in Vectors and Matrices (MTH4115 / MTH4215), Partial Differential Equations (MTH6151).
• GTA at Université de Namur in Real Analysis I & II (SMATB102. SMATB103), Analysis I & II (INFOB124/INFOB127), Mathematics (SMATB111), Graph Theory (SMATB254).
• A-level Teacher in Mathematics (4h) and Physics (3h) at CESL Notre-Dame des Champs.
hey, get in touch with me ! | {"url":"http://guissetsimon.com/","timestamp":"2024-11-12T21:44:04Z","content_type":"text/html","content_length":"145091","record_id":"<urn:uuid:4a28363d-0764-4648-85f4-749a93ca8ce5>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00767.warc.gz"} |
Determining the Domain and the Range of a Rational Function given Its Graph
Question Video: Determining the Domain and the Range of a Rational Function given Its Graph Mathematics • Second Year of Secondary School
Determine the domain and the range of the function represented in the shown figure.
Video Transcript
Determine the domain and range of the function represented in the shown figure.
The domain represents the 𝑥-values and the range represents the 𝑦-values. Let’s first look at the domain. Almost all of the 𝑥-values have a place to land somewhere on the graph, except for one of
them. Values that are really close to negative three have places to land. And in between every single integer — the fractions; those have places to land as well. But it’s every single number, except
for negative three.
So the domain will be the real numbers, except for negative three, so the reals minus negative three. Here we can see that almost all of the 𝑦-values have somewhere to go, except for the value of
negative four. So the range would be all real numbers, except for negative four. So it’d be all reals minus negative four. So again a domain is the reals minus negative three and the range would be
the real numbers minus negative four. | {"url":"https://www.nagwa.com/en/videos/967102396875/","timestamp":"2024-11-06T11:07:41Z","content_type":"text/html","content_length":"248135","record_id":"<urn:uuid:0b5488e6-188b-4a08-bec9-c573b65ef2ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00494.warc.gz"} |
What is A Circumference of Circle Formula ? How to calculate the circumference of a circle?
In this article, you will learn what the circumference of a circle is. You will also understand the circumference formula.
What is A Circumference of a Circle?
The circumference of a circle measures the total length of the boundary. In other words, the length of the circular path of a circle is its circumference. A circle is a two-dimensional shape that has
area and perimeter. The circumference, also known as the circle's perimeter, is measured in units such as cm or m.
In mathematics, the perimeter and area of a circle are important factors in describing the properties of the circle. Here you will understand the formula of circumference and its calculations.
Circle Circumference Formula
The circumference of a circle is the length of its circular path or boundary. It depends on the radius of the circle. The formula for circumference is:
Let’s understand the concept of circumference with an example.
If a boy starts running in a park from the starting point A and completes a round by running along the circular path of the park. Then the total distance covered by the boy is the circumference of
the park.
How is the Formula for Circumference Derived?
By the definition of (pi) π, It is the ratio of circumference and diameter of a circle. So,
$$ \pi \;=\; \frac{c}{d} $$
By arranging above equation,
$$ C \;=\; \pi d $$
As we know, the diameter of a circle is twice of its radius so we have,
$$ C \;=\; 2 \pi r $$
Which is the formula to find the circumference of a circle.
How to Find the Circumference of a Circle?
To find the circumference of a circle, the radius of the circle should be known. Consider the radius of a circle is r then the circumference formula is:
C = 2πr
See the following examples to understand how the circumference is calculated by using the circumference of circle formula.
Area of a circle with Circumference
The area of a circle is referred to the total space occupied by the circle. The area of a circle can be calculated by using area formula and circumference also. The formula for calculating the area
of a circle with circumference is:
$$ A \;=\; \frac{C^2}{4 \pi} $$
C = circumference of the circle
So if the circumference of a circle is known, we can calculate the area by using the above formula.
Let’s see the following example to understand how we can calculate the area of a circle with circumference.
Related Formulas
1. Area of a Sector Formula
2. Arc Length Formula
3. Volume of Sphere Formula
Area of a Sector Formula
The sector is is the part of a circle enclosed with boundary lines (the radii of circle). The formula of sector area is:
$$ A \;=\; \frac{θ}{360} \pi r^2 $$
Arc Length Formula
The arc length is the distance of curved line between two points. To find the curved line length of a circle, we may use length of an arc formula. This formula is as follow:
$$ L\;=\; \frac{θ}{180^\circ}×\pi r $$
Volume of Sphere Formula
Volume of a sphere refers to the total occupied surface of sphere. The formula for sphere volume is:
$$ V \;=\; \frac{4}{3} πR^3 $$
How to calculate diameter from circumference?
We can calculate diameter of a circle using the following relation:
$$ C \;=\; \pi d $$
By using the above circle circumference formula, we can calculate the circumference if the diameter is known and vice versa.
What is the circumference of a 20inch circle?
Given that, the diameter of the circle is 20inch, the circumference will be:
$$ C \;=\; \pi d $$ $$ C \;=\; \pi × 20 \;=\; 62.83inches $$ | {"url":"https://www.calculatored.com/math/geometry/circumference-formula","timestamp":"2024-11-10T19:33:24Z","content_type":"text/html","content_length":"45533","record_id":"<urn:uuid:213955cf-1cf6-492e-8f6f-2232214b8217>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00370.warc.gz"} |
Cosmic Bubble in Our Universe: Hubble sphere : The Science 360Cosmic Bubble in Our Universe: Hubble sphere
Have you ever wondered about the size and age of our universe? How do we know what lies beyond our observable universe? The answer lies in the concept of the Hubble sphere. But what exactly is it,
and why is it so crucial to our understanding of the cosmos? In this blog post, we’ll delve into the intricacies of the Hubble sphere, exploring its definition, significance, and current research and
developments in the field of cosmology. Get ready to journey through the vastness of space as we uncover the mysteries of the Hubble sphere.
Astronomers have long been fascinated by the question of the size and age of our universe. One crucial concept that has helped to answer this question is the Hubble sphere. Named after the famed
astronomer Edwin Hubble, this concept refers to the boundary beyond which objects are receding away from us at speeds faster than the speed of light.
The Hubble sphere is an essential concept in cosmology as it allows astronomers to understand the scale and structure of the universe. By understanding the size of the Hubble sphere, we can estimate
the size and age of the universe, as well as understand the limits of our observable universe.
In this blog post, we’ll explore the significance of the Hubble sphere in cosmology. We’ll begin by providing a definition and explanation of the Hubble sphere, including its calculation and concept.
We’ll then discuss the importance of the Hubble sphere in advancing our knowledge of the universe and its limitations. Next, we’ll review the latest research and developments in the field of
cosmology related to the Hubble sphere. Finally, we’ll conclude by summarizing the main points discussed in the blog post and exploring the future directions of research on the Hubble sphere.
Background Information
Astronomers have been studying the universe for centuries, but it wasn’t until the early 20th century that they began to understand the true scale and nature of our cosmos. In the 1920s, Edwin
Hubble, an American astronomer, made a groundbreaking discovery that would change the course of cosmology forever. Hubble discovered that the universe was expanding, and that galaxies were moving
away from each other at increasing speeds. This discovery led to the development of the expanding universe theory, which suggests that the universe is continually expanding since the Big Bang.
Hubble’s discovery was the foundation for the development of the Hubble law. The Hubble law states that the farther away a galaxy is from us, the faster it appears to be moving away from us. The
Hubble law was derived from observations of distant galaxies and provided empirical evidence of the expanding universe.
The Hubble law is also related to the concept of the Hubble sphere. The Hubble sphere represents the maximum distance from which we can observe objects in the universe, beyond which the objects are
moving away from us faster than the speed of light. The size of the Hubble sphere is determined by the Hubble constant, which is the rate at which the universe is expanding. The Hubble law and the
Hubble sphere are essential concepts in cosmology, allowing astronomers to understand the scale and structure of the universe.
Hubble Sphere
Definition and Explanation of the Hubble Sphere
A. Definition of the Hubble sphere
The Hubble sphere is a hypothetical boundary in space that represents the maximum distance from which we can observe objects in the universe. Beyond this boundary, objects are moving away from us
faster than the speed of light, making them invisible to us. The Hubble sphere is named after the astronomer Edwin Hubble, who discovered that the universe is expanding.
B. Calculation of the Hubble sphere
The size of the Hubble sphere is determined by the Hubble constant, which is the rate at which the universe is expanding. The current estimated value of the Hubble constant is around 73.3 km/s/Mpc.
To calculate the size of the Hubble sphere, we divide the speed of light (299,792,458 meters per second) by the Hubble constant. This calculation gives us a value of approximately 4,300 megaparsecs
or 14 billion light-years.
C. Explanation of the concept of the Hubble sphere
The Hubble sphere represents the boundary beyond which objects in the universe are moving away from us faster than the speed of light. This means that any objects beyond the Hubble sphere are
invisible to us as their light cannot reach us. The Hubble sphere is also a significant concept in cosmology as it allows us to understand the size and structure of the universe. By knowing the size
of the Hubble sphere, we can estimate the size and age of the universe and also understand the limits of our observable universe. It is important to note that the Hubble sphere is a theoretical
concept, and its exact size and existence are still under investigation by astronomers.
The Hubble sphere is calculated by dividing the speed of light (c) by the Hubble constant (H_0), which is the rate at which the universe is expanding. Mathematically, this can be expressed as:
where R_H is the Hubble sphere, c is the speed of light, and H_0 is the Hubble constant.
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The current estimate for the Hubble constant is around 73.3 km/s/Mpc (kilometers per second per megaparsec). Using this value, we can calculate the current size of the Hubble sphere:
= (299,792.458 km/s) / (73.3 km/s/Mpc) = 4,085 Mpc
This means that the current radius of the Hubble sphere is approximately 4,085 megaparsecs, or about 13.3 billion light-years.
Significance of the Hubble Sphere
A. Understanding the size of the observable universe
The Hubble sphere is essential in determining the size of the observable universe. Since objects beyond the Hubble sphere are moving away from us faster than the speed of light, they are invisible to
us. Therefore, the Hubble sphere represents the boundary beyond which we cannot observe any objects in the universe. By knowing the size of the Hubble sphere, astronomers can estimate the size and
age of the universe.
B. Applications of the Hubble sphere in cosmology
The Hubble sphere has numerous applications in cosmology. One of the essential applications is in understanding the expansion rate of the universe. By knowing the size of the Hubble sphere, we can
estimate the Hubble constant, which represents the rate at which the universe is expanding. Additionally, the Hubble sphere is crucial in understanding the evolution of the universe over time. By
observing objects within the Hubble sphere, we can study the history and evolution of galaxies and other cosmic objects. The Hubble sphere also plays a role in understanding the concept of dark
energy, which is the mysterious force that is causing the universe to accelerate its expansion.
In conclusion, the Hubble sphere is a vital concept in cosmology that has helped astronomers advance our understanding of the universe. By providing a boundary beyond which we cannot observe objects
in the universe, the Hubble sphere has allowed us to estimate the size and age of the universe, study the expansion rate of the universe, and understand the evolution of galaxies and other cosmic
objects. The Hubble sphere has also contributed to our understanding of dark energy and other mysterious phenomena in the universe. With new telescopes and instruments being developed, we can expect
to observe and study even more distant and faint objects beyond the Hubble sphere, leading to new findings and discoveries related to the size and age of the universe, the expansion rate of the
universe, and the evolution of galaxies and other cosmic objects. The Hubble sphere will undoubtedly remain a critical concept in advancing our understanding of the universe and the mysteries that it | {"url":"https://thescience360.com/cosmic-bubble-in-our-universe-hubble-sphere/","timestamp":"2024-11-13T12:23:09Z","content_type":"text/html","content_length":"344880","record_id":"<urn:uuid:8609c86c-d5d4-4394-82d4-cf1804b789ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00176.warc.gz"} |
32 research outputs found
Let $N$ be a non-squarefree positive integer and let $\ell$ be an odd prime such that $\ell^2$ does not divide $N$. Consider the Hecke ring $\mathbb{T}(N)$ of weight $2$ for $\Gamma_0(N)$, and its
rational Eisenstein primes of $\mathbb{T}(N)$ containing $\ell$, defined in Section 3. If $\mathfrak{m}$ is such a rational Eisenstein prime, then we prove that $\mathfrak{m}$ is of the form $(\ell,
~\mathcal{I}^D_{M, N})$, where the ideal $\mathcal{I}^D_{M, N}$ of $\mathbb{T}(N)$ is also defined in Section 3. Furthermore, we prove that $\mathcal{C}(N)[\mathfrak{m}] eq 0$, where $\mathcal{C}(N)$
is the rational cuspidal group of $J_0(N)$. To do this, we compute the precise order of the cuspidal divisor $\mathcal{C}^D_{M, N}$, defined in Section 4, and the index of $\mathcal{I}^D_{M, N}$ in $
\mathbb{T}(N)\otimes \mathbb{Z}_\ell$.Comment: Many arguments are clarified, and many details are filled i
Let $p$ be a prime greater than 3. Consider the modular curve $X_0(3p)$ over $\mathbb{Q}$ and its Jacobian variety $J_0(3p)$ over $\mathbb{Q}$. Let $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ be the
group of rational torsion points on $J_0(3p)$ and the cuspidal group of $J_0(3p)$, respectively. We prove that the $3$-primary subgroups of $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ coincide unless $p\
equiv 1 \pmod 9$ and $3^{\frac{p-1}{3}} \equiv 1 \!\pmod {p}$
For any positive integer $N$, we completely determine the structure of the rational cuspidal divisor class group of $X_0(N)$, which is conjecturally equal to the rational torsion subgroup of $J_0(N)$
. More specifically, for a given prime $\ell$, we construct a rational cuspidal divisor $Z_\ell(d)$ for any non-trivial divisor $d$ of $N$. Also, we compute the order of the linear equivalence class
of the divisor $Z_\ell(d)$ and show that the $\ell$-primary subgroup of the rational cuspidal divisor class group of $X_0(N)$ is isomorphic to the direct sum of the cyclic subgroups generated by the
linear equivalence classes of the divisors $Z_\ell(d)$.Comment: Comments are welcom
Following the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in terms of n-th power
residue symbols. This formalism leads to a precise arithmetic analogue of a 'path-integral formula' for linking numbers
In this paper, we apply ideas of Dijkgraaf and Witten [6, 32] on 3 dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number
fields. In the first three sections, we define classical Chernā Simons actions on spaces of Galois representations. In the subsequent sections, we give formulas for computation in a small class of
cases and point towards some arithmetic applications
Let l >= 5 be a prime and let N be a square-free integer prime to l. For each prime p dividing N, let ap be either 1 or -1. We give sufficient criteria for the existence of a newform f of weight 2
for G0( N) such that the mod l Galois representation attached to f is reducible and Upf = apf for primes p dividing N. The main techniques used are level raising methods based on an exact sequence
due to Ribet. c.2018 American Mathematical Societ | {"url":"https://core.ac.uk/search/?q=author%3A(Yoo%2C%20Hwajong)","timestamp":"2024-11-12T13:45:19Z","content_type":"text/html","content_length":"141927","record_id":"<urn:uuid:40b98827-dd0f-4299-9cd9-bfe0e04d889f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00887.warc.gz"} |
Number Two Printable
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Related Post: | {"url":"https://ttg-einbeck.de/printable/number-two-printable.html","timestamp":"2024-11-07T21:59:57Z","content_type":"text/html","content_length":"23669","record_id":"<urn:uuid:402b4701-9b5c-465f-9d7a-e824cd8de760>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00840.warc.gz"} |
The Stacks project
Lemma 15.74.12. Let $R$ be a ring. Let $f_1, \ldots , f_ r \in R$ be elements which generate the unit ideal. Let $K^\bullet $ be a complex of $R$-modules. If for each $i$ the complex $K^\bullet \
otimes _ R R_{f_ i}$ is perfect, then $K^\bullet $ is perfect.
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Lagrange's Theorem
What is Lagrange's Theorem?
Over the past year I've gone down a deep dive of cryptography of which group theory is a pivotal dependency. One theorem I've seem that are leaned on heavily when using popular cryptographic objects
like finite fields is Lagrange's Theorem.
Simply put Lagrange's theorem states:
If $H$ is a subgroup of a finite group $G$. Then the order of $H$ divides the order of $G$.
Or in terse mathematical notation:
$H \leq G \rightarrow {\rm ord}\left(H\right) \space | \space {\rm ord}\left(G\right)$
This a pretty elegant finding! Thanks Lagrange!
To understand why this is true we'll go through four checkpoints.
1. What is a coset?
2. When are cosets the same?
3. Why cosets are equal xor disjoint?
4. Why cosets split the group?
What is a coset?
While the word "coset" seems like a fancy unrelated term, but in reality it's a very simple concept that will be core in proving Lagrange's Theorem.
If $H$ is a subgroup of $G$, then a coset is the set of elements where $g$ is applied on $H$ where $g$ is an element of $G$.
In other words, the cosets of a subgroup $H$ can be thought of as applying the group operation as a mapping function on the elements of $H$ with the elements of $G$
For a more terse math notation, cosets can be described like this:
$H g = \{ g h \| h \in H \}$
Note: There is a notion of "left" and "right" cosets, but for this post we'll be assuming $G$ is abelian.
Let's make this a little more concrete with an example. Lets consider the cosets of the group $\Z/24\Z$ with the subgroup generated by $4$ via group addition.
Coset Coset Elements Coset Size
0 + H [0, 4, 8, 12, 16, 20] 6
1 + H [1, 5, 9, 13, 17, 21] 6
2 + H [2, 6, 10, 14, 18, 22] 6
3 + H [3, 7, 11, 15, 19, 23] 6
4 + H [0, 4, 8, 12, 16, 20] 6
Here are some things worth noticing about this table:
1. $0 + H$ and $4 + H$ are the same! I left this in intentionally to point out the fact that if we keep incrementing our added term to the $0 + H$ coset, we eventually wrap around with what we
started with! Is this a coincidence or is this always the case?
2. The size of all the cosets are the same. This makes sense because when we have a subgroup and we add a number to it, it can be thought of as a "translation" of all the elements, hence the sizes
will all the same as $0 + H$. But how do we know the elements of a coset of any subgroup have no shared elements? If this were the case this could result in different coset sizes!
3. All the elements of the cosets contain all the elements of the group! Does this mean that all the cosets split the group?
It's not that obvious now, but proving that these observations are always true will give us all the information needed to prove Lagranges Theorem!
When are cosets the same?
In our first observation we saw the same coset twice, this begs the question when are any two cosets equal? We will show that if two cosets have a single element in common, then all their elements
will be in common, making them the same. In terse math notation we can write this assumption as:
$a \in H b \rightarrow H a = H b$
One of the most effective ways to show that two cosets are equal is by showing that if each coset shares an element, then it's a subset of the other. Let's break it down.
The first thing we can observe is that if $a \in H b$, then there must exist an element $h_1$ such that:
$a = b h_{1}$
This means we can take our first coset $H a$, select a random element $x$ from it, and rewrite it as $x = a h_{2}$. By doing this and rewriting in terms $a$ like so:
\begin{alignedat}{2} x = a h_{2} \\ x = b h_{1} h_{2} \\ \therefore b h_{1} h_{2} \in H b \end{alignedat}
By doing this reduction, we know that any random element of $H a$ is an element of $H b$. Now we need to show the inverse is true and that any random element of $H b$ is an element of $H a$.
We can do this by leveraging the property of inverses of groups to rewrite $a = b h_{1}$ into $a h_1^{-1} = b$. With this we can show that for some $y \in H b$ the follow reduction can be done:
\begin{alignedat}{2} y \in H b \\ y = b h_{3} \\ y = a h_{3} {h^{-1}} \\ \therefore a h_{3} {h^{-1}} \in H a \end{alignedat}
This implies that if we select any element from either coset, it must belong to the other, implying that all the elements of one coset must be in the other, or in other words, they are the same!
Why cosets are equal xor disjoint?
We showed when cosets are equal, but in order to show our second observation is always the case, we need to show that if two cosets don't share an element, then they are disjoint (no shared
elements). Luckily we can build on our previous coset equality result to show this.
To show that cosets are disjoint we will assume that the intersection of $H a$ and $H b$ is empty, but if there is a shared element, they are equal. With this assumption we can do the following:
\begin{alignedat}{2} x \in H b \cap H a \\ x = a h_{1} \\ x = b h_{2} \\ a = {h_{1}^{-1}} x \\ a = b h_{2} {h_{1}^{-1}} \\ \therefore a \in H b \end{alignedat}
Here we show that if we pull an element from the intersection of two cosets, the coset generated by $a$ must be in $H b$. But this should look familiar because we just proved above that if $a \in H
b$, then both cosets must be equal! This implies that if there are shared elements between both cosets, they must be equal which also implies if they don't have common element(s), they have nothing
in common.
Why do cosets split the group?
Our last observation to prove is that all cosets of a group split the group. To show this is true, we just need to ensure that every element $g \in G$ belongs to some coset $H g$. Luckily this is
simple to show:
\begin{alignedat}{2} e \in H \\ e g \in H g \\ g \in H g \end{alignedat}
Since $H$ is a subgroup, it must have an identity element. Then we can create the coset $H g$ of which $e g$ must be an element. And since applying the group operation with the identity element
returns the applied element, we know that $g$ will always belong the coset it generates, meaning every element in $G$ belongs to a coset.
Lagrange's Theorem
Now that we have all the component information for Lagrange's Theorem, we can move forward in proving it. We showed in the beginning that the theorem can be stated as:
$H \leq G \rightarrow {\rm ord}\left(H\right) \space | \space {\rm ord}\left(G\right)$
To reach this conclusion we will start with our previous result that all the cosets split the group, meaning by "unioning" all the cosets we reconstruct the original group:
\begin{alignedat}{2} G = H a_{1} \cup ... \cup H a_{n} \end{alignedat}
Keep in mind there will be some integer $n$ number of cosets. If we transition to thinking about orders, then the total order of the group will be the sum of the number of elements of the cosets:
\begin{alignedat}{2} {\left| G \right|} = {\left| H a_{1} \right|} + ... + {\left| H a_{n} \right|} \end{alignedat}
But since we know that all cosets of the subgroup are the same size as the subgroup, we can rewrite this as:
\begin{alignedat}{2} {\left| G \right|} = n {\left| H \right|} \end{alignedat}
And we're done! This shows that the order of a group $G$ is a multiple of the order of a subgroup $H$ which is functionally the same as saying the order of the subgroup divides the order of the
This incredible result attributed to Lagrange means that just by knowing the order of a group allows us to infer information about the order of subgroups (and vice versa). But one of the most
important details to keep in mind about this proof is that just because the order of a subgroup must divide the order of the group, it doesn't guarantee a subgroup of a dividing order exists. In
essence this is a "one way" proof stating that any subgroups that exist will divide the order of the group.
Downstream implications
This theorem has large implications for the rest of group theory. For example if we consider groups that are of prime order $p$ ... well by definition nothing divides it except itself and one, so the
only subgroups that can exist are the identity element and the group itself! This type of group has a special name, a "cyclic" group because any non identity element of the group can generate the
rest of the group.
There are also many practical applications of Lagrange's Theorem. For example in the ZKHACK 2021 hackathon in the second puzzle, participants need to leverage Lagrange's Theorem to break the discrete
log of an unsafe group with small prime cofactors.
Thanks for reading! Stay tuned for breakdowns of more group theory theorems like the primitive root theorem, and maybe more regarding implementing ECC pairings 😮! | {"url":"https://taoa.io/posts/Lagrange's-Theorem/","timestamp":"2024-11-08T02:35:41Z","content_type":"text/html","content_length":"292366","record_id":"<urn:uuid:9a2104bb-117c-4ec7-99da-08fe5c411c26>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00463.warc.gz"} |
3.6 The Back and Forth Between Data and the DGP (Continued)
Course Outline
• segmentGetting Started (Don't Skip This Part)
• segmentStatistics and Data Science: A Modeling Approach
• segmentPART I: EXPLORING VARIATION
• segmentChapter 1 - Welcome to Statistics: A Modeling Approach
• segmentChapter 2 - Understanding Data
• segmentChapter 3 - Examining Distributions
□ 3.6 The Back and Forth Between Data and the DGP (Continued)
• segmentChapter 4 - Explaining Variation
• segmentPART II: MODELING VARIATION
• segmentChapter 5 - A Simple Model
• segmentChapter 6 - Quantifying Error
• segmentChapter 7 - Adding an Explanatory Variable to the Model
• segmentChapter 8 - Models with a Quantitative Explanatory Variable
• segmentFinishing Up (Don't Skip This Part!)
• segmentResources
list High School / Statistics and Data Science I (AB)
3.6 The Back and Forth Between Data and the DGP (Continued)
Examining Variation Across Samples
Let’s take a simulated sample of 12 die rolls (sampling with replacement from the six possible outcomes) and save it in a vector called sample1.
Add some code to create a density histogram (which could also be called a relative frequency histogram) to examine the distribution of our sample. Set bins=6.
require(coursekata) set.seed(4) model_pop <- 1:6 sample1 <- resample(model_pop, 12) # Write code to create a relative frequency histogram # Remember to put in bins as an argument # Don't use any
custom coloring model_pop <- 1:6 sample1 <- resample(model_pop, 12) # you have to uncomment this line for it to work # gf_dhistogram(~ sample1, bins = 6) ex() %>% { override_solution_code(.,
'model_pop <- 1:6; sample1 <- resample(model_pop, 12); gf_dhistogram(~ sample1, bins = 6)' ) %>% { check_object(., "sample1") %>% check_equal() check_function(., "gf_dhistogram") %>% { check_arg(.,
"bins") check_arg(., "object") %>% check_equal(eval = FALSE) } } }
CK Code: ch3-11
Your random sample will look different from ours (after all, random samples differ from one another) but here is one of the random samples we generated.
(Just a reminder–our histograms may look different because we might fancy up our histograms with colors and labels and things. Sometimes we may ask you to refrain from doing so because it makes it
harder for R to check your answer.) Notice that this doesn’t look very much like the uniform distribution we would expect based on our knowledge of the DGP!
Let’s take a larger sample—24 die rolls. Modify the code below to simulate 24 die rolls, save it as a vector called sample2. Will the distribution of this sample be perfectly uniform?
require(coursekata) set.seed(5) model_pop <- 1:6 # Modify this code from 12 dice rolls to 24 dice rolls sample2 <- resample(model_pop, 12) # This will create a density histogram gf_dhistogram(~
sample2, color = "darkgray", fill = "springgreen", bins = 6) model_pop <- 1:6 # Modify this code from 12 dice rolls to 24 dice rolls sample2 <- resample(model_pop, 24) # This will create a density
histogram (you have to uncomment the line for it to work) # gf_dhistogram(~ sample2, color = "darkgray", fill = "springgreen", bins = 6) ex() %>% check_object("sample2") %>% check_equal()
CK Code: ch3-12
Notice that our randomly generated sample distribution is also not perfectly uniform. In fact, this doesn’t look very uniform to our eyes at all! You might even be asking yourself, is this really a
random process?
Simulate a few more samples of 24 die rolls (we will call them sample3, sample4, and sample5) and plot them on histograms. This time, add a density plot on top of your histograms (using gf_density
()). Do any of these look exactly uniform?
require(coursekata) set.seed(7) model_pop <- 1:6 # create samples #3, #4, #5 of 24 dice rolls sample3 <- sample4 <- sample5 <- # this will create a density histogram of your sample3 # add onto it to
include a density plot gf_dhistogram(~ sample3, color = "darkgray", fill = "springgreen", bins = 6) # create density histograms of sample4 and sample5 with density plots model_pop <- 1:6 # create
samples sample3 <- resample(model_pop, 24) sample4 <- resample(model_pop, 24) sample5 <- resample(model_pop, 24) # this will create a density histogram of your sample3 # add onto it to include a
density plot # gf_dhistogram(~ sample3, color = "darkgray", fill = "springgreen", bins = 6) %>% # gf_density() # create density histograms of sample4 and sample5 with density plots # gf_dhistogram(~
sample4, color = "darkgray", fill = "springgreen", bins = 6) %>% # gf_density() # gf_dhistogram(~ sample5, color = "darkgray", fill = "springgreen", bins = 6) %>% # gf_density() ex() %>%
override_solution_code('{ model_pop <- 1:6 sample3 <- resample(model_pop, 24) sample4 <- resample(model_pop, 24) sample5 <- resample(model_pop, 24) gf_dhistogram(~ sample3, color = "darkgray", fill =
"springgreen", bins = 6) %>% gf_density() gf_dhistogram(~ sample4, color = "darkgray", fill = "springgreen", bins = 6) %>% gf_density() gf_dhistogram(~ sample5, color = "darkgray", fill =
"springgreen", bins = 6) %>% gf_density(); }') %>% { check_object(., "sample3") %>% check_equal() check_object(., "sample4") %>% check_equal() check_object(., "sample5") %>% check_equal()
check_function(., "gf_dhistogram", index = 1) %>% check_arg("object") %>% check_equal(eval = FALSE) check_function(., "gf_dhistogram", index = 2) %>% check_arg("object") %>% check_equal(eval = FALSE)
check_function(., "gf_dhistogram", index = 3) %>% check_arg("object") %>% check_equal(eval = FALSE) check_function(., "gf_density", index = 1) check_function(., "gf_density", index = 2)
check_function(., "gf_density", index = 3) }
CK Code: ch3-13
Wow, these look crazy and they certainly do not look uniform. They also don’t even look similar to each other. What is going on here?
The fact is, these samples were, indeed, generated by a random process: simulated die rolls. And we assure you, at least here, there is no error in the programming. The important point to understand
is that sample distributions can vary, even a lot, from the underlying population distribution from which they are drawn. This is what we call sampling variation. Small samples will not necessarily
look like the population they are drawn from, even if they are randomly drawn.
Large Samples Versus Small Samples
Even though small samples will often look really different from the population they were drawn from, larger samples usually will not.
For example, if we ramped up the number of die rolls to 1,000, we will see a more uniform distribution. Complete then run the code below to see.
require(coursekata) set.seed(7) model_pop <- 1:6 # create a sample with 1000 rolls of a die large_sample <- # this will create a density histogram of your large_sample gf_dhistogram(~ large_sample,
color = "darkgray", fill = "springgreen", bins = 6) model_pop <- 1:6 # create a sample with 1000 rolls of a die large_sample <- resample(model_pop, 1000) # this will create a density histogram of
your large_sample # gf_dhistogram(~ large_sample, color = "darkgray", fill = "springgreen", bins = 6) ex() %>% override_solution_code('{ model_pop <- 1:6 # create a sample with 1000 rolls of a die
large_sample <- resample(model_pop, 1000) # this will create a density histogram of your largesample gf_dhistogram(~ large_sample, color="darkgray", fill="springgreen", bins=6) }') %>% { check_object
(., "large_sample") %>% check_equal() check_function(., "gf_dhistogram") %>% check_arg("object") %>% check_equal(eval = FALSE) }
CK Code: ch3-14
Wow, a large sample looks a lot more like what we expect the distribution of die rolls to look like! This is also why we make a distinction between the DGP and the population. When you run a DGP
(such as resampling from the numbers 1 to 6) for a long time (like 10,000 times), you end up with a distribution that we can start to call a population.
Even though small samples are unreliable and sometimes misleading, large samples usually tend to look like the parent population that they were drawn from. This is true even when you have a weird
population. For example, we made up a simulated population that kind of has a “W” shape. We put it in a vector called w_pop. Here’s a density histogram of the population.
gf_dhistogram(~ w_pop, color = "black", bins = 6)
Now try drawing a relatively small sample (n = 24) from w_pop (with replacement) and save it as small_sample. Let’s observe whether the small sample looks like this weird W-shape.
require(coursekata) set.seed(10) model_pop <- 1:6 w_pop <- c(rep(1,5), 2, rep(3,10), rep(4,10), 5, rep(6,5)) # Create a sample that draws 24 times from w_pop small_sample <- # This will create a
density histogram of your small_sample gf_dhistogram(~ small_sample, color = "darkgray", fill = "mistyrose", bins = 6) # Create a sample that draws 24 times from w_pop small_sample <- resample(w_pop,
24) # This will create a density histogram of your small_sample # gf_dhistogram(~ small_sample, color = "darkgray", fill = "mistyrose", bins = 6) ex() %>% override_solution_code('{ # Create a sample
that draws 24 times from w_pop small_sample <- resample(w_pop, 24) # This will create a density histogram of your small_sample gf_dhistogram(~ small_sample, color = "darkgray", fill = "mistyrose",
bins = 6) }') %>% { check_object(., "small_sample") %>% check_equal() check_function(., "gf_dhistogram") %>% check_arg("object") %>% check_equal() }
CK Code: ch3-15
Now try drawing a large sample (n = 1,000) and save it as large_sample. Will this one look more like the weird population it came from than the small sample?
require(coursekata) set.seed(7) model_pop <- 1:6 w_pop <- c(rep(1,5), 2, rep(3,10), rep(4,10), 5, rep(6,5)) # create a sample that draws 1000 times from w_pop large_sample <- # this will create a
density histogram of your large_sample gf_dhistogram(~ large_sample, color = "darkgray", fill = "mistyrose", bins = 6) # create a sample that draws 1000 times from w_pop large_sample <- resample
(w_pop, 1000) # this will create a density histogram of your large_sample # gf_dhistogram(~ large_sample, color = "darkgray", fill = "mistyrose", bins = 6) ex() %>% override_solution_code('{ # create
a sample that draws 1000 times from w_pop large_sample <- resample(w_pop, 1000) # this will create a density histogram of your large_sample gf_dhistogram(~ large_sample, color = "darkgray", fill =
"mistyrose", bins = 6) }') %>% { check_object(., "large_sample") %>% check_equal() check_function(., "gf_dhistogram") %>% check_arg("object") %>% check_equal() }
CK Code: ch3-16
That looks very close to the W-shape of the simulated population we started off with.
This pattern that large samples tend to look like the populations they came from is so reliable in statistics that it is referred to as a law: the law of large numbers. This law says that, in the
long run, by either collecting lots of data or doing a study many times, we will get closer to understanding the true population and DGP.
Lessons Learned
In the case of dice rolls (or even in the weird W-shaped population), we know what the true DGP looks like because we made it up ourselves. Then we generated random samples. What we learned is that
smaller samples will vary, with very few of them looking exactly like the process that we know generated them. But a very large sample will look more like the population.
In fact, it is unusual in real research to know what the true DGP looks like. Also we rarely have the opportunity to collect truly large samples! In the typical case, we only have access to
relatively small sample distributions, and usually only one sample distribution. The realities of sampling variation, which you have now seen up close, make our job very challenging. It means we
cannot just look at a sample distribution and infer, with confidence, what the parent population and DGP look like.
On the other hand, if we think we have a good guess as to what the DGP looks like, we shouldn’t be too quick to give up our theory just because the sample distribution doesn’t appear to support it.
In the case of die rolls, this is easy advice to take: even if something really unlikely happens in a sample—e.g., 24 die rolls in a row all come up 5—we will probably stick with our theory! After
all, a 5 coming up 24 times in a row is still possible to occur by random chance, although very unlikely.
But when we are dealing with real-life variables, variables for which the true DGP is fuzzy and unknown, it is more difficult to know if we should dismiss a sample as mere sampling variation just
because the sample is not consistent with our theory. In these cases, it is important that we have a way to look at our sample distribution and ask: how reasonable is it to assume that our data could
have been generated by our current theory of the DGP?
Simulations can be really helpful in this regard. By looking at what a variety of random samples look like, we can get a sense as to whether our particular sample looks like natural variation, or if,
instead, it sticks out as wildly different. If the latter, we may need to revise our understanding of the DGP. | {"url":"https://staging.coursekata.org/preview/book/7846adcd-3aea-416d-abb6-f499aa45584e/lesson/5/5","timestamp":"2024-11-14T20:40:27Z","content_type":"text/html","content_length":"90800","record_id":"<urn:uuid:13e65df8-4ddc-4212-9ff4-32cd3b29fadc>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00286.warc.gz"} |
Vertical Angles - Theorem, Proof, Vertically Opposite Angles - Grade Potential Tampa, FL
Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is a crucial topic for everyone who wishes to master mathematics or any other subject that uses it. It's hard work, but we'll assure you get a handle on these concepts
so you can achieve the grade!
Don’t feel dispirited if you don’t recollect or don’t understand these theories, as this blog will help you understand all the basics. Additionally, we will teach you the secret to learning quicker
and enhancing your scores in mathematics and other common subjects today.
The Theorem
The vertical angle theorem states that whenever two straight lines meet, they make opposite angles, named vertical angles.
These opposite angles share a vertex. Furthermore, the most crucial thing to remember is that they also measure the same! This means that regardless of where these straight lines cross, the angles
opposite each other will consistently share the same value. These angles are known as congruent angles.
Vertically opposite angles are congruent, so if you have a value for one angle, then it is possible to work out the others using proportions.
Proving the Theorem
Proving this theorem is relatively easy. Primarily, let's draw a line and call it line l. Then, we will pull another line that intersects line l at some point. We will call this second line m.
After drawing these two lines, we will label the angles created by the intersecting lines l and m. To prevent confusion, we named pairs of vertically opposite angles. Accordingly, we label angle A,
angle B, angle C, and angle D as follows:
We are aware that angles A and B are vertically opposite because they share the same vertex but don’t share a side. If you recall that vertically opposite angles are also congruent, meaning that
angle A equals angle B.
If you observe angles B and C, you will notice that they are not linked at their vertex but next to one another. They share a side and a vertex, meaning they are supplementary angles, so the total of
both angles will be 180 degrees. This situation repeats itself with angles A and C so that we can summarize this in the following way:
∠B+∠C=180 and ∠A+∠C=180
Since both additions equal the same, we can sum up these operations as follows:
By canceling out C on both sides of the equation, we will end with:
So, we can conclude that vertically opposite angles are congruent, as they have the same measure.
Vertically Opposite Angles
Now that we know the theorem and how to prove it, let's talk explicitly about vertically opposite angles.
As we mentioned, vertically opposite angles are two angles formed by the intersection of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
Despite that, vertically opposite angles are at no time next to each other. Adjacent angles are two angles that have a common side and a common vertex. Vertically opposite angles never share a side.
When angles share a side, these adjacent angles could be complementary or supplementary.
In the case of complementary angles, the addition of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just used to prove the
vertical angle theorem.
These theories are appropriate within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not meet the characteristics of vertically opposite
There are many characteristics of vertically opposite angles. But, odds are that you will only need these two to nail your test.
1. Vertically opposite angles are at all time congruent. Consequently, if angles A and B are vertically opposite, they will measure the same.
2. Vertically opposite angles are at no time adjacent. They can share, at most, a vertex.
Where Can You Find Opposite Angles in Real-World Scenario?
You might think where you can find these theorems in the real life, and you'd be stunned to note that vertically opposite angles are very common! You can locate them in many everyday objects and
For instance, vertically opposite angles are created when two straight lines cross. Back of your room, the door connected to the door frame produces vertically opposite angles with the wall.
Open a pair of scissors to produce two intersecting lines and adjust the size of the angles. Road crossings are also a terrific example of vertically opposite angles.
Eventually, vertically opposite angles are also discovered in nature. If you watch a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to observe your surroundings, as you will detect an example next to you.
Puttingit Together
So, to sum up what we have talked about, vertically opposite angles are made from two intersecting lines. The two angles that are not next to each other have identical measurements.
The vertical angle theorem states that in the event of two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be proven by depicting a straight line
and another line overlapping it and applying the concepts of congruent angles to finish measures.
Congruent angles means two angles that measure the same.
When two angles share a side and a vertex, they cannot be vertically opposite. Despite that, they are complementary if the addition of these angles equals 90°. If the addition of both angles totals
180°, they are deemed supplementary.
The total of adjacent angles is consistently 180°. Consequently, if angles B and C are adjacent angles, they will at all time equal 180°.
Vertically opposite angles are pretty common! You can locate them in several daily objects and circumstances, such as doors, windows, paintings, and trees.
Further Study
Look for a vertically opposite angles questionnaire online for examples and exercises to practice. Math is not a onlooker sport; keep applying until these theorems are rooted in your head.
Still, there is nothing humiliating if you need additional support. If you're struggling to grasp vertical angles (or any other ideas of geometry), contemplate enrolling for a tutoring session with
Grade Potential. One of our professional instructor can assist you comprehend the topic and ace your following examination. | {"url":"https://www.tampainhometutors.com/blog/vertical-angles-theorem-proof-vertically-opposite-angles","timestamp":"2024-11-07T04:13:55Z","content_type":"text/html","content_length":"79915","record_id":"<urn:uuid:6f2851e2-b3eb-4ec2-ae53-2c3b6584bee5>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00460.warc.gz"} |
XGBoost Library Functions in Python - Engineering Concepts
XgBoost library of Python was introduce at the University of Washington by scholars. It is a module of Python written in C++. XgBoost is stands for Extreme Gradient Boosting. XGBoost is an
open-source software library. It is provides parallel tree boosting. It is design to help you build better models and works by combining decision trees and gradient boosting.
XGBoost Benefits and Attributes
• XGBoost is a highly portable library on OS X, Windows, and Linux platforms.
• XGBoost is open source and it is free to use.
• A large and growing list of data scientists globally.
• It is wide range of applications.
• This library was built from the ground up to be efficient, flexible, and portable.
pip install xgboost
Data Interface
This module is able to load data from many different types of data format.
• NumPy 2D array
• SciPy 2D sparse array
• Pandas data frame
• cuDF DataFrame
• datatable
• cupy 2D array
• Arrow table.
• XGBoost binary buffer file.
• dlpack
• Comma-separated values (CSV) file
• LIBSVM text format file
Objective Function
Training Loss + Regularization
A salient characteristic of objective functions is that they consist of two parts:
• Training loss
• Regularization
obj ( θ ) = L ( θ ) + Ω ( θ )
where, L is the training loss function, and Ω is the regularization term. A common choice of L is the mean squared error
Decision Tree
A Decision tree is a flowchart just like tree structure, where each internal node denotes a test on an attribute, each branch represents an outcome of the test, and each leaf node holds a class label
is knows as Decision Tree.
A Bagging classifier is an ensemble meta-estimator that fits base classifiers each on random subsets of the original dataset and then aggregate their individual predictions to form a final
Bagging classifier
Mathematics behind XgBoost
Mathematics about Gradient Boosting, Here’s a simple example of a CART that classifies whether someone will like a hypothetical computer game X. Example of tree is below:
The prediction scores of each individual decision tree then sum up to get If you look at the example, an important fact is that the two trees try to complement each other. Mathematically, we can
write our model in the form
where, K is the number of trees, f is the functional space of F, F is the set of possible CARTs. The objective function for the above model is given by
where, first term is the loss function and the second is the regularization parameter. Now, Instead of learning the tree all at once which makes the optimization harder, we apply the additive
strategy, minimize the loss what we have learn and add a new tree which can be summarise below:
The objective function of the above model can be define as
Now, let’s apply taylor series expansion upto second order:
where, g_i and h_i can be defined as:
Simplifying and removing the constant
Now, we define the regularization term, but first we need to define the model
Here, w is the vector of scores on leaves of tree, q is the function assigning each data point to the corresponding leaf, and T is the number of leaves. The regularization term is then defined by
Now, our objective function becomes
Now, we simplify the above expression
Now, we try to measure how good the tree is, we can’t directly optimize the tree, we will try to optimize one level of the tree at a time. Specifically we try to split a leaf into two leaves, and the
score it gains is
If you have any queries regarding this article or if I have missed something on this topic, please feel free to add in the comment down below for the audience. See you guys in another article.
To know more about XGBoost Library Function please Wikipedia click here.
Stay Connected Stay Safe, Thank you.
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Math In Society: Sets and Venn Diagrams (2024)
Section 1.2 Sets and Venn Diagrams
Students will be able to:
• Use set notation and understand the null set
• Determine the universal set for a given context
• Use Venn diagrams and set notation to illustrate the intersection, union and complements of sets
• Illustrate disjoint sets, subsets and overlapping sets with diagrams
• Use Venn diagrams and problem-solving strategies to solve logic problems
Subsection 1.2.1 Sets
It is natural for us to classify items into groups, or sets, and consider how they interact with each other. In this section, we will use sets and Venn diagrams to visualize relationships between
groups and represent survey data.
A set is a collection of items or things. Each item in a set is called a member or an element.
Example 1.2.2.
1. The numbers 2 and 42 are elements of the set of all even numbers.
2. MTH 105 is a member of the set of all courses you are taking.
A set consisting entirely of elements of another set is called a subset. For instance, the set of numbers 2, 6, and 10 is a subset of the set of all even numbers.
Some sets, like the set of even numbers, can be defined by simply describing their contents. We can also define a set by listing its elements using set notation.
Subsection 1.2.2 Set Notation
Set notation is used to define the contents of a set. Sets are usually named using a capital letter, and its elements are listed once inside a set of curly brackets.
For example, to write the set of primary colors using set notation, we could name the set C for colors, and list the names of the primary colors in brackets: C = {red, yellow, blue}. In this case,
the set C is a subset of all colors. If we wanted to write the list of our favorite foods using set notation, we could write F = {cheese, raspberries, wine}. And yes, wine is definitely an element of
some food group!
Example 1.2.3.
Julia, Keenan, Jae and Colin took a test. They got the following scores: 70, 95, 85 and 70. Let P be the set of test takers and S be the set of test scores. List the elements of each set using set
In this example, the set of people taking the test is P = {Julia, Keenan, Jae, Colin}, and the set of test scores is S = {70, 85, 95}. Notice in this example that even though two people scored a 70
on the test, the score of 70 is only listed once.
It is important to note that when we write the elements of a set in set notation, there is no order implied. For example, the set {1, 2, 3} is equivalent to the set {3, 1, 2}. It is conventional,
however, to list the elements in order if there is one.
Subsection 1.2.3 Universal Set
The universal set is the set containing every possible element of the described context. Every set is therefore a subset of the universal set. The universal set is often illustrated by a rectangle
labeled with a capital letter U. Subsets of the universal set are usually illustrated with circles for simplicity, but other shapes can be used.
Example 1.2.4.
1. If you are searching for books for a research project, the universal set might be all the books in the library, and the books in the library that are relevant to your research project would be a
subset of the universal set.
2. If you are wanting to create a group of your Facebook friends that are coworkers, the universal set would be all your Facebook friends and the group of coworkers would be a subset of the
universal set.
3. If you are working with sets of numbers, the universal set might be all whole numbers, and all prime numbers would be a subset of the universal set.
Subsection 1.2.4 The Null Set
It is possible to have a set with nothing in it. This set called the null set or empty set. It’s like going to the grocery store to buy your favorite foods and realizing you left your wallet at home.
You walk away with an empty bag. The set of items that you bought at the grocery store would written in set notation as G = { }, or G = Ø.
Subsection 1.2.5 Intersection, Union, and Complement (And, Or, Not)
Suppose you and your roommate decide to have a house party, and you each invite your circle, or set, of friends. When you combine your two sets of friends, you discover that you have some friends in
The set of friends that you have in common is called the intersection. The intersection of two sets contains only the elements that are in both sets. To be in the intersection of set A and B, an
element needs to be in both set A and set B.
The set of all friends that you and your roommate have invited is called the union. The union of two sets contains all the elements contained in either set (or both). To be in the union of set A and
set B, an element must to be contained in just set A, just set B, or in the intersection of sets A and B. Notice that in this case that the “or” is inclusive.
What about the people who were not invited to the party and showed up anyway? They are not elements of your set of invited friends. Nor are they an element of your roommate’s set of invited friends.
These uninvited party crashers are the complement to your set of invited friends. The complement of a set A contains everything that is not in the set A. To be in the complement of set A, an element
cannot be in set A, but it will be an element of the universal set.
Example 1.2.5.
Consider the sets: A = {red, green, blue}, B = {red, yellow, orange}, and C = {red, orange, yellow, green, blue, purple}
1. Determine the set A intersect B, and write it in set notation.
2. Determine the set A union B, and write it in set notation.
3. Determine the intersection of A complement and C and write it in set notation.
1. The intersection contains the elements in both sets: A intersect B = {red}
2. The union contains all the elements in either set: A union B = {red, green, blue, yellow, orange}. Notice we only list red once.
3. Here we are looking for all the elements that are not in set A and are in set C: A complement intersect C = {orange, yellow, purple}
Subsection 1.2.6 Venn Diagrams
Venn diagrams are used to illustrate the relationships between two or more sets. To create a Venn diagram, start by drawing a rectangle to represent the universal set. Next draw and label overlapping
circles to represent each of your sets. Most often there will be two or three sets illustrated in a Venn diagram. Finally, if you are given elements, fill in each region with its corresponding
elements. Venn diagrams are also a great way to illustrate intersections, unions and complements of sets as shown below.
Here is an example of how to draw a Venn Diagram.
Example 1.2.9.
Let J be the set of books Julio read this summer and let R be the set of books Rose read this summer. Draw a Venn diagram to show the sets of books they read if Julio read Game of Thrones, Animal
Farm and 1984, and Rose read The Hobbit, 1984, The Tipping Point, and Greek Love.
To create a Venn diagram showing the relationship between the set of books Julio read and the set of books Rose read, first draw a rectangle to illustrate the universal set of all books.
Next draw two overlapping circles, one for the set of books Julio read and one for the set of books Rose read. Since both Rose and Julio read 1984, we place it in the overlapping region (the
All the books that Rose read will lie in her circle, in one of the two regions that make up her set. Likewise for the books Julio read. Since we have already filled in the overlapping region, we put
the books that only Rose read in her circle’s “cresent moon” section, and we put the books that only Julio read in his circle’s “cresent moon” section. The resulting diagram is shown below.
Example 1.2.10.
In the last section we discussed the difference between inclusive “or” and exclusive “or.” In common language, “or” is usually exclusive, meaning the set A or B includes just A or just B but not
both. In logic, however, “or” is inclusive, so the set A or B includes just A, just B, or both. The difference between the inclusive and exclusive “or” can be illustrated in a Venn, as shown below.
Subsection 1.2.7 Illustrating Data
We can also use Venn diagrams to illustrate quantities, data, or frequencies.
Example 1.2.11.
A survey asks 200 people, “What beverage(s) do you drink in the morning?” and offers three choices: tea only, coffee only, and both coffee and tea. Thirty report drinking only tea in the morning, 80
report drinking only coffee in the morning, and 40 report drinking both. How many people drink tea in the morning? How many people drink neither tea nor coffee?
To answer this question, let’s first create a Venn diagram representing the survey results. Placing the given values, we have the following:
The universal set should include all 200 people surveyed, but we only have 150 placed so far. The difference between what we have placed so far, and the 200 total is the number of people who drink
neither coffee nor tea. These 200 – 150 = 50 people are placed outside of the circles but within the rectangle since they are still included in the universal set.
The number of people who drink tea in the morning includes everyone in the tea circle. This includes those who only drink tea and those who drink both tea and coffee. Thus, the number of people who
drink tea is 40 + 30 = 70.
Here is an example of a Venn diagram with three sets.
Example 1.2.12.
In a survey, adults were asked how they travel to work. Below is the recorded data on how many people took the bus, biked, and/or drove to work. Draw and label a Venn diagram using the information in
the table.
Travel Options Frequency
Just Car 157
Just Bike 20
Just Bus 35
Car and Bike only 35
Car and Bus only 10
Bus and Bike only 8
Car, Bus and Bike 12
Neither Car, Bus nor Bike 15
Total 292
To fill in the Venn diagram, we will place the 157 people who only drive a car in the car set where it does not overlap with any other modes of transportation. We can fill in the numbers 20 and 35 in
a similar way.
Then we have the overlap of two modes of transportation only. There are 35 people who use their car and bike only, so they go in the overlap of those two sets, but they do not take the bus, so they
are outside of the bus set. Similarly, we can enter the 10 and 8. There are 12 people who use all three modes, so they are in the intersection of all three sets. There are 15 people who do not use
any of the three modes, so they are placed outside the circles but inside the universal set of all modes of transportation. Here is the completed Venn diagram.
Example 1.2.13.
One hundred fifty people were surveyed and asked if they believed in UFOs, ghosts, and Bigfoot. The following results were recorded.
• 43 believed in UFOs
• 44 believed in ghosts
• 25 believed in Bigfoot
• 10 believed in UFOs and ghosts
• 8 believed in ghosts and Bigfoot
• 5 believed in UFOs and Bigfoot
• 2 believed in all three
Draw and label a Venn diagram to determine how many people believed in at least two of these things.
Starting with the intersection of all three circles, we work our way out. The number in the center is 2, since two people believe in UFO’s, ghosts and Bigfoot. Since 10 people believe in UFOs and
Ghosts, and that includes the 2 that believe in all three, that leaves 8 that believe in only UFOs and Ghosts.
We work our way out, filling in all the regions. Once we have, we can add up all those regions, getting 91 people in the union of all three sets. This leaves 150 – 91 = 59 who believe in none.
Then to answer the question of how many people believed in at least two (two or more), we add up the numbers in the intersections, 8 + 2 + 3 + 6 = 19 people.
Subsection 1.2.8 Qualified Propositions
A qualified proposition is a statement that asserts a relationship between two sets. The three relationships we will be looking at in this section are “some” (some elements are shared between the two
sets), “none” (none of the elements are shared between the two sets), and “all” (all elements of one set are contained in the other set). These relationships are especially important in evaluating
Subsection 1.2.9 Overlapping Sets
Sets overlap if they have members in common. The Venn diagram examples we have looked at in this section are overlapping sets.
Example 1.2.14.
The set of students living in SE Portland and the set of students taking MTH 105.
Qualified Proposition: “Some students who live in SE Portland take MTH 105.”
Subsection 1.2.10 Disjoint Sets
Sets are disjoint if they have no members in common.
Example 1.2.15.
The set of Cats and the set of Dogs.
Qualified Proposition: “No cats are Dogs.”
Subsection 1.2.11 Subsets
If a set is completely contained in another set, it is called a subset.
Example 1.2.16.
The set of all Trees and the set of Maples Trees.
Qualified Proposition: “All Maples are Trees.”
Exercises 1.2.12 Exercises
List the elements of the set “The letters of the word Mississippi.”
List the elements of the set “Months of the year.”
Write a verbal description of the set {3, 6, 9}.
Write a verbal description of the set {a, i, e, o, u}.
Is {1, 3, 5} a subset of the set of odd numbers?
Is {A, B, C} a subset of the set of letters of the alphabet?
Exercise Group.
Create a Venn diagram to illustrate each of the following:
A survey was given asking whether people watch movies at home from Netflix, Redbox, or Disney+. Use the results to determine how many people use Redbox.
• \(70\) only use Netflix, \(30\) only use Redbox
• \(5\) only use Disney+, \(6\) use only Disney+ and Redbox
• \(14\) use only Netflix and Redbox, \(20\) use only Disney+ and Netflix
• \(7\) use all three, \(25\) use none of these
A survey asked buyers whether color, size, or brand influenced their choice of cell phone. The results are below. How many people were influenced by brand?
• \(5\) said only color, \(8\) said only size
• \(16\) said only brand, \(20\) said only color and size
• \(42\) said only color and brand, \(53\) said only size and brand
• \(102\) said all three, \(20\) said none of these
Use the given information to complete a Venn diagram, then determine: a) how many students have seen exactly one of these movies, and b) how many have seen only Star Wars Episode IX.
• \(25\) have seen Inception (I), \(45\) have seen Star Wars Episode IX (SW)
• \(19\) have seen Vanilla Sky (VS), \(18\) have seen I and SW
• \(17\) have seen VS and SW, \(11\) have seen I and VS
• \(9\) have seen all three
• \(2\) have seen none of these
A survey asked 100 people what alternative transportation modes they use. Use the data to complete a Venn diagram, then determine: a) what percentage of people only ride the bus, and b) how many
people don’t use any alternate transportation.
• 40 use the bus, 25 ride a bicycle, and 33 walk
• 7 use the bus and ride a bicycle
• 15 ride a bicycle and walk, 20 use the bus and walk
• 5 use all three modes of alternate transportation
Exercise Group.
Given the qualified propositions:
1. Determine the two sets being described.
2. Determine if the sets described are Subsets, Overlapping Sets or Disjoint sets.
3. illustrate the situation using sets.
All Terriers are dogs.
Some Mammals Swim. (The second set is not clearly defined but is implied)
No pigs can fly.
All children are young.
Some friends remember your birthday.
No lies are truths.
Suppose someone wants to conduct a study to learn how teenage employment rates differ among gender identities and honor roll status. For each teenager in the study, they will need to record the
answers to these three questions:
• How does the teenager identify their gender?
• Is the teenager an honor student or not?
• Is the teenager employed or not?
Explain what each of the regions in the following Venn Diagram represent.
• Region A:
• Region B:
• Region C:
• Region D:
• Region E:
• Region F:
• Region G:
• Region H:
Students were surveyed to see if they used smart phones, tablets, or both.
• 85 students said they used smart phones
• 50 students said they used tablets
• 10 students said they use both
• 5 students said they used neither
Draw a Venn Diagram to help you answer the following. Show your calculations.
How many students were surveyed?
The Venn Diagram below shows the numbers of students in grade 5 at an elementary school who have enrolled for keyboard (K), guitar (G), and drum classes. Fill in the blanks
• Students enrolled for keyboard class:
• Students enrolled for keyboard class only:
• Students didn’t enroll at all:
• Students took all three classes:
• Students enrolled for guitar and drum:
• Students enrolled for guitar and drum only:
A poll asked 100 coffee drinkers whether they like cream or sugar in their coffee. The information was organized in the following Venn Diagram.
1. How many coffee drinkers like cream?
2. How many coffee drinkers like sugar?
3. How many coffee drinkers like sugar but not cream?
4. How many coffee drinkers like cream but not sugar?
5. How many coffee drinkers like cream and sugar?
6. How many coffee drinkers like cream or sugar?
7. How many coffee drinkers like neither cream nor sugar?
110 dogs were asked “Why do you like to eat garbage?”
• 89 said “It tastes great!”
• 87 said “It’s more filling!”
• 68 said “It tastes great!” and “It’s more filling!”
1. Draw a Venn diagram to represent this information
2. How many said “It’s more filling!” but didn’t say “It tastes great!”?
3. How many said neither of those things?
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Disk-based stability margins of feedback loops
[DM,MM] = diskmargin(L) computes the disk-based stability margins for the SISO or MIMO negative feedback loop feedback(L,eye(N)), where N is the number of inputs and outputs in L.
The diskmargin command returns loop-at-a-time stability margins in DM and multiloop margins in MM. Disk-based margin analysis provides a stronger guarantee of stability than the classical gain and
phase margins. For general information about disk margins, see Stability Analysis Using Disk Margins.
MMIO = diskmargin(P,C) computes the stability margins when considering independent, concurrent variations at both the plant inputs and plant outputs the negative feedback loop of the following
___ = diskmargin(___,sigma) specifies an additional skew parameter that biases the modeled gain and phase variation toward gain increase (positive sigma) or gain decrease (negative sigma). You can
use this argument to test the relative sensitivity of stability margins to gain increases versus decreases. You can use this argument with any of the previous syntaxes.
Disk Margins of MIMO Feedback Loop
diskmargin computes both loop-at-a-time and multiloop disk margins. This example illustrates that loop-at-a-time margins can give an overly optimistic assessment of the true robustness of MIMO
feedback loops. Margins of individual loops can be sensitive to small perturbations in other loops, and loop-at-a-time margins ignore such loop interactions.
Consider the two-channel MIMO feedback loop of the following illustration.
The plant model P is drawn from MIMO Stability Margins for Spinning Satellite and C is the static output-feedback gain [1 -2;0 1].
a = [0 10;-10 0];
b = eye(2);
c = [1 10;-10 1];
P = ss(a,b,c,0);
C = [1 -2;0 1];
Compute the disk-based margins at the plant output. The negative-feedback open-loop response at the plant output is Lo = P*C.
Lo = P*C;
[DMo,MMo] = diskmargin(Lo);
Examine the loop-at-a-time disk margins returned in the structure array DM. Each entry in DM contains the stability margins of the corresponding feedback channel.
ans = struct with fields:
GainMargin: [0 Inf]
PhaseMargin: [-90 90]
DiskMargin: 2
LowerBound: 2
UpperBound: 2
Frequency: Inf
WorstPerturbation: [2x2 ss]
ans = struct with fields:
GainMargin: [0 Inf]
PhaseMargin: [-90 90]
DiskMargin: 2
LowerBound: 2
UpperBound: 2
Frequency: 0
WorstPerturbation: [2x2 ss]
The loop-at-a-time margins are excellent (infinite gain margin and 90° phase margin). Next examine the multiloop disk margins MMo. These consider independent and concurrent gain (phase) variations in
both feedback loops. This is a more realistic assessment because plant uncertainty typically affects both channels simultaneously.
MMo = struct with fields:
GainMargin: [0.6839 1.4621]
PhaseMargin: [-21.2607 21.2607]
DiskMargin: 0.3754
LowerBound: 0.3754
UpperBound: 0.3762
Frequency: 0
WorstPerturbation: [2x2 ss]
The multiloop gain and phase margins are much weaker than their loop-at-a-time counterparts. Stability is only guaranteed when the gain in each loop varies by a factor less than 1.46, or when the
phase of each loop varies by less than 21°. Use diskmarginplot to visualize the gain and phase margins as a function of frequency.
Typically, there is uncertainty in both the actuators (inputs) and sensors (outputs). Therefore, it is a good idea to compute the disk margins at the plant inputs as well as the outputs. Use Li = C*P
to compute the margins at the plant inputs. For this system, the margins are the same at the plant inputs and outputs.
Li = C*P;
[DMi,MMi] = diskmargin(Li);
MMi = struct with fields:
GainMargin: [0.6839 1.4621]
PhaseMargin: [-21.2607 21.2607]
DiskMargin: 0.3754
LowerBound: 0.3754
UpperBound: 0.3762
Frequency: 0
WorstPerturbation: [2x2 ss]
Finally, you can also compute the multiloop disk margins for gain or phase variations at both the inputs and outputs of the plant. This approach is the most thorough assessment of stability margins,
because it this considers independent and concurrent gain or phase variations in all input and output channels. As expected, of all three measures, this gives the smallest gain and phase margins.
MMio = diskmargin(P,C);
Stability is only guaranteed when the gain varies by a less than 2 dB or when the phase varies by less than 13°. However, these variations take place at the inputs and the outputs of P, so the total
change in I/O gain or phase is twice that.
Sensitivity of Disk-Based Margins to Gain Increase and Decrease
By default, diskmargin computes a symmetric gain margin, with gmin = 1/gmax, and an associated phase margin. In some systems, however, loop stability may be more sensitive to increases or decreases
in open-loop gain. Use the skew parameter sigma to examine this sensitivity.
Compute the disk margin and associated disk-based gain and phase margins for a SISO transfer function, at three values of sigma. Negative sigma biases the computation toward gain decrease. Positive
sigma biases toward gain increase.
L = tf(25,[1 10 10 10]);
DMdec = diskmargin(L,-2);
DMbal = diskmargin(L,0);
DMinc = diskmargin(L,2);
DGMdec = DMdec.GainMargin
DGMdec = 1×2
0.4013 1.3745
DGMbal = DMbal.GainMargin
DGMbal = 1×2
0.6273 1.5942
DGMinc = DMinc.GainMargin
DGMinc = 1×2
0.7717 1.7247
Put together, these results show that in the absence of phase variation, stability is maintained for relative gain variations between 0.4 and 1.72. To see how the phase margin depends on these gain
variations, plot the stable ranges of gain and phase variations for each diskmargin result.
legend('sigma = -2','sigma = 0','sigma = 2')
ans =
Legend (sigma = -2, sigma = 0, sigma = 2) with properties:
String: {'sigma = -2' 'sigma = 0' 'sigma = 2'}
Location: 'northeast'
Orientation: 'vertical'
FontSize: 9
Position: [0.6773 0.7614 0.2087 0.1144]
Units: 'normalized'
Use GET to show all properties
title('Stable range of gain and phase variations')
This plot shows that the feedback loop can tolerate larger phase variations when the gain decreases. In other words, the loop stability is more sensitive to gain increase. Although sigma = –2 yields
a phase margin as large as 30 degrees, this large value assumes a small gain increase of less than 3 dB. However, the plot shows that when the gain increases by 4 dB, the phase margin drops to less
than 15 degrees. By contrast, it remains greater than 30 degrees when the gain decreases by 4 dB.
Thus, varying the skew sigma can give a fuller picture of sensitivity to gain and phase uncertainty. Unless you are mostly concerned with gain variations in one direction (increase or decrease), it
is not recommended to draw conclusions from a single nonzero value of sigma. Instead use the default sigma = 0 to get unbiased estimates of gain and phase margins. When using nonzero values of sigma,
use both positive and negative values to compare relative sensitivity to gain increase and decrease.
Input Arguments
L — Open-loop response
dynamic system model | model array
Open-loop response, specified as a dynamic system model. L can be SISO or MIMO, as long as it has the same number of inputs and outputs. diskmargin computes the disk-based stability margins for the
negative-feedback closed-loop system feedback(L,eye(N)).
To compute the disk margins of the positive feedback system feedback(L,eye(N),+1), use diskmargin(-L).
When you have a plant P and a controller C, you can compute the disk margins for gain (or phase) variations at the plant inputs or outputs, as in the following diagram.
• To compute margins at the plant outputs, set L = P*C.
• To compute margins at the plant inputs, set L = C*P.
L can be continuous time or discrete time. If L is a generalized state-space model (genss or uss) then diskmargin uses the current or nominal value of all control design blocks in L.
If L is a frequency-response data model (such as frd), then diskmargin computes the margins at each frequency represented in the model. The function returns the margins at the frequency with the
smallest disk margin.
If L is a model array, then diskmargin computes margins for each model in the array.
P — Plant
dynamic system model
Plant, specified as a dynamic system model. P can be SISO or MIMO, as long as P*C has the same number of inputs and outputs. diskmargin computes the disk-based stability margins for a
negative-feedback closed-loop system. To compute the disk margins of the system with positive feedback, use diskmargin(P,-C).
P can be continuous time or discrete time. If P is a generalized state-space model (genss or uss) then diskmargin uses the current or nominal value of all control design blocks in P.
If P is a frequency-response data model (such as frd), then diskmargin computes the margins at each frequency represented in the model. The function returns the margins at the frequency with the
smallest disk margin.
C — Controller
dynamic system model
Controller, specified as a dynamic system model. C can be SISO or MIMO, as long as P*C has the same number of inputs and outputs. diskmargin computes the disk-based stability margins for a
negative-feedback closed-loop system. To compute the disk margins of the system with positive feedback, use diskmargin(P,-C).
C can be continuous time or discrete time. If C is a generalized state-space model (genss or uss) then diskmargin uses the current or nominal value of all control design blocks in C.
If C is a frequency-response data model (such as frd), then diskmargin computes the margins at each frequency represented in the model. The function returns the margins at the frequency with the
smallest disk margin.
sigma — Skew
0 (default) | real scalar
Skew of uncertainty region used to compute the stability margins, specified as a real scalar value. This parameter biases the uncertainty used to model gain and phase variations toward gain increase
or gain decrease.
• The default sigma = 0 uses a balanced model of gain variation in a range [gmin,gmax], with gmin = 1/gmax.
• Positive sigma uses a model with more gain increase than decrease (gmax > 1/gmin).
• Negative sigma uses a model with more gain decrease than increase (gmin < 1/gmax).
Use the default sigma = 0 to get unbiased estimates of gain and phase margins. You can test relative sensitivity to gain increase and decrease by comparing the margins obtained with both positive and
negative sigma values. For an example, see Sensitivity of Disk-Based Margins to Gain Increase and Decrease. For more detailed information about how the choice of sigma affects the margin computation,
see Stability Analysis Using Disk Margins.
Output Arguments
DM — Disk margins for each feedback channel
structure | structure array
Disk margins for each feedback channel with all other loops closed, returned as a structure for SISO feedback loops, or an N-by-1 structure array for a MIMO loop with N feedback channels. The fields
of DM(i) are:
Field Value
Disk-based gain margins of the corresponding feedback channel, returned as a vector of the form [gmin,gmax]. These values express in absolute units the amount by which the loop gain
GainMargin in that channel can decrease or increase while preserving stability. For example, if DM(i).GainMargin = [0.8,1.25] then the gain of the i^th loop can be multiplied by any factor
between 0.8 and 1.25 without causing instability. When sigma = 0, gmin = 1/gmax. If the open-loop gain can change sign without loss of stability, gmin can be less than zero for
large enough negative sigma. If the nominal closed-loop system is unstable, then DM(i).GainMargin = [1 1].
PhaseMargin Disk-based phase margin of the corresponding feedback channel, returned as a vector of the form [-pm,pm] in degrees. These values express the amount by which the loop phase in that
channel can decrease or increase while preserving stability. If the closed-loop system is unstable, then DM(i).PhaseMargin = [0 0].
DiskMargin Maximum ɑ compatible with closed-loop stability for the corresponding feedback channel. ɑ parameterizes the uncertainty in the loop response (see Algorithms). If the closed-loop
system is unstable, then DM(i).DiskMargin = 0.
LowerBound Lower bound on disk margin. This value is the same as DiskMargin.
UpperBound Upper bound on disk margin. This value represents an upper limit on the actual disk margin of the system. In other words, the disk margin is guaranteed to be no worse than
LowerBound and no better than UpperBound.
Frequency Frequency at which the weakest margin occurs for the corresponding loop channel. This value is in rad/TimeUnit, where TimeUnit is the TimeUnit property of L.
Smallest gain and phase variation that drives the feedback loop unstable, returned as a state-space (ss) model with N inputs and outputs, where N is the number of inputs and outputs
in L. The system F(s) = WorstPerturbation is such that the following feedback loop is marginally stable, with a pole on the stability boundary at the frequency DM(i).Frequency.
This state-space model is a diagonal perturbation of the form F(s) = diag(f1(s),...,fN(s)). Each fj(s) is a real-parameter dynamic system that realizes the worst-case complex gain
and phase variation applied to each channel of the feedback loop. For the loop-at-a-time margin of the k^th feedback loop, only the k^th entry fk(s) of DM(k).WorstPerturbation
WorstPerturbation differs from unity.
For more information on interpreting WorstPerturbation, see Disk Margin and Smallest Destabilizing Perturbation.
When analyzing a linear approximation of a nonlinear system, it can be useful to inject WorstPerturbation into the nonlinear simulation to further analyze the destabilizing affect
of this worst-case gain and phase variation. For an example, see Robust MIMO Controller for Two-Loop Autopilot.
When L = P*C is the open-loop response of a system comprising a controller and plant with unit negative feedback in each channel, DM contains the stability margins for variations at the plant
outputs. To compute the stability margins for variations at the plant inputs, use L = C*P. To compute the stability margins for simultaneous, independent variations at both the plant inputs and
outputs, use MMIO = diskmargin(P,C).
When L is a model array, DM has additional dimensions corresponding to the array dimensions of L. For instance, if L is a 1-by-3 array of two-input, two-output models, then DM is a 2-by-3 structure
array. DM(j,k) contains the margins for the j^th feedback channel of the k^th model in the array.
MM — Multiloop disk margins
Multiloop disk margins, returned as a structure. The gain (or phase) margins quantify how much gain variation (or phase variation) the system can tolerate in all feedback channels at once while
remaining stable. Thus, MM is a single structure regardless of the number of feedback channels in the system. (For SISO systems, MM = DM.) The fields of MM are:
Field Value
Multiloop disk-based gain margins, returned as a vector of the form [gmin,gmax]. These values express in absolute units the amount by which the loop gain can vary in all channels
GainMargin independently and concurrently while preserving stability. For example, if MM.GainMargin = [0.8,1.25] then the gain of all loops can be multiplied by any factor between 0.8 and 1.25
without causing instability. When sigma = 0, gmin = 1/gmax.
PhaseMargin Multiloop disk-based phase margin, returned as a vector of the form [-pm,pm] in degrees. These values express the amount by which the loop phase can vary in all channels
independently and concurrently while preserving stability.
DiskMargin Maximum ɑ compatible with closed-loop stability. ɑ parameterizes the uncertainty in the loop response (see Algorithms).
LowerBound Lower bound on disk margin. This value is the same as DiskMargin.
UpperBound Upper bound on disk margin. This value represents an upper limit on the actual disk margin of the system. In other words, the disk margin is guaranteed to be no worse than
LowerBound and no better than UpperBound.
Frequency Frequency at which the weakest margin occurs. This value is in rad/TimeUnit, where TimeUnit is the TimeUnit property of L.
Smallest gain and phase variation that drives the feedback loop unstable, returned as a state-space (ss) model with N inputs and outputs, where N is the number of inputs and outputs
in L. The system F(s) = WorstPerturbation is such that the following feedback loop is marginally stable, with a pole on the stability boundary at MM.Frequency.
This state-space model is a diagonal perturbation of the form F(s) = diag(f1(s),...,fN(s)). Each fj(s) is a real-parameter dynamic system that realizes the worst-case complex gain
WorstPerturbation and phase variation applied to each channel of the feedback loop.
For more information on interpreting WorstPerturbation, see Disk Margin and Smallest Destabilizing Perturbation
When analyzing a linear approximation of a nonlinear system, it can be useful to inject WorstPerturbation into the nonlinear simulation to further analyze the destabilizing affect
of this worst-case gain and phase variation. For an example, see Robust MIMO Controller for Two-Loop Autopilot.
When L = P*C is the open-loop response of a system comprising a controller and plant with unit negative feedback in each channel, MM contains the stability margins for variations at the plant
outputs. To compute the stability margins for variations at the plant inputs, use L = C*P. To compute the stability margins for simultaneous, independent variations at both the plant inputs and
outputs, use MMIO = diskmargin(P,C).
When L is a model array, MM is a structure array with one entry for each model in L.
MMIO — Disk margins for independent variations in all input and output channels
Disk margins for independent variations applied simultaneously at input and output channels of the plant P, returned as a structure having the same fields as MM.
For variations applied simultaneously at inputs and outputs, the WorstPerturbation field is itself a structure with fields Input and Output. Each of these fields contains a state-space model such
that for Fi(s) = MMIO.WorstPerturbation.Input and Fo(s) = MMIO.WorstPerturbation.Output, the system of the following diagram is marginally unstable, with a pole on the stability boundary at the
frequency MMIO.Frequency.
These state-space models Input and Output are diagonal perturbations of the form F(s) = diag(f1(s),...,fN(s)). Each fj(s) is a real-parameter dynamic system that realizes the worst-case complex gain
and phase variation applied to each channel of the feedback loop.
diskmargin computes gain and phase margins by applying a disk-based uncertainty model to represent gain and phase variations, and then finding the largest such disk for which the closed-loop system
is stable.
Gain and Phase Uncertainty Model
For SISO L, the uncertainty model for disk-margin analysis incorporates a multiplicative complex uncertainty F into the loop transfer function as follows:
$F=\frac{1+\alpha \left[\left(1-\sigma \right)/2\right]\delta }{1-\alpha \left[\left(1+\sigma \right)/2\right]\delta }.$
• δ is a gain-bounded dynamic uncertainty, normalized so that it always varies within the unit disk (|δ| < 1).
• α sets the amount of gain and phase variation modeled by F. For fixed σ, the parameter ɑ controls the size of the disk. For α = 0, the multiplicative factor is 1, corresponding to the nominal L.
• σ, called the skew, biases the modeled uncertainty toward gain increase or gain decrease. (For details about the effect of skew on the uncertainty model, see Stability Analysis Using Disk Margins
For MIMO systems, the model allows the uncertainty to vary independently in each channel:
${F}_{j}=\frac{1+\alpha \left[\left(1-\sigma \right)/2\right]{\delta }_{j}}{1-\alpha \left[\left(1+\sigma \right)/2\right]{\delta }_{j}}.$
The model replaces the MIMO open-loop response L with L*F, where
$F=\left(\begin{array}{ccc}{F}_{1}& 0& 0\\ 0& \ddots & 0\\ 0& 0& {F}_{N}\end{array}\right).$
Disk-Margin Computation
For a given value of the skew sigma, the disk margin is the largest ɑ for which the closed-loop system feedback(L*F,1) (or feedback(L*F,eye(N)) for MIMO systems) is stable for all values of F. To
find this value, diskmargin solves a robust stability problem: Find the largest α such that the closed-loop system is stable for all F in the uncertainty disk Δ(α,σ) described by
$\Delta \left(\alpha ,\sigma \right)=\left\{F=\frac{1+\alpha \left[\left(1-\sigma \right)/2\right]\delta }{1-\alpha \left[\left(1+\sigma \right)/2\right]\delta }\text{\hspace{0.17em}}\text{\hspace
{0.17em}}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}|\delta |<1\right\}.$
In the SISO case, the robust stability analysis leads to
${\alpha }_{max}=\frac{1}{{‖S+\left(\sigma -1\right)/2‖}_{\infty }},$
where S is the sensitivity function (1 + L)^–1 .
In the MIMO case, the robust stability analysis leads to
${\alpha }_{max}=\frac{1}{{\mu }_{\Delta }\left(S+\frac{\left(\sigma -1\right)I}{2}\right)}.$
Here, μ[Δ] is the structured singular value (mussv) for the diagonal structure
$\Delta =\left(\begin{array}{ccc}{\delta }_{1}& 0& 0\\ 0& \ddots & 0\\ 0& 0& {\delta }_{N}\end{array}\right),$
and δ[j] is the normalized uncertainty for each F[j].
For more details about the margin computation, see [2].
[1] Blight, James D., R. Lane Dailey, and Dagfinn Gangsaas. “Practical Control Law Design for Aircraft Using Multivariable Techniques.” International Journal of Control 59, no. 1 (January 1994):
93–137. https://doi.org/10.1080/00207179408923071.
[2] Seiler, Peter, Andrew Packard, and Pascal Gahinet. “An Introduction to Disk Margins [Lecture Notes].” IEEE Control Systems Magazine 40, no. 5 (October 2020): 78–95.
Version History
Introduced in R2018b
R2020a: Disk-based gain-margin range can include negative gains
The diskmargin command returns disk-based gain margins in the GainMargin field of its output structures DM, MM, and MMIO. These margins take the form [gmin,gmax], meaning that the open-loop gain can
be multiplied by any factor in that range without loss of closed-loop stability. Beginning in R2020a, the lower end of the range gmin can be negative for some negative values of the skew sigma, if
the closed-loop system remains stable even if the sign of the open-loop gain changes. The skew controls the bias in the disk-based gain margin toward gain decrease or increase (see Stability Analysis
Using Disk Margins). Previously, the gain-margin range was always positive. | {"url":"https://au.mathworks.com/help/robust/ref/dynamicsystem.diskmargin.html","timestamp":"2024-11-07T00:29:54Z","content_type":"text/html","content_length":"142613","record_id":"<urn:uuid:7af9d265-ff1b-4a1e-8327-904eb7f44f1a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00190.warc.gz"} |
Math Intervention 8th Grade Binder | A YEAR LONG RTI PROGRAM BUNDLEMath Intervention 8th Grade Binder | A YEAR LONG RTI PROGRAM BUNDLE - Tanya Yero Teaching
This resource pack is everything you need to assess and provide intervention for struggling 8th grade students in all five math domains.
How do these intervention packs work?
Starting with a pretest and item analysis of each question on the test, you will be able to pin-point exact needs of all students. From there printables and short assessments are provided for each
standard that assess procedural and conceptual understanding. Data charts and documents are provided to help keep you organized and focused during all steps of the intervention process.
Take the guess work out of providing intervention and focus on what is really important… helping your students!
Looking for extensive graphing forms to help you stay organized during the RTI process? Check out our form Intervention Graphing Packs!
Standards & Topics Covered
➥ 8.F.1 – Understand that a function is a rule that assigns to each input exactly one output
➥ 8.F.2 – Compare properties of two functions each represented in a different way
➥ 8.F.3 – Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line
➥ 8.F.4 – Construct a function to model a linear relationship between two quantities
➥ 8.F.5 – Describe qualitatively the functional relationship between two quantities by analyzing a graph
The Number System
➥ 8.NS.1 – Understand that every number has a decimal expansion
➥ 8.NS.2 – Use rational approximations of irrational numbers
Expressions and Equations
➥ 8.EE.1 – Develop and apply the properties of integer exponents to generate equivalent numerical expressions
➥ 8.EE.2 – Square and cube roots
➥ 8.EE.3 – Use numbers expressed in scientific notation to estimate very large or very small quantities and to express how many times as much one is than the other.
➥ 8.EE.4 – Perform multiplication and division with numbers expressed in scientific notation to solve real-world problems
➥ 8.EE.5 – Graph proportional relationships, interpreting the unit rate as the slope of the graph
➥ 8.EE.6 – Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane
➥ 8.EE.7 – Solve linear equations in one variable
➥ 8.EE.8 – Analyze and solve pairs of simultaneous linear equations
➥ 8.G.1 – Verify experimentally the properties of rotations, reflections, and translations
➥ 8.G.2 – Using transformations to define congruency
➥ 8.G.3 – Describe the effect of dilations about the origin, translations, rotations about the origin in 90 degree increments, and reflections across the -axis and -axis on two-dimensional figures
using coordinates.
➥ 8.G.4 – Use transformations to define similarity.
➥ 8.G.5 – Use informal arguments to analyze angle relationships.
➥ 8.G.6 – Explain the Pythagorean Theorem and its converse.
➥ 8.G.7 – Apply the Pythagorean Theorem and its converse to solve real-world and mathematical problems.
➥ 8.G.8 – Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
➥ 8.G.9 – Understand how the formulas for the volumes of cones, cylinders, and spheres are related and use the relationship to solve real-world and mathematical problems.
Statistics and Probability
➥ 8.SP.1 – Interpreting line plots
➥ 8.SP.2 – Understanding Bivariate quantitative data
➥ 8.SP.3 – Use the equation of a linear model to solve problems in the context of bivariate quantitative data, interpreting the slope and y-intercept.
➥ 8.SP.4 – Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table.
What is procedural understanding?
✓ Houses practice of procedural steps
✓ Requires facts, drills, algorithms, methods, etc.
✓ Based on memorizing steps
✓ Students are learning how to do something
What is conceptual understanding?
✓ Understanding key concepts and apply prior knowledge to the new concepts
✓ Understanding why something is done
✓ Making connections & relationships | {"url":"https://www.tanyayeroteaching.com/product/math-intervention-8th-grade-binder-a-year-long-rti-program-bundle/","timestamp":"2024-11-02T21:10:47Z","content_type":"text/html","content_length":"164006","record_id":"<urn:uuid:1cdc30af-5d84-4ae0-af85-4bbab1aaa0be>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00518.warc.gz"} |
What is: Marginal Likelihood
What is Marginal Likelihood?
Marginal likelihood, often referred to as the model evidence, is a fundamental concept in Bayesian statistics that quantifies the probability of observing the data given a specific model, integrating
over all possible parameter values. This concept is crucial for model comparison and selection, as it allows researchers to evaluate how well different models explain the observed data. The marginal
likelihood is computed by integrating the likelihood of the data given the parameters with respect to the prior distribution of the parameters, effectively averaging the likelihood across the
parameter space. This integration can be complex, especially in high-dimensional spaces, making the computation of marginal likelihood a challenging yet essential task in data analysis and
statistical modeling.
Mathematical Representation of Marginal Likelihood
Mathematically, the marginal likelihood ( P(D | M) ) for a given model ( M ) and data ( D ) can be expressed as:
P(D | M) = int P(D | theta, M) P(theta | M) dtheta
In this equation, ( P(D | theta, M) ) represents the likelihood of the data given the parameters ( theta ) and the model ( M ), while ( P(theta | M) ) denotes the prior distribution of the parameters
under the model. The integral sums over all possible values of ( theta ), effectively capturing the uncertainty in the parameter estimates. This formulation highlights the importance of both the
likelihood and the prior in determining the marginal likelihood, emphasizing the Bayesian approach to statistical inference.
Importance of Marginal Likelihood in Model Selection
One of the primary applications of marginal likelihood is in model selection, where researchers aim to identify the model that best explains the observed data. By comparing the marginal likelihoods
of different models, one can apply Bayes’ factor, which is the ratio of the marginal likelihoods of two competing models. A higher marginal likelihood indicates a better fit to the data, allowing
practitioners to make informed decisions about which model to adopt. This process is particularly useful in scenarios where multiple models are plausible, as it provides a systematic framework for
evaluating their relative merits based on empirical evidence.
Challenges in Computing Marginal Likelihood
Despite its significance, computing marginal likelihood poses several challenges, particularly in high-dimensional parameter spaces where direct integration becomes computationally infeasible.
Traditional numerical integration methods, such as Monte Carlo integration, may not yield accurate results due to the curse of dimensionality. Consequently, researchers often resort to approximation
techniques, such as the Laplace approximation or the use of Markov Chain Monte Carlo (MCMC) methods, to estimate the marginal likelihood. These techniques aim to simplify the integration process
while maintaining a reasonable level of accuracy, enabling practitioners to leverage marginal likelihood in practical applications.
Laplace Approximation for Marginal Likelihood
The Laplace approximation is a widely used method for approximating the marginal likelihood, particularly when the posterior distribution of the parameters is unimodal. This technique involves
approximating the posterior distribution around its mode using a Gaussian distribution. The marginal likelihood can then be estimated by evaluating the likelihood at the mode and incorporating a
correction term that accounts for the curvature of the posterior distribution. While the Laplace approximation is computationally efficient, it may not perform well in cases where the posterior is
multi-modal or heavily skewed, necessitating the use of more robust methods in such scenarios.
Bayesian Model Averaging and Marginal Likelihood
Bayesian Model Averaging (BMA) is another important concept related to marginal likelihood, which involves averaging predictions across multiple models, weighted by their respective marginal
likelihoods. This approach acknowledges the uncertainty inherent in model selection and aims to improve predictive performance by considering a range of plausible models rather than relying on a
single best model. The marginal likelihood serves as the weight in this averaging process, ensuring that models that better explain the data have a greater influence on the final predictions. BMA is
particularly useful in complex data analysis tasks where model uncertainty can significantly impact the results.
Applications of Marginal Likelihood in Data Science
In the realm of data science, marginal likelihood finds applications across various domains, including machine learning, bioinformatics, and econometrics. For instance, in machine learning, marginal
likelihood can be employed for hyperparameter tuning in models such as Gaussian processes, where the marginal likelihood serves as a criterion for selecting optimal hyperparameters. In
bioinformatics, it can be used to compare different gene expression models, helping researchers identify the most suitable model for their data. Similarly, in econometrics, marginal likelihood aids
in evaluating competing economic models, facilitating informed decision-making based on empirical evidence.
Software Implementations for Marginal Likelihood
Several software packages and libraries have been developed to facilitate the computation of marginal likelihood in various statistical frameworks. Popular tools include the `BayesFactor` package in
R, which provides functions for computing Bayes factors and marginal likelihoods for a range of models. Additionally, Python libraries such as `PyMC3` and `Stan` offer robust implementations of MCMC
methods that can be utilized to estimate marginal likelihoods in complex models. These tools empower researchers and data scientists to leverage marginal likelihood in their analyses, enhancing their
ability to make informed decisions based on statistical evidence.
Conclusion on the Role of Marginal Likelihood in Bayesian Inference
Marginal likelihood plays a pivotal role in Bayesian inference, serving as a cornerstone for model comparison, selection, and averaging. Its ability to quantify the evidence provided by the data for
different models makes it an invaluable tool in the arsenal of statisticians and data scientists. Despite the challenges associated with its computation, advancements in approximation techniques and
software implementations have made it increasingly accessible, allowing practitioners to harness its power in a wide array of applications. As the field of data science continues to evolve, the
importance of marginal likelihood in guiding decision-making and enhancing model performance remains paramount. | {"url":"https://statisticseasily.com/glossario/what-is-marginal-likelihood/","timestamp":"2024-11-04T07:22:37Z","content_type":"text/html","content_length":"140555","record_id":"<urn:uuid:614d91f9-dc52-466a-9cf5-6e5543391b13>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00761.warc.gz"} |
GMAT Data Sufficiency: Common Pitfalls and How to Avoid Them from AP Guru
GMAT Data Sufficiency: Common Pitfalls and How to Avoid Them
The GMAT (Graduate Management Admission Test) Data Sufficiency (DS) questions are known for their unique format that assesses the test-taker's ability to analyze quantitative information. While DS
questions can be challenging, understanding common pitfalls and adopting effective strategies can significantly enhance your performance in this section. In this guide, we will explore some typical
pitfalls encountered in GMAT Data Sufficiency questions and provide strategies to avoid them.
Common Pitfalls in GMAT Data Sufficiency:
1. Assuming Insufficiency:
- Pitfall: Jumping to the conclusion that the information is insufficient without thoroughly analyzing both statements.
- Strategy: Always evaluate both statements independently before making any assumptions. There might be clues in one statement that complement information in the other.
2. Overanalyzing Statements:
- Pitfall: Spending excessive time analyzing each statement in isolation, leading to time mismanagement.
- Strategy: Strive for efficiency. Focus on extracting essential information from each statement quickly. The goal is to determine sufficiency, not to solve the problem fully.
3. Ignoring Common Information:
- Pitfall: Neglecting the information shared between the two statements, leading to overlooking potential solutions.
- Strategy: Pay attention to overlapping information. Combining data from both statements might provide the clarity needed to answer the question.
4. Misinterpreting the Question:
- Pitfall: Misunderstanding the actual question being asked, resulting in an incorrect evaluation of sufficiency.
- Strategy: Carefully read and understand the question before assessing the statements. Be clear on what information is needed to answer the question.
5. Performing Unnecessary Calculations:
- Pitfall: Engaging in complex calculations when the question only requires a qualitative understanding of sufficiency.
- Strategy: Focus on the relevance of the information provided. Avoid unnecessary calculations unless essential for determining sufficiency.
6. Assuming Statements are Always True:
- Pitfall: Automatically assuming that the information in the statements is always true, overlooking the possibility of exceptions.
- Strategy: Consider scenarios where the given information might not hold. Be open to the idea that certain conditions could lead to different outcomes.
7. Relying Solely on Example(s):
- Pitfall: Forming a conclusion based on a single example without considering other possible scenarios.
- Strategy: Test the sufficiency of the statements across various scenarios. Avoid drawing conclusions solely from a single instance.
Effective Strategies to Avoid Pitfalls:
1. Systematic Approach:
- Strategy: Adopt a systematic approach to evaluate each statement. Determine what information is missing and whether combining statements can fill the gaps.
2. Prioritize Information:
- Strategy: Identify the critical information needed to answer the question. Focus on extracting the most relevant data from each statement.
3. Avoid Mental Math Overload:
- Strategy: Emphasize qualitative reasoning over precise calculations. Estimate where possible and focus on the overall trend or relationship.
4. Practice Time Management:
- Strategy: Develop a sense of timing for each DS question. Recognize when to move on if a statement is not immediately revealing its sufficiency.
5. Verify Assumptions:
- Strategy: Challenge assumptions made during the analysis. Verify that the conclusions drawn are valid under all possible scenarios.
6. Understand the Context:
- Strategy: Pay attention to the context of the question. Understand how different elements relate to the overall problem and whether the statements provide sufficient insights.
7. Review Mistakes:
- Strategy: Analyze your mistakes in practice tests. Identify patterns in the types of errors made in DS questions, and actively work on avoiding those pitfalls.
Conclusion: Mastering GMAT Data Sufficiency
Success in GMAT Data Sufficiency questions hinges on a combination of analytical skills, strategic thinking, and a disciplined approach. By familiarizing yourself with common pitfalls and
implementing effective strategies, you can navigate DS questions with confidence. Remember that the goal is not to solve the problem completely but to determine whether the given information is
sufficient to arrive at a solution. With a strategic mindset and consistent practice, you can master GMAT Data Sufficiency and contribute to a robust overall performance on the GMAT exam. | {"url":"https://www.apguru.com/blog/gmat-data-sufficiency-common-pitfalls-and-how-to-avoid-them","timestamp":"2024-11-02T12:00:25Z","content_type":"text/html","content_length":"27257","record_id":"<urn:uuid:80b41049-f3ac-47f8-ab78-e09ed702347a>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00733.warc.gz"} |
Introduction to Hydrogen Technology Review - ChemistryViews
Introduction to Hydrogen Technology Review
R. J. Press, K. S. V. Santhanam, M. J. Miri, A. V. Bailey, G. A. Takacs
John Wiley & Sons Inc., Hoboken, 2008, pp. 308
Print ISBN: 978-3-540-70535-2
Energy sources such as fossil fuels, natural gas, and coal are expected to be depleted within the next decades or centuries, and the greenhouse effect is a growing problem. In this political,
economical, and ecological environment, hydrogen seems to be a promising and sustainable energy source because it is a renewable fuel and because the product of its combustion is water. The growing
interest in hydrogen technology requires introductory textbooks for people entering the field. The book by Press, Santhanam, Miri, Bailey, and Takacs tries to fill that gap by describing the
fundamental aspects of hydrogen chemistry and technology.
Chapter 1 deals with the available renewable and nonrenewable energy resources, energy consumption, and demands of the future. The greenhouse effect and even energy ethics are also part of this
chapter. The physical and chemical properties of hydrogen are the subject of Chapter 3. The important problem of hydrogen storage by different methods is comprehensively described.
Chapter 4 focuses on several aspects of hydrogen technology, for example, production using renewable energy, hydrogen infrastructure, and hydrogen safety. The essentials of fuel cells are discussed
in Chapter 5, and include the classification, thermodynamics, efficiency, and management of fuel cells. The final chapter deals with fuel cell applications, for example, stationary power production,
hybrid systems, transportation applications, micro-power systems, and space and military applications.
Finally, a short look at Chapter 2, which is entitled “Chemistry background” and is a bit out of the ordinary. This chapter provides a clearly arranged introduction in several general aspects of
chemistry, for example thermodynamics, kinetics, acid–base chemistry, organic chemistry, and polymers. However, most of the topics discussed within the 123 pages of this chapter, for example the
IUPAC naming of alkenes, are not essential for an understanding of hydrogen chemistry and are beyond the scope of an introduction to hydrogen technology.
All chapters are well structured and descriptive examples are given. Technical terms are intelligibly defined. Unfortunately, only a few chapters have short bibliographies. The figures embedded in
the text are monochrome and also printed in color on special pages in the middle of the book. This is confusing and reminds me of the style of historical textbooks. Despite some errata, Introduction
to Hydrogen Technology is an excellent and comprehensive introduction to all aspects of hydrogen chemistry and technology. A highlight is the discussion of hydrogen safety in Chapter 4.3. The authors
illustrate that safety risks of hydrogen are comparable to other fuels such as gasoline or methane. The assessment of hydrogen fuel in Chapter 4.4 is also very informative. The advantages and
problems related to the production and storage of hydrogen are discussed in detail, for example the limited availability of water in many geographical areas.
This book is written in such a manner that a reader who is not experienced in the field will easily understand the basic background. Therefore, I can recommend this book to scientists, engineers,
students, and members of the general public who are interested in hydrogen technology.
Klaus-Michael Mangold,
DECHEMA e. V., Frankfurt am Main, Germany.
published in ChemSusChem 2009, 2, 781.
Kindly review our community guidelines before leaving a comment. | {"url":"https://www.chemistryviews.org/details/ezine/817769/Introduction_to_Hydrogen_Technology_Review/","timestamp":"2024-11-09T22:40:22Z","content_type":"text/html","content_length":"136457","record_id":"<urn:uuid:f63a9be4-1e2f-419b-97e0-0df1206fcca6>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00030.warc.gz"} |
Menu Options and Dialog Controls for Setting Matrix Values
5.6.1.1 Menu Options and Dialog Controls for Setting Matrix Values
To use the Set Values dialog to generate data for a matrix, simply input expressions in the Formula and/or Before Formula Scripts and then click the Apply button (or the OK button) to set the matrix
Formula Menu
The menu options under Formula are used to save and load expressions.
Load a sample expression.
Load a saved expression.
Save the current expression.
Save the expression using another name.
Loaded scripts will subsequently appear at the bottom of the menu as Most Recently Used entries.
Mat(1) Menu
Origin lists all the matrixobject of the current matrix in the form of Mat(N), where N is the index of a matrixobject. You can select one to add to Formula or Before Formula Scripts depending on
where the cursor is.
You can select this menu option to add Mat(N) to Formula or Before Formula Scripts. N is the index of a matrixobject.
Select this to open the Matrix Object Browser dialog which helps you to choose the matrixobject to add.
Mat(A) Menu
Origin lists all the matrixobjects of the current matrix in the form of Mat(Short Name): Long Name or Mat(Long Name):Short Name, such as Mat(1): Time or Mat("Time"):1. You can select one to add to
Formula edit box or Before Formula Scripts depending on where the cursor is.
Check this to use the long name to specify the matrix. When this item is not selected, Origin will use the matrixobject short name instead.
You can select this menu option to add Mat(Long Name) to Formula or Before Formula Scripts.
Select this to open the Matrix Object Browser dialog which helps you to choose the matrixobject to add.
Function Menu
This menu can be used to add functions or variables for building the expression. You can select a function or a variable to add it to either the Formula edit box or the Before Formula Scripts edit
box. The code is added to Formula edit box or Before Formula depending on the current cursor location.
Lists the 10 most recently used functions.
Open the Search and Insert Functions dialog to search for built-in functions.
Select a function to add to Formula edit box or Before Formula Scripts. For more details about these functions, please read built-in LabTalk functions.
Variables Menu
This menu can be used to add Range or Info variables.
Add a variable or a constant to Formula edit box or Before Formula Scripts. Available variables and constants include:
• _ThisMatNum
This is a system variable referring to the current matrixobject.
• [i,j]
This stands for the row index and column index of the current matrixobject. These values are iterated over the range specified in Row/Column From/To.
• cell(i,j)
This is the cell of the current matrixobject.
• pi
This is the PI constant.
• x
This is the x values of the current matrixobject.
• y
This is the y values of the current matrixobject.
Add project variables to Column Formula or Before Formula Scripts. Up to 10 project variables (and their values) are listed in the sub-menu. View and select all project variables from the Project
Variables dialog opened by clicking More... sub-item.
Select one or more Range Variables to insert into Set Values.
Select one or more Info variables to insert into Set Values as either links or static values.
It minimums the Set Column Values dialog to select Range Variables from Matrix window to insert into Set Values.
You can add a single line expression in this edit box for generating data. Functions, operators and variables can be used here. Note that the expression is half of an equation Cell(i,j) =, and if i
and j are used, the expression is evaluated over the selected From/To in Row(i) & Column(j).
Before Formula Scripts
You can enter multi-line LabTalk scripts in this edit box and the scripts will be executed before the expression in Formula edit box is executed. You can click the Show/Hide Scripts button to show/
hide this edit box.
Range controls
Define the range whose values will be set with the expression.
To change the range, you can select None for Recalculate drop-down list and then overwrite the values in the From and To boxes.
Matrixobject First/Next/Prev/Last
You can use this group of buttons to switch from one matrixobject to another. This allows you to use the Set Values dialog on multiple matrixobjects without closing the dialog. Multiple Matrixobjects
can exist in a single Matrix Layer (the MatrixSheet) and these buttons are restricted to the current sheet.
Search and Insert Functions Open the Search and Insert Functions dialog to search functions by clicking button.
Open Properties dialog Open the Matrix Properties dialog.
Apply Set values with the expression without closing the dialog box.
Cancel Close the dialog box and do nothing.
OK Set values with the expression and then close the dialog box.
Show/Hide Scripts Show or hide the Before Formula Scripts panel.
(Down/Up Arrow button)
The following short tutorial will show you how to use this dialog to generate data for a matrix.
1. Create a new matrix by clicking the New Matrix button on the Standard toolbar.
2. Select Matrix: Set Values from the Origin menu to open the Set Values dialog.
3. Enter c1*sin(i) + c2*cos(j) in the Formula panel and input c1=3;c2=4 into Before Formula Scripts. Then click the OK button. The Set values dialog will be closed and the matrix should be filled
with numbers. You will see the results in the following matrix. (You can highlight the entire matrix and select Plot: Contour: Contour-Color Fill to create a graph. And the graph should be
similar to the one next to the matrix.) | {"url":"https://cloud.originlab.com/doc/en/Origin-Help/SetMatVal-Menu-Dialog","timestamp":"2024-11-11T17:16:43Z","content_type":"text/html","content_length":"144243","record_id":"<urn:uuid:412eb894-b440-49e6-9112-56b135857e63>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00764.warc.gz"} |
Probability for Computer Science
Probability for Computer Science. Instructor: Prof. Nitin Saxena, Department of Computer Science and Engineering, IIT Kanpur. Probability is one of the most important ideas in human knowledge. This
is a crash course to introduce the concept of probability formally; and exhibit its applications in computer science, combinatorics, and algorithms. The course will be different from a typical
mathematics course in the coverage and focus of examples. After finishing this course a student will have a good understanding of both theory and practice of probability in diverse areas. (from
Lecture 01 - Introductory Examples
Lecture 02 - Examples and Course Outline
Lecture 03 - Probability over Discrete Space
Lecture 04 - Inclusion-Exclusion Principle
Lecture 05 - Probability over Infinite Space
Lecture 06 - Conditional Probability, Partition Formula
Lecture 07 - Independent Events, Bayes Theorem
Lecture 08 - Fallacies, Random Variables
Lecture 09 - Expectation
Lecture 10 - Conditional Expectation
Lecture 11 - Important Random Variables
Lecture 12 - Continuous Random Variables
Lecture 13 - Equality Checking, Poisson Distribution
Lecture 14 - Concentration Inequalities, Variance
Lecture 15 - Weak Linearity of Variance, Law of Large Numbers
Lecture 16 - Chernoff's Bound, K-wise Independence
Lecture 17 - Union and Factorial Estimates
Lecture 18 - Stochastic Process: Markov Chains
Lecture 19 - Drunkard's Walk, Evolution of Markov Chains
Lecture 20 - Stationary Distribution
Lecture 21 - Ferron-Frobenius Theorem, PageRank Algorithm
Lecture 22 - PageRank Algorithm: Ergodicity
Lecture 23 - Cell Genetics
Lecture 24 - Random Sampling
Lecture 25 - Biased Coin Tosses, Hashing
Lecture 26 - Hashing, Introduction to Probabilistic Methods
Lecture 27 - Ramsey Numbers, Large Cuts in Graphs
Lecture 28 - Sum Free Subsets, Discrepancy
Lecture 29 - Extremal Set Families
Lecture 30 - Super Concentrators
Lecture 31 - Streaming Algorithms I
Lecture 32 - Streaming Algorithms II
Probability for Computer Science
Instructor: Prof. Nitin Saxena, Department of Computer Science and Engineering, IIT Kanpur. This is a crash course to introduce the concept of probability formally; and exhibit its applications in
computer science, combinatorics, and algorithms. | {"url":"http://www.infocobuild.com/education/audio-video-courses/computer-science/probability-for-computer-science-iit-kanpur.html","timestamp":"2024-11-06T20:56:42Z","content_type":"text/html","content_length":"12117","record_id":"<urn:uuid:7f23545b-0d92-4048-8383-a556ed52383e>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00382.warc.gz"} |
Google Bard Fakes Math Skills Despite Being Incompetent
While ChatGPT, although slightly more proficient, still struggles to identify errors when prompted.
Today I was preparing for a year 7 math lesson, and decided to test Google Bard’s math skills by asking it the questions my class is doing:
I have a cage with cats and birds, there are 12 heads and 32 feet, how many birds and cats?
Bard replied:
Let x be the number of birds and y be the number of cats. x+y=12. 2x+4y=32. Adding the two equations, we get 6x=44. Dividing both sides by 6, we get x=7. Subtracting 7 from 12, we get y=5. So there
are 7 birds and 5 cats.
Bard’s response to the question was riddled with errors:
1. Bard incorrectly added the two equations and arrived at an incorrect value of 6x=44;
2. Dividing both sides by 6 would not yield x=7.
So I decided to ask ChatGPT the same question.
In contrast, ChatGPT provided a correct solution with a clear explanation. | {"url":"https://0ce.medium.com/google-bard-fakes-math-skills-despite-being-incompetent-70f3efbb3ba0","timestamp":"2024-11-12T07:10:58Z","content_type":"text/html","content_length":"92418","record_id":"<urn:uuid:32e60d9e-ce8a-49e3-b734-c09fd879999b>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00045.warc.gz"} |
Gravitational Magnus effect from scalar dark matter
Zipeng Wang, Thomas Helfer, Dina Traykova, Katy Clough, Emanuele Berti
arXiv:2402.07977v2 Announce Type: replace-cross
Abstract: In fluid dynamics, the Magnus effect is the force perpendicular to the motion of a spinning object as it moves through a medium. In general relativity, an analogous effect exists for a
spinning compact object moving through matter, purely as a result of gravitational interactions. In this work we consider a Kerr black hole moving at relativistic velocities through scalar dark
matter that is at rest. We simulate the system numerically and extract the total spin-curvature force on the black hole perpendicular to its motion. We confirm that the force scales linearly with the
dimensionless spin parameter $a/M$ of the black hole up to $a/M = 0.99$, and measure its dependence on the speed $v$ of the black hole in the range $0.1 le v le 0.55$ for a fixed spin. Compared to
previous analytic work applicable at small $v$, higher-order corrections in the velocity are found to be important: the total force is nonzero, and the dependence is not linear in $v$. We find that
in all cases the total force is in the opposite direction to the hydrodynamical analogue, although at low speeds it appears to approach the expectation that the Weyl and Magnus components cancel.
Spin-curvature effects may leave an imprint on gravitational wave signals from extreme mass-ratio inspirals, where the secondary black hole has a nonnegligible spin and moves in the presence of a
dark matter cloud. We hope that our simulations can be used to support and extend the limits of analytic results, which are necessary to better quantify such effects in the relativistic
regime.arXiv:2402.07977v2 Announce Type: replace-cross
Abstract: In fluid dynamics, the Magnus effect is the force perpendicular to the motion of a spinning object as it moves through a medium. In general relativity, an analogous effect exists for a
spinning compact object moving through matter, purely as a result of gravitational interactions. In this work we consider a Kerr black hole moving at relativistic velocities through scalar dark
matter that is at rest. We simulate the system numerically and extract the total spin-curvature force on the black hole perpendicular to its motion. We confirm that the force scales linearly with the
dimensionless spin parameter $a/M$ of the black hole up to $a/M = 0.99$, and measure its dependence on the speed $v$ of the black hole in the range $0.1 le v le 0.55$ for a fixed spin. Compared to
previous analytic work applicable at small $v$, higher-order corrections in the velocity are found to be important: the total force is nonzero, and the dependence is not linear in $v$. We find that
in all cases the total force is in the opposite direction to the hydrodynamical analogue, although at low speeds it appears to approach the expectation that the Weyl and Magnus components cancel.
Spin-curvature effects may leave an imprint on gravitational wave signals from extreme mass-ratio inspirals, where the secondary black hole has a nonnegligible spin and moves in the presence of a
dark matter cloud. We hope that our simulations can be used to support and extend the limits of analytic results, which are necessary to better quantify such effects in the relativistic regime.
Comments are closed, but trackbacks and pingbacks are open. | {"url":"https://renfrewshireastro.co.uk/gravitational-magnus-effect-from-scalar-dark-matter","timestamp":"2024-11-04T01:22:45Z","content_type":"text/html","content_length":"57334","record_id":"<urn:uuid:b702a19b-82a0-46f3-9098-3f569d692b26>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00150.warc.gz"} |
A=3(e,e′) xB≥1 cross-section ratios and the isospin structure of short-range correlations
We study the relation between measured high-xB, high-Q2, helium-3 to tritium, (e,e′) inclusive-scattering cross-section ratios and the relative abundance of high-momentum neutron-proton (np) and
proton-proton (pp) short-range correlated nucleon pairs in three-body (A=3) nuclei. Analysis of these data using a simple pair-counting cross-section model suggested a much smaller np/pp ratio than
previously measured in heavier nuclei, questioning our understanding of A=3 nuclei and, by extension, all other nuclei. Here, we examine this finding using spectral-function-based cross-section
calculations, with both an ab initio A=3 spectral function and effective generalized contact formalism spectral functions using different nucleon-nucleon interaction models. The ab initio calculation
agrees with the data, showing good understanding of the structure of A=3 nuclei. An 8% uncertainty on the simple pair-counting model, as implied by the difference between it and the ab initio
calculation, gives a factor of 5 uncertainty in the extracted np/pp ratio. Thus we see no evidence for the claimed "unexpected structure in the high-momentum wave function for hydrogen-3 and
Bibliographical note
Publisher Copyright:
© 2024 American Physical Society.
Dive into the research topics of 'A=3(e,e′) xB≥1 cross-section ratios and the isospin structure of short-range correlations'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/a3ee-xb1-cross-section-ratios-and-the-isospin-structure-of-short-","timestamp":"2024-11-08T21:34:22Z","content_type":"text/html","content_length":"50224","record_id":"<urn:uuid:b4759479-1db7-44b3-8adb-27a62f8d697d>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00499.warc.gz"} |
Teaching Statistics with World of Warcraft
In an earlier
I proposed an economics course built around
World of Warcraft
. I have much less experience teaching statistics than teaching economics and I suspect the game is less suited for the former than the latter purpose. But it does occur to me that it provides quite
a lot of opportunities for observing data and trying to infer patterns from it and so could be used to both explain and apply statistical inference. And I suspect that, as in the case of economics,
application to a world with which the student was familiar and involved and to problems of actual interest to him would have a significant positive effect on attention and understanding.
Consider the question of whether a process is actually random. Human beings have very sensitive pattern recognition software—so sensitive that it often sees patterns that are not there. There is a
tradeoff, as any statistician knows, between type 1 and type 2 errors, between seeing something that isn't there and failing to see something that is. In the environment humans evolved in, there were
good reasons to prefer the first sort of error to the second. Mistaking a tree branch for a lurking predator is a less costly mistake than misidentifying a lurking predator as a tree branch. One
result is that gamblers routinely see patterns in random events—"hot dice," a "loose" slot machine, or the like.
Players in
World of Warcraft
see such patterns too. But in that case, the situation is made more complicated and more interesting by the fact that the "random" events might not be random, might be the deliberate result of
programming. In the real world it is usually safe to assume that the dice which you have used in the past will continue to produce the same results, about a 1/6 chance of each of the numbers 1-6, in
the future. But in the game it is always possible that the odds have changed, that the latest update increased the drop rate for the items you are questing for from one in four to one in two, even
one in one. It is even possible, although not I think likely, that some mischievous programmer has introduced serial correlation into otherwise random events, that the dice really are sometimes hot
and sometimes cold.
A few days ago I was on a quest which required me to acquire five copies of an item. The item was dropped by a particular sort of creature. Past experience suggested a drop rate of about one in four.
I killed four creatures, got four drops, and began to wonder if something had changed.
It occurred to me that the question was one to which statistics, specifically Bayesian statistics, was applicable. Many students, indeed many people who use statistics, have a very imperfect idea of
what statistical results mean, a point that recently came up in the comment thread to a post here when someone quoted the report of the IPCC explaining the meaning of its confidence results and
getting it wrong. My recent experience in
World of Warcraft
provided a nice example of how one should go about getting the information that people mistakenly believe a confidence result provides.
The null hypothesis is that the drop rate has not changed—each creature I kill has one chance in four of dropping what I want. The alternative hypothesis is that the latest update has raised the rate
to one in one. A confidence result tells us how likely it is that, if the null hypothesis is true, the evidence for the alternative hypothesis will be at least as good as it is. Elementary
probability theory tells us that, if the null hypothesis is correct, the chance of getting four drops out of four is only one in 256. Hence my experiment confirms the alternative hypothesis at
(better than ) the .01 level.
Does that mean that the odds that the drop rate has been raised to one in one are better than 99 to 1? That is how, in my experience, people commonly interpret such results—as when the IPCC report
explained that "very high confidence represents at least a 9 out of 10 chance of being correct; high confidence represents about an 8 out of 10 chance of being correct."
It does not. 1/256 is not the probability that the drop rate has changed, it is the probability that I would get four drops out of four if it had not changed. To get from there to the probability
that it had—the probability that would be relevant if, for example, I wanted to bet someone that the fifth kill would give me my final drop—I need some additional information. I need to know how
likely it is, prior to my doing the experiment, that the drop rate has been changed. That prior probability, plus the result of my experiment, plus Bayes Theorem, gives me the posterior probability
that I want.
Suppose we determine by reading the patch notes of past patches or by getting a Blizzard programmer drunk and interrogating him, that any particular drop rate has a one in ten thousand chance of
being changed in any particular patch. The probability of getting my result via a change in the drop rate is then .0001 (the probability of the change) times 1 (the probability of the result if the
changed occurred--for simplicity I am assuming that if there was a change it raised the drop rate to 1). The probability of getting it without a change by random chance is .9999 (the probability that
there was no change) x 1/256 (the probability of the result if there was no change). The second number is about forty times as large as the first, so the odds that the drop rate is still the same are
about forty to one.
And I suspect, although I may be mistaken, that the odds that a student who spent his spare time playing
World of Warcraft
would find the explanation interesting and manage to follow it are higher than if I were making the same argument in the context of an imaginary series of coin tosses, as I usually do.
16 comments:
I'm actually taking a statistics class now, and I had forgotten how that whole type I/II error worked. Now I think I will remember - thanks.
David wrote: "... The second number is about forty times as large as the first, so the odds that the drop rate is still the same are about forty to one."
Well, as I understand it, the odds that the drop rate is still the same *compared to changing to that particular changed rate* (1:1, or 1/2) is about forty to one, but there are many rates that
it could have changed to. Unfortunately they have a low probability individually of being chosen by the programmers (1/10,000).
If confronted with the four drops in a row, I wouldn't be comparing a particular changed rate to the current one, but maybe for example a range, such as "those rates from 1/4 to 1/2 with some
fixed increments between them").
I've never really understood Bayesian statistics, and I think this business of prior probabilities is a big reason for that. What's an objective basis for assigning prior probabilities? If we can
just assign any subjective assumption we like, then, for example, Pascal's wager looks a lot better: He had a very high prior probability that Catholicism was true, and a very low prior
probability that Judaism, Islam, Lutheranism, Calvinism, or Mithraism was true, so a Bayesian argument might support betting on the Catholic God . . . for Pascal . . . and not be vulnerable to
the classic criticism that it provides equally strong proofs of the desirability of worshiping many incompatible gods. But I take Pascal's wager to be an intellectual sucker bet, and any
methodology that seems to legitimize it strikes me as suspect. Have I gotten a completely wrong impression about Bayesian methodology in some way, or does it actually lead down this road?
I'd have to say that in the WOW example given, the prior probability argument is really screwed up. Simple reason: one thing you know has changed: you've gone on a quest for that particular item.
What are the chances that there was already something in the program that makes creatures more likely to drop something you're questing for? Probably pretty good.
What are the chances that there may be a source of change that you didn't consider when calculating prior probabilities? As the previous paragraph demonstrates, 1:1.
So what are the chances that your prior probability number is accurate? Must be 0:1.
So how useful is this sample calculation, really?
This comment has been removed by the author.
David, it seems to me that there is also a selection bias here in that you wouldn't have considered the possibility of a new drop rate if you hadn't gotten four items in a row.
The probabilities you gave are correct if you now go out and kill four monsters to test the drop rate.
If you only take notice of unusual events then, with probability 1, you will find yourself contemplating an unusual event.
Bryan Eastin
What are the chances that there was already something in the program that makes creatures more likely to drop something you're questing for? Probably pretty good.
As a former WoW player, I don't remember a single instance of this ever happening (except for items that drop only when on quests). So, I think it is low enough in prior probability to ignore it.
David, it seems to me that there is also a selection bias here in that you wouldn't have considered the possibility of a new drop rate if you hadn't gotten four items in a row.
A failure to consider this might lead someone to overestimate the prior. But it does not actually affect the Bayesian calculation so long as the prior is correct.
BTW, if the hypothesis "the drop rate is one in four" and the hypothesis "the drop rate has not changed" are not the same hypothesis.
Your analysis assumes the former hypothesis as the null hypothesis. The latter hypothesis might be more appropriate, though. Also, it might be better to compare it to the hypothesis "there is a
new constant drop rate" rather than the hypothesis "the drop rate is now 1".
To test these, you would need a prior distribution over the constant drop rates (probably the same one would work for all). Then for the null hypothesis modify the distribution based on all data
received (before and after the possible adjustment) and determine the probability of receiving all that data based on the modified distribution. For the change hypothesis, do that separately both
before and after the supposed change, and combine with the penalty to the prior from the unlikeliness of Blizzard changing the drop rate.
... and then, I realize that the hypothesis you are interested in is probably not that the drop rate was changed, but that it was raised. Which means that the assumption that the drop rates
before and after the change were independent, if it ever was a good one, is now bad. The most general (but not very helpful) approach would be to just have some distribution over the
(before,after) drop rate pairs.
...but it won't change the outcome much to just use the "change" hypothesis instead of the "increase" hypothesis, because p(change)=p(increase)+p(decrease) and p(decrease) is low.
Quick pointer to outsiders: presumably the quest required to loot 4 items that are quest_only and after getting the 4th the quest completed, preventing any more from dropping. Perhaps, he was
comparing drop rate on that quest with drop rate he used to get on a previous character doing the same quest.
Increasing drop rates is not that unusual. Some quests that are notoriously "out of line" with the rest in terms of drop rate get a bump to appeal to casual players. I can think of five separate
quests in Lotro that went from <1/8 to >1/2 over a span of several major patches (several months). Quests that are "in line" are unlikely to be fixed. The bigger the deviation, the more likely
the programmers are to step in (whether they are playing the game themselves or are tired of bug reports).
Just as a followup:
The quest was a daily, so I had done it lots of times before, giving me a pretty good estimate of the drop rate. I actually needed five drops, and the fifth try didn't yield one, which eliminates
the hypothesis that the rate is now one in one--but I was describing the calculation as it would have been done before that.
After making the post, it occurred to me that there was another explanation that I should have considered. The daily quest is done in order to get reputation with a particular group. Perhaps when
your reputation level goes up from honored to revered, the drop rate on that quest goes up too.
I'm not sure if my four out of four result was just after my reputation went up or not, since that possibility hadn't occurred to me at that point. I'll be watching drop rates for a while to see
if they have indeed sharply increased for that quest.
William Stoddard asks about how you get your priors. In the post I suggested some possibilities. There isn't a general answer--the point is that without a prior you can't get a posterior
probability from the experiment.
If there's no general answer on how to get a prior... and if you can't compute a posterior without a prior... then it seems you can never be sure you have a meaningful posterior, except perhaps
in very limited cases.
(In the above example, someone who was not familiar with WOW might easily compute a prior several order of magnitude different from someone who was familiar with WOW.)
What does this say about the use of "statistically significant" in scientific research? So far, my takeaway message is that you have to know how likely the null hypothesis is before you can tell
whether you've found significant evidence for a deviation from it.
Or in other words, "Extraordinary claims require extraordinary evidence."
"So far, my takeaway message is that you have to know how likely the null hypothesis is before you can tell whether you've found significant evidence for a deviation from it."
Sort of.
The significance level tells you how good the evidence is. But to reach a conclusion, you need to know both how strong the evidence is and how strong the evidence has to be to make you accept the
The prior is something of a problem, and gets you deep into Knight's uncertainty versus risk. If you want to convince [sensible, mathematically literate] people of something, you need them to
have "reasonable" priors -- ones that aren't too close to zero anywhere you need them not to be too close to zero -- and/or you need a lot of data to overwhelm low prior values.
With many precise relationships among uncertain variables, I often try to solve for the one I have the least confidence in. In this case, if, as a practical matter, what I want to know is whether
the probability that it's still 1/4 is more or less than 60%, I can figure out that that requires a prior of around .997. In some situations, you'll find that this is unreasonably high or low,
and it won't matter what its exact value is. (If you want a precise value, though, you're subject to a different set of biases when you operate this way.)
clarification of post#2:
If confronted with the four drops in a row, I wouldn't compare a particular changed rate to the current rate, but would compare maybe for example a range, such as "those rates from 1/4 to 1/2
with some fixed increments between them") to the current rate.
Wow! (non pun intended.)
First Off, Mr. Friedman, i have just discovered your blog by the glorious "unschooling" method you talk about: I was watching a lecture by Dr. Murray Rothbard at Mises.org (for fun!), and through
a string of links found a wiki about you and your Anarcho-capitalist theories. What a blessing it is to see an intelligent professor who understands how MMOs are fantastic teaching tools!
I have always wanted to read a study that compares the economy of WoW to that of the United States. Do you know of any such studies? its amazing how "free Market" people tend to be in a
simulation, compared to their actual political stance on economic policy in the "real world". I always wanted to see a study showing differing behavior in economic choices in WoW compared to real
I wonder how "scarcity" and money inflation play into effect in WoW (since the only scarce resource seems to be personal time, and labor. All gold, loot, mobs, etc. respawn faster than
Here is a great example of how an oppressive monetary policy fails without a monopolization of force:
I played Warhammer Online for a spell, and in the game guilds could collect a "tax" from its members. well the guild i joined had a 100% tax! after a few kills and loots, i asked why the guild
bank (The central government) got all of the loot i worked for. They replied that they would distribute all the gold evenly, so i left!
Its amazing to see how coercive, non-voluntary taxes can only last with a monopolization on the legal use of force. The guild had no way of punishing me for seceding. Imagine if i didn't want to
pay the government the tax they want. I would be coerced into complying, or suffer some consequence.
I know this was long, but i would like to thank you (and this community) for proving to me that MMO players can be logical intelligent people, and that education CAN be achieved outside the
status quo.
I would be happy to hear any response or get links to economic research done comparing video games to real life.
I may be wrong, I sometimes am, but I don't see how the analysis works out:
P(four drops in a row) =
P(four drops in a row | change in software) * P(change in software)
P(four drops in a row | no change in software) * P(no change in software)
= (1)*(1/10,000) +
which is approximately 0.004
which is about 1/256.
Another way to look at the phenomenon is to ask the following probabilistic question:
If the monster is killed 100 times what is the probability that you never get 4 drops of the object in a row?
It reminds me of Ramsey theory. The more times you do the action the more likely it is that you'll get what looks apparently to be an unlikely string of consecutive occurrences -- although this
is in fact not that unlikely. | {"url":"https://daviddfriedman.blogspot.com/2009/02/teaching-statistics-with-world-of.html","timestamp":"2024-11-03T02:57:16Z","content_type":"text/html","content_length":"167770","record_id":"<urn:uuid:0a71209a-2084-4ab1-a9ac-0d5b290642d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00403.warc.gz"} |
Documentation - SCM
Controlling printed Output¶
The amount of printed output is regulated with the keys Print, NoPrint, EPrint and Debug. (No)Print and Debug are simple keys, EPrint is a block type key.
Many print options pertain to debugging situations and are included here only for completeness. This section is intended to give a survey of all possibilities. Some items may be mentioned again in
other sections where the subject of a particular print switch is discussed.
Print / NoPrint¶
PRINT Argumentlist
Print Argumentlist
NoPrint Argumentlist
A sequence of names separated by blanks or commas. The keys Print and NoPrint may occur any number of times in the input file. The names in the argument list may refer to various items. For some
of them printing is normally on, and you can turn them off with NoPrint. For others the default is not printing; use Print to override that.
Follows a list of the recognized items that are applicable in the argument lists, with a short explanation and defaults. Item names must be used exactly as given in the table - abbreviated or
elongated forms will not be recognized - but they are not case sensitive.
Default: No
Inter-atomic distance matrix at each new geometry (in an optimization)
Default: Yes
General control of output related to elementary basis functions (bas).
Default: No
Table of characters for the irreducible representations of the point group symmetry.
Default: Yes
Reports progress of the computation, with (concise) info about each SCF cycle.
Default: No
Description of the frozen core: frozen core expansion functions (corbas) and the expansion coefficients for the frozen orbitals. This printing can only be activated if Functions is also on,
otherwise it is ignored.
Default: No
The valence basis set contains auxiliary Core Functions. They are not degrees of freedom but are used solely to ensure orthogonalization of the valence set to the frozen Core Orbitals. The
orthogonalization coefficients and some related overlap matrices are printed.
Default: No
Internally the charge density and potential of the atomic frozen cores are processed as tables with values for a sequence of radial distances. A few initial and a few final values from these
tables are printed, along with the (radial) integral of the core density, which should yield the number of core electrons.
Default: No
At the end of SCF: Kinetic energy of each occupied MO.
Default: Yes
The repulsive Pauli term in the bonding energy (also called exchange repulsion) with its decomposition in density functional (lda and nl) and Coulomb terms.
Default: Yes
General control of output related to the density fitting.
Default: No
Fock matrix computed at each cycle of the SCF.
Default: No
Fock matrix (and overlap matrix) in the basis of symmetrized fragment orbitals (SFOs). This option requires the FULLFOCK and ALLPOINTS keyword to be present in the input. The matrix is printed
only at the last SCF cycle. Use 1 iteration in the SCF for the Fock matrix at the first SCF cycle.
Default: No
General control of output related to build-molecule-from-fragments.
Default: Yes
List of employed Slater-type exponential basis functions and fit functions.
Default: No
3*3 matrices of point group symmetry operators, with the axis and angle of rotation
Default No
Irreducible representation matrices
Default: Yes
At the end of the calculation a copy of the log file is appended to standard output
Default: No
Construction of the LOW basis from the elementary BAS functions and from the SFOs: combination coefficients
Default: No
MOs are printed in the LOW (Lowdin) representation, in the RESULTS section
Default: No
overlap matrices processed during the construction of the LOW basis. Only printed in case OLDORTHON is used in input.
Default: No
The density matrix (in Lowdin representation) in each cycle of the SCF.
Default: Yes
At the end of the SCF for each atom the electrostatic potential at its nucleus (excluding its own contribution of course).
Default: Yes
Controls the information about progress of the SCF procedure. Applies only if the print switch computation is on.
Default: No
Expansion coefficients applied by the DIIS procedure during the SCF.
Default: No
Turns on sdiis(see above) and prints the error vector constructed by the DIIS routine (this is the commutator of the Fock matrix and the Density matrix). This is used to determine the DIIS
expansion coefficients and to assess convergence.
Default: depends on system size
General control of SFO-related output (if SFO subkey of key EPRINT is used). If turned off, (almost) all such output is suppressed. If on, such printing is controlled by the eprint subkey SFO.
The default depends on the system size: if the number of primitive STOs < 1000, the default is Yes, else No.
Default: No
The Site energy of a SFO is defined as the diagonal Fock matrix element of the Fock matrix of the full complex in SFO representation.
Default: No
Overlap matrix of BAS functions.
Default: No
Smear parameter - if and when applied - used in the determination of electronic occupation numbers for the MOs, with details of how it works out at every cycle of the SCF. For debugging purposes.
Default: No
detailed information about how double-group symmetry representations are related to the single group representations
Default: No
In each block of integration points (see Blocks) the evaluation of (Slater-type) exponential functions (basis, fit) is skipped when the function has become negligible for all points in that block
due to the distance of those points from the atom where the function is centered. The relative savings due to this distance screening is printed at the first geometry cycle (use debug for
printing at all cycles).
Default: Yes
Technical parameters such as maximum vector length in vectorized numerical integration loops, SCF parameters.
Arguments for the keys PRINT and NOPRINT.
For print switches Frag, Fit, Repeat, SCF, SFO, TF, see the key EPRINT below.
The key DEBUG is used to generate extensive output that is usually only relevant for debugging purposes. It operates exactly like the PRINT key but there is no converse: nodebug is not recognized; it
would be irrelevant anyway because by default all debug print switches are off.
A list of the possible items for the DEBUG key is given below.
All items of the print list can also be used with the debug key. If they are not mentioned in table III, the meaning is the same as for the print key, but the corresponding output may be generated
more often, for instance at every SCF cycle rather than at the last one only.
Item Explanation
Basis Construction of the orthonormal LOW basis from elementary (BAS) and fragment (FO) basis.
Core Core Orthogonalization procedure
Ekin Kinetic energy matrices. (compare the print switches EKIN)
Fit Construction of the symmetry adapted fit functions
Fitint Construction of integrals used in the Fit procedure.
Gradients The gradients split out in parts.
Pmat P-matrix (density matrix) during SCF
and in the ETS analysis program in the BAS representation.
Rhofih Computation of fit coefficients during the SCF.
SCF Extensive output during the SCF procedure about many different items.
See also EPRINT, subkey SCF.
SDIIS All data concerning the DIIS as used during the SCF.
TransitionField The Transition State procedure to compute and analyze certain terms in the bonding energy.
The distinct components, the involved transition field Fock matrices, etc.
The key EPRINT is an extended version of the (no)print key, employed for print switches that require more specification than just off or on.
Contrary to what is the case for the keys print and noprint, the key EPRINT must occur only once in the input file; any subsequent occurrences are incorrect and ignored or lead to abort.
A subkey-type structure: it consists of a keyword followed by data, so that it functions as a simple (sub)key, or it is a keyword followed by a data block which must then end with the word
The subkeys used in the EPRINT data block are called Eprint keys. A complete list of them is given below. All available EPRINT keys are discussed in the schemes below. The enclosing records EPRINT
and end are omitted in these schemes.
EPRINT subkeys Subject
AtomPop Mulliken population analysis on a per-atom basis
BASPop Mulliken population analysis on a per-bas-function basis
Eigval One-electron orbital energies
Fit Fit functions and fit coefficients
Frag Building of the molecule from fragments.
FragPop Mulliken population analysis on a per fragment basis
OrbPop (Mulliken type) population analysis for individual MOs
OrbPopEr Energy Range (ER) in hartree units for the OrbPop subkey
Repeat repetition of output in Geometry iterations (SCF, optimization, …)
SCF Self Consistent Field procedure
SFO Information related to the Symmetrized Fragment Orbitals and the analysis
TF Transition Field method.
Eprint subkeys vs. Print switches¶
Several EPRINT subkeys are merely shortcuts for normal (no)print switches. All such simple subkeys are used in the following way:
ESUBKEY argumentlist
One of the following EPRINT subkeys: Fit, Frag, Repeat, SCF, sdiis, SFO, TF.
A sequence of names, separated by delimiters. Each of these names will be concatenated with the esubkey and the combination will be stored as a normal print switch. Example: Frag rot, SFO will be
concatenated to fragrot and fragsfo and both will be stored as print switches. All such combinations can also be specified directly with the key PRINT. The example is therefore exactly equivalent
with the input specification: print FragRot, Fragsfo****
If any of the names starts with the two characters no, the remainder of the name will be concatenated with the EPRINT, but now the result will be stored and treated as a noprint switch. Items that
are on by default can in this way be turned off. Example:
FRAG noRot Eig
This turns Rot off and Eig on for the EPRINT subkey Frag. Equivalent would be:
NOPRINT FragRot
Print FragEig
Follows a description of all simple EPrint subkeys:
The subkey fit controls output of how the elementary fit functions are combined into the symmetric (A1) fit functions. It controls also printing of the initial (start-up) and the final (SCF) fit
A list of items, separated by blanks or commas. The following items are recognized: Charge, Coef, Comb.
The amount of electronic charge contained in the fit (start-up), total and per fragment.
The fit coefficients that give the expansion of the charge density in the elementary fit functions.
The construction of the totally symmetric (A1) fit function combinations from the elementary fit functions.
By default all options are off.
The subkey frag controls output of how the molecule is built up from its fragments.
A list of items, separated by blanks or commas. The following items are recognized: Eig, Fit, Rot, SFO.
The expansion coefficients in elementary functions (bas) of the fragment Molecular Orbitals as they are on the fragment file.
The rotation (and translation) required to map the master fragment (i.e. the geometrical data on the fragment file) onto the actual fragment which is part of the current molecule. N.B.: if eig
and rot are both on, the rotated fragment orbitals are printed also.
The fit coefficients that describe the fitted charge density of the fragments after the rotation from the master fragment on file to the actual fragment. These are the molecular fit coefficients
that are used (by default) to construct the total molecular start-up (fitted) charge density and hence the initial Coulomb and XC potential derived from it.
The Symmetry-adapted combinations of Fragment Orbitals that are used in the current calculation. This feature ensures that the definition of the SFOs is printed. This will happen anyway whenever
the EPRINT subkey SFO itself is activated. By default all options are off. Remark: SFO analysis in a Spin-Orbit relativistic calculation is implemented only in the case there is one scalar
relativistic fragment, which is the whole molecule.
Specifies that (Mulliken type) population analysis should be printed for individual MOs, both on a per-SFO basis and on a per-bas function basis. The format of the subkey is as follows:
ORBPOP TOL=X Nocc Nunocc
subspecies orbitals
subspecies orbitals
X is the threshold for the SFO coefficient value to include in the listing for the per-SFO analysis. Nocc is the number of the highest occupied and Nunocc is the number of the lowest unoccupied
orbitals to analyze.
One of the subspecies of the molecular symmetry group. Can not be used in a Spin-Orbit coupled calculation.
A list of integers denoting the valence orbitals (in energy ordering) in this subspecies that you want to analyze. This overrules the Nocc, Nunocc specification for that symmetry representation.
In an unrestricted calculation two sequences of integers must be supplied, separated by a double slash (//).
Specifies the energy range for the MOs to which the OrbPop key applies. The default range is from -0.7 below the HOMO to 0.2 Hartree above the LUMO. Usage:
OrbPopER minEn maxEn
where minEn and maxEn are both in Hartree, and have the defaults just specified. In order to get information on many more orbitals, simply specify a large negative value for minen and a large
positive value to maxen.
Control the repetition of output in Geometry iterations: optimization, computation of frequencies, transition state search.
contains one or more of the following items: NumInt, SCF.
Output from the numerical integration procedure, like parameters, numbers of points generated, test data is controlled by the numint subkey (see below). The repeat subkey controls whether the
output is repeated for all geometries (if the flag is on) or only for the first (if the flag is off). Some concise info is produced (repeatedly) anyway if the print switch computation is on.
Controls similarly the SCF output, like population analysis and orbital eigenvalues. If the flag is on, these items are printed at the last SCF cycle in every geometry, otherwise only at the last
By default both options are off.
Output during the SCF procedure.
is a list of items, separated by blanks or commas. The following items are recognized: Eigval, Eigvec, Err, Fmat, Keeporb, MOPop, Occ, Pmat, Pop, Start.
Eigenvalues of the one-electron orbitals at the last SCF cycle. In a run with multiple SCF runs (Geometry Optimization,..) this printing occurs only for the last SCF procedure. See also the
eigval subkey of EPRINT. (Use the repeat subkey of EPRINT to get output for the last SCF procedure at each SCF run, use DEBUG SCFEIGVAL to get output on all SCF cycles).
MO eigenvector coefficients in the BAS representation. Only printed on the last SCF cycle.
SCF error data which are checked for convergence. By default this takes effect after cycle 25 of the SCF. If the key is set it takes effect at the first cycle. Optionally one may type ErrN,where
n is an integer (written directly after Err without a blank in between), in which case the key takes effect at cycle n.
Fock matrix in the low representation.
If the KeepOrbitals option is activated (see the key SCF), output is generated whenever this option actually results in a change of occupation numbers as regards the energy ordering.
concise output of SCF occupation numbers on last SCF cycle if no eigenvalues are printed (see: Eigval).
Mulliken populations in terms of the elementary basis functions (bas), per MO, for input-specified MOs (see the EPRINT subkey orbpop)
Density matrix
General control of bas Mulliken populations. This supervises all printing (whether populations are printed or not) according to the EPRINT subkeys atompop, fragpop, orbpop (the latter only as
regards the bas population analysis at the end of the SCF procedure).
Data pertaining to the first SCF cycle (of the first SCF procedure, in case of an optimization; use repeat to get this for all SCFs).
By default Eigval, Keeporb, Occ, and Pop are on, the others off.
Information pertaining to the use of Symmetrized Fragment Orbitals (for analysis purposes).
A list of items, separated by blanks or commas. The following items are recognized: eig, eigcf, orbpop, grosspop, fragpop, ovl.
The MO coefficients in terms of the SFOs.
idem, but now also containing the coefficients pertaining to the CoreFunctions.
population analysis of individual orbitals. The orbitals analyzed are set with the EPRINT subkey orbpop.
Gross populations of the SFOs, split out in symmetry representations. GrossPop is automatically turned on when OrbPop is activated.
Population analysis on a per-FragmentType basis. This analysis does in fact not depend on the SFOs (ie, the result does not depend on how the SFOs are defined), but the computation of these
populations takes place in the SFO-analysis module, which is why it is controlled by the SFO print option. FragPop output is given per orbital when OrbPop is activated, per symmetry
representation when GrossPop is activated, and as a sum-over-all-orbitals-in-all-irreps otherwise (if FragPop is active).
Overlap matrix of the SFO basis, separately for each symmetry representation.
By default orbpop is on, the other options off.
In a Spin-Orbit calculation the SFO analysis is not yet implemented completely.
Remark: the options eig and eigcf replace the previous (now disabled) simple print options eigsfo and eigsfo.
Note that the simple print key SFO controls whether or not the EPRINT subkey sfo is effective at all.
Part of the bonding energy is computed and analyzed by the so-called Transition State procedure 2, 3. This has nothing to do with physical transition states, but is related to the Fock operator
defined by an average charge density, where the average is taken of the initial (sum-of-orthogonalized-fragments) and the final (SCF) charge density. There is also an analogous term where the average
is taken of the sum-of-fragments and the sum-of-orthogonalized-fragments. Various terms, Fock operators and Density Matrices used in this approach may be printed. To avoid confusion with real
Transition States (saddle points in the molecular Energy surface) the phrase TransitionField is used here.
A list of items, separated by blanks or commas. The following items are recognized: Energy, Fmat, DiagFmat, FragPmat, DiagFragPmat, F*dPmat, DiagF*dPmat, OrbE.
Energy terms computed from the TransitionField.
TransitionField Fock matrices.
Idem, but only the diagonal elements.
The molecular P-matrix constructed from the sum-of-fragments.
idem, but only the diagonal elements.
The TransitionField energy term can be expressed as a Fock operator times the difference between two P-matrices (initial and final density).
only diagonal elements
Orbital energies in the TransitionField.
By default all options are off.
Other Eprint subkeys¶
We discuss now the remaining EPRINT sub keys that are not simple shortcuts for print switches.
Eigval noccup {nvirtual}
This specifies the number of one-electron orbitals for which in the SCF procedure energies and occupation numbers are printed whenever such data is output: the highest noccup occupied orbitals and
the lowest nvirtual empty orbitals. Default values are noccup=10, nvirtual=10. If only one integer is specified it is taken as the noccup value and nvirtual is assumed to retain its standard value
(10). Printing can be turned off completely with the EPRINT sub key SCF, see above.
Mulliken Population Analysis
All population subkeys of EPRINT refer to Mulliken type populations.
Populations accumulated per atom.
level must be none, gross or matrix. none completely suppresses printing of the populations; gross yields the gross populations; matrix produces the complete matrix of net and overlap populations.
Default value: matrix.
Populations are printed per elementary (bas) basis function. The level options are none, short, gross, matrix. none, gross and matrix are as for atompop.
short yields a summary of BAS gross populations accumulated per angular momentum (l) value and per atom.
Default value: gross.
Completely similar to the atompop case, but now the populations per fragment. Of course in the case of single-atom fragments this is the same as atompop and only one of them is printed. Default:
For all three population keys atompop, fragpop and baspop, specification of a higher level implies that the lower-level data, which are in general summaries of the more detailed higher level options,
are also printed.
Printing of any populations at the end of the SCF procedure is controlled with the EPRINT sub key SCF (pop).
Population Analysis per MO
A very detailed population analysis tool is available: the populations per orbital (MO). The printed values are independent of the occupation numbers of the MOs, so they are not populations in a
strict sense. The actual populations are obtained by multiplying the results with the orbital occupations.
The analysis is given in terms of the SFOs and provides a very useful characterization of the MOs at the end of the calculation, after any geometry optimization has finished. This feature is now also
available in a Spin-Orbit coupled relativistic calculation, in the case there is one scalar relativistic fragment, which is the whole molecule.
The same analysis is optionally (see EPRINT subkey SCF, option mopop also provided in terms of the elementary basis functions (bas).
OrbPop {noccup {nvirtual}} {tol=tol}
subspecies orbitals
subspecies orbitals
Determines how many of the highest occupied orbitals are analyzed in each irrep. Default noccup=10.
Determines in similar fashion how many of the lowest virtual orbitals are analyzed in each irrep. Default nvirtual=4.
Tolerance parameter. Output of SFO contributions smaller than this tolerance may be suppressed. Default: 1e-2.
One of the subspecies of the molecular symmetry group. Can not be used (yet) in a Spin-Orbit coupled calculation.
A list of integers denoting the valence orbitals (in energy ordering) in this subspecies that you want to analyze. This overrules the noccup,nvirtual specification for that symmetry
representation. In an unrestricted calculation two sequences of integers must be supplied, separated by a double slash (//).
Any subset of the subspecies can be specified; it is not necessary to use all of them. No subspecies must occur more than once in the data block. This can not be used in a Spin-Orbit coupled equation
A total SFO gross populations analysis (from a summation over the occupied MOs) and an SFO population analysis per fragment type are preformed unless all MO SFO-populations are suppressed.
Reduction of output¶
One of the strong points of ADF is the analysis in terms of fragments and fragment orbitals (SFOs) that the program provides. This aspect causes a lot of output to be produced, in particular as
regards information that pertains to the SFOs.
Furthermore, during the SCF and, if applicable, geometry optimizations, quite a bit of output is produced that has relevance merely to check progress of the computation and to understand the causes
for failure when such might happen.
If you dislike the standard amount of output you may benefit from the following suggestions:
If you are not interested in info about progress of the computation:
If you’d like to suppress only the SCF-related part of the computational report:
If you don’t want to see any SFO stuff:
To keep the SFO definitions (in an early part of output) but suppress the SFO-mo coefficients and the SFO overlap matrix:
SFO noeig, noovl
Note: the SFO-overlap matrix is relevant only when you have the SFO-MO coefficients: the overlap info is needed then to interpret the bonding/anti-bonding nature of the various SFO components in an
If you are not interested in the SFO populations:
L. Versluis, The determination of molecular structures by the HFS method, PhD thesis, University of Calgary, 1989
T. Ziegler and A. Rauk, On the calculation of Bonding Energies by the Hartree Fock Slater method. I. The Transition State Method, Theoretica Chimica Acta 46, 1 (1977)
T. Ziegler and A. Rauk, A theoretical study of the ethylene-metal bond in complexes between copper(1+), silver(1+), gold(1+), platinum(0) or platinum(2+) and ethylene, based on the
Hartree-Fock-Slater transition-state method, Inorganic Chemistry 18, 1558 (1979) | {"url":"https://www.scm.com/doc/ADF/Input/Printed_Output.html","timestamp":"2024-11-14T17:07:19Z","content_type":"text/html","content_length":"598808","record_id":"<urn:uuid:8a28a75f-9646-4f9e-9bb9-b2a228d273dc>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00347.warc.gz"} |
Di Vittorio Lab - Final Problem Solving Lab Example
Final Problem Solving Lab Example
The following provides an example for the final version of the Problem Solving Lab Assignments in EGR 312. The problem explained and described below refers to the "falling parachutist" problem that
is described in the course text book (Numerical Methods for Engineers 8th Edition - Chapra and Canale) and in class. Please use this as a guide when completing your final problem solving lab as it
includes all key elements that you are expected to complete as part of your submission. Your final assignment should be in the form of a webpage. Remember that you are encouraged to present your
results in a way that seems most appropriate to your problem and you should not exactly replicate the format and content you see here. The content below was developed by Dr. Di Vittorio, Dr. Lauren
Lowman (co-instructor), and Nick Corak (EGR 312 TA). An example of an Initial Problem Solving Lab is also provided.
This problem explores the concept of terminal velocity through the modeling of the velocity of a falling parachutist before they release their parachute. The parachutist has a mass of 68.1 kg and
their initial velocity is assumed to be equal to zero. The drag coefficient is assumed constant and equal to 12.5kg/s.
In this example, a free body diagram is used to set up the model in terms of acceleration, or the derivative of velocity. The model takes the form of a differential equation with an analytical
solution, so the “true” solution can be calculated. For the numerical solution, Euler’s method will be applied, which is a first order method for solving differential equations. Euler's method
introduces some error in the calculations; however, these errors are largest in the first few iterations. Because the goal of this simulation is to calculate terminal velocity, these initial errors
are not concerning. Later in the course we will learn about higher order methods that are more accurate.
First, a mathematical model needs to be formulated to describe the net forces on the parachutist. The result is a differential equation describing velocity change over time, as shown in Figure 2. In
these equations, the following parameters are used:
F = external forces (N) (Fg = gravitational force; Fd = drag force)
m = mass (kg)
a = acceleration (m/s^2)
v = velocity (m/s)
c_d = drag coefficient
g = gravity (m/s^2)
Note: Sometimes you will be provided with the mathematical model and will not need to show the full derivation, but you should at least explain the model and how it was derived. You can lump this in
with "Numerical" or "Alternative" or you can place it in its own section.
Figure 2: Mathematical Model
Euler’s method can be applied to solve the differential equation numerically. Euler’s is a first order approximation to the derivative and can be applied to our problem according to the steps shown
in Figure 3, where "i" is the iterator for time (t).
Figure 3: Explanation of Numerical Approach
Sample Calculations
Sample calculations for Euler's method applied to the falling parachutist problem are provided in Figure 4, starting with a velocity = 0 at time = 0. A time step of 1 second was used for three
iterations. The calculations show that velocity is still increasing after 4 seconds, although the rate of increase appears to be slowing down. The simulation will have to be extended to reach
terminal velocity, but this can be done most efficiently with a computer.
The following pseudocode was used to implement the simulation in MATLAB. The MATLAB code is included at the end of this webpage.
Define constant parameters
g = 9.81
c = 12.5
m = 68.1
Calculate true velocity using analytical solution and plot over time
Set initial conditions
t0 = 0 (time)
v0 = 0 (initial velocity)
h = 1 (step size)
Begin FOR loop - go out to 150 seconds
approximate v(t+1) using Euler's (refer to sample calcs)
store time and velocity in variable for each step size
END loop
Calculate true error and approximate error
comparison plot of true and approximate velocity for different step sizes
Plot error for different step sizes together
Figure 4: Sample Calculations for Numerical Approach
Instead of simulating the velocity using Euler's method, the exact equation for velocity can be derived by taking the integral of the ODE, using concepts and strategies from calculus. The analytical
derivation is shown in Figure 5. u-substitution is used to simplify the integral and then initial conditions of t = 0 and v = 0 are used to solve for the integrating constant. Sample hand
calculations using the resulting analytical equation are shown in Figure 6 below. These sample calculations can be directly compared to the numerical since the initial conditions and time step are
the same. They can also be checked against the MATLAB output to ensure no coding errors were made.
Figure 6: Sample Calculations for Analytical Approach
Figure 5: Analytical Derivation of Velocity
The numerical and analytical velocity calculations were implemented in MATLAB for different step sizes, from 0.5 seconds to 5 seconds. All velocity simulations are plotted together in Figure 7, where
v_a is the analytical velocity and v_n is the numerical simulation.
Figure 7: Analytical and Numerical Results
The absolute percent error was calculated for each time step using MATLAB. A sample error calculation is shown below and the error is plotted for each step size in Figure 8.
Figure 8: Error for Each Numerical Simulation
The results in Figure 7 show that the numerical simulations that used Euler's method overestimate the true velocity. However, as the step size is reduced, the numerical simulations more closely
approximate the true values. In addition, all of the simulations eventually reach the terminal velocity. Therefore, if we are only concerned with achieving the terminal velocity, then a large step
size is sufficient as long as the velocity converges (stops changing). However, if we want to know how the velocity changes over time before reaching terminal velocity, then a smaller step size will
greatly reduce the numerical error.
Figure 8 displays the true percent relative error over time for each time step. A step size of 5 seconds contains errors greater than 50% at the beginning of the simulation. If the step size is
reduced by a factor of 10 (0.5s) then the maximum error is less than 5%. Both Figures 7 and 8 show that the velocity simulation converges around t = 25 seconds. Therefore, the parachutist will reach
terminal velocity about 25 seconds after jumping from the plane.
When truncation error is reduced from decreasing the step size, the round-off error increases because more computations are performed. However, round-off error appears to be minor in this simulation,
as the error is still drastically reduced for a step size of 0.5 seconds. The error could be further reduced for a smaller step size, but considering the simplicity of this model and processes that
we have likely neglected the error reduction is likely small compared to overall model uncertainties. For instance, there could be wind currents that exert a horizontal force on the parachutist. The
parachutists would also likely move and adjust their position, which would create a non-constant drag coefficient.
Regardless of the larger model uncertainties, this simple model is still helpful for answering questions such as the following: What is the maximum velocity for which we should design the parachute
dimensions and material properties? If we are given an initial elevation, when should the parachutist open the chute to ensure they have enough time to safely land? What is the equivalent "wind
speed" that the parachutist feels while falling? Numerical simulations of complex systems allow us to perform a variety of quick analyses at a low cost to develop safer engineering designs. | {"url":"https://divittorio.engineering.wfu.edu/home/teaching/egr-312/final-problem-solving-lab-example","timestamp":"2024-11-06T21:09:57Z","content_type":"text/html","content_length":"143876","record_id":"<urn:uuid:f6ddbce3-946a-4502-8bd4-eab96663c67e>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00401.warc.gz"} |
Three-Dimensional Geometry
Read the Headline Story to the students. Encourage them to make interesting statements using information from the story.
Possible student responses
• If she made a long and think 1-by-6 rectangle, then the perimeter was 14 inches.
• If she made a 2-by-3 rectangle, then the perimeter was 10 inches.
• She could have made a shape that looks like steps with 3 squares in bottom row, 2 in the middle row, and 1 in the top row, in which case the perimeter would have been 12 inches. | {"url":"https://elementarymath.edc.org/mindset/three-dimensional-geometry-lesson-4/","timestamp":"2024-11-09T17:46:00Z","content_type":"text/html","content_length":"123822","record_id":"<urn:uuid:966bf82f-99af-4e9a-b2fc-60142604910c>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00278.warc.gz"} |
Moving from IBM® SPSS® to R and RStudio®
A Statistics Companion
March 2021 | 312 pages | SAGE Publications, Inc
Are you a researcher or instructor who has been wanting to learn R and RStudio®, but you don't know where to begin? Do you want to be able to perform all the same functions you use in IBM® SPSS® in
R? Is your license to IBM® SPSS® expiring, or are you looking to provide your students guidance to a freely-available statistical software program?
Moving from IBM® SPSS® to R and RStudio®: A Statistics Companion is a concise and easy-to-read guide for users who want to know learn how to perform statistical calculations in R. Brief chapters
start with a step-by-step introduction to R and RStudio, offering basic installation information and a summary of the differences. Subsequent chapters walk through differences between SPSS and R, in
terms of data files, concepts, and structure. Detailed examples provide walk-throughs for different types of data conversions and transformations and their equivalent in R. Helpful and comprehensive
appendices provide tables of each statistical transformation in R with its equivalent in SPSS and show what, if any, differences in assumptions factor to into each function. Statistical tests from t
-tests to ANOVA through three-factor ANOVA and multiple regression and chi-square are covered in detail, showing each step in the process for both programs. By focusing just on R and eschewing
detailed conversations about statistics, this brief guide gives adept SPSS® users just the information they need to transition their data analyses from SPSS to R.
About the Author
Chapter 1. Introduction to R
1.2 Why are Some Features of R?
1.3 Installing R and Getting Help Learning R
1.4 Conducting Statistical Analyses in Spss Versus R: A First Example
Chapter 2. Preparing to Use R and Rstudio
2.1 Tasks to Perform Before Your First R Session
2.2 Tasks to Perform Before Any R Session
2.3 Tasks To Perform During Any R Session
Chapter 3. R Terms, Concepts, and Command Structure
3.2 Command-Related Terms
Chapter 4. Introduction to Rstudio
4.3 Components of RStudio
4.4 Writing and Executing R Commands in RStudio
Chapter 5. Conducting Rstudio Sessions: A Detailed Example
5.2 2. Create a New Script File (Optional)
5.3 3. Define the Working Directory
5.4 4. Import CSV File to Create a Data Frame
5.5 5. Change Any Missing Data in Data Frame to NA
5.6 6. Save Data Frame With NAs As CSV File in the Working Directory
5.7 7. Read the Modified CSV File to Create a Data Frame
5.8 8. Download and Install Packages (If Not Already Done)
5.9 9. Load Installed Packages (As Needed)
5.10 10. Conduct Desired Statistical Analyses
5.11 11. Open a New Markdown File
5.12 12. Copy Commands and Comments into the Markdown File
5.13 13. Knit the Markdown File to Create a Markdown Document
5.14 Exiting Rstudio (Save the Workspace Image?)
Chapter 6. Conducting Rstudio Sessions: A Brief Example
6.2 2. Create a New Script File (Optional)
6.3 3. Define the Working Directory
6.4 4. Import CSV File to Create a Data Frame
6.5 5. Change Any Missing Data in Data Frame to NA
6.6 6. Save Data Frame with NAs as CSV File in the Working Directory
6.7 7. Read the Modified CSV File to Create a Data Frame
6.8 8. Download and Install Packages (If Not Already Done)
6.9 9. Load Installed Packages (As Needed)
6.10 10. Conduct Desired Statistical Analyses
6.11 11. Open a New Markdown File
6.12 12. Copy Commands and Comments into the Markdown File
6.13 13. Knit the Markdown File to Create a Markdown Document
Chapter 7. Conducting Statistical Analyses Using This Book: A Detailed Example
7.2 2. Copy and Paste an Example Script into a Script File
7.3 3. Modify the Example Script as Needed for the Desired Statistical Analysis
7.4 4. Execute the Script to Confirm It Works Properly
7.5 5. Copy and Paste the Script into a Markdown File
7.6 6. Knit the Markdown File to Create a Markdown Document
Chapter 8. Conducting Statistical Analyses Using This Book: A Brief Example
8.2 2. Copy and Paste an Example Script into a Script File
8.3 3. Modify the Example Script as Needed for the Desired Statistical Analysis
8.4 4. Execute the Script to Confirm it Works Properly
8.5 5. Copy and Paste the Script into a Markdown File
8.6 6. Knit the Markdown File to Create a Markdown Document
Chapter 9. Working With Data Frames and Variables in R
9.1 Working with Data Frames
9.2 Working With Variables
Chapter 10. Conducting Statistical Analyses Using SPSS Syntax
10.1 Conducting Analyses in SPSS Using Menu Choices
10.2 Conducting Analyses in Spss Using Syntax Commands
10.3 Editing SPSS Output Files
Appendix A: Data Transformations
Reverse Score a Variable (Recode)
Reduce the Number of Groups in a Categorical Variable (Recode)
Create a Categorical Variable from a Continuous Variable (Recode)
Create a Variable from Other Variables (Minimum Number of Valid Values) (Compute)
Create a Variable from Occurrences of Values of Other Variables (Count)
Perform Data Transformations When Conditions are Met (IF)
Perform Data Transformations Under Specified Conditions (DO IF/END IF)
Perform Data Transformations Under Different Specified Conditions (DO IF/ELSE IF/END IF)
Use Numeric Functions in Data Transformations (ABS, RND, TRUNC, SQRT)
Appendix B: Statistical Procedures
Descriptive Statistics (All Variables)
Descriptive Statistics (Selected Variables)
Descriptive Statistics (Selected Variables) by Group
Frequency Distribution Table
Confidence Interval for the Mean
T-Test for Independent Means
T-Test for Dependent Means (Repeated-Measures T-Test)
One-Way Anova and Tukey Post-Hoc Comparisons
One-Way Anova and Trend Analysis
Single-Factor Within-Subjects (Repeated Measures) Anova
Two-Factor Between-Subjects Anova
Two-Factor Between-Subjects Anova (Simple Effects)
Two-Factor Between-Subjects Anova (Simple Comparisons)
Two-Factor Between-Subjects Anova (Main Comparisons)
Two-Factor Mixed Factorial Anova
Two-Factor Within-Subjects Anova
Three-Factor Between-Subjects Anova
Pearson Correlation (One Correlation)
Pearson Correlation (Correlation Matrix)
Internal Consistency (Cronbach’s Alpha)
Principal Components Analysis (Varimax Rotation)
Principal Components Analysis (Oblique Rotation)
Factor Analysis (Principal Axis Factoring)
Multiple Regression (Standard)
Multiple Regression (Hierarchical With Two Steps)
Multiple Regression (Hierarchical With Three Steps)
Multiple Regression (Testing Moderator Variables Using Hierarchical Regression)
Multiple Regression (Portraying A Significant Moderating Effect)
Multiple Regression (Stepwise)
Multiple Regression (Backward)
Multiple Regression (Forward)
Canonical Correlation Analysis
Discriminant Analysis (Two Groups)
Discriminant Analysis (Three Groups)
Cross-Tabulation and the Chi-Square Test of Independence
Further Resources
Student Study SiteVisit the companion website to download data files and code to accompany this book. | {"url":"https://www.sagepub.com/en-us/cab/moving-from-ibm%C2%AE-spss%C2%AE-to-r-and-rstudio%C2%AE/book273758","timestamp":"2024-11-01T19:34:04Z","content_type":"text/html","content_length":"122812","record_id":"<urn:uuid:93cd6b91-7cdc-492a-aab3-6b71f2deed7a>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00849.warc.gz"} |
Hey, guys. In this video, we're going to start looking at projectiles that are thrown upwards rather than flat or downwards, and then we're going to start solving a specific case of this upward
launch. But before we do that, I just want to give a general point that whenever you launch something upwards, the initial velocity is always going to be positive. And this is because, for example,
in this specific example here, we've got a football that's launched with some initial velocity at some angle here. So if you break this up into its \( y \) and \( x \) components, then our \( v_{0y}
\) is always going to be positive as well as our \( v_{0x} \). Now, I'm going to skip this second point just for now. We're going to get back to it later on in the video.
Now let's start talking about this specific kind of example that we'll be solving in this video. So if we have this football, that's going to be kicked up at some angle like this. It's going to go up
to its height and then it's going to later return back to the ground again. So whenever this happens, whenever you have an object that returns to the initial height from which it was thrown or
launched, meaning that the \( y \) final is equal to your \( y \) initial, its trajectory is going to be perfectly symmetrical. So notice how we can just basically draw a line that splits this
parabola in half. So we're going to start talking about how to solve symmetrical launch problems. Symmetrical launches are a special case of upward launches, and they have some special properties
that are going to make our equations simpler.
Alright. So we're going to get right to the example. We've got a football that's kicked upwards and we're going to calculate in this first part the time that it takes to reach its maximum height.
Before we do that, let's just go ahead and stick to the steps. We're going to draw the paths in \( x \) and \( y \) and then figure out the points of interest. So if you were only traveling in the \(
x \) axis, you would basically just go straight along the ground like this, and the \( y \) axis you would be going up like this, and then you would have returning back down to where you started
from. So what are our points of interest? Well, the first one is just going to be the initial, that's point \( a \). And then what happens is, and it's going to hit the ground at some later time, but
there's something that happens in between which is it goes up and it reaches its maximum height over here, which is always going to be a point of interest. This is point \( b \) over here like this.
So it's going to go up to point \( b \) and then back down again towards point \( c \). So those are our paths in the \( x \) and \( y \) axis and our points of interest. So we've got initial, final,
and the maximum height. So now we're going to figure out the target variable. Let's go ahead and do that.
In part \( a \), we're looking for the time, so that's variable \( t \) that it takes to reach its maximum height. Now we just need to figure out the interval and then start working through our
equations. The interval we are looking at is the time that it takes to go from the point where it just launched up to its maximum height, which is point \( b \). So this is the interval that we're
going to use for our equations, the one from \( a \) to \( b \).
Okay. So that means we're looking for \( T_{AB} \), and remember we're looking for time. The equation that we're going to use first is always going to be the \( x \) axis equation because it's the
easiest. So here in the \( x \), we've got \( \Delta x \) from \( a \) to \( b \) equals \( v_x \) times \( t \) from \( a \) to \( b \). So if we're looking for time, then we're going to need both
of these other variables here. What about the initial velocity or the \( x \) axis velocity? Well, that's actually the simplest one because remember we have the magnitude, we know that \( v_0 \) is
20 and we have the angle which is 53 degrees. So our \( v_{0x} \) which is just \( v_{ax} \), which is just \( v_x \) throughout the whole motion is going to be \( 20 \times \cos(53^\circ) \), and so
you'll get 12. You do the same thing for the \( y \) axis, \( v_{0y} \) which is \( v_{ay} \) is just equal to \( 20 \times \sin(53^\circ) \), and that's 16. So we know what the \( y \) velocity or
so the \( x \) velocity is, so we have this. Unfortunately, we don't have the horizontal displacement from \( a \) to \( b \). That would be the horizontal distance that's covered from \( a \) to \(
b \). We don't know anything about that. So unfortunately, we're a little bit stuck here in the \( x \) axis and so therefore, I'm going to have to go into the \( y \) axis. So in the \( y \) axis,
remember, I need my five variables. I'm looking for \( a \), I have \( a_y \) which is always negative \( 9.8 \) regardless of the interval that you're using. Then I've got the initial velocity which
is just \( v_{ay} \), which I know is 16. The final velocity, it's going to be velocity at \( b \). Then I've got my \( \Delta y \) from \( a \) to \( b \), and my \( t \) from \( a \) to \( b \).
Remember, I came over to this axis here because I'm looking for \( T_{AB} \). So really, in order to solve this equation or just pick one of my equations, I'm going to need either the final velocity
or I'm going to need the vertical displacement. So unfortunately, for \( \Delta y \) from \( a \) to \( b \), I don't know what the vertical displacement is from \( a \) to \( b \). I don't know what
the height of that peak is. So I don't know what \( \Delta y \) from \( a \) to \( b \) is. But what about \( V_{by} \)? What's the final velocity? Well, once you're going from \( a \) to \( b \)
here, then just like we just what we did for vertical motion, what we can say is that when the object reaches its maximum height, then the velocity in the \( y \) axis is momentarily 0. So think
about this projectile as it's moving through the air, it's, being, you know, it's \( y \) velocity is be | {"url":"https://www.pearson.com/channels/physics/learn/patrick/projectile-motion/projectile-motion-symmetric-launch?chapterId=8fc5c6a5","timestamp":"2024-11-14T01:33:15Z","content_type":"text/html","content_length":"497884","record_id":"<urn:uuid:7c20be09-2f15-49b8-a2ab-a690a58e412d>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00198.warc.gz"} |
NCERT Solutions for Class 6 Maths Chapter 7 Exercise 7.8 Fractions PDF
Class 6 Maths NCERT Solutions for Fractions Chapter 7 Exercise 7.8 - FREE PDF Download
Vedantu provides NCERT Solutions for Class 6 Maths Chapter 7 Exercise 7.8, which focuses on Addition and Subtraction of Fractions. This exercise helps students learn how to add and subtract like and
unlike fractions according to the latest Class 6 Maths Syllabus. With clear, step-by-step solutions, students can easily understand the methods and practice solving various problems. These Class 6
Maths NCERT Solutions are designed to improve students’ problem-solving skills and build a strong foundation in fractions, preparing them for future maths topics and exams. Download the free PDF and
start practising today!
1. Class 6 Maths NCERT Solutions for Fractions Chapter 7 Exercise 7.8 - FREE PDF Download
2. Glance on NCERT Solutions Maths Chapter 7 Exercise 7.8 Class 6 | Vedantu
3. Access NCERT Solutions for Maths Class 6 Chapter 7 - Fractions
3.2Subtract as indicated:
3.3Solve the following problems:
4. Benefits of NCERT Solutions for Class 6 Maths Chapter 7 Fractions Exercise 7.8
5. Class 6 Maths Chapter 7: Exercises Breakdown
6. Important Study Material Links for Class 6 Maths Chapter 7 - Fractions
8. Chapter-Specific NCERT Solutions for Class 6 Maths
9. Related Important Links for Class 6 Maths
Glance on NCERT Solutions Maths Chapter 7 Exercise 7.8 Class 6 | Vedantu
• Exercise 7.8 in Class 6 Maths focuses on the Addition and Subtraction of Fractions, covering both like and unlike fractions.
• Students learn how to add and subtract fractions by finding common denominators, simplifying fractions, and solving word problems.
• Vedantu's NCERT Solutions provides detailed explanations and step-by-step guidance, ensuring students understand each process clearly.
• This exercise helps students strengthen their fraction skills, build problem-solving abilities, and prepare for more advanced mathematical topics in future classes.
FAQs on NCERT Solutions for Class 6 Maths Chapter 7 - Fractions Exercise 7.8
1. What is covered in Exercise 7.8 of Class 6 Maths?
Exercise 7.8 focuses on the Addition and Subtraction of Fractions, teaching students how to perform these operations with like and unlike fractions.
2. How does Vedantu's NCERT Solution help with Exercise 7.8?
Vedantu provides clear, step-by-step explanations, making it easy for students to understand and solve problems related to adding and subtracting fractions.
3. Are both like and unlike fractions covered in Exercise 7.8?
The exercise includes addition and subtraction of both like fractions (same denominator) and unlike fractions (different denominators).
4. How does Vedantu explain the process of adding unlike fractions?
Vedantu's solutions guide students through finding the least common denominator (LCD) to add unlike fractions and simplify them.
5. Does the Class 6 Maths Exercise 7.8 include word problems?
Yes, the exercise includes word problems to help students apply the concept of fraction addition and subtraction in real-life situations.
6. Can I download the PDF of Class 6 Maths Exercise 7.8 solutions for FREE?
Vedantu offers a free PDF download of the NCERT Solutions for Class 6 Maths Chapter 7 Exercise 7.8.
7. Will Class 6 Maths Exercise 7.8 solutions help me prepare for exams?
These solutions offer thorough practice and explanations, helping students gain confidence for exams.
8. Are visual aids used in the Class 6 Maths Exercise 7.8 solutions for better understanding?
Diagrams and visual aids are used where necessary to help students visualize the addition and subtraction of fractions.
9. How does Vedantu help with solving word problems involving fractions?
Vedantu provides clear explanations and step-by-step solutions to help students easily understand and solve word problems involving fractions.
10. Are there practice questions included in Class 6 Maths Exercise 7.8 Vedantu’s PDF?
The PDF includes plenty of practice questions to help students strengthen their understanding and master the topic. | {"url":"https://www.vedantu.com/ncert-solutions/ncert-solutions-class-6-maths-chapter-7-exercise-7-8","timestamp":"2024-11-05T06:39:34Z","content_type":"text/html","content_length":"402630","record_id":"<urn:uuid:8d0e78ab-4968-4746-be50-1bfc8692bda8>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00807.warc.gz"} |
Mortar Requirement for Wall
`V = f( "length" * "height" , 8 , 2.625 , 4 , 0.375 )`
Enter a value for all fields
The Mortar for a Wall calculator estimates the number of bricks or blocks needed for a wall based on the wall dimensions and the size of the bricks or blocks and the associated amount of mortar.
MASONRY WALL
INSTRUCTIONS: Choose units and enter the following:
• (l) Length of Wall
• (h) Height of Wall
• (bL) Brick or Block Length
• (bH) Brick or Block Height
• (bW) Brick or Block Width
• (MJ) Mortar Joint Thickness
• (OVR) Overage (percent added to total estimate)
Mortar for a Brick or Block Wall (V): The calculator returns the volume of concrete mortar in cubic yards. However, this can be automatically converted to compatible units via the pull-down menu.
The Math / Science
The Mortar for a Brick or Block Wall calculator estimates the number of brick or block needed for a wall based on the dimensions of the wall and the size of the brick or block to compute the
associated amount of mortar needed. The step to calculate are as follows:
1. Compute the number of rows, which is the wall height divided by the brick or block height, with the results rounded up.
2. Compute the number of units per row, which is the wall length divided by the brick or block length, with the results rounded up.
3. Apply safety margin, which is the addition of the overage percentage to the total.
4. Compute the mortar needed per unit and apply it to the total number.
• Bricks or Blocks for a Wall: Computes the number of bricks or blocks needed for a wall based on the wall's dimensions and the size of the bricks or block.
• Brick Block Wall Cost: Computes the number of brick or block needed for a wall based on the wall dimensions and the brick or block size, and then applies a cost per unit to the number to estimate
the total cost of bricks or blocks.
• Mortar Requirement for Wall: Computes the amount of mortar for a wall based on the dimensions of the wall and the size of the brick or block used.
• Bricks for a House: Estimates the number of bricks needed for a four walled structure (e.g., house) based on the building dimensions, the size of the bricks or blocks and the square footage
dedicated to doors and windows.
• Cost of Bricks for a House: Estimates the number of bricks for a four walled structure and applies a unit price per brick or block to estimate the total cost.
• Block for a Foundation: Computes the number of blocks needed for the walls of a foundation based on the dimensions of the foundation and the size of the blocks.
• Cost of Blocks for a Foundation: Estimates the number of block for the wall of a foundation based on the dimensions of the foundation and the size of the blocks, and then applies the unit price
per block to provide cost estimate.
• Mortar Needed for Foundation: Estimates the amount of mortar needed for the cinder block walls of a foundation based on the dimensions of the walls, size of blocks, and thickness of the mortar
• Bricks per Row:
• Bricks High: Computes the number of rows of bricks to achieve a height based on the height and the size of the bricks.
• Custom Stone in a Wall: Computes the number of masonry units (stone, block or stone) in a wall based on the dimensions of the wall and the dimensions of the units and the thickness of the mortar
joints. It also returns the number of rows needed and the number of stone per row.
• Concrete, Rebar and Forms for Wall: Computes the volume of concrete, amount of rebar and the surface area of forms for a concrete wall.
• Price of Delivered Concrete: Computes the price of delivered concrete based on the volume, price per cubic yard, pumping cost, delivery distance, and mileage.
• Water Needed for Concrete: Compute the amount of water needed to make a volume of concrete.
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Patent application title: COMMUNICATION SYSTEM AND TECHNIQUE USING QR DECOMPOSITION WITH A TRIANGULAR SYSTOLIC ARRAY
Inventors: Alexander Maltsev (Nizhny Novgorod, RU) Vladimir Pestretsov (Nizhny Novgorod, RU) Alexey Khoryaev (Dzerzhinsk, RU) Roman Maslennikov (Nizhny Novgorod, RU)
IPC8 Class: AH04B138FI
USPC Class: 375219
Class name: Pulse or digital communications transceivers
Publication date: 2009-12-17
Patent application number: 20090310656
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Patent application title: COMMUNICATION SYSTEM AND TECHNIQUE USING QR DECOMPOSITION WITH A TRIANGULAR SYSTOLIC ARRAY
An apparatus, system, and method to perform QR decomposition of an input complex matrix are described. The apparatus may include a triangular systolic array to load the input complex matrix and an
identity matrix, to perform a unitary complex matrix transformation requiring three rotation angles, and to produce a complex unitary matrix and an upper triangular matrix. The upper triangular
matrix may include real diagonal elements. Other embodiments are described and claimed.
An apparatus, comprising:a node to perform QR decomposition of an input complex matrix using three angle complex rotation, said node comprising:a three angle complex rotation triangular systolic
array to load said input complex matrix and an identity matrix, to perform a unitary matrix transformation using three rotation angles, and to produce a complex unitary matrix and an upper triangular
matrix, said upper triangular matrix comprising real diagonal elements.
The apparatus of claim 1, wherein said triangular systolic array comprises multiple processing modules including one more delay unit modules, one or more processing element modules, and one or more
rotational unit modules.
The apparatus of claim 2, wherein said processing modules comprise CORDIC-based processing modules able to operate in a vectoring mode and a rotation mode.
The apparatus of claim 3, wherein diagonal elements of said input matrix comprise control signals for determining an operation mode of said processing modules.
The apparatus of claim 1, wherein said triangular systolic array is to perform an additional unitary matrix transformation to eliminate complex diagonal elements of said upper triangular matrix.
The apparatus of claim 1, wherein said node is to determine an inverse matrix of said complex unitary matrix and an inverse matrix of said upper triangular matrix having real diagonal elements.
The apparatus of claim 1, wherein said node is to perform multiple-input multiple output (MIMO) signal processing algorithms.
The apparatus of claim 7, wherein said multiple-input multiple output signal processing comprises one or more of a zero-forcing algorithm and a minimum mean square error algorithm.
The apparatus of claim 7, wherein said node comprises a multiple-input multiple output transceiver.
A system, comprising:at least one antenna; anda node to couple to said at least one antenna over a multicarrier communication channel and to perform to perform QR decomposition of an input matrix,
said node comprising:a triangular systolic array to load said input matrix and an identity matrix, to perform a unitary matrix transformation using three rotation angles, and to produce a complex
unitary matrix and an upper triangular matrix, said upper triangular matrix comprising real diagonal elements.
The system of claim 10, wherein said triangular systolic array comprises multiple processing modules including one more delay unit modules, one or more processing element modules, and one or more
rotational unit modules.
The system of claim 11, wherein said processing modules comprise CORDIC-based processing modules able to operate in a vectoring mode and a rotation mode.
The system of claim 12, wherein diagonal elements of said input matrix comprise control signals for determining a operation mode of said processing modules.
The system of claim 10, wherein said triangular systolic array is to perform an additional unitary matrix transformation to eliminate complex diagonal elements of said upper triangular matrix.
The system of claim 10, wherein said node is to determine an inverse matrix of said complex unitary matrix and an inverse matrix of said upper triangular matrix having real diagonal elements.
The system of claim 10, wherein said node is to perform multiple-input multiple output (MIMO) signal processing algorithms.
The system of claim 16, wherein said multiple-input multiple output signal processing comprises one or more of a zero-forcing algorithm and a minimum mean square error algorithm.
The system of claim 16, wherein said node comprises a multiple-input multiple output transceiver.
The system of claim 16, wherein said node comprises a processing unit of a multiple-input multiple output receiver to implement complex matrix operations including matrix QR decomposition.
A method to perform QR decomposition of an input complex matrix, comprising:sequentially loading said input complex matrix to be decomposed and an identity matrix to obtain a complex unitary matrix
into a node comprising a triangular systolic array;performing a unitary complex matrix transformation using three rotation angles; andproducing a complex unitary matrix and an upper triangular
matrix, said upper triangular matrix comprising only real diagonal elements.
The method claim 20, wherein said triangular systolic array comprises multiple processing modules including one more delay unit modules, one or more processing element modules, and one or more
rotational unit modules.
The method of claim 21, wherein said processing modules comprise CORDIC-based processing modules able to operate in a vectoring mode and a rotation mode.
The method of claim 22, further comprising determining an operation mode of said processing modules based on control signals passed through said array jointly with diagonal elements of said input
The method of claim 21, further comprising performing an additional unitary matrix transformation to eliminate complex diagonal elements of said upper triangular matrix using said processing modules
of said triangular systolic array.
The method of claim 20, further comprising:determining an inverse matrix of said complex unitary matrix; anddetermining an inverse matrix of said upper triangular matrix having real diagonal
BACKGROUND [0001]
Modern wireless communication systems may operate according to Institute of Electrical and Electronics Engineers (IEEE) standards such as the 802.11 standards for Wireless Local Area Networks (WLANs)
and the 802.16 standards for Wireless Metropolitan Area Networks (WMANs). Worldwide Interoperability for Microwave Access (WiMAX) is a wireless broadband technology based on the IEEE 802.16 standard
of which IEEE 802.16-2004 and the 802.16e amendment are Physical (PHY) layer specifications. IEEE 802.16-2004 supports several multiple-antenna techniques including Alamouti Space-Time Coding (STC),
Multiple-Input Multiple-Output (MIMO) antenna systems, and Adaptive Antenna Systems (AAS).
Future wireless communication systems are expected to support multiple antenna techniques such as MIMO and spatial division multiple access (SDMA) modes of transmission, which allow spatial
multiplexing of data streams from one or multiple users. The performance and complexity of such systems will strictly depend on the number of antennas used. There is a need, therefore, to develop
highly efficient architectures for realization of different signal processing algorithms in MIMO-OFDM systems having a large number of antenna elements.
BRIEF DESCRIPTION OF THE DRAWINGS [0003]FIG. 1
illustrates one embodiment of a communications system.
[0004]FIG. 2
illustrates one embodiment of a signal processing system.
FIGS. 3A-3C illustrate one embodiment of signal processing modules.
FIGS. 4A-5B illustrate one embodiment of data flows.
[0007]FIG. 6
illustrates one embodiment of a logic flow.
DETAILED DESCRIPTION [0008]FIG. 1
illustrates one embodiment of a system.
FIG. 1
illustrates a block diagram of a communications system 100. In various embodiments, the communications system 100 may comprise multiple nodes. A node generally may comprise any physical or logical
entity for communicating information in the communications system 100 and may be implemented as hardware, software, or any combination thereof, as desired for a given set of design parameters or
performance constraints. Although
FIG. 1
may show a limited number of nodes by way of example, it can be appreciated that more or less nodes may be employed for a given implementation.
In various embodiments, a node may comprise, or be implemented as, a computer system, a computer sub-system, a computer, an appliance, a workstation, a terminal, a server, a personal computer (PC), a
laptop, an ultra-laptop, a handheld computer, a personal digital assistant (PDA), a set top box (STB), a telephone, a mobile telephone, a cellular telephone, a handset, a wireless access point, a
base station (BS), a subscriber station (SS), a mobile subscriber center (MSC), a radio network controller (RNC), a microprocessor, an integrated circuit such as an application specific integrated
circuit (ASIC), a programmable logic device (PLD), a processor such as general purpose processor, a digital signal processor (DSP) and/or a network processor, an interface, an input/output (I/O)
device (e.g., keyboard, mouse, display, printer), a router, a hub, a gateway, a bridge, a switch, a circuit, a logic gate, a register, a semiconductor device, a chip, a transistor, or any other
device, machine, tool, equipment, component, or combination thereof. The embodiments are not limited in this context.
In various embodiments, a node may comprise, or be implemented as, software, a software module, an application, a program, a subroutine, an instruction set, computing code, words, values, symbols or
combination thereof. A node may be implemented according to a predefined computer language, manner or syntax, for instructing a processor to perform a certain function. Examples of a computer
language may include C, C++, Java, BASIC, Perl, Matlab, Pascal, Visual BASIC, assembly language, machine code, micro-code for a network processor, and so forth. The embodiments are not limited in
this context.
The nodes of the communications system 100 may be arranged to communicate one or more types of information, such as media information and control information. Media information generally may refer to
any data representing content meant for a user, such as image information, video information, graphical information, audio information, voice information, textual information, numerical information,
alphanumeric symbols, character symbols, and so forth. Control information generally may refer to any data representing commands, instructions or control words meant for an automated system. For
example, control information may be used to route media information through a system, or instruct a node to process the media information in a certain manner. The media and control information may be
communicated from and to a number of different devices or networks.
In various implementations, the nodes of the communications system 100 may be arranged to segment a set of media information and control information into a series of packets. A packet generally may
comprise a discrete data set having fixed or varying lengths, and may be represented in terms of bits or bytes. It can be appreciated that the described embodiments are applicable to any type of
communication content or format, such as packets, cells, frames, fragments, units, and so forth.
The communications system 100 may communicate information in accordance with one or more standards, such as standards promulgated by the IEEE, the Internet Engineering Task Force (IETF), the
International Telecommunications Union (ITU), and so forth. In various embodiments, for example, the communications system 100 may communicate information according to one or more IEEE 802 standards
including IEEE 802.11 standards (e.g., 802.11a, b, g/h, j, n, and variants) for WLANs and/or 802.16 standards (e.g., 802.16-2004, 802.16.2-2004, 802.16e, 802.16f, and variants) for WMANs. The
communications system 100 may communicate information according to one or more of the Digital Video Broadcasting Terrestrial (DVB-T) broadcasting standard and the High performance radio Local Area
Network (HiperLAN) standard. The embodiments are not limited in this context.
In various embodiments, the communications system 100 may employ one or more protocols such as medium access control (MAC) protocol, Physical Layer Convergence Protocol (PLCP), Simple Network
Management Protocol (SNMP), Asynchronous Transfer Mode (ATM) protocol, Frame Relay protocol, Systems Network Architecture (SNA) protocol, Transport Control Protocol (TCP), Internet Protocol (IP), TCP
/IP, X.25, Hypertext Transfer Protocol (HTTP), User Datagram Protocol (UDP), and so forth.
The communications system 100 may include one or more nodes arranged to communicate information over one or more wired and/or wireless communications media. Examples of wired communications media may
include a wire, cable, printed circuit board (PCB), backplane, switch fabric, semiconductor material, twisted-pair wire, co-axial cable, fiber optics, and so forth. An example of a wireless
communication media may include portions of a wireless spectrum, such as the radio-frequency (RF) spectrum. In such implementations, the nodes of the system 100 may include components and interfaces
suitable for communicating information signals over the designated wireless spectrum, such as one or more transmitters, receivers, transceivers, amplifiers, filters, control logic, antennas and so
The communications media may be connected to a node using an input/output (I/O) adapter. The I/O adapter may be arranged to operate with any suitable technique for controlling information signals
between nodes using a desired set of communications protocols, services or operating procedures. The I/O adapter may also include the appropriate physical connectors to connect the I/O adapter with a
corresponding communications medium. Examples of an I/O adapter may include a network interface, a network interface card (NIC), a line card, a disc controller, video controller, audio controller,
and so forth.
In various embodiments, the communications system 100 may comprise or form part of a network, such as a WiMAX network, a broadband wireless access (BWA) network, a WLAN, a WMAN, a wireless wide area
network (WWAN), a wireless personal area network (WPAN), an SDMA network, a Code Division Multiple Access (CDMA) network, a Wide-band CDMA (WCDMA) network, a Time Division Synchronous CDMA (TD-SCDMA)
network, a Time Division Multiple Access (TDMA) network, an Extended-TDMA (E-TDMA) network, a Global System for Mobile Communications (GSM) network, an Orthogonal Frequency Division Multiplexing
(OFDM) network, an Orthogonal Frequency Division Multiple Access (OFDMA) network, a North American Digital Cellular (NADC) network, a Universal Mobile Telephone System (UMTS) network, a third
generation (3G) network, a fourth generation (4G) network, a local area network (LAN), a wide area network (WAN), a metropolitan area network (MAN), the Internet, the World Wide Web, a cellular
network, a radio network, a satellite network, and/or any other communications network configured to carry data. The embodiments are not limited in this context.
The communications system 100 may employ various modulation techniques including, for example: OFDM modulation, Quadrature Amplitude Modulation (QAM), N-state QAM (N-QAM) such as 16-QAM (four bits
per symbol), 32-QAM (five bits per symbol), 64-QAM (six bits per symbol), 128-QAM (seven bits per symbol), and 256-QAM (eight bits per symbol), Differential QAM (DQAM), Binary Phase Shift Keying
(BPSK) modulation, Quadrature Phase Shift Keying (QPSK) modulation, Offset QPSK (OQPSK) modulation, Differential QPSK (DQPSK), Frequency Shift Keying (FSK) modulation, Minimum Shift Keying (MSK)
modulation, Gaussian MSK (GMSK) modulation, and so forth. The embodiments are not limited in this context.
The communications system 100 may form part of a multi-carrier system such as a MIMO system. The MIMO system may employ one or more multi-carrier communications channels for communicating
multi-carrier communication signals. A multi-carrier channel may comprise, for example, a wideband channel comprising multiple sub-channels. The MIMO system may be arranged to communicate one or more
spatial data streams using multiple antennas. Examples of an antenna include an internal antenna, an omni-directional antenna, a monopole antenna, a dipole antenna, an end fed antenna, a circularly
polarized antenna, a micro-strip antenna, a diversity antenna, a dual antenna, an antenna array, and so forth.
In various embodiments, the system 100 may comprise a physical (PHY) layer component for WLAN devices either hardware or software based on IEEE standards 802.11n, 802.16-2004, and/or 802.16e, for
example. In one embodiment, the communications system 100 may comprise a transceiver for a MIMO-OFDM system. The embodiments are not limited in this context.
As shown in
FIG. 1
, the communications system 100 may be illustrated and described as comprising several separate functional elements, such as modules and/or blocks. In various embodiments, the modules and/or blocks
may be connected by one or more communications media. Communications media generally may comprise any medium capable of carrying information signals. For example, communication media may comprise
wired communication media, wireless communication media, or a combination of both, as desired for a given implementation.
The modules and/or blocks may comprise, or be implemented as, one or more systems, sub-systems, processors, devices, machines, tools, components, circuits, registers, applications, programs,
subroutines, or any combination thereof, as desired for a given set of design or performance constraints. Although certain modules and/or blocks may be described by way of example, it can be
appreciated that a greater or lesser number of modules and/or blocks may be used and still fall within the scope of the embodiments. Further, although various embodiments may be described in terms of
modules and/or blocks to facilitate description, such modules and/or blocks may be implemented by one or more hardware components (e.g., processors, DSPs, PLDs, ASICs, circuits, registers), software
components (e.g., programs, subroutines, logic) and/or combination thereof.
The communications system 100 may comprise a transmitter node 102. In one embodiment, for example, the transmitter node 102 may comprise a MIMO transmitter to transmit one or more spatial data
streams over a multicarrier communication channel.
The transmitter node 102 may comprise an encoder block 104. In various embodiments, the encoder block 104 may be arranged to generate an encoded bit sequence from input data flow. The encoder block
104 may use various coding rates (e.g., 1/2, 2/3, 3/4) depending on the puncturing pattern. In one embodiment, for example, the encoder block 104 may comprise an error-correcting encoder, such as a
forward error correcting (FEC) encoder, and may generate a bit sequence encoded with an FEC code. In other embodiments, the encoder block 104 may comprise a convolutional encoder. The embodiments are
not limited in this context.
The transmitter node 102 may comprise an interleaver block 106. In various embodiments, the interleaver block 106 may perform interleaving on the bits of the encoded bit sequence. In one embodiment,
for example, the interleaver block 106 may comprise a frequency interleaver. The embodiments are not limited in this context.
The transmitter node 102 may comprise a mapper block 108. In various embodiments, the mapper block 108 may map the interleaved bit sequence into a sequence of transmit symbols. In one embodiment, for
example, the mapper block 108 may map the interleaved bit sequence into a sequence of OFDM symbols. Each OFDM symbol may comprise N frequency symbols, with N representing a positive integer (e.g.,
16, 64). In various implementations, the mapper block 108 may map the transmit symbols to subcarrier signals of a multicarrier communication channel.
The transmitter node 102 may comprise a transmit (TX) MIMO signal processing block 110. In various embodiments, the TX MIMO signal processing block 110 may be arranged to perform various multiple
antenna signal processing techniques such as such as space-time coding (STC), TX beamforming, MIMO coding, and/or other MIMO processing techniques, for example. In various implementations, the TX
MIMO signal processing block 110 may be arranged to apply beamformer and/or equalizer weights to transmit symbols (e.g., OFDM symbols). In various implementations, one or more of the MIMO signal
processing techniques may involve the calculation of weight matrices for every subcarrier and/or or group of adjacent subcarriers and the multiplication of OFDM subcarrier symbols in the frequency
domain by a weighting matrix. The embodiments are not limited in this context.
The transmitter node 102 may comprise inverse fast Fourier transform (IFFT) blocks 112-1-n, where n represents a positive integer value. In various embodiments, the IFFT blocks 112-1-n may be
arranged to convert OFDM symbols to time-domain signals. In various implementations, the IFFT blocks 112-1-n may perform guard interval (GI) insertion. In such implementations, GI insertion may
comprise inserting a time-domain guard interval between OFDM symbols to reduce inter-symbol interference.
The transmitter node 102 may comprise digital-to-analog conversion (DAC) and radio-frequency (RF) processing blocks 114-1-n, where n represents a positive integer value. In various embodiments, the
DAC and RF processing blocks 114-1-n may be arranged to perform DAC processing and to generate RF signals for transmission on the spatial channels of a multicarrier communication channel.
The transmitter node 102 may comprise transmit antennas 116-1-n, where n represents a positive integer value. In various embodiments, each of the transmit antennas 116-1-n may correspond to one of
the spatial channels of a multicarrier communications channel.
The transmitter node 102 may transmit information over communication channel 118. In various embodiments, the communication channel 118 may comprise a multicarrier communication channel (e.g., MIMO
channel) for communicating multicarrier communication signals (e.g., OFDM signals). The MIMO channel may comprise, for example, a wideband channel comprising multiple subchannels. Each subchannel may
comprise closely spaced orthogonal data subcarriers allowing a single OFDM symbol to be transmitted together by the data subcarriers. The embodiments are not limited in this context.
The communications system 100 may comprise a receiver node 120 for receiving information over communication channel 118. In various embodiments, the receiver node 120 may comprise receive antennas
122-1-n, where n represents a positive integer value. In various implementations, each of the receive antennas 122-1-n may correspond to one of the spatial channels of a multicarrier communications
The transmitter node 102 may comprise RF and analog-to-digital conversion (ADC) processing blocks 124-1-n, where n represents a positive integer value. In various embodiments, the RF and ADC
processing blocks 124-1-n may be arranged to perform RF and ADC processing on signals received on the spatial channels of a multicarrier communication channel.
The receiver node 122 may comprise fast Fourier transform (FFT) blocks 126-1-n, where n represents a positive integer value. In various embodiments, the FFT blocks 126-1-n may be arranged to convert
time-domain signals to frequency-domain signals. In various implementations, the FFT blocks 126-1-n may perform GI removal. In such implementations, GI removal may comprise removing a time-domain
guard interval between OFDM symbols.
The receiver node 122 may comprise a receive (RX) MIMO signal processing block 128. In various embodiments, the RX MIMO signal processing block 128 may be arranged to perform various multiple antenna
signal processing techniques including, for example: channel estimation, frequency domain equalization, space-time decoding, RX beamforming, MIMO decoding, and/or other MIMO processing techniques
such as MIMO detection schemes used in 802.11n and 802.16e transceivers. In various implementations, the RX MIMO signal processing block 128 may be arranged to calculate beamformer and/or
equalization weights and to apply the beamformer and/or equalizer weights to receive symbols (e.g., OFDM symbols). In various implementations, one or more of the MIMO signal processing techniques may
involve the calculation of weight matrices for every subcarrier and/or or group of adjacent subcarriers and the multiplication of OFDM subcarrier symbols in the frequency domain by a weighting matrix
to produce linear estimates of the transmitted signal. The embodiments are not limited in this context.
The receiver node 120 may comprise a demapper block 130. In various embodiments, the demapper block 130 may be arranged to demap a sequence of symbols, such as a sequence of OFDM symbols. The
embodiments are not limited in this context.
The receiver node 120 may comprise a deinterleaver block 132. In various embodiments, the deinterleaver block 132 may perform deinterleaving on the bits of the encoded bit sequence. In one
embodiment, for example, the deinterleaver block 132 may comprise a frequency deinterleaver. The embodiments are not limited in this context.
The receiver node 120 may comprise a decoder block 134. In various embodiments, the decoder block 134 may be arranged to decode an encoded bit sequence into an output data flow. The decoder block 134
may use various coding rates (e.g., 1/2, 2/3, 3/4) depending on the puncturing pattern. In one embodiment, for example, the decoder block 134 may comprise an error-correcting encoder, such as an FEC
decoder, and may generate an output data flow from a bit sequence encoded with an FEC code. In other embodiments, the decoder 134 may comprise a convolutional decoder. The embodiments are not limited
in this context.
In various embodiments, the communications system 100 may be arranged to implement a Triangular Systolic Array (TSA) Three Angle Complex Rotation (TACR) architecture and technique. The TSA/TACR
architecture and technique may be used, for example, to implement various MIMO signal processing techniques requiring complex matrix calculations (e.g., computation, manipulation, inversion,
multiplication, division, addition, subtraction, etc.). In various implementations, the TSA/TACR architecture and technique may be used to perform QR Decomposition (QRD) of complex matrices. In some
embodiments, the TSA/TACR architecture and technique may employ processing elements based on the Coordinate Rotation Digital Computer (CORDIC) algorithm. In comparison with conventional QRD
architectures and techniques, the TSA/TACR architecture and technique may provide significant latency reduction. The embodiments are not limited in this context.
[0040]FIG. 2
illustrates a signal processing system.
FIG. 2
illustrates one embodiment of a signal processing system 200. In various embodiments, the signal processing system 200 may comprise, or be implemented as, a MIMO signal processing system such as the
TX MIMO signal processing block 110 and/or RX MIMO signal processing block 128, for example. The embodiments are not limited in this context.
In various embodiments, the signal processing system 200 may be arranged to perform various MIMO signal processing techniques, such as MIMO detection schemes used in 802.11n and 802.16e transceivers,
for example. The MIMO signal processing techniques may employ algorithms, such as Zero-Forcing (ZF) or Minimum Mean Square Error (MMSE) linear algorithms, due to the efficiency and relatively low
computational complexity of such algorithms. In various implementations, the realization of the linear ZF and MMSE algorithms includes the calculation of weight matrices for every subcarrier (or
group of adjacent subcarriers) and the multiplication of OFDM subcarrier symbols (in the frequency domain) from different antennas by a weighting matrix to obtain estimates of transmitted data
The weight matrices for ZF and MMSE algorithms, respectively, can be calculated as follows:
-ZF (1)
-MMSE (2)
i is a subcarrier index,
W(i) is a weight matrix for the i-th subcarrier,
H(i) is a channel transfer matrix for the i-th subcarrier,
I is the identity matrix, and
ρ(i) is a reciprocal to signal-to-noise ration (SNR) for the i-th subcarrier.
From the above equations (1) and (2), it can be seen that computational complexity for the ZF and MMSE algorithms arises from matrix inversion, which should be done for every subcarrier (or group of
several adjacent subcarriers).
In various embodiments, the signal processing system 200 may be arranged to perform a QRD technique to solve the matrix inversion problem. The QRD technique may be implemented, for example, in MIMO
and SDMA systems having a large number of transmit and receive antenna elements.
The QRD technique for matrix V finds matrices Q and R such that V=QR, where Q is a unitary matrix and R is a triangular matrix. If QR decomposition is performed, then the inverse matrix can be found
as follows:
The inverse matrix for unitary matrix Q can be found as a Hermitian transposed matrix and the inversion of the triangular matrix R is straightforward using a back substitution algorithm.
In various embodiments, such as in 802.11n and 802.16e systems, the channel is measured using training symbols and/or pilots, which are followed by data symbols with spatial multiplexing. As such,
the calculation of weight matrices should be done with a low latency so that the overall receive latency is not significantly impacted.
In various embodiments, the signal processing system 200 may employ dedicated hardware for matrix inversion to meet the stringent requirements of 802.11n and 802.16e systems. The dedicated hardware
for implementing ZF or MMSE algorithms and QRD realization may comprise, for example, a weight calculation unit and a combiner unit to combine data subcarriers from different antennas using the
weights. In various implementations, the signal processing system 200 may comprise, or be implemented as, a key processing unit for MIMO equalizer/beamformer weight calculation that may perform QRD
of complex matrices. The embodiments are not limited in this context.
In various embodiments, the signal processing system 200 may comprise a TSA to perform QRD realization for matrix inversion. In various implementations, the TSA may comprise processing elements based
on the CORDIC algorithm. Implementing the TSA using CORDIC-based processing elements may allow total QRD to be performed without multiplications. The CORDIC algorithm is designed for rotation of a
two-dimensional vector in circular coordinate system, using only add and shift operations. The CORDIC algorithm is also designed for computation of different trigonometric functions (e.g., sin, cos,
arctan, square roots, etc.). In general, computations of the CORDIC algorithm are based on the execution of add and shift operations and realized by consecutive iterations. An exemplary iteration of
the CORDIC algorithm can be defined by the following equations:
The CORDIC iteration above describes a rotation of an intermediate plane vector v
, y
to v
+1, y
. The third iteration variable z
keeps track of the rotation angle θ
The CORDIC algorithm may operate in different modes including rotation and vectoring modes of operation. The vectoring mode can be used to determine the magnitude and phase of a complex number, and
the rotation mode can be applied to rotate a given complex number on a desired angle. The CORDIC modes differ in the way the direction of rotation μ
is determined. For example, in rotation mode, the rotation direction (that is steered by the variable μ
, can be determined by the following rule:
μ i = sign ( z i ) = { - 1 , if z i < 0 + 1 , otherwise ( 5 ) ##EQU00001##
In vectoring mode, the rotation direction is chosen to drive the y coordinate to zero while keeping x coordinate positive. Such approach results in an output angle equal to arctangent of inputs
quotient, namely, arctan(y/x). To achieve this, μ
is chosen to be:
μ i = - sign ( x i ) sign ( y i ) , where function sign ( A ) = { - 1 , if A < 0 + 1 , otherwise ( 6 ) ##EQU00002##
In various embodiments, the signal processing system 200 may be arranged to implement a TSA/TACR architecture and technique for QRD of complex matrices. In various implementations, the TSA/TACR
architecture and technique may employ both modes of CORDIC operations to perform QRD realization. In comparison with conventional QRD architectures and techniques, the TSA/TACR technique for QRD of
complex matrices may provide a significant reduction of latency in terms of systolic operation time, for example.
One example of a conventional QRD technique is the Complex Givens Rotations (CGR) technique. In general, the CGR technique is used to selectively introduce a zero into a matrix. For QRD realization,
the CGR technique can reduce an input matrix to triangular form by applying successive rotations to matrix elements below the main diagonal.
The CRG approach can be illustrated using a 2×2 square complex matrix V
×2 defined by the expression:
2 × 2 = [ A jθ a C jθ c B jθ b D jθ d ] ( 7 ) ##EQU00003##
where j= {square root over (-1)}; A, B, C, D--are magnitudes and θ
, θ
, θ
, θ
--are angles.
The CGR is described by two rotation angles θ
, θ
through the following matrix transformation:
[ cos θ 1 sin θ 1 j θ 2 - sin θ 1 - jθ 2 cos θ 1 ] [ A jθ a C jθ c B jθ b D jθ d ] = [ X θ x Y θ y 0 Z θ z ] ( 8 ) ##EQU00004##
where angles θ
, θ
are chosen to zero the matrix element below the main diagonal and defined by equations:
It is easy to verify from equations (8) and (9) that the CGR technique leads to an upper triangular matrix having complex diagonal elements.
In contrast, the TSA/TACR technique for QRD may produce an upper triangular matrix having only real diagonal elements. In various embodiments, the TSA/TACR technique may comprise a unitary matrix
transformation described by three rotation angles. In one embodiment, for example, the transformation for the QRD of the square complex matrix V
×2 of equation (7) may be presented by following equation:
[ cos θ 1 jθ 2 sin θ 1 j θ 3 - sin θ 1 jθ 2 cos θ 1 jθ 3 ] . ( 10 ) ##EQU00005##
In this embodiment, to obtain the upper triangular matrix, the unitary transformation requires three angles θ
, θ
, θ
determined by the following equations:
The TSA/TACR technique results in the following upper triangular matrix.
[ cos θ 1 jθ 2 sin θ 1 j θ 3 - sin θ 1 jθ 2 cos θ 1 jθ 3 ] [ A jθ a C jθ c B jθ b D jθ d ] = [ X Y θ y 0 Z θ z ] ( 12 ) ##EQU00006##
It is noted that the matrix transformation of the TSA/TACR technique introduces the real element X on the matrix diagonal. As such, application of the TSA/TACR technique to a square matrix with an
arbitrary N×N size will lead to the appearance of all real elements on the matrix diagonal except the lowest one. For further inversion of the upper triangular matrix, it is advantageous to eliminate
the complex lowest diagonal element in order to avoid complex division. To produce an upper triangular matrix having only real diagonal elements, the TSA/TACR technique may comprise an additional
unitary transformation. In one embodiment, for example, the additional matrix transformation for a square 2×2 matrix may comprise:
[ 1 0 0 - jθ z ] [ X Y θ y 0 Z θ z ] = [ X Y θ y 0 Z ] ( 13 ) ##EQU00007##
It is noted that the additional unitary transformation at the last phase of QRD does not complicate the hardware realization and may be embedded into a CORDIC-based TSA/TACR architecture.
In various embodiments, the signal processing system 200 may comprise a TSA/TACR architecture for complex matrices. As shown in FIG. 2, for example, the signal processing system 200 may comprise a
TSA/TACR architecture including various processing modules having different functionality. In one embodiment, the TSA/TACR architecture may comprise Delay Unit (DU) modules 202-1-3, Processing
Element (PE) modules 204-1-6, and Rotational Unit (RU) module 206. In various implementations, the DU modules 202-1-3, the PE modules 204-1-6, and the RU module 206 may comprise CORDIC-based
processing blocks to perform the TSA/TACR technique for QRD. The embodiments are not limited in this context.
[0072]FIG. 2
illustrates one embodiment of input/output signal organization of the TSA/TACR architecture for performing QRD of an exemplary 4×4 input matrix V. According to the TSA/TACR technique, the input
matrix V is loaded in the temporally (skewed) triangular shape and followed by an identity matrix to produce the complex unitary matrix Q
at the output. It is noted that the diagonal elements of the input matrix V are marked by token `*`, which represents the control signal propagated together with a data element and controls the mode
and/or functionality of the PE modules 204-1-6 and RU module 206.
In operation, the TSA/TACR architecture and technique for QRD results in an upper triangular matrix R having only real diagonal elements instead of complex elements, which result when using the CGR
approach. As such, the TSA/TACR architecture and technique may significantly simplify the following inversion of matrix R when using a back substitution algorithm, which requires many divisions on
diagonal elements of matrix R.
FIGS. 3A-3C illustrate one embodiment of processing modules.
FIG. 3A
illustrates one embodiment of a DU module 302. In various embodiments, the DU module 302 may comprise, or be implemented as, the DU modules 202-1-3 of the processing system 200. The embodiments are
not limited in this context.
In various implementations, the DU module 302 may be arranged to read complex samples and/or control signals from a north input port N, to delay (e.g., for a period equal to PE operation time), and
then to pass the complex samples and/or control signals to the east output port E.
[0076]FIG. 3B
illustrates one embodiment of a PE module 304. In various embodiments, the PE module 304 may be implemented as, the PE modules 204-1-6 of the processing system 200. The embodiments are not limited in
this context.
In various implementations, the PE module 304 may be arranged as a main signal processing element of a TSA. The structure of the PE module 304 may be based on a CORDIC processor comprising one or
more hardwired CORDIC blocks. The number of CORDIC blocks and the associated architecture may be chosen in accordance with TSA design constraints that is tradeoff between TSA area and computation
In various embodiments, the PE module 304 may operate in vectoring and rotation modes. The mode of the PE module 304 may be controlled by a flag *. In one embodiment, for example, the PE module 304
operates in vectoring mode if a data sample carries a flag * and enters the PE module 304 from a west input port W. In all other cases, the PE module 304 operates in rotation mode.
The PE module 304 may comprise CORDIC blocks configured to calculate angle and amplitude of input complex samples in vectoring mode. In various embodiments, in vectoring mode, the PE module 304 may
receive complex samples a and b from a west input port W and a north input port N, respectively. The PE module 304 may be arranged to compute three angles θ
=arg(a), θ
=arg(b), θ
=arctg(|b|/|a|) and to store the results into three internal angle registers.
In various embodiments, the PE module 304 may produce vectoring mode outputs comprising magnitude {square root over (|a|
)} at the east output port E and zero sample at the south output port S. It is noted that the vectoring mode may require the use of three CORDIC operations (see
FIG. 4B
). In vectoring mode, an input control signal * present in the west input port W is to be asserted to the east output port E (from West to East).
The PE module 304 may comprise CORDIC blocks configured to perform vector rotation in polar coordinates in rotation mode. In various embodiments, in rotation mode, the samples a and b (taken from the
west input port W and the north input port N, respectively) may be transformed to complex samples x and y according to the equation:
[ x y ] = [ cos ( θ 1 ) sin ( θ 1 ) - sin ( θ 1 ) cos ( θ 1 ) ] [ exp ( - θ a ) 0 0 exp ( - θ b ) ] [ a b ] , ( 14 ) ##EQU00008##
where θ
, θ
, θ
are angles that were computed and stored during the vectoring mode of operation.
In various embodiments, the PE module 304 may produce rotation mode outputs comprising the transformed complex samples x and y at the east output port E and the south output port S, respectively. It
is noted that four CORDIC operations may be required to perform the transformation described by equation (14). If the control signal * is present at the north input port N of the PE module 304, then
it passes to the south output port S (from North to South).
[0084]FIG. 3c
illustrates one embodiment of an RU module 306. In various embodiments, the RU module 306 may comprise, or be implemented as, the RU module 206 of the processing system 200. The embodiments are not
limited in this context.
In various implementations, the RU 306 module may be arranged to eliminate complex lowest diagonal element of the output upper triangular matrix R by additional rotation. The RU module 306 may
comprise one or more CORDIC blocks and may operate in vectoring or rotation modes. In one embodiment, for example, the RU module 306 may operate in vectoring mode if a control signal * appears at the
north input port N.
In vectoring mode, the RU module 306 may receive a complex sample b from the north input port N and apply the CORDIC algorithm to compute angle θ
=arg(b) and magnitude |b|. The RU module 306 may store the angle θ
=arg(b) to an internal angle register θ
=arg(b) and send the magnitude |b| to the east output port E.
In rotation mode, the RU module 306 may rotate the complex sample b received at the north input port N by the angle θ
stored in the internal angle register. As a result, the RU module 306 may output a complex sample x=exp(-iθ
)b at the output port E.
In various embodiments, the TSA/TACR architecture and technique for QRD may achieve latency reduction in comparison to conventional QRD architectures and techniques. To illustrate the advantages of
the TSA/TACR architecture and technique, the latencies of PE modules based on the TSA/TACR and CGR approaches were compared. It is evident that TSA operation time required for the QRD is directly
determined by the latency of PE modules and PE time complexity. It is noted that for both approaches, the PE modules may be realized using one or more hardwired CORDIC blocks.
FIGS. 4A-5B illustrate one embodiment of data flows. FIGS. 4A and 4B illustrate a comparison between a CGR data flow 400A and a TACR data flow 400B required by PE modules in vectoring mode. FIGS. 5A
and 5B illustrate a comparison between a CGR data flow 500A and a TACR data flow 500B required by PE modules in rotation mode. In this embodiment, the PE modules were realized using two hardwired
CORDIC blocks for both vectoring and rotation modes of operation.
As shown in FIGS. 4A and 4B, the vectoring mode for the CGR data flow 400A requires the usage of four CORDIC operations, while the vectoring mode for the TACR data flow 400B requires the usage of
three CORDIC operations. As shown in FIGS. 5A and 5B, the rotation mode for the CGR data flow 500A requires the usage of six CORDIC operations, while the rotation mode for the TACR data flow 500B
requires the usage of four CORDIC operations. Accordingly, in both vectoring and rotation modes, the internal architecture of the TACR PE modules outperforms CGR in terms of required CORDIC
operations and provides full parallel usage of all available hardware CORDIC resources. In contrast to CGR data flows 400A and 500A, there are no idle CORDIC modules on any step of signal processing
in TACR data flows 400B and 500B.
Table 1 further summarizes latency estimates for PE modules based on the CGR and TACR approaches, respectively. Table 1 illustrates two possible embodiments for PE modules, namely, one and two
hardwired CORDIC modules. As demonstrated by Table 1, the TACR approach may reduce the TSA operation time up to 40 percent.
In Table 1, the PE latency is determined by taking into account two main factors:
N--number of clock cycles required by CORDIC.
--number of clock cycles to make CORDIC Scale Factor Correction (SFC).
It is noted, for latency calculation, that the SFC may be implemented using the same
CORDIC hardware resources.
-US-00001 TABLE 1 PE Latency for CGR and TACR QRD Latency of PE based on Latency of PE based on one hardwired CORDIC two hard-wired CORDIC module modules Vectoring Rotation Vectoring Rotation QRD
approach mode mode mode mode CGR approach 4N + 2N
6N + 4N
3N + 2N
3N + 3N
TACR 3N + N
4N + 2N
2N + N
2N + N
approach Latency N + N
2N + 2N
N + N
N + 2N
Operations for various embodiments may be further described with reference to the following figures and accompanying examples. Some of the figures may include a logic flow. It can be appreciated that
an illustrated logic flow merely provides one example of how the described functionality may be implemented. Further, a given logic flow does not necessarily have to be executed in the order
presented unless otherwise indicated. In addition, a logic flow may be implemented by a hardware element, a software element executed by a processor, or any combination thereof. The embodiments are
not limited in this context.
[0098]FIG. 6
illustrates one embodiment of a logic flow.
FIG. 6
illustrates a logic flow 600 for a TSA/TACR technique to perform signal processing. In various embodiments, the logic flow 600 may be performed by various systems, nodes, and/or modules. It is to be
understood that the logic flow 600 may be implemented by various other types of hardware, software, and/or combination thereof. The embodiments are not limited in this context.
The logic flow 600 may comprise sequential loading of an input matrix and an identity matrix at block 602. In various embodiments, the input matrix may comprise a square complex matrix. In one
embodiment, for example, the input matrix may comprise a 2×2 square complex matrix V
×2, that may be defined by the expression:
2 × 2 = [ A jθ a C jθ c B jθ b D jθ d ] ##EQU00009##
where j= {square root over (-1)}; A, B, C, D--are magnitudes and θ
, θ
, θ
, θ
--are angles. The embodiments are not limited in this context.
In various implementations, the input matrix and the identity matrix may be loaded into a TSA. The TSA may comprise processing modules including, for example, one or more DU modules, PE modules, and
RU modules. The processing modules may comprise CORDIC-based processing modules that may operate in a vectoring mode and a rotation mode. The diagonal elements of the input matrix may comprise
control signals for determining the mode of the CORDIC-based processing modules. The embodiments are not limited in this context.
The logic flow 600 may comprise performing a unitary matrix transformation using three rotation angles at block 604. In one embodiment, for example, the unitary matrix transformation for QRD of the
input matrix V
×2 may comprise:
[ cos θ 1 jθ 2 sin θ 1 j θ 3 - sin θ 1 jθ 2 cos θ 1 jθ 3 ] . ##EQU00010##
In this embodiment, the unitary transformation requires three angles θ
, θ
, θ
determined by the following equations:
[b] [0104]
The embodiments are not limited in this context.
In various implementations, the unitary matrix transformation introduces real elements on a matrix diagonal of an upper triangular matrix. In one embodiment, for example, the unitary matrix
transformation leads to the appearance of all real elements on the main matrix diagonal except the lowest one.
The logic flow 600 may comprise performing an additional unitary matrix transformation to eliminate complex diagonal elements at block 606. In various embodiments, the additional unitary matrix
transformation may eliminate the remaining (lowest) complex diagonal elements on the matrix diagonal of the upper triangular matrix. In one embodiment, for example, the additional unitary matrix
transformation for a square 2×2 matrix may comprise:
[ 1 0 0 - jθ z ] [ X Y θ y 0 Z θ z ] = [ X Y θ y 0 Z ] . ##EQU00011##
The embodiments are not limited in this context.
The logic flow 600 may comprise producing a complex unitary matrix and an upper triangular matrix having only real diagonal elements at block 608. In one embodiment, for example, the upper triangular
matrix may comprise:
[ 1 0 0 - jθ z ] [ cos θ 1 jθ 2 sin θ 1 j θ 3 - sin θ 1 jθ 2 cos θ 1 jθ 3 ] [ A jθ a C jθ c B jθ b D jθ d ] = [ 1 0 0 - jθ z ] [ X Y θ y 0 Z θ z ] = [ X Y θ y 0 Z ] ##EQU00012##
The embodiments are not limited in this context.
The logic flow 600 may comprise determining the inverse matrix of the complex unitary matrix and the inverse matrix of the upper triangular matrix that were produced by QRD of the input matrix. These
steps are depicted at block 610 of the logic flow 600. In various embodiments, the upper triangular matrix comprises only real diagonal elements, which significantly simplifies the inversion of the
upper triangular matrix (for example applying back substitution algorithm). The inversion of complex unitary matrix is straightforward and may be found as a simple Hermitian transposition. Further,
the inverted unitary and upper triangular matrices may be used to calculate inverse input matrix applying matrix multiplication. The embodiments are not limited in this context.
In various implementations, the described embodiments may provide a TSA/TACR architecture and technique for QRD of large size complex matrixes. The developed parallel TSA/TACR architecture may speed
up existing very large scale integration (VLSI) approaches for QRD. In various embodiments, the TSA/TACR architecture may improve TSA hardware utilization efficiency by avoiding idle CORDIC modules
awaiting data. The TSA/TACR architecture may reduce PE latency and computational complexity leading to a significant time savings for QRD. For example, the TSA/TACR architecture for complex matrix
QRD may provide up to a 40% reduction in latency as compared to a conventional CGR architecture.
In various implementations, the described embodiments may provide a CORDIC-based TSA/TACR architecture that produces an upper triangular matrix R having only real diagonal elements that is
advantageous for further inversion of matrix R.
In various implementations, the described embodiments may improve the effectiveness of QRD systolic arrays for performing MIMO signal processing and multiple antenna techniques in WiMAX systems. In
various embodiments, the TSA/TACR architecture and technique may implement various MIMO algorithms for multiple antenna systems. For example, multiple antenna designs may use up to twelve antennas at
a base station, but computational complexity may limit optimization between only four antennas. The TSA/TACR architecture and technique may allow mutual optimization between all twelve antennas
leading to a significant performance gain.
In various implementations, the described embodiments may provide a TSA/TACR architecture used as a key QRD signal-processing block in various other high-speed matrix inverse applications to meet the
challenge of real-time data processing.
Numerous specific details have been set forth herein to provide a thorough understanding of the embodiments. It will be understood by those skilled in the art, however, that the embodiments may be
practiced without these specific details. In other instances, well-known operations, components and circuits have not been described in detail so as not to obscure the embodiments. It can be
appreciated that the specific structural and functional details disclosed herein may be representative and do not necessarily limit the scope of the embodiments.
Some embodiments may be implemented, for example, using a machine-readable medium or article which may store an instruction or a set of instructions that, if executed by a machine, may cause the
machine to perform a method and/or operations in accordance with the embodiments. Such a machine may include, for example, any suitable processing platform, computing platform, computing device,
processing device, computing system, processing system, computer, processor, or the like, and may be implemented using any suitable combination of hardware and/or software. The machine-readable
medium or article may include, for example, any suitable type of memory unit, memory device, memory article, memory medium, storage device, storage article, storage medium and/or storage unit, for
example, memory, removable or non-removable media, erasable or non-erasable media, writeable or re-writeable media, digital or analog media, hard disk, floppy disk, Compact Disk Read Only Memory
(CD-ROM), Compact Disk Recordable (CD-R), Compact Disk Rewriteable (CD-RW), optical disk, magnetic media, magneto-optical media, removable memory cards or disks, various types of Digital Versatile
Disk (DVD), a tape, a cassette, or the like. The instructions may include any suitable type of code, such as source code, compiled code, interpreted code, executable code, static code, dynamic code,
and the like. The instructions may be implemented using any suitable high-level, low-level, object-oriented, visual, compiled and/or interpreted programming language, such as C, C++, Java, BASIC,
Perl, Matlab, Pascal, Visual BASIC, assembly language, machine code, and so forth. The embodiments are not limited in this context.
Some embodiments may be implemented using an architecture that may vary in accordance with any number of factors, such as desired computational rate, power levels, heat tolerances, processing cycle
budget, input data rates, output data rates, memory resources, data bus speeds and other performance constraints. For example, an embodiment may be implemented using software executed by a
general-purpose or special-purpose processor. In another example, an embodiment may be implemented as dedicated hardware, such as a circuit, an ASIC, PLD, DSP, and so forth. In yet another example,
an embodiment may be implemented by any combination of programmed general-purpose computer components and custom hardware components. The embodiments are not limited in this context.
Unless specifically stated otherwise, it may be appreciated that terms such as "processing," "computing," "calculating," "determining," or the like, refer to the action and/or processes of a computer
or computing system, or similar electronic computing device, that manipulates and/or transforms data represented as physical quantities (e.g., electronic) within the computing system's registers and/
or memories into other data similarly represented as physical quantities within the computing system's memories, registers or other such information storage, transmission or display devices. The
embodiments are not limited in this context.
It is also worthy to note that any reference to "one embodiment" or "an embodiment" means that a particular feature, structure, or characteristic described in connection with the embodiment is
included in at least one embodiment. The appearances of the phrase "in one embodiment" in various places in the specification are not necessarily all referring to the same embodiment.
While certain features of the embodiments have been illustrated as described herein, many modifications, substitutions, changes and equivalents will now occur to those skilled in the art. It is
therefore to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the embodiments.
User Contributions:
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Patent applications by Alexander Maltsev, Nizhny Novgorod RU
Patent applications by Alexey Khoryaev, Dzerzhinsk RU
Patent applications by Roman Maslennikov, Nizhny Novgorod RU
Patent applications in class TRANSCEIVERS
Patent applications in all subclasses TRANSCEIVERS
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Structure and Interpretation of Classical Mechanics - Free Computer, Programming, Mathematics, Technical Books, Lecture Notes and Tutorials
Structure and Interpretation of Classical Mechanics
• Title: Structure and Interpretation of Classical Mechanics
• Author(s) Gerald Jay Sussman (Author), Jack Wisdom (Author)
• Publisher: The MIT Press (November 19, 2024); eBook (Creative Commons Licensed)
• License(s): Creative Commons License (CC)
• Hardcover: 578 pages
• eBook: HTML and PDF
• Language: English
• ISBN-10: 0262553457
• ISBN-13: 978-0262553452
• Share This:
Book Description
The new edition of a classic text that concentrates on developing general methods for studying the behavior of classical systems, with extensive use of computation.
We now know that there is much more to classical mechanics than previously suspected. Derivations of the equations of motion, the focus of traditional presentations of mechanics, are just the
beginning. This innovative textbook, now in its second edition, concentrates on developing general methods for studying the behavior of classical systems, whether or not they have a symbolic
It focuses on the phenomenon of motion and makes extensive use of computer simulation in its explorations of the topic. It weaves recent discoveries in nonlinear dynamics throughout the text, rather
than presenting them as an afterthought. Explorations of phenomena such as the transition to chaos, nonlinear resonances, and resonance overlap to help the student develop appropriate analytic tools
for understanding.
The book uses computation to constrain notation, to capture and formalize methods, and for simulation and symbolic analysis. The requirement that the computer be able to interpret any expression
provides the student with strict and immediate feedback about whether an expression is correctly formulated.
This second edition has been updated throughout, with revisions that reflect insights gained by the authors from using the text every year at MIT. In addition, because of substantial software
improvements, this edition provides algebraic proofs of more generality than those in the previous edition; this improvement permeates the new edition.
About the Authors
• Gerald Jay Sussman is Panasonic Professor of Electrical Engineering at MIT.
• Jack Wisdom is Professor of Planetary Science at MIT.
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Triangle Inequality
A triangle is a three-sided polygon. It has three sides and three angles. The three sides and three angles share an important relationship. In Mathematics, the term “inequality” represents the
meaning “not equal”. Let us consider a simple example if the expressions in the equations are not equal, we can say it as inequality. In this article, let us discuss what is triangle inequality in
Maths, activities for explaining the concept of the triangle inequality theorem, and so on.
What is Triangle Inequality?
In Mathematics, the term “triangle inequality” is meant for any triangles. Let us take a, b, and c are the lengths of the three sides of a triangle, in which no side is being greater than the side c,
then the triangle inequality states that,
c ≤ a+b
This states that the sum of any two sides of a triangle is greater than or equal to the third side of a triangle.
Activity For Triangle Inequality Theorem
Let us discuss the relationship and equality and inequality of triangle, through an activity.
Activity 1: On a paper mark two points Y and Z and join them to form a straight line. Mark another point X outside the line lying on the same plane of the paper. Join XY as shown.
Now mark another point X’ on the line segment XY, join X’Z. Similarly, mark X’’ and join X’’Z with dotted lines as shown.
From the above figure we can easily deduce that if we keep on decreasing the length of side XY such that XY> X’Y> X’’Y> X’’’Y the angle opposite to side XY also decreases i.e. ∠XZY >∠X’ZY >∠X’’ZY >
∠X’’’ZY. Thus, from the above activity, we can infer that if we keep on increasing one side of a triangle then the angle opposite to it increases.
Now let us try out another activity.
Activity: Draw 3 scalene triangles on a sheet of paper as shown.
Let us consider fig. (i). In ∆ABC, AC is the longest side and AB is the shortest. We observe that ∠B is the largest in measure and ∠C is the smallest. Similarly, in ∆XYZ, XY is the largest side and
XZ is the smallest and ∠Z is the largest in measure and ∠Y is the smallest. In the last figure also the same kind of pattern is followed i.e. side PR is largest and so is the ∠Q opposite to it.
Triangle Inequality Theorem
Let us consider the triangle. The following are the triangle inequality theorems.
Theorem 1: In a triangle, the side opposite to the largest side is greatest in measure.
The converse of the above theorem is also true according to which in a triangle the side opposite to a greater angle is the longest side of the triangle.
In the above fig., since AC is the longest side, the largest angle in the triangle is ∠B.
Another theorem which follows can be stated as:
Theorem 2: The sum of lengths of any two sides of a triangle is greater than the length of its third side.
According to triangle inequality, AB + BC > AC
To learn more about triangle inequality proof and other properties please download BYJU’S- The Learning App. | {"url":"https://mathlake.com/Triangle-Inequality","timestamp":"2024-11-06T23:17:18Z","content_type":"text/html","content_length":"11045","record_id":"<urn:uuid:85b6f593-ef6f-4d6b-b326-36c578cbad47>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00551.warc.gz"} |
Because D is the midpoint of AB and E is the midpoint of BC, it follows that the length of AC is exactly twice that of DE. We then construct perpendiculars (shown in green) from E and B to AC. By
similar triangles it is clear that the length of BK is twice that of EJ.
The area of triangle ABC is one-half the length of the base (AC) times the height (BK). The area of parallelogram DEFG is the length of the base (DE) times the height (EJ). But (AC)=2(DE) and (BK)=2
(EJ), and so it follows that the area of the triangle ABC is twice that of the quadrilateral DEFG.
Clearly, we can carry out the same construction on the bottom half of the quadrilateral, from which it follows that the area of the midpoint polygon is half that of the original quadrilateral. | {"url":"http://www.math.brown.edu/tbanchof/midpoint/nonquad.html","timestamp":"2024-11-11T10:29:13Z","content_type":"text/html","content_length":"3510","record_id":"<urn:uuid:d9df7c2b-da75-4e5c-ba78-0eb2bc683712>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00543.warc.gz"} |
How do you write ln 4= 1.386.. in exponential form? | HIX Tutor
How do you write #ln 4= 1.386..# in exponential form?
Answer 1
#ln(4)# is just another way of writing #log_e(4)# and #log_b(a)=c color(white)("X")hArrcolor(white)("X")b^c=a#
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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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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-write-ln-4-1-386-in-exponential-form-8f9afa6c50","timestamp":"2024-11-12T19:02:59Z","content_type":"text/html","content_length":"572734","record_id":"<urn:uuid:77e19c19-a6e3-4b8e-950a-c467c74e7863>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00402.warc.gz"} |
APL Hacking; Project Euler (#27)
Problem #27
⍝ Find the product of the coefficients A and B
⍝ that have the longest consecutive n values from 0
⍝ for the quadratic formula.
⍝ P is the prime table used by CONS∆PRMS.
⍝ Seed it with the primes we use for B since we compute
⍝ those anyways.
⍝ Use the full range for A
⍝ We only need to use the primes for B
X←,A∘.CONS∆PRMS B
∇N←A CONS∆PRMS B;R;⎕IO
⍝ Given (N2)+(A×N)+B, how many consecutive values of N
⍝ starting at N=0 result in primes?
⍝ Assume a global P that stores the primality of each
⍝ value R, where P[R] is as follows:
⍝ ¯1 Uninitialized
⍝ 0 Not prime
⍝ 1 Prime
LP:N←N+1 ⋄ R←|(N2)+(A×N)+B
⍝ Grab the memoized result if available,
→(0=R)/0 ⋄ →(0=P[R])/0 ⋄ →(1=P[R])/LP
⍝ Memoize the result, otherwise.
I did some profiling on this one to discover that the prime number table is actually best done with a small amount of initial primes, because the table ends up being a little sparse, and you don't
want to waste time doing the prime computations if you don't have to.
Of course, having the prime table at all basically doubles the speed of the program. The biggest inefficiency now is the search for the highest prime N in the CONS∆PRMS function. I think I could do
better if I did some sort of shortcutting or found some sort of mathematical property that let me reduce the search, but unless I find something like that, the looping there accounts for over 60% of
the total running time. | {"url":"https://www.sacrideo.us/apl-hacking-project-euler-27/","timestamp":"2024-11-14T15:15:22Z","content_type":"text/html","content_length":"35601","record_id":"<urn:uuid:77fd9ab4-e764-4992-8cc0-0c38285bfead>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00838.warc.gz"} |
Constructive response answers/grading - HELP PLEASE
I’m sure there are threads on this, but couldn’t find anything. I’ve been working through CFAI prior exams, and today did a Schweser mock. I’m concerned about how the AM is graded. In Schweser
suggested responses and grading guidelines, it makes it sound like you have to write out the formula for each problem you work to get full credit.
For instance, Equity Q = MV of Equity/(MV of Assets-MV of Liab), then actually show your inputs such as Equity Q = 2/(3-2) =1.
This example is easy, but when you start talking about formulas for relative value of gift vs. bequest or some other more complex formulas, do you have to state all the correct nomenclature, etc?
Seems a bit ridiculous that you get graded on your format.
Or for instance, what if instead of putting in a formula for the required return you use calculator inputs, but you list out what inputs you used. Am I going to get dinged for that even if I get the
same outcome?
Thanks for the input folks…just wondering how hard I need to study “proper” formulas to memory this late in the game.
i don’t think you need to write out the words and names of the terms in the formula. But it would be wise to write the set-up for your calculations so that if you get it wrong when you’re inputting
in your calculator – the grader can at least see you knew the calculation.
Don’t stress on the “proper formula” just make sure you write down what you input into your calculator – my $0.02 maybe others feel differently
There isn’t a single person on this forum that knows exactly how the exams are graded. Obvioulsy there is a universal guideline process that all of the graders adhere to, but there is plenty of room
for ambiguity amongst humans. That being said, I would think that if you show all of your work and get to the right answer you should be awarded full points regardless of whether or not you used
their exact foruma verbatim. The questions do not state: “You must list the formula X to receive all credit”. Maybe they dock you a point, but I feel like that should be the most. In the case of a
wrong answer, if your calculations make sense and you simply transposed something or incorrectly rounded then I still think you should recevie a majority of the credit for the question.
Most of the formula’s simply provide a short-hand way of solving for X. If you show a different method I think the CFAI should be accepting of it.
At the end of the day, the CFAI is not testing whether or not you can plug and chug, but if you understand the curriculum. If you can solve the questions in a different manner then I would think you
have fulfilled your duty as a candidate and derseve full marks. | {"url":"https://www.analystforum.com/t/constructive-response-answers-grading-help-please/83938","timestamp":"2024-11-02T02:24:20Z","content_type":"text/html","content_length":"22967","record_id":"<urn:uuid:68bcfda1-6737-4948-a642-8fb69185bba3>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00292.warc.gz"} |
What is a Matrix? - Max 8 Documentation
What is a Matrix?
A matrix is a grid, with each location in the grid containing some information. For example, a chess board is a matrix in which every square contains a specific item of information: a particular
chess piece, or the lack of a chess piece.
White has just moved a pawn from matrix location e2 to location e4.
For the sake of this discussion, though, let's assume that the "information" at each location in a matrix is numeric data (numbers). Here's a matrix with a number at each grid location.
A spreadsheet is an example of a two-dimensional matrix.
We'll call each horizontal line of data a row, and each vertical line of data a column. On roadmaps, or on chessboards, or in spreadsheet software, one often labels columns with letters and rows with
numbers. That enables us to refer to any grid location on the map by referring to its column and its row. In spreadsheets, a grid location is called a cell. So, in the example above, the numeric
value at cell C3 is 0.319.
The two pictures shown above are examples of matrices that have two dimensions, (horizontal) width and (vertical) height. In Jitter, a matrix can have any number of dimensions from 1 to 32. (A
one-dimensional matrix is comparable to what programmers call an array. Max already has some objects that are good for storing arrays of numbers, such as table and multislider. There might be cases,
though, when a one-dimensional matrix in Jitter would be more useful.) Although it's a bit harder to depict on paper, one could certainly imagine a matrix with three dimensions, as a cube having
width, height, and depth. (For example, a matrix 3 cells wide by 3 cells high by 3 cells deep would have 3x3x3=27 cells total.)
A 3x3x3 matrix has 27 cells.
And although it challenges our visual imagination and our descriptive vocabulary, one can even have matrices of four or more dimensions. For this tutorial, however, we'll restrict ourselves to
two-dimensional matrices.
A Video Screen is One Type of Matrix
A video screen is made up of tiny individual pixels (picture elements), each of which displays a specific color. On a computer monitor, the resolution of the screen is usually some size like 1024
pixels wide by 768 pixels high, or perhaps 800x600 or 640x480. On a television monitor (and in most conventional video images), the resolution of the screen is approximately 640x480, and on computers
is typically treated as such. Notice that in all of these cases the so-called aspect ratio of width to height is 4:3. In the wider DV format, the aspect ratio is 3:2, and the image is generally
720x480 pixels. High-Definition Television (HDTV) specifies yet another aspect ratio—16:9. In these tutorials we'll usually work with an aspect ratio of 4:3, and most commonly with
smaller-than-normal pixel dimensions 320x240 or even 160x120, just to save space in the Max patch.
Relative sizes of different common pixel dimensions
A single frame of standard video (i.e., a single video image at a given moment) is composed of 640x480=307,200 pixels. Each pixel displays a color. In order to represent the color of each pixel
numerically, with enough variety to satisfy our eyes, we need a very large range of different possible color values.
There are many different ways to represent colors digitally. A standard way to describe the color of each pixel in computers is to break the color down into its three different color components —red,
green, and blue ( a.k.a. RGB)—and an additional transparency/opacity component (known as the alpha channel). Most computer programs therefore store the color of a single pixel as four separate
numbers, representing the alpha, red, green, and blue components (or channels). This four-channel color representation scheme is commonly called ARGB or RGBA, depending upon how the pixels are
arranged in memory.
Jitter is no exception in this regard. In order for each cell of a matrix to represent one color pixel, each cell actually has to hold four numerical values (alpha, red, green, and blue), not just
one. So, a matrix that stores the data for a frame of video will actually contain four values in each cell.
Each cell of a matrix may contain more than one number.
A frame of video is thus represented in Jitter as a two-dimensional matrix, with each cell representing a pixel of the frame, and each cell containing four values representing alpha, red, green, and
blue on a scale from 0 to 255. In order to keep this concept of multiple-numbers-per-cell (which is essential for digital video) separate from the concept of dimensions in a matrix, Jitter introduces
the idea of planes.
What is a Plane?
When allocating memory for the numbers in a matrix, Jitter needs to know the extent of each dimension—for example, 320x240—and also the number of values to be held in each cell. In order to keep
track of the different values in a cell, Jitter uses the idea of each one existing on a separate plane. Each of the values in a cell exists on a particular plane, so we can think of a video frame as
being a two-dimensional matrix of four interleaved planes of data.
The values in each cell of this matrix can be thought of as existing on four virtual planes.
Using this conceptual framework, we can treat each plane (and thus each channel of the color information) individually when we need to. For example, if we want to increase the redness of an image, we
can simply increase all the values in the red plane of the matrix, and leave the others unchanged.
The normal case for representing video in Jitter is to have a 2D matrix with four planes of data—alpha, red, green, and blue. The planes are numbered from 0 to 3, so the alpha channel is in plane 0,
and the RGB channels are in planes 1, 2, and 3.
The Data in a Matrix
Computers have different internal formats for storing numbers. If we know the kind of number we will want to store in a particular place, we can save memory by allocating only as much memory space as
we really need for each number. For example, if we are going to store Western alphabetic characters according to the ASCII standard of representation, we only need a range from 0 to 255, so we only
need 8 bits of storage space to store each character (because 2^8 = 256 different possible values). If we want to store a larger range of numbers, we might use 32 bits, which would give us integer
numbers in a range from -2,147,483,648 to 2,147,483,647. To represent numbers with a decimal part, such as 3.1416, we use what is called a floating point binary system, in which some of the bits of a
32-bit or 64-bit number represent the mantissa of the value and other bits represent the exponent.
Much of the time when you are programming in Max (for example, if you're just working with MIDI data) you might not need to know how Max is storing the numbers. However, when you're programming
digital audio in MSP it helps to be aware that MSP uses floating point numbers. (You will encounter math errors if you accidentally use integer storage when you mean to store decimal fractions.) In
Jitter, too, it is very helpful to be aware of the different types of number storage the computer uses, to avoid possible math errors.
A Jitter matrix can store numbers as 64-bit floating-point (known to programmers as a double-precision float, or double), 32-bit floating point (known simply as float), 32-bit integers (known as long
int, or just int), and 8-bit characters (known as char). Some jit objects store their numerical values in only one of these possible formats, so you will not have to specify the storage type. But
other Jitter objects can store their values in various ways, so the storage type must be typed in as an argument in the object, using the words , , , or .
Important concept: In cases where we're using Jitter to manipulate video, perhaps the most significant thing to know about data storage in Jitter matrices is the following. When a matrix is holding
video data—as in the examples in the preceding paragraphs—it assumes that the data is being represented in ARGB format, and that each cell is thus likely to contain values that range from 0 to 255
(often in four planes). For this reason, the most common data storage type is char. Even though the values being stored are usually numeric (not alphabetic characters), we only need 256 different
possible values for each one, so the 8 bits of a char are sufficient. Since a video frame contains so many pixels, and each cell may contain four values, it makes sense for Jitter to conserve on
storage space when dealing with so many values. Since manipulation of video data is the primary activity of many of the Jitter objects, most matrix objects use the char storage type by default. For
monochrome (grayscale) images or video, a single plane of char data is sufficient. | {"url":"https://docs.cycling74.com/legacy/max8/tutorials/jitterchapter00a_whatisamatrix","timestamp":"2024-11-07T21:56:30Z","content_type":"text/html","content_length":"57519","record_id":"<urn:uuid:16bb70e7-f70c-4815-8964-636feac9945c>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00771.warc.gz"} |
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On Mon, 20 Mar 2000 11:49:14 +0200, Eli Zaretskii <eliz AT is DOT elta DOT co DOT il>
>I don't think we are *that* desperate to support NT and W2K quirks to
>go to those lengths. I'm quite sure that eventually, someone will
>come up with a much easier and less painful solution to tell NT from
Isn't there a "Get version of Windows" interrupt call? Wouldn't it
return 4.x for NT 4.x and 5.x for 2000? Allegro seems to be able to
get the version number of the running GUI quite nicely. Or is it
implemented on the 3.1/9x fork only?
Damian Yerrick http://yerricde.tripod.com/
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Copyright © 2019 by DJ Delorie Updated Jul 2019 | {"url":"https://delorie.com/archives/browse.cgi?p=djgpp/2000/03/20/15:49:57","timestamp":"2024-11-04T04:21:54Z","content_type":"text/html","content_length":"8198","record_id":"<urn:uuid:bf08bd8d-5c3c-4cc6-a57e-11c9b26856cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00426.warc.gz"} |
Adding Additional Parameters to Fluxes
Adding Additional Parameters to Fluxes
By default simweights makes PDGID, energy, and zenith available to flux models passed to simweights.Weighter.get_weights(). Normally that is all you need for most models. But if you have a more
complex model that depends on additional parameters you can add them with simweights.Weighter.add_weight_column().
In the example below, azimuth angle from PolyplopiaPrimary is added to so that a model with azimuth angle can be applied. In this case the data was taken from the same data table but in principle it
could be from any source as long as it is a numpy array of the correct shape.
import simweights
# load the hdf5 file and make the weigher
hdffile = pd.HDFStore("Level2_IC86.2016_NuMu.021217.N100.hdf5", "r")
weighter = simweights.NuGenWeighter(hdffile, nfiles=100)
# add a non-standard weighting column
weighter.add_weight_column("azimuth", weighter.get_column("PolyplopiaPrimary", "azimuth"))
def simple_model(energy: ArrayLike) -> ArrayLike:
"""This function only depends on energy can be used as a flux models
Note that the units are GeV^-1 * cm^-2 * sr^-1 * s^-1 per particle type.
return 1e-8 * energy**-2
def azimuthal_model(energy: ArrayLike, azimuth: ArrayLike) -> ArrayLike:
"""This function that takes azimuth as a parameter. get_weights() will use the name
of the function parameter to know which weighting column to access.
return 1e-8 * plt.cos(azimuth) ** 2 * energy**-2
for flux_function in (simple_model, azimuthal_model):
# get the weights by passing the flux function to the weighter
weights = weighter.get_weights(flux_function)
# We can access our recently created weight column with get_weight_column()
azi = weighter.get_weight_column("azimuth")
# histogram the primary energy with the weights
plt.hist(azi, weights=weights, bins=50, histtype="step", label=flux_function.__name__) | {"url":"https://software.icecube.wisc.edu/simweights/main/additional_flux_params.html","timestamp":"2024-11-11T04:46:18Z","content_type":"text/html","content_length":"13372","record_id":"<urn:uuid:8e5c694d-9a2c-4ddc-8c8f-8acd40f9aeaa>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00554.warc.gz"} |
*intro flashs /a mish mash of sorts*
Christchurch airport.
SLOWHATCH, I don't know whether your signpost is the other side of #2 of the screencaps (I think it probably isn't). Anyway, for #2, there is no possibility that the signpost is in Christchurch.
Follow this logic:
1. Given the larger distance to Sydney(2169 km) than Melbourne(1682 km), the place must be south of Melbourne to make that so.
2. The only place in Australia of consequence south of Melbourne is Hobart and that is far too close.
3. That leaves only New Zealand as a possibility.
4. My world distance calculator gives these distances:
Auckland to Sydney 2151 km Auckland to Melbourne 2614 km
Christchurch to Sydney 2123 km Christchurch to Melbourne 2387 km
5. To Melbourne is too high, so we need something southwest of Christchurch. There is only one major airport that fits, Queenstown, where AR2 did some bungee jumping and other extreme sports. I
haven't found a distance calculator that includes Queenstown but a rough approximation of how much the distance is to Melbourne should be in the right ballpark.
I have discovered a signpost in Auckland airport that looks totally different and it states Sydney 2159 km.
I have also not been able to discover any signpost at the Queenstown airport so far.
I have done an approximation of the lower distance that Queenstown would bring from Melbourne compared to Christchurch. It would be less than 400 km and I need a reduction of about 700km.
For purposes of validating what I was doing I tried Dunedin airport, southeast of Queenstown. It gave results of 2087 km to Sydney and 2253 km to Melbourne.
What I believe this shows is that there isn't going to be any airport on New Zealand that will be able to approximate the required 2169/1682 from the signpost. The only spot which would have those
distances would be in the Tasman Sea off the west coast of the South Island of New Zealand. | {"url":"https://forum.realityfanforum.com/index.php?topic=13733.300","timestamp":"2024-11-08T13:51:28Z","content_type":"application/xhtml+xml","content_length":"102825","record_id":"<urn:uuid:9c049c62-d13e-4354-865d-71c5150b2f6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00558.warc.gz"} |
Number Properties on the GMAT
How long has it been since you learned about odd and even numbers? What about the differences between positives and negatives? And when did you learn the definition of an integer or a prime number?
Most of us learned all of these things before we were 10 years old. But as you may have noticed, the GMAT tests these concepts—collectively known as number properties—in ways that make them seem
absolutely foreign. Here is some example question stems drawn from Data Sufficiency problems:
• Is x even?
• Is x < 0?
• Is x an integer?
In Problem Solving, algebra problems are frequently set up such that you will answer incorrectly if you assume x must be an integer or that it must be positive.
Number properties are all about categories and rules; certain kinds of numbers behave the same way in all cases. The GMAT will reward you for using the Core Competencies of Pattern Recognition and
Critical Thinking to draw inferences about how numbers behave, based on certain characteristics or “properties” they possess. That’s why number properties questions appear on the Quant section with
greater frequency than other topics.
Here are some definitions to brush up on and some tips for mastering the number properties concepts tested most frequently on the GMAT.
The term integer refers to positive whole numbers, negative whole numbers, and zero. When an integer is added to, subtracted from, or multiplied by another integer, the result is always an integer.
(An integer divided by an integer may or may not result in an integer; it depends on whether the first number is a multiple of the second.) Picking Numbers makes questions about integers and
non-integers easier to tackle.
Zero is a special case. It is an integer, and it is even, but it is neither positive nor negative. Zero has no sign. One is also somewhat special; 0 and 1 behave differently from all other integers
when you multiply them, with one important exception: odd and even rules. Zero behaves the same as all other even integers when adding, subtracting, or multiplying, and 1 behaves the same as all
other odds in these situations.
When “integer” is a central word in a question, you know have a number properties question. Questions that focus on the rules governing integers force test-takers to discriminate between different
categories of numbers (whole numbers versus fractions or decimals). These questions also contain an important trap that you must learn to avoid: Never assume a number is an integer unless you’re told
that it is. The absence of information in a GMAT question can be just as important as its inclusion.
The terms odd and even apply only to integers. Even numbers are integers that are divisible by 2, and odd numbers are integers that are not. Odd and even numbers may be negative; 0 is even. The
product of an even number and any integer will always be even. Here are the rules that govern odd and even:
Notice that there are no universal rules for dividing odds and evens.
Positive numbers are greater than zero, falling to the right of 0 on a number line. Negative numbers are less than zero, falling to the left on a number line.
When multiplying or dividing two numbers that have the same “sign” (positive or negative), the result is always positive. When multiplying or dividing two numbers with different signs, the result is
always negative.
Some GMAT questions hinge on whether the numbers involved are positive or negative. These properties are especially important to keep in mind when Picking Numbers on a Data Sufficiency question. If
both positives and negatives are permissible for a given question, make sure you test both possibilities, since doing so will often yield significant (different) results. Take the same approach that
you’ve been learning to use for other number properties: Spend some time memorizing the rules, but always keep your eye out for strategic opportunities to pick numbers.
The special properties of −1, 0, and 1 make them important numbers to consider in Data Sufficiency questions, as well as for the “could be/must be” kinds of Problem Solving questions. Because numbers
between −1 and 1 behave differently than do other numbers, they are good numbers to pick when testing whether one expression always has to be less than or greater than another.
Because the GMAT tests math concepts most of us learned in elementary school, it can be challenging to recall some of these basic quantitative building blocks. Boost your ability to do mental math
with these tips for remembering how to divide numbers in your head.
Rules for Twos, Fives, and Tens
The easiest of the divisibility rules are the rules for 2, 5, and 10:
• If a number is even, it is divisible by 2.
• If a number ends in 0 or 5, it is divisible by 5.
• If a number ends in 0, it is divisible by 10.
Rules for Threes, Sixes, and Nines
If the sum of a number’s digits is divisible by 3, the number is divisible by 3. For example, 432 → 4+3+2=9, so 432 is divisible by 3. But 253 → 2+5+3=10, so 253 is not divisible by 3.
If a number is even AND the sum of its digits is divisible by 3, it is divisible by 6: 432 → 4+3+2=9, and 432 is even (divisible by 2), so it is divisible by 6.
If the sum of a number’s digits is divisible by 9, the number is divisible by 9: 432 → 4+3+2=9, so 432 is divisible by 9; 837 → 8+3+7=18, so 837 is divisible by 9.
An interesting side note about 9: All multiples of 9, when their digits are summed, eventually yield 9. For example, 837 → 8+3+7=18, and 1+8=9.
If the last two digits of a number are a multiple of 4, the entire number is a multiple of 4. You don’t add the digits together; if the last two are a multiple of 4 that you recognize, you can trust
you have a multiple of 4. Because 100 is divisible by 4, all that matters is the last two digits. For example, 2,348,632 is divisible by 4 because the last two digits, 32, is divisible by 4.
I’m not aware of any useful ways to determine divisibility by 7 or 8, so your best bet for those is to learn their multiples instead. Most of us have forgotten our multiplication tables, so there is
no shame in needing to refresh them. Take care to learn the multiples of 13 as well. Because most people brush up on multiples through 12, the GMAT loves to throw in 13 to catch you off guard.
Spend some time reviewing number properties. Being aware of and comfortable with the behaviors of numbers will take you a long way toward landing your best GMAT score on Test Day.
Jennifer Mathews Land has taught for Kaplan since 2009. She prepares students to take the GMAT, GRE, ACT, and SAT and was named Kaplan’s Alabama-Mississippi Teacher of the Year in 2010. Prior to
joining Kaplan, she worked as a grad assistant in a university archives, a copy editor for medical websites, and a dancing dinosaur at children’s parties. Jennifer holds a Ph.D. and a master’s in
library and information studies (MLIS) from the University of Alabama, and an AB in English from Wellesley College. When she isn’t teaching, she enjoys watching Alabama football and herding cats.
0 0 admin http://wpapp.kaptest.com/wp-content/uploads/2020/09/kaplan_logo_purple_726-4.png admin2022-08-30 12:35:282022-08-30 17:26:03Number Properties on the GMAT | {"url":"https://wpapp.kaptest.com/study/gmat/land-score-number-properties-gmat/","timestamp":"2024-11-05T10:13:19Z","content_type":"text/html","content_length":"199562","record_id":"<urn:uuid:4abb0ba1-26ef-436a-9a84-58c4773b1768>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00629.warc.gz"} |
Transient error mitigation by means of approximate logic circuits
abstract = "The technological advances in the manufacturing of electronic circuits have allowed to greatly improve their performance, but they have also increased the sensitivity of electronic
devices to radiation-induced errors. Among them, the most common effects are the SEEs, i.e., electrical perturbations provoked by the strike of high-energy particles, which may modify the internal
state of a memory element (SEU) or generate erroneous transient pulses (SET), among other effects. These events pose a threat for the reliability of electronic circuits, and therefore fault-tolerance
techniques must be applied to deal with them.
The most common fault-tolerance techniques are based in full replication (DWC or TMR). These techniques are able to cover a wide range of failure mechanisms present in electronic circuits. However,
they suffer from high overheads in terms of area and power consumption. For this reason, lighter alternatives are often sought at the expense of slightly reducing reliability for the least critical
circuit sections. In this context, a new paradigm of electronic design is emerging, known as approximate computing, which is based on improving circuit performance in exchange for slight
modifications of the intended functionality. This is an interesting approach for the design of lightweight fault-tolerant solutions, which has not been studied in depth yet.
The main goal of this thesis consists in developing new lightweight fault-tolerant techniques with partial replication by means of approximate logic circuits. These circuits can be designed with
great flexibility. This way, the level of protection as well as the overheads can be adjusted at will depending on the necessities of each application. However, finding optimal approximate circuits
for a given application is still a challenge.
In this thesis a method for approximate circuit generation is proposed, denoted as fault approximation, which consists in assigning constant logic values to specific circuit lines. On the other hand,
several criteria are developed to generate the most suitable approximate circuits for each application, by using this fault approximation mechanism. These criteria are based on the idea of
approximating the least testable sections of circuits, which allows reducing overheads while minimising the loss of reliability. Therefore, in this thesis the selection of approximations is linked to
testability measures.
The first criterion for fault selection developed in this thesis uses static testability measures. The approximations are generated from the results of a fault simulation of the target circuit, and
from a user-specified testability threshold. The amount of approximated faults depends on the chosen threshold, which allows to generate approximate circuits for different tradeoffs. Although this
approach was initially intended for combinational circuits, an extension to sequential circuits has been performed as well, by considering the flip-flops as both inputs and outputs of the
combinational part of the circuit. The experimental results show that this technique achieves a wide scalability and an acceptable tradeoff between reliability and overheads. In addition, its
computational complexity is very low.
However, the selection criterion based in static testability measures has some drawbacks. Adjusting the trade-off of the generated approximate circuits by means of the approximation threshold is not
intuitive, and the static testability measures do not take into account the changes as long as faults are approximated. Therefore, an alternative criterion is proposed, which is based on dynamic
testability measures. With this criterion, the testability of each fault is computed by means of an implication-based probability analysis. The probabilities are updated with each new approximated
fault, in such a way that in each iteration the most beneficial approximation is chosen, that is, the fault with the lowest probability. In addition, the computed probabilities allow to estimate the
level of protection against faults that the generated approximate circuits provide. Therefore, it is possible to generate circuits which stick to a target error rate. By modifying this target,
circuits for different trade-offs can be obtained. The experimental results show that this new approach is able to stick to the target error rate with reasonably good precision. In addition, the
approximate circuits generated with this technique show better characteristics than with the approach based in static testability measures. Finally, the fault implications have been reused too in
order to implement a new type of logic transformation, which consists in substituting functionally similar nodes.
Once the fault selection criteria have been developed, they are applied to different scenarios. First, an extension of the proposed techniques to FPGAs is performed, taking into account the
specificities of this kind of circuits. This approach has been validated by means of radiation experiments, which show that a partial replication with approximate circuits can be even more robust
than a full replication approach, because a smaller area reduces the probability of SEE occurrence. Besides, the proposed techniques have been applied to a real application circuit as well, in
particular to the microprocessor ARM Cortex M0. A set of software benchmarks is used to generate the required testability measures. Finally, a comparative study of the proposed approaches with
approximate circuit generation by means of evolutionary techniques have been performed. These approaches are able to generate multiple circuits by trial and error, thus reducing the possibility of
falling into local minima. The experimental results demonstrate that the circuits generated with evolutionary approaches present slightly better trade-offs than the circuits generated with the
techniques here proposed, although with a much higher computational effort.
In summary, several original error mitigation techniques with approximate logic circuits are proposed. These approaches are demonstrated in various scenarios, showing that the scalability and
adaptability to the requirements of each application are their main virtues.", | {"url":"http://gpbib.pmacs.upenn.edu/gp-html/Thesis_AJSC.html","timestamp":"2024-11-03T20:28:36Z","content_type":"text/html","content_length":"9992","record_id":"<urn:uuid:6175fe0d-f7e8-4c03-bc43-03a4c2f2f4f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00469.warc.gz"} |
Anomalous scaling at non-thermal fixed points of the sine-Gordon model - IsoQuant
We extend the theory of nonthermal fixed points to the case of anomalously slow universal scaling dynamics according to the sine-Gordon model. This entails the derivation of a kinetic equation for
the momentum occupancy of the scalar field from a nonperturbative two-particle irreducible effective action, which resums a series of closed loop chains akin to a large-N expansion at next-to-leading
order. The resulting kinetic equation is analyzed for possible scaling solutions in space and time that are characterized by a set of universal scaling exponents and encode self-similar transport to
low momenta. Assuming the momentum occupancy distribution to exhibit a scaling form we can determine the exponents by identifying the dominating contributions to the scattering integral and power
counting. If the field exhibits strong variations across many wells of the cosine potential, the scattering integral is dominated by the scattering of many quasiparticles such that the momentum of
each single participating mode is only weakly constrained. Remarkably, in this case, in contrast to wave turbulent cascades, which correspond to local transport in momentum space, our results suggest
that kinetic scattering here is dominated by rather nonlocal processes corresponding to a spatial containment in position space. The corresponding universal correlation functions in momentum and
position space corroborate this conclusion. Numerical simulations performed in accompanying work yield scaling properties close to the ones predicted here.
P. Heinen, A. N. Mikheev, T. Gasenzer, “Anomalous scaling at non-thermal fixed points of the sine-Gordon model”, Phys. Rev. A 107 (2023).
Related to Project A04, A07 | {"url":"https://www.isoquant-heidelberg.de/anomalous-scaling-at-non-thermal-fixed-points-of-the-sine-gordon-model/","timestamp":"2024-11-09T14:11:42Z","content_type":"text/html","content_length":"28019","record_id":"<urn:uuid:23687ab9-e558-43ad-b2a3-26af3d2f1d90>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00616.warc.gz"} |
In this page you will find educational resources related to Biostistics. That includes announcements about events, material for online workshops, material related to workshops that we held in
person, and other resources.
Meta-Analysis Workshop (Nov 2024)
R & SPSS Workshop Series (Jan-Feb 2024)
R & SPSS Workshop Series (Feb-Apr 2023)
Past Events
SPSS and R Workshop Series (Apr-Jun 2022)
SPSS Workshop Series (Feb-Apr 2021)
11 workshops held on Webex You can access the recordings and do the assignments!
Computing Fundamentals - Introduction to R (Oct 2020)
Video 1 - Introducting myself and some quick stats background (7m19s)
Video 2 - Introduction to R and RStudio. Split in three parts: Part 1, Part 2 and Part 3, each with around 25 minutes.
CSV Dataset - This is the dataset we used in the workshop.
RMarkdown - This is the markdown file with instructions and script that we used.
HTML Output - This is the output from the RMarkdown file in HTML format, which you can read in your web browse.
Online Workshops and Tutorials
In this section you will find material that helps you to learn software and statistics related to the work you do in your research at CAMH.
SPSS Online Workshops
Since 2016 we have had frequent series of SPSS workshops. These workshops are delivered in person, in the CAMH computer lab. They expanded from the initial introductory courses to the current series
that inovlves more than 10 different topics of interest to CAMH researchers.
To the extent of possible, we will try to make available here the material of those workshops, or online versions of them.
Introduction to SPSS
SPSS is probably the most popular software for Statistical Analysis at CAMH, and one that is accessible to researchers. The material available here will allow you to have a first and basic experience
with SPSS, from opening the software, to importing data, doing some simple data cleaning and data analysis tasks. The material is currently in PDF format but it is in our priority list to have it
also in video format.
1. Download this PDF file and follow it by actively using SPSS.
What is SPSS?
Video 1 - You will learn:
1. What is SPSS and how it store data
2. General concepts of rows and columns in a dataset
3. Elements of data and metadata
Typing in data
Vido 2 - You will learn:
1. Define variable names, labels and values
2. Type of variables
3. How to enter data into SPSS manually
Importing Data into SPSS
Video 3 - You will learn:
1. How to open an SPSS dataset
2. How to read Excel datasets
3. How to read CSV (comma separated values) datasets
Download this CSV Dataset, this SPSS Dataset and this Excel Dataset, and then follow the video!
SPSS Frequency and Crosstab
Video 4 - You will learn:
1. How to use Freuqencies
2. How to use Crosstabs
3. Type of variables.
The activities in this video uses this SPSS Dataset.
SPSS Means
Video 5 - You will learn:
1. How to calculate statistics for continuous variables
2. Use of Frequency, Means and Descriptives
The activities in this video uses this SPSS Dataset.
Introduction to SPSS Syntax
In order for your data analysis to be reproducible, you need to keep a record of all you did. Ideally, that means everything from reading the initial dataset to your published results. Doing data
cleaning and data analysis using script is the perfect way to make your research reproducible.
SPSS Syntax here does not mean SPSS programming. We just want you to know that is easy to do anything in SPSS using Syntax, and get you comfortable with it. As such, this course is very introductory,
we don't expect you to have a high level of SPSS or programing knowledge in order to understand it. We do expect that you know SPSS at the introductory level, that you already know how to do some
basica analysis using the point & click menu.
1. Watch this video. Here I just give you a quick introduction to syntax and some tips.
2. Download this SPSS Dataset, a this SPSS Syntax File.
R Online Workshops
Introduction to R
Here you will find a collection of videos and auxiliary material to help you get started with R.
There are not pre
What is R and RStudio?
Video 1 - You will learn:
1. What is R
2. What is R Studio
Installing R and RStudio
Video 2 - You will learn:
1. How to find and download R and R Sutdio
2. How to install R and R Studio
Opening RStudio and creating a project
Video 3- You will learn:
1. Opening R Studio
2. What is a R Studio Project
3. Creating a R Studio Project
R Script and R Markdown
Video 4 - You will learn:
1. What is R Script and R Markdown
2. Why you should use them
3. How to Create a new R Script and R Markdown
4. How to run scripts in R Script and R Markdown
R Packages
Video 5- You will learn:
1. What is a R Package
2. How to find the package you need
3. How to Install and Load a package
4. How to use a package
Dataset Format CSV and Dataset Format SPSS - Please, download these datasets so that you can follow the activities in the video.
Importing a dataset into R
Video 6 - You will learn:
1. How to import your data into R
2. Inspecting the data
3. Different types of R objects
P-values and the Scientific Method.
Video 1- You will learn:
1. The scientific method
2. How not to use p-value
3. Definition of p-value | {"url":"https://kcniconfluence.camh.ca/display/ED/Learning","timestamp":"2024-11-07T09:03:19Z","content_type":"text/html","content_length":"70749","record_id":"<urn:uuid:582c1638-4f4b-4215-bdbe-dab0e15ba51c>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00258.warc.gz"} |
Programming a(b+c)
• Assume a, b and c are declared variables and that the result is saved in $v0
lw $t0,a # Get value of a
lw $t1,b # Get value of b
lw $t2,c # Get value of c
add $t1,$t1,$t2 # Add b and c
mult $v0,$t0,$t1 # Multiply result times a
This 3-operand multiply pseudoinstruction might be generated as ...
mult $t0,$t1 # Do multiply
mflo $v0 # Get result assuming < 2x109
How can one test to see if the number was small enough? | {"url":"https://courses.cs.washington.edu/courses/cse378/00sp/CSE378-00.Lec7/tsld002.htm","timestamp":"2024-11-08T08:23:13Z","content_type":"text/html","content_length":"1630","record_id":"<urn:uuid:1157ecdb-538c-458e-b162-9ef70217a55a>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00176.warc.gz"} |
Sequence logos
1 General =====
• Type: - Matrix Analysis
• Heading: - Misc. (Analysis)
• Source code: not public.
2 Brief description
Create and display sequence logos based on a column containing protein sequence windows centered around sites of interest.
3 Parameters
3.1 Sequences
Selected categorical column that contains the amino acid sequences for which a Sequence logo should be generated (default: first categorical column in the matrix).
Hint: The sequences need to have the same length.
3.2 Column
Selected categorical column that groups the rows according to their value in that column and generates one sequence logo for each value (default: <None>). If <None> is selected one Sequence logo is
generated for all sequences in the column defined in the parameter “Sequences”.
3.3 Compute position-specific p-values
Specifies the input to calculate the position-specific scoring matrix (PSSM) containing the p-values for each position (default: global occurrence). For each Sequence logo one PSSM is calculated
containing the p-value for each amino acid at each position in the sequence. The PSSM can be obtained by clicking on the “Export aa p-values” button in the “Sequence logos” tab of the matrix that was
used to generate the Sequence logo(s). | {"url":"https://cox-labs.github.io/coxdocs/sequencelogo.html","timestamp":"2024-11-12T22:41:10Z","content_type":"application/xhtml+xml","content_length":"26468","record_id":"<urn:uuid:1fa028cc-c865-41fd-ba0e-494df74b6251>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00288.warc.gz"} |
slartg.f - Linux Manuals (3)
slartg.f (3) - Linux Manuals
slartg.f -
subroutine slartg (F, G, CS, SN, R)
SLARTG generates a plane rotation with real cosine and real sine.
Function/Subroutine Documentation
subroutine slartg (realF, realG, realCS, realSN, realR)
SLARTG generates a plane rotation with real cosine and real sine.
SLARTG generate a plane rotation so that
[ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1.
[ -SN CS ] [ G ] [ 0 ]
This is a slower, more accurate version of the BLAS1 routine SROTG,
with the following other differences:
F and G are unchanged on return.
If G=0, then CS=1 and SN=0.
If F=0 and (G .ne. 0), then CS=0 and SN=1 without doing any
floating point operations (saves work in SBDSQR when
there are zeros on the diagonal).
If F exceeds G in magnitude, CS will be positive.
F is REAL
The first component of vector to be rotated.
G is REAL
The second component of vector to be rotated.
CS is REAL
The cosine of the rotation.
SN is REAL
The sine of the rotation.
R is REAL
The nonzero component of the rotated vector.
This version has a few statements commented out for thread safety
(machine parameters are computed on each entry). 10 feb 03, SJH.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 98 of file slartg.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-slartg.f/","timestamp":"2024-11-10T15:59:23Z","content_type":"text/html","content_length":"8205","record_id":"<urn:uuid:e6d805f7-e3b5-4509-949b-2f50f8ac47e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00205.warc.gz"} |
Inquiry Maths - Sums of series inquiry
Mathematical inquiry processes: Interpret and verify; reason; generate examples; prove. Conceptual field of inquiry: Sigma notation; sum of a series; summation formulae.
The prompt is suitable for the A-level Further Maths course. It invites students to compare the sums of two series.
The sums of the series are equal and the prompt is true. The teacher might show this by making n = 5, for example, when both sides give 0 + 2 + 6 + 12 + 20 = 40. To show that explicitly, the
following illustration is part of the PowerPoint in the resources section.
Students would then test other values of n.
As with all inquiries on the website, the prompt does not require discrete knowledge to be taught beforehand. Indeed, students are intrigued by the sigma notation and, after attempts to deduce its
meaning, a teacher's explanation based on those deductions is more meaningful and powerful.
In the notice and wonder phase at the start of the inquiry, students often compare the two sides of the equation by saying what is the same and what is different:
• "The symbol is the same as are the variables r and n."
• "The r and terms in n are one less on the left-hand side."
• "The algebraic expressions (general terms) are similar, but the (r + 1) is replaced by (r - 1) on the right-hand side."
Why is the prompt true? The teacher should start an explanation by pointing out that the general terms are deceptive. The r on the left-hand side has become (r - 1) on the right-hand side and (r + 1)
has changed to r - that is, both parts are one less on the right-hand side. In other words, f(r) is transformed to f(r - 1).
At the same time the series on the right-hand side starts and finishes at terms that are one more than the first and last term on the left-hand side. The two changes 'counteract' or cancel each other
out. Such an explanation opens the way for students to create more general terms with their limits that sum to the same amount.
In a structured inquiry, the class could follow the lines of inquiry below; in a more open inquiry, students might select an approach from the regulatory cards.
November 2023
Firstly, students verify that the sums of the series are equal, being 1(0) + 2(1) + 3(2) + 4(3) + 5(4) + 6(5) + 7(6) + 8(7) + 9(8) + 10(9) in the three cases. They can then extend the list for n = 10
or change the value of n.
Secondly, students find a series equivalent to those in the prompt by choosing between two options (see table below).
One option is correct and the other shows the misconception that if the limits increase or decrease by k, then f(r) becomes f(r + k) or f(r - k) respectively.
Both tasks reinforce the explanation about the changes to the general terms and limits.
2. Generate examples
Students aim to generate general terms with limits that are equivalent to the ones in the list.
3. Prove
To end the inquiry, students prove the prompt is true using summation formulae for the sum of n natural numbers and the sum of the squares of n natural numbers. They can also use the summation
formulae for higher powers on their own general terms. See the mathematical notes for examples of proof. | {"url":"https://www.inquirymaths.com/home/algebra-prompts/sums-of-series-inquiry","timestamp":"2024-11-01T22:12:11Z","content_type":"text/html","content_length":"169609","record_id":"<urn:uuid:623eb80e-0b07-418a-a708-c1ddf5166414>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00186.warc.gz"} |
Sven Leyffer's Publications
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Published Papers
1. Minseok Ryu, Ahmed Attia, Arthur Barnes, Russell Bent, Sven Leyffer, Adam Mate. Heuristic algorithms for placing geomagnetically induced current blocking devices. Electric Power Systems Research.
Volume 234, September 2024, 110645. DOI:10.1016/j.epsr.2024.110645.
2. Tyler H. Chang, Layne T. Watson, Sven Leyffer, Thomas C. H. Lux, Hussain M. J. Almohri. Remark on Algorithm 1012: Computing projections with large data sets. ACM Trans. Math. Softw. April 2024.
3. Jongeun Kim , Sven Leyffer , Prasanna Balaprakash. Learning Symbolic Expressions: Mixed-Integer Formulations, Cuts, and Heuristics. INFORMS J. Computing, 2023. DOI:10.1287/ijoc.2022.0050
4. A. Attia, S. Leyffer, and T. Munson. Stochastic Learning Approach to Binary Optimization for Optimal Design of Experiments. SIAM J. Scientific Computing, 44(2), 2022. DOI:10.1137/21M1404363.
5. Anna Thuenen, Sven Leyffer, and Sebastian Sager. State Elimination for Mixed-Integer Optimal Control of PDEs by Semigroup Theory. Optimal Control Applications and Methods, 2022. DOI:10.1002/
6. Wenjing Wang, Mohan Krishnamoorthy, Juliane Muller, Stephen Mrenna, Holger Schulz, Xiangyang Ju, Sven Leyffer, and Zachary Marshall. BROOD: Bilevel and Robust Optimization and Outlier Detection
for Efficient Tuning of High-Energy Physics Event Generators. SciPost Phys. Core 5, 001, DOI:10.21468/SciPostPhysCore.5.1.001, 2022.
7. Ryan H. Vogt, Sven Leyffer, and Todd Munson. A Mixed-Integer PDE-Constrained Optimization Formulation for Electromagnetic Cloaking. SIAM J. Scientific Compututing, 44(1), B29–B50. January 2022.
DOI:10.1137/20M1315993 Supplementary Material: A Mixed-Integer PDE-Constrained Optimization Formulation for Electromagnetic Cloaking
8. Mirko Hahn, Sven Leyffer, and Sebastian Sager. Binary Optimal Control by Trust-Region Steepest Descent. Mathematical Programming (online first). DOI:10.1007/s10107-021-01733-z. January 2022.
9. Fabian Gnegel, Armin Fügenschuh, Michael Hagel, Sven Leyffer, and Marcus Stiemer. A Solution Framework for Linear PDE-Constrained Mixed-Integer Problems. Mathematical Programming 118:695-728.
DOI010.1007/s10107-021-01626-1, 2021.
10. Selin Aslan, Zhengchun Liu, Viktor Nikitin, Tekin Bicer, Sven Leyffer, and Doğa Gürsoy. Joint ptycho-tomography with deep generative priors. Machine Learning: Science and Technology, 2 045017,
2021. DOI:10.1088/2632-2153/ac1d35.
11. Ron Shepard, Scott R. Brozell, Jeffrey Larson, Paul Hovland, and Sven Leyffer. Wave function analysis with a maximum flow algorithm. Molecular Physics, 119:13(e1861351), 2021. DOI:10.1080/
12. Noam Goldberg, Steffen Rebennack, Youngdae Kim, Vitaliy Krasko, and Sven Leyffer. MINLP formulations for continuous piecewise linear function fitting. Computational Optimization and Applications
(2021). DOI:10.1007/s10589-021-00268-5.
13. Sven Leyffer, Paul Manns, and Malte Winckler. Convergence of Sum-Up Rounding Schemes for the Electromagnetic Cloak Problem. Computational Optimization and Applications (2021). DOI:10.1007/
14. Ashutosh Mahajan, Sven Leyffer, Jeff Linderoth, James Luedtke, and Todd Munson. Minotaur: A Mixed-Integer Nonlinear Optimization Toolkit. Mathematical Programming Computation, November 2020.
15. Anthony P. Austina, Mohan Krishnamoorthy, Sven Leyffer, Stephen Mrenna, Juliane Muller, and Holger Schulz. Multivariate Rational Approximation. Computer Physics Communications, October 2020.
16. Meenarli Sharma, Mirko Hahn, Sven Leyffer, Lars Ruthotto, and Bart van Bloemen Waanders. Inversion of Convection-Diffusion Equation with Discrete Sources. Optimization and Engineering, online
first, July 2020. doi.org/10.1007/s11081-020-09536-5
17. Selin Aslan, Zhengchun Liu, Viktor Nikitin, Tekin Bicer, Sven Leyffer, and Doga Gursoy. Deep Priors for Ptycho-tomography. Microscopy and Microanalysis" 26 (S2) (2020): 2466-2466. DOI:10.1017/
18. Sven Leyffer and Charlie Vanaret. An Augmented Lagrangian Filter Method. Mathematical Methods of Operations Research, 92(2), 343-376, 2020. DOI:10.1007/s00186-020-00713-x.
19. Sven Leyffer, Matt Menickelly, Todd Munson, Charlie Vanaret, and Stefan M. Wild. Nonlinear Robust Optimization. INFOR: Information Systems and Operational Research, 58(2):342-373, 2020.
20. Youngdae Kim, Sven Leyffer, and Todd Munson. MPEC methods for bilevel optimization problems. Argonne National Laboratory, MCS Division Preprint, ANL/MCS-P9195-0719. July 2019. to appear in
"Bilevel optimization: advances and next challenges," S. Dempe and A. Zemkoho, editors, 2020.
21. Martinek, Janna, Wagner, Michael, Zolan, Alexander, Boyd, Matthew, Newman, Alexandra, Morton, David, Leyffer, Sven, Larson, Jeffrey, and USDOE Office of Energy Efficiency and Renewable Energy.
Design, Analysis, and Operations Toolkit (DAO-Tk). Computer software. USDOE Office of Energy Efficiency and Renewable Energy (EERE), Solar Energy Technologies Office (EE-4S) (2019). DOI:10.11578/
22. Viktor Nikitin, Selin Aslan, Yudong Gao, Tekin Bicer, Sven Leyffer, Rajmund Mokso, and Doga Gursoy. Photon-limited ptychography of 3D objects via Bayesian reconstruction. OSA Continuum 2
(10):2948-2967 (2019). DOI:10.1364/OSAC.2.002948
23. Wendy Di, Si Chen, Doga Gursoy, Tatjana Paunesku, Sven Leyffer, Stefan Wild, and Stefan Vogt. Optimization-based simultaneous alignment and reconstruction in multi-element tomography. Optics
Letters 44(17):4331-4334 (2019) DOI:10.1364/OL.44.004331
24. Anthony P. Austin, Zichao Wendy Di, Sven Leyffer, Stefan M. Wild. Simultaneous Sensing Error Recovery and Tomographic Inversion Using an Optimization-based Approach. SIAM Journal on Scientific
Computing Vol. 41, Issue 3, pp.B497--B521, 2019.
25. Selin Aslan, Viktor Nikitin, Daniel J. Ching, Tekin Bicer, Sven Leyffer, and Doğa Gürsoy. Joint ptycho-tomography reconstruction through alternating direction method of multipliers. Optics
Express Vol. 26, Issue 6, pp.9128-9143 (2019) DOI:10.1364/OE.27.009128.
26. Zichao (Wendy) Di, Si Chen, Young Pyo Hong, Chris Jacobsen, Sven Leyffer, and Stefan M. Wild. Joint Reconstruction of X-Ray Fluorescence and Transmission Tomography. Optics Express Vol. 25, Issue
12, pp.13107-13124 (2017). DOI:10.1364/OE.25.013107.
27. Michael S. Scioletti, Alexandra M. Newman, Johanna K. Goodman, Alexander J. Zolan, and Sven Leyffer. Optimal Design and Dispatch of a System of Diesel Generators, Photovoltaics and Batteries for
Remote Locations. Optimization and Engineering DOI 10.1007/s11081-017-9355-4, May, 2017.
28. Fu Lin, Sven Leyffer, and Todd Munson. A Two-Level Approach to Large Mixed-Integer Programs with Application to Cogeneration in Energy-Efficient Buildings. Computational Optimization and
Applications 65(1):1-46, DOI: 10.1007/s10589-016-9842-0 (online first).
29. Zichao (Wendy) Di, Sven Leyffer, and Stefan M. Wild. Optimization-Based Approach for Tomographic Inversion from Multiple Data Modalities. SIAM J. Imaging Sciences, 9(1), 1-23, 2016.
30. Noam Goldberg, Sven Leyffer, and Ilya Safro. Optimal Response to Epidemics and Cyber Attacks in Networks. Networks, 66(2), June 2015. DOI: 10.1002/net.21619.
31. Ashish Tripathi, Sven Leyffer, Todd Munson, and Stefan M. Wild. Visualizing and Improving the Robustness of Phase Retrieval Algorithms. Procedia Computer Science, 51:815-824, 2015. DOI:10.1016/
32. Noam Goldberg and Sven Leyffer. Active Set Method for Second-Order Conic-Constrained Quadratic Programming. SIAM J. Optimization, 25(3), 2015. DOI:10.1137/140958025
33. Noam Goldberg, Youngdae Kim, Sven Leyffer, and Thomas Veselka. Adaptively Refined Dynamic Program for Linear Spline Regression. Computational Optimization and Applications, 58(3):523-541, 2014.
34. Rachel Mak, Mirna Lerotic, Holger Fleckenstein, Stefan Vogt, Stefan M. Wild, Sven Leyffer, Yefim Sheynkin, and Chris Jacobsen. Non-negative matrix analysis for effective feature extraction in
x-ray spectromicroscopy. Faraday Discussions, DOI: 10.1039/C4FD00023D, 2014, 171, 357.
35. Marc Snir, Robert W. Wisniewski, Jacob A. Abraham, Sarita V Adve, Saurabh Bagchi, Pavan Balaji, Jim Belak, Pradip Bose, Franck Cappello, Bill Carlson, Andrew A. Chien, Paul Coteus, Nathan A.
Debardeleben, Pedro Diniz, Christian Engelmann, Mattan Erez, Saverio Fazzari, Al Geist, Rinku Gupta, Fred Johnson Sriram Krishnamoorthy, Sven Leyffer, Dean Liberty, Subhasish Mitra, Todd Munson,
Rob Schreiber, Jon Stearley, and Eric Van Hensbergen. Addressing Failures in Exascale Computing. International Journal of High Performance Computing Applications 1094342014522573, first published
on March 21, 2014 as doi:10.1177/1094342014522573.
36. Siwei Wang, Jesse Ward, Sven Leyffer, Stefan M. Wild, Chris Jacobsen, and Stefan Vogt. Unsupervised Cell Identification on Multidimensional X-Ray Fluorescence Datasets. Journal of Synchrotron
Radiation, 21, 568-579, 2014. https://doi.org/10.1107/S1600577514001416
37. Pierre Bonami, Jon Lee, Sven Leyffer, Andreas Waechter. On branching rules for convex mixed-integer nonlinear optimization. Journal of Experimental Algorithmics (JEA), 18, 2013.
38. Kristopher A. Pruitt, Sven Leyffer, Alexandra M. Newman, and Robert J. Braun. A mixed-integer nonlinear program for the optimal design and dispatch of distributed generation systems. Optimization
and Engineering, DOI 10.1007/s11081-013-9226-6, August 2013.
39. Sven Leyffer and Ilya Safro. Fast Response to Infection Spread and Cyber Attacks on Large-Scale Networks, Journal of Complex Networks, DOI: 10.1093/comnet/cnt009, July 2013.
40. Christian Kirches and Sven Leyffer. TACO-A Toolkit for AMPL Control Optimization. Mathematical Programming Computation, 1-39, DOI 10.1007/s12532-013-0054-7, April 2013.
41. Pietro Belotti, Christian Kirches, Sven Leyffer, Jeff Linderoth, Jim Luedtke, and Ashutosh Mahajan. Mixed-Integer Nonlinear Optimization. Acta Numerica 22:1-131, 2013. DOI: http://dx.doi.org/
42. Christian Kirches, Hans Georg Bock, and Sven Leyffer, Modeling Mixed-Integer Constrained Optimal Control Problems in AMPL, Mathematical Modelling, 7(1):1124-1129, 2012.
43. Chungen Chen, Roger Fletcher, and Sven Leyffer. A Nonmonotone Filter Method for Nonlinear Optimization. Computational Optimization and Applications, 52(3):583-607, 2012. DOI: 10.1007/
44. Andres Guerra, Alexandra M. Newman, and Sven Leyffer. Concrete Structure Design Using Mixed-Integer Nonlinear Programming with Complementarity Constraints. SIAM J. Optimization, 21(3):833-863,
45. Mine Altunay, Sven Leyffer, Jeffrey T. Linderoth, and Zhen Xie. Optimal Security Response to Attacks on Open Science Grids. Computer Networks, 55(1):61-73, 2011. DOI:10.1016/j.comnet.2010.07.012.
46. Kumar Abhishek, Sven Leyffer, and Jeffrey T. Linderoth. FilMINT: An Outer-Approximation-Based Solver for Nonlinear Mixed Integer Programs. INFORMS Journal on Computing, 22: 555 - 567, 2010.
47. Sven Leyffer and Ashutosh Mahajan. Foundations of Constrained Optimization. In Wiley Encyclopedia of Operations Research and Management Science, editors Cochran, James J. and Cox, Louis A. and
Keskinocak, Pinar and Kharoufeh, Jeffrey P. and Smith, J. Cole. John Wiley & Sons, Inc. 2010. DOI: 10.1002/9780470400531.eorms0630.
48. Sven Leyffer and Ashutosh Mahajan. Software For Nonlinearly Constrained Optimization. In Wiley Encyclopedia of Operations Research and Management Science, editors Cochran, James J. and Cox, Louis
A. and Keskinocak, Pinar and Kharoufeh, Jeffrey P. and Smith, J. Cole. John Wiley & Sons, Inc. 2010. DOI: 10.1002/9780470400531.eorms0570.
49. Donald A. Hanson, Yaroslav Kryukov, Sven Leyffer, and Todd S. Munson. Optimal control model of technology transition. International Journal of Global Energy Issues, 33(3-4):154-175, 2010. DOI:
50. Fengqi You and Sven Leyffer, Oil Spill Response Planning with MINLP, SIAG/OPT Views-and-News, 21(2):1-8, 2010.
51. Fengqi You and Sven Leyffer, Mixed-Integer Dynamic Optimization for Oil-Spill Response Planning with Integration of a Dynamic Oil Weathering Model, Preprint ANL/MCS-P1794-1010. AIChe Journal, 57
(12):3555–3564, 2011. DOI: 10.1002/aic.12536.
52. Ryan Miller, Zhen Xie, Sven Leyffer, Michael Davis, Stephen Gray. Surrogate-Based Modeling of the Optical Response of Metallic Nanostructures, J. Phys. Chem. C, 114 (48), 20741-20748, 2010.
53. Haw-ren Fang, Sven Leyffer, and Todd S. Munson. A Pivoting Algorithm for Linear Programs with Complementarity Constraints. Optimization Methods and Software, 27(1):89-114, 2012. DOI: 10.1080/
54. Kumar Abhishek, Sven Leyffer, and Jeffrey T. Linderoth. Modeling without Categorical Variables: A Mixed-Integer Nonlinear Program for the Optimization of Thermal Insulation Systems. Optimization
and Engineering, 11(2):185-212, 2010.
55. S. Leyffer and T. S. Munson. Solving multi-leader-common-follower games. Optimization Methods and Software, 25(4):601-623, 2010.
56. Joana Maria, Tu T. Truong, Jimin Yao, Tae-Woo Lee, Ralph G. Nuzzo, Sven Leyffer, Stephen K. Gray, and John A. Rogers. Optimization of 3D Plasmonic Crystal Structures for Refractive Index Sensing.
Journal of Physical Chemistry C, 113 (24):10493–10499, 2009.
57. S. Leyffer. A Complementarity Constraint Formulation of Convex Multiobjective Optimization Problems. INFORMS Journal on Computing, 21(2):257-267, 2009.
58. M.P. Friedlander and S. Leyffer. Global and Finite Termination of a Two-Phase Augmented Lagrangian Filter Method for General Quadratic Programs. SIAM Journal on Scientific Computing, 30
(4):1706-1729, 2008.
59. R. Fletcher, S. Leyffer and Ph. L. Toint. A Brief History of Filter Methods. SIAG/Optimization Views-and-News, 18(1):2-12, 2007.
60. B. Addis and S. Leyffer. A trust-region algorithm for global optimization. Computational Optimization and Applications, 35(3):287-304, 2006.
61. S. Leyffer, G. Lopez-Calva and J. Nocedal. Interior methods for mathematical programs with complementarity constraints. SIAM Journal on Optimization, 17(1): 52–77, 2006.
62. Y. Chen, B. F. Hobbs, S. Leyffer, and T. S. Munson. Leader-follower equilibria for electric power and NOx allowances markets. Computational Management Science, 3(4):307-330, 2006.
63. S. Leyffer. Complementarity constraints as nonlinear equations: Theory and numerical experience. In S. Dempe and V. Kalashnikov, editors, Optimization and Multivalued Mappings, pages 169–208.
Springer, 2006.
64. R. Fletcher, S. Leyffer, D. Ralph, and S. Scholtes. Local convergence of SQP methods for mathematical programs with equilibrium constraints. SIAM Journal Optimization, 17(1):259–286, 2006.
65. S. Leyffer. The penalty interior point method fails to converge. Optimization Methods and Software, 20(4-5):559-568, 2005.
66. N. I. M. Gould, S. Leyffer, and Ph. L. Toint. A multidimensional filter algorithm for nonlinear equations and nonlinear least squares. SIAM J. Optimization, 15:17–38, 2004.
67. R. Fletcher and S. Leyffer. Solving mathematical program with complementarity constraints as nonlinear programs. Optimization Methods and Software, 19(1):15–40, 2004.
68. J. S. Pang and S. Leyffer. On the global minimization of the value-at-risk. Optimization Methods and Software, 19(5):611–631, 2004.
69. N. I. M. Gould and S. Leyffer. An introduction to algorithms for nonlinear optimization. In J. F. Blowey, A. W. Craig, and T. Shardlow, Frontiers in Numerical Analysis, pages 109-197. Springer
Verlag, Berlin, 2003.
70. A. Altay-Salih, M. C. Pinar, and S. Leyffer. Constrained nonlinear programming for volatility estimation with GARCH models. SIAM Review, 45(3):485 – 503, 2003.
71. R. Fletcher and S. Leyffer. Filter-type algorithms for solving systems of algebraic equations and inequalities. In G. di Pillo and A. Murli, editors, High Performance Algorithms and Software for
Nonlinear Optimization, pages 259–278. Kluwer, 2003.
72. S. Leyffer. Mathematical programs with complementarity constraints. SIAG/OPTViews-and-News, 14(1):15–18, 2003.
73. R. Fletcher, N. I. M. Gould, S. Leyffer, Ph. L. Toint, and A. W¨achter. Global convergence of trust-region SQP-filter algorithms for general nonlinear programming. SIAM J. Optimization, 13
(3):635–659, 2002.
74. R. Fletcher, S. Leyffer, and Ph. L. Toint. On the global convergence of a filter-SQP algorithm. SIAM J. Optimization, 13(1):44–59, 2002.
75. R. Fletcher and S. Leyffer. Nonlinear programming without a penalty function. Mathematical Programming, 91:239–270, 2002.
76. J.-P. Goux and S. Leyffer. Solving large MINLPs on computational grids. Optimization and Engineering, 3:327–346, 2002.
77. S. Leyffer. Generalized outer approximation. In C. A. Floudas and P. M. Pardalos, editors, Encyclopedia of Optimization, volume 2, pages 247–254. Kluwer, 2001.
78. S. Leyffer. Integrating SQP and branch-and-bound for mixed integer nonlinear programming. Computational Optimization & Applications, 18:295–309, 2001.
79. H. Skrifvars, S. Leyffer and T. Westerlund. Comparison of Certain MINLP Algorithms When Applied to a Model Structure Determination and Parameter Estimation Problem, Computers & Chemical
Engineering 22(12), pp. 1829-1835, 1998.
80. R. Fletcher and S. Leyffer. Numerical Experience with lower bounds for MIQP branch--and--bound, SIAM J. Optimization 8(2), pp. 604-616, 1998.
81. R. Fletcher A. Grothey and S. Leyffer. Computing sparse Hessian and Jacobian approximations with optimal hereditary properties, in Large-Scale Optimization with Applications, Part II: Optimal
Design and Control, editors L.T. Biegler, T.F. Coleman, A.R. Conn and F.N. Santosa, 1997.
82. R. Fletcher and S. Leyffer. Solving Mixed Integer Nonlinear Programs by outer approximation, Mathematical Programming 66, pages 327--349, 1994
Conference Proceedings
1. Mohan Krishnamoorthy, Holger Schulz, Xiangyang Ju, Wenjing Wang, Sven Leyffer, Zachary Marshall, Stephen Mrenna, Juliane Müller and James B. Kowalkowski. Apprentice for Event Generator Tuning. In
25th International Conference on Computing in High Energy and Nuclear Physics (CHEP 2021). EPJ Web Conf. Volume 251, August 2021. DOI:doi.org/10.1051/epjconf/202125103060
2. Sven Leyffer, Bart van Bloemen Waanders, Mirko Hahn, Todd Munson, Lars Ruthotto, Meenarli Sharma, Ryan Vogt. Mixed-Integer PDE-Constrained Optimization. Oberwolfach Report 2019-26, June 2019.
3. Anthony Austin, Sven Leyffer, Steven Mrenna, Juliane Mueller, and Holger Schulz. Teaching PROFESSOR new math. CHEP 2018. 23rd International Conference on Computing in High Energy and Nuclear
Physics, 2018.
4. Zhengchun Liu, Rajkumar Kettimuthu, Sven Leyffer, Prashant Palkar, and Ian Foster. A mathematical programming- and simulation-based framework to evaluate cyberinfrastructure design choices. 13th
IEEE International Conference on eScience, Auckland NZ, October 24-17, 2017. Published in thirteenth IEEE eScience Conference, IEEE.
5. Mike Fagan, Jeremy Schlachter, Kazutomo Yoshii, Sven Leyffer, Krishna Palem, Marc Snir, Stefan M. Wild, and Christian Enz. Overcoming the Power Wall by Exploiting Inexactness and Emerging COTS
Architectural Features. IEEE SOCC 2016.
6. Sven Leyffer, Pelin Cay, Drew Kouri, and Bart van Bloemen Waanders, Mixed-Integer PDE-Constrained Optimization. Argonne National Laboratory, MCS Division Preprint ANL/MCS-P5429-1015, November
2015. Oberwolfach Reports (OWR).
7. Preeti Malakar, Venkatram Vishwanath, Todd Munson, Christopher Knight, Mark Hereld, Sven Leyffer, Michael Papka, Optimal Scheduling of In-Situ Analysis for Large-Scale Scientific Simulations.
SuperComputing, November 2015.
8. Yuri Alexeev, Sheri Mickelson, Sven Leyffer, Robert Jacob, and Anthony Craig. The Heuristic Static Load-Balancing Algorithm Applied to CESM. In SuperComputing 2013, 2013.
9. Siwei Wang, Jesse Ward, Sven Leyffer, Stefan Wild, Chris Jacobsen, and Stefan Vogt. Unsupervised cell identification on multidimensional X-ray fluorescence datasets. In ACM SIGGRAPH 2013 Posters
(SIGGRAPH '13). ACM, New York, NY, USA. DOI=10.1145/2503385.2503481, 2013.
10. Noam Goldberg, Sven Leyffer, and Todd Munson. A New Perspective on Convex Relaxations of Sparse SVM, in Proceedings of SDM 2013, J. Ghosh, Z. Obradovic, C. Kamath, and S. Partasarthy (eds.), pp.
450-457, SIAM, 2013.
11. Yuri Alexeev, Ashutosh Mahajan, Sven Leyffer, Graham Fletcher, and Dmitri G. Fedorov. Heuristic Static Load-Balancing Algorithm Applied to the Fragment Molecular Orbital Method, SC12, November
10-16, 2012, Salt Lake City, Utah, USA.
12. Victor M. Zavala, Jianhui Wang, Sven Leyffer, Emil M. Constantinescu, Mihai Anitescu, and Guenter Conzelmann. Proactive Energy Management for Next-Generation Building Systems. SimBuild 2010,
August 11-13, 2010.
13. Sven Leyffer Experiments with MINLP Branching Techniques. European Workshop on Mixed Integer Nonlinear Programming, April 2010.
14. Wei Guan, Alexander Gray, and Sven Leyffer. Mixed-Integer Support Vector Machine, OPT 2009: 2nd NIPS Workshop on Optimization for Machine Learning, 2009.
15. Sven Leyffer, Jeff Linderoth, James Luedtke, Andrew Miller, and Todd Munson. Applications and Algorithms for Mixed Integer Nonlinear Programming. Journal of Physics: Conference Series,
180:012014, SciDAC 2009.
16. S. Leyffer. The Return of the Active Set Method, Oberwolfach Reports 2(1), 2005.
17. J.P. Bardhan, J.H. Lee, M.D. Altman, S. Leyffer, S. Benson, B. Tidor and J.K. White. Biomolecule Electrostatic Optimization with an Implicit Hessian, Nanotech 2004 Vol. 1, 2004.
White Papers
1. Sven Leyffer, Daniel R. Reynolds, Daniel M Tartakovsky, and Carol S. Woodward. Control and Design of Multiscale Systems, DOE-ASCR Applied Mathematics PI Meeting White Paper, September 2017.
2. Mihai Anitescu, Sven Leyffer, Todd Munson, and Stefan Wild. Design, Optimization, and Control of Complex Interconnected Systems (DocSis), DOE-ASCR Applied Mathematics PI Meeting White Paper,
September 2017.
3. Di, Zichao, Sven Leyffer, M. Otten and Stefan Wild. Department of Energy, [Exposing Latent Hierarchies for Large-Scale Design and Discovery https://doi.org/10.6084/m9.figshare.5339416],
September, 2017.
Blogs and News
1. Jon Lee and Sven Leyffer (Eds). Mixed Integer Nonlinear Programming, in The IMA Volumes in Mathematics and its Applications, Vol. 154, Springer, 2012.
Edited Volumes
Google Scholar Profile
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Why receivers and accumulators in refrigeration cycles?
More refrigerant leads to higher pressures
A refrigeration cycle is a closed system with constant total fluid mass, apart from unwanted leakages. Like any closed thermodynamic system, the mass of the fluid stored in it has a direct influence
on its state, described by state variables such as pressure and temperature. Just like a bicycle tire, the pressure increases when more mass is added.
A special feature of the refrigeration cycle is that fluid is not only present as gas, as in a bicycle tire. Instead, there are both liquid and gaseous phases at different points in the cycle. This
makes it a bit more complicated, you can even construct cases where the pressure initially drops even though you add refrigerant mass. But as a general rule of thumb, more mass means higher
pressures. It should be mentioned again that resulting pressure levels depend not only on the refrigerant charge, but also on all other degrees of freedom listed above.
The two pressure levels (evaporation and condensing) have a significant influence on the efficiency of a refrigeration cycle. This leads to the fact that for given boundary conditions (especially
secondary inlet temperatures) there is an optimal refrigerant charge at which energy efficiency is at its maximum. Unfortunately, this optimal refrigerant charge also changes when boundary conditions
change. This means that a refrigerant cycle that is optimally filled at 20°C outside temperature is anything but optimally filled at 30°C. In this case, the energy efficiency is unnecessarily low.
Self-regulating buffer vessels
In order to optimize efficiency of refrigeration cycles under varying boundary conditions, buffer vessels are often installed to store not needed refrigerant and release it when needed.
The basic functional principle is separation of gaseous and liquid phase by gravity. The buffer vessel can either be installed on the high-pressure side after the condenser, where they are called
receiver. Or on the low pressure side after the evaporator, where they are called accumulator. These components are designed to provide saturated liquid at receiver outlet and saturated vapor at
accumulator outlet. Both cycle variants are shown here:
Depending on the state of incoming refrigerant, the level of liquid refrigerant in the buffer vessel increases or decreases. And since liquid density is much greater compared to that of gas, stored
refrigerant mass changes with the filling level.
These buffer vessels are self-regulating in a closed refrigeration cycle, provided only one buffer vessel is installed.
The stationary state
In order to understand the self-regulating behavior, some considerations about the stationary state of the entire system are helpful. Due to the separation behaviour of buffer vessels, there is
always saturated liquid or saturated vapor at its outlet. If we assume a stationary state, total energy and mass of the buffer vessel must remain constant. That means the incoming mass flow rate is
equal to the outgoing mass flow rate. And – neglecting pressure losses – refrigerant state must be the same at inlet and outlet. From this follows directly:
• At receiver inlet and condenser outlet must be saturated liquid in a stationary state (exactly 0 K subcooling).
• At separator inlet and evaporator outlet must be saturated vapor in a stationary state (exactly 0 K superheating).
The following conclusion can be drawn from these considerations: By installing a buffer vessel in a refrigeration cycle, one loses the degree of freedom “refrigerant charge” and at the same time
fixes one of the unknowns superheating or subcooling to zero.
This shows, for example, that it makes little sense to use the expansion valve to control superheat if a suction-side accumulator is installed in a cycle.
Influence on dynamics
The previous considerations refer exclusively to the stationary state of a refrigeration cycle. But a buffer vessel also has a decisive influence on the dynamics.
When changing from one stationary operating point to another, for example by changing the external recooling temperature at the condenser, a shift of mass in the refrigeration cycle takes place.
Refrigerant is stored in or removed from the buffer vessel. This dynamic is strongly dependent on the operating point or, expressed mathematically, is strongly non-linear.
As an example we consider the case that mass has to be removed from a suction side accumulator. As always, we have saturated vapor at the outlet. The buffered refrigerant is present as a liquid. To
get it out of the accumulator, it must evaporate. However, the required evaporation enthalpy must be supplied somehow. If no significant heat flow enters the component from the outside, the only
possibility is to supply the required evaporation enthalpy via an increased enthalpy flow and thus superheated refrigerant at the inlet. However, the maximum possible superheating is limited by other
boundary conditions (e.g. minimum suction pressure). Therefore, the needed internal mass transfer can only take place very slowly. It is often the slowest and thus for the control design most
important dynamic in a refrigeration cycle.
Refrigeration cycle simulation
A receiver or accumulator has a variety of effects on a refrigeration cycle – both on the stationary operating points that are set and on the dynamics. In order to take these effects fully into
account when designing a cycle and controls, computer simulation is very helpful. With the model library TIL we at TLK offer a professional tool to perform such calculations. Please contact us if you
want to learn more about it! | {"url":"https://tlk-energy.de/blog-en/why-are-there-receivers-and-accumulators-in-refrigeration-cycles","timestamp":"2024-11-09T22:43:18Z","content_type":"text/html","content_length":"48180","record_id":"<urn:uuid:3f96d752-4915-4edb-a528-77b0848268cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00094.warc.gz"} |
Fall 2022
CMSC/Math/ENE 456 -- Fall 2022
• Lectures: Tuesday, Thursday 12:30-1:45 PM, IRB 0318
• Instructor: Daniel Gottesman (e-mail: dgottesm@umd.edu, office hours Tuesday 10:30-11:30 AM, Atlantic 3251)
• Teaching Assistants:
□ Amadeo David De La Vega Parra (e-mail: adelaveg@umd.edu, office hours Thursday 9:15-10:45 AM, AVW 4122)
□ Samira Goudarzi (e-mail: samirag@umd.edu, office hours Wednesday 2:00-3:30 PM, AVW 4122)
□ Mahathi Vempati (e-mail: mahathi@umd.edu, office hours Monday 1:30-2:30 PM, AVW 4122)
• Textbook: Katz and Lindell, Introduction to Modern Cryptography, 3rd ed.
It may be possible to join some of these office hours by Zoom. Please e-mail the person conducting the office hours in advance to arrange a Zoom office hour if needed.
Important Dates
• Mid-term: Thursday, October 20 (in class)
• Thanksgiving: Thursday, November 24 (no class)
• Last lecture: Thursday, December 8
• Final exam: Monday, Dec. 19, 1:30 - 3:30 PM (location TBA)
Slides and Homeworks
Problem Sets: (To be turned in on Gradescope)
Solution sets are available on ELMS roughly 1 week after the due date for the assignment.
If you are reading these slides before you see the lecture and you see a "Vote" on the slide, stop and think about your answer before proceeding. The point of those votes is to get you to think about
the material during the lecture.
Topics Covered
• Classical cryptography
• Modern private-key cryptography (including one-time pad, pseudorandom generators and functions, security definitions and proofs, DES, AES)
• Public key encryption (including purpose and applications, RSA)
• Authentication (including message authentication codes, digital signatures)
• Additional advanced topics, as time permits (possibilities include post-quantum cryptography, quantum key distribution, secure multiparty computation, homomorphic encryption, blockchain)
Learning Objectives
• Terminology, types, and techniques of cryptographic protocols
• What makes a protocol secure or insecure
• Basic understanding of particular protocols used in real world, such as AES and RSA.
Your grade will have 3 components:
• Problem sets (30%)
• Mid-term exam (30%)
• Final exam (40%)
Additional notes on grading and assignments:
• The problem sets will be available on this web page.
• The problem sets will be turned in on Gradescope.
• The problem sets will be a mix of theory-focused problems and programming assignments.
• The problem set grade will be determined by dropping the highest and lowest grades and then averaging the remaining scores.
• By default, the scores will not be curved. However, I may curve up the grades for any problem set or exam if I decide it was substantially harder than I expected. I will not curve down grades if
the assignment is easier than expected.
• For the problem sets, if you use any external material to solve it (other than the lectures and textbook), cite the source and indicate what you took from it.
• You may discuss problem sets with other students, but you must understand and write up your solution or code by yourself. If you do collaborate, indicate who you talked to on your assignment.
• Late problem sets will not be accepted unless an extension is granted by me or one of the TAs before the problem set is due.
□ Note that the extension must be granted before the deadline, not requested before the deadline. Be sure to leave enough time to get a response (24 hours should be sufficient).
□ Extension requests should specify a valid reason and how long an extension you are requesting. Medical, religious, family emergency are examples of valid reasons (not an exhaustive list). "I
have an assignment due in another class" is not a valid reason: Plan ahead!
□ Maximum extension is 1 week, so that we can distribute solutions. If you have a valid reason for a longer extension, discuss with me.
General Information
• Lectures will be recorded and available through the course's page on ELMS. However, I strongly recommend that you attend class and not rely on the recordings to follow the class.
• There will be a Piazza for asking questions on the class. Unless you have a question that is very specific to you personally, please use the Piazza to ask questions. This includes questions about
both the content and administration of the course. | {"url":"https://www.cs.umd.edu/class/fall2022/cmsc456/","timestamp":"2024-11-10T03:31:47Z","content_type":"text/html","content_length":"11189","record_id":"<urn:uuid:b377d3c1-162d-4e18-8cb2-172953ec0a57>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00149.warc.gz"} |
First Angle vs Third Angle
Collection of 2D drawings which enables you to have a complete representation of an object is called orthographic projection. Collection of 2D drawings consist of six orthographic views (Top, Bottom,
Right, Left, Front and Back view) also called as six principle views.
From these six orthographic views front, right side and top view are most commonly used to represent the orthographic projection of an object. For getting the orthographic views from a 3D object
there are two main types of projections used one is First angle projection and other is Third angle projection.
Main objective of both of these projections angle is to provide detail 2D drawings of 3D object but there are some differences in the way one can get the projects by using one on these methods.
Following is a comparison of first angle vs third angle and a comprehensive differences between First angle and third angle projection.
Quadrant (first angle vs third angle)
To start getting projection first divides the plane into four quadrants. For the first angle projection object is paced in the first quadrant and for the third angle projection object is placed is
third quadrant.
Quadrants of First Angle vs Third Angle Projection
Quadrants of First Angle Projection
Quadrants of Third Angle Projection
Object Placement (difference between first and third angle projection)
To get first angle projection object is place in between the plane of projection and observer and for the third angle projection plane of projection is placed between the object and observer.
Object Placement of First Angle and Third Angle Projection
State of Projection Plane (first and third angle projection)
In first angle of projection plane of projection is taken solid while in third angle of projection it is taken as transparent.
Views Sequence (First Angle vs Third Angle)
In first angle projection right view come in left of front view and top view come at the bottom of front view while in third angle projection right view come on right side of front view and top view
comes at the top of front view.
View Sequence of First Angle vs Third Angle Projection
Symbols First Angle vs Third Angle
Symbol of First Angle vs Third Angle Projection
First Angle vs Third Angle
5 comments:
1. First Angle difination
2. Thanks
3. Thanks
4. THANK YOU SO MUCH...
5. this is one of the best ways to remember, first angle projection - planes are assumed opaque, third angle projection - planes are assume transparent | {"url":"https://www.green-mechanic.com/2016/01/difference-between-first-third-angle.html","timestamp":"2024-11-11T17:52:13Z","content_type":"text/html","content_length":"71359","record_id":"<urn:uuid:1572a608-60b1-4293-88a4-2b09c24ff1d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00249.warc.gz"} |
How to Use SUBTOTAL with SUMIF in Excel
To use SUBTOTAL with SUMIF in Excel, you need to follow these steps:
1. First, create a helper column in which you calculate the subtotals for each group of data you want to sum up. To do this, use the SUBTOTAL function in the following format:
=SUBTOTAL(9, range)
Where 9 is the function number for SUM in the SUBTOTAL function, and range is the range of cells you want to sum up.
2. Next, use the SUMIF function to sum up the values in the helper column based on your given criteria. The SUMIF function should be in the following format:
=SUMIF(range, criteria, sum_range)
Where range is the range of cells you want to apply the criteria to, criteria is the condition you want to match, and sum_range is the range of cells containing the subtotals from the helper column.
Let's say you have the following data in Excel:
A B C
1 Group Value Subtotal
2 A 10
3 A 20
4 B 30
5 A 40
6 B 50
You want to calculate the subtotal for each group (A and B) and then use SUMIF to sum up the subtotals for group A only.
1. First, create a helper column in column C to calculate the subtotals for each group. Use the SUBTOTAL function in cell C2:
=SUBTOTAL(9, B$2:B2)
2. Drag this formula down to fill the other cells in column C. Your data will now look like this:
A B C
1 Group Value Subtotal
2 A 10 10
3 A 20 30
4 B 30 30
5 A 40 70
6 B 50 80
3. Now use the SUMIF function to sum up the subtotals for group A only. In a new cell (e.g., E2), enter the following formula:
=SUMIF(A2:A6, "A", C2:C6)
4. Press Enter, and Excel will return the sum of the subtotals for group A, which is 70 in this example.
Remember to adjust the cell ranges according to your data set.
Did you find this useful? | {"url":"https://sheetscheat.com/excel/how-to-use-subtotal-with-sumif-in-excel","timestamp":"2024-11-09T10:20:56Z","content_type":"text/html","content_length":"13053","record_id":"<urn:uuid:02933942-20ec-42b6-af9b-b6ea2ba9b66c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00460.warc.gz"} |
BMI Calculator - toolverse
BMI Calculator
Calculate your BMI - Body Mass Index, the measure to determine if you're at a healthy weight.
Fill in the two fields above.
BMI Table
BMI Des
Less than 17 Very below ideal weight
Between 17 and 18.5 Below ideal weight
Between 18.5 and 24.9 Considered normal weight
Between 25.0 and 29.9 Above ideal weight
Between 30.0 and 39.9 Obesity level II
Greater than 40 Obesity level III
About BMI
BMI stands for Body Mass Index and it is a measure of body fat based on height and weight. BMI is important because it can help determine if a person is at a healthy weight or if they are overweight
or underweight. Knowing your BMI can also help assess your risk for certain health conditions, such as heart disease, diabetes, and some cancers. | {"url":"https://toolverse.app/bmi-calculator/","timestamp":"2024-11-08T18:35:32Z","content_type":"text/html","content_length":"13361","record_id":"<urn:uuid:d4cc7d10-a7cd-4b2f-b4fd-ab9be14041a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00727.warc.gz"} |
How good is Good-Turing for Markov samples?
Abstract: The Good-Turing (GT) estimator for the missing mass (i.e., total probability of missing symbols) in $n$ samples is the number of symbols that appeared exactly once divided by $n$. For
i.i.d. samples, the bias and squared-error risk of the GT estimator can be shown to fall as $1/n$ by bounding the expected error uniformly over all symbols. In this work, we study convergence of the
GT estimator for missing stationary mass (i.e., total stationary probability of missing symbols) of Markov samples on an alphabet $\mathcal{X}$ with stationary distribution $[\pi_x:x\in\cX]$ and
transition probability matrix (t.p.m.) $P$. This is an important and interesting problem because GT is widely used in applications with temporal dependencies such as language models assigning
probabilities to word sequences, which are modelled as Markov. We show that convergence of GT depends on convergence of $(P^{\sim x})^n$, where $P^{\sim x}$ is $P$ with the $x$-th column zeroed out.
This, in turn, depends on the Perron eigenvalue $\lambda^{\sim x}$ of $P^{\sim x}$ and its relationship with $\pi_x$ uniformly over $x$. For randomly generated t.p.ms and t.p.ms derived from New York
Times and Charles Dickens corpora, we numerically exhibit such uniform-over-$x$ relationships between $\lambda^{\sim x}$ and $\pi_x$. This supports the observed success of GT in language models and
practical text data scenarios. For Markov chains with rank-2, diagonalizable t.p.ms having spectral gap $\beta$, we show minimax rate upper and lower bounds of $1/(n\beta^5)$ and $1/(n\beta)$,
respectively, for the estimation of stationary missing mass. This theoretical result extends the $1/n$ minimax rate for i.i.d. or rank-1 t.p.ms to rank-2 Markov, and is a first such minimax rate
result for missing mass of Markov samples. We also show, through experiments, that the MSE of GT decays at a slower rate as the rank of the t.p.m increases.
Submission Length: Regular submission (no more than 12 pages of main content)
Assigned Action Editor: ~Bryon_Aragam1
Submission Number: 1843 | {"url":"https://openreview.net/forum?id=KokkP2nQ24","timestamp":"2024-11-12T19:36:56Z","content_type":"text/html","content_length":"42322","record_id":"<urn:uuid:b9f32271-510b-44a8-8415-24d68a0eaaf3>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00193.warc.gz"} |
Practices for simply solving pesky probability problems in grades 6-12
Practices for simply solving pesky probability problems in grades 6-12
By Susan A. Peters
Did you learn probability as a set of rules to be followed? If you did, the rules may not have made much sense to you. One of the main problems with probability is its counterintuitive nature, yet
approaching probability with rules does little to build intuition.
If we truly wish to have students persevere in solving problems – particularly problems in probability – then we need to provide them with the tools that position them for success. One way to do so
is by approaching many probability problems with what statistician Roxy Peck calls “hypothetical 1,000” tables.
The hypothetical 1,000 tables are two-way frequency tables for which we assume a population of 1,000, use given probability information to complete some cells in the table, use arithmetic to complete
the remaining cells in the table, and use the cell values in the tables to accurately estimate probabilities.
This approach can be used to address grade 7 probability standards related to compound events (7.SP) and high school probability standards related to conditional probabilities and compound events
(S.CP). It also engages students with the mathematical practices of problem solving (MP1) and attending to precision (MP6). Let’s look at some problems to examine this approach.
Typical Probability Problem and Solution
First, consider the following scenario. In 2017, the Pew Research Center published a report by Kenneth Olmstead and Aaron Smith with results from a 2016 survey of adult internet users to investigate
what the public knows about cybersecurity. The center used surveying methods known to produce representative samples and collected data about respondents, including education level. They also asked
respondents to answer questions about a variety of topics related to cybersecurity.
The study’s authors found that 35 percent of respondents were college graduates, and 65 percent of the college graduates knew that email is not encrypted by default. Among all of the respondents,
only 46 percent of respondents knew this fact.
Now answer the following question using the information from this scenario. What is the probability that an adult internet user would be both college educated and know that email is not encrypted by
How did you approach this problem? If you are like many people, you began by recording what you know and what you want to find, using symbols such as the following.
Let C represent college graduate.
Let E represent knowing email is not encrypted by default.
Then P(C) = 0.35, P(E) = 0.46, and P(E|C) = 0.65.
Find P(E and C).
At this point, you might consider different probability formulas that you may remember. There are two different multiplication rules: one for independent events – events in which the occurrence of
one event is not affected by knowledge about the other event – and one for dependent events.
E and C are not independent. We know that P(E) is not the same as P(E|C) from the information given to us, so we could use the rule for dependent events to solve the problem – P(E and C) = P(C)*P(E|
C) = 0.35*0.65 = 0.2275.
A Problem-Solving Tool for Typical Probability Problems
Apart from the multiplication rule, you could approach the problem by setting up a table using the given probabilities to consider results for a hypothetical 1,000 people. The scenario describes two
variables related to education and email knowledge, and each variable has two possible outcomes. A 2×2 table with these variables might look something like the following. With middle school students,
we likely would use only the words displayed in the table, but with high school students, we likely would use the symbols instead.
C Not C (C^C) Totals
College Graduate Not College Graduate
Know about email encryption
Not E (E^C)
Not know about email encryption
We begin completing the table by considering results for a hypothetical 1,000 people. We know that 35 percent of the people are college graduates. We can complete the totals row in our table because
we know there are 1,000*0.35 = 350 college graduates and 1,000 – 350 = 650 non-college graduates. We also know that 46 percent of the people know that email is not encrypted by default. We can
complete the totals column of our table because 1,000*0.46 = 460 people would know this information about email and 1,000 – 460 = 540 would not.
C Not C (C^C) Totals
College Graduate Not College Graduate
E 460
Know about email encryption
Not E (E^C) 540
Not know about email encryption
Totals 350 650 1,000
We also know that 65 percent of the 350 college graduates, or approximately 0.65*350 = 228 people, would know that email is not encrypted by default. That means that 350 – 228 = 122 college graduates
would not know this fact about email.
Notice that without even completing this table, we have enough information to determine the probability that an adult internet user would be both college educated and know that email is not encrypted
by default. From the table, we can see that 228 of the 1,000 adult internet users – or 22.8 percent – meet both conditions.
C Not C (C^C) Totals
College Graduate Not College Graduate
E 228 460
Know about email
Not E (E^C) 122 540
Not know about email encryption
Totals 350 650 1,000
We also have enough information to complete the table (see below) and to answer other probability questions. We can answer questions such as the probability that an adult internet user would be
college educated or know that email is not encrypted by default P(C or E) = (122 + 228 + 232)/1000 = 582/1000 = 0.582, the probability that an individual who is not a college graduate would know that
email is not encrypted by default [P(E given not C) = P(E│C^C ) = 232/650≈0.35], and the probability that an individual who does not know that email is not encrypted by default is not a college
graduate [P(not C given not E) = P(C^C│E^C) = 418/540≈0.774], among others.
C Not C (C^C) Totals
College Graduate Not College Graduate
E 228 232 460
Know about email
Not E (E^C) 122 418 540
Not know about email encryption
Totals 350 650 1,000
Attending to Precision
You might have noticed that we achieved a slightly different answer for the probability that an adult internet user would be both college educated and know that email is not encrypted by default from
using the 2×2 table than we did from using the formulas to solve the problem. We need greater precision in our answer from the 2×2 table.
How do we achieve greater precision? We simply increase the number of hypothetical people we consider. For this problem, the difference in our answers resulted from our approximated number of college
graduates who know about email encryption, and the differences disappear when we consider a hypothetical 10,000 people. In general, we would keep increasing the number of hypothetical people we
consider until we reach our desired level of precision.
C Not C (C^C) Totals
College Graduate Not College Graduate
E 2,275 2,325 4,600
Know about email
Not E (E^C) 1,225 4,175 5,400
Not know about email encryption
Totals 3,500 6,500 10,000
Middle School and High School Applications
Middle School
How might the hypothetical 1,000 approach be used in middle school? One of the standards in the Statistics and Probability domain for 7th-grade students relates to finding probabilities for compound
events: “Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.” We can use the approach to help students calculate probabilities for compound events.
A common problem given to middle school students in relation to this standard is to find the probability that when two coins are tossed, both coins will land on heads. We can use the hypothetical
1,000 approach to solve this problem by considering results for 1,000 tosses of two coins! We would begin with a table such as the following.
Coin 1 Totals
Heads Tails
Coin 2 Heads
Totals 1,000
Assuming that both coins are fair, we would expect half of each coin’s tosses to land on heads and half to land on tails. As a result, we can complete the Totals row and column in the table. Not only
can we complete the totals entries, however, we can complete the entire table because what remains is to now consider the outcome of 500 tosses for each coin.
Again, if each coin is fair, we would expect half of the tosses to land on heads and half to land on tails. From the completed table, then, we can see that of the 1,000 tosses of two coins, 250
result in both coins landing on heads, or 25 percent. Thus, the probability that both coins will land on heads is 1/4.
Coin 1 Totals
Heads Tails
Coin 2 Heads 250 250 500
Tails 250 250 500
Totals 500 500 1,000
Students can use the hypothetical 1,000 approach to solve many of the compound probability problems that we ask middle levels students to solve, although we would need to expand the table if more
than two outcomes are possible for either variable. The approach provides an intuitive means for students to calculate probabilities without memorizing formulas and use of this approach positions
students for success with meeting high school probability standards.
High School
How might the hypothetical 1,000 table be used in high school? One of the standards in the Conditional Probability and the Rules of Probability domain for high school students relates to using
two-way tables to find conditional probabilities: “Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table
as a sample space to decide if events are independent and to approximate conditional probabilities.”
We examined several examples of using a two-way table to find conditional probabilities when we first considered the hypothetical 1,000 approach. In addition to using the approach to find conditional
and compound probabilities, we can use the approach to develop the addition rule and the multiplication rule for students to gain greater understanding of the rules and to increase their probability
of success with solving problems using the rules.
To consider development of the rule for calculating conditional probabilities and the general multiplication rule, we will refer to the table we created using data for the variables of college
graduation and knowing about email encryption, displayed again below. Also, consider the probability that an individual who is not a college graduate would know that email is not encrypted by default
[P(E given not C) = P(E│C^C) = 232/650≈0.35].
From the table, the two values we use to calculate the probability are P(E and C^C) and P(C^C), and we see that P(E│C^C) = P(E and C^C)/P(C^C). If we use the table to calculate additional conditional
probabilities such as P(E│C) and P(C│E), we would see that P(E│C) = (P(E and C)/(P(C)) and P(C│E) = (P(E and C))/(P(E)).
From these probabilities, we can also see that P(E and C) = P(C)P(E│C) = P(E)P(C│E). Calculating additional probabilities using this table and others should lead students to generalize their
observations and develop the rule for calculating conditional probabilities and the General Multiplication rule: P(A and B) = P(A)P(B│A) = P(B)P(A|B).
C Not C (C^C) Totals
College Graduate Not College Graduate
E 228 232 460
Know about email
Not E (E^C) 122 418 540
Not know about email encryption
Totals 350 650 1,000
To consider development of the addition rule using these same data, remember that the probability that an adult internet user would be college educated or know that email is not encrypted by default
is P(C or E) = (122 + 228 + 232)/1000 = 582/1000. The values used in this calculation are in the table above.
Notice, however, that we could rewrite the probability as P(C or E) = (122 + 228 + 232)/1000 = 122/1000 + 228/1000 + 232/1000 = P(C and E^C) + P(C and E) + P(C^C and E) = P(C) + P(C^C and E). An
alternative way of writing P(C^C and E) is P(E) – P(C and E). The resulting formula then becomes P(C or E) = P(C) + P(E) – P(C and E). If we look at adding P(C) and P(E) from the table, we can see
that the value of 228 is added twice in our calculations, which why we need to subtract P(C and E). Although the addition rule, P(A or B) = P(A) + P(B) – P(A and B), might be more difficult for
students to develop, the image of the table and solving probability problems using the table likely will help them to remember the rule.
Conditions for Using the Hypothetical 1,000 Approach
When can we use the hypothetical 1,000 approach? In general, to use the approach with a 2×2 table, we need to know three of the probabilities associated with two events with two possible outcomes.
With respect to events C and E from our original example, we would need to know the probabilities of each event occurring [P(C) and P(E)] and one of the compound probabilities that one or both of the
events would occur [either P(C or E) or P(E and C)].
If, however, the two events are mutually exclusive or independent, we only would need to know the probability of each event occurring [P(C) and P(E)]. Technically, if the events are mutually
exclusive, we already know the probability of both events occurring because the two events cannot occur at the same time [P(E and C) = 0].
In the case of the two independent events, we can find the probability that both events will occur by using the multiplication rule [P(E and C) = P(E)P(C)]. Students should have sufficient
information to use the approach and find probabilities using the information provided in most traditional probability problems.
Probability does not need to be a word that evokes images of complicated formulas or that prompts nightmares. The hypothetical 1,000 approach shifts the focus from formulas to calculate probabilities
to the meanings of the probabilities being calculated. The approach arms students with a tool that they can use to persist in solving complex probability problems while providing opportunities for
students to attend to precision. The approach can be used to address standards at both the middle school and high school levels to allow students to naturally transition from solving relatively
simple probability problems to solving complex problems.
Susan A. Peters is an associate professor in the Department of Middle and Secondary Education at the University of Louisville. She teaches prospective middle and high school mathematics teachers and
is interested in statistics education and mathematics teacher education.
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Share This Story | {"url":"https://www.kentuckyteacher.org/subjects/mathematics/2017/09/practices-for-simply-solving-pesky-probability-problems-in-grades-6-12/","timestamp":"2024-11-10T21:07:31Z","content_type":"text/html","content_length":"98733","record_id":"<urn:uuid:69c2ce22-9776-4aed-8e97-88e8f53ccf5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00815.warc.gz"} |
MATLAB - Find the error on polynomial fit parameters of experimental data
• MATLAB
• Thread starter SK1.618
• Start date
In summary, the attached PDF provides information on calculating errors for fit parameters using polyfit and polyval. The function polyfit returns polynomial coefficients and a structure S, which
includes the triangular factor, degrees of freedom, and norm of the residuals. The function polyval uses this structure to generate error estimates, delta, which represents the standard deviation of
the error in predicting a future observation. If the coefficients are least squares estimates and the errors in the data are independent, normal, and have constant variance, the predictions will fall
within the range of y±delta at least 50% of the time. A helpful example can also be found on the polyfit page.
See attached PDF for details: How do I calculate errors on the fit parameters, p?
Check out the doc on polyfit and polyval. In particular,
[p,S] = polyfit(x,y,n) returns the polynomial coefficients p and a structure S for use with polyval to obtain error estimates or predictions. Structure S contains fields R, df, and normr, for the
triangular factor from a QR decomposition of the Vandermonde matrix of x, the degrees of freedom, and the norm of the residuals, respectively. If the data y are random, an estimate of the
covariance matrix of p is (Rinv*Rinv')*normr^2/df, where Rinv is the inverse of R. If the errors in the data y are independent normal with constant variance, polyval produces error bounds that
contain at least 50% of the predictions.
[y,delta] = polyval(p,x,S) uses the optional output structure S generated by polyfit to generate error estimates delta. delta is an estimate of the standard deviation of the error in predicting a
future observation at x by p(x). If the coefficients in p are least squares estimates computed by polyfit, and the errors in the data input to polyfit are independent, normal, and have constant
variance, then y±delta contains at least 50% of the predictions of future observations at x.
There is also a good example on the polyfit page.
FAQ: MATLAB - Find the error on polynomial fit parameters of experimental data
1. What is MATLAB?
MATLAB is a software platform commonly used by scientists, engineers, and analysts to conduct data analysis, visualization, and algorithm development. It allows for easy manipulation of data,
creation of complex mathematical models, and provides a user-friendly interface for programming.
2. How can I use MATLAB to find errors in polynomial fit parameters?
To find errors in polynomial fit parameters using MATLAB, you can use the built-in function polyfiterror. This function takes in the experimental data, as well as the polynomial degree, and returns
the estimated error in the polynomial fit parameters.
3. Can I plot the errors of the polynomial fit parameters in MATLAB?
Yes, you can plot the errors of the polynomial fit parameters in MATLAB using the function errorbar. This function takes in the x and y values of the data, as well as the error values, and plots them
as vertical lines on the data points.
4. How does MATLAB calculate the errors in polynomial fit parameters?
MATLAB uses a statistical method called least squares to calculate the errors in polynomial fit parameters. This method minimizes the sum of the squares of the differences between the data points and
the polynomial curve, and provides the best fit for the data.
5. What should I do if I encounter an error while using MATLAB to find errors in polynomial fit parameters?
If you encounter an error while using MATLAB, try checking your code for any mistakes or typos. You can also refer to MATLAB's documentation or seek help from the MATLAB community to troubleshoot the
issue. Additionally, make sure you have the correct data format and parameters specified for the function you are using. | {"url":"https://www.physicsforums.com/threads/matlab-find-the-error-on-polynomial-fit-parameters-of-experimental-data.669583/","timestamp":"2024-11-10T15:17:06Z","content_type":"text/html","content_length":"79188","record_id":"<urn:uuid:d2995770-0151-43bb-990c-86556a50bf9c>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00882.warc.gz"} |
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The Brainbox Tutorials has specially designed this online maths quiz for Class 10 students. This Class 10 Maths Quadratic Equations MCQ Mock Test covers all the important topics from the capter
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Data for "On-grid compressive sampling for spherical field measurements in acoustics" in The Journal of the Acoustical Society of America.
This dataset contains CSV files for the figures in the paper titled "On-grid compressive sampling for spherical field measurements in acoustics" in The Journal of the Acoustical Society of America.
In this paper, we derive a compressive sampling method for spherical harmonic/spherical wavefunction or Wigner D-function series with sparse coefficients. Applications of these sparse expansions
include spherical field measurements in acoustics and spherical near-field antenna measurements, to name a couple. The figures that this dataset is for are examples demonstrating the following:
example acoustic field coefficients in the spherical harmonic/spherical wavefunction basis; relationships between spherical harmonic/spherical wavefunction/Wigner D-function coefficient sparsity and
spatial Fourier coefficient sparsity; example compressive sampling reconstruction using our proposed compressive sampling method with and without noise; and comparisons between classical Nyquist
sampling and our proposed compressive sampling method.
About this Dataset
Updated: 2024-02-22
Metadata Last Updated: 2022-11-09 00:00:00
Date Created: N/A
Data Provided by:
Dataset Owner: N/A
Table representation of structured data
Title Data for "On-grid compressive sampling for spherical field measurements in acoustics" in The Journal of the Acoustical Society of America.
This dataset contains CSV files for the figures in the paper titled "On-grid compressive sampling for spherical field measurements in acoustics" in The Journal of the Acoustical Society
of America. In this paper, we derive a compressive sampling method for spherical harmonic/spherical wavefunction or Wigner D-function series with sparse coefficients. Applications of
Description these sparse expansions include spherical field measurements in acoustics and spherical near-field antenna measurements, to name a couple. The figures that this dataset is for are
examples demonstrating the following: example acoustic field coefficients in the spherical harmonic/spherical wavefunction basis; relationships between spherical harmonic/spherical
wavefunction/Wigner D-function coefficient sparsity and spatial Fourier coefficient sparsity; example compressive sampling reconstruction using our proposed compressive sampling method
with and without noise; and comparisons between classical Nyquist sampling and our proposed compressive sampling method.
Modified 2022-11-09 00:00:00
Publisher National Institute of Standards and Technology
Contact mailto:[email protected]
Keywords compressive sensing , compressive sampling , sparse signal processing , far-field pattern , near-field pattern , antenna characterization , Wigner D-functions , spherical harmonics ,
acoustic fields;
"identifier": "ark:\/88434\/mds2-2842",
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"contactPoint": {
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"title": "Data for \"On-grid compressive sampling for spherical field measurements in acoustics\" in The Journal of the Acoustical Society of America.",
"description": "This dataset contains CSV files for the figures in the paper titled \"On-grid compressive sampling for spherical field measurements in acoustics\" in The Journal of the Acoustical Society of America. In this paper, we derive a compressive sampling method for spherical harmonic\/spherical wavefunction or Wigner D-function series with sparse coefficients. Applications of these sparse expansions include spherical field measurements in acoustics and spherical near-field antenna measurements, to name a couple. The figures that this dataset is for are examples demonstrating the following: example acoustic field coefficients in the spherical harmonic\/spherical wavefunction basis; relationships between spherical harmonic\/spherical wavefunction\/Wigner D-function coefficient sparsity and spatial Fourier coefficient sparsity; example compressive sampling reconstruction using our proposed compressive sampling method with and without noise; and comparisons between classical Nyquist sampling and our proposed compressive sampling method.",
"language": [
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"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/supp_figure_1_sorted_coefficients_case_2a.csv",
"description": "This file contains the data for supplemental figure 1. It shows the relative magnitude in dB of the sorted Fourier and Wigner D function coefficients for case 2a as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "supp_figure_1_sorted_coefficients_case_2a.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/supp_figure_2_sorted_coefficients_case_2a.csv",
"description": "This file contains the data for supplemental figure 2. It shows the coefficient normalized error in dB of the sorted Fourier and Wigner D function coefficients for case 2a as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "supp_figure_2_sorted_coefficients_case_2a.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/supp_figure_3_sorted_coefficients_case_2b.csv",
"description": "This file contains the data for supplemental figure 3. It shows the relative magnitude in dB of the sorted Fourier and Wigner D function coefficients for case 2b as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "supp_figure_3_sorted_coefficients_case_2b.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/supp_figure_4_sorted_coefficients_case_2b.csv",
"description": "This file contains the data for supplemental figure 4. It shows the coefficient normalized error in dB of the sorted Fourier and Wigner D function coefficients for case 2b as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "supp_figure_4_sorted_coefficients_case_2b.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/supp_figure_5_sorted_coefficients_case_2c.csv",
"description": "This file contains the data for supplemental figure 5. It shows the relative magnitude in dB of the sorted Fourier and Wigner D function coefficients for case 2c as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "supp_figure_5_sorted_coefficients_case_2c.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_4a_SW_coefs_DirPat_CUBE_1098Hz.csv",
"description": "This file contains the data for figure 4a. It shows the magnitude of the spherical wavefunction coefficients for the DirPat CUBE driver 1 loudspeaker at 1098Hz. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_4a_SW_coefs_DirPat_CUBE_1098Hz.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_4b_SW_coefs_DirPat_CUBE_1400Hz.csv",
"description": "This file contains the data for figure 4b. It shows the magnitude of the spherical wavefunction coefficients for the DirPat CUBE driver 1 loudspeaker at 1400Hz. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_4b_SW_coefs_DirPat_CUBE_1400Hz.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/supp_figure_6_sorted_coefficients_case_2c.csv",
"description": "This file contains the data for supplemental figure 6. It shows the coefficient normalized error in dB of the sorted Fourier and Wigner D function coefficients for case 2c as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "supp_figure_6_sorted_coefficients_case_2c.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/supp_figure_7_sorted_coefficients_case_3a.csv",
"description": "This file contains the data for supplemental figure 7. It shows the relative magnitude in dB of the sorted Fourier and Wigner D function coefficients for case 3a as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "supp_figure_7_sorted_coefficients_case_3a.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/supp_figure_8_sorted_coefficients_case_3a.csv",
"description": "This file contains the data for supplemental figure 8. It shows the coefficient normalized error in dB of the sorted Fourier and Wigner D function coefficients for case 3a as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "supp_figure_8_sorted_coefficients_case_3a.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/supp_figure_9_sorted_coefficients_case_3b.csv",
"description": "This file contains the data for supplemental figure 9. It shows the relative magnitude in dB of the sorted Fourier and Wigner D function coefficients for case 3b as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "supp_figure_9_sorted_coefficients_case_3b.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/supp_figure_10_sorted_coefficients_case_3b.csv",
"description": "This file contains the data for supplemental figure 10. It shows the coefficient normalized error in dB of the sorted Fourier and Wigner D function coefficients for case 3b as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "supp_figure_10_sorted_coefficients_case_3b.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_4c_SW_coefs_DirPat_CUBE_1895Hz.csv",
"description": "This file contains the data for figure 4c. It shows the magnitude of the spherical wavefunction coefficients for the DirPat CUBE driver 1 loudspeaker at 1895Hz. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_4c_SW_coefs_DirPat_CUBE_1895Hz.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_3_data_s_D_verus_average_s_F_100_trials.csv",
"description": "This file contains the data for Figure 3, which shows the sorted concentrations of the Fourier basis sparsity as a function of Wigner D-function sparsity for different sparsity levels where the Wigner D-function coefficients are set to 1 at random positions. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_3_data_s_D_verus_average_s_F_100_trials.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_5a_sorted_coefficients_case_1a.csv",
"description": "This file contains the data for figure 5a. It shows the relative magnitude in dB of the sorted Fourier and Wigner D function coefficients for case 1a as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_5a_sorted_coefficients_case_1a.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_5b_sorted_coefficients_case_1a.csv",
"description": "This file contains the data for figure 5b. It shows the coefficient normalized error in dB of the sorted Fourier and Wigner D function coefficients for case 1a as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_5b_sorted_coefficients_case_1a.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_6a_sorted_coefficients_case_1b.csv",
"description": "This file contains the data for figure 6a. It shows the relative magnitude in dB of the sorted Fourier and Wigner D function coefficients for case 1b as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_6a_sorted_coefficients_case_1b.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_6b_sorted_coefficients_case_1b.csv",
"description": "This file contains the data for figure 6b. It shows the coefficient normalized error in dB of the sorted Fourier and Wigner D function coefficients for case 1b as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_6b_sorted_coefficients_case_1b.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_11_classical_Nyquist_RelativeError_vs_n_meas_with_noise.csv",
"description": "This file contains the data for figure 11. It shows the relative error (dB) for classical Fourier sampling as a function of sample grid density for cases 1a, 2a, and 3a from the paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_11_classical_Nyquist_RelativeError_vs_n_meas_with_noise.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_12a_RelativeError_vs_n_phys_meas_vs_grid_dens.csv",
"description": "This file contains the data for figure 12a. It shows the relative error (dB) for Fourier based compressive sampling as a function of sample grid density and number of measurements for case 1a from the paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_12a_RelativeError_vs_n_phys_meas_vs_grid_dens.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_12b_RelativeError_vs_n_phys_meas_vs_grid_dens.csv",
"description": "This file contains the data for figure 12b. It shows the relative error (dB) for Fourier based compressive sampling as a function of sample grid density and number of measurements for case 2a from the paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_12b_RelativeError_vs_n_phys_meas_vs_grid_dens.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_12c_RelativeError_vs_n_phys_meas_vs_grid_dens.csv",
"description": "This file contains the data for figure 12c. It shows the relative error (dB) for Fourier based compressive sampling as a function of sample grid density and number of measurements for cases 3a from the paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_12c_RelativeError_vs_n_phys_meas_vs_grid_dens.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_13_coherence_vs_sample_dense_vs_grid_dens.csv",
"description": "This file contains the data for figure 13. It shows the average coherence of the 2DDFT CS measurment matrix as a function of grid density and average sample number. The average is over 25 trials. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_13_coherence_vs_sample_dense_vs_grid_dens.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/supp_figure_11_sorted_coefficients_case_3c.csv",
"description": "This file contains the data for supplemental figure 11. It shows the relative magnitude in dB of the sorted Fourier and Wigner D function coefficients for case 3c as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "supp_figure_11_sorted_coefficients_case_3c.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/supp_figure_12_sorted_coefficients_case_3c.csv",
"description": "This file contains the data for supplemental figure 12. It shows the coefficient normalized error in dB of the sorted Fourier and Wigner D function coefficients for case 3c as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "supp_figure_12_sorted_coefficients_case_3c.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/README.txt",
"description": "This is a \"read me\" file that contains and overview of the dataset.",
"mediaType": "text\/plain",
"title": "README.txt"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_7a_sorted_coefficients_case_1c.csv",
"description": "This file contains the data for figure 7a. It shows the relative magnitude in dB of the sorted Fourier and Wigner D function coefficients for case 1c as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_7a_sorted_coefficients_case_1c.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_7b_sorted_coefficients_case_1c.csv",
"description": "This file contains the data for figure 7b. It shows the coefficient normalized error in dB of the sorted Fourier and Wigner D function coefficients for case 1c as described in the file header and paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_7b_sorted_coefficients_case_1c.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_8a_actual_vs_CS_field_C1a.csv",
"description": "This file contains the data for figure 8a. It shows the near-field reconstruction (relative magnitude in dB) using Fourier based compressive sensing for an acoustic field (case 1a from the paper). The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_8a_actual_vs_CS_field_C1a.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_8b_CS_field_RelativeError_C1a.csv",
"description": "This file contains the data for figure 8b. It shows the relative error (dB) of the near-field reconstruction using Fourier based compressive sensing for an acoustic field (case 1a from the paper). The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_8b_CS_field_RelativeError_C1a.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_9_rel_err_vs_n_phys_meas_Fourier_vs_WingerD.csv",
"description": "This file contains the data for figure 9. It shows the coefficient relative error as a function of measurement number for Fourier and Wigner D function based compressive sensing for acoustic fields (case 1a, 2a, 3a from the paper). The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_9_rel_err_vs_n_phys_meas_Fourier_vs_WingerD.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_10_RelativeError_vs_n_phys_meas_c1a_Fourier_vs_WingerD_grid_dens_1_to_4.csv",
"description": "This file contains the data for figure 10. It shows the coefficient relative error as a function of measurement number for Fourier and on-grid Wigner D function based compressive sensing for the acoustic fields in case 1a as the sampling grid density is increased. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_10_RelativeError_vs_n_phys_meas_c1a_Fourier_vs_WingerD_grid_dens_1_to_4"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_14a_RelativeError_vs_sample_dense_vs_grid_dens.csv",
"description": "This file contains the data for figure 14a. It shows the relative error (dB) for Fourier based compressive sampling as a function of sample grid density and sample density for cases 1a from the paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_14a_RelativeError_vs_sample_dense_vs_grid_dens.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_14b_RelativeError_vs_sample_dense_vs_grid_dens.csv",
"description": "This file contains the data for figure 14b. It shows the relative error (dB) for Fourier based compressive sampling as a function of sample grid density and sample density for cases 2a from the paper. The data is organized as described in the header (first row) of the csv file.",
"mediaType": "text\/csv",
"title": "Figure_14b_RelativeError_vs_sample_dense_vs_grid_dens.csv"
"downloadURL": "https:\/\/data.nist.gov\/od\/ds\/mds2-2842\/Figure_14c_RelativeError_vs_sample_dense_vs_grid_dens.csv",
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Fundamental Skills
As part of Pacific’s undergraduate and first professional degree graduation requirements, all students must satisfy two fundamental skills: quantitative analysis (math) and writing. These
requirements must be met before a student graduates with a bachelor’s degree or a first professional degree. Failure to make progress toward fulfilling Pacific’s fundamental skills requirements
during the first year of study is grounds for being placed on academic probation. Failure to satisfy the fundamental skills requirements by the end of four semesters of full-time study at the
University is grounds for academic disqualification.
Students can fulfill the math and writing requirements in one of four ways:
1. Completion of Pacific's highest level developmental skills course;
2. Completion of an appropriately articulated course at an accredited college or university;
3. Satisfactory performance on an approved, nationally administered examination; or
4. Satisfactory performance on Pacific's placement examinations.
Students with documented disabilities that directly affect their mastery of these skills or students concurrently enrolled in an approved English-as-a-Second-Language (ESL) Program of instruction in
reading and writing may seek a written extension of the deadline for demonstrating competence.
The Developmental Math Program consists of courses designed to help students be successful in all levels of math or quantitative reasoning courses.
University of the Pacific students are required to demonstrate fundamental competency in quantitative analysis (math). The requirement must be met before a student graduates with a bachelor’s degree
or a first professional degree. Failure to make progress toward fulfilling Pacific’s fundamental math skills requirements during the first year of study is grounds for being placed on academic
probation. Failure to satisfy the fundamental math skills requirements by the end of four semesters of full-time study at the University is grounds for academic disqualification.
To satisfy the University's quantitative analysis (math) fundamental skills requirement, a student must complete one of the following:
• SAT math score of 570 or above
• ACT math score of 24 or above
• SAT Math Subject Test Level 1 score of 540 or above
• SAT Math Subject Test Level 2 score of 520 or above
• AP Calculus AB score of 3 or higher
• AP Calculus BC score of 3 or higher
• AP Statistics exam score of 3 or higher
• IB Math HL (Higher Level) score of 4 or higher
• Pacific's Intermediate Algebra Math Placement Exam score of 56 or higher
• ALEKS PPL score of 61 or higher
• Successfully complete MATH 5 (Intermediate College Algebra) or MATH 35 (Elementary Statistical Inference) with a grade of C- or higher (or an equivalent course from another college or university
with a grade of C or better).
The Developmental Writing Program helps students build the writing skills required for success in college-level writing.
University of the Pacific students are required to demonstrate fundamental competency in writing. The requirement must be met before a student graduates with a bachelor’s degree or a first
professional degree. Failure to make progress toward fulfilling Pacific’s fundamental skills requirements during the first year of study is grounds for being placed on academic probation. Failure to
satisfy the fundamental skills requirements by the end of four semesters of full-time study at the University is grounds for academic disqualification.
To satisfy the University's fundamental skills writing requirement, a student must
• Score 550 or higher on the SAT writing exam
• Score 22 or higher on the ACT English/Writing exam
• Score 26 or higher on TOEFL, writing sub-score
• Score 7.5 or higher on IELTS, writing sub-score
• Score 3 or higher on an AP English language or literature exam
• Score 4 or higher on an IB English literature exam
• Complete WRIT 010 with a C- or higher
• Complete a transferable course equivalent to a College Writing Course with a C or higher
• Score 5.0 or higher on Pacific’s Writing Diagnostic Exam
Mathematics Courses
MATH 001. Pre-algebra and Lab. 3 Units.
This course is designed for students whose Mathematics Placement Test score indicates a need to review arithmetic skills and Pre-algebra material. Topics covered include fractions, decimals,
percents, basic area and volume formulas, signed numbers, use of variables in mathematical statements, translating statements in English to mathematical equations, solving linear equations and ratio
and proportion. The course is taught using a Personalized System of Instruction. Neither the course credit nor course grade applies towards graduation. Prerequisite is an appropriate test score or
permission of instructor.
MATH 003. Elementary Algebra and Lab. 3 Units.
Topics covered include signed numbers, linear equations, polynomials, factoring, algebraic fractions, radicals, quadratic equations, inequalities and systems of linear equations. This is an
introductory course for students with limited high school background in mathematics. This course is taught using a Personalized System of Instruction. This course is inappropriate for students who
have passed the Elementary Algebra placement exam or any higher level placement exam. Neither the course credit nor course grade applies towards graduation. Prerequisite: MATH 001 with a "C" or
better or an appropriate test score or permission of instructor.
MATH 004. Math Literacy for College. 3 Units.
This course is designed to help students develop mathematical reasoning skills and the foundational algebra skills needed to be successful in an introductory statistics or intermediate algebra
course. The topics in the course are selected around the central goals for developing numeracy, proportional reasoning, algebraic reasoning, and an understanding of functions. An emphasis is placed
on performing, explaining, and applying relevant skills to new situations. Problems are generally presented in context of real world situations. The course is not appropriate for students who have
placed into MATH 005 or above. There is no prerequisite for this course. Students passing this course with a C- or better are eligible to take MATH 005, 005E, 035, or 161.
MATH 005. Intermediate College Algebra. 3 Units.
This course is taught in a traditional lecture format. Topics covered in this course include the real number system, solution of linear equations and inequalities, word problems, factoring, algebraic
equations, exponents and radicals, quadratic equations, relations, functions, graphs, systems of equations and logarithmic and exponential functions. This course is not appropriate for students who
have passed the Intermediate Algebra placement test of any higher level test. Pass/No Credit (P/NC) grading option is not allowed for this course. A grade of C- or better is required to satisfy the
University’s Fundamental Skills requirement in quantitative analysis/math. Prerequisite: MATH 003 with a “C-“ or better or an appropriate test score or permission of instructor. (MATH)
MATH 005E. Intermediate College Algebra and Lab. 3 Units.
This course is taught using the emporium model in which students use technology to drive their learning in a lab setting with on-demand support from the instructor and tutors. Topics covered in this
course include the real number system, solution of linear equations and inequalities, word problems, factoring, algebraic equations, exponents and radicals, quadratic equations, relations, functions,
graphs, systems of equations and logarithmic and exponential functions. This course is not appropriate for students who have passed the Intermediate Algebra placement test of any higher level test.
Pass/No Credit (P/NC) grading option is not allowed for this course. A grade of C- or better is required to satisfy the University’s Fundamental Skills requirement in quantitative analysis/math.
Prerequisite: MATH 003 with a “C-“ or better or an appropriate test score or permission of instructor. (MATH)
MATH 007. Trigonometry and Lab. 2 Units.
Topics in this course include angle measure, trigonometric functions, applications of trigonometry, graphs of trigonometric functions, trigonometric identities, inverse functions and complex numbers.
This course is designed for students who have not studied trigonometry in high school. Prerequisites include a satisfactory score on the Intermediate Algebra placement test. This course is taught
using a Personalized System of Instruction and meets three hours per week. Pass/No credit (P/NC) grading option is not allowed for this course. Students who complete MATH 005 and MATH 007 with a C-
or better may enroll in MATH 051. Prerequisite: MATH 005 with a "C-" or better, an appropriate test score, or permission of instructor. (MATH)
MATH 011. Chair's Seminar. 1 Unit.
The learning objective of this course is for students to gain insight into the Math department and university resources. Throughout the semester students will be exposed to various facets of the Math
department and the university. Math professors will make presentations, introducing students to their research areas and upper division math courses they teach. Some of these presentations will
introduce students to opportunities to get involved in undergraduate research. Students will also be exposed to university resources including, but not limited to, the Math Tutoring Center, Career
Resource Center, Counselling and Psychological Services.
MATH 033. Elements of Calculus. 4 Units.
This course covers polynomial, rational, exponential and logarithmic functions as well as differentiation, integration and maxima/minima of functions of several variables. Elementary differential
equations are studied and applications to natural sciences, social sciences and other fields are covered. Credit is not given for this course if a students has received credit for MATH 051 or AP
credit in Calculus. Prerequisites: Two years of high school algebra and an appropriate score on either the Intermediate Algebra placement test or the Pre-Calculus placement test; or MATH 005 or MATH
041 with a "C-" or better. (GE3B, GEQR, MATH)
MATH 035. Elementary Statistical Inference. 3 Units.
Sampling, simple experimental designs, descriptive statistics, confidence intervals & hypothesis tests for means and proportions, Chi-square tests, linear & multiple regression, analysis of variance.
Use of statistical software and/or online statistical calculators. Credit is not given for this course if a student has received credit for MATH 037 or MATH 131 or has AP credit in statistics.
Prerequisite: MATH 004 or exemption by placement. GE IIIB. (GE3B, GEQR, MATH, PLAW)
MATH 037. Introduction to Statistics and Probability. 4 Units.
Students will develop mathematical tools for collecting, summarizing, analyzing, and drawing inferences from data. Topics covered include elements of descriptive statistics, such as graphs, tables,
measures of central tendency and dispersion; discrete and continuous probability models for experiments and sampling distributions including the normal, t-, and chi-square distributions; and basic
concepts of inferential statistics including confidence intervals, p-values, hypothesis tests for both one-and two-sample problems, ANOVA, and linear regression. The use of statistical software is
required. This course is not recommended for first semester freshmen. Credit will not given for this course if a student has received credit for MATH 035 or has AP credit in Statistics.
Prerequisites: MATH 033 or MATH 041 or MATH 045 or MATH 051 or MATH 053 with a "C-" or better or appropriate score on the calculus placement test. (GE3B, GEQR, MATH, PLAW)
MATH 039. Probability with Applications to Statistics. 4 Units.
Probability concepts in discrete and continuous spaces is explored in some depth as well as important probability models (e.g., binomial, Poisson, exponential, normal, etc.), mathematical expectation
and generating functions. Applications to statistical inference includes maximum likelihood, moment and least squares estimation. Confidence intervals and hypothesis testing is also covered. Credit
is not given for both MATH 039 and MATH 131. Prerequisite: MATH 053 with a "C-" or better. (GE3B)
MATH 041. Pre-calculus. 4 Units.
The algebraic and trigonometric concepts which are necessary preparation for Calculus I are studied. Topics include the real number system, algebraic, trigonometric, exponential and logarithmic
functions. Emphasis is on the function concept; graphing functions; solving equations, inequalities and linear systems; and applied problems. Credit for this course is not given if a student has AP
Calculus credit. Prerequisite: MATH 005 with a "C-" or better or an appropriate score on either the Intermediate Algebra placement test, the Pre-calculus placement test or the calculus placement
test. (GE3B, GEQR, MATH)
MATH 045. Introduction to Finite Mathematics and Calculus. 3 Units.
Applications of finite math & calculus to problems in business, economics, and related fields through the study of systems of equations, elementary functions, elementary linear programming, the
derivative, and the integral. Credit for this course is not given if a student has credit for MATH 051 or AP Calculus credit. Prerequisites: One of the following: (1) MATH 005 or MATH 041 with a
grade of C- or higher (2) Math Placement (3) Exemption from Math Placement. (GE3B, GEQR, MATH)
MATH 049. Introduction to Abstract Mathematics. 4 Units.
An introduction to the spirit and rigor of mathematics is the focus of the course. The content may vary with instructor, but the objective is to develop the skills required to read and write
mathematics and prove theorems. Concepts include elementary logic, sets and functions, cardinality, direct and indirect proofs, mathematical induction. Prerequisite: MATH 053 with a "C-" or better or
permission of the instructor.
MATH 051. Calculus I. 4 Units.
Students study differential calculus of algebraic and elementary transcendental functions, anti-derivatives, introductory definite integrals, and the Fundamental Theorem of Calculus. Applications
include the first and second derivative tests and optimization. Credit is not given for this course if a student has AP Calculus I credit. Prerequisites: MATH 007 or MATH 041 with a "C-" or better, a
score of 3 on either AP Calculus AB or BC exam, or an appropriate score on the placement test for calculus. (GE3B, GEQR, MATH)
MATH 053. Calculus II. 4 Units.
This course covers techniques and applications of integration, sequences and series, convergence of series, and Taylor Polynomials. Credit is not given for this course if a student has AP Calculus II
credit. (GE3B, GEQR, MATH)
MATH 055. Calculus III. 4 Units.
This course introduces multivariable calculus. Topics covered include vector geometry of the plane and Euclidean 3-space; differential calculus of real-valued functions of several variables, as well
as partial derivatives, gradient, max-min theory, quadratic surfaces, and multiple integrals. Prerequisite: MATH 053 with a "C-" or better or AP Math BC credit.
MATH 057. Applied Differential Equations I: ODEs. 4 Units.
Students study ordinary differential equations, first-order equations, separable and linear equations. Also covered are direction fields, second order linear equations with constant coefficients,
method of undetermined coefficients, laplace transforms, and unit impulse response and convolutions. Homogeneous systems of first order linear equations and matrix algebra determinants, eigenvalues,
eigenvectors are also studied. Existence and uniqueness theorems are discussed and calculators or computers are used to display solutions and applications. Prerequisites: MATH 055 with a "C-" or
better, or MATH 053 and MATH 075, both with a "C-" or better, or permission of instructor.
MATH 064. Ancient Arithmetic. 4 Units.
This course traces mathematical and historical developments throughout the ancient world, ending with the Scientific Revolution. Students will gain mathematical knowledge through the analysis of
historical problems and solution methods, while contextualizing these endeavors into a larger historical context. Students will read mathematical primary sources, and will learn to think about the
development of mathematical primary sources, and will learn to think about the development of mathematics as an intellectual pursuit over time. This course is cross-listed with HIST 066.
Prerequisite: Fundamental Skills.
MATH 072. Operations Research Models. 4 Units.
Operations Research (OR) is concerned with scientific design and operation of systems which involve the allocation of scarce resources. This course surveys some of the quantitative techniques used in
OR. Linear Programs are solved using graphical techniques and the simplex algorithm. Among the other models studied is the transportation, assignment, matching, and knapsack problems. Prerequisite:
MATH 033 or MATH 045 or MATH 051 with a "C-" or better or the appropriate score on the calculus placement test.
MATH 074. Discrete and Combinatorial Mathematics. 4 Units.
The fundamental principles of discrete and combinatorial mathematics are covered. Topics include the fundamental principles of counting, the Binomial Theorem, generating functions, recurrence
relations and introductory graph theory, that includes trees and connectivity. Prerequisite: MATH 033 or MATH 045 or MATH 051 with a "C-" or better, or an appropriate score on the calculus placement
MATH 075. Introduction to Linear Algebra. 4 Units.
Linear algebra is the generalized study of solutions to systems of linear equations. The study of such systems dates back over 2000 years and now is foundational in the design of computational
algorithms for many modern applications. This course will serve as an introduction to basic computational tools in linear algebra including the algebra and geometry of vectors, solutions to systems
of linear equations, matrix algebra, linear transformations, determinants, eigenvalue-eigenvector problems, and orthogonal bases. Prerequisite: MATH 051 with a “C-“ or better.
MATH 081. Writing Math Problems. 1 Unit.
This course is an introduction to LaTeX math typesetting software commonly used by mathematicians including document creation, special document classes, mathematics commands and terminology. Writing
problems for contests in multiple content areas and proofreading math problems. Practicum aspect: students will provide the content and grading for Pacific’s Avinash Raina High School Math
Competition. Prerequisite may be taken concurrently: MATH 051. (Spring).
MATH 093. Special Topics. 1-4 Units.
MATH 093E. Math Literacy for College. 3 Units.
MATH 093F. Special Topics. 4 Units.
MATH 095. Problem Solving Seminar. 1 Unit.
The objective of this course is to learn mathematics through problem solving. Students in mathematics courses are often given the impression that to solve a problem, one must imitate the solution to
a similar problem that has already been solved. This course will attempt to develop student creativity in solving problems by considering problems not commonly encountered in other mathematics
courses. Students enrolled in this course are expected to participate in the William Lowell Putnam Mathematical Competition on the first Saturday in December. Students may take this course for credit
at most four times. Prerequisite: MATH 053 with a "C-" or better.
MATH 101. Introduction to Abstract Mathematics. 4 Units.
An introduction to the spirit and rigor of mathematics is the focus of the course. The content may vary with instructor, but the objective is to develop the skills required to read and write
mathematics and prove theorems. Concepts include elementary logic, sets and functions, cardinality, direct and indirect proofs, mathematical induction. Prerequisites: MATH 053 with a "C-" or better
or permission of the instructor.
MATH 110. Numerical Analysis. 4 Units.
Numerical analysis deals with approximation of solutions to problems arising from the use of mathematics. The course begins with a necessary but brief discussion of floating point arithmetic, and
then proceeds to discuss the computer solution of linear algebraic systems by elimination and iterative methods, the algebraic eigenvalue problem, interpolation, numeric integration, that includes a
discussion of adaptive quadrature, the computation of roots of nonlinear equations and the numerical solution of initial value problems in ordinary differential equations. Prerequisite: MATH 055 with
a "C-" or better.
MATH 121. Financial Mathematics I. 3 Units.
This course provides understanding of fundamental concepts in financial mathematics and how those concepts are applied in calculating present and accumulated values for various streams of cash flows
as a basis for future use in reserving, valuation, pricing, asset/liability management, investment income, capital budgeting, and valuing contingent cash flows. Topics include interest rates,
determinants of interest rates, and interest-related concepts, annuities involving both level and varying payments, and varying interest rates, projects appraisal evaluation, loans and loan payment
methods, bonds and bond evaluations. This course, together with MATH 122, prepares students for the Society of Actuaries Financial Mathematics examination. Prerequisite: MATH 053 with a “C-“ or
better or permission of instructor.
MATH 122. Financial Mathematics II. 3 Units.
This course is the second semester of one-year financial mathematics. The course starts with reviewing bonds and bond evaluations. New topics include: discount model in common stock evaluation,
analysis of term structure of interest rates, concepts of duration and convexity, and using and convexity to approximate bond price changes with respect to interest rate change, cash flow matching,
immunization (including full immunization), Redington immunization, interest rate swaps. This course, together with MATH 121, prepares students for the Society of Actuaries Financial Mathematics
examination. Prerequisite: MATH 121 with a “C-“ or better or permission of instructor.
MATH 122P. Problem Solving in Financial Mathematics. 1 Unit.
This 1 unit course is designed to prepare students for actuarial professional Exam FM. The course will review basic concepts in theory of interest and interest rate swaps (material covered in both
MATH 121 and MATH 122). The course is entirely problem driven. Prerequisite: MATH 122 with a “C-“ or better.
MATH 124. Advanced Financial Mathematics. 4 Units.
This course is designed to develop student’s knowledge of the theoretical basis of certain actuarial models and the application of those models to insurance and other financial risks. The primary
topics are: Option relations, binomial option pricing, Black-Scholes equation, market-making and delta hedging, exotic options, and Lognormal Distribution. Prerequisites: BUSI 123 and MATH 131 with a
“C-“ or better.
MATH 125. Actuarial Models I. 3 Units.
Actuaries put a price on risk, and this course considers constructing and analyzing actuarial loss models (risk theory, severity and ruin models). This is the first part of a two-course series that
covers the theory and applications of actuarial modeling. Actuarial Models I covers topics in probability theory relevant to the construction of actuarial models. After a review of random variables
and basic probability distributional properties, the course examines severity and frequency loss models. Aggregate loss models, risk measures and the impact of coverage modifications on both
frequency and severity will also be discussed. Finally, we will explore various ways of simulating random variables. Prerequisite: MATH 132 with a “C-“ or better or Permission of Instructor.
MATH 126. Actuarial Models II. 3 Units.
This course is the second part of a two-course series that covers the theory and applications of actuarial modeling. The course continues a study of the loss modeling processes introduced in
Actuarial Models I. The primary topics the course cover are: (1) Estimation for complete data: empirical distributions for complete, individual data and grouped data. (2) Estimation for modified
data: point estimation, Mean, variance, and interval estimation, kernel density models, approximations for large data sets. (3) Frequentist estimation: method of moments and percentile matching,
maximum likelihood estimation, variance and interval estimation, Bayesian estimation, estimation for discrete distribution. (4) Frequentist estimation for discrete distribution. (5) Model selection:
representations of the data and model, hypothesis tests, two types of selection criteria, extreme value models, copula models, models with covariates. (6) Simulation. Prerequisite: MATH 125 with a
“C-“ or better or Permission of Instructor.
MATH 127. Models of Life Contingencies I. 4 Units.
This course is an introduction to life contingencies as applied in actuarial practice. This course is the first semester of two-semester course sequence, and it is designed to develop knowledge of
the theoretical basis of life-contingent actuarial models and the application of those models to insurance and other financial risks. It covers the mathematical and probabilistic topics that underlie
life contingent financial instruments like life insurance, pensions and lifetime annuities. Topics include life tables, present value random variables for contingent annuities and insurance, their
distributions and actuarial present values, equivalence principle, and other principles for determining premiums and reserves. Prerequisites: MATH 122; MATH 131 with a “C-“ or better or Permission of
MATH 128. Models of Life Contingencies II. 4 Units.
This course is a continuation of the study of life contingencies. It is designed to develop the student’s knowledge of the theoretical basis of life-contingent actuarial models and the application of
those models to insurance and other financial risks. Topics include insurance and annuity reserves, characterization of discrete and continuous multiple decrement models in insurance, employee
benefits, benefit reserves, and multiple life models. Prerequisite: MATH 127 with a “C-“ or better or Permission of Instructor.
MATH 130. Topics in Applied Statistics. 3 Units.
This course covers topics in applied statistics not normally covered in an introductory course. Students study multiple regression and correlation, analysis of variance of one- and two-way designs
and other topics selected from non-parametric methods, time series analysis, discriminant analysis, factor analysis, that depend upon student interest. There is extensive use of packaged computer
programs. Prerequisites: MATH 035 or MATH 037 with a "C-" or better.
MATH 131. Probability and Mathematical Statistics I. 4 Units.
This course covers counting techniques, discrete and continuous random variables, distribution functions, special probability densities such as binomial, hypergeometric, geometric, negative binomial,
Poisson, Uniform, Gamma, Exponential, Weibull, and Normal. Students study joint distributions, marginal and conditional distributions, mathematical expectations, moment generating functions,
functions of random variables, sampling distribution of the mean, and the Central Limit Theorem. Credit is not given for both MATH 039 and MATH 131. Prerequisite: MATH 053 with a "C-" or better.
MATH 131P. Problem Solving in Probability. 1 Unit.
This course is designed to prepare students for actuarial professional Exam P. This course will review basic concepts in theory of probability. The primary focus is problem solving; applying
fundamental probability tools in assessing risks. Prerequisite: MATH 131 or permission of instructor.
MATH 132. Probability and Mathematical Statistics II. 4 Units.
Sampling distributions such as Chi-square, t and F are studied as estimation methods such as methods of moments, maximum likelihood and least squares. The course covers properties of estimators such
as unbiasedness, consistency, sufficiency, tests of hypothesis concerning means, difference between means, variances, proportions, one and two-way analysis of variance. Prerequisite: MATH 131 with a
"C-" or better.
MATH 133. Statistical Learning Methods. 3 Units.
This course will describe, implement and compare statistical models for classification and regression problems including ordinary least squares regression, logistic regression, K-nearest neighbors,
shrinkage methods, decision trees, random forests, clustering algorithms, principal component analysis, random walks, and autoregressive models. Common methods for the selection and validation of
models such as stepwise selection, cross-validation, training/testing sets and data visualization will also be discussed. The use of statistical software will be emphasized. Prerequisites: MATH 037
with a “C-“ or better or permission of instructor. Some background in programming is also recommended.
MATH 141. Linear Algebra. 4 Units.
Fundamental linear algebra concepts from an abstract viewpoint, with the objective of learning the theory and writing proofs. Concepts include: vector spaces, bases, linear transformations, matrices,
invertibility, eigenvalues, eigenvectors, invariant subspaces, inner product spaces, orthogonality, and the spectral theorem. Prerequisites: MATH 049, MATH 075 with a "C-" or better.
MATH 143. Abstract Algebra I. 4 Units.
This is an introductory course to groups, rings and fields, with an emphasis on number theory and group theory. Students study finite groups, permutation groups, cyclic groups, factor groups,
homomorphisms, and the isomorphic theorem. The course concludes with an introduction to polynomial rings. Prerequisite: MATH 049 with a "C-" or better or permission instructor.
MATH 144. Abstract Algebra II. 4 Units.
This course is a continuation of MATH 143, and it emphasizes field theory and the application of groups to geometry and field extensions. Students study algebraic and separable field extensions,
dimension, splitting fields, Galois theory, solvability by radicals, and geometric constructions. Prerequisite: MATH 143 with a "C-" or better or permission of instructor.
MATH 145. Applied Linear Algebra. 4 Units.
This is the second semester course in linear algebra with an emphasis on the theory and application of matrix decompositions. Topics include methods for solving systems of equations, QR
factorization, the method of least squares, diagonalization of symmetric matrices, singular value decomposition, and applications. Prerequisites: MATH 053, MATH 075 with a “C-“ or better.
MATH 148. Cryptography. 3 Units.
Cryptography and cryptanalysis from historical cryptosystems through the modern use of cryptology in computing are studied. Topics include public and symmetric key cryptosystems, digital signatures,
modular arithmetic and other topics in number theory and algebra. Possible additional topics include error correcting codes, digital cash, and secret sharing techniques. Prerequisite: MATH 053 with a
"C-" or better or permission of instructor.
MATH 152. Vector Analysis. 4 Units.
Vector analysis and topics for students of applied mathematics, physics and engineering are studied. Topics include vector fields, gradient, divergance and curl, parametiric surfaces, line integrals,
surface integrals, and integral theorems. Formulations of vector analysis in cylindrical and spherical coordinates are also included. Prerequisites: MATH 055 with a "C-" or better.
MATH 154. Topology. 4 Units.
This course introduces general topology and its relation to manifold theory. Topics include metric spaces, general spaces, continuous functions, homeomorphisms, the separation axioms, connectedness,
compactness, and product spaces. Prerequisite: MATH 049 with a "C-" or better.
MATH 155. Real Analysis I. 4 Units.
This course focuses on properties of real numbers, sequences and series of real numbers, limits, continuity and differentiability of real functions. Prerequisites: MATH 049 and MATH 055 with a "C-"
or better.
MATH 156. Real Analysis II. 4 Units.
This course covers integration, series of real numbers, sequences and series of functions, and other topics in analysis. Prerequisite: MATH 155 with a "C-" or better.
MATH 157. Applied Differential Equations II. 4 Units.
This course covers partial differential equations, derivation and solutions of the Wave, Heat and Potential equations in two and three dimensions as well as Fourier series methods, Bessel functions
and Legendre polynomials, and Orthogonal functions. Additional topics may include Fourier integral transform methods, the Fast Fourier Transform and Sturm-Liouville theory. Computer exercises that
use MATLAB are included. Prerequisite: MATH 057 with a "C-" or better.
MATH 161. Elementary Concepts of Mathematics I. 3 Units.
Concepts and principles underlying elementary and middle school programs in mathematics. Laboratory materials will be used to reinforce understanding of concepts. Prerequisite: MATH 004, suitable
score on placement test, or exemption from placement test. Not open to freshman. This course does not count as an elective for a B.S. degree.
MATH 162. Elementary Concepts of Mathematics II. 3 Units.
Continuation of MATH 161. Concepts and principles of elementary and middle school mathematics. Prerequisites: MATH 161 (concurrency allowed) or permission of instructor.
MATH 164. Topics in History of Mathematics. 3 Units.
Topics in mathematics are studied from a historical perspective. Topics are chosen from: numeration systems; mathematics of the ancient world, especially Greece; Chinese, Hindu and Arabic
mathematics; the development of analytic geometry and calculus; and modern axiomatic mathematics. Students solve problems using historical and modern methods. Students read and report on the
biography of a mathematician. Prerequisite: MATH 053 with a "C-" or better. Junior standing or permission of the instructor.
MATH 166. Mathematical Concepts for Secondary Education. 3 Units.
This course covers secondary school mathematics from an advanced viewpoint and pedagogical perspective. Content is aligned with the mathematics subject matter requirements from the California
Commision on Teacher Credentialing. Prerequisite: MATH 053 with a "C-" or better.
MATH 168. Modern Geometries. 4 Units.
Selected topics in this course are from Euclidean, non-Euclidean and transformational geometry in additionto both analytic and synthetic methods. The history of the development of geometries and
axiomatic systems is covered. The course uses laboratory materials and computer packages used to reinforce understanding of the concepts. The course is required for high school teacher candidates.
Prerequisite: MATH 049 with a "C-" or better or permission of instructor.
MATH 174. Graph Theory. 4 Units.
This course is an in-depth consideration of discrete structures and their applications. Topics include connectivity, Eulerian and Hamiltonian paths, circuits, trees, Ramsey theory, digraphs and
tournaments, planarity, graph coloring, and matching and covering problems. Applications of graph theory to fields such as computer science, engineering, mathematics, operations research, social
sciences, and biology are considered. Prerequisites: MATH 051 or MATH 074 or COMP 047 with a "C-" or better or an appropriate score on the calculus placement test.
MATH 189A. Statistical Consulting Practicum. 2 Units.
While working under close faculty supervision, students gain valuable practical experience in applying statistical methods to problems presented by University researchers, business and industry.
Students enrolled in MATH 189A ordinarily participate in more sophisticated projects and take a more responsible role than students in MATH 089A. Pass/No credit. Prerequisites: for MATH 089A, MATH
130 with a "C-" or better or permission of the instructor; for MATH 189A, 089A with a "C-" or better and permission of the instructor.
MATH 191. Independent Study. 2-4 Units.
Student-initiated projects cover topics not available in regularly scheduled corses. A written proposal that outlines the project and norms for evaluation must be approved by the department
MATH 197. Undergraduate Research. 2-4 Units.
Writing Courses
WRIT 001. Academic Writing I. 2 Units.
This course includes approximately 4,000 words of edited composition. During the semester, students will accrue points on essays, assignments, classwork and research projects. Students will engage in
higher-level writing and will cover the essay writing process, note taking, outlining, summarizing, and editing. It also focuses on development of vocabulary, comprehension, concentration, memory and
fluency skills. Critical thinking, analysis and evaluation are emphasized as students engage with themed materials. Students will develop research skills in the use of outside reference materials
including locating and evaluating sources and properly documenting source information. Students are expected to progress in a variety of academic writing forms including, but not limited to, reports,
short term papers, essays and journal writing, incorporating increasingly complex rhetoric. This course is part of a sequence designed for those students who need to meet the university fundamental
skills requirement. Pre-requisites for placement are determined by qualifying standardized or diagnostic test scores. Pass/No credit (P/NC) grading option is not allowed for this course. Students
taking this course are required to take WRIT 002 the following semester and must earn a “C-“ or better to be eligible for advancement.
WRIT 002. Academic Writing II. 2 Units.
This course will include approximately 4,000 words of edited composition. Students will develop advanced writing projects as they locate, evaluate, and synthesize source material from various
disciplines and compose research papers using APA, MLA, CMS and CSE documentation as needed. Special emphasis is placed on the skills related to vocabulary development, critical thinking and
interpretation of scholarly material for the purpose of in-class discussions, expository writing assignments and literary analysis. This course is part of a sequence designed for those students who
need to meet the university fundamental skills requirement. Pass/No credit (P/NC) grading option is not allowed for this course. Students taking this course are required to take PACS Plus in the
upcoming fall semester and must earn a “C-“ or better to be eligible for advancement. Prerequisite: WRIT 001 with a “C-“ or better.
WRIT 010. Academic Writing. 2 Units.
This course is intended for students who need to fulfill the university's fundamental skills requirement in writing. This course will include approximately 5,000 words of edited composition. Students
will work on various writing projects as they develop strong written and oral communication skills, critical thinking, and reading skills necessary for success in their majors and will gain
information literacy by locating, evaluating, and synthesizing source material from various disciplines. Students will also learn how to appropriately document papers, using APA and MLA citation
styles as needed. Pass/No credit (P/NC) grading option is not allowed for this course. A grade of C- or better is required to satisfy the university’s fundamental skills requirement in writing. WRIT
010 cannot be repeated if a grade of C- or better is earned. Students who repeat the course must choose a new topic for their research paper.
WRIT 093I. Academic Writing Bridge. 1-4 Units.
WRIT 093L. Reading & Writing Lab. 1 Unit.
WRIT 093W. Academic Writing Intensive. 4 Units.
This course is designed as a transition into college-level writing and will include approximately 5,000 words of edited composition. During the session, students will accrue points on essays,
assignments, classwork and research projects. Students will engage in the higher-level reading and writing skills necessary for university work. The course primarily focuses on academic expository
writing and covers the essay writing process, note taking, outlining, summarizing, and editing. Critical thinking, analysis and evaluation is emphasized as students engage with themed materials.
Students will also begin to develop research skills in the use of outside reference materials including locating and evaluating sources and properly documenting source information. Students will be
exposed to a variety of academic writing forms including but not limited to reports, short term papers, essays and journal writing. This course is part of a sequence designed for those students who
need to meet the university fundamental skills requirement. Pass/No credit (P/NC) grading option is not allowed for this course. Students taking this course are required to take PACS 1 Plus in the
upcoming fall semester and must earn a C- or better to be eligible for advancement.
WRIT 093X. Academic Reading and Writing I. 1-4 Units.
WRIT 093Y. Academic Reading and Writing II. 1-4 Units.
WRIT 093Z. Accelerated Academic Reading and Writing. 1-4 Units.
WRIT 191. Independent Study. 1-4 Units.
Fundamental Skills Faculty
Emily Brienza-Larsen, Instructor, Developmental Writing, BA, English, University of the Pacific, 2002; MA, Education, National University, 2004; MA, English, National University, 2017.
Andrew Pitcher, Instructor, Developmental Math, BS, Mathematics, University of the Pacific, 2000; MA, Mathematics, UC Davis, 2002. | {"url":"https://catalog.pacific.edu/stocktongeneral/fundamentalskills/","timestamp":"2024-11-05T08:58:44Z","content_type":"text/html","content_length":"89577","record_id":"<urn:uuid:15e59684-a9b1-4b6a-854d-6907e27f0713>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00111.warc.gz"} |
Brain Challenge: Which Number Should Be in Place of the Question Mark? Brain Challenge: Which Number Should Be in Place of the Question Mark?
Brain Challenge: Which Number Should Be in Place of the Question Mark?
Want to determine how much focus your brain is capable of achieving? This math test will be an excellent method to train your mind to do just that.
Contrary to popular belief, solving math problems is a lot of fun. And that’s what this quiz brings to the table! If you can solve it, you will be able to distract yourself from your everyday
schedule. That’s pretty much the push your brain needs to gain some much-needed focus.
Take your time with this puzzle:
As you can see, the first three equations have exact solutions. Can you get the answer to the fourth one without having to scroll down?
How to Find the Solution
• The first equation shows a summation in which the three lipsticks give a result equal to 30.
• The next equation sums up one lipstick and two mirrors giving a result of 20.
• Equation 3 represents a summation of two pairs of nail polish and one mirror to give a result of 9
• The final equation is the one with a question mark. It is a representation of one mirror, one nail polish, and a lipstick.
Did you get an answer?
The Solution
To solve the fourth equation, you will need to use the first three so that you can determine what each component stands for.
• In the first equation, all three lipstick represent the same number. You find this number by dividing the result by 3 to get 10. Therefore, one lipstick represents the number 10.
• Taking 10 as your base value for the lipstick, you find that both mirrors represent 10 in equation number 2. Every mirror is then determined to be equal to a value of 5.
• Taking 5 and adding two pairs of nail polishes will give a value of 9. You can tell that every pair of lipstick represents a value of 2.
• This only leaves you with one lipstick representing a value of 1.
• The final equation can then be represented by 5 + 1 x 10
The equation simplifies to 5 + 1 x 10, which gives a value of 15.
The answer is, therefore, 15.
Did you successfully find the solution to this quiz? Invite your friends to try it out as well and see who gets it right! | {"url":"https://mydailybrain.me/brain-challenge-which-number-should-be-in-place-of-the-question-mark/","timestamp":"2024-11-05T13:50:48Z","content_type":"text/html","content_length":"272448","record_id":"<urn:uuid:b8c7e5a5-1ec5-49e7-ac92-601b2a949895>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00084.warc.gz"} |
Library Guides: MATH 4350 & 4360 Abstract Algebra: Find Books
Galois Theory for Beginners by Galois theory is the culmination of a centuries-long search for a solution to the classical problem of solving algebraic equations by radicals. In this book,
Bewersdorff follows the historical development of the theory, emphasizing concrete examples along the way. As a result, many mathematical abstractions are now seen as the natural consequence of
particular investigations. Few prerequisites are needed beyond general college mathematics, since the necessary ideas and properties of groups and fields are provided as needed.Results in Galois
theory are formulated first in a concrete, elementary way, then in the modern form. Each chapter begins with a simple question that gives the reader an idea of the nature and difficulty of what lies
ahead. The applications of the theory to geometric constructions, including the ancient problems of squaring the circle, duplicating the cube, and trisecting an angle, and the construction of regular
$n$-gons are also presented. This book is suitable for undergraduates and beginning graduate students.
Call Number: QA214 .B49 2006
ISBN: 9780821838174
Publication Date: 2006-09-05 | {"url":"https://libguides.nova.edu/cnso-abstractalgebra/books","timestamp":"2024-11-01T22:31:28Z","content_type":"text/html","content_length":"126706","record_id":"<urn:uuid:84a6f649-30e6-45a2-bed0-62c683736992>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00456.warc.gz"} |
Title : Asymptotic prime divisors - II
Abstract : Consider a Noetherian ring R and an ideal I of R. Ratliff asked
a question that what happens to Ass(R/I^n) as n gets large ? He was able
to answer that question for the integral closure of I. Meanwhile Brodmann
answered the original question, and proved that the set Ass(R/I^n)
stabilizes for large n.
We will discuss the proof of stability of Ass(R/I^n). We will also
give an example to show that the sequence is not monotone. The aim of
this series of talks to present the first chapter of S. McAdam,
Asymptotic prime divisors, Lecture Notes in Mathematics 1023,
Springer-Verlag, Berlin, 1983.
Sun, October 22, 2017 12:27pm IST | {"url":"https://www.math.iitb.ac.in/webcal/view_entry.php?id=183&date=20171024&friendly=1","timestamp":"2024-11-08T04:27:13Z","content_type":"text/html","content_length":"4247","record_id":"<urn:uuid:0c883890-8139-408a-b987-4107695ddcb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00618.warc.gz"} |