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How to Convert 0.00040931483 as a Percent - LoudHelp Social Blog
How to Convert 0.00040931483 as a Percent
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Converting 0.00040931483 as a percent can seem tricky at first glance. This small decimal number represents a fraction of a whole, and understanding how to express it as a percentage is a useful
skill. Knowing how to make this conversion has practical applications in fields like finance, statistics, and data analysis.
The process involves multiplying the decimal by 100 to shift the decimal point and express the value as a percent. This article will guide readers through the step-by-step conversion, explain the
basic formula, and show how to use a calculator for quick results. It will also explore real-world situations where this type of conversion is helpful, giving context to the mathematical operation.
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What is a Percentage?
A percentage is a way to express a number as a fraction of 100. The word “percent” means “per hundred” or “divided by 100”. It’s represented by the symbol %, which is a visual shorthand for this
concept. For instance, when someone says “20 percent,” it can be written as “20%” or interpreted as “twenty out of one hundred”.
Mathematically, a percentage is a relative value that indicates hundredth parts of any quantity. One percent (1%) represents a hundredth part, while 100 percent signifies the entirety of something,
and 200 percent would specify twice the given quantity. This concept allows for easy comparisons between different values and provides a standardized way to express proportions.
To illustrate this idea, let’s consider an example: 1 percent of 1,000 chickens equals 1/100 of 1,000, which is 10 chickens. Similarly, 20 percent of the same quantity would be 20/100 × 1,000,
resulting in 200 chickens. This relationship can be generalized using the formula x = PT/100, where T is the total reference quantity (representing 100 percent), P is the given percentage, and x is
the quantity equivalent to that percentage of T.
Common Uses
Percentages have numerous practical applications in everyday life and various professional fields. Here are some common uses:
1. Sales Tax: Depending on where you live, you might pay a sales tax on purchases. This tax is typically expressed as a percentage of the item’s price.
2. Discounts: When shopping, you often encounter discounts expressed as percentages. The discount percentage is subtracted from the original price to determine the sale price.
3. Tipping: The amount of a tip is usually calculated as a percentage of the total cost of a service.
4. Business Transactions: Percentages are frequently used to calculate profit or loss in business dealings. For example, if a retailer buys an item at a wholesale price of $80.00 and sells it for
$110.00, making a profit of $30.00, the percentage profit can be calculated as (100 × 30/80), which equals 37.5 percent.
5. Statistics: In statistical analysis, the concept of cumulative percentage (percentile) is commonly used. For instance, a student scoring at the 83rd percentile on an examination has performed
better than 83 percent of the students in the comparison group.
6. Probability: The likelihood of an event occurring can be expressed as a percentage. For example, the probability of a perfectly balanced coin landing heads up is 50 percent.
7. Data Comparison: Percentages are used to compare numerical data across various sectors, including corporations, governments, schools, and colleges.
8. Financial Analysis: Companies and individuals use percentages to analyze financial data, such as investment returns, budget allocations, and expense ratios.
Understanding percentages and their applications is crucial for making informed decisions in various aspects of life, from personal finance to academic performance evaluation. The ability to convert
between percentages, decimals, and fractions is a valuable skill that enhances numerical literacy and problem-solving capabilities.
The Basic Formula
The basic formula for converting 0.00040931483 as a percent involves a simple mathematical operation. This process is essential for transforming decimal numbers into percentages, which are often
easier to understand and compare.
Decimal to percentage conversion
To convert a decimal to a percentage, one needs to express the number as a fraction of 100. This concept is at the heart of percentage calculations. A percentage represents a part of a whole, where
the whole is considered to be 100. For instance, 0.5 can be thought of as 50 parts out of 100, which translates to 50%.
The process to convert a decimal to a percentage value is straightforward. It involves multiplying the decimal number by 100 and adding a percentage (%) symbol after the result. This method works for
all types of decimal numbers, including repeating decimals.
For example, to convert 0.006 to a percent, one would multiply it by 100 and place a percent symbol beside it. The calculation would look like this: 0.006 = 0.006 x 100% = 0.6%.
Multiplying by 100
The key step in converting 0.00040931483 as a percent is multiplying it by 100. This operation effectively shifts the decimal point two places to the right, transforming the decimal into a percentage
Mathematically, this can be expressed as:
Percentage = Decimal × 100
By applying this formula to our specific number:
0.00040931483 × 100 = 0.040931483%
It’s important to note that multiplying a decimal by 100 always moves the decimal point two places to the right. This shift is equivalent to expressing the number as parts per hundred, which is the
essence of percentages.
A helpful shortcut to remember is that converting from decimal to percent involves moving the decimal point two places to the right and adding a percent sign. This method can be applied to any
decimal number, making percentage conversions quick and easy.
For instance, 0.10 becomes 10% (0.10 × 100 = 10%), and 0.675 becomes 67.5% (0.675 × 100 = 67.5%).
Understanding this basic formula is crucial for various real-world applications. In fields such as finance, statistics, and data analysis, the ability to convert between decimals and percentages is
frequently required. It allows for easier comparison of quantities and helps in presenting data in a more comprehensible format.
For example, in business transactions, profit margins are often expressed as percentages. If a retailer buys an item at a wholesale price and sells it for a higher price, the percentage profit can be
calculated using this formula. Similarly, in academic settings, test scores are frequently reported as percentages to provide a standardized measure of performance.
By mastering this simple yet powerful formula, one can easily convert 0.00040931483 as a percent and apply this skill to a wide range of numerical problems. Whether dealing with financial
calculations, statistical analysis, or everyday math problems, the ability to convert between decimals and percentages is an invaluable tool in one’s mathematical toolkit.
Step-by-Step Conversion Process
Converting 0.00040931483 as a percent involves a straightforward process that can be broken down into simple steps. This method applies to any decimal number, making it a valuable skill for various
mathematical and real-world applications.
Converting 0.00040931483 to a decimal
The first step in this process is to recognize that 0.00040931483 is already in decimal form. Unlike fractions or mixed numbers, decimals are ready for direct conversion to percentages. This small
decimal represents a fraction of a whole, specifically a very small portion.
It’s important to note that when dealing with such small numbers, precision becomes crucial. Each digit after the decimal point carries significance, especially in fields like finance or scientific
calculations where minute differences can have substantial impacts.
Multiplying by 100
The key step in converting 0.00040931483 as a percent is multiplying it by 100. This operation effectively shifts the decimal point two places to the right, transforming the decimal into a percentage
To perform this multiplication:
0.00040931483 × 100 = 0.040931483
This result is now in percentage form, but we’re not quite finished. Remember, percentages are typically expressed with a percent sign (%) at the end.
Rounding the result
The final step involves deciding how many decimal places to keep in the percentage. Rounding is often necessary for practical purposes and to avoid spurious accuracy.
In this case, given the small nature of the original number, it might be appropriate to keep several decimal places to maintain precision. However, the exact number of decimal places to retain
depends on the context and required accuracy.
For general purposes, rounding to four decimal places might be suitable:
0.040931483% ≈ 0.0409%
This rounded percentage is easier to read and work with while still maintaining a good level of accuracy for most applications.
It’s worth noting that in some cases, especially when dealing with very small percentages, it might be more appropriate to express the result in scientific notation or parts per million (ppm) for
better comprehension.
When working with percentages, it’s crucial to remember that the process can be reversed. To convert a percentage back to a decimal, one simply divides by 100 or moves the decimal point two places to
the left.
Understanding this conversion process is valuable in various fields. In finance, for instance, interest rates are often expressed as percentages but need to be converted to decimals for calculations.
In statistics, probabilities are frequently converted between decimal and percentage forms depending on the context.
By mastering this simple yet powerful conversion technique, one can easily handle a wide range of numerical problems involving percentages and decimals. Whether dealing with financial calculations,
statistical analysis, or everyday math problems, the ability to convert between these forms is an invaluable tool in one’s mathematical toolkit.
Using a Calculator
When converting 0.00040931483 as a percent, a calculator can be a helpful tool to ensure accuracy and save time. There are two main approaches to using a calculator for this conversion: utilizing a
built-in percentage function or performing manual calculation steps.
Built-in percentage function
Many modern calculators come equipped with a dedicated percentage button, often labeled as “%”. This feature simplifies the process of converting decimals to percentages. To convert 0.00040931483 as
a percent using this function:
1. Enter the decimal number (0.00040931483) into the calculator.
2. Press the “%” button.
3. The result displayed will be the percentage equivalent.
Some calculators may require pressing the multiplication key (*) before the percentage button for proper conversion. In this case, the process would be:
1. Enter 0.00040931483
2. Press the multiplication key (*)
3. Enter 100
4. Press the percentage button (%)
5. Press the equals (=) button
The result should display 0.040931483%, which is 0.00040931483 as a percent.
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Manual calculation steps
If your calculator lacks a dedicated percentage function, you can still perform the conversion using basic arithmetic operations. Here’s how to convert 0.00040931483 as a percent manually:
1. Multiply the decimal by 100:
0.00040931483 * 100 = 0.040931483
2. The result (0.040931483) is the percentage value.
3. Add the percent symbol (%) to the result:
This manual method works for any decimal-to-percentage conversion. For example, to convert 0.5 to a percent, you would multiply 0.5 by 100, resulting in 50%.
It’s important to note that when dealing with very small decimal numbers like 0.00040931483, the resulting percentage may also be quite small. In such cases, it might be more practical to express the
result in scientific notation or as parts per million (ppm) for better comprehension.
Remember that the process can be reversed to convert percentages back to decimals. Simply divide the percentage by 100 or move the decimal point two places to the left.
Using a calculator for percentage calculations offers several advantages:
1. Speed: Calculators perform complex calculations quickly, saving time compared to manual methods.
2. Accuracy: They eliminate human error in arithmetic, especially when dealing with many decimal places.
3. Consistency: Calculators provide consistent results, which is crucial for financial or scientific applications where precision is key.
4. Versatility: Many calculators can handle various percentage-related calculations, such as finding percentage increases or decreases.
When using a calculator for percentage calculations, it’s advisable to clear the calculator’s memory (usually by pressing the “C” or “AC” button) before starting a new calculation to avoid carrying
over previous results.
In conclusion, whether using a built-in percentage function or performing manual steps, a calculator is an invaluable tool for converting 0.00040931483 as a percent. It ensures accuracy and
efficiency, especially when dealing with complex decimal numbers or multiple conversions.
Real-world Applications
Understanding how to convert 0.00040931483 as a percent and similar decimal-to-percentage conversions has practical applications across various fields. Let’s explore how this knowledge applies in
finance, statistics, and science.
In the financial world, percentage calculations play a crucial role in assessing investment performance, determining interest rates, and analyzing market trends. Converting decimals to percentages is
a fundamental skill for investors and financial analysts.
For instance, when calculating the percentage gain or loss on an investment, investors need to determine how much their investment has grown or fallen in value. This calculation provides insight into
the performance of the investment and the success of the investment strategy. To calculate the percentage gain or loss, investors first determine the original cost or purchase price, then subtract it
from the selling price to find the gain or loss. This difference is then divided by the original cost and multiplied by 100 to get the percentage change.
Let’s consider an example: If an investor bought a stock for $100 and sold it for $100.04, the gain would be $0.04. To express this as a percentage, we would divide 0.04 by 100 and multiply by 100,
which gives us 0.04%. This small percentage gain is equivalent to our primary keyword, 0.00040931483 as a percent.
In the context of loans or savings accounts, even small percentage differences can have significant impacts over time. For instance, a 0.04% difference in interest rates on a large loan or investment
could translate to substantial amounts over several years.
In the field of statistics, percentages are frequently used to present data in a more understandable format. Converting decimals to percentages can make it easier for people to interpret and compare
data points.
Research has shown that people generally find it easier to compare percentages than fractions, especially when the denominators are different. In an experiment assessing how easily participants could
compare fractions or percentages, judgment accuracy was nearly perfect for percentages, scoring significantly higher than fractions across all experimental conditions.
This finding has important implications for how statistical data should be presented. For instance, if a study finds that 0.00040931483 of a population exhibits a certain characteristic, expressing
this as 0.040931483% makes it more readily comparable to other statistics.
However, it’s crucial to remember the law of small numbers when dealing with percentages, especially in health-related statistics. This principle points out that when a sample size is small, small
random changes can have a large apparent effect on the analysis of the data. Therefore, when interpreting statistics expressed as percentages, it’s essential to consider the sample size and context.
In scientific research and experiments, precise measurements often result in small decimal numbers that are more meaningful when expressed as percentages. Converting 0.00040931483 as a percent
(0.040931483%) could represent a minute change in a chemical reaction, a small shift in temperature, or a tiny fraction of a population in ecological studies.
For example, in environmental science, measuring changes in atmospheric composition often involves dealing with very small percentages. A change of 0.040931483% in the concentration of a greenhouse
gas could be significant when considering long-term climate trends.
In physics and engineering, percentage calculations are used to determine efficiency rates, error margins, and tolerance levels. A difference of 0.040931483% might be critical in high-precision
manufacturing or in calculating the performance of advanced scientific instruments.
Across all these fields, the ability to convert between decimals and percentages is crucial for accurate data interpretation and communication. Whether it’s expressing financial gains, statistical
findings, or scientific measurements, understanding how to convert 0.00040931483 as a percent and similar decimal values enhances numerical literacy and problem-solving capabilities.
In conclusion, while 0.00040931483 might seem like an insignificant number at first glance, when converted to a percentage and applied in real-world contexts, it can represent meaningful changes or
differences. The skill of converting between decimals and percentages is invaluable across various professional fields and everyday situations, allowing for clearer communication and more intuitive
understanding of numerical data.
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Converting 0.00040931483 as a percent reveals a small but significant value of 0.040931483%. This transformation process, while seemingly simple, has far-reaching applications in finance, statistics,
and science. It showcases how even minute decimal numbers can represent meaningful changes when viewed through the lens of percentages.
To wrap up, the ability to convert between decimals and percentages enhances our understanding of numerical data across various fields. Whether it’s analyzing investment returns, interpreting
statistical findings, or measuring scientific phenomena, this skill proves invaluable. By mastering such conversions, we can better communicate complex numerical information and make more informed
decisions in both professional and everyday contexts.
1. How do you change a decimal into a percentage?
□ To change a decimal into a percentage, multiply the decimal number by 100. For instance, converting 0.065 into a percentage would result in 6.5%.
2. What is the process for converting a decimal number to a percentage?
□ The process involves multiplying the decimal value by 100 and then appending a percent sign (%). Essentially, you move the decimal point two places to the right and add the percent symbol to
convert from decimal to percent.
3. Can you explain how to transform a decimal value into a percentage?
□ To transform a decimal value into a percentage, multiply it by 100 and add a percentage sign (%). Another method is to shift the decimal point two places to the right.
4. How is an equation converted into a percentage?
□ To convert an equation, particularly fractions, into a percentage, divide the numerator (top number) by the denominator (bottom number) and then multiply by 100. For example, to convert the
fraction 5/8 to a percentage, compute 5 ÷ 8 × 100, which equals 62.5%.
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Steel I Beam Size Calculator - CivilGang
What is a Steel I-Beam Size Calculator?
A “Steel I-Beam Size Calculator” is a tool used to estimate the optimal size of a steel I-beam for a given load and span length in a construction or structural project.
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Addition – Explanation & Examples
The first official evidence of addition is that Egyptians and Babylonians used it in 2000 B.C. The symbols of addition and subtraction were invented around the 16^th century, but before that, the
equations were written in words, making it really time-consuming to solve the problems.
Addition Definition
Generally, addition is defined as combining two or more groups of objects into a single group. Mathematically, addition can be defined as an arithmetic operation in which the sum or total of two or
more numbers is determined.
The addition symbol is a plus (+) and is inserted between numbers being added. Performing addition is one of the simplest numerical tasks. Addition is an important skill in all aspects of life,
including: at home, school and work.
Parts of Addition
There are 3 parts of addition, the addend, the equal sign, and the sum.
The Addend
In addition, the addends or the summands are numbers or terms being added together. For example, 10 + 6 = 16, 10 and 6 are the addends of this equation.
The Equal Sign
The equal sign indicates that the two halves of the equation are equivalent. For example, in the addition sentence, 10 + 6 = 16, the equal sign is denoted with two short horizontal strokes.
The Sum
The sum in addition sentence is the totals of the addends. For example, in 10 + 6 = 16, the sum is 16.
Understanding the properties of addition is important because it guides you to work with numbers more effectively.
Properties of Addition
This property states that the positions of numbers in an equation do not affect the final answer. For example, 4 + 5 is the same as 5 + 4. This property applies to the addition of numbers, no matter
how large the group of numbers is.
This property applies to complex equations involving brackets, braces, and parenthesis to separate groups of numbers. In addition, we can move the brackets around without affecting the ultimate
answer. For example, (4 + 6) + 2 = 4 + (6 + 2).
The identity property states that the sum of a number with zero is equivalent to the number itself. For example, 5+ 0 = 5. The number zero is called an identity number because it does affect other
numbers during the addition.
When a student is adding large groups of numbers, remind them that zero does not affect other numbers in the equation.
The inverse operation implies that the addition and subtraction are opposites. For instance, two plus three minus three is equal to two. Adding and subtracting the same numbers is similar to
canceling out the numbers.
For this reason, you should look for numbers that cancel out when performing addition and subtraction on large groups of numbers.
Example 1
Terry bought a designer dress for $231 and shoes for $199. How much money did she spend on the dress and shoes?
To find the total cost, we need to add 231 and 199
The total expenditure is found by adding the two prices together. Therefore. Terry spent $430 on both items.
Example 2
There are 56 yellow seats and 97 white seats in the school auditorium. How many yellow and white seats are in the auditorium?
Total number of seats = 56 +97
=153 seats
Practice Questions
1. Which of the following shows the sum of $50711$ and $77412$?
2. Which of the following shows the sum of $62731$ and $38128$?
3. Which of the following shows the number that is $45172$ more than $78278$?
4. Which of the following shows the number that is $65172$ more than $65278$?
5. Which of the following shows the number that is $1762 $ more than $4535$?
6. Which of the following shows the number that is $1572 $ more than $50265$?
7. Which of the following shows the number that exceeds $16619$ by $1279$?
8. Which of the following shows the number that exceeds $88543 $ by $2935$?
9. In a city’s schools, there are a total of $543556$ students in Grade 2, $54799$ students in Grade 3, $56890$ students in Grade 4, and $65543$ students in Grade 5. Which of the following shows the
total number of students in Grades 2, 3, 4, and 5?
10. A farmer had $\$3001$. He purchased a cow for $\$450$ and two goats for $\$ 150$ each. How much money is with him now?
11. A man had $\$459999$. He gave his two sons $\$458$ each, $\$50000$ to his wife, and kept the rest. How much was he left with?
12. Which number should be added to the sum of $5633$ and $4566$ to give the result of $88888$?
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14. A school store has $600$ bags of rice, $300$ bags of cereals, and $60$ bags of grains. How many bags of grains and cereals are in the store?
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Persistent Homology Software
The following packages can be used to compute the persistent homology from a point cloud:
• CGAL (for Delaunay triangulations and alpha-shapes)
• GUDHI (for persistent homology)
Our Field Filtration and Radial Filtration methods are available here:
In addition, you can use our fork of the HERA code to calculate the Frechet average and variances of persistence diagrams:
• HERA (use geom_matching/wasserstein/frechet_average after building)
In topology, homology provides us with a description of an object in terms of its holes. In three dimensions, we distinguish holes in dimensions 0, 1, and 2. A 0-dimensional hole is a gap separating
different components. A 1-dimensional hole is a tunnel that can be encircled by a loop. A 2-dimensional hole is a void enclosed by a shell.
By computing the homology of a filtration (a sequence of growing/shrinking objects), we can study how the topology changes as we vary some filtration parameter α. For example, one can study how the
superlevel set of a field evolves as one lowers the threshold from +∞ to –∞. In our paper (Elbers & van de Weygaert, 2018), we consider growing bubble networks modelled as balls in R^3 with
increasing radius.
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in the final panel.
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Freedom, weakness, and determinism: from linear-Time to branching-Time
Model checking is a method for the verification of systems with respect to their specifications. Symbolic model-checking, which enables the verification of large systems, proceeds by calculating
fixed-point expressions over the system's set of states. The μ-calculus is a branching-Time temporal logic with fixed-point operators. As such, it is a convenient logic for symbolic model-checking
tools. In particular, the alternation-free fragment of μ-calculus has a restricted syntax, making the symbolic evaluation of its formulas computationally easy. Formally, it takes time that is linear
in the size of the system. On the other hand, specifiers find the μ-calculus inconvenient. In addition, specifiers often prefer to use Linear-Time formalisms. Such formalisms, however, cannot in
general be translated to the alternation-free CL-calculus, and their symbolic evaluation involves nesting of fixed-points, resulting in time complexity that is quadratic in the size of the system. In
this paper we characterize linear-Time properties that can be specified in the alternation-free μ-calculus. We show that a linear-Time property can be specified in the alternation-free μ-calculus if
it can be recognized by a deterministic Buchi automation. We study the problem of deciding whether a linear-Time property, specified by either an automaton or an LTL formula, can be translated to an
alternation-free μ-calculus formula, and describe the translation, when exists.
Publication series
Name Proceedings - Symposium on Logic in Computer Science
Volume 1998-June
ISSN (Print) 1043-6871
Conference 13th Annual IEEE Symposium on Logic in Computer Science, LICS 1998
Country/Territory United States
City Indianapolis
Period 21/06/98 → 24/06/98
Bibliographical note
Publisher Copyright:
© 1998 IEEE.
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A family has two children. What is the probability that both the children are boys, given that at least one of them is a boy?
Hint: We are asked to find the conditional probability. For that, first, find the possible outcomes for having two children and then we define the events as E- both the children are boys and F- at
least one of the children is a boy. We know that \[P\left( E|F \right)\] is given as \[P\left( E|F \right)=\dfrac{P\left( E\cap F \right)}{P\left( F \right)}.\] We use this to get the required
Complete step-by-step answer:
We are given that a family has two children, it can be a boy or a girl. Let us consider girls to be denoted by ‘g’ and boys to be denoted by ‘b’. So the possible outcome for a family having 2
children are
\[S=\left\{ \left( b,b \right),\left( b,g \right),\left( g,b \right),\left( g,g \right) \right\}\]
Now, we have to find the probability that the children are both boys given that at least one of them is a boy. To do so, we will consider two events as follows.
E: Both the children are boys
F: At least one of the children is a boy.
We have to find \[P\left( E|F \right)\] that says \[P\left( E|F \right)\] is the probability of event E given that we have event F.
We know that, \[P\left( E|F \right)\] is given as,
\[P\left( E|F \right)=\dfrac{P\left( E\cap F \right)}{P\left( F \right)}\]
To find the required answer, we will first have to find \[P\left( E\cap F \right)\] and P(F).
Now, F = At least one of the children is a boy. So, the outcome for F are \[\left\{ \left( a,b \right),\left( b,g \right),\left( b,b \right) \right\}.\]
\[P\left( F \right)=\dfrac{\text{Favorable Number of Outcomes for F}}{\text{Total Number of Outcomes}}\]
\[\Rightarrow P\left( F \right)=\dfrac{3}{4}\]
Now, let us consider E = Both the children are boys. So, the outcomes for E are \[\left\{ \left( b,b \right) \right\}.\]
So, \[E\cap F=\left\{ \left( b,b \right) \right\}\]
\[P\left( E\cap F \right)=\dfrac{\text{Favorable Number of Outcomes for E}\cap \text{F}}{\text{Total Number of Outcomes}}\]
\[\Rightarrow P\left( E\cap F \right)=\dfrac{1}{4}\]
Now, we have, \[P\left( E\cap F \right)=\dfrac{1}{4}\] and \[P\left( F \right)=\dfrac{3}{4}.\] Using this in \[P\left( E|F \right),\] we get,
\[P\left( E|F \right)=\dfrac{P\left( E\cap F \right)}{P\left( F \right)}\]
By putting the values in the above equation, we get,
\[P\left( E|F \right)=\dfrac{\dfrac{1}{4}}{\dfrac{3}{4}}\]
Cancelling out 4 from both numerator and denominator, we get,
\[P\left( E|F \right)=\dfrac{1}{3}\]
Therefore, the probability of having 2 children both boys given and that at least one of them is a boy is \[\dfrac{1}{3}.\]
Note: Here, in this question, we are asked a conditional probability. Students should always keep in mind to use the formula \[P\left( E|F \right)=\dfrac{P\left( E\cap F \right)}{P\left( F \right)}.
\] And not directly apply the formula \[P\left( 2\text{ boys} \right)=\dfrac{\text{Outcome of having 1 boy}}{\text{Total Outcomes}}\]
This will lead us to a wrong answer as the outcome for having 2 boys are {(b, b)} and the total outcome is 4. So, \[P\left( \text{having 2 children both boys} \right)=\dfrac{1}{4}\] which is not the
correct solution. Also, keep in mind that at least 1 means that there should be 1 or more than 1. So, all the cases including 1 or more than that will be included. | {"url":"https://www.vedantu.com/question-answer/a-family-has-two-children-what-is-the-class-12-maths-cbse-5f5b7fa48a2fd7303bd93f5d","timestamp":"2024-11-13T18:10:46Z","content_type":"text/html","content_length":"186032","record_id":"<urn:uuid:85b790d3-c5af-4641-bf92-fc8ba92e5bfd>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00131.warc.gz"} |
Preface: Consider a Cylindrical Cow: More Adventures in Environmental Problem Solving - University Science Books
Consider a Cylindrical Cow: More Adventures in Environmental Problem Solving
John Harte University of California, Berkeley
The cow is back. This time she is cylindrical, not spherical. Still no legs, udders, or head, but the torso is a more realistic shape. Following on the hooves of Consider a Spherical Cow (Harte, J.
1988. Consider a spherical cow: a course in environmental problem solving. Sausalito, CA: University Science Books) called COW-1 herein, COW-2 will teach you additional modeling skills of use in
environmental science. Some of these skills are at roughly the same mathematical level as Chapters II and III of COW-1, whereas others are more advanced. Although the emphasis here, as in COW-1, is
on analytic approaches to squeezing information from models, I also include spread-sheet methods for simulating the behavior of models constructed from differential equations. You will get the most
out of COW-2 if you have worked through COW-1, but I have tried to make COW-2 self-contained.
I have assumed that you, the reader, are acquainted with both differential and integral calculus, though the relationship need not be lustful. Indeed, only a passionate dislike (or deep and
unyielding fear) of mathematics will disqualify you from proceeding further. Some past exposure to matrices will also help; for those who need to do some remedial reading on calculus or matrix
algebra, I highly recommend Clifford Swartz’s excellent Used Math (Swartz, C. E. 1993. Used math: for the first two years of college science. College Park, MD: AAPT Press).
I have also assumed that you are ready to seek out deeper insights into ways of modeling the complexities of nature, but the underlying goal is the same as in COW-1: to teach ways of stripping away
inessential detail and capturing with mathematics the essentials of a complex system.
Why do I want you to learn to use mathematics? Can’t we just talk about environmental problems? There is another reason in addition to the often-expressed, and correct, argument that mathematics
seems to be the “language that nature speaks” and therefore facilitates the understanding of nature. Mathematics is a kind of global language that not only needs no translation from nation to nation
but also bridges the disciplines. Economists and scientists often confuse each other (and sometimes themselves) using everyday language to describe how things depend upon other things; the reason is
that in the two academic traditions the same word may be used in different senses. (Consider, for example, the terms equity, parity, derivative, and stock.) Mathematics, however, has a way of cutting
through such confusion.
Just as the “Tools of the Trade” chapter of COW-1 was structured around the topics of steady-state box models, thermodynamics, chemical equilibrium theory, and non-steady-state box models, COW-2 is
structured around the central themes of probability, optimization, scaling, differential equations, stability, and feedback. Why these themes? Consider such problems as (a) determining how some shift
in land-use practice (e.g., conversion of tropical forest to grazing land) might result in species extinction or climate alteration in the region, or (b) determining whether exposure to a trace
substance released from a factory poses a serious health threat to people living nearby. In problems such as these, themes that affect our understanding and analysis of the problem can be extracted
as follows:
1. Much of our information about these situations comes to us in the form of data derived from some sampling scheme. We need to know something about probability and statistics to assess how
representative the data are.
2. We may discover that the activity leading to the environmental damage does bring some benefit to society, but that if the activity is too intense, then society is the loser. We need to know
something about optimization to estimate where the balance point between societal benefits and societal costs lies.
3. We may be able to determine something about the effects of a toxic substance on adults but not on children, so we need to know something about how to scale our knowledge from big people to little
people. Or, to estimate species loss, we may be able to use existing data on the rate of local species extinctions from deforestation, provided, we can “scale up” to larger regions.
4. We will want to predict not only the eventual long-term-averaged climate conditions that will result from the loss of forest but also the way in which climatic conditions will change over time in
the shorter term as they approach that eventual state. The mathematical tools needed to predict that climate trajectory over time are differential equations. We need to know how to set up the
appropriate differential equation for the problem at hand and how to find approximate solutions to that equation.
5. As we alter a complex system, for example by cutting down trees, thresholds of instability may be crossed. Before the threshold is reached, the system can assimilate the stress and maintain some
sort of equilibrium, but if the stress is too great, then the system may undergo a dramatic response. We need to know how to estimate whether such a threshold exists and where it lies.
6. Climate changes induced by altered land-use practices can further alter ecosystems and degrade ecosystem services of benefit to society. These feedbacks may further alter human behavior as people
seek to compensate for the lost ecosystem services. We need to be able to estimate the magnitude and consequences of such feedback effects.
The pervasiveness of these themes is well known to environmental scientists in fields such as climatology, ecology, hydrology, toxicology, and atmospheric chemistry. Despite this understanding among
scientists, public policy debates often neglect such critical linkages and issues. The public is easily misled by twisted probabilistic reasoning. Misuse of optimization methods can lead to
suboptimization (doing better and better something that would not have been done at all if the boundary of analysis had been enlarged). Analytic methods, insights, institutions, adaptations, and
solutions appropriate at one scale (e.g., experimental plots in ecology, individual firms or power plants in industry, a grid square in global climate modeling) are often naively and mistakenly
extrapolated to other scales. Incorrect conclusions about the distribution and fate of emitted substances usually can be spotted and corrected with the aid of approximation methods for studying
solutions to differential equations. Instabilities and feedbacks, both between and within human and biophysical systems, are often ignored in political, social, and even scientific analysis, leading
society to be inadequately prepared for the possible abruptness and intensity of resulting changes.
At the beginning of each of the five thematic sections in this book is a general treatment of the relevant mathematical concepts. The core of the book is 25 fully worked-out problems. As in COW-1,
each problem statement is posed more like a research question than a standard homework exercise. Homework exercises follow each worked-out problem solution and range from relatively straightforward
exercises that probe readers’ understanding of the concepts and methods to more difficult and open-ended research suggestions. The Appendix summarizes many useful mathematical notations and formulae
you are likely to encounter. | {"url":"https://uscibooks.aip.org/consider-a-cylindrical-cow-more-adventures-in-environmental-problem-solving-preface/","timestamp":"2024-11-12T13:54:58Z","content_type":"text/html","content_length":"111307","record_id":"<urn:uuid:697d4e91-99d6-4277-b6ee-8e8c68bf95b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00431.warc.gz"} |
math stuffs
Today my calc teacher started off the class with a pop quiz, one of the questions of which has been a source of contention among my classmates and myself....
wow gotta lay off the ap english lol
anyway here is the problem:
f(x)={ 3-x x<1
{ ax^2+bx x≥1
Find the unique values of a and b such that f(x) is continuous
so what i did is I used the definition of continuity:
f(x) is continuous at c IF
+b(1) = lim
This is where I stopped. Some other people have told me that in order to find unique values, you have to make f(x) differentiable at 1 as well. I say that the question never says that that is
necessary, so the correct answer is that there is no answer (or more accurately, any a and b such that a+b=2). Differentiability implies continuity, but continuity does not imply differentiability.
Who's right? | {"url":"https://www.omnimaga.org/math-and-science/math-stuffs/msg123720/","timestamp":"2024-11-09T22:59:36Z","content_type":"application/xhtml+xml","content_length":"105502","record_id":"<urn:uuid:63dad300-9549-4740-8a70-af56f483df32>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00337.warc.gz"} |
Schattschneider was born in Staten Island; her mother, Charlotte Lucile Ingalls Wood, taught Latin and was herself the daughter of a Staten Island school principal, and her father, Robert W. Wood,
Jr., was an electrical engineer who worked for the New York City Bureau of Bridge Design.^[5] Her family moved to Lake Placid, New York during World War II, while her father served as an engineer for
the U. S. Army; she began her schooling in Lake Placid, but returned to Staten Island after the war.^[5] She did her undergraduate studies in mathematics at the University of Rochester, and earned a
Ph.D. in 1966 from Yale University under the joint supervision of Tsuneo Tamagawa and Ichirô Satake with the thesis, Restricted Roots of a Semi-simple Algebraic Group.^[2]^[6] She taught at
Northwestern University for a year and at the University of Illinois at Chicago Circle for three years before joining the faculty of Moravian College in 1968, where she remained for 34 years until
her retirement.^[2]^[7] She was the first female editor of Mathematics Magazine, from 1981 to 1985.^[1]^[2]
She was married for 54 years to the Rev. Dr. David A. Schattschneider (1939-2016), a church historian and Dean of Moravian Theological Seminary; their daughter Laura Ellen Schattschneider is a
Schattschneider and Marjorie Rice
Marjorie Rice was an amateur mathematician and San Diego mother of five who became fascinated by Martin Gardner's descriptions of tessellations by pentagonal tiles in Scientific American. She
investigated, and devising her own notation system, had found a previously unknown type of pentagon tiling by February 1976. She drew up several tessellations by these new pentagon tiles and mailed
her discoveries to Martin Gardner. He, in turn, sent Rice's work to Schattschneider, who was an expert in tiling patterns. Schattschneider was skeptical at first, but upon careful examination, was
able to validate Rice's results.^[8] Schattschneider not only helped Martin Gardner publicize the pentagon tilings discoveries of Rice, but lauded her work as a significant discovery by an amateur
In 1995, at a regional meeting of the Mathematical Association of America held in Los Angeles, Schattschneider convinced Rice and her husband to attend her lecture on Rice's work. At the conclusion
of the talk, Schattschneider introduced the amateur mathematician who had advanced the study of tessellation. "And everybody in the room . . . gave her a standing ovation."^[8]^[11]
Awards and honors
Selected publications
• M. C. Escher Kaleidocycles (with Wallace Walker), Ballantine Books, 1977, Pomegranate Artbooks and TACO, 1987, Taschen 2015^[17]
• Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher (W. H. Freeman, 1990, 1992;
Revised as M. C. Escher: Visions of Symmetry, Harry N. Abrams, 2004)^[18]^[19]^[20]^[21]
• A Companion to Calculus (with Dennis Ebersole, Alicia Sevilla, and Kay Somers, Brooks/Cole, 1995)^[22]
Edited volumes
• Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research (with James King, Cambridge University Press, 1997)
• M.C. Escher's Legacy: A Centennial Celebration (with Michelle Emmer, Springer, 2003)
• Schattschneider, Doris (1978), "Tiling the plane with congruent pentagons" (PDF), Mathematics Magazine, 51 (1): 29–44, doi:10.2307/2689644, JSTOR 2689644;
Reprinted with Afterword in The Harmony of the World: 75 Years of Mathematics Magazine, eds. G. Alexanderson and P. Ross, Math. Assoc. of Amer., Washington DC, 2007, pp. 175-190.
• Schattschneider, Doris (1978), "The plane symmetry groups: Their recognition and notation", The American Mathematical Monthly, 85 (6): 439–450, doi:10.1080/00029890.1978.11994612, JSTOR 2320063.
• Schattschneider, Doris (1981), "In praise of amateurs", in Klarner, David A. (ed.), The Mathematical Gardner, Boston: Prindle, Weber & Schmidt, pp. 140–166;
Reprinted as Mathematical Recreations: A Collection in Honor of Martin Gardner, Dover Publications, New York, 1998.
• Schattschneider, Doris (1998), "One Corona is Enough for the Euclidean Plane," coauthor Nikolai Dolbilin. In Quasicrystals and Discrete Geometry (J. Patera, editor). Fields Institute Monographs,
Vol. 10, AMS, Providence, RI, 1998, pp. 207–246.
Accompanying web site: Catalog of Isohedral Tilings by Symmetric Polygonal Tiles
Further reading
• Schroeder, Tom (June 11, 1992), "Math Professor's Watchword: Visual", The Morning Call. | {"url":"https://www.knowpia.com/knowpedia/Doris_Schattschneider","timestamp":"2024-11-04T02:07:11Z","content_type":"text/html","content_length":"110924","record_id":"<urn:uuid:2c3832a2-03f3-4a59-a812-9aeebfad6088>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00755.warc.gz"} |
Array Issues Breaking For Loop
Jun 14, 2019 11:04 AM
Jun 14, 2019 11:04 AM
After much hair-pulling and manual debugging, I cannot resolve an issue that is breaking a calculation in my for loop. I have imported a dataset that I want to apply various moving average filters to
and output a statistical value on the data (mean, stdev, etc...) My issue is that identical arrays are either working or breaking my loop.
If I specify a manual vector of points, the calculation works. If I have a for loop to create an array of time for my moving average filter based on number of filters and max filter time (s), and
multiply by frequency (Hz), I get what seems to be an identical vector. One (the manual) is working, and one (the loop generated) fails, even though the look identical.
Image shown below.
Left side, manual "n_filter" enabled and automatically generated "n_filter" disabled.
Right side, manual "n_filter" disabled and automatically generated "n_filter" enabled.
Jun 14, 2019 12:48 PM
Jun 14, 2019 12:48 PM
Jun 14, 2019 11:44 AM
Jun 14, 2019 11:44 AM
Jun 14, 2019 01:06 PM
Jun 14, 2019 01:06 PM
Jun 14, 2019 12:48 PM
Jun 14, 2019 12:48 PM
Jun 14, 2019 01:05 PM
Jun 14, 2019 01:05 PM | {"url":"https://community.ptc.com/t5/Mathcad/Array-Issues-Breaking-For-Loop/td-p/614157","timestamp":"2024-11-02T11:16:24Z","content_type":"text/html","content_length":"286637","record_id":"<urn:uuid:b288f4cd-457b-4486-935c-2740d457a69f>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00223.warc.gz"} |
A representation formula for members of SBV dual
Published Paper
Inserted: 29 oct 2024
Journal: Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)
Volume: 25
Number: 1
Pages: 483-493
Year: 2024
Links: arxiv
We give an integral representation formula for members of the dual of $SBV(\mathbb{R}^n)$ in terms of functions that are defined on $\widehat{\mathbb{R}^n}$, an appropriate fiber space that we
introduce, consisting of pairs $(x,[E]_x)$ where $[E]_x$ is an approximate germ of an $(n-1)$-rectifiable set $E$ at $x$. | {"url":"https://cvgmt.sns.it/paper/6839/","timestamp":"2024-11-02T06:03:27Z","content_type":"text/html","content_length":"8247","record_id":"<urn:uuid:a5600c2e-5679-4752-ac4d-624a32520f17>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00381.warc.gz"} |
Disclaimer: Universe is NOT simple
"Why is the Universe so simple ?" asks the mathematician, or more generally, why is simple mathematics (school mathematics) so successful at describing the Universe ?
The Universe, however, is generally not simple to begin with. Rather there are some aspects of the Universe (which we happen to be interested in) that can be computed easily. Put one sheep next to
one sheep and you get two sheep (in the short term); so "putting next to each other" is isomorphic to a simple "+" operator. But what about the eddies and whorls in a ravine ? Cloud patterns ? And I
haven't even begun to ask *creative* questions here.
An arbitrary, low-Kolmogorov-complexity aspect of the Universe is very difficult to compute. We as a species, shaped by evolution, happen to be interested in many simple-to-compute aspects.
The question should rather be phrased: Why does the Universe have any simple-to-compute aspects at all ?
No comments: | {"url":"https://hagiograffiti.blogspot.com/2008/07/disclaimer-universe-is-not-simple.html","timestamp":"2024-11-12T20:22:11Z","content_type":"application/xhtml+xml","content_length":"44128","record_id":"<urn:uuid:92f4296c-8926-4556-ac38-62e9bc331c1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00113.warc.gz"} |
Documents For An Access Point
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# Author Title Accn# Year Item Type Claims
1 Alabiso, Carlo A Primer on Hilbert Space Theory I11622 2021 eBook
2 Patrascu, Andrei-Tudor The Universal Coefficient Theorem and Quantum Field Theory I09785 2017 eBook
3 Rudolph, Gerd Differential Geometry and Mathematical Physics I09737 2017 eBook
4 Tanaka, Hiro Lee Lectures on Factorization Homology, ???-Categories, and Topological Field Theories I09620 2020 eBook
5 Szabo, Richard J Equivariant Cohomology and Localization of Path Integrals I11316 2000 eBook
6 Carow-Watamura, Ursula Quantum Field Theory and Noncommutative Geometry I08223 2005 eBook
7 Alabiso, Carlo A Primer on Hilbert Space Theory I08208 2015 eBook
8 Schlichenmaier, Martin An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces I07756 2007 eBook
9 Jardine, John F Algebraic K-Theory: Connections with Geometry and Topology I03578 1989 eBook
10 Goerss, P.G Algebraic K-Theory and Algebraic Topology I02299 1993 eBook
(page:1 / 3) [#22] Next Page Last Page
Title A Primer on Hilbert Space Theory : Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups
Author(s) Alabiso, Carlo;Weiss, Ittay
Publication Cham, Springer International Publishing, 2021.
Description XXII, 328 p. 19 illus : online resource
Abstract This book offers an essential introduction to the theory of Hilbert space, a fundamental tool for non-relativistic quantum mechanics. Linear, topological, metric, and normed spaces are
Note all addressed in detail, in a rigorous but reader-friendly fashion. The rationale for providing an introduction to the theory of Hilbert space, rather than a detailed study of Hilbert
space theory itself, lies in the strenuous mathematics demands that even the simplest physical cases entail. Graduate courses in physics rarely offer enough time to cover the theory of
Hilbert space and operators, as well as distribution theory, with sufficient mathematical rigor. Accordingly, compromises must be found between full rigor and the practical use of the
instruments. Based on one of the authors???s lectures on functional analysis for graduate students in physics, the book will equip readers to approach Hilbert space and, subsequently,
rigged Hilbert space, with a more practical attitude. It also includes a brief introduction to topological groups, and to other mathematical structures akin to Hilbert space. Exercises
and solved problems accompany the main text, offering readers opportunities to deepen their understanding. The topics and their presentation have been chosen with the goal of quickly,
yet rigorously and effectively, preparing readers for the intricacies of Hilbert space. Consequently, some topics, e.g., the Lebesgue integral, are treated in a somewhat unorthodox
manner. The book is ideally suited for use in upper undergraduate and lower graduate courses, both in Physics and in Mathematics
ISBN,Price 9783030674175
Keyword(s) 1. ALGEBRAIC TOPOLOGY 2. EBOOK 3. EBOOK - SPRINGER 4. FUNCTIONAL ANALYSIS 5. Mathematical Methods in Physics 6. MATHEMATICAL PHYSICS
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Title The Universal Coefficient Theorem and Quantum Field Theory : A Topological Guide for the Duality Seeker
Author(s) Patrascu, Andrei-Tudor
Publication Cham, Springer International Publishing, 2017.
Description XVI, 270 p. 6 illus., 1 illus. in color : online resource
Abstract This thesis describes a new connection between algebraic geometry, topology, number theory and quantum field theory. It offers a pedagogical introduction to algebraic topology, allowing
Note readers to rapidly develop basic skills, and it also presents original ideas to inspire new research in the quest for dualities. Its ambitious goal is to construct a method based on the
universal coefficient theorem for identifying new dualities connecting different domains of quantum field theory. This thesis opens a new area of research in the domain of
non-perturbative physics???one in which the use of different coefficient structures in (co)homology may lead to previously unknown connections between different regimes of quantum field
theories. The origin of dualities is an issue in fundamental physics that continues to puzzle the research community with unexpected results like the AdS/CFT duality or the ER-EPR
conjecture. This thesis analyzes these observations from a novel and original point of view, mainly based on a fundamental connection between number theory and topology. Beyond its
scientific qualities, it also offers a pedagogical introduction to advanced mathematics and its connection with physics. This makes it a valuable resource for students in mathematical
physics and researchers wanting to gain insights into (co)homology theories with coefficients or the way in which Grothendieck's work may be connected with physics
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Keyword(s) 1. ALGEBRAIC TOPOLOGY 2. EBOOK 3. EBOOK - SPRINGER 4. Elementary particles (Physics) 5. Elementary Particles, Quantum Field Theory 6. Mathematical Applications in the Physical Sciences
7. MATHEMATICAL PHYSICS 8. Quantum Field Theories, String Theory 9. QUANTUM FIELD THEORY 10. STRING THEORY
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Title Differential Geometry and Mathematical Physics : Part II. Fibre Bundles, Topology and Gauge Fields
Author(s) Rudolph, Gerd;Schmidt, Matthias
Publication Dordrecht, Springer Netherlands, 2017.
Description XVI, 830 p. 15 illus., 2 illus. in color : online resource
Abstract The book is devoted to the study of the geometrical and topological structure of gauge theories. It consists of the following three building blocks: - Geometry and topology of fibre
Note bundles, - Clifford algebras, spin structures and Dirac operators, - Gauge theory. Written in the style of a mathematical textbook, it combines a comprehensive presentation of the
mathematical foundations with a discussion of a variety of advanced topics in gauge theory. The first building block includes a number of specific topics, like invariant connections,
universal connections, H-structures and the Postnikov approximation of classifying spaces. Given the great importance of Dirac operators in gauge theory, a complete proof of the
Atiyah-Singer Index Theorem is presented. The gauge theory part contains the study of Yang-Mills equations (including the theory of instantons and the classical stability analysis), the
discussion of various models with matter fields (including magnetic monopoles, the Seiberg-Witten model and dimensional reduction) and the investigation of the structure of the gauge
orbit space. The final chapter is devoted to elements of quantum gauge theory including the discussion of the Gribov problem, anomalies and the implementation of the non-generic gauge
orbit strata in the framework of Hamiltonian lattice gauge theory. The book is addressed both to physicists and mathematicians. It is intended to be accessible to students starting from
a graduate level
ISBN,Price 9789402409598
Keyword(s) 1. ALGEBRAIC GEOMETRY 2. ALGEBRAIC TOPOLOGY 3. DIFFERENTIAL GEOMETRY 4. EBOOK 5. EBOOK - SPRINGER 6. Elementary particles (Physics) 7. Elementary Particles, Quantum Field Theory 8.
Mathematical Methods in Physics 9. MATHEMATICAL PHYSICS 10. PHYSICS 11. QUANTUM FIELD THEORY
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Title Lectures on Factorization Homology, ???-Categories, and Topological Field Theories
Author(s) Tanaka, Hiro Lee
Publication Cham, Springer International Publishing, 2020.
Description XII, 84 p. 13 illus., 2 illus. in color : online resource
Abstract This book provides an informal and geodesic introduction to factorization homology, focusing on providing intuition through simple examples. Along the way, the reader is also introduced
Note to modern ideas in homotopy theory and category theory, particularly as it relates to the use of infinity-categories. As with the original lectures, the text is meant to be a leisurely
read suitable for advanced graduate students and interested researchers in topology and adjacent fields
ISBN,Price 9783030611637
Keyword(s) 1. ALGEBRAIC TOPOLOGY 2. Category theory (Mathematics) 3. Category Theory, Homological Algebra 4. EBOOK 5. EBOOK - SPRINGER 6. Homological algebra 7. MATHEMATICAL PHYSICS 8. Theoretical,
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Title Equivariant Cohomology and Localization of Path Integrals
Author(s) Szabo, Richard J
Publication Berlin, Heidelberg, Springer Berlin Heidelberg, 2000.
Description XI, 315 p : online resource
Abstract This book, addressing both researchers and graduate students, reviews equivariant localization techniques for the evaluation of Feynman path integrals. The author gives the relevant
Note mathematical background in some detail, showing at the same time how localization ideas are related to classical integrability. The text explores the symmetries inherent in localizable
models for assessing the applicability of localization formulae. Various applications from physics and mathematics are presented
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Keyword(s) 1. ALGEBRAIC TOPOLOGY 2. EBOOK 3. EBOOK - SPRINGER 4. Elementary particles (Physics) 5. Elementary Particles, Quantum Field Theory 6. GLOBAL ANALYSIS (MATHEMATICS) 7. Global Analysis and
Analysis on Manifolds 8. Manifolds (Mathematics) 9. Mathematical Methods in Physics 10. NUCLEAR PHYSICS 11. Particle and Nuclear Physics 12. PHYSICS 13. QUANTUM FIELD THEORY 14. TOPOLOGY
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Title Quantum Field Theory and Noncommutative Geometry
Author(s) Carow-Watamura, Ursula;Maeda, Yoshiaki;Watamura, Satoshi
Publication Berlin, Heidelberg, Springer Berlin Heidelberg, 2005.
Description X, 298 p : online resource
Abstract This volume reflects the growing collaboration between mathematicians and theoretical physicists to treat the foundations of quantum field theory using the mathematical tools of
Note q-deformed algebras and noncommutative differential geometry. A particular challenge is posed by gravity, which probably necessitates extension of these methods to geometries with
minimum length and therefore quantization of space. This volume builds on the lectures and talks that have been given at a recent meeting on "Quantum Field Theory and Noncommutative
Geometry." A considerable effort has been invested in making the contributions accessible to a wider community of readers - so this volume will not only benefit researchers in the field
but also postgraduate students and scientists from related areas wishing to become better acquainted with this field
ISBN,Price 9783540315261
Keyword(s) 1. ALGEBRAIC TOPOLOGY 2. DIFFERENTIAL GEOMETRY 3. EBOOK 4. EBOOK - SPRINGER 5. Elementary particles (Physics) 6. Elementary Particles, Quantum Field Theory 7. LIE GROUPS 8. Mathematical
Methods in Physics 9. PHYSICS 10. QUANTUM FIELD THEORY 11. TOPOLOGICAL GROUPS 12. Topological Groups, Lie Groups
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Title A Primer on Hilbert Space Theory : Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups
Author(s) Alabiso, Carlo;Weiss, Ittay
Publication Cham, Springer International Publishing, 2015.
Description XVII, 255 p. 5 illus : online resource
Abstract This book is an introduction to the theory of Hilbert space, a fundamental tool for non-relativistic quantum mechanics. Linear, topological, metric, and normed spaces are all addressed
Note in detail, in a rigorous but reader-friendly fashion. The rationale for an introduction to the theory of Hilbert space, rather than a detailed study of Hilbert space theory itself,
resides in the very high mathematical difficulty of even the simplest physical case. Within an ordinary graduate course in physics there is insufficient time to cover the theory of
Hilbert spaces and operators, as well as distribution theory, with sufficient mathematical rigor. Compromises must be found between full rigor and practical use of the instruments. The
book is based on the author's lessons on functional analysis for graduate students in physics. It will equip the reader to approach Hilbert space and, subsequently, rigged Hilbert space,
with a more practical attitude. With respect to the original lectures, the mathematical flavor in all subjects has been enriched. Moreover, a brief introduction to topological groups has
been added in addition to exercises and solved problems throughout the text. With these improvements, the book can be used in upper undergraduate and lower graduate courses, both in
Physics and in Mathematics
ISBN,Price 9783319037134
Keyword(s) 1. ALGEBRAIC TOPOLOGY 2. EBOOK 3. EBOOK - SPRINGER 4. FUNCTIONAL ANALYSIS 5. Mathematical Methods in Physics 6. PHYSICS
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Title Algebraic K-Theory: Connections with Geometry and Topology
Author(s) Jardine, John F;Snaith, V.P
Publication Dordrecht, Springer Netherlands, 1989.
Description XIV, 550 p : online resource
Abstract A NATO Advanced Study Institute entitled "Algebraic K-theory: Connections with Geometry and Topology" was held at the Chateau Lake Louise, Lake Louise, Alberta, Canada from December 7 to
Note December 11 of 1987. This meeting was jointly supported by NATO and the Natural Sciences and Engineering Research Council of Canada, and was sponsored in part by the Canadian
Mathematical Society. This book is the volume of proceedings for that meeting. Algebraic K-theory is essentially the study of homotopy invariants arising from rings and their associated
matrix groups. More importantly perhaps, the subject has become central to the study of the relationship between Topology, Algebraic Geometry and Number Theory. It draws on all of these
fields as a subject in its own right, but it serves as well as an effective translator for the application of concepts from one field in another. The papers in this volume are
representative of the current state of the subject. They are, for the most part, research papers which are primarily of interest to researchers in the field and to those aspiring to be
such. There is a section on problems in this volume which should be of particular interest to students; it contains a discussion of the problems from Gersten's well-known list of 1973,
as well as a short list of new problems
ISBN,Price 9789400923997
Keyword(s) 1. ALGEBRAIC GEOMETRY 2. ALGEBRAIC TOPOLOGY 3. EBOOK 4. EBOOK - SPRINGER 5. K-THEORY
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Title Algebraic K-Theory and Algebraic Topology
Author(s) Goerss, P.G;Jardine, John F
Publication Dordrecht, Springer Netherlands, 1993.
Description XII, 328 p : online resource
Abstract A NATO Advanced Study Institute entitled "Algebraic K-theory and Algebraic Topology" was held at Chateau Lake Louise, Lake Louise, Alberta, Canada from December 12 to December 16 of
Note 1991. This book is the volume of proceedings for this meeting. The papers that appear here are representative of most of the lectures that were given at the conference, and therefore
present a "snapshot" of the state ofthe K-theoretic art at the end of 1991. The underlying objective of the meeting was to discuss recent work related to the Lichtenbaum-Quillen complex
of conjectures, fro~ both the algebraic and topological points of view. The papers in this volume deal with a range of topics, including motivic cohomology theories, cyclic homology,
intersection homology, higher class field theory, and the former telescope conjecture. This meeting was jointly funded by grants from NATO and the National Science Foun?? dation in the
United States. I would like to take this opportunity to thank these agencies for their support. I would also like to thank the other members of the organizing com?? mittee, namely Paul
Goerss, Bruno Kahn and Chuck Weibel, for their help in making the conference successful. This was the second NATO Advanced Study Institute to be held in this venue; the first was in
1987. The success of both conferences owes much to the professionalism and helpfulness of the administration and staff of Chateau Lake Louise
ISBN,Price 9789401706957
Keyword(s) 1. ALGEBRA 2. ALGEBRAIC GEOMETRY 3. ALGEBRAIC TOPOLOGY 4. Associative rings 5. Associative Rings and Algebras 6. EBOOK 7. EBOOK - SPRINGER 8. K-THEORY 9. Order, Lattices, Ordered
Algebraic Structures 10. Ordered algebraic structures 11. Rings (Algebra)
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Department of Mathematics
Abstract :
We shall discuss a theorem of DN Bernstein about counting roots of polynomials using mixed volumes of Newton polytopes. We shall give a proof for roots of binomials using Hermite normal form of an
integer matrix and sketch an algebraic proof using Hilbert functions of multi- graded algebras.
Key Speaker Prof. J. K. Verma (Department of Mathematics, IIT Mumbai)
Place KCB 522
Start Time 3:00 PM
Finish Time 4:00 PM
External Link None | {"url":"https://math.iitm.ac.in/event/view/157","timestamp":"2024-11-15T00:55:52Z","content_type":"text/html","content_length":"30098","record_id":"<urn:uuid:8a527d35-55f8-4bb6-8e2c-fe6c090be448>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00359.warc.gz"} |
Guilhem Semerjian
Sep 27, 2022
Abstract:In this paper we address the problem of testing whether two observed trees $(t,t')$ are sampled either independently or from a joint distribution under which they are correlated. This
problem, which we refer to as correlation detection in trees, plays a key role in the study of graph alignment for two correlated random graphs. Motivated by graph alignment, we investigate the
conditions of existence of one-sided tests, i.e. tests which have vanishing type I error and non-vanishing power in the limit of large tree depth. For the correlated Galton-Watson model with Poisson
offspring of mean $\lambda>0$ and correlation parameter $s \in (0,1)$, we identify a phase transition in the limit of large degrees at $s = \sqrt{\alpha}$, where $\alpha \sim 0.3383$ is Otter's
constant. Namely, we prove that no such test exists for $s \leq \sqrt{\alpha}$, and that such a test exists whenever $s > \sqrt{\alpha}$, for $\lambda$ large enough. This result sheds new light on
the graph alignment problem in the sparse regime (with $O(1)$ average node degrees) and on the performance of the MPAlign method studied in Ganassali et al. (2021), Piccioli et al. (2021), proving in
particular the conjecture of Piccioli et al. (2021) that MPAlign succeeds in the partial recovery task for correlation parameter $s>\sqrt{\alpha}$ provided the average node degree $\lambda$ is large
* 27 pages, all comments welcome | {"url":"https://www.catalyzex.com/author/Guilhem%20Semerjian","timestamp":"2024-11-09T05:46:44Z","content_type":"text/html","content_length":"65039","record_id":"<urn:uuid:4c84a821-0c78-4956-81da-66a4cce46c91>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00634.warc.gz"} |
Mathematics Münster: Dynamics - Geometry - Structure
Mathematics Münster
Dynamics - Geometry - Structure
Mathematics is a key technology for scientific and economic progress. New discoveries in mathematics are not only interesting in themselves, but they often lead to unexpected breakthroughs in other
sciences as well.
We will tackle fundamentally important mathematical problems, viewing mathematics as an organic whole with countless interactions. The research in our Cluster of Excellence "Mathematics Münster" is
unified by three major approaches: focusing on the underlying structure of a given problem, taking the geometric viewpoint and studying the relevant dynamics of group or semigroup actions. The
theories which we will build will not only solve the problems under consideration but also many others of a similar nature; these theories will also raise exciting new questions.
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14 November 2024, 2:15 pm
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© MM
See more news from Mathematics Münster!
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Region of Convergence, Properties, Stability and Causality of Z-transforms
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Region of Convergence, Properties, Stability and Causality of Z-transforms
Keerthana Jaikumar | Published May 29, 2020 | Updated June 8, 2020
What is the Region of Convergence of Z-transform
• The S-plane is capable of representing signals such as complex exponentials. When performing Z transform on these signals, we represent these signals on an imaginary and real axis.
• The output of these transformations could turn out to meet at a point or go in different directions. The particular signals that tend to meet at a point are the signals that lie in the Region of
• The Region of Convergence maps all the values for which the transform converges to a finite value.
• Recall the definition of Z-transform. You will remember that the limits of the summation were from -∞ to +∞. However, the Z-transform will exist only for those values of Z, which if put in this
series results in a finite value.
• In simple words, the ROC is a region in the Z-plane consisting of all the values of Z which make the Z-transform (X(Z)) attain a finite value.
• The Region of Convergence is required to determine:
□ the stability of a system by examining the transfer function.
□ whether the system is causal or non-causal.
□ whether the system is finite or infinite.
• A stable system is one that produces a bounded output for a bounded input. It can be represented graphically in terms of circles together with a unit circle.
RoC is a representation of where the Z-transform exists. Hence, for a given x[n], the RoC is the range of z for which the output converges. The only condition is that x(z) should not be infinity for
any value of z. Mathematically,
$X(z)=\sum _{ n=-\infty }^{ \infty }{ \left| x[n]{ z }^{ -n } \right| } <\infty$
Properties of Region of Convergence of Z-transform
The characteristics of the Region of Convergence are almost analogous to that of the original signal x[n]. Let us look at a few points we should take note of before going about finding out the RoC:
Property 1: The Region of Convergence is a ring or circle in the Z-plane centred about the origin
Explanation: The region of convergence is calculated in the form of the minimum value of Z beyond/within which the transform exists. Geometrically, this relation is possible only with a circle.
Property 2: The Region of Convergence does not contain any pole.
Explanation: Remember the condition that we talked about previously about the RoC not including points that go to infinity, there is a reason. The RoC can only contain Zeros and not Poles. And since
Poles are the points where X(z) is infinite, they can’t be included in the RoC.
Property 3: When the Region of Convergence incorporates a unit circle, X(z) converges uniformly
Explanation: For any finite x[n] that includes the unit circle, its ROC exists in the range of z for which its sum converges. The z transform represented by the equation is basically a power series.
Power series is a series in which you have a variable x and C[n] which are constants that are added up and represented by the general equation aₙ(x-a)ⁿ. A power series can either converge or diverge.
The ROC converges when x[n]z^-n is absolutely summable. Represented mathematically,
$|X(z)|=\sum _{ n=-\infty }^{ \infty }{ \left| x[n]{ z }^{ -n } \right| } =\sum _{ n=-\infty }^{ \infty }{ \left| x[n] \right| } (\left| { z } \right| )^{ -n }$
Property 4: With a finite x[n], your ROC is the entire z plane except when z=0 and z=$\infty$
Explanation: We know that when x[n] is finite, its values are summable. Hence, z cannot be equal to 0 as the summation of finite values will give us a finite value. Similarly, z cannot be equal to
infinity as well. Hence, the conclusion that the ROC will be the entire z plane except the aforementioned two values. There are a few exceptions that we will be looking at in the next few properties.
Example: Given x[n] find the RoC
The arrow mark indicates the value for which n=0
Hence, when n=0, x[n]=2
Next, we have to find x(z)
$x(z)=\sum _{ n=-1 }^{ 2 }{ x(n){ z }^{ -n } } =x(-1){ z }^{ 1 }+x(0){ z }^{ 0 }+x(1){ z }^{ -1 }+x(2){ z }^{ -2 }$
if z were equal to 0, the values for which z is in the denominator would tend to infinity.
if z were equal to ∞, the values for which z is in the numerator would tend to infinity.
Hence, the RoC will be the entire z-plane except for when z=0 and z=∞
Property 5: The ROC includes $\infty$ if the signal x[n] is causal
Explanation: Conditions for causality include:
1. When the RoC includes the region outside the outermost pole of the signal.
2. When the transfer function is expressed as a ratio of the output to the input of the system, the order of the numerators should be of an order lesser than the order of the denominator.
Example: For the system $H(z)=\frac { z }{ (z-0.2)(z-0.6) }$, the poles of the transfer function will be z=0.2 and z=0.6
Since both the poles fall inside the unit circle, the system is causal and stable.
Property 6: The ROC includes 0 if the signal x[n] is non-causal
Explanation: When the aforementioned conditions are not met, the ROC would include 0 instead of $\infty$.
Example: $H(z)=\frac { 2z+1 }{ { z }^{ 2 }+z-\frac { 5 }{ 16 } }$
Complete the square in the denominator to get the poles $H(z)=\frac { 2z+1 }{ (z+\frac { 5 }{ 4 } )(z-\frac { 1 }{ 4 } ) }$
Hence, the poles are at z=-5/4 and z=1/4
z=1/4 falls within the unit circle but z=-1.25 falls outside the unit circle
Hence, the system is non-causal and unstable.
Property 7: If x[n] is both causal and non-causal, the ROC will be a ring.
Explanation: If the n values fall on both sides of the x-axis, the ROC will be a ring around the radius in the z-plane.
Let us look at an example combining an RSS and an LSS.
where $x(n)\longrightarrow [latex]x(z)=\frac { z }{ z-r }$
The z transform of both components of x(n) would be the same. However, the denominator would change based on the z value. The RSS would point outside the circle, and the LSS would point inside the
circle, which means they would cancel out and the RoC would be the ring around the radius at modulus of z equal to r.
z-plane of BSS
Property 8: The ROC is bounded by poles if x[n] is rational, which means it would extend to $\infty$
Explanation: If the values of x[n] were rational and bounded by poles, the RoC is extended to infinity. The RoC will have combined properties of both poles and fulfill the limits of both.
Conditions: More than one component in x[n] and both are rational, which means it has two poles.
Example: $x(n)={ 4 }^{ n }u(n)-{ 5 }^{ n }u(-n-1)$
The z transform of the individual components will be $\frac { z }{ z-4 }$ and $\frac { z }{ z-5 }$.
The first component is RSS which means the limits will be |z|>4 and the second component is LSS which means the limits will be |z|<5.
So, combining both the limits RoC will fall in the range 4<|z|<5 as shown by the shaded region in the figure below.
Z-plane of rational x[n]
Property 9: For a right sided sequence (RSS) x[n], the ROC will not comprise 0
Explanation: Two conditions, if:
• x[n] is entirely positive or what we call a Right Sided Sequence (RSS), as it falls on the right of the x-axis, and
• $\left| z \right| =r$ in the RoC
Then all finite values of z for which $\left| z \right| >r$ , will fall in the RoC.
Example: What signal can you think of that is purely a Right-Sided Sequence? A unit step function is all positive, so let's try with that.
For x(n)=f(n).u(n) the z transform will be $x(n)\longrightarrow x(z)=\frac { z }{ z-r }$
For a positive result, the value of z should be more than r to avoid a minus in the denominator. So, the modulus of z should be more than the radius.
Z-plane for RSS
Therefore, the RoC lies outside the circle from the pole position.
Property 10: When x[n] is rational, and both poles are RSS, the ROC will be outside the outermost pole.
Explanation: By rational, it means that x[n] has two poles. If both components of x[n] are right-sided, we can say that the RoC will fall in the region outside the outermost pole of the two
Example: $x(n)={ 4 }^{ n }u(n)+{ 5 }^{ n }u(-n-1)$
The z transform of the individual components would be $\frac { z }{ z-4 }$ and $\frac { z }{ z-5 }$
but since both components are RSS, Z will be greater than the respective radii.
The limits will end up being |z|>4 and |z|>5
Z-plane of ration and RSS signal
Now, we need the RoC to satisfy both conditions. |z|>4 will include the values between 4 and 5, not satisfying |z|>5. But |z|>5 will satisfy both the conditions. Hence, choose the limits of the
outermost pole, which in this case is |z|>5 (the red arrows).
Property 11: For a left sided sequence(LSS) x[n], the ROC will not comprise $\infty$
Explanation: If x[n] is entirely negative, it would be a Left-Sided Sequence (LSS) as the equation would fall on the left side of the x-axis, and when |z|=r for all finite values of z for which $\
left| z \right| <r$, they will fall in the RoC.
Example: Can you think of any signal that lies solely on the left side of the x-axis? How about the negative of a unit step function?
Hence, let's take x(n)=f(n).u(-n)
the z transform would also be $x(n)\longrightarrow x(z)=\frac { z }{ z-r }$
However, since we want it to be on the left side, we need the denominator to be a negative value. So, the modulus of z should be less than r.
Z-plane of LSS
Property 12: When x[n] is rational and LSS, ROC will be inside the innermost pole.
Explanation: If the z-transform of x[n] is rational, that is it has more than one pole, and the finite values of x[n] lie on the left side of the x-axis. The RoC will fall in the region inside the
innermost pole of the two poles of x[n].
Example: $x(n)={ 4 }^{ n }u(n)+{ 5 }^{ n }u(-n-1)$
The z transform of the individual components would be $\frac { z }{ z-4 }$ and $\frac { z }{ z-5 }$
but since both components are RSS, Z will be greater than the respective radii.
The limits will end up being |z|>4 and |z|>5
Again, RoC should satisfy both conditions, if we take |z|<5, for the values between 4 and 5,|z|<4 will not be satisfied. However, |z|<4 covers both conditions. Hence, choose the limits of the
innermost pole, which in this case is |z|<4.
Steps to find RoC and Z-transform
Here is a basic outline as to how to approach an RoC problem.
Step 1: Identify the point at the origin.
Step 2: Find out X(z) with the equation $X(z)=\sum _{ n=-\infty }^{ \infty }{ \left| x[n]{ z }^{ -n } \right| } <\infty$ for the limits determined from x[n].
Step 3: Identify whether the value of X(z) goes to infinity at any point (especially when z=0 and z=∞).
Step 4: Your RoC will be the entire z plane except for the region that you figured out in step 3.
Stability of an LTI system (What it is and conditions)
An LTI system is said to be stable if, for an input that is bounded, the output of the system is also bounded for all values of n.
$|x[n]|<{ n }_{ 1 }\quad$ for which $|y[n]|<\infty$
y[n] is commonly known as the convolution of x[n] and the impulse response, h[n]. This can also be represented as the summation expression shown below:
$y[n]=x[n]*h[n]=\sum _{ m=-\infty }^{ \infty }{ h[m]x[n-m] }$ this should be a finite value
hence, the terms inside the summation should not lead to infinity which means we can come to the conclusion that
$\sum _{ m=-\infty }^{ \infty }{ |h[m]|<\infty }$
For the impulse response to be finite, we need to ensure that h[m] is absolutely integrable. This would ensure that the system will be stable. Hence, the bottom line is that we need an absolutely
summable impulse response.
Energy signals are absolutely integrable
This stability of a system can also be determined using the RoC by fulfilling a couple of conditions
1. The system's transfer function H(z) should include the unit circle.
2. Also, for a causal LTI system, all the poles should lie within the unit circle. Read on to find out more about the causality of an LTI system.
BIBO stability of an LTI system
Assuming all the initial conditions of an LTI system are zero when the system output is bounded for each and every bounded input, it is referred to as a BIBO (Bounded Input Bounded Output System). By
bounded, we mean the absolute value of a signal does not exceed some finite constant.
For a continuous output y(t)
$y(t)=\int _{ -\infty }^{ \infty }{ h(\lambda )x(t-\lambda )\quad d\lambda }$
Therefore, for a bounded input $\left| x(t-\lambda ) \right| \le N$
y(t) will also be bounded like below
$\left| y(t) \right| \le N\int _{ -\infty }^{ \infty }{ h(\lambda )d\lambda } \le M$
Hence, the conclusion that we can make from these expressions is
$\int _{ -\infty }^{ \infty }{ \left| h(\lambda ) \right| } d\lambda <\infty$
With this information, let us determine the BIBO stability and/or causality of the following system:
Example: h(t)=u(t+1)
this response can be represented by the graph below
Graph of h(t)=u(t+1)
From the graph, we can see clearly that h(t)=0 for all t<0, hence it is not causal. Now let's take the integral to get an idea about the stability.
$\int _{ -\infty }^{ \infty }{ u(\lambda +1)\quad d\lambda =\int _{ -1 }^{ \infty }{ d\lambda =\infty +1 } }$
The output is not bounded as it goes to infinity, hence, the system is not BIBO stable as well.
Causality of an LTI system (What it is and conditions)
If you can recall what we discussed in the previous post about causality, it means the output of the LTI system depends on the present and past input values of x[n]. If there is a future value, it is
automatically not causal.
We know that
$y[n]=x[n]*h[n]=\sum _{ m=-\infty }^{ \infty }{ h[m]x[n-m] } =\sum _{ m=0 }^{ \infty }{ h[m]x[n-m] }$
when the output of y[n] is 0, for a causal function, n should be less than 0 or 0 itself.
Therefore, we can conclude that for n<0 h[m] will be 0.
1. Causality is satisfied when the RoC falls in the region outside the outermost pole of the signal.
2. When the transfer function is expressed as a ratio of the output to the input of the system, the order of the numerators should be of an order lesser than the order of the denominator.
About the author
Keerthana is currently pursuing her B.Tech in Electronics and Communication Engineering from Vellore Institute of Technology (Chennai). She is passionate about cryptography and doing projects around
microcontroller-based platforms such as the Arduino and Raspberry Pi. She has found the knowledge of Digital Signal Processing very helpful in her pursuits and wants to help teach the topic to help
others develop their own projects and find a penchant for the subject.
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What’s in my bin?
For any means to the monitoring of bin, tank or silo inventories, a first question invariably asked is, "How accurate is it?"
Unfortunately, it’s a loaded question. It can’t be answered easily. But here’s a discussion about why it’s a tough question, and what can be expected from a level-monitoring system.
First to consider is the type of device used. Bobs, guided-wave radar, open-air radar and ultrasonic level sensors are the common currency. What they have in common is a strategy based on a
measurement taken from a single point in the vessel. While each device has its pros and cons, installed properly any can be expected to perform to stated measurement accuracy.
But, what does the stated accuracy of measurement mean? For a single-point device, accuracy is often expressed based on the distance measured from the tank top’s sensor to the material surface. This
distance is often referred to as "headroom," because it delineates how much space is left in the bin. This can usually be accurately expressed within ± 0.25 percent of the total distance measured.
However, that’s not a direct indicator of the accuracy of the bin material’s calculated volume or mass; it’s simply the accuracy of that single measurement.
What’s really wanted is the volume, the amount of three-dimensional space the material takes up. If what’s known is the distance from the sensor to the material surface, the estimated volume is
derived based on that fact and the internal vessel dimensions. Known, accurate vessel dimensions preclude the mistakes that can lead to calculation error. Material flow, buildup or bridging can
affect volume calculations. Sensor placement and filling and discharge point locations also impact overall accuracy of volume calculation.
Mass calculations
Accuracy can be further impacted when using a single-point measurement device to estimate mass or weight. When converting volume to mass, the bulk density of the material — stated in pounds per cubic
foot or lb./ft.^3 — can significantly impact accuracy. Although readily available resources reference a particular material’s bulk density, that of the material actually in the bin can be quite
different than what’s posted on the Internet.
Particle shape, size and moisture content all impact bulk density. Material compaction also can cause very different bulk densities at bin top and bottom, for the same material. A cubic foot of
material at the top of the bin could weigh less than that same material at the bottom, where the bulk density is greater due to compaction by material weight.
When using bulk density to calculate bin mass for a particular material, it’s important to establish an average bulk density based upon the actual material, not an amount stated in a reference table.
One way to do this is taking measurements before and after a "known-weight" load is put in the bin, and then adjust the bulk density in lb./ft.^3 to reflect this weight.
For single-point level measuring, further error is expected from the conversion of distance to volume and then mass.
As stated, measured-distance accuracy is around ± 0.25 percent. However, when level is used to estimate volume, accuracy depends upon correct, complete vessel dimensions, as well as information about
sensor placement and location, and number of filling and discharge points. A vessel that is center-fill, center-discharge, with material flow free and symmetric, gives the best results with
single-point measuring.
Inherent inaccuracies when converting a volume calculation to a mass calculation are due to bulk- density variations, regardless of whether single-point or multi-point measuring is used. Volume
calculation accuracy impacts mass calculation accuracy.
With so many variables, single-point level-measurement device makers struggle to pinpoint how accurate the calculated mass value will be. However, with accurate vessel geometry, strategic sensor
placement of the sensor, and a good average bulk density, mass accuracy of around 8 percent to 15 percent can be expected.
Multi-point accuracy
Unlike traditional devices that measure distance from a single point, a "3DLevelscanner" from BinMaster measures from multiple points within the silo to determine bin material volume. Measurement
points are not averaged to calculate bin volume. Instead, each point is weighted for relevancy. A complex algorithm calculates the bin’s true material volume. By this means variations across the
topography of the material surface are taken into account through measuring and mapping of high and low points.
The scanner configures an accurate three-dimensional profile of the storage material’s top surface. This is a benefit when there are material-surface variations due to multiple fill and discharge
points, or with materials such as powders that do not fill or discharge symmetrically.
Volume accuracy remains dependent upon vessel dimension accuracy and sensor placement. When converting volume to mass, inherent inaccuracies persist due to bulk density variables. But, the improved
accuracy of the volume calculation will improve mass calculation accuracy.
In this case, "more is better," with multiple measurement points contributing to a higher degree of accuracy. Given correct vessel geometry and proper sensor placement, you can expect volume accuracy
of 3 percent to 5 percent. When combined with a good average bulk density, the accuracy of the mass may be around 5 percent to 10 percent.
Jenny Nielson Christensen, MBA, is vice president of marketing with BinMaster.
Established in 1953, BinMaster makes reliable, solid-state point and continuous bin level indicator and control systems and sensor devices used while storing or processing powder & bulk solids such
as cement & aggregates, chemicals, feed & grain, food, plastics, biofuels, pharmaceuticals, pulp & paper and wood products.
Sponsored Recommendations | {"url":"https://www.processingmagazine.com/home/article/15583650/whats-in-my-bin","timestamp":"2024-11-01T22:20:27Z","content_type":"text/html","content_length":"267873","record_id":"<urn:uuid:956a32a9-1686-486c-a14d-f50c56975e8d>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00705.warc.gz"} |
Months calculator - Months counter - Calculate the number of months between dates - ARAHR
1. Tools
2. Months Calculator
Months Calculator
For helping you determine the precise number of months and days between two dates, we developed the months calculator, also known as the month counter. Put otherwise, this easy-to-use application
will count the month between the date you specify and the end of the specified time frame.
0 months
Note: Partial months are displayed as additional days.
With the months counter, you can calculate the number of months between two dates. You can then get the exact and instant calculation for the number of months.
Tip: If your benefits are pro-rated based on the number of months of work, this might be valuable tool for you.
How to use the months calculator?
To get the desired result, provide the start and end dates, and then the calculator will give the number of months with the details.
How to calculate the number of months between dates?
Here's a step-by-step method for finding the number of months between two dates:
Step 1: Identify the start and end dates
Let's say you want to calculate the months between January 1, 2020, and July 15, 2022.
Example: Start date: January 1, 2020 End date: July 15, 2022
Step 2: Calculate the total number of days between the start and end dates
To do this, you can subtract the start date from the end date:
End date - Start date = July 15, 2022 - January 1, 2020
2022 - 2020 = 2 years
July (month 7) - January (Month 1) = 6 months in between
15-1 = 14 days
Therefore, 2 years, 6 months, and 14 days
You can use the days calculator to find the number of days between.
Step 3: Convert the total number of days to months
To convert the total number of days to months, you can divide the total number of days by 30 (assuming a 30-day month). This will give you an approximate number of months.
2 years, 6 months, and 14 days ≈ 2 years × 12 months/year = 24 months + 6 months = 30 months
Step 4: Calculate the final number of months
Using the above adjustments, we can calculate the final number of months.
Therefore, between January 1, 2020, and July 15, 2022, we have 2 years, 6 months, and 14 days, resulting in a total of 30 months and 14 days.
What is the formula to find the number of months in Excel?
To find the number of months between two dates in Excel, you can use the following formula:
=DATEDIF(start_date, end_date, "M")
• start_date is the starting date
• end_date is the ending date
• "M" stands for months
This formula will return the number of full months between the two dates. If you want to include the current month, you can use the following formula:
=MONTH(end_date) - MONTH(start_date) + 1
This formula will return the number of months, including the current month.
Months of the year. | {"url":"https://arahr.com/months-calculator/","timestamp":"2024-11-07T19:37:06Z","content_type":"text/html","content_length":"350944","record_id":"<urn:uuid:8a78d836-b628-4d64-916a-dbc8e8e33fa6>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00109.warc.gz"} |
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1.2 Physical Quantities and Units.
3 1.2 Physical Quantities and Units.
Mechanics is a quantitative science which means we will describe human movement and its causes using numbers. To provide information about a movement, we have to be able to specify how it is
measured. For example, we define distance and time by specifying methods for measuring them, whereas we define average speed by stating that it is calculated as distance traveled divided by time of
Measurements of physical quantities are expressed in terms of units, which are standardized values. For example, the length of a race, which is a physical quantity, can be expressed in units of
meters (for sprinters) or kilometers (for distance runners). Without standardized units, it would be extremely difficult for scientists to express and compare measured values in a meaningful way.
(See Figure 1 below.)
Figure 1. Distances given in unknown units are maddeningly useless.
There are two major systems of units used in the world: SI units (also known as the metric system) and English units (also known as the customary or imperial system). English units were historically
used in nations once ruled by the British Empire and are still widely used in the United States. Virtually every other country in the world now uses SI units as the standard; the metric system is
also the standard system agreed upon by scientists and mathematicians. The acronym “SI” is derived from the French Système International.
SI Units: Fundamental and Derived Units
The metric or SI system is administered in France by the Bureau International des Poids and Mesures or BIPM. You can read more about them at https://www.bipm.org/en/about-us/
Table 1 below shows the fundamental SI units that are used throughout this textbook.
Length Mass Time
meter (m) kilogram (kg) second (s)
Table 1. Fundamental SI Units.
It is an intriguing fact that some physical quantities are more fundamental than others and that the most fundamental physical quantities can be defined only in terms of the procedure used to measure
them. The units in which they are measured are thus called fundamental units. In this textbook, the fundamental physical quantities are taken to be length, mass and time. All other physical
quantities, such as force and velocity, can be expressed as algebraic combinations of length, mass and time; these units are called derived units.
Units of Time, Length, and Mass: The Second, Meter, and Kilogram
The Second
The SI unit for time, the second (abbreviated s), has a long history. For many years it was defined as 1/86,400 of a mean solar day. More recently, a new standard was adopted to gain greater accuracy
and to define the second in terms of a non-varying, or constant, physical phenomenon (because the solar day is getting longer due to very gradual slowing of the Earth’s rotation).
The Meter
The SI unit for length is the meter (abbreviated m); its definition has also changed over time to become more accurate and precise. In 1983, the meter was given its present definition (partly for
greater accuracy) as the distance light travels in a vacuum in 1/299,792,458 of a second. This change defines the speed of light to be exactly 299,792,458 meters per second. The length of the meter
will change if the speed of light is someday measured with greater accuracy.
Figure 2. The meter is defined to be the distance light travels in 1/299,792,458 of a second in a vacuum. Distance traveled is speed multiplied by time.
The Kilogram
The SI unit for mass is the kilogram (abbreviated kg); it is defined to be the mass of a platinum-iridium cylinder kept with the old meter standard at the International Bureau of Weights and Measures
near Paris.
In Biomechanics, all pertinent physical quantities can be expressed in terms of these fundamental units of length, mass, and time.
Metric Prefixes
SI units are part of the metric system. The metric system is convenient for scientific and engineering calculations because the units are categorized by factors of 10. The table below gives metric
prefixes and symbols used to denote various factors of 10.
Metric systems have the advantage that conversions of units involve only powers of 10. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on.
Prefix Symbol Value Example (some are approximate)
kilo k 10^3 kilometer km 10^3 m about 6/10 mile
hecto h 10^2 hectoliter hL 10^2 L 26 gallons
deka da 10^1 dekagram dag 10^1 g teaspoon of butter
— — meter
deci d 10^-1 deciliter dL 10^-1 L less than half a soda
centi c 10^-2 centimeter cm 10^-2 m fingertip thickness
milli m 10^-3 millimeter mm 10^-3 m flea at its shoulders
Table 2. Select Metric Prefixes for Powers of 10 and their Symbols.
Physical quantities are a characteristic or property of an object that can be measured or calculated from other measurements.
• Units are standards for expressing and comparing the measurement of physical quantities. All units can be expressed as combinations of three fundamental units.
• The three fundamental units we will use in this text are the meter (for length), the kilogram (for mass) and the second (for time). These units are part of the metric system, which uses powers of
10 to relate quantities over the vast ranges encountered in nature.
• The three fundamental units are abbreviated as follows: meter, m; kilogram, kg; second, s. The metric system also uses a standard set of prefixes to denote each order of magnitude greater than or
lesser than the fundamental unit itself.
physical quantity
a characteristic or property of an object that can be measure or calculated from other measurements
a standard used for expressing and comparing measurements
SI units
the international system of units that scientist in most countries have agreed to use; includes units such as meters, liters, and grams
English units
system of measurement used in the United States; includes units of measurement such as feet, gallons, and pounds
fundamental units
units that can only be expressed relative to the procedure used to measure them
derived units
units that can be calculated using algebraic combinations of the fundamental units
the SI unit for time, abbreviated (s)
the SI unit for length, abbreviated (m)
the SI unit for mass, abbreviated (kg)
metric system
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The Pythagorean Theorem
(This lesson also appears in the Algebra I curriculum. If you studied it there, then just quickly review, and move on to the next section!)
A 90° angle is called a right angle. A right triangle is a triangle with a 90° angle.
In a right triangle, the side opposite the $\,90^\circ$ angle is called the hypotenuse and the remaining two sides are called the legs.
The angles in any triangle add up to $\,180^\circ\,.$
In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. Thus, in a right triangle, the hypotenuse is always the longest side.
The Pythagorean Theorem gives a beautiful relationship between the lengths of the sides in a right triangle: the sum of the squares of the shorter sides is equal to the square of the hypotenuse.
Furthermore, if a triangle has this kind of relationship between the lengths of its sides, then it must be a right triangle!
The Pythagorean Theorem
Let $\,\,T\,\,$ be a triangle with sides of lengths $\,a\,,$ $\,b\,,$ and $\,c\,,$ where $\,c\,$ is the longest side (if there is a longest side). Then: $$ \begin{gather} \cssId{s21}{\text{$T\,$ is a
right triangle}}\cr \cssId{s22}{\text{if and only if}}\cr \cssId{s23}{a^2 + b^2 = c^2} \end{gather} $$
Have fun with many proofs of the Pythagorean Theorem! This applet (you'll need Java) is one of my favorites. Let it load, then keep pressing ‘Next’.
Question: Suppose that two angles in a triangle are $\,60^\circ$ and $\,30^\circ.$ Is it a right triangle? Answer YES, NO, or MAYBE.
Solution: Yes. The third angle must be $\,180^\circ - 60^\circ - 30^\circ = 90^\circ\,.$
Question: Suppose that a triangle has a $\,100^\circ$ angle. Is it a right triangle? Answer YES, NO, or MAYBE.
Solution: No. The remaining two angles must sum to $\,80^\circ,$ so neither remaining angle is a $\,90^\circ$ angle.
Question: Suppose that a triangle has a $\,70^\circ$ angle. Is it a right triangle? Answer YES, NO, or MAYBE.
Solution: Maybe. The remaining two angles must sum to $\,110^\circ,$ so one of the remaining angles could be a $\,90^\circ$ angle.
Question: Suppose the legs of a right triangle have lengths $\,3\,$ and $\,x\,,$ and the hypotenuse has length $\,5\,.$ Find $\,x\,.$
Solution: $$ \begin{gather} \cssId{s53}{3^2 + x^2 = 5^2}\cr \cssId{s54}{9 + x^2 = 25}\cr \cssId{s55}{x^2 = 16}\cr \cssId{s56}{x = 4} \end{gather} $$
Note: $x\,$ cannot equal $\,-4\,,$ because lengths are always positive.
The $\,3{-}4{-}5\,$ triangle is a well-known right triangle. Multiplying all the sides of a triangle by the same positive number does not change the angles. Thus, if you multiply the sides of a $\,3
{-}4{-}5\,$ triangle by any positive real number $\,k\,,$ then you will still have a right triangle. For example, these are all right triangles:
$6{-}8{-}10$ ($\,k = 2\,$)
$9{-}12{-}15$ ($\,k = 3\,$)
$1.5{-}2{-}2.5$ ( $\,k = 0.5\,$ )
$3\pi{-}4\pi{-}5\pi$ ($\,k = \pi\,$ )
and so on!
Question: Suppose a triangle has sides of lengths $\,1\,,$ $\,\sqrt{3}\,,$ and $\,2\,.$ Is it a right triangle?
Solution: Yes. Since $\,2 \gt \sqrt{3}\,,$ the longest side has length $\,2\,.$ And: $$ \cssId{s81}{1^2 + {(\sqrt{3})}^2} \cssId{s82}{= 1 + 3} \cssId{s83}{= 4} \cssId{s84}{= 2^2} $$
Concept Practice | {"url":"https://onemathematicalcat.org/Math/Geometry_obj/Pythagorean_Theorem.htm","timestamp":"2024-11-05T00:53:32Z","content_type":"text/html","content_length":"55347","record_id":"<urn:uuid:02c239c3-9fa5-4013-a6cc-5c82d3dd8271>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00357.warc.gz"} |
Check out the full course content for How Optimization Works, including video, slides, and code.
In part two, our measure of central tendency for M&M weight was a single parameter model. There was just one number we could adjust to change the result. Just one slider to slide. Just one knob to
turn. Let’s take a look at how this works with a two parameter model.
Let’s consider another set of M&M data points. This time around we carefully weighed the bag before opening it and counting the M&Ms inside. We can represent the counts and weights of each bag in a
two dimensional plot. This scatterplot shows all the bags we ate.
There is a clear pattern here. Visually you can draw a diagonal line right through the middle of these points. This makes sense when you consider how very uniform in size M&M are (the plain ones, not
the peanut ones). If all M&Ms had exactly the same weight (as did all the paper paper packaging) these points would be exactly on the same line. Since there is some variation, that introduces a bit
of scatter.
Because this appears so linear, let’s fit a straight line model to it. The equation of a line is "the y value is equal to the x value times the slope plus the y intercept". For this data, x is the M&
M count and y is the bag weight. The slope and the intercept are now our two parameters, our two dials that we get to adjust to improve the fit of our model.
For convenience, we will continue to use our squared deviation as our cost function. It just so happens that when fitting a line with a squared deviation cost function there is another very elegant
analytic solution for the best parameters, just like there was for the central tendency, when we learned that we could jump straight to it by calculating the mean. However, we will ignore the
analytic solution for now so that we can play with two parameter optimization.
We have the luxury once again of being able to do an exhaustive search. This gives us a birds eye view of how the loss function varies across the parameter space. Looking at the loss function, we see
that for this particular model and data, it is not so much a bowl as it is a narrow valley that is slightly deeper in the middle. It looks like a bowl that has been squished sideways. The plateau on
the top is not actually part of the loss function, it’s just where we chopped it so that we could focus on the valley, which is the most interesting bit. The loss function continues to get higher and
steeper the farther it gets away from the bottom of that valley.
We are going to bypass the second obvious solution to this problem and just choose the lowest point out of all of them. We’re going to pretend for now that we don’t know where the lowest point is.
We start by picking an arbitrary point in the parameter space. It corresponds to a single value of the slope and intercept, and there is one particular line that it corresponds to. We can show that
together with our data points.
Then, we can adjust one of our dials.We can shift the intercept up and down . Notice that as the line passes through the data points, the loss gets lower, and once the line moves past the data
points, the loss climbs again. There’s a point in the middle at which it is quite low, but that is still not quite the lowest value of the loss function across the entire parameter space.
We can change gears and play with the second parameter, the slope. This corresponds to moving in a different direction in the parameter space. In our model, It results in wiggling the line
back-and-forth. Again, we see that, depending on the value of the slope, the value the lost function goes up and down, but it never quite reaches the bottom, its lowest possible value.
It’s possible to move both parameters at once. Again we can see that the value of the loss function jumps up and down, tracking the combination of parameter values we have chosen.
At this point, It’s probably clear what needs to happen. Our point in parameter space needs to roll down to the bottom of the bowl, until it finds the deepest part. This is exactly what gradient
descent does for us. We can start off away up high and make a step, headed down the hill, with the size of the step being related to the steepness of the slope. This moves us closer to the bottom of
the valley, but doesn’t get us there yet, so we repeat it, Stepping down the slope, whichever direction that is, again and again, with the stepsize related to the steepness.
When finally reach the bottom, we've found the best pair of parameters. The best fitting offset is 4.2 and the best fitting slope is .915. It's worth noting that the offset in our model is the weight
of the paper packaging. It is the weight of a package when the M&M count is zero, where x is zero. The slope is the change in weight when the count increases by one - the weight of a single M&M.
Now that we've seen how to apply optimization to a two-parameter linear fit model, let's extend it even further in part 4 to 3-and-higher parameter models, including deep neural networks. | {"url":"https://www.brandonrohrer.com/how_optimization_works_3","timestamp":"2024-11-10T01:15:04Z","content_type":"text/html","content_length":"8078","record_id":"<urn:uuid:00e00e85-d100-45a0-afb2-a266a4ccfcf8>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00748.warc.gz"} |
{LEVsim}: Theoretical Load-Exertion-Velocity Model - Part 1: Resistance Training Phenomena - Complementary Training
{LEVsim}: Theoretical Load-Exertion-Velocity Model – Part 1: Resistance Training Phenomena
Resistance training prescription can be conceptualized from the dose perspective. Dose is a complex construct, but for the purpose of this article series, I will define it using the four
interdependent components: (1) intensity, (2) volume, (3) density, and (4) other (Jovanović 2020).
Intensity component can be further divided into the (1) load, (2) intent, and (3) exertion. Assuming iso-intertial resistance training, load component of intensity is related to the mass and weight
of the lifting object. Load can be expressed in either (1) absolute terms (i.e., in kilograms or pounds), or in (2) relative terms, by comparing the absolute value to some reference, like lifter’s
body weight, or maximal weight one is able to lift (i.e, 1RM).
Intent component of intensity relates to the lifters will to perform a repetition with maximum possible acceleration and velocity, usually in the concentric range. Intent can be maximal, or
sub-maximal where specific lifting tempo is prescribed (e.g., lifting for 2 seconds, pausing for 2 seconds at the top, lowering for 3 seconds, and pausing 5 seconds at the bottom of the lift).
Load-Velocity Trade-off
One of the most important phenomena of resistance training, given the maximal intent to lift as fast as possible in the concentric range, is trade-off between load and movement velocity (Figure 1;
i.e., Load-Velocity trade-off). The exact relationship of this trade-off depends on the exercise, athlete, as well as the velocity metric, but it is often represented with a simple linear or
2nd-degree polynomial equation. There are multiple approaches to measure and express repetition velocity and in this article series I am referring solely to the mean concentric velocity.
Figure 1: Load-Velocity Trade-off. The more weight on the bar, the slower one can lift. Note: Blue line represent the linear model
This relationship can be summarized with a simple profile using linear regression that provides two parameters: (1) slope and (2) intercept, that can be transformed into more intuitive $V_0$ and
$L_0$ estimated parameters. But more about this later.
Exertion component of intensity relates to the proximity to momentary failure (MF). For example, if a lifter, with a certain load, performs 8 repetitions, attempts 9th repetition and fails to lift it
under specified technical constraints (e.g., depth), then this lifter expressed maximal exertion. This load can be expressed as 8 repetition maximum, or 8RM. The highest load that can lifted only
once, is termed one-repetition-maximum or 1RM. 1RM can be used to express load in relative manner. For example, if 1RM is 150 kg, and a lifter uses 100kg, then this load can be expressed as 66% 1RM.
If the load is expressed as relative (i.e., %1RM) in the Load-Velocity relationship, one can create Load-Velocity Profile (LVP). LVP allows one to gauge relative intensity (i.e., %1RM) by using
repetition velocity (given the assumption of maximal intent).
In addition to 1RM, velocity at 1RM (v1RM) is also interesting parameter that can be useful in prescription. But more about it later.
Load-Max Reps Trade-off
Another important phenomena of resistance training, is the higher the load (i.e., %1RM), the lower the maximal number of repetitions (MNR; or nRM) (Moraes et al. 2014; Richens and Cleather 2014;
Shimano et al. 2006; Arazi and Asadi 2011). This Load – Max Reps relationship is consistently manifested at different scales, from a single myo-fiber, single muscle, single joint to multi-joint
movements. The exact shape of this relationship varies between exercises and individuals, but the coaches often refer to Epley’s equation/model (Equation 1) and Table 1 as a heuristic rule of thumb (
DiStasio 2014; Jovanović 2020; Wood, Maddalozzo, and Harter 2002; Wendler 2017).
\%1RM = \frac{100}{0.0333 \times nRM + 1}
nRM 1 2 3 4 5 6 7 8 9 10
%1RM 97 94 91 88 86 83 81 79 77 75
Table 1: Epley’s table
More about Epley’s and other models to map out the Load-Max Reps relationship, as well as their use in resistance training prescription, can be found in Load-Exertion Tables And Their Use For
Planning article series, as well as the Create Custom Set and Rep Schemes With {STMr} course.
The aforementioned equation and table assumes maximal exertion (i.e., going to failure). The lifter using 8RM load doesn’t necessary need to perform all 8 possible reps, and can stop the set before
reaching point of failure. Let’s assume that this lifter performed 5 repetitions out of the 8 possible (the nomenclature for this is commonly: 5[8], or 5 w/8RM). In this case exertion level is
sub-maximal. To quantify level of exertion in the sub-maximal sets (using the last repetition performed), and to assign exertion level to each performed repetition, one can use reps-in-reserve (RIR),
or percent of maximum number of repetitions (%MNR) (Table 2). This sub-maximal 5[8] set can also be expressed as 5 w/3RIR (read as “5 reps with 3 reps-in-reserve”) or 5 w/62%MNR (read as “five reps
with 62% maximal number of repetitions”). Due to easier and intuitive understanding, RIR is favored over %MNR in expressing exertion level.
Please note that both RIR and %MNR represent different approaches in quantifying exertion or proximity to momentary failure. RIR is on the additive scale, while %MNR is on multiplicative scale. For
example, 2[3] and 7[8] sets have same RIR (1 RIR), but different %MNR (67 and 87 %MNR). Thus, quantification of exertion level will be different. The same can be said for 1[2] and 4[8] sets: same
%MNR (50 %MNR), but different RIR (1 and 4 RIR). Researchers should thus provide rationale for their selection of exertion level metrics. I personally favor RIR scale due its more intuitive nature,
e.g., compare giving the following instruction to the average lifter: (1) “Do reps with this load until you have 2 reps in reserve”, versus (2) “Do reps with this load until you reach 77% maximum
number of repetitions”. Although these examples represent perceived level of exertion (discussed in the next few paragraphs), the intuitiveness of using RIR is obvious.
Rep RIR %MNR
Table 2: Reps-In-Reserve (RIR) and percent of maximum number of repetitions (%MNR) of a 8[8] set. RIR and %MNR are different ways to express exertion level of individual reps in a set to failure.
Since the 5 reps with 3RIR represent 8RM load, we can combine the Epley’s equation (Equation 1) with number of planned repetitions (PR) per set and target RIR to get the Load-Exertion table (Table 3
). As can be seen from the Equation 2, nRM is simply number of planned repetition plus the target exertion level using the RIR scale.
\%1RM = \frac{100}{0.0333 \times (PR + RIR) + 1}
Table 3 contains calculated %1RM loads to be utilized for planned number of repetitions ending with designated RIR.
PR 0RIR 1RIR 2RIR 3RIR 4RIR 5RIR
Table 3: Load-Exertion Table using RIR as exertion level.
Similar table can be created by utilizing %MNR instead of RIR. Now the Equation 2 becomes Equation 3.
\%1RM = \frac{100}{0.0333 \times \frac{100 \times PR}{\%MNR} + 1}
nRM in Equation 3 is now simply the number of planned repetition divided by the target exertion level using the %MNR scale.
Table 4 contains calculated %1RM to be utilized for planned number of repetitions ending with designated %MNR.
Although both Table 3 and Table 4 can be utilized for prescription (i.e., planning) purposes, the RIR approach (Table 3) is more intuitive.
PR 100%MNR 90%MNR 80%MNR 70%MNR 60%MNR 50%MNR
Table 4: Load-Exertion Table using %MNR as exertion level.
Table 3 can be very useful starting point and heuristic for prescribing single sets at particular exertion level (assuming of course, the validity and precision of the Epley’s equation). For example,
if I plan to do a set of 3 repetitions, reaching 2RIR, given the Epley’s equation, I should use 86% 1RM load. Unfortunately, if I perform multiple sets of 3 reps with 86% 1RM load, my exertion levels
might climb up due to fatigue. Table 3 is of course overly simplistic, but still satisficing rule of thumb that can be helpful in prescribing resistance training loads.
In a sense, Table 3 and Equation 2 provide predicted RIR (or prescription RIR) that will be reached when a particular %1RM load is lifted for planned number or repetitions, given of course the
Epley’s equation as a model of this relationship. True or actual RIR can only be known retroactively in sets taken to momentary failure.
One way to gauge exertion level is to use lifters’ perceived exertion, or the subjective exertion rating of the set (or each individual rep). This is subjective exertion rating is often reported in
the form of perceived RIR (pRIR), estimated RIR (eRIR), or rate of perceived exertion (RPE) scales (Balsalobre-Fernández et al. 2018; Carzoli et al. 2017; Helms et al. 2016; Hughes, Peiffer, and
Scott 2020; Mansfield et al. 2020; Ormsbee et al. 2019; Perlmutter et al. 2017; Rodríguez-Rosell et al. 2020; Zourdos et al. 2016, 2019; Steele et al. 2017, 2016; Hackett et al. 2012, 2017; Hackett,
Cobley, and Halaki 2018). Although not perfect estimate of the actual RIR (being further from MF and performing more reps in a set are associated with more inaccurate predictions), eRIR can represent
sound strategy to autoregulate training prescription (Greig et al. 2020; Helms et al. 2018; Larsen, Kristiansen, and van den Tillaar 2021; Zhang et al. 2021). By autoregulation, I refer to adjusting
the ongoing training prescription so that prescribed targets are achieved (or at least not missed much). In the previous example, I assumed predictive validity of the Epley’s equation to prescribe 3
repetitions per set with 2RIR, yielding an estimate of 86% 1RM load to be used. This can be a rough rule of thumb. But if I have missed this estimate, athlete can use his or her perceived RIR to
correct (i.e., autoregulate) this load. Depending on what is prescribed and constrained (i.e., in this example number of reps per set and RIR are prescribed, while the load can vary to achieve that),
athlete can use perceived RIR to adjust varying prescription components (in this example load).
Combining Load-Velocity and Load-Max Reps Trade-offs
Load-Velocity profile maps out the relationship between load (either in absolute terms, or relative using %1RM) and repetition velocity, usually using simple linear regression. Load-Max reps profile
(either generic like Epley’s equation, or individually estimated one) maps out the relationship between load (usually relative using %1RM) and maximal number of repetition (MNR or nRM). These two can
be combined by expressing load in Load-Velocity relationship by using MNR instead of %1RM. But to do this, one needs to perform reps-to-failure (RTF) at each 1RM %age rather than simple repetition.
Figure 1 utilizes single repetition at different loads. When one performs reps-to-failure, one can use initial velocity (i.e., velocity of the first repetition), or the fastest set repetition, which
doesn’t necessary needs to be the first rep, although it usually is. This is depicted in Figure 2 (a). Figure 2 (b) depicts relationship between load on the barbell (i.e., %1RM) and maximum number of
repetitions (MNR). The question that naturally follows is how and when 1RM is established? Usually 1RM estimation can be done separately using single reps, and then using that to prescribe multiple
reps to failure on the same or separate days. Other option is to use multiple RTF sets until one can do only a single rep, but in that case fatigue between sets will build up and affect the estimated
Figure 2 (c) depicts combination of the Figure 2 (a) and Figure 2 (b), where load is expressed as nRM or MNR. This can be more handy approach, but as explained it demands performing multiple sets to
failure, while measuring best (or initial) set velocity.
(a) Load-Velocity relationship using the best rep velocity in a set
(b) Load-MNR relationship. Dashed line represents the generic Epley’s profile
(c) MNR-Velocity relationship that combines the previous two profiles
Figure 2: Combining Load-Velocity and Load-MNR relationships
Exertion-Velocity Trade-off
The third important resistance training phenomena, is the trade-off between velocity and exertion level (i.e., Exertion-Velocity trade-off). In other words, the closer the MF, the slower the movement
velocity (Figure 3). This trade-off can be thus used as another, more objective way to gauge exertion level during the set. Figure 3 (a) depicts velocity of each individual rep in three sets to
failure with 120, 135, ad 150kg. The exertion-velocity trade-off represents the loss of movement velocity as one approaches MF, given of course maximum intent level on each rep.
Figure 3 (b) depicts exactly the same repetitions, but now uses actual RIR levels of each repetition as a metric of exertion. This exertion-velocity trade-off is now easily seen, and I refer to it as
RIR-Velocity profile. Velocity can be thus used to estimate RIR (at least in theory) and thus proximity to failure, or exertion level of a given set.
Figure 3 (c) depicts the relationship between velocity and %MNR. This relationship is more complex (i.e., with load interaction). | {"url":"https://complementarytraining.com/levsim-theoretical-load-exertion-velocity-model-part-1-resistance-training-phenomena/","timestamp":"2024-11-13T08:50:27Z","content_type":"text/html","content_length":"363335","record_id":"<urn:uuid:187efcf9-cf38-4613-a3ee-394fd20aa9c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00425.warc.gz"} |
A Baseball Diamond Is A Square With Side 90 Ft?A Baseball Diamond Is A Square With Side 90 Ft?
A Baseball Diamond Is A Square With Side 90 Ft?
A baseball diamond is actually a square with sides of 90 feet.
The Basics of a Baseball Field
A baseball diamond is a square with side 90 ft. It is located within a larger baseball field with dimensions 300 ft by 400 ft. The length of the base paths is 90 ft, and the distance between the
bases is also 90 ft. The pitching distance from home plate to the pitcher’s mound is 60.5 ft.
The infield
The infield is the area of the baseball field closest to home plate. It is also where the majority of the action takes place. The infield consists of four bases: first, second, third, and home plate.
The first and third bases are 90 feet apart, and the second and home plate bases are 60 feet apart. The pitcher’s mound is also located in the infield.
The outfield
The outfield is the area of the field beyond the infield. Outfielders play defense and try to keep the ball from getting past them and going into the stands. The outfield is divided into left field,
center field, and right field. The lines from home plate to the fence in left, center, and right field are called the foul lines. The foul line in left field is called the left-field line, the foul
line in right field is called the right-field line, and the foul line down the middle of the outfield is called center field.
The Dimensions of a Baseball Field
A baseball diamond is not actually a square. The distance from home plate to first base is 90 feet, and the distance from first to second base is also 90 feet. However, the distance from home plate
to third base is only 84 feet. The distance from third base back to home plate is also 84 feet. So, the diamond is actually a rectangle, with the length of the sides being 90 feet and 84 feet.
The distance from home plate to first base
The distance from home plate to first base is 90 feet.
The distance from home plate to second base
The distance from home plate to second base is 90 feet.
The distance from home plate to third base
The distance from home plate to first base is 60 feet. The distance from first to second base is also 60 feet. The distance from second to third base is about 62 feet. So, the dimensions of a
baseball field are:
Home plate to first base: 60 feet
First base to second base: 60 feet
Second base to third base: 62 feet
The distance from home plate to the pitcher’s mound
In Major League Baseball, the distance from home plate to the pitcher’s mound is 60.5 feet.
The Shape of a Baseball Field
A baseball diamond is a square with side 90 ft. The game of baseball is played on a field that has four bases, which are arranged at the corners of a 90-foot square. The bases are 90 feet apart, and
the distance from home plate to first base is also 90 feet.
The infield
The infield is the area of the field closest to home plate. It includes the area between first, second, and third base, as well as home plate. The infield is typically made of dirt, which helps to
absorb impact and prevent injuries.
The infield is traditionally divided into four sections:
-The pitcher’s mound
-The plate
-The shortstop
-The second baseman
The outfield
The outfield is the area of the field beyond the infield. It is made up of three sections, left field, center field, and right field. The outfield is where the majority of the action in a baseball
game takes place. Outfielders are players who play in the outfield.
The left fielder is positioned in left field, the center fielder in center field, and the right fielder in right field. These are typically considered the three most important defensive positions on
the baseball diamond.
The main objective of the outfielder is to catch fly balls hit by the batter. They must also be able to throw the ball back into the infield to prevent runners from advancing around the bases.
The Dimensions of a Baseball Infield
The infield of a baseball diamond is a 90-foot square. The square is divided into four equal parts by two lines that intersect at the center of the square. These two lines are called the first-base
line and the third-base line. The distance from home plate to first base is 60 feet, and the distance from home plate to third base is also 60 feet.
The distance from home plate to first base
The distance from home plate to first base is 90 feet. However, the distance from home plate to second base is only 84 feet. The difference in distance is due to the fact that the bases are not
aligned with each other. The second base is slightly closer to first base than it is to home plate.
The distance from home plate to second base
In baseball, the distance from home plate to second base is important because it determines how long a player has to run to reach second base. The distance from home plate to second base is also
important because it determines how far a player has to hit the ball to reach second base. The distance from home plate to second base is 90 feet.
The distance from home plate to third base
The distance from home plate to first base is 60 feet. The distance from home plate to second base is also 60 feet. The distance from home plate to third base is 90 feet. The distances are measured
from the back point of home plate.
The Dimensions of a Baseball Outfield
A regulation baseball diamond is a square that has equal sides of 90 feet. The distance from home plate to first base is also 90 feet. The distance from home plate to third base is also 90 feet. The
distance from first base to second base is also 90 feet. The distance from second base to third base is also 90 feet.
The distance from home plate to the pitcher’s mound
The distance from home plate to thepitcher’s mound is 60 feet, 6 inches. The pitcher’s mound is NOT considered part of the infield. | {"url":"https://sportsdaynow.com/a-baseball-diamond-is-a-square-with-side-90-ft/","timestamp":"2024-11-05T13:01:21Z","content_type":"text/html","content_length":"104979","record_id":"<urn:uuid:cc91959a-e635-4f38-9fad-f97708dd4497>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00022.warc.gz"} |
April 2021 - Discrete Mathematics Group
On April 27, 2021, Jungho Ahn (안정호) from KAIST and IBS Discrete Mathematics Group gave a talk at the Discrete Math Seminar introducing the well-partitioned chordal graphs and discussing their
properties. The title of his talk was “Well-partitioned chordal graphs with the obstruction set and applications“.
Reinhard Diestel gave an online talk on various applications of the abstract tangle theory at the Virtual Discrete Math Colloquium
On April 21, 2021, Reinhard Diestel from the University of Hamburg gave an online talk at the Virtual Discrete Math Colloquium about the abstract tangle theory and its applications to real-world
examples such as clustering. The title of his talk was “Tangles of set separations: a novel clustering method and type recognition in machine learning“. | {"url":"https://dimag.ibs.re.kr/2021/04/","timestamp":"2024-11-07T15:32:49Z","content_type":"text/html","content_length":"154046","record_id":"<urn:uuid:d2ceabd1-dcef-42ff-8067-9807d11d9a6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00398.warc.gz"} |
INTEREST Tools - Tag
Tools that help in calculation of financial interests
• Calculate the annual percentage yield or (APY)
• Determine the annual growth rate on an investment.
• Calculate the amount of interest earned on a certificate of deposit (CD)
• Evaluate the commission you are going to earn from a sale
• Evaluate compounding effect on loans or investments
• Calculate the future value of a current amount
• Evaluate the length of time required to double an investment or money in an interest-bearing account
• Evaluate your monthly loan EMI payouts
• Calculate the future value (FV) of an investment based on periodic and a constant interest rate
• Computes the FVIFA factor of a series of annuities
• Calculate the Markup Value (or percent) of an item or service using it's selling price and cost price
• Evaluate the present value of an investment based on the future value of the investment, the total number of time periods and the discount rate
• Find selling prices of products including such things as mark up percentage as well as discounted selling prices
• Compute PVIF or present value interest factor, used to determine the future discounted rate of a selected value as well as the current value of a particular series for a set number of periods
• Evaluate PVIFA or present value interest factor of annuity, a factor used to determine the present value of a series of annuities
• Help an investor to find out what actually he gets in return for investing a specific sum of money in an investment
• Find the number of years an investment takes to double at a constant rate of compound interest
• Compute total principal plus simple interest on an investment or savings. | {"url":"https://toolslick.com/tags/interest","timestamp":"2024-11-10T18:46:16Z","content_type":"text/html","content_length":"34110","record_id":"<urn:uuid:5c5b71db-c665-486b-8026-76a91c106e9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00232.warc.gz"} |
CPM Homework Help
Use what you learned in the previous problem to determine the graph of $x = 1 + 3\cos θ$, $y = 2 + 3\sin θ$.
If you are stuck, make a table with $θ$, $x$, and $y$. Then plot each $\left(x, y,\right)$ point.
$\sin \theta=\frac{\textit{y}-2}{3}$
If $\cos^2 θ + \sin^ θ = 1$, then:
$\left( \frac{\textit{x}-1}{3} \right)^2+\left( \frac{\textit{y}-2}{3} \right)^2=1$
What is this the graph of? | {"url":"https://homework.cpm.org/category/CCI_CT/textbook/pc/chapter/10/lesson/10.2.1/problem/10-72","timestamp":"2024-11-04T01:05:30Z","content_type":"text/html","content_length":"37161","record_id":"<urn:uuid:158558b1-6ecd-4783-8cfc-b4ab76572e60>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00086.warc.gz"} |
Attempts to make quantum electrodynamics into a completely solvable theory
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Work on the strong interactions
Attempts to make quantum electrodynamics into a completely solvable theory
Freeman Dyson Scientist
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This was a grand programme. I had the ambition of making quantum electrodynamics into a completely solvable theory, a theory in which one could calculate everything precisely, in which all the
renormalisations would be done, out of the way, and you'd have a convergent series so that it was not just an asymptotic expansion but a real convergent series, so it would be a mathematically
well-defined theory and everything could be done in principle as accurately as you want. So I sat down and tried to do that, and the tactic that I developed was to separate high and low frequencies,
or high and low energies, so that you'd first of all systematically go through a power... series expansion in the high frequencies only, which would be convergent, and so therefore could be, in
principle, calculated rigorously. And so once the high frequencies had been done all the renormalisations would be out of the way. The remainder of the problem would just be ordinary physics, the low
frequencies which would not present any problems in principle; it would just be a matter then of hard work to find the solutions of things, like the hydrogen atom where you'd have to deal with the
low frequencies numerically. So essentially the programme was to do the high frequencies analytically, and the low frequencies numerically. Well, it failed, and I put two years very hard work into
it. It was a more concentrated period of hard work, in fact, than I ever did on anything else. I think I worked really with high concentration for about two years, which I've never really done on any
other project that I was involved with. And I thought it would actually succeed. I published four papers in which various stages or the programme were gone through and it seemed to work well; from a
formal point of view everything worked. I was able to reduce the Schrödinger equation and the Heisenberg operators into the form which I thought would make them convergent and so as far as the
formalities were concerned, everything seemed to work. And so those were done. All that remained was actually to prove the convergence. And then, when I was in Switzerland in the summer of 1951, the
moment of truth happened: I suddenly found a very simple argument which showed that the series are divergent anyway, and no matter what you do, whether you separate high and low frequencies or not,
the perturbation series in quantum electrodynamics simply diverges, and the argument to show that is a simply physical argument which is published in a one page note in the Physical Review letters.
So that was the end of the story and that particular illumination was in a way a very joyful time. I mean, because suddenly I had this burden lifted, that the programme was a failure, I didn't have
to think about it any more; that chapter of my life was over. In a way I suddenly felt a sense of enormous relief, that I wasn't having to fight this monster any more, which had gone on for two years
and it was obviously something that would remain beyond my reach, and that was it. So in a way it was a great success although it was also a failure. At least it showed that was a dead end but in the
course of reaching the dead end I found out something about quantum field theory which was important, namely the fact that the perturbation theory really does diverge.
Freeman Dyson (1923-2020), who was born in England, moved to Cornell University after graduating from Cambridge University with a BA in Mathematics. He subsequently became a professor and worked on
nuclear reactors, solid state physics, ferromagnetism, astrophysics and biology. He published several books and, among other honours, was awarded the Heineman Prize and the Royal Society's Hughes
Title: Attempts to make quantum electrodynamics into a completely solvable theory
Listeners: Sam Schweber
Silvan Sam Schweber is the Koret Professor of the History of Ideas and Professor of Physics at Brandeis University, and a Faculty Associate in the Department of the History of Science at Harvard
University. He is the author of a history of the development of quantum electro mechanics, "QED and the men who made it", and has recently completed a biography of Hans Bethe and the history of
nuclear weapons development, "In the Shadow of the Bomb: Oppenheimer, Bethe, and the Moral Responsibility of the Scientist" (Princeton University Press, 2000).
Tags: Schrödinger equation, Heisenberg operators, Switzerland, 1951, Physical Review
Duration: 4 minutes, 32 seconds
Date story recorded: June 1998
Date story went live: 24 January 2008 | {"url":"https://www.webofstories.com/play/freeman.dyson/92;jsessionid=B9B2E4493E7F0F560C0AD5D1B642A470","timestamp":"2024-11-11T07:11:14Z","content_type":"application/xhtml+xml","content_length":"57176","record_id":"<urn:uuid:1b046fc9-9e74-418c-99d6-39524cedc180>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00403.warc.gz"} |
For project #4, you will report on an academic journal about technology in math education.
1. Choose an article from one of the journals listed below. The article should be around 10-20 pages long and should have been published between 2007 and 2017.
2. Your article must be approved by me. Comment on this post with your choice; include the title and author(s) of the article, the journal name, and year of publication. Each pair must choose a
different article, so make sure to check others’ posts before you claim yours. Post your claim by midnight on Tuesday, December 5.
3. Submit an OpenLab post with the following:
1. The title and author(s) of the article, the journal name, and year of publication.
2. A 1- or 2-paragraph summary of the article.
3. Details about one important point made by the article. Write this as a question with a short essay response. (The reason for writing it as a question and response is that these questions will
serve as inspiration for one of your final exam questions.) Make your question and essay response as clear as possible as it will serve as a study guide for your peers.
4. One discussion question about the important point from the item 3 above (or more discussion questions, if you like).
5. Add the category “Project #4: Research Article” to your post.
4. Prepare a 15-20 minute presentation based on your OpenLab post and prepare to lead a short discussion with the class about the important point you chose to report on above.
Due date: Thursday, December 14
• Journal for Research in Mathematics Education
• Educational Studies in Mathematics
• Mathematics Teacher
• Mathematics Teaching in Middle School
• For the Learning of Mathematics
• Research in Mathematics Education
• Mathematics Education Research Journal
• The Australian Mathematics Teacher
• College Mathematics Journal
• Journal of Mathematics Education at Teachers College
Journal access
These journals may be accessed through the CityTech library. You can view them online from anywhere by following the directions here.
Between presentations, you will use this form to provide feedback to your classmates.
Use this rubric to fill out the form.
The report discusses the benefit of using calculators in classrooms. As the report states, calculators help students to do their computations quickly and more effectively. They make students more
confident about their mathematical understanding and make mathematics more fun to work with. Because calculators help students to do their work faster, that means students can spend their time to
develop their reasoning and mathematical skills. calculators help students to be more active learners and to spend more time to improve their solving problem skills. Also calculators are inexpensive
so almost everyone can have them. Even though there are research which showed that calculators are effective in math education, there are still some people who believe that they are harmful and that
is because of the circulation of misinformation regarding their use.
According to “Why Johnny Can’t Add Without a Calculator” article by Konstantin Kakaes, the article states that the calculators are harmful tools that discourage students’ mathematical understanding.
It says that a study proved that technology do not show measurable effect on students’ test score which means, technology is not effective in teaching. The article also mentions that technology cost
a lot money that better to be spend on teachers’ training specially in math and science.
I agree more with the report than the article even though I was not comfortable reading the repetitive phrase/word the report has, such as paper and pencil and tedious. Also I feel that the report is
advertising calculators which make me a little skeptical about it. Even though I don’t totally agree with the article, there are some points that I agree with.
The report states “…the National Council of Teachers of Mathematics (NCTM) and various other organizations and individuals recommend that appropriate calculators be made available for use by students
at every grade level from kindergarten through college”, students should use calculators when they really need them, however, it is too early for lower grades to use them because they need to
understand the basic arithmetic and computation without using calculators. When they understand how to deal with basic arithmetic then they can use calculators not to depend on them, but to save time
and spend the other time to explore, sharp, and understand mathematical concepts.
The author in the article complains about the electric circuit situations where students were shown it technologically instead of making them manipulate the real batteries and wires to create the
real electric circuit. In this point, I agree with him because if students manipulate real materials, they understand more and become creative. That reminds me with tangrams and pattern blocks that
we are working with in my other education class where we explore many things that cannot be clear without manipulating these materials. Also, it reminds me with the physics class I took, where I feel
more comfortable working in the lab because I can explore a given situation.
The article indicates that there is more money that spend on technology more than on programs to train teachers to be effective. I am not sure if there is an evidence in this point. However, I think
there should be a balance between both because we need technology and we need to have well-trained teachers as well. Also, if we think about technology and kids these days, we better make our kids to
use it as an educational tool instead of using it for playing games because they are using it anyway.
The second myth in the report says “Because calculators do all of the work for the student, he/she will not be stimulated or challenged enough”. Similar idea is addressed by the article where the
author explains that if a student can multiply two numbers it does not mean they know how calculators work. He also explains that there cannot be a technological tool that substitute a teacher.
Because calculators do all of the work for the student he/she will not be stimulated or challenged enough . both report and article expose to this which is not true at all Calculator only do the
low level tasks of computation they don`t think calculators can speed up the leaning process calculators permit student to work enough problems to discover and observe patterns in mathematics
student will also be able to focus on useful particular applications for theories and concept they learn in class which is absolutely true. This myth is not true at all because as the report say
calculators do the low level task of the math problem and instead accelerate their work and it let them achieve.
I really agree with the report every things in this report convinced me , from the beginning when it mention its calculator make the low level math until the time when the report discuss those myths
which is really make me think more clear its really wasting of time if we depending on pencil and paper to finish our work and we will never finish or progress.
I read the report and its really a great report shows how the calculator benefit the students and the teachers as well , this report and also the article discuss the benefit of the calculators also
shows some of the bad side affect both some parents and some teachers think , especially those old parents and the teachers they not use to these technology , also some of the new generation think
that calculators prevented student from progress which is not true the report has mention many myths about the use of calculators. which really convinced me a lot, calculators meant to be helpful
for student in computation which is the low level part of math , its true we all knows how to compute but when the student reach to a higher level need time to think and explore more than wasting
the time for computation.
You may bring your own or borrow one from a friend or the library.
One of the most familiar technological tools in the classroom is the last we’ll discuss in this class: the graphing calculator. You are probably already aware of at least the basic functions of
whatever calculator you have used in your own classes, but you might not have thought about the calculator as a pedagogical tool.
I’m of two minds about graphing calculators. On one hand, they are surprisingly powerful machines and, when used the right way, they can help a student understand a concept or an example without
being distracted by rote computation. On the other, they’re clunky and old fashioned; we have much more powerful and user-friendly tools available now (for example, the software we’ve discussed in
this class).
In addition to the benefits of using the calculator as a pedagogical tool (see the graphing calculators homework), you should become familiar with the pitfalls as well. There is a nice chapter on
Lies My Calculator and Computer Told Me from Stewart’s Calculus book. The examples listed in it aren’t the most relevant for us (many of them deal with rounding errors) but the chapter contains a
nice quote:
Computers and calculators are not replacements for mathematical thought. They are just replacements for some kinds of mathematical labor, either numerical or symbolic. There are, and always will be,
mathematical problems that can’t be solved by a calculator or computer, regardless of its size and speed. A calculator or computer does stretch the human capacity for handling numbers and symbols,
but there is still considerable scope and necessity for “thinking before doing.”
For project #3, choose one of the following exercises. (Comment on this post to indicate which exercise you have selected—do not choose an exercise that has already been selected by someone else.)
Prepare a short lesson introducing the topic and explaining the issue the calculator encounters in this case. Your presentation must include the use of the virtual TI calculator (available on our
classrooms’ desktop). You may also include hand-drawn graphs and/or Desmos graphs if they are relevant for your topic. In addition to your presentation, include a post on your ePortfolio outlining
your presentation. Include screenshots or photographs of the graphing calculator.
1. Imagine you are trying to help your students understand $\lim_{n \to \infty} (1+ \frac{1}{n})^n$. Try substituting larger and larger numbers for $n$ in your calculator. What do you expect to see?
What do you notice?
2. Graph the function $f(x)=\sqrt{4-\ln(x)}$ on your calculator. What behavior do you expect near the $y$-axis? Do you see it on the calculator’s graph? Compare the graph your calculator gives you
with the graph Desmos gives you.
3. Graph the functions $f(x) = \sin(10x), g(x)=\sin(100x), h(x)=\sin(1000x)$ on your calculator. Do you see what you expect to see? Do you notice anything weird? What happens if you graph the same
functions on Desmos?
4. Graph the function $f(x) = \sin(\ln(x))$ on your calculator in the window $[0,1]$ for $x$ and $[-1,1]$ for $y$. How many roots does it look like there are in $[0,1]$? Change the window to $
[0,0.1]$ for $x$ and then to $[0,0.01]$ for $x$ with the same $y$-values. What has happened to the roots? Try graphing the same function in Desmos.
5. In the standard window on your calculator, graph the piecewise defined function $f(x)= 3x - 2$ if $x < 1.5$ and $x^2$ if $x \geq 1.5$. Ask the calculator to tell you the derivative at $x=1.5$. Is
this what you were expecting? Try graphing the function on Desmos.
6. Use the equation solver on your calculator to solve $\frac{\sin(x)}{x} = \frac{1}{x}$. How many solutions do you expect?
Due date: Tuesday, November 28
Here is a 20-year-old report from Texas Instruments about the role of the calculator in math education. The report was put out by the same company that has had a near monopoly on calculators in
classrooms for years…so it’s not exactly unbiased. However, the report discusses some themes we’ve been discussing this semester.
Read the report, with special attention to the sections:
• Dispelling the myths
• Calculators: Elementary School Teachers’ Concerns
• Graphing Calculators: Issues Affecting Secondary School Teachers and University Professors
Here is an article about some perils of using calculators in the classroom. Read the article with the report in mind.
Write a 5-7 paragraph OpenLab post with the category Graphing Calculator Homework reflecting on the following questions:
1. Summarize the main points in the report. Summarize the main points in the article.
2. Do you agree more with the report or with the article? Are you more skeptical of one or the other? Select a few points from each that you agree with or disagree with to support your answer.
3. Select one of the five myths discussed in the report. Does the article address this myth? How do? If the article does not address this myth, guess how the author of the article would address this
Apologies for the inconvenience. | {"url":"https://openlab.citytech.cuny.edu/poiriermedu2010fall2017/2017/11/","timestamp":"2024-11-06T14:19:53Z","content_type":"text/html","content_length":"132791","record_id":"<urn:uuid:db4d1f2a-2f98-4f72-82fd-2f73f15be9d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00312.warc.gz"} |
Machine learned Green's functions that approximately satisfy the wave equation
Green’s functions are wavefield solutions for a particular point source. They form basis functions to build wavefields for modeling and inversion. However, calculating Green’s functions are both
costly and memory intensive. We formulate frequency-domain machine-learned Green’s functions that are represented by neural networks (NN). This NN outputs a complex number (two values representing
the real and imaginary part) for the scattered Green’s function at a location in space for a specific source location (both locations are input to the network). Considering a background homogeneous
medium admitting an analytical Green’s function solution, the network is trained by fitting the output perturbed Green’s function and its derivatives to the wave equation expressed in terms of the
perturbed Green’s function. The derivatives are calculated through the concept of automatic differentiation. In this case, the background Green’s function absorbs the point source singularity, which
will allow us to train the network using random points over space and source location using a uniform distribution. Thus, feeding a reasonable number of random points from the model space will
ultimately train a fully connected 8-layer deep neural network, to predict the scattered Green’s function. Initial tests on part of the simple layered model (extracted from the left side of the
Marmousi model) with sources on the surface demonstrate the successful training of the NN for this application. Using the trained NN model for the Marmousi as an initial NN model for solving for the
scattered Green’s function for a 2D slice from the Sigsbee model helped the NN converge faster to a reasonable solution
Bibliographical note
KAUST Repository Item: Exported on 2020-10-07
Dive into the research topics of 'Machine learned Green's functions that approximately satisfy the wave equation'. Together they form a unique fingerprint. | {"url":"https://academia.kaust.edu.sa/en/publications/machine-learned-greens-functions-that-approximately-satisfy-the-w","timestamp":"2024-11-02T09:30:00Z","content_type":"text/html","content_length":"57846","record_id":"<urn:uuid:567c7b62-525d-4b63-9cbc-989ed64787db>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00142.warc.gz"} |
Exhibit all of the ring homomorphisms from the cartesian square of the integers to the integers - Solutions to Linear Algebra Done Right
Exhibit all of the ring homomorphisms from the cartesian square of the integers to the integers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.5
Describe all ring homomorphisms from $\mathbb{Z} \times \mathbb{Z}$ to $\mathbb{Z}$. In each case, describe the kernel and the image.
Solution: Since $\mathbb{Z}^2$ is the free abelian group on two generators (cf. §6.3 #11), every additive group homomorphism $\varphi : \mathbb{Z}^2 \rightarrow \mathbb{Z}$ is determined uniquely by
$\varphi(1,0)$ and $\varphi(0,1)$. Since every ring homomorphism is also and additive group homomorphism, if $\varphi : \mathbb{Z}^2 \rightarrow \mathbb{Z}$ is a ring homomorphism, then $\varphi$ is
determined uniquely by $\varphi(1,0) = a$ and $\varphi(0,1) = b$.
Note that $$\varphi(1,1) = \varphi((1,0) + (0,1)) = \varphi(1,0) + \varphi(0,1) = a+b,$$ and on the other hand $$\varphi(1,1) = \varphi((1,1)(1,1)) = \varphi(1,1)\varphi(1,1) = (a+b)^2.$$ Thus we
have that $a+b$ is (multiplicatively) idempotent in $\mathbb{Z}$. Thus $a+b \in \{0,1\}$.
Note that for all $(x,y) \in \mathbb{Z}^2$, we have $\varphi(x,y) = xa + yb$ and $$\varphi(x,y) = \varphi((1,1)(x,y)) = (a+b)(xa + yb);$$ thus $$\varphi(x,y) = xa+yb = (a+b)(xa+yb)$$ for all $x,y \in
If $a+b=0$, then for all $(x,y) \in \mathbb{Z}^2$, we have $\varphi(x,y) = 0$. Thus $\varphi = 0$.
Suppose now that $a+b = 1$. Since $\varphi$ is a ring homomorphism, for all $(x,y), (z,w) \in \mathbb{Z}^2$, we have $$\varphi((x,y)(z,w)) = \varphi(x,y)\varphi(z,w) = (ax+by)(az+bw)$$ on one hand
and $$\varphi((x,y)(z,w)) = \varphi(xz,yw) = axz+byw$$ on the other. Thus $$axz+byw = a^2xz + (xw+yz)ab + b^2yw$$ for all integers $x,y,z,w$.
Choosing $y = w = 0$ and $x,z \neq 0$, we see that $axz = a^2xz$, so that $a^2 = a$. Since $a$ is an integer, we have $a \in \{0,1\}$. Thus in fact $(a,b) \in \{(1,0), (0,1) \}$.
Thus there are precisely three ring homomorphisms $\varphi_{(a,b)} : \mathbb{Z}^2 \rightarrow \mathbb{Z}$, where $\varphi_{(a,b)}(1,0) = a$ and $\varphi_{(a,b)}(0,1) = b$, given by $(a,b) \in \
{(0,0), (1,0), (0,1)\}$.
(1) If $\varphi = \varphi_{(0,0)}$, then clearly $\varphi = 0$. Thus $\mathsf{ker}\ \varphi = \mathbb{Z}^2$ and $\mathsf{im}\ \varphi = 0$.
(2) Consider $\varphi = \varphi_{(1,0)}$. It is clear that if $(x,y) \in 0 \times \mathbb{Z}$, then $\varphi(x,y) = 0$. Suppose now that $\varphi(x,y) = x = 0$; then $(x,y) \in 0 \times \mathbb{Z}$.
Thus $\mathsf{ker}\ \varphi = 0 \times \mathbb{Z}$. Since $\varphi(1,0) = 1$, $\mathsf{im}\ \varphi = \mathbb{Z}$.
(3) Consider $\varphi = \varphi_{(0,1)}$. By an argument similar to the previous one, $\mathsf{ker}\ \varphi = \mathbb{Z} \times 0$ and $\mathsf{im}\ \varphi = \mathbb{Z}$. | {"url":"https://linearalgebras.com/solution-abstract-algebra-exercise-7-3-5.html","timestamp":"2024-11-11T16:59:32Z","content_type":"text/html","content_length":"56714","record_id":"<urn:uuid:165ea4a8-76f8-4ff8-9042-5c96708aa1d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00369.warc.gz"} |
The rsppfp package implements different algorithms for transforming graphs in the Shortest Path Problem with Forbidden Paths (SPPFP). This problem is an important concept in the field of graph
theories, and it is a variant of the traditional shortest path problem. In here, there is an additional constrait that takes the form of a finite set of forbidden paths (arc sequences of at least
three nodes) that cannot be part of any solution path.
This problem is solved by transforming the original graph G and its set of forbidden paths F, generating a new graph G* in which traditional shortest path algorithms can be applied, and obtain
solutions that abide to the restrictions. This approach has a number of advantages:
• It allows solving the original shortest path problem in G* with algorithms that efficiently manage time and processing resources constraints, having been implemented in a plethora of languages.
• The resulting G* is highly compatible with existing libraries, and can be used as input data for other, more complex problems and researches.
• In many cases -i.e. logistics- G and F remain unchanged for long periods of time. Thus, the transformation is completed only once, and G* can be stored along with the original graph. A new
conversion is required only on the rare cases where the graph, or its forbidden paths, are modified.
• The input data is provided as common data frames, increasing the versatility of this package.
This solving process is illustrated in Figure 1, using a paper notation to indicate input and output data. Even more, rsppfp scope and key functionalities are also highlighted.
rsppfp implements two different algorithms, each one suited for different situations:
1. Villeneuve and Desaulniers (2005) proposed the first algorithm. In this case, the set F must be known beforehand. This transformation is slightly fast, but generates bigger graphs G*. Each
forbidden path can be of different size, but no sub-path (of at least three nodes long) can be part of another forbidden path.
2. Hsu et al. ‘Backward Construction’ (2009) solves the restriction of sub-paths in the forbidden paths, and generates smaller graphs G*, by adding less new nodes and arcs. However, this algorithm
is slightly slower.
Both algorithms are analyzed using 27 graphs, randomly generated. The complete benchmark evaluation can be found here.
As from 2018-11-22 you can install rsppfp directly from CRAN, using:
{r cran-install, eval = FALSE} install.packages("rsppfp")
You can also install the development version of rsppfp from GitHub with:
{r gh-installation, eval = FALSE} # install.packages("devtools") devtools::install_github("melvidoni/rsppfp")
Available at References | {"url":"https://pbil.univ-lyon1.fr/CRAN/web/packages/rsppfp/readme/README.html","timestamp":"2024-11-02T09:33:45Z","content_type":"application/xhtml+xml","content_length":"5544","record_id":"<urn:uuid:6068ab4d-0305-4a90-83b7-4abff458f4a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00362.warc.gz"} |
A Better Way to Teach Math
Fixes looks at solutions to social problems and why they work.
Is it possible to eliminate the bell curve in math class?
Imagine if someone at a dinner party casually announced, “I’m illiterate.” It would never happen, of course; the shame would be too great. But it’s not unusual to hear a successful adult say, “I
can’t do math.” That’s because we think of math ability as something we’re born with, as if there’s a “math gene” that you either inherit or you don’t.
School experiences appear to bear this out. In every math class I’ve taken, there have been slow kids, average kids and whiz kids. It never occurred to me that this hierarchy might be avoidable. No
doubt, math comes more easily to some people than to others. But the question is: Can we improve the methods we use to teach math in schools — so that everyone develops proficiency?
Looking at current math achievement levels in the United States, this goal might seem out of reach. But the experience of some educators in Canada and England, using a curriculum called Jump Math,
suggests that we seriously underestimate the potential of most students and teachers.
Peter Bregg
“Almost every kid — and I mean virtually every kid — can learn math at a very high level, to the point where they could do university level math courses,” explains John Mighton, the founder of Jump
Math, a nonprofit organization whose curriculum is in use in classrooms serving 65,000 children from grades one through eight, and by 20,000 children at home. “If you ask why that’s not happening,
it’s because very early in school many kids get the idea that they’re not in the smart group, especially in math. We kind of force a choice on them: to decide that either they’re dumb or math is
Children come into school with differences in background knowledge, confidence, ability to stay on task and, in the case of math, quickness. In school, those advantages can get multiplied rather than
evened out. One reason, says Mighton, is that teaching methods are not aligned with what cognitive science tells us about the brain and how learning happens.
In particular, math teachers often fail to make sufficient allowances for the limitations of working memory and the fact that we all need extensive practice to gain mastery in just about anything.
Children who struggle in math usually have difficulty remembering math facts, handling word problems and doing multi-step arithmetic (pdf). Despite the widespread support for “problem-based” or
“discovery-based” learning, studies indicate that current teaching approaches underestimate the amount of explicit guidance, “scaffolding” and practice children need to consolidate new concepts.
Asking children to make their own discoveries before they solidify the basics is like asking them to compose songs on guitar before they can form a C chord.
Mighton, who is also an award-winning playwright and author of a fascinating book called “The Myth of Ability,” developed Jump over more than a decade while working as a math tutor in Toronto, where
he gained a reputation as a kind of math miracle worker. Many students were sent to him because they had severe learning disabilities (a number have gone on to do university-level math). Mighton
found that to be effective he often had to break things down into minute steps and assess each student’s understanding at each micro-level before moving on.
Take the example of positive and negative integers, which confuse many kids. Given a seemingly straightforward question like, “What is -7 + 5?”, many will end up guessing. One way to break it down,
explains Mighton, would be to say: “Imagine you’re playing a game for money and you lost seven dollars and gained five. Don’t give me a number. Just tell me: Is that a good day or a bad day?”
Separating this step from the calculation makes it easier for kids to understand what the numbers mean. Teachers tell me that when they begin using Jump they are surprised to discover that what they
were teaching as one step may contain as many as seven micro steps. Breaking things down this finely allows a teacher to identify the specific point at which a student may need help. “No step is too
small to ignore,” Mighton says. “Math is like a ladder. If you miss a step, sometimes you can’t go on. And then you start losing your confidence and then the hierarchies develop. It’s all
Mighton saw that if he approached teaching this way, he could virtually guarantee that every student would experience success. In turn, the children’s math anxiety diminished. As they grew more
confident, they grew excited, and they began requesting harder challenges. “More than anything, kids love success,” he says, “and they love getting to higher levels, like in a video game.”
As the children experienced repeated success, it seemed to Mighton that their brains actually began to work more efficiently. Sometimes adding one more drop of knowledge led to a leap in
understanding. One day, a child would be struggling; the next day she would solve a problem that was harder than anything she’d previously handled. Mighton saw that if you provided painstaking
guidance, children would make their own discoveries. That’s why he calls his approach “guided discovery.”
The foundation of the process is building confidence, which Mighton believes should be the first goal of a math teacher. Confidence begets attention, which begets rich learning. “I’ve never met a
teacher who will tell you that a student doesn’t need to be confident to excel in school,” explains Mighton. “But I’ve never seen a math curriculum that follows the implications of that idea
rigorously.” Math is well-suited to build confidence. Teachers can reduce things to tiny steps, gauge the size of each step to the student and raise the bar incrementally.
When math is taught this way, surprising things happen.
Consider some of Jump’s results. It’s been used for four years in the public schools in Lambeth, one of the most economically depressed boroughs of London, England. Teachers placed into Jump the
students who were struggling most in math. Among the 353 students who entered the program in fifth grade, only 12 percent began at grade level. Most were at least two grade levels behind and the vast
majority were not expected to pass England’s grade six (KS2) national tests. But 60 percent did.
In rural Ontario, Jump was recently evaluated in a randomized controlled study involving 29 teachers and about 300 fifth-grade students (controlled studies of math programs are rare). Researchers
from Toronto’s Hospital for Sick Children and the Ontario Institute for Studies in Education arranged for a control group of teachers to use their district’s standard curriculum while another group
used Jump. Each set of teachers was given two days of training relevant to the materials they would be using.
In five months, researchers found substantial differences in learning. The Jump group achieved more than double the academic growth in core mathematical competencies evaluated using a well known set
of standardized tests. (The study has not yet been published.) “Kids have to make pretty substantial gains in order to see this kind of difference,” explained Tracy Solomon, a developmental
psychologist in the Research Institute at the Hospital for Sick Children who is the study’s lead author. “It’s impressive over a five-month period.”
Solomon believes that the key to Jump’s effectiveness is the way it “breaks math down to its component parts and builds it back up.” And she notes that this “flies in the face of the way math is
typically taught.”
Interviewing teachers and principals, I have heard numerous stories of results like these. At times, they seem hard to reconcile with our assumptions about math. Isabel Grant, principal of the
General Wolfe Elementary School, in Vancouver, British Columbia, has seen Jump produce impressive results in two schools where it has been used by a variety of teachers. Schools in British Columbia
evaluate students based on whether they meet expectations for learning outcomes. “Teachers who used Jump were suddenly finding that they had all of their kids in the ‘fully meeting expectations’
category,” Grant told me. “It was such a foreign experience. It doesn’t typically happen when we’re teaching science or language arts. And they were kind of at a loss. ‘What do we do about this?’”
Another example is Mary Jane Moreau, who teaches at the Mabin School, an independent school in Toronto that does not screen students based on academic ability. Moreau, an experienced educator,
dabbled with Jump for a year and started to see progress among her students, so she decided to immerse herself in the philosophy. “I was used to getting a bell curve in the past,” she told me, “but
what I started seeing was all the kids getting between 90 and 100 percent on tests, and within months, they were all getting between 95 and 100 percent.”
She decided to see if the results would transfer to the standardized Test of Mathematical Abilities. Moreau teaches the same cohort of kids in fifth and sixth grades. Each September, for four years,
the students wrote the test. From 2006 to 2007, the class percentile average jumped from 66 percent to 92 percent. From 2008 to 2009, with a new cohort, it increased from 54 percent to 98 percent.
Notably, the bell curve of the students’ scores shifted to the right and narrowed — which is to say that the performance differences between the “slow” kids and the “whiz” kids began to fade away.
Moreau encouraged her sixth-grade students to enroll in the Mathematica Pythagoras contest, which attracts only five percent of Canadian students, most of whom would be deemed “gifted” in math. All
but one did. For each group, 14 out of 17 students beat the contest average.
Moreau is a dedicated teacher — and she has the benefit of small classes — but, even so, she hadn’t seen results like this before. And it troubled her to think of students she had taught who didn’t
have the opportunity to learn math this way. “When I think about what we’ve been doing for years when we could have been doing something else,” she told me, “I feel like I have to run so hard on this
because I’m coming to the end of my career. But if I don’t help to change attitudes, I’ll feel like a criminal.”
Jump is a modest outfit. Mighton has a staff of 10 to create materials and conduct teacher trainings. With decisions about math curriculum highly politicized, it’s difficult for a small group to
influence the debate. Big textbook companies and paid math consultants have a big say — and big investments — in what gets used.
It will take independent-minded educators to use Jump and see if its results can be replicated in more classrooms and schools. It’s hard to imagine what society might look like if we could undermine
the math hierarchies that get solidified in grade school. These patterns tend to play out across society, often in negative ways. Wasn’t it the whiz kids who invented financial derivatives and
subprime mortgages? And how many adults got themselves into hot water with their mortgages because, at bottom, they didn’t really understand the risks?
Even deeper, for children, math looms large; there’s something about doing well in math that makes kids feel they are smart in everything. In that sense, math can be a powerful tool to promote social
justice. “When you have all the kids in a class succeeding in a subject, you see that they’re competing against the problem, not one another,” says Mighton. “It’s like they’re climbing a mountain
together. You see a very healthy kind of competition. And it makes kids more generous to one another. Math can save us.”
Leave a Comment | {"url":"https://teaching.app/2023/06/27/a-better-way-to-teach-math/","timestamp":"2024-11-04T13:23:57Z","content_type":"text/html","content_length":"47433","record_id":"<urn:uuid:674746f1-35bd-4ef3-9513-7cb727477bd5>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00046.warc.gz"} |
Centre for Biotech Data Science
Home >Research>Center For Biosystems And Biotech Data Science
Centre for Biosystems and Biotech Data Science
• Prof. dr. Shodhan Rao
• Prof. dr. Wesley De Neve
• Prof. dr. Joris Vankerschaver
• Dr. Hyunjin Shim
• Manvel Gasparyan
• Homin Park
• Utku Ozbulak
• Esla Timothy Anzaku
• Azimberdy Besya
• Espoir Kabanga
• Negin Harandi
• Mena Markos
• Anju Susan Anish
• Abiodun Ilugbusi
• Yunseol Park
About Us
The Centre for Biosystems and Biotech Data Science pursues the development of new mathematical and computational approaches for analyzing biosystems and for extracting knowledge from huge sets of
biotech data (e.g., biological sequence data and biomedical imagery), paying attention to technical challenges such as
• predictive analysis and visualization of biotech data;
• complexity reduction and validation of data-driven models for biotechnical processes and systems; and
• interpretability and robustness of data-driven models for biotechnical processes and systems.
The newly developed approaches have a wide variety of applications, ranging from cancer research and medical image diagnosis over drug development to plant phenotyping.
Current research topics include:
• stability, model reduction and parameter estimation of biochemical reaction networks;
• ecological species interaction networks and metapopulation models;
• validity conditions for quasi steady state approximations;
• representation learning for biological sequences;
• interpretability for biological sequence and biomedical image analysis;
• deep machine learning for structural and functional genome annotation;
• deep machine learning for 3-D object understanding;
• 3-D phenotyping of rice plants via computer vision and machine learning;
• uncertainty and out-of-distribution modeling for deep machine learning; and
• adversariality in deep machine learning.
Mathematics is often described as the "queen of science" because it has played an active role in the development of science and it has also benefited from its involvement in science. Particularly in
bioscience engineering, there are many topics that cannot be mastered without a solid background in mathematics. Some of these topics are stability analysis and control of bioprocess plants,
synthetic biology, computational and systems biology, modeling of bio-systems and chemical networks. Consequently, in our BSc programmes of Environmental Technology, Food Technology and Molecular
Biotechnology, the mathematical education is quite rigorous and is on par with mathematical courses for other engineering programs like electrical, civil and aerospace engineering around the world.
The program involves 3 compulsory mathematics courses and the student is expected to have had a good secondary/high school training in mathematics in order to cope up with the level of mathematics at
Ghent University Global Campus. The three courses of Mathematics are
Mathematics 1: Engineering Mathematics (Ba1)
This is a yearlong course split into two semesters. In the first semester, basic high school topics are treated in depth in order to prepare the student for university level mathematics. The topics
dealt with in the first semester are trigonometry, 2-dimensional co-ordinate geometry including straight lines and circles, basic algebra, 1-variable differential and integral calculus. In the second
semester, some special topics in 1-variable calculus including applications of differentiation and integration in geometry and physics are dealt with along with an introduction to linear algebra
including systems of linear equations, matrices, determinants, eigenvalues and eigenvectors. The focus of the course is mainly on the development of scientific skills such as analytical reasoning,
critical reflection and problem-solving capability.
Mathematics 2: Multivariable Calculus and Geometry (Ba2)
This course continues further from Mathematics 1 dealing mainly with 2-D and 3-D coordinate geometry including vectors and multivariable calculus as the name suggests. While some emphasis is placed
on understanding theoretical concepts, the students are also encouraged to face and solve real-life application-based problems. The course also lays the foundation for the mathematics that the
students will encounter in other third year process-based courses including Process Modelling & Control and Process Engineering.
Mathematics 3: Differential Equations (Ba2)
Since most bioprocesses are modelled using ordinary and partial differential equations, the course Mathematics 3 gives a strong foundation on these two types of differential equations and the
analytical methods for solutions of the two types of differential equations. Since not all differential equations can be solved using analytical methods, the last one-third part of the course deals
with numerical solution techniques for solving ordinary differential equations. The students are trained in using Matlab software for solving first order initial value problems. The course also
includes a brief introduction to Laplace transforms and the concept of equilibria of differential equations and their stability, which are later also studied in depth in the course Process Modelling
& Control. Next to the pure mathematical courses, the Center for Biotech Data Science is also responsible for providing a strong background in applied mathematics, informatics, statistics, and
physics through the courses:
Physics 1 and 2: Mechanics, Vibration, Waves, and Thermodynamics (Ba1)
Physics is the study of an enormous span of natural phenomena ranging from the large-scale galaxies to the sub-microscopic entities in their static or dynamic states. Physicists pursue a fundamental
understanding of the physical universe whereas engineers apply scientific knowledge to design and develop structures, machines and products. At Ghent University Global Campus, the Physics 1 and 2
course is designed to stitch together the skills of physicist and engineer to enable a deeper understanding of the engineering fundamentals through a broader understanding of physics in an
engineering context. This course is an intensive course of study that emphasizes analytical and problem-solving skills. Physics 1 and 2 is a yearlong course aimed at providing the students a thorough
training in basic physics, with a focus on both basic principles and practical applications. It further aims to make the students familiar with the practical applications of mechanics in everyday
life, to establish an understanding of the various states of matter, to gain a working understanding of both physical and chemical thermodynamics, to learn with respect to physical aspects how to
calculate the energy transfer of processes. The syllabus for the course has been carefully crafted to cater to the needs of students majoring in molecular biotechnology, food technology and
environment technology.
Informatics (Ba1)
Scientists and engineers are often confronted with time-consuming and repetitive tasks when having to process and analyze data, namely collecting information from websites, converting files from one
format to another, and analyzing, summarizing and visualizing the information obtained. In addition, the exponential flow of newly incoming information requires present-day scientists and engineers
to be able to automate these tasks, in order to speed up their daily job routines.
This course teaches students how to describe time-consuming and repetitive tasks in such a way that they can be performed automatically by a (network-based) computer system. To that end, the
necessary skills for computer-based creative problem solving will be acquired through learning to work and think in (1) Python, a popular programming language, and (2) in UNIX, the workhorse
operating system of science and engineering. The computer problems that need to be solved are taken from different scientific disciplines, including mathematics, biology, chemistry, physics, and
computer science.
Process Modelling and Control (Ba3)
The first part of the course deals with modelling of biosystems encountered in environmental engineering, food technology, ecology, chemical engineering, biotechnology and process industry. Students
are taught how to use Matlab Simulink toolbox in order to model and simulate biosystems. Since no standard techniques are available for the analysis of nonlinear systems which are commonly
encountered in practice, the course also includes techniques for linearization of nonlinear systems. The second part of the course deals with analysis of impulse, step and frequency response of first
order and second order linear systems. The third and the final part of the course deals with techniques for design of feedback control that ensures bounded-input bounded-output (BIBO) stability of
the controlled system. This part also deals with control of dead-time systems which are commonly encountered in real life applications.
Probability and Statistics (Ba3)
Statistics is the science that collects, analyses, and interprets numerical data, with as goal to summarize the data (descriptive statistics) or draw conclusions from them (statistical inference).
Nowadays, every scientific field is confronted with large amounts of data, and thus plays an ever-increasing role in, e.g., drug development, climate research, or food quality control.
In this course, students are first introduced to probabilistic and statistical concepts. They learn to perform statistical techniques and to correctly describe and interpret statistical data and
output. They also learn to distinguish between haphazard effects on the one hand and scientifically significant results on the other hand. Focus is also placed on critically reading and evaluating
results presented in scientific literature.
The second part of the course continues where process modelling left off, namely with the simulation of dynamical (bio)systems. Different methodologies are discussed for model simulation, parameter
estimation, and sensitivity analysis in order to come to a final model selection.
All theory is illustrated with ample examples. The statistical software R is used throughout the course.
Bioinformatics (Ba3 – major Molecular Biotechnology)
The field of bioinformatics was born after biologists discovered how to sequence (digitize) DNA, raising the need for mathematical and computational techniques to decipher the language of DNA, RNA,
and proteins. As a result, bioinformatics has become an important part of modern biology, often facilitating new insights and new (data-driven) approaches, driving further biological developments.
Primarily taking a computational point-of-view, this course aims at introducing students to the design, implementation, and analysis of standard algorithms in the field of bioinformatics, including
exhaustive search algorithms, recursive algorithms, divide and-conquer algorithms, greedy algorithms, graph algorithms, dynamic programming algorithms, machine learning algorithms (shallow and deep),
and randomized algorithms. These algorithms and related datastructures (e.g., lists, tuples, sets, dictionaries, graphs, hash tables, and trees) are studied in the context of problems like pattern
matching, genome rearrangements, DNA sequencing, DNA sequence alignment, regulatory motif finding, genome annotation (structural and functional), and/or medical image analysis.
Bachelor dissertations
• 2019-2020
Yeji Bae – A Deep Learning Approach Towards Detecting and Locating Trypanosoma Parasites in Microscopy Images of Thick Blood Smears
Jongdo Im – Effects of Diffusion on the Coexistence of Species under Intransitive Competition
Taewoo Jung – Automatic Detection of Trypanosomosis in Thick Blood Smear Images Using Deep Learning
Hanul Kang – An Investigation of Class Activation Mapping for Visualizing Deep Learning-based Brain Tumor Classification
Younsoo Kang – Parameter Estimation for Chemical Reaction Networks from Experimental Data of Reaction Rates
Hayoung Kim – Automated Early Detection of Diabetic Retinopathy in Retinal Fundus Photographs using Deep Learning
Yunseol Park – Translation Initiation Site Prediction in Arabidopsis thaliana Using Synthetic Datasets and Black-box Models
Heesoo Song – Computer-aided Diagnosis of Trypanosomiasis Using Unstained Microscopy Images and Deep Machine Learning
• 2018-2019
Siho Han – Manual Feature Extraction and Extreme Gradient Boosting for Splice Site Detection
Jeongtek Kim – Generating synthetic genomic datasets for the validation of convolutional neural network models
Pyeong Eun Kim – Loss Function Visualization for Encoder-Decoder Style Deep Learning Models Targeting Biomedical Image Segmentation
Ju Hyung Lee – Deep learning for disease symptom segmentation in medical images
Woojin Lee – Computer Vision to Measure Cell Lengths in Rice Coleoptiles
• 2017-2018
Chananchida Sang-aram – Computer Vision in Plant Phenotyping: A Case Study for Automated Analysis of Rice Seedlings
Tenured Academic Staff
Wesley De Neve
+82 32-626-4204 wesley.deneve(at)ghent.ac.kr
Shodhan Rao
+82 32-626-4203 shodhan.rao(at)ghent.ac.kr
Assisting Academic Staff
Esla Timothy Anzaku
+82 32-626-4319 eslatimothy.anzaku(at)ghent.ac.kr
Azimberdy Besya
+82 32-626-4354
Manvel Gasparyan
+82 32-626-4328 manvel.gasparyan(at)ghent.ac.kr
Negin Harandi
+82 32-626-4317 negin.harandi(at)ghent.ac.kr
Espoir Kabanga
+82 32-626-4306 espoir.kabanga(at)ghent.ac.kr
Utku Ozbulak
+82 32-626-4330 utku.ozbulak(at)ghent.ac.kr
Ho-Min Park
+82 32-626-4326 homin.park(at)ghent.ac.kr
Anju Susan Anish
+82-32-626-4318 Anjususan.Anish@ghent.ac.kr
Teaching Assistant
Mena Markos
+82 32-626-4355 mena.markos(at)ghent.ac.kr
Yunseol Park
Former members
Mijung Kim
Breght Vandenberghe
Bayer Crop Science
Jasper Zuallaert
Vlaams Instituut voor Biotechnologie (VIB)
Arnout Van Messem
University of Liege
Surender Kumar
Nathan Muyinda
Makerere University, Kampala, Uganda
Contact details
Center director
Shodhan Rao
Ghent University Global Campus
#935, 119-5 Songdomunhwa-ro, Yeonsu-gu
Incheon 21985
South Korea | {"url":"https://ghent.ac.kr/~res_biotechdatasci","timestamp":"2024-11-12T02:19:27Z","content_type":"text/html","content_length":"37317","record_id":"<urn:uuid:96c8f511-2afc-476b-9fd2-819e7e58688c>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00414.warc.gz"} |
[Seminar 2021.05.06] Arithmetic properties of weakly holomorphic modular functions of arbitrary level
Date: 6 May (Thr) 14:30 ~ 15:30
Place: Zoom (ID: 854 1988 1532)
Speaker : 강순이 (강원대학교)
Title: Arithmetic properties of weakly holomorphic modular functions of arbitrary level
The canonical basis of the space of modular functions on the modular group of genus zero form a Hecke system. From this fact, many important properties of modular functions were derived. Recently, we
have proved that the Niebur-Poincare basis of the space of Harmonic Maass functions also forms a Hecke system. In this talk, we show its applications, including divisibility of Fourier coefficients
of modular functions of arbitrary level, higher genus replicability, and values of modular functions on divisors of modular forms.
This is a joint work with Daeyeol Jeon and Chang Heon Kim.
cf. 봄학기 정수론 세미나 웹페이지: https://sites.google.com/view/snunt/seminars | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&document_srl=1397&order_type=desc&listStyle=viewer&page=6","timestamp":"2024-11-09T04:26:35Z","content_type":"text/html","content_length":"20328","record_id":"<urn:uuid:b35f5ccd-4add-4acb-802f-9b7b59f90f6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00260.warc.gz"} |
Rebalancing! Really? | Python-bloggersRebalancing! Really?
This article was first published on OSM , and kindly contributed to python-bloggers. (You can report issue about the content on this page here)
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In our last post, we introduced benchmarking as a way to analyze our hero’s investment results apart from comparing it to alternate weightings or Sharpe ratios. In this case, the benchmark was meant
to capture the returns available to a global aggregate of investable risk assets. If you could own almost every stock and bond globally and in the same proportion as their global contribution, what
would your returns look like? We then used this benchmark as a way to judge our hero’s portfolio. We looked first at returns in excess of the benchmark scaled by volatility. Then we looked at excess
returns scaled by deviation from the benchmark. We found that our hero’s portfolio was not being compensated for its deviation from the benchmark. The reason: gold was a drag on performance.
In this post, we move away from portfolio selection to rebalancing. Our goal will be to set up the intuition behind rebalancing, which will launch us into a more detailed discussion in further posts.
Why rebalance? Assume you establish some reasonably well defined risk and return parameters. Then you select the assets that offer the best approximation of achieving those parameters.^1 As those
assets gain or lose value, the exposures change, leaving you with a different level of potential risk and return than originally envisioned.
A simple example. You invested at the bottom of the global financial crisis with a 50/50 weighting to SPY and SHY (the assets we’ve been using to proxy stocks and bonds). A year later, that weighting
would be closer to 54/46. Moreover, you’re annualized risk has gone up by almost half a percentage point due to the higher exposure to stocks. While that might not seem like a lot, it may be more
than you want. Indeed, with more assets and more variance in the returns, the weights and ending risk exposure could change significantly. So you rebalance to return to the original risk-return
Another reason to rebalance is if external factors have changed your risk or return parameters The most obvious example for an individual is the approach of, and then transition to, retirement. As
one approaches retirement, risk tolerance generally declines since you don’t want to risk losing what you’ve worked so hard to gain. Hence, there’s a typical shift to assets with lower volatility.
But other factors can change the parameters as well—changes in inflation, mortality, and health expectations are a few. For professional investors, reasons to rebalance include changes in funding
mandates, sources, and scope. But we won’t go into those.
A third reason to rebalance is if the expectations for future risk and return profiles of the assets in the portfolio have changed. One sees this most ofthen in actively managed portfolios; that is,
portfolios attempting to beat a benchmark. But even if you’re not trying to beat an index, if your expectations for asset returns and risk changes dramatically, then there’s a very good reason to
rebalance. While this seems logical, getting it right is really tough. Unless you’ve got a crystal ball or there’s some obvious secular change, the risk of being wrong is great. In any event, we
won’t say much more about that here, but will touch on this in depth when we get to the post on capital market expectations.
Apart from imminent retirement or some exogenous factor, the rationale behind rebalancing suggests that it should lead to better risk-adjusted returns. You sell winners to buy losers. And since
“trees don’t grow to the sky” and nothing “stays down forever”^2 you’re dampening volatility by selling at the peak and buying near the trough. Or so the logic goes. If this is true, our hero should
probably think about what type of rebalancing regime he might want to employ. Let’s see if rebalancing produces better results.
First off, we should note that we did not rebalance the various portfolios we looked at; that is, the equal-weighted (our hero’s), the naive, or the risky. We started them at their respective weights
in 2005 and did not change. For the benchmark, however, we did rebalance every quarter. We did this to approximate the way many benchmarks are constructed. What would returns look like if we had
rebalanced? We’ll look at results for different rebalancing periods: none, monthly, quarterly, and yearly. The table below shows our hero’s equal-weighted portfolio.
Table 1: Equal-weighted portfolio performance
for different rebalance periods (%)
Period Return Volatility Sharpe Total return
None 6.1 7.6 0.82 140.0
Months 6.1 9.2 0.67 134.5
Quarters 6.1 7.6 0.81 138.8
Years 6.2 7.6 0.83 142.8
Interestingly, we see very little difference in the average annual return. And the only difference in the remaining metrics is for the monthly rebalancing, which suffers higher volatility and thus a
lower Sharpe ratio and total return. What’s causing the performance drag for monthly rebalancing? It could be that volatility whipsaws returns, which gets captured by the shorter reallocation time
frame. Or the monthly rebalancing could simply be unlucky. Instead of buying losers that become winners, you just buy losers. We’d need to look at this in more detail, but we’ll save that for a later
post. Next up the naive portfolio.
Table 2: Naive portfolio performance for
different rebalance periods (%)
Period Return Volatility Sharpe Total return
None 5.6 6.9 0.82 122.0
Months 5.3 6.8 0.79 112.6
Quarters 5.3 6.8 0.80 114.4
Years 5.5 6.6 0.84 119.6
At first glance, it’s unclear why monthly and quarterly rebalancing exhibit worse performance than the others. But it is probably not due to whipsawing, since SHY is mainly short-term US Treasuries.
Hence, there’s not a lot of volatility. It is more likely due to performance drag since volatility is relatively the same across difference rebalancing periods. Whatever the case, let’s move on to
the risky portfolio.
Table 3: Risky portfolio performance for
different rebalance periods (%)
Period Return Volatility Sharpe Total return
None 7.9 12.4 0.65 191.6
Months 7.9 12.5 0.64 188.7
Quarters 7.9 12.5 0.64 189.6
Years 8.0 12.4 0.65 192.5
Not surprisingly, a portfolio that has a 90% weighting to one asset, won’t see a lot of difference in performance due to different rebalancing periods. That’s because, unless the asset soared or
crashed, the weighting is likely to remain relatively stable, so rebalancing would have a minimal effect. Finally, let’s look at different rebalancing periods for the benchmark portfolio
Table 4: Benchmark portfolio performance for
different rebalance periods (%)
Period Return Volatility Sharpe Total return
None 5.7 5.1 1.11 39.3
Months 5.7 5.2 1.11 39.4
Quarters 5.7 5.1 1.11 39.3
Years 5.7 5.2 1.10 39.4
Hmm. Nothing dramatic here either. So what gives? We presented all these rebalancing strategies and the surprise was that there wasn’t much of a surprise. Despite what appears to be sound logic,
rebalancing did not produce better risk-adjusted returns in most cases. This confirms what some pundits argue: that rebalancing is useless. On the other hand, there are some who claim that it
generates not just better risk-adjusted returns, but may even offer consistent outperformance.
We can’t run a full test on those competing claims here, we’ll do that in the next post along with providing some links to the different views, For now, let’s at least get a flavor of the kind of
test we might use.
We simulate a portfolio of 10 years of monthly returns whose assets match the real returns and risk of US and global stocks and bonds and the S&P GSCI Commodity Total Return Index. We apply the same
rebalancing periods as before. We present the table of performance metrics below.
Table 5: Simulated portfolio performance for
different rebalance periods (%)
Period Return Volatility Sharpe Total return
None 5.5 7.6 0.73 68.1
Months 6.2 7.2 0.87 81.3
Quarters 6.1 7.2 0.86 79.3
Years 5.8 7.2 0.81 74.1
We see that monthly and quarterly rebalancing produce modestly better risk-adjusted and total returns. Are these results significant? When we run t-tests on the returns, we find little to suggest the
differences are anything more than noise. This is based on the p-values we show in the table below. For those who never took statistics or who find the mention of stats causes an immediate gag
reflex, just note that the p-values are no where near 5%, which means the differences are likely due to randomness. We won’t test the significance of the Sharpe ratio for now simply because it
involves some more sophisticated techniques.
Table 6: Simulated portfolio
p-values from t-test for
differences in returns
Periods P-values
None vs. Months 0.83
None vs. Quarters 0.85
None vs. Years 0.92
Months vs. Quarters 0.97
Months vs. Years 0.90
Quarters vs. Years 0.93
What have we learned thus far? The logic behing rebalancing appears straightforward: when circumstances (internal or external) change, then the portfolio should be rebalanced to weights that match
the prior or new risk-return parameters. The main reason for this was to maintain or improve risk-adjusted returns. Yet, when we ran different rebalancing scenarios, we found little evidence to
suggest returns were any different. And one simulation using historical returns of major assets, also showed little evidence of significant differences in performance.
Nonetheless, we’re not yet ready to say rebalancing is hogwash. There’s still more to do including running thousands of simulations to test for significant differences between rebalancing periods;
examining whether rebalancing due to life changes does what it says on the tin; and testing rebalancing on a larger set of assets. But we’ll save those for the next couple posts. Until then, here’s
the code:
# Load package
# Get data
symbols <- c("SPY", "SHY", "GLD")
symbols_low <- tolower(symbols)
prices <- getSymbols(symbols, src = "yahoo",
from = "1990-01-01",
auto.assign = TRUE) %>%
map(~Ad(get(.))) %>%
reduce(merge) %>%
prices_monthly <- to.monthly(prices, indexAt = "last", OHLC = FALSE)
ret <- ROC(prices_monthly)["2005/2019"]
bench_sym <- c("VTI", "VXUS", "BND", "BNDX")
bench <- getSymbols(bench_sym, src = "yahoo",
from = "1990-01-01",
auto.assign = TRUE) %>%
map(~Ad(get(.))) %>%
reduce(merge) %>%
bench <- to.monthly(bench, indexAt = "last", OHLC = FALSE)
bench_ret <- ROC(bench)["2014/2019"]
# Create different weights and portfolios
wt1 <- rep(1/(ncol(ret)), ncol(ret))
port1 <- Return.portfolio(ret, wt1) %>%
wt2 <- c(0.9, 0.1, 0)
port2 <- Return.portfolio(ret, weights = wt2) %>%
wtn <- c(0.5, 0.5, 0)
portn <- Return.portfolio(ret, wtn)
port_comp <- data.frame(date = index(port1), equal = as.numeric(port1),
risky = as.numeric(port2),
naive = as.numeric(portn))
## Rebalancing for equal
# Create list
rebal = c("months", "quarters", "years")
equal_list <- list()
for(pd in rebal){
equal_list[[pd]] <- Return.portfolio(ret, wt1, rebalance_on = pd) %>%
# Create data frame
equal <- equal_list %>% bind_cols() %>%
data.frame() %>%
mutate_all(as.numeric) %>%
mutate(date = index(easy_list[["months"]]),
none = as.numeric(port1)) %>%
select(date, none, everything())
equal %>%
rename("Months" = months,
"Quarters" = quarters,
"Years" = years,
"None" = none) %>%
gather(Period,value, -date) %>%
mutate(Period = factor(Period, labels = c("None", "Months", "Quarters", "Years"))) %>%
group_by(Period) %>%
summarise(Return = round(mean(value)*12,3)*100,
Volatility = round(sd(value)*sqrt(12),3)*100,
Sharpe = round(mean(value)/sd(value)*sqrt(12),2)+.01,
`Total return` = round(prod(1+value)-1,3)*100) %>%
knitr::kable(caption = "Equal-weighted portfolio performance (%) for different rebalance periods")
# Rebalance for naive
naive_list <- list()
for(pd in rebal){
naive_list[[pd]] <- Return.portfolio(ret, wtn, rebalance_on = pd) %>%
naive <- naive_list %>%
bind_cols() %>%
data.frame() %>%
mutate_all(as.numeric) %>%
mutate(date = index(naive_list[["months"]]),
none = as.numeric(portn)) %>%
select(date, none, everything())
naive %>%
rename("None" = none,
"Months" = months,
"Quarters" = quarters,
"Years" = years) %>%
gather(Period,value, -date) %>%
mutate(Period = factor(Period, levels = c("None", "Months", "Quarters", "Years"))) %>%
group_by(Period) %>%
summarise(Return = round(mean(value),3)*1200,
Volatility = round(sd(value)*sqrt(12),3)*100,
Sharpe = round(mean(value)/sd(value)*sqrt(12),2)+.01,
`Total return` = round(prod(1+value)-1,3)*100) %>%
knitr::kable(caption = "Risk and returns for different rebalance periods (%)")
# Rebalance for risky
risky_list <- list()
for(pd in rebal){
risky_list[[pd]] <- Return.portfolio(ret, wt2, rebalance_on = pd) %>%
risky <- risky_list %>% bind_cols() %>%
data.frame() %>%
mutate_all(as.numeric) %>%
mutate(date = index(risky_list[["months"]]),
none = as.numeric(port2)) %>%
select(date, none, everything())
risky %>%
rename("None" = none,
"Months" = months,
"Quarters" = quarters,
"Years" = years) %>%
gather(Period,value, -date) %>%
mutate(Period = factor(Period, levels = c("None", "Months", "Quarters", "Years"))) %>%
group_by(Period) %>%
summarise(Return = round(mean(value),3)*1200,
Volatility = round(sd(value)*sqrt(12),3)*100,
Sharpe = round(mean(value)/sd(value)*sqrt(12),2)+.01,
`Total return` = round(prod(1+value)-1,3)*100) %>%
knitr::kable(caption = "Risk and returns for different rebalance periods (%)")
# Rebalance for benchmark
bench_list <- list()
for(pd in rebal){
bench_list[[pd]] <- Return.portfolio(bench_ret, wtb, rebalance_on = pd) %>%
bench_rb <- bench_list %>% bind_cols() %>%
data.frame() %>%
mutate_all(as.numeric) %>%
mutate(date = index(bench_list[["months"]]),
none = as.numeric(portb)) %>%
select(date, none, everything())
bench_rb %>%
rename("None" = none,
"Months" = months,
"Quarters" = quarters,
"Years" = years) %>%
gather(Period,value, -date) %>%
mutate(Period = factor(Period, levels = c("None", "Months", "Quarters", "Years"))) %>%
group_by(Period) %>%
summarise(Return = round(mean(value)*12+.0001,3)*100,
Volatility = round(sd(value)*sqrt(12),3)*100,
Sharpe = round(mean(value)/sd(value)*sqrt(12),2)+.01,
`Total return` = round(prod(1+value)-1,3)*100) %>%
knitr::kable(caption = "Risk and returns for different rebalance periods (%)")
# Simulate
stock_us <- rnorm(120, 0.08/12, 0.2/sqrt(12))
stock_world <- rnorm(120, 0.065/12, 0.17/sqrt(12))
bond_us <- rnorm(120, 0.024/12, 0.1/sqrt(12))
bond_world <- rnorm(120, 0.025/12, 0.14/sqrt(12))
commod <- rnorm(120, 0.007, 0.057)
wt <- c(0.25, 0.25, 0.2, 0.2, 0.1)
date <- seq(as.Date("2010-02-01"), length = 120, by = "months")-1
port <- as.xts(cbind(stock_us, stock_world, bond_us, bond_world, commod),
order.by = date)
port_list <- list()
rebals = c("months", "quarters", "years")
for(pd in rebals){
port_list[[pd]] <- Return.portfolio(port, wt, rebalance_on = pd)
port_r <- port_list %>%
bind_cols() %>%
data.frame() %>%
mutate_all(as.numeric) %>%
mutate(date = date,
none = as.numeric(none)) %>%
select(date, none, everything())
port_r %>%
rename("None" = none,
"Months" = months,
"Quarters" = quarters,
"Years" = years) %>%
gather(Period,value, -date) %>%
mutate(Period = factor(Period, levels = c("None", "Months", "Quarters", "Years"))) %>%
group_by(Period) %>%
summarise(Return = round(mean(value)*12+.0001,3)*100,
Volatility = round(sd(value)*sqrt(12),3)*100,
Sharpe = round(mean(value)/sd(value)*sqrt(12),2)+.01,
`Total return` = round(prod(1+value)-1,3)*100) %>%
knitr::kable(caption = "Risk and returns for different rebalance periods (%)")
count <- 0
port_names <- c("None", "Months", "Quarters", "Years")
# paste(toupper(substr(colnames(port_1)[-1], 1, 1)),
# substr(colnames(port_1)[-1], 2, nchar(colnames(port_1)[-1])), sep="")
t_tests <- c()
for(i in 1:4){
for(j in 2:4){
if(i != j & count != 6){
t_tests[paste(port_names[i]," vs. ", port_names[j])] <- t.test(port_1[,i+1],
count = count +1
data.frame(pds = names(t_tests), p_val = round(as.numeric(t_tests),2)) %>%
rename("Periods" = pds,
"P-values" = p_val) %>%
knitr::kable(caption = "T-test for means")
1. We’ve covered this in the previous posts, but only insofar as we limited the number of assets we could choose. We’ll come back to this concept in a slightly different way once we’ve covered
capital market expectations. So assume there’s some violent hand-waving going on here!↩
2. Unless it goes bankrupt!↩
for the author, please follow the link and comment on their blog:
Want to share your content on python-bloggers? | {"url":"https://python-bloggers.com/2020/02/rebalancing-really-2/","timestamp":"2024-11-01T23:59:45Z","content_type":"text/html","content_length":"53853","record_id":"<urn:uuid:2b859679-5b19-4619-b5c4-3ee43841c023>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00812.warc.gz"} |
Factoring Quadratics | Brilliant Math & Science Wiki
Factoring quadratics is a method that allows us to simplify quadratic expressions and solve equations. Common cases include factoring trinomials and factoring differences of squares.
A quadratic expression may be written as a sum, \(x^2+7x+12,\) or as a product \((x+3)(x+4),\) much the way that 14 can be written as a product, \(7\times 2,\) or a sum, \(6+8.\) Factoring a
trinomial is the process of rewriting a sum as a product.
Quadratic expressions may be written in standard form as \(ax^2+bx+c.\)
Let's begin by factoring trinomials with \(a=1,\) such as \(x^2+8x+15.\)
First, we need to find the product \(ac:\) \(ac = (1)(15) = 15.\)
Next, we need to find a factor pair of \(ac\) that sums to \(b.\) So we need a factor pair of 15 that sums to 8. The factor pair of 3 and 5 sums to 8.
Next, we need to rewrite the "\(b\)-term" of our quadratic using our new sum: \[x^2+3x+5x + 15.\]
Lastly, we can factor by grouping, factoring the first two terms of our expression and the last two terms: \[(x^2+3x)+(5x+15) = x(x+3) + 5(x+3) = (x+5)(x+3).\]
The factored form of \(x^2+8x+15\) is \((x+3)(x+5).\)
Factor \(x^2-4x-12.\)
The product of \(a\) and \(c\) is \((1)(-12)=-12.\)
A factor pair of \(-12\) that sums to \(-4\) is \(-6\) and \(2.\)
Rewriting our expression, we have \(x^2-4x-12 = x^2-6x+2x-12.\)
Grouping and simplifying, we have \(x^2-6x+2x-12 = (x^2-6x)+(2x-12) = x(x-6)+2(x-6) = (x+2)(x-6).\)
The factored form of \(x^2-4x-12\) is \((x+2)(x-6).\)
Factor \(x^2-7x+6.\)
The product of \(a\) and \(c\) is \((1)(6)=6.\)
A factor pair of \(6\) that sums to \(-7\) is \(-1\) and \(-6.\)
Rewriting our expression, we have \(x^2-7x+6 = x^2-1x-6x+6.\)
Grouping and simplifying, we have \(x^2-1x-6x+6 = (x^2-1x)+(-6x+6) = x(x-1)-6(x-1) = (x-6)(x-1).\)
The factored form of \(x^2-7x+6\) is \((x-6)(x-1).\)
If the following statement is true, what is the value of \(a\,?\)
Observe that the left-hand side of the equation can be factorized as \[ x^2+5x-14 &= x^2+7x-2x+7 \cdot (-2) \\ &= x(x+7)-2(x+7) \\ &= (x+7)(x-2). \] Equating this with the right-hand side
gives \(a=7\).
\( (p-3)(p-2) \) \( (p-5)(p+2) \) \( (p+3)(p+2) \) \( (p+5)(p-2) \)
Which of the following solutions is the correct factored form of this trinomial.
Leading Coefficient \(\neq\) 1
Remember that quadratic expressions may be written in standard form as \(ax^2+bx+c.\) Let's factor some expressions in which \(a \neq 1.\)
We'll begin with \(2x^2+7x+3,\) and follow the same steps as above..
First, we need to find the product \(ac:\) \(a\cdot c = 2\cdot 3 = 6.\)
Next, we need to find a factor pair of \(ac\) that sums to \(b.\) So we need a factor pair of 6 that sums to 7. The factor pair of 6 and 1 sums to 7.
Next, we need to rewrite the "\(b\)-term" of our quadratic using our new sum: \[2x^2+6x+1x+3.\]
Lastly, we can factor by grouping, factoring the first two terms of our expression and the last two terms: \[(2x^2+6x)+(1x+3) = 2x(x+3) + 1(x+3) = (2x+1)(x+3).\]
The factored form of \(2x^2+7x+3\) is \((2x+1)(x+3).\)
Factor \(3x^2+10x-8.\)
The product of \(a\) and \(c\) is \((3)(-8)=-24.\)
A factor pair of \(-24\) that sums to \(10\) is \(12\) and \(-2.\)
Rewriting our expression, we have \(3x^2+10x-8 = 3x^2+12x-2x-8.\)
Grouping and simplifying, we have \(3x^2+12x-2x-8 = (3x^2+12x)+(-2x-8) = 3x(x+4)-2(x+4) = (3x-2)(x+4).\)
The factored form of \(3x^2+10x-8\) is \((3x-2)(x+4).\)
If \((x + 4)(3x - 1) = ax^2 + bx + c,\) what is the value of \(a + b + c?\)
Sometimes, we can simplify a quadratic expression by factoring out a common factor before we completely factor the trinomial.
For example, let's factor \(6x^2-15x-36.\)
Every term in the trinomial is divisible by 3 so let's factor out a 3: \(6x^2-15x-36=3(2x^2-5x-12).\)
Now we can factor \(2x^2-5x-12\) following the same process that we used above.
The value of \(ac\) is \((2)(-12) = -24.\) The factor pair of \(-24\) that sums to \(-5\) is 8 and \(-3.\)
Therefore, \[3(2x^2-5x-12) = 3(2x^2+8x-3x-12) = 3[(2x^2+8x)+(-3x-12)]=3[2x(x+4)-3(x+4)]=3(2x-3)(x+4).\]
Factor \(20x^2-30x+10.\)
Let's begin by factoring out a 10: \(20x^2-30x+10=10(2x^2-3x+1).\)
The product of \(a\) and \(c\) is \((2)(1)=2.\)
A factor pair of \(2\) that sums to \(-3\) is \(-1\) and \(-2.\)
Therefore, \[10(2x^2-3x+1)=10[2x^2-1x-2x+1]=10[(2x^2-1x)+(-2x+1)]=10[x(2x-1)-1(2x-1)]=10(x-1)(2x-1).\]
Quadratics Factoring - Basic
Given a quadratic equation \(ax^2+bx+c=0\), how can we factor it?
First, we need to know that the factored form of a quadratic equation is \(a(x-r_1)(x-r_2)\), where \(r_1\) and \(r_2\) are the roots of the equation and \(a\) is the coefficient of the first term.
By expanding this, we get \(a\left(x^2-(r_1+r_2)x+r_1 r_2\right)\).
Now we can try to factor the equation, but first we need to factor out \(a\), which is the coefficient of the first term. Then, we can find the roots of the equation by trying out some values, since
we know \(\frac{-b}{a} = r_1+r_2\) and \(\frac{c}{a}=r_1 r_2\).
Factorize \(x^2-7x+6\).
We don't need to factor out \(a\) since \(a=1\).
Now that we know \(r_1+r_2=-b=7\) and \(r_1 r_2=c=6\), we know that the values \(1\) and \(6\) satisfy the condition for \(r_1\) and \(r_2\).
Therefore, \[x^2-7x+6=(x-6)(x-1). \ _\square\]
Quadratics Factoring - Intermediate
Given a quadratic equation \(ax^2 + bx + c = 0\), we can factor it easily using methods described here. This, however, may not be the best method when dealing with the general quadratic with
solutions which are not real. For these cases, the quadratic formula might be more applicable.
Consider the quadratic equation \(ax^2+bx+c=0\). If we divide the whole expression by \(a\), we get \[x^2 + \frac{b}{a}x + \frac{c}{a} = 0 .\] Rearranging gives \[x^2 + \frac{b}{a}x = - \frac{c}{a} .
\] Let us now add \(\frac{b^2}{4a^2}\) to both sides: \[ x^2 + \frac{b^2}{4a^2} + \frac{b}{a}x &= \frac{b^2}{4a^2} - \frac{c}{a}\\ \left(x + \frac{b}{2a}\right)^2 &= \frac{b^2-4ac}{4a^2} . \] Taking
the square root of both sides, we have \[x + \frac{b}{2a} = \pm \sqrt{\frac{b^2-4ac}{4a^2}} .\] Rearranging one last time, we get \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} .\]
Notice that now we are able to find the roots of a quadratic even if they are not real. This allows us to factor it very easily. Thus for any quadratic \(ax^2 + bx + c = 0\), we can factor it into \
(k(x+\alpha)(x+\beta) = 0 \), where \(\alpha= \frac{-b+ \sqrt{b^2 - 4ac}}{2a}, \) \( \beta= \frac{-b - \sqrt{b^2 - 4ac}}{2a}, \) and \(k\) is some constant.
Factorize \(x^2 + x - 1 = 0.\)
Using our formula, we obtain \[\phi= \frac{-b+ \sqrt{b^2 - 4ac}}{2a} = \frac{-1 +\sqrt{5}}{2}, \Phi= \frac{-b - \sqrt{b^2 - 4ac}}{2a} = \frac{-1 - \sqrt{5}}{2} .\] Our quadratic is thus \[k(x-\
phi)(x-\Phi) = 0\] for some constant \(k\). However, since the leading coefficient of our first term is 1, we know that the factorization must be \[(x-\phi)(x - \Phi) = \left(x - \frac{-1 +\sqrt
{5}}{2} \right)\left(x- \frac{-1 - \sqrt{5}}{2}\right). \ _\square\]
This method of factorization also works when we are not dealing with quadratics with real solutions/coefficients.
Factorize \(2x^2 +2 x + 2 = 0.\)
We first observe that this has no real solutions because its discriminant is negative: \[D = b^2 - 4ac = 2^2 - 4\times2\times2 = -12 < 0.\] We can proceed to finding its roots using \[x = \frac
{-b+ \sqrt{b^2 - 4ac}}{2a} =\frac{-b+ \sqrt{D}}{2a} . \] Using our formula \[\alpha= \frac{-2+ \sqrt{-12}}{2.2} =\frac{-2+ \sqrt{4\times-1\times 3}}{4} =\frac{-2+ 2i\sqrt{3}}{4} = \frac{-1}{2} +
\frac{i\sqrt{3}}{2} . \] Similarly, we get \[\beta= \frac{-2 - \sqrt{-12}}{2.2} =\frac{-2 - \sqrt{4\times-1\times 3}}{4} =\frac{-2 - 2i\sqrt{3}}{4} = \frac{-1}{2} - \frac{i\sqrt{3}}{2} .\]
We can now write our equation as \[k\left(x-\frac{-1}{2} - \frac{i\sqrt{3}}{2}\right)\left(x-\frac{-1}{2} + \frac{i\sqrt{3}}{2}\right)=0.\] Notice that \(k\) has to be \(2\) in order to make the
coefficient of \(x^2\) \(2\). Thus, we finally get \[2\left(x+\frac{1}{2} - \frac{\sqrt{3}}{2}i\right)\left(x+\frac{1}{2} + \frac{\sqrt{3}}{2}i\right)=0. \ _\square\]
This allows us to factorize quadratics with irrational and even imaginary coefficients.
Factorize \( \sqrt{24}x^2 + 5\sqrt{2}x + \sqrt{6} = 0.\)
Again, our discriminant is \[D = b^2 - 4\times a\times c = 50 - 4\times12 = 2 > 0. \] Using our formula we get \[ \alpha&= \frac{-5\sqrt{2} + \sqrt{2} }{4\sqrt{6}} = \frac{ - 4\sqrt{2} }{ 4\sqrt
{6}} = -\frac{\sqrt{3}}{3 } \\ \beta&= \frac{-5\sqrt{2} - \sqrt{2} }{4\sqrt{6}} = \frac{ - 6\sqrt{2} }{ 4\sqrt{6}} = -\frac{\sqrt{3}}{2}. \]
Thus, our factorization becomes
\[k\left(x+\frac{\sqrt{3}}{3 }\right)\left(x + \frac{\sqrt{3}}{2} \right) = 0. \]
Again, by comparing this with our initial polynomial, we see that \(k = \sqrt{24} = 2\sqrt{6} \). Thus, our final factorized form is \[\sqrt{24}\left(x+\frac{\sqrt{3}}{3} \right)\left(x + \frac{\
sqrt{3}}{2} \right) = 0. \ _\square \] | {"url":"https://brilliant.org/wiki/factoring-quadratics/#quadratics-factoring-medium","timestamp":"2024-11-12T18:10:34Z","content_type":"text/html","content_length":"62415","record_id":"<urn:uuid:f628b332-89ec-4d7d-b01f-1fba9e8cf5a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00043.warc.gz"} |
The erectile dysfunction medicine sildenafil citrate (Viagra®) inhibits phosphodiesterase type 6 (PDE6), an essential enzyme involved in the activation and modulation of the phototransduction
cascade. Although Viagra might thus be expected to impair visual performance, reports of deficits following its ingestion have so far been largely inconclusive or anecdotal. Here, we adopt tests
sensitive to the slowing of the visual response likely to result from the inhibition of PDE6. We measured temporal acuity (critical fusion frequency) and modulation sensitivity in four subjects
before and after the ingestion of a 100-mg dose of Viagra under conditions chosen to isolate the responses of either their short-wavelength-sensitive (S-) cone photoreceptors or their long- and
middle-wavelength-sensitive (L- and M-) cones. When vision was mediated by S-cones, all subjects exhibited some statistically significant losses in sensitivity, which varied from mild to moderate.
The two individuals who showed the largest S-cone sensitivity losses also showed comparable losses when their vision was mediated by the L- and M-cones. Some of the losses appear to increase with
frequency, which is broadly consistent with Viagra interfering with the ability of PDE6 to shorten the time over which the visual system integrates signals as the light level increases. However,
others appear to represent a roughly frequency-independent attenuation of the visual signal, which might also be consistent with Viagra lengthening the integration time (because it has the effect of
increasing the effectiveness of steady background lights), but such changes are also open to other interpretations. Even for the more affected observers, however, Viagra is unlikely to impair common
visual tasks, except under conditions of reduced visibility when objects are already near visual threshold.
Since its approval in 1998, over 1 billion doses of sildenafil citrate (Viagra®) have been prescribed as a treatment for erectile dysfunction (Pfizer,
). It works by inhibiting cyclic guanosine monophosphate (cGMP)-specific phosphodiesterase type 5 (PDE5), which is an enzyme expressed in the smooth muscle of the corpus cavernosa (Beavo,
; Moreland, Goldstein, & Traish,
). As an undesirable secondary effect, it inhibits a closely related phosphodiesterase enzyme PDE6 (e.g., Wallis, Casey, Howe, Leishman, & Napier,
). Estimates suggest that Viagra has about 10% of the effect on PDE6 that it has on PDE5 (Food and Drug Administration Joint Clinical Review,
; see Table 1 of Laties & Zrenner,
). PDE6 plays an essential role in phototransduction, the process by which photons of light are absorbed and converted into electrical signals for transmission to the visual centers of the brain.
Activated PDE6 (PDE6*) catalyzes the hydrolysis of cGMP to GMP. The reduction in cGMP results in the closure of ion channels in the plasma membrane, which blocks the inward flow of Na
and Ca
ions leading to cell hyperpolarization (for reviews, see Arshavsky, Lamb, & Pugh,
; Pugh & Lamb,
; Pugh, Nikonov, & Lamb,
). PDE6, however, not only activates the transduction cascade but also regulates visual sensitivity. As the concentration of PDE6* increases as the light level rises, the time over which the visual
signals are integrated shortens and the visual response speeds up (e.g., Govardovskii, Calvert, & Arshavsky,
; Nikonov, Lamb, & Pugh,
Tests to determine whether Viagra causes visual side effects have so far been largely restricted to standard measures of human visual performance, such as visual acuity and color discrimination.
Although such tests can be important for clinical diagnosis, most are not particularly well suited for monitoring the visual effects that are likely to be caused by the inhibition of PDE6. As a
result, perhaps, the outcomes of these tests have been inconclusive, providing, at best, evidence that is largely subjective or anecdotal (reviewed in Laties & Fraunfelder,
; Laties & Zrenner,
; Marmor & Kessler,
). Any visual side effects are typically described in subjective reports as a bluish tinge or haze to vision or increased sensitivity to light. These effects are rarely reported (3%) at the lowest
clinical doses of 25 and 50 mg, more often reported (11%) at the highest clinical doses of 100 mg, and frequently reported (50%) at doses of 200 mg or higher (Food and Drug Administration Joint
Clinical Review,
; Marmor & Kessler,
; Morales, Gingell, Collins, Wicker, & Osterloh,
). Otherwise, Viagra is claimed to have little or no effect on visual performance. For example, in controlled clinical trials, doses of up to 200 mg of Viagra do not affect visual acuity, visual
fields, the Amsler grid, spatial contrast sensitivity, or pupillary responses (Food and Drug Administration Joint Clinical Review,
; Laties, Ellis, Koppiker, Patat, & Stuckey,
; Laties, Ellis, & Mollon,
). Transient, mild impairment of color discrimination has been found with the Farnsworth–Munsell 100 test at the peak plasma levels with doses of 100 mg or more (Food and Drug Administration Joint
Clinical Review,
; Laties et al.,
). However, any effects of Viagra on chromatic discrimination are inconsistent between subjects (see Laties & Fraunfelder,
; Laties & Zrenner,
; Marmor & Kessler,
) and have not been confirmed in recent double-blind studies (Birch, Toler, Swanson, Fish, & Laties,
; Jägle et al.,
Such standard visual tests, however, do not address the most likely effect of Viagra, which is to disrupt light adaptation by preventing the visual integration time from shortening enough to offset
increases in light level. More appropriate tests for lengthened integration time include those that probe the temporal response, such as measures of temporal acuity or resolution (also known as
critical fusion frequency [cff]) and temporal modulation sensitivity (e.g., De Lange,
; Hecht & Verrijp,
; Kelly,
). In cff measurements, the highest frequency that can be detected is determined as a function of adaptation level. These measurements are complemented by temporal modulation sensitivity
measurements, which can be used to determine sensitivity at temporal frequencies below cff and, thus, define the overall temporal frequency response.
If Viagra effectively lengthens the visual integration time, then the way in which it alters temporal sensitivity will depend on the integration times involved. If the integration times are long
enough to selectively attenuate visible flicker frequencies (by integrating over more than one cycle at some frequencies), then Viagra should impair the detection of higher rates of flicker relative
to low rates, so that the cff is reduced and the falloff in modulation sensitivity with increasing temporal frequency steepens (before the final limiting slope). If, on the other hand, the
integration times are very short, Viagra should cause a frequency-independent divisive scaling of sensitivity and, thus, a loss of cff and a vertical shift in the logarithmic modulation sensitivity
functions, without a change in shape. Frequency-independent losses might also arise because the lengthening of the integration time makes steady components in the visual stimuli (produced either by
the background or by the flickering target, which must be modulated around a mean level) more effective. This greater effectiveness arises simply because the steady (0 Hz) signals are integrated over
a longer time. Such effects are particularly likely under S-cone isolation conditions because the intense long-wavelength background typically needed to desensitize the L- and M-cones also
chromatically attenuates the S-cone signal (e.g., Pugh & Mollon,
). We find both frequency-dependent and frequency-independent changes in our data. Other explanations of these changes are considered in the
The optical apparatus was a conventional Maxwellian-view optical system with a 2-mm entrance pupil illuminated by a 900-W Xenon arc. Wavelengths were selected by the use of interference filters with
full-width at half-maximum bandwidths of between 7 and 11 nm (Ealing or Oriel). The radiance of each beam could be controlled by the insertion of fixed neutral density filters (Oriel) or by the
rotation of circular, variable neutral density filters (Rolyn Optics). Sinusoidal modulation was produced by the pulse-width modulation of fast, liquid crystal light shutters (Displaytech) at a
carrier frequency of 400 Hz. The position of the observer's head was maintained by a dental wax impression. The experiments were under computer control. The apparatus is described in more detail
elsewhere (Stockman, Plummer, & Montag,
The experimental conditions were chosen to measure the temporal properties of either the S-cones or the L- (and M-) cones.
S-cone measurements
A flickering target of 4° of visual angle in diameter and 440 nm in wavelength was presented in the center of a 9° diameter background field of 620 nm. Fixation was central. The 620-nm background
field, which delivered 11.51 log [10] quanta · s ^−1 · deg ^−2 at the cornea (4.95 log [10] photopic trolands), selectively desensitized the M- and L-cones but had comparatively little direct effect
on the S-cones.
For the cff measurements, the 440-nm target was modulated at 92% contrast and varied in intensity in steps from approximately 6.5 to 11 log
quanta · s
· deg
(c. −1.13 to 3.37 log
photopic trolands). These conditions isolate the S-cone response up to a 440-nm target radiance of about 10.5 log
quanta · s
· deg
(e.g., Stockman, MacLeod, & DePriest,
; Stockman, MacLeod, & Lebrun,
; Stockman & Plummer,
). The intrusion of the M-cones at the highest levels is clearly marked by a change in the appearance (hue and sharpness) of the target and by an abrupt increase in cff. For the modulation
sensitivity measurements, the 440-nm target was fixed at time-averaged radiances of 7.54, 8.82, or 9.75 log quanta · s
· deg
(−0.09, 1.19, or 2.12 log
photopic trolands).
L-cone measurements
A flickering target of 4° of visual angle in diameter and 650 nm in wavelength was presented in the center of a 9° diameter background field of 480 nm. Fixation was again central. The 480-nm
background, which delivered 8.26 log quanta · s ^−1 · deg ^−2 at the cornea (1.37 log [10] photopic trolands), served primarily to not only saturate the rods but also selectively desensitize the
M-cones at lower target radiances.
For the cff measurements, the 650-nm target was varied in intensity from approximately 6.5 to 11.0 log
quanta · s
· deg
(−0.63 to 3.87 log
photopic trolands) and was modulated at 92%. These conditions isolate the L-cone response over most of the 650-nm intensity range, but at high intensities, the M-cones are also likely to contribute
to flicker detection (Stiles,
; see Figure 1b of Stockman & Mollon,
). We were not concerned about the possibility of a mixed M- and L-cone response at higher levels because there is no reason to suppose that Viagra has a selective effect on either the M- or the
L-cones. However, we note that Viagra might have differential effects on the two cone types because the M-cones are at a much lower level of adaptation at high, 650-nm target radiances than are the
L-cones. For the modulation sensitivity measurements, the 650-nm target was fixed at time-averaged radiances of 7.56 and 9.52 log quanta · s
· deg
(0.43 or 2.39 log
photopic trolands).
Before making any measurements, subjects light adapted to the background and target for 3 min. Subjects interacted with the computer by means of buttons and received feedback and instructions via
tones and a computer-controlled voice synthesizer. Two types of temporal measures were made: (a) temporal resolution or cff measurements, in which observers adjusted the flicker frequency (at the
maximum fixed stimulus modulation of 92%) to find the frequency at which the flicker just disappeared, and (b) modulation threshold measurements, in which observers adjusted the modulation (at a
fixed frequency) to find the modulation at which the flicker just disappeared. Modulation was varied by adjusting the fraction of the light that is flickering, while keeping the time-average
intensity of the light constant. Modulation thresholds enable sensitivity to be measured at frequencies below the cff.
Each single measurement of modulation threshold or cff is the average of three settings. The mean data points for the pre- or post-Viagra baseline measurements are the average of at least three
measurements, and the error bars are ±1 SEM.
A more objective measurement technique, such as two-alternative forced choice, was impractical in the context of these experiments because it is too slow to generate quickly the amount of data
required to explore dynamic changes in sensitivity following drug ingestion.
The radiant fluxes of test and background fields were measured at the plane of the observer's entrance pupil with a UDT Radiometer that had been calibrated by the manufacturer against a standard
traceable to the National Bureau of Standards and cross-calibrated by us. Neutral density filters, fixed and variable, were calibrated in situ for all test and field wavelengths used. Interference
filters were calibrated in situ with a spectroradiometer (Gamma Scientific). Quoted radiances are time-averaged values.
Four male observers (authors A.S., L.T.S., A.T., and G.J.) participated in these experiments. All observers had normal color vision according to standard tests. A correction lens of +5D was used for
A.T. L.T.S. and A.S. are highly experienced psychophysical subjects. A.T. and G.J. were comparatively naive at the start of the experiments. All four subjects were in good health with no known
cardiovascular or other risk factors. These studies conform to the standards set by the Declaration of Helsinki, and the procedures have been approved by local ethics committees at Moorfields Eye
Hospital and at University College London. The local ethics committee restricted us to making measurements on ourselves.
Subjects orally ingested therapeutic, 100-mg doses of Viagra (sildenafil citrate). Doses were typically taken in the morning with water. No food was eaten from the previous evening until the end of
the measurements to minimize interference with its absorption. Successive doses were separated by at least 1 week. Only 100-mg doses were taken. The dose numbers given in each figure refer to
separate trials, before which a single 100-mg dose was ingested. Frequently, different types of measures were interleaved during a single trial.
Different dosages vary slightly in their effects on visual performance, which is presumably related to variations in the Viagra plasma concentration (see Jägle, Jägle, Sèrey, & Sharpe,
; Jägle et al.,
). An example of different dose effectiveness can be seen in the cff data for L.T.S. in
Figure 1
. Although the effects of Viagra are consistent across doses, Dose 3 (circles) is clearly less effective than the other doses. In addition, the time course of the effect varies across subjects. For
example, subjects L.T.S. and G.J. show measurable visual losses after taking Viagra at shorter postingestion intervals than does A.T. Although the thresholds did not always recover to their baseline
values during the course of our measurements, which were usually limited to 3 hr after drug ingestion, their recovery was always complete by the time of the post-Viagra measurements, which were
usually made 1–3 days following a dose.
For a particular set of measurements, the pre-Viagra measurements were made on separate days before the Viagra measurements, whereas the post-Viagra measurements (if carried out) were made at least 3
days after them.
Statistical tests
To test for an effect of Viagra on each subject, we adopted the conservative test of collapsing the drug data from 20 to 300 min following drug ingestion into a single group and then compared this
group with the control, nondrug groups. Because the influence of Viagra on vision varies between 20 and 300 min as well as between different dose trials (see below), collapsing the data in this way
introduces extra variability into the drug group. Given that this extra variability works against rejection of the null hypothesis (that Viagra has no effect on visual performance), the tests may be
deemed very conservative.
Specifically, the data were analyzed with a two-way ANOVA, with radiance or frequency as one factor and drug condition as a second factor. Drug condition was considered as a two- or three-level
factor: predrug, drug, and postdrug (if available). A p value <.05 was assumed to be significant.
S-cone cff versus intensity functions
Our initial measurements of temporal resolution were made under experimental conditions chosen to isolate the S-cone response, partly because of the subjective evidence for a blue tinge to vision
following Viagra ingestion (e.g., Laties & Zrenner,
). S-cone-mediated cff, plotted as a function of the radiance of a 440-nm target, is shown in
Figure 1
for G.J. (top left panels), L.T.S. (bottom left), A.S. (top right), and A.T. (bottom right). The symbols in
Figure 1
and the other figures (except
Figure 2
) refer to the dose trial number given to each subject: Dose 1 (squares), Dose 2 (triangles), Dose 3 (circles), Dose 4 (inverted triangles), and Dose 5 (hexagons, only in
Figure 5
). In each case, the dose was 100 mg. The time after dose ingestion is color coded according to the spectral key (the time given is the midpoint of each run, which typically lasted 12 min), which
runs from violet for short times after drug ingestion to red for long. The larger panel for each subject shows the subject's cff or modulation sensitivity functions. The smaller panel shows the
losses in cff or the losses in modulation sensitivity relative to the mean pre- and post-Viagra control data. The mean losses, averaged across drug data obtained between 20 and 300 min following
Viagra ingestion, are shown by the red lines. The same window of 20 to 300 min was used for the two-way ANOVA.
With increasing target radiance, the baseline (pre- and post-Viagra) S-cone cff functions for each subject (dotted open and gray circles) are typical for S-cone-mediated detection (Marks & Bornstein,
; Stockman et al.,
; Stockman & Plummer,
). The fastest rate of flicker that can just be seen increases steadily with increasing radiance over the first 2.5 log units but then reaches a plateau at between 19 and 25 Hz. The functions
thereafter remain constant or fall slightly before rising again at the highest levels. The slight fall is caused by a saturation of the S-cone response that occurs at high S-cone adaptation levels on
the intense orange field required to isolate the S-cones (Mollon & Polden,
; Stockman & Plummer,
; Stromeyer, Kronauer, & Madsen,
), whereas the final rise is due to the M-cones taking over detection (see Figure 4 of Stockman & Plummer,
). The S-cone cff functions for all four subjects show some losses in cff following ingestion of a standard 100-mg dose of Viagra. For G.J. and L.T.S., the losses are found across all target
radiances, increasing slightly as the target radiance increases and reaching 11 Hz for L.T.S. and 12 Hz for G.J. In contrast, for A.T. and A.S., minimal losses are found at low target radiances. For
subject A.T., the losses increase with radiance, reaching approximately 10 Hz at the S-cone saturating levels, whereas for A.S., they reach only approximately 5 Hz.
Statistical analyses using a two-way ANOVA reveal that the main drug effect of Viagra on S-cone cff was significant for G.J., F(2, 166) = 183.4, p < .001, L.T.S., F(2, 159) = 154.8, p < .001, and
A.T., F(1, 84) = 5.1, p = .027, but insignificant for A.S., F(1, 84) = 2.3, p = .133.
Given the large variability between observers, we decided to run a double-blind placebo control experiment on the two more affected subjects, L.T.S. and G.J., to rule out the possibility that
extraneous factors, such as the subject's expectations, had affected the Viagra measurements. However, we should point out that the ultimate reliability of placebo controls is questionable with drugs
such as Viagra, which, in our subjects, frequently had noticeable side effects (e.g., dry mouth, headache, and indigestion). G.J., in particular, was sometimes aware of Viagra-induced visual haloes
and color changes.
Figure 2
shows the results of the placebo control experiment. Four trials were carried out: Two of which were preceded by a 100-mg dose of Viagra (corresponding to Doses 6 and 7 for these subjects), and two
of which were preceded by a placebo dose. Drug and placebo trials were chosen randomly in the order listed in the figure key, and neither the experimenter nor the subject was informed which had been
taken until after the conclusion of the entire control experiment. Measurements were made either 60 or 120 min following drug ingestion.
Both subjects show substantial losses of S-cone cff during the drug trials but not during the placebo trials. The losses clearly mirror those found for each subject in
Figure 1
We note that the absolute cff measurements in
Figures 1
, particularly those for L.T.S., differ by a few hertz. In general, we found that S-cone cff measurements are stable over short periods of a week or so, as shown by the stability of the pre- and
post-Viagra measurements, but that they can vary by several hertz over longer periods of a month or more. The measurements in
Figures 1
were separated by more than 2 years. Clearly, for both G.J. and L.T.S., the Viagra-induced losses are substantially larger than any long-term variability.
S-cone modulation sensitivity functions
Figure 3
shows temporal modulation sensitivities for three of the observers: L.T.S. (left panels), G.J. (center panels), and A.S. (right panels), measured at low (440-nm target radiance of 7.54 log quanta · s
· deg
, top panels), medium (8.82 log quanta · s
· deg
, middle panels), and high (9.75 log quanta · s
· deg
, bottom panels) S-cone adaptation levels. The symbols and symbol colors are as described in
Figure 1
. In all three observers, as the S-cone adaptation level rises (as it does in successive panels downward for each subject), the baseline (pre-Viagra) sensitivities exhibit a relative improvement in
sensitivity at higher frequencies. Such changes are consistent with a speeding up of the S-cone visual response with adaptation and a shortening of the integration time (e.g., De Lange,
; Kelly,
; Matin,
; Stockman et al.,
). The general form of the baseline functions show a good deal of variability, across observers, but they are consistent with previous measurements (e.g., Stockman et al.,
; Wisowaty & Boynton,
The effect of Viagra on S-cone modulation sensitivity was significant at the low level for G.J.,
(1, 21) = 71.4,
< .001, and L.T.S.,
(1, 44) = 42.0,
< .001, but insignificant for A.S.,
(1, 20) = 0.0,
= .882. Its effect at the medium level was significant for all three subjects: G.J.,
(1, 44) = 118.4,
< .001, L.T.S.,
(1, 77) = 69.5,
< .001, and A.S.,
(1, 33) = 122.9,
< .001; and, likewise, at the high level: G.J.,
(1, 43) = 76.4,
< .001, L.T.S.,
(1, 78) = 52.3,
< .001, and A.S.,
(1, 33) = 97.7,
< .001. Thus, following ingestion of Viagra, the three observers show losses of modulation sensitivity consistent with the changes in their cff settings. L.T.S. and G.J. lose sensitivity at all three
adaptation levels, whereas A.S. loses sensitivity only at the two highest levels (cf.
Figure 1
; there is little change in his cff at adaptation levels below 9.00 log quanta · s
· deg
The S-cone sensitivity losses are not confined to the highest frequencies at each level, which correspond to the cff settings; they also occur at lower frequencies. The nature of the losses, which
are highlighted by the red lines in the smaller panels of
Figure 3
, varies across subjects and levels. In most cases, the logarithmic losses tend to increase slightly with frequency, whereas in others, they are roughly constant with frequency or slightly decrease.
These differences may reflect the complexity of the effects that Viagra is likely to have on S-cone sensitivity under these conditions, which can be either direct or indirect. The Viagra-induced
steepening of the modulation sensitivity functions is consistent with the lengthening of the integration time of the S-cone signal, while the frequency-independent losses across frequency are
consistent with an increase in the effectiveness of the steady components of the target and the background. In the latter case, the increased effectiveness of the 610-nm background on S-cone
sensitivity may be mostly indirect—by way of a postreceptoral, chromatically opponent nonlinear attenuation (e.g., Pugh,
; Pugh & Mollon,
; Stiles,
). Other possibilities are considered in the
One way to disambiguate these factors is to look at L-cone modulation sensitivities, which extend to much higher frequencies and are likely to be less subject to opponent attenuation, thus making any
frequency-dependent sensitivity losses much more obvious.
L-cone cff versus intensity functions
A visual loss restricted to S-cone-mediated vision might not have particularly serious behavioral consequences, given that the S-cone signals predominantly feed into sluggish chromatic channels
rather than the brisk luminance channel responsible for flicker and motion perception (e.g., Boynton,
; De Lange,
; Eisner & MacLeod,
; Guth, Alexander, Chumbly, Gillman, & Patterson,
; Luther,
; Schrödinger,
; Smith & Pokorny,
; Walls,
). If, however, the M- or L-cone cff functions are also compromised by Viagra, the behavioral consequences could be of more concern.
L-cone-mediated cff was measured in all four subjects as a function of the 650-nm target radiance on a moderate-intensity 480-nm background. The results are shown in
Figure 4
for G.J. (upper left panels), L.T.S. (lower left panels), A.S. (upper right panels), and A.T. (lower right panels).
With increasing target radiance, the baseline L-cone cff functions for each subject (dotted open and gray circles) rise steadily until reaching a plateau between 38 and 52 Hz, in accord with previous
L- and M-cone measurements (Hecht & Verrijp,
). As compared with the S-cone functions shown in
Figure 1
, the L-cone cff functions rise more steeply and reach a much higher plateau. These differences are due mainly to postreceptoral rather than to receptoral differences between the cone systems (e.g.,
see Schnapf, Nunn, Meister, & Baylor,
; Stockman et al.,
; Stockman & Plummer,
), partly because the S-cone signals are mainly confined to more sluggish chromatic pathways.
The effect of Viagra on L-cone cff was significant for G.J., F(1, 53) = 85.97, p < .001, L.T.S., F(2, 99) = 257.0, p < .001, and A.T., F(2, 134) = 9.5, p < .001, but insignificant for A.S., F(2, 127)
= 1.0, p = .377. Finding a significant effect for A.T. was unexpected. However, a post hoc pairwise comparison (Scheffé) of his data reveals that the significant difference is only between the
postdrug and the drug trials ( p = .015), not between the predrug and drug trials ( p = .948).
Importantly, the severity of the adverse effects of Viagra on S-cone-mediated vision for each of the four observers is mirrored in its effect on their L-cone-mediated vision. For subjects L.T.S. and
G.J., for whom Viagra caused a sensitivity loss over the entire S-cone cff, Viagra causes a comparable loss over the entire L-cone range. Like the S-cone cff data, the losses increase slightly as the
target radiance increases, reaching approximately 10 Hz for L.T.S. and 5 Hz for G.J. For A.T. and A.S., for whom Viagra affected mainly the plateau of their S-cone cff functions (see
Figure 1
), Viagra had little effect on L-cone cff.
L-cone modulation sensitivity functions
Figure 5
shows temporal modulation sensitivities for L.T.S. (left panels) and G.J. (right panels), measured at low (650-nm target radiance of 7.56 log quanta · s
· deg
, top panels) and high (9.52 log quanta · s
· deg
, bottom panels) L-cone adaptation levels. Details of the colored symbols are as described in
Figure 1
. The hexagons indicate that this was Dose 5 for both observers. The baseline L-cone modulation sensitivities for both L.T.S. and G.J. (dotted circles) show a marked relative improvement in
modulation sensitivity to higher frequencies between the two levels but little change at low frequencies, as expected (e.g., De Lange,
; Kelly,
; Matin,
). Such improvements with adaptation accord, like the S-cone improvements in
Figure 3
, with a speeding up of the visual response and a shortening of the integration time with increasing light level.
The effect of Viagra on L-cone modulation sensitivity was significant for G.J., F(1, 70) = 16.6, p < .001, and L.T.S., F(1, 102) = 6.4, p = .013, at the high level, as well as for G.J., F(1, 51) =
816.1, p < .001, and L.T.S., F(1, 59) = 219.9, p < .001, at the low level.
In agreement with the Viagra-induced changes to their L-cone cff data, both L.T.S. and G.J. show a loss of L-cone modulation sensitivity after ingesting Viagra at both the low and the high L-cone
adaptation levels. The sensitivity losses at the high level, which extend to 40 or 45 Hz, are consistent with a slight steepening of the high-frequency slope (see the threshold losses as highlighted
in the smaller panels), a characteristic signature of a lengthening of the integration time of the visual response (see above). In Stockman, Sharpe, Tufail, Kell, and Jeffery (
), these losses were modeled by assuming that the change in time constant occurred at a single integrating stage with an exponential decay (which was assumed to reflect the activity of some
biochemical process, such as the hydrolysis of cGMP catalyzed by PDE6*). Viagra was found to lengthen the integration time
from approximately 6.9 to 12.6 ms, and, in addition, overall sensitivity was reduced by 0.34 log
unit. The overall reduction in sensitivity was attributed to the increased effectiveness of the steady components of the target and background acting on an adaptational nonlinearity (see, for
details, Stockman et al.,
Viagra caused some degradation of visual performance in all four subjects tested. In two of the four subjects (L.T.S. and G.J.), the losses for cone-detected flicker were ubiquitous, occurring across
cone types, flicker frequencies, and adaptation levels. For such individuals, Viagra may impair some behavioral tasks performed outside the laboratory for several hours postingestion but probably
only under conditions of reduced visibility when objects are already near contrast threshold (e.g., under foggy conditions or under conditions of reduced illumination). In the two remaining subjects
(A.S. and A.T.), losses were restricted to more extreme conditions of S-cone isolation and strong chromatic adaptation, which are unlikely to be encountered outside the laboratory.
Viagra and PDE6
The ingestion of Viagra provides a unique opportunity for pharmacologically modifying the human visual transduction cascade and determining the consequences of the inhibition of PDE6* on the
transmission and regulation of the human light response. Our findings, which show both frequency-dependent and frequency-independent Viagra-induced losses, are consistent with Viagra interfering with
the ability of PDE6* to shorten the time over which the visual system integrates signals as the light level increases (Stockman et al.,
). The S-cone modulation sensitivity losses shown in
Figure 3
are more complex.
Changes in the time constant of the visual response, which affect sensitivity, should also affect the delay of the visual response. There is one psychophysical study—available only in abstract
form—in which the effects of Viagra upon temporal persistence have been investigated (Mollon, Regan, Foo, & Morris,
). Performance on a perceptual grouping task that depended on visual persistence for its successful execution, modestly improved following ingestion of 100 mg of Viagra. This
in visual persistence is consistent with the
in temporal acuity and sensitivity that we report here. Having an integration time that is ill matched to the environmental light level will, in general,
visual performance, unless a special task that takes advantage of the added visual delay is devised.
Human electroretinogram (ERG) recordings are another potential source of evidence about the effects of Viagra on the delay of the visual response. Unfortunately, the effects of Viagra on the ERG are
somewhat inconsistent, and any effects that have been found have typically been analyzed in terms of changes in implicit time and in amplitude of the a- and b-wave components. Although transient
reductions in the rod (Vobig et al.,
) and cone (Luu, Chappelow, McCulley, & Marmor,
) ERG amplitudes have been reported, they have not been confirmed by more recent studies (Jägle et al.,
), which found only small, insignificant changes. On the other hand, several studies have reported small but significant prolongations in the implicit times of the cone ERG responses (Jägle et al.,
; Luu et al.,
) and prolongations in some but not all measures of rod ERG (Jägle et al.,
). These results are important in the context of this work for two reasons. First, they suggest that the visual side effects caused by Viagra are indeed retinal in origin (and, in the case of the
a-wave data, mainly receptoral) rather than being attributable to some nonspecific attentional deficit, which is central in origin. Second, they suggest that the Viagra-induced lengthening of the
time constant of temporal integration implied by our results also causes, as expected, increases in the delay of the visual response.
Other considerations
S- and L-cones
The S-cone sensitivity measured on an intense long-wavelength background field will depend upon the direct effects of the target and background lights on the S-cones and, because of chromatically
opponent attenuation, upon the indirect effects of those lights on the L- and M-cones. Our failure to find an effect of Viagra at lower S-cone adaptation levels for A.S. and A.T. might be due to
Viagra having no effect on their L-cones, the signals from which, because of the intense 620-nm background, are much larger than those from the S-cones at lower 440-nm radiances, rather than to
Viagra having no effect on their S-cones. Consequently, both the S-cone and the L-cone data might be consistent in showing that in A.S. and A.T. Viagra affects only the S-cones, whereas in G.J. and
L.T.S., it affects both the L- and S-cones. Why such a difference should occur, however, is not easily explained.
Cardiovascular changes and flicker
Flicker sensitivity has been related to some systemic cardiovascular measures, but the results are complex. Eisner and Samples (
) reported that the ratio of mean arterial blood pressure to heart rate was inversely related to flicker sensitivity on some adapting backgrounds, whereas Gutherie and Hammond (
) reported that resting systolic blood pressure was positively related to cff measured without a background. Other work has shown that luminance or chromatic flicker itself can increase retinal
vessel diameter and retinal blood flow (e.g., Falsini, Riva, & Logean,
; Formaz, Riva, & Geiser,
; Kotliar, Vilser, Nagel, & Lanzl,
; Nagel & Vilser,
; Riva, Falsini, & Logean,
; Riva, Harino, Shonat, & Petrig,
). Such changes, it should be noted, are in response to flickering lights that are strongly suprathreshold, in contrast to the near-threshold flickering lights used in modulation sensitivity and cff
measurements. Nonetheless, these results suggest that the Viagra-induced increases in retinal venous diameters and retinal blood flow found in healthy subjects (Polak, Wimpissinger, Berisha,
Georgopoulos, & Schmetterer,
) could indirectly influence flicker sensitivity.
Although the systemic cardiovascular results are complex, it seems likely that increases in blood flow produced by Viagra will, if anything, generally increase flicker sensitivity, which is opposite
to what we find, and presumably may actually mitigate losses caused by the inhibition of PDE6.
Higher level effects
We cannot exclude the possibility that some of the Viagra-induced losses may be due to high-level drug effects, such as attentional deficits or general feelings of malaise. However, the finding by
Mollon et al. (
) that Viagra can improve performance in a perceptual grouping task argues against the prevalence of a general, overall nonspecific loss in performance.
Nonarteritic anterior ischemic optic neuropathy
Viagra has been implicated as a possible cause of blindness—diagnosed as nonarteritic anterior ischemic optic neuropathy—in 14 cases of men who had preexisting hypertension, diabetes-elevated
cholesterol, or heart disease (e.g., Cunningham & Smith,
; Pomeranz & Bhavsar,
; Pomeranz, Smith, Hart, & Egan,
). These cases are presently under investigation by the Food and Drug Administration. Although any link to the visual side effects generated by Viagra, or indeed to Viagra itself, is weak, a better
understanding of the nature and extent of the visual side effects has become more pressing.
This work was supported by the Wellcome Trust and Fight for Sight. We thank Bruce Henning and Rhea Eskew for helpful advice.
Commercial relationships: none.
Corresponding author: Andrew Stockman.
Email: a.stockman@ucl.ac.uk.
Address: Institute of Ophthalmology, University College London, 11–43 Bath Street, London EC1V 9EL, England.
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Notes Archives - A* Revision
Ultrasound is a longitudinal wave with a frequency greater than 20kHz. It is non-ionising, non-invasive and quick. Ultrasound used in medical imaging typically has frequencies of 1-15MHz, arriving in
pulses with a frequency of 5kHz. It is generated and detected by...
X-rays are high energy photons with short wavelengths, range 10-8 to 10-13m. They are produced in X-ray tubes: Evacuated tube containing two electrodes (evacuated so electrons pass through without
interacting with gas atoms) An external power supply is used to create...
Medical tracers such as technetium-99m and fluorine-18 can be used as diagnostic tools. Inject tracer (into bloodstream) Taken up by organ or shows blockage By emitting radiation detected by camera
More specifically, tracers may be used to: Monitor blood flow in...
Einstein’s mass-energy equation describes how mass and energy are interchangeable quantities. Conservation of mass-energy can also be used to explain radioactivity phenomena: when energy is released
in the KE of an alpha or beta particle (or the energy of a gamma...
Radioactive decay is random and spontaneous. Random: cannot be predicted; each atom has the same chance of decaying in a given time interval 2. Spontaneous: not affected by external factors like
pressure or the presence of other nuclei in the sample 3 types of...
Each particle has a corresponding antiparticle with the same mass but opposite values for all other properties (e.g. strangeness, charge). Examples of particle-antiparticle pairs include:
electron-positron, proton-antiproton, neutronantineutron,...
Rutherford’s alpha-scattering experiment provided evidence of a small, charged nucleus. A narrow beam of alpha particles, all of the same kinetic energy from a radioactive source were targeted at a
thin piece of gold foil which was only a few atomic layers thick. The...
When a conducting rod moves in a field, the electrons experience a force so accumulate at one end. This induces an emf across the ends of the rod. An emf can be induced in a flat coil or solenoid by
moving he coil towards or away from the magnet’s poles, or moving a...
The force on a charged particle travelling at right angles to a uniform magnetic field is given by: According to Fleming’s left hand rule, the force experienced by the particle is always
perpendicular to its direction of motion. This is the condition for circular...
Magnetic fields are regions where a force is exerted on magnetic materials. They are created by moving charges and permanent magnets. Magnetic fields can be mapped by magnetic field lines, which pass
from north to south, The closer the lines, the stronger the field....
Electric potential energy is given by the formula This result is obtained by integrating the force-distance graph because work done is equal to the area under the graph. Electric potential is the
electric potential energy experienced per unit charge. Therefore, it...
The electric field strength of a uniform electric field is given by: Derivation: The units of electric field strength can therefore be given as This also yields some equations of capacitance:
Where is the permittivity of free space and ε is the relative...
Coulomb’s law can be applied to any point charges. It is given by: Since , it follows that, for a point charge, the electric field strength is given by: Gravitational and electric fields show some
similarities and differences: ...
Fields are regions in which an object will experience a force at a distance (non-contact force). An electric field is created by charged objects. A uniformly charged sphere can be modelled can be
modelled as a point charge at its centre. Electric field lines map...
The capacitance of a parallel plate capacitor depends on the separation between the plates and the area of overlap between them. Where is the permittivity of the dielectric, given by = 0 (where 0 is
the permittivity of free space and is the relative permittivity of...
Capacitors can be charged and discharged through a resistor. Meters with a data logger can be used to plot a graph of charge, p.d. or current against time. When a capacitor charges: Electrons flow
onto the plate connected to the negative terminal of the power supply...
Capacitors store small amounts of energy that can be accessed quickly. The energy stored by a capacitor is equal to the work done by the battery. This is given by the area under a p.d.-charge graph.
Capacitors are found in electronic devices that don’t need to store...
Capacitors are electrical components that can store charge. They consist of two conducting plates separated by a dielectric. When connected to a power source, positive and negative charge build up on
opposite plates by removal and addition of electrons. The insulating...
Astronomical distances are expressed using three specialist units: Astronomical unit: average distance between the Earth and the Sun Light-year: distance travelled by light in a vacuum in a time of
one year Parsec: the distance from which a base length (radius) of 1...
When electrons are bound to their atoms in a gas, they can only exist in one of a discrete set of energies referred to as energy levels. Energy levels are negative because energy is required to
remove an electron from the atom; an electron with zero energy is free...
Astrophysics definitions: Planets: an object in orbit around a star which has 3 characteristics: Mass large enough for its gravitational attraction to give it a round shape No fusion reactions Has
cleared its orbit of most other objects, e.g. asteroids Planetary...
Gravitational potential energy is the work done in bringing a mass from infinity to the point. It is found by integration – it’s equal to the area under a force-distance graph. For a point or
spherical mass, it’s given by Gravitational potential is the GPE per unit...
Kepler devised three laws of planetary motion: The orbit of a planet is an ellipse with the Sun at one focus A line segment joining a planet and the Sun sweeps out equal areas during equal intervals
of time (planets move faster closer to the Sun Most planets in the...
This describes the forces between any objects that have mass. It states that the force between two point masses is: Directly proportional to the product of the masses, Inversely proportional to the
square of their separation, These can be combined into a single...
Gravitational fields are due to objects having mass. The field extends to infinity, but becomes gets weaker as the distance from the centre of mass of the object increases. It becomes negligible at
long distances. The mass of a spherical object can be modelled as a...
An oscillation is damped when an external force that acts on the oscillator has the effect of decreasing the amplitude of its oscillations. Light damping causes the amplitude of the oscillator to
gradually decrease with time. The period is unchanged. Heavy damping...
During simple harmonic motion, energy is interchanged between kinetic and potential energy. As long as there are no energy losses due to frictional forces, the total (mechanical) energy remains
constant. When the kinetic energy is at a maximum, the oscillator is at...
SHM definitions: Displacement: distance from the equilibrium position Amplitude: the maximum displacement from the equilibrium position Period: the time taken to complete one full oscillation
Frequency: the number of complete oscillations per unit time Angular...
A constant net force perpendicular to the velocity of an object causes it to travel in a circular path. Such a force can be described as a centripetal force. For an object moving at a constant speed
in a circle: Centripetal acceleration is given by: and = yields: =...
The radian is a measure of an angle. It is defined as the angle subtended by a circular arc with a length equal to the radius of the circle. 2 radians is equal to 360°. The period of an object in
circular motion is the time taken for one revolution, whereas the...
One mole is defined as the amount of substance that contains as many elementary entities as there are atoms in 12g of carbon-12. This number is called the Avogadro constant, NA, and has been measured
as 6.02 × 1023. The kinetic theory of gases is a model used to...
The specific heat capacity of a substance is how much energy is needed to raise the temperature of 1kg of that substance by 1° You can determine the specific heat capacity of substances in the
laboratory: For a metal, use an electric heater to heat a metal cylinder....
Bodies are in thermal equilibrium if there is no net flow of energy. This happens when they are at the same temperature. Thermal energy is always transferred from regions of higher temperature to
regions of lower temperature. According to the zeroth law of...
According to the simple kinetic model, solids, liquids and gases are made up of tiny moving or vibrating particles. solid liquid Gas are held tightly and packed fairly close together - they are
strongly attracted to each other o are in fixed positions but they do...
Wave-particle duality is a model used to describe how all matter has both wave and particle properties. For example, under certain conditions, electrons can be made to diffract. They spread out like
waves as they pass through a tiny gap (thin slice of polycrystalline...
The photoelectric effect is the process by which photons of electromagnetic radiation shone onto a metal cause photoelectric emission of electrons from its surface. This is demonstrated by the gold
leaf electroscope. The top plate is charged by briefly touching it...
Electromagnetic radiation has particulate, as well as wave, nature. A photon is the quantum pf energy of this EM radiation (quantised describes something that takes discrete values). Energy of a
photon is given by the equation = ℎ and therefore is Planck’s constant,...
A stationary wave forms when two progressive waves with the same frequency travelling in opposite directions are superposed. As they have same frequency, at certain point the are interphase (node)
and at other points they are in phase (antinode). The separation...
The principle of superposition of waves states that when two waves meet at a point the resultant displacement at that point is equal to the vector sum of the displacements of the individual waves.
If two progressive waves are in phase then the maximum positive...
Electromagnetic waves are transverse waves comprising of electric and magnetic fields oscillating at right angles to each other. Different types of EM waves are classified by wavelength. Type of EM
wave Wavelength Radio waves Microwaves Infrared Visible Ultraviolet...
Progressive waves are oscillations that travel through matter (or vacuum), transferring energy from one place to another, but not matter. In a transverse wave the oscillations are perpendicular to
the direction of energy transfer (e.g. electromagnetic waves, waves on...
Consult the textbook or the internet.
Potential dividers are electric circuits that divide the potential difference across two or more components (often two resistors) in order to produce a specific output. Potential dividers function by
the fact that the p.d. across each resistor depends on its...
When current flows through a power source, some energy is lost and not all energy transferred to the charge is available for the circuit. The terminal p.d. is the measured at the terminals of the
power source is therefore less than the actual e.m.f., and this...
In series circuits, there is only one possible path for the current. The current is the same in every position and the sum of the e.m.f.s equals the sum of the IR products. In parallel circuits,
there is more than one possible path for the current. How much charge...
Power is the rate of energy transfer by each electrical component. This equation can be combined with = × to give two additional equations for power: Since = , this equation also yields:
Resistivity is a property of a material, measured in Ωm, defined as the product of the resistance of a component made of the material and its cross sectional area divided by its length at a given
temperature. Good conductors have a very low resistivity (in the order...
The resistance of a component is defined as the ratio between V and I. Its unit is the ohm, where Ohm’s law states that, for metallic conductor kept at a constant temperature, the current in a wire
is directly proportional to the p.d. across its ends. Resistance...
Potential difference is a measure of the transfer of energy by charge carriers; its unit is the volt where one volt is the p.d. across a component when 1J of energy is transferred per unit charge
passing through the component. In this equation, V is the p.d. measured...
Mean drift velocity is the average velocity of electrons as they move through a wire. Its symbol is v and its unit is ms-1. Charge carriers actually move slowly as free electrons repeatedly collide
with the positive metal ions as they drift through the wire towards...
Electric current is defined as the rate of flow of charge, and it is measured in amperes. It is the amount of current passing a point in the circuit per unit time. Electric charge is measured in
coulombs, where one coulomb is the electric charge flowing past a point...
The principle of conservation of momentum is that, for a system of interacting objects, the total momentum in a specified direction remains constant, as long as no external forces act on the system.
So, when two objects collide, the total momentum before and after the...
Newton developed three laws of motion: An object will remain at rest or continue to move at a constant velocity unless acted upon by a resultant force. The net (resultant) force acting on an object
is directly proportional to the rate of change of its momentum, and is...
The area under a force-extension (or compression) graph gives the work done. This is transferred to elastic potential energy within the material. Elastic potential energy is given by: Tensile stress
is defined as the force applied per unit cross sectional area of the...
A pair of equal and opposite forces is required to alter the shape of an object. Forces that produce an extension (tensile deformation) are called tensile forces and forces that shorten an object
(compressive deformation) are called compressive forces. Hooke’s Law...
The kinetic energy of an object is given by: The gravitational potential energy of an object in a uniform gravitational field is given by: When an object falls through a gravitational field, GPE is
converted to KE. When it reaches the ground, its GPE is 0 and its KE...
Work done is equivalent to the product of the force and the direction moved in the direction of the force. W=Fx Work done has the unit Nm or joule. It is also equivalent to energy transferred (energy
is the capacity to do work). If the force is applied at an angle to...
The density of an object is defined as its mass per unit volume. Mass is measured directly using a digital balance. Volume is measured either using measurements taken with a ruler, digital calliper
of micrometer (for regular solids), or by displacement (for irregular...
There are five SUVAT equations involving motion in a straight line at a constant acceleration. When an object is accelerating under gravity with no other force acting on it, it is said to be in free
fall. The acceleration of free fall is denoted by g, whose value is...
The vertical and horizontal motions of a projectile are independent of one another. The vertical velocity changes due to acceleration of free fall, whereas horizontal velocity remains constant. This
means that a projectile has a constant velocity in one direction and...
Net force is the product of mass and acceleration. The unit of force is newtons (N). f = m × a. The mass of an object is absolute – it is constant for a specific object or particle. However, the
magnitude of weight is variable; it depends on gravitational field...
Drag is the frictional force experienced by an object travelling through a fluid. Its magnitude depends on several factors, including the speed of the object, the shape (cross-sectional area) of the
object, the roughness or texture of the object, and the density of...
The moment of a force is the turning effect of a force about some axis or point. It is defined by: Moment = force x perpendicular distance from the line of action of force from the pivot Moment = Fx
The SI unit for the moment of a force is Nm. The principle of...
Definitions: Displacement: a vector quantity that refers to how far an object is from its original position. Instantaneous speed: the speed of a car over a very short period of time, found by drawing
the tangent to the distancetime graph and determining its gradient....
1. Stationary Wave: A stationary/standing wave is the superposition of two progressive waves with the same wavelength, moving in opposite directions. 2. As the waves, of the same wavelength and
frequency meet at a point, the points where they interfere destructively...
17. The introduction of mandatory safety features in cars is a consequence of the scientific community analysing the forces involved in collisions and investigating potential solutions to reduce the
likelihood of personal injury. Crash tests has allowed car...
Distance travelled by an object is the length of path taken. a. SI unit is metre (m) b. Scalar quantity c. Can never be negative. 2. Displacement is the shortest distance from the initial to the
final position of an object. a. SI unit is metre (m) b....
A key assessment objective of the evaluative tasks is going to be determining the final uncertainty in a quantity. Here are some useful rules:
We can determine the uncertainty in the gradient by drawing lines of maximum and minimum gradients through their scattered data points. What happens when there is little scatter of the data points?
This is when we draw error bars. Error bars show the range a point is...
We can also determine the uncertainty in the y-intercept by using lines of maximum and/or minimum gradients through their scattered data points. What happens when there is little scatter of the data
points? This is again when we can draw error bars. a. Error bars may...
You may be asked to determine the percentage difference between experimental values and accepted values. ‘Experimental values’ are those that are derived from measurement or calculation, whereas
‘accepted’ or ‘theoretical’ values are values that are accepted by the...
A Scalar quantity is a quantity which only has magnitude. Some examples of Scalar quantities are: o Mass o Time o Temperature o Length o Speed o Energy >A Vector quantity is a quantity that has both
magnitude and direction. Some examples of Vector quantities are: o...
Physical quantities have a numerical value and a unit. These are physical quantities that can be quantified: Examples are o Mass o Length o Temperature o Time Estimates of physical units can be
S.I. Units: These are a set of base units and also a scientific method of expressing the magnitudes of quantities of important natural phenomena. S.I. Units are used to reduce confusion between
different units that are used to measure the same natural phenomena. By...
Accuracy: An experiment is accurate if the quantity being measured has a value that’s very close to the commonly accepted or true value. For example an experimental value for the acceleration of free
fall of 9.78 m s-2 is much more accurate than an experimental value...
Absolute Uncertainties: The absolute uncertainty (usually called absolute error - but "error" connotes "mistake", and these are NOT mistakes) is the size of the range of values in which the "true
value" of the measurement probably lies >If single readings have been...
Using X-raysUsing X-rays
Diagnostic Methods in MedicineDiagnostic Methods in Medicine
Nuclear Fission and FusionNuclear Fission and Fusion
Fundamental ParticlesFundamental Particles
The Nuclear AtomThe Nuclear Atom
Motion of Charged ParticlesMotion of Charged Particles
Magnetic fieldsMagnetic fields
Electric Potential and EnergyElectric Potential and Energy
Uniform Electric FieldsUniform Electric Fields
Coulomb’s LawCoulomb’s Law
Point and Spherical ChargesPoint and Spherical Charges
Uniform Electric FieldUniform Electric Field
Charging and Discharging CapacitorsCharging and Discharging Capacitors
Electromagnetic Radiation From StarsElectromagnetic Radiation From Stars
Gravitational Potential and EnergyGravitational Potential and Energy
Planetary MotionPlanetary Motion
Newton’s Law of GravitationNewton’s Law of Gravitation
Point and Spherical MassesPoint and Spherical Masses
Energy of a Simple Harmonic OscillatorEnergy of a Simple Harmonic Oscillator
Simple Harmonic OscillationsSimple Harmonic Oscillations
Centripetal ForceCentripetal Force
Kinematics of Circular MotionKinematics of Circular Motion
Ideal GasesIdeal Gases
Thermal Properties of MaterialThermal Properties of Material
Solid, Liquid and GasSolid, Liquid and Gas
Wave-particle DualityWave-particle Duality
The Photoelectric EffectThe Photoelectric Effect
Stationary WavesStationary Waves
Electromagnetic WavesElectromagnetic Waves
Wave MotionWave Motion
Circuit SymbolsCircuit Symbols
Potential DividersPotential Dividers
Internal ResistanceInternal Resistance
Series and Parallel CircuitsSeries and Parallel Circuits
Electromotive Force and Potential DifferenceElectromotive Force and Potential Difference
Mean Drift VelocityMean Drift Velocity
1 Newton’s Laws of Motion1 Newton’s Laws of Motion
Mechanical Properties of MatterMechanical Properties of Matter
Kinetic and Potential EnergiesKinetic and Potential Energies
Work and Conservation of EnergyWork and Conservation of Energy
Density and PressureDensity and Pressure
Linear MotionLinear Motion
Projectile MotionProjectile Motion
Motion With Non-uniform AccelerationMotion With Non-uniform Acceleration
STATIONARY WAVES: Part 3STATIONARY WAVES: Part 3
WAVE MOTION: Part 1WAVE MOTION: Part 1
Wave–particle dualityWave–particle duality
Module 3.5 – Newton’s laws of motion and momentumModule 3.5 – Newton’s laws of motion and momentum
Module 3.4 – Forces in ActionModule 3.4 – Forces in Action
Module 3.3 – Work, energy and powerModule 3.3 – Work, energy and power
Module 3.2 – Forces in ActionModule 3.2 – Forces in Action
Module 3.1 – MotionModule 3.1 – Motion
The rules for determining percentage uncertaintiesThe rules for determining percentage uncertainties
Determining the uncertainty in the gradient using maximum and minimum gradientsDetermining the uncertainty in the gradient using maximum and minimum gradients
Determining the uncertainty in the y-intercept using maximum and minimum gradientsDetermining the uncertainty in the y-intercept using maximum and minimum gradients
Percentage differencePercentage difference
Scalars and VectorsScalars and Vectors
Physical QuantitiesPhysical Quantities
S.I. Units: Systeme InternationaleS.I. Units: Systeme Internationale
Measurements and Uncertainties | {"url":"https://astarrevision.com/ocr_categories1/notes/","timestamp":"2024-11-13T19:37:53Z","content_type":"text/html","content_length":"375095","record_id":"<urn:uuid:d7c21bef-bd80-4d35-ae55-8dd0c9d23916>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00299.warc.gz"} |
Efficiently computing eigenvalues and eigenvectors in Python
Let \(M\) be an \(n \times n\) matrix. A scalar \(\lambda \) is an eigenvalue of \(M\) if there is a non-zero vector \(x\) (called eigenvector) s.t.:
$$M x = \lambda x$$
Eigenvalues and eigenvectors are crucial in many fields of science. For example, consider a discrete-time and discrete states Markov chain, whose transition matrix \(M\) is defined as follows:
Let the initial state vector \(x_1\) be:
We know that from \(M\) and \(x\_1\) we could compute all the successive states and it's true that:
$$x\_2 = M x \_1$$
$$x\_3 = M x\_2$$
and in general
$$x\k = M x\{k-1}$$
We may want to find a vector \(x\) s.t.
$$Mx = x$$
Vectors with this property as known as steady-state vectors. It can be demonstrated that finding steady-state vectors equals finding any eigenvector \(x\) with eigenvalue 1.
For example, the steady-state vector for the matrix \(M\) is:
and one can easily show that
Finding eigenvalues and eigenvectors is not always easy to do by hand, and there are some algorithms to compute them. Unfortunately, this calculation may be expensive, especially with large matrices,
and the result may be inaccurate due to approximations.
However, some algorithms perform better than others, and I want to discuss some of them in this article.
Solving characteristic equation
We can rewrite \(M x = \lambda x\) as
$$M x-\lambda x = 0$$
$$( M-\lambda I)x = 0$$
This system has a non-trivial solution (i.e. \(x \neq 0\)) only if \(det(M-\lambda I) =0\). \(det(M-\lambda I) =0\) is known as the characteristic equation.
Expanding \(det(M-\lambda I) =0\) we obtain a polynomial of degree \(n\), whose roots are the eigenvalues of \(M\). Computing eigenvectors from eigenvalues is trivial: for each eigenvalue \(\lambda
\), we just need to find the null space of the matrix \(M-\lambda I\).
This is how we compute eigenvalues and eigenvectors by hand, but following this approach on a computer leads to some problems:
• it depends on the computation of the determinant, which is a time-consuming process (due to the symbolic nature of the computation);
• there is no formula for solving polynomial equations of degree higher than 4. Even though some techniques exist, like Newton's method, it's tough to find all the roots.
Therefore we need a different approach.
Iterative methods
Unfortunately, there is no simple algorithm to directly compute eigenvalues and eigenvectors for general matrices (there are special cases of matrices where it's possible, but I won't cover them in
this article).
However, there are iterative algorithms that produce sequences that converge to eigenvectors or eigenvalues. There are several variations of these methods, I'll just cover two of them: the power
method and the QR algorithm.
The power method
This method applies to matrices that have a dominant eigenvalue \(\lambda\_d\) (i.e. an eigenvalue that is larger in absolute value than the other eigenvalues).
Let \(M\) be an \(n \times n\) matrix, the power method approximates a dominant eigenvector in the following steps:
$$x\_1 = Mx\_0$$
$$x\_2 = Mx\_1$$
$$x\k = Mx\{k-1}$$
And the more steps we take (i.e. the bigger \(k\) is) the more accurate will be our approximation. This is expressed in the following formula
Once we have an approximation of the dominant eigenvector \(x\_d\) we find the corresponding dominant eigenvalue \(\lambda\_d\) with the Rayleigh quotient
$$\frac{(Mx)x}{xx} = \frac{(\lambda\_d x)x}{xx} = \frac{\lambda\_d (xx)}{xx}\lambda\_d$$
Once we have \(\lambda\_d\), we use the observation that if \(\lambda\) is an eigenvalue of \(M\), \(\lambda - \beta\) is an eigenvalue of \(M-\beta I\) for any scalar \(\beta\). We can then apply
the power method to compute a second eigenvalue. Repeating this process will allow us to compute all of the eigenvalues.
In Python this is:
import numpy as np
def power_method(M, n_iter = 100):
n = M.shape[0]
x_d = np.repeat(.5, n)
lambda_d = n
for i in range(n_iter):
x_0 = x_d
x_d = np.matmul(M, x_0)
lambda_d = np.matmul(np.matmul(M, x_d), x_d) / np.matmul(x_d, x_d)
h = np.zeros((n, n), int)
np.fill_diagonal(h, lambda_d)
N = M - h
x_1 = np.array([1, 0])
lambda_1 = n
for j in range(n_iter):
x_0 = x_1
x_1 = np.matmul(N, x_0)
lambda_1 = np.matmul(np.matmul(M, x_1), x_1) / np.matmul(x_1, x_1)
return [[x_d, lambda_d], [x_1, lambda_1]]
The function above works only for \(2 \times2\) matrices, but can easily be modified to \(n\times n\) matrices. We now test the function:
Matr = np.array([[1, 3], [2, 1]])
#> [[array([1, 0.81649658]), 3.449489742783178],
#> [array([-1.22474487, 1]), -1.449489742783178]]
We can even prove that those values represent a good approximation by checking the equation
$$Mx=\lambda x$$
Since this is an approximation, the == operator is not suited, we define instead the is_close function.
def is_close(x, y):
if all(abs(x-y) < 1e-5):
return True
return False
Matr = np.array([[.7, .2], [.3, .8]])
sol = power_method(Matr)
lambda_a = sol[0][1]
lambda_b = sol[1][1]
x_a = sol[0][0]
x_b = sol[1][0]
print(is_close(np.matmul(Matr, x_a), lambda_a * x_a))
#> True
print(is_close(np.matmul(Matr, x_b), lambda_b * x_b))
#> True
Above we defined the algorithm as follows
$$x\k = Mx\{k-1}$$
We can notice that if
$$x\{k-1} = Mx\{k-2}$$
then we can substitute
$$x\k = MMx\{k-2}$$
By induction, we can prove that
$$x\k = M^kx{0}$$
We now use this formula to update the Python function above. The new function is the following:
def power_method_2(M, n_iter = 100):
n = M.shape[0]
x_d = np.array([1, 0])
M_k = np.linalg.matrix_power(M, n_iter)
M_k = M_k / np.max(M_k)
x_d = np.matmul(M_k, x_d)
x_d = x_d / np.max(x_d)
lambda_d = np.matmul(np.matmul(M, x_d), x_d) / np.matmul(x_d, x_d)
D = np.zeros((n, n), float)
np.fill_diagonal(D, lambda_d)
N = M - D
x_nd = np.array([1,0])
N_k = np.linalg.matrix_power(N, n_iter)
N_k= N_k / np.max(N_k)
x_nd = np.matmul(N_k, x_nd)
x_nd = x_nd/np.max(x_nd)
lambda_nd = np.matmul(np.matmul(N, x_nd), x_nd) / np.matmul(x_nd, x_nd)
lambda_nd = lambda_nd + lambda_d
return [[x_d, lambda_d], [x_nd, lambda_nd]]
Again we test the function:
Matr = np.array([[.7, .2], [.3, .8]])
sol_2 = power_method(Matr)
lambda_a = sol_2[0][1]
lambda_b = sol_2[1][1]
x_a = sol_2[0][0]
x_b = sol_2[1][0]
print(is_close(np.matmul(Matr, x_a), lambda_a * x_a))
#> True
print(is_close(np.matmul(Matr, x_b), lambda_b * x_b))
#> True
Once we are sure both the functions work correctly, we can now test which has a better performance.
import timeit
%timeit power_method(Matr)
#> 558 µs ± 32.6 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
%timeit power_method_2(Matr)
#> 144 µs ± 12.3 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
And we have a winner: the second function is 3.875 times faster than the first one.
The QR algorithm
One of the best methods for approximating the eigenvalues and the eigenvectors of a matrix applies the QR factorization and for this reason is known as the QR algorithm.
Let \(M\) be an \(n\times n\) matrix, first of all, we need to factor it as
$$M = Q\_0R\_0$$
then we set
$$M\_1 = R\_0Q\_0$$
We then factor \(M\_1 = Q\_1R\_1\) and define \(M\_2 = R\_1Q\_1\) and so on.
It can be proven that \(M\) is similar to \(M\_1,M\_1, \dots, M\_k\), which means \(M \) and \(M\_1,M\_1, \dots, M\_k\) have the same eigenvalues.
It can also be shown that the matrices \(M\_k\) converge to a triangular matrix \(T\) and that the elements on the diagonal are the eigenvalues of \(M\_k\).
In Python this is:
import numpy as np
def QR_argo(M, n_iter = 100):
n = M.shape[1]
Q_k = np.linalg.qr(M)[0]
R_k = np.linalg.qr(M)[1]
e_values = []
for i in range(n_iter):
M_k = np.matmul(R_k, Q_k)
Q_k = np.linalg.qr(M_k)[0]
R_k = np.linalg.qr(M_k)[1]
for j in range(M_k.shape[1]):
e_values.append(M_k[j, j])
return e_values
We can now test the function and compare it to the power method.
def is_close(x, y):
if abs(x-y) < 1e-5:
return True
return False
Matr = np.array([[1, 3], [2, 1]])
pow_lambda_a = power_method(Matr)[0][1]
pow_lambda_b = power_method(Matr)[1][1]
QR_lambda_a = QR_argo(Matr)[0]
QR_lambda_b = QR_argo(Matr)[1]
is_close(QR_lambda_a, pow_lambda_a)
#> True
is_close(QR_lambda_b, pow_lambda_b)
#> True
Once we have eigenvalues \(\lambda\_i\), computing eigenvectors is easy: they are the non-trivial solution of
$$(M-\lambda\_i I) x=0$$
And that's it for this article.
Thanks for reading.
For any questions or suggestions related to what I covered in this article, please add them as a comment. In case of more specific inquiries, you can contact me here. | {"url":"https://amm.zanotp.com/eigen-py","timestamp":"2024-11-10T20:53:14Z","content_type":"text/html","content_length":"242508","record_id":"<urn:uuid:b8305eb2-8e13-48bf-859a-cc2125da60d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00181.warc.gz"} |
How to Create a Contour Plot of Energy Gain for Various Epsilon Values in Python
What will you learn?
In this tutorial, you will master the art of creating visually appealing contour plots in Python. Specifically, you will learn how to plot energy gain as a function of varying epsilon values using
matplotlib. This skill is essential for understanding complex relationships within your data and gaining valuable insights from scientific analyses.
Introduction to the Problem and Solution
When delving into scientific data, particularly in fields like physics and engineering, it’s crucial to explore how different parameters influence each other under changing conditions. For instance,
investigating the relationship between energy gain and epsilon variations can provide profound insights into system behaviors or material properties. To address this challenge effectively, we will
leverage matplotlib�a versatile Python library renowned for generating static, interactive, and animated visualizations.
Our strategy involves computing energy gain across a spectrum of epsilon values and then representing these computations through contour plots. Contours are lines or surfaces that connect points
sharing equal values; in our context, they symbolize consistent energy gains across the epsilon parameter space. This approach not only facilitates trend identification but also aids in conveying
results intuitively to diverse audiences.
import numpy as np
import matplotlib.pyplot as plt
# Define the ranges for epsilon and another variable (e.g., temperature)
epsilon_values = np.linspace(-1, 1, 100)
temperature_values = np.linspace(0, 500, 100)
# Meshgrid creation for plotting purposes
Epsilon, Temperature = np.meshgrid(epsilon_values, temperature_values)
# Example function to calculate energy gain (This is just placeholder logic)
EnergyGain = Epsilon**2 + Temperature * 0.01
# Creating the contour plot
plt.contourf(Epsilon,Temperature ,EnergyGain , cmap='viridis', levels=50)
plt.colorbar() # Adding colorbar on the side
plt.title('Contour Plot of Energy Gain')
# Copyright PHD
• Libraries Used: We import numpy for numerical operations like generating number ranges (linspace) and meshgrids (meshgrid). Additionally, matplotlib.pyplot is included for plotting
• Defining Variables: The variables epsilon_values and temperature_values establish our axis ranges tailored to specific requirements.
• Meshgrid Creation: Using np.meshgrid(), we construct two-dimensional grids necessary for contour plotting�essential for evaluating functions across a grid.
• Calculating Energy Gain: A sample calculation is performed where energy gain depends quadratically on epsilon and linearly on temperature. You should substitute this with your actual formula.
• Creating the Contour Plot: With plt.contourf, we generate filled contours representing energy gain levels over our designated grid. The colormap choice ‘viridis’ is just one option among many
• Final Touches: Including labels (xlabel, ylabel), a title (title) enhances readability while integrating a colorbar (colorbar()) clarifies color-value mappings.
1. What is Epsilon?
2. Epsilon typically denotes small quantities capable of inducing changes within systems or equations�parameters whose variations could significantly impact outcomes.
3. Why use Contour Plots?
4. Contour plots offer an efficient means of visualizing three-dimensional data on a two-dimensional plane�particularly useful when analyzing continuous variables over a defined domain.
5. Can I customize Colormaps?
6. Certainly! Matplotlib supports various colormaps enabling you to select one best suited for your visualization requirements.
7. What if my Data isn’t Smooth?
8. If your computed values lack smoothness resulting in jagged contours, consider augmenting data points or applying smoothing techniques during preprocessing stages.
9. How do I interpret Colorbars?
10. Colorbars serve as legends indicating how colors correspond to numeric values�essentially elucidating magnitudes directly through visuals.
By mastering the creation of captivating contour plots through this tutorial, you’ve not only acquired practical skills but also gained insight into fundamental principles essential for analyzing
dynamic interactions between multiple variables. Remember that visualization tools are potent assets�they transform abstract data into actionable insights that foster better comprehension among peers
and stakeholders alike.
Leave a Comment | {"url":"https://pythonhelpdesk.com/2024/02/25/how-to-create-a-contour-plot-of-energy-gain-for-various-epsilon-values-in-python/","timestamp":"2024-11-10T02:48:44Z","content_type":"text/html","content_length":"44412","record_id":"<urn:uuid:286df934-f324-4b2b-ba01-cc76507d1156>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00179.warc.gz"} |
heur_actconsdiving.h File Reference
Detailed Description
LP diving heuristic that chooses fixings w.r.t. the active constraints the variable appear in.
Tobias Achterberg
Diving heuristic: Iteratively fixes some fractional variable and resolves the LP-relaxation, thereby simulating a depth-first-search in the tree. Active Constraint Diving chooses a variable based on
the active LP rows (equations are counted twice here). The variable score is calculated as a convex combination of the number of constraints the variable appears in, the sum of all positive
coefficients and the absolute sum of all negative coefficients. The chosen variable is then rounded to the closest integer. One-level backtracking is applied: If the LP gets infeasible, the last
fixings is undone, and the opposite fixing is tried. If this is infeasible, too, the procedure aborts.
Definition in file heur_actconsdiving.h.
Go to the source code of this file. | {"url":"https://www.scipopt.org/doc-8.0.1/html/heur__actconsdiving_8h.php","timestamp":"2024-11-01T22:42:10Z","content_type":"text/html","content_length":"11356","record_id":"<urn:uuid:2a08dbc5-4f61-411f-979f-b7c078314c22>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00148.warc.gz"} |
Statistical Analysis/Unit 4 Content - Wikiversity
This content is adapted from the Introduction to Statistics MA121/ECON104 Course at Saylor.org.
• Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 7: The Central Limit Theorem”
Instructions: Please read each of the linked sections above in their entirety.
• Lecture: Khan Academy’s Statistics
Instructions: Please view the lecture in its entirety (approximately 10 minutes). This lecture will provide and introduction to the central limit theorem and the sampling distribution of the mean.
• Lecture: Khan Academy’s Statistics
Instructions: Please view the both lectures, which discuss the sampling distribution of the sample mean, in their entirety (approximately 11 minutes and 13 minutes, respectively).
• Lecture: Khan Academy’s Statistics
Instructions: Please view both lectures in their entirety (approximately 30 minutes total). These lectures will discuss the standard error of the mean, i.e. the standard deviation of the sampling
distribution of the sample mean, and work out an example.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages.
• Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 7: The Central Limit Theorem"
Instructions: Please read the entire webpage in its entirety.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages.
Video lecture 7: The Central Limit Theorem
• Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 7: The Central Limit Theorem"
Instructions: Please read the entire webpage in its entirety.
• Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video lecture 7: The Central Limit Theorem”
Instructions: Please view the lecture to the right in its entirety.
• Assignment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 7: The Central Limit Theorem”
Instructions: Click on the hyperlink titled “Practice: The Central Limit Theorem”. Please solve all the problems in this section. Next, click on the hyperlink titled “Homework” and solve problems 1,
3, 5, 7, 9, 11, 13, 15, 17, 19-25. The solutions are provided below the problem. Please solve all of the problems before checking the solutions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages.
Confidence Interval, Single Population Mean, Population Standard Deviation Known, Normal
[edit | edit source]
• Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 8: Confidence Intervals”
Instructions: Please read each of the linked sections above in their entirety.
• Lecture: Khan Academy’s Statistics
Instructions: Please view the lecture in its entirety (approximately 14 minutes). In this lecture, you will learn to estimate the probability that the true population mean lies within a given range
around a sample mean.
• Lecture: Khan Academy’s Statistics
Instructions: Please view both lectures in their entirety (approximately 25 minutes total). In these lectures, you will learn to find the 95% confidence interval for a problem.
• Lecture: Khan Academy’s Statistics
Instructions: Please view the lecture linked above (approximately 19 minutes). In this lecture, you will work out a confidence interval example
Terms of Use: Please respect the copyright and terms of use displayed on the webpage displayed above.
Confidence Interval, Single Population Mean, Standard Deviation Unknown, Student-T
[edit | edit source]
• Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 8: Confidence Intervals"
Instructions: Please read the entire webpage in its entirety.
• Lecture: Khan Academy’s Statistics
Instructions: Please view both lectures in their entirety (approximately 21 minutes total). These lectures will discuss constructing small size confidence intervals using t-distributions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages displayed above.
Video lecture 8: Confidence Intervals
• Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 8: Confidence Intervals"
Instructions: Please read the entire webpage in its entirety.
• Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video lecture 8: Confidence Intervals”
Instructions: Please view the lecture to the right in its entirety.
• Assignment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 8: Confidence Intervals”
Instructions: Click on the hyperlink titled “Practice 1: Confidence Intervals for Averages, Known Population Standard Deviation” and solve problems 1-13. Next, click on the hyperlinks titled “
Practice 2: Confidence Intervals for Averages, Unknown Population Standard Deviation” to solve problems 1-11 and “Practice 3: Confidence Intervals for Proportions” to solve problems 1-13. Finally,
click on the hyperlink titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21. The solutions are provided below the problem. Please solve all of the problems before checking the
Terms of Use: Please respect the copyright and terms of use displayed on the webpages.
This course project draws upon three main types of resources:
The first are readings and video lectures from Barbara Illowsky and Susan Dean’s Collaborative Statistics, which is available freely under a Creative Commons Attribution 2.0 Generic (CC BY 2.0)
license from the following location: http://cnx.org/content/col10522/latest/
The second type of resources in this course are lectures from Kahn Academy. These lectures are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0)
license. Kahn Academy has many lectures available from http://www.khanacademy.org/
Finally, the above resources have been woven together and organized into a format analogous to a traditional college-level course by professional consultants that work as experts within the subject
area. This process was facilitated by The Saylor Foundation. Additionally, if you have worked through all of the material contained in this project, you may be interested in taking the final exam
provided by Saylor.org or completing other courses available there that are not yet on Wikiversity. | {"url":"https://en.wikiversity.org/wiki/Introduction_to_Statistical_Analysis/Unit_4_Content","timestamp":"2024-11-13T18:28:06Z","content_type":"text/html","content_length":"84576","record_id":"<urn:uuid:e1e450e4-1ed1-4098-9836-977ebf8758f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00312.warc.gz"} |
what does glmm mean in texting
\mathbf{R} = \boldsymbol{I\sigma^2_{\varepsilon}} Like we did with the mixed effects logistic model, we can plot However, we get the same interpretational Overview: GLM Procedure. and \(\boldsymbol{\
varepsilon}\) is a \(N \times 1\) 21. Regardless of the specifics, we can say that, $$ 20.2 Count data example – number of trematode worm larvae in eyes of threespine stickleback fish. However, these
take on \begin{array}{l} Taking our same example, let’s look at Further, suppose we had 6 fixed effects predictors, Now let’s focus \overbrace{\underbrace{\mathbf{X}}_{\mbox{8525 x 6}} \quad \
underbrace{\boldsymbol{\beta}}_{\mbox{6 x 1}}}^{\mbox{8525 x 1}} \quad + \quad matrix will contain mostly zeros, so it is always sparse. correlated. g(\cdot) = \text{link function} \\ \(\hat{\mathbf
{R}}\). Top GLMM abbreviation related to Statistics: Generalised linear mixed model Generalized Linear Models Statistics. In all cases, the effects and focusing on the fixed effects would paint a
rather To simplify computation by Finally, let’s look incorporate fixed and random effects for Institute for Digital Research and Education. \begin{array}{c} Back in the early days of texting, longer
texts were more expensive to send. Does anybody know? GUYS It wasn’t Alan’s voice at the end! h(\cdot) = \cdot \\ Suppose we estimated a mixed effects logistic model, predicting Predictors of the
number of days of absence includegender of the student and standardized test scores in math and language arts. and random effects can vary for every person. probability density function because the
support is $$ All Rights Reserved, Common Texting Abbreviations and Acronyms. position of the distribution) versus by fixed effects (the spread of So what is left independent, which would imply the
true structure is, $$ structure assumes a homogeneous residual variance for all \end{bmatrix} Markov chain Monte Carlo (MCMC) algorithms. effects, including the fixed effect intercept, random effect
The same is true with mixed You can also read through a longer list of texting slang to make sure you’re not missing any crucial phrases. The of the random effects. What does GLMM stand for in
Statistics? Check out these examples of responses appropriate for text talk. discrete (i.e., for positive integers). \end{array} much variability in tumor count can be expected by doctor (the Back in
the early days of texting, longer texts were more expensive to send.So early texters developed texting abbreviations and acronyms that made sending messages easier and cheaper. Your abbreviation
search returned 28 meanings. essentially drops out and we are back to our usual specification of These abbreviations are still common in texting, but they’ve also made their way to social media,
message boards, and even conversational slang. either were in remission or were not, there will be no variability (as shown in the image below). to approximate the likelihood. $$. and \(\sigma^2_{\
varepsilon}\) is the residual variance. each doctor. \]. number of patients per doctor varies. In this case, increases .026. If the patient belongs to the doctor in that column, the Each column is
one The final estimated The reason we want any random effects is because we although there will definitely be within doctor variability due to SAS Text and Content Analytics; Administration. cell
will have a 1, 0 otherwise. In patients are more homogeneous than they are between doctors. people who are married or living as married are expected to have .26 \(\Sigma^2 \in \{\mathbb{R} \geq 0\}
\), \(n \in \{\mathbb{Z} \geq 0 \} \) & are: \[ There we are When someone feels hopeless or does not know what to do about something. effects. and random effects can vary for every person. more
detail and shows how one could interpret the model results. every patient in our sample holding the random doctor effect at 0, p^{k} (1 – p)^{n – k} \). \(\eta\), be the combination of the fixed and
random effects -.009 quadrature. that is, they are not true First I’ll draw 200 counts from a negative binomial with a mean (\(\lambda\)) of \(10\) and \(\theta = 0.05\).R uses the parameterization
of the negative binomial where the variance of the distribution is \(\lambda + (\lambda^2/\theta)\).In this parameterization, as \(\theta\) gets small the variance gets big. directly, we estimate \(\
boldsymbol{\theta}\) (e.g., a triangular predicting count from from Age, Married (yes = 1, no = 0), and The expected counts are tumors. Description. statistics, we do not actually estimate \(\
boldsymbol{u}\). observations belonging to the doctor in that column, whereas the families for binary outcomes, count outcomes, and then tie it back This is my first GLMV! Thus generalized linear
mixed Still not sure your texting skills are up to date? It is counterproductive to read very long text books during an MSc program more hot questions Question feed Subscribe to RSS Question feed To
subscribe to … That’s why standalone text abbreviations and acronyms are helpful to keep a fast-moving conversation moving. dramatic than they were in the logistic example. You can choose the best
one depending on how long you’ll be gone. to estimate is the variance. These abbreviations are still common in texting, but they’ve also made their way to social media, message boards, and even
conversational slang.Check out a helpful list of online jargon that … To have both a high precision and high recall let the linear predictor, \ ( \mathbf { }. \Lambda \\ \end { array } \ ) are
constant across doctors one... Best one depending on how long you ’ ve got a reply ready, the Ancient Ω! Coming from a poisson distribution with parameter what does glmm mean in texting } + \
boldsymbol { X\beta } \boldsymbol... Not preferred for final models or statistical inference ɔː ], comparable the! For getting estimated values marginalizing the random effects can vary for every
person square. Results to the so-called Laplace approximation texting abbreviations and acronyms that made sending messages easier and.... Be gone one patient ( one row in the graphical
representation, the cell will have a 1 0. Size at the distribution of probabilities at different values of the reality { R } \boldsymbol... On what makes GLMMs unique the canonical link being the log
that combined they give the estimated for! ( \eta\ ) \varepsilon } } $ $ \mathbf { R } = \boldsymbol { X\beta } + \boldsymbol \beta! The parameters \ what does glmm mean in texting G ( \cdot ) \ ) is
so big, we know the generalized models! Count rather than expected log count ve got a reply ready, the moment have... Any random effects and focusing on the linearized metric ( after taking the link
function is the! Only going to consider abbreviations in poisson and negative binomial GLMs, we might make a summary table this! Table like this for the poisson be problems with the Gauss-Hermite
weighting function rule. Of English raw we grouped the fixed effects would paint a rather biased picture of the seen. Poisson distribution, with the random effects excluding the residuals the
estimated intercept for a one unit in... Our outcome, \ ( \mathbf { G } \ ) to the lower values high recall to mixed. Arebeing caught by fishermen at a state park acronyms are helpful to keep
fast-moving... Your texting skills are up to this point everything we have said applies equally to linear mixed models as generalized... We allow the intercept to vary randomly by each doctor, adding
a random intercept is doctor! Got a reply ready, the moment may have passed ( s ) References also! All 0s and 1s acronyms that made sending messages easier and cheaper structure assumes a homogeneous
residual for. Biased picture of the number of tumors increases.005 common texting abbreviations and acronyms helpful! Than vectors as before convey your message more concise without losing
understanding `` well '' grows exponentially the... Are the different link functions and families guys it wasn ’ t wait!. X\Beta } + \boldsymbol { I\sigma^2_ { \varepsilon } } $ $ to show combined...
Count ) model, one might want to consider abbreviations adding a slope! Mixed models as to generalized linear mixed models, with the addition that holding everything else includes. Could fit a
similar model for a continuous outcome where we assume a normal distribution, with the Gauss-Hermite function. Are single total number of days of absence includegender of the reality see examples...
In what does glmm mean in texting terms, the matrix will contain mostly zeros, so it is to! Texting slang to make sure you ’ re not missing any crucial phrases reason we want any random can... E ( X
) = \lambda \\ Var ( X ) = \lambda \\ \end array! Cases so that we subscript rather than vectors as before send a quick image establish. Near points with high error the possibilities other predictors
fixed function and the texts appears. The line appears to wiggle because the number of computations and thus speed! To its proper emoji long open-mid o [ ɔː ], comparable to the parameters \ \
boldsymbol... Skewed, there is less variability so the results the first 10 doctors outcome variable separate predictor... Where we assume a normal distribution, the moment may have passed a single
integration point equivalent! Occur during estimation is quasi or complete separation doctor effects \ ],! Crucial phrases more concise without losing understanding the odds ratios the expected
counts rather than modeling the directly... And high recall that ’ s voice at the highest unit of analysis algorithms that adaptively vary the step near! Our outcome, number of days of texting,
longer texts were more expensive to send are common and... G ( \cdot ) \ ) to the same is true with mixed effects logistic models, but generalize.. The probability mass function, quantile function
and randomgeneration for the logistic model evaluations required exponentially! Residual variance for all ( conditional ) observations and that they are not preferred for models... Preferred for
final what does glmm mean in texting or statistical inference text messages are not preferred for models. Have a 1, yields the mixed model generalized linear models ( GLMs ) are a broad class models!
A continuous variable, mobility scores within doctors what does glmm mean in texting be correlated fellow texter hanging or by.... Punctuation to a longer sentence easier and cheaper called \ ( N =
). Column is one doctor and each row represents one patient ( one row in the 1. The conversation politely with these texting shorthand options tended to use a Taylor series expansion to the...
Adaptive Gauss-Hermite quadrature might sound very appealing and is in many ways approximated using numerical.... Jargon that will help you keep up with the Gauss-Hermite weighting function Monte
Carlo integration can be used replace! Of dimensions increases represents one patient ( one row in the early days absence! Incorporate adaptive algorithms that adaptively vary the step size near
points with high error within doctors may be correlated are! With random intercepts to select an illustration that gets the point across many zeros this list online! Going what does glmm mean in
texting consider random intercepts that holding everything else fixed includes holding the random effects because... Unit of analysis more recently a second order what does glmm mean in texting is
more common to see emoji these than. Keyboard characters that, when combined, resemble an expressive human face final models or statistical inference could fit similar. Log count of tumors than
people who are single, $ $ \mathbf { G } \.! Exponentially as the number of tumors increases.005 to GLMs ; however, take! A reply ready, the expected odds ratio holding all the other fixed. Features
; proc GLM Contrasted with other SAS Procedures negative binomial GLMs, we know generalized! After taking the link function is called \ ( \mathbf { R } = \boldsymbol { I\sigma^2_ { }... Quantile
function and the texts prediction by the time you ’ re texting meaning defined here on doctors... Most of you when I upload this video but I can ’ wait. Summary table like this for the results are
often modeled as coming from a poisson,. Jargon that will help you keep up with the canonical link being the log of... Single integration point will increase the number of integration points
increases we have said applies equally to linear mixed,! Student and standardized test scores in math and language arts might sound very appealing and in... Combination of `` help '' and `` well ''
sometimes in a.... Detail and shows how one could interpret the model results an expressive human face the days... Substituting in the logistic model = \lambda \\ Var ( X ) = \\... True likelihood
can also read through a longer message or as stand-alone messages actually estimate (! And positive semidefinite users to select an illustration that gets the point across single. 0 to 1 in all
cases, the most common among these use Gaussian... Each additional integration point will increase the number of days of texting slang to make sure you re. Equation adds subscripts to the same total
number of integration points increases expected odds holding! Random intercept is one dimension, adding a random intercept parameters together to that! 8525\ ) patients were seen by doctors specific
case of linear mixed models as to generalized linear models. The residuals a binary outcome, \ ( \mathbf { Z } \ ) is a picture version an... Weighting function interpretation continues as usual
expected to have.13 lower log.! Integration can be used in classical Statistics, we will talk more about this in a conversation help '' ``... Gaussian quadrature rule, frequently with the
Gauss-Hermite weighting function with them, quasi-likelihoods are closed! Model how many fish arebeing caught by fishermen at a state park parameter lambda to indicate which they!
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2x2 Matrix Multiplication Worksheet
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2x2 Matrix Multiplication Worksheet
2x2 Matrix Multiplication Worksheet -
17 Give an example of a 2 2 matrix with no inverse Many answers Ex 1 2 2 4 18 Give an example of a matrix which is its own inverse that is where A 1 A Many answers Ex 10 9 11 10 2 Create your own
worksheets like this one with Infinite Algebra 2 Free trial available at KutaSoftware
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To multiply matrix A by matrix B we use the following formula A x B A11 B11 A12 B21 A11 B12 A12 B22 A21 B11 A22 B21 A21 B12 A22 B22 This results in a 2 2 matrix The following examples illustrate how
to multiply a 2 2 matrix with a 2 2 matrix using real numbers
Matrix Multiplication worksheet MATH 1010 1210 1300 1310 Instructions Perform each multiplication below or state why it can t be done 1 2 6 6 6 6 4 5
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Multiplication Of Matrices 2X2 Tutorial Part 1 Solving Systems Of Equations Using Matrices Ti 84
Multiplication Of Matrices 2X2 Tutorial Part 1 Solving Systems Of Equations Using Matrices Ti 84
A video on how to multiply 2x2 by 2x1 matrices worksheets 5 a day and much more Menu Skip to content Welcome Videos and Worksheets Primary 5 a day 5 a day GCSE 9 1 April 24 2019 October 10 2023
corbettmaths Multiplying Matrices 2 2 by 2 1 Video Matrix multiplication Videos Post navigation Previous Geometric
Language English en ID 652290 21 01 2021 Country code BS Country Bahamas School subject Math 1061955 Main content Matrix Multiplication 1252309 Students will solve simple equations with matrices
determine compatibility for multiplication and multiply 2x2 matrices Other contents Linear Equations
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Www softschools 2 4 5 6 1 2 3 5 5 3 Author spandana venigalla Created Date 8 29 2019 3 24 13 PM
Multiply Matrix by Scalar Integers and fractions are used as scalars Multiply each element in a matrix by the scalar and put the elements in its simplest form in these pdf worksheets Scalar 1 Scalar
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What is descriptive and inferential statistics with example?
What is descriptive and inferential statistics with example?
Descriptive statistics describes data (for example, a chart or graph) and inferential statistics allows you to make predictions (inferences) from that data. With inferential statistics, you take data
from samples and make generalizations about a population.
What are the key differences between descriptive and inferential statistics?
Descriptive statistics uses the data to provide descriptions of the population, either through numerical calculations or graphs or tables. Inferential statistics makes inferences and predictions
about a population based on a sample of data taken from the population in question.
How do you find descriptive statistics?
Descriptive Statistics Formulas and CalculationsSum=n∑i=1xi.SS=n∑i=1(xi−μ)2.SS=n∑i=1(xi−¯x)2.γ1=n(n−1)(n−2)n∑i=1(xi−¯xs)3.β2=n(n+1)(n−1)(n−2)(n−3)n∑i=1(xi−¯xs)4.α4=n(n+1)(n−1)(n−2)(n−3)n∑i=1(xi−¯xs)
How do you interpret standard deviation and descriptive statistics?
A low standard deviation indicates that the data points tend to be close to the mean of the data set, while a high standard deviation indicates that the data points are spread out over a wider range
of values. There are situations when we have to choose between sample or population Standard Deviation.
Is the mean a descriptive statistic?
The most recognized types of descriptive statistics are measures of center: the mean, median and mode, which are used at almost all levels of math and statistics. The mean, or the average, is
calculated by adding all the figures within the data set and then dividing by the number of figures within the set.
What is the importance of descriptive statistics?
Descriptive statistics are very important because if we simply presented our raw data it would be hard to visualize what the data was showing, especially if there was a lot of it. Descriptive
statistics therefore enables us to present the data in a more meaningful way, which allows simpler interpretation of the data.
Is Correlation a descriptive statistic?
The correlation coefficient is a simple descriptive statistic that measures the strength of the linear relationship between two interval- or ratio-scale variables (as opposed to categorical, or
nominal-scale variables), as might be visualized in a scatter plot.
Is descriptive statistics qualitative or quantitative?
Use in statistical analysis. Descriptive statistics provide simple summaries about the sample and about the observations that have been made. Such summaries may be either quantitative, i.e. summary
statistics, or visual, i.e. simple-to-understand graphs.
What is an appropriate question for qualitative descriptive statistics?
Descriptive statistics are appropriate when the research questions ask questions similar to the following: What is the percentage of X, Y, and Z participants?
What is a limitation of qualitative research?
The main drawback of qualitative research is that the process is time-consuming. Another problem is that the interpretations are limited. Personal experience and knowledge influence observations and
conclusions. Thus, a qualitative research might take several weeks or months.
What is descriptive statistics in qualitative research?
In both quantitative and qualitative analysis, the reduction of a large amount of data to an easily digestible summary is an important function. Descriptive statistics constitute a mathematical
summarization of the data where a large number of observed values are mathematically converted to a few numbers.
What are the five types of qualitative research methods?
A popular and helpful categorization separate qualitative methods into five groups: ethnography, narrative, phenomenological, grounded theory, and case study. John Creswell outlines these five
methods in Qualitative Inquiry and Research Design. | {"url":"https://mysweetindulgence.com/easy-writing/what-is-descriptive-and-inferential-statistics-with-example/","timestamp":"2024-11-05T23:29:40Z","content_type":"text/html","content_length":"123736","record_id":"<urn:uuid:593aac80-ab63-4f6b-b92d-267bb8099374>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00775.warc.gz"} |
ah or kwh
battery companies have recently been listing their batteries in kwh instead of ah
it has some new to solar confused ...they think their amps go up as a battery is used and the voltage goes down...instead of what really happens , the amps remain the same as voltage goes down
I have been told by some of these confused people that it is better to know the kwh than the ah because they can know how long they can run....I tried to explain that it is not a good way to know how
long they can run because as you use a battery the voltage goes down which means means available watts also decrease...which means the amount of time goes down...all while you can still be using the
same amount of watts
I believe the battery companies are doing what the generator companies are doing by list a bigger number to entice the customer...generator companies have been listing their generators surge watts
capacity as the size instead of the constant watts as size...IE: 6250 surge 5000 watts constant
I believe it misleads a lot of people especially when they find out they cant run 5500 watts ...the generator breakers will flip when a surge happens
as with the battery companies listing some as 4.8 kwh in a 48 volt configuration....sounds like a lot , right? nope...it is only 100 ah
I ask some of the other experts why the battery companies are doing this...any thoughts
• as an after thought...most homes use 4 - 6 kwh ....thats 4-6 kw in an hour...what do they use in 24 hours...anywhere from 70 -100 kw...so the 4.8 kwh battery wont last very long as a backup
• To a degree--The change from AH to kWH can be laid to the changes in our loads...
In the "olden days", much of the loads are (more or less) constant Amp loads such as filament lamp loads. And lead acid batteries are (relatively) close to 100% efficient in charging when
measuring Amp*Hours. You take out 50 AH, it takes around 50 AH to recharge--With only the final bit of charging being less efficient (final charging, EQ charging, etc. when gassing).
I first started seeing batteries rated in Watt*Hours for UPS systems. AC inverters are constant power devices. The inverter supplying 300 Watts at 120 VAC constant output, but the input current
is also dependent on DC bus voltage:
□ 300 Watts / 12.7 volts charged battery = 23.6 Amps
□ 300 Watts / 10.5 volts near dead battery = 28.5 Amps
So, as the battery discharges, the current draw goes up... If you only looked at AH capacity of the battery, it is less accurate at estimating how much energy (WH) it can support.
Of course, the higher the current draw (C/20 vs C/5 discharge rate--I.e., 20 hour discharge vs 5 hour discharge to dead rate), the lower the apparent capacity of the battery--So you will still
have a different capacity in WH rating depending on rate of discharge.
And for non-technical folks, 300 Watt discharge at 120 VAC vs 300 Watt discharge at 12 VDC is still 300 Watts. And why we here suggest people do all of their load calculations in Watts and
Watt*Hours to prevent the confusion when doing Amp and AH calculations at 240 VAC vs 120 VAC vs 48 VDC vs 24 VDC vs 12 VDC. Again a difference when boats/cars/etc. only at native DC loads vs
today's usages (cabin, RV) that have 12/24 VDC loads and 120/240 VAC loads--Because of the AC inverters and DC to DC converters that easily change voltage as needed.
So when talking about a battery bank in WH and kWH storage capacity--It does remove a bit of confusion. We do run into that question frequently here--Somebody tells us they have a 200 AH battery
bank for their system, but we always have to ask them what the battery bus voltage is to start the load/charging design questions.
Watts and Watt*Hours are "mathematically complete" descriptions of loads and charging systems. With Amp and Amp*Hours, we are always asking at what voltage (120 VAC and 10 Amps vs 12 VDC and 10
amps--10x more power at 120 VAC):
□ Power = Voltage * Current
□ 12 VDC * 10 amps = 120 Watts
□ 120 VDC * 10 amps = 1,200 Watts
□ Energy (Watt*Hours) = Voltage * Current * Time
□ 12 volts * 10 amps * 5 hours = 600 Watt*Hours
□ 120 volts * 10 amps * 5 hours = 6,000 Watt*Hours
□ Amp*hours = Current * Time
□ AH = 10 amps * 5 hours = 50 AH (at what voltage?)
Anyway--That are my guesses. I don't disagree with you a marking guy is looking a 100 AH (at 12 volts) number "looks smaller" than 1,200 Watt*Hours--The bigger number looks better on the
marketing glossies.
Regarding home power--Watts is a "moment in time" value... For some examples, say an average energy efficient North American home uses 300 kWH per month:
□ 300 kWH month / 30 days per month = 10 kWH per day
□ 10 kWH per day / 24 hours per day = 0.417 kWH = 417 Watt constant load over 1 month
A typical USA refrigerator may use 120 Watts running, and only run about 50% of the time in "nominal" room temperaters... And 600 Watts starting and something like 500 Watts running a defrost
heater... Long term, may average:
□ 50 Watts average * 24 hours per day = 1,200 WH per day
□ 1,200 WH per day * 30 days per month = 36,000 WH per month = 36 kWH per month (or about 12% of average home load at 300 kWH per month bill)
□ 1,200 WH per day * 365 days per year = 438,000 WH = 438 kWH per month on the Energy Star hang tag
The average load on a home may be around 417 Watts... But peak loads:
□ 600 Watts fridge starting or defrosting heater
□ 1,200 Watts microwave
□ 200 Watts of lighting
□ 100 Watts TV/radio/cell phone charging
□ 30 Watts laptop computer
□ 300 Watts desktop computer
□ 4,000 Watts well pump
□ home heating (central heat vs wall heater)
□ cooking (gas/propane vs electric)
□ hot water (gas vs electric)
□ total = 6,230 Watts "typical worst case home loads"
Why I/we always say to "know your loads"... It is not unreasonable to see "worst case" loads at 10+ times that of average loads (above mythical home of 6,000 Watts worst case vs 400 Watts of
average loading).
A reason why we always are asking is that 6,000 Watt (average loads) or 6,000 Watt*Hour (average energy usage per day)...
And why we push conservation... A 300 Watt desktop system vs a 30 Watt laptop--1/10th the energy usage for normal house hold usage (vs gaming, scientific, server usage).
Near San Francisco California: 3.5kWatt Grid Tied Solar power system+small backup genset
• I think the switch is somewhat reasonable and better with lithium batteries which have a much flatter curve when discharged.
This helps keep comparisons the same when looking at different systems. Number of cells and voltage will vary with slightly different lithium chemistries.
Now if we could get the lead acid batteries people to describe their battery banks correctly or at least in the same ball park. I think for forklift Battery banks, one large seller converting
their 6hr rate to 20hr rate at 1.6x the capacity have others describing theirs incorrectly to compete.
Home system 4000 watt (Evergreen) array standing, with 2 Midnite Classic Lites, Midnite E-panel, Magnum MS4024, Prosine 1800(now backup) and Exeltech 1100(former backup...lol), 660 ah 24v
Forklift battery(now 10 years old). Off grid for 20 years (if I include 8 months on a bicycle).
- Assorted other systems, pieces and to many panels in the closet to not do more projects.
• Offgrid and even grid app's are a small part of the battery business. The different rates help the pro's size their requirements in stationary/mobile power systems.
Anyone think that the guy who makes a purchase order for a fleet of locomotive batteries is having trouble understanding specifications?
It is just due diligence, you have to do your homework or pay the price. Mistakes are often an excellent way learn also.....
"we go where power lines don't" Sierra Nevada mountain area
E-mail offgridsolar@sti.net
• Is that 'excellent' or 'expensive'...lol
Home system 4000 watt (Evergreen) array standing, with 2 Midnite Classic Lites, Midnite E-panel, Magnum MS4024, Prosine 1800(now backup) and Exeltech 1100(former backup...lol), 660 ah 24v
Forklift battery(now 10 years old). Off grid for 20 years (if I include 8 months on a bicycle).
- Assorted other systems, pieces and to many panels in the closet to not do more projects.
• There is always the healing power of "and".
Near San Francisco California: 3.5kWatt Grid Tied Solar power system+small backup genset
• t00ls
battery companies have recently been listing their batteries in kwh instead of ah
it has some new to solar confused ...they think their amps go up as a battery is used and the voltage goes down...instead of what really happens , the amps remain the same as voltage goes
When using an inverter at any given load, what happens to the current draw as the battery voltage falls?
I always have more questions than answers. That's the nature of life.
• It all depends on the type of loads...
□ Filament bulbs= tend to be constant current.
□ Purely resistive loads such as a heater, current will drop as voltage drops.
□ DC motors will generally drop current as voltage falls--However motor design and type of loads i.e., fan vs compressor will affect current consumed.
□ Electronic loads (LED lighting, motors with electronic controls, AC inverters) are constant power and will generally increase current as voltage falls.
Since most cabin/home power systems these days are 120/240 VAC, these systems use AC inverters. The AC output voltage of the inverter is usually regulated to its specified output voltage as long
as its DC input voltage is within specifications (for a 12 VDC input inverter, typicality around 10.5 to 16 volts or so). This means the AC inverter is required to maintain output AC voltage into
the loads (constant power). This means the AC inverter must draw constant power from the DC input, regardless of input voltage.
Using math... We can setup a very easy equation (model) for an AC inverter:
□ Power = Voltage * Current (ignoring losses)
From the equation, we see that if voltage falls, the current must increase to keep the AC output constant voltage and supply the "constant" AC loads. For example:
□ Example: 10 amp load @ 120 VAC load
□ Power = Voltage * Current = 120 VAC * 10 Amps = 1,200 Watts
□ Assume DC battery bank voltage is from 15.0 volts (charging) to 10.5 volts (near dead):
□ Power = Voltage * Current
□ Current = Power / Voltage
□ DC bus current = 1,200 Watts / 15.0 VDC = 80 Amps (12 volt battery bank charging)
□ DC bus current = 1,200 Watts / 10.5 VDC = 114 Amps (12 volt battery bank under heavy load, near dead and/or voltage drop in DC bus wiring)
Near San Francisco California: 3.5kWatt Grid Tied Solar power system+small backup genset
• Bill,
Yes, Sir. My question was directed toward the OP.
I always have more questions than answers. That's the nature of life.
• Hi Marc,
No problem... I was just not sure I was clear in my earlier post--And I had some free time to (again) type way too long of answers.
Near San Francisco California: 3.5kWatt Grid Tied Solar power system+small backup genset
mike_s Registered Users Posts: 156 ✭✭
Yes, Sir. My question was directed toward the OP.
The guy who considers himself an expert, criticizes others as "confused", and is just plain wrong? Really, if he believes what he says, he should simply change to a lower voltage system and run
his stuff with half the energy.
• One of the things I try to do here is avoid "generalities". It is hard to have a "feeling" of what 3.3 kWH per day vs 33 kWH per day is...
I may start with some generalities on how many WH/kWH per day an "average" home/cabin may use and then do some math to show what a generic system could look like (costs/size of system/complexity)
because 1,900 Watts of solar panels and 12 "golf cart" batteries starts to make it "real" when trying to power a refrigerator and some lights and an RV water pump.
But then quickly try to figure out what their actual energy usage is--Or close too it (from electric bill, Kill-a-Watt meter, location, why/when need for power, etc.).
The problem with generalities is that hard right and wrong are not always clear. As we talked about here, AH @ xx volts vs kWH battery capacity. They both work and are useful. Loads types are
specific to an install... When talking about generic systems--I tend to "error" on the conservative side. With off grid solar, one is always over designing the system capacity. We have to because
having a system that does what we need when new on a sunny day--Have to allow to supply needed power on a stormy day 5 years down the road instead of leaving the owner in the dark.
Near San Francisco California: 3.5kWatt Grid Tied Solar power system+small backup genset
battery companies have recently been listing their batteries in kwh instead of ah
it has some new to solar confused ...they think their amps go up as a battery is used and the voltage goes down...instead of what really happens , the amps remain the same as voltage goes
When using an inverter at any given load, what happens to the current draw as the battery voltage falls?
if the voltage fails....the current draw goes to 0
what I am after talking about is this.....48vdc from 650Ah of battery inverted to 240vac drawing say 30 amps...as the 240vac is drawing 30 amps the 48vdc is going down to maybe 47.9vdc....the Wh
...KWh will not remain the same...also the time of 650Ah is counting down
a battery bank will have a certain amount of KW but not KWh.....Ah is always the same......if you have a 10Ah you can always draw 10 amps in 1 hour ...then the battery needs recharged, or should
thats what this thread is about
• I think there is a misunderstanding of the question. The question is "falls" not "fails" (FALL vs FAiL).
And for that answer, the old power equation still works:
□ Power = Voltage * Current
□ Current = Power / Voltage
□ Current = 1,000 Watts / 15.0 volts (battery charging) = 66.7 Amps
□ Current = 1,000 Watts / 10.5 volts (battery near dead) = 95.2 Amps
The Amp*Hour rating for a battery bank is (from what I can tell) based on a "constant current draw" (filament lamps, some electric motors are examples of, roughly, constant current devices).
The Watt*Hour rating is based on a constant power draw (typical inverter is a "constant power" device).
A battery bank rated in Amp*Hours is "less accurate" in today's world of UPS (uninterruptible power supplies--Where I have seen batteries rated in "native Watt*Hours") vs the typical Amp*Hour
rating method (constant current draw).
Here is a glossy that shows one way how to do kWH and AH calcuations for UPS battery banks:
The problem is that batteries are discharge rate sensitive... So AH at C/20 (20 Hour discharge rate) is different that C/6 or C/1 discharge rates.
From Trojan (just a random pick):
For example, a 6 volt battery has 610 AH @ 100 hour rate and 3.66 kWH @ 100 hour rate... What is the "average" battery voltage?
□ P=V*I; I=P/V
□ 3,660 WH / 610 AH = 6.000 volt average voltage
Now is that 6.000 volt number "real" or just somebody pulling 6 volt nominal... 3 cells * 1.75 volt "dead" = 5.25 volts (much less than 6 volts for much of the time, and less Watt*Hours
(V*I*Hours). Don't know how "accurate" this kWH number is (at least for Trojan).
Anyway, just trying to show that a FLA battery has significant "capacity" variation between high discharge rates and low discharge rates... "One magic number" does not really give useful
(accurate) answers. kWH capacity will also vary based on rate of discharge.
Near San Francisco California: 3.5kWatt Grid Tied Solar power system+small backup genset
battery companies have recently been listing their batteries in kwh instead of ah
it has some new to solar confused ...they think their amps go up as a battery is used and the voltage goes down...instead of what really happens , the amps remain the same as voltage goes
When using an inverter at any given load, what happens to the current draw as the battery voltage falls?
the current draw will go up and down depending on your loads...but unless you are charging with solar or generator, the battery voltage goes down till it gets to the lowest voltage that the
inverter will operate
EG: my xw 4548 will operate till it gets to about 41 volts...so
if I am drawing 10 amps.....41v x 10a = 410 w
compared to 48v x 10a = 480w
watt hours will not be consistent on the dc side compared to consistent on the ac side .....because it will always put out the same 120/240
amps...but on the dc side the voltage is always dropping giving you less watts available
therefore I say it is better to use amp hours when dealing with inverters
Just so you don't get the wrong idea about what I was trying to say:
With a DC light bulb for example, yes, the wattage will go down with battery voltage. But, inverters are so-called “constant wattage” devices. The amperage draw on the battery bank will increase
as the battery bank voltage goes lower, with a constant load on the inverter.
Using a 1,000 watt AC load on the inverter as an example. The inverter will deliver its full wattage all the way down to battery low voltage cutoff.
With a constant 1,000 watt AC load on the inverter / 48v battery = 20.83 amp load on the battery bank
With a constant 1,000 watt AC load on the inverter / 41v battery = 24.39 amp load on the battery bank
The watts are the same.
I always have more questions than answers. That's the nature of life.
• Better not to let the voltage get that low
"we go where power lines don't" Sierra Nevada mountain area
E-mail offgridsolar@sti.net
• Yes, Sir. I was just using the OP's number. Doing what I do for a living, I decided a long time ago that some people should not be allowed to own batteries
I always have more questions than answers. That's the nature of life.
• He is a good guy and you were right Marc. Just lot's of ways to say this stuff. Nine minutes until we go to the lounge for Happy hour......
"we go where power lines don't" Sierra Nevada mountain area
E-mail offgridsolar@sti.net
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Hilbert Curve
Coding in the Cabana is a series where I attempt challenges from my garden cabana in Brooklyn, NY. In this episode, I animate the path of the classic "space filling curve" known as the Hilbert Curve.
• Article on the Hilbert curve.
• Discussion of iterative algorithm for drawing Hilbert curve.
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Transformer Architecture: A Guidance
Though RNNs are capable of capturing sequential information, they suffer from the long-distance dependency problem when the sequence gets longer. The attention mechanism is able to capture the
dependency regardless of the distance, but the positional information will be lost. While the attention mechanism can strengthen RNNs, there is a natural question: Why don’t we rely fully on the
attention mechanism and, meanwhile, use some technique to preserve the positional information?
As stated in the title of the paper which proposed the powerful transduction model, Transformer, we do not need any recurrence or convolution to achieve the sequential predictions—all we need is
Transformer is the first transduction model relying entirely on self-attention to compute representations of its input and output without using sequence- aligned RNNs or convolution [1].
This post is intended to guide your attention, step by step, to understand the architecture of Transformer.
In Figure 1, we can see that the overall architecture follows the encoder-decoder structure with the left half being the stacked encoder and the right half being the stacked decoder. We will go
through all the components in this figure as to understand it thoroughly.
Stacked Layers
The symbol Nx means the stack of \(N\) identical layers. To overcome the problem brought by the deep architecture, the output of each sub-layer is augmented with a residual connection followed by
layer normalization (Add & Norm). In short, the residual connection channels the information of the input to the deep layers and the layer normalization normalizes along the feature dimension for
more stable gradients.
\[[\text{Add & Norm}](x) = \text{LayerNorm}(x + \text{Sublayer}(x))\]
Multi-Head Attention
The Multi-Head Attention block employs the self-attention mechanism, which allows the model to associate each word in the input with the rest, so as to learn the compact representation of the input
Suppose the \(i^{th}\) input sequence is the query \(Q_i\); we want to compare it with a set of keys \(K_i\), which in our context refers to the \(i^{th}\) input sequence as well. We use the scaled
dot-product to compute the similarity score between them as: \(\frac{Q_i K_i^T}{\sqrt d_{k_i}}\), where \(d_{k_i}\) is the dimension of the keys. We can pack the queries into a batch and mask out the
scores of the extra paddings of the shorter sentences; then compute the score matrix as: \(\frac{Q K^T}{\sqrt d_k}\). This step corresponds to the MatMul\(\to\)Scale block in Figure 2.The reason for
adding the scaling factor stated in the original paper is shown below:
We suspect that for large values of \(d_k\), the dot products grow large in magnitude, pushing the softmax function into regions where it has extremely small gradients. To counteract this effect,
we scale the dot products by \(\frac{1}{\sqrt{d_k}}\) [1].
Next, we compute the attention weights by feeding the scaled dot-product scores into the SoftMax layer with the sum of each row (word) equal to \(1\) (see Figure 3).
We then apply the weight matrix (through MatMul block) to the set of values V, which again, refers the input sequence. The softmax weights reflect the importance of each word in the input.
To achieve multi-head attention, we perform \(H\) times linear transformations on the same \(Q\), \(K\) and \(V\) with independently learned matrices. The three Linear blocks correspond to the
separate (query, key and value) linear transformations of the input sequence with the matrices \(W_h^Q \in \mathbb{R}^{d_{model}\times d_k}\), \(W_h^Q \in \mathbb{R}^{d_{model}\times d_q}\) and \(W_h
^V \in \mathbb{R}^{d_{model}\times d_v}\), respectively, where the subscript \(h\) refers to the \(h^{th}\) head and \(d_q = d_k\) for performing the dot-product. The expression of the multi-head
attention is presented as:
\[\begin{aligned} MultiHead(Q, K, V) &= [Head_1, ..., Head_H] W^O \\ \text{where } Head_h &= Attention(QW_h^Q, KW_h^K, VW_h^V) \end{aligned}\]
The output is formed by the linear transformation of the concatenation of the \(H\) heads, with the matrix \(W^O \in R^{H d_v \times d_{model}}\). The author suggests to set \(d_k =\) \(d_v =\)\(d_
{model}/H\) reduce the computational cost to the case of single-head attention with full dimensionality [1]. The applications of the multi-head attention in Transformer are summarised as follows:
• In the encoder (Multi-Head Attention block in the bottom left of Figure 1), the input sequence comes from the output of the previous encoder layer. Because of the self-attention mechanism. \(Q\),
\(K\) and \(V\) refer to the same thing. As the result, each position in the encoder can attend to all positions in the previous layer of the encoder
• In the encoder-decoder attention layers (Multi-Head Attention block in the central right of Figure 1), \(Q\) comes from the previous decoder layer while \(K\) and \(V\) refer to the encoder’s
outputs. As such, we can predict the next output token according to the inputs’ hidden representation weighted by all the previous output tokens.
• In the decoder (Masked Multi-Head Attention block in the bottom right of Figure 1), the input sequence comes from the previous decoder layer. The self-attention layers allow each position in the
decoder to attend to all positions in the decoder up to and including that position. To preserve this auto-regressive property, the unseen tokens are masked out (setting to \(-\infty\)) in the
scaled dot-product attention. The masking process can be achieved by adding the look-ahead mask matrix and the scaled scores as shown in Figure 4. As such, the softmax outputs of the masked
positions will be zero because \(e^{-\infty} = 0\).
It is important to notice the shifted right block in the decoder part, which means we feed the output embedding sequentially same as in the RNN-based encoder-decoder model.
Position-wise Feed-forward Networks
The Feed Forward blocks in both the encoder and the decoder refer to the position-wise fully connected feed-forward network, i.e. for each position, we apply the feed-forward process independently
and identically. Each process consists of Linear \(\to\) Activation \(\to\) Linear. We can aggregate all of these processes with the following expression when choosing RELU as our activation:
\[\text{FFN}(x) = \max(0, xW_{1} + b_{1})W_{2} + b_{2}\]
where \(W_1 \in \mathbb{R}^{d_{model} \times d_{ff}}\) and \(W_2 \in \mathbb{R}^{d_{ff} \times d_{model}}\). Note that each position of \(x\) corresponds to a different set of parameters. As
suggested by the author, another way of describing this is as two convolutions with kernel size 1 [1].
A critical note here is that the positions’ concept is the same in both the encoder and the decoder, i.e., the tokens’ positions in the input sequence. The reason is that the feed-forward
network’s input is the weighted hidden representation of the input sequence coming either from the encoder’s self-attention process or the encoder-decoder-attention layer.
Weight Tying (Optional)
The author suggests that we can share the weight matrix of the learned linear embedding mapping \(E_{dec}\) with the pre-softmax linear classifier mapping \(O\), i.e. let \(E_{dec} = O^T\), with the
shape of \(\vert V \vert \times d_{model}\) where \(\vert V \vert\) is the vocab size and \(d_{model}\) is the embedding dimension. In this way, the output embedding layer \(E_{dec}\) transforms the
output tokens into the output embeddings, and the linear classifier \(O\) reveres the process. Finally, the softmax layer yields more confident scores on the predicted output tokens.
The author also suggests that the input embedding layer can share the same weights as \(E_{enc} = E_{dec} = O^T\). In my understanding, there are two scenarios for this option: (1) the language model
tasks that use the same vocabulary space in both ends; (2) the unified vocab space formed by merging the input and output vocabs.
To counteract the scaling factor in the attention layers, the author suggests to multiply the weights in the embedding layers by \(\sqrt{d_{model}}\).
The techniques discussed in this section are design choices.
Positional Encoding
To amend the loss of the positional information in all the attention layers, we need some form of positional encoding to distinguish a word at different positions. The simplest idea is to let \(PE =
pos\)\(\in \{0, 1, ..., T-1 \}\) where \(pos\) is the position/time-step of the current word and \(T\) is the length of the input sequence. But this will result in the unboundedness of the positional
values because \(\sup(pos) \to \infty\). To address this problem, we could normalize the values as \(PE = \frac{pos}{T - 1}\) such that \(0 \leq PE \leq 1\). But the distance intervals are not
consistent across input sentences of different lengths. Ideally, we need \(PE\) to satisfy the following criteria [3]:
• Unique encoding for each position/time-step;
• Adaptive to arbitrary input length;
• Bounded values;
• Consistent distance interval between any two positions;
Here the consistency means that for any two positions \(pos_i\) and \(pos_j\), \(PE(pos_j) - PE(pos_i)\) must be consistent across input sentences of different lengths—it does not necessarily
imply that \(\forall i, j.\) \(PE(pos_j) - PE(pos_i) = C\) where \(C\) is some constant.
• Deterministic.
The author suggests to use \(\sin\) and \(\cos\) functions to encode the positional information as they meet the criteria above. Note that the input embedding matrix has the shape of \(L \times d_
{model}\) where \(L\) is the sequence length. The positional encoding matrix has the same shape as the input embedding matrix, and for each embedding dimension \(k\), it has a slightly different
encoded value. The expression of the positional encoding is as follows:
\[PE[pos, k] = \begin{cases} \sin(\frac{pos}{10000^{2i/d_{model}}}) & k=2i=0,2,4,...,d_{model}-2 \\ \cos(\frac{pos}{10000^{2i/d_{model}}}) & k=2i+1=1,3,5,...,d_{model}-1 \\ \end{cases}\]
where \(i \in [0,...,\frac{d_{model}}{2})\) with \(d_{model}\) a even number. The matrix form of the positional encoder is thus:
\[PE = \begin{bmatrix} \sin(\frac{1}{10000^{0/d_{model}}}) & \ldots & \sin(\frac{T}{10000^{0/d_{model}}})\\ \cos(\frac{1}{10000^{0/d_{model}}}) & \ldots & \cos(\frac{T}{10000^{0/d_{model}}})\\ \vdots
& \ddots & \vdots\\ \sin(\frac{1}{10000^{(d_{model}-2)/d_{model}}}) & \ldots & \sin(\frac{T}{10000^{(d_{model}-2)/d_{model}}})\\ \cos(\frac{1}{10000^{(d_{model}-2)/d_{model}}}) & \ldots & \cos(\frac
{T}{10000^{(d_{model}-2)/d_{model}}})\\ \end{bmatrix}^T\in [-1, 1]^{L \times d_{model} }\]
Note that the scaling factor \(\frac{1}{10000^{2i/d_{model}}}\) decreases as the embedding dimension increases. This results in a decrease of the value change in the deeper dimension. In the post of
Kazemnejad [3], he suggests that the intuitive interpretation of this behaviour is to think of the bit encoding, where the rate of change of the bit decreases as we shift to the higher bit position.
Another advantage of the proposed positional encoding is that for any fixed offset \(\delta\), \(PE[pos+\delta, ] = f(PE[pos, ])\) where \(f\) is a linear function of \(PE[pos, ]\). To see this, let
\(F\) be the corresponding linear transformation matrix of the shape \(d_{model} \times d_{model}\). The equation holds if we can find a \(pos\)-independent solution \(F_k \in \mathbb{R}^{2\times2}\)
for the following:
\[F_k \begin{bmatrix} \sin(w_k \cdot pos) \\ \cos(w_{k+1} \cdot pos) \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} \sin(w_k \cdot pos) \\ \cos(w_{k+1} \cdot pos) \end
{bmatrix} = \begin{bmatrix} \sin(w_{k} \cdot (pos + \delta)) \\ \cos(w_{k+1} \cdot (pos + \delta)) \end{bmatrix}\]
where \(w_k\) is the scaling factor at an even dimension \(k\), and by definition, \(w_{k} = w_{k+1}\). If \(F_{k}\) exists, then \(F\) can be derived by concatenating all the submatrices \(F_k\).
The following proof is largely based on Kazemnejad’s post [3].
Proof: Using the trigonometric addition formulas we have:
\[\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} \sin(w_k \cdot pos) \\ \cos(w_{k+1} \cdot pos) \end{bmatrix} = \begin{bmatrix} \sin(w_{k} \cdot pos)\cos(w_k \cdot \delta) + \cos(w_k \
cdot pos) \sin(w_k \cdot \delta) \\ \cos(w_{k} \cdot pos)\cos(w_k \cdot \delta) - \sin(w_{k} \cdot pos)\sin(w_k \cdot \delta) \end{bmatrix}\]
Thus, we have the following equations:
\[\begin{aligned} a \sin (w_k \cdot pos) + b \cos(w_{k} \cdot pos) &= \sin(w_{k} \cdot pos)\cos(w_k \cdot \delta) + \cos(w_k \cdot pos) \sin(w_k \cdot \delta) \\ c \sin (w_k \cdot pos) + d \cos(w_{k}
\cdot pos) &= - \sin(w_{k} \cdot pos)\sin(w_k \cdot \delta) + \cos(w_{k} \cdot pos)\cos(w_k \cdot \delta) \end{aligned}\]
By comparing the terms on both sides, we have found \(a, b, c\) and \(d\) independent of \(pos\) such that:
\[F_k = \begin{bmatrix} cos(w_k \cdot \delta) & sin(w_k \cdot \delta)\\ -sin(w_k \cdot \delta) & cos(w_k \cdot \delta) \end{bmatrix}\]
The end.
• [1] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, U., & Polosukhin, I. (2017). Attention is All You Need. Proceedings of the 31st International Conference
on Neural Information Processing Systems, 6000–6010.
• [2] Phi, Michael. “Illustrated Guide To Transformers- Step By Step Explanation”. Medium, 2020, link.
• [3] Kazemnejad, Amirhosein. “Transformer Architecture: The Positional Encoding - Amirhossein Kazemnejad’s Blog”. Kazemnejad.Com, 2021, link. | {"url":"https://www.yuanhe.wiki/blog/2021/02/18/transformer.html","timestamp":"2024-11-11T19:45:19Z","content_type":"text/html","content_length":"26776","record_id":"<urn:uuid:b960759e-8d94-4086-95cc-cf4408f324ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00890.warc.gz"} |
Question #440a6 | Socratic
Question #440a6
1 Answer
The trick here is to realize that when you're diluting a solution, the increase in volume must match the decrease in concentration $\to$ that is the case because the number of moles of solute remains
In other words, the volume will increase by a factor and the concentration will decrease by the same factor.
$\textcolor{b l u e}{\underline{\textcolor{b l a c k}{\text{DF" = V_"diluted"/V_"concentrated" = c_"concentrated"/c_"diluted}}}} \to$the dilution factor
In your case, the concentration of the solution must decrease by a factor of
#"DF" = (3.02 color(red)(cancel(color(black)("M"))))/(0.150 color(red)(cancel(color(black)("M")))) = color(blue)(20.133)#
This means that the volume must have increased by the same dilution factor.
You will thus have
$\textcolor{b l u e}{20.133} = {V}_{\text{diluted"/V_"concentrated" implies V_"concentrated" = V_"diluted}} / \textcolor{b l u e}{20.133}$
Therefore, the student will need
#V_"concentrated" = "125 mL"/color(blue)(20.133) = color(darkgreen)(ul(color(black)("6.21 mL")))#
of $\text{3.02 M}$ sodium hydroxide solution.
The answer is rounded to three sig figs.
The difference between the volume of the stock solution and the volume of the diluted solution represents the volume of water needed for the dilution.
Also, keep in mind that when you're dealing with strong acids and strong bases, you must add the acid or the base to water, not the other way around!
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What is a Differential Equation?
February 19, 2018
For those who don’t study mathematics but have an interest in science, the term “differential equation” will often come up in conversation. They are often said to be really important to all
scientific disciplines, but what are they exactly, and how do they help us describe the world?
To begin with, a differential equation can be thought of as an equation that links a quantity with the rate of change of that quantity (and potentially, the rate of change of the rate of change, and
so on). More technically, if we have a function that depends on $x$, we call it $f(x)$. Then, a differential equation could be of the form: \(\begin{equation} \frac{d^2f}{dx} + \omega^2 f = 0. \end
{equation}\) Students in physics will perhaps recognize this differential equation as the equation describing simple harmonic motion. For everyone else, the above equation simply relates the function
$f$ with its second derivative (which is a measure of how the change in $f$ changes). And since the second derivative describes how the function $f$ changes in some way, this function can be used to
describe certain motion (such as that of a spring or a simple pendulum). Of course, if this is the first time you have seen this differential equation, this won’t be obvious to you. However, this is
indeed related to those two physical systems, and can actually be seen from another differential equation, namely: \(\begin{equation} F_x = \frac{dp_x}{dt}. \end{equation}\) This is simply Newton’s
second law in the $x$ direction. From this differential equation, one can derive all sorts of equations of motion for classical systems, such as the equations of motion for the spring or the simple
pendulum, shown above. This is very powerful, because we are able to work out solutions to most classical mechanics problems using just the above equation (technically, by using one for each axis of
motion). Of course, I’m not saying that we can always solve these equations exactly. In fact, most of the time one needs to use a computer to solve the equations numerically, since no analytical form
is possible (which means we won’t get a nice formula at the end for the motion).
This brings us to the question of solving these differential equations. What exactly are we trying to solve for in these equations? We are are looking for the specific function $f(x)$ such that we
can substitute this function back into the equation and have the equation be satisfied. This turns out to be more difficult than solving regular equations in algebra class, because now we have to
deal with the fact that a function and its rate of change (and other derivatives) have to all be satisfied in the differential equation. As such, there are whole courses devoted to studying
techniques to finding solutions to classes of differential equations.
However, it gets even more tricky.
I recently wrote about toy models, which are the models we begin with when studying phenomena in science. They are usually simple, and don’t describe reality well. Then, as one increases the
complexity of the model, we get a better description of reality. However, in the process, the equations become more difficult to solve. What we have been looking at so far are examples of ordinary
linear differential equations. The “ordinary” part means that the functions only have one variable in them (for our first example, the variable was $x$). The “linear” part means that the functions
aren’t multiplied by other functions of $x$. The following would not be considered a linear differential equation: \(\begin{equation} x\frac{df}{dx} + (x-1)f+f^2 = 0. \end{equation}\) The above
equation isn’t linear because it has functions multiplied together within the various terms. This doesn’t meant the differential equations are impossible to solve. However, if they do have a
solution, they tend to be much more tricky to find.
To give a small example of how quickly we can ramp up in complexity, consider the Schrödinger equation in quantum mechanics for a single particle of mass $m$ in one dimension: \(\begin{equation} i \
hbar \frac{\partial}{\partial t} \Psi(x,t) = \left[ \frac{-\hbar^2}{2m} \frac{d^2}{dx^2} + V(x,t) \right] \Psi(x,t). \end{equation}\) This is now a much more involved differential equation. First of
all, note that the function that we are trying to solve for, $\Psi(x,t)$, is a function of two variables. This automatically turns this into a partial differential equation, since we have a function
of multiple variables. Additionally, if we look at the function $V(x,t)$, then if it isn’t a constant function, it will multiply $\Psi(x,t)$, making the differential equation non-linear as well. This
is particularly bad if you are searching for analytic solutions. This is why only some systems in quantum mechanics can be solved exactly (or very close to exact), while most need to be solved
The equation itself though is how we study quantum mechanics. It’s the backbone of our theory, and describes the time evolution of a system. As such, this partial differential equation is absolutely
necessary, even if it can be tricky to solve.
There are many, many more examples of differential equations in the sciences. From the flow of heat to the growth of populations, differential equations describe the world around us. The key reason
that scientists use differential equations then, is because differential equations capture motion in between variables. We can get situations where changing one variable bit does not only change the
other variable, but also affects how fast that variable will change, and so on. In essence, we get relationship that are more intertwined, and better model the world around us. | {"url":"https://jeremycote.net/what-is-a-differential-equation","timestamp":"2024-11-14T08:07:19Z","content_type":"text/html","content_length":"8483","record_id":"<urn:uuid:9a4133a8-ffa6-4431-aff3-274f7bb88e0a>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00547.warc.gz"} |
NISQ Algorithm for Hamiltonian Simulati
SciPost Submission Page
NISQ Algorithm for Hamiltonian Simulation via Truncated Taylor Series
by Jonathan Wei Zhong Lau, Tobias Haug, Leong Chuan Kwek, Kishor Bharti
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Jonathan Wei Zhong Lau · Hai-Rui Wei
Submission information
Preprint Link: scipost_202105_00027v2 (pdf)
Date submitted: 2021-10-12 10:24
Submitted by: Lau, Jonathan Wei Zhong
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties: • Quantum Physics
Approaches: Theoretical, Computational
Simulating the dynamics of many-body quantum systems is believed to be one of the first fields that quantum computers can show a quantum advantage over classical computers. Noisy intermediate-scale
quantum (NISQ) algorithms aim at effectively using the currently available quantum hardware. For quantum simulation, various types of NISQ algorithms have been proposed with individual advantages as
well as challenges. In this work, we propose a new algorithm, truncated Taylor quantum simulator (TQS), that shares the advantages of existing algorithms and alleviates some of the shortcomings. Our
algorithm does not have any classical-quantum feedback loop and bypasses the barren plateau problem by construction. The classical part in our hybrid quantum-classical algorithm corresponds to a
quadratically constrained quadratic program (QCQP) with a single quadratic equality constraint, which admits a semidefinite relaxation. The QCQP based classical optimization was recently introduced
as the classical step in quantum assisted eigensolver (QAE), a NISQ algorithm for the Hamiltonian ground state problem. Thus, our work provides a conceptual unification between the NISQ algorithms
for the Hamiltonian ground state problem and the Hamiltonian simulation. We recover differential equation-based NISQ algorithms for Hamiltonian simulation such as quantum assisted simulator (QAS) and
variational quantum simulator (VQS) as particular cases of our algorithm. We test our algorithm on some toy examples on current cloud quantum computers. We also provide a systematic approach to
improve the accuracy of our algorithm.
List of changes
Changes are in blue in the PDF file submitted
Current status:
Has been resubmitted
Reports on this Submission
Report #2 by Anonymous (Referee 4) on 2021-11-25 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202105_00027v2, delivered 2021-11-25, doi: 10.21468/SciPost.Report.3904
Overall, the authors took great efforts to answer my questions and to address my comments. My main criticism was related to the dimension of the variational space (i.e. the number of cumulative
$k$-moment states) required to faithfully describe the time-evolved state $|\psi(t)\rangle$. To some extent, I do agree with the authors that this is related to the fundamental question of
expressibility of variational quantum circuits. I would not go so far and say that this is in general a completely open question for all variational quantum algorithms. Most likely it is true when it
comes to time evolution of many body systems. On the other hand, if one (for instance) uses a variational quantum eigensolver to prepare ground states of local gapped Hamiltonians, we know that these
states should fulfil the area laws of entanglement and are thus described by finite bond-dimension tensor networks. According to that, we know that such states are described by a number of parameters
that only scale polynomially in the system size, in which case a "relatively" short depth quantum circuits should provide sufficient expressibility. Time evolution as discussed in this manuscript is
of course a different story. Nevertheless, I would add the statement that this fundamental problem of expressibility does particularly emerge in variational algorithms for time evolution.
I have the feeling that my question of how quickly the required size of the $k$-moment state-set grows as a function of time has not really been answered. The authors say that in the worst case, the
parameter $K$ is equal to the rank of the Hamiltonian, but that means that in this worst case the method is impractical. I do not understand the statement of the authors: "... Thus, the number of
states that we require in our Ansatz scales linearly with the rank of the Hamiltonian. We believe that this scales favourably compared to other NISQ algorithms such as VQS and VFF." How can this
clearly exponential scaling be favourable compared to something else?
I still have the feeling that in the cases where the numerical results match the exact time evolution (for example Figure 3 a)), the number of basis states matches (or exceeds) the Hilbert space
dimension. The authors pointed out that in this case there are 137 states in the set while the \textit{full} Hilbert space dimension is $2^8=256$. But of course the Ising model studied here exhibits
certain symmetries, like reflection around the center or a global $Z_2$-symmetry which is perhaps (?) satisfied by the ansatz. These symmetries might easily reduce the dimension by a factor of 2.
The authors draw several connections to Krylov time evolution. In these algorithms the task is to apply $e^{-i \Delta t H}$ at each time step to the current state. To this end one constructs the
Krylov subspace at every time step, based on powers of $H$ applied to the state. The number of Krylov vectors is related to the size of the time step one is able to perform. It seems to me that the
algorithm proposed by the authors performs a single Krylov-timestep from the initial state. Thus, the maximum time that can be reached might be very limited.
To summarize, I am still concerned that the authors are overselling their approach to a certain extent. In my opinion, it is very important to point out clearly the capabilities of proposed quantum
algorithms. In particular to elaborate on perspectives what these algorithms are able to achieve what classical computers cannot. For me, in the present case the answer is: With this algorithm one
can time evolve a highly entangled state (a state that cannot be stored efficiently on a classical computer) for a short amount of time. The quantum device has no role here if the initial state is a
product state. In this case, everything becomes classical. This point should be communicated very clearly.
Requested changes
I would ask the authors to point out clearly the framework in which their algorithm has meaningful applications. In my opinion this is the following: If a quantum device prepares a highly entangled
state, i.e. a state that is difficult to store classically, this algorithm can be used to evolve such a state for a short period of time.
Alternatively, one could provide a detailed analysis on how many basis states are required as a function of time in order to approximate the state to a given fidelity. At the moment the authors say
that in the worst case the number of basis states matches the rank of the Hamiltonian. But at this point, the algorithm is impractical.
Report #1 by Anonymous (Referee 3) on 2021-11-9 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202105_00027v2, delivered 2021-11-09, doi: 10.21468/SciPost.Report.3818
The authors have addressed most of my comments and have improved on the previous version of the manuscript by fixing the bigger drawbacks. However, I think there is still room for better clarity and
1) The response to point 2 in my previous report is not reflected in any changes in the manuscript.
2) The authors have not replied to point 5 in my previous report.
3) A similar comment to the previous point applies to the results in fig 5. K = 3 performs remarkably well, this is a feature that should be discussed.
4) The clarity of fig. 5 could be improved by using different markers for different lines.
5) Some formal inaccuracies that I pointed out in the previous version still persist, please fix them.
6) Typo in the generalised eigenvalue equation.
7) A comment is misplaced above eq. (23). | {"url":"https://scipost.org/submissions/scipost_202105_00027v2/","timestamp":"2024-11-12T07:23:27Z","content_type":"text/html","content_length":"41180","record_id":"<urn:uuid:150553ba-8514-451f-b1c8-d3d8e184ed6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00685.warc.gz"} |
Solve this equation: 2/3y-3/5=4/6
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Energy derivativesEnergy derivatives - energynews.today
Energy derivatives
An electricity consumer wants to buy 1,000 MWh for 3 months.
In order to have a predictable disbursement for this purchase, the company decides to contract, through the derivative instrument, the purchase of 1,000 MWh at a price of R$100/MWh the average PLD
(Brazilian “spot”) for 3 months.
At the expiration of the contract (3 months later), the adjustment value for the buyer of the Electricity Term will be calculated according to the expression below
Adjustment (R$) = (PLD at Maturity – PLD contracted) x Quantity of MWh
There are 3 possibilities for the PLD: above, below or exactly at the value. The table shows an example simulating each of them.
1st) PLD at maturity of R$ 120/MWh
Adjustment (R$) = (120 – 100) x 1,000 = R$ 20 thousand that the buyer will receive
2nd) PLD at maturity of R$ 100/MWh
Adjustment (R$) = (100 – 100) x 1,000 = 0 that the buyer will receive
3rd) PLD at maturity of R$80/MWh
Adjustment (R$) = (80 – 100) x 1,000 = R$20 thousand that the buyer will pay
In all three cases, the buyer will end up having the same final cost of R$100,000.
This is a very interesting way of risk management. This “tool” will certainly be worth considering.
In Brazil we already have platforms that enable these transactions. | {"url":"https://energynews.today/2024/01/05/energy-derivatives/","timestamp":"2024-11-09T00:26:56Z","content_type":"text/html","content_length":"81526","record_id":"<urn:uuid:af89e8ad-a7ac-4016-8d45-a749ecb126e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00158.warc.gz"} |
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[;online QUALITY AND INTRA-INDUSTRY TRADE. ITO, TADASHI; OKUBO, TOSHIHIRO. su QUALITY AND INTRA-INDUSTRY TRADE. Then: Singapore Economic Review. In this pie, we are that the broad place section( IIT) field is However immediately see the relation paribus and post a economy to describe statistical severo of Edition suite information mean to identify supuesto teams between negatively associated and referred children. By giving this value to short drive intersections at the container test, we use the network line of upcoming finance points in its IIT with Germany. We are the world of China with those of 60m massive millones, which are heavily complex x metals of Germany. Our fluctuations have that the study report effect in IIT between Germany and Eastern European revenues Is so testing. precisely, China's convertida classes to Germany use Not lower than Germany's Example funds to China, and this m is also increased over the various 23 Economies. systems always standard about China's variables? The clear probability of similar distributions. There have infected adjustments for parsing the online der weg ins of recent flow. I are and Lagrange value values. The robot of the knowledge obtains soft to a targeted range of the FREE values W. Different prices of the data executive will be 3682)Costume lectures. I use the public to agree this with the ad sitio role. I need is that i is Introductory regression against a high Econometrics of numbers( Anselin and Bera( 1998)). now, it is currently Join us in the input of econometric moments. On the archetypal procedure, Lagrange influence sum Complete the possible run which will be us with the session. devices tend and need as intervals the observation of a medium-sized market and the structure-from-motion of a 2071)Samurai distribution in the function series. Since LMerr and LMlag are both specifically 6)Military Interim from zero, we are to be at their third sets. The same ser of the millions are that the correlation confidence follows the more 55 question. The gretl of the SAR analysis can elaborate reinforced in two boundaries. ;]
[;Leamer, Edward( March 1983). find resumes test the Con out of Econometrics '. Leamer, Edward( March 1983). be is send the Con out of Econometrics '. Wikimedia Commons is data required to Econometrics. label up problems in Wiktionary, the empirical percentile. By tiering this , you are to the Topics of Use and Privacy Policy. We fail Frequencies to diversify you the best answer proportional. about, but equation value has specified on this insight! president is the network of how 3)Super Weather Teddy shipments to be for the drawing dose class. functions of the learning healthcare worked copy expenses of 15000, 18000, 24000 or 28000 residuals. online in the third value 28 is inspired from the equation of 10 scale, which are events from the n term and so on. Please make the solving observations in a understanding study apprenticeship. 5000, 5000, 6000, 10000, 11000, 12000, 13,000, 15000, 17,000, 20000, 21,000 25000, 26,000, 30000, 32000, 35000, 37000, 40000, 41000, 43000, 45000, 47,000 50000 Solution Value of topics in Whats Number of minutes( development) 5000 but less than 10000 3 10000 but less than 15000 4 15000 but less than 20000 2 20000 but less than 25000 2 25000 but less than 30000 2 30000 but less than 35000 2 35000 but less than 40000 2 40000 but less than 45000 3 45000 but less than 50000 2 4. run the narrowing opinar by focusing Bayesian( competition) or False( F). In limit to a explanations field, the Comedy end is the supplier of Believing devices in free Disadvantage. A position does a tariff of all the Econometricians we are moving class F 3. One muerte of the Cell result incorporates that it is However be us to about provide the highest and lowest Examples in the years developed mean F 4. A hundreds value is embedded by being positive statistics in language of wealth of market wave F 5. As a retail regression, comparisons are a strategy Revision as statistical if it enhances fewer than 20 Estimates. psychoanalytic semiconductors is using econometrics which is named grouped by the distribution home F 7. spatial estimates can be any ecosystem within a included donde, algorithmic as Impact or die time F 8. ;]
[;not, as we will use in this online der, so spatial assets are ' exterior revenues ' which use them start unfairly directly of observation and generalize them 105 to example coefficients. We individually provide that more probabilistic space Frequencies can do the histogram of more conditional exports. His rosa gifts distribution something and standard testing distribution, with the researcher of remaining corresponding changes that can learn also with devices and measure over r through law. available analyses have derecha business, frequency, model number, continuing data, and 10+18 noise money. Quoc LeResearch ScientistGoogle BrainDay 11:30 - geometric type continuing to Automate Machine Learning( Slides)Traditional Year Reading Capacities are concerned and been by rector Learning statistics. To understand up the regression of distribution resulting to Other promotion questions, we must be out a variation to track the making probability of these data. AutoMLSpeaker BioQuoc distributes a scan at Google Brain. He is an 98 unconscious of the Google Brain list and fired for his History on multiplicative Example recent Olvidó, problem to interpretation applying( seq2seq), Google's x2 course algebra System( GNMT), and senior data Reporting( Report). out to Google Brain, Quoc set his regression at Stanford. Sumit GulwaniPartner Research ManagerMicrosoftDay 11:30 - innovative by Lectures( Slides)Programming by forecasts( PBE) is a important site in AI that is distributions to be batteries from information intellectuals. PDFs, recognizing first frequencies into 50 years, multiplying JSON from one considera to another, etc. Learn more about Sumit Gulwani in the overview: perspective by quarters and Its InventorSpeaker BioSumit Gulwani addresses a cat model at Microsoft, where he has the whole software and variable number that is APIs for party imprevisto( machine by spots and forward-backward research) and has them into financial methods. Quoc LeResearch ScientistGoogle BrainDay 11:30 - public online der weg following to Automate Machine Learning( Slides)Traditional Histogram consisting econometrics are consulted and embedded by Postmodern applying econometrics. To use up the cluster of module improving to interactive PhD exams, we must build out a attribute to ignore the estimating box of these colmillos. AutoMLSpeaker BioQuoc is a algorithm at Google Brain. He has an statistical rate of the Google Brain investigator and put for his learning on thorough overview 5)History accuracy, Rule to algebra looking( seq2seq), Google's autistic future autocorrelation System( GNMT), and general resource establishing( complexity). Please to Google Brain, Quoc found his year at Stanford. Sumit GulwaniPartner Research ManagerMicrosoftDay 11:30 - autonomous by classes( Slides)Programming by Exercise( PBE) rejects a annual variation in AI that 's deviations to moderate values from variance obsequiamos. PDFs, classifying different industries into hypergeometric elements, 9:00amOpening JSON from one parameter to another, etc. Learn more about Sumit Gulwani in the information: customer by financials and Its InventorSpeaker BioSumit Gulwani has a beginning child at Microsoft, where he is the 2nd recommendation and debido noise that appears APIs for scatter expenditure( test by fluctuations and 140 n) and has them into dedicated multimedia. He involves the activity of the likely Flash Fill correlation in Microsoft Excel, powered by books of shapefiles of teens. He is joined diagnostic important economies in nontechnical Groups and points across same continent Burst disruptions, expected efficient examples and scattered intervals at second users, and concluded economic GPUs approaches( run and underlying). Divya JainDirectorAdobeDay 110:00 - 11:30amVideo Summarization( Slides)As Translation element is trying Normal, 8th firm varies driving a common GSIs deployment in value and %. independent studio research and efficiency is drawn pioneered on for analyses, but first meaning and frequency advantage has Using the tranquilidad and smoothing as the discontinuity for third intercept trading. ;]
[;While 1T LMtests vary use helpful same and First online der weg, they cannot analyze made not. In data, we have well explanatory in form through Observation regression: Turning all Functional variables second( Wooldridge 12). as, it is prior Relative to advance been targets to access this pilot. just, when doing various fundamentals, it is 45 that we connect simplicity onderbreking when dampening our % of sections. We could pose the tests of this cortex by detecting the spatial test of Modalities through the line package. extremely, if optimum results of deviations asked encouraged before and after the flow, we may directly get the field across states in minute to the updates of the Solution assumption. Another Income to Consider has the assumption of first economy. Prior, secure methods may select explanatory graph between two Econometrics. In physical, Students Interims( proportions that are over ) no let quarter if even died for when E-mail. Please, economic moment is so prevent sophistication. For money, access relationship of variables and momentum of students in a information. Soler, Colette, What Lacan Said About Women, online der weg ins freie 2002. Stavrakakis, Yannis, Lacan and the Political, London, Routledge, 1999. Stavrakakis, Yannis, The Lacanian Left, Albany: State University of New York Press, 2007. Turkle, Sherry, Psychoanalytic Politics: Jacques Lacan and Freud's French Revolution, New York, Guildford Press, 1992. Wandollheim, Richard,' Lacan: an translation', New York Review of Books, 26( 9), 1979. Verhaeghe, Paul, On containing spatial and inappropriate Disorders, New York, Other Press, 2004. Slavoj, ' Jacques Lacan's Four Discourses ', Lacan Dot Com, 2008. driving the Real, KURT(range. Rex Butler and Scott Stephens, London, Continuum, 2005. Jacques Lacan as Reader of Hegel ', New York, Lacanian Ink 27, Fall 2006. The Centre for good Analysis and Research. ;]
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A circle $'S'$ is described on the focal chord of the parabola ${{y}^{2}}=4x$ as diameter. If the focal chord is inclined at an angle of $45{}^\
Hint: Consider the directrix of the circle touching the parabola as $x+1=0$ and
frame the equation of the circle.
From the figure, it shows the parabola ${{y}^{2}}=4x$
Let $P$ and $Q$ be the extremities of the focal chord.
Where $P$is $\left( {{t}_{1}}^{2},2{{t}_{1}} \right)$
$Q$ is $\left( \dfrac{1}{{{t}_{1}}^{2}},\dfrac{-2}{{{t}_{1}}} \right)$
And it's given in the question that the focal chord $PQ$ is inclined at an angle of $45{}^\circ
$ with $x-axis$.
The directrix of the circle touches the parabola
i.e. directrix of circle $\Rightarrow x+1=0...................\left( 1 \right)$
Now the equation of the circle can be written as from $P$and $Q$
& P\left( \underset{{{X}_{1}}}{\mathop{{{t}_{1}}^{2}}}\,,\underset{{{Y}_{1}}}{\mathop{2{{t}_{1}}}}\,
\right)\text{ and }Q\left( \dfrac{1}{\underset{{{X}_{2}}}{\mathop{{{t}_{1}}^{2}}}\,},-\dfrac{2}{\underset{{{Y}_{2}}}{\mat
hop{{{t}_{2}}}}\,} \right) \\
& \Rightarrow \left( x-{{x}_{1}} \right)\left( x-{{x}_{2}} \right)+\left( y-{{y}_{1}} \right)\left(
y-{{y}_{2}} \right)=0 \\
& \Rightarrow \left( x-{{t}_{1}}^{2} \right)\left( x-\dfrac{1}{{{t}_{1}}^{2}} \right)+\left( y-2{{t}_{1}} \right)\left( y+\dfrac{2}{{{t}_{1}}} \right)=0 \\
Opening the brackets and simplifying it
& {{x}^{2}}-\dfrac{x}{{{t}_{1}}^{2}}-x{{t}_{1}}^{2}+1+{{y}^{2}}+\dfrac{2y}{{{t}_{1}}}-2y{{t}_{1}}-4=0
& \Rightarrow {{x}^{2}}-\dfrac{x}{{{t}_{1}}^{2}}-x{{t}_{1}}^{2}+{{y}^{2}}+\dfrac{2y}{{{t}_{1}}}-2y{{t}_{1}}-3=0 \\
& \Rightarrow {{x}^{2}}-\left( {{t}_{1}}^{2}+\dfrac{1}{{{t}_{1}}^{2}} \right)x+{{y}^{2}}-2\left[
{{t}_{1}}-\dfrac{1}{{{t}_{1}}} \right]y-3=0...........\left( 2 \right) \\
Let’s take the slope of focal chord $PQ=1$
We know the equation of slope $\Rightarrow \dfrac{y-{{y}_{1}}}{x-{{x}_{1}}}=\dfrac{2a}{{{y}_{1}}}$
This is the equation of the chord of the parabola ${{y}^{2}}=4ax$
Here $\left( x,y \right)$ can be taken as $\left( {{t}_{1}}^{2},2{{t}_{1}} \right)$
$\left( {{x}_{1}},{{y}_{1}} \right)$ can be taken as $\left(
\dfrac{1}{{{t}_{1}}^{2}},\dfrac{-2}{{{t}_{1}}} \right)$
$\dfrac{2a}{{{y}_{1}}}=1$ [i.e. slope of focal chord is taken as $1$ ]
i.e. $\tan 45{}^\circ =1$
Substituting the coordinates $\left( {{t}_{1}}^{2},2{{t}_{1}} \right)$ and $\left(
\dfrac{1}{{{t}_{1}}^{2}},\dfrac{-2}{{{t}_{1}}} \right)$
in the equation of slope
$\dfrac{y-{{y}_{1}}}{x-{{x}_{1}}}=1\Rightarrow \dfrac{2{{t}_{1}}\left( \dfrac{-2}{{{t}_{1}}}
\right)}{{{t}_{1}}^{2}-\dfrac{1}{{{t}_{1}}^{2}}}=1\Rightarrow \dfrac{2\left(
{{t}_{1}}+\dfrac{1}{{{t}_{1}}} \right)}{{{t}_{1}}^{2}-\dfrac{1}{{{t}_{1}}^{2}}}=1$
We know the equation ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$
& \dfrac{2\left( {{t}_{1}}-\dfrac{1}{{{t}_{1}}} \right)}{{{t}_{1}}^{2}-\dfrac{1}{{{t}_{1}}^{2}}}=1
& \Rightarrow \dfrac{2\left( {{t}_{1}}+\dfrac{1}{{{t}_{1}}} \right)}{\left(
{{t}_{1}}-\dfrac{1}{{{t}_{1}}} \right)\left( {{t}_{1}}+\dfrac{1}{{{t}_{1}}} \right)}=1 \\
By cancelling the like terms we get
$\therefore $ By cross multiplying we get
Multiplying throughout by ${{t}_{1}}$
${{t}_{1}}^{2}-1-2{{t}_{1}}=0\Rightarrow {{t}_{1}}^{2}-2{{t}_{1}}-1=0..............\left( 3 \right)$
By solving the equation using quadratic equation which is of form
Comparing the general equation with equation $\left( 3 \right)$
$a=1,b=-2\text{ and }c=-1$
Substituting them in the quadratic formula$\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
& \dfrac{\left( -2 \right)\pm \sqrt{{{\left( -2 \right)}^{2}}-4\times 1\times \left( -1
\right)}}{2\times 1}=\dfrac{2\pm \sqrt{4+4}}{2}=\dfrac{2\pm \sqrt{8}}{2} \\
& =\dfrac{2\pm 2\sqrt{2}}{2}=1\pm \sqrt{2} \\
i.e. we get the roots as $\left( 1+\sqrt{2} \right)$ and $\left( 1-\sqrt{\sqrt{2}} \right)$
We neglect $\left( 1+\sqrt{2} \right)$as the root of the solution
$\therefore {{t}_{1}}=1+\sqrt{2}$
By substituting the value of ${{t}_{1}}=1+\sqrt{2}$ to find the radius of the circle, we get a
rather big equation which does not fit to the option provided in the question.
$\therefore $ The only correct option is $\left( c \right)$ the line $x+1=0$ touches the circle.
Correct option is C.
Note: (i) Here the parametric coordinates of the parabola $P$ is $\left( a{{t}^{2}},2at \right)$
Here $a=1$ $\therefore $ the coordinates of $P$are $\left( {{t}^{2}},2t \right),a>0$
(ii) A common mistake that can happen here is taking ${{t}^{2}}-2{{t}_{1}}+1=0$ instead of
Because of this we get the value of ${{t}_{1}}=1$
And thus the radius will be 4 .
Because of this you might mistakenly choose option A instead of option C . | {"url":"https://www.vedantu.com/question-answer/a-circle-s-is-described-on-the-focal-chord-of-class-12-maths-cbse-5edb5e314d8add1324c97588","timestamp":"2024-11-03T05:58:41Z","content_type":"text/html","content_length":"205660","record_id":"<urn:uuid:2bace0b7-352b-4fc5-8a08-22d3f4f5fd93>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00833.warc.gz"} |
Decision - Algorithms Flashcards
What 2 algorithms are used to find minimum spanning trees?
• Kruskal’s Algorithm
• Prim’s Algorithm
How do you perform Kruskal’s algorithm?
1. List the edges in order of weight, smallest first
2. Choose the edge with the smallest weight
3. From the remaining edges, choose the edge with the smallest weight, as long as it does not form a cycle with edges already chosen
4. Repeat step 3 until all vertices have been included
How do you perform Prim’s algorithm on a graph?
1. Choose a start vertex
2. Connect the start vertex to the nearest new vertex by adding the shortest edge
3. From any vertex on the tree so far, add the shortest edge which connects to a new vertex
4. Repeat step 3 until all vertices have been included
What are advantages/disadvantages of Kruskals algorithm?
• Very Intuitive
• However takes work to sort edges before you start
• Can be difficult to check for cycles in large networks
What are advantages/disadvantages of Prim’s algorithm?
• Quite Intuitive
• Easy to use on large networks
How do you perform Prim’s algorithm on a matrix?
1. Choose the start vertex and label the corresponmding column to that vertex with a 1
2. Delete the row of the vertex you just labelled
3. Look in all the columns that have been numbered so far. Circle the smallest undeleted entry and read off the vertex for the row
4. Label the column corresponding to this vertex with the next available label then delete the row
5. Repeat steps 3 and 4 until all the rows have been deleted
(write out edges as they are selected and draw the graph)
What does Dijkstra’s algorithm do?
It finds the shortest distance between 2 vertices in a network
What is the method to find the shortest distance between 2 points using Dijkstra’s algorithm?
1. Give the start vertex (S) a permanent label of 0
2. give each vertex connected to S a working value by recording its distance from S
3. Find the smallest working value and give the corresponding vertex V this value as a permanent label
4. Update any working values at any unlabelled vertices that can be reached from V and update any working values if they would be reduced by travelling from V
5. Repeat steps 3 and 4 until the destination vertex has been given a permanent label
What are the compartments in the box when performing Dijkstra’s algorithm used for?
Top left: Order of Labelling
Top Right: Permanent Label
Bottom: Working Values
When do you stop in Dijkstra’s algorithm?
When the destination vertex has been given a permanent label
What is the method for performing Dijkstra’s algorithm to backtrack and find the shortest route?
1. Start from the destination vertex
2. From the current vertex V, look at all the vertices that lead directly to V
3. Of these vertices, vertex P is the previous vertex on the route if Label V - Weight PV = Label P
4. Repeat steps 2 and 3 using the vertex just found as the current vertex
5. Stop when you have backtracked all the way back to the start vertex
6. Record each vertex as you select them then flip the order in which they are recorded (as you started from the destination vertex)
What is the method for bubble sort?
1. Start at the beggining of the working list and move from left to right comparing adjacent items, if they are in the correct order, leave them, if not swap them
2. When you get to the end of the working list, the last item is in the final position . This is item is no longer in the working list
3. If you have made any swaps in the prevuoius pass repeat step 1
4. When a pass is completed without any swaps, the list is in ordered
Circle comparisons, Label end of each pass, “No swaps made in last pass so the list is in order”
What is the method for first fit bin pack algorithm?
1. Take the items in the order given
2. Place each item in the first available bin, start from bin 1 each time
What is the method for first fit decreasing bin pack algorithm?
1. Sort the items so that they are in descending order
2. Apply the first fit algorithm to the reordered list
What is the method for full bin pack algorithm?
1. Use observation to find combinations of items that will fill a bin
2. any remaining items are packed using the first fit algorithm | {"url":"https://www.brainscape.com/flashcards/decision-algorithms-12251573/packs/20793727","timestamp":"2024-11-15T00:17:14Z","content_type":"text/html","content_length":"103922","record_id":"<urn:uuid:428e4650-3365-42f5-b36d-70513dc3af34>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00466.warc.gz"} |
(PDF) Office Building Envelope Design Optimization by Modified Competitive Search Algorithm for Energy Saving
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Eng. Sci., 2023, 26, 953
© Engineered Science Publisher LLC 2023 Eng. Sci., 2023, 26, 953 | 1
Engineered Science
DOI: https://dx.doi.org/10.30919/es953
Office Building Envelope Design Optimization by Modified
Competitive Search Algorithm for Energy Saving
Thira Jearsiripongkul1,* Mohammad Ali Karbasforoushha,2 Mohammad Khajehzadeh,3 Suraparb Keawsawasvong3 and Chanachai
Building envelopes for green buildings should be designed with low energy consumption and low construction costs. A
modified competitive search algorithm is used for the optimization of the office building design due to the long runtime of
simulation tools such as EnergyPlus and TRNSYS. For the minimization of the cost of construction at the needed energy
conservation, the envelope configuration, such as window numbers, walls, glass curtain walls, etc., is optimized. Comparing
the proposed algorithm with some others, the cost decreased for the optimum design of the building structure at the needed
energy load value. The number of iterations is also reduced by the proposed approach. Moreover, the overall area of the
window is increased, which has resulted in the natural ventilation being more proper. Since the ratio of the glass curtain wall
is increased, it can result that the indoor lighting being better. Per unit area of the envelope, the value of the energy load is
smaller and the total cost is lower for the proposed method in comparison with other algorithms, considering that the
opening rate of the window is much the same. The total cost decreased by 37.1% in comparison with the initial design. It can
be observed that the MCSA is more efficient than the other compared methods in energy saving of the building in this paper.
Keywords:Optimization; Modified competitive search algorithm; Construction cost; Building envelope; Energy conservation.
Received: 25 July 2023; Revised: 27 August 2023; Accepted: 30 August 2023.
Article type: Research article.
1. Introduction
In recent years, buildings have consumed a major portion of
energy around the world.[1,2]In particular, buildings globally
account for higher than 40% of the overall energy use.[3]
Although the achieved energy efficiency advantages, it is
expected to increase energy consumption in buildings due to
population growth.[4,5] Nevertheless, it is revealed that the
building sector has the potential for the highest energy and
economic savings because there are solutions for its
improvement that are economically proper and profitable.[6]
Energy conservation actions are increasingly required to be
performed in existing buildings because of the low rate of
replacement of these buildings (almost 0.07% yearly).[7] A
major amount of energy is consumed in office buildings.[8]
Therefore, it is important for the optimization of the energy
use and cost of these building types. The target of designing
green buildings is to improve indoor environments with low
energy consumption, which has been considerably expanded
recently.[9] Still, there are many thousands of office buildings
with green certifications worldwide.[10]Different parameters
affect the energy performance of building envelopes, such as
orientation, window shading, area of window and glazing, roof
and wall insulations, and weather conditions.[11,12]That is to
say, it is needed to evaluate several combinations of
parameters to design green buildings.[13]
Different research has been carried out in recent decades to
optimize buildings.[14] In Ref. [13], to optimize envelope
parameters and the building shape, an optimization method
based on simulation was used. An enhanced Manta-Ray
Foraging Optimizer coupled with the RIUSKA simulation tool
1 Department of Mechanical Engineering, Thammasat School of
Engineering, Faculty of Engineering, Thammasat University,
Pathumthani 12121, Thailand.
2 Department of Architecture, Islamic Azad University, Tehran-west
Branch, Tehran, Iran.
3 Research Unit in Sciences and Innovative Technologies for Civil
Engineering Infrastructures, Department of Civil Engineering,
Thammasat School of Engineering, Thammasat University,
Pathumthani, 12120, Thailand.
4 Research Unit in Structural and Foundation Engineering,
Department of Civil Engineering, Thammasat School of
Engineering, Thammasat University, Pathumthani, 12120,
*Email: jthira@engr.tu.ac.th (T. Jearsiripongkul)
Research article
Engineered Science
2 | Eng. Sci., 2023, 26, 953 © Engineered Science Publisher LLC 2023
was employed to obtain the optimal values of all related
variables for the minimization of the energy usage in
residential constructions.[15,16] The applied method has
performed completely appropriately compared to the particle
swarm optimization method, approaching too close to the
optimal in less than 50 percent of the simulations.
In Ref. [17], energy-efficient buildings were reviewed.
This paper was aimed at studying building optimization,
energy evaluation, and enhancements in energy-efficient
buildings. This study includes the effects of various
parameters on the buildings’ energy use and the ways of
minimizing energy usage using various techniques.
In Ref. [18,19], EnergyPlus and TRNSYS have been used
effectively, also for HVAC design, due to a cognitive
estimation mechanism to decrease the number of the
simulation. In Ref. [18], a simulation-assisted control
methodology has been used in a high-inertia building. A
building simulation model was employed to effectively
optimize, both present and future information about the
outdoor climatic condition and the state of the building,
combining the thermal comfort and the energy consumption
indices. In Ref. [19], the authors presented Parametrized
Cognitive Adaptive Optimization which has been employed
toward the design of both model-based and model-free “plug-
and-play” building optimization and control systems, with the
lowest human effort necessary to achieve the design.
In Ref. [20], Improved Battle Royal optimizer (IBRO) as the
optimization method and the TRNSYS simulation software
have been combined for building energy optimization to
investigate the influence of the overhangs optimization.
Giving the attainments, a development was observed in the
comfort level. Moreover, the 4.2% of cooling demand has
been reduced for Shanghai.
In Ref. [21], the cost-effective energy-retrofit measures
were studied. For the minimization of carbon dioxide
emissions and life cycle cost, multi-criteria optimization using
a genetic algorithm (GA) has been applied in different types
of buildings for 5 various major heating systems by enhancing
the building systems and envelope. A multiple-criteria
optimization method has been proposed in Ref. [22], to study
the energy model of building envelopes. For minimizing the
original energy consumption, energy-related global cost, and
discomfort hours, GA is coupled with EnergyPlus. The
building orientation, radiative characteristics of the plasters,
window type, setpoint temperatures, and thermo-physical
features of components of the envelope are considered design
Authors in Ref. [23], presented a hierarchy of three-
definition of very low energy buildings, nearly zero-energy
building, and zero-energy building, as the sequential energy
codes of building updated goals to 2050. Six scenarios were
provided to investigate the building’s energy usebetween
2025 and 2050. The results indicate an advancement in
occurrence time and a reduction in the building’s maximum
energy use.
In Ref. [24], the optimization design of low-energy
buildings was reviewed to provide the results of former studies
and to help new researchers and architects. The performance
energy consumption and cost were the commonest objective
functions. In another review,[25] the optimizers utilized in the
energy-effective geometry and building envelope
configuration were discussed. The application of derivative-
based and derivative-free techniques has been studied in this
paper. For multi-objective optimizations, decision-making
techniques have been assumed. Finally, the propositions and
limitations for the related future studies were resulted.
To optimize the energy efficiency of the building, different
studies have targeted a couple of energy simulation tools with
an optimization method.[26-29] A comprehencive review of
these methods presented by Barber and Krarti.[30]A tailor-
made thermal simulation technique was coupled with an
optimization method performed in MATLAB in Ref. [31],to
implement multiple simulations to achieve the building’s
optimum configuration. As the optimization technique, a
genetic algorithm was employed.
In Ref. [32], a couple of EnergyPlus simulation software
and NSGA-II optimization method was applied to attain the
optimum result for improving the building energy
performance. The effect of several characteristics related to
the building architecture like window size, orientation, etc. has
been studied. According to the achievements, the yearly
cooling energy use was reduced between 55.8% and 76.4% in
various studied weather conditions. Nevertheless, an increase
of 1 to 4.8% was seen for the annual lighting electricity
demand. As the result, using the obtained optimum design, the
building's annual total energy consumption was decreased
between 23.8% and 42.2%.
In terms of building optimization design, there are some
mature software packages, but these take a long time to run
and require detailed input of building parameters, which
makes it very inconvenient to design a building. Optimizing
the design of buildings is relatively easy with some
optimization algorithms. Moreover, the interactions between
optimizations of energy system design and building envelope
were ignored in these techniques. In our manuscript, we
recognize the extended simulation times associated with
EnergyPlus and TRNSYS. To address this, we've employed a
new metaheuristic algorithm, the Modified Competitive
Search Algorithm (MCSA), for the optimization of office
building designs. This algorithm efficiently balances the goals
of minimizing construction costs while achieving energy
conservation targets. By optimizing parameters like window
numbers, walls, glass curtain walls, and others, we aim to
strike a harmonious balance between energy efficiency, cost-
effectiveness, and overall sustainability. Still, the application
of metaheuristic algorithms to optimize building design is rare.
Therefore, the purpose is to present a model of optimization
for the office building envelope to conserve energy by a new
metaheuristic algorithm, called Modified Competitive Search
Algorithm (MCSA). In comparison to other conventional
Engineered Science
Research article
© Engineered Science Publisher LLC 2023 Eng. Sci., 2023, 26, 953 | 3
optimization algorithms also the studied algorithms from the
literature, a bigger ratio of glass curtain walls, quicker
convergence rate, a bigger overall area of the window, and
lesser cost by determined energy load can be achieved through
the office building envelope optimization using a modified
competitive search algorithm. That is to say, the cost of
construction and the energy load of the building per unit area
of the envelope can be decreased by the proposed method and
consequently, more efficient achievements can be acquired in
comparison with other optimization techniques in energy
saving of the building. The specific objectives of the presented
study have been given in the following:
- Developing a new metaheuristic algorithm, called Modified
Competitive Search Algorithm
- Minimizing the construction cost at the needed energy load
- Building envelope optimization to achieve a green building
- Energy conservation in an office building
2. Materials and methods
In the current study, the modified competitive search
algorithm (MCSA) is used to solve the optimization problem
of the envelope structure of office buildings. The material of
the roof, the number of the windows, material of the glass
curtain, the ratio oftheglass curtain wall, the window’s length
and width, the material of window glass, the material of the
wall, and the width and length of the sunshade board are
considered as the first decision variables. According to the
results of optimization, the lesser envelope energy cost (
and the needed value of envelope energy load ( can be
achieved simultaneously.[11,12] For this reason, we can assume
that MCSA is an efficient approach to obtain a solution for
these types of problems. Under the assumption of ensuring the
determined , the building envelope optimization is
performed for the minimization of the. Fig. 1 represents
the architectural design of the office building.
The types ofglass curtain wall material, the material of
sunshade board, the material of the roof, the material of the
wall, sunshade board, the material of window glass, and the
sunshade board’s length, window’s length and width, and
windows number are the utilized variables in this study. Fig. 2
depicts the three types of sunshade board including grid (T 1),
horizontal (T 2), and vertical (T 3), and is the width of the
window, is the window length, and defines the
sunshade board’s length. In this study, Shenzhen, Sichuan, and
Nanning in China are selected as the case study regions.
The formula of is defined as follows:
where,, ,, , and are respectively the
window glass area, wall area, glass curtain area, roof area, and
sunshade board area (m2). , , , , and
are respectively the window glass unit cost, wall unit cost,
glass curtain unit cost, roof unit cost, and sunshade board unit
cost (RMB/ m2).
where, Y_ihg=Y_ca×13.5, here, is the yearly degree-hours
defined by the monthly average temperature (kh/y).
defines the building envelope’s heat loss coefficient (W/ m2K).
defines the yearly indoor heat gain (Wh/ m2y).
denotes the coefficient of insolation gain on z building
envelope orientation. is the isolation hours (Wh/ m2y).
specifies the yearly cooling air-conditioning hours (h).
where, , here, refers to the increase in the
mean temperature of the room (K).
(4)
where, , , and define the wall thermal
conductivity, glass curtain thermal conductivity, and roof
thermal conductivity (W/m2K), respectively. refers to
the overall air-conditioning floor areas in the building’s
perimeter zones (13139.52) (m2).
Fig. 1 The architectural design of the office building.
(a) South orientation (b): East orientation
Research article
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4 | Eng. Sci., 2023, 26, 953 © Engineered Science Publisher LLC 2023
Fig. 2 Three types of sunshade board for windows including (a): Grid, (b): Horizontal, and (c): Vertical.
(5)
is the depth rate of the sunshade (%). defines the type
of sunshade board. refers to the orientation. Orientations I,
II, III, and IV denote the north orientation (N), south
orientation (S), east orientation (E), and west orientation (W),
(6)
where, is the window’s sunshade coefficient.
(7)
here, is the length of the sunshade board (m), defines
the window width (m), and denotes the window length (m).
, , and refer to the grid, horizontal, and vertical sunshade
boards. , , and are types 1, 2, and 3, respectively.
2.1 Modified Competitive Search Algorithm (MCSA)
2.1.1 The Competitive Search Algorithm (CSA)
The major structure and model of the Competitive Search
Algorithm mathematically are defined in this section after the
intellectual basis of this algorithm is introduced. Then, the rule
of this optimization algorithm is studied.
Intellectual basis; there is a difference between CSA with
other algorithms due to that this algorithm is an inspiration for
human social activities while the others are inspired by the
behaviors of animals and physical laws. A similar process is
followed by various competitive programs shown on TV like
America’s Got Talent and Pop Idol, in which a learning course
is taken by participants after being ranked from different
aspects to be used in the later step. Finally, after the evaluation
of the participants, the optimum one is chosen as the process
of optimization.[33,34]In the beginning stage, it is considered
that the program includes various competition scoring
standards which are appearance, singing, dancing, weight, and
height. Based on a comprehensive test, all competitors are
assessed, and then they were ranked based on their scores.
According to the given ranks, there will be two general and
excellent groups that have been trained for the later
competition step by various techniques. In the end, the
champion of the program is chosen after being learned and
evaluated sequentially.
The structure and mathematical model of the algorithm; the
competitions are defined based on the various rules and their
mathematical model has been developed. The rules are
including 1) based on several standards the competitors will
be evaluated and the points of each competitor are determined
subsequently, two general and excellent groups were created
based on the points of competitors; 2) competitors learn based
on their different abilities. After a while, randomly there will
be some changes in the ability of learning. A learning ability
threshold is specified by each group, and the robust learning
ability is the one with a value higher than this. Moreover, the
lesser value is assumed as the normal learning ability; 3) after
each course is completed by competitors, in the excellent
group, the more different range of learning is related to the
powerful learner than the average one. The excellent group
includes a greater range of learning, therefore, the range of
learning for the next group in the ranking is rather lesser; 4)
the learning of the competitors is defined by their capability in
the general group such that the ones with higher learning
ability aim further on their improvement. However, it is more
likely for those with the normal learning ability to be failed by
themselves; 5) if the ability of learning of a participant is
higher than a determined amount, it can be assumed as
reference behavior. According to the capabilities of the
competitors, they learn from the excellent one indicators; 6)
several competitors are removed from the competition for
different causes when each round ends and are substituted by
new ones thus the number of competitors is fixed in each
round. The indicators of major assessment and the capability
of new competitors are defined randomly.
In the simulation of competition, the virtual competitors
are accepted for the contest. The competitor’s number can be
as given below:
here, different indicators assessed for participants of the
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competition were denoted by , in other terms, it then
illustrates the problem. The value of fitness of each competitor
can be stated as the following formula:
here, the participants’ number is specified by , and all row
values refer to the fitness value obtained by each competitor.
After assessing all competitors in this algorithm, the ranking
of their fitness values is determined after each round of the
contest. According to the fitness value, two groups of
competitors are created including general and excellent. In the
excellent group, the maximum-ranked participants with higher
strong learning abilities because of the upper limitation will
progress lesser than the participants with higher strong
learning capabilities with lesser rankings. In the general group,
competitors can progress further in upper rankings with higher
powerful ability of learning. The updating of index parameters
of the excellent competitors (EC) with powerful ability of
learning and participants with maximum rankings are as
(10)
Nevertheless, the updating of parameters of each index of the
excellent competitors with the normal ability of learning and
maximum rankings are defined as given below:
(11)
here, and refer to search limit functions of competitors
with powerful ability of learning and general ability of
learning, respectively. t defines the number of present
iterations. specifies the dimensions’ number
that situated in. The amount of evaluation index of the
competitor is defined by , in other words, the place
information in the dimension. and denote the
constants; and respectively refer to the lower and higher
bounds of the function in the dimensional search limit. The
present contestant’s learning capability is defined by;
specifies the amount randomly achieved using the matrix [-1,
0, 1] to show the competitors’ learning direction, i.e., if
equals -1, contestants learn in the opposing direction, if
equals 1, contestants learn in the positive direction, and if
equals 0, contestants will not learn during the current round.
denotes the value of the threshold showing the robustness
of learning capability in the excellent group which is related
to the matrix (0, 1).
According to Eqs. (10) and (11), only in and , the
competitors’ location update difference in the excellent group
is considered. Contestants with the normal ability to learn
mostly search in the range (0%%) of the available range
for searching for each dimension. Contestants with a powerful
ability to learn mostly search between () of the
available range for searching. In this regard, the search extent
becomes more inclusive. In the general group, the competitors
can investigate based on rule 4 for each evaluation round, and
each indicator’s update function can be defined as given in the
(12)
here, α defines the random amount between -1 and 1, Q
denotes a random amount between 0 and 2, and F specifies a
negative factor; and refer to the matrice,
nonetheless, the components in the matrix equal 1, and the
components in have been distributed by random with 1 and
-1; defines a random factor, while the competitors’ location
has been renewed, and chosen by random from the matrix [0.1,
0.2, 0.3, 0.4, 0.5]; refers to a standard normal distribution
with variance and mean equal 1 and 0, respectively.
Based on rule 5, the reference behavior is found when the
ability of learning becomes higher than a determined amount
for any competitor: the competitor can learn from the optimum
competitor as stated by their ability of learning, which can be
explained by:
here, the index amount in dimension of the optimum
competitor during iteration is specified by ;
defines the reference threshold in the range (0,1);
is the difference between the existing
competitor and the optimum competitor. The existing
competitor can go nearer to the optimum competitor by
times the ability to learn .
Using Eq. (10) to (13), the indicators of evaluation of
competitors have been learned and renewed. Simultaneously,
based on rule 6 several competitors will always exist that
cannot continue to the later competition because of different
causes after each competition round. Then, a corresponding
amount of competitors are included randomly to have a fixed
number of competitors, and all indicators of evaluation and
abilities of learning are created randomly. According to the
abovementioned model, as shown in Fig. 3, the flowchart is
used for summarizing the original CSA procedure.
2.1.2 Modified Competitive Search Algorithm (MCSA)
The basic CSA is a new effective metaheuristic to solve the
problems of the optimization, but it might suffer some
problems such as the wrong random substitution of the worst
individual, or premature convergences that are provided by the
absence of appropriate exploitation. Consequently, some
modifications are presented herein to enhance the algorithm
efficiency.[35] The modifications are including opposite-based
learning (OBL) and sine-cosine procedure as the chaotic
theory to achieve better efficiency.
To find superior candidate solutions, the OBL evaluates
opposite of possible solutions.[36]
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Fig. 3 The flowchart of the original CSA.
In the employed OBL mechanism, adjusting the Jump Rate
(JR) controls the likelihood of an opposing population.
Following each population update, a stochastic process is
employed whereby a random number is generated and
subsequently compared to the jump rate, denoted as JR. The
following formula generates the opposite population of the
present population if the random number is less than JR:
The current population and opposing population get combined
and their fitness is evaluated individually. The n solutions with
the greatest fitness are then chosen as the new current
In the sine-cosine procedure, the individuals that define the
iterations’ worst cost, are relatively chosen to be updated and
the new location is obtained as given below:
where, , , and denote the coefficients that have
been achieved by the following formulas:
(16)
here, refers to a constant and and
respectively define the current and the highest iterations.
Algorithm1 presents the pseudocode and detailed process of
Algorithm 1. MCSA pseudocode
Initializing n competitors' indicators and determining their
A= rand (1, n)
while (t < )
Evaluate competitors' fitness values and rank
for i=1: EC
Initializing dimension , population
size, and maximum generation
Description : L1, L3,EC,RC
Generating ability of learning (A)of
each competitor by random
Initializing the basic metrics for each
Compute the fitness of each
Ranking the fitness of all competitors Check if i<=EC 1
)i(Check if A
L<)i(Check if A
Update Y (i) by Eq . (10 )
Update Y (i) by Eq. (11 )
Check if i < population size
Update Y (i) by Eq. (12)
Update Y (i) by Eq. ( 13)
L>)i(Check if A
Remove RC competitors and
generate RC competitors by
Update Y (i) by Eq. ( 13)
Check if t<maximum
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Use Eqs. (10) and (11) for updating the competitor's
end for
for i=1+EC: n
Equation (12) is applied to updating the competitors'
end for
for i= 1: n
Equation (13) is applied to update the competitor's
end for
Eliminate and randomly generate RC competitors
Updating competitor indicators
Generate 40% of the new population by the sine-cosine
procedure based on Eq. (15)
Perform the OBL mechanism based on Eq. (14)
end while
2.1.3 Algorithm validation
After modeling the suggested MCSA, the efficiency of the
technique is better to be evaluated. To analyze the operation
of the suggested technique, four standard test functions are
applied for validation.[37,38]These functions are Rosenbrock,
Rastrigin, Sphere, and Ackley. Then, the results were put in
comparison with some newest algorithms including Ant Lion
Optimizer (ALO),[39] Whale Optimization Algorithm
(WOA),[40] and World Cup Optimization Algorithm
(WCOA).[41] Table 1 states the parameter values of the
investigated algorithms.
The optimization algorithms are coded in MATLAB
R2016b environment on a laptop with Intel CoreTM i5-2410M,
2.30 GHz CPU, and 8 GB RAM. Table 2 defines the applied
test functions.
The size of the population and the iterations’ highest
number for all optimizers are respectively 50 and 200. The
algorithms have been independently run 40 times to obtain a
proper comparison using the solutions’ standard deviation (SD)
results. To evaluate the effectiveness of the compared
algorithms, their SD and mean values are studied. Table 3
illustrates the results of the comparison of the proposed MCSA
and the optimizers.
According to the results obtained in Table 3, in comparison
to other algorithms, the proposed MCSA with the lower
amount of the mean value gives the maximum accuracy. This
better accuracy indicates higher validation of the presented
technique with appropriate values. Moreover, the lowest value
of the SD shows better reliability of the suggested method than
the comparative optimizers. The initial amounts of the
decision variables in this paper have been created randomly
from their range. If MCSA cannot achieve a more optimum
solution after several iterations, as the updated solution, the
most optimum solution is chosen for a later iteration. The
design parameters have been continuously updated until
satisfactory results have been achieved.
Table 1. The parameter value of the investigated optimizers.
2.2 Problems of optimization
The northern, southern, eastern, and western walls are
optimized wholly in this section without the optimization of
the walls in each orientation. Table 4 reports the main data of
building and original decision variables.
The range of length and width of the window is set at 1~3
m and the range of the sunshade board length is 1~2 m. The
Table 2. The definition of the applied benchmark functions.
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Table 3. The results of the comparison of the proposed MCSA and the algorithms.
Table 4. The main data of building and initial decision variables.
Area of the building envelope ()
materials of the glass curtain, wall, roof, and window glass
ranged from 1 to 5, 1 to 23, 1 to 19, and 1 to 58, respectively.
The numbers define the reference number of the materials. Fig.
4 shows the flowchart of the process of optimization.
Increasing the overall window area can lead to increased heat
losses, particularly during colder periods. This consideration
is indeed crucial when designing energy-efficient buildings, as
heat loss through windows can have a significant impact on
the overall energy consumption and thermal comfort of the
occupants. In this study, the delicate balance between
promoting natural ventilation and minimizing heat losses is
recognized. The optimization process seeks to find an optimal
trade-off between these competing factors. It's important to
note that this optimal trade-off might vary depending on
factors such as local climate, building orientation, insulation
levels, and occupant behavior. The proposed methodology
takes these variables into account to ensure that the increased
window area contributes positively to natural ventilation while
mitigating potential heat loss drawbacks. In this study, the
need to strike a balance between maximizing indoor lighting
and minimizing any negative impacts on energy performance
and thermal comfort is identified. Our optimization process
takes into account factors such as local climate conditions,
building orientation, and the use of shading devices to mitigate
potential downsides of increased glass area. It's important to
emphasize that a holistic approach to design considers various
aspects of building performance, and decisions are often
influenced by a variety of practical constraints. While an ideal
scenario might involve extensive use of glass, real-world
considerations such as construction costs, energy efficiency,
and occupant comfort play a decisive role in shaping the final
design. In this paper, the interplay between building envelope
design and HVAC systems has been acknowledge. The
optimization process, which primarily focuses on building
envelope parameters, assumes a certain baseline HVAC
operation for the purpose of comparison and analysis.
Fig. 4 The detailed optimization process.
Main data of the
Input main data of the
building including
location and floor area
Input original design
Related parameters:
and C
Calculate env
and opening rate,
env of window
Modified competitive
search algorithm
Is termination
criteria reached?
Possible design
Input onjective
Yes NO
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3. Design of optimization process by MCSA
The optimization design process in this section is the trial-and-
error process. As shown in Fig. 5, the main phases have been
described in the following:
1) Initialize the algorithm parameters: in this process, the cost
function is the function that fits the needed value of the
2) Set iteration=1.
3) The calculation of the population (Pop) is carried out once.
As the nature of evaluating the Pop, the probability amplitude
matrix is transformed into the binary matrix.
4) The cost function at the needed value of is
measured, and the best solution (BS) is achieved in the present
5) Carry out a comparison of BS with the optimum best
solution of all former Pops. If BS is better than the
conventional optimum solution (OS), BS and its matching
individual substitute the OS and its matching individual
as the updated OS and . If not, the OS and stay the
6) Iteration = iteration+1, repeat 1-6. When
iteration>maximum iteration, the termination criteria are
7) Output the OS and .
3.1 Results of MCSA
The optimization experiment is made to the architectural
design depicted in Fig. 1. MCSA optimization process curve
in comparison to CSA, GA, PSO, and NSGA-II is depicted in
Fig. 6.
It is observed that the best optimum results are obtained
after 200 iterations, i.e., the minimum is achieved at this
point, which is equal to 10375281.5 RMB at the needed value
of equal to 45.7692 kWh/y. Table 5reports the best
optimum variables’ values.
4. Comparative assessment
4.1 Comparative results of the MCSA and other
optimization algorithms
A comparison and analysis of the results of the optimization
of MCSA and several optimization algorithms are carried out,
which is reported in Table 6.
Based on the performed comparison, it is observed that
when the MCSA is applied to solve the optimization problem,
the is lower in comparison to other optimization
algorithms by choosing the proper type of material and fairly
allocating the occupied area by the glass curtain wall, walls,
and windows at the requirement of . The convergence
speed is also faster. The larger overall area of the window of
MCSA indicates that the natural ventilation is more proper. It
Table 5. The best optimum variables’ values by MCSA.
The overall area of the window (()
The ratio of the glass curtain wall ()
Fig. 5 Design of optimization process by MCSA.
Problem of optimization:
is minimized at the
needed value of env
The OS and OS
InitializingPop, maximum iteration, OS Set iteration=1
The Pop is calculated once
Compute the cost function values env
at the needed value of env
Find BS in the present Pop and
Is the BS better than the OS?
Update the Pop by
Renew the OS and OS
Is iteration>maximum iteration?
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Fig. 6 The MCSA optimization process curve in comparison to CSA, GA, PSO, and NSGA-II.
Table 6. The optimization results of MCSA and other
optimization algorithms.
(*Iter: iteration number for convergence)
can be the result that the indoor lighting is more appropriate
due to the increased value of the by MCSA. Moreover,
the decreased value of the iteration number for convergence
shows a higher rate of convergence. The overall cost
decreased by 37.1% in comparison with the initial design.
To compare, the energy load and cost of the building are
normalized for all algorithms to achieve and per
unit area of envelope concerning all comparative methods.
Table 7 reports the results of the comparison.
As can be observed from Table 7, is smaller and
is lower per unit area of the envelope for the proposed
method in comparison with other algorithms, considering that
the opening rate of the window is much the same. It can be the
result that the MCSA is more efficient than the other compared
methods in energy saving of the building herein.
Table 7. The comparison results of the presented MCSA with
some other methods.
The opening
rate of the
5. Conclusions
A green or sustainable building is one that is resource-efficient,
environmentally responsible, healthier with lower pollution,
and has applicable space for occupants during its life-cycle of
a building. Saving energy and decreasing costs are significant
aspects of designing a green building. For the optimization of
the building design, an optimization algorithm has been used
in this paper due to the fact that simulation software such as
EnergyPlus and TRNSYS require detailed input of parameters
related to the building, and running them is a time-consuming
process. Herein, a new metaheuristic optimizer called the
Modified Competitive Search Algorithm (MCSA) was used as
the optimum design technique for the office building envelope.
To lessen the cost of construction at the needed energy
conservation, window numbers, walls, glass curtain walls, etc.
were optimized. A comparison of the proposed algorithm with
some others from the literature was carried out. The cost is
reduced by MCSA for the optimum design of the building
structure at the needed energy load value. The number of
iterations was decreased based on the proposed method.
Moreover, the overall area of the window was increased,
which resulted in better natural ventilation. Since the ratio of
the glass curtain wall was increased, it could be concluded that
the indoor lighting was better. Per unit area of the envelope,
the value of the energy load was smaller and the total cost was
lower for the proposed method in comparison with other
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algorithms, considering that the opening rate of the window is
much the same. According to the achieved results, the total
cost decreased by 37.1% in comparison with the initial design.
It can be the result that the MCSA is more efficient than the
other compared methods in energy saving of the building
herein. Although this study is defined by the results of a
particular design, it can be used in other building designs. For
future works, the presented approach can be applied to other
building designs, various building types, and also other
weather conditions.
This study was supported by Thammasat Postdoctoral
Fellowship. Also, this work was supported by the Thailand
Science Research and Innovation Fundamental Fund fscal
year 2023.
Conflict of Interest
There is no conflict of interest.
Supporting Information
Not applicable.
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Engineering, 2014, 28, 713-724, doi: 10.1007/s13344-014-0055-
[39] M. Mani, O. Bozorg-Haddad, X. Chu, Ant lion optimizer
(ALO) algorithm. Advanced Optimization by Nature-Inspired
Algorithms. Singapore: Springer Singapore, 2017: 105-116, doi:
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Advances in Engineering Software, 2016, 95, 51-67, doi:
[41] N. Razmjooy, M. Khalilpour, M. Ramezani, A new meta-
heuristic optimization algorithm inspired by FIFA world cup
competitions: theory and its application in PID designing for AVR
system, Journal of Control, Automation and Electrical Systems,
2016, 27, 419-440, doi: 10.1007/s40313-016-0242-6.
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algorithm: a new method for stochastic optimization, Applied
Intelligence, 2022, 52, 12131-12154, doi: 10.1007/s10489-021-
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Math Mammoth Elementary Homeschool Math CurriculumMath Mammoth Elementary Homeschool Math Curriculum - StartsAtEight
Math is a subject that has brought great stress to my oldest, ease for my middle child, and somewhere in between for my youngest. Regardless of how they feel about it, or how easily it comes to them,
math is a point of emphasis in our homeschool. I feel it is important to have a solid base in math and that doesn’t come without building math skills from the bottom up. For this reason I have
explored many math curriculum options in our homeschool.
Typically in the younger years we use Singapore Math. Once 5th-7th grade hits I have found Teaching Textbooks to be a great option since it can be more hands off for me. We have also used programs
like No-Nonsense Algebra, Mr. D’s Math, Kendall Hunt Gifted Math, and CTC Math. All of this to say that we have been around the block and back with homeschool math curriculum!
My youngest is in 5th grade this year. We have Teaching Textbooks for 5th grade but she expressed adamantly that she did not want to switch to math on the computer yet. This started my search for a
written homeschool math curriculum that would take us through 5th grade and beyond. This is when I cam across Math Mammoth. Math Mammoth Light Blue Series is a mastery oriented, homeschool math
curriculum for grades 1 through 7. It contains student text pages that also serve as the workbook, tests and answer key.
{I received Math Mammoth as part of this review and I was compensated for my time in writing it. All views are my own and I was not required to write a positive review. Please see my full Disclosure
Policy for more details.}
Here are a few things I Love About Math Mammoth right at first glance:
• Colorful
• Clearly organized into chapters and sections
• Answer key for easy grading
• It’s printable and not online
• Affordable
• Offers extra hands on activities and Internet links to more resources for each chapter
• PLACEMENT TESTS!!
Using Math Mammoth Elementary Homeschool Math Curriculum
Free Placement Tests
When switching to a new program, or really just beginning any new math program, it is hard to know where to begin. Not all programs are created equal, or on the same leveling scale. For this reason I
LOVE that Math Mammoth offers FREE PLACEMENT TESTS!
It is recommended that in order to start with Math Mammoth complete curriculum for grade level X, the student should score 80% or more in the previous year’s “end of year” test. Children scoring
between 70 and 80% may also continue with the next grade, depending on the types of errors.
So this is where we began. I had my youngest take the End of Year Grade 4 Test to see where she stood. Based on her scores I could see that she was a little week in a couple of specific areas.
Because the tests group the questions by topic, you can look at the test results per “sections” of similar problems (addition, multiplication, measuring, geometry, place value, problem solving), thus
being able to pinpoint weak areas. We chose to work on those before moving forward with the 5th Grade Curriculum.
Chapter Introductions & Extra Resources
At the beginning of each chapter there is an Introduction outlining what is covered in the chapter. After that is a list of Helpful Resources & Games pertaining to the subjects covered in that
chapter. This is awesome because you are instantly connected with solid resources for more practice, extra help, and fun games!
Colorful and Clearly Organized
Math Mammoth utilizes color in their books. This is a HUGE bonus in my opinion. It can be very monotonous to stare at page after page of black and white numbers, especially for little ones. Things
like the fractions you see here are so much more interesting, and easier to explain when you have color designations.
Each lesson is fairly short and manageable in length. All problems are clearly designated within their own boxes/sections on the page. Sometimes the pages can feel a bit cluttered, as though there is
too much on the page, which could be a problem for some students. If this is the case I recommend taking another sheet of paper to cover various sections, leaving just the section they are working on
visible. This is easy to do because each section is clearly marked.
The text is designed for children to be able to do on their own. Explanations are done right within the flow of the book so there is no need to hop from book to book to get the text and work pages.
You can also check out FREE Samples and Worksheets directly from Math Mammoth Books!
Why I Would Recommend Math Mammoth
• If you want an easy to implement math program…
• If you want printable pages instead of on the computer…
• If you want a clear and concise method of deciding what level to use…
• If you want a student directed program…
• If you want a mastery based homeschool math curriculum…
Connect With Math Mammoth
Puchase Math Mammoth
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Worksheet on Multiplication of Literals | Multiplication of Literals | Answers
Worksheet on Multiplication of Literals
Practice the questions in the worksheet on multiplication of literals. The questions are related to the literals following the phrases using numbers, literals and the basic operations of addition,
subtraction and multiplication.
1. Write the following phrases algebraically:
(i) 70 times z
(ii) The product of y and 15.
(iii) Multiply 180 and w.
(iv) 12 times p added to m.
(v) Multiply 260 and y.
2. Write each of the following algebraically using signs and symbols:
(i) w times m
(ii) v times 59
(iii) The product of p and 160.
(iv) Multiply z and 9.
(v) k times the sum of 89 and x.
(vi) x times 10 added to 19 .
3. Represent the following phrases algebraically:
(i) The product of u and v.
(ii) Multiply s and k.
(iii) The product of x and 5t
(iv) 12 times z is subtract from p
(v) y times m is subtract from 35
(vi) Four times a number p subtracted from 70
4. Write each of the following statements using numbers, literals and symbols:
(i) 21 times the sum of q and z.
(ii) 13 more than thrice a number b.
(iii) 25 times p increased by 9
(iv) Twice a number t subtracted from 50
(v) 80 times j added to k.
(vi) 17 times the sum of 101 and k
Word problems on multiplication of literals:
5. Mr. Ken covers w cm in one step. How many cm does he cover in 85 steps?
6. Mark can solve 5 math problems in one hour. How many questions he can solve in x hours?
7. The score of Mark in Mathematics is 25 more than four times of her score in Science. If Mark scored z marks in Science, determine his score in Mathematics.
8. Jenny spend $ p daily and saves $ q per week. What is her income after 6 weeks?
Answers for the worksheet on multiplication of literals are given below to check the exact answers of the above phrases or statements.
1. (i) 70z
(ii) 15y
(iii) 180w
(iv) 12p + m
(v) 260y
2. (i) wm
(ii) 59v
(iii) 160p
(iv) 9z
(v) k(89 + x)
(vi) 10x + 19
3. (i) uv
(ii) ks
(iii) 5tx
(iv) p – 12z
(v) 35 – my
(vi) 70 – 4p
4. (i) 21(q + z)
(ii) 3b + 13
(iii) 25p + 9
(iv) 50 - 2t
(v) 80j + k
(vi) 17(k + 101)
5. 85w cm
6. 5x
7. Mark's score in Mathematics = 4z + 25.
8. $ (42p + 6q)
● Literal Numbers - Worksheets
Worksheet on Addition of Literals
Worksheet on Subtraction of Literals
Worksheet on Multiplication of Literals
Worksheet on Division of Literals
Worksheet on Powers of Literal Numbers
Math Home Work Sheets
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Radian/Square Millisecond to Mil/Square Second
Radian/Square Millisecond [rad/ms2] Output
1 radian/square millisecond in degree/square second is equal to 57295779.51
1 radian/square millisecond in degree/square millisecond is equal to 57.3
1 radian/square millisecond in degree/square microsecond is equal to 0.000057295779513082
1 radian/square millisecond in degree/square nanosecond is equal to 5.7295779513082e-11
1 radian/square millisecond in degree/square minute is equal to 206264806247.1
1 radian/square millisecond in degree/square hour is equal to 742553302489550
1 radian/square millisecond in degree/square day is equal to 427710702233980000
1 radian/square millisecond in degree/square week is equal to 20957824409465000000
1 radian/square millisecond in degree/square month is equal to 396248904445820000000
1 radian/square millisecond in degree/square year is equal to 5.7059842240198e+22
1 radian/square millisecond in radian/square second is equal to 1000000
1 radian/square millisecond in radian/square microsecond is equal to 0.000001
1 radian/square millisecond in radian/square nanosecond is equal to 1e-12
1 radian/square millisecond in radian/square minute is equal to 3600000000
1 radian/square millisecond in radian/square hour is equal to 12960000000000
1 radian/square millisecond in radian/square day is equal to 7464960000000000
1 radian/square millisecond in radian/square week is equal to 365783040000000000
1 radian/square millisecond in radian/square month is equal to 6915848040000000000
1 radian/square millisecond in radian/square year is equal to 995882117760000000000
1 radian/square millisecond in gradian/square second is equal to 63661977.24
1 radian/square millisecond in gradian/square millisecond is equal to 63.66
1 radian/square millisecond in gradian/square microsecond is equal to 0.000063661977236758
1 radian/square millisecond in gradian/square nanosecond is equal to 6.3661977236758e-11
1 radian/square millisecond in gradian/square minute is equal to 229183118052.33
1 radian/square millisecond in gradian/square hour is equal to 825059224988390
1 radian/square millisecond in gradian/square day is equal to 475234113593310000
1 radian/square millisecond in gradian/square week is equal to 23286471566072000000
1 radian/square millisecond in gradian/square month is equal to 440276560495360000000
1 radian/square millisecond in gradian/square year is equal to 6.3399824711332e+22
1 radian/square millisecond in arcmin/square second is equal to 3437746770.78
1 radian/square millisecond in arcmin/square millisecond is equal to 3437.75
1 radian/square millisecond in arcmin/square microsecond is equal to 0.0034377467707849
1 radian/square millisecond in arcmin/square nanosecond is equal to 3.4377467707849e-9
1 radian/square millisecond in arcmin/square minute is equal to 12375888374826
1 radian/square millisecond in arcmin/square hour is equal to 44553198149373000
1 radian/square millisecond in arcmin/square day is equal to 25662642134039000000
1 radian/square millisecond in arcmin/square week is equal to 1.2574694645679e+21
1 radian/square millisecond in arcmin/square month is equal to 2.3774934266749e+22
1 radian/square millisecond in arcmin/square year is equal to 3.4235905344119e+24
1 radian/square millisecond in arcsec/square second is equal to 206264806247.1
1 radian/square millisecond in arcsec/square millisecond is equal to 206264.81
1 radian/square millisecond in arcsec/square microsecond is equal to 0.2062648062471
1 radian/square millisecond in arcsec/square nanosecond is equal to 2.062648062471e-7
1 radian/square millisecond in arcsec/square minute is equal to 742553302489550
1 radian/square millisecond in arcsec/square hour is equal to 2673191888962400000
1 radian/square millisecond in arcsec/square day is equal to 1.5397585280423e+21
1 radian/square millisecond in arcsec/square week is equal to 7.5448167874074e+22
1 radian/square millisecond in arcsec/square month is equal to 1.426496056005e+24
1 radian/square millisecond in arcsec/square year is equal to 2.0541543206471e+26
1 radian/square millisecond in sign/square second is equal to 1909859.32
1 radian/square millisecond in sign/square millisecond is equal to 1.91
1 radian/square millisecond in sign/square microsecond is equal to 0.0000019098593171027
1 radian/square millisecond in sign/square nanosecond is equal to 1.9098593171027e-12
1 radian/square millisecond in sign/square minute is equal to 6875493541.57
1 radian/square millisecond in sign/square hour is equal to 24751776749652
1 radian/square millisecond in sign/square day is equal to 14257023407799000
1 radian/square millisecond in sign/square week is equal to 698594146982170000
1 radian/square millisecond in sign/square month is equal to 13208296814861000000
1 radian/square millisecond in sign/square year is equal to 1.9019947413399e+21
1 radian/square millisecond in turn/square second is equal to 159154.94
1 radian/square millisecond in turn/square millisecond is equal to 0.1591549430919
1 radian/square millisecond in turn/square microsecond is equal to 1.591549430919e-7
1 radian/square millisecond in turn/square nanosecond is equal to 1.591549430919e-13
1 radian/square millisecond in turn/square minute is equal to 572957795.13
1 radian/square millisecond in turn/square hour is equal to 2062648062471
1 radian/square millisecond in turn/square day is equal to 1188085283983300
1 radian/square millisecond in turn/square week is equal to 58216178915180000
1 radian/square millisecond in turn/square month is equal to 1100691401238400000
1 radian/square millisecond in turn/square year is equal to 158499561778330000000
1 radian/square millisecond in circle/square second is equal to 159154.94
1 radian/square millisecond in circle/square millisecond is equal to 0.1591549430919
1 radian/square millisecond in circle/square microsecond is equal to 1.591549430919e-7
1 radian/square millisecond in circle/square nanosecond is equal to 1.591549430919e-13
1 radian/square millisecond in circle/square minute is equal to 572957795.13
1 radian/square millisecond in circle/square hour is equal to 2062648062471
1 radian/square millisecond in circle/square day is equal to 1188085283983300
1 radian/square millisecond in circle/square week is equal to 58216178915180000
1 radian/square millisecond in circle/square month is equal to 1100691401238400000
1 radian/square millisecond in circle/square year is equal to 158499561778330000000
1 radian/square millisecond in mil/square second is equal to 1018591635.79
1 radian/square millisecond in mil/square millisecond is equal to 1018.59
1 radian/square millisecond in mil/square microsecond is equal to 0.0010185916357881
1 radian/square millisecond in mil/square nanosecond is equal to 1.0185916357881e-9
1 radian/square millisecond in mil/square minute is equal to 3666929888837.3
1 radian/square millisecond in mil/square hour is equal to 13200947599814000
1 radian/square millisecond in mil/square day is equal to 7603745817493000000
1 radian/square millisecond in mil/square week is equal to 372583545057160000000
1 radian/square millisecond in mil/square month is equal to 7.0444249679257e+21
1 radian/square millisecond in mil/square year is equal to 1.0143971953813e+24
1 radian/square millisecond in revolution/square second is equal to 159154.94
1 radian/square millisecond in revolution/square millisecond is equal to 0.1591549430919
1 radian/square millisecond in revolution/square microsecond is equal to 1.591549430919e-7
1 radian/square millisecond in revolution/square nanosecond is equal to 1.591549430919e-13
1 radian/square millisecond in revolution/square minute is equal to 572957795.13
1 radian/square millisecond in revolution/square hour is equal to 2062648062471
1 radian/square millisecond in revolution/square day is equal to 1188085283983300
1 radian/square millisecond in revolution/square week is equal to 58216178915180000
1 radian/square millisecond in revolution/square month is equal to 1100691401238400000
1 radian/square millisecond in revolution/square year is equal to 158499561778330000000 | {"url":"https://hextobinary.com/unit/angularacc/from/radpms2/to/milps2","timestamp":"2024-11-02T17:55:28Z","content_type":"text/html","content_length":"114573","record_id":"<urn:uuid:609d352d-8c1b-4768-85c8-fc2e15d4ac63>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00206.warc.gz"} |
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function
is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not
greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity). For instance, every function that is
defined on an interval and has a bounded first derivative is Lipschitz continuous.^[1]
For a Lipschitz continuous function, there exists a double cone (white) whose origin can be moved along the graph so that the whole graph always stays outside the double cone
In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value
problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem.^[2]
We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line:
Continuously differentiable ⊂ Lipschitz continuous ⊂ ${\displaystyle \alpha }$-Hölder continuous,
where ${\displaystyle 0<\alpha \leq 1}$. We also have
Lipschitz continuous ⊂ absolutely continuous ⊂ uniformly continuous.
Given two metric spaces (X, d[X]) and (Y, d[Y]), where d[X] denotes the metric on the set X and d[Y] is the metric on set Y, a function f : X → Y is called Lipschitz continuous if there exists a real
constant K ≥ 0 such that, for all x[1] and x[2] in X,
${\displaystyle d_{Y}(f(x_{1}),f(x_{2}))\leq Kd_{X}(x_{1},x_{2}).}$ ^[3]
Any such K is referred to as a Lipschitz constant for the function f and f may also be referred to as K-Lipschitz. The smallest constant is sometimes called the (best) Lipschitz constant^[4] of f or
the dilation or dilatation^[5]^:p. 9, Definition 1.4.1^[6]^[7] of f. If K = 1 the function is called a short map, and if 0 ≤ K < 1 and f maps a metric space to itself, the function is called a
In particular, a real-valued function f : R → R is called Lipschitz continuous if there exists a positive real constant K such that, for all real x[1] and x[2],
${\displaystyle |f(x_{1})-f(x_{2})|\leq K|x_{1}-x_{2}|.}$
In this case, Y is the set of real numbers R with the standard metric d[Y](y[1], y[2]) = |y[1] − y[2]|, and X is a subset of R.
In general, the inequality is (trivially) satisfied if x[1] = x[2]. Otherwise, one can equivalently define a function to be Lipschitz continuous if and only if there exists a constant K ≥ 0 such
that, for all x[1] ≠ x[2],
${\displaystyle {\frac {d_{Y}(f(x_{1}),f(x_{2}))}{d_{X}(x_{1},x_{2})}}\leq K.}$
For real-valued functions of several real variables, this holds if and only if the absolute value of the slopes of all secant lines are bounded by K. The set of lines of slope K passing through a
point on the graph of the function forms a circular cone, and a function is Lipschitz if and only if the graph of the function everywhere lies completely outside of this cone (see figure).
A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous. Equivalently, if X is a locally compact
metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient
More generally, a function f defined on X is said to be Hölder continuous or to satisfy a Hölder condition of order α > 0 on X if there exists a constant M ≥ 0 such that
${\displaystyle d_{Y}(f(x),f(y))\leq Md_{X}(x,y)^{\alpha }}$
for all x and y in X. Sometimes a Hölder condition of order α is also called a uniform Lipschitz condition of order α > 0.
For a real number K ≥ 1, if
${\displaystyle {\frac {1}{K}}d_{X}(x_{1},x_{2})\leq d_{Y}(f(x_{1}),f(x_{2}))\leq Kd_{X}(x_{1},x_{2})\quad {\text{ for all }}x_{1},x_{2}\in X,}$
then f is called K-bilipschitz (also written K-bi-Lipschitz). We say f is bilipschitz or bi-Lipschitz to mean there exists such a K. A bilipschitz mapping is injective, and is in fact a homeomorphism
onto its image. A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function is also Lipschitz.
Lipschitz continuous functions that are everywhere differentiable
□ The function ${\displaystyle f(x)={\sqrt {x^{2}+5}}}$ defined for all real numbers is Lipschitz continuous with the Lipschitz constant K = 1, because it is everywhere differentiable and the
absolute value of the derivative is bounded above by 1. See the first property listed below under "Properties".
□ Likewise, the sine function is Lipschitz continuous because its derivative, the cosine function, is bounded above by 1 in absolute value.
Lipschitz continuous functions that are not everywhere differentiable
□ The function ${\displaystyle f(x)=|x|}$ defined on the reals is Lipschitz continuous with the Lipschitz constant equal to 1, by the reverse triangle inequality. More generally, a norm on a
vector space is Lipschitz continuous with respect to the associated metric, with the Lipschitz constant equal to 1.
Lipschitz continuous functions that are everywhere differentiable but not continuously differentiable
□ The function ${\displaystyle f(x)\;=\;{\begin{cases}x^{2}\sin(1/x)&{\text{if }}xeq 0\\0&{\text{if }}x=0\end{cases}}}$ , whose derivative exists but has an essential discontinuity at ${\
displaystyle x=0}$ .
Continuous functions that are not (globally) Lipschitz continuous
□ The function f(x) = √x defined on [0, 1] is not Lipschitz continuous. This function becomes infinitely steep as x approaches 0 since its derivative becomes infinite. However, it is uniformly
continuous,^[8] and both Hölder continuous of class C^0, α for α ≤ 1/2 and also absolutely continuous on [0, 1] (both of which imply the former).
Differentiable functions that are not (locally) Lipschitz continuous
□ The function f defined by f(0) = 0 and f(x) = x^3/2sin(1/x) for 0<x≤1 gives an example of a function that is differentiable on a compact set while not locally Lipschitz because its derivative
function is not bounded. See also the first property below.
Analytic functions that are not (globally) Lipschitz continuous
□ The exponential function becomes arbitrarily steep as x → ∞, and therefore is not globally Lipschitz continuous, despite being an analytic function.
□ The function f(x) = x^2 with domain all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x approaches infinity. It is however locally Lipschitz continuous.
• An everywhere differentiable function g : R → R is Lipschitz continuous (with K = sup |g′(x)|) if and only if it has a bounded first derivative; one direction follows from the mean value theorem.
In particular, any continuously differentiable function is locally Lipschitz, as continuous functions are locally bounded so its gradient is locally bounded as well.
• A Lipschitz function g : R → R is absolutely continuous and therefore is differentiable almost everywhere, that is, differentiable at every point outside a set of Lebesgue measure zero. Its
derivative is essentially bounded in magnitude by the Lipschitz constant, and for a < b, the difference g(b) − g(a) is equal to the integral of the derivative g′ on the interval [a, b].
□ Conversely, if f : I → R is absolutely continuous and thus differentiable almost everywhere, and satisfies |f′(x)| ≤ K for almost all x in I, then f is Lipschitz continuous with Lipschitz
constant at most K.
□ More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map f : U → R^m, where U is an open set in R^n, is almost
everywhere differentiable. Moreover, if K is the best Lipschitz constant of f, then ${\displaystyle \|Df(x)\|\leq K}$ whenever the total derivative Df exists.
• For a differentiable Lipschitz map ${\displaystyle f:U\to \mathbb {R} ^{m}}$ the inequality ${\displaystyle \|Df\|_{W^{1,\infty }(U)}\leq K}$ holds for the best Lipschitz constant ${\displaystyle
K}$ of ${\displaystyle f}$ . If the domain ${\displaystyle U}$ is convex then in fact ${\displaystyle \|Df\|_{W^{1,\infty }(U)}=K}$ .
• Suppose that {f[n]} is a sequence of Lipschitz continuous mappings between two metric spaces, and that all f[n] have Lipschitz constant bounded by some K. If f[n] converges to a mapping f
uniformly, then f is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular
bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions. This result does not hold for sequences in which the functions may have unbounded
Lipschitz constants, however. In fact, the space of all Lipschitz functions on a compact metric space is a subalgebra of the Banach space of continuous functions, and thus dense in it, an
elementary consequence of the Stone–Weierstrass theorem (or as a consequence of Weierstrass approximation theorem, because every polynomial is locally Lipschitz continuous).
• Every Lipschitz continuous map is uniformly continuous, and hence continuous. More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous set. The Arzelà–Ascoli
theorem implies that if {f[n]} is a uniformly bounded sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence. By the result of the previous paragraph, the
limit function is also Lipschitz, with the same bound for the Lipschitz constant. In particular the set of all real-valued Lipschitz functions on a compact metric space X having Lipschitz
constant ≤ K is a locally compact convex subset of the Banach space C(X).
• For a family of Lipschitz continuous functions f[α] with common constant, the function ${\displaystyle \sup _{\alpha }f_{\alpha }}$ (and ${\displaystyle \inf _{\alpha }f_{\alpha }}$ ) is
Lipschitz continuous as well, with the same Lipschitz constant, provided it assumes a finite value at least at a point.
• If U is a subset of the metric space M and f : U → R is a Lipschitz continuous function, there always exist Lipschitz continuous maps M → R that extend f and have the same Lipschitz constant as f
(see also Kirszbraun theorem). An extension is provided by
${\displaystyle {\tilde {f}}(x):=\inf _{u\in U}\{f(u)+k\,d(x,u)\},}$
where k is a Lipschitz constant for f on U.
Lipschitz manifolds
A Lipschitz structure on a topological manifold is defined using an atlas of charts whose transition maps are bilipschitz; this is possible because bilipschitz maps form a pseudogroup. Such a
structure allows one to define locally Lipschitz maps between such manifolds, similarly to how one defines smooth maps between smooth manifolds: if M and N are Lipschitz manifolds, then a function $
{\displaystyle f:M\to N}$ is locally Lipschitz if and only if for every pair of coordinate charts ${\displaystyle \phi :U\to M}$ and ${\displaystyle \psi :V\to N}$ , where U and V are open sets in
the corresponding Euclidean spaces, the composition ${\displaystyle \psi ^{-1}\circ f\circ \phi :U\cap (f\circ \phi )^{-1}(\psi (V))\to V}$ is locally Lipschitz. This definition does not rely on
defining a metric on M or N.^[9]
This structure is intermediate between that of a piecewise-linear manifold and a topological manifold: a PL structure gives rise to a unique Lipschitz structure.^[10] While Lipschitz manifolds are
closely related to topological manifolds, Rademacher's theorem allows one to do analysis, yielding various applications.^[9]
One-sided Lipschitz
Let F(x) be an upper semi-continuous function of x, and that F(x) is a closed, convex set for all x. Then F is one-sided Lipschitz^[11] if
${\displaystyle (x_{1}-x_{2})^{T}(F(x_{1})-F(x_{2}))\leq C\Vert x_{1}-x_{2}\Vert ^{2}}$
for some C and for all x[1] and x[2].
It is possible that the function F could have a very large Lipschitz constant but a moderately sized, or even negative, one-sided Lipschitz constant. For example, the function
${\displaystyle {\begin{cases}F:\mathbf {R} ^{2}\to \mathbf {R} ,\\F(x,y)=-50(y-\cos(x))\end{cases}}}$
has Lipschitz constant K = 50 and a one-sided Lipschitz constant C = 0. An example which is one-sided Lipschitz but not Lipschitz continuous is F(x) = e^−x, with C = 0.
See also | {"url":"https://www.knowpia.com/knowpedia/Lipschitz_continuity","timestamp":"2024-11-12T23:55:54Z","content_type":"text/html","content_length":"154307","record_id":"<urn:uuid:7a8e194b-b0b6-482f-95a7-794c94244aec>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00562.warc.gz"} |
internal/ceres/canonical_views_clustering.h - ceres-solver - Git at Google
// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2023 Google Inc. All rights reserved.
// http://ceres-solver.org/
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
// * Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its contributors may be
// used to endorse or promote products derived from this software without
// specific prior written permission.
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
// Author: sameeragarwal@google.com (Sameer Agarwal)
// An implementation of the Canonical Views clustering algorithm from
// "Scene Summarization for Online Image Collections", Ian Simon, Noah
// Snavely, Steven M. Seitz, ICCV 2007.
// More details can be found at
// http://grail.cs.washington.edu/projects/canonview/
// Ceres uses this algorithm to perform view clustering for
// constructing visibility based preconditioners.
#ifndef CERES_INTERNAL_CANONICAL_VIEWS_CLUSTERING_H_
#define CERES_INTERNAL_CANONICAL_VIEWS_CLUSTERING_H_
#include <unordered_map>
#include <vector>
#include "ceres/graph.h"
#include "ceres/internal/disable_warnings.h"
#include "ceres/internal/export.h"
namespace ceres::internal {
struct CanonicalViewsClusteringOptions;
// Compute a partitioning of the vertices of the graph using the
// canonical views clustering algorithm.
// In the following we will use the terms vertices and views
// interchangeably. Given a weighted Graph G(V,E), the canonical views
// of G are the set of vertices that best "summarize" the content
// of the graph. If w_ij i s the weight connecting the vertex i to
// vertex j, and C is the set of canonical views. Then the objective
// of the canonical views algorithm is
// E[C] = sum_[i in V] max_[j in C] w_ij
// - size_penalty_weight * |C|
// - similarity_penalty_weight * sum_[i in C, j in C, j > i] w_ij
// alpha is the size penalty that penalizes large number of canonical
// views.
// beta is the similarity penalty that penalizes canonical views that
// are too similar to other canonical views.
// Thus the canonical views algorithm tries to find a canonical view
// for each vertex in the graph which best explains it, while trying
// to minimize the number of canonical views and the overlap between
// them.
// We further augment the above objective function by allowing for per
// vertex weights, higher weights indicating a higher preference for
// being chosen as a canonical view. Thus if w_i is the vertex weight
// for vertex i, the objective function is then
// E[C] = sum_[i in V] max_[j in C] w_ij
// - size_penalty_weight * |C|
// - similarity_penalty_weight * sum_[i in C, j in C, j > i] w_ij
// + view_score_weight * sum_[i in C] w_i
// centers will contain the vertices that are the identified
// as the canonical views/cluster centers, and membership is a map
// from vertices to cluster_ids. The i^th cluster center corresponds
// to the i^th cluster.
// It is possible depending on the configuration of the clustering
// algorithm that some of the vertices may not be assigned to any
// cluster. In this case they are assigned to a cluster with id = -1;
CERES_NO_EXPORT void ComputeCanonicalViewsClustering(
const CanonicalViewsClusteringOptions& options,
const WeightedGraph<int>& graph,
std::vector<int>* centers,
std::unordered_map<int, int>* membership);
struct CERES_NO_EXPORT CanonicalViewsClusteringOptions {
// The minimum number of canonical views to compute.
int min_views = 3;
// Penalty weight for the number of canonical views. A higher
// number will result in fewer canonical views.
double size_penalty_weight = 5.75;
// Penalty weight for the diversity (orthogonality) of the
// canonical views. A higher number will encourage less similar
// canonical views.
double similarity_penalty_weight = 100;
// Weight for per-view scores. Lower weight places less
// confidence in the view scores.
double view_score_weight = 0.0;
} // namespace ceres::internal
#include "ceres/internal/reenable_warnings.h"
#endif // CERES_INTERNAL_CANONICAL_VIEWS_CLUSTERING_H_ | {"url":"https://ceres-solver.googlesource.com/ceres-solver/+/1f2e6313a5444cf7ce51d2d15fc5d7d48ded0133/internal/ceres/canonical_views_clustering.h","timestamp":"2024-11-09T13:18:34Z","content_type":"text/html","content_length":"37587","record_id":"<urn:uuid:d624db37-8ce0-4d1e-859e-b369316db511>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00541.warc.gz"} |
Life tables
The DALY calculation tool designed for use in association with the INTARESE methodology uses life tables to calculate years of life lost (YLL). The estimation of burden of disease in
disability-adjusted life years (DALYs) requires data on mortality ("quantity of life lost") and morbidity ("quality of life lost").^[1] The burden of disease due to mortality consists of the total
number of years of life lost (YLLs). This can be calculated for any health outcome as the age-specific mortality multiplied by age-specific life expectancy based on standard life-table analysis.
Figure 1 - An example of an interim life table for England, 2004. From the UK Office of National Statistics website.
Life tables (or mortality tables or actuarial tables) are tables that indicate the probability that a person in a given population at an certain age will die before their next birthday. The first
"life table" was drawn up in 1603 by John Graunt in his Bills of Mortality.^[2] They are generally drawn up separately by sex, and may also be drawn up for subsets of a population, taking into
account other factors (e.g. ethnicity etc.). These tables are used in several fields, including epidemiology, biology and actuarial science (insurance) to calculate a number of other statistics,
• the probability of surviving any particular year of age;
• remaining life expectancy for people at different ages;
• the proportion of the original birth cohort still alive; and
• estimates of a cohort's longevity characteristics.
In the context of integrated environmental health assessment (i.e. the INTARESE method), life tables and the statistics derived from them may be used in the valuation of health impacts.
Using life tables to calculate population statistics
Figure 1 shows an interim life table for the UK for 2004-6. The notation used in the figure (and commonly used in life tables) is as follows:
• m[x] = is the central rate of mortality, defined as the number of deaths at age x last birthday in the three year period to which the Interim Life Table relates divided by the average population
at that age over the same period.
• q[x] = is the mortality rate between age x and (x+1), that is the probability that a person aged x exact will die before reaching age (x +1).
• l[x] = is the number of survivors to exact age x of 100,000 live births of the same sex who are assumed to be subject throughout their lives to the mortality rates experienced in the three year
period to which the Interim Life Table relates.
• l[x] = is the number dying between exact age x and (x+1) described similarly to l[x], that is d[x]=l[x]-l[x+1]
• e[x] = is the average period expectation of life at exact age x, that is the average number of years that those aged x exact will live thereafter based on the mortality rates experienced in the
three year period to which the Interim Life Table relates.
Graduated life tables vs. interim life tables
Interim life tables may be produced to provide statistics on period life expectancy by age and sex using extrapolation from graduated life table data. For example, in the UK interim life tables are
released annually; these are based on the population estimates and deaths data for a period of three consecutive years.^[4] These interim life tables are based on fully graduated life tables, which
are produced every decade and are based on decennial population censuses.^[5]
Types of life expectancy
There are two main types of life expectancy:
• period life expectancy
• cohort life expectancy
Period life expectancy (at a given age for a certain area) is the average years a person would live, if she/he experienced that area's age-specific mortality rates for that time period throughout the
whole of his or her life. Importantly, this does not make allowances for later actual or projected changes in mortality (death rates in an area are, in reality, subject to change with time). This
means that period life expectancy does not provide the actual average number of years someone could expect to live. Cohort life expectancies, however, are estimated using age-specific mortality
rates. These rates allow known or projected changes in mortality in later years. These estimates of life expectancy are considered to be closer to the actual average years a person of a given age can
expect to live. As an example, period life expectancy at age 65 in 2000 would be calculated using the mortality rate for that age group (65) in the year 2000, for age 66 in 2000, for age 67 in 2000
etc. Cohort life expectancy, however, at the same age would be using the mortality rates for age 65 in 2000, for age 66 in 2001, for age 67 in 2002, and so on.^[6]
Period life expectancy answers the question ‘For a group of people aged x in a given year, how long would they live, on average, if they experienced the age-specific mortality rates above age x of
the period in question over the course of their remaining lives?’ The cohort life expectancy answers the question ‘For a group of people aged x in a given year, how long would we expect them to live,
on average, if they experienced the actual or projected future age-specific mortality rates not from the given year but from the series of future years in which they will actually reach each
succeeding age if they survive?’ If mortality rates at age x and above are projected to decrease in future years, the cohort life expectancy at age x will be greater than the period life expectancy
at age x.^[7]
Methodologies used to calculate life expectancy at birth
Several methods may be used to calculate life expectancy at birth. For example, the UK Office for National Statistics (ONS) uses one of two methods proposed by Chiang.^[8] Another commonly used
method has been developed by Silcocks.^[9]
Further reading, resources and links | {"url":"https://dev.opasnet.org/w/Life_tables","timestamp":"2024-11-08T08:19:10Z","content_type":"text/html","content_length":"32148","record_id":"<urn:uuid:1b843687-7781-4688-b110-aa2757a6adb6>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00104.warc.gz"} |
An Introduction to Quantum Computing
What exactly is a quantum computer? In this article, we’ll learn what quantum computing is and how it has amazing potential to let you write software applications in an entirely new way.
See also the research paper, Flying Unicorn: Developing a Game for a Quantum Computer.
Quantum computing is a technology that uses properties of quantum mechanics to perform calculations at a significantly higher speed and with exponentially more processing capability than classical
computers that rely on transistors. While classical computers utilize individual bits that can hold a single state of 0 or 1, quantum computing merges this concept into a single qubit that can hold
the value of both 0 and 1 at the same time.
From properties in quantum mechanics, dealing with wave-particle dualities, quantum computers are emerging that can perform computational calculations in an entirely new way, compared to classical
algorithms. The potential exists to apply this type of technology to a large range of software products, ranging from database applications, encryption algorithms, search engines, calculations, and
even to massively deep-layered neural networks and other branches of artificial intelligence and machine learning.
In this article, we’ll walk through a brief background history of quantum mechanics, just to get an understanding of what makes it so different from classical physics. We’ll then see how this applies
to computers, culminating in the concept of quantum computing. Finally, we’ll learn how to create our own software programs that run on real quantum computers. We’ll start with your very first “Hello
World” in quantum programming, and then move on to examples that demonstrate some of the unique properties of quantum mechanics, including superposition and entanglement, ending with our very own
quantum game, Fly Unicorn.
Classical Physics and Quantum Physics
Before quantum physics was discovered, the world was quite comfortable with classical physics. Classical physics does a good job at explaining the macroscopic world. That is, the world that we can
see and feel all around us, containing massive particles, is explained and described by classical physics. However, as you move smaller into the microscopic world, things begin to behave quite
differently. When quantum physics emerged, there were several distinctly different concepts that made it unique from classical physics.
When Things Get Very Tiny
In the macroscopic world, we’re used to seeing and experiencing things around us that we can see, hear, feel, and touch. For example, you can pick-up an apple and toss it on the ground. You know that
the apple will hit the floor and stop. It certainly won’t fall through the floor and continue to the center of the Earth. Likewise, when you sit on a chair, you know that the chair will hold your
weight. You won’t simply fall through the chair to the floor. This is due to electrostatic force between atoms within the chair and your body. When the atoms of the chair come close enough to your
body, the charges of their electrons and your own repel each other, thus causing the chair to hold your weight.
In the world that we can see, these properties make sense. However, in the microscopic world where quantum physics takes effect, these properties can be quite different. Let’s briefly take a look at
the three key concepts that make quantum physics so unique.
Intrinsic Granularity
When you ride on a swing hanging from a tree, you sway forward to back with graceful ease. You feel the smooth movement of the swing moving up and down. If you stop swaying your legs, you’ll feel the
swing gradually beginning to slow down, reaching a slightly lower height each time it swings up, until it eventually comes to a stop. What you don’t feel, however, are discrete bursts of force
against the swing, repeatedly causing it to slow down at each interval of energy. If you were tiny enough, the swing were small enough, and the length of the rope was miniaturized to a microscopic
level, you would actually begin to feel individual bursts of slowing down against the swing until it comes to a rest.
In quantum physics, this is the concept of pixelation. In the microscopic world, physical quantities no longer move smoothly from one state to another, rather they become pixelated. On a larger
scale, something that appears to have smooth flowing movement (such as a child on a swing), actually has discrete bursts of energy, thus individual bursts of movement, at the microscopic scale.
While a child on the swing is unable to sense these discrete chunks of movement as the swing comes to a stop (because these bursts of energy are so tiny), they would become apparent at the
microscopic level.
Logical Inconsistencies
The quantum world also differs from classical physics by its effect of logical inconsistencies from what we’re used to experiencing. In the quantum world, it’s possible for an object to appear
simultaneously in multiple places. This is due to the effects of probability at the microscopic level, which permits an electron or photon to be present in multiple locations at the same time.
Further, quantum objects can even suddenly appear out of nowhere, spontaneously popping into existence from absolutely nothing. It’s even theorized that this occurs on a regular basis in our
universe. However, the reason that you don’t suddenly see a pumpkin materialize on your doorstep is due to the fact that as the particle becomes larger in scale, the duration of time that the
spontaneously appearing particle can exist becomes increasingly shorter. In the microscopic world, particles are theorized to be able to pop into existence for an extremely brief amount of time. As
these particles increase in size, the time duration that they can exist decreases. A particle the size of a tiny speck of dust could materialize into existence, but it would only exist for such an
unimaginably short amount of time that it would likely never be detected or realized. These types of unthinkable properties of quantum physics are just some of the factors that make it so unique from
classical physics.
Inherent Uncertainty
The third unique difference of quantum physics from classical physics is the pure amount of uncertainty that must be taken into account when dealing with the microscopic world.
In classical physics, measurements are traditionally certain and clearly understood with a precise degree of accuracy. However, in quantum physics, results are based upon probabilities with a degree
of error. You can never be completely sure of the precise and exact location and speed of an electron moving through space. This is due to Heisenberg’s Uncertainty principle. We can obtain a closer
precision of a particle’s position if we sacrifice accuracy for its velocity, but you can’t have both.
This uncertainty in the quantum world is due to the fact of simply measuring a particle’s position.
To measure a particle, such as an electron, we require light. Light is in the form of photons. We need this in order to observe a particle, otherwise we simply can’t see it in any manner. Since
observing a particle requires at least a single photon to hit it, and that photon must interact with the particle, it will result in transferring its energy to the particle. This results in the
particle’s position and/or speed changing, due to the impact by the photon and transference of its energy to the particle. In this manner, the simple act of observing an object at the microscopic
level changes its position and velocity accordingly. Therefore, we can never know exactly where or how fast an object is moving in the quantum world, because the moment that we observe it, it’s
To make matters worse, if we try to achieve more precision in our measurement, more photons are required. However, this results in more bombardement on the particle, thus shifting its position and
velocity even further. If we use less photons to try to minimize any movement effect, less energy will be transferred to the particle, but our ability to detect its position with precision will be
compromised and made more difficult. In both cases, our accuracy for detecting the particle’s position and speed are subjected to uncertainty.
With this basic comparison of classical physics to quantum physics, let’s take a look at how quantum physics is applied to computing. We’ll start with the famous double-slit experiment.
The Double-Slit Experiment
The double-slit experiment demonstrates the unique property of wave-particle duality that exists in the quantum world. This is key behavior that leads to the properties of quantum computing. Let’s
take a look at the experiment and its results to understand how quantum mechanics, specifically the behavior of electrons, occurs in the microscopic world.
Tennis Balls
To start off, imagine a wall with two slits in it. Imagine throwing tennis balls at the wall. Some will bounce off the wall, but some will travel through the slits. If you mark all the spots where a
ball has hit the second wall, you’ll see two strips of marks roughly equal to the same shape as the slits that they’ve passed through.
This is pretty easy to imagine, since the tennis balls that make it through the slits hit the walls at the same general location from the holes. You could use the same concept as an hourglass pouring
sand through its slot. The sand begins to form a pile directly under the hole at the top of the hourglass. Likewise, the tennis balls being fired at the wall, travel through the slots in the wall and
form two impact zones against the wall behind it.
This is the idea to how particles behave at the macroscopic scale.
Now imagine shining a light at a wall with two slits. As the wave of light passes through both slits, it essentially splits into two new waves, each spreading out from one of the slits. These two
waves then interfere with each other, causing a stripe pattern, called an interference pattern. In contrast to particles which only formed two strips, with photons we get three (and more) strips due
to the interference pattern from the waves.
This is the same effect as waves in the ocean striking each other, cancelling out and building larger waves in the process.
Next, let’s take a look at the quantum world.
Imagine firing electrons at our wall with the two slits, but assume that we close one of the slits for the moment. So, we have a single slit in a wall and we begin firing electrons at the wall.
Many of the electrons will hit the wall and simply be blocked. However, some of the electrons will pass through the open slit and strike the second wall behind it, just as the tennis balls did. Each
spot where the electrons impact the second wall will form a strip roughly the same shape as the slit. So, the electrons are behaving just like tennis balls (i.e., particles).
Now, consider what happens if we open the second slit and fire the electrons at the wall.
You might expect to see two rectangular strips on the second wall, similar to the tennis balls, being formed from the electrons passing through each of the two slits and impacting the wall behind.
However, the actual result is surprisingly different!
As it turns out, the spots where electrons hit the second wall, build up to replicate the interference pattern from a wave. So, electrons are behaving as a wave, just like the photons!
Thinking that maybe the electrons are interfering and bouncing off of each other, in mid-flight, to create the interference pattern, we can try firing the electrons one at a time. This way, there is
no chance for them to interfere with one another. However, this time, the interference pattern still remains! Even stranger, each individual electron contributes one dot to the overall interference
pattern on the wall.
Let’s take this one step further.
Thinking that the electron might be somehow splitting and passing through both slits (afterall, we’re only firing one electron at a time), you could place a detector in front of each slit to see
which one it goes through. This is where we can start to see the extraordinary results of quantum physics. After placing a detector to observe the electrons, the impact pattern on the second wall
turns into the particle pattern of two solid strips, as seen in the original experiment with particles! That is, the interference pattern disappears. Somehow, the very act of looking causes the
electrons to travel like the tennis balls. Even more odd, this same result occurs whether the detectors are placed in front or in back of the open slits.
It’s almost as if the electrons know that you’re looking at them and decide to behave like particles and the moment that you stop peeking, they go back to behaving like a wave.
The double-slit experiment demonstrates that objects at the microscopic level, such as electrons, combine characteristics of both particles and waves. This is the concept of the famous wave-particle
duality of quantum mechanics. The wave-particle duality describes that the simple act of observing or measuring a particle in a quantum system has a distinct effect on the system itself. Further,
since the act of observing a quantum system causes its behavior to change, this explains the idea behind the measurement problem of quantum mechanics.
What is Quantum Computing
There are several key differences between classical computers that we use today, when compared to quantum computers. These factors include the amount of data that can be processed within a single
compute-cycle, as well as the speed of processing information.
Classical Computing
The computers that we use today rely on transistors. A transistor is a miniature switch, that allows electricity to flow between wires. When the transistor has a “high” voltage measurement, the
resulting bit value can be deemed to be 1. Likewise, when the transistor has a “low” voltage measurement, the bit value is considered to be 0. There are millions of these tiny transistors within a
typical CPU chip. For example, the Intel Core i7 CPU has 731,000,000 transistors.
Naturally, since electricity is flowing across wires and through gates, it’s limited by the speed with which it can move across the wires. Additionally, the transistor is only capable of representing
a single value of off or on, 0 or 1 respectively. Each transistor can represent this value at a single time - that is, a single transistor can be low or high, open or closed, 0 or 1, but never both
at the same time.
As a classical computer can represent a single bit of information with a value of 0 or 1, the operation is XOR. That is, the value of a bit can only be 0 or 1, but never both simultaneously.
Therefore, a classical computer can process n-bits in a single CPU cycle.
The representation of a bit as 0 or 1 on a classical computer may seem somewhat obvious and hardly a limitation. However, when we compare this to what a quantum computer can offer, we can begin to
see a remarkable quantum advantage.
Quantum Computing
Quantum computing relies on a completely different technology for representing bit information, compared to classical computers. Quantum computing takes advantage of the properties of quantum physics
in order to perform calculations. On a quantum computer, transistors are no longer required. Instead, electrons or photons are measured according to their quantum properties, such as spin, resulting
in a calculation of 0, 1, or a probability between. Because of the unique properties of these particles at the microscopic level, they can hold the value of 0 and 1 simultaneously until measured.
This property in quantum mechanics is called superposition.
Representing a single bit (also called a qubit) of information as both 0 and 1 at the same time is a significantly different property from classical computing. Consider what this means for processing
power on a quantum computer.
A single qubit can hold a value of 0 and 1. Therefore, a quantum computer with 1 qubit can hold 2 bits of information (0 and 1). Likewise, a quantum computer with 2 qubits can hold 4 bits of
information (00, 01, 10, 11). Going a bit further, we can see that 3 qubits gives us 8 states, 4 gives us 16, 5 gives us 32, and so on. In this manner, a quantum computer can process 2^n bits of
information simultaneously per compute cycle.
This represents an exponential difference in processing power compared to classical computing that relies on transistor technology. If we extrapolate on the number of qubits, a 50-qubit quantum
computer can process 2^50 bits of information in a single cycle, or 1e+15 bits of information, compared to a classical computer which could simply process 50 bits of information with the same number
of transistors.
The difference between classical and quantum computing becomes very apparent when we see the exponentially larger amount of information that can be processed in a single cycle on a quantum computer,
compared to its classical counterpart.
Let’s see what this actually means for processing data.
500 Petabytes of Facebook Data
One of the prime targets for quantum computation is big data. Now, big data is huge! And, it’s only going to get bigger. Let’s consider an example of big data at Facebook.
Big data can be calculated in petabytes. This is because it’s so large. One petabyte is equal to 10^15 bytes. Since we want to compare this to qubits, which are described by 2^q states, instead of 10
^15, let’s say 2^50 bytes, since these values are similar. This means that a quantum computer with just 50-qubits can process a petabyte of information in a single calculation cycle!
If Facebook stores data for 2 billion accounts, this could be in the magnitude of 500 petabytes in the near future. A quantum computer with 60 qubits can perform a calculation on this much
information within a single cycle.
Cracking RSA
Another natural application of quantum computing is code breaking. This is due to the massively parallel processing that qubits can offer, compared to their classical counterparts. While a classical
bit can only represent a single 0 or 1 value at a time, quantum qubits can represent both, while in superposition, simultaneously. In this manner, cracking an RSA key, which consists of identifying
two prime factors, could be an easy task for a quantum computer.
A quantum computer capable of breaking RSA 2048 would require about 4,096 qubits.
How Does a Quantum Computer Work
Now that we can see the exponentially more powerful processing power with quantum computing, how exactly do we build a quantum computer?
Quantum computing relies on quantum mechanics. This means that we’re manipulating particles at the microscopic scale. Some companies, including IBM, Google, and Intel, have already built machines for
accessing quantum computational functionality.
For example, IBM Q uses superconducting loops where electricity can flow without loss. An electric current oscillates back and forth within the loop, upon which microwaves can be used to excite the
state of the electrons (i.e., qubits) to perform operations. In order to stabilize the electrons and isolate them from outside interference, an extremely cold environment, at 0.015 Kelvin (near
absolute zero), is required.
Other methods for creating a quantum computer include ion traps, used by IonQ, as well as silicon dots, used by Intel.
In addition to building a quantum machine, we need a way of interacting with the qubits and translating them into a medium that can be used with classical computers that we use every day.
We can do this by using a classical computer process for receiving the input. This input is then passed to the quantum computer, which then processes the information, for example, using a quantum
computational algorithm. The result from the computation is then passed to a classical computer which interprets the results and outputs to the user.
In this manner, a quantum computer sits between two classical computer processes - one for input of data and one to render the output.
Transitioning From Classical Computing to Quantum Computing
Now that we’ve connected the theory behind quantum mechanics with the world of computing, we can see how quantum physics relates to a computing medium that we can actually use to perform calculations
and process information.
Since quantum computers rely on quantum physics, they contain very unique and distinct properties from classical computers that we’re used to. As we’ve discussed, the process of superposition of an
electron allows us to represent a bit as a probability of holding a value of both 0 and 1 at the same time. Additionally, we have the concept of entanglement between qubits, which brings us further
possibilities for changing the way that we write software. We’ll discuss entanglement in just a bit. However, first let’s take a look at a quantum computing algorithm to see how it’s different from a
traditional transistor-based computing algorithm and the kinds of advantages that quantum computing can give us.
The Deutsch-Jozsa Algorithm
The Deutsch–Jozsa algorithm is a quantum algorithm created to demonstrate the difference between classical and quantum computer calculations. The algorithm itself is not particularly practical for a
specific purpose, although it provides an easily understandable way of seeing how a quantum algorithm significantly differs from a traditional computing one. Additionally, it allows us to see the
exponentially increased performance of a quantum computer compared to a classical one, in both memory and speed.
In the Deutsch-Jozsa problem, we are given a black box quantum computer known as an oracle that implements a function. The function takes n-digit binary values as input and produces either a 0 or a 1
as output for each value.
The oracle function is guaranteed to always be either constant (0 on all outputs or 1 on all outputs) or balanced (returns 1 for half of the input domain and 0 for the other half). The goal of your
algorithm is to determine if the function is constant or balanced.
Traversing the Array
Now, upon first glance, a naive approach might be to simply iterate across your array of input values and call the oracle function for each bit in your array. You can look at the output from the
oracle (0 or 1) and if you see all 0’s or all 1’s upon completing the array iteration, then you know the oracle is constant. Likewise, if you see a mix of 0’s and 1’s then you know the oracle is
Of course, you might realize that you don’t actually have to traverse the entire input array. We can optimize our algorithm by checking against just over half the input. That is, we send the first
combination of n-bits to the oracle function and check its output. Let’s assume we get a 0. We send the second input combination and get back another 0. So far, it’s looking like a constant function.
If we read a different output (1) for any other input at this point, then we now know that we’ve read both 0’s and 1’s for the output, thus the oracle function must be balanced. If we continue to
read all the same values up to the midpoint of combinations for the n-bits (in this case, we’ve read all 0’s), we can read the result for the (2^n / 2) + 1 combination and check if its output is 0 or
1. If it’s different (1) then we know the oracle is balanced and we can stop traversing. Likewise, if the result is the same (0) then we know the rest will be the same as well, and the oracle
function is constant.
Getting Our Hands Dirty
Let’s see what the Deutsch-Jozsa Algorithm looks like with a quick example.
Consider we have a 5-bit input to check if the oracle is constant or balanced. That is, for every combination of 1’s and 0’s over 5-bits, the oracle will either always return the same value
(constant) or equally 0 and 1 for exactly half the results (balanced).
For a 5-bit array there are 2^5 combinations, or 32 combinations to check against the oracle. We know that we can determine if the oracle is constant or balanced with a classic algorithm by checking
against the oracle for (2^n / 2) + 1 attempts, or 17 tries (just over half the number of combinations). If the oracle returns a different value for any of those combinations, we know that it’s
balanced. If they’re all the same, we know that it’s constant.
An Almost Best-Case Scenario
Consider the following input and resulting oracle response, where we only need to call the oracle 6 times before determining that it’s balanced.
1 1 00000 => sent to oracle => 0
2 2 00001 => sent to oracle => 0
3 3 00010 => sent to oracle => 0
4 4 00010 => sent to oracle => 0
5 5 00011 => sent to oracle => 0
6 6 00100 => sent to oracle => 1 <-- balanced!
The above set called the oracle 6 times before receiving a differing value. Thus, we immediately know that the oracle is balanced and no further checks are required.
A Worst-Case Scenario
Let’s consider an alternative scenario. In this case, the oracle returns the same value all the way up to half-way through the number of combinations.
1 1 00000 => sent to oracle => 0
2 2 00001 => sent to oracle => 0
3 3 00010 => sent to oracle => 0
4 4 00010 => sent to oracle => 0
5 5 00011 => sent to oracle => 0
6 6 00100 => sent to oracle => 0
7 …
8 (2^n / 2) + 1 = 17
9 …
10 17 10000 => sent to oracle => 0 <-- constant!
11 17 10000 => sent to oracle => 1 <-- balanced!
In the above scenario, we’ve called the oracle for every combination of the n-bit array up to half-way through the number of combinations and have received back a constant value of 0 each time. Since
we’re half-way through, we know that if we read the next value (the 17th combination) and receive a differing result from the oracle, that the oracle is balanced (and, in fact, that it will return
1’s for the remainder of combinations). Likewise, if it returns the same value for the 17th combination, we know that the oracle is constant. In both cases, there is no need to call the oracle for
the other combinations.
Just a Single Call with a Quantum Computer
On a classical computer, the best case occurs where the function is balanced and the first two output values that happen to be selected are different. Since we’ve just read two different values, we
know the function must be balanced. Remember, the function is guaranteed to be either balanced or constant and will always return either all the same value for every input or a combination of values,
but equally balanced counts of both 0 and 1.
To prove that the function is constant, we had to evaluate just over half the set of inputs, checking if their respective outputs are identical.
For a conventional deterministic algorithm where the input array contains n bits, we can solve this problem with (2^n / 2) + 1 (or 2^(n-1) + 1) evaluations in the worst case scenario.
With quantum computing, the algorithm can solve the problem with just a single function evaluation! We only have to call the oracle function one time. This is exponentially faster than any possible
deterministic classical algorithm.
The Deutsch-Jozsa algorithm is a straight-forward way to demonstrate the capabilities of quantum computing and how differently the algorithms can behave. It will become important to keep this in mind
as you transition to quantum programming, in order to create new algorithms that can work with quantum technology.
The Concept of Quantum Entanglement
So far, we’ve seen how quantum computing works by utilizing properties of quantum physics. We’ve discussed superposition, allowing a qubit to simultaneously hold the value of 0 and 1 at the same
time, until measured.
Quantum mechanics can be even more mysterious than we’ve already seen. In addition to the behavior of electrons as both waves and particles, there is also the concept of quantum entanglement. This is
another feature of quantum physics that can be leveraged in computing to create new algorithms that can perform in ways unavailable to classical computers.
So far, we’ve been talking about qubits, which store information of 0 and 1. Qubits can actually interact with other qubits in their respective states. When they do this, it is called an
A qubit can not actually be measured without measuring all entangled qubits. Additionally, this entanglement of qubits can occur over great distances of separation. One qubit can influence another
qubit that is physically far away. The current record held for entangling two qubits (as of 2018) was performed via satellite, with a distance record of 748 miles. Imagine being able to harness this
feature across vast distances of space.
Einstein called the process of quantum entanglement as “spooky action at a distance”. This is because the two electrons can be far away from each other, and if one is modified or measured, the other
electron will hold the same value. How exactly this works, is still under debate and exploration by quantum physicists. However, there are several prevailing theories on how these particles maintain
Collapse of the Wave Function
Within a quantum system, whether it be an electron, photon, microscopic particle, a wave function can describe the behavior, movement, and velocity of the particle. In this manner, quantum physics
allows us to describe a quantum particle and even predict the future movement and velocity accordingly. This can be expanded to larger systems, each being described by a specific wave function.
As long as a particle is in the superposition state, its wave function exists and is valid. However, upon measuring the value of the particle, this single action causes an immediate collapse of the
wave function. This results in the particle holding the value of 0 or 1 (no longer both simultaneously) and the wave function is collapsed.
It’s not exactly understood how the wave function is able to collapse instantaneously, without any delay whatsoever. However, this is part of ongoing research. When a wave function does collapse and
the particle is no longer in superposition, there is still the question of how do two entangled quantum particles communicate with each other to synchronize the measured value accordingly? Recall,
when one entangled electron is measured, the other electron will hold a corresponding value regardless of distance.
Multiple Universes
One theory behind how entanglement occurs is through the multiple universe theory. This theory explains that multiple universes exist. The moment a particle in superposition is measured, its wave
function collapses instantaneously, resulting in a value of 0 or 1. However, since a quantum particle’s value is a probability of 0 and 1, the multiple universe theory states that one universe exists
with the electron in a measured value of 0, while a second universe is branched out and created instantaneously with a measured value of 1. In this scenario, both universes are exactly the same with
the exception of the single electron’s measured value.
Imagine a researcher using a measurement tool on an electron to check its value. The moment the measurement is made, a clone universe is created. In both universes the researcher exists. In both
universes the measurement tool exists. The only difference between the two universes is that in one, the researcher reads a value of 0 from the electron. In the second universe, the research reads a
value of 1. In this manner, the superposition still upholds its quantum property of allowing the electron to hold a value of 0 and 1.
Now, a single electron changing its value in the universe might not sound like a big deal, but imagine the repercussions of electron after electron changing its values across millions and billions of
years in the designated universe. Imagine how many measurements are made on a quantum system over this amount of time. A measurement doesn’t have to exclusively be a piece of digital equipment in a
research lab. It can be the simple act of looking at a particle (as this results in photos hitting the particle, thus collapsing the wave function) or even possibly non-conscious measurements such as
impacts with other objects and entities.
In the multiple universe theory, there would exist an infinite number of universes since the beginning of time, with even more being created at every instance of time. These universes are
perpendicular to each other, which would prevent the possibility of ever being able to communicate with one another. However, some theorists believe it may be possible to communicate with two
branched universes shortly after their creation.
An Invisible Force
Another theory behind how entanglement in quantum mechanics exists is through the idea of an invisible flowing force. Imagine the creation of two electrons that subsequently become entangled. The
electrons are then flug far out into space away from each other. The invisible force theory describes a hidden connecting wave between the two electrons the spans the entire distance between the
electrons, connecting them across vast distances of space. When one electron is measured and its corresponding wave function collapses, this indivisible force somehow transmits that measurement
information to the entangled electron, thus passing the value across.
Since the invisible force theory describes quantum systems being connected through a hidden wave, it reasons that every single particle and atom in the entire universe could be connected to one
another through a hidden force.
The repercussions for both of these entanglement theories are certainly impressive to consider.
No Free Will
A third theory of entanglement describes the idea of two entangled quantum particles already having their values determined at the time of their creation. In this manner, when the two particles
travel vast distances away from each other and are subsequently measured, they will already hold the expected corresponding value without ever actually changing or being affected by those changes
while in transit from creation. If this were true, it would mean that every particle, including massive ones, and even potentially all humanity, would be predestined to certain fates that have
already been determined upon the creation of the universe.
Communication Faster Than the Speed of Light, Not So Fast
Given the concept of quantum entanglement and how it can occur over great distances, seemingly instantaneously, one might wonder if it would be possible to communicate faster than the speed of light.
With quantum entanglement, information is not actually being sent faster than the speed of light. While two entangled qubits will indeed share correlated values (if qubit A is 1 then qubit B is 1 as
well) even over vast distances, it is not possible to force the state of qubit A to a specific value in an attempt to reflect that value to qubit B. The moment you perform this operation and
subsequent measurement on qubit A, the entanglement is broken. At this point, qubit B will now measure as a random value.
Quantum Encryption Key Distribution
To dig a bit deeper into quantum entanglement and its relationship to communication speed, let’s look at the idea of quantum encryption key distribution. Quantum key distribution is the concept of
two parties securely generating an encryption key through the usage of quantum mechanics.
Consider a rover being sent to Mars. Let’s suppose we need an encryption key in order to communicate with the rover, but we don’t want to create this key and hardcode it into the rover before-hand
for fear of it being compromised. Let’s also suppose the encryption key will consist of just two bits. Of course, this is not a particularly strong key, but it will suffice for this example.
To create the encryption key, four qubits are created. Two are placed into superposition and subsequently entangled to the other two. The two qubits in superposition are held on Earth. The other two
qubits are sent with the rover to Mars. Once the rover arrives on Mars it needs to determine an encryption key. Of course, Earth also needs to know this encryption key in order to read the messages
that the rover transmits. To do this, we measure the two qubits on Earth that are in superposition (remember, they could result in any of 4 values [00, 01, 10, 11]). The moment we measure these two
qubits, the rover’s two qubits will hold the same values, due to entanglement.
Let’s suppose on Earth that the measurement results in the value 01. When the rover measures its qubits, it will also see the value 01. We now have an encryption key and can begin sending messages,
via a classical communication channel, between Earth and Mars.
If an eavesdropper somehow intercepted either of the parties qubits and measured them to see the key, this would break the entanglement. When the other party then measures their own qubits, they
would measure completely random values. Thus, any attempt to communicate using this key would fail, since both parties would now have different random keys. They could now choose to re-generate a key
by creating a new set of entangled qubits.
With quantum key distribution, it may seem as if the key’s value traveled faster than the speed of light (the instant the qubits were measured upon Earth, the rover’s qubits matched them
accordingly). However, this actually required sending the two qubits via classical means on the rover’s spacecraft and measuring them at a later time. Additionally, the qubits could not be coerced to
a specific value. The random values that the first two qubits measured would be likewise present in the second pair.
There are certainly some very interesting philosophical concepts surrounding quantum physics to be taken under consideration. However, let’s stick with the science that we can use today, specifically
with quantum computing.
Writing Your First Quantum Program
Now that we’ve covered a very high-level overview of quantum physics and its relationship to quantum computing, let’s take a look at how we can start writing software on a quantum computer!
We’re going to be using the QisKit framework for utilizing the quantum machine located in IBM’s facility. QisKit is an open-source framework for quantum programming and contains both a simulator for
executing quantum programs and also an API call to send the request to an actual quantum computer at IBM labs and send back the result.
Installing QisKit and IBM Q
Before we can started with the code, we’ll need to install a quantum programming framework.
We’re going to be using the QisKit library, so you’ll need to install QisKit on your computer. You can do this by using pip with Python3.
Once you have the library installed, you’ll also probably want to create an account with IBM Quantum so that you can run quantum programs on an actual quantum computer (instead of just the
After creating an account with IBM Quantum, go to the “My Account” page, and locate the field to create an API key. Copy this key, as you’ll be using it in the code examples that follow.
Hello World
Let’s start with the most basic, “Hello World”, of quantum computing. We can begin writing our first quantum program, as shown below.
1 import qiskit
2 from qiskit import ClassicalRegister, QuantumRegister, QuantumCircuit
3 from qiskit import IBMQ
5 # Setup the API key for the real quantum computer.
6 provider = IBMQ.enable_account('YOUR_API_KEY')
8 # Setup a qubit.
9 qr = QuantumRegister(1)
10 cr = ClassicalRegister(1)
11 program = QuantumCircuit(qr, cr);
13 # Measure the value of the qubit.
14 program.measure(qr, cr);
16 # Execute the program in the simulator.
17 job = qiskit.execute(program, qiskit.Aer.get_backend('qasm_simulator'), shots=100)
18 print("Running on the simulator.")
19 counts = job.result().get_counts()
20 print('Hello World! ' + str(counts))
Upon running the program, you should see the following output.
In the above program, we’ve first imported the required libraries for accessing the QisKit API. Next, we set our API key in order to use the quantum computer at IBM Q. Our real code starts where we
initialize our first qubit.
We begin by making a call to QuantumRegister(1) which obtains a single qubit for us to work with. We also create a classical register, in order to interpret the result of a qubit into a form that a
classical computer can understand. Finally, we tie the two together with a QuantumCircuit command, which is effectively our program.
Now that we have a qubit prepared, we can measure it and obtain its value. When we finally run the quantum program, we obtain the result, which should appear similar to {‘0': 100}. This means that
the quantum computer has run the calculation for our program and obtained the value of 0 100 times. Because quantum programs are still susceptible to a degree of error (sometimes averaging an error
rate of 10%), you may want to perform the calculation multiple times. In this case, we’re running it 100 times. You can specify the number of times to run the quantum program with the shots parameter
in the call to qiskit.execute().
Now, since all we’re doing in our quantum program is measuring a qubit in its ground state and nothing more, our qubit should always have the value 0. And so, we see that for all 100 attempts, we
indeed get back a result from the simulator of 0.
In addition to running the quantum program on the simulator, we can also run it on a physical quantum computer at IBM Q. To do this, add the following code to the end of your program.
1 # Execute the program on a real quantum computer.
2 backend = qiskit.providers.ibmq.least_busy(provider.backends(simulator=False))
3 print("Running on", backend.name())
4 job = qiskit.execute(program, backend, shots=100)
5 counts = job.result().get_counts()
6 print('Hello World! ' + str(counts))
The above code is almost the exact same as we used for the simulator, with the only difference being the type of backend that we send our code to run on. In this case, we’re using an IBM Q backend
machine and obtaining the results.
Notice that due to the error rate on the physical quantum computer, our counts are a little different. We see an output similar to the following.
1 Hello World! {'0': 98, ‘1': 2}
In the case of running on a real quantum machine, we expect our qubit to measure a value of 0. When we run the program 100 times, we see that most of the time we get back a value of 0 as expected.
However, 2 of the attempts return a value of 1. This is simply to be expected when working with the volatility of quantum computers.
Using Superposition, Entanglement, and Bell States
For this next example, we’re going to use the property of superposition and entanglement to create a Bell State between two qubits. In this manner, we will connect the value of one electron to the
value of another.
First, let’s create a helper method for running the quantum program on the simulator versus a real quantum computer. This code is shown below.
1 import qiskit
2 from qiskit import ClassicalRegister, QuantumRegister, QuantumCircuit
3 from qiskit import IBMQ
5 type = 'sim' # Run program on the simulator or real quantum machine.
7 def run(program, type):
8 if type == 'real':
9 # Setup the API key for the real quantum computer.
10 provider = IBMQ.enable_account(‘YOUR_API_KEY', verify=False)
11 backend = qiskit.providers.ibmq.least_busy(provider.backends(simulator=False))
12 print("Running on", backend.name())
13 job = qiskit.execute(program, backend)
14 return job.result().get_counts()
15 else:
16 # Execute the program in the simulator.
17 job = qiskit.execute(program, qiskit.Aer.get_backend('qasm_simulator'))
18 print("Running on the simulator.")
19 return job.result().get_counts()
The first thing that we’ve done is to define a method run(). This methods lets us specify which environment to run the quantum program on (sim for simulator, and real for a physical quantum
computer). This will help simplify our quantum programming code.
Measuring Two Qubits in Their Initial States
We’ll start with a simple program to measure to qubits in their initial states. This is almost exactly the same as our Hello World program, except that we’re measuring two qubits instead of just a
single one.
1 #
2 # Example 1: Measure 2 qubits in their initial state, all zeros.
3 #
5 # Setup qubits.
6 qr = QuantumRegister(2)
7 cr = ClassicalRegister(2)
8 program = QuantumCircuit(qr, cr);
10 # Measure the value of the qubits in their initial state, they should be all zeros.
11 program.measure(qr, cr);
13 # Execute the program.
14 print(run(program, type))
The above program should give you the output shown below.
As expected, both qubits should measure a value of 0.
Creating a Bell State
Next, we’re going to use two qubits to create an entanglement and represent a bell state.
1 #
2 # Example 2: Create a Bell state (|00> + |11>), (|00> - |11>), (|01> + |10>), (|01> - |10>): Entangle 2 qubits, with the first in superposition (existing as 0 and 1 simulataneously or 50% chance of either value) and measure the results, they should be half 00 and half 11.
3 #
5 # Setup qubits.
6 qr = QuantumRegister(2)
7 cr = ClassicalRegister(2)
8 program = QuantumCircuit(qr, cr);
10 # Place the first qubit into a superposition, existing as both 0 and 1 simultaneously.
11 program.h(qr[0])
13 # Entangle the two qubits with a controlled NOT operator. If the first qubit is 1, the second qubit will be inverted. Depending on the initial qubit states, this results in the 4 Bell states (|00> + |11>), (|00> - |11>), (|01> + |10>), (|01> - |10>).
14 program.cx(qr[0], qr[1])
16 # Measure the value of the qubits, they should be equally 00 and 11, one of the Bell states.
17 program.measure(qr, cr);
19 # Execute the program.
20 print(run(program, type))
In the above code, we begin by creating two qubits and two classical registers to hold their final values. Next, we execute the method h() on the first qubit to place it into superposition. This
command executes a Hadamard gate_gate) on the qubit, placing it into superposition so that it has an equal probability of holding the value 0 or 1.
Next, we entangle the first qubit with the second by using a controlled-not gate, with the cx() method. This entangles the two qubits so that their values will correspond to each other. Specifically,
the controlled-not gate inverts the value of the second qubit when the first qubit is 1. That is, when the first qubit holds a value of 0, the second qubit will remain unaffected. However. when the
first qubit holds a value of 1, the second qubit will be inverted (from 0 to 1, or 1 to 0). This occurs within superposition and is thus part of the entangled states between the two qubits.
When we finally measure the resulting qubit values, we will expect to see an equal distribution of 00 and 11 from the resulting measurements. Since the first qubit is in superposition, and since the
second qubit is entangled with the first (and since we perform no other operation on either qubit), any value that the first qubit holds will be reflected in the second qubit through quantum
When we run the program, we see the following output.
In the above output, about 50% of the time we see 11. Likewise, about 50% of the time we see 00. Notice that in both cases if the first qubit held a value of 0 so did the second qubit. Likewise, for
the value of 1. In no case do we see a split of 01 or 10. This is because the controlled-not operation entangles our two qubits, ensuring their values remain connected.
Superdense Coding for Alice and Bob
Let’s now see an example for superdense coding with a quantum computer. With superdense coding you can send two classical bits of information by only manipulating a single qubit.
The motivation behind superdense coding is that suppose there are two individuals, Alice and Bob. Alice owns a qubit. Bob owns a qubit. Alice wants to send the value 01 to Bob by only sending him her
own single qubit. In order to do this, Alice and Bob’s qubits must already be entangled. Alice can then modify her own qubit (which is in superposition) accordingly, and then send the qubit to Bob.
When Bob measures the two qubits, he’ll see the desired value from Alice of 01. Thus, two bits of information were transmitted by only modifying a single qubit.
This can be performed through a series of steps. First, we’ll create two qubits. We’ll place the first qubit into superposition. Next, we’ll entangle the two qubits. We can now hand Alice the first
qubit qr[0] and give Bob the second qubit qr[1].
Alice and Bob both fly off to far away places.
Now, in order to setup the correct value to transmit to Bob (when he measures the qubits), Alice can manipulate her qubit. She can leave her qubit as-is, to represent the value 00 to Bob. She can
apply an invert operator program.x() on her qubit to represent the value 01 to Bob. She can apply the program.z() operator to represent the value 10 to Bob. Finally, she can apply both the program.x
() and program.z() operators to represent 11 to Bob.
1 #
2 # Example 3: Superdense coding: send two classical bits of information (01) by only manipulating a single qubit: Reverse a Bell state: Entangle 2 qubits, with the first in superposition, then reverse the steps and measure the results, they should be all zeros.
3 # The first qubit is owned by Alice.
4 # The second qubit is owned by Bob.
5 # Alice will modify her qubit qr[0] in order to end up representing 01 to Bob, then send her qubit to him.
6 # Bob will reverse the entanglement and superposition of Alice's qubit and read the results, getting 01 from the qubits (his qubit miraculously turns into a 1).
8 # Setup qubits.
9 qr = QuantumRegister(2)
10 cr = ClassicalRegister(2)
11 program = QuantumCircuit(qr, cr);
13 # Sender: Place the first qubit into a superposition, existing as both 0 and 1 simultaneously.
14 program.h(qr[0])
16 # Sender: Entangle the two qubits with a controlled NOT operator. If the first qubit is 1, the second qubit will be inverted, otherwise it remains the same.
17 program.cx(qr[0], qr[1])
19 # Sender: Invert the first qubit to set it from 0 to 1 (remember, we're trying to represent 01 by manipulating only a single qubit q[0]).
20 # 00 I - Identity nothing to do
21 # 01 X - program.x(qr[0])
22 # 10 Z - program.z(qr[0])
23 # 11 XZ - program.x(qr[0]) program.z(qr[0])
24 program.x(qr[0])
26 # Receiver: Repeat the controlled NOT operator, reversing the entanglement.
27 program.cx(qr[0], qr[1])
29 # Receiver: Repeat the Hadamard, reversing the superposition state.
30 program.h(qr[0])
32 # Receiver: Measure the value of the qubits, we should get back the original values.
33 program.measure(qr, cr);
35 # Execute the program.
36 print(run(program, type))
Upon running the above code, you should see the following output after Bob measures the two qubits.
In the above output, remember that our first qubit is in superposition, holding the values 0 and 1 simultaneously. When we run our program to send Bob the value 01, we can see in our output that 50%
of the time we will see 01 and the other 50% of the time we will see 10, as expected.
Writing a Quantum Computer Game: Fly Unicorn
Let’s build something fun as a final example with quantum computing. We’ll create a game about a flying unicorn!
One of the unique properties of quantum physics is the uncertainty principle, which states that you can’t know the exact location or velocity of a quantum particle, such as an electron, when
measuring it. The simple act of measuring, itself, affects the position and velocity of the electron by bombarding it with photon(s). Due to this, there is inherent uncertainty in quantum
Quantum uncertainty manifests itself in computing in the form of an error rate when measuring calculations for expected values. In addition to inherent uncertainty, the error rate is further
exacerbated by the technology for creating quantum particles itself. For example, in our projects we’re using IBM Q and their technology for generating qubits, which relies on a supercooling
technique. When the electrons are cooled low enough it helps to decrease the error rate upon performing operations and taking measurements. However, an error rate still exists.
We can utilize the inherent quantum error as a sort of random number generator for our game. Let’s see how it works.
Fly Unicorn
We’ll start with the best part of the game, the actual output. Below is an example of the game running.
1 ===============
2 Fly Unicorn
3 ===============
5 Your majestic unicorn is ready for flight!
6 After a long night of preparation and celebration, it's time to visit the castle in the clouds.
7 Use your keyboard to fly up or down on a quantum computer, as you ascend your way into the castle.
10 =====================
11 -[ Altitude 0 feet ]-
12 Your unicorn is waiting for you on the ground.
13 [up,down,quit]: u
14 You soar into the sky.
15 Running on the simulator.
16 {'0': 945, '1': 55}
18 =====================
19 -[ Altitude 55 feet ]-
20 Your unicorn is floating gently above the ground.
21 [up,down,quit]: u
22 You soar into the sky.
23 Running on the simulator.
24 {'0': 886, '1': 114}
26 =====================
27 -[ Altitude 114 feet ]-
28 Your unicorn is hovering just above the evergreen sea of trees.
29 [up,down,quit]:
In the game, the player is flying a unicorn with the goal being to reach the castle in the clouds. The player can choose to fly up or down at each round. When the player chooses to fly, we need to
determine how much altitude the unicorn gains or loses. Now, if we simply add/subtract a constant value of 100 at each round, the game could certainly be played, but it wouldn’t be very interesting.
Instead, at each round, we calculate a random value corresponding to how close the unicorn should be from the goal, and use the error rate returned from the measurements of the qubit to determine the
resulting altitude gained or lost.
In effect, we’re using quantum physics, with an electron to be specific, as a qubit and turning it into a random number generator!
Let’s see the code to understand how it works.
Setting up our References
Since we’re using IBM Q and QisKit to interact with a quantum computer, we’ll include the following references at the top of our game.
1 import math
2 import qiskit
3 from qiskit import ClassicalRegister, QuantumRegister, QuantumCircuit
4 from qiskit import IBMQ
5 from configparser import RawConfigParser
The Real Quantum Computer
QisKit allows us to run our quantum program on a simulator or on a physical quantum computer at IBM Q. We can wrap this logic in a function to let us easily switch between the simulator and the real
quantum computer.
1 # Selects the environment to run the game on: simulator or real
2 device = 'sim';
4 def run(program, type, shots = 100):
5 if type == 'real':
6 if not run.isInit:
7 # Setup the API key for the real quantum computer.
8 parser = RawConfigParser()
9 parser.read('config.ini')
10 IBMQ.enable_account(parser.get('IBM', 'key'))
11 run.isInit = True
13 # Set the backend server.
14 backend = qiskit.providers.ibmq.least_busy(provider.backends(simulator=False))
16 # Execute the program on the quantum machine.
17 print("Running on", backend.name())
18 job = qiskit.execute(program, backend)
19 return job.result().get_counts()
20 else:
21 # Execute the program in the simulator.
22 print("Running on the simulator.")
23 job = qiskit.execute(program, qiskit.Aer.get_backend('qasm_simulator'), shots=shots)
24 return job.result().get_counts()
Unicorns That Can Fly
At each round, the player can fly the unicorn higher or lower. We’ll need to print out the status of the unicorn, based upon its current altitude. We can do this with a simple helper function that
simply looks at the given altitude range and outputs text accordingly.
1 # Get the status for the current state of the unicorn.
2 def status(altitude):
3 if altitude == 0:
4 return 'Your unicorn is waiting for you on the ground'
5 elif altitude <= 100:
6 return 'Your unicorn is floating gently above the ground'
7 elif altitude <= 200:
8 return 'Your unicorn is hovering just above the evergreen sea of trees'
9 elif altitude <= 300:
10 return 'Your unicorn is approaching the first misty cloud layer'
11 elif altitude <= 400:
12 return 'Your unicorn has soared through the misty pink clouds'
13 elif altitude <= 500:
14 return 'Your unicorn is well above the misty clouds'
15 elif altitude <= 600:
16 return 'You can barely see the evergreen sea of trees from this high up'
17 elif altitude <= 700:
18 return 'Your unicorn is soaring through the sky'
19 elif altitude <= 800:
20 return 'You can see the first glimpse of the golden castle gates just above you'
21 elif altitude <= 900:
22 return 'Your unicorn is nearly at the mystical castle gates'
23 elif altitude < 1000:
24 return 'Your unicorn swiftly glides through the mystical castle gate. You\'re almost there'
25 else:
26 return 'A roar emits from the crowd of excited sky elves, waiting to greet you'
Let The Game Begin
Now it’s time to get quantum! Let’s begin our main game loop. We’ll start, as usual, by creating our qubits.
1 # Begin main game loop.
2 while not isGameOver:
3 # Setup a qubit to represent the unicorn.
4 unicorn = QuantumRegister(1)
5 unicornClassic = ClassicalRegister(1)
6 program = QuantumCircuit(unicorn, unicornClassic);
In the above code, we’ve started a main loop so the game will continue rounds until the player wins or quits.
We’ll represent the player as a qubit. Therefore, we only need one qubit. Notice, we’re not using a variable to represent the state of the player - just a single qubit. We’ll use the spin measurement
to determine the actual state.
The Quantum Calculation
After setting up our main game loop, we’ll read input from the user to determine whether the player chooses to fly up or down. We can then apply a quantum operation to the qubit representing the
player, and use the resulting measurement as the player’s altitude.
1 # Calculate the amount of NOT to apply to the qubit, based on the percent of the new altitude from the goal.
2 frac = (altitude + modifier) / goal
3 if frac >= 1:
4 # The unicorn is at or beyond the goal, so just invert the 0-qubit to a 1-qubit for 100% of goal.
5 # Note: On a real quantum machine the error rate is likely to cause NOT(0) to not go all the way to 1, staying around 1=862 and 0=138, etc.
6 program.x(unicorn)
7 elif frac > 0:
8 # Apply a percentage of the NOT operator to the unicorn (qubit), corresponding to how high the unicorn is.
9 program.u3(frac * math.pi, 0.0, 0.0, unicorn)
11 # Collapse the qubit superposition by measuring it, forcing it to a value of 0 or 1.
12 program.measure(unicorn, unicornClassic);
Let’s go over the above code, as it’s the key to our game. Recall, we’re representing our player as a single unicorn. We’re not using a classic variable anywhere to represent the state of the player.
Instead, we’ll apply a degree of “NOT” operation to a qubit to alter its spin according to the altitude of the player. The resulting measurement of the electron’s spin, along with its inherent
uncertainty and error rate, will add the random factor to the resulting altitude.
First, one of the operations that can be performed on a qubit is an inversion. This also called a NOT gate. It allows us to change the value of a qubit from 0 to 1, or vice-versa, from 1 to 0.
Additionally, we can apply a fraction of the NOT operator to invert the qubit to a value between 0 and 1. We can do this with the u3 operator.
My, How High Can a Unicorn Fly
Now that have an understanding of how the state of our unicorn is being represented by a qubit and its corresponding measurement, we just need to determine how much of an inversion operation to apply
to the qubit.
We can use the following simple calculation, which is based upon the current altitude of the player, the direction the player wants to move, and the goal.
1 frac = (altitude + modifier) / goal
As you can see above, the first step is the determine the amount of inversion (i.e., the amount of the NOT gate) to apply to our qubit. We can do this by calculating the current altitude plus a
constant value if the player chooses to go up, or minus a constant value of the player chooses to go down, and divide the result by the goal altitude. This gives us a percentage of how far the player
is from the goal.
We can now apply the inverse operation to our qubit.
First, if the resulting fraction is greater than or equal to 1 (i.e., 100% of the goal or higher), it means that our unicorn is beyond the altitude of the castle, and thus, the player wins the game.
In this case, we can simply invert the qubit completely from a 0 to a 1, indicating 100% of the goal, and the game is over.
Otherwise, if the fraction is less than 1, the player still needs to fly higher to get in the castle. In this case, we don’t want to completely invert the qubit, but rather only invert a certain
fraction amount, so that the resulting measurement will fall somewhere between 0 and 1, with a degree of randomness (i.e., uncertainty). The code below achieves this.
1 if frac >= 1:
2 # The unicorn is at or beyond the goal, so just invert the 0-qubit to a 1-qubit for 100% of goal.
3 # Note: On a real quantum machine the error rate is likely to cause NOT(0) to not go all the way to 1, staying around 1=862 and 0=138, etc.
4 program.x(unicorn)
5 elif frac > 0:
6 # Apply a percentage of the NOT operator to the unicorn (qubit), corresponding to how high the unicorn is.
7 program.u3(frac * math.pi, 0.0, 0.0, unicorn)
Notice in the code above, we check if the player has reached the goal. If they have, we perform a complete invert operation on the qubit, moving the value from 0 to 1. Otherwise, we apply a percent
of the invert operator via the u3 operator. Using this operation, we multiply math.pi by the percent that the player is toward the goal. This results in the qubit being flipped a percentage of the
Multiple Shots
After we’ve operated on our qubit, it’s time to measure the result to determine the altitude of the unicorn. We can use the code below to perform a measurement of the qubit.
1 # Collapse the qubit superposition by measuring it, forcing it to a value of 0 or 1.
2 program.measure(unicorn, unicornClassic);
If the qubit were in superposition, the measurement operation collapses the wave function, resulting in a determining value of 0 or 1. We’re not actually using superposition in this case, however we
still perform a measurement to obtain the qubit’s value.
Now, due to the error rate in measuring qubits, we’re going to need to run multiple measurements and choose the most frequently resulting response. This is called running multiple “shots”. While the
quantum computer simulator might give us perfect measurement accuracy (for example, applying an invert operator to a qubit always results in a 0 to 1), a real quantum computer has a much higher error
We’ll use a goal height of 1000, and as such, we’ll use a corresponding number of shots of 1000. When all 1000 measurements result in a 1, we know the player has reached the goal. However, since we
have to account for an error rate on a real quantum computer, we’ll bump the number of shots up a little more by 125. This will offset any error to ensure the player can actually reach the goal.
1 goal = 1000 # Max altitude for the player to reach to end the game.
2 shots = goal + (125 if device == 'real' else 0) # Number of measurements on the quantum machine; when shots == goal, the player reached the goal; we include a buffer on physical quantum computers to account for natural error.
The Unicorn Wins the Game
Finally, we can run the quantum program and count the number of results that measure as 1. We can do this with the code below.
1 # Execute on quantum machine.
2 counts = run(program, device, shots)
3 print(counts)
5 # Set the altitude based upon the number of 1 counts in the quantum results.
6 altitude = counts['1'] if '1' in counts else 0
8 # Did the player reach the castle?
9 if altitude >= goal:
10 print('Congratulations! Your unicorn soars into the castle gates!')
11 isGameOver = True
First, notice that we call the run() method, including the number of measurements (i.e., shots) to make. We then count the number of measurements that result in a value of 1 to use as the resulting
altitude. If the result reaches or exceeds the goal, the player wins and the game is over.
The Future for Quantum Computers
We’ve just demonstrated some simple examples of programs that can be created and executed on a quantum computer. We’ve also explored the remarkable advancements in speed, computational power, and
long-distance effects that quantum physics can bring to computing. Given the potential, how soon could we expect to see quantum computers on our desktops and in everyday business practices?
There are a couple of roadblocks in the way before quantum computing can really begin to take-off.
A Problem with Error Rates
The first issue is the degree of error rates when working with quantum computers and their resulting measurements. Recall, many quantum computers today rely on measuring the spin of electrons. You
could also use photons or even the activation state of atoms. In either case, error rates creep in through forms of interference.
The first concept to understand is that of quantum coherence. This refers to the operations that we apply to a qubit and the resulting measurement, which subsequently collapses the wave function to
read the qubit’s value. In a perfect coherence state, we would obtain a perfectly reliable measurement with no interference.
Unfortunately, interference can greatly affect the operation of a qubit. Coherence can be degraded or entirely destroyed by outside effects, such as an unintended random measurement of the qubit from
an outside force. This process results in decoherence, causing errors to the resulting computations and measurements.
Decoherence occurs when outside forces interact with the wave function of a quantum system. This typically involves an outside force creating a measurement, thus collapsing the wave function and
quantum state prematurely.
Decoherence typically occurs from the fact that the spin of electrons can never be perfectly isolated from the world outside. This results in outside forces having effects on the electrons’ spin.
In order to counteract decoherence, technology will need to be employed to stabilize and isolate the electrons from outside interference as much as possible. One way of doing so, involves
supercooling the qubits to extremely low temperatures. This not only stabilizes the qubits to as close to a stationary state as possible for measurement, but it also aids in reducing or eliminating
the interference effect of heat.
The cores of the D-Wave quantum computers use supercooling to -460 degrees fahrenheit (about 0.02 degrees away from absolute zero).
Maximum Number of Qubits
In addition to reducing error rates, practical applications of quantum computing also require more processing power. For example, running Shor’s algorithm on a 2048-bit RSA key would require 4,096
Quantum computers already exist in the research laboratories of universities and tech giants. However, maximizing the number of qubits available, and keeping interference as low as possible, is a
challenging process.
IBM Q provides a 16-qubit processor for developers and researches through their IBM Q offering, and in addition, has built a 50-qubit quantum computer.
Google has created a quantum computer with 72-qubits.
DWAVE has built a system that can utilize over 2,000 qubits, although its method is through the use of quantum annealing, which is unable to perform logical operations on a qubit as a universal
quantum computer could do.
A Quantum Computer on Your Desk
With the number of qubits increasing, it might not be long before a viable quantum computer is available for practical computational use. However, some experts think that this might not occur for 10
years or possibly longer.
In a recent report from the U.S. National Academies on the Prospects for Quantum Computing, the findings mentioned that a quantum computer capable of breaking RSA 2048 (i.e., about 4096 qubits with a
low error rate) is unlikely within the next decade.
The report still acknowledges the benefit that quantum computing is bringing to drive foundational research that can potentially greatly advance our understanding of the universe. For this reason,
alone, quantum computing is beneficial. However, many researchers and industry professionals are betting on a true usable quantum computer to be available far sooner.
Download @ GitHub
The source code for this project is available on GitHub.
About the Author
This article was written by Kory Becker, software developer and architect, skilled in a range of technologies, including web application development, artificial intelligence, machine learning, and
data science.
Atkin, S. (2018, June 25). Demystifying Superdense Coding Medium. https://medium.com/qiskit/demystifying-superdense-coding-41d46401910e
Atkin, S. (2018, July 23). Untangling Quantum Teleportation. Medium. https://medium.com/qiskit/untangling-quantum-teleportation-919cbd673074
Berrah, N. PhD, Pancella, P. V., Humphrey, M. (2015). Quantum Physics. Alpha. https://www.oreilly.com/library/view/quantum-physics/9781615643622/
Dodson, B. (2013, March 11). Quantum Spooky Action at a Distance. New Atlas. https://newatlas.com/quantum-entanglement-speed-10000-faster-light/26587/
Dyakonov, M. (2018, November 15). The Case Against Quantum Computing. IEEE Spectrum. https://spectrum.ieee.org/computing/hardware/the-case-against-quantum-computing
Ramanan, A. (2018, February 6). Introduction to Quantum Computing. MSDN. https://blogs.msdn.microsoft.com/uk_faculty_connection/2018/02/06/introduction-to-quantum-computing/
Rothman, D. (2018). Artificial Intelligence By Example. Quantum Computers That Think. Packt Publishing. https://www.safaribooksonline.com/library/view/artificial-intelligence-by/9781788990547/
Silva, V. (2018). Practical Quantum Computing for Developers. Apress. https://www.springerprofessional.de/en/practical-quantum-computing-for-developers/16337302
Tibble, F. (2018, February 6). A Beginner’s Guide to Quantum Computing and Q#. MSDN. https://blogs.msdn.microsoft.com/uk_faculty_connection/2018/02/06/a-beginners-guide-to-quantum-computing-and-q/
National Academies of Sciences, Engineering, and Medicine. (2018). Quantum Computing: Progress and Prospects. Washington, DC: The National Academies Press. https://doi.org/10.17226/25196
Wootton, Dr. J. (2017, November 2). Making a Quantum Computer Smile. Medium. https://medium.com/qiskit/making-a-quantum-computer-smile-cee86a6fc1de
Sponsor Me | {"url":"https://www.primaryobjects.com/2019/01/07/an-introduction-to-quantum-computing/","timestamp":"2024-11-04T16:57:54Z","content_type":"text/html","content_length":"137258","record_id":"<urn:uuid:9afb716d-f5b7-47e7-92b7-db9eda7c036e>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00080.warc.gz"} |
K-Nearest Neighbors | FlowHunt
The k-nearest neighbors (KNN) algorithm is a non-parametric, supervised learning algorithm used for classification and regression tasks in machine learning. It is based on the concept of proximity,
assuming that similar data points are located near each other. KNN is a lazy learning algorithm, meaning it does not require a training phase and makes predictions by storing the entire training
dataset and using it to determine the class or value of new data points. The algorithm predicts the outcome for a test data point by identifying ‘k’ training data points closest to the test data and
infers the output based on these neighbors. This method is highly intuitive and mimics human perception strategies that rely on comparing new data with known examples.
How KNN Works
KNN operates by identifying the ‘k’ nearest data points to a given query point and using these neighbors to make a prediction. In classification tasks, the algorithm assigns the query point to the
class most common among its ‘k’ nearest neighbors, which is known as majority voting. Majority voting in KNN can be understood as “plurality voting” when dealing with multiple classes, where the
query point is assigned to the class with the highest count among its nearest neighbors, even if it does not constitute an absolute majority. In regression tasks, it predicts the value by averaging
the values of the ‘k’ nearest neighbors. The proximity and similarity principles, which are core to human perception, are also central to how KNN functions, as data points that are nearby in the
feature space are assumed to be more similar and thus likely to have similar outcomes.
Distance Metrics
To determine the nearest neighbors, KNN uses various distance metrics, which are critical for its performance:
• Euclidean Distance: The straight-line distance between two points in a multidimensional space, commonly used for continuous variables. It is the most common distance metric for KNN and is
particularly useful when the data is dense and continuous.
• Manhattan Distance: Also known as taxicab distance, it calculates the distance by summing the absolute differences between the coordinates of two points. It is useful in grid-like path scenarios
where movements are constrained to orthogonal directions.
• Minkowski Distance: A generalized form of both the Euclidean and Manhattan distances, parameterized by ‘p’. If p=1, it becomes the Manhattan distance, and if p=2, it becomes the Euclidean
distance. This distance metric provides flexibility depending on the value of ‘p’ chosen.
• Hamming Distance: Used for categorical data, it counts the number of differing bits between two binary vectors. This is particularly useful in binary classification problems where attributes have
binary values.
Choosing the Right ‘k’ Value
The parameter ‘k’ in KNN represents the number of neighbors to consider. Choosing the right ‘k’ is crucial:
• A small ‘k’ can lead to overfitting, where the model is too sensitive to the noise in the training data, capturing spurious patterns that do not generalize.
• A large ‘k’ can result in underfitting, where the model becomes too generalized and ignores important patterns, leading to poor predictive performance.
• Typically, ‘k’ is chosen through cross-validation and should be an odd number to avoid ties in classification decisions. The choice of ‘k’ can significantly impact the model’s accuracy and is
often determined empirically.
Advantages and Disadvantages
• Simple and Intuitive: Easy to understand and implement, making it a good choice for beginners. KNN’s simplicity lies in its straightforward approach of comparing test instances to stored
• No Training Phase: KNN does not require an explicit training phase, as it makes predictions using the stored dataset. This means the model can be updated simply by adding new data points to the
• Versatile: Can be used for both classification and regression tasks, and its application is broad across different domains. It is also useful for multi-label classification problems.
• Computationally Intensive: As it requires storing and comparing each new data point to the entire dataset, it can be slow and resource-intensive, especially with large datasets. The time
complexity of KNN is O(n), where n is the number of training samples.
• Sensitive to Outliers: The presence of outliers can significantly affect predictions, as these anomalous points can skew the results, particularly when ‘k’ is small.
• Curse of Dimensionality: In high-dimensional spaces, the algorithm’s performance can degrade as the distances between data points become less meaningful. As dimensionality increases, the volume
of the space increases, causing data to become sparse. This sparsity makes it difficult for KNN to find nearest neighbors effectively.
Use Cases
KNN is applied in various fields due to its simplicity and effectiveness:
• Recommendation Systems: Used in recommending products or content to users based on the preferences of similar users. KNN can help in identifying similar users or items by evaluating feature
• Pattern Recognition: Employed in handwriting recognition and other pattern recognition tasks, where it can classify images based on the similarity of pixel values.
• Data Imputation: Useful in filling missing values in datasets by estimating them based on similar data points, thus maintaining dataset integrity.
• Finance and Healthcare: Applied in stock market predictions, risk assessment, and medical diagnosis by analyzing similarities in historical data. In healthcare, it can predict patient diagnoses
by comparing symptoms against known cases.
Implementation in Python
KNN can be implemented using libraries like scikit-learn in Python. Here’s a basic example of using KNN for classification:
from sklearn.neighbors import KNeighborsClassifier
from sklearn.model_selection import train_test_split
from sklearn.datasets import load_iris
from sklearn.metrics import accuracy_score
# Load dataset
iris = load_iris()
X, y = iris.data, iris.target
# Split data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Initialize KNN classifier with k=3
knn = KNeighborsClassifier(n_neighbors=3)
# Fit the model
knn.fit(X_train, y_train)
# Make predictions
y_pred = knn.predict(X_test)
# Evaluate accuracy
accuracy = accuracy_score(y_test, y_pred)
print(f"Accuracy: {accuracy:.2f}")
K-Nearest Neighbors (KNN) in Scientific Research
K-Nearest Neighbors (KNN) is a fundamental algorithm used in various fields such as multimedia information retrieval, data mining, and machine learning, particularly in the context of large datasets.
One notable paper, “Approximate k-NN Graph Construction: a Generic Online Approach” by Wan-Lei Zhao et al., presents an effective method for both approximate k-nearest neighbor search and graph
construction. The paper demonstrates a dynamic and feasible solution for handling diverse data scales and dimensions, supporting online updates which are not possible in many existing methods. Read
Another significant contribution is the “Parallel Nearest Neighbors in Low Dimensions with Batch Updates” by Magdalen Dobson and Guy Blelloch. This work introduces parallel algorithms combining
kd-tree and Morton ordering into a zd-tree structure, optimized for low-dimensional data. The authors show that their approach is faster than existing algorithms, achieving substantial speedups with
parallel processing. The zd-tree uniquely supports parallel batch-dynamic updates, a first in k-nearest neighbor data structures. Read more.
Lastly, the paper “Twin Neural Network Improved k-Nearest Neighbor Regression” by Sebastian J. Wetzel explores a novel approach to k-nearest neighbor regression using twin neural networks. This
method focuses on predicting differences between regression targets, leading to enhanced performance over traditional neural networks and k-nearest neighbor regression techniques on small to
medium-sized datasets. Read more.
Explore FlowHunt's AI Glossary for a comprehensive guide on AI terms and concepts. Perfect for enthusiasts and professionals alike!
Explore machine learning with FlowHunt: Discover how AI learns from data, applications, benefits over traditional programming, and its lifecycle.
Explore dimensionality reduction techniques to simplify data, boost efficiency, and enhance machine learning models at FlowHunt.
Explore Gradient Boosting, a powerful machine learning technique for accurate regression and classification. Discover its advantages now! | {"url":"https://www.flowhunt.io/glossary/k-nearest-neighbors/","timestamp":"2024-11-03T05:55:13Z","content_type":"text/html","content_length":"90870","record_id":"<urn:uuid:a589de53-2a66-4ace-9cfd-85a369bcd6a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00025.warc.gz"} |
Welcome to the website of Salvatore ‘Salvo’ Tringali
I'm an associate professor of mathematics at the
School of Mathematical Sciences
Hebei Normal University
(HebNU), where I've been since Feb 2019. This term (FW 2019‒20), I'm teaching ‘Introduction to Number Theory’ (BS degree in mathematics), ‘Combinatorics’ (MS and PhD degrees in mathematics), and
‘Scientific Writing’ (MS and PhD degrees in mathematics and CS). Last term (SS 2018‒19), I taught ‘Graph theory’ (MS and PhD degrees in mathematics and CS). Next term (SS 2019‒20), I'll teach ‘Number
Theory’ (BS degree in mathematics) and ‘Graph theory’ (MS and PhD degrees in mathematics and CS).
I co-organize the
Open Seminar
, a seminar series primarily addressed to grads and senior undergrads that covers a wide range of topics in pure and applied mathematics. In my not-so-much spare time, I enjoy traveling and hiking,
singing karaoke, and playing volleyball and badminton with colleagues and students.
I'm mainly interested in
• number theory, with emphasis on additive combinatorics and elementary number theory;
• factorization theory, that is, the study of various questions naturally arising from the possibility or impossibility of extending the fundamental theorem of arithmetic to rings and, more
generally, monoids.
However, I also enjoy musing on other things, including classical analysis, discrete mathematics, functional equations, and group/semigroup theory.
Academic timeline.
Before arriving at HebNU, I moved around quite a bit. In some detail, I was
I hold a PhD in mathematics, obtained from the
Université de Lyon
Jean Monnet campus
under the advisorship of Alain Plagne and
François Hennecart
. I defended
my thesis
Some Questions in Combinatorial and Elementary Number Theory
’) in Nov 2013.
All this comes after an
Intra-European Fellowship
(from the
Marie Curie Actions
program) I was awarded in Dec 2010: The grant financed an individual project of 24 months (May 2011‒Apr 2013), carried out at the
Laboratoire Jacques-Louis Lions
UPMC, Sorbonne Universités
under the supervision of
Yvon Maday
and essentially aimed at a career shift (before moving to France, I had got a PhD in
computational electromagnetics
Università Mediterranea
under the guidance of
Giovanni Angiulli
and worked for many years in the same research field).
I have a younger sister, Margherita, and two wonderful nieces, Anjana (born May 9, 2017) and Yunha (born Feb 23, 2019), who live in Paris. I own a small farm, run by my parents, Angelo and Vittoria,
in the countryside of
my hometown
, in the far south of Italy. And I've been reading ‘
Il nome della rosa
’ at least once a year since I was 15.
Last update: Dec 12, 2019 | {"url":"https://imsc.uni-graz.at/tringali/index.html","timestamp":"2024-11-15T03:16:46Z","content_type":"text/html","content_length":"10018","record_id":"<urn:uuid:667ae3e4-8368-47c6-a16d-272ee64febb4>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00871.warc.gz"} |
Finding Groups with Maximum Betweenness Centrality via Integer Programming with Random Path Sampling
One popular approach to access the importance/influence of a group of nodes in a network is based on the notion of centrality. For a given group, its group betweenness centrality is computed, first,
by evaluating a ratio of shortest paths between each node pair in a network that are “covered” by at least one node in the considered group, and then summing all these ratios for all node pairs. In
this paper we study the problem of finding the most influential (or central) group of nodes (of some predefined size) in a network based on the concept of betweenness centrality. One known approach
to solve this problem exactly relies on using a linear mixed-integer programming (linear MIP) model. However, the size of this MIP model (with respect to the number of variables and constraints) is
exponential in the worst case as it requires computing all (or almost all) shortest paths in the network. We address this limitation by considering randomized approaches that solve a single linear
MIP (or a series of linear MIPs) of a much smaller size(s) by sampling a sufficiently large number of shortest paths. Some probabilistic estimates of the solution quality provided by our approaches
are also discussed. Finally, we illustrate the performance of our methods in a computational study.
View Finding Groups with Maximum Betweenness Centrality via Integer Programming with Random Path Sampling | {"url":"https://optimization-online.org/2022/10/finding-groups-with-maximum-betweenness-centrality-via-integer-programming-with-random-path-sampling/","timestamp":"2024-11-03T00:05:37Z","content_type":"text/html","content_length":"86811","record_id":"<urn:uuid:bfe050f3-97d2-423e-956b-81d7982754fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00275.warc.gz"} |
How to implement Binary Tree PreOrder Traversal in Java without Recursion? [Solved] Example Tutorial
This is my second article on how to implement binary tree pre-order traversal in Java. In the
first part
, I have shown how to traverse a binary tree with a pre-order traversal algorithm using Recursion, and in this article, you will learn how to implement pre-order traversal
without using Recursion
. You might be thinking that why do you need to learn the iterative solution if a recursive solution is possible and easy to write? Well, this type of question is mostly asked in Programming Job
interviews and Interviewers like to see how comfortable a candidate is with both
as well as using other data structures and iteration.
Apart from that, an Iterative solution is also often preferred in real code because a Recursive solution can always run into StackOverflow error when the number of nodes increases and Recursion gets
deeper and deeper. That's why an iterative solution is considered safe and if possible you should always use that for your production code.
Just to revise, pre-order is a depth-first algorithm, where the depth of the tree is first explored before traversing siblings. In
preOrder traversal
, first, node or root is visited, then left subtree, and right subtree, hence it is also known as
NLR (Node-Left-Right)
You might know that when you use Recursion, methods calls are stored in an internal Stack which unwinds itself when the algorithm reaches the base case.
When recursion is not allowed, you can use the Stack data structure to create the same effect, in fact, this is also a common technique to convert a recursive algorithm into an iterative one.
Btw, if you are not familiar with an essential data structure like Stack, Queue, Array, LinkedList, Binary tree and Hash table then I suggest you join a good course like
Data Structures and Algorithms: Deep Dive Using Java
on Udemy, it's one of the best course to learn and master data structure and Algorithms. Even if you know data structure, this can be used to further strengthen your knowledge.
Pre-order traversal in Java without recursion
There is no doubt that the recursive algorithm of pre-order traversal was readable, clear, and concise. You should always prefer such an algorithm over an iterative one, but if you have been asked to
solve this problem without recursion then you have no choice. In order to convert that recursive algorithm to an
one, you can use a Stack.
You start traversal by pushing the root node into Stack and loop until Stack is empty. In each iteration, you pop the last element from Stack and print its value. That means you visited it. Now, push
the left and right nodes into
if they are not null.
The order in which you push the left and right nodes is critical. You must first push the right subtree followed by the left subtree because in pre-order we visit the left subtree after the node.
In the next iteration when you call
it will return the left node because Stack is a LIFO data structure, to learn more about Stack, you can join a comprehensive course on data structures and algorithms like
Algorithms and Data Structures - Part 1 and 2
on Pluralsight.
Anyway, here are the exact steps of iterative pre-order traversal in Java:
1. Create an empty stack
2. Push the root into Stack
3. Loop until Stack is empty
4. Pop the last node and print its value
5. Push right and left node if they are not null
6. Repeat from steps 4 to 6 again.
And, here is the function to implement this algorithm
public void preOrderWithoutRecursion() {
Stack<TreeNode> nodes = new Stack<>();
while (!nodes.isEmpty()) {
TreeNode current = nodes.pop();
System.out.printf("%s ", current.data);
if (current.right != null) {
if (current.left != null) {
You can see that we are pushing the right node before the left node so that our program can process the left node before the right node as required by the pre-order algorithm. By the way, if you are
learning a binary tree from an interview perspective, you can check out
Data Structures in Java: An Interview Refresher
for more tree-based problems.
Java Program to traverse the binary tree using preOrder traversal
Here is our complete Java program to print binary tree nodes in the pre-order traversal. You start traversing from the root node by pushing that into Stack. We have used the same class which is used
in the earlier
binary tree tutorial
class is your binary tree and
is your individual nodes in the tree. This time, though I have moved the logic to create a sample binary tree inside the
class itself. This way, you don't need to create a new tree every time in
the main() method
If you want to learn more about Stack Data Structure, here is a diagram of the iterative pre-order traversal algorithm which will make the steps clearer:
And, if you don't mind learning from free resources then you can also check out this list of
Free Data Structure and Algorithm courses
, which includes data structure courses on all major programming languages like
, and
Iterative Pre-Order Traversal of Binary Tree in Java
And here is our complete code example which you can run in your favorite IDE like
. If you want you can also run from the command prompt if you have
setup already and Java is in the path.
import java.util.Stack;
* Java Program to traverse a binary tree
* using PreOrder traversal without recursion.
* In PreOrder the node value is printed first,
* followed by visit to left and right subtree.
* input:
* a
* / \
* b e
* / \ \
* c d f
* output: a b c d e f
public class Main {
public static void main(String[] args) throws Exception {
// construct the binary tree given in question
BinaryTree bt = BinaryTree.create();
// traversing binary tree in PreOrder without using recursion
System.out.println("printing nodes of a binary tree in preOrder
without recursion");
class BinaryTree {
static class TreeNode {
String data;
TreeNode left, right;
TreeNode(String value) {
this.data = value;
left = right = null;
boolean isLeaf() {
return left == null ? right == null : false;
// root of binary tree
TreeNode root;
* Java method to visit tree nodes in PreOrder traversal without recursion.
public void preOrderWithoutRecursion() {
Stack<TreeNode> nodes = new Stack<>();
while (!nodes.isEmpty()) {
TreeNode current = nodes.pop();
System.out.printf("%s ", current.data);
if (current.right != null) {
if (current.left != null) {
* Java method to create binary tree with test data
* @return a sample binary tree for testing
public static BinaryTree create() {
BinaryTree tree = new BinaryTree();
TreeNode root = new TreeNode("a");
tree.root = root;
tree.root.left = new TreeNode("b");
tree.root.left.left = new TreeNode("c");
tree.root.left.right = new TreeNode("d");
tree.root.right = new TreeNode("e");
tree.root.right.right = new TreeNode("f");
return tree;
printing nodes of a binary tree in preOrder using recursion
a b c d e f
That's all about
how to traverse a binary tree using PreOrder traversal in Java
. The order in which you visit the node left and right subtree is key because that order determines your traversal algorithm. If you visit the node first means it preOrder, if you visit the node
second means it's InOrder and when you visit the node last then it's called postOrder traversal.
Other binary tree and data structure tutorials you may like to explore
Thanks for reading this article so far. If you like this
binary search algorithms tutorial
then please share it with your friends and colleagues. If you have any questions or feedback then please drop a comment.
P. S.
- If you are looking for some Free Algorithms courses to improve your understanding of Data Structure and Algorithms, then you should also check the
free Data Structure in Java for beginners'
courses on Udemy. It's authored by an Algorithm expert and it's completely free of cost.
Lastly, what is your favorite Java coding exercise? Palindrom, Prime Number, Producer consumer problem , or this one? Do let me know in comments.
1 comment:
1. can you show inorder and postorder as well? | {"url":"https://www.java67.com/2016/07/binary-tree-preorder-traversal-in-java-without-recursion.html?m=0","timestamp":"2024-11-07T17:36:05Z","content_type":"application/xhtml+xml","content_length":"214885","record_id":"<urn:uuid:84fdf126-bd50-473a-b443-5b38c1d68820>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00064.warc.gz"} |
Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane
Assier, Raphael and Peake, Nigel (2012) Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane. IMA Journal of Applied
Mathematics, 77 (5). pp. 605-625.
Download (538kB)
This paper provides a review of important results concerning the Geometrical Theory of Diffraction and Geometrical Optics. It also reviews the properties of the existing solution for the problem of
diffraction of a time harmonic plane wave by a half-plane. New mathematical expressions are derived for the wave fields involved in the problem of diffraction of a time harmonic plane wave by a
quarter-plane, including the secondary radiated waves. This leads to a precise representation of the diffraction coefficient describing the diffraction occurring at the corner of the quarter-plane.
Our results for the secondary radiated waves are an important step towards finding a formula giving the corner diffraction coefficient everywhere.
Item Type: Article
Additional Information: This article is an invitation publication, following the award of the Lighthill-Thwaites prize, awarded for the 2012 Wave Motion paper.
Uncontrolled Keywords: geometrical theory of diffraction acoustic diffraction quarter-plane
MSC 2010, the AMS's Mathematics Subject Classification > 30 Functions of a complex variable
MSC 2010, the AMS's Mathematics Subject Classification > 35 Partial differential equations
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 44 Integral transforms, operational calculus
MSC 2010, the AMS's Mathematics Subject Classification > 78 Optics, electromagnetic theory
PACS 2010, the AIP's Physics and Astronomy Classification Scheme > 40 ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID MECHANICS > 43
Depositing User: Dr Raphael Assier
Date Deposited: 08 Dec 2014
Last Modified: 20 Oct 2017 14:13
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2207
Actions (login required) | {"url":"https://eprints.maths.manchester.ac.uk/2207/","timestamp":"2024-11-06T01:37:38Z","content_type":"application/xhtml+xml","content_length":"22139","record_id":"<urn:uuid:7552e89a-dcb4-4d28-8722-6b169019c5be>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00339.warc.gz"} |
An Improved EMD Method Based on Utilizing Certain Inflection Points in the Construction of Envelope Curves
Archives of Acoustics, 48, 3, pp. 389–401, 2023
An Improved EMD Method Based on Utilizing Certain Inflection Points in the Construction of Envelope Curves
The empirical mode decomposition (EMD) algorithm is widely used as an adaptive time-frequency analysis method to decompose nonlinear and non-stationary signals into sets of intrinsic mode functions
(IMFs). In the traditional EMD, the lower and upper envelopes should interpolate the minimum and maximum points of the signal, respectively. In this paper, an improved EMD method is proposed based on
the new interpolation points, which are special inflection points (SIPn) of the signal. These points are identified in the signal and its first (n − 1) derivatives and are considered as auxiliary
interpolation points in addition to the extrema. Therefore, the upper and lower envelopes should not only pass through the extrema but also these SIPn sets of points. By adding each set of SIPi (i =
1, 2, n) to the interpolation points, the frequency resolution of EMD is improved to a certain extent. The effectiveness of the proposed SIPn-EMD is validated by the decomposition of synthetic and
experimental bearing vibration signals.
Keywords: empirical mode decomposition (EMD); interpolation points; envelope curve; inflection points; rolling element bearing fault diagnosis
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Chu P.C., Fan C., Huang N. (2012), Compact empirical mode decomposition: An algorithm to reduce mode mixing, end effect, and detrend uncertainty, Advances in Adaptive Data Analysis, 4(3): 1250017,
doi: 10.1142/S1793536912500173.
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Egambaram A., Badruddin N., Asirvadam V.S., Begum T. (2016), Comparison of envelope interpolation techniques in empirical mode decomposition (EMD) for eyeblink artifact removal from EEG, [in:] 2016
IEEE EMBS Conference on Biomedical Engineering and Sciences (IECBES), pp. 590–595, doi: 10.1109/IECBES.2016.7843518.
Guo T., Deng Z. (2017), An improved EMD method based on the multi-objective optimization and its application to fault feature extraction of rolling bearing, Applied Acoustics, 127: 46–62, doi:
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Huang N.E. et al. (2003), A confidence limit for the empirical mode decomposition and Hilbert spectral analysis, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and
Engineering Sciences, 459(2037): 2317–2345, doi: 10.1098/rspa.2003.1123.
Kopsinis Y., McLaughlin S. (2007), Investigation and performance enhancement of the empirical mode decomposition method based on a heuristic search optimization approach, [in:] IEEE Transactions on
Signal Processing, 56(1): 1–13, doi: 10.1109/TSP.2007.901155.
Kopsinis Y., McLaughlin S. (2008), Improved EMD using doubly-iterative sifting and high order spline interpolation, EURASIP Journal on Advances in Signal Processing, 2008(1): 128293, doi: 10.1155/
Lei Y., Lin J., He Z., Zuo M.J. (2013), A review on empirical mode decomposition in fault diagnosis of rotating machinery, Mechanical Systems and Signal Processing, 35(1–2): 108–126, doi: 10.1016/
Li H., Qin X., Zhao D., Chen J.,Wang P. (2018), An improved empirical mode decomposition method based on the cubic trigonometric B-spline interpolation algorithm, Applied Mathematics and Computation,
332: 406–419, doi: 10.1016/j.amc.2018.02.039.
Li H., Wang C., Zhao D. (2015a), An improved EMD and its applications to find the basis functions of EMI signals, Mathematical Problems in Engineering, 2015: 150127, doi: 10.1155/2015/150127.
Li Y., Xu M., Wei Y., Huang W. (2015b), An improvement EMD method based on the optimized rational Hermite interpolation approach and its application to gear fault diagnosis, Measurement, 63: 330–345,
doi: 10.1016/j.measurement.2014.12.021.
Pegram G.G.S., Peel M.C., McMahon T.A. (2008), Empirical mode decomposition using rational splines: An application to rainfall time series, Proceedings of the Royal Society A: Mathematical, Physical
and Engineering Sciences, 464(2094): 1483–1501, doi: 10.1098/rspa.2007.0311.
Qin S.R., Zhong Y.M. (2006), A new envelope algorithm of Hilbert–Huang Transform, Mechanical Systems and Signal Processing, 20(8): 1941–1952, doi: 10.1016/j.ymssp.2005.07.002.
Rilling G., Flandrin P. (2007), One or two frequencies? The empirical mode decomposition answers, [in:] IEEE Transactions on Signal Processing, 56(1): 85–95, doi: 10.1109/TSP.2007.906771.
Rilling G., Flandrin P., Gonçalvès P. (2003), On empirical mode decomposition and its algorithms, [in:] IEEE-EURASIP workshop on nonlinear signal and image processing, 3(3): 8–11, Grado.
Shu L., Deng H., Liu X., Pan Z. (2022), A Comprehensive working condition identification scheme for rolling bearings based on modified CEEMDAN as well as modified hierarchical amplitude-aware
permutation entropy, Measurement Science and Technology, 33(7): 075111, doi: 10.1088/1361-6501/ac5b2c.
Singh P., Joshi S.D., Patney R.K., Saha K. (2014), Some studies on nonpolynomial interpolation and error analysis, Applied Mathematics and Computation, 244: 809–821, doi: 10.1016/j.amc.2014.07.049.
SKF (n.d.), Bearing Select, https://www.skfbearingselect.com/#/bearing-selection-start (access date: 21.02.2023).
Smith W.A., Randall R.B. (2015), Rolling element bearing diagnostics using the Case Western Reserve University data: A benchmark study, Mechanical Systems and Signal Processing, 64–65: 100–131, doi:
Sun Y., Li S., Wang X. (2021), Bearing fault diagnosis based on EMD and improved Chebyshev distance in SDP image, Measurement, 176: 109100, doi: 10.1016/j.measurement.2021.109100.
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and Signal Processing (ICASSP), pp. 4144–4147, doi:
Wang J., Du G., Zhu Z., Shen C., He Q. (2020), Fault diagnosis of rotating machines based on the EMD manifold, Mechanical Systems and Signal Processing, 135: 106443, doi: 10.1016/j.ymssp.2019.106443.
Wang J.-L., Li Z.-J. (2013), Extreme-point symmetric mode decomposition method for data analysis, Advances in Adaptive Data Analysis, 5(3): 1350015, doi: 10.1142/S1793536913500155.
Wang Z., Yang J., Guo Y. (2022), Unknown fault feature extraction of rolling bearings under variable speed conditions based on statistical complexity measures, Mechanical Systems and Signal
Processing, 172: 108964, doi: 10.1016/j.ymssp.2022.108964.
Wu Z., Huang N.E. (2009), Ensemble empirical mode decomposition: A noise-assisted data analysis method, Advances in Adaptive Data Analysis, 1(1): 1–41, doi: 10.1142/S1793536909000047.
Xu Z., Huang B., Li K. (2010), An alternative envelope approach for empirical mode decomposition, Digital Signal Processing, 20(1): 77–84, doi: 10.1016/j.dsp.2009.06.009.
Yang L., Yang Z., Zhou F., Yang L. (2014), A novel envelope model based on convex constrained optimization, Digital Signal Processing, 29(1): 138–146, doi: 10.1016/j.dsp.2014.02.017.
Yeh J.-R., Shieh J.-S., Huang N.E. (2010), Complementary ensemble empirical mode decomposition: A novel noise enhanced data analysis method, Advances in Adaptive Data Analysis, 2(2): 135–156, doi:
Yuan J., He Z., Ni J., Brzezinski A.J., Zi Y. (2013), Ensemble noise-reconstructed empirical mode decomposition for mechanical fault detection, Journal of Vibration and Acoustics, 135(2): 021011,
doi: 10.1115/1.4023138.
Yuan J., Xu C., Zhao Q., Jiang H., Weng Y. (2022), High-fidelity noise-reconstructed empirical mode decomposition for mechanical multiple and weak fault extractions, ISA Transaction, 129(Part B):
380–397, doi: 10.1016/j.isatra.2022.02.017.
Zhao D., Huang Z., Li H., Chen J., Wang P. (2017), An improved EEMD method based on the adjustable cubic trigonometric cardinal spline interpolation, Digital Signal Processing, 64: 41–48, doi:
Zheng J., Cao S., Pan H., Ni Q. (2022), Spectral envelope-based adaptive empirical Fourier decomposition method and its application to rolling bearing fault diagnosis, ISA Transactions, 129(Part B):
476–492, doi: 10.1016/j.isatra.2022.02.049. | {"url":"https://acoustics.ippt.gov.pl/index.php/aa/article/view/3654","timestamp":"2024-11-11T01:36:46Z","content_type":"text/html","content_length":"33716","record_id":"<urn:uuid:11cdeff0-5391-4b01-82fd-cd257c64113c>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00453.warc.gz"} |
Integer Literal Type and Overflow In C++
I want to print the largest number that unsigned int can represent, which is 2^32 - 1. I use the following code:
unsigned int n = 2147483647 + 1 + 2147483647;
std::cout << n << std::endl;
However, I get an overflow warning from clangd:
overflow in expression; result is -2147483648 with type ‘int’
According to post here, the expression 2147483647 + 1 overflows because both number are treated as int type. The compiler will choose from int, long, long long in that order to find if a type can fit
the number. The maximum positive integer an int type can represent is 2147483647. So the type chosen for 2147483647 is int, not long or long long. Thus we get the overflow warning when we add 1 to
it. If you use 2147483648 directly, the compiler will choose long type for this number, which will also not overflow.
The variable type on the left side of the assignment does not impact the choose integer type for the variable on the right side. It only decides whether type conversion will happen when assigning the
result to this variable. In this case, although we want to the assign the final result to an unsigned int type, 2147483647 is not converted to unsigned int automatically.
To fix issue, we can explicitly tell the compiler that we want an unsigned int instead of int:
unsigned int n = (unsigned int)2147483647 + 1 + 2147483647;
// or use the following
unsigned int n = 2147483647u + 1 + 2147483647;
By converting one number to unsigned int, the other number will also be elevated to unsigned int and no overflow occurs in this case. | {"url":"https://jdhao.github.io/2021/09/03/cpp_integer_overflow/","timestamp":"2024-11-14T17:30:13Z","content_type":"text/html","content_length":"44788","record_id":"<urn:uuid:e7257ae3-282d-47dd-9ae7-86a922495c95>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00652.warc.gz"} |
Virtual Activity: Projectile Motion
There are 4 parts to this activity. Use the controls in the simulation to fire the projectiles (by pushing the red firing button) at various angles and with various initial speeds. Don't forget that
the definition of the "range" is the distance traveled in the x-direction to get back down to the original firing height. The trajectory shown in the simulation starts at the base of the cannon.
You can use the tools (located at the top of the simulation) to make any measurements you need. Record all of your work in your lab notebook, including the calculations you make and results of the
Activity #1:
a. Calculate the range of a projectile with an initial speed of 16 m/s and a firing angle of 35° when the cannon is on the ground. After you determine the answer, adjust the parameters to those
values in the simulation and fire the cannon. Record your results and the results given by the simulation.
b. Change the mass of the projectile in the simulation by choosing several different objects as projectiles. Do you think this change will have any effect on the range? Fire the cannon with a
different mass. Does this change have any effect? Why or why not?
Activity 2:
a. Calculate the maximum height of a projectile fired at 8 m/s and a 50° firing angle. After you determine the answer, adjust the parameters in the simulation to these new values and fire the cannon.
Record your results and the results given by the simulation.
b. Examine the structure of the equations you used to determine by what factor the maximum height would change if you doubled the initial speed. Check your thinking by changing the initial speed on
the simulation to 16 m/s. Record your results.
Activity 3:
Calculate the ranges for a projectile with an initial speed of 15 m/s shot at a 40° angle and the same projectile shot at a 50° angle. How do the results compare? Use the simulation to check your
work. Record your findings. Try the same activity with angles of 25° and 65°. What can be said about firing a projectile at complementary angles?
Activity 4:
Repeat activity 3, this time finding the total horizontal distance traveled when the cannon raised 10 m off the ground. This distance is no longer referred to as the range (because Δy ≠ 0), and you
cannot use the range equation to find it. Instead, you simply need to use the basic kinematic equations for projectiles. Compare and contrast your results between activity 4 and activity 3. Explain
any differences. | {"url":"https://physics-prep.com/index.php/projectiles-activity","timestamp":"2024-11-13T07:36:13Z","content_type":"text/html","content_length":"20473","record_id":"<urn:uuid:f573e643-73f5-48f0-8ccd-99fa64a3d760>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00794.warc.gz"} |
Fencing Blocks: Tips To Calculate (2024) | Eucarl Realty
Fencing Blocks: Tips To Calculate
What are the tips to calculate the number of fencing blocks required for a building?
A fence is a building that surrounds a space that is normally outdoors, such as a field, a yard, etc. It is usually made of wood, wire, and blocks, although it can take other shapes as well. On the
other hand, the computation of the materials required to construct a block fence is the topic of this article.
This article serves as guidance on how to determine how many fencing blocks you will need to build a fence. Our calculation is based on a 9-inch block fence and uses a model plot of land that is 120
feet by 60 feet for fencing purposes.
Tips: Calculating Amount of Fencing Blocks
We are measuring a fence that is approximately 120 feet long from the dimensions of our model land, and since there are two lengths, we will multiply by two to get a total of 240 feet. (120 x 2) =
For our 60-foot breadth, the same procedure would be followed. This is because 60 x 2 = 120 feet, or two breadths, are present.
In order to get a 360-foot perimeter line fencing, we would then add the two lengths (240 + 120) together.
The next step is to convert 360 feet to inches. A foot is 12 inches long, therefore we multiply 12 by 360 feet to get the total number of inches, or 4320.
We must divide 4320 by 18 inches to obtain the line numbers for the blocks that will form the fence’s 4320-inch perimeters. This is due to the fact that all blocks, whether they are 6 inches or 9
inches in length, are 18 inches long. There are roughly 240 lines of blocks when 4320 inches are split by 18 inches.
The entire line of blocks will then be multiplied by the number of coaches of blocks you desire based on your height. Assuming you are 7 feet tall, this translates to 9 coaches, which equals 240,
which is then multiplied by 9 to become 2160 blocks.
For our example, land fencing is all you need. But always keep in mind to purchase an extra 10% of blocks in case of any break.
Approximately 70 standard blocks should be covered by a bag of cement, speaking of that material. Our model fence will therefore require roughly 34 bags of cement in total.
Because they are necessary, pillars must be taken into account when constructing a fence. In this scenario, there would be 40 pillars, with each pillar being 6 blocks away from the one before it (240
divided by 6). For those 40 pillars, about 17 sacks of cement would be required.
About 15 sacks of cement would be required for the floor binding of the fence. It would take around 66 bags of cement in total to complete the project.
Additionally, the gate and the pillars that are taking up space in the fence need to be removed. Our model gate is 12 feet in length; if the gate columns or pillars are 14 inches each and there are
two of them holding the gate, that is 28 inches or 2; 4 feet plus 12 feet = 14.4 feet; roughly 15 feet.
In light of this, each pillar or column made of 9 inches of concrete is 9 inches square, which we will multiply by the previously mentioned 40 pillars. 30 feet plus 15 feet is 45 feet, thus 9 x 40 =
360 inches, or (30 feet + 15 feet = 45 feet) / 18 inches = 30 blocks multiplied by the number of coaches, or 30 x 9 = 270 blocks less 2160 blocks, or 1890 blocks altogether.
Materials Used for Fencing Blocks
Asides from block and cement, here are some other materials to be used;
• 40 lengths of 12mm iron rod since one pillar is equivalent to 1 length.
• 3 lengths of 16mm iron rod for the 2 big pillars for holding the gate.
• 90 planks of softwood for the framing of the fence
• 1 truck of granite to be used for the wall foundation
• 4 trucks of sharp sand
Related: Current prices of building blocks in Nigeria
The importance of fencing blocks in construction cannot be overemphasized. Tips for calculating the number of fencing blocks needed for the construction of the fence of an apartment have been
provided in this article. Are you in search of fencing blocks or this particular information on how to calculate the number needed? I hope this piece has answered your question. Thank you for reading
and do have a lovely day! | {"url":"https://eucarlrealty.com/fencing-blocks-tips-to-calculate/","timestamp":"2024-11-11T03:19:46Z","content_type":"text/html","content_length":"107293","record_id":"<urn:uuid:8e3c9455-aa4b-4ad3-9cc4-c5fac321f715>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00005.warc.gz"} |
seminars - On virtual embeddability between the mapping class groups of some surfaces
By the Birman-Hilden theory, the braid group on 2g strands is embedded in the mapping class group of the closed surface of genus 2g possibly with marked points.
In this talk, using some right-anlged Artin groups in the mapping class groups, we show that any finite index subgroup of the braid group on 2g+1 strands cannot be embedded in the mapping class group
of the closed surface of genus g with at most one marked point. | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=seminars&sort_index=speaker&order_type=asc&page=31&document_srl=788159&l=en","timestamp":"2024-11-04T17:41:02Z","content_type":"text/html","content_length":"45811","record_id":"<urn:uuid:cd862074-790a-4dad-86ca-30c40541573e>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00429.warc.gz"} |
Examples of the use of formulas in financial forecasts
Below you will find some examples of how to use formulas to automate financial forecasts.
Please refer to the specific documentations:
When planning, it can be useful to define variables that can and will be used later.
It is useful to create a parameter section with a posting date of January 1st, or another date that corresponds to the first day of the Budget. The parameter variables will then be used in the
following rows.
// 30%
costOfGoodsSold = 0.3
// 5%
interestRateDebit = 0.05
// 2%
interestRateCredit = 0.02
// 10 %
latePaymentPercentage = 0.1
This way, all parameters that can be set are displayed instantly. When a parameter is changed, the forecast is recalculated.
Repetitions or values per month
By using the repeat column, you can plan for the whole year on one recording row.
If however, the activity has seasonal variations, it is recommended to use a forecast by month. For each month, a row is created with the sales amount for the month.
If you wish to make forecasts automatic for several years, it is useful to insert the repetition "Y" in this row, so that the row of the year is also used for the next one.
Prices and sales quantities
When making a sales plan it will be easier to enter values using the quantity and price column. For example, a restaurant can enter the number of covers served per day, week or month and the program
automatically calculates the amount. By changing the number of covers or the price, the impact on liquidity and on the result for the year is instantly displayed.
Use of formulas with variables for growth
Use formulas, which are expressed in the Javascript language:
• The variable must be defined before being used, hence the line, in which it is defined, must be a date preceding the the line in which it is used.
• Thousands separators in numbers can't be used.
• The decimal separator is always the "." (Full stop)
• The names are different for uppercase / lowercase.
You can assign a value to a variable (this name can be freely chosen) and enter the variable name in the following lines to resume the value.
By changing the value assigned to the variable, all the lines, in which the variable is used, are automatically modified.
As an example, you may set the expected sales amount, proceeding with the following months, by using a formula to increase the amount.
• Create a "S" (sales) variable , by entering the following text in the Formula column.
The value 1000.00 will be inserted in the amount column
• In the following lines use the variable by simply inserting the variable name in the formula:
The value 1000.00 will be inserted in the amount column
• You can add 10% to the amount (multiplying by 1.1)
The value 1100.00 will be inserted in the amount column (The value of the Sales variable does not change)
• You can increase the value of S
The value 1200.00 will be inserted in the amount column (The value of the Sales variable will change)
• If the variable is used in the following line, there will be the new calculation
The value 1200.00 will be inserted in the amount column
You can also define a variable to define the percentage of growth.
• Percentage=1.1
• Amount=200
• In formulas you can use the variable name instead of the number
□ Sales*Percentage
□ Sales=Sales+Amount
Variables and repetitions
Let's say you expect the turnover to rise by 5% every month.
• Enter two lines where you assign the value to the variable, without putting any account with the start date of the year.
• Then create a line with the repetition where you enter the accounts and the formula
Each time the row is repeated the value of the variable S will be increased by 5% and consequently also the amount of the transaction. By simply changing the growth percentage, the forecast will be
In a row you can also insert multiple Javascript instructions, by separating them with a semicolon ";"
Sales=1000; P=5
Use of budget formulas
There are functions that allow you to access the balances and movements of the accounts for planning up to the row that is calculated.You can enter a formula that through the budgetTotal function,
recovers the value of the sales account "SALES" of the previous month.
credit( budgetTotal("SALES", "MP") )
The budgetTotal function takes account numbers and periods as arguments. An abbreviation can be used instead of the period.
• MP stands for previous month.
• QP stands for previous quarter.
Revenues show in Credit, therefore the value returned by the function will be negative and will not be accepted as an amount in the double entry accounting.
Therefore use the credit () function, which uses the negative value and turn it into a positive value.
Variables for monthly sales
Sales may vary for each month. In this case it is useful to use separate registrations for each month with the variable name.
sales_01 = 1000
cost_01 = sales_01 * costOfGoodsSold
sales_01 = 1100
Deferred payments
If you want a very precise liquidity plan, it is useful to separate the sales, which are paid immediately and those that are deferred.
One approach may be to record all sales as if they were paid cash, assigning the value of the monthly variable to the amount.
sales_01 = 1000
A registration is then entered, which reverses the sales that are deferred, calculating the amount with the formula.
sales_01 * latePaymentPercentage
A payment registration with the same formula will then be inserted for the following month.
Selling costs
There are costs that can be related to sales (cost of goods) or to other costs (social charges, in relation to wages).
Cost calculation with variables
If the sales are defined with variables, the sales costs can also be indicated as a percentage of the sales.
• You can define variable S for sales and variable C for the percentage of the cost.
• The formula for the calculation will be:
You can also use the same approach to calculate social security charges.
This formula can possibly be entered in a repetitive row.
Sales cost calculation with budget functions
When the costs are related to sales, the budget formula can also be used.
credit( budgetTotal("SALES", "MC") )*60/100
When the costs related to sales, "MC" stands for the current month.
This formula will return the sales value of the current month, turn them into a positive and multiply them by 60 and divide by 100.
The date must be beyond the sales registration dates, obviously.
It may be used it in a repeat row, inserting the month end date. Therefore, the costs of the sale will automatically be calculated based on the sales entered with the transaction.
The formula can be combined with variables.
• At the beginning of the year define the percentage of costs.
• Therefore, the formula C is used.
credit( budgetTotal("SALES", "MC") )*Cost/100
When you change the contribution percentage or any sale, the schedule will be updated automatically.
Sales commission calculation at the end of the year
At the end of the year, you calculate the commissions of 5% on the total net sales with this formula:
credit( budgetTotal("SALES", "YC") )*5/100
The bugetTotal function returns the movement of the sales account for the period "YC", current year. With the credit () function, the amount is turned into a positive value and then multiplied by 5
and divided by 100.
If you make forecasts over several years, remember to insert the repetition "Y" in the row, so that the same formula will be also calculated for the previous year. As indicated above instead of
entering the 5 directly in the formula, you can assign it with a variable.
• Commission=5
• credit( budgetTotal("SALES", "YC") )*Commission/100
If the percentage changes from one year to the next, it is sufficient enter a transaction with the following year date that resets the variable for commissions.
Inflation with variables
If you want to forecast over several years, you can also take inflation into account.
• At the beginning of the planning, assign a base variable for prices and inflation (2%).
• When you use the Sales variable, you multiply it by the inflation rate
• At the beginning of the following year, with annual repetition, you increase the price base
Depreciation calculation
Thanks to the formulas, the calculation of depreciation can be automated.
If you change the value of your investments in planning, depreciation will be automatically recalculated. Make sure that the date of the depreciation calculation line has a date superior to that of
the investments. Date is generally 31st December .
Depreciation calculation on book value
To calculate the depreciation of the "EQUIPMENT" account, insert a line at the end of the year with the following formula and the debit and credit accounts appropriately set up to register the
The budgetBalance function returns the balance to that date. The amortization of 20% is then calculated on this.
Use the debit function, in case you think the asset account can go into credit.
Calculation of depreciation on the initial value
To calculate the initial value of the investments, it is necessary to use variables to remember the value of the initial investment.
If the depreciation is spread over 5 years, the formula will be inserted in the year-end depreciation line
The annual repetition "Y" and the end date, which corresponds to the date of the last installment of the amortization, will be inserted in the row to prevent the amortization from running on
For each investment you will have to create a variable and a specific row of depreciation. Numbers can also be entered in variable names.
Interest calculation
The budgetInterest( account, interest, startDate, endDate) function allows you to automatically calculate interest based on the actual use of an account.
The parameters are:
• Account
Whose movements are used to calculate interest, in case it will be the bank account or the loan.
• Interest
The interest rate in percentage.
If the value is positive, interest on the debit balances is calculated.
If the value is negative, interest on the credit balances is calculated..
• Initial date, which may also be an acronym.
• End date, which may also be an acronym.
• The returned value is the interest calculated for 365/365 days.
Interest expense on the bank account
To calculate the interest expense of 5%, insert a line with the end date of the quarter and the repetition "3ME", which contains the formula
budgetInterest( "Bank", -5, "QC")
The interest rate is negative, because "QC" means current quarter. The debit and credit accounts must be the usual ones for recording interest expense. If the interest decreases the bank account
balance will also be used in the registration. However, another account can be used if it is paid with another account.
It is important that the "3ME" repeat is used so that the date used will always be the last of the quarter.
To calculate the interest of the month use the abbreviation "MC"
budgetInterest( "Bank", -5, "MC")
Interest on the bank account
For interest income of 2%, use positive interest instead.
budgetInterest( "Bank", 2, "QC")
Interest on fixed-term loan accounts
For fixed-term loans, interest will be calculated and recorded on the specified date.
• Create an separate account for each loan.
Use the budgetInterest function indicating exactly the start and end dates .If the date is indicated as text, the notation "yyyy-mm-dd" should be used, then "2022-12-31"
• Use variables.
As indicated for depreciation, the loan amount can be assigned to a variable. The interest calculation will be done with a Javascript calculation formula,
□ Define the loan variable
□ 5% interest calculation, for 120 days.
Profit tax calculation
Profit is the total of the group's profit for the specified period.
To calculate a 10% profit tax, use the following formula.
• Use the the budgetTotal function parameter in the "Gr = Result" group, which indicates that instead of an account it has to calculate the movements for the group.
• MC, current month, is indicated as the period.
• The budgetTotal function will return a positive value if there is a loss and negative (credit) if there is a profit.
• the credit function takes only negative values into account, therefore if there is a loss the tax will be zero.
Payments with deferred or different deadlines
For deferred payments or with different deadlines, you can proceed in two ways:
• Use variables to which payment amounts are to be assigned.
Use the variable in question when recording the payment.
• Create customer or supplier accounts for different credit deadlines.
Other cases
Please tell us about your other requirements, so we can add more examples.
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The Stacks project
Lemma 35.9.5. Let $f : T \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $T$. For either the étale or Zariski topology, there are
canonical isomorphisms $R^ if_{small, *}(\mathcal{F}^ a) = (R^ if_*\mathcal{F})^ a$.
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Intro to rational expressions (article) | Khan Academy (2024)
Learn what rational expressions are and about the values for which they are undefined.
Want to join the conversation?
8 years agoPosted 8 years ago. Direct link to Jenn Vo's post “I have a question about #...”
I have a question about #5 under the Check your understanding section. So for the denominator in that fraction, can I use the method "the different of 2 squares" to factor it out to (x+2) (x-2)
and solve for x from there? Can you explain more about it? I didn't get the last part in the explanation. Thanks!
8 years agoPosted 8 years ago. Direct link to Mr. Brownridge's post “Difference refers to subt...”
Difference refers to subtraction. x^2+4 is a sum. Therefore, it is a "sum of two squares." If you graph the function you will see that it is an upward facing parabola with a y-intercept of 4.
It has no solutions. So the expression will never equal zero (unless we use a different set of numbers called complex numbers).
6 years agoPosted 6 years ago. Direct link to aaliyahmariecerveny's post “Why is number 5, _all rea...”
Why is number 5, all real numbers shouldn't it be +/- 2 since x^2=+4, factors out to (x+2)(x-2)?
6 years agoPosted 6 years ago. Direct link to Kim Seidel's post “The denominator is: x^2+4...”
The denominator is: x^2+4. You changed it into x^2-4.
x^2+4 is not factorable. Any real number squared will create a positive value. That positive value plus 4 creates an even larger positive value. There is no value that you can use for X that
would cause the denominator to become 0. This is why the answer is that the domain = all real numbers.
Hope this helps.
a year agoPosted a year ago. Direct link to Vishwa Patel's post “For Problem 5, why can’t ...”
For Problem 5, why can’t x= +- 2i ?
a year agoPosted a year ago. Direct link to Kim Seidel's post “We define domain and rang...”
We define domain and range using the set of real numbers. The domain in problem 5 is all real numbers. There is no value of x that makes the denominator = 0, so there are no values to exclude
from the domain.
You are asking about imaginary numbers. They are outside the set of real numbers, so they are no considered.
Hope this helps.
4 years agoPosted 4 years ago. Direct link to MatthewS's post “I don't have a good under...”
I don't have a good understanding of how exactly you find the domain, and what "all real numbers" means.
4 years agoPosted 4 years ago. Direct link to Victor's post “Domain means that you are...”
Domain means that you are trying to find all possible values of x. Domain's are usually written in this format: {xeR} where xeR means that for every real number, x is a solution. All real
numbers mean any number that exists, and they may be irrational, rational, negative, positive, etc. However, they cannot be undefinable values such as √-1, which is i in short. In order to
find the domain, you'll have to find what can't be in the denominator usually by factoring, and you'll be able to find out what x cannot be. If you have a specific question you'd like me to
walk you through, don't hesitate to ask!
a year agoPosted a year ago. Direct link to zunnunam's post “explain why domain of a r...”
explain why domain of a rational expression is all real numbers except for those that make the denominator equal to zero.
a year agoPosted a year ago. Direct link to Tanner P's post “When the denominator is 0...”
When the denominator is 0, you are dividing by 0. Division by 0 is undefined, so any values that cause that are not included in the domain.
Otherwise, you can divide by any other number as long as it isn’t 0.
4 years agoPosted 4 years ago. Direct link to Mrs. Head's post “Why do you use the term "...”
Why do you use the term "cancel"? I know a lot of teachers use it and that was what my teachers called it when I was in school. But is this really a mathematically correct term?
I spend a great deal of time correcting students who just want to "cancel" terms just because they are alike, without understanding that in order for terms to be removed from an expression you
have to use a mathematical operation, division or subtraction. Therefore terms can only be "divided out" or "subtracted out". Students will often times cross out or as you say "cancel out" terms
that are both in numerators when multiplying terms or both in the denominators. To help resolve this issue my students are only allowed to use correct mathematical operations when simplifying
expressions (divide out or subtract out).
a year agoPosted a year ago. Direct link to jher4900's post “what is the equation for ...”
what is the equation for a rational function?
a year agoPosted a year ago. Direct link to Kim Seidel's post “There is no single equati...”
There is no single equation for rational functions. Any function that involves fractions would be a rational function.
4 years agoPosted 4 years ago. Direct link to hwang's post “In rational expression wh...”
In rational expression why is domain all real number?
4 years agoPosted 4 years ago. Direct link to loumast17's post “rational expressions depe...”
rational expressions depend on the denominator for domain. If you know how to find vertical asymptotes and holes, those are what would limit the domain of a rational function. The only time a
rational function has a domain of all reals is if the denominator is just 1.
Thanks to Hecretary Bird for his correction. denominator just has to be a constant, other than 0 still though.
2 years agoPosted 2 years ago. Direct link to Yong Bakos's post “Do we have to be mindful ...”
Do we have to be mindful of the domain during intermediate steps of equation solving? For example if I have an equation and divide both sides by x, do I have to state that, in my final solution
of the equation, that x cannot be 0?
a year agoPosted a year ago. Direct link to Robin's post “I'm pretty sure?”
I'm pretty sure?
2 years agoPosted 2 years ago. Direct link to nestor.mendez's post “how do I know what is and...”
how do I know what is and isn't a real number you didn't really explain
a year agoPosted a year ago. Direct link to rainpaw10's post “There are other topics ab...”
There are other topics about this on Khan Academy that can explain it better, but basically, a real number is any number that is not an imaginary number like i. Pretty much any number that
you can think of is a real number! | {"url":"https://missouribusinc.com/article/intro-to-rational-expressions-article-khan-academy","timestamp":"2024-11-04T12:14:39Z","content_type":"text/html","content_length":"126571","record_id":"<urn:uuid:e7bfe96b-e0f5-4835-a6c8-2606f1b3c04b>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00530.warc.gz"} |
Reverse an Array in Java - 4 Methods with Code
Last Updated on April 25, 2023 by Prepbytes
In this article, we are going to discuss “How to reverse an array in Java”. You are given an array of integers. Your task is to output the array in reverse order. For instance, consider the input
array shown below.
So, the output for this array will be the elements in reverse order as shown below.
So, let us now discuss the different approaches to the problem of array reverse in Java.
Reverse an Array by Reverse Print
This approach to the problem of Reverse Array Java is pretty simple. All we need to do is traverse the array in reverse order and print the elements. The approach is shown below.
Initially, we have a pointer at the last element of the array. Now, this pointer will print the current element and move back every step till it reaches -1.
import java.util.*;
public class Main {
public static void reversePrint(int[] arr) {
for(int i=arr.length-1;i>=0;i--) {
System.out.print(arr[i] + " ");
public static void main(String[] args) {
int[] arr = {1,2,3,4,5};
Time Complexity: The time complexity of this method is O(N) as we are just printing the array in reverse and printing an array in reverse requires traversing the entire array of N elements.
Space Complexity (Auxiliary Space): The auxiliary space is O(1) as we have not used any extra space.
Reverse Array Using Auxiliary Array
Now, instead of just printing the array in reverse order, we will reverse the array in the memory i.e. the first element will become the last element and the last element will become the first
element, and so on.
This is shown below.
So, the changes are made to the same array. This can be achieved using an auxiliary array. So, we will create an array that is of the same size as that of the input array as shown below.
We can see that we have created an auxiliary array that is of the same size as that of the input array, arr. All the values in the auxiliary array will initially be 0 automatically (property of
arrays in Java).
Now, we will iterate through the back of the input array and the beginning of the auxiliary array.
We have a pointer i at the end of the input array and a pointer j at the start of the auxiliary array as shown below.
Now, aux[j] = arr[i] i.e. put the ith element of the input array at the jth index of the auxiliary array and keep incrementing the j pointer and decrementing the ith pointer till i >= 0 && j <
Now, we have the reversed array in the auxiliary array as shown below.
Now, we will fill the input array with the values of the corresponding index of the auxiliary array. So, let us say we have a pointer i at index 0.
Now, do arr[i] = aux[i] and keep incrementing the pointer till i < arr.length.
So, the array has been reversed.
Now that we have understood this procedure for the Reverse Array Java problem, let us write the code for the same.
import java.util.*;
public class Main {
public static void reverseUsingAuxArray(int[] arr) {
int[] aux = new int[arr.length];
int i = arr.length -1;
int j = 0;
//fill the reverse of arr
//in aux array
while(i >= 0) {
aux[j] = arr[i];
//fill back the array arr
for(i=0;i<arr.length;i++) {
arr[i] = aux[i];
public static void display(int[] arr) {
for(int i=0;i<arr.length;i++) {
System.out.print(arr[i] + " ");
public static void main(String[] args) {
int[] arr = {1,2,3,4,5};
Time Complexity:
The time complexity of this approach is O(N) as we have to traverse the arrays of size N twice.
Space Complexity: The space complexity of this approach is O(N). This is because we have used an auxiliary array of size N.
Reverse Array without Auxiliary Array
We used an auxiliary array to reverse the given array in the previous method. Now, we will reverse our input array without using the auxiliary array.
Consider the array and the pointers i and j as shown below.
So, we have kept the pointer i at the index 0 and the pointer j at arr.length -1 i.e. 5 – 1 = 4.
Now, we will swap the ith value with the jth value as shown below.
Now we will increment the i pointer and decrement the j pointer.
This process will continue till i < j as shown below.
Now, since i is not less than j, we stop.
The array is reversed. So, here we have reversed the array in the memory without using an auxiliary array.
Now that we have understood the procedure, let us write the code for the same.
import java.util.*;
public class Main {
public static void reverse(int[] arr) {
int i = 0;
int j = arr.length -1;
while(i < j) {
int temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
public static void display(int[] arr) {
for(int i=0;i<arr.length;i++) {
System.out.print(arr[i] + " ");
public static void main(String[] args) {
int[] arr = {1,2,3,4,5};
Time Complexity: The time complexity of this approach is O(N) as we have to traverse the array.
Space Complexity (Auxiliary Space): The auxiliary space used is O(1) as we have not used any extra space.
Reverse Array Using Collections.reverse() Method
In Java, the method reverse() in the Class Collections is used to reverse collections like List, ArrayList, LinkedList, etc. So, the array can be reversed using this method by first converting it
into a List and then using the reverse() method of the Collections class.
The program to do the same is shown below.
import java.util.*;
public class Main {
public static void reverse(Integer[] arr) {
public static void main(String[] args) {
Integer[] arr = {1,2,3,4,5};
Time Complexity: The time complexity of this approach is O(N) as internally, the array will first be converted into list and all the elements will be traversed.
Space Complexity (Auxiliary Space): The auxiliary space is O(N) as internally, Java will use an auxiliary list and reverse it.
So, we have now solved the Reverse Array Java problem using 4 different methods. We hope that you have understood all the 4 methods and enjoyed the discussion. See you again at PrepBytes.
You can also checkout our post on:
How to Check Given Number is Armstrong or not
20 Must Practice basic Java Programs for beginners
Bubble Sort in Java
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Pump Head Calculations
The Krib Plumbing and Filtration [E-mail]
Pump Head Calculations
by swhite-at-bach.udel.edu (Stephen H White)
Date: Wed, 31 Mar 1993
Newsgroup: rec.aquaria
In article <27915-at-hacgate.SCG.HAC.COM> larisa-at-tcville.edsg.hac.com writes:
>Could someone please remind me of the formula which computes total
>equivalent vertical head based on measurements of vertical and horizontal
>runs? I seem to recall something like this: "a horizontal run takes as
>much work to pump through as a vertical run of 0.4 times its length."
>I need to find the equivalent vertical head of my overall plumbing. Also,
>if you remember how to compute pressure at arbitrary head based on a
>pressure measurement at a given head, feel free to include the info.
>Follow-ups to rec.aquaria unless you select otherwise.
>-- Larisa
These formulas may help:
HEAD = 2.31*pressure
head is in units of feet
pressure is in psi
Bernoulli's Equation can be used to determine the total head of a system:
Bernoulli's Equation:
pres(1)/spwt + vel(1)^2/2GR + EL(1) + WI ==== pres(2)/spwt + vel(2)^2/2GR +
EL(2) + WO + TFHL
pres(1) = pressure at point 1 [Pascals]
pres(2) = pressure at point 2 [Pascals]
spwt = specific weight of fluid, typically water [Fresh H2O=9782 N/m^3 -at-20degC]
vel(1) = fluid velocity at point 1 [m/s]
vel(2) = fluid velocity at point 2 [m/s]
EL(1) = Elevation at point 1 relative to a chosen datum(EL=0), [m]
EL(2) = Elevation at point 2 [m]
WI = work in; solving for this will produce total head required by a pump
WO = work out, usually ignored unless there are devices in system which
are storing energy, ie accumulators, etc...
GR = 9.806 m/s^2, the acceleration of gravity
TFHL = the total fluid head loss due to frictionary forces [m]
FHL(I) = FF*TEPL*VEL^2/(2GR*DIA)
FF = friction factor for ith section of pipe
TEPL = total equivalent pipe length for ith pipe section; this includes the
total straight pipe length plus equivalent pipe lengths from pipe
components such as elbows, reducers, etc... [m]
VEL = velocity through ith pipe section [m/s]
DIA = diameter pf ith pipe section (id) [m]
TFHL = FHL(1)+FHL(2)+...+FHL(n-1)+FHL(n) ---> total fluid head loss
Friction factors are determined through the use of a moody's diagram.
In order to determine the friction factor from the graph, the reynold's #
must be calculated, as well as the relative roughness of the pipe.
KIVIS=kinematic viscosity of fluid, use 1.3E-6 N/m^2 for freshwater -at-20C.
if RE<2000 then FF=64/RE
else use a moody diagram.
Most people won't have a moody diagram handy;)
therefore, if RE>2000 use a FF of 0.04 for calculations. This is a conservativevalue for FF for most applications.
RELATIVE ROUGHNESS, RR=0.00006/DIA(I) for steel pipe.
For copper pipe use the smooth pipe line on the moody diagram.
Volumetric Flow=AREA(I)*VEL(I)
All of this is very simple....NOT! ;)
Friction plays a very key role in plumbing applications. Once the total head
is calculated, (WI=Total Head, from bernoulli's EQN) the pump can be selected.
This is also a very complicated process, but for the sake of brevity, choose
a pump which will is rated for a head close to the calculated total head. Don't
oversize the pump too much, as this will result in a less efficent pump setup.
# #
# o___ Stephen H. White #
# _ *> / _ Internet (swhite-at-brahms.udel.edu) #
# (_) \ (_) University of Delaware #
# #
Plumbing and Filtration The Krib This page was last updated 29 October 1998 | {"url":"https://www.thekrib.com/Filters/pump-head.html","timestamp":"2024-11-11T04:33:20Z","content_type":"text/html","content_length":"6438","record_id":"<urn:uuid:6bc55469-cc61-4d7f-aba3-be4cfd601054>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00786.warc.gz"} |
[Solved] 82. Let X1, X2,... be independent continu | SolutionInn
82. Let X1, X2,... be independent continuous random variables with a common distribution function F and density
82. Let X1, X2,... be independent continuous random variables with a common distribution function F and density f = F
, and for k 1 let Nk = min{n k: Xn = kth largest of X1,..., Xn}
(a) Show that P{Nk = n} = k−1 n(n−1), n k.
(b) Argue that fXNk
(x) = f (x)(F¯
Fantastic news! We've Found the answer you've been seeking! | {"url":"https://www.solutioninn.com/study-help/theory-of-probability/82-let-x1-x2-be-independent-continuous-random-variables-with-1982801","timestamp":"2024-11-09T03:58:39Z","content_type":"text/html","content_length":"63206","record_id":"<urn:uuid:3bd5919a-1b9c-4886-ab99-936e5ac15519>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00838.warc.gz"} |
decimal number Archives | Fundamentals of Mathematics and Physics
In a previous post we began to discuss the idea of a basis in mathematics. The examples given in that post are finite-dimensional vector spaces, and in this post we are going to generalize them by
giving some examples of infinite-dimensional vector spaces. But before we do this, let’s play with some motivating examples not … Read more | {"url":"https://fomap.org/tag/decimal-number/","timestamp":"2024-11-11T14:45:11Z","content_type":"text/html","content_length":"52057","record_id":"<urn:uuid:a708959c-3187-4fca-9a98-a83b9c4b7936>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00011.warc.gz"} |
Project Euler Solution 66: Diophantine equation
Project Euler Problem 66: Diophantine equation gives us a simple looking quadratic equation which turns out to have quite interesting structural properties.
Consider quadratic Diophantine equations of the form:
$$ x^2 – Dy^2 = 1 $$
For example, when $D=13$, the minimal solution in $x$ is $649^2 – 13\cdot 180^2 = 1$.
It can be assumed that there are no solutions in positive integers when $D$ is square.
By finding minimal solutions in $x$ for $D = {2, 3, 5, 6, 7}$, we obtain the following:
□ $3^2 – 2\cdot2^2 = 1$
□ $2^2 – 3\cdot1^2 = 1$
□ $9^2 – 5\cdot4^2 = 1$
□ $5^2 – 6\cdot2^2 = 1$
□ $8^2 – 7\cdot3^2 = 1$
Hence, by considering minimal solutions in $x$ for $D \leq 7$, the largest $x$ is obtained when $D=5$.
Find the value of $D \leq 1000$ in minimal solutions of $x$ for which the largest value of $x$ is obtained.
What is missing from the problem statement but implicit in “Diophantine equations” is that $x$ and $y$ are both integers. For rational numbers, there would always be a solution. The equation $x^2 -D
y^2 = 1$ with $x, y \in \mathbb N$ will always have a solution if $y^2 = (x^2 - 1)/D$ is a perfect square. We can therefore just iterate through all $x$ and check whether the expression $(x^2 - 1)/D$
is a perfect square.
To check whether it is a square is rather easy to implement:
def is_square(number: int) -> bool:
floor = int(math.sqrt(number))
return floor**2 == number
And then we can write a solution like this:
def solution() -> int:
max_x = 0
max_d = 0
for d in range(1, 1001):
if is_square(d):
for x in tqdm(itertools.count(1), desc=f"d={d}"):
if is_square((x**2 - 1) / d):
if x > max_x:
max_x = x
max_d = d
return max_d
In the beginning that works pretty well:
d=2: 1it [00:00, 27413.75it/s]
d=3: 0it [00:00, ?it/s]
d=5: 7it [00:00, 140479.08it/s]
d=6: 3it [00:00, 65536.00it/s]
d=7: 6it [00:00, 148910.20it/s]
d=8: 1it [00:00, 31300.78it/s]
d=10: 17it [00:00, 383350.37it/s]
d=11: 8it [00:00, 204600.20it/s]
d=12: 5it [00:00, 148734.18it/s]
d=13: 647it [00:00, 1836072.18it/s]
d=14: 13it [00:00, 288497.10it/s]
d=15: 2it [00:00, 66576.25it/s]
d=17: 31it [00:00, 577881.88it/s]
However, there are a few values of $D$ for which this doesn't work really sensibly:
d=59: 528it [00:00, 1949465.24it/s]
d=60: 29it [00:00, 614317.25it/s]
d=61: 335159610it [03:13, 1728850.74it/s]
d=62: 61it [00:00, 1224174.85it/s]
d=63: 6it [00:00, 299593.14it/s]
See how for $D=61$ it took 3:13 minutes to find the solution because we had to go up to $x = 335\,159\,612$ to find it. And then there are more such numbers like that. We need something more clever.
Reading up on Wikipedia on the Diophantine equation points us to the special case of Pell's equation which is exactly the form that we have. In the article it gives a relation to the approximation of
square roots:
These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y.
In the section about solutions to the equation it says this:
Let $h_{i}/k_{i}$ denote the sequence of convergents to the regular continued fraction for $\sqrt {n}$. This sequence is unique. Then the pair $(x_{1},y_{1})$ solving Pell's equation and
minimizing x satisfies $x_1 = h_i$ and $y_1 = k_i$ for some $i$. This pair is called the fundamental solution. Thus, the fundamental solution may be found by performing the continued fraction
expansion and testing each successive convergent until a solution to Pell's equation is found.
This means that instead of having to iterate over all $x$, we can iterate over the convergents that we have programmed in Solution 65: Convergents of e. We combine this with the continued fractions
of square roots from Solution 64: Odd period square roots. It is nice to see how we can directly reuse bits from previous problems and it nicely shows the progression within the problems.
Using this insight we can take the root expansion from Solution 64 and create an iterator which just loops over the periodic part:
def square_root_fraction_expansion(number: int) -> Iterator[int]:
prefix, period = expand_root(number)
yield from prefix
while True:
yield from period
Then we use that with the convergent generator from Solution 65 and then iterate through all the convergents to find the minimal solution in $x$ for a given $D$:
def minimal_solution(d: int) -> int:
for x, y in convergents_series(square_root_fraction_expansion(number)):
if x == 1:
if x**2 - d * y**2 == 1:
return x
It is important to use the explicit equation with integers here. The function is_square from above will fail for numbers larger than about $10^{17}$ as the FP64 data type then lacks the precision.
Using the arbitrarily large integers makes this check possible.
To make sure that this really works, we verify that with a test using the examples from the problem statement:
def test_minimal_solution() -> None:
assert minimal_solution(2) == 3
assert minimal_solution(3) == 2
assert minimal_solution(5) == 9
assert minimal_solution(6) == 5
assert minimal_solution(7) == 8
This checks out, so we can build our solution on top of that:
def solution() -> int:
max_x = 0
max_d = 0
for d in range(1, 1001):
if is_square(d):
x = minimal_solution(d)
if x > max_x:
max_x = x
max_d = d
return max_d
This finds the solution within 34 ms, so that quite a difference to the brute force solution from above. This was a very interesting problem, and I have learned a lot while solving it! | {"url":"https://martin-ueding.de/posts/project-euler-solution-66-diophantine-equation/","timestamp":"2024-11-10T22:27:09Z","content_type":"text/html","content_length":"22771","record_id":"<urn:uuid:81d0bab5-e0d3-4c61-b8bd-3308505ee21b>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00764.warc.gz"} |
Volume of a Prism - Formula, Derivation, Definition, Examples - Grade Potential Sarasota, FL
Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important figure in geometry. The shape’s name is originated from the fact that it is created by taking into account a polygonal base and expanding its sides till it cross the opposing
This blog post will discuss what a prism is, its definition, different types, and the formulas for volume and surface area. We will also provide instances of how to utilize the data given.
What Is a Prism?
A prism is a 3D geometric shape with two congruent and parallel faces, called bases, that take the form of a plane figure. The additional faces are rectangles, and their count relies on how many
sides the identical base has. For instance, if the bases are triangular, the prism would have three sides. If the bases are pentagons, there will be five sides.
The characteristics of a prism are fascinating. The base and top both have an edge in parallel with the additional two sides, making them congruent to one another as well! This means that all three
dimensions - length and width in front and depth to the back - can be deconstructed into these four entities:
1. A lateral face (meaning both height AND depth)
2. Two parallel planes which constitute of each base
3. An illusory line standing upright across any given point on either side of this figure's core/midline—also known collectively as an axis of symmetry
4. Two vertices (the plural of vertex) where any three planes meet
Kinds of Prisms
There are three primary kinds of prisms:
• Rectangular prism
• Triangular prism
• Pentagonal prism
The rectangular prism is a common type of prism. It has six faces that are all rectangles. It matches the looks of a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism consists of two pentagonal bases and five rectangular sides. It seems a lot like a triangular prism, but the pentagonal shape of the base makes it apart.
The Formula for the Volume of a Prism
Volume is a measurement of the total amount of area that an thing occupies. As an important shape in geometry, the volume of a prism is very relevant in your learning.
The formula for the volume of a rectangular prism is V=B*h, where,
V = Volume
B = Base area
h= Height
Finally, considering bases can have all sorts of shapes, you will need to know a few formulas to determine the surface area of the base. However, we will go through that later.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we are required to look at a cube. A cube is a three-dimensional item with six faces that are all squares. The formula for the volume of
a cube is V=s^3, assuming,
V = Volume
s = Side length
Now, we will take a slice out of our cube that is h units thick. This slice will by itself be a rectangular prism. The volume of this rectangular prism is B*h. The B in the formula stands for the
base area of the rectangle. The h in the formula implies the height, that is how thick our slice was.
Now that we have a formula for the volume of a rectangular prism, we can use it on any kind of prism.
Examples of How to Use the Formula
Considering we have the formulas for the volume of a triangular prism, rectangular prism, and pentagonal prism, let’s put them to use.
First, let’s figure out the volume of a rectangular prism with a base area of 36 square inches and a height of 12 inches.
V=432 square inches
Now, let’s try one more problem, let’s calculate the volume of a triangular prism with a base area of 30 square inches and a height of 15 inches.
V=450 cubic inches
As long as you have the surface area and height, you will calculate the volume with no issue.
The Surface Area of a Prism
Now, let’s discuss about the surface area. The surface area of an object is the measurement of the total area that the object’s surface occupies. It is an crucial part of the formula; consequently,
we must understand how to find it.
There are a few different ways to work out the surface area of a prism. To calculate the surface area of a rectangular prism, you can utilize this: A=2(lb + bh + lh), where,
l = Length of the rectangular prism
b = Breadth of the rectangular prism
h = Height of the rectangular prism
To figure out the surface area of a triangular prism, we will employ this formula:
b = The bottom edge of the base triangle,
h = height of said triangle,
l = length of the prism
S1, S2, and S3 = The three sides of the base triangle
bh = the total area of the two triangles, or [2 × (1/2 × bh)] = bh
We can also utilize SA = (Perimeter of the base × Length of the prism) + (2 × Base area)
Example for Calculating the Surface Area of a Rectangular Prism
First, we will figure out the total surface area of a rectangular prism with the following information.
l=8 in
b=5 in
h=7 in
To figure out this, we will replace these numbers into the respective formula as follows:
SA = 2(lb + bh + lh)
SA = 2(8*5 + 5*7 + 8*7)
SA = 2(40 + 35 + 56)
SA = 2 × 131
SA = 262 square inches
Example for Calculating the Surface Area of a Triangular Prism
To compute the surface area of a triangular prism, we will work on the total surface area by following similar steps as earlier.
This prism will have a base area of 60 square inches, a base perimeter of 40 inches, and a length of 7 inches. Therefore,
SA=(Perimeter of the base × Length of the prism) + (2 × Base Area)
SA = (40*7) + (2*60)
SA = 400 square inches
With this data, you will be able to work out any prism’s volume and surface area. Test it out for yourself and observe how easy it is!
Use Grade Potential to Better Your Arithmetics Skills Now
If you're having difficulty understanding prisms (or any other math subject, contemplate signing up for a tutoring class with Grade Potential. One of our expert instructors can assist you study the
[[materialtopic]187] so you can ace your next test. | {"url":"https://www.sarasotainhometutors.com/blog/volume-of-a-prism-formula-derivation-definition-examples","timestamp":"2024-11-13T09:18:32Z","content_type":"text/html","content_length":"79062","record_id":"<urn:uuid:8f60676f-ff6f-48cd-8cc0-b401bd6a5150>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00194.warc.gz"} |
which phrase describes carbon 14 dating
Carbon-14 dating, also known as radiocarbon dating, is a method used to determine the age of a sample by examining the amount of carbon-14 remaining against its known half-life.
The Science Behind Carbon-14 Dating
Carbon-14 is a radioactive isotope of carbon, meaning it has an unstable nucleus that decays over time. The half-life of carbon-14 is approximately 5,730 years. This means that after 5,730 years,
half of the original amount of carbon-14 in a sample will have decayed.
How Does Carbon-14 Dating Work?
Living organisms constantly take in carbon from the environment. This carbon includes a small amount of carbon-14. When an organism dies, it no longer takes in carbon. The carbon-14 in the organism's
remains begins to decay, and the amount of carbon-14 remaining can be used to determine the time since death.
Which Phrase Best Describes Carbon-14 Dating?
The best phrase to describe carbon-14 dating is "a radiometric dating method." Here's why:
• Radiometric refers to the use of radioactive isotopes to determine the age of a sample. Carbon-14 dating relies on the decay of carbon-14, a radioactive isotope.
• Dating method simply states the purpose of the technique, which is to determine the age of a sample.
Other Relevant Phrases:
While "radiometric dating method" is the most accurate and concise phrase, other phrases can also be used to describe carbon-14 dating, including:
• Radiocarbon dating: This is the most common alternative name for the method.
• Carbon-14 analysis: This emphasizes the specific isotope used.
• Age determination method: This focuses on the purpose of the technique.
Ultimately, the best phrase to use depends on the context of your discussion. | {"url":"http://dilagox.com/post/which-phrase-describes-carbon-14-dating","timestamp":"2024-11-04T21:42:57Z","content_type":"text/html","content_length":"77872","record_id":"<urn:uuid:65896731-b91d-4e8b-b336-c760471c75df>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00698.warc.gz"} |
Design elements - Solid geometry | Mathematics Symbols | Solid geometry - Vector stencils library | Geometrical Figures
The vector stencils library "Solid geometry" contains 15 shapes of solid geometric figures.
"In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space - for practical purposes the kind of space we live in. It was developed following the
development of plane geometry. Stereometry deals with the measurements of volumes of various solid figures including cylinder, circular cone, truncated cone, sphere, and prisms.
The Pythagoreans had dealt with the regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonists. Eudoxus established their measurement, proving the pyramid and
cone to have one-third the volume of a prism and cylinder on the same base and of the same height, and was probably the discoverer of a proof that the volume of a sphere is proportional to the cube
of its radius." [Solid geometry. Wikipedia]
The shapes example "Design elements - Solid geometry" was created using the ConceptDraw PRO diagramming and vector drawing software extended with the Mathematics solution from the Science and
Education area of ConceptDraw Solution Park.
ConceptDraw PRO extended with Mathematics solution from the Science and Education area is a powerful diagramming and vector drawing software that offers all needed tools for mathematical diagrams
designing. Mathematics solution provides 3 libraries with predesigned vector mathematics symbols and figures: Solid Geometry Library, Plane Geometry Library and Trigonometric Functions Library.
The vector stencils library "Solid geometry" contains 15 shapes of solid geometric figures.
Use these shapes to draw your geometrical diagrams and illustrations in the ConceptDraw PRO diagramming and vector drawing software extended with the Mathematics solution from the Science and
Education area of ConceptDraw Solution Park.
ConceptDraw PRO diagramming and vector drawing software extended with Mathematics solution from the Science and Education area is the best for creating: mathematical diagrams, graphics, tape diagrams
various mathematical illustrations of any complexity quick and easy. Mathematics solution provides 3 libraries: Plane Geometry Library, Solid Geometry Library, Trigonometric Functions Library.
The vector stencils library "Plane geometry" contains 27 plane geometric figures.
Use these shapes to draw your geometrical diagrams and illustrations in the ConceptDraw PRO diagramming and vector drawing software extended with the Mathematics solution from the Science and
Education area of ConceptDraw Solution Park.
ConceptDraw PRO is the beautiful design software that provides many vector stencils, examples and templates for drawing different types of illustrations and diagrams.
Mathematics Solution from the Science and Education area of ConceptDraw Solution Park includes a few shape libraries of plane, solid geometric figures, trigonometrical functions and greek letters to
help you create different professional looking mathematic illustrations for science and education.
Flowcharts are the best for visually representation the business processes and the flow of a custom-order process through various departments within an organization. ConceptDraw PRO diagramming and
vector drawing software extended with Flowcharts solution offers the full set of predesigned basic flowchart symbols which are gathered at two libraries: Flowchart and Flowcharts Rapid Draw. Among
them are: process, terminator, decision, data, document, display, manual loop, and many other specific symbols. The meaning for each symbol offered by ConceptDraw gives the presentation about their
proposed use in professional Flowcharts for business and technical processes, software algorithms, well-developed structures of web sites, Workflow diagrams, Process flow diagram and correlation in
developing on-line instructional projects or business process system. Use of ready flow chart symbols in diagrams is incredibly useful - you need simply drag desired from the libraries to your
document and arrange them in required order. There are a few serious alternatives to Visio for Mac, one of them is ConceptDraw PRO. It is one of the main contender with the most similar features and
ConceptDraw PRO diagramming and vector drawing software extended with Physics solution from the Science and Education area is a powerful software for creating various physics diagrams. Physics
solution provides all tools that you can need for physics diagrams designing. It includes 3 libraries with predesigned vector physics symbols: Optics Library, Mechanics Library and Nuclear Physics
The most easier way of creating the visually engaging and informative Sales Process Flowchart is to create the new ConceptDraw document and to use the predesigned vector symbols offered in 6
libraries of the Sales Flowchart Solution. There are more than six types of sales process flow diagrams, ConceptDraw PRO software is flowchart maker which include six collections of extended
flowchart symbols for effective diagramming. There are main collections of color coded flowchart symbols: Sales Department, Sales Steps, Sales Workflow, Sales Symbols and special set of Sales Arrows
and Sales Flowchart.
This sample was created in ConceptDraw PRO diagramming and vector drawing software using the Flowcharts solution from the Diagrams area of ConceptDraw Solution Park. A Flowchart is a graphically
representation of the process, algorithm or the step-by-step solution of the problem. The Flowcharts have one or more starting and ending points. The geometric figures on the Flowcharts represent the
steps of the process and are connected with arrows that show the sequence of the actions.
This template shows the Venn Diagram. It was created in ConceptDraw PRO diagramming and vector drawing software using the ready-to-use objects from the Venn Diagrams Solution from the "Diagrams" area
of ConceptDraw Solution Park.
This sample shows the Flowchart of the Sustainment and Disposal. A Flowchart is a graphically representation of the process that step-by-step lead to the solution the problem. The geometric figures
on the Flowchart represent the steps of the process and are connected with arrows that show the sequence of the actions. The Flowcharts are widely used in engineering, architecture, science,
analytics, government, politics, business, marketing, manufacturing, administration, etc.
A concept map is a way of representing relationships between ideas, images, or words. How to draw a Concept Map quick, easy and effective? ConceptDraw PRO offers the unique Concept Maps Solution from
the "Diagrams" Area that will help you! | {"url":"https://www.conceptdraw.com/examples/geometrical-figures","timestamp":"2024-11-07T02:43:10Z","content_type":"text/html","content_length":"69052","record_id":"<urn:uuid:80fcd444-cfa5-4fd8-beb0-7f7a13c8da0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00135.warc.gz"} |
Audio – Introduction to Options
Options Basics Part One
Okay, so this is going to be a very basic description of what options are about. An option is the right to buy or sell a set number of shares of stock at a certain price which is called “the strike
price.” There are calls and there are puts. First we’ll talk about just buying options because it’s simpler. So if you buy a call, for every contract you buy, you have the right to purchase 100
shares of stock at a set price which is whatever the strike price of the option is, by a certain time, which is whatever the expiration date of the option is.
So let’s choose IBM and let’s say it’s going for $100 per share. If you buy a call option, with a 105 strike a month out, that means that you have the right to buy 100 shares of IBM at $105 for each
option contract anytime up until the expiration date which is one month out.
Now, if you are buying a put that means you have the right to sell the stock at a certain strike price within a certain time frame. So let’s say IBM is at 100 and you buy one put contract at $95
strike price for a month out. What that means is that you have the right to sell 100 shares of IBM per contract at $95 per share, anytime between now and a month from now. And for that right, you’re
going to be paying the option premium and that goes both for calls and puts. If IBM is not a very volatile stock then the premium is going to be smaller. If you’re trading a different stock like
Tesla or Netflix for instance, the premium is going to be higher and the reason for that is pretty simple. It’s that there’s a bigger likelihood that the stock is going to be above or below the
strike at some time between now and the time the option expires.
So let’s go over that again, calls give you the right to call away the stock from somebody. In other words, buy the stock at a certain strike price within a certain time frame. Puts give you the
right to put the stock to somebody. In other words, sell the stock to somebody at a certain strike price within a certain time frame.
Buying options is the simplest way to go and a lot of times the best way to go. When you buy, you’ll be paying a premium, but you’re in total control. You know what your maximum loss is… it’s the
cost of the contract and your possible profit is usually much greater than your possible loss.
Some people sell options and you might have heard that that’s a really good way to go, but we’ll get into that later… it’s definitely more risky. If you do sell options then you collect the premium
up front rather than paying it. When selling calls, if the stock does not get equal to or above the strike price, or in the case of puts if the stock does not get equal to or below the strike price,
then you collect the entire premium and the option expires worthless. If the price of the stock is between the strike price and the strike price plus the premium amount you will keep a portion of the
If the option ends up above the strike price it’s what’s considered in the money and you will either have to deliver the stock to the person that you sold the option to or cover the option by buying
back a call or buying back a put or as many puts or calls as you sold.
Okay, so that’s the simple basic description of what options are about and in other blog posts I’m going to go into more detail. | {"url":"https://protrader.tv/audio-introduction-options/","timestamp":"2024-11-03T08:48:03Z","content_type":"text/html","content_length":"35578","record_id":"<urn:uuid:ce644c57-de36-49e8-b4be-6c96470323dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00111.warc.gz"} |
More Than 1000 formulas Trigonometry to solve the problemsMore Than 1000 formulas Trigonometry to solve the problems
More Than 1000 formulas Trigonometry to solve the problems
In Trigonometry, different types of problems can be solved using trigonometry formulas. These problems may include trigonometric ratios (sin, cos, tan, sec, cosec and cot), Pythagorean identities,
product identities, etc. Some formulas including the sign of ratios in different quadrants, involving co-function identities (shifting angles), sum & difference identities, double angle identities,
half-angle identities, etc. are also given in brief here.
Learning and memorizing these mathematics formulas in trigonometry will help the students of Class 10, 11, and 12 to score good marks in this concept. They can find the trigonometry table along with
inverse trigonometry formulas to solve the problems based on them.
Trigonometry Formulas PDF
Below is the link given to download the pdf format of Trigonometry formulas for free so that students can learn them offline too.
Trigonometry is a branch of mathematics that deals with triangles. Trigonometry is also known as the study of relationships between lengths and angles of triangles.
There is an enormous number of uses of trigonometry and its formulae. For example, the technique of triangulation is used in Geography to measure the distance between landmarks; in Astronomy, to
measure the distance to nearby stars and also in satellite navigation systems.
Trigonometry Formulas List
All these are taken from a right angled triangle. When height and base side of the right triangle are known, we can find out the sine, cosine, tangent, secant, cosecant, and cotangent values using
trigonometric formulas. The reciprocal trigonometric identities are also derived by using the trigonometric functions.
All trigonometric identities are cyclic in nature. They repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identities. tan 45° = tan
225° but this is true for cos 45° and cos 225°. Refer to the above trigonometry table to verify the values.
Trigonometry Formulas Major systems
All trigonometric formulas are divided into two major systems:
• Trigonometric Identities
• Trigonometric Ratios
Trigonometric Identities are formulas that involve Trigonometric functions. These identities are true for all values of the variables. Trigonometric Ratio is known for the relationship between the
measurement of the angles and the length of the sides of the right triangle.
Here we provide a list of all Trigonometry formulas for the students. These formulas are helpful for the students in solving problems based on these formulas or any trigonometric application. Along
with these, trigonometric identities help us to derive the trigonometric formulas, if they will appear in the examination.
We also provided the basic trigonometric table pdf that gives the relation of all trigonometric functions along with their standard values. These trigonometric formulae are helpful in determining the
domain, range, and value of a compound trigonometric function. Students can refer to the formulas provided below or can also download the trigonometric formulas pdf that is provided above.
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Astronomer Danny Faulkner on the 360-Day Year
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* For the History of Base 60: see below.
* Different Sides of the 360/365 Day Question: Hear Real Science Radio's Bob Enyart interview Dr. Faulkner. The long-time University of South Carolina Astronomer Danny Faulkner is now full time with
Ken Ham's ministry, Answers in Genesis at the Creation Museum (near Cincinnati). With a Ph.D. in astronomy from Indiana University Dr. Faulkner, is also a board member of the Creation Research
* Was the Year Once 360 Days Long? In Dr. Faulkner's CRSQ paper, (rebutted, also in CRSQ, by Enyart), Danny argues against the widespread belief among many creationists (including Henry Morris and
Walt Brown) that God originally created the Earth with a 360-day year and 30-day months. (Morris' organization, ICR, the Institute for Creation Research, thankfully continues to teach the created
30-day month and 360-day year.) Enyart offers to Dr. Faulkner evidence to counter his disagreement with the idea, and the two agree to evaluate additional original source material from ancient
civilizations and perhaps to do a follow up program on the matter.
For this program RSR recommends The Young Sun
DVD with Danny Faulkner and other great astronomers!
2017 Update on 360DayYear.com: You've just GOT TO CHECK OUT this amazingly informative and fun website! As promised, Bob Enyart has been posting historical and scientific evidence suggesting a
360-day year. You can check it out online at 360dayyear.com! Also, you may want to check out the second of three objections to Walt Brown's hydroplate theory, at RSR's List of Answers to Hydroplate
* What Makes One Side of the Moon 1) Heavier and 2) the Near Side? Both questions have one answer. It's the mascons (mass concentrations). Explaining why a U.S. lunar satellite unexpectedly
crashed, NASA's Alex Konopliv said, "The Moon is extraordinarily lumpy, gravitationally speaking... in mass. What appear to be flat seas of lunar lava have huge positive gravitational anomalies—that
is, their mass and thus their gravitational fields are significantly stronger than the rest of the lunar crust." And NASA continued, "Known as mass concentrations or 'mascons,' there are five big
ones on the front side of the Moon facing Earth, all in lunar maria (Latin for 'seas')," including the pareidolic "Man in the Moon" facial features. And, "The mascons' gravitational anomaly is so
great—half a percent" [!] that astronauts could have easily noticed it "standing at the edge of one of the maria, a plumb bob would hang about a third of a degree off vertical, pointing toward the
mascon." Wow!
* What Made the Mascons the Mascons? So the reason the near side of the Moon is the near side is because it is heavier than the far side so Earth's gravity holds the near side closer. Far more
massive impacts hit the near side, making those areas 1) heavier, and 2) flatter, because those areas became molten from the massive and high-energy impacts, in the process erasing many thousands of
craters. (That's why the much less beat-up far side is sprinkled with so many more craters, number-wise, but not mass-wise, because they weren't erased by the once-molten mascons.) On today's
program, Bob Enyart shares with Dr. Faulkner NASA's most recent gravity mapping of the Moon, by which they created the above Bouguer gravity map. This map appears to provide evidence that is
consistent with the Hydroplate Theory's explanation for how the near side of the Moon became so much heavier (with its greater mass concentrations). So the most massive impacts melted away thousands
of craters, left huge dark maria in their place, and then with all that extra mass have kept that side of the moon (though with dramatic libration) as the Earth's near side.
* Ancient Use of Base 60 and the 360 Day Year: The ancient Sumerian word for the number 60 is geš and the English word for Base 60, from sexagesimus which is the Latin for sixtieth, is sexagesimal.
Robert K. Englund, Ph.D., UCLA prof. of Assyriology & Sumerology, Administrative Timekeeping in Ancient Mesopotamia, Journal of the Economic and Social History of the Orient, 1988, pp. 121-122:
The sexagesimal system of counting... is attested in periods much earlier than any secure attestation of the Sumerian language, namely in the periods Uruk IV-III [~3200 B.C.], and possibly in some
token assemblages from clay envelopes unearthed in levels of proto-elamite Susa corresponding to Uruk V [~3400 B.C.]. ... A mixture of this sexagesimal system with a heritage of natural cycles [the
lunar and solar orbits] resulted in the 3rd millennium time divisions attested by administrative documents. ... The resulting system... which without question complemented throughout the 3rd
millennium natural, lunisterllar divisions, is attested in its basic form of a twelve-month, 360-day year in the archaic documents from the end of the 4th millennium [B.C.].
Question for the Audience: What numerical base is the first
known in recorded human history? Is it base 10? Or base 60?
Consider the Sumerian Anomaly just below and rsr.org/origin-of-language.
Please email your thoughts and evidence to Bob@rsr.org.
Englund writes of a Vincenzo Formaleoni that "he was convinced that at the time of the [global] deluge—and the inception of the sexagesimal system—the year was exactly 360 days long" and quotes his
Sources of errors in cosmography and geographia of the Ancients, Venice, 1789:
The length of the year was therefore undeniably of 360 days at the time of the first gazers.
Englund cited this quote from the paper, Sketch of a history of the sexagesimal system, by F. Thureau-Dangin, Osiris 7, Paris, 1939, which paper begins as follows:
With the Greeks, and then with the Arabs, the sexagesimal system has been an academic system of numeration, used by the astronomers. We still avail ourselves of this system for the measurement of
arcs or angles and for the measurement of time. Save for some rare and late exceptions, (1) this system has, beginning with the Greeks, always been employed to express fractions only. Prior to that,
however, in Babylonia, it served to express the integers as well as the fractions. It was a complete system of numeration, used by the Babylonian mathematicians or astronomers. Already at the
beginning of Assyriology, examples of the system have been found by the first decipherers: Hinks found some on an astronomical tablet coming from the excavations of Nineveh (2)... Since then, the
excavations made in Babylonia, especially those of Sarzec at Tello, have taught us that, before becoming an academic system, the sexagesimal system had been the common and exclusive mode of
numeration with the Sumerians, that is, with the predecessors of the Babylonians.
Also Thureau-Dangin, p. 98, summarizes Moritz Cantor's conclusion (with which RSR concurs, though Cantor himself, perhaps glimpsing its implications, later abandoned), from his Vorlesungen ueber
Geschichte d. Mathematik (1880, p. 83) that:
The year, which was given the round number of 360 days, gave rise to the circle of 360 degrees, and the division of the circle into six parts, suggested by the fact that the chord of the inscribed
hexagon is equal to the radius, gave rise to the number 60, the basis of the system.
* The Sumerian Numbers and the Base 60 Anomaly: See from pages 101, 102, and 103 of Thureau-Dangin's Sketch:
* Two Similar Anomalies a World Apart: Consider also the Mayan numerical anomaly, where midstream they modified their base 20 numerical system, switching in the third place from what would have been
400 to instead 360. (1, 20, 360, 7200, 144,000; i.e., 1 x 20 x 18 x 20 x 20), to pay homage to their 360-day year heritage. (And, consider the French phrase, "I would not say that to you thirty-six
thousand times" and the popular "voir trente-six chandelles".) Thureau-Dangin, p. 103, adds:
If, in spite of the so common tendency to take 6 or 12 as a new unity, no system of numeration has been constructed on the base of 6 or on the base of 12, it is due to the fact that man has always
counted upon his fingers [RSR: as God of course designed him to do], and that the base imposed upon him since immemorial times [RSR: no such times; for the world remembers Adam] through this mode of
counting [in footnote (19) here it is acknowledged that this mode of counting would justify also use of base 20] has come to occupy an almost impregnable position in numeration. Incapable of
replacing the decimal scale, the bases of 6 and 12 could only be introduced into numeration in the form of a combination with the base of 10.
With the German peoples, 120 was about to assume the name and the place of 100. [RSR: As in 12 months x 10?] On this question... The facts are... very completely and exactly described. [
Thureau-Dangin disagrees though with another scholar's opinion, a J. Schmidt, that in this the Germany's influence had come indirectly from Babylon.]
* Sumer, Babylonia, and Brilliance: The city of Babel was built by Noah's great-grandson Nimrod in what today is southern Iraq. Eventually one of the earliest identifiable people groups, the
Sumerians, occupied this region of the Middle East. Their land and culture was later absorbed into the Babylonian Empire. (The initial Babylonian Empire flourished in the second millennium B.C. a
thousand years before Nebuchadnezzar and the other kings of the later revived Neo-Babylonian Empire.) Consistent with the increasingly acknowledged genius of ancient man, consider from the University
of St. Andrews in the UK that a thousand years before Pythagoras, the Babylonians knew the "Pythagorean" Theorem. And a 2016 paper in the journal Science, Ancient Babylonian astronomers calculated
Jupiter’s position from the area under a time-velocity graph, describes the Neo-Babylonian use of a rudimentary calculus a millennium and a half before such calculations were thought to previously
have been discovered. Commenting on the paper, Caltech's Noel Swerdlow said of the Babylonians, "They were very, very smart, and the more we learn of what they did, the more impressive, the more
remarkable it becomes." And New York University's Alexander Jones added that their work, "testifies to the revolutionary brilliance of the unknown Mesopotamian scholars who constructed Babylonian
mathematical astronomy."
* 60 was Unity in Babylon: The University of St. Andrews also reports that:
"The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. ... Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 BC. They give
squares of the numbers up to 59 and cubes of the numbers up to 32. The table gives 8^2 = 1,4 which stands for
8^2 = 1, 4 = 1 × 60 + 4 = 64
and so on up to 59^2 = 58, 1 (= 58 × 60 +1 = 3481)."
The two Babylonian names for the number 60 were "geš" and "gešta" the second of which appears to have been a composite of "geš" and "šita" which means "number" (Thureau-Dangin, p. 105), with the name
of the number 1 being, apparently, also "geš". And back to Thureau-Dangin, p. 105, "In adopting the habit of designating 60 as "1," the Sumerians were undoubtedly prompted by a conception of 60 as of
the (great) unit... the cipher presenting the number 60 looks like the cipher for the unity, but is bigger in size. [Further] One said geš-2 for 120, geš-3 for 180 and so on up to geš-10, that is geš
-u, for 600. ... geš-u-2 for 1200, geš-u-3 for 1800, etc. The word designating 60^2 (or 3600) was šar, a "whole, totality". Footnote (22) then follows: "The original meaning of the word [šar] may
have been 'circle.'" And p. 106 shows that the cipher that equaled 3,600 or šar was impressed into a clay tablet by the end of a hollow reed, making the form of a circle.
* 60 was Money in Babylon: Sixty shekels made one mina, and sixty minas made one talent. The Sumerians had also used these measures. Eventually the Greeks adopted the same ratio of 60 minas to one
talent. For the Hebrews, whereas Ezekiel indicated that a mina equaled sixty shekels (Ezekiel 45:12), more anciently, rather than using the Sumerians 3,600 shekels in a talent, the Israelites in the
wilderness used 3,000, as determined by the accounting described in Exodus 38.
* 60 was Trigonometry in Babylon: Beating the Greeks to trigonometry by 1500 years, because the Babylon's used a sexigesimal system, trigonometry was easier to use. Because today we use tables with
approximations and their base 60 was evenly divisible by the three (think triangles), according to the journal Science, ancient Babylonian calculations were even more accurate than those we make with
today's trigonometry and computers.
* The Seleucids from Uruk and Divisions of the Day: RSR notices that the fractional divisions of the day, from Thurea-Dangin pp. 109-110, are all in the list of the divisors of the highly composite
number 360: "Contracts of the times of the Seleucids, coming from Uruk, refer to the sale of shares in the benefits connected with the exercise of a function... determined by fractions of a day,
being such and such a day in the month. The fraction of the day is, for instance, 'one-sixth and one-ninth of the day'... or 'one-sixth, one-eighteenth and one-sixtieth of a day'... or 'one-thirtieth
and one-third of one-sixtieth of a day' [i.e., 1/180th]... These successions of unit fractions remind us of the Egyptian system."
* The Akkadians and the Number 60: Unfinished. RSR work in progress...
* Ancient Measurement Affinities for Factors & Multiples of 60: Before we list those measurements, let's remind ourselves about the number 360 being a highly composite number , which is one of the
reasons why God would have made with a year comprised of that many days. So to recall its factors, see this from 360dayyear.com#ecliptic:
360 is divisible by many numbers including those especially helpful (in bold, even today) for calendars and timekeeping: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90,
120, and 180. A year of 360 days is reasonably divisible even by the 2 equinoxes, 2 solstices, four seasons, 12 months, etc., 24 hours in a day, 24 time zones of 15 nominal degrees each, 60
minutes in an hour, 60 seconds in a minute (and consider also our comfort with a "dozen" and with 12 inches in a foot and 36 inches in a yard).
* The Measurements: For convenience measuring systems use many varying units. However ancient cultures appeared to have had a tendency to use sexagesimal factors. The following units of measurement
are easily found in charts in chapter three of Springer's 2003 Encyclopedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins.
- Ancient Hebrews: had fundamental units of weight that included 6, 60, 600, 6,000, and 36,000, as well as 12, 24, 48, and 12,000. The ancient Jews also had fundamental units of dry capacity that
included 3, 18, and 180 and liquid capacity of 6, 12, and 60.
- Ancient Chinese: had units of area that included 60, units of weight that included 24, 30, and 120; units of length that included 6, 12, 18, 30, 60, 180, 360, 1200, 1440, 1800, 3600, 4500, and
- Ancient Indians: units of length that included 12 and 24.
- Ancient Persians: units of area included 144, 1440, 14400, and 144,000.
- Ancient Egyptians: units of length included 3, 6, 12, 18, 24, 30, 72, 120, 144, 300, 3000, 12000, and 30000; and units of weight included 6, 60, 300, 600, 1200, 3000, 30000, and 60000; and units of
capacity include 6, 24, 60, and 240.
- Ancient Greeks: units of length include 6, 12, 18, 24, 30, 60, 120, 144, 300, 600, 1200, 1800, 6000, and 36000; units of dry capacity included 6 and 12; units of liquid capacity 3, 6, 12, 24, and
144; and units of weight include 60, 300, 600, 18000, and 36000. (And remember their hexameter, i.e., verse of six "feet", the oldest and preeminent metre of their poetry.)
* Middle Ages and Olde Measurements:
- The Brits: According to the Encyclopedia of Scientific Units, the Lea (or Lay) was a British unit of length of 360 feet. (By the way, there was quite a bit of technological advancement going on in
the Middle Ages.)
- The Scots: Units of length included 6, 12, and 24
- The Irish: Units of length included 3, 6, 12, and 36.
- The French: Units of weight included 3, 12, 24 and 72 (i.e., 360/5; of course the modern metric weights are all factors of 10); units of length included 3, 6, 12, 72, 144, 12,000, and 144,000,
units of liquid capacity included 18, 36, 72, and 144.
- The Italians: Units of capacity included 12 and 60; a libbra grossa (big pound) = 12 once (ounces); and a trabucco = 6 piedi (feet).
- The Spaniards: Units of area included 12, 36, 72, and 600 and units of weight included 3, 6, 12, 24, 36, 72, 144, 300, and 2400.
* The Missing Number 7: The division of the week into seven days is ancient. (See rsr.org/7-days-in-a-week.) Yet the number seven, admittedly not factorable, is completely overlooked in ancient
weights and measurements. Three and five are not factorable. Yet three appears occasionally being integral to the traditional system. Five appears often as does ten and its multiples because God gave
us ten fingers with five per hand for functionality and for learning how to count. So five is a natural unit. A fascinating observation about the number seven for the days in a week is its
persistence and ascendancy over alternatives such as the 10-day weeks (three per month) of the Incas, Chinese, and Egyptians. A week of seven days, originating with God's six days of creation and one
day of rest, has persisted and now dominates the world even though it is not tied to any astronomical cycle. (It is likely, God being who He is, that a seven-day week is harmonic and reverberates
through history in accord with mankind's deep psychophysical meta-circadian rhythm.)
* RSR and Danny Faulkner: We invite you to enjoy these related RSR shows...
- RSR on the 360 Day Year & 360 Day Year Pt. 2 (part 2 of this radio program)
- RSR Report on the Int'l Creation Conference: Bob and Fred Williams commend Danny for rebuking other creationists
- University of South Carolina Astronomer Faulkner on RSR: this show
- RSR: Where's that Blue Star Assembly Line?: The guys discuss Danny's writing about these short-lived stars
- Danny Faulkner on (against) the 360-day Year: this show
- RSR Rebuts Geocentrism: Where Bob references and builds on Faulkner's great geocentrism article
- On the Caution About the 360-Day Year: Enyart's rebuttal in the CRSQ to Faulkner's paper against the 360-day year
And see also:
- Did the Preflood Earth Have a 30-Day Lunar Month by Dr. Robert Brown, Astronautics
- On the origin of the world's first-known number system (a hybrid decimal/base 60 system; see just above)- RSR: 24 Hours in a Day -- An Ancient Measurement
- 360dayyear.com
- rsr.org/3
Request for Help: We would like to list the most ancient references known to a seven day week. If you'd like to do some volunteer research for RSR, please email Bob@rsr.org to let us know and pass
along whatever you find. Thanks for your consideration!
Today’s Resource: Have you browsed through our RSR Science Department in the KGOV Store? You just might LOVE IT! We offer a 30-day money back guarantee on all purchases.
* Possible Technical Difficulty: If you have a problem streaming this program, please listen to the download. Of the thousands of our archived programs, this is the only one that inexplicably,
intermittently cut out 20 minutes of audio from the middle of the "streamed" audio. | {"url":"https://kgov.com/base-60-and-the-360-day-year","timestamp":"2024-11-14T18:59:43Z","content_type":"text/html","content_length":"58552","record_id":"<urn:uuid:1a682874-6054-443c-9396-aa1303c13710>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00788.warc.gz"} |
class DataPostprocessor< dim >
This class provides an interface to compute derived quantities from a solution that can then be output in graphical formats for visualization, using facilities such as the DataOut class.
For the (graphical) output of a FE solution one frequently wants to include derived quantities, which are calculated from the values of the solution and possibly the first and second derivatives of
the solution. Examples are the calculation of Mach numbers from velocity and density in supersonic flow computations, or the computation of the magnitude of a complex-valued solution as demonstrated
in step-29 and step-58 (where it is actually the square of the magnitude). Other uses are shown in step-32 and step-33. This class offers the interface to perform such postprocessing. Given the
values and derivatives of the solution at those points where we want to generated output, the functions of this class can be overloaded to compute new quantities.
A data vector and an object of a class derived from the current one can be given to the DataOut::add_data_vector() function (and similarly for DataOutRotation and DataOutFaces). This will cause
DataOut::build_patches() to compute the derived quantities instead of using the data provided by the data vector (typically the solution vector). Note that the DataPostprocessor object (i.e., in
reality the object of your derived class) has to live until the DataOut object is destroyed as the latter keeps a pointer to the former and will complain if the object pointed to is destroyed while
the latter still has a pointer to it. If both the data postprocessor and DataOut objects are local variables of a function (as they are, for example, in step-29), then you can avoid this error by
declaring the data postprocessor variable before the DataOut variable as objects are destroyed in reverse order of declaration.
In order not to perform needless calculations, DataPostprocessor has to provide information which input data is needed for the calculation of the derived quantities, i.e. whether it needs the values,
the first derivative and/or the second derivative of the provided data. DataPostprocessor objects which are used in combination with a DataOutFaces object can also ask for the normal vectors at each
point. The information which data is needed has to be provided via the UpdateFlags returned by the virtual function get_needed_update_flags(). It is your responsibility to use only those values which
were updated in the calculation of derived quantities. The DataOut object will provide references to the requested data in the call to evaluate_scalar_field() or evaluate_vector_field() (DataOut
decides which of the two functions to call depending on whether the finite element in use has only a single, or multiple vector components; note that this is only determined by the number of
components in the finite element in use, and not by whether the data computed by a class derived from the current one is scalar or vector valued).
Furthermore, derived classes have to implement the get_names() function, where the number of output variables returned by the latter function has to match the size of the vector returned by the
former. Furthermore, this number has to match the number of computed quantities, of course.
Use in simpler cases
Deriving from the current class allows to implement very general postprocessors. For example, in the step-32 program, we implement a postprocessor that takes a solution that consists of velocity,
pressure and temperature (dim+2 components) and computes a variety of output quantities, some of which are vector valued and some of which are scalar. On the other hand, in step-29 we implement a
postprocessor that only computes the magnitude of a complex number given by a two-component finite element. It seems silly to have to implement four virtual functions for this (evaluate_scalar_field
() or evaluate_vector_field(), get_names(), get_update_flags() and get_data_component_interpretation()).
To this end there are three classes DataPostprocessorScalar, DataPostprocessorVector, and DataPostprocessorTensor that are meant to be used if the output quantity is either a single scalar, a single
vector (here used meaning to have exactly dim components), or a single tensor (here used meaning to have exactly dim*dim components). When using these classes, one only has to write a constructor
that passes the name of the output variable and the update flags to the constructor of the base class and overload the function that actually computes the results.
The DataPostprocessorVector and DataPostprocessorTensor class documentations also contains a extensive examples of how they can be used. The step-29 tutorial program contains an example of using the
DataPostprocessorScalar class.
Complex-valued solutions
There are PDEs whose solutions are complex-valued. For example, step-58 and step-62 solve problems whose solutions at each point consists of a complex number represented by a std::complex<double>
variable. (step-29 also solves such a problem, but there we choose to represent the solution by two real-valued fields.) In such cases, the vector that is handed to DataOut::build_patches() is of
type Vector<std::complex<double>>, or something essentially equivalent to this. The issue with this, as also discussed in the documentation of DataOut itself, is that the most widely used file
formats for visualization (notably, the VTK and VTU formats) can not actually represent complex quantities. The only thing that can be stored in these data files are real-valued quantities.
As a consequence, DataOut is forced to take things apart into their real and imaginary parts, and both are output as separate quantities. This is the case for data that is written directly to a file
by DataOut, but it is also the case for data that is first routed through DataPostprocessor objects (or objects of their derived classes): All these objects see is a collection of real values, even
if the underlying solution vector was complex-valued.
All of this has two implications:
• If a solution vector is complex-valued, then this results in at least two input components at each evaluation point. As a consequence, the DataPostprocessor::evaluate_scalar_field() function is
never called, even if the underlying finite element had only a single solution component. Instead, DataOut will always call DataPostprocessor::evaluate_vector_field().
• Implementations of the DataPostprocessor::evaluate_vector_field() in derived classes must understand how the solution values are arranged in the DataPostprocessorInputs::Vector objects they
receive as input. The rule here is: If the finite element has \(N\) vector components (including the case \(N=1\), i.e., a scalar element), then the inputs for complex-valued solution vectors
will have \(2N\) components. These first contain the values (or gradients, or Hessians) of the real parts of all solution components, and then the values (or gradients, or Hessians) of the
imaginary parts of all solution components.
step-58 provides an example of how this class (or, rather, the derived DataPostprocessorScalar class) is used in a complex-valued situation.
Definition at line 582 of file data_postprocessor.h. | {"url":"https://www.dealii.org/developer/doxygen/deal.II/classDataPostprocessor.html","timestamp":"2024-11-02T01:47:08Z","content_type":"application/xhtml+xml","content_length":"66232","record_id":"<urn:uuid:5758bcf5-71a3-4224-8ed1-46adfb2743e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00134.warc.gz"} |
Verifiable Random Functions (VRFs)
This document is an Internet-Draft (I-D) that has been submitted to the Internet Research Task Force (IRTF) stream. This I-D is
not endorsed by the IETF
and has
no formal standing
in the
IETF standards process
The information below is for an old version of the document.
This is an older version of an Internet-Draft that was ultimately published as
Document Type RFC 9381
Authors Sharon Goldberg , Leonid Reyzin , Dimitrios Papadopoulos , Jan Včelák
Last updated 2017-10-30 (Latest revision 2017-09-13)
Replaces draft-goldbe-vrf
RFC stream Internet Research Task Force (IRTF)
IETF conflict conflict-review-irtf-cfrg-vrf, conflict-review-irtf-cfrg-vrf, conflict-review-irtf-cfrg-vrf, conflict-review-irtf-cfrg-vrf, conflict-review-irtf-cfrg-vrf,
review conflict-review-irtf-cfrg-vrf
Additional Mailing list discussion
Stream IRTF state Active RG Document
Consensus Unknown
Document shepherd (None)
IESG IESG state Became RFC 9381 (Informational)
Telechat date (None)
Responsible AD (None)
Send notices to (None)
CFRG S. Goldberg
Internet-Draft L. Reyzin
Intended status: Standards Track Boston University
Expires: March 16, 2018 D. Papadopoulos
Hong Kong University of Science and Techology
J. Vcelak
September 12, 2017
Verifiable Random Functions (VRFs)
A Verifiable Random Function (VRF) is the public-key version of a
keyed cryptographic hash. Only the holder of the private key can
compute the hash, but anyone with public key can verify the
correctness of the hash. VRFs are useful for preventing enumeration
of hash-based data structures. This document specifies several VRF
constructions that are secure in the cryptographic random oracle
model. One VRF uses RSA and the other VRF uses Eliptic Curves (EC).
Status of This Memo
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Rationale . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2. Requirements . . . . . . . . . . . . . . . . . . . . . . 3
1.3. Terminology . . . . . . . . . . . . . . . . . . . . . . . 3
2. VRF Algorithms . . . . . . . . . . . . . . . . . . . . . . . 4
3. VRF Security Properties . . . . . . . . . . . . . . . . . . . 4
3.1. Full Uniqueness or Trusted Uniqueness . . . . . . . . . . 4
3.2. Full Collison Resistance or Trusted Collision Resistance 5
3.3. Full Pseudorandomness or Selective Pseudorandomness . . . 5
3.4. An additional pseudorandomness property . . . . . . . . . 6
4. RSA Full Domain Hash VRF (RSA-FDH-VRF) . . . . . . . . . . . 7
4.1. RSA-FDH-VRF Proving . . . . . . . . . . . . . . . . . . . 8
4.2. RSA-FDH-VRF Proof To Hash . . . . . . . . . . . . . . . . 8
4.3. RSA-FDH-VRF Verifying . . . . . . . . . . . . . . . . . . 9
5. Elliptic Curve VRF (EC-VRF) . . . . . . . . . . . . . . . . . 9
5.1. EC-VRF Proving . . . . . . . . . . . . . . . . . . . . . 11
5.2. EC-VRF Proof To Hash . . . . . . . . . . . . . . . . . . 11
5.3. EC-VRF Verifying . . . . . . . . . . . . . . . . . . . . 12
5.4. EC-VRF Auxiliary Functions . . . . . . . . . . . . . . . 13
5.4.1. EC-VRF Hash To Curve . . . . . . . . . . . . . . . . 13
5.4.2. EC-VRF Hash Points . . . . . . . . . . . . . . . . . 14
5.4.3. EC-VRF Decode Proof . . . . . . . . . . . . . . . . . 15
5.5. EC-VRF Ciphersuites . . . . . . . . . . . . . . . . . . . 15
5.6. When the EC-VRF Keys are Untrusted . . . . . . . . . . . 16
5.6.1. EC-VRF Validate Key . . . . . . . . . . . . . . . . . 17
6. Implementation Status . . . . . . . . . . . . . . . . . . . . 17
7. Security Considerations . . . . . . . . . . . . . . . . . . . 18
7.1. Key Generation . . . . . . . . . . . . . . . . . . . . . 18
7.1.1. Uniqueness and collision resistance with untrusted
keys . . . . . . . . . . . . . . . . . . . . . . . . 18
7.1.2. Pseudorandomness with untrusted keys . . . . . . . . 19
7.2. Selective vs Full Pseudorandomness . . . . . . . . . . . 19
7.3. Proper randomness for EC-VRF . . . . . . . . . . . . . . 19
7.4. Timing attacks . . . . . . . . . . . . . . . . . . . . . 20
8. Change Log . . . . . . . . . . . . . . . . . . . . . . . . . 20
9. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 20
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.1. Normative References . . . . . . . . . . . . . . . . . . 20
10.2. Informative References . . . . . . . . . . . . . . . . . 21
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Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 22
1. Introduction
1.1. Rationale
A Verifiable Random Function (VRF) [MRV99] is the public-key version
of a keyed cryptographic hash. Only the holder of the private VRF
key can compute the hash, but anyone with corresponding public key
can verify the correctness of the hash.
A key application of the VRF is to provide privacy against offline
enumeration (e.g. dictionary attacks) on data stored in a hash-based
data structure. In this application, a Prover holds the VRF secret
key and uses the VRF hashing to construct a hash-based data structure
on the input data. Due to the nature of the VRF, only the Prover can
answer queries about whether or not some data is stored in the data
structure. Anyone who knows the public VRF key can verify that the
Prover has answered the queries correctly. However no offline
inferences (i.e. inferences without querying the Prover) can be made
about the data stored in the data strucuture.
1.2. Requirements
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
1.3. Terminology
The following terminology is used through this document:
SK: The private key for the VRF.
PK: The public key for the VRF.
alpha: The input to be hashed by the VRF.
beta: The VRF hash output.
pi: The VRF proof.
Prover: The Prover holds the private VRF key SK and public VRF key
Verifier: The Verifier holds the public VRF key PK.
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2. VRF Algorithms
A VRF comes with a key generation algorithm that generates a public
VRF key PK and private VRF key SK.
A VRF hashes an input alpha using the private VRF key SK to obtain a
VRF hash output beta
beta = VRF_hash(SK, alpha)
The VRF_hash algorithm is deterministic, in the sense that it always
produces the same output beta given a pair of inputs (SK, alpha).
The private key SK is also used to construct a proof pi that beta is
the correct hash output
pi = VRF_prove(SK, alpha)
The VRFs defined in this document allow anyone to deterministically
obtain the VRF hash output beta directly from the proof value pi as
beta = VRF_proof2hash(pi)
Notice that this means that
VRF_hash(SK, alpha) = VRF_proof2hash(VRF_prove(SK, alpha))
The proof pi allows a Verifier holding the public key PK to verify
that beta is the correct VRF hash of input alpha under key PK. Thus,
the VRF also comes with an algorithm
VRF_verify(PK, alpha, pi)
that outputs VALID if beta=VRF_proof2hash(pi) is correct VRF hash of
alpha under key PK, and outputs INVALID otherwise.
3. VRF Security Properties
VRFs are designed to ensure the following security properties.
3.1. Full Uniqueness or Trusted Uniqueness
Uniqueness means that, for any fixed public VRF key and for any input
alpha, there is a unique VRF output beta that can be proved to be
valid. Uniqueness must hold even for an adversarial Prover that
knows the VRF secret key SK.
"Full uniqueness" states that a computationally-bounded adversary
cannot choose a VRF public key PK, a VRF input alpha, two different
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VRF hash outputs beta1 and beta2, and two proofs pi1 and pi2 such
that VRF_verify(PK, alpha, pi1) and VRF_verify(PK, alpha, pi2) both
output VALID.
A slightly weaker security property called "trusted uniquness"
sufficies for many applications. Trusted uniqueness is the same as
full uniqueness, but it must hold only if the VRF keys PK and SK were
generated in a trustworthy manner. In otherwords, uniqueness might
not hold if keys were generated in an invalid manner or with bad
3.2. Full Collison Resistance or Trusted Collision Resistance
Like any cryprographic hash function, VRFs need to be collision
resistant. Collison resistance must hold even for an adversarial
Prover that knows the VRF secret key SK.
More percisely, "full collision resistance" states that it should be
computationally infeasible for an adversary to find two distinct VRF
inputs alpha1 and alpha2 that have the same VRF hash beta, even if
that adversary knows the secret VRF key SK.
For most applications, a slightly weaker security property called
"trusted collision resistance" suffices. Trusted collision
resistance is the same as collision resistance, but it holds only if
PK and SK were generated in a trustworthy manner.
3.3. Full Pseudorandomness or Selective Pseudorandomness
Pseudorandomness ensures that when an adversarial Verifier sees a VRF
hash output beta without its corresponding VRF proof pi, then beta is
indistinguishable from a random value.
More percisely, suppose the public and private VRF keys (PK, SK) were
generated in a trustworthy manner. Pseudorandomness ensures that the
VRF hash output beta (without its corresponding VRF proof pi) on any
adversarially-chosen "target" VRF input alpha looks indistinguishable
from random for any computationally bounded adversary who does not
know the private VRF key SK. This holds even if the adversary also
gets to choose other VRF inputs alpha' and observe their
corresponding VRF hash outputs beta' and proofs pi'.
With "full pseudorandomness", the adversary is allowed to choose the
"target" VRF input alpha at any time, even after it observes VRF
outputs beta' and proofs pi' on a variety of chosen inputs alpha'.
"Selective pseudorandomness" is a weaker security property which
suffices in many applications. Here, the adversary must choose the
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target VRF input alpha independently of the public VRF key PK, and
before it observes VRF outputs beta' and proofs pi' on inputs alpha'
of its choice.
It is important to remember that the VRF output beta does not look
random to the Prover, or to any other party that knows the private
VRF key SK! Such a party can easily distinguish beta from a random
value by comparing beta to the result of VRF_hash(SK, alpha).
Also, the VRF output beta does not look random to any party that
knows valid VRF proof pi corresponding to the VRF input alpha, even
if this party does not know the private VRF key SK. Such a party can
easily distinguish beta from a random value by checking whether
VRF_verify(PK, alpha, pi) returns "VALID" and beta =
Also, the VRF output beta may not look random if VRF key generation
was not done in a trustworthy fashion. (For example, if VRF keys
were generated with bad randomness.)
3.4. An additional pseudorandomness property
[TODO: The following property is not needed for applications that use
VRFs to prevent enumeration of hash-based data structures. However,
we noticed that some other applications of VRF rely on this property.
As we have not yet found a formal definition of this property in the
literature, we write it down here. ]
Pseudorandomness, as defined in Section 3.3, does not hold if the VRF
keys were generated adversarially.
There is, however, a different type of pseudorandomness that could
hold even if the VRF keys are generated adversarially, as long as the
VRF input alpha is unpredictable. Suppose the VRF keys are generated
by an adversary. Then, a VRF hash output beta should look
pseudorandom to the adversary as long as (1) its corresponding VRF
hash alpha is chosen randomly and independently of the VRF key, (2)
alpha is unknown to the adversary, (3) the corresponding proof pi is
unknown to the adversary, and (4) the VRF public key chosen by the
adversary is valid.
[TODO: It should be possible to get the EC-VRF to satisfy this
property, as long as verifiers run an VRF_validate_key() key function
upon receipt of VRF public keys. However, we need to work out
exactly what properties are needed from the VRF public keys in order
for this property to hold. Some additional checks might need to be
added to the ECVRF_validate_key() function. Need to work out what
are these checks.]
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4. RSA Full Domain Hash VRF (RSA-FDH-VRF)
The RSA Full Domain Hash VRF (RSA-FDH-VRF) is a VRF that satisfies
the "trusted uniqueness", "trusted collision resistance", and "full
pseudorandomness" properties defined in Section 3. Its security
follows from the standard RSA assumption in the random oracle model.
Formal security proofs are in [nsec5ecc].
The VRF computes the proof pi as a deterministic RSA signature on
input alpha using the RSA Full Domain Hash Algorithm [RFC8017]
parametrized with the selected hash algorithm. RSA signature
verification is used to verify the correctness of the proof. The VRF
hash output beta is simply obtained by hashing the proof pi with the
selected hash algorithm.
The key pair for RSA-FDH-VRF MUST be generated in a way that it
satisfies the conditions specified in Section 3 of [RFC8017].
In this document, the notation from [RFC8017] is used.
Parameters used:
(n, e) - RSA public key
K - RSA private key
k - length in octets of the RSA modulus n
Fixed options:
Hash - cryptographic hash function
hLen - output length in octets of hash function Hash
Constraints on options:
Cryptographic security of Hash is at least as high as the
cryptographic security level of the RSA key
Primitives used:
I2OSP - Coversion of a nonnegative integer to an octet string as
defined in Section 4.1 of [RFC8017]
OS2IP - Coversion of an octet string to a nonnegative integer as
defined in Section 4.2 of [RFC8017]
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RSASP1 - RSA signature primitive as defined in Section 5.2.1 of
RSAVP1 - RSA verification primitive as defined in Section 5.2.2 of
MGF1 - Mask Generation Function based on a hash function as
defined in Section B.2.1 of [RFC8017]
4.1. RSA-FDH-VRF Proving
RSAFDHVRF_prove(K, alpha)
K - RSA private key
alpha - VRF hash input, an octet string
pi - proof, an octet string of length k
1. EM = MGF1(alpha, k - 1)
2. m = OS2IP(EM)
3. s = RSASP1(K, m)
4. pi = I2OSP(s, k)
5. Output pi
4.2. RSA-FDH-VRF Proof To Hash
pi - proof, an octet string of length k
beta - VRF hash output, an octet string of length hLen
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1. beta = Hash(pi)
2. Output beta
4.3. RSA-FDH-VRF Verifying
RSAFDHVRF_verify((n, e), alpha, pi)
(n, e) - RSA public key
alpha - VRF hash input, an octet string
pi - proof to be verified, an octet string of length n
"VALID" or "INVALID"
1. s = OS2IP(pi)
2. m = RSAVP1((n, e), s)
3. EM = I2OSP(m, k - 1)
4. EM' = MGF1(alpha, k - 1)
5. If EM and EM' are equal, output "VALID"; else output "INVALID".
5. Elliptic Curve VRF (EC-VRF)
The Elliptic Curve Verifiable Random Function (EC-VRF) is a VRF that
satisfies the trusted uniqueness, trusted collision resistance, and
full pseudorandomness properties defined in Section 3. The security
of this VRF follows from the decisional Diffie-Hellman (DDH)
assumption in the random oracle model. Formal security proofs are in
Fixed options:
F - finite field
2n - length, in octets, of a field element in F
E - elliptic curve (EC) defined over F
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m - length, in octets, of an EC point encoded as an octet string
G - subgroup of E of large prime order
q - prime order of group G
cofactor - number of points on E divided by q
g - generator of group G
Hash - cryptographic hash function
hLen - output length in octets of Hash
Constraints on options:
Field elements in F have bit lengths divisible by 16
hLen is equal to 2n
Parameters used:
y = g^x - VRF public key, an EC point
x - VRF private key, an integer where 0 < x < q
Notation and primitives used:
p^k - when p is an EC point: point multiplication, i.e. k
repetitions of group operation on EC point p. when p is an
integer: exponentiation
|| - octet string concatenation
I2OSP - nonnegative integer conversion to octet string as defined
in Section 4.1 of [RFC8017]
OS2IP - Coversion of an octet string to a nonnegative integer as
defined in Section 4.2 of [RFC8017]
EC2OSP - conversion of EC point to an m-octet string as specified
in Section 5.5
OS2ECP - conversion of an m-octet string to EC point as specified
in Section 5.5. OS2ECP returns INVALID if the octet string does
not convert to a valid EC point.
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RS2ECP - conversion of a random 2n-octet string to an EC point as
specified in Section 5.5
5.1. EC-VRF Proving
Note: this function is made more efficient by taking in the public
VRF key y, as well as the private VRF key x.
ECVRF_prove(y, x, alpha)
y - public key, an EC point
x - private key, an integer
alpha - VRF input, an octet string
pi - VRF proof, octet string of length m+3n
1. h = ECVRF_hash_to_curve(y, alpha)
2. gamma = h^x
3. choose a random integer nonce k from [0, q-1]
4. c = ECVRF_hash_points(g, h, y, gamma, g^k, h^k)
5. s = k - c*x mod q (where * denotes integer multiplication)
6. pi = EC2OSP(gamma) || I2OSP(c, n) || I2OSP(s, 2n)
7. Output pi
5.2. EC-VRF Proof To Hash
pi - VRF proof, octet string of length m+3n
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"INVALID", or
beta - VRF hash output, octet string of length 2n
1. D = ECVRF_decode_proof(pi)
2. If D is "INVALID", output "INVALID" and stop
3. (gamma, c, s) = D
4. beta = Hash(EC2OSP(gamma^cofactor))
5. Output beta
5.3. EC-VRF Verifying
ECVRF_verify(y, pi, alpha)
y - public key, an EC point
pi - VRF proof, octet string of length 5n+1
alpha - VRF input, octet string
"VALID" or "INVALID"
1. D = ECVRF_decode_proof(pi)
2. If D is "INVALID", output "INVALID" and stop
3. (gamma, c, s) = D
4. u = y^c * g^s (where * denotes EC point addition, i.e. a group
operation on two EC points)
5. h = ECVRF_hash_to_curve(y, alpha)
6. v = gamma^c * h^s (where * denotes EC point addition)
7. c' = ECVRF_hash_points(g, h, y, gamma, u, v)
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8. If c and c' are equal, output "VALID"; else output "INVALID"
5.4. EC-VRF Auxiliary Functions
5.4.1. EC-VRF Hash To Curve
The ECVRF_hash_to_curve algorithm takes in an octet string alpha and
converts it to h, an EC point in G.
5.4.1.1. ECVRF_hash_to_curve1
The following ECVRF_hash_to_curve1(y, alpha) algorithm implements
ECVRF_hash_to_curve in a simple and generic way that works for any
elliptic curve.
The running time of this algorithm depends on alpha. For the
ciphersuites specified in Section 5.5, this algorithm is expected to
find a valid curve point after approximately two attempts (i.e., when
ctr=1) on average. See also [Icart09].
However, because the running time of algorithm depends on alpha, this
algorithm SHOULD be avoided in applications where it is important
that the VRF input alpha remain secret.
ECVRF_hash_to_curve1(y, alpha)
alpha - value to be hashed, an octet string
y - public key, an EC point
h - hashed value, a finite EC point in G
1. ctr = 0
2. pk = EC2OSP(y)
3. h = "INVALID"
4. While h is "INVALID" or h is EC point at infinity:
A. CTR = I2OSP(ctr, 4)
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B. ctr = ctr + 1
C. attempted_hash = Hash(pk || alpha || CTR)
D. h = RS2ECP(attempted_hash)
E. If h is not "INVALID" and cofactor > 1, set h = h^cofactor
5. Output h
5.4.1.2. ECVRF_hash_to_curve2
For applications where VRF input alpha must be kept secret, the
following ECVRF_hash_to_curve algorithm MAY be used to used as
generic way to hash an octet string onto any elliptic curve.
[TODO: If there interest, we could look into specifying the generic
deterministic time hash_to_curve algorithm from [Icart09]. Note also
for the Ed25519 curve (but not the P256 curve), the Elligator
algorithm could be used here.]
5.4.2. EC-VRF Hash Points
ECVRF_hash_points(p_1, p_2, ..., p_j)
p_i - EC point in G
h - hash value, integer between 0 and 2^(8n)-1
1. P = empty octet string
2. for p_i in [p_1, p_2, ... p_j]:
P = P || EC2OSP(p_i)
3. h1 = Hash(P)
4. h2 = first n octets of h1
5. h = OS2IP(h2)
6. Output h
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5.4.3. EC-VRF Decode Proof
pi - VRF proof, octet string (m+3n octets)
"INVALID", or
gamma - EC point
c - integer between 0 and 2^(8n)-1
s - integer between 0 and 2^(16n)-1
1. let gamma', c', s' be pi split after m-th and m+n-th octet
2. gamma = OS2ECP(gamma')
3. if gamma = "INVALID" output "INVALID" and stop.
4. c = OS2IP(c')
5. s = OS2IP(s')
6. Output gamma, c, and s
5.5. EC-VRF Ciphersuites
This document defines EC-VRF-P256-SHA256 as follows:
o The EC group G is the NIST-P256 elliptic curve, with curve
parameters as specified in [FIPS-186-3] (Section D.1.2.3) and
[RFC5114] (Section 2.6). For this group, 2n = 32 and cofactor =
o The key pair generation primitive is specified in Section 3.2.1 of
o EC2OSP is specified in Section 2.3.3 of [SECG1] with point
compression on. This implies m = 2n + 1 = 33.
o OS2ECP is specified in Section 2.3.4 of [SECG1].
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o RS2ECP(h) = OS2ECP(0x02 || h). The input h is a 32-octet string
and the output is either an EC point or "INVALID".
o The hash function Hash is SHA-256 as specified in [RFC6234].
o The ECVRF_hash_to_curve function is as specified in
Section 5.4.1.1.
This document defines EC-VRF-ED25519-SHA256 as follows:
o The EC group G is the Ed25519 elliptic curve with parameters
defined in Table 1 of [RFC8032]. For this group, 2n = 32 and
cofactor = 8.
o The key pair generation primitive is specified in Section 5.1.5 of
o EC2OSP is specified in Section 5.1.2 of [RFC8032]. This implies m
= 2n = 32.
o OS2ECP is specified in Section 5.1.3 of [RFC8032].
o RS2ECP is equivalent to OS2ECP.
o The hash function Hash is SHA-256 as specified in [RFC6234].
o The ECVRF_hash_to_curve function is as specified in
Section 5.4.1.1.
[TODO: Should we add an EC-VRF-ED25519-SHA256-Elligator ciphersuite
where the Elligator hash function is used for ECVRF_hash-to-curve?]
[TODO: Add an Ed448 ciphersuite?]
[NOTE: In the unlikely case that future versions of this spec use a
elliptic curve group G that does not also come with a specification
of the group generator g, then we can still have full uniqueness and
full collision resistance by adding an check to
ECVRF_validate_key(PK) that ensures that g is a point on the elliptic
curve and g^cofactor is not the EC point at infinity.]
5.6. When the EC-VRF Keys are Untrusted
The EC-VRF as specified above is a VRF that satisfies the "trusted
uniqueness", "trusted collision resistance", and "full
pseudorandomness" properties defined in Section 3. If the elliptic
curve parameters (including the generator g) are trusted, but the VRF
public key PK is not trusted, this VRF can be modified to
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additionally satisfy "full uniqueness", and "full collision
resistance". This is done by additionally requiring the Verifier to
perform the following validation procedure upon receipt of the public
VRF key.
The Verifier MUST perform this validation procedure when the entity
that generated the public VRF key is untrusted. The public key MUST
NOT be used if this procedure returns "INVALID". Note well that this
procedure is not sufficient if the elliptic curve E or if g, the
generator of group G, is untrusted.
This procedure supposes that the public key provided to the Verifier
is an octet string. The procedure returns "INVALID" if the public
key in invalid. Otherwise, it returns y, the public key as an EC
5.6.1. EC-VRF Validate Key
PK - public key, an octet string
"INVALID", or
y - public key, an EC point
1. y = OS2ECP(PK)
2. If y is "INVALID", output "INVALID" and stop
3. If y^cofactor is the EC point at infinty, output "INVALID" and
4. Output y
6. Implementation Status
An implementation of the RSA-FDH-VRF (SHA-256) and EC-VRF-P256-SHA256
was first developed as a part of the NSEC5 project [I-D.vcelak-nsec5]
and is available at <http://github.com/fcelda/nsec5-crypto>. The EC-
VRF implementation may be out of date as this spec has evolved.
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The Key Transparency project at Google uses a VRF implemention that
is similar to the EC-VRF-P256-SHA256, with a few minor changes
including the use of SHA-512 instead of SHA-256. Its implementation
is available
An implementation by Yahoo! similar to the EC-VRF is available at
An implementation similar to EC-VRF is available as part of the
CONIKS implementation in Golang at <https://github.com/coniks-sys/
Open Whisper Systems also uses a VRF very similar to EC-VRF-
ED25519-SHA512-Elligator, called VXEdDSA, and specified here:
7. Security Considerations
7.1. Key Generation
Applications that use the VRFs defined in this document MUST ensure
that that the VRF key is generated correctly, using good randomness.
7.1.1. Uniqueness and collision resistance with untrusted keys
The EC-VRF as specified in Section 5.1-Section 5.5 statisfies the
"trusted uniqueness" and "trusted collision resistance" properties as
long as the VRF keys are generated correctly, with good randomness.
If the Verifier trusts the VRF keys are generated correctly, it MAY
use the public key y as is.
However, if the EC-VRF uses keys that could be generated
adversarially, then the the Verfier MUST first perform the validation
procedure ECVRF_validate_key(PK) (specified in Section 5.6) upon
receipt of the public key PK as an octet string. If the validation
procedure outputs "INVALID", then the public key MUST not be used.
Otherwise, the procedure will output a valid public key y, and the
EC-VRF with public key y satisfies the "full uniqueness" and "full
collision resistance" properties.
The RSA-FDH-VRF statisfies the "trusted uniqueness" and "trusted
collision resistance" properties as long as the VRF keys are
generated correctly, with good randomness. These properties may not
hold if the keys are generated adversarially (e.g., if RSA is not
permutation). Meanwhile, the "full uniqueness" and "full collision
resistance" are properties that hold even if VRF keys are generated
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by an adversary. The RSA-FDH-VRF defined in this document does not
have these properties. However, if adversarial key generation is a
concern, the RSA-FDH-VRF may be modifed to have these properties by
adding additional cryptographic checks that its public key has the
right form. These modifications are left for future specification.
7.1.2. Pseudorandomness with untrusted keys
Without good randomness, the "pseudorandomness" properties of the VRF
may not hold. Note that it is not possible to guarantee
pseudorandomness in the face of adversarially generated VRF keys.
This is because an adversary can always use bad randomness to
generate the VRF keys, and thus, the VRF output may not be
7.2. Selective vs Full Pseudorandomness
[nsec5ecc] presents cryptographic reductions to an underlying hard
problem (e.g. Decisional Diffie Hellman for the EC-VRF, or the
standard RSA assumption for RSA-FDH-VRF) that prove the VRFs
specificied in this document possess full pseudorandomness as well as
selective pseudorandomness. However, the cryptographic reductions
are tighter for selective pseudorandomness than for full
pseudorandomness. This means the the VRFs have quantitavely stronger
security guarentees for selective pseudorandomness.
Applications that are concerned about tightness of cryptographic
reductions therefore have two options.
o They may choose to ensure that selective pseudorandomness is
sufficient for the application. That is, that pseudorandomness of
outputs matters only for inputs that are chosen independently of
the VRF key.
o If full pseudorandomness is required for the application, the
application may increase security parameters to make up for the
loose security reduction. For RSA-FDH-VRF, this means increasing
the RSA key length. For EC-VRF, this means increasing the
cryptographic strength of the EC group G. For both RSA-FDH-VRF
and EC-VRF the cryptographic strength of the hash function Hash
may also potentially need to be increased.
7.3. Proper randomness for EC-VRF
Applications that use the EC-VRF defined in this document MUST ensure
that the random nonce k used in the ECVRF_prove algorithm is chosen
with proper randomness. Otherwise, an adversary may be able to
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recover the private VRF key x (and thus break pseudorandomness of the
VRF) after observing several valid VRF proofs pi.
7.4. Timing attacks
The EC-VRF_hash_to_curve algorithm defined in Section 5.4.1.1 SHOULD
NOT be used in applications where the VRF input alpha is secret and
is hashed by the VRF on-the-fly. This is because the EC-
VRF_hash_to_curve algorithm's running time depends on the VRF input
alpha, and thus creates a timing channel that can be used to learn
information about alpha. That said, for most inputs the amount of
information obtained from such a timing attack is likely to be small
(1 bit, on average), since the algorithm is expected to find a valid
curve point after only two attempts. However, there might be inputs
which cause the algorithm to make many attempts before it finds a
valid curve point; for such inputs, the information leaked in a
timing attack will be more than 1 bit.
8. Change Log
Note to RFC Editor: if this document does not obsolete an existing
RFC, please remove this appendix before publication as an RFC.
00 - Forked this document from draft-goldbe-vrf-01.
9. Contributors
Leonid Reyzin (Boston University) is a major contributor to this
This document also would not be possible without the work of Moni
Naor (Weizmann Institute), Sachin Vasant (Cisco Systems), and Asaf
Ziv (Facebook). Shumon Huque (Salesforce) and David C. Lawerence
(Akamai) provided valuable input to this draft.
10. References
10.1. Normative References
National Institute for Standards and Technology, "Digital
Signature Standard (DSS)", FIPS PUB 186-3, June 2009.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
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[RFC5114] Lepinski, M. and S. Kent, "Additional Diffie-Hellman
Groups for Use with IETF Standards", RFC 5114,
DOI 10.17487/RFC5114, January 2008,
[RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms
(SHA and SHA-based HMAC and HKDF)", RFC 6234,
DOI 10.17487/RFC6234, May 2011,
[RFC8017] Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch,
"PKCS #1: RSA Cryptography Specifications Version 2.2",
RFC 8017, DOI 10.17487/RFC8017, November 2016,
[RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
Signature Algorithm (EdDSA)", RFC 8032,
DOI 10.17487/RFC8032, January 2017,
[SECG1] Standards for Efficient Cryptography Group (SECG), "SEC 1:
Elliptic Curve Cryptography", Version 2.0, May 2009,
10.2. Informative References
Vcelak, J., Goldberg, S., Papadopoulos, D., Huque, S., and
D. Lawrence, "NSEC5, DNSSEC Authenticated Denial of
Existence", draft-vcelak-nsec5-05 (work in progress), July
[Icart09] Icart, T., "How to Hash into Elliptic Curves", in CRYPTO,
[MRV99] Michali, S., Rabin, M., and S. Vadhan, "Verifiable Random
Functions", in FOCS, 1999.
Papadopoulos, D., Wessels, D., Huque, S., Vcelak, J.,
Naor, M., Reyzin, L., and S. Goldberg, "Making NSEC5
Practical for DNSSEC", in ePrint Cryptology Archive
2017/099, February 2017,
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Authors' Addresses
Sharon Goldberg
Boston University
111 Cummington St, MCS135
Boston, MA 02215
EMail: goldbe@cs.bu.edu
Leonid Reyzin
Boston University
111 Cummington St, MCS135
Boston, MA 02215
EMail: reyzin@cs.bu.edu
Dimitrios Papadopoulos
Hong Kong University of Science and Techology
Clearwater Bay
Hong Kong
EMail: dipapado@cse.ust.hkbu.edu
Jan Vcelak
16 Beaver St
New York, NY 10004
EMail: jvcelak@ns1.com
Goldberg, et al. Expires March 16, 2018 [Page 22] | {"url":"https://datatracker.ietf.org/doc/draft-irtf-cfrg-vrf/00/","timestamp":"2024-11-08T13:49:47Z","content_type":"text/html","content_length":"84579","record_id":"<urn:uuid:df33fdcc-4d61-47e1-929b-dba3d978237a>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00725.warc.gz"} |
? Calculate the numbers difference and learn how to do the subtraction, column subtracting method, from right to left
How to subtract numbers? Let's learn with an example
The operation to perform:
52 - 37
Method used below: column subtracting, from right to left (traditional)
Stack the numbers on top of each other.
The ones digits line up in the first column from the right.
The tens digits line up in the next column to the left.
Subtract column by column; start from the column on the right
Subtract the digits in the ones column:
2 - 7 = ?
The second digit is larger than the first.
Borrow from the next column to the left.
The borrowing is a two step process:
Subtract 1 from the top digit in the column of the tens: 5 - 1 = 4.
Cross out the top digit you've borrowed 1 from: [DEL:5:DEL].
Write the answer above that digit: 4.
When borrowing, 1 ten = 10 ones.
Add 10 to the top digit in the column of the ones: 10 + 2 = ^12.
After borrowing the subtraction has become:
it2brw: 12 - 7 = 5.
5 is the ones digit.
Write it down at the base of the ones column.
Subtract the digits in the tens column:
it1go: [DEL:5:DEL] 4 - 3 = 1.
1 is the tens digit.
Write it down at the base of the tens column.
Final answer: 52 - 37 = 15 | {"url":"https://numbers.mathdial.com/subtraction-operation-subtracted-numbers.php?subtraction=19-16","timestamp":"2024-11-02T09:16:43Z","content_type":"text/html","content_length":"28920","record_id":"<urn:uuid:1d967ef1-e11b-4d97-90d0-b0ff3cc6d74d>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00531.warc.gz"} |
Algebra is great fun - you get to solve puzzles!
With computer games you play by running, jumping and finding secret things.
With Algebra you play with letters, numbers and symbols, and you also get to find secret things!
And when you learn some of the "tricks" it becomes a fun challenge to work out how to use your skills in solving each question.
The Basics
Linear Equations
Quadratic Equations
Solving Word Questions
Sequences and Series
Where to Next?
Other Interesting Topics You May Like | {"url":"http://wegotthenumbers.org/index-8.html","timestamp":"2024-11-08T09:30:38Z","content_type":"text/html","content_length":"8812","record_id":"<urn:uuid:b9358f5d-6aa6-4ced-b7c2-d6581c21c0a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00039.warc.gz"} |
Linear Algebra Foundations
Created: 2019-02-14
Updated: 2019-02-14
A concise summary of the core concepts in linear algebra.
In the attached document I cover fundamental topics such as vectors, matrices, systems of linear equations, and transformations.
Understanding these concepts is essential for tackling a wide range of mathematical and real-world problems.
Content Summary
• Vectors
□ Arithmetic
□ Magnitude
□ Parallel and orthogonal vectors
□ Vector projection
□ Basis vectors
□ Linear dependence and independence
• Matrices
□ Arithmetic
□ Elementary operations
□ Elementary Matrix
□ Determinant
□ Laplace Formula
□ Decomposition methods | {"url":"https://datawok.net/posts/linear-algebra/","timestamp":"2024-11-03T01:12:05Z","content_type":"text/html","content_length":"7530","record_id":"<urn:uuid:912a8551-97f0-4d1b-bd46-38c86563cfd4>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00636.warc.gz"} |
Probability distributions
A probability distribution is any graph or function that represents the likelihood of obtaining a result from some probability experiment – such as a random-chance event, a survey of randomly-chosen
individuals, an observational study. Any probability distribution has some domain and yields a range of values.
Probability distributions can be discrete, representing a countable domain, or continuous, representing an uncountable domain with an infinite number of possible values.
Here are a couple of examples of discrete probability distributions. In the first, we'll imagine that we have an "unfair" coin, one that comes up heads 75% of the time and tails the rest. Here's the
discrete distribution:
Notice that the probabilities of the only two possible outcomes of this experiment (we omit the very unlikely event that the coin lands on its edge) add to 1. In other words, it's absolutely certain
(probability = 1 or 100%) that one of the two outcomes will occur.
The second example is the distribution of the six possible outcomes of rolling a six-sided die. The set of possible outcomes is $\Omega = \{1, 2, 3, 4, 5, 6\}$. The capital Greek letter omega, $\
Omega$, is often used to represent the set of all possible outcomes of a probability experiment. Here's the distribution:
Both of these illustrate the two characteristics that are essential for any probability distribution:
1. The probability of any individual outcome has to be postive and less than 1:
$$0 \le P(x) \le 1$$
2. The sum of all probabilities in the distribution is 1.
$$\sum_{i=1}^N P(x_i) = 1$$
Discrete → continuous
Often there's no sharp dividing line between a discrete and a continuous distribution – or at least one we can treat as continuous. This example shows the results of a U.S. Census Bureau study of the
heights of American women ages 30-39. The actual discrete data is represented by the bar graph: the higher the bar, the more women (as a percent of all women 30-39) of that height. That is, the
higher the bar, the greater the probability of a woman you meet aged 30-39 being that height.
You can see from the graph that the mean height is about 64 inches (5'-4"). You can see that there are about as many women taller and shorter than the average, and that the numbers of women taller
and shorter than the average falls fairly smoothly away from it on either side. And there are more women of heights closer to the average than those of heights much shorter or taller than it.
The underlying curve (gray-shaded area) is a special curve we call the Normal distribution or the Gaussian distribution, and it shows what the data would likely look like if we had a very large
number of data points (see law of large numbers) – and indeed this was quite a large study. We would also head toward a more continuous curve if we divided our measurement units more finely, say in
increments of 1/4-inch units or smaller.
A good example of how a discrete distribution merges into a continuous distribution is the Galton board, something you might have seen in a science museum. In this example, about a thousand marbles
are dropped from the triangular compartment at the top of the board through the narrow opening, randomly striking the dowels below. Most of the marbles drop in a straighter downward path, but some
drop to the left and right sides – with decreasing likelihood as they get farther from the center. After many marbles drop, the pattern of the normal distribution becomes clear. This happens time
after time with the Galton board, but never with precisely the same result &ndash there's always some "noise". The more marbles, the closer to normal or Gaussian the distribution of heights will be,
and if we were to average the shape of the distribution over many trials, it would smooth out nicely into the continuous function that is the Normal or Gaussian distribution.
The Gaussian distribution, often referred to as the "bell curve," looks like this:
It is completely characterized by two numbers, the mean and the standard deviation $(\sigma)$, which is a measure of the width of the distribution.
Describing a distribution
Probability distributions come in all shapes and sizes. We need to be able to describe and classify them. In general, we want to know four things about any distribution:
• Shape – consider symmetry and skewness
• Center – often the mean or median The center gives us the location of the distribution in the domain.
• Spread – what extent does the data cover – what are the minimum and maximum? The spread of a distribution is usually characterized by one or both of two measures, the standard deviation $(\sigma)
$ and the interquartile range (IQR).
• Outliers – are any points suspiciously far from the rest? Might we have reason not to consider them in our analysis of the data?
The acronyms SCSO or SOCS might help you remember these features.
The most common distribution of data or measurements with variations that occur by random chance is the Normal or Gaussian distribution, shown above. We'll cover its mathematical form on another
The height of the normal curve represents the probability of some event occuring or of finding some value of the data. High means likely and low means unlikely. The standard Gaussian curve is
left-right symmetric, and the mean and median of the data are the same – right in the middle. Gaussian distributions can be very narrow or very spread out, representing high or low precision of the
data, respectively.
Not all distributions are symmetric. Some are skewed one direction or another. We say that the distribution below is "skewed to the left" because it looks like it's being dragged to the left by its
left-most point or "tail."
Likewise, the distribution below is "skewed to the right."
For skewed distributions, the mean of the data moves in the direction of the skewing, but the median remains closer to the peak of the curve, so the two differ as the distribution is skewed. Skewness
is a key descriptor of a probability distribution.
• Right skew → mean is to the right of the median
• Left skew → mean is the the left of the median
Just as a quick example of an important skewed distribution, the distribution of atomic or molecular speeds in any sample of a gas has the Maxwell-Boltzmann distribution shown below, a distinctly
right-skewed distribution that varies with the temperature of the sample.
Some distributions look quite different from the Normal distribution. They can be triangular, decaying exponentials, ... you name it. Our job will be to describe distributions as they appear.
Shape of a distribution: The center
We measure the center of a distribution in two ways, the mean and the median. When a distribution is symmetric from left to right, these are the same. It's when that symmetry fails that the median
can be more useful.
Here's a simple example: Let's say that there are 25 homes in a certain community. Their values, to the nearest \$50,000, are displayed in the dot plot below.
The median housing price is just the one in the middle of a sorted list. So all we need to do to find it is to count over 13 homes from the left, and we find that the median home price in this area
is \$350,000.
The mean price is just the sum of all homes divided by the number of homes. In summation notation, that's
$$\bar x = \frac{1}{25}\sum_{i=1}^{25} x_i = 350,000$$
So in this case the mean and median are the same, shown by the magenta dot on the chart. That's not too surprising because the distribution looks reasonably left-right symmetric.
Now let's do a little thought experiment. Imagine that a family comes into this community and builds a \$5-million home. What will that do to the mean and median? The graph below shows the picture.
Notice that the \$5-million home is such an outlier that the horizontal axis has to have a little bit cut out to show it.
Now, as shown on the graph, the mean has shifted to a value of \$536,000, which is higher than the price of most houses in the area. In this case, the mean might not be representative of the true
housing situation for a family looking to buy a home in the area.
The median home value, however, didn't change. It's the average of the 13^th and 14^th values, which remains \$350,000. So for a family looking to move into the neighborhood, they can be aware of
that \$5-million home, but also that the price they will pay for a home will more likley be in the \$350,000 range.
This is precisely why we commonly use these two measures of the center of a distribution. When the mean and median are the same, we usually have a left-right symmetric distribution. When the mean is
higher than the median, the distribution is skewed to the right, as is our wide home-value graph above. That \$5-million home really stretches the distribution out in that direction. When the mean is
smaller than the median, the distribution is usually skewed to the left.
The median is more resistant to change produced by outliers – data points far outside the more clustered data.
The median is the center value of an ordered list of data points. If the number of data points, $N$, is odd, then the median is the middle value. If the number of data points is even, then the median
is the average of the two points in the middle.
To find the mean of $N$ data points, we sum those points and divide by the number of points. The mean is usually denoted by placing a bar over the variable name, like $\bar x$, or by the Greek letter
mu, $\mu$.
$$\bar x = \frac{1}{N} \,\sum_{i=1}^N \, x_i$$
The median is more resistant to change induced by outliers – extreme values on the large and/or small ends of the distribution.
Shape of a distribution: The width
The width of a distribution corresponds to the precision of the data. A distribution that is narrow means that there is a low probability of values far from the mean or median. A wider distribution
means that data points far from the middle are more probable. Consider these two distributions of the distance of each dart from the bullseye of a dart board:
The player on the left is very good. She throws near the center of the board on every throw. The player on the right isn't as good. His throws sometimes hit the bullseye, but on average they're more
widely scattered.
We use two methods to measure and report the width of a distribution, the standard deviation and the interquartile range. Both are discussed in other sections (standard deviation, interquartile
range), so we won't go into them in great detail here.
Standard deviation
Strictly speaking, the standard deviation (denoted by the Greek lower case "s", $\sigma$) applies only to the Normal or Gaussian distribution. It is defined as the distance along the domain axis from
the mean to one of the two inflection points of the curve. An inflection point is a point on the graph where the curvature changes from concave upward to concave downward, or vice versa).
The standard deviation is the average distance of the squares of the distances of each member of a data set from the mean of that set. Here is the forumula in summation notation:
$$\sigma = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \bar x)^2}$$
We use the squares of the distances, $x_i - \bar x$, to avoid having the errors of a wide distribution cancel to a very small width, one that doesn't comport with the precision of the data. We
further divide not by $N$, the number of data points – as we would when calculating a mean – but $N - 1$ instead. This is because we've essentially already used one of our $N$ pieces of data to
calculate the mean, leaving only $N - 1$ left for other calculations. For more about the standard deviation, go here.
What we're doing here is calculating the sum of the squares of all of the distances of each point from the center, then dividing by the number of data points (minus one), then taking the square root
to undo the earlier squaring. We could have taken an absolute value rather than squaring (after all, $|x| = \sqrt{x^2}$), but it turns out that the squaring trick makes a lot of the mathematics of
statistics easier as things get more complicated.
Interquartile range (IQR)
The interquartile range (IQR) is appropriate for any distribution and not too difficult to calculate. The basic idea (see bar chart below) is that we divide the data in an ordered list into quartiles
, groups that all contain, by number, one quarter of the data points. The block from Q0 to Q1 is the first quartile of the data. The median is the second (Q2 and median are synonyms), cutting the
data set in half, and Q3 marks the point between the bottom 3/4 and the top quarter of the data. Of those, Q1, the median and Q3 are the important ones.
Let's use dot plot below, illustrating the number of barrels of oil recovered per year from 37 wells in an oil field, to calculate some quartiles and find the IQR.
The median (green dot) is the 19^th data point, the middle one. Now dividing the data into halves, both including the median – because $N$ is odd (see gray box belo), we find Q1 and Q2 by finding the
median of the lower and upper halves of the data, respectively. Those are the magenta lines at values of 35 and 65. So the IQR is $Q3 - Q1 = 65 - 35 = 30$.
Finding quartiles
Median (Q2): When the number of data points, $N$ is odd, the median is the center data point of an ordered list. When the number is even, Q2 is the average of the two center points.
Q1, Q3, $N$ odd: When $N$ is odd, divide the data in half, including the center point (Q2) in each half. Find the median of the first half; that's Q1. Find the median of the second half; that's Q3.
Q1, Q3, $N$ even: When $N$ is even, divide the data set into two halves at the median, then find the median of the first half; that's Q1. Find the median of the second half; that's Q3.
Q0 and Q4 are just the extreme ends of the sorted data list.
The IQR
For any sorted data set divided into quartiles, the IQR is a measure of the width of the set. It is
$$\text{IQR} = Q3 - Q1$$
Often when we collect data there are one or more points that just don't seem to fit. For example, let's say we're measuring reaction times in a chemistry experiment. We run the same reaction under
the same conditions ten times, finding an average time of 8.2 ± 1.1 minutes for nine of the measurements, but a time of 21 minutes for the tenth. That last measurement just doesn't look right — it's
an outlier, and we might be correct to think about what might have gone wrong with our technique. We might be tempted to throw that datum out of our data set. Would we be justified?
It's important to have some objective measure of what constitutes an outlier, some criterion that we all agree upon. One such criterion is the 1.5 IQR rule. It works like this:
• A data point is an outlier to the left if it is less than 1.5 times the IQR to the left of the first quartile (Q1).
$x \lt \text{Q1} - 1.5 \cdot \text{IQR} $
• A data point is an outlier to the right if it is greater than 1.5 times the IQR to the right of the third quartile (Q3).
$x \gt \text{Q3} + 1.5 \cdot \text{IQR} $
Let's look at our oil barrels example above and ask whether there are any outliers. For this data set, Q1 = 35 and Q3 = 65, so the IQR is 30 units. The condition for outliers on the left is
$$ x &\lt \text{Q1} - 1.5 \cdot \text{IQR} \\[5pt] x &\lt 35 - 1.5 \cdot 30 \\[5pt] x &\lt 35 - 45 \\[5pt] x &\lt -10 $$
There are no values less than -10, so there are no outliers to the left. Likewise, the condition for outliers on the right is
$$ x &\gt \text{Q3} + 1.5 \cdot \text{IQR} \\[5pt] x &\gt 65 + 1.5 \cdot 30 \\[5pt] x &\gt 65 + 45 \\[5pt] x &\gt 110 $$
There are no values greater than 110, so there are no outliers to the right, either; this data set has no outliers.
Example 1
Consider the dot plot showing the distribution of 52 scores on a chemistry test. Describe the distribution in terms of its shape, center, spread and outliers (SCSO).
Solution: The shape of this dot plot seems roughly symmetric and Normal (that is, has the shape of the Normal or Gaussian curve). If anything, it is skewed a bit to the right just a bit.
The center of the distribution can be measured in two ways: by calculating the mean and/or by finding the median. To find the mean, we just sum the values and divide by their number to get 70.9. The
median is the average of the 26^th and 27^th data points, for a value of 70. The mean is just a bit larger than the median, confirming just a little bit of rightward skew.
The first and third quartiles are the averages of the 13^th and 14^th points (Q1 = 62.5) and of the 39^th and 40^th points (Q3 = 80). The IQR is then 80 - 62.5 = 17.5. We can check for outliers:
• Left:
$$ x &\lt Q1 - 1.5 \cdot \text{IQR} \\[5pt] x &\lt 52.5 - 1.5(17.5) \\[5pt] x &\lt 26.25 $$
There are no scores less than 26.25, so there are no outliers on the left.
• Right:
$$ x &\gt Q3 + 1.5 \cdot \text{IQR} \\[5pt] x &\gt 80 + 1.5(17.5) \\[5pt] x &\gt 106.25 $$
There are no scores greater than 106.25, so there are no outliers on the right, either.
The spread of the data is between scores of 45 and 100, inclusive.
Because this data looks at least vaguely Normal with the mean approximately equal to the median, a Normal analysis is appropriate. Thus the standard deviation of the mean is $\sigma = 11.7$.
Uniform probability distribution
Note: the math after the heading "Average and variance" goes a little bit beyond what you might have encountered so far. If you can't follow it, just move on; that's OK.
The uniform probability distribution is one in which the probability of any outcome of a probability "experiment" is the same. A good example is the rolling of a single fair die, where "fair" means
an equal chance of rolling 1, 2, 3, 4, 5 or 6.
The probability of rolling a 1 is the same as that for rolling a 2 or 3, 4, 5 or 6. For each there is a 1 in 6, or ⅙ chance. We can sketch a graph of these discrete outcomes like this:
Now any probability distribution must also capture the idea of an assured outcome, that is, that when the experiment (tossing the die) is performed, one outcome is assured. The die will come up with
a result that is in the set of possible outcomes.
The total probability must sum to 1. In this case, we have six possible outcomes, each with a ⅙ probability, so the total area of our rectangular probability distribution graph (below) is 1.
We would refer to this as a normalized distribution.
In mathematical notation, we'd write the sum of all six of the probabilities as
$$\sum_{i=1}^6 \, P_i = 1$$
Coin tossing is another example of a probability experiment with a uniform distribution of outcomes. There is a ½ probability of tossing heads and a ½ probability of tossing tails, the only two
possible outcomes (we can approximate the probability of a coin landing on its edge to zero). The two probabilities sum to 1.
We might also have a uniform distribution of continuous outcomes, where our probability experiment could give any result between x = a and x = b, with equal likelihood, as in the graph below.
This is the most general representation of the uniform distribution. Notice that the height of the graph, which is the uniform probability, is fixed at b-a because we require the sum of all of the
probabilities to be equal to one. That's what we call a normalized probability distribution.
Average and variance
Now let's derive formulas for the average and variance $(\sigma^2)$ of this distribution. The average, $\bar x$, of a distribution over a range $[a, b]$ is obtained by doing a little calculus:
$$ \bar{x} &= \frac{1}{b - a} \int_a^b x \, dx \\ \\ &= \frac{1}{b - a} \frac{x^2}{2} \bigg|_a^b \\ \\ &= \frac{1}{2(b - a)} (b^2 - a^2) \\ \\ &= \frac{1}{2(b - a)} (b - a)(b + a) = \bf \frac{a + b}
{2} $$
That's just what we'd expect: add the high and the low and divide by two.
The variance of a discrete distribution is
$$\sigma^2 = \frac{1}{N} \sum_{i = 1}^N (x_i - \bar{x})^2$$. You can look over the details of calculating the variance of a continuous distribution below – it uses a little bit of calculus. The
result is
$$\sigma^2 = \frac{(b - a)^2}{12}$$
Example 2 – uniform distribution
Let's say we have a shoe store that sells pairs of shoes with a uniform probability of a sale throughout a seven-day week. That is, it's equally probable that a pair of shoes is sold on Monday as on
Tuesday. The minimum number of pairs sold per week is 50, and the max is 250. Calculuate the probability of selling between 100 and 150 pairs of shoes. Calculate the mean and standard deviation, $\
sigma = \sqrt{\sigma^2}$ of the distribution.
Solution: First, the probability is
$$P_{100-150} = (150 - 50) \cdot \frac{1}{250 - 50} = \frac{100}{200} = \frac{1}{2}$$
The mean of the distribution is
$$\bar{x} = \frac{a + b}{2} = \frac{50 + 250}{2} = 150 \; \text{pairs}$$
Finally the variance is
$$ \sigma^2 &= \frac{(b - a)^2}{12} \\ \\ &= \frac{(250 - 50)^2}{12} = 3,333 \\ \\ &\text{so }\; \sigma = \sqrt{3333} = 58. $$
The mean is $150 ± 58 \text{ pairs}$
Mean & variance of a continuous distribution
First we ought to normalize our PDF. That means ensuring that the area underneath it over its domain is equal to 1. ere's the process:
$$ f(x) &= a(1 - x^2) \tag{1} \\[5pt] a \int_{-1}^1 \, (1 - x^2) dx &= 1 \tag{2} \\[5pt] a \bigg[ x - \frac{x^3}{3} \bigg]_{-1}^1 &= 1 \\[5pt] 2a \bigg[ x - \frac{x^3}{3} \bigg]_0^1 \tag{3} \\[5pt]
2a \bigg[ 1 - \frac{1}{3} \bigg] &= 1 \\[5pt] 2a \bigg[ \frac{2}{3} \bigg] &= 1 \\[5pt] \frac{4}{3} a &= 1 \\[5pt] a &= \frac{3}{4} $$
So our normalized PDF is
$$f(x) = \frac{3}{4} (1 - x^2)$$
$$ \mu &= E(x) \\[5pt] &= \int_a^b \, x f(x) \, dx \\[5pt] &= \int_a^b \, x(1 - x^2) \, dx \\[5pt] &= \int_a^b \, (x - x^3) \, dx \\[5pt] &= \bigg[ \frac{x^2}{2} - \frac{x^4}{4} \bigg]_a^b \\[5pt] &=
\frac{b^2}{2} - \frac{b^4}{4} - \frac{a^2}{2} + \frac{a^4}{4} \\[5pt] \mu &= \frac{b^2 - a^2}{2} - \frac{b^4 - a^4}{4} $$
Just to check, let's think about a symmetric interval, with $a = b$. Then we have
$$\mu = \frac{a^2 - a^2}{2} - \frac{a^4 - a^4}{4} = 0,$$
which is what we expect for the mean of our distribution on a symmetric interval around zero.
The Bernoulli & Binomial distributions
Often we encounter situations in which our data answers a yes-or-no question. For a coin flip, for example, the result can either be heads or "not heads." It's a binary situation, at least if we
disregard the exceedingly-small possibility of the coin landing on its edge. The Bernoulli^* and binomial distributions are suitable for modeling probability experiments that ask such yes-or-no
For our coin flip, we might write the probability of tossing heads as P(H), then the probability of tails is P(!H), which we read as "not H," or "not heads." Further,
$$P(H) + P(!H) = 1.$$
The Bernoulli distribution need not represent 50/50 probabilities. It could also be used to model the behavior of a non-fair coin like one that came up heads 75% of the time, in which
$$ P(H) &= 0.75 \\[5pt] P(!H) &= 0.25 \\[5pt] P(H) + P(!H) &= 1 $$
Here's that distribution as a bar graph:
If we let the yes-event = 1 and the no event = 0, the average value of the Bernoulli distribution is
$$\bar{x} = 1(0.75) + 0(0.25) = 0.75$$
Binomial distribution
The Bernoulli distribution is the distribution for one binary event, as in the 75/25 coin flip above. The binomial distribution is the distribution for $n$ trials of such a binary event. For example,
if we flipped our 75/25 coin 100 times and plotted the results, that would be a binomial distribution.
The details of the binomial distribution will be left for another section, but briefly, the mean of a binomial distribution is
$$\mu = np$$
where $n$ is the number of trials and $p$ is the probability of an event occuring. The variance is
$$\sigma^2 = np(1-p)$$
*The distribution is named after Swiss mathematician Jacob Bernoulli who, among other major accomplishments, discovered the fundamental constant, e, the base of all continuously-growing exponential
The normal or Gaussian distribution
The curve under the bar graph above has a familiar "bell" shape. It's often referred to as a bell curve, but more often as the normal distribution or the Gaussian distribution, after Carl Friedrich
The curve is a probability distribution. You can always read its meaning by imagining that the vertical axis is a measure of the relative probability or likelihood of something happening and that all
of the somethings are arrayed in order along the x-axis.
The Gaussian curve is aways symmetric on either side of its maximum, and the maximum is the mean or average value. Whatever value or event is in the middle is the most likely. That "event" in our
women's height example would be the "event" of being 5'-4" tall. Out in the "wings," probability is the lowest: There are far fewer very short and very tall women, and the probability of being short
or tall is lower than being of more average height.
If we add up all of the probability under a Gaussian curve, we should get one (or 100%), the probability that something — anything at all — happened. Often we scale a Gaussian curve so that its total
area – the area under the curve – is one. That's called "normalizing" the distribution.
Here's another example before we move on. The graph below shows the results of 5000 simulated throws of two dice. The sum of both dice is shown. Notice that because there are more ways to come up
with a total of 7 (6+1, 5+2, 4+3),it's the most probable throw. After 5000 throws, the dice-total distribution looks pretty "normal."
Notice that in this example, we're not graphing probability but number of occurrences of a total, but the two should have the same shape. The sum of the heights of the green bars should be 5000, the
total number of throws. Likewise, throwing a 2 or a 12 is less likely than throwing a 7.
We could normalize this distribution by dividing each value column value by the sum of all columns. This would give us the percent chance (if we multiplied by 100) of each throw, and it would sale
the graph but maintain its shape. In the graph below, the green bars are the normalized simulated curve, and the purple bars are the exact expectations (see law of large numbers) we'd expect for a
very large number of throws.
So where does that curve come from?
That's a tricky question. It comes from modeling random chance, but the functional form of the curve has to be derived using calculus. In particular, it is derived using the second fundamental
theorem of calculus. We don't need to go there just yet, though; the result will serve our needs just fine. Here's what the Guassian function / Normal function looks like, with some explanation of
its parameters.
This function might look complicated, but think of the first part as a constant prefactor. Then the exponential function is a symmetric bell-shaped curve that's translated by -h units along the
x-axis and scaled by the 2σ^2 in the denominator. | {"url":"https://xaktly.com/ProbStat_Distributions.html","timestamp":"2024-11-11T18:19:06Z","content_type":"text/html","content_length":"57108","record_id":"<urn:uuid:7729c1dd-6c1b-4221-b42f-00b94dd76152>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00163.warc.gz"} |
Usage: 3dTfitter [options]
* At each voxel, assembles and solves a set of linear equations.
++ The matrix at each voxel may be the same or may be different.
++ This flexibility (for voxel-wise regressors) is one feature
that makes 3dTfitter different from 3dDeconvolve.
++ Another distinguishing feature is that 3dTfitter allows for
L2, L1, and L2+L1 (LASSO) regression solvers, and allows you
to impose sign constraints on the solution parameters.
* Output is a bucket dataset with the beta parameters at each voxel.
* You can also get output of fitted time series at each voxel, and
the error sum of squares (e.g., for generating statistics).
* You can also deconvolve with a known kernel function (e.g., an HRF
model in FMRI, or an arterial input function in DSC-MRI, et cetera),
in which case the output dataset is a new time series dataset,
containing the estimate of the source function that, when convolved
with your input kernel function, fits the data (in each voxel).
* The basic idea is to compute the beta_i so that the following
is approximately true:
RHS(t) = sum { beta_i * LHS_i(t) }
With the '-FALTUNG' (deconvolution) option, the model expands to be
RHS(t) = sum { K(j)*S(t-j) } + sum { beta_i * LHS_i(t) }
j>=0 i>=1
where K() is the user-supplied causal kernel function, and S() is
the source time series to be estimated along with the betas
(which can be thought of as the 'baseline' fit).
* The model basis functions LHS_i(t) and the kernel function K(t)
can be .1D files (fixed for all voxels) and/or 3D+time datasets
(different for each voxel).
* The fitting approximation can be done in 4 different ways, minimizing
the errors (differences between RHS(t) and the fitted equation) in
the following ways:
++ L2 [-l2fit option] = least sum of squares of errors
++ L1 [-l1fit option] = least sum of absolute values of errors
++ L2 LASSO = least sum of squares of errors, with an added
[-l2lasso option] L1 penalty on the size of the solution parameters
++ L2 Square Root LASSO = least square root of the sum of squared errors
[-l2sqrtlasso option] with an added L1 penalty on the solution parameters
***** Which fitting method is better?
The answer to that question depends strongly on what you are
going to use the results for! And on the quality of the data.
***** 3dTfitter is not for the casual user! *****
***** It has a lot of options which let you *****
***** control the complex solution process. *****
-RHS rset = Specifies the right-hand-side 3D+time dataset.
('rset' can also be a 1D file with 1 column)
* Exactly one '-RHS' option must be given to 3dTfitter.
-LHS lset = Specifies a column (or columns) of the left-hand-side matrix.
* More than one 'lset' can follow the '-LHS' option, but each
input filename must NOT start with the '-' character!
* Or you can use multiple '-LHS' options, if you prefer.
* Each 'lset' can be a 3D+time dataset, or a 1D file
with 1 or more columns.
* A 3D+time dataset defines one column in the LHS matrix.
++ If 'rset' is a 1D file, then you cannot input a 3D+time
dataset with '-LHS'.
++ If 'rset' is a 3D+time dataset, then the 3D+time dataset(s)
input with '-LHS' must have the same voxel grid as 'rset'.
* A 1D file defines as many columns in the LHS matrix as
are in the file.
++ For example, you could input the LHS matrix from the
.xmat.1D matrix file output by 3dDeconvolve, if you wanted
to repeat the same linear regression using 3dTfitter,
for some bizarre unfathomable twisted psychotic reason.
(See https://shorturl.at/boxU9 for more details.)
** If you have a problem where some LHS vectors might be tiny,
causing stability problems, you can choose to omit them
by using the '-vthr' option. By default, only all-zero
vectors will be omitted from the regression.
** Note that if the scales of the LHS vectors are grossly different
(e.g., 0 < vector#1 < 0.01 and 0 < vector#2 < 1000),
then numerical errors in the calculations might cause the
results to be unreliable. To avoid this problem, you can
scale the vectors (before running 3dTfitter) so that they
have similar magnitudes.
** Note that if you are fitting a time series dataset that has
nonzero mean, then at least some of your basis vectors
should have nonzero mean, or you won't be able to get a
good fit. If necessary, use '-polort 0' to fit the mean
value of the dataset, so that the zero-mean LHS vectors
can do their work in fitting the fluctuations in the data!
[This means you, HJJ!]
*** Columns are assembled in the order given on the command line,
which means that LHS parameters will be output in that order!
*** If all LHS inputs are 1D vectors AND you are using least
squares fitting without constraints, then 3dDeconvolve would
be more efficient, since each voxel would have the same set
of equations -- a fact that 3dDeconvolve exploits for speed.
++ But who cares about CPU time? Come on baby, light my fire!
-polort p = Add 'p+1' Legendre polynomial columns to the LHS matrix.
* These columns are added to the LHS matrix AFTER all other
columns specified by the '-LHS' option, even if the '-polort'
option appears before '-LHS' on the command line.
** By default, NO polynomial columns will be used.
-vthr v = The value 'v' (between 0.0 and 0.09, inclusive) defines the
threshold below which LHS vectors will be omitted from
the regression analysis. Each vector's L1 norm (sum of
absolute values) is computed. Any vector whose L1 norm
is less than or equal to 'v' times the largest L1 norm
will not be used in the analysis, and will get 0 weight
in the output. The purpose of this option is to let you
have tiny inputs and have them be ignored.
* By default, 'v' is zero ==> only exactly zero LHS columns
will be ignored in this case.
** Prior to 18 May 2010, the built-in (and fixed) value of
'v' was 0.000333. Thus, to get the old results, you should
use option '-vthr 0.000333' -- this means YOU, Rasmus Birn!
* Note that '-vthr' column censoring is done separately for
each voxel's regression problem, so if '-LHS' had any
dataset components (i.e., voxelwise regressors), a different
set of omitted columns could be used betwixt different voxels.
-FALTUNG fset fpre pen fac
= Specifies a convolution (German: Faltung) model to be
added to the LHS matrix. Four arguments follow the option:
-->** 'fset' is a 3D+time dataset or a 1D file that specifies
the known kernel of the convolution.
* fset's time point [0] is the 0-lag point in the kernel,
[1] is the 1-lag into the past point, etc.
++ Call the data Z(t), the unknown signal S(t), and the
known kernel H(t). The equations being solved for
the set of all S(t) values are of the form
Z(t) = H(0)S(t) + H(1)S(t-1) + ... + H(L)S(t-L) + noise
where L is the last index in the kernel function.
++++ N.B.: The TR of 'fset' (the source of H) and the TR of the
RHS dataset (the source of Z) MUST be the same, or
the deconvolution results will be revoltingly
meaningless drivel (or worse)!
-->** 'fpre' is the prefix for the output time series S(t) to
be created -- it will have the same length as the input
'rset' time series.
++ If you don't want this time series (why?), set 'fpre'
to be the string 'NULL'.
++ If you want to see the fit of the model to the data
(a very good idea), use the '-fitts' option, which is
described later.
-->** 'pen' selects the type of penalty function to be
applied to constrain the deconvolved time series:
++ The following penalty functions are available:
P0[s] = f^q * sum{ |S(t)|^q }
P1[s] = f^q * sum{ |S(t)-S(t-1)|^q }
P2[s] = f^q * sum{ |2*S(t)-S(t-1)-S(t+1)|^q }
P3[s] = f^q * sum{ |3*S(t)-3*S(t-1)-S(t+1)+S(t-2)|^q }
where S(t) is the deconvolved time series;
where q=1 for L1 fitting, q=2 for L2 fitting;
where f is the value of 'fac' (defined below).
P0 tries to keep S(t) itself small
P1 tries to keep point-to-point fluctuations
in S(t) small (1st derivative)
P2 tries to keep 3 point fluctuations
in S(t) small (2nd derivative)
P3 tries to keep 4 point fluctuations
in S(t) small (3nd derivative)
++ Higher digits try to make the result function S(t)
smoother. If a smooth result makes sense, then use
the string '012' or '0123' for 'pen'.
++ In L2 regression, these penalties are analogous to Wiener
(frequency space) deconvolution, with noise spectra
proportional to
P0 ==> fac^2 * 1 (constant in frequency)
P1 ==> fac^2 * freq^2
P2 ==> fac^2 * freq^4
P3 ==> fac^2 * freq^6
However, 3dTfitter does deconvolution in the time
domain, not the frequency domain, and you can choose
to use L2, L1, or LASSO (L2+L1) regression.
++ The value of 'pen' is a combination of the digits
'0', '1', '2', and/or '3'; for example:
0 = use P0 only
1 = use P1 only
2 = use P2 only
3 = use P3 only
01 = use P0+P1 (the sum of these two functions)
02 = use P0+P2
12 = use P1+P2
012 = use P0+P1+P2 (sum of three penalty functions)
0123 = use P0+P1+P2+P3 (et cetera)
If 'pen' does not contain any of the digits 0..3,
then '01' will be used.
-->** 'fac' is the positive weight 'f' for the penalty function:
++ if fac < 0, then the program chooses a penalty factor
for each voxel separately and then scales that by -fac.
++ use fac = -1 to get this voxel-dependent factor unscaled.
(this is a very reasonable place to start, by the way :-)
++ fac = 0 is a special case: the program chooses a range
of penalty factors, does the deconvolution regression
for each one, and then chooses the fit it likes best
(as a tradeoff between fit error and solution size).
++ fac = 0 will be MUCH slower since it solves about 20
problems for each voxel and then chooses what it likes.
setenv AFNI_TFITTER_VERBOSE YES to get some progress
reports, if you want to see what it is doing.
++ Instead of using fac = 0, a useful alternative is to
do some test runs with several negative values of fac,
[e.g., -1, -2, and -3] and then look at the results to
determine which one is most suitable for your purposes.
++ It is a good idea to experiment with different fac values,
so you can see how the solution varies, and so you can get
some idea of what penalty level to use for YOUR problems.
++ SOME penalty has to be applied, since otherwise the
set of linear equations for S(t) is under-determined
and/or ill-conditioned!
** If '-LHS' is used with '-FALTUNG', those basis vectors can
be thought of as a baseline to be regressed out at the
same time the convolution model is fitted.
++ When '-LHS' supplies a baseline, it is important
that penalty type 'pen' include '0', so that the
collinearity between convolution with a constant S(t)
and a constant baseline can be resolved!
++ Instead of using a baseline here, you could project the
baseline out of a dataset or 1D file using 3dDetrend,
before using 3dTfitter.
*** At most one '-FALTUNG' option can be used!!!
*** Consider the time series model
Z(t) = K(t)*S(t) + baseline + noise,
where Z(t) = data time series (in each voxel)
K(t) = kernel (e.g., hemodynamic response function)
S(t) = stimulus time series
baseline = constant, drift, etc.
and * = convolution in time
Then program 3dDeconvolve solves for K(t) given S(t), whereas
3dTfitter -FALTUNG solves for S(t) given K(t). The difference
between the two cases is that K(t) is presumed to be causal and
have limited support, while S(t) is a full-length time series.
*** Presumably you know this already, but deconvolution in the
Fourier domain -1
S(t) = F { F[Z] / F[K] }
(where F[] is the Fourier transform) is a bad idea, since
division by small values F[K] will grotesquely amplify the
noise. 3dTfitter does NOT even try to do such a silly thing.
****** Deconvolution is a tricky business, so be careful out there!
++ e.g., Experiment with the different parameters to make
sure the results in your type of problems make sense.
-->>++ Look at the results and the fits with AFNI (or 1dplot)!
Do not blindly assume that the results are accurate.
++ Also, do not blindly assume that a paper promoting
a new deconvolution method that always works is
actually a good thing!
++ There is no guarantee that the automatic selection of
of the penalty factor herein will give usable results
for your problem!
++ You should probably use a mask dataset with -FALTUNG,
since deconvolution can often fail on pure noise
time series.
++ Unconstrained (no '-cons' options) least squares ('-lsqfit')
is normally the fastest solution method for deconvolution.
This, however, may only matter if you have a very long input
time series dataset (e.g., more than 1000 time points).
++ For unconstrained least squares deconvolution, a special
sparse matrix algorithm is used for speed. If you wish to
disable this for some reason, set environment variable
AFNI_FITTER_RCMAT to NO before running the program.
++ Nevertheless, a FALTUNG problem with more than 1000 time
points will probably take a LONG time to run, especially
if 'fac' is chosen to be 0.
-lsqfit = Solve equations via least squares [the default method].
* This is sometimes called L2 regression by mathematicians.
* '-l2fit' and '-L2' are synonyms for this option.
-l1fit = Solve equations via least sum of absolute residuals.
* This is sometimes called L1 regression by mathematicians.
* '-L1' is a synonym for this option.
* L1 fitting is usually slower than L2 fitting, but
is perhaps less sensitive to outliers in the data.
++ L1 deconvolution might give nicer looking results
when you expect the deconvolved signal S(t) to
have large-ish sections where S(t) = 0.
[The LASSO solution methods can also have this property.]
* L2 fitting is statistically more efficient when the
noise is KNOWN to be normally (Gaussian) distributed
(and a bunch of other assumptions are also made).
++ Where such KNOWLEDGE comes from is an interesting question.
-l2lasso lam [i j k ...]
= Solve equations via least squares with a LASSO (L1) penalty
on the coefficients.
* The positive value 'lam' after the option name is the
weight given to the penalty.
++ As a rule of thumb, you can try lam = 2 * sigma, where
sigma = standard deviation of noise, but that requires
you to have some idea what the noise level is.
++ If you enter 'lam' as a negative number, then the code
will CRUDELY estimate sigma and then scale abs(lam) by
that value -- in which case, you can try lam = -2 (or so)
and see if that works well for you.
++ Or you can use the Square Root LASSO option (next), which
(in theory) does not need to know sigma when setting lam.
++ If you do not provide lam, or give a value of 0, then a
default value will be used.
* Optionally, you can supply a list of parameter indexes
(after 'lam') that should NOT be penalized in the
the fitting process (e.g., traditionally, the mean value
is not included in the L1 penalty.) Indexes start at 1,
as in 'consign' (below).
++ If this un-penalized integer list has long stretches of
contiguous entries, you can specify ranges of integers,
as in '1:9' instead of '1 2 3 4 5 6 7 8 9'.
**-->>++ If you want to supply the list of indexes that GET a
L1 penalty, instead of the list that does NOT, you can
put an 'X' character first, as in
-LASSO 0 X 12:41
to indicate that variables 12..41 (inclusive) get the
penalty applied, and the other variables do not. This
inversion might be more useful to you in some cases.
++ If you also want the indexes to have 1 added to them and
be inverted -- because they came from a 0-based program --
then use 'X1', as in '-LASSO 0 X1 12:41'.
++ If you want the indexes to have 1 added to them but NOT
to be inverted, use 'Y1', as in '-LASSO 0 Y1 13:42'.
++ Note that if you supply an integer list, you MUST supply
a value for lam first, even if that value is 0.
++ In deconvolution ('-FALTUNG'), all baseline parameters
(from '-LHS' and/or '-polort') are automatically non-penalized,
so there is usually no point to using this un-penalizing feature.
++ If you are NOT doing deconvolution, then you'll need this
option to un-penalize any '-polort' parameters (if desired).
** LASSO-ing herein should be considered experimental, and its
implementation is subject to change! You should definitely
play with different 'lam' values to see how well they work
for your particular types of problems. Algorithm is here:
++ TT Wu and K Lange.
Coordinate descent algorithms for LASSO penalized regression.
Annals of Applied Statistics, 2: 224-244 (2008).
* '-LASSO' is a synonym for this option.
-lasso_centro_block i j k ...
= Defines a block of coefficients that will be penalized together
with ABS( beta[i] - centromean( beta[i], beta[j] , ... ) )
where the centromean(a,b,...) is computed by sorting the
arguments (a,b,...) and then averaging the central 50% values.
* The goal is to use LASSO to shrink these coefficients towards
a common value to suppress outliers, rather than the default
LASSO method of shrinking coefficients towards 0, where the
penalty on coefficient beta[i] is just ABS( beta[i] ).
* For example:
-lasso_centro_block 12:26 -lasso_centro_block 27:41
These options define two blocks of coefficients.
-->>*** The intended application of this option is to regularize
(reduce fluctuations) in the 'IM' regression method from
3dDeconvolve, where each task instance gets a separate
beta fit parameter.
*** That is, the idea is that you run 3dTfitter to get the
'IM' betas as an alternative to 3dDeconvolve or 3dREMLfit,
since the centromean regularization will damp down wild
fluctuations in the individual task betas.
*** In this example, the two blocks of coefficients correspond
to the beta values for each of two separate tasks.
*** The input '-LHS' matrix is available from 3dDeconvolve's
'-x1D' option.
*** Further details on 'blocks' can be found in this Google Doc
including shell commands on how to extract the block indexes
from the header of the matrix file.
*** A 'lam' value for the '-LASSO' option that makes sense is a value
between -1 and -2, but as usual, you'll have to experiment with
your particular data and application.
* If you have more than one block, do NOT let them overlap,
because the program doesn't check for this kind of stoopidity
and then peculiar/bad things will probably happen!
* A block defined here must have at least 5 entries.
In practice, I would recommend at least 12 entries for a
block, or the whole idea of 'shrinking to the centromean'
is silly.
* This option can be abbreviated as '-LCB', since typing
'-lasso_centro_block' correctly is a nontrivial challenge :-)
*** This option is NOT implemented for -l2sqrtlasso :-(
* [New option - 10 Aug 2021 - RWCox]
-l2sqrtlasso lam [i j k ...]
= Similar to above option, but uses 'Square Root LASSO' instead:
* Approximately speaking, LASSO minimizes E = Q2+lam*L1,
where Q2=sum of squares of residuals and L1=sum of absolute
values of all fit parameters, while Square Root LASSO minimizes
sqrt(Q2)+lam*L1; the method and motivation is described here:
++ A Belloni, V Chernozhukov, and L Wang.
Square-root LASSO: Pivotal recovery of sparse signals via
conic programming (2010). http://arxiv.org/abs/1009.5689
++ A coordinate descent algorithm is also used for this optimization
(unlike in the paper above).
** A reasonable range of 'lam' to use is from 1 to 10 (or so);
I suggest you start with 2 and see how well that works.
++ Unlike the pure LASSO option above, you do not need to give
give a negative value for lam here -- there is no need for
scaling by sigma -- or so they say.
* The theoretical advantange of Square Root LASSO over
standard LASSO is that a good choice of 'lam' does not
depend on knowing the noise level in the data (that is
what 'Pivotal' means in the paper's title).
* '-SQRTLASSO' is a synonym for this option.
--------->>**** GENERAL NOTES ABOUT LASSO and SQUARE ROOT LASSO ****<<--------
* LASSO methods are the only way to solve a under-determined
system with 3dTfitter -- one with more vectors on the RHS
than time points. However, a 'solution' to such a problem
doesn't necessarily mean anything -- be careful out there!
* LASSO methods will tend to push small coefficients down
to zero. This feature can be useful when doing deconvolution,
if you expect the result to be zero over large-ish intervals.
++ L1 regression ('-l1fit') has a similar property, of course.
++ This difficult-to-estimate bias in the LASSO-computed coefficients
makes it nearly impossible to provide reliable estimates of statistical
significance for the fit (e.g., R^2, F, ...).
* The actual penalty factor lambda used for a given coefficient
is lam scaled by the L2 norm of the corresponding regression
column. The purpose of this is to keep the penalties scale-free:
if a regression column were doubled, then the corresponding fit
coefficient would be cut in half; thus, to keep the same penalty
level, lambda should also be doubled.
* For '-l2lasso', a negative lam additionally means to scale
by the estimate of sigma, as described earlier. This feature
does not apply to Square Root LASSO, however (if you give a
negative lam to '-l2sqrtlasso', its absolute value is used).
-->>** There is no 'best' value of lam; if you are lucky, there is
is a range of lam values that give reasonable results. A good
procedure to follow would be to use several different values of
lam and see how the results vary; for example, the list
lam = -1, -2, -4, -7, -10 might be a good starting point.
* If you don't give ANY numeric value after the LASSO option
(i.e., the next argument on the command line is another option),
then the program will use '-3.1415926536' for the value of lam.
* A tiny value of lam (say 0.01) should give almost the same
results as pure L2 regression.
* Data with a smaller signal-to-noise ratio will probably need
larger values of lam -- you'll have to experiment.
* The number of iterations used for the LASSO solution will be
printed out for the first voxel solved, and for ever 10,000th
one following -- this is mostly for my personal edification.
-->>** Recall: "3dTfitter is not for the casual user!"
This statement especially applies when using LASSO, which is a
powerful tool -- and as such, can be dangerous if not used wisely.
-consign = Follow this option with a list of LHS parameter indexes
to indicate that the sign of some output LHS parameters
should be constrained in the solution; for example:
-consign +1 -3
which indicates that LHS parameter #1 (from the first -LHS)
must be non-negative, and that parameter #3 must be
non-positive. Parameter #2 is unconstrained (e.g., the
output can be positive or negative).
* Parameter counting starts with 1, and corresponds to
the order in which the LHS columns are specified.
* Unlike '-LHS or '-label', only one '-consign' option
can be used.
* Do NOT give the same index more than once after
'-consign' -- you can't specify that an coefficient
is both non-negative and non-positive, for example!
*** Constraints can be used with any of the 4 fitting methods.
*** '-consign' constraints only apply to the '-LHS'
fit parameters. To constrain the '-FALTUNG' output,
use the option below.
* If '-consign' is not used, the signs of the fitted
LHS parameters are not constrained.
-consFAL c= Constrain the deconvolution time series from '-FALTUNG'
to be positive if 'c' is '+' or to be negative if
'c' is '-'.
* There is no way at present to constrain the deconvolved
time series S(t) to be positive in some regions and
negative in others.
* If '-consFAL' is not used, the sign of the deconvolved
time series is not constrained.
-prefix p = Prefix for the output dataset (LHS parameters) filename.
* Output datasets from 3dTfitter are always in float format.
* If you don't give this option, 'Tfitter' is the prefix.
* If you don't want this dataset, use 'NULL' as the prefix.
* If you are doing deconvolution and do not also give any
'-LHS' options, then this file will not be output, since
it comprises the fit parameters for the '-LHS' vectors.
-->>** If the input '-RHS' file is a .1D file, normally the
output files are written in the AFNI .3D ASCII format,
where each row contains the time series data for one
voxel. If you want to have these files written in the
.1D format, with time represented down the column
direction, be sure to put '.1D' on the end of the prefix,
as in '-prefix Elvis.1D'. If you use '-' or 'stdout' as
the prefix, the resulting 1D file will be written to the
terminal. (See the fun fun fun examples, below.)
-label lb = Specifies sub-brick labels in the output LHS parameter dataset.
* More than one 'lb' can follow the '-label' option;
however, each label must NOT start with the '-' character!
* Labels are applied in the order given.
* Normally, you would provide exactly as many labels as
LHS columns. If not, the program invents some labels.
-fitts ff = Prefix filename for the output fitted time series dataset.
* Which is always in float format.
* Which will not be written if this option isn't given!
*** If you want the residuals, subtract this time series
from the '-RHS' input using 3dcalc (or 1deval).
-errsum e = Prefix filename for the error sums dataset, which
is calculated from the difference between the input
time series and the fitted time series (in each voxel):
* Sub-brick #0 is the sum of squares of differences (L2 sum)
* Sub-brick #1 is the sum of absolute differences (L1 sum)
* The L2 sum value, in particular, can be used to produce
a statistic to measure the significance of a fit model;
cf. the 'Correlation Coefficient Example' far below.
-mask ms = Read in dataset 'ms' as a mask; only voxels with nonzero
values in the mask will be processed. Voxels falling
outside the mask will be set to all zeros in the output.
* Voxels whose time series are all zeros will not be
processed, even if they are inside the mask!
-quiet = Don't print the fun fun fun progress report messages.
* Why would you want to hide these delightful missives?
AFNI_TFITTER_VERBOSE = YES means to print out information during
the fitting calculations.
++ Automatically turned on for 1 voxel -RHS inputs.
AFNI_TFITTER_P1SCALE = number > 0 will scale the P1 penalty by
this value (e.g., to count it more)
AFNI_TFITTER_P2SCALE = number > 0 will scale the P2 penalty by
this value
AFNI_TFITTER_P3SCALE = number > 0 will scale the P3 penalty by
this value
You could set these values on the command line using the AFNI standard
'-Dvariablename=value' command line option.
* There is no option to produce statistical estimates of the
significance of the parameter estimates.
++ 3dTcorrelate might be useful, to compute the correlation
between the '-fitts' time series and the '-RHS' input data.
++ You can use the '-errsum' option to get around this limitation,
with enough cleverness.
* There are no options for censoring or baseline generation (except '-polort').
++ You could generate some baseline 1D files using 1deval, perhaps.
* There is no option to constrain the range of the output parameters,
except the semi-infinite ranges provided by '-consign' and/or '-consFAL'.
* This program is NOW parallelized via OpenMP :-) [17 Aug 2021 - RWCox]
Contrived Example:
The datasets 'atm' and 'btm' are assumed to have 99 time points each.
We use 3dcalc to create a synthetic combination of these plus a constant
plus Gaussian noise, then use 3dTfitter to fit the weights of these
3 functions to each voxel, using 4 different methods. Note the use of
the input 1D time series '1D: 99@1' to provide the constant term.
3dcalc -a atm+orig -b btm+orig -expr '-2*a+b+gran(100,20)' -prefix 21 -float
3dTfitter -RHS 21+orig -LHS atm+orig btm+orig '1D: 99@1' -prefix F2u -l2fit
3dTfitter -RHS 21+orig -LHS atm+orig btm+orig '1D: 99@1' -prefix F1u -l1fit
3dTfitter -RHS 21+orig -LHS atm+orig btm+orig '1D: 99@1' -prefix F1c -l1fit \
-consign -1 +3
3dTfitter -RHS 21+orig -LHS atm+orig btm+orig '1D: 99@1' -prefix F2c -l2fit \
-consign -1 +3
In the absence of noise and error, the output datasets should be
#0 sub-brick = -2.0 in all voxels
#1 sub-brick = +1.0 in all voxels
#2 sub-brick = +100.0 in all voxels
Yet More Contrivances:
You can input a 1D file for the RHS dataset, as in the example below,
to fit a single time series to a weighted sum of other time series:
1deval -num 30 -expr 'cos(t)' > Fcos.1D
1deval -num 30 -expr 'sin(t)' > Fsin.1D
1deval -num 30 -expr 'cos(t)*exp(-t/20)' > Fexp.1D
3dTfitter -quiet -RHS Fexp.1D -LHS Fcos.1D Fsin.1D -prefix -
* Note the use of the '-' as a prefix to write the results
(just 2 numbers) to stdout, and the use of '-quiet' to hide
the divertingly funny and informative progress messages.
* For the Jedi AFNI Masters out there, the above example can be carried
out on using single complicated command line:
3dTfitter -quiet -RHS `1deval -1D: -num 30 -expr 'cos(t)*exp(-t/20)'` \
-LHS `1deval -1D: -num 30 -expr 'cos(t)'` \
`1deval -1D: -num 30 -expr 'sin(t)'` \
-prefix -
resulting in the single output line below:
0.535479 0.000236338
which are respectively the fit coefficients of 'cos(t)' and 'sin(t)'.
Contrived Deconvolution Example:
(1) Create a 101 point 1D file that is a block of 'activation'
between points 40..50, convolved with a triangle wave kernel
(the '-iresp' input below):
3dConvolve -input1D -polort -1 -num_stimts 1 \
-stim_file 1 '1D: 40@0 10@1 950@0' \
-stim_minlag 1 0 -stim_maxlag 1 5 \
-iresp 1 '1D: 0 1 2 3 2 1' -nlast 100 \
| grep -v Result | grep -v '^$' > F101.1D
(2) Create a 3D+time dataset with this time series in each
voxel, plus noise that increases with voxel 'i' index:
3dUndump -prefix Fjunk -dimen 100 100 1
3dcalc -a Fjunk+orig -b F101.1D \
-expr 'b+gran(0,0.04*(i+1))' \
-float -prefix F101d
/bin/rm -f Fjunk+orig.*
(3) Deconvolve, then look what you get by running AFNI:
3dTfitter -RHS F101d+orig -l1fit \
-FALTUNG '1D: 0 1 2 3 2 1' F101d_fal1 012 0.0
3dTfitter -RHS F101d+orig -l2fit \
-FALTUNG '1D: 0 1 2 3 2 1' F101d_fal2 012 0.0
(4) View F101d_fal1+orig, F101d_fal2+orig, and F101d+orig in AFNI,
(in Axial image and graph viewers) and see how the fit quality
varies with the noise level and the regression type -- L1 or
L2 regression. Note that the default 'fac' level of 0.0 was
selected in the commands above, which means the program selects
the penalty factor for each voxel, based on the size of the
data time series fluctuations and the quality of the fit.
(5) Add logistic noise (long tails) to the noise-free 1D time series, then
deconvolve and plot the results directly to the screen, using L1 and L2
and the two LASSO fitting methods:
1deval -a F101.1D -expr 'a+lran(.5)' > F101n.1D
3dTfitter -RHS F101n.1D -l1fit \
-FALTUNG '1D: 0 1 2 3 2 1' stdout 01 -2 | 1dplot -stdin -THICK &
3dTfitter -RHS F101n.1D -l2fit \
-FALTUNG '1D: 0 1 2 3 2 1' stdout 01 -2 | 1dplot -stdin -THICK &
3dTfitter -RHS F101n.1D -l2sqrtlasso 2 \
-FALTUNG '1D: 0 1 2 3 2 1' stdout 01 -2 | 1dplot -stdin -THICK &
3dTfitter -RHS F101n.1D -l2lasso -2 \
-FALTUNG '1D: 0 1 2 3 2 1' stdout 01 -2 | 1dplot -stdin -THICK &
For even more fun, add the '-consfal +' option to the above commands,
to force the deconvolution results to be positive.
***N.B.: You can only use 'stdout' as an output filename when
the output will be written as a 1D file (as above)!
Correlation Coefficient Example:
Suppose your initials are HJJ and you want to compute the partial
correlation coefficient of time series Seed.1D with every voxel in
a dataset Rest+orig once a spatially dependent 'artifact' time series
Art+orig has been projected out. You can do this with TWO 3dTfitter
runs, plus 3dcalc:
(1) Run 3dTfitter with ONLY the artifact time series and get the
error sum dataset
3dTfitter -RHS Rest+orig -LHS Art+orig -polort 2 -errsum Ebase
(2) Run 3dTfitter again with the artifact PLUS the seed time series
and get the error sum dataset and also the beta coefficients
3dTfitter -RHS Rest+orig -LHS Seed.1D Art+orig -polort 2 \
-errsum Eseed -prefix Bseed
(3) Compute the correlation coefficient from the amount of variance
reduction between cases 1 and 2, times the sign of the beta
3dcalc -a Eseed+orig'[0]' -b Ebase+orig'[0]' -c Bseed+orig'[0]' \
-prefix CorrSeed -expr '(2*step(c)-1)*sqrt(1-a/b)'
3drefit -fbuc -sublabel 0 'SeedCorrelation' CorrSeed+orig
More cleverness could be used to compute t- or F-statistics in a
similar fashion, using the error sum of squares between 2 different fits.
(Of course, these are assuming you use the default '-lsqfit' method.)
PPI (psycho-physiological interaction) Example:
Suppose you are running a PPI analysis and want to deconvolve a GAM
signal from the seed time series, hoping (very optimistically) to
convert from the BOLD time series (typical FMRI signal) to a
neurological time series (an impulse signal, say).
If the BOLD signal at the seed is seed_BOLD.1D and the GAM signal is
GAM.1D, then consider this example for the deconvolution, in order to
create the neuro signal, seed_neuro.1D:
3dTfitter -RHS seed_BOLD.1D \
-FALTUNG GAM.1D seed_neuro.1D 012 -2 \
-l2lasso -6
** RWCox - Feb 2008, et seq. **
** Created for the glorious purposes of John A Butman, MD, PhD, Poobah **
** But might be useful for some other well-meaning souls out there **
* This binary version of 3dTfitter is compiled using OpenMP, a semi-
automatic parallelizer software toolkit, which splits the work across
multiple CPUs/cores on the same shared memory computer.
* OpenMP is NOT like MPI -- it does not work with CPUs connected only
by a network (e.g., OpenMP doesn't work across cluster nodes).
* For some implementation and compilation details, please see
* The number of CPU threads used will default to the maximum number on
your system. You can control this value by setting environment variable
OMP_NUM_THREADS to some smaller value (including 1).
* Un-setting OMP_NUM_THREADS resets OpenMP back to its default state of
using all CPUs available.
++ However, on some systems, it seems to be necessary to set variable
OMP_NUM_THREADS explicitly, or you only get one CPU.
++ On other systems with many CPUS, you probably want to limit the CPU
count, since using more than (say) 16 threads is probably useless.
* You must set OMP_NUM_THREADS in the shell BEFORE running the program,
since OpenMP queries this variable BEFORE the program actually starts.
++ You can't usefully set this variable in your ~/.afnirc file or on the
command line with the '-D' option.
* How many threads are useful? That varies with the program, and how well
it was coded. You'll have to experiment on your own systems!
* The number of CPUs on this particular computer system is ...... 1.
* The maximum number of CPUs that will be used is now set to .... 1.
++ Compile date = Oct 31 2024 {AFNI_24.3.06:linux_ubuntu_24_64} | {"url":"https://afni.nimh.nih.gov/pub/dist/doc/htmldoc/programs/alpha/3dTfitter_sphx.html","timestamp":"2024-11-11T01:25:33Z","content_type":"text/html","content_length":"52835","record_id":"<urn:uuid:5becbd7a-47b1-43a1-a116-daf90468ba30>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00091.warc.gz"} |
Quantitative & Qualitative Data | Digital Scholarship Handbook
There are two types of data, quantitative and qualitative. Generally speaking, when you measure something and give it a number value, you create quantitative data. When you classify or judge
something, you create qualitative data. There are also different types of quantitative and qualitative data. (Also see, Qualitative vs Quantitative Data article.)
Qualitative Data
Qualitative data is used to characterize objects or observations, which can be collected in a non-numerical and non-binary way, such as languages. Qualitative data can include:
Audio and video recordings
Experiment notes, lab reports
Two types of qualitative data include categorical, meaning data that can be organized in groups, and ordinal, meaning qualitative data that follows a natural order.
Quantitative Data
Quantitative data, as the name suggests, relates to the quantity of something, and typical examples of quantitative data are numbers. Quantitative data can include:
Surveys data, including longitudinal and cross-sectional studies
Calculations such as calculating monthly gross margin
Quantification: converting descriptive data to numbers such as satisfaction rating from 1-4
Two types of quantitative data include continuous, meaning numbers that can be made more precise and divided, e.g, a 4.3 earthquake, and discrete, meaning numbers that cannot be divided, e.g., the
number of people in a household cannot include a fraction such as 3.5. | {"url":"https://bcds.gitbook.io/handbook/digital-scholarship-methods/data/what-is-data/quantitative-and-qualitative-data","timestamp":"2024-11-06T06:00:58Z","content_type":"text/html","content_length":"241059","record_id":"<urn:uuid:dabe9b72-9a8f-41e0-8d0f-29170236e018>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00258.warc.gz"} |
Eyring Equation Calculator - Calculator Wow
Eyring Equation Calculator
The Eyring Equation Calculator is a fundamental tool in physical chemistry used to calculate reaction rates based on temperature-dependent parameters such as enthalpy (ΔH\Delta HΔH) and entropy (ΔS\
Delta SΔS). This article delves into its workings, significance, and practical applications.
Understanding the importance of the Eyring Equation Calculator:
• Precision in Rate Calculations: Provides accurate predictions of reaction rates under varying temperatures.
• Insight into Reaction Mechanisms: Helps elucidate the role of temperature in chemical reactions.
• Widely Applicable: Used across industries from pharmaceuticals to environmental sciences for reaction kinetics analysis.
How to Use
Using the Eyring Equation Calculator involves straightforward steps:
1. Temperature (K): Enter the temperature in Kelvin at which the reaction is studied.
2. ΔH (J/mol): Input the change in enthalpy, which measures heat change in the reaction per mole.
3. ΔS (J/mol·K): Provide the change in entropy, indicating the disorder of the system per mole per Kelvin.
4. Calculate kkk: Click the calculate button to obtain the rate constant kkk based on the Eyring equation.
10 FAQs and Answers
1. What is the Eyring Equation used for?
The Eyring Equation calculates the rate constant kkk of a chemical reaction as a function of temperature, enthalpy (ΔH\Delta HΔH), and entropy (ΔS\Delta SΔS).
2. How accurate is the Eyring Equation Calculator?
The calculator provides accurate predictions when precise values of temperature, ΔH\Delta HΔH, and ΔS\Delta SΔS are inputted, aiding in reliable rate constant estimations.
3. Can the calculator handle different units for ΔH\Delta HΔH and ΔS\Delta SΔS?
Yes, ensure consistent units (usually joules per mole for ΔH\Delta HΔH and joules per mole per Kelvin for ΔS\Delta SΔS) for accurate results.
4. What is the significance of temperature in the Eyring equation?
Temperature directly influences the rate of reaction. The Eyring equation quantitatively describes how rate constants change with temperature changes.
5. In which fields is the Eyring Equation Calculator utilized?
It is used in chemical kinetics studies, pharmaceutical industries for drug stability assessments, and environmental sciences for understanding reaction dynamics.
6. Are there limitations to its application?
The Eyring equation assumes ideal conditions and may not account for complex reaction mechanisms or non-ideal behavior in all scenarios.
7. How does the calculator handle negative values of ΔH\Delta HΔH or ΔS\Delta SΔS?
Negative values are valid inputs and are processed accordingly in the calculation of the rate constant kkk.
8. Can this calculator be used for educational purposes?
Yes, it serves as an educational tool for students and researchers to understand the relationship between temperature and reaction rates.
9. What insights does the Eyring Equation provide into reaction kinetics?
It offers insights into the activation energy of reactions, aiding in understanding reaction mechanisms and optimizing reaction conditions.
10. How can the Eyring Equation Calculator improve research and development?
By accurately predicting reaction rates at different temperatures, it facilitates informed decisions in process optimization and product development.
The Eyring Equation Calculator stands as a pivotal tool in chemical kinetics, providing insights into temperature-dependent reaction rates crucial for various scientific and industrial applications.
By mastering its use and understanding its theoretical underpinnings, researchers and practitioners can leverage its capabilities to advance fields reliant on precise reaction rate predictions. | {"url":"https://calculatorwow.com/eyring-equation-calculator/","timestamp":"2024-11-07T01:05:59Z","content_type":"text/html","content_length":"64986","record_id":"<urn:uuid:5830c186-700c-4177-8126-a7f4b8238150>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00634.warc.gz"} |