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Math, Grade 6, Rate, Using Rate To Determine Speed & Distance
Material Type:
Lesson Plan
Middle School
Media Formats:
Using Rate To Determine Speed & Distance
In this lesson, students watch a video of a runner and express his speed as a rate in meters per second. Students then use the rate to determine how long it takes the runner to go any distance.
Key Concepts
Speed is a rate that is expressed as distance traveled per unit of time. Miles per hour, laps per minute, and meters per second are all examples of units for speed. The measures of speed, distance,
and time are all related. The relationship can be expressed in three ways: d = rt,r = dt,t = dr.
Goals and Learning Objectives
• Explore speed as a rate that measures the relationship between two aspects of a situation: distance and time.
• In comparing distance, speed, and time, understand how to use any two of these measures to find the third measure.
Jason Running
Lesson Guide
Have students watch the video and try to answer the questions. Students may need to watch it more than once to figure out the two aspects of the situation tracked by the double number line.
Discuss the questions as a class.
ELL: Help ELLs build their mathematical vocabulary by continually modeling the use of new terms in the context of classroom work and activities. If time allows, provide an opportunity for ELLs to
build their own sentences using the new vocabulary.
To successfully approach the Work Time problems, students need to be able to answer and understand all three of the following questions:
1. What two quantities do you see in this video (tracked by the number lines)?
□ One number line tracks the distance Jason runs.
□ The other number line tracks the time that he runs.
2. How does the double number line show the relationship between the two quantities?
□ The double number line shows how much time has passed for each given distance, and it shows the distance Jason has run for each given time.
□ Dividing any distance in meters on the double number line by the corresponding time in seconds gives us the following rate: Jason runs at a speed of 5 meters per second.
3. What is Jason's speed in meters per second?
Discuss these questions with the class. Keep the discussion short, but make sure that all students understand that Jason's speed is 5 meters per second. They will use this rate during Work Time.
Jason Running
Watch the video and think about the following questions.
• What two quantities in this video can be measured?
• How does the double number line track the relationship between the quantities?
VIDEO: Jason Running
Math Mission
Lesson Guide
Discuss the Math Mission.
Students will use Jason's speed, expressed as the rate 5 meters per second, to find how long it takes him to go any distance.
SWD: Students with disabilities may have difficulty with the complexity of this Math Mission. Consider how you may provide all students access to this information in a manageable way. One option is
to break down the task into discreet, sequential steps (first, students will find the ratio; then, they will use that ratio to help them calculate the time to travel a given distance).
Explain how to use Jason’s speed to find how long it takes him to go any distance.
Jason’s Running Speed
Lesson Guide
Have students who finish early:
• Prepare to present their work to the class.
• Find another way to solve this problem, or show their solution using another representation.
ELL: For the Work Time activity, make sure you demonstrate and orally explain the activity step by step to ensure that ELLs understand what they are being asked to do.
Mathematical Practices
Mathematical Practice 1: Make sense of problems and persevere in solving them.
• As students work, look for students who make sense of the problem situation and understand the relationship between speed, distance, and time.
Mathematical Practice 5: Use appropriate tools strategically.
• Identify students who appropriately use the double number line tool or a table to represent the relationship between distance and time.
Mathematical Practice 6: Attend to precision.
• Watch for students who attend to precision when calculating or when labeling their double number lines and tables.
Student has difficulty getting started.
• Describe the problem in your own words to your partner.
• What information do you know?
• What are you trying to find?
• What does the rate 5 meters per second mean?
• Can you use a double number line or a table to help you?
• Can you try solving a simpler problem?
Student has an incorrect solution.
• Multiplies 5 meters per second by 240 meters:
□ Does your answer 1,200 seconds make sense? How many minutes are equal to 1,200 seconds?
□ Can you use a double number line or a table to support your answer?
□ Explain why you multiplied the rate 5 meters per second by 240 meters.
Student has a solution.
• Explain your strategy for solving the problem.
• Where do you see the rate in your [diagram, double number line, table]?
• Where do you see the answer in your [diagram, double number line, table]?
• Could you have used a different method to solve this problem? Explain.
• It will take Jason 48 seconds to run 240 meters.
• Jason can run 275 meters in 55 seconds.
• Students' answers should include an estimate, the quantities involved, a representation of their solution, equations, and two complete sentences.
Work Time
Jason’s Running Speed
Jason runs at a constant speed of 5 meters per second.
• How much time does it take Jason to run 240 meters?
• How many meters can Jason run in 55 seconds?
• Include the following in your answers:
□ An estimate of what you think the answers might be
□ The quantities involved in the problem
□ A diagram, table, double number line, or other representation that shows why your solution makes sense
□ Your equations, your work, and your solutions
□ Two complete sentences that answer the questions
Multiply the rate 5 meters per second by the amount of time to find the distance traveled. For example, to find the distance traveled in 20 seconds, multiply 5 meters per second by 20 seconds:
20 seconds • 5 meters/second = 100 meters
Prepare a Presentation
Preparing for Ways of Thinking
As students work, look for responses in which students:
• Multiply speed by distance. 5 meters 1 second⋅240 meters = 1,200 seconds
• Correctly divide distance by speed. 240 meters ÷ 5 meters1 second = 48 seconds
• Find the rate 0.2 seconds per meter and use it to solve the first problem.
1 second 5 meters = 0.2 seconds per meter
0.2 seconds 1 meters⋅240 meters = 48 seconds
Challenge Problem
• Michael Duane Johnson ran the 200-meter race at a speed of 10.35 meters per second and the 400-meter race at a speed of 9.20 meters per second.
• Possible solution: His 200-meter speed was faster because he ran a longer distance—10.35 meters compared to 9.20 meters—in the same time (1 second).
Work Time
Prepare a Presentation
Prepare a presentation about your method for solving the problem.
• Explain what you did differently to find the time compared with finding the distance.
• Use your work to support your explanation.
Challenge Problem
At the 1996 Olympics, Michael Duane Johnson set world records for the 200-meter and 400-meter races. He ran the 200-meter race in 19.32 seconds and the 400-meter race in 43.49 seconds.
• Calculate his speed for each race using a rate.
• In which race did he have the fastest speed? Justify your thinking.
Make Connections
Lesson Guide
Select a variety of student work with an eye to drawing out correspondences between sensible methods. By the end of the discussion, students should understand each other's methods and why some
methods did not work.
SWD: The idea that three different presentations of data represent the same information may be challenging for students with disabilities. Review and reinforce this idea whenever possible to help
students comprehend and recall this important concept.
Ask students who used different methods (for example, a table, a number line, or division by the rate) to present. As part of the discussion, challenge students to compare their representations of
the problem.
• How did [Names] make sense of the problem, and how did they model the problem situation mathematically? What is similar about their approaches? What is different?
• Which method helps you to see the structure of the mathematics the most clearly? Which method makes the most sense to you?
• [Name] multiplied 0.2 seconds per meter by 240 meters. Why does this method work? How does this method compare to dividing 240 meters by 5 meters per second?
• How can you find the distance using the rate 0.2 meter per second? How does this approach compare to using the rate 5 meters per second to find the distance?
• If no students notice, you might point out that dividing any number on the distance number line by 5 results in the corresponding number on the time number line.
At the end of the discussion, remind students that when a rate is something we use many times, sometimes it has a special name. Speed is the rate that represents distance over time. Population
density is a rate that represents population per area.
Performance Task
Ways of Thinking: Make Connections
• Take notes during the class discussion about how to solve for both distance and time.
• Answer questions about the methods you used to solve the problem.
As your classmates present, ask questions such as:
• Where do you see the rate 5 meters per second in your methods? Where do you see the answer?
• What operation did you use and why?
• Can you explain your solution in terms of the unit: what are the units of the rate, and how did the seconds cancel out?
Distance Equals Rate Times Time
Student has difficulty getting started.
• Describe the problem in your own words to your partner.
• What information do you know?
• What are you trying to find?
• Can you use a double number line or a table to help you?
• Can you try solving a simpler problem?
Student has a solution.
• Explain your strategy for solving the problem.
• Where do you see the rate in your [diagram, double number line, table]?
• Where do you see the answer in your [diagram, double number line, table]?
• Could you have used a different method to solve this problem? Explain.
• Emma's walking rate is 1/20 of a mile per minute, or 20 minutes per mile.
• Emma can walk 4.5 miles in 90 minutes.
• Emma can walk 0.75 mile in 15 minutes.
• Students will make a double number line to show the relationship between the numbers.
Work Time
Distance Equals Rate Times Time
Emma walked 3 miles in 60 minutes.
• What was her walking speed in terms of a rate?
• If she walks 4.5 miles at the same rate, how long will she walk?
• If she walks for 15 minutes at the same rate, how far will she walk?
• Make a double number line to show the relationships between the numbers.
• What are the quantities in the situation?
• How can you find the walking speed in terms of a rate?
• Did you use the expression d =rt to help you solve for time or distance?
Understanding Speed
A Possible Summary
Speed is a rate that is calculated by dividing distance by time. You can multiply time by speed to find any distance. The quantities speed, distance, and time are all related. The relationship can be
expressed in three ways: d = rt,r = dt,t = dr.
Additional Discussion Points
Speed is a rate that is expressed as the distance traveled in a unit of time. Miles per hour, laps per minute, and meters per second are all examples of units for speed.
Formative Assessment
Summary of the Math: Understanding Speed
Summarize the mathematics of speed and how speed relates to rate.
Check your summary:
• Does your summary explain what speed means, and does it give an example?
• Does your summary include the term rate ?
• Does your summary show the relationship among speed, distance, and time in three different ways?
Reflect On Your Work
Lesson Guide
Have each student write a brief reflection before the end of class. Review students' reflections.
If you find some reflections interesting enough to pursue later, you can save them and share them with the class when appropriate.
Work Time
Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.
Something I learned about speed that really helps me solve problems is … | {"url":"https://openspace.infohio.org/courseware/lesson/2136/overview","timestamp":"2024-11-02T21:12:46Z","content_type":"text/html","content_length":"71018","record_id":"<urn:uuid:5f5aaf4f-907a-4e08-b9b3-8343bccdb6e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00723.warc.gz"} |
Determining Speed and Wavelength of Red Light in Plastic
How can we calculate the speed and wavelength of red light in a medium like plastic?
What equation can we use to determine the speed of light in plastic? How can we find the wavelength of light in a medium?
Calculating Speed and Wavelength of Red Light in Plastic
In order to calculate the speed and wavelength of red light in a medium like plastic, we can use the equations involving the speed of light in a medium and the wavelength of light in a medium.
The speed of light in a medium, such as plastic, is given by the equation v = c/n, where:
v is the speed of light in the medium,
c is the speed of light in vacuum (approximately 3 x 10^8 m/s), and
n is the refractive index of the medium.
To determine the speed of red light in plastic, we first need to know the refractive index of the specific plastic material.
The wavelength of light in a medium can be determined using the equation λ = λ0/n, where:
λ is the wavelength of light in the medium,
λ0 is the wavelength of light in vacuum, and
n is the refractive index of the medium.
Given that the red laser has a wavelength of 633 nm (633 x 10^-9 m) in vacuum, we can calculate the wavelength of red light in plastic using the refractive index of the plastic material. | {"url":"https://bsimm2.com/physics/determining-speed-and-wavelength-of-red-light-in-plastic.html","timestamp":"2024-11-07T06:36:31Z","content_type":"text/html","content_length":"21251","record_id":"<urn:uuid:f23142f7-37a7-4153-ae5a-1c603e48d034>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00467.warc.gz"} |
Math Forum :: View topic - Binomial theorem - www.mathdb.org
Math Forum :: View topic – Binomial theorem
It sounds strange. the coefficient of x^n is actually an expression consisting of fibonacci sequence and nCr. I constructed the generating function of it, that is, x(1+x)^n/(x^2+x-1), and that
expression cannot be 0… That expression is actually nC1 * f1 + nC2* f2 +… + nCn*fn | {"url":"https://www.mathdb.org/phpbb2/viewtopicphpp3871amp/","timestamp":"2024-11-06T01:45:36Z","content_type":"text/html","content_length":"26940","record_id":"<urn:uuid:6657a9f8-2842-4fde-a01d-7889107d1a8a>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00775.warc.gz"} |
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Translation Math - Math Lessons
Translation Math
Greeting math peeps! Today we are going to talk about translation math in reference to transformations and geometry! Translations are a type of transformation where we take a point, line, or shape
and move it up, down, left, or right on a coordinate plane. A shape retains its angles, lengths, and direction after a translation, all we are doing is simply moving it around the coordinate plane.
Think you are ready to master the topic!? Keep reading for more explanation, video lesson, and step by step examples. Happy calculating!
Translations Math Formula:
P(x,y) -> P^| (x+h, y+k)
h=Horizontal Shift (add (+) when moving right, subtract (-) when moving left)
k= Vertical Shift (add (+) when moving up, subtract (-) when moving down)
What are Translations?
Translations are a type of transformation in geometry where we take a point, line, or shape and move it up, down, left, or right on a coordinate plane. Let’s take a look at what a translation looks
like below.
In the translation example above, we go from square ABCD to square A^|B^|C^|D^| translating each coordinate of the original square 6 units to the right and 2 units up.
Properties of a Translation + Rigid Motion:
When a shape, line, or point is translated, the shape retains Rigid Motion, meaning its length, area, and angles are unaffected by the transformation, as well as, maintaining its same direction.
1. Angle values remain the same.
2. Parallel and perpendicular lines remain the same. (shape/line maintains same direction)
3. Length, area, and perimeter remain the same.
There are two main types of translations: horizontal translations and vertical translations, let’s take a look at what each mean below.
Horizontal Translation:
When we translate a point, line, or shape left or right, it is undergoing a horizontal translation along the x-axis. Any type of left or right movement on a coordinate plane is a horizontal
How does this affect the x-coordinate? If the shape is being translated to the right, then we are adding units to the x-coordinate, and if the shape is shifting left then we are subtracting units
from the x-coordinate.
Vertical Translation:
When we translate a point, line, or shape up or down, it is undergoing a vertical translation along the y-axis. Any type of up and down movement on a coordinate plane is a vertical translation.
How does this affect the y-coordinate? If the shape is being translated up, then we are adding units to the y-coordinate, and if the shape is being shifted down then we subtract from the
Even though a horizontal shift or a vertical shift can happen when we move a shape, line, or point, many translations have a combo of the two!
How do Coordinates Change after a Translation?
The truth is there is no one unique rule for translations, but numbers will always be added or subtracted from the x and/or y coordinate values. If something is translated to the right, then we add
units to the x-value. On the other had if something is translated to the left, we subtract units from the x-value. The same can be said for moving a shape up, we then add units to the y-value, and
if a shape is translated down, we subtract units from the y-value. This gives us the following translation formula below:
If we look at our example, when we translate original square ABCD to square A^|B^|C^|D^| we end up translating each coordinate of original square ABCD 6 units to the right and 2 units up. What we
are really doing when we translate is adding 6 units to each x-coordinate as well as adding 2 units to each y-coordinate of the original figure square ABCD. Check it out below:
Now that we understand more about translations, let’s jump into some examples!
Translation Math Examples
Example #1: Translating Triangles
The Game Plan: To solve this problem, we’re going to need to compare two corresponding coordinate points before and after the translation. We can look at any of the following pairs to find our
answer A & A^|, B & B^|, C & C^|. Still confused? If so, no worries, because we will take this step by step!
Step 1: Identify a pair of coordinates to compare before and after the translation. In this case we compare Points A and A^|.
Step 2: Now we need to measure how many units to the right and how many units up Point A^| is from Point A.
We’ve gone over translating shapes, alas, shapes aren’t the only things that can be translated. Fun surprise: lines can be translated too! The same rules apply to translating lines as they do to
shapes. The nice thing about translating a line vs. a shape is that we now have even less work to do (as there are less points to translate).
Example #2: Translating Lines
The Game Plan: This question is asking us to find the translation from line RT to line R^|T^| . We’re going to need to compare corresponding coordinate points before and after the translation. We
can look at a set of points before and after the translation including Point R & R^| or Point T & T^|. Nothing we can’t handle! Let’s do this!
Step 1: Identify a pair of corresponding points before and after the translation, for this problem let’s focus on Point R and Point R^|.
Step 2: Now we measure how many units to the left and how many units down Point R^| is from Point R.
Practice Questions:
Other Transformations:
If you’re looking for more Transformations other than on the translation math you see here, check out these lessons below! Or, click this link here!
Also, be on the lookout for the latest MathSux lessons and study tips on Twitter, Facebook, YouTube, and TikTok! Happy calculating!
Translation math
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3 thoughts on “Translation Math”
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Lesson 3
Types of Transformations
3.1: Why is it a Dilation? (10 minutes)
In this warm-up, students analyze the results of the transformation rule \((x,y)\rightarrow (3x,3y)\). They connect the geometric definition of a dilation to this coordinate rule that produces a
Student Facing
Point \(B\) was transformed using the coordinate rule \((x,y) \rightarrow (3x,3y)\).
1. Add these auxiliary points and lines to create 2 right triangles: Label the origin \(P\). Plot points \(M=(2,0)\) and \(N=(6,0)\). Draw segments \(PB',MB,\) and \(NB’\).
2. How do triangles \(PMB\) and \(PNB’\) compare? How do you know?
3. What must be true about the ratio \(PB:PB'\)?
Activity Synthesis
The goal of the discussion is to connect the geometric definition of a dilation to the coordinate rule \((x,y) \rightarrow (3x,3y)\). Here are some questions for discussion:
• “Why does it make sense that the ratios of the legs of the triangles were both \(1:3\)?” (This is because we multiplied each coordinate by 3, and the legs are just the vertical and horizontal
distances to the points.)
• “Look at the definition of a dilation on your reference chart. How does it match what’s happening the coordinate rule \((x,y) \rightarrow (3x,3y)\)?” (For scale factor \(k\), a dilation takes a
point \(k\) times farther away from the center of dilation. Here, the rule \((x,y) \rightarrow (3x,3y)\) took the image of \(B\) 3 times farther from the origin than the original.)
3.2: Congruent, Similar, Neither? (10 minutes)
In this activity students match graphs to rules in coordinate transformation notation. Then they analyze both the rules and images to decide which represent similarity transformations and which
represent rigid transformations.
Students will use a variety of strategies to explain which figures are similar or congruent (counting, Pythagorean Theorem, recognizing right angles, properties of isosceles triangles, and
trigonometry). Monitor for students who use these different methods.
Student Facing
Match each image to its rule. Then, for each rule, decide whether it takes the original figure to a congruent figure, a similar figure, or neither. Explain or show your reasoning.
1. \((x,y) \rightarrow \left(\frac{x}{2}, \frac{y}{2}\right)\)
2. \((x,y) \rightarrow (y, \text-x)\)
3. \((x,y) \rightarrow (\text-2x, y)\)
4. \((x,y) \rightarrow (x-4, y-3)\)
Student Facing
Are you ready for more?
Here is triangle \(A\).
1. Reflect triangle \(A\) across the line \(x=2\).
2. Write a single rule that reflects triangle \(A\) across the line \(x=2\).
Anticipated Misconceptions
Some students may not be sure how to work with the rule \((x,y)\rightarrow (y,\text-x)\). Ask these students, “What are the coordinates of point \(A\)?” (\((3,\text-4)\).) “Which of those is the \(x
\)-coordinate?” (3.) “In the transformation rule, where does the \(x\) land, and what happens to it?” (The \(x\)-coordinate lands in the \(y\) spot, and its sign is opposite.) Another option is to
suggest students write out \(x=3\) and \(y=\text-4\), then substitute each value into the transformation rule.
If students suggest that figures are congruent or similar simply because they look that way, tell them that they need to provide more backing for their answer. What are some ways we can verify that 2
figures are congruent or similar? Remind students that for triangles, they’ve learned some shortcuts that they can use here.
Activity Synthesis
Select previously identified students to share their reasoning. Ask them how they calculated side lengths or angle measures. The key point to emphasize is that similar figures have congruent angles
and proportional sides, while congruent figures have congruent angles and sides. Students know some shortcuts for triangles, but for the rectangles there is no shortcut.
Conversing, Representing: MLR8 Discussion Supports. Use this routine to amplify mathematical uses of language to justify whether the figures are congruent, similar, or neither. After students share a
response, invite them to repeat their reasoning using mathematical language relevant to the lesson, such as angles, sides, congruent, and proportional. For example, ask students, “Can you say that
again, using the term ‘proportional’?” Consider inviting the remaining students to repeat these phrases to provide additional opportunities for all students to produce this language.
Design Principle(s): Support sense-making
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display
sentence frames to support student conversation, such as: “I matched _____ to _____ because . . . .”, “I noticed _____ so I . . . .”, “Why did you . . .?”, and “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills
3.3: You Write the Rules (15 minutes)
In this activity students work backwards to figure out a transformation rule. They are given the opportunity to come up with their own method of organizing their work. Monitor for students who have
methods of clearly documenting information they are given as well as students who record their attempts at a rule. Once students write a rule, they analyze the figures to decide if the transformation
takes shapes to congruent shapes.
Give students 2–3 minutes of quiet work time. If most students haven’t come up with a method to organize their information at that point, invite a student who has made a table to share.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the question about whether triangles \(ABC\) and \(A’B’C’\)
are congruent, similar, or neither. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their
ideas and clarify their language. For example, “How do you know that \(AC\) and \(A’C’\) are not congruent?”, and “How do you know that the corresponding sides of the triangle and its image are not
proportional?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for why the triangle and its
image are neither similar nor congruent.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Student Facing
1. Write a rule that will transform triangle \(ABC\) to triangle \(A’B’C’\).
2. Are \(ABC\) and \(A’B’C’\) congruent? Similar? Neither? Explain how you know.
3. Write a rule that will transform triangle \(DEF\) to triangle \(D’E’F’\).
4. Are \(DEF\) and \(D’E’F’\) congruent? Similar? Neither? Explain how you know.
Anticipated Misconceptions
If students struggle to write a rule, ask them to start by writing out the pattern they see in words. For example, they may write, “The \(x\)-coordinate stays the same and the \(y\)-coordinate
doubles.” Then ask how they could put those words into coordinate transformation notation.
Activity Synthesis
The goal is to use the language of distance and angle preserving moves to describe the 2 transformations. Here are some questions for discussion:
• “Look at the corresponding side lengths and angles in the 2 pairs of triangles. How do they compare?” (In triangles \(DEF\) and \(D'E'F'\), all sets of corresponding sides and all sets of
corresponding angles are congruent. In triangles \(ABC\) and \(A'B'C'\), neither the corresponding sides nor the corresponding angles are congruent.)
• “What would a transformation look like if it kept the angles the same but not the side lengths?” (It could be a dilation. The corresponding sides' lengths would need to be proportional for this
to be true.)
• “Is it possible for a transformation to keep side lengths the same but not keep the angles the same?” (This isn’t possible in a triangle. For a square, it could be transformed into a rhombus.
However, none of our standard transformations (translation, reflection, rotation, or dilation) would accomplish this.)
Lesson Synthesis
Invite students to decide which of the following rules represent rigid transformations, which represent similarity transformations, and which represent neither.
1. \((x,y) \rightarrow (x+3, y+1)\)
2. \((x,y) \rightarrow (x+12,y-2)\)
3. \((x,y) \rightarrow (2x,2y)\)
4. \((x,y) \rightarrow (x-3,y+8)\)
5. \((x,y) \rightarrow (2x,y)\)
6. \((x,y) \rightarrow (y,2x)\)
7. \((x,y) \rightarrow (y,\text-x)\)
8. \((x,y) \rightarrow (\text-x,y)\)
9. \((x,y) \rightarrow \left(\frac{x}{3}, \frac{y}{3}\right)\)
10. \((x,y) \rightarrow (\text-x, \text-y)\)
11. \((x,y) \rightarrow (\text-2x, y)\)
12. \((x,y) \rightarrow (x-4, y-3)\)
All of these rules are ones students have seen before (and they are welcome to look back in their notes to decide), but seeing them all at once will allow students to make generalizations. (Rules 1,
2, 4, 7, 8, 10, and 12 are rigid transformations. Rules 3 and 9 are similarity transformations but not rigid ones. The remaining rules are neither rigid nor similarity transformations.)
Invite students to share any patterns they notice. Ideas that may surface include:
• Adding and subtracting from the coordinates produce rigid transformations.
• Multiplying or dividing both coordinates by the same value \(k\) produces a similarity transformation. If \(k\) is 1 or -1, then the transformation is also a rigid transformation.
• If one coordinate is multiplied by some value and the other is multiplied by a different value, and the 2 values aren’t 1 and -1, then the result is neither a similarity transformation nor a
rigid transformation.
3.4: Cool-down - Write a Rule (5 minutes)
Student Facing
Triangle \(ABC\) has been transformed in two different ways:
• \((x,y) \rightarrow (\text-y,x)\), resulting in triangle \(DEF\)
• \((x,y) \rightarrow (x,3y)\), resulting in triangle \(XYC\)
Let’s analyze the effects of the first transformation. If we calculate the lengths of all the sides, we find that segments \(AB\) and \(DE\) each measure \(\sqrt5\) units, \(BC\) and \(EF\) each
measure 5 units, and \(AC\) and \(DF\) each measure \(\sqrt{20}\) units. The triangles are congruent by the Side-Side-Side Triangle Congruence Theorem. That is, this transformation leaves the lengths
and angles in the triangle the same—it is a rigid transformation.
Not all transformations keep lengths or angles the same. Compare triangles \(ABC\) and \(XYC\). Angle \(X\) is larger than angle \(A\). All of the side lengths of \(XYC\) are larger than their
corresponding sides. The transformation \((x,y) \rightarrow (x,3y)\) stretches the points on the triangle 3 times farther away from the \(x\)-axis. This is not a rigid transformation. It is also not
a dilation since the corresponding angles are not congruent. | {"url":"https://curriculum.illustrativemathematics.org/HS/teachers/2/6/3/index.html","timestamp":"2024-11-11T06:54:04Z","content_type":"text/html","content_length":"122477","record_id":"<urn:uuid:49583bee-9c05-4a81-9fd1-ae54c57c8f96>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00145.warc.gz"} |
Important Statistics for Data Science and Their Uses in 2023
Statistics as used in data science, refers to the branch of mathematics that involves collecting, analyzing, interpreting, and presenting data. It encompasses a range of techniques and methodologies
to extract meaningful insights from datasets and make data-driven decisions. In this blog, you would be learning about different types of important statistics for data science.
Data science involves large amounts of data to be processed, analyzed and deriving data driven decisions. Meaningful insights can be drawn from the data by identifying the patterns and trends. The
unstructured data needs to be converted into a structured, numeric form which presents boundless prospects of extracting information from it. Statistics is the method to extract the information for
useful insights. So, in this blog, you will explore how statistics used in data science, important statistics for data science and their uses.
What are the important statistics for data science?
Important Statistics for Data Science
In data science, several important statistics are commonly used to gain insights and draw meaningful conclusions from data. Some of these important statistics include:
1. Mean: Using this parameter, we can obtain the arithmetic average of values in a set. It provides a measure of central tendency and is often used to represent the typical value of a variable.
2. Median: The middle value in a dataset when it is arranged in ascending or descending order. It is less affected by extreme values and provides a robust measure of central tendency.
3. Mode: The most frequently occurring value(s) in a dataset. It is useful for categorical or discrete variables and can indicate the most common category or value.
4. Variance: A measure of the spread or dispersion of a dataset. It quantifies the average squared difference between each data point and the mean, providing an understanding of the variability of
the data.
5. Standard Deviation: Square root of the variance gives standard deviation. It provides a measure of the average distance between each data point and the mean, giving an indication of how spread
out the data is around the mean.
6. Correlation: This statistical value specifies the strength and direction of the linear relationship between two variables. It is often represented by the correlation coefficient, which ranges
from -1 to 1.
7. Hypothesis Testing: A statistical method used to assess whether observed data supports or contradicts a specific hypothesis or claim about a population. It involves formulating null and
alternative hypotheses and performing statistical tests to make conclusions.
8. Confidence Interval: This statistical parameter gives the probability of a population parameter falling in a range of values for a certain proportion of times. It provides an estimate of the
uncertainty associated with a sample statistic.
9. Regression Analysis: A set of statistical processes used to estimate the relationship between a dependent variable, known as outcome, and one or more independent variables, known as predictors.
It is used for predicting and understanding the impact of independent variables on the dependent variable.
10. Probability Distributions: These are the mathematical functions which provide the probability of different occurrences in an experiment or event. Common distributions used in data science include
the normal distribution, binomial distribution, and Poisson distribution.
The above mentioned are the best statistics for data science which form the foundation of data analysis in data science and are essential for understanding data, identifying patterns, making
predictions, and drawing valid conclusions.
How statistics used in data science?
Statistics is a fundamental tool in data science, serving several crucial purposes. There is a widespread usage of statistics in data science. Statistics is a fundamental tool for collecting,
evaluating, analyzing and inferring conclusions from data. Quantitative mathematical models for relevant variables are executed using important statistics for data science.
Here are some ways to see how statistics is used in data science:
1. Descriptive Statistics: Data scientists employ various descriptive statistics, such as measures of central tendency (mean, median, mode) and dispersion (variance, standard deviation), to
summarize and describe datasets.
2. Inferential Statistics: Statistical inference is used to make predictions and draw conclusions about a larger population based on a sample of data. Techniques like hypothesis testing and
confidence intervals help data scientists make statistically supported inferences.
3. Probability Theory: Probability theory is a cornerstone of data science, allowing data scientists to model uncertain events and quantify their likelihood. Probability distributions, such as the
normal distribution, are frequently used to describe and analyze data.
4. Experimental Design: Statistics helps in designing experiments and A/B tests to assess the impact of changes and interventions. It guides the selection of sample sizes, randomization techniques,
and control groups, ensuring statistically valid and reliable results.
5. Regression Analysis: Regression models are employed to understand and quantify relationships between variables. These models help data scientists identify patterns, make predictions, and assess
the significance of predictor variables.
6. Machine Learning: Statistics provides the theoretical foundation for many machine learning algorithms. Techniques like linear regression, logistic regression, decision trees, and neural networks
leverage statistical concepts to build predictive models.
7. Data Exploration and Visualization: Statistical techniques facilitate exploratory data analysis, where data scientists use graphical representations and summary statistics to understand the
structure and patterns within datasets.
8. Anomaly Detection: Statistical methods play a crucial role in identifying outliers and anomalies in data. By comparing data points to statistical distributions or using time-series analysis, data
scientists can detect unusual or suspicious observations.
Overall, you have learnt how statistics used in data science contributes to analyze, interpret, and draw meaningful insights from data, ultimately enabling informed decision-making.
How data processing make important statistics for Data Science
In the data science life cycle , a large amount of time is invested in pre-processing the data. Statistics is a powerful mechanism to remove the undesired information and organize the useful data in
a straightforward way making the gigantic task of arranging inputs seem so effortless when you use best statistics for data science to analyze datasets.
Let us understand how important is statistics in data science.
1. Data Organization: Using unstructured data can not generate any useful insights for an organization. Data needs to be classified and organized in order to work upon market plans. Statistics can
help in organizing the data efficiently. If data is well ordered and categorized, it can boost the services of the company and improve the products as well.
2. Identification of trends: The process of data collection can be an exhaustive process. Using statistics, it is easy for data scientists for recognizing patterns, trends in the data, thus saving
huge amounts of time and cost.
3. Estimation and probability distribution: Based on the real time problem, machine learning models use different types of algorithms to predict and to derive decisions. Statistics play a major role
in estimation and probability distribution.
4. Effective Data Visualization: Data Visualization is an essential stage in the Data Science life cycle. The large amount of complex data needs to be presented in a readable format for better
interpretation of data and drawing insights. Statistical tools like histograms, pie charts, bar graphs help in analyzing the patterns and trends in the data and depiction of the results.
5. Reduced dependence on assumptions: Mathematics forms the strong basis for machine learning, artificial intelligence and data analytics. Factors like differentiation and continuity help in
deriving the outcomes based on precise inferences and not on assumptions. Thus, statistics enhances the strength of the model which in turn produces accurate predictions.
6. Reporting variability in data: Statistics helps in reporting different variables like clusters, time, space in the data used for analysis. If statistical methods are not used, then variability
might not be reported in the data analysis, which might result in incorrect estimates.
Statistics is a vital component of data science, providing the necessary tools and techniques to explore, analyze, and interpret data. It enables data scientists to draw meaningful insights, validate
models, and make informed decisions based on statistical evidence. Descriptive statistics, inferential techniques, predictive analytics and experimental design are few of the important statistics for
data science that form the backbone of data science methodologies. Its integration with machine learning algorithms enhances predictive capabilities, while decision-making under uncertainty is
facilitated by statistical approaches. For aspirants who aim to grow as data scientists, it is best statistics for data science as it empowers them to extract valuable information from data, drive
data-driven solutions, and shape the future of various industries.
7 thoughts on “Important Statistics for Data Science and Their Uses in 2023”
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1. Aakash Jha
Hello there! I'm glad you enjoyed my writing. I'm always happy to help out with questions or provide more information.
Connect with me at aakash@clinilaunchresearch.in
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1. Aakash Jha
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Leave a Comment | {"url":"https://wininlifeacademy.com/important-statistics-for-data-science/","timestamp":"2024-11-05T15:36:58Z","content_type":"text/html","content_length":"233814","record_id":"<urn:uuid:009f60a8-004c-418f-b5e2-3652da2077ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00748.warc.gz"} |
Part 1: Given a list of integer ranges (a la 5-8), what is the first value not in any given range?
Most of this problem actually came down to making a relatively complicated data structure that can take an arbitrary number of overlapping ranges and simplify them, also allowing the addition and
removal of more ranges. It’s more complicated than it strictly needs to be, but it makes this problem much easier. :)
class IntRange(object):
Represents a list of integers.
Specific values can be allowed (included) / denied (excluded) from the range.
def __init__(self, min, max):
'''Create a new int range with the given values initially allowed.'''
self._ranges = [(min, max)]
def __repr__(self):
'''Pretty print a range (this can get long).'''
return 'IntRange<{}>'.format(self._ranges)
def __in__(self, value):
'''Test if a value is in this int range.'''
# Slower version
# return any(lo <= value <= hi for (lo, hi) in self._ranges)
index = bisect.bisect(self._ranges, (value, value))
lo, hi = self._ranges[index]
return lo <= value <= hi
def __iter__(self):
'''Return all values in this int range.'''
for (lo, hi) in self._ranges:
yield from range(lo, hi + 1)
def __len__(self):
'''Return how many values are in this IP range.'''
return sum(hi - lo + 1 for (lo, hi) in self._ranges)
def _simplify(self):
'''Go through current ranges and remove/collapse overlapping ranges.'''
i = 0
while i + 1 < len(self._ranges):
range1_lo, range1_hi = self._ranges[i]
range2_lo, range2_hi = self._ranges[i + 1]
# Only guarantee: lo1 is <= lo2
# There is an overlap, combine and remove range2
# Continue without incrementing since another range might be collapsed
if range2_lo <= range1_hi:
self._ranges[i] = (range1_lo, max(range1_hi, range2_hi))
del self._ranges[i + 1]
i += 1
def allow(self, allow_min, allow_max):
'''Add a new range of allowed values.'''
# Insert sorted (using bisect) then simplify
bisect.insort(self._ranges, (allow_min, allow_max))
def deny(self, deny_min, deny_max):
'''Remove a range of (possibly) previously allowed values.'''
i = 0
while i < len(self._ranges):
lo, hi = self._ranges[i]
# Range is completely denied
if deny_min <= lo <= hi <= deny_max:
del self._ranges[i]
# Denial is completely within the range, split it
elif lo <= deny_min <= deny_max <= hi:
del self._ranges[i]
self._ranges.insert(i, (lo, deny_min - 1))
self._ranges.insert(i + 1, (deny_max + 1, hi))
# Partial overlap, adjust the range
elif lo <= deny_min <= hi:
self._ranges[i] = (lo, deny_min - 1)
elif lo <= deny_max <= hi:
self._ranges[i] = (deny_max + 1, hi)
i += 1
The interesting functions there are:
• allow where I use the bisect module to insert elements into a sorted position in a list
• _simplify where I take a sorted list and combine overlapping adjacent elements (as part of allow)
• deny where I iterate through the list and remove/combine existing ranges that have been deleted or broken apart
deny took the most time, but it was still relatively straight forward.
lo, hi = map(int, args.range.split('-'))
ips = IntRange(lo, hi)
for line in fileinput.input(args.files):
if line:
lo, hi = map(int, line.split('-'))
ips.deny(lo, hi)
for ip in ips:
print('First allowed IP: {}'.format(ip))
Part 2: How many numbers are there between 0 and 2^32-1 (inclusive) are not in any range?
We already solved this with the __len__ method of IntRange:
print('Number of allowed IPs: {}'.format(len(ips)))
Interesting aside: You could probably do this with Python’s built in ipaddress module, specifically using a list of ipaddress.IPv4Network . I thought it was interesting to work out myself though. | {"url":"https://blog.jverkamp.com/2016/12/20/aoc-2016-day-20-filter-table/","timestamp":"2024-11-13T02:56:08Z","content_type":"text/html","content_length":"25561","record_id":"<urn:uuid:b3d3ae4c-f78f-4b03-ad60-3b7d47953bac>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00137.warc.gz"} |
Practice Writing Numbers 1-100 Worksheets - Printable Tracing Numbers
Practice Tracing Numbers 1 100 – There are several fantastic choices to choose from if you’ve got children at home and are looking for a fun approach to teaching your children to trace. Worksheets
are readily available to aid your child in improving the control of their pencils, strengthening their understanding of numbers and also … Read more | {"url":"https://www.tracingnumbers.net/tag/practice-writing-numbers-1-100-worksheets/","timestamp":"2024-11-02T08:24:12Z","content_type":"text/html","content_length":"47306","record_id":"<urn:uuid:4cf56cdd-94be-469f-8e3d-add8394c0f00>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00058.warc.gz"} |
Book Review: Proofiness: The Dark Arts of Mathematical Deception by Charles Seife
Review by Alexandra Witze
The 2000 U.S. presidential election should have been decided by a coin flip.
Or so argues Seife, a mathematician-turned-journalist who tackles some of society’s biggest math problems in his new book. The race between George W. Bush and Al Gore was, mathematically speaking,
too close to call. So, Seife suggests, instead of counting chads, the contested state of Florida should have relied on an age-old procedure for breaking a tie: drawing lots.
Seife is somewhat obsessed with the flaws in the country’s electoral system, but he makes an eloquent case that all citizens should be so concerned. What he dubs “proofiness” — the manipulation of
mathematics for untrue ends — permeates modern culture.
He gives plenty of examples. One flawed study suggests that women who have had an abortion have a 30 percent increased risk of breast cancer. Another argues that tobacco is a gateway to harder drug
use. Statisticians can spend all day eviscerating the math behind these studies, but proofiness nonetheless trickles deep into social policy.
Even mathphobes will appreciate Seife’s clear explanations of why polls are so flawed and how risks are routinely exaggerated to justify a particular decision. Seife is trying to do the admirable and
the impossible — educate the public so people can understand when they are being manipulated by bogus numbers. If only those doing the manipulation would believe that the public is too smart to be
Viking, 2010, 295 p., $25.95. | {"url":"https://www.sciencenews.org/article/book-review-proofiness-dark-arts-mathematical-deception-charles-seife","timestamp":"2024-11-06T17:12:45Z","content_type":"text/html","content_length":"260101","record_id":"<urn:uuid:23c69e9d-79d3-4e51-83bc-3cb1808b1b1d>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00240.warc.gz"} |
Mix C and matlab in Linux and mix linuxmatlab
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Recovering social networks from contagion information
Many algorithms for analyzing social networks assume that the structure of the network is known, but this is not always a reasonable assumption. We wish to reconstruct an underlying network given
data about how some property, such as disease, has spread through the network. Properties may spread through a network in different ways: for instance, an individual may learn information as soon as
one of his neighbors has learned that information, but political beliefs may follow a different type of model. We create algorithms for discovering underlying networks that would give rise to the
diffusion in these models.
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 6108 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Other 7th Annual Conference on Theory and Applications of Models of Computation, TAMC 2010
Country/Territory Czech Republic
City Prague
Period 6/7/10 → 6/11/10
• Contagion
• Diffusion
• Graph Algorithms
• Social Networks
ASJC Scopus subject areas
• Theoretical Computer Science
• General Computer Science
Dive into the research topics of 'Recovering social networks from contagion information'. Together they form a unique fingerprint. | {"url":"https://experts.syr.edu/en/publications/recovering-social-networks-from-contagion-information","timestamp":"2024-11-11T18:06:38Z","content_type":"text/html","content_length":"44210","record_id":"<urn:uuid:92e13379-516d-4ed9-bdb2-c9a0e7411150>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00585.warc.gz"} |
Optimal Moebius Band Suite
This webpage has links that relate to my solution of the Halpern-Weaver Conjecture about optimal paper Moebius bands. Here is the main paper.
Here is a friendlier and less formal version written for undergraduate math majors and/or advanced high school students.
Here are 3 sequel papers
The original paper got some press: | {"url":"http://www.math.brown.edu/reschwar/moebius_suite.html","timestamp":"2024-11-10T08:30:40Z","content_type":"text/html","content_length":"2651","record_id":"<urn:uuid:b35499c0-ab36-453d-9bfc-396fdcbeb165>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00177.warc.gz"} |
January 6, 2022
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CloseFor all a, b, c e R and if a > b and c < 0 then—-A) ac > bcB) ...
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Software 3D rendering in JavaScript, Part 1: Wireframe model
The other day I stumbled onto Dmitry Sokolov’s ‘How OpenGL works’ series, about implementing your own software 3D renderer which functions kind of like OpenGL. I decided to give it a go myself, and
maybe write some posts detailing the journey. In part 1 we get set up and get some pseudo wireframe rendering going. You can check out the results, then read on to find out how it’s done.
Source Code
If you want to follow along, just download and unzip the code for this part. I’ll briefly go over the contents before we proceed:
• index.html - The file containing the main code for this part. Our source code will be inside this HTML in an inline script tag, at least for now.
• teapot.obj - A 3D teapot model I nabbed off the internet.
• serve.bat - A batch file which, assuming you have Python installed and in your PATH, starts a server at http://127.0.0.1:12345, necessary because otherwise using XMLHttpRequest to load files
won’t work. If you have Python 2.x, this will work, but if you’re running 3.x you should replace ‘SimpleHTTPServer’ with ‘http.server’.
• misc.js - This contains some helper code that isn’t that interesting to cover for what we’re doing, including:
□ IO module - Contains the File class which lets us load files using XMLHttpRequest, in bulk and using a queue if necessary.
□ debugMode function - Lets us turn alerts for errors with nicely formatted stack traces on and off.
□ usingModule function - Takes an object and adds all its properties to the window object so we can use for instance new File("teapot.obj") instead of new IO.File("teapot.obj").
And because I’m a crotchety C# programmer at heart I’ll be using ECMAScript 6’s new class keyword and features, which means that this will run in recent versions of Firefox, Chrome, and Edge 13 and
basically nothing below that. Sorry not sorry.
Preparing the Canvas
First things first, we need a canvas to draw into, so we get that ready.
<canvas id="canvas" moz-opaque style="border: 1px solid black"></canvas>
Pretty simple. Just a canvas element, with the mind bogglingly original id of canvas, the property moz-opaque for some performance enhancements in browsers that support it since we don’t care about
seeing what’s below the canvas, and a 1 pixel black border.
First we want a program entry point. In my experience it’s best to run your JavaScript program on load, so I create the following code:
// main function called on load
function run() {
// set our canvas size
var canvas = document.getElementById("canvas");
canvas.width = 800;
canvas.height = 480;
// clear our canvas to opaque black
var context = canvas.getContext("2d");
context.fillStyle = "black";
context.fillRect(0, 0, canvas.width, canvas.height);
// get image data for direct pixel access
var imageData = context.getImageData(0, 0, canvas.width, canvas.height);
// turn on debug mode so errors throw up an alert
// containing error information and stack trace
// run our program when the document's loaded
document.body.onload = run;
Inside the run function we get our canvas by its id, and set the width and height, as well as create a 2d rendering context. Then, for good measure and since not all browsers support moz-opaque, we
clear it to black just to be sure. Finally we store our canvas’ image data so we can pass it around to our custom drawing functions later.
Note that the debugMode function is one of our helper functions from misc.js. If you’re following along, make sure to include this in the head section of your file using <script src="misc.js"></
Bresenham’s Line Drawing Algorithm
This is what part 1 of the lessons I’m following covers. However, I already had this implemented, and just reused/rewrote the C# code I already had to implement this. Other people are far more
qualified to cover Bresenham’s algorithm, so I won’t go into detail on how it actually works. I will however, provide the code that does the drawing.
// draws a line using Bresenham's line algorithm (integer version)
function drawLine(imageData, x0, y0, x1, y1) {
// convert all coordinates to integers
x0 |= 0;
y0 |= 0;
x1 |= 0;
y1 |= 0;
// get binary pixel data array
var data = imageData.data;
var steep = Math.abs(y1 - y0) > Math.abs(x1 - x0);
if (steep) {
var tmp;
tmp = x0; x0 = y0; y0 = tmp;
tmp = x1; x1 = y1; y1 = tmp;
var yStep;
if (x0 > x1) {
var tmp;
tmp = x0; x0 = x1; x1 = tmp;
tmp = y0; y0 = y1; y1 = tmp;
if (y0 < y1) {
yStep = 1;
else {
yStep = -1;
var deltaX = x1 - x0;
var deltaY = Math.abs(y1 - y0);
var error = 0;
var index = 0;
var y = y0;
for (var x = x0; x <= x1; x++) {
// index is vertical coordinate times width, plus horizontal coordinate,
// times 4 because every pixel consists of 4 bytes
if (steep) {
index = (x * imageData.width + y) * 4; // y, x
else {
index = (y * imageData.width + x) * 4; // x, y
// set RGBA values to 255 producing opaque white pixel
data[index] = data[index + 1] = data[index + 2] = data[index + 3] = 255;
error += deltaY;
if ((error << 1) >= deltaX) {
y += yStep;
error -= deltaX;
We pass it an ImageData instance, which we got from our canvas already, as well as the coordinates to draw from and to, and this draws an aliased 1 pixel thick white line. Note that Bresenham’s line
algorithm can use only integers for added performance, which this version does. In order to make that work we have to convert the coordinates to integers, so we do that. Then we get the pixel data
from our imageData using its data property, which is a Uint8ClampedArray instance, or an array of width times height times 4 unsigned 8-bit integers with values between 0 and 255 inclusive. We’ll be
using this to write directly to the pixels.
When we want to draw a pixel we determine its starting position inside this array and set all 4 bytes to 255, which creates a full opaque white pixel. Ideally you’d want to be able to specify the
color of the pixels, but right now we’re just trying to get up and running so this will do. Now we can render a line to our canvas using the following code in our run function:
// draw line from (10, 10) to (100, 30)
drawLine(imageData, 10, 10, 100, 30);
At this point we could theoretically render a model, if we had one, so that’s our next step.
Loading and Parsing the Model
Loading files inside an HTML file is a right pain in the ass. The way its usually done is using XMLHttpRequest, which can fetch other documents either synchronously or asynchronously, although the
former is discouraged, so we don’t want to do that. Fortunately our IO module in misc.js has us covered with the IO.File class, which lets us queue up any number of files to load, and letting us know
when it’s done. We modify our earlier run function to do this like so:
// add all properties on the IO object to the window object
// so they can be used without prefixing "IO."
var model = null;
// main function called on load
function run() {
// loads list of files asynchronously, calls function when done
function(files) {
// parse our teapot obj model file
model = parseObj(files[0].value);
// start main program
function start() {
// set our canvas size
var canvas = document.getElementById("canvas");
canvas.width = 800;
canvas.height = 480;
// clear our canvas to opaque black
var context = canvas.getContext("2d");
context.fillStyle = "black";
context.fillRect(0, 0, canvas.width, canvas.height);
// get image data for direct pixel access
var imageData = context.getImageData(0, 0, canvas.width, canvas.height);
The IO.File.loadFiles function takes a function to call when done loading the files, and then a list of items which can either be an URI string, or an array containing an URI string and a type
string. Since the default way to load is as a plain-text file, we can just supply our "teapot.obj" string directly. Once our file is loaded our function will be called with a list of files in the
order we supplied them, so files[0] will be an instance of IO.File, and if all is well its value property will have the text we need, which we pass to a parseObj function that we will create shortly.
Once everything’s loaded and the model’s been parsed we call a new start function, which contains the code that was in our run function before. Of course we need to actually parse the model file.
Fortunately the obj format is pretty easy to follow. For example:
v 0 0 0
v 2 0 0
v 2 3 0
f 1 2 3
This describes a 3D ‘model’, with three vertices and one face (or triangle). Every line that starts with ‘v’ describes a vertex, with its X, Y, then Z coordinates. Every line that starts with ‘f’
describes a single triangle, with the three numbers describing the indices of the vertices in order. For some reason the obj file format decided that the first declared vertex starts at index 1, not
0, so be careful of that. A face declaration can also have negative indices, in which case -1 refers to the last vertex specified so far, -2 to the second to last, and so on.
With this knowledge we can write our parseObj function to get the vertices and faces that make up our model:
// parses an obj file from a text string
function parseObj(text) {
var verts = [];
var faces = [];
// split the text into lines
var lines = text.replace('\r', '').split('\n');
var count = lines.length;
for (var i = 0; i < count; i++) {
var line = lines[i];
if (line[0] == 'v') {
// lines that start with 'v' are vertices
var tokens = line.split(' ');
x: parseFloat(tokens[1]),
y: parseFloat(tokens[2]),
z: parseFloat(tokens[3])
else if (line[0] == 'f') {
// lines that start with 'f' are faces
var tokens = line.split(' ');
var face = [
parseInt(tokens[1], 10),
parseInt(tokens[2], 10),
parseInt(tokens[3], 10)
if (face[0] < 0) {
face[0] = verts.length + face[0];
if (face[1] < 0) {
face[1] = verts.length + face[1];
if (face[2] < 0) {
face[2] = verts.length + face[2];
// return an object containing our vertices and faces
return {
verts: verts,
faces: faces
There’s really nothing shocking here. We loop through all the lines in our text, if a line starts with ‘v’ we push a vertex into our array of vertices, if it starts with ‘f’ we push a face into the
faces array containing the vertex indices. Once we’ve gone through all the lines we return an object containing our vertex and face arrays.
Rendering the Model
Now we’re finally ready to render our model! We add the following code to the end of our start function:
// some drawing positioning
var centerX = canvas.width / 2.0;
var centerY = canvas.height / 2.0;
var scale = 100;
centerY += 150;
// draw our model
for (var i = 0; i < model.faces.length; i++) {
var face = model.faces[i];
var v0 = model.verts[face[0] - 1];
var v1 = model.verts[face[1] - 1];
var v2 = model.verts[face[2] - 1];
if (v0 && v1 && v2) {
drawLine(imageData, centerX + v0.x * scale, centerY - v0.y * scale, centerX + v1.x * scale, centerY - v1.y * scale);
drawLine(imageData, centerX + v1.x * scale, centerY - v1.y * scale, centerX + v2.x * scale, centerY - v2.y * scale);
drawLine(imageData, centerX + v2.x * scale, centerY - v2.y * scale, centerX + v0.x * scale, centerY - v0.y * scale);
else {
if (!v0) { console.log("Vertice " + (face[0] - 1) + " not found!"); }
if (!v1) { console.log("Vertice " + (face[1] - 1) + " not found!"); }
if (!v2) { console.log("Vertice " + (face[2] - 1) + " not found!"); }
// write our new pixels to the canvas
context.putImageData(imageData, 0, 0);
First we set up some variables that lets us position where we draw onto the canvas. We want to center our drawing, so we calculate the center X and Y coordinates. The teapot we’re rendering is kinda
small though, so we’ll just multiply all coordinates by a scale variable, and it sits on top of zero on the Y axis, so we add an offset to that as well.
Next we loop through all the faces on our model, and using the indices we stored earlier grab the corresponding three vertices. Because JavaScript arrays start with the first item at index 0 and the
obj format starts at index 1 we decrease all indices by 1 to get the right vertex. If you don’t do this, the model will look very messed up. Then we check to see if all our vertices have an actual
value, and print some debug info if something went wrong.
If all vertices look good we draw a line from vertex 0, to vertex 1, to vertex 2, and back to vertex 0. You might notice that we’re subtracting the vertices’ Y coordinates from the center Y
coordinate, instead of adding them. That’s because in 3D space a bigger value for Y means the position is higher up, but we’re drawing in 2D space where a higher value means the position is actually
lower, so we subtract the value instead of adding it. This isn’t a problem for now, and it’ll fix itself when we start doing proper 3D transformations later on.
Finally, we update our canvas with our new pixel data to see the results. And viola, we have a teapot!
Backface Culling
The astute among you might notice something doesn’t seem quite right. Every quad on the model is divided like a sort of X shape, into what appears to be 4 triangles. What is actually happening is
that we’re drawing the model’s wireframe without backface culling, that is we are drawing the back of the teapot as well as the front, and they line up and overlap perfectly because of how the model
is made, but the diagonal faces the opposite way, creating the X shape.
The lessons I was following saved backface culling for the 2nd lesson but I figured I’d get a head start and implement it straight away. Face culling relies on something called the polygon’s winding
order; the clockwise or counterclockwise direction you get if you follow the order of the vertices inside the polygon on the screen from one, to the next, to the next, to the first. In OpenGL the
convention is that if the order of vertices is counterclockwise, then we’re looking at the front of a polygon, if its clockwise then we’re looking at the back of a polygon and (unless specified
otherwise by the programmer) we don’t need to draw it.
Unfortunately, this does involve some math. Feel free to skip most of the explanations below and skip straight to the code, but for those interested in understanding how to do it, I’ll explain. To
figure out the winding order, we need to perform the following steps:
1. Transform all the vertices to screen space.
2. Calculate the cross product of two of the polygon’s edges.
3. Check the Z component of the result to see if the polygon is front or back facing.
Since we’re not actually using 3D space yet, we can skip step 1, since we already have our screen space coordinates. Step 2 involves the cross product of two vectors, which takes two vectors and
returns a vector which is at a right angle to both. If both vectors start at the origin (0, 0, 0) then the cross product becomes quick and easy to calculate. In JavaScript it’d look something like
function crossProduct(a, b) {
return {
x: a.y * b.z - a.z * b.y,
y: a.z * b.x - a.x * b.z,
z: a.x * b.y - a.y * b.x
Given that we have our vertices in screen space, we know that X is the horizontal axis, Y is the vertical axis, and Z is the axis that points into and out of the screen. The X and Y axes aren’t going
to tell us if we’re looking at the back or front, we want the axis that either points towards us, or away from us, to tell us if we’re looking at the polygon’s back or front. That means we don’t
actually need the full cross product, but just the Z component, defined as a.x * b.y - a.y * b.x, which saves us a lot of calculations.
Unfortunately our vertices are not defined relative to the origin. Fortunately, it’s easy to fix that. Imagine a line from a point at (1, 1) to another point at (5, 1), the red line in the image
above. If you subtract the former point from the latter point, you get a point that’s at (5-1, 1-1) or (4, 0) in space. If you then draw the green line from (0, 0) to (4, 0), you’d get the same line
as the red one, just eminating from the origin instead of from somewhere else.
We can do the same with 3D points, and we happen to have three of them; our vertices. If we subtract our first vertex’s positions from the 2nd and 3rd, we get two vectors that describe the edges of
our triangle, starting at the origin, which lets us do our cross product cheaply and easily. So if we define our vertices as v0, v1, and v2 in order we get the following:
a.x = v1.x - v0.x
a.y = v1.y - v0.y
b.x = v2.x - v0.x
b.y = v2.y - v0.y
z = a.x * b.y - a.y * b.x
z = (v1.x - v0.x) * (v2.y - v0.y) - (v1.y - v0.y) * (v2.x - v0.x)
You might notice we’re not calculating a.z or b.z, but that’s just because the calculation we’re doing doesn’t require those. If we try this out we wind up getting a negative value for Z if the
vertices are counterclockwise, which lets us write our function as follows:
// returns true if vertices are in counterclockwise order
function isCcw(v0, v1, v2) {
return (v1.x - v0.x) * (v2.y - v0.y) - (v1.y - v0.y) * (v2.x - v0.x) >= 0;
And we can add a call to this function to our drawing code to only draw polygons which are facing towards us:
if (v0 && v1 && v2) {
if (isCcw(v0, v1, v2)) {
drawLine(imageData, centerX + v0.x * scale, centerY - v0.y * scale, centerX + v1.x * scale, centerY - v1.y * scale);
drawLine(imageData, centerX + v1.x * scale, centerY - v1.y * scale, centerX + v2.x * scale, centerY - v2.y * scale);
drawLine(imageData, centerX + v2.x * scale, centerY - v2.y * scale, centerX + v0.x * scale, centerY - v0.y * scale);
This section was very heavy on the maths, but if you don’t understand the cross product you can just use the code as provided or read more about the cross product on Math Is Fun.
Note: In an earlier version of this article I said that isCcw should return a negative value for front facing polygons even though OpenGL specifies that a negative Z value points away from us, into
the screen. I argued that this was correct because our screen coordinates are reversed from OpenGL’s where 0 is at the bottom, and Y increases towards the top. This explanation would have been
correct if the cross product we were calculating was the cross product of the screen coordinates, unfortunately I failed to recognize that we were passing it the model’s vertices, which do not have a
flipped Y axis, and thus would function completely as normal. Oops?
End Result
And this is what our final result looks like. Not bad! Next we’ll be adding triangle rasterization and some other things. You can find the source code download and relevant links below, and if you
liked this post maybe give it a share using one of the buttons below or check out options to support our work. | {"url":"https://kitsunegames.com/post/development/2016/07/11/canvas3d-3d-rendering-in-javascript/","timestamp":"2024-11-11T14:17:58Z","content_type":"text/html","content_length":"74909","record_id":"<urn:uuid:46ee145e-b57c-4ad3-a229-e26d23d40a7f>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00709.warc.gz"} |
Tips to score maximum marks in Maths in JEE Mains exam - DailybnTips to score maximum marks in Maths in JEE Mains exam
Getting 90+ Marks out of 120 in JEE Mains Maths is anything but troublesome work. There needs to be concentration and hard work for the exams. It just relies upon the Candidates how they get ready
for the assessments. Maths is about Practice. So Candidates should revise an ever-increasing number of questions. The more they practice the more they foster critical thinking abilities. Maths is the
troublesome area in JEE Mains from all that of 3 areas: Physics, Chemistry and Maths. It incorporates ideas as well as so many practice questions to tackle. Candidates need to clear all the doubts
and get a stronghold of speed. Assuming Candidates need to score 90+ Marks in JEE Mains they need to follow just one stage that is ‘PRACTICE’.
1. Build your concepts
Attend Class routinely, Make notes/short notes appropriately, Revise Questions, build your Concepts and Confidence, Mark Important Questions. Just these simple four steps will help students to build
their concepts appropriately as well as solve their doubts.
Aspirants should initially endeavour and understand questions rather than watching answers first. Numerous Students tend to read the questions and when they find it troublesome, they leave those
questions and watch the appropriate response. This ought not to become a habit. On the off chance that they are doing, they will become dependent on the Solutions and not on their knowledge. Students
ought to have a battling soul to tackle the troublesome questions more and more. Competitors should endeavour questions at least 15 minutes on each question then, at that point, allude to the
2. Choose the chapters with more weightage.
Competitors should focus on the Chapters of Maths that have more weightage in JEE Mains. A few Chapters are vital from the exam perspective. Applicants should think about the test pattern,
Chapterwise weightage of Maths in JEE Mains. This will assist them with getting a thought about what kind of questions normally arrive in a test. What’s more, What are the examples of questions as
well as the language and so forth which will help them in preparation for the test. A few points will be given more weightage as per the significance while learning. Students will become more
acquainted with which points are more significant for tests and which one is less significant and can focus on the themes by investigating the difficulty level and weightage. You can refer to IIT JEE
Mains maths important questions online.
A few parts are vital according to JEE Mains exam perspective. These parts incorporate — Integrals, 3-D Geometry, Probability, Conic segment, Vector Algebra, Functions, Limits, Continuity and
Derivability, Matrices and determinants, Probability, Permutation and Combination. Class ’12 and Class 11 both have an equivalent weightage of marks in the test.
3. Speed=Logic
In case Students are focusing on 90+ marks in tests. Then, at that point, alongside the question solving they ought to think legitimately. The test is 3 hours long. So equivalent time ought to be
given to every one of the segments in tests. Maths is for an hour. So consistent reasoning ought to be of cutting edge level then, at that point, Candidates will want to get more marks in tests.
Attempt to focus more on comprehension and building applied clarity rather than only learning the stuff.
Never pass up simple marks. Go through the entire paper once, before endeavouring any question. Mark the questions as indicated by your perspective in three classes, for example simple, moderate and
intense. To start with, tackle every one of the simple ones and afterwards move on to others.
4. Go with the trend
Examine the pattern from earlier year papers, allot time to parts as needs are. Themes like Coordinate Geometry, Calculus are of most weightage and ought to be engaged upon steadily.
Gain proficiency with the graphical way to deal with difficult questions. It is less tedious than the traditional methodology. Getting to know the diagrams of some simple equations we face will be
profoundly useful.
Give mock test papers consistently in the month before. Make your list of scores and mark the points wherein you submitted mistakes in the test. Analyze and work on them, so it doesn’t get repeated
in upcoming tests. Ensure that you have perceived the questions concerning what it is requesting and what you are deciphering.
Customary self-examination and following progress that you are making ought to be done consistently.
5. Ample material is required
Start with the NCERT, do every one of the questions to some degree once. Try not to miss the various questions at the end of every part. Allude to Objective Mathematics by RD Sharma or standard books
of JEE Main Maths for calculated approach and practice questions.
Practice as much as you can. A few students even get a count of registers and pens that they have used. The more questions you endeavour, the better you get with calculations. It diminishes the shot
at committing silly errors during the test.
Mathematics is normally thought to be lengthy and time consuming by students. The justification for this is how we approach this segment. You don’t have to settle every single question routinely. A
few questions can undoubtedly be solved in less time by utilizing option elimination measures. Have a go at embracing and dealing with these hacks during your preparation stage.
6. Give sufficient time to Maths
The main thing that you ought to do is foster a positive outlook towards the subject. Quit fearing Maths as a subject. Be certain and prepared to learn new things. You should begin your preparations
for JEE from the beginning phase. As a rule, students start from the eleventh standard itself. During this, you go through the JEE Math syllabus for JEE Advanced and JEE Main test and comprehend the
learning goals of the question makers. Look into the significant themes and make a preparation procedure appropriately.
Give enough time to the subject in your daily timetable and constantly go through the topics you have learnt so far. At last, you mustn’t get nervous or over-confident. Just ace your exam with
confidence and belief.
While planning for profoundly important tests like JEE Main, Mathematics is quite possibly the main subject a candidate needs to manage. We can proceed to say that Math shapes a critical piece of our
development. It is the main consideration toward the end. The establishment of Physics and Chemistry is made by arithmetic. Practically every one of the mathematical issues of Physics and Chemistry
utilizes points like quadratic conditions, differential conditions, and analytics. So, it becomes more important. | {"url":"https://dailybn.com/tips-to-score-maximum-marks-in-maths-in-jee-mains-exam/","timestamp":"2024-11-04T06:00:41Z","content_type":"text/html","content_length":"202429","record_id":"<urn:uuid:4800b275-d633-4ca8-837f-497dd24cd42e>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00465.warc.gz"} |
Dark Buzz
There is a
discussion on St ormfront
my book
. The most detailed argument
appeared in 2009
under the name Karl Radl, and is being repeated under the name LionAxe.
also on this blog
, under the name Hans Blackthorne.
There are too many points to answer here, but I just mention some major ones. It quotes
that I have answered on this blog.
First of all: Poincaré did maintain the mechanist aether as crucial.
Absolutely not. You will never see a quote to back this up, as Poincare always said that the aether "is only a convenient hypothesis ... will be thrown aside as useless." His theory did not depend on
the aether.
The space and time transformations improved by Poincaré from Lorentz were therefore based on a set of "fictitious" transformations: since they had been obtained based on systematic errors during
their measurements. Einstein's theories differed greatly.
I don't know whom he is quoting, but Poincare was the first (in 1905) to say that the transformations form a symmetry group of space and time. That view was adopted by Minkowski in 1907, by most
other physicists in 1908, and finally by Einstein in 1909.
The fact that Einstein's work on Special Relativity put the pieces together and revealed the complete theory in a coherent, correct and provable formulation is not really disputable.
Lorentz and Poincare developed most of the math used, but never fully embraced the principles behind it.
This is nonsense. Ever single principle was adopted by Lorentz and Poincare years ahead of Einstein. Einstein added nothing.
You have been given the facts of how Einstein's and Poincaré's work are not interchangable, how they differed, how they did not accept each others work et al.
Einstein told the story dozens of times about how he invented special relativity, but he was never able to explain how his theory was better than Poincare's. Einstein sometimes attempted to explain
how his 1905 work was better than Lorentz's, but always gave arguments based on what Lorentz had done ten or more years earlier, and avoided Lorentz's recent work.
Poincare did explain how his theory differed from Lorentz's. After Einstein's 1905 paper appeared, it was called the "Lorentz-Einstein theory" as everyone agreed that Einstein's paper was just a
recapitulation of Lorentz. Within a couple of years, it was universally recognized that Poincare's approach was superior.
Here is a timeline if the major concepts of special relativity.
length contraction (FitzGerald 1889, Lorentz 1892)
aether is just a convention (Poincare 1889)
first-order Lorentz transformations (Lorentz 1895)
relativistic time (Lorentz 1895)
relativity principle (Poincare 1895)
relativistic mass (Lorentz 1899, experimentally tested in 1902)
constant speed of light (Maxwell, Lorentz, Poincare pre-1900)
light synchronization of clocks (Poincare 1900)
E = mc^2 (Poincare 1900)
full Lorentz transformations (Lorentz 1904)
4-dimensional spacetime geometry (Poincare 1905)
electromagnetic covariance (Poincare 1905)
Einstein's first relativity paper was in 1905. He had an exposition of most of the above concepts, but did not have the most crucial ones -- the 4-dimensional spacetime geometry and the
electromagnetic covariance. He had no new ideas or formulas. He did not have the essence of the theory as it is known today, or even as it was known in 1908.
The attacks on Lorentz and Poincare are mostly based on allegations about their beliefs, and not what they said. This is especially true about the aether. Their theories did not depend on any
properties of the aether, and they explicitly said so. While they occasionally later used the word, so did Einstein.
I don't know how anyone can understand relativity, read the above timeline, and still credit Einstein. If you read an article by someone defending Einstein, I suggest trying to figure out whether he
agrees with the above timeline.
Update: LionAxe has it backwards about the aether. Lorentz wrote in 1895 that "It is not my intention to ... express assumptions about the nature of the aether. ... we cannot speak about an absolute
rest of the aether". Einstein wrote in 1905 that "the view here to be developed will not require an 'absolutely stationary space' provided with special properties". Einstein always admitted that he
based his 1905 paper on Lorentz's 1895, and they were saying essentially the same thing about the aether.
Poincare proved in 1905 that there was a symmetry group making all moving frames equivalent, and that there was therefore no privileged frame. Poincare never said that there was a privileged frame.
If you think that he did, then show me the quote, and show me just what privilege that frame had in the theory.
LionAxe continues to complain about things that Poincare got completely correct, such as clocks in a moving frame showing the local time in that frame. That is an essential concept of special
relativity due to Lorentz and Poincare, and it has been taught in the textbooks ever since. Einstein never had such an original and brilliant idea in his whole life. Lorentz got the Nobel Prize in
1902, and one of the arguments for it was his invention of local time.
Update: I asked LionAxe to show me where a mechanist aether was ever crucial to Poincare's work. He supplied "We know nothing as to what the aether is" (1904 lecture) "Does our aether actually exist?
We know the origin of our belief in the aether." (1902 book) "Thus Lorentz's hypothesis is the only one consistent with the inability to demonstrate absolute motion" (1905 long paper) I have
corrected the quotes using my sources.
In none of these does Poincare say that the aether exists, or that a privileged frame exists, or that the aether is needed for his theory. On the contrary, he repeatedly says that the aether is just
a convention, that it will eventually be considered useless, and that a symmetry group makes all frames equivalent.
On the other hand, Einstein frequently says things like "It is essential to have time defined by means of stationary clocks in the stationary system". Einstein uses a privileged frame throughout his
famous 1905 paper, and he is never able to show independence of that frame. Poincare proves independence using the symmetry group.
Update: I did not realize that Bjerknes had already refuted LionAxe.
Skeptic Michael Shermer writes:
Why is there something rather than nothing? The question is usually posed by Christian apologists as a rhetorical argument meant to pose as the drop-dead killer case for God that no scientist can
possibly answer. Those days are over. Even though scientists are not in agreement on a final answer to the now non-rhetorical question, they are edging closer to providing logical and even
potentially empirically testable hypotheses to account for the universe. ...
According to the “many worlds” interpretation of quantum mechanics, there are an infinite number of universes in which every possible outcome of every possible choice that has ever been
available, or will be available, has happened in one of those universes. This many-worlds multiverse is grounded in the bizarre findings of the famous “double-slit” experiment, in which light is
passed through two slits and forms an interference pattern of waves on a back surface ...
Carroll then cautioned: “Obviously the entire set of ideas is controversial and speculative, and should be presented as such, but it’s taken very seriously by a large number of extremely smart
and respectable people.” For example: Leonard Susskind, Alex Vilenkin and Alan Guth (on the pro-multiverse side) and David Gross, Paul Steinhardt, and Edward Farhi (skeptical of the multiverse
No, the many-worlds multiverse is not grounded in the double-slit experiment. That experiment is not bizarre at all, if you believe in the wave nature of light (or electrons).
Instead of just saying that some light goes thru each slit, many-worlds says that the light goes thru differenct universes. Because the light could have gone thru either slit in the past, many-worlds
says that the light is split into different universes in the future. The theory is bizarre and does not explain the simplest experiments.
Many-worlds says that all possibilities happen in alternate universes. That deflates the predictive power of quantum mechanics. It is hard to say that an experiment is contrary to theory, because we
might be in the alternate universe where the improbable happens.
The belief in many-worlds is largely based on a belief in a certain sort of time symmetry called unitarity. People argue that if there are multiple possibilities for events in the past, then there
should be multiple possibilities in the future. In the case of the double-slit, the two slits are not just possibilities, but understanding the interference requires that the light really goes thru
both slits. So the many-worlds advocates make the leap to saying that future possibilities must really happen also, even if in alternate universes.
Shermer points to smart people instead of giving evidence, and uses this as an example of how science is better than religion. That is foolish. There are plenty of smart people who believe in
religion also.
Michael Brooks was on Stossel TV plugging his book, Free Radicals: The Secret Anarchy of Science. He talked about 10% of scientists taking drugs, and Einstein didn't really prove his famous equation.
A review says:
According to Brooks, in Science anything goes. The competition is so tough and the prizes so valuable that no punches are pulled. Drugs, lies, fraud, politics - all are part of the game. He
exposes famed personalities from Newton to Einstein - showing how human they all are; and how the successful ones never hesitated to break the rules. Most of us have heard of Newton's famous
statement on '..standing on the shoulders of giants', but we would not have heard of his skill of stomping down other scientists!. Any literate person would have heard of Einstein and his E=MC2
equation, but it is unbelievable to hear that he could not fully prove it in spite of eight attempts!!
I explain this history in my book.
Physicist and many-worlds guru David Deutsch had a new book last year, The Beginning of Infinity, where he argues for judging a scientific theory by its explanatory power. He is known for arguing
that multiple universe quantum theory explains quantum computing.
Everyone is all in favor of scientific explanation, so with whom is he disagreeing? It seems that he wants to carve out a position that is opposite to mathematical Finitism and Logical positivism. He
also dislikes the Copenhagen interpretation of quantum mechanics.
I defend logical positivism and Copenhagen. I should defend finitism also, as most math can be done from a finitary viewpoint.
Jared Diamond got famous for giving grand explanations for the history of civilization, but, as Deutsch points out, the evidence for those explanations is very weak.
Deutsch's book is interesting for his view of the world. However his specific arguments are not substantiated well enough to take seriously.
Einstein scholar Galina Weinstein just posted this paper:
In this paper I present a personal and scientific biographical sketch of Poincare,... He was so encyclopedic that he dealt with the outstanding questions in the different branches of physics and
mathematics; he had altered whole fields of science such as non-Euclidean geometry, Arithmetic, celestial mechanics, thermodynamics and kinetic theory, optics, electrodynamics, Maxwell's theory,
and other topics from the forefront of Fin de Siecle physical science. It is interesting to note that as opposed to the prosperity of biographies and secondary papers studying the life and
scientific contributions of Albert Einstein, one finds much less biographies and secondary sources discussing Poincare's life and work. As opposed to Einstein, Poincare was not a cultural icon.
Beginning in 1920 Einstein became a myth and a world famous figure.
She makes several comparisons, including this:
In 1900 Poincaré was indeed in his highest ranks, and he was the most successful scientist in France and maybe in the whole world. However, Poincaré felt deep inside a very big crisis. The
contents of his lectures, which he presented in the two international conferences (of physics and philosophy), and the talk presented in the Lorentz Festschrift celebrations, reveal this crisis
pertaining to reconciling Lorentz's theory with the principle of relativity and the principle of reaction.Q
It is interesting to note that Einstein had a crisis at about the same time. Einstein appeared to have been trying to solve the conflict between the principle of Galilean relativity and that of
the constancy of the velocity of light in Maxwell's theory; and the conflict between the principle of Galilean relativity and Maxwell's theory and Faraday's law. Although both Einstein and
Poincaré were feeling a crisis at about the same time, they followed completely different routes.
Yes, they had different routes. Poincare's approach was to use the constant speed of light to synchronize clocks and define space and time; insist on the relativity principle as the best explanation
of the Michelson-Morley experiment; invent a non-Euclidean geometry for four-dimensional spacetime using the Lorentz group and Minkowski metric; prove the covariance of Maxwell's equations for
electromagnetism; and use symmetry invariance to find new laws of physics so that relativity becomes a spacetime theory that applies to everything.
Einstein's approach was to make Lorentz's theorem of corresponding states into a postulate, and to use Poincare synchronization to give an exposition of Poincare's physical interpretation of
Lorentz's local time. He is thus able to give a presentation of Lorentz's electron theory. Lorentz explained, "Einstein simply postulates what we have deduced".
Poincare's approach quickly became the backbone of special relativity. This is all explained in my book.
I got some good comments on my FQXi essay. Jonathan Burdick reduced me to 6 words:
Our friend Rog needs 10 pages to opine that "the map is not the territory"? :-)
That's right, the mathematics is the map that describes how to observe the physics, but it is not identical to the physics. His slogan would have made a great title.
Another said:
I enjoyed reading your essay. It is really clearly written and thoroughly accessible, even to someone without a maths or physics background. You have set out your arguments very clearly and I
might have been convinced had I not previously given this subject quite a bit of thought.
In other words: Good argument, but my mind was already made up!
I'm somewhat in sympathy with Jonathan Burdick's pithy response, but of course in ten pages you do more that say that the map is not the territory. I take you also to say that the territory
(reality) is not mathematical. ...
What is left to do is very hard, in the usual story of all the low-hanging fruit having been picked, but we have made better tools than our forebears. It is also possible that there is some part
of the territory that only ever happens once, so that it cannot be subject of Physics taken to be a repeatable experimental subject. ...
In any case, there has been a constant interplay between Mathematics and Physics, ...
I like his phrase "construction of a new systematization of experimental data." Yes, that is a laudable goal and mathematics is a terrific tool. I also "accept that Physics is the systematic
description of reproducible experimental results."
My purpose is to better understand the limits to mathematical reasoning in physics. For example, consider the No-cloning theorem. If a physical state is perfectly representable by some numbers or
other mathematical objects, then it is very hard to understand why a perfect copy cannot be made. Perfect cloning of mathematical objects is axiomatic. I say that the quantum state is great for
systematizing experimental data, but when you take it too literally as being reality then paradoxes result. It is better to step back, and admit that our mathematical models may be necessarily
My essay's public rating is currently a meager 4.4 out of 10. My essay goes against conventional wisdom, but I don't expect a high rating, but I hope that it is good enough to qualify for judging
this fall. At least my essay answers the contest question:
Questioning the Foundations: Which of Our Basic Physical assumptions are Wrong?
What assumptions are ripe for rethinking? ...
What are the implicit assumptions we tend to forget we have postulated, or that have become so ingrained that they have become unquestioned dogma? ...
Note: Successful and interesting essays will not use this topic as an opportunity to trot out their pet theories simply because those theories reject assumptions of some other or established
theory. Rather, the challenge here is to create new and insightful questions or analysis about basic, often tacit, assumptions that can be questioned but often are not.
Galina Weinstein posted Genesis of general relativity - Discovery of general relativity
The intermediate stage of the development of general relativity is inseparable of Marcel Grossmann's mathematical assistance. Einstein acknowledges Grossmann's help during 1912-1914 to the
development of general relativity. ...
Einstein and Grossmann's first joint paper entitled, "Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation" ("Outline of a Generalized Theory of Relativity and of
a Theory of Gravitation") is called by scholars the "Entwurf" paper. ... Grossmann wrote the mathematical part of this paper and Einstein wrote the physical part. ...
The "Entwurf" theory was already very close to Einstein's general theory of relativity that he published in November 1915. The gravitational field is represented by a metric tensor, the
mathematical apparatus of the theory is based on the work of Riemann, Christoffel, Ricci and Levi-Civita on differential covariants, and the action of gravity on other physical processes is
represented by generally covariant equations (that is, in a form which remained unchanged under all coordinate transformations). ...
Scholars asked: Why did Einstein discard in winter 1912-1913 what appears in hindsight to be essentially the correct gravitational field equation, and what made his field equation acceptable in
late 1915? Why did he reject equations of much broader covariance in 1912-1913? ... His rejection of the Ricci tensor need not be explained in terms of simple error. He was rather not prepared to
accept generally covariant equations as a result of a number of misconceptions. ...
He introduced an ingenious argument – the Hole Argument – to demonstrate that generally covariant field equations were not permissible. The Hole Argument seemed to cause Einstein great
satisfaction, or else he persuaded himself that he was satisfied. Having found the Hole argument, Einstein spent two years after 1913 looking for a non-generally covariant formulation of
gravitational field equations.
The Einstein-Grossmann collaboration was one where Einstein wrote the wrong stuff and Grossmann wrote the correct stuff. Einstein argued for two years that Grossmann was wrong to require generally
covariant field equations, even tho Levi-Civita explained the advantages of covariant equations, until David Hilbert eventually convinced Einstein of those equations.
Einstein's famous 1916 general relativity paper starts:
The theory which is sketched in the following pages forms the most wide-going generalization conceivable of what is at present known as "the theory of Relativity;" this latter theory I
differentiate from the former "Special Relativity theory," and suppose it to be known. The generalization of the Relativity theory has been made much easier through the form given to the special
Relativity theory by Minkowski, which mathematician was the first to recognize clearly the formal equivalence of the space like and time-like co-ordinates, and who made use of it in the building
up of the theory. The mathematical apparatus useful for the general relativity theory, lay already complete in the "Absolute Differential Calculus", which were based on the researches of Gauss,
Riemann and Christoffel on the non-Euclidean manifold, and which have been shaped into a system by Ricci and Levi-Civita, and already applied to the problems of theoretical physics. I have in
part B of this communication developed in the simplest and clearest manner, all the supposed mathematical auxiliaries, not known to Physicists, which will be useful for our purpose, so that, a
study of the mathematical literature is not necessary for an understanding of this paper. Finally in this place I thank my friend Grossmann, by whose help I was not only spared the study of the
mathematical literature pertinent to this subject, but who also aided me in the researches on the field equations of gravitation. [1920 translation]
About half the paper is an explanation of tensor analysis. There is no reference to the Entwurf theory or Hilbert. The only papers cited were those by Einstein and K. Schwarzschild showing how
Grossmann's covariant equations affect the precession of Mercury's orbit. The assistance from Levi-Civita and Grossmann is understated.
The Wikipedia article on general relativity mentions Einstein about 50 times, and says nothing about Grossmann or Hilbert. There is a lot more historical info at Relativity priority dispute.
There are many interpretations of quantum mechanics, and no experiment to prove that any one is better than any other. So choosing one is a matter of convention. You might even prefer one
interpretation for some problem, and another interpretation for others. Medieval astronomers sometimes used a geocentric model for some planets, and a heliocentric model for others, even tho the
models conflict.
The founders of quantum mechanics were believers in positivism, a philosophy that has since gone out of favor. Positivists believe in what is observable, and avoid giving opinions on what is not. I
believe that the more the interpretations stray from positivism, the harder it is to make sense out of them. Therefore I propose what I call the positivist interpretation as the core minimalist way
to understand the theory.
The positivist interpretation is instrumentalist. However the terms are confusing because a lot of physicists talk as if they are instrumentalists, but they are not positivists. For example, Max
Tegmark (MIT) writes:
I advocate an extreme "shut-up-and-calculate" approach to physics, where our external physical reality is assumed to be purely mathematical.
But that assumption is extremely dubious, and not substantiated by any observation. I cannot prove him wrong, but a positivist would reject it just because there is no observational support for it,
and because it is not even particularly useful in modern physics. I have written a FQXi essay against it.
The original Copenhagen interpretation was positivist, but it is widely misunderstood. The Ensemble interpretation is supposed to be minimalist, but it is not truly positivist because it does not
predict single experiments. Other interpretations assume all sorts of things that can never be observed.
English science writer Philip Ball writes:
Most physicists accept something like Bohr and Heisenberg’s Copenhagen interpretation. This holds that there is no essential reality beyond the quantum description, nothing more fundamental and
definite than probabilities.
This is close to being positivist, but a true positivist would not say that there is no essential reality. He would accept the observable realities, and dismiss talk of other realities as being
meaningless until someone relates them to observables. He also would not say that probabilities are fundamental, as they are interpretational and not observable.
Here are characteristics of the positivist interpretation.
It is local. No action-at-a-distance has ever been observed, nor is it possible in a relativistic theory. Lumo gives recent explanation of why nonlocality research is pseudoscience.
It rejects hidden variables, as they are not observed and all evidence is to the contrary. Even the wave function is not real, as it is not directly observable.
Positivism also rejects Counterfactual definiteness. As A. Peres said, Unperformed Experiments Have no Results.
I am neutral on Determinism. I am convinced that no quantum probabilities are needed. If the theory does not need them, and they are not observable, then they are not part of the positivist
I am also neutral on what I call the weak mathematical universe hypothesis. When a positivist says he is neutral, that means that he rejects it as extraneous. I have never seen anyone explicitly
reject it, but I believe that it will eventually be seen to be false.
Explanations of quantum mechanics often get hung up on trying to attach some meaning to reality that is independent of what is observed. Physicists will even say that quantum mechanics proves that
there is no such thing as reality. What they are really saying is that non-positivist interpretations are hard to understand. Adopt a positivist philosophy, and the problems disappear.
A good explanation of positivist quantum mechanics is this essay on Quantum Reality. The author favors a positivist variant of the Copenhagen interpretation that he calls the London (Ticker-Tape)
Interpretation. He says "Bohr got it pretty much right" and positivism has the virtue of "no deeper meaning than that obtained through measurement".
Positivists are sometimes criticized for saying that there is no deeper meaning, when they cannot prove that there is no deeper meaning. But that criticism misunderstands positivism. The more correct
statement is that positivists admit that there might be a deeper meaning involving determinism or probability, mathematical or physical universe, waves or particles, etc. But quantum mechanics
experiments are unable to resolve these issues, so they are not worthy of scientific discussion.
Adding to the confusion, modern philosophers not only reject positivism, they also deny that Bohr was a positivist.
Yes, Bohr did get it right with his positivism, and he was considered the winner of the Bohr–Einstein debates. But as positivism has gone out of fashion, so has Bohr's view. Probably a lot of
physicists and philosophers today would say they prefer Einstein's view because it is more realist. They cause a lot of confusion. It would be better if quantum mechanics were taught with the
positivist interpretation.
The main virtue of the positivist interpretation is that it only requires you to believe in the core physics, and does not require you to take a position on determinism, many-worlds, consciousness,
or anything like that.
Lumo writes today:
In the last 85+ years since the discovery of quantum mechanics, all people opposing quantum mechanics have lost, all of their predictions differing from the predictions of quantum mechanics have
been proved wrong, and the whole philosophy of trying to find and promote "problems" with the proper Copenhagen quantum mechanics – and all these efforts are always driven by the desire to undo
the quantum revolution and return physics to the age when the classical framework was dominant – has been an utter failure, an embarrassing pseudointellectual catastrophe, a huge pile of stinky
junk that no sensible scientist would associate herself with.
I am amazed that even this modest and balanced summary of the situation may be considered controversial by some physicists in 2012. I am amazed that Brian Greene may be on the evil side, too.
Niels Bohr treated the theories about many worlds as garbage bringing nothing new and correct to physics for a simple reason: they were garbage that was bringing nothing that was both new and
I agree with this. I believe that the richer interpretations of quantum mechanics are wrong for reasons explained in my FQXi essay. I differ from him in that I subscribe to an even more minimalist
interpretation, as I say that no quantum probabilities are needed, not even for the double slit.
A reader tells me that she found my book, How Einstein Ruined Physics, in a bookstore next to the new book, How The Beatles Destroyed Rock 'n' Roll: An Alternative History of American Popular Music.
That's funny. I am going to contact the publisher and suggest a series of books. I will propose How Babe Ruth Ruined Baseball, How Mark Twain Ruined Literature, How Pablo Picasso Ruined Art, How
Margaret Mead Ruined Anthropology, and How Steve Jobs Ruined Computers.
With all the publicity about the discovery of the Higgs boson, hardly anyone is explaining that it is a confirmation of modern Aether theories. Usually any mention of the aether is followed by saying
that it was a quaint 19th century concept that was disproved by Einstein's relativity. But that is not true, and was not even Einstein's view.
The concept of the luminiferous aether dates back to ancient times, and refers to whatever fills outer space that allows us to see the light of the stars. It is sometimes said that the vacuum is
empty space, and that no such aether is needed to explain the propagation of light. But that is not true either, as modern theories of light require a nonempty vacuum. Quantum electrodynamics is a
perturbation theory of the aether.
The authoritative description of the 19th century aether is J.C. Maxwell's 1878 encyclopedia article. That concluded:
No theory of the constitution of the aether has yet been invented which will account for such a system of molecular vortices being maintained for an indefinite time without their energy being
gradually dissipated into that irregular agitation of the medium which, in ordinary media, is called heat.
Whatever difficulties we may have in forming a consistent idea of the constitution of the aether, there can be no doubt that the interplanetary and interstellar spaces are not empty, but are
occupied by a material substance or body, which is certainly the largest, and probably the most uniform body of which we have any knowledge.
Whether this vast homogeneous expanse of isotropic matter is fitted not only to be a medium of physical interaction between distant bodies, and to fulfil other physical functions of which,
perhaps, we have as yet no conception, but also, as the authors of the Unseen Universe seem to suggest, to constitute the material organism of beings exercising functions of life and mind as high
or higher than ours are at present, is a question far transcending the limits of physical speculation.
Maxwell's view was that the aether was pervasive, uniform, invisible, frictionless, and permeating matter. It is sometimes said that the aether presupposed some sort of fixed coordinate system, but
Maxwell does not say that.
What I say here is the consensus view, except that not everyone uses the word "aether". Frank Wilczek, the 2004 Nobel Prize winner in physics, wrote the book, The Lightness of Being: Mass, Ether, and
the Unification of Forces:
In the first part of the twentieth century, the upheavals of relativity and (especially) quantum theory shattered the foundations beneath classical physics. Existing theories of matter and light
were reduced to rubble. That process of creative destruction made it possible to construct, over the second part of the twentieth century, a new and deeper theory of matter/light that removed the
ancient separation. The new theory sees a world based on a multiplicity of space-filling ethers, a totality I call the Grid. The new world-model is extremely strange, but also extremely
successful and accurate.
He details:
What is Space? Is it an empty stage, where the physical world of matter acts out its drama -- an equal participant, like the classical Ether, that both provides background and has a life of its
own -- or the primary reality, of which matter is a secondary manifestation? Today, the third view is triumphant. Where our eyes see nothing our brains, pondering the revelations of sharply tuned
experiments, discover the Grid that powers physical reality.
The Higgs boson is not just some isolated particle. It is the quantization of an aether that is pervasive, uniform, invisible, frictionless, and permeating matter. And that aether is completely
essential to modern physics.
The Higgs aether gives mass to the electrons and quarks, the basic constituents of matter. All electrons are identical, and have the same mass. So the Higgs is the same everywhere. The aether is the
largest and most uniform body, just as Maxwell said.
You could also say that there is an electron field in a vacuum, with fluctuations making virtual electrons. But there is no net number of electrons. The Higgs aether is different in that the Higgs
field is nonzero in the vacuum.
The Higgs boson discovery is being announced:
On Wednesday (July 4), scientists heading two major experiments at the LHC plan to announce their most recent findings ...
The Higgs boson is the last piece of the physics Standard Model, a collection of theories that underpin all modern physics. The Higgs particle is theorized to mediate mass -- like a photon (also
a boson) mediates the electromagnetic force, i.e., light -- and creates the "Higgs field" that must pervade the entire Universe, endowing matter with mass.
Dennis Overbye writes:
Known as the Higgs boson, it explains why things in the universe have mass, and is a cornerstone of modern physics despite never being seen.
Here is why we believe in the Higgs, and why it was really discovered decades ago.
Geometry symmetry is the most important concept in 20th century physics. Poincare introduced it with his 1905 geometric version of special relativity, and his search for physical laws that obey
symmetries. Its crucial importance to classical mechanics was shown by Noether, and the importance to quantum mechanics by Hermann Weyl. The concept can be used to generalize electromagnetism to
other fields (like strong and weak forces), as shown by Weyl, Higgs, 'tHooft, and others.
The Standard Model is based on geometric symmetries, but if there are too many of them, then all particles are massless like photons and nothing interesting happens. So there has to be a field that
breaks the symmetry. All fields are quantized, so there has to be a particle also. That is what is being found at 125 GeV.
In a sense, the Higgs is like an aether that is uniform, everywhere, and invisible. You could say that mass is just a measure of resistance to passing thru the Higgs aether. This description is a
little misleading because most of the proton mass comes from the binding energy of the quarks, but the quark mass is believed to be derived from the Higgs breaking the symmetry. This is all explained
in my book. For a recent survey by an expert, see Wilczek, Origins of Mass.
If you have been watching PBS TV science shows, you might have been expecting LHC announcements on string theory, supersymmetry, and the multiverse. However, all of the evidence has been against
those misguided concepts.
Update: Overbye adds in the July 4 NY Times:
Physicists working at CERN’s Large Hadron Collider said Wednesday that they had discovered a new subatomic particle that looks for all the world like the Higgs boson, a potential key to
understanding why elementary particles have mass and indeed to the existence of diversity and life in the universe. ...
Confirmation of the Higgs boson or something very like it would constitute a rendezvous with destiny for a generation of physicists who have believed in the boson for half a century without ever
seeing it. And it reaffirms a grand view of a universe ruled by simple and elegant and symmetrical laws, but in which everything interesting in it, such as ourselves, is due to flaws or breaks in
that symmetry.
According to the Standard Model, which has ruled physics for 40 years now, the Higgs boson is the only visible and particular manifestation of an invisible force field, a cosmic molasses that
permeates space and imbues elementary particles that would otherwise be massless with mass. Particles wading through it would gain heft.
Without this Higgs field, as it is known, or something like it, physicists say all the elementary forms of matter would zoom around at the speed of light, flowing through our hands like
moonlight. There would be neither atoms nor life. ...
Although they have never been seen, Higgs-like fields play an important role in theories of the universe and in string theory. Under certain conditions, according to the strange accounting of
Einsteinian physics, they can become suffused with energy that exerts an anti-gravitational force. Such fields have been proposed as the source of an enormous burst of expansion, known as
inflation, early in the universe, and, possibly, as the secret of the dark energy that now seems to be speeding up the expansion of the universe.
The Higgs boson is the most unusual known particles, being spin 0 and playing a unique symmetry-breaking role. Its discovery is the most dramatic achievement of theoretical physics in history. It is
part of the aether that is completely essential to modern physics. The aether is everywhere, uniform, and invisible.
The aether probably explains dark energy, but none of this has anything to do with string theory. String theory does not predict an aether, Higgs boson, or dark energy. After 30 years of work by the
world's smartest physicists, it has never had a success like the Higgs. The Higgs is a confirmation of the theory that string theory was supposed to replace.
SciAm blogger John Horgan writes:
Hearing all the hoopla about the Higgs, the public might understandably assume that it represents a crucial step toward a unified theory–and perhaps at least tentative confirmation of the
existence of strings, branes, hyperspaces, multiverses and all the other fantastical eidolons that Kaku, Stephen Hawking, Brian Greene and other unification enthusiasts tout in their bestsellers.
But the Higgs doesn’t take us any closer to a unified theory than climbing a tree would take me to the Moon.
That's right. The LHC spent $10B to confirm the high-energy physics of the 1970s, but all hopes for a grander theory have been a total failure.
Horgan also says:
Physicists have already produced theories –- Newtonian mechanics, quantum mechanics, general relativity, nonlinear dynamics –- that work extraordinarily well in certain domains, and there is no
reason why there should be a single theory that accounts for all the forces of nature. The quest for a unified theory will come to be seen not as a branch of science, which tells us about the
real world, but as a kind of mathematical theology.
I agree with that, and I go further in my FQXi essay. I say that there is no reason that this mathematical theology should even be valid for those domains like quantum mechanics. Theoretical
physicists are chasing the impossible.
This is from The Argyle Sweater by Scott Hilburn. Sometimes people assume that Copernicus was particularly novel or courageous by saying that the Sun does not revolve around the Earth. Maybe not. | {"url":"https://blog.darkbuzz.com/2012/07/","timestamp":"2024-11-02T17:40:16Z","content_type":"text/html","content_length":"173120","record_id":"<urn:uuid:4bd28238-bd1e-492e-817d-8970153921ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00359.warc.gz"} |
Unit testing with R | R-bloggersUnit testing with R
[This article was first published on
Econometrics and Free Software
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I've been introduced to unit testing while working with colleagues on quite a big project for which we use Python.
At first I was a bit skeptical about the need of writing unit tests, but now I must admit that I am seduced by the idea and by the huge time savings it allows. Naturally, I was wondering if the same
could be achieved with R, and was quite happy to find out that it also possible to write unit tests in R using a package called testthat.
Unit tests (Not to be confused with unit root tests for time series) are small functions that test your code and help you make sure everything is alright. I'm going to show how the testthat packages
works with a very trivial example, that might not do justice to the idea of unit testing. But you'll hopefully see why writing unit tests is not a waste of your time, especially if your project gets
very complex (if you're writing a package for example).
First, you'll need to download and install testthat. Some dependencies will also be installed.
Now, you'll need a function to test. Let's suppose you've written a function that returns the nth Fibonacci number:
Fibonacci <- function(n){
a <- 0
b <- 1
for (i in 1:n){
temp <- b
b <- a
a <- a + temp
You then save this function in a file, let's call it fibo.R. What you'll probably do once you've written this function, is to try it out:
## [1] 5
You'll see that the function returns the right result and continue programming. The idea behind unit testing is write a bunch of functions that you can run after you make changes to your code, just
to check that everything is still running as it should.
Let's create a script called test_fibo.R and write the following code in it:
test_that("Test Fibo(15)",{
phi <- (1 + sqrt(5))/2
psi <- (1 - sqrt(5))/2
expect_equal(Fibonacci(15), (phi**15 - psi**15)/sqrt(5))
The code above uses Binet's formula, a closed form formula that gives the nth Fibonacci number and compares it our implementation of the algorithm. If you didn't know about Binet's formula, you could
simply compute some numbers by hand and compare them to what your function returns, for example. The function expect_equal is a function from the package testthat and does exactly what it tells. We
expect the result of our implementation to be equal to the result of Binet's Formula. The file test_fibo.R can contain as many tests as you need. Also, the file that contains the tests must start
with the string test, so that testthat knows with files it has to run.
Now, we're almost done, create yet another script, let's call it run_tests.R and write the following code in it:
test_results <- test_dir("path/to/tests", reporter="summary")
After running these lines, and if everything goes well, you should see a message like this:
> library(testthat)
> source("path/to/fibo.R")
> test_results <- test_dir("path/to/tests", reporter="summary")
Your tests are dandy!
Notice the small . over the message? This means that one test was run successfully. You'll get one dot per successful test. If you take a look at test_results you'll see this:
> test_results
file context test nb failed skipped error user system real
1 test_fibo.R Test Fibo(15) 1 0 FALSE FALSE 0.004 0 0.006
You'll see each file and each function inside the files that were tested, and also whether the test was skipped, failed etc. This may seem overkill for such a simple function, but imagine that you
write dozens of functions that get more and more complex over time. You might have to change a lot of lines because as time goes by you add new functionality, but don't want to break what was
working. Running your unit tests each time you make changes can help you pinpoint regressions in your code. Unit tests can also help you start with your code. It can happen that sometimes you don't
know exactly how to start; well you could start by writing a unit test that returns the result you want to have and then try to write the code to make that unit test pass. This is called test-driven
I hope that this post motivated you to write unit tests and make you a better R programmer! | {"url":"https://www.r-bloggers.com/2016/03/unit-testing-with-r/","timestamp":"2024-11-10T21:41:16Z","content_type":"text/html","content_length":"95576","record_id":"<urn:uuid:8009377c-c966-46ef-9654-fcabdfa5be11>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00191.warc.gz"} |
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• long division crossword polynomial | {"url":"https://softmath.com/algebra_stats/basic-concept-in-algebra.html","timestamp":"2024-11-14T20:18:51Z","content_type":"text/html","content_length":"36970","record_id":"<urn:uuid:b1a5d668-b3b1-4ef1-ac67-0a1ac6e355b0>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00752.warc.gz"} |
Researching | Analysis of extinction characteristics of non-spherical biological particle aggregates [Invited]
As a novel extinction material, biological materials have attracted wide attention due to their advantages such as low preparation cost, environmental protection, non-toxic, easy degradation, and
wide extinction band^[1 -3]. Gu et al.^[4] studied the effect of the activity of biological materials on their average mass extinction coefficient (MEC) and presented a practical method to
discriminate the viable and dead microbes. Wang et al.^[5] found that when the relative humidity of air exceeds 70%, the ratio of attenuation rate to concentration increases with relative humidity.
Generally, biological materials are composed of artificially prepared biological particles, and biological particles have abundant sources and different morphologies^[6], such as spherical,
ellipsoidal, cylindrical, and irregular shapes. When released into the air, biological particles exist in the form of aggregates with complex spatial structures^[7] and can attenuate light. With the
widespread application of photoelectric detection systems, biomaterials with strong extinction performance have aroused great interest in research and practical applications. | {"url":"https://m.researching.cn/articles/OJ5a1b7ee91005084f","timestamp":"2024-11-08T19:06:47Z","content_type":"text/html","content_length":"103449","record_id":"<urn:uuid:744a7d1a-7d58-4b56-95be-84388bf1de6a>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00082.warc.gz"} |
Practice Exam - Question #6 - Discrete Distributions – Q&A Hub – 365 Data Science
Practice Exam - Question #6 - Discrete Distributions
Hi team,
Thanks for your hard work! Great material.
I do not understand why the solution to this problem is not simply 3 + 8. The way it's worded, it seems like the answer should be 3 + 8. Claude, the AI tool, also thought the answer should be 11.
Please clarify this for me. Thanks!
Here's what Claude said when I provided it the solution to the problem:
I apologize, but there appears to be an error in the explanation provided in the image. The explanation is incorrect and inconsistent with the principles of the Poisson distribution.
In a Poisson distribution with parameter λ, the mean (expected value) is equal to λ. Therefore:
For family cars with λ = 3, the mean is 3, not 1/3.
For single-member cars with λ = 8, the mean is 8, not 1/8.
The correct solution should be:
Mean of family cars: 3 Mean of single-member cars: 8 Sum of means: 3 + 8 = 11
The explanation in the image incorrectly inverts the λ values, which is not mathematically valid for Poisson distributions. The sum should indeed be 11, but it's obtained by adding 3 and 8 directly,
not by adding their reciprocals.
This mistake in the explanation likely caused your confusion. You were correct in your earlier reasoning when you added 3 and 8 to get 11. That is the correct approach for finding the sum of the
expected values (means) of these two Poisson distributions.
1 answers ( 0 marked as helpful)
Hey Justin,
Thank you for bringing this to our attention! The question is now replaced.
Let us know if you encounter other issues.
Kind regards,
365 Hristina | {"url":"https://365datascience.com/question/practice-exam-question-6-discrete-distributions/","timestamp":"2024-11-07T10:30:26Z","content_type":"text/html","content_length":"112912","record_id":"<urn:uuid:de29dec8-dbb9-4137-ba92-175389b8d025>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00719.warc.gz"} |
Frequency response data or model
An idfrd object stores frequency response data over a range of frequency values. You can use an idfrd object in two ways. You can use the object as estimation data for estimating a time-domain or
frequency-domain model, similarly to an iddata object. Or, you can use the object as a linear model, similarly to how you use an idss state-space model or any other identified linear model. Use the
idfrd command to encapsulate frequency response data or to convert a linear time-domain or frequency-domain dynamic model into a frequency response model.
Commands that accept iddata objects, such as the model estimation command ssest, generally also accept idfrd objects. However, an idfrd object can contain data from only one experiment. It does not
have the multiexperiment capability that an iddata object has.
Commands that accept identified linear models, such as the analysis and validation commands compare, sim, and bode, generally also accept idfrd models.
For a model of the form
the transfer function estimate is $G\left({e}^{i\omega }\right)$ and the additive noise spectrum Φ[v] at the output is
${\Phi }_{v}\left(\omega \right)=\lambda T{|H\left(e{}^{i\omega T}\right)|}^{2}$
Here, λ is the estimated variance of e(t) and T is the sample time.
For a continuous-time system, the noise spectrum is
${\Phi }_{v}\left(\omega \right)=\lambda {|H\left(e{}^{i\omega }\right)|}^{2}$
An idfrd object stores $G\left({e}^{i\omega }\right)$ and Φ[v].
You can obtain an idfrd model in one of three ways.
• Create the model from frequency response data using the idfrd command. For example, create an idfrd model that encapsulates frequency response data taken at specific frequencies using the sample
time Ts.
sysfr = idfrd(ResponseData,Freq,Ts)
For an example, see Create idfrd Object from Frequency Response Data.
• Estimate the model using a frequency response estimation command such as spa, using time-domain, frequency-domain, or frequency response data.
For more information about frequency response estimation commands, see spa, spafdr, and etfe.
• Convert a linear model such as an idss model into an idfrd model by computing the frequency response of the model.
For an example of linear model conversion, see Convert Time-Domain Model to Frequency Response Model.
For information on functions you can use to extract information from or transform idfrd model objects, see Object Functions.
Create Frequency Response Object
sysfr = idfrd(ResponseData,Frequency,Ts) creates a discrete-time idfrd object that stores the frequency response ResponseData of a linear system at frequency values Frequency. Ts is the sample time.
For a continuous-time system, set Ts to 0.
sysfr = idfrd(___,Name,Value) sets additional properties using one or more name-value arguments. Specify the name-value arguments after the first three arguments. For instance, to specify the
frequency units as MHz, use sysfr = idfrd(ResponseData,Frequency,Ts,'FrequencyUnits','MHz').
Convert Linear Identified Model to Frequency Response Model
sysfr = idfrd(sys) converts a System Identification Toolbox™ or Control System Toolbox™ linear model to frequency response data at default frequencies, including the output noise spectra and spectra
sysfr = idfrd(sys,Frequency) computes the frequency response at frequencies Frequency, where Frequency is expressed in radians/TimeUnit.
sysfr = idfrd(sys,Frequency,FrequencyUnits) interprets frequencies in the Frequency vector in the units specified by FrequencyUnit.
Input Arguments
sys — Linear dynamic system model
linear dynamic system model
Linear dynamic system model, specified as a System Identification Toolbox or Control System Toolbox linear model.
ResponseData — Frequency response data
3-D array of complex numbers
Frequency response data, specified as a 3-D array of complex numbers.
• For SISO systems, ResponseData is a vector of frequency response values at the frequency points specified in the Frequency property.
• For MIMO systems with N[u] inputs and N[y] outputs, ResponseData is an N[y]-by-N[u]-by-N[f] array, where N[f] is the number of frequency points.
ResponseData(ky,ku,kf) represents the frequency response from the input ku to the output ky at the frequency Frequency(kf).
Frequency — Frequency points
column vector
Frequency points corresponding to ResponseData, specified as a column vector that contains N[f] points in the units specified by FrequencyUnit.
FrequencyUnit — Units for frequency vector
'rad/TimeUnit' (default) | 'cycles/TimeUnit' | 'rad/s' | 'Hz' | 'kHz' | 'MHz' | 'GHz' | 'rpm'
Units of the frequency vector in the Frequency property, specified as one of the following values:
• 'rad/TimeUnit'
• 'cycles/TimeUnit'
• 'rad/s'
• 'Hz'
• 'kHz'
• 'MHz'
• 'GHz'
• 'rpm'
The units 'rad/TimeUnit' and 'cycles/TimeUnit' are relative to the time units specified in the TimeUnit property.
Changing this property does not resample or convert the data. Modifying the property changes only the interpretation of the existing data. Use chgTimeUnit to convert the data to different frequency
SpectrumData — Power spectra and cross spectra
vector of complex numbers | 3-D array of complex numbers
Power spectra and cross spectra of the system output disturbances (noise), specified as a vector (single-output system) or a 3-D array of complex numbers (multiple-output system). For response data
with N[y] outputs and N[f] frequency points, specify SpectrumData as an N[y]-by-N[y]-by-N[f] array.
SpectrumData(ky1,ky2,kf) is the cross spectrum between the noise at output ky1 and the noise at output ky2 at the frequency Frequency(kf). The power spectrum is the subset of the cross spectrum where
ky1 and ky2 are equal.
CovarianceData — Covariance of response
5-D numeric array
Covariance of SpectrumData, specified as a 5-D array with dimensions N[y]-by-N[u]-by-N[f]-by-2-by-2, where N[y] is the number of outputs, N[u] is the number of inputs, and N[f] is the number of
frequency points.
CovarianceData(ky,ku,kf,:,:) is the 2-by-2 covariance matrix of SpectrumData(ky,ku,kf). The (1,1) element is the variance of the real part, the (2,2) element is the variance of the imaginary part,
and the (1,2) and (2,1) elements are the covariance between the real and imaginary parts. squeeze(CovarianceData(ky,ku,kf,:,:)) thus gives the covariance matrix of the corresponding response.
If you obtain sysfr by converting a model sys, the value of CovarianceData depends on how you obtained sys.
• If you obtained sys by identification, the software computes the estimated covariance for sysfr from the uncertainty information in sys. The software uses Gauss' approximation formula for this
calculation for all model types, except grey-box models. For grey-box models (idgrey), the software applies numerical differentiation.
• If you created sys by using commands such as idss, idtf, idproc, idgrey, or idpoly, then the software sets CovarianceData for sysfr to [].
NoiseCovariance — Power spectra variance
numeric vector | 3-D numeric array | 0
Power spectra variance, specified as a vector (single-output system) or a 3-D array (multiple-output system). For response data with N[y] outputs and N[f] frequency points, specify NoiseCovariance as
an N[y]-by-N[y]-by-N[f] array. NoiseCovariance(ky1,ky2,kf) is the variance of the corresponding power spectrum.
To eliminate the influence of the noise component from the model, specify NoiseCovariance as 0. With zero covariance, the predicted output is the same as the simulated output.
InterSample — Intersample behavior
'zoh' | 'foh' | 'bl' | cell array of character vectors
Intersample behavior of the input signal for transformations between discrete time and continuous time, specified as a character vector or as an N[u]-by-1 cell array of character vectors, where N[u]
is the number of input channels. This property is meaningful only when you are estimating continuous-time models (sample time Ts > 0) from discrete-time data.
For each input channel, the possible values of InterSample are:
• 'zoh' — Zero-order hold maintains a piecewise-constant input signal between samples.
• 'foh' — First-order hold maintains a piecewise-linear input signal between samples.
• 'bl' — Band-limited behavior specifies that the continuous-time input signal has zero power above the Nyquist frequency (pi/sys.Ts rad/s). This behavior typically occurs when the input signal is
measured experimentally using an antialiasing filter and a sampler. Ideally, treat the data as continuous-time. That is, if the signals used for the estimation of the frequency response were
subject to anti-aliasing filters, set sys.Ts to zero.
If you obtain sysfr by conversion of a model sys, then InterSample is equal to the Intersample property of the iddata object that you used to estimate sys.
For more information on this property, see Effect of Input Intersample Behavior on Continuous-Time Models.
IODelay — Transport delays
0 (default) | numeric array
Transport delays, specified as a numeric array containing a separate transport delay for each input-output pair.
For continuous-time systems, transport delays are expressed in the time unit stored in the TimeUnit property. For discrete-time systems, transport delays are expressed as integers denoting delays of
a multiple of the sample time Ts.
For a MIMO system with N[y] outputs and N[u] inputs, set IODelay as an N[y]-by-N[u] array. Each entry of this array is a numerical value representing the transport delay for the corresponding
input-output pair. You can set IODelay to a scalar value to apply the same delay to all input-output pairs.
InputDelay — Input delay for each input channel
0 (default) | scalar | vector
Input delay for each input channel, specified as a scalar value or numeric vector. For continuous-time systems, specify input delays in the time unit stored in the TimeUnit property. For
discrete-time systems, specify input delays in integer multiples of the sample time Ts. For example, setting InputDelay to 3 specifies a delay of three sample times.
For a system with N[u] inputs, set InputDelay to an N[u]-by-1 vector. Each entry of this vector is a numerical value that represents the input delay for the corresponding input channel.
You can also set InputDelay to a scalar value to apply the same delay to all channels.
OutputDelay — Output delay for each output channel
0 (default)
For identified systems such as idfrd, OutputDelay is fixed to zero.
Ts — Sample time
1 (default) | 0 | positive scalar | -1
Sample time, specified as one of the following.
• Discrete-time model with a specified sampling time — a positive scalar representing the sampling period expressed in the unit specified by the TimeUnit property of the model
• Continuous-time model — 0
• Discrete-time model with an unspecified sample time — -1
Changing this property does not discretize or resample the model. Use c2d and d2c to convert between continuous- and discrete-time representations. Use d2d to change the sample time of a
discrete-time system.
TimeUnit — Units for time variable
'seconds' (default) | 'nanoseconds' | 'microseconds' | 'milliseconds' | 'minutes' | 'hours' | 'days' | 'weeks' | 'months' | 'years'
Units for the time variable, the sample time Ts, and any time delays in the model, specified as one of the following values.
• 'nanoseconds'
• 'microseconds'
• 'milliseconds'
• 'seconds'
• 'minutes'
• 'hours'
• 'days'
• 'weeks'
• 'months'
• 'years'
Changing this property does not resample or convert the data. Modifying the property changes only the interpretation of the existing data. Use chgTimeUnit (Control System Toolbox) to convert data to
different time units.
InputName — Input channel names
'' (default) | character vector | cell array
Input channel names, specified as a character vector or cell array.
• Single-input model — Character vector. For example, 'controls'.
• Multi-input model — Cell array of character vectors.
Alternatively, use automatic vector expansion to assign input names for multi-input models. For example, if sys is a two-input model, enter:
sys.InputName = 'controls';
The input names automatically expand to {'controls(1)';'controls(2)'}.
When you estimate a model using an iddata object data, the software automatically sets InputName to data.InputName.
You can use the shorthand notation u to refer to the InputName property. For example, sys.u is equivalent to sys.InputName.
You can use input channel names in several ways, including:
• To identify channels on model display and plots
• To extract subsystems of MIMO systems
• To specify connection points when interconnecting models
InputUnit — Input channel units
'' (default) | character vector | cell array
Input channel units, specified as a character vector or cell array:
• Single-input model — Character vector
• Multi-input Model — Cell array of character vectors
Use InputUnit to keep track of input signal units. InputUnit has no effect on system behavior.
InputGroup — Input channel groups
structure with no fields (default) | structure
Input channel groups, specified as a structure. The InputGroup property lets you divide the input channels of MIMO systems into groups so that you can refer to each group by name. In the InputGroup
structure, set field names to the group names, and field values to the input channels belonging to each group.
For example, create input groups named controls and noise that include input channels 1 and 2 and channels 3 and 5, respectively.
sys.InputGroup.controls = [1 2];
sys.InputGroup.noise = [3 5];
You can then extract the subsystem from the controls inputs to all outputs using the following syntax:
OutputName — Output channel names
'' (default) | character vector | cell array
Output channel names, specified as a character vector or cell array.
• Single-input model — Character vector. For example, 'measurements'
• Multi-input model — Cell array of character vectors
Alternatively, use automatic vector expansion to assign output names for multi-output models. For example, if sys is a two-output model, enter:
sys.OutputName = 'measurements';
The output names automatically expand to {'measurements(1)';'measurements(2)'}.
When you estimate a model using an iddata object data, the software automatically sets OutputName to data.OutputName.
You can use the shorthand notation y to refer to the OutputName property. For example, sys.y is equivalent to sys.OutputName.
You can use output channel names in several ways, including:
• To identify channels on model display and plots
• To extract subsystems of MIMO systems
• To specify connection points when interconnecting models
OutputUnit — Output channel units
'' (default) | character vector | cell array
Output channel units, specified as a character vector or cell array.
• Single-input model — Character vector. For example, 'seconds'.
• Multi-input model — Cell array of character vectors.
Use OutputUnit to keep track of output signal units. OutputUnit has no effect on system behavior.
OutputGroup — Output channel groups
structure with no fields (default) | structure
Output channel groups, specified as a structure. The OutputGroup property lets you divide the output channels of MIMO systems into groups and refer to each group by name. In the OutputGroup
structure, set field names to the group names, and field values to the output channels belonging to each group.
For example, create output groups named temperature and measurement that include output channel 1, and channels 3 and 5, respectively.
sys.OutputGroup.temperature = [1];
sys.OutputGroup.measurement = [3 5];
You can then extract the subsystem from all inputs to the measurement outputs using the following syntax.
Name — System name
'' (default) | character vector
System name, specified as a character vector. For example, 'system_1'.
Notes — Notes on system
0-by-1 string (default) | string | character vector
Any text that you want to associate with the system, specified as a string or a cell array of character vectors. The property stores whichever data type you provide. For instance, if sys1 and sys2
are dynamic system models, you can set their Notes properties as follows.
sys1.Notes = "sys1 has a string.";
sys2.Notes = 'sys2 has a character vector.';
ans =
"sys1 has a string."
ans =
'sys2 has a character vector.'
UserData — Data to associate with system
[] (default) | any MATLAB^® data type
Data to associate with the system, specified as any MATLAB data type.
SamplingGrid — Sampling grid
[] (default) | struct
Sampling grid for model arrays, specified as a structure.
For arrays of identified linear (IDLTI) models that you derive by sampling one or more independent variables, this property tracks the variable values associated with each model. This information
appears when you show or plot the model array. Use this information to trace results back to the independent variables.
Set the field names of the data structure to the names of the sampling variables. Set the field values to the sampled variable values associated with each model in the array. All sampling variables
must be numeric and scalar valued, and all arrays of sampled values must match the dimensions of the model array.
For example, suppose that you collect data at various operating points of a system. You can identify a model for each operating point separately and then stack the results together into a single
system array. You can tag the individual models in the array with information regarding the operating point.
nominal_engine_rpm = [1000 5000 10000];
sys.SamplingGrid = struct('rpm', nominal_engine_rpm)
Here, sys is an array containing three identified models obtained at 1000, 5000, and 10,000 rpm, respectively.
For model arrays that you generate by linearizing a Simulink^® model at multiple parameter values or operating points, the software populates SamplingGrid automatically with the variable values that
correspond to each entry in the array.
Report — Summary report
report field values
This property is read-only.
Summary report that contains information about the estimation options and results when the frequency-response model is obtained using estimation commands, such as spa, spafdr, and etfe. Use Report to
query a model for how it was estimated, including its:
• Estimation method
• Estimation options
The contents of Report are irrelevant if the model was created by construction.
f = logspace(-1,1,100);
[mag,phase] = bode(idtf([1 .2],[1 2 1 1]),f);
response = mag.*exp(1j*phase*pi/180);
sysfr = idfrd(response,f,0.08);
If you obtain the frequency-response model using estimation commands, the fields of Report contain information on the estimation data, options, and results.
load iddata3;
sysfr = spa(z3);sysfr.Report.Method
For more information on this property and how to use it, see the Output Arguments section of the corresponding estimation command reference page and Estimation Report.
Object Functions
Many functions applicable to Dynamic System Models are also applicable to an idfrd model object. These functions are of three general types.
• Functions that operate on and return idfrd model objects, such as chgTimeUnit and chgFreqUnit
• Functions that perform analytical and simulation functions on idfrd objects, such as bode and sim
• Functions that retrieve or interpret model information, such as getcov
Unlike other identified linear models, you cannot directly convert an idfrd model into another model type using commands such as idss or idtf. Instead, use the estimation command for the model you
want, using the idfrd object as the estimation data. For instance, use sys = ssest(sysfr,2) to estimate a second-order state-space model from the frequency response data in idfrd model sysfr. For an
example of using an idfrd object as estimation data, see Estimate Time-Domain Model Using Frequency Response Data.
The following lists contain a representative subset of the functions that you can use with idss models.
Transformation and Manipulation
chgTimeUnit Change time units of dynamic system
chgFreqUnit Change frequency units of frequency-response data model
fselect Select frequency points or range in FRD model
frdata Access data for frequency response data (FRD) object
fcat Concatenate FRD models along frequency dimension
Analysis and Simulation
bode Bode frequency response of dynamic system
spectrum Plot or return output power spectrum of time series model or disturbance spectrum of linear input/output model
Information Extraction and Interpretation
Create idfrd Object from Frequency Response Data
Create an idfrd object from frequency response data.
Load the magnitude data AMP, the phase data PHA, and the frequency vector W. Set sample time Ts to 0.1.
load demofr AMP PHA W
Ts = 0.1;
Use the values of AMP and PHA to compute the complex-valued response response.
response = AMP.*exp(1j*PHA*pi/180);
Create an idfrd object to store response in the idfrd object frdata.
frdata = idfrd(response,W,Ts)
frdata =
IDFRD model.
Contains Frequency Response Data for 1 output(s) and 1 input(s).
Response data is available at 1000 frequency points, ranging from 0.03142 rad/s to 31.42 rad/s.
Sample time: 0.1 seconds
Created by direct construction or transformation. Not estimated.
Plot the data.
frdata is a complex idfrd object with object properties that you can access using dot notation. For example, confirm the value of Ts.
You can also set property values. Set the Name property to 'DC_Converter'.
frdata.Name = 'DC_Converter';
If you import frdata into the System Identification app, the app names this data DC_Converter, and not the variable name frdata.
Use get to obtain the full set of property settings.
FrequencyUnit: 'rad/TimeUnit'
Report: [1x1 idresults.frdest]
SpectrumData: []
CovarianceData: []
NoiseCovariance: []
InterSample: {'zoh'}
ResponseData: [1x1x1000 double]
IODelay: 0
InputDelay: 0
OutputDelay: 0
InputName: {''}
InputUnit: {''}
InputGroup: [1x1 struct]
OutputName: {''}
OutputUnit: {''}
OutputGroup: [1x1 struct]
Notes: [0x1 string]
UserData: []
Name: 'DC_Converter'
Ts: 0.1000
TimeUnit: 'seconds'
SamplingGrid: [1x1 struct]
Frequency: [1000x1 double]
Convert Time-Domain Model to Frequency Response Model
Convert a state-space model to a frequency response model using the idfrd command.
Load the data z2 and estimate a second-order state-space model sys.
load iddata2 z2
sys = ssest(z2,2);
Convert sys to the idfrd model frsys.
frsys =
IDFRD model.
Contains Frequency Response Data for 1 output(s) and 1 input(s), and the spectra for disturbances at the outputs.
Response data and disturbance spectra are available at 68 frequency points, ranging from 0.1 rad/s to 1000 rad/s.
Output channels: 'y1'
Input channels: 'u1'
Created by conversion from idss model.
Plot frsys.
frsys is an idfrd model that you can use as a dynamic system model or as estimation data for a time-domain or frequency-domain model.
Create idfrd Object from Frequency Response of Time-Domain Model
Obtain the frequency response of a transfer function model and convert the response into an idfrd object.
Construct a transfer function model with one zero and three poles.
systf = idtf([1 .2],[1 2 1 1]);
Use bode to obtain the frequency response of systf, in terms of magnitude and phase, for the frequency vector f.
f = logspace(-1,1,100);
[mag,phase] = bode(systf,f);
Use the values of mag and phase to compute the complex-valued response response.
response = mag.*exp(1j*phase*pi/180);
Create an idfrd object frdata to store response, specifying a sample rate Ts of 0.8.
Ts = 0.8;
frdata = idfrd(response,f,Ts)
frdata =
IDFRD model.
Contains Frequency Response Data for 1 output(s) and 1 input(s).
Response data is available at 100 frequency points, ranging from 0.1 rad/s to 10 rad/s.
Sample time: 0.8 seconds
Created by direct construction or transformation. Not estimated.
Plot the data.
frdata is a complex idfrd object.
Estimate Time-Domain Model Using Frequency Response Data
Estimate a transfer function model from time-domain data and convert the resulting idtf model to an idfrd model. Estimate a new transfer function model from the frequency response data in the idfrd
model. Compare the model responses with the original data.
Load time-domain data z2 and use it to estimate a transfer function sys that has two poles and one zero.
load iddata2 z2
sys = tfest(z2,2,1);
Convert sys to an idfrd model and plot the frequency response.
frsys = idfrd(sys);
Estimate a new transfer function sys1 using the data from frsys as the estimation data.
Compare the responses of sys and sys1 with the original estimation data z2.
The model responses are identical.
Version History
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Polynom calculator
Algebra Tutorials! Saturday 2nd of November
polynom calculator
Related topics:
Home maths test for ks2 free printable | multiply and simplify calculator | ''answers to algebra with pizzazz'' | ged past papers online | trig value chart | solving
Rotating a Parabola differential equations with excel | revise maths what is an integer | java interpolation equation source example | solving square root multivariable derivative |
Multiplying Fractions solving simultaneous second order equations online | hard equations | what is lineal metre | algebra ii two variable simple linear programming problem |
Finding Factors factorizing cubed
Miscellaneous Equations
Mixed Numbers and
Improper Fractions Author Message
Systems of Equations in
Two Variables txree_bimeapplec Posted: Wednesday 27th of Dec 10:06
Literal Numbers Hi, I am a freshman in high school and I am having trouble with my homework. One of my problems is dealing with polynom calculator; can
Adding and Subtracting anyone help me understand what it is all about? I need to complete this asap. Thanks for helping.
Subtracting Integers
Simplifying Complex Registered:
Fractions 15.11.2003
Decimals and Fractions From: Australia
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
Mixed AllejHat Posted: Wednesday 27th of Dec 14:19
The Square of a Binomial There are several inside the whole category of polynom calculator, such as perpendicular lines, subtracting exponents or linear algebra. I
Factoring Trinomials am acquainted with various folks who abandoned the high-ticket alternatives for aid too. Nevertheless , do not give up because I found an
The Pythagorean Theorem alternative resolution that is not so outrageously priced, easy to utilize and to a greater extent more useful than I could have ever
Solving Radical imagined . Following my trials with instructive mathematic software programs and nearly surrendering, I acquired Algebrator. This software
Equations in One Registered: system has accurately supplied answers to every mathematics issue I have supplied to it . But just as essential , Algebrator also furnishes
Variable 16.07.2003 all of the temporal strides needed to derive the supreme solution . Even though anyone could employ the software only to complete
Multiplying Binomials From: Odense, Denmark assignments , I dubiousness about anyone will be permitted to employ the program for examinations .
Using the FOIL Method
Imaginary Numbers
Solving Quadratic
Equations Using the Momepi Posted: Friday 29th of Dec 14:05
Quadratic Formula Algebrator has helped students all over the globe. It is a very nice piece of software and I would recommend it to every student who has
Solving Quadratic problems with their homework.
Order of Operations Registered:
Dividing Complex Numbers 22.07.2004
Polynomials From: Ireland
The Appearance of a
Polynomial Equation
Standard Form of a Line
Positive Integral zatias8318 Posted: Friday 29th of Dec 16:11
Divisors Sounds like something I’ve been looking for all this time! Thanks guys, just one final question, can someone please provide me a website
Dividing Fractions address where I can order my copy of this program?
Solving Linear Systems
of Equations by
Elimination Registered:
Factoring 26.10.2001
Multiplying and Dividing From: Somewhere over
Square Roots the rainbow
Functions and Graphs
Dividing Polynomials
Solving Rational
Equations Admilal`Leker Posted: Sunday 31st of Dec 07:08
Numbers I remember having often faced problems with binomials, function range and linear inequalities. A really great piece of math program is
Use of Parentheses or Algebrator software. By simply typing in a problem homework a step by step solution would appear by a click on Solve. I have used it
Brackets (The through many math classes – Remedial Algebra, Pre Algebra and Pre Algebra. I greatly recommend the program.
Distributive Law)
Multiplying and Dividing Registered:
by Monomials 10.07.2002
Solving Quadratic From: NW AR, USA
Equations by Graphing
Multiplying Decimals
Use of Parentheses or
Brackets (The daujk_vv7 Posted: Tuesday 02nd of Jan 08:53
Distributive Law) You can order this software online: https://gre-test-prep.com/combining-like-radical-terms.html. You won’t regret buying it, besides it’s
Simplifying Complex so cheap considering the depth of knowledge you gain from using it. They even offer an unconditional money back guarantee. All the best for
Fractions 1 your test .
Adding Fractions
Simplifying Complex Registered:
Fractions 06.07.2001
Solutions to Linear From: I dunno, I've
Equations in Two lost it.
Quadratic Expressions
Completing Squares
Dividing Radical
Rise and Run
Graphing Exponential
Multiplying by a
The Cartesian Coordinate
Writing the Terms of a
Polynomial in Descending
Quadratic Expressions
Solving Inequalities
Solving Rational
Inequalities with a Sign
Solving Linear Equations
Solving an Equation with
Two Radical Terms
Simplifying Rational
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step
Solving Quadratic
Equations Graphically
and Algebraically
Collecting Like Terms
Solving Equations with
Radicals and Exponents
Percent of Change
Powers of ten
(Scientific Notation)
Comparing Integers on a
Number Line
Solving Systems of
Equations Using
Factoring Out the
Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing
Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting
Rational Expressions
with Different
Expressions with
Variables as Exponents
The Quadratic Formula
Writing a Quadratic with
Given Solutions
Simplifying Square Roots
Adding and Subtracting
Square Roots
Adding and Subtracting
Rational Expressions
Combining Like Radical
Solving Systems of
Equations Using
Dividing Polynomials
Graphing Functions
Product of a Sum and a
Solving First Degree
Solving Equations with
Radicals and Exponents
Roots and Powers
Multiplying Numbers
polynom calculator
Related topics:
Home maths test for ks2 free printable | multiply and simplify calculator | ''answers to algebra with pizzazz'' | ged past papers online | trig value chart | solving
Rotating a Parabola differential equations with excel | revise maths what is an integer | java interpolation equation source example | solving square root multivariable derivative |
Multiplying Fractions solving simultaneous second order equations online | hard equations | what is lineal metre | algebra ii two variable simple linear programming problem |
Finding Factors factorizing cubed
Miscellaneous Equations
Mixed Numbers and
Improper Fractions Author Message
Systems of Equations in
Two Variables txree_bimeapplec Posted: Wednesday 27th of Dec 10:06
Literal Numbers Hi, I am a freshman in high school and I am having trouble with my homework. One of my problems is dealing with polynom calculator; can
Adding and Subtracting anyone help me understand what it is all about? I need to complete this asap. Thanks for helping.
Subtracting Integers
Simplifying Complex Registered:
Fractions 15.11.2003
Decimals and Fractions From: Australia
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
Mixed AllejHat Posted: Wednesday 27th of Dec 14:19
The Square of a Binomial There are several inside the whole category of polynom calculator, such as perpendicular lines, subtracting exponents or linear algebra. I
Factoring Trinomials am acquainted with various folks who abandoned the high-ticket alternatives for aid too. Nevertheless , do not give up because I found an
The Pythagorean Theorem alternative resolution that is not so outrageously priced, easy to utilize and to a greater extent more useful than I could have ever
Solving Radical imagined . Following my trials with instructive mathematic software programs and nearly surrendering, I acquired Algebrator. This software
Equations in One Registered: system has accurately supplied answers to every mathematics issue I have supplied to it . But just as essential , Algebrator also furnishes
Variable 16.07.2003 all of the temporal strides needed to derive the supreme solution . Even though anyone could employ the software only to complete
Multiplying Binomials From: Odense, Denmark assignments , I dubiousness about anyone will be permitted to employ the program for examinations .
Using the FOIL Method
Imaginary Numbers
Solving Quadratic
Equations Using the Momepi Posted: Friday 29th of Dec 14:05
Quadratic Formula Algebrator has helped students all over the globe. It is a very nice piece of software and I would recommend it to every student who has
Solving Quadratic problems with their homework.
Order of Operations Registered:
Dividing Complex Numbers 22.07.2004
Polynomials From: Ireland
The Appearance of a
Polynomial Equation
Standard Form of a Line
Positive Integral zatias8318 Posted: Friday 29th of Dec 16:11
Divisors Sounds like something I’ve been looking for all this time! Thanks guys, just one final question, can someone please provide me a website
Dividing Fractions address where I can order my copy of this program?
Solving Linear Systems
of Equations by
Elimination Registered:
Factoring 26.10.2001
Multiplying and Dividing From: Somewhere over
Square Roots the rainbow
Functions and Graphs
Dividing Polynomials
Solving Rational
Equations Admilal`Leker Posted: Sunday 31st of Dec 07:08
Numbers I remember having often faced problems with binomials, function range and linear inequalities. A really great piece of math program is
Use of Parentheses or Algebrator software. By simply typing in a problem homework a step by step solution would appear by a click on Solve. I have used it
Brackets (The through many math classes – Remedial Algebra, Pre Algebra and Pre Algebra. I greatly recommend the program.
Distributive Law)
Multiplying and Dividing Registered:
by Monomials 10.07.2002
Solving Quadratic From: NW AR, USA
Equations by Graphing
Multiplying Decimals
Use of Parentheses or
Brackets (The daujk_vv7 Posted: Tuesday 02nd of Jan 08:53
Distributive Law) You can order this software online: https://gre-test-prep.com/combining-like-radical-terms.html. You won’t regret buying it, besides it’s
Simplifying Complex so cheap considering the depth of knowledge you gain from using it. They even offer an unconditional money back guarantee. All the best for
Fractions 1 your test .
Adding Fractions
Simplifying Complex Registered:
Fractions 06.07.2001
Solutions to Linear From: I dunno, I've
Equations in Two lost it.
Quadratic Expressions
Completing Squares
Dividing Radical
Rise and Run
Graphing Exponential
Multiplying by a
The Cartesian Coordinate
Writing the Terms of a
Polynomial in Descending
Quadratic Expressions
Solving Inequalities
Solving Rational
Inequalities with a Sign
Solving Linear Equations
Solving an Equation with
Two Radical Terms
Simplifying Rational
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step
Solving Quadratic
Equations Graphically
and Algebraically
Collecting Like Terms
Solving Equations with
Radicals and Exponents
Percent of Change
Powers of ten
(Scientific Notation)
Comparing Integers on a
Number Line
Solving Systems of
Equations Using
Factoring Out the
Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing
Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting
Rational Expressions
with Different
Expressions with
Variables as Exponents
The Quadratic Formula
Writing a Quadratic with
Given Solutions
Simplifying Square Roots
Adding and Subtracting
Square Roots
Adding and Subtracting
Rational Expressions
Combining Like Radical
Solving Systems of
Equations Using
Dividing Polynomials
Graphing Functions
Product of a Sum and a
Solving First Degree
Solving Equations with
Radicals and Exponents
Roots and Powers
Multiplying Numbers
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and
Improper Fractions
Systems of Equations in
Two Variables
Literal Numbers
Adding and Subtracting
Subtracting Integers
Simplifying Complex
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical
Equations in One
Multiplying Binomials
Using the FOIL Method
Imaginary Numbers
Solving Quadratic
Equations Using the
Quadratic Formula
Solving Quadratic
Order of Operations
Dividing Complex Numbers
The Appearance of a
Polynomial Equation
Standard Form of a Line
Positive Integral
Dividing Fractions
Solving Linear Systems
of Equations by
Multiplying and Dividing
Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational
Use of Parentheses or
Brackets (The
Distributive Law)
Multiplying and Dividing
by Monomials
Solving Quadratic
Equations by Graphing
Multiplying Decimals
Use of Parentheses or
Brackets (The
Distributive Law)
Simplifying Complex
Fractions 1
Adding Fractions
Simplifying Complex
Solutions to Linear
Equations in Two
Quadratic Expressions
Completing Squares
Dividing Radical
Rise and Run
Graphing Exponential
Multiplying by a
The Cartesian Coordinate
Writing the Terms of a
Polynomial in Descending
Quadratic Expressions
Solving Inequalities
Solving Rational
Inequalities with a Sign
Solving Linear Equations
Solving an Equation with
Two Radical Terms
Simplifying Rational
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step
Solving Quadratic
Equations Graphically
and Algebraically
Collecting Like Terms
Solving Equations with
Radicals and Exponents
Percent of Change
Powers of ten
(Scientific Notation)
Comparing Integers on a
Number Line
Solving Systems of
Equations Using
Factoring Out the
Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing
Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting
Rational Expressions
with Different
Expressions with
Variables as Exponents
The Quadratic Formula
Writing a Quadratic with
Given Solutions
Simplifying Square Roots
Adding and Subtracting
Square Roots
Adding and Subtracting
Rational Expressions
Combining Like Radical
Solving Systems of
Equations Using
Dividing Polynomials
Graphing Functions
Product of a Sum and a
Solving First Degree
Solving Equations with
Radicals and Exponents
Roots and Powers
Multiplying Numbers
polynom calculator
Related topics:
maths test for ks2 free printable | multiply and simplify calculator | ''answers to algebra with pizzazz'' | ged past papers online | trig value chart | solving differential equations with
excel | revise maths what is an integer | java interpolation equation source example | solving square root multivariable derivative | solving simultaneous second order equations online |
hard equations | what is lineal metre | algebra ii two variable simple linear programming problem | factorizing cubed
Author Message
txree_bimeapplec Posted: Wednesday 27th of Dec 10:06
Hi, I am a freshman in high school and I am having trouble with my homework. One of my problems is dealing with polynom calculator; can anyone help me understand what
it is all about? I need to complete this asap. Thanks for helping.
From: Australia
AllejHat Posted: Wednesday 27th of Dec 14:19
There are several inside the whole category of polynom calculator, such as perpendicular lines, subtracting exponents or linear algebra. I am acquainted with various
folks who abandoned the high-ticket alternatives for aid too. Nevertheless , do not give up because I found an alternative resolution that is not so outrageously
priced, easy to utilize and to a greater extent more useful than I could have ever imagined . Following my trials with instructive mathematic software programs and
nearly surrendering, I acquired Algebrator. This software system has accurately supplied answers to every mathematics issue I have supplied to it . But just as
Registered: essential , Algebrator also furnishes all of the temporal strides needed to derive the supreme solution . Even though anyone could employ the software only to complete
16.07.2003 assignments , I dubiousness about anyone will be permitted to employ the program for examinations .
From: Odense, Denmark
Momepi Posted: Friday 29th of Dec 14:05
Algebrator has helped students all over the globe. It is a very nice piece of software and I would recommend it to every student who has problems with their homework.
From: Ireland
zatias8318 Posted: Friday 29th of Dec 16:11
Sounds like something I’ve been looking for all this time! Thanks guys, just one final question, can someone please provide me a website address where I can order my
copy of this program?
From: Somewhere over
the rainbow
Admilal`Leker Posted: Sunday 31st of Dec 07:08
I remember having often faced problems with binomials, function range and linear inequalities. A really great piece of math program is Algebrator software. By simply
typing in a problem homework a step by step solution would appear by a click on Solve. I have used it through many math classes – Remedial Algebra, Pre Algebra and Pre
Algebra. I greatly recommend the program.
From: NW AR, USA
daujk_vv7 Posted: Tuesday 02nd of Jan 08:53
You can order this software online: https://gre-test-prep.com/combining-like-radical-terms.html. You won’t regret buying it, besides it’s so cheap considering the
depth of knowledge you gain from using it. They even offer an unconditional money back guarantee. All the best for your test .
From: I dunno, I've
lost it.
Author Message
txree_bimeapplec Posted: Wednesday 27th of Dec 10:06
Hi, I am a freshman in high school and I am having trouble with my homework. One of my problems is dealing with polynom calculator; can anyone help me understand what it is all
about? I need to complete this asap. Thanks for helping.
From: Australia
AllejHat Posted: Wednesday 27th of Dec 14:19
There are several inside the whole category of polynom calculator, such as perpendicular lines, subtracting exponents or linear algebra. I am acquainted with various folks who
abandoned the high-ticket alternatives for aid too. Nevertheless , do not give up because I found an alternative resolution that is not so outrageously priced, easy to utilize
and to a greater extent more useful than I could have ever imagined . Following my trials with instructive mathematic software programs and nearly surrendering, I acquired
Algebrator. This software system has accurately supplied answers to every mathematics issue I have supplied to it . But just as essential , Algebrator also furnishes all of the
Registered: temporal strides needed to derive the supreme solution . Even though anyone could employ the software only to complete assignments , I dubiousness about anyone will be permitted
16.07.2003 to employ the program for examinations .
From: Odense, Denmark
Momepi Posted: Friday 29th of Dec 14:05
Algebrator has helped students all over the globe. It is a very nice piece of software and I would recommend it to every student who has problems with their homework.
From: Ireland
zatias8318 Posted: Friday 29th of Dec 16:11
Sounds like something I’ve been looking for all this time! Thanks guys, just one final question, can someone please provide me a website address where I can order my copy of
this program?
From: Somewhere over
the rainbow
Admilal`Leker Posted: Sunday 31st of Dec 07:08
I remember having often faced problems with binomials, function range and linear inequalities. A really great piece of math program is Algebrator software. By simply typing in a
problem homework a step by step solution would appear by a click on Solve. I have used it through many math classes – Remedial Algebra, Pre Algebra and Pre Algebra. I greatly
recommend the program.
From: NW AR, USA
daujk_vv7 Posted: Tuesday 02nd of Jan 08:53
You can order this software online: https://gre-test-prep.com/combining-like-radical-terms.html. You won’t regret buying it, besides it’s so cheap considering the depth of
knowledge you gain from using it. They even offer an unconditional money back guarantee. All the best for your test .
From: I dunno, I've
lost it.
Posted: Wednesday 27th of Dec 10:06
Hi, I am a freshman in high school and I am having trouble with my homework. One of my problems is dealing with polynom calculator; can anyone help me understand what it is all about? I need to
complete this asap. Thanks for helping.
Posted: Wednesday 27th of Dec 14:19
There are several inside the whole category of polynom calculator, such as perpendicular lines, subtracting exponents or linear algebra. I am acquainted with various folks who abandoned the
high-ticket alternatives for aid too. Nevertheless , do not give up because I found an alternative resolution that is not so outrageously priced, easy to utilize and to a greater extent more useful
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How to Benchmark Optimization Algorithms
How to Benchmark Optimization Algorithms#
Benchmarking optimization algorithms is an important step when developing a new algorithm or when searching for an algorithm that is good at solving a particular problem.
In general, benchmarking constists of the following steps:
1. Define the test problems (or get pre-implemented ones)
2. Define the optimization algorithms and the tuning parameters you want to try
3. Run the benchmark
4. Plot the results
optimagic helps you with all of these steps!
1. Get Test Problems#
optimagic includes the problems of Moré and Wild (2009) as well as Cartis and Roberts.
Each problem consist of the inputs (the criterion function and the start parameters) and the solution (the optimal parameters and criterion value) and optionally provides more information.
Below we load a subset of the Moré and Wild problems and look at one particular Rosenbrock problem that has difficult start parameters.
problems = om.get_benchmark_problems("example")
2. Specify the Optimizers#
To select optimizers you want to benchmark on the set of problems, you can simply specify them as a list. Advanced examples - that do not only compare algorithms but also vary the algo_options - can
be found below.
optimizers = [
3. Run the Benchmark#
Once you have your problems and your optimizers set up, you can simply use run_benchmark. The results are a dictionary with one entry for each (problem, algorithm) combination. Each entry not only
saves the solution but also the history of the algorithm’s criterion and parameter history.
results = om.run_benchmark(
4a. Profile plots#
Profile Plots compare optimizers over a whole problem set.
The literature distinguishes data profiles and performance profiles. Data profiles use a normalized runtime measure whereas performance profiles use an absolute one. The profile plot does not
normalize runtime by default. To do this, simply set normalize_runtime to True. For background information, check Moré and Wild (2009).
fig = om.profile_plot(
The x axis shows runtime per problem. The y axis shows the share of problems each algorithm solved within that runtime. Thus, higher and further to the left values are desirable. Higher means more
problems were solved and further to the left means, the algorithm found the solutions earlier.
You can choose:
• whether to use n_evaluations or walltime as runtime_measure
• whether to normalize runtime such that the runtime of each problem is shown as a multiple of the fastest algorithm on that problem
• how to determine when an evaluation is close enough to the optimum to be counted as converged. Convergence is always based on some measure of distance between the true solution and the solution
found by an optimizer. Whether distiance is measured in parameter space, function space, or a combination of both can be specified.
Below, we consider a problem to be solved if the distance between the parameters found by the optimizer and the true solution parameters are at most 0.1% of the distance between the start parameters
and true solution parameters.
fig = om.profile_plot(
4b. Convergence plots#
Convergence Plots look at particular problems and show the convergence of each optimizer on each problem.
fig = om.convergence_plot(
problem_subset=["rosenbrock_good_start", "box_3d"],
The further to the left and the lower the curve of an algorithm, the better that algorithm performed.
Often we are more interested in how close each algorithm got to the true solution in parameter space, not in criterion space as above. For this. we simply set the distance_measure to parameter_space.
fig = om.convergence_plot(
problem_subset=["rosenbrock_good_start", "box_3d"],
5a. Convergence report#
The Convergence Report shows for each problem and optimizer which problems the optimizer solved successfully, failed to do so, or where it stopped with an error. The respective strings are “success”,
“failed”, or “error”. Moreover, the last column of the pd.DataFrame displays the number of dimensions of the benchmark problem.
df = om.convergence_report(
│ │nag_dfols│scipy_neldermead│scipy_truncated_newton│dimensionality│
│ problem │ │ │ │ │
│ bard_good_start │success │success │success │3 │
│ bdqrtic_8 │success │success │success │8 │
│ box_3d │success │success │success │3 │
│ brown_dennis_good_start │success │success │success │4 │
│ chebyquad_6 │success │success │success │6 │
│freudenstein_roth_good_start │success │success │success │2 │
│ helical_valley_good_start │success │success │success │3 │
│ mancino_5_good_start │success │success │success │5 │
│ powell_singular_good_start │success │success │success │4 │
│ rosenbrock_good_start │success │success │success │2 │
5b. Rank report¶#
The Rank Report shows the ranks of the algorithms for each problem; where 0 means the algorithm was the fastest on a given benchmark problem, 1 means it was the second fastest and so on. If an
algorithm did not converge on a problem, the value is “failed”. If an algorithm did encounter an error during optimization, the value is “error”.
df = om.rank_report(
│ │nag_dfols│scipy_neldermead│scipy_truncated_newton│dimensionality│
│ problem │ │ │ │ │
│ bard_good_start │1 │2 │0 │3 │
│ bdqrtic_8 │1 │2 │0 │8 │
│ box_3d │0 │2 │1 │3 │
│ brown_dennis_good_start │1 │2 │0 │4 │
│ chebyquad_6 │0 │2 │1 │6 │
│freudenstein_roth_good_start │1 │2 │0 │2 │
│ helical_valley_good_start │0 │2 │1 │3 │
│ mancino_5_good_start │1 │2 │0 │5 │
│ powell_singular_good_start │0 │2 │1 │4 │
│ rosenbrock_good_start │0 │2 │1 │2 │
5b. Traceback report¶#
The Traceback Report shows the tracebacks returned by the optimizers if they encountered an error during optimization. The resulting pd.DataFrame is empty if none of the optimizers terminated with an
error, as in the example below.
df = om.traceback_report(problems=problems, results=results)
│ │ │traceback│dimensionality │
│algorithm│problem│ │ │ | {"url":"https://estimagic.org/en/latest/how_to/how_to_benchmarking.html","timestamp":"2024-11-05T15:32:05Z","content_type":"text/html","content_length":"47723","record_id":"<urn:uuid:d8d31e87-82db-494c-80b9-61484712645b>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00863.warc.gz"} |
Blog: Logs for the Monero Research Lab Meeting Held on 2019-02-04
Logs for the Monero Research Lab Meeting Held on 2019-02-04
February 04, 2019
<sarang> OK, let's begin
<sarang> 1. GREETINGS
<sarang> hello everyone
<ArticMine> Hi
<sgp_> hello!
<Xeagu> Hey!
<chron0> oi
<sarang> a special ping to moneromooo as well, since he has a PR that I wish to discuss
<sarang> Since it approaches, let's discuss 2. NETWORK UPGRADE REVIEW
<xmrmatterbridge> <serhack> Hi!
<sgp_> we know the topic is important since it's in ALL CAPS
<suraeNoether> hi guys!
<sarang> As linked in the agenda, the current plan for payment IDs has been subject to much discussion, so please read through the notes at that link
<sarang> I don't want to discuss opinions on that at this meeting, only to remind of what the timeline has been
<needmoney90> Hey!
<sarang> Of more pressing interest is the block size algorithm
<Mochi101> You guys going to introduce more tools for sub-addresses too?
<Mochi101> Like being able to query last sub-address generated?
<sarang> You may recall that we were looking at a few different options for block size scaling, to allow for growth while protecting from cheap spam
<suraeNoether> for the audience: The current pull request for changing the dynamic block size adjustment method is proposed by ArticMine (see PR 5124 here: https://github.com/monero-project/monero/
pull/5124/files ) and it occurs to me that we are intricately in the weeds of fees, incentives, and block sizes right now.
<sarang> Mochi101: probably better for #monero-dev
<Mochi101> ah ok
<suraeNoether> I have concerns about this proposal for a few reasons.
<sarang> The reason I bring this up is because we have essentially two methods on the table: a simple dual-median approach, and the one linked in the PR which uses a long-term median weighting
<sarang> The PR contains moneromooo's implementation of the latter
<ArticMine> Actually There were 3 option for block weight scaling
<ArticMine> That were formalized
<ArticMine> 1) The double median
<ArticMine> 2) The double median with smooths change
<sarang> Please remind the room of the change for (2)
<suraeNoether> ^ thanks sarang :P
<ArticMine> 3) My last proposal that moneromooo implemented in PR 5124
<suraeNoether> i far prefer the dual median approach than PR 5124
<ArticMine> In the first option the full blockweight including the burst portion was used to scale the long term median
<ArticMine> This allowed the burst to work with scaling but had the problem of using the entire burst to scale the long term median
<suraeNoether> PR 5124 concerns me for a couple of reasons. firstly, demonstrating the long-term stability of it… proving that stability… is difficult, and finding parameters that seem to be good is
an exercise in hand-tuning. additionally, it's not clear what we gain from using the more complicated method in the short term. if we are pushing more complicated code, it should be because we have a
clear advantage from
<suraeNoether> using it, some clear efficiency or security gain
<sarang> Given an unbounded adversary, all proposals allow chain bloat of O(10-100 GB) over, say, a weeklong sustained max-bloat attack
<sarang> With resulting block sizes 15-30 MB
<suraeNoether> how much does it cost to do that in all the proposals?
<ArticMine> 2) smooth's modification was to not use the 50x burst in scaling the long term median
<suraeNoether> if all proposals have the same total consequence for the blockchain and the same approximate cost for the attacker, there is zero reason to use something complicated instead of
something simple.
<sarang> O($1-10 M) depending on whether you assume the miners are purely breaking even or not
<sarang> Under ArticMine's 4x-rational-miner assumption, they scale 4x from that
<ArticMine> This had the serious problem of killing the burst over time.
<moneromooo> If all proposals have the same total consequence for the blockchain and the same approximate cost for the attacker, then you use "normal use" behaviour to select which one is the best.
<moneromooo> It's not enough to resist attacks, it must also be useful for normal use.
<ArticMine> My final proposal addressed the problem in both 1) and 2) b only using the portion of the block weight that would have been allowed under the long term median to scale the long term
<sarang> ArticMine: does moneromooo's code accurately reflect your proposal?
<sarang> And does my Python code?
<sarang> I want to make absolutely sure we are all operating on the same assumptions for bloat and cost possibilities
<sarang> Otherwise we are debating unknowns
<suraeNoether> ^
<moneromooo> The python code I saw from… sarang I assume, can't recall now :) and my code agreed on the results when pushing blocks with the max allowed weight.
<moneromooo> Also agreed with some python code I made.
<sarang> For example, a recent change in minimum fee computation to include the long-term weight (and not short-term median) switches from applying the 4x penalty to using the min fee, for example
<ArticMine> sarang Your python code uses the penalty rather than 4x the penalty
<sarang> and that spiked the total cost
<suraeNoether> i am so wildly uncomfortable with this conversation, tbh
<ArticMine> But it was switched to 4x penalty
<sarang> Let's assume we are all talking about this paste: http://paste.debian.net/hidden/292e473f/
<sarang> Note line 56 please
<Xeagu> Whatever is implemented should have a clear way to explain to the rest of the ecosystem.
<sarang> It used the short-term median in the min fee computation, but was changed in line 57 to the lt-weight
<sarang> This spikes the total cost by order of magnitude, so clearly it means the min fee is applying, rather than the 4x penalty
<sarang> We should ensure that we are clear of this
<sarang> *clear on this
<sarang> So my current questions are: ArticMine is this your intention? moneromooo is the current paste (not the commented-out line) what is in the PR?
<moneromooo> AFAIK, yes.
<ArticMine> The past reflect the rational miner so yes
<ArticMine> paste
<moneromooo> I do not agre with the rational miner comment though :P
<sarang> ArticMine: what about the difference between lines 56 and 57?
<moneromooo> A rational miner would also accept smaller fees, unless I can be convinced otherwise.
<sarang> That has a _massive_ effect on the cost assumption
<ArticMine> The difference is between 4x penalty and 1x
<sarang> No
<sarang> The difference between 56 and 57 is in how the min fee is scaling
<sarang> the penalty applies in the next line
<suraeNoether> there has not been an effort to formalize the design goals of our blocksize adjustment algorithm, so even if we have a new proposal, all we do is say "hmm, how fast can the blockchain
grow? hmm how much does that cost?"
<suraeNoether> we may as well be picking random functions and seeing if they do what we want them to do, if we aren't going to start with design principles or urgent needs first and move forward from
<suraeNoether> With that in mind, our primary urgent need is to prevent a bloatocalypse before the next fork. Keeping next block size at something like R*Median(N) is still rational, we just need to
boost N and decrease R to make it so that 6 months can't give us more than a certain growth over the trendline, selecting N to dictate how long the attacker needs to blow fees.
<oneiric_> what is the practical urgency of implementing one of the suggested modifications to blocksize adjustment?
<suraeNoether> in the meantime, I am willing to make a formal bet that the only reason we could possibly need a 50x boost in block size in the next 180 days is a bloat attack. i see no good reason to
take the short-term xmas day factor into account for this immediate next hard fork, for a lot of design reasons
<ArticMine> No it is not
<suraeNoether> ArticMine: ??
<ArticMine> The urgency here is questionable at most
<suraeNoether> ArticMine: i absolutely disagree
<oneiric_> so postponing for 1-2 forks is not the worst-case scenario?
<suraeNoether> i don't want to be chicken little here
<suraeNoether> the sky is falling yada yada
<suraeNoether> but guys, for essentially the cost of an apartment building, someone could take the monero blockchain and make it bigger than ethereum's
<suraeNoether> i find this to be an intolerably insecure property of our block size adjustment algorithm
<suraeNoether> it's a whim for the winklevii
<ArticMine> If we rush into this without taking into consideration the implication on Monero ability to scale we can destroy a critical component of Monero's social covenant
<ArticMine> Or cripple it
<suraeNoether> ArticMine: our social contract is to ensure that our currency remains functional
<suraeNoether> a 160GB blockchain is not functional
<ArticMine> It has been functional for 4 years
<moneromooo> The fees were just dropped to ~nothing recently though.
<oneiric_> is there any indication that 160GB is near?
<suraeNoether> "the attack hasn't happened yet" != "the attack will not happen or is not incentivized to happen"
<xmrmatterbridge> <serhack> ArticMine: Yes, but that does not mean that could work for the next years.
<moneromooo> Actually I take that back. It was still the same penalty.
<suraeNoether> since any change we make in this fork can be undone in the next fork, your argument about long term scaling and the social contract doesn't hold up
<ArticMine> This is a theoretical attack that is more expensive than a 51% attack
<Xeagu> Do we have simulations of upper bound organic growth in x time? Say Monero gets added to Open Bazaar or something - what is the realistic transaction growth we might expect?
<suraeNoether> it's about as expensive as a 51% attack, with the difference that it requires no hardware
<suraeNoether> xeagu no, we have no data on adoption rate into new markets, although we could ballpark some estimates of what we think are reasonable, we are still just hand-tuning our approach
<ArticMine> a 515 attack include the cost of hardware
<ArticMine> 51%
<suraeNoether> ArticMine: in order to pull off a 51% attack, i need i) the money and ii) some miners willing to take my bribe at the cost of the value of their xmr
<suraeNoether> in order to do a bloat attack, i just need a bunch of xmr outputs
<ArticMine> You have to pay for both
<suraeNoether> so the cost may be equivalent in terms of USD, but the social friction to one attack is much lower compared to the other
<Xeagu> One is permissionless
<ArticMine> So it the other
<ArticMine> is
<suraeNoether> ArticMine: nah, you need to go find miners to work with
<ArticMine> Rent the hard ware
<suraeNoether> even if you have to spin up AWS, that means you have to go through amazon
<suraeNoether> that isn't permissionless
<suraeNoether> anyway
<ArticMine> or buy it
<suraeNoether> we are going down a rabbit hole
<sarang> OK, so
<suraeNoether> I'm advocating that we merely modify our growth rate and sample size and then spend the next 6 months making a non-rushed decision
<sarang> What is on the table right now is a proposed PR, which you may or may not think is too complex
<sarang> I want to know that PR 5124 matches the linked paste
<sarang> and that the cost computation in the paste is accurate
<sarang> then we _at least_ know what the cost and bloat effects of 5124 are
<ArticMine> My point is we have created artificial urgency and as a consequence could destroy a critical component of Monero
<suraeNoether> articmine: tweaking R and N can not destroy a critical component of monero
<suraeNoether> but implementing a complicated blocksize adjustment algorithm with 18 parantheses that is hard to intuit or reason about? that could.
<ArticMine> That is the trouble with the artificial urgency
<suraeNoether> okay, i said my piece
<moneromooo> Having ArticMine's change in does not prevent it being changed later if something more appropriate is found. Which may be the case if suraeNoether does the study he advocated earlier.
<sarang> I am currently advocating for nothing, since I have received no satisfactory guidance about whether 5124 matches the numbers we have been basing assumptions on
<ArticMine> I am in the process of evaluating just that
<moneromooo> It matches the numbers for max sized blocks.
<moneromooo> I did not try non-max size blocks though. I will try that.
<sarang> ty ArticMine. In particular: please try switching between lines 56 and 57 and note the cost change
<sarang> and ensure we are comfortable with that
<ArticMine> In hour paste
<ArticMine> your
<sarang> moneromooo: right, the question is fee and cost, not blocksize, between those lines' changes
<Xeagu> If we implement ArticMine's change, can we also come up with a ELI5 explanation of what is going on?
<sarang> ArticMine: yes, 56-57 in my paste, linked again here: http://paste.debian.net/hidden/292e473f/
<sarang> Xeagu: yes
<moneromooo> Oh OK. I did not do anything with fee with my code.
<sarang> moneromooo: you had specifically requested the line 57 change
<sarang> IIRC
<moneromooo> s/do/test, log, etc/
<moneromooo> Yes, the change is what ArticMine suggested a while back to make the attack more expensive.
<moneromooo> (and I do not agree it does make it more expensive fwiw)
<moneromooo> That change is what the second commit in 5124 does.
<ArticMine> Yes 57 addresses maintenance cost of keeping the locks bloated
<ArticMine> Which is what I recommended
<sarang> OK, so in your opinion ArticMine, running that paste in its current form will provide the cost to an attacker, assuming rational miners
<sarang> ?
<ArticMine> Yes
<sarang> thank you
<sarang> note to moneromooo ^
<moneromooo> who still does not think it works
<sarang> At least we agree on the final bloat and blocksize :/
<sarang> (using the "attacker with infinite money and rage" model)
<ArticMine> This all comes back to suraeNoether point what are our design parameters?
<sarang> We have not formally defined them
<suraeNoether> after the meeting i'm going to host an impromptu brain session on that i think articmine
<Xeagu> Fwiw none of these measures seem to prevent slower bloat with min fee txs flooding the mempool and eventually getting confirmed over time
<ArticMine> Yes
<suraeNoether> i have some formal ideas
<sarang> OK, so let's move on for now perhaps. Right now we have the current scheme, and the PR, and a dual-median idea that's not in a PR
<sarang> After meeting, more discussion
<sarang> Output selection is also being worked on, with no final changes yet
<sarang> The idea being to weight block selection for ring decoys by the number of outputs, consistent with our gamma spend distribution
<suraeNoether> isthmus and i had some discussion on output selection
<ArticMine> If we are trying to minimize the risk of bloat without any other considerations then we end up destroying the adaptive blocksize
<sarang> ArticMine: let's discuss after meeting
<ArticMine> Sure
<sarang> Please go ahead with output selection suraeNoether
<suraeNoether> one thing isthmus pointed out is the classic problem with multi-input transactions
<suraeNoether> for example, if I produce a ring signature with ring {A, B, C} and a ring signature with ring {D, E, F} in the same transaction, and A and D both have the same age, then it's "obvious"
who the true spender is
<suraeNoether> or rather, allows for heuristic'in
<sarang> monerocoinjoin helps to break this, fwiw
<suraeNoether> and after more and more discussion, i'm finally coming around on the idea of binning inputs
<suraeNoether> sarang ^ yes
<sarang> s/helps/could help someday
<sarang> The major issue at hand _that was recently introduced_ is biased coinbase selection (and biased selection in general)
<Xeagu> Input age heuristic for multi ring transactions is an interesting consideration
<Inge-> 18:25 < ArticMine> is
<sarang> IMO we should iteratively first focus on removing some of the selection bias, and then later move to address other multi-in heuristics
<Xeagu> Got it
<sarang> Is there opposition to a move to a weighted selection that accounts for this bias?
<Xeagu> What is the biased coinbase selection problem?
<sarang> If you select a block and then an output within it, you are likely to have bad distributions that aren't representative of the whole chain, due to variance in block sizes
<moneromooo> The fewer outputs in a block, the more likely they are to be chosen as fake outs.
<sarang> If you weight based on # of outputs, you can remove this
<sarang> doing it efficiently is the tricky part, due to the gamma requirement
<sarang> I only bring it up here to see if there is opposition to moving to such an approach, and then later continuing to iterate if needed
<Xeagu> It's almost like you need extra decoy outputs as padding in each block so there is not much variance in total outputs between blocks
<suraeNoether> i can spend about 30 minutes today writing up some pseudocode if that helps with the distribution stuff…
<sgp_> I think it's a good stopgap
<suraeNoether> xeagu no, extra decoy outputs have their own security problems
<moneromooo> No. Do not spam the chain.
<suraeNoether> ^
<sarang> Xeagu: we really don't
<sarang> weighting accounts for variance
<sarang> that's the point
<Xeagu> Noted
<sarang> suraeNoether: please write up your pseudocode ASAP, and I will as well
<sarang> this is not consensus, but the upgrade is a good time to do it if possible
<suraeNoether> yeah, essentially, for the stats nerds in the audience
<moneromooo> If someone tells me how to calculate the gamma CDF (or something that's close enough), I can do that code. It's the only thing I'm missing.
<sarang> moneromooo: this assumes you're recalculating the product (this block's gamma factor)*(this block's weight) for each block, every time a new block arrives?
<sarang> that was the inefficiency I wanted to address
<moneromooo> Every time someone wants to make a tx. Which is hopefully less often.
<sarang> OK, but it's for the entire chain?
<moneromooo> OK. Good to have if you can, sure.
<moneromooo> Yes.
<sarang> got it
<sarang> there was talk about a cutoff point, but that renders old outputs detectable
<moneromooo> Well, technically the whole chain since rct started.
<sarang> heh sure
<suraeNoether> we are transitioning from "drawing randomly from an ideal distribution ignoring the current blockchain state (in terms of density of txns per block)" to "drawing randomly from a
partially empirical distribution based both on an ideal distribution and the observation of the current blockchain state.
<sarang> so in fact that does allow a cutoff of sorts…
<sarang> Cool, we shall prioritize that to get it in before freeze
<sarang> Let's move to 3. OTHER WORK
<sarang> I continue to work on a few things listed in the agenda
<moneromooo> I'll use some interpolation as smooth suggested, so you'll have to calc gamma for 1/100 points or whatever in practice.
<sarang> in addition to the upgrade items
<suraeNoether> but i do want to go back to the multi-input thing this week, because i think it's a critical security concern that we have merely gotten used to because we've known about it for awhile
and didn't have a solution for… but we have more pressing stuff to take care of immediately
<sarang> Yes we do
<sarang> an interesting point is Dandelion, which I've been looking into more deeply
<oneiric_> ^
<sarang> there was talk among btc devs about possible DoS avenues in the stem phase of routing
<suraeNoether> the more i read about dandelion++ the more i like it, but hte more i wish they stuck with a single +
<sarang> having to do with the mempool
<sarang> These may not apply to us, but I'm reading up on their concerns
<sarang> They also had issues relating to unconfirmed txs relying on each other that we don't care about
<sarang> Any specific questions for me regarding my agenda-listed items?
<sarang> My first priorities are upgrade items, of course
<Xeagu> Not mentioned is mining algo. Is that planning on being changed next fork?
<sarang> yes
<moneromooo> Yes.
<sarang> details tbd
<Xeagu> Got it
<oneiric_> are the candidates known atm?
<sarang> I also consider that outside the scope of this room, for the most part
<sarang> I hear CN-R with tbd tweak(s)
<oneiric_> thanks sarang
<sarang> OK, suraeNoether ?
<sarang> your updates?
<suraeNoether> Well, I finally realized how to formally hypothesize about the EAE attack in the bipartite graph setting for MRL11, which has allowed me to start writing up simulations
<sarang> neat!
<suraeNoether> basically: i'm assuming blocks arrive one at a time linearly for the sake of simplicity, i assume transaction density in each block is poisson distributed, and i assume each
transaction is M-in, N-out according to some distribution I haven't selected yet (but will probably be empirical from our blockchain)
<suraeNoether> I do this for a number of blocks T, to simulate a fake blockchain that exists before the game begins
<suraeNoether> then, in addition to the random block density I just described, a single user is going to churn K times
<sarang> What timing distribution?
<suraeNoether> the wallet distribution
<sarang> define plz
<sarang> I mean, how often do they churn?
<suraeNoether> oh, i'm going to explore that as a parameter with K=1 to K=15 or so
<sarang> roger
<suraeNoether> then i'm going to run the matching algorithm and see the probability that the person "gets caught" meaning their transactions are labeled/matched
<sarang> sweet
<suraeNoether> we already have ideas on timing, so we can figure out how this scales to a big parallel process, and we can generate a confusion table
<moneromooo> Is this the same thing as a confusion matrix ?
<suraeNoether> yeash
<suraeNoether> yes*
<suraeNoether> so the idea is to ask the question 'under certain user behavior, which is what we recommend for churn, what sort of security can someone practically expect?'
<suraeNoether> concretely
<sarang> Awesome, please keep us updated on the sim results. This will be very useful to those needing to churn safely
<suraeNoether> i also have an alternative model where only coinbase outputs are added to the system but they get sent with wait-times equal to the wallet distribution also
<suraeNoether> will do
<Xeagu> You testing if there is a limit to churning that begins to weaken security?
<suraeNoether> xeagu trying to formalize "how much should you churn for the following security level"
<Xeagu> Got it
<suraeNoether> it'd be interesting to see if churning too much gives it away
<suraeNoether> which is an old hypothesis of mine :D
<sarang> Anything else you're working on?
<suraeNoether> the blocksize dynamic adaptability problem, of course, and my December + January end of month report
<sarang> Neat
<suraeNoether> i like giving plenty of time for comments. *cough*
<sarang> Before we review action items, any specific directed questions, or other work to share?
<suraeNoether> oh i read a paper
<suraeNoether> oh oh
<needmoney90> Have you considered the alternative funding avenue discussed in Palo alto?
<suraeNoether> https://eprint.iacr.org/2018/379
<sarang> our old friend Liu
<sarang> needmoney90: remind the room of this funding avenue
<needmoney90> Patreon or a similar system
<suraeNoether> oh yes
<needmoney90> With the unfunded part going to the MFS
<needmoney90> Gives more assurance of future funding
<sarang> It would increase the funding timeline
<suraeNoether> needmoney90 brought up the idea of doing a patreon for MRL so that researchers can get a little bit of reliability in getting paid in USD
<sarang> Keep in mind we can now also request immediate MFS payout if this is clearly stated
<suraeNoether> i honestl haven't thought about it much, but the centralization risk of going through patreon is a concern… even though it's mitigated by the MFS
<sarang> suraeNoether: nobody has to use patreon
<sarang> they can donate via mfs
<needmoney90> Yup, this prevents the whole 'Will we get funded' thing, where you know you have a baseline level of contribution even if the MFS falls through
<moneromooo> What is MFS ?
<sarang> Nice. I'm not opposed to it as another option
<sarang> apparently the renaming for NewFFS
<needmoney90> Monero funding system
<needmoney90> It's no longer on a forum
<sgp_> I thought it was the CCS
<sarang> -_____-
<needmoney90> No one told me :(
<Xeagu> For F*** Sake I liked FFS
<sarang> Anyway
<oneiric_> lol
<sarang> To keep things tight, 5. ACTION ITEMS
<sarang> Deciding on block size algorithm, if any, should be a priority
<sarang> otherwise nothing will happen
<sarang> After meeting will be a chance to discuss formal requirements for this
<sarang> Next
<sarang> suraeNoether and I have different ideas on weighted output selection that need to be tested and shared for possible inclusion in the upgrade
<sarang> suraeNoether will be working on churn sims
<sarang> I will be trying my darndest to get back to the bulletproofs MPC, and understanding DoS risks in Dandelion routing
<sarang> Any last comments before we officially adjourn?
<oneiric_> what are the chances of something like the btc key-reuse in monero?
<sarang> what does this mean
<oneiric_> there was that paper recently about computing btc addresses iirc
<moneromooo> About weighted output selection: if I have the code for calculating gamma before any fancy opt is found, I can add the code already. Then add the fancy opts on top.
<sarang> ty moneromooo
<sarang> oneiric_: linkylinky
<sarang> We can at least get a series approximation for later interpolation
<oneiric_> https://eprint.iacr.org/2019/023.pdf
<sarang> any bias in that approximation will be a wash
<sarang> Ah yes. Don't use silly RNGs
<moneromooo> I'm fine with a way to compute an approximation.
<oneiric_> right, but is monero vulnerable to that style of attack?
<sarang> Thanks, everyone. Meeting adjourned
Post tags : Dev Diaries, Cryptography, Monero Research Lab | {"url":"https://www.getmonero.org/2019/02/04/logs-for-the-Monero-Research-Lab-meeting-held-on-2019-02-04.html","timestamp":"2024-11-07T09:35:58Z","content_type":"text/html","content_length":"66977","record_id":"<urn:uuid:bdcbe2b6-9360-46bf-8700-05932993f0b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00443.warc.gz"} |
Learn Using Related Subtraction Facts to Solve Addition Sentences - Grade 1 - E Singapore Math
G1.1.8 - Using Related Subtraction Facts to Solve Addition Sentences
• Recognize related addition and subtraction sentences
• Write fact families
• Use fact families to solve word problems
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General Maths & Science
General Maths & Science tutors in Bulwer
Personalized Tutoring Near You General Maths & Science lessons for online or at home learning in Bulwer
General Maths & Science tutors in Bulwer near you
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4.1.2.1.1 Map Projection Name
Definition: Name of the map projection.
Format: Free text.
Map Projection Name: Albers Conical Equal Area
Map Projection Name: Azimuthal Equidistant
Map Projection Name: Equidistant Conic
Map Projection Name: Equirectangular
Map Projection Name: General Vertical Near-sided Projection
Map Projection Name: Gnomonic
Map Projection Name: Lambert Azimuthal Equal Area
Map Projection Name: Lambert Conformal Conic
Map Projection Name: Mercator
Map Projection Name: Modified Stereographic for Alaska
Map Projection Name: Miller Cylindrical
Map Projection Name: Oblique Mercator
Map Projection Name: Orthographic
Map Projection Name: Polar Stereographic
Map Projection Name: Polyconic
Map Projection Name: Robinson
Map Projection Name: Sinusoidal
Map Projection Name: Space Oblique Mercator (Landsat)
Map Projection Name: Stereographic
Map Projection Name: Transverse Mercator
Map Projection Name: Van der Grinten
4.1.2.1.2 Albers Conical Equal Area
Definition: Contains parameters for the Albers Conical Equal Area projection.
4.1.2.1.3 Azimuthal Equidistant
Definition: Contains parameters for the Azimuthal Equidistant projection.
4.1.2.1.4 Equidistant Conic
Definition: Contains parameters for the Equidistant Conic projection.
4.1.2.1.5 Equirectangular
Definition: Contains parameters for the Equirectangular projection.
4.1.2.1.6 General Vertical Near-sided Perspective
Definition: Contains parameters for the General Vertical Near-sided Perspective projection.
4.1.2.1.7 Gnomonic
Definition: Contains parameters for the Gnomonic projection.
4.1.2.1.8 Lambert Azimuthal Equal Area
Definition: Contains parameters for the Lambert Azimuthal Equal Area projection.
4.1.2.1.9 Lambert Conformal Conic
Definition: Contains parameters for the Lambert Conformal Conic projection.
4.1.2.1.10 Mercator
Definition: Contains parameters for the Mercator projection.
4.1.2.1.11 Modified Stereographic for Alaska
Definition: Contains parameters for the Modified Stereographic for Alaska projection.
4.1.2.1.12 Miller Cylindrical
Definition: Contains parameters for the Miller Cylindrical projection.
4.1.2.1.13 Oblique Mercator
Definition: Contains parameters for the Oblique Mercator projection.
4.1.2.1.14 Orthographic
Definition: Contains parameters for the Orthographic projection.
4.1.2.1.15 Polar Stereographic
Definition: Contains parameters for the Polar Stereographic projection.
4.1.2.1.16 Polyconic
Definition: Contains parameters for the Polyconic projection.
4.1.2.1.17 Robinson
Definition: Contains parameters for the Robinson projection.
4.1.2.1.18 Sinusoidal
Definition: Contains parameters for the Sinusoidal projection.
4.1.2.1.19 Space Oblique Mercator (Landsat)
Definition: Contains parameters for the Space Oblique Mercator projection.
4.1.2.1.20 Stereographic
Definition: Contains parameters for the Stereographic projection.
4.1.2.1.21 Transverse Mercator
Definition: Contains parameters for the Transverse Mercator projection.
4.1.2.1.22 Van der Grinten
Definition: Contains parameters for the Van der Grinten projection.
4.1.2.1.23 Map Projection Parameters
Definition: A complete parameter set of the projection that was used for the dataset. The information provided shall include the names of the parameters and values used for the dataset that
describe the mathematical relationship between the Earth and the plane or developable surface for the projection.
4.1.2.1.23.1 Standard Parallel
Definition: Line of constant latitude at which the surface of the Earth and the plane or developable surface intersect.
Format: Real; Range from -90.0 to 90.0 latitude.
4.1.2.1.23.2 Longitude of Central Meridian:
Definition: The line of longitude at the center of a map projection generally used as the basis for constructing the projection.
Format: Real; Range from -180.0 to 180.0 longitude.
4.1.2.1.23.3 Latitude of Projection Origin
Definition: Latitude chosen as the origin of rectangular coordinates for a map projection.
Format: Real; Range from -90.0 to 90.0 latitude.
4.1.2.1.23.4 False Easting
Definition: The value added to all "x" values in the rectangular coordinates for a map projection. This value frequently is assigned to eliminate negative numbers and is expressed in the unit
of measure identified in Planar Coordinate Units.
Format: Real.
4.1.2.1.23.5 False Northing
Definition: The value added to all "y" values in the rectangular coordinates for a map projection. This value frequently is assigned to eliminate negative numbers and is expressed in the unit
of measure identified in Planar Coordinate Units.
Format: Real.
4.1.2.1.23.6 Scale Factor at Equator
Definition: A multiplier for reducing a distance obtained from a map by computation or scaling to the actual distance along the equator.
Format: Real; Factor > 0.0
4.1.2.1.23.7 Height of Perspective Point Above Surface
Definition: Height of viewpoint above the Earth, expressed in meters.
Format: Real; Height > 0.0
4.1.2.1.23.8 Longitude of Projection Center
Definition: Longitude of the point of projection for azimuthal projections.
Format: Real; Range from -180.0 to 180.0 longitude.
4.1.2.1.23.9 Latitude of Projection Center
Definition: Latitude of the point of projection for azimuthal projections.
Format: Real; Range from -90.0 to 90.0 latitude.
4.1.2.1.23.10 Scale Factor at Center Line
Definition: A multiplier for reducing a distance obtained from a map computation or scaling to the actual distance along the center line.
Format: Real; Factor > 0.0
4.1.2.1.23.11 Oblique Line Azimuth
Definition: Method used to describe the line along which an oblique mercator map projection is centered using the map projection origin and an azimuth.
4.1.2.1.23.11.1 Azimuthal Angle
Definition: Angle measured clockwise from north, and expressed in degrees.
Format: Real; Range from 0.0 to 360.0 degrees.
4.1.2.1.23.11.2 Azimuth Measure Point Longitude
Definition: Longitude of the map projection origin.
Format: Real; Range from -180.0 to 180.0 longitude.
4.1.2.1.23.12 Oblique Line Point
Definition: Method used to describe the line along which an oblique mercator map projection is centered using two points near the limits of the mapped region that define the center line.
4.1.2.1.23.12.1 Oblique Line Latitude
Definition: Latitude of a point defining the oblique line.
Format: Real; Range from -90.0 to 90.0 latitude.
4.1.2.1.23.12.2 Oblique Line Longitude
Definition: Longitude of a point defining the oblique line.
Format: Real; Range from -180.0 to 180.0 longitude.
4.1.2.1.23.13 Straight Vertical Longitude from Pole
Definition: Longitude to be oriented straight up from the North or South Pole.
Format: Real; Range from -180.0 to 180.0 longitude.
4.1.2.1.23.14 Scale Factor at Projection Origin
Definition: Multiplier for reducing a distance obtained from a map by computation or scaling to the actual distance at the projection origin.
Format: Real; Factor > 0.0
4.1.2.1.23.15 Landsat Number
Definition: Number of the Landsat satellite. (Note: This data element exists solely to provide a parameter needed to define the space oblique mercator projection. It is not used to identify
data originating from a remote sensing vehicle.)
Format: Integer; Range from 1 to 7.
4.1.2.1.23.16 Path Number
Definition: Number of the orbit of the Landsat satellite. (Note: This data element exists solely to provide a parameter needed to define the space oblique mercator projection. It is not used
to identify data originating from a remote sensing vehicle.)
Format: Integer; Range from 1 to 251 (Landsat 1, 2, 3) and 1 to 233 (Landsat 4, 5, 7).
4.1.2.1.23.17 Scale Factor at Central Meridian
Definition: A multiplier for reducing a distance obtained from a map by computation or scaling to the actual distance along the central meridian.
Format: Real; Factor > 0.0
4.1.2.1.23.18 Other Projection's Definition
Definition: A complete description of a projection, not described elsewhere in the standard, that was used for the dataset. The information provided shall include the name of the projection,
names of the parameters and values used for the dataset, and the citation of the specification for the algorithms that describe the mathematical relationship between Earth and plane or
developable surface for the projection.
Format: Free text.
4.1.2.2.1 Grid Coordinate System Name
Definition: Name of the grid coordinate system.
Format: Text; Select from list.
Grid Coordinate System Name: Universal Transverse Mercator
Grid Coordinate System Name: Universal Polar Stereographic
Grid Coordinate System Name: State Plane Coordinate System 1927
Grid Coordinate System Name: State Plane Coordinate System 1983
Grid Coordinate System Name: ARC Coordinate System
Grid Coordinate System Name: Other Grid System
4.1.2.2.2 Universal Transverse Mercator (UTM)
Definition: A grid system based on the transverse mercator projection, applied between latitudes 84 degrees North and 80 degrees South on the Earth's surface.
4.1.2.2.2.1 UTM Zone Number
Definition: Identifier for the UTM zone.
Format: Integer; Range from 1 to 60 (Northern Hemisphere) and -1 to -60 (Southern Hemisphere).
4.1.2.2.3 Universal Polar Stereographic (UPS)
Definition: A grid system based on the polar stereographic projection, applied to the Earth's polar regions north of 84 degrees North and south of 80 degrees South.
4.1.2.2.3.1 UPS Zone Identifier
Definition: Identifier for the UPS zone.
Format: Text; Select from list.
UPS Zone Identifier: A
UPS Zone Identifier: B
UPS Zone Identifier: Y
UPS Zone Identifier: Z
4.1.2.2.4 State Plane Coordinate System
Definition: A plane-rectangular coordinate system established for each state in the United States by the National Geodetic Survey.
4.1.2.2.4.1 SPCS Zone Identifier
Definition: Identifier for the SPCS zone.
Format: Text; Four-digit numeric codes for the State Plane Coordinate Systems based on the North American Datum of 1927^1 and codes for the State Plane Coordinate Systems based on the North
American Datum of 1983^2.
4.1.2.2.5 ARC Coordinate System
Definition: The Equal Arc-second Coordinate System, a plane-rectangular coordinate system^3.
4.1.2.2.5.1 ARC System Zone Identifier
Definition: Identifier for the ARC Coordinate System zone.
Format: Integer; Range from 1 to 18.
4.1.2.2.6 Other Grid System's Definition
Definition: A complete description of a grid system, not defined elsewhere in this standard, that was used for the dataset. The information provided shall include the name of the grid system, the
names of the parameters and values used for the dataset, and the citation of the specification for the algorithms that describe the mathematical relationship between the Earth and the coordinates
of the grid system.
Format: Free text.
"A local planar coordinate system is any planar coordinate system for which the relationship between the planar coordinates and geographic (latitude and longitude) coordinates is not known. In these
cases, the standards ask the producer to describe the coordinate system (the orientation of the axis, etc.) and any means that can be used to link the local system to geographic coordinates."^4
4.1.2.4.1 Planar Coordinate Encoding Method
Definition: The means used to represent horizontal positions.
Format: Text; Select from list.
Planar Coordinate Encoding Method: Coordinate Pair
Planar Coordinate Encoding Method: Distance and Bearing
Planar Coordinate Encoding Method: Row and Column
4.1.2.4.2 Coordinate Representation
Definition: The method of encoding the position of a point by measuring its distance from perpendicular reference axes (the "coordinate pair" and "row and column" methods).
4.1.2.4.2.1 Abscissa Resolution
Definition: The (nominal) minimum distance between the "x" or column values of two adjacent points, expressed in Planar Distance Units of measure.
Format: Real; Resolution > 0.0
4.1.2.4.2.2 Ordinate Resolution
Definition: The (nominal) minimum distance between the "y" or row values of two adjacent points, expressed in Planar Distance Units of measure.
Format: Real; Resolution > 0.0
4.1.2.4.3 Distance and Bearing Representation
Definition: A method of encoding the position of a point by measuring its distance and direction (azimuthal angle) from another point.
4.1.2.4.3.1 Distance Resolution
Definition: The minimum distance measurable between two points, expressed in Planar Distance Units of measure.
Format: Real; Resolution > 0.0
4.1.2.4.3.2 Bearing Resolution
Definition: The minimum angle measurable between two points, expressed in Bearing Units of measure.
Format: Real; Resolution > 0.0
4.1.2.4.3.3 Bearing Units
Definition: Units of measure used for angles.
Format: Text; Select from list.
Bearing Units: Decimal Degrees
Bearing Units: Decimal Minutes
Bearing Units: Decimal Seconds
Bearing Units: Degrees and Decimal Minutes
Bearing Units: Degrees, Minutes, and Decimal Seconds
Bearing Units: Radians
Bearing Units: Grads
4.1.2.4.3.4 Bearing Reference Direction
Definition: Direction from which the bearing is measured.
Format: Text; Select from list.
Bearing Reference Direction: North
Bearing Reference Direction: South
4.1.2.4.3.5 Bearing Reference Meridian
Definition: Axis from which the bearing is measured.
Format: Text; Select from list.
Bearing Reference Meridian: Assumed
Bearing Reference Meridian: Grid
Bearing Reference Meridian: Magnetic
Bearing Reference Meridian: Astronomic
Bearing Reference Meridian: Geodetic
4.1.2.4.4 Planar Distance Units
Definition: Units of measure used for distances.
Format: Free text.
Planar Distance Units: Meters
Planar Distance Units: International Feet
Planar Distance Units: Survey Feet | {"url":"https://sedac.ciesin.columbia.edu/metadata/guide/spatref.html","timestamp":"2024-11-02T23:37:34Z","content_type":"text/html","content_length":"53459","record_id":"<urn:uuid:981a8c72-37c7-4fbf-8e85-5dd63f2ca4ef>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00647.warc.gz"} |
R tutorial: clique percolation to detect communities in networks | Psych Networks
R tutorial: clique percolation to detect communities in networks
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In two previous blog posts, we identified a fundamental challenge to community detection in psychometric network analysis: Commonly used algorithms assign each node to one particular community. One
of these blog posts was an R tutorial I wrote, the other a guest blog by Tessa Blanken and Marie Deserno on a new method to identify communities.
What is the problem? Let’s look at this empirical network of 20 PTSD symptoms:
While the walktrap algorithm—a commonly used method that I described previously—detects 3 communities, we can see that node D2 in the bottom sort of belongs to several communities: while it is
assigned to the large red community, it also has a strong connection to D4, which in turn connects to D3. So arguably, D2 is, at least to some degree, also part of that other community.
Many of you will know by now that network models are equivalent to latent variable models under a set of conditions. If you generate data under a 1-factor model, you will find a fully connected
network with one giant community. And if you simulate data from a network model that has 3 strong communities that are unconnected, a 3-factor model will describe the data very well. Denny Borsboom
has described this from a more theoretical perspective in this guest blog, and Joost Kruis described his statistical paper on the topic in this guest blog.
The advantage of factor models is that they can deal with cross-loadings, i.e. when an item loads on 2 or more factors^1:
Current community detection methods that do not allow such “cross-loadings” are therefore akin to simple structure in confirmatory factor models where an item can only load on one factor (i.e. belong
to one community). And if there is any agreement among psychometricians, then that simple structure is rarely found in psychological data.
Clique Percolation
Enter clique percolation, a well-established method e.g. described in chapter 9 of the free network science book by Barabasi^2. Usually, the program cfinder is used for that, which was also used in
the network psychometrics paper by Tessa and Marie. Clique percolation allows to identify nodes that belong to multiple communities.
A few days ago, Jens Lange published the R package CliquePercolation on CRAN, and finally we can use clique percolation in R! Jens provided a beautiful and detailed explanation of the package’s
functionalities, and I will use some of his code and explanations here in this short tutorial, and apply it to an open clinical dataset.
Clique percolation tutorial
So let’s use empirical data and see how the method performs. Code and data for this tutorial are available here. We use a dataset of 221 military veterans for whom we have 20 PTSD symptoms based on
the DSM-5; our empirical paper on this dataset is available here where you can also find more information on the sample composition, measurement of symptoms, etc.
1. PTSD network based on the DSM-5
In a first step, we simply estimate the network structure, using a regularized gaussian graphical model, and use 4 communities based on theory: the 4 communities of symptoms described in the DSM-5.
This can be easily done in R:
### estimate network
n1 <- estimateNetwork(data, default="EBICglasso")
### plot network
names<-c('B1','B2', 'B3', 'B4', 'B5', 'C1', 'C2', 'D1', 'D2', 'D3', 'D4', 'D5', 'D6', 'D7', 'E1', 'E2', 'E3', 'E4', 'E5', 'E6')
longnames <- c('Intrusive thoughts', 'Nightmares', 'Flashbacks', 'Emotional cue reactivity', 'Physiological cue reactivity', 'Avoidance of thoughts', 'Avoidance of reminders', 'Trauma-related amnesia', 'Negative beliefs', 'Blame of self or others', 'Negative trauma-related emotions', 'Loss of interest', 'Detachment', 'Restricted affect', 'Irritability/anger', 'Self-destructive/reckless behavior', 'Hypervigilance', 'Exaggerated startle response', 'Difficulty concentrating', 'Sleep disturbance')
gr1 <- list('Intrusions'=c(1:5), 'Avoidance'=c(6:7), 'Cognition & mood alterations'=c(8:14),'Arousal & reactivity alterations'=c(15:20))
pdf("Network1.pdf", width=8.5, height=5)
g1<-plot(n1, labels=names, layout="spring", vsize=6, cut=0, border.width=1.5, border.color='black', title="DSM-5 communities",
groups=gr1, color=c('#a8e6cf', '#dcedc1', '#ffd3b6', '#ff8b94'), nodeNames = longnames,legend.cex=.35)
As you can see, the theoretical relations do not seem to map well onto the empirical relations. Especially E2 and E3 do not really closely inter-relate with other nodes from their DSM-5 community.
2. PTSD network based on the walktrap algorithm
So let’s estimate the network using the walktrap algorithm, which is implemented in the R package EGAnet:
comm1<- EGA(data, plot.EGA = TRUE); comm1
gr2 <- list('Intrusions & Avoidance'=c(1:7), 'Cogn. & mood alterations, arousal'=c(8,9,12:20), 'Blame/emotions'=c(10:11))
pdf("Network2.pdf", width=8.5, height=5)
g2<-plot(n1, labels=names, vsize=6, cut=0, border.width=1.5, border.color='black', title="Walktrap communities",
groups=gr2, color=c('#a8e6cf', '#ff8b94', '#ffd3b6'), nodeNames = longnames,legend.cex=.35)
Walktrap identifies 3 communities:
Note that there are several other ways to identify communities, such as spinglass or simply eigenvalue decomposition, which I described in my previous tutorial. I use walktrap here because it’s
implemented in EGAnet, and more convenient to estimate.
3. Clique percolation by optimizing I and k
Third, we compare this to the results of clique percolation, for which we use Jens’ CliquePercolation package. You can find more details on the methodology here, which I only summarize briefly. There
are several ways of running the algorithm, and I showcase 2 here.
To run the algorithm for weighted networks, one option is to optimize k and I, where I determines how strong the average relations among a community need to be to be detected as a community, and k
determines the minimum clique size. CliquePercolation requires a minimum k of 3, so we cannot use k=2, which would have made sense based on the results of the walktrap algorithm that identified a
community of 2 nodes (D3 & D4). In the example below, I asked the program to search through ranges of I from 0.01 to 0.20, given that an average partial correlation of 0.20 would appear to be very
large. This is usually done for larger networks than we have here, so might not be entirely appropriate.
We then identify the optimal value of I based on the rule that with increasing I, we should extract the solution for which the ratio threshold crosses to values above 2, in the best case accompanied
by a large χ value. In our case, this is I=0.12, as you can see in the output below. Missing values (“NA”) are based on the fact that certain metrics can only be computed for networks with at least 2
or 3 communities.
Here the accompanying R code:
### use Clique Percolation
W <- qgraph(n1$graph)
thresholds <- cpThreshold(W, method = "weighted", k.range = 3,
I.range = c(seq(0.20, 0.01, by = -0.01)),
threshold = c("largest.components.ratio","chi")); thresholds
results <- cpAlgorithm(W, k = 3, method = "weighted", I = 0.12)
Now we plot the network:
pdf("Network3.pdf", width=8.5, height=5)
g3 <- cpColoredGraph(W, list.of.communities = results$list.of.communities.numbers, layout=L, theme='colorblind',
color=c('#a8e6cf', '#ff8b94', '#ffd3b6', '#444444'), labels=names, vsize=6, cut=0, border.width=1.5,
border.color='black', nodeNames = longnames,legend.cex=.35,
edge.width = 1, title ="PTSD communities based on Clique Percolation")
We see that clique percolation also groups a lot of the initial nodes into one group, like the DSM-5 and walktrap; nodes E2 and D1 are not assigned to any community; and 4 nodes are assigned to 2
communities. Overall, results seem sensible, including the assignment of D6 to partially red, given its relations to other red and half-red nodes. Interestingly, E2 is not assigned to any community.
Unlike node D1, which has the lowest centrality (i.e. interconnectedness), where no assignment makes sense, E2 is at least moderately connected, to both the blue and red community. EDIT: Turns out we
can visualize this via Fruchterman-Reingold easily, so I updated the section above a bit (thanks Jens for your email).
4. Clique percolation by optimizing entropy
A second way that may be better suited for smaller networks is using entropy, based on Shannon information. Jens has a detailed tutorial on that near to the bottom of the package vignette for
CliquePercolation, so I will not describe this in detail here. For the PTSD dataset, this method leads us to choose an intensity value of I=0.14 rather than 0.12 above, and the final network solution
looks like this:
The solution seems overall a bit more conservative, with 5 unassigned nodes. Extracted solutions appear to make sense, although not assigning D3 to a community does not align with prior results, and
seems inconsistent given the strong edges the node shows.
Conclusions and ways forward
Data and code for the tutorial are available here. It’s awesome to see commonly used community detection algorithms from other fields being implemented into the free open source software environment,
and I see several avenues for future work.
1. Model validation: how well does clique percolation recover the true network structure of weighted graphs, under which set of conditions.
2. What specific guidelines should we follow to decide what solution to extract? The current guidelines, e.g. for optimizing I and k simultaneously, leave considerable researcher degrees of freedom,
which could be exploited if researchers pursue data analysis with certain goals in mind. These lose rules would also make a preregistration of Clique Percolation challenging.
3. Power analysis: how many datapoints are required for the method to reliably recover communities, given what specific method?
4. Extensions to cases where nodes can belong to more than two communities (akin to cross loadings in the CFA context).
Thanks again Jens for translating cfinder into R^3! And if someone finds a way to make the code colorblind friendly (i.e. by assigning different shades/backgrounds to parts of nodes), please do send
it over and I am happy to upload it here (with full attribution of course).
1. example from here
2. Chrome gives me an “unsafe” warning but I have used this resource a lot in the past so I think it should be ok
5 Comments
1. Hi there,
Thank you for the informative tutorial. I’ve been finding it useful as I attempt to apply network analysis to my work with fisheries dynamics. I am very new to network analysis and was wondering
if this method of clique percolation can be applied to temporal networks. As a crude analysis I know I could apply this methods to the network at each time step, but I’d like to have the network/
community structure of the previous time step inform the network/community structure of the next time step if possible. I really appreciate how this method allows for cross loadings and can’t
seem to find a temporal community detection package/method (in R) that allows for cross loadings. Do you know of any packages/methods that utilizes clique percolation in a temporal framework?
1. Community detection algorithms are usually applied to matrices, not data (e.g. spinglass and walktrap in igraph will want a matrix as input). So I don’t see principled concerns to use
matrices derived from specific methods. If you use a temporal matrix, e.g. from a VAR model, I would consider omitting autoregressive effects though, given that they really differ from the
cross-lagged effects. Or maybe do both, with and without those AR coefficients.
2. I really enjoyed the tutorial! It actually helped me address some of the comments that the peer reviewers raised about the clustering analysis I used (it was the EGA and they recommended using
clique percolation).
The question I have however is – does this method work on networks using binary variables or mixed (binary and ordinal) variables?
All the best!
1. Glad to hear it was useful Marcin!
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The Stacks project
Lemma 15.27.1. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $A$ be a set. Assume $R$ is Noetherian and complete with respect to $I$. There is a canonical map
\[ \left(\bigoplus \nolimits _{\alpha \in A} R\right)^\wedge \longrightarrow \prod \nolimits _{\alpha \in A} R \]
from the $I$-adic completion of the direct sum into the product which is universally injective.
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Discretization of Continuous Variables in Pymc
Hello! I would like to discretize a continuous variable in PyMC or create a discrete Categorical node that depends on a continuous parent node. My goal is to assign the parent node’s value to a
specific interval and map it to a corresponding discrete category. How can I achieve this?
This notebook would probably be a good place to start.
But I want to create discrete sub-nodes, is it possible to achieve this through regression?
What does “discrete sub-nodes” mean?
I’m concerned this question is an XY problem, perhaps you could elaborate more on what you’re modeling, and how this discretization fits into your larger plan?
For example, if continuous nodes a and b have a discrete node c as their child, and the CPT of node c is manually specified.
Can you provide a specific numerical example? Using numpy with random draws is fine
The CPT to be defined for the “combination” node:
file_path = 'Combination.xlsx'
df = pd.read_excel(file_path, sheet_name = 0, header=None)
Combination_cpt = df.to_numpy().T
node definition：
PoA = pm.Triangular('PoA', lower=0, c=0.3, upper=1)
CF = pm.Lognormal('CF', mu=4.5, sigma=0.2)
TEF = pm.Deterministic('TEF', CF * PoA)
MPLEF = pm.Deterministic('MPLEF', TEF * vulnerability)
CoSL = pm.Normal('CoSL', mu=0.7, sigma=0.2)
PLF = pm.Poisson('PLF', mu=MPLEF)
SLF = pm.Binomial('SLF', n=PLF, p=CoSL)
# The "combination" node is a child node of both PLF and SLF, and the CPT data is imported from Excel.
# It is desired to divide both PLF and SLF into 12 intervals each to match the dimensions of CPT
Perhaps have a look into the OrderedLogistic / Probit distributions? Those allow you to map a continuous latent quantity into uneven categorical bins
file_path = 'Combination.xlsx'
df = pd.read_excel(file_path, sheet_name = 0, header=None)
Combination_cpt = df.to_numpy().T
bins = np.array([0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120])
num_states_plf = len(bins) - 1
num_states_slf = len(bins) - 1
lookup_table_COM = {}
for i, j in itertools.product(range(num_states_plf), range(num_states_slf)):
idx = i * num_states_slf + j
lookup_table_COM[(i, j)] = Combination_cpt[idx]
with pm.Model() as model:
PoA = pm.Triangular('PoA', lower=0, c=0.3, upper=1)
CF = pm.Lognormal('CF', mu=4.5, sigma=0.2)
TEF = pm.Deterministic('TEF', CF * PoA)
vulnerability = pm.Normal('vulnerability', mu=1, sigma=0.5)
MPLEF = pm.Deterministic('MPLEF', TEF * vulnerability)
CoSL = pm.Normal('CoSL', mu=0.7, sigma=0.2)
PLF = pm.Poisson('PLF', mu=MPLEF)
SLF = pm.Binomial('SLF', n=PLF, p=CoSL)
plf_bin = np.digitize(PLF.eval(), bins) - 1
slf_bin = np.digitize(SLF.eval(), bins) - 1
current_state = (plf_bin, slf_bin)
p_combination = lookup_table_COM.get(current_state)
combination = pm.Categorical('combination', p=p_combination)
trace = pm.sample(1000, tune=1000)
Combination.csv (37.2 KB)
I tried the OrderedLogistic / Probit distributions as you suggested, but it doesn’t seem to achieve the effect I want. I tried another method, but there are issues with sampling in the “combination”
node. Could it be a problem with the node definition?
The computational backend for pymc is pytensor. You cannot use numpy functions on symbolic PyMC objects, because they aren’t arrays of numerical values, they are symbols representing computation. If
you call .eval(), you will execute the computation the node represents, and freeze the result at a constant value. Importantly, if the computation depends on random variables, it will not be updated
when new values are drawn.
What should I do to achieve the desired effect? I’d really appreciate any help
If you are overwhelmed you will probably have to start way simpler instead or jumping directly to your idea of the final model.
Have a look at some of the studies in pymc-examples to get a feeling for how models of different complexity are built: PyMC Example Gallery — PyMC example gallery
1 Like
Okay, thank you for your help!
You need to replace np. operations with pt. operations, and remove .eval() calls. For example:
plf_bin = np.digitize(PLF.eval(), bins) - 1
is equivalent to:
import pytensor.tensor as pt
plf_bin = pt.searchsorted(bins, PLF) - 1
(This equivalence between digitize and searchsorted is described in the documentation for np.digitize)
You will also not be able to use a dictionary for lookup_table_COM, because current_state will not be a tuple of integers, it will be a tuple of symbolic computations. Instead, you can make it an
array, and you can make it an array:
lookup_table_COM = np.zeros((num_states_plf), num_states_slf))
for i, j in itertools.product(range(num_states_plf), range(num_states_slf)):
idx = i * num_states_slf + j
lookup_table_COM[i, j] = Combination_cpt[idx]
lookup_table_pt =pt.as_tensor_variable(lookup_table_COM)
which you can then symbolically index into just like a numpy array:
p_combination = lookup_table_pt[plf_bin, slf_bin]
import pytensor.tensor as pt
import numpy as np
import pymc as pm
file_path = 'Combination.xlsx'
df = pd.read_excel(file_path, sheet_name = 0, header=None)
Combination_cpt = df.to_numpy().T
bins = np.array([0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120])
# num_states_plf = len(bins) - 1
# num_states_slf = len(bins) - 1
Combination_cpt_tensor = pt.as_tensor_variable(Combination_cpt)
with pm.Model() as model:
PoA = pm.Triangular('PoA', lower=0, c=0.3, upper=1)
CF = pm.Lognormal('CF', mu=4.5, sigma=0.2)
TEF = pm.Deterministic('TEF', CF * PoA)
vulnerability = pm.Normal('vulnerability', mu=1, sigma=0.5)
MPLEF = pm.Deterministic('MPLEF', TEF * vulnerability)
CoSL = pm.Normal('CoSL', mu=0.7, sigma=0.2)
PLF = pm.Poisson('PLF', mu=MPLEF)
SLF = pm.Binomial('SLF', n=PLF, p=CoSL)
plf_bin = pt.searchsorted(bins, PLF) - 1
slf_bin = pt.searchsorted(bins, SLF) - 1
def get_cpt_value_combination(plf_state, slf_state, cpt_tensor):
idx = plf_state * 12 + slf_state
return cpt_tensor[idx]
p_Combination = get_cpt_value_combination(plf_bin, slf_bin, Combination_cpt_tensor)
Combination = pm.Categorical('Combination', p=p_Combination)
trace = pm.sample(10000, tune=1000)
Thank you for your suggestions!! I tried converting the CPTs into tensors in advance and also followed your advice. However, sampling from the Combination node is still not working. Are there any
other issues that could be causing this?
I’m afraid you’re going to need to be a lot more specific than “it’s not working”
Can you open an issue in PyTensor to add the helper?
1 Like
I realized that the issue might not be related to how PyTensor is used, but rather to the fact that the CPT of my combination node consists of 0 and 1, with a large number of 0. I found that in PyMC,
categorical no\des defined with CPTs of 0 and 1 often face difficulties in sampling
I also don’t know what this array means, but the summary you posted suggests your p in the categorical is 100% on the 8th category, and 0% everywhere else. I suggest you inspect the values your are
obtaining for p and make sure they make sense.
To be clear, for the p argument, Categorical expects a vector of length n_categories that sums to 1. | {"url":"https://discourse.pymc.io/t/discretization-of-continuous-variables-in-pymc/16006","timestamp":"2024-11-02T11:29:07Z","content_type":"text/html","content_length":"61747","record_id":"<urn:uuid:b275595c-fee7-4bea-832c-89bec6926614>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00216.warc.gz"} |
Section: Scientific Foundations
High performance solvers for large linear algebra problems
Participants : Emmanuel Agullo, Mikko Byckling, Mathieu Chanaud, Olivier Coulaud, Iain Duff, Luc Giraud, Abdou Guermouche, Andra Hugo, Yan-Fei Jing, Matthieu Lecouvez, Yohan Lee-Tin-Yien, Jean Roman,
Pablo Salas Medina, Stojce Nakov, Xavier Vasseur, Mawussi Zounon.
Starting with the developments of basic linear algebra kernels tuned for various classes of computers, a significant knowledge on the basic concepts for implementations on high-performance scientific
computers has been accumulated. Further knowledge has been acquired through the design of more sophisticated linear algebra algorithms fully exploiting those basic intensive computational kernels. In
that context, we still look at the development of new computing platforms and their associated programming tools. This enables us to identify the possible bottlenecks of new computer architectures
(memory path, various level of caches, inter processor or node network) and to propose ways to overcome them in algorithmic design. With the goal of designing efficient scalable linear algebra
solvers for large scale applications, various tracks will be followed in order to investigate different complementary approaches. Sparse direct solvers have been for years the methods of choice for
solving linear systems of equations, it is nowadays admitted that such approaches are not scalable neither from a computational complexity nor from a memory view point for large problems such as
those arising from the discretization of large 3D PDE problems. Although we will not contribute directly to this activity, we will use parallel sparse direct solvers as building boxes for the design
of some of our parallel algorithms such as the hybrid solvers described in the sequel of this section. Our activities in that context will mainly address preconditioned Krylov subspace methods; both
components, preconditioner and Krylov solvers, will be investigated.
Hybrid direct/iterative solvers based on algebraic domain decomposition techniques
One route to the parallel scalable solution of large sparse linear systems in parallel scientific computing is the use of hybrid methods that combine direct and iterative methods. These techniques
inherit the advantages of each approach, namely the limited amount of memory and natural parallelization for the iterative component and the numerical robustness of the direct part. The general
underlying ideas are not new since they have been intensively used to design domain decomposition techniques; those approaches cover a fairly large range of computing techniques for the numerical
solution of partial differential equations (PDEs) in time and space. Generally speaking, it refers to the splitting of the computational domain into sub-domains with or without overlap. The splitting
strategy is generally governed by various constraints/objectives but the main one is to express parallelism. The numerical properties of the PDEs to be solved are usually intensively exploited at the
continuous or discrete levels to design the numerical algorithms so that the resulting specialized technique will only work for the class of linear systems associated with the targeted PDE.
In that context, we attempt to apply to general unstructured linear systems domain decomposition ideas. More precisely, we will consider numerical techniques based on a non-overlapping decomposition
of the graph associated with the sparse matrices. The vertex separator, built by a graph partitioner, will define the interface variables that will be solved iteratively using a Schur complement
techniques, while the variables associated with the internal sub-graphs will be handled by a sparse direct solver. Although the Schur complement system is usually more tractable than the original
problem by an iterative technique, preconditioning treatment is still required. For that purpose, the algebraic additive Schwarz technique initially developed for the solution of linear systems
arising from the discretization of elliptic and parabolic PDE's will be extended. Linear systems where the associated matrices are symmetric in pattern will be first studied but extension to
unsymmetric matrices will be latter considered. The main focus will be on difficult problems (including non-symmetric and indefinite ones) where it is harder to prevent growth in the number of
iterations with the number of subdomains when considering massively parallel platforms. In that respect, we will consider algorithms that exploit several sources and grains of parallelism to achieve
high computational throughput. This activity may involve collaborations with developers of sparse direct solvers as well as with developers of run-time systems and will lead to the development to the
library MaPHyS (see Section 5.2 ). Some specific aspects, such as mixed MPI-thread implementation for the computer science aspects and techniques for indefinite system for the numerical aspects will
be investigated in the framework of a France Berkeley Fund project granted that started last year.
Full geometric multigrid method for 3D Maxwell equations
The multigrid methods are among the most promising numerical techniques to solve large linear system of equations arising from the discretization of PDE's. Their ideal scalabilities, linear growth of
memory and floating-point operations with the number of unknowns, for solving elliptic equations make them very appealing for petascale computing and a lot of research works in the recent years has
been devoted to the extension to other types of PDE.
In this work (Ph. D. of Mathieu Chanaud in collaboration with CEA/CESTA), we have considered a full geometric multigrid solver for the solution of methodology for solving large linear systems arising
from Maxwell equations discretized with first-order Nédelec elements on fully unstructued meshes. This solver combines a parallel sparse direct solver and full multigrid cycles. The goal of this
method is to compute the solution for problems defined on fine irregular meshes with minimal overhead costs when compared to the cost of applying a classical direct solver on the coarse mesh. Mathieu
Chanaud defended his PhD in October 2011.
The direct solver can handle linear systems with up to a few tens of million unknowns, but this size is limited by the computer memory, so that finer problem resolutions that often occur in practice
cannot be handled by this direct solver.
Linear Krylov solvers
Preconditioning is the main focus of the two activities described above. They aim at speeding up the convergence of a Krylov subspace method that is the complementary component involved in the
solvers of interest for us. In that framework, we believe that various aspects deserve to be investigated; we will consider the following ones:
Preconditioned block Krylov solvers for multiple right-hand sides. In many large scientific and industrial applications, one has to solve a sequence of linear systems with several right-hand sides
given simultaneously or in sequence (radar cross section calculation in electromagnetism, various source locations in seismic, parametric studies in general, ...). For “simultaneous" right-hand
sides, the solvers of choice have been for years based on matrix factorizations as the factorization is performed once and simple and cheap block forward/backward substitutions are then performed. In
order to effectively propose alternative to such solvers, we need to have efficient preconditioned Krylov subspace solvers. In that framework, block Krylov approaches, where the Krylov spaces
associated with each right-hand sides are shared to enlarge the search space will be considered. They are not only attractive because of this numerical feature (larger search space), but also from an
implementation point of view. Their block-structures exhibit nice features with respect to data locality and re-usability that comply with the memory constraint of multicore architectures. For
right-hand sides available one after each other, various strategies that exploit the information available in the sequence of Krylov spaces (e.g. spectral information) will be considered that include
for instance technique to perform incremental update of the preconditioner or to built augmented Krylov subspaces. In that context, Yan-Fei Jing, who joint HiePACS as post-doc, is investigating how
reliable block Arnoldi procedure can be combined with deflated restarted block GMRES technique.
Flexible Krylov subspace methods with recycling techniques. In many situations, it has been observed that significant convergence improvements can be achieved in preconditioned Krylov subspace
methods by enriching them with some spectral information. On the other hand effective preconditioning strategies are often designed where the preconditioner varies from one step to the next (e.g. in
domain decomposition methods, when approximate solvers are considered for the interior problems, or more generally for block preconditioning technique where approximate block solution are used) so
that a flexible Krylov solver is required. In that context, we intend to investigate how numerical techniques implementing subspace recycling and/or incremental preconditioning can be extended and
adapted to cope with this situation of flexible preconditioning; that is, how can we numerically benefit from the preconditioning implementation flexibility.
Krylov solver for complex symmetric non-Hermitian matrices. In material physics when the absorption spectrum of a molecule due to an exterior field is computed, we have to solve for each frequency a
dense linear system where the matrix depends on the frequency. The sequence of matrices are complex symmetric non-Hermitian. While a direct approach can be used for small molecules, a Krylov subspace
solver must be considered for larger molecules. Typically, Lanczos-type methods are used to solve these systems but the convergence is often slow. Based on our earlier experience on preconditioning
techniques for dense complex symmetric non-Hermitian linear system in electromagnetism, we are interested in designing new preconditioners for this class of material physics applications. A first
track will consist in building preconditioners on sparsified approximation of the matrix as well as computing incremental updates, eg. Sherman-Morrison type, of the preconditioner when the frequency
varies. This action will be developed in the framework of the research activity described in Section 4.2 .
Approximate factoring of the inverse. When the matrix of a given sparse linear system of equations is known to be nonsingular, the computation of approximate factors for the inverse constitutes an
algebraic approach to preconditioning. The main aim is to combine standard preconditioning ideas with sparse approximate inverse approximation to have implicitly dense approximate inverse
approximations. Theory has been developed and encouraging numerical experiments have been obtained on a set of sparse matrices of small to medium size. We plan to propose a parallel implementation of
the construction of the preconditioner and to investigate its efficiency on real-life problems. Extension of this technique to build a sparse approximation of the Schur complement for algebraic
domain decomposition has also been investigated and could be integrated in the MaPHyS package in the future.
Extension or modification of Krylov subspace algorithms for multicore architectures. Finally to match as much as possible to the computer architecture evolution and get as much as possible
performance out of the computer, a particular attention will be paid to adapt, extend or develop numerical schemes that comply with the efficiency constraints associated with the available computers.
Nowadays, multicore architectures seem to become widely used, where memory latency and bandwidth are the main bottlenecks; investigations on communication avoiding techniques will be undertaken in
the framework of preconditioned Krylov subspace solvers as a general guideline for all the items mentioned above.
Eigensolvers. Many eigensolvers also rely on Krylov subspace techniques. Naturally some links exist between the Krylov subspace linear solvers and the Krylov subspace eigensolvers. We plan to study
the computation of eigenvalue problems with respect to the following three different axes:
• Exploiting the link between Krylov subspace methods for linear system solution and eigensolvers, we intend to develop advanced iterative linear methods based on Krylov subspace methods that use
some spectral information to build part of a subspace to be recycled, either though space augmentation or through preconditioner update. This spectral information may correspond to a certain part
of the spectrum of the original large matrix or to some approximations of the eigenvalues obtained by solving a reduced eigenproblem. This technique will also be investigated in the framework of
block Krylov subspace methods.
• In the framework of an FP7 Marie project (MyPlanet), we intend to study parallel robust nonlinear quadratic eigensolvers. It is a crucial question in numerous technologies like the stability and
vibration analysis in classical structural mechanics. The first research action consists in enhancing the robustness of the linear eigensolver and to consider shift invert technique to tackle
difficult problems out of reach with the current technique. One of the main constraint in that framework is to design matrix-free technique to limit the memory consumption of the complete solver.
For the nonlinear part different approaches ranging from simple nonlinear stationary iterations to Newton's type approaches will be considered.
• In the context of the calculation of the ground state of an atomistic system, eigenvalue computation is a critical step; more accurate and more efficient parallel and scalable eigensolvers are
required (see Section 4.2 ). | {"url":"https://radar.inria.fr/report/2011/hiepacs/uid11.html","timestamp":"2024-11-08T02:57:35Z","content_type":"text/html","content_length":"54101","record_id":"<urn:uuid:b84713fb-dd06-4926-8d28-4a7d38687eb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00771.warc.gz"} |
Bicentric heptagons
A bicentric heptagon is one for which all vertices lie on a circle, and for which all edges are tangential to another circle. If \(R\) and \(r\) are the radii of the outer and inner circles
respectively, and \(d\) is the distance between their centres, there is an expression which relates the three values when a bicentric heptagon can be formed.
To start, define
\[ a = \frac{1}{R+d},\quad b = \frac{1}{R-d},\quad c = \frac{1}{r} \]
and then:
\[ E_1 = -a^2+b^2+c^2,\quad E_2 = a^2-b^2+c^2,\quad E_3 = a^2+b^2-c^2 \]
The expression we want is:
\[ E_1E_2E_3+2abE_1E_2 -2bcE_2E_3-2acE_1E_3=0. \]
See the page at Wolfram Mathworld for details.
However, a bicentric heptagon can exist in three forms: a convex polygon, and two stars.
The above expression, impressive though it is (even more so when it is rewritten in terms of \(R\), \(r\) and \(d\)), doesn't give any hint as to which values give rise to which form of polygon.
However, suppose we scale the heptagon by setting \(R=1\). We can then rewrite the above expression as a polynomial is \(r\), whose coefficients are functions of \(d\):
\begin{multline*} 64d^2r^6-32(d^2+1)(d^4-1)r^5-16d^2(d^2-1)^2r^4+8(d^2-1)^3(3d^2+1)r^3\\ -4(d^2-1)^4r^2-4(d^2-1)^5r+(d^2-1)^6=0. \end{multline*}
and this can be simplified with the substitutions \(u=d^2-1\) and \(x=2r\):
\[ (u+1)x^6-u(u+1)(u+2)^2x^5-u^2(u+1)x^4+u^3(3u+4)x^3-u^4x^2-2u^5x+u^6=0. \]
Since \(R=1\), it follows that \(d\) (and so also \(u\)) is between 0 and 1, and it turns out that in this range the sextic polynomial equation above has four real roots, of which only three can be
used. For the other root \(d+r>1\), which would indicate the inner circle not fully contained in the outer circle.
You can play with this polynomial here:
Then the different forms of the bicentric heptagon correspond with the different roots; the root with the largest absolute value produces a convex polygon, the root with the smallest absolute value
produces the star with Schläfli symbol \({7:3}\) (which is the "pointiest" star), and the other root to the star with symbol \({7:2}\). Look at the table on the Wikipedia page just linked, and the
column for heptagons.
Here are the heptagons, which because of Poncelet's Porism, can be dragged around (if the diagram doesn't update, refresh the page; it should work): | {"url":"https://numbersandshapes.net/posts/bicentric_heptagons/","timestamp":"2024-11-05T17:19:21Z","content_type":"text/html","content_length":"27233","record_id":"<urn:uuid:e4f8514b-3719-480c-84d7-7f8de4e5ef73>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00613.warc.gz"} |
Variable variational success
11 Jan 2019
In connection with my new project OPENFLUX, I have been experimenting with bayesian inference for twin models. Why? To find some procedure that allows flexible, direct comparisons of posterior
distributions of twin study results across groups (for example across cohorts or social groups). What I am looking for is a framework for specifying and estimating large sets of various types of
biometric twin models with and without extended pedigrees and obtain a posterior for each model in the set.
There is already an R package called BayesTwin, but it seems like it is more tailor-made for multiple-response item psychology measures. Maybe there are other packages out there, I must admit I
haven’t checked very carefully.
What I have done is to estimate classical twin models (the ACE type) that decomposes the variance in the outcome (or phenotype, as the geneticists say) into three components: The genetic variance
(A), the shared environmental variance (C), and the non-shared environmental (or error term/measurement error) variance (E). For these experiments I used some simulated data on 300 twin pairs. It
sounds like a lot, and it is, but in the project we’ll have many more.
Typically, behavior geneticists would use OpenMx or other specialized tools for estimating parameters in a twin model. But, the models are just regular SEMs that can also be specified as “mixed
models” (or random effects models). Statisticians Gjessing, Rabe-Hesketh and Skrondal has a paper in Biometrics in 2008 where they demonstrate this approach).
This means that any bayesian engine that lets you estimate mixed models, also lets you estimate twin models. I chose to examine Stan, the (at least for me) fairly new tool, associated (again at least
for me) with Andrew Gelman.
Quickly I ran into problems with speed with the Stan’s Hamiltonian MC method. So I tried the variational bayes method (which I don’t really know anything about) as well. It is much faster!
Variational bayes is about ten times faster than the MCMC procedure. But it is also quite a bit off! The figure below show 1000 draws from the posterior distributions of variance components A, C and
E obtained with regular Stan Hamiltonian MC and variational bayes. The trade-off between speed and quality is there for everyone to see. The A component seems reasonably well estimated in VB (but it
is known for underestimating the variance), but for the C component it is quite off.
The scatterplot below shows the same draws, with A and C plotted on the x and y axes respectively. The VB method clearly yields results very different from the HMC, and the likely underestimation of
the variance in the posterior means it is a no, no for future use (except perhaps for debugging). | {"url":"https://torkildl.github.io/2019/01/bayesian-twin-models/","timestamp":"2024-11-06T02:26:43Z","content_type":"text/html","content_length":"26482","record_id":"<urn:uuid:45191dad-2f7b-4c60-baeb-0988df143775>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00365.warc.gz"} |
A Two-Stage Siamese Network Model for Offline Handwritten Signature Verification
School of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
Education Information Technology Center, Southwest University of Political Science and Law, Chongqing 401120, China
Author to whom correspondence should be addressed.
Submission received: 18 April 2022 / Revised: 9 June 2022 / Accepted: 10 June 2022 / Published: 12 June 2022
Offline handwritten signature verification is one of the most prevalent and prominent biometric methods in many application fields. Siamese neural network, which can extract and compare the writers’
style features, proves to be efficient in verifying the offline signature. However, the traditional Siamese neural network fails to represent the writers’ writing style fully and suffers from low
performance when the distribution of positive and negative handwritten signature samples is unbalanced. To address this issue, this study proposes a two-stage Siamese neural network model for
accurate offline handwritten signature verification with two main ideas: (a) adopting a two-stage Siamese neural network to verify original and enhanced handwritten signatures simultaneously, and (b)
utilizing the Focal Loss to deal with the extreme imbalance between positive and negative offline signatures. Experimental results on four challenging handwritten signature datasets with different
languages demonstrate that compared with state-of-the-art models, our proposed model achieves better performance. Furthermore, this study tries to extend the proposed model to the Chinese signature
dataset in the real environment, which is a significant attempt in the field of Chinese signature identification.
1. Introduction
Fingerprints, irises, faces, voices, and handwritten signatures are five prevalent biometric recognition in many practical fields such as financial payment, attendance, computer vision, and contract
signature [
]. Biometric recognition was started initially with the body measurements. Later with time and necessity, it involved many biometric properties related to the human body to provide authentication.
Among these, the handwritten signature is the most commonly accepted symbol [
]. Verifying a person’s identity using one’s handwritten signatures is challenging, where a forger can access a person’s handwritten signature and deliberately attempts to imitate it [
]. The main difficulties of offline handwritten signature verification are the high internal variability of individuals, scarcity of skilled forgery samples, and a limited number of training samples.
Moreover, the discrepancy between genuine signature and skilled forgery is subtle since forgers attempt to imitate genuine signatures. With small inter-class variation between genuine signatures and
skilled forgery while the larger intra-class variation of genuine signatures from the same person, it is very hard to differentiate between inter-personal forgery and intra-person handwritten
signature genuineness.
Recently, powered by the rapid development of pattern recognition and image processing technology [
], it has become possible to verify handwritten signatures automatically. However, in the offline handwritten signature verification process, the dynamic information of the signature writing process
is lost, and it is difficult to design a good feature extractor that can distinguish genuine signatures from skilled forged signatures, which makes the problem more challenging.
Up to now, the Siamese neural network model [
] is one of the most popular and powerful approaches to address this issue and has greatly promoted the development of image identification. Siamese neural network transforms the features extracted
from handwriting verification from traditional image texture features into convolution features [
], greatly improving the performance of handwriting verification. However, most of the existing methods regard offline handwritten signature [
] as an image recognition problem and cannot fully represent the writers’ writing style. When the distribution of positive and negative handwritten signatures is very unbalanced [
], the performance of these models is unsatisfactory.
From this point of view, is it possible to improve the Siamese neural network model by considering the imbalance distribution of positive and negative signatures? To address this issue, this study
proposes a two-stage Siamese neural network model. First, feature extraction is carried out by a two-stage convolutional neural network, which contains both the verification of the original two
handwritten signatures and the verification of the handwritten signature after data enhancement. Second, this study introduces Focal loss as a loss function of the proposed network model, which can
fully take into account the extremely unbalanced distribution of positive and negative signatures, making the neural network focus on stroke information rather than background information of the
handwritten signatures. Extensive experimental results on four challenging handwritten signature datasets of different languages demonstrate the effectiveness of the proposed model.
The main contributions of this paper are as follows:
• This is a significant attempt to study Chinese signature identification.
• A two-stage Siamese network model is proposed to verify the offline handwritten signature.
• Visualization of the process of feature representation is analyzed.
The remaining of this paper is organized as follows:
Section 2
reviews the preliminaries on offline handwriting signature verification.
Section 3
presents the proposed model. In
Section 4
, four experiments have been carried out for evaluating the proposed approach, and the results are explained and discussed.
Section 5
concludes the findings and states the future works.
2. Preliminaries
2.1. Related Work
Offline handwritten signature verification can be considered as a two-class classification problem: to decide whether two offline handwritten signatures are signed by the same person or not, and to
judge whether a handwritten signature is genuine for a specific user or not. Up to now, many methods have been proposed for offline handwritten signature verification [
]. Many studies often use texture features extraction such as gray-level co-occurrence matrix [
] and Local Binary Patterns [
]; directional-based features such as directional-pdf [
] and histogram of oriented gradients [
]; feature extractors specifically designed for offline handwritten signatures, such as the estimation of strokes by fitting Bezier curves [
]. Moreover, an inverse discriminative network [
] is proposed for writer-independent handwritten signature verification. Li, L et al. [
] proposed a region-based deep convolutional Siamese network for feature and metric learning. Wei et al. [
] proposed an inverse discriminative network that is capable of intensifying the effective information of signatures. Mustafa et al. [
] utilized a two-channel CNN as a feature extractor, where the two channels represent the reference and query signatures, respectively. A multi-task architecture based on R-SigNet architecture [
] is proposed, which exploits relaxed loss to learn a reduced feature space for writer-independent signature verification. All these methods have a good effect on signature identification to a
certain extent. Despite the remarkable progress, signature verification is still very challenging due to the high intra-class variety and low inter-class variety among signatures from different
writers. Note that these methods have limits. They treat the offline handwritten signature as an image processing problem and fail to represent the writers’ writing style fully. When the distribution
of positive and negative handwritten signatures is very unbalanced, the performance of these models is unsatisfactory.
Despite great achievements in offline handwritten signature verification, existing models still have some limits as follows:
• Most of them only treat handwriting signature as a picture and do not mine deep signature style.
• They commonly ignore the imbalance distribution of positive and negative signatures that often occurs in real scenarios.
• The signature samples of each writer are usually small and the similarity between real signature and forged signature is high in real scenarios. The existing models usually generate synthetic
data that are quite different from the real ones.
In comparison, our proposed model is significantly different from the existing models because:
• It has a two-stage Siamese network module to verify the offline-handwritten signature. This network includes both traditional original handwriting recognition and data-enhanced handwriting
recognition to mine the writers’ deep signature style.
• It employs the Focal loss to deal with the extreme imbalance between positive and negative offline signatures, which is quite different from previous studies.
• It is the first attempt to study the Chinese signatures with a real Chinese signature dataset.
2.2. CNN and Siamese Neural Network
Convolutional Neural Networks (CNN) are multilayer neural networks consisting of several convolutional layers with different kernel sizes interleaved by pooling layers, which summarize and
downsamples the output of its convolutional layers before feeding to the next layers. The structure of the classical Convolutional Neural Network is shown in
Figure 1
. To obtain nonlinear correction, an activation function is also used. With the gradual increase of the number of convolutional layers, the range of the receptive field is gradually expanding. The
closer it is to the subsequent output, the affected range of pixels of the image is wider. The convolutional neural network can learn different features in each volume base by advancing layer by
layer and finally realize related recognition and classification functions.
Siamese network architecture was first introduced into the field of signature verification by Bromley et al. [
]. Since then, it has been widely used in many different fields such as one-time learning as well as text recognition and face recognition [
]. It consists of two identical subnetworks, maps inputs to higher-dimensional Spaces, and computes distance measures between high-level feature representations. The structure of the Siamese neural
network is shown in
Figure 2
. Two CNNS share the same network and parameters but input different data. Two samples are taken as inputs, and then their representations are embedded into the output in the high-dimensional space
to compare the similarity of the two inputs. Through the forward processing of a convolutional neural network, the data that is difficult to distinguish in the original space can be represented in a
specified dimension, making it easy to distinguish. The Siamese neural network is widely used in face verification, signature verification, and other tasks, in which samples are not directly
classified, but compared with known patterns to determine whether they belong to the same category. The Siamese neural network model generally adopts the cross-entropy loss function [
] with regularization term. As a typical binary classification problem, it is given by:
$L ( x 1 , x 2 ) = y ( x 1 , x 2 ) log P ( x 1 , x 2 ) + ( 1 − y ( x 1 , x 2 ) ) log ( 1 − p ( x 1 , x 2 ) ) + λ T | w | 2$
y(x[1], x[2]) = 1
represents when
belong to the same kind of object or
y(x[1], x[2])
= 0 represents when
belong to the different kind of object.
2.3. Focal Loss
In handwriting signature identification cases, the handwriting signature that can be obtained is usually very limited. The positive signature is usually from a credit card consumption signature, file
data signature, or contract signature, while the same document can only provide one positive signature at the same time. A forged signature often comes from someone else’s imitation, which can be
multiple imitations. In the existing open dataset, the handwriting signature of the positive sample pair is often much fewer than that of the negative sample pair, and the distribution of positive
and negative signatures used for comparison in the data set is imbalanced.
Dealing with unbalanced data has always been a challenge in deep learning and machine learning. Based on the classical cross-entropy loss function, Focal Loss [
] was first proposed to handle the object detection scenario where the unbalance exists between foreground and background classes. To handle the imbalanced data of different classes, a weighting
factor and modulating factor are used to adjust the loss function. The sharing weight to the total loss can be adjusted by changing the value of the weighting factor
and modulation factor
. Compared with the classical cross-entropy loss, the Focal Loss focuses more on the difficult and misclassified cases and plays a well-regulating role in the extremely unbalanced distribution of
positive and negative samples in the dataset. Formally, the Focal Loss [
] adds the factor
$− α ( 1 − y ^ ) γ$
$− ( 1 − α ) y ^ γ$
to the standard cross entropy criterion. Setting
> 0 reduces the relative loss for well-classified examples, putting more focus on hard, misclassified examples, it is given by:
$FL = { − α ( 1 − y ^ ) γ ⋅ log ( y ^ ) , y = 1 − ( 1 − α ) y ^ γ ⋅ log ( 1 − y ^ ) , y = 0$
is a weighting factor of [0, 1],
denotes the predicted value, and
is an adjustable focus parameter to prevent easy samples from contributing too much.
3. Model
3.1. Problem Formulation
Assume x is a verification sample signature and s is a genuine signature sample. This study aims to distinguish whether x is forged or genuine compared to s. The output predicted value
∈ [0, 1] is the label for validating the decision where
= 1 means the verification sample signature is genuine to s signature image and
= 0 indicates the verification sample signature is forged to
signature image. Thus, this signature verification problem can be represented as
where the decision function
(·) maps the input handwriting signature images
to the predicted value
. After training and learning, all parameters are saved in parameter set
. Note that the problem defined in Equation (3) is similar to but different from the regression problem in the neural network machine learning tradition. First, traditional regression problems in
neural network machine learning generally have one input, while handwritten signature verification problems have two original inputs. Second, in most traditional binary classification problems,
especially in the field of image processing, the main object of feature extraction is based on color, texture, intensity, etc. In the problem of handwriting signature verification, the key part is to
distinguish the difference in handwriting style. Writing style, as an abstract feature, is an indescribable attribute defined by the strokes of a handwritten signature rather than color or
3.2. Architecture of the Two-Stage Network
The architecture of the proposed two-stage connected neural network is shown in
Figure 3
. In the proposed model architecture, the upper and lower layers of the model are completely symmetric, and the left and right sides of the model are relatively symmetric. The model contains a
two-stage Siamese neural network and consists of three modules: the convolutional neural feature extractor module, image enhancement module, and the objective function module. The structure and
function of each module will be described in detail in later chapters. By adopting a two-stage siamese neural network, the original input handwritten signature verification and the image enhanced
signature verification are realized simultaneously, and the output results of the two-stage networks are combined to verify with the label Simultaneously. The two-stage network is beneficial to the
extraction and verification of shallow and deep features of handwriting and improves the accuracy of handwriting signature verification.
The Focal Loss function was adopted to adjust the extreme imbalance between positive signature and negative signature samples. In this model,
$x ˜$
$s ˜$
are the new images of the original input signature image after A series of image transformations and image enhancement, respectively. This idea is shown in
Figure 3
. The original input of the check sample signature and the genuine sample signature will obtain the corresponding validation decision tag
. A new inspection sample pair is generated by enhancing the image data of the original inspection sample image and the sample image. These newly generated inspection sample image pairs should have
the same validation labels, for the image enhancement processes do not change the signature structure and writing style.
3.3. The Feature Extractor
The architecture of the feature extractor module is illustrated in
Figure 4
. The proposed networks take two original handwritten signatures as inputs and take the feature of the signatures as outputs. The two input images share the same network parameters. Firstly, the
image signature is preprocessed. Since the neural network requires all input sizes to be consistent, all signature sizes are unified as 115 × 220 dimensions of the gray image in this study. After the
signature is input into the network structure, some convolutional neural network layers are used to extract features from the signature image. The feature extraction process of the convolutional
neural network includes a convolution layer, nonlinear activation layer, max-pooling layer, batch normalization layer, etc. The convolution layer and full connection layer have learnable parameters,
and the parameters are continuously optimized in the training process. After each learnable layer, we apply batch normalization, followed by ReLU nonlinear activation. The last layer adopts Softmax
nonlinear activation and interprets the output as a probability.
There are four cascaded convolutional operations inspired by the Visual Geometry Group net, and each operator group consists of two convolutional layers followed by Rectified Linear Units function, a
normalized layer, and a pooling layer. Generally, through four cascades, the global and local features in the handwritten signature images can be fully represented. The channel number of four
cascaded convolutional layers in the corresponding operator group is 32, 64, 96, and 128, respectively. The main function of the batch normalization layer is to standardize the input of each layer
and prevent the gradient explosion and gradient disappearance in the subsequent calculation. The batch normalization layer only works for training, not for validation sets. The function of the
activation layer is to add nonlinear factors and map features to high-dimensional nonlinear intervals for interpretation. As shown in
Figure 4
, all the input image processing processes use the same training parameters, that is, all CNN modules share the same network and parameters. By learning these training parameters to extract effective
signature author style features, the computational complexity is reduced, and the performance of the model is improved effectively.
3.4. The Signature Image Data Enhancement
The architecture of the image enhancement module is illustrated in
Figure 5
. In this module, it receives the original input signature image x and the feature map F(x), which comes from the input signature through the output of the feature extraction module as the inputs of
this module. The feature F(x) already contains the writing features of part of the signature handwriting after a series of convolution processes. To well represent the writing style features of
signature handwriting pictures, this module further refines the features. After a GAP layer (global average of pooling layer) processing and two FC layers (fully-connected layers), the output feature
is reshaped to 57 × 110 dimensions. The extracted features are restored to the matrix form, and then the nearest neighbor up-sampling and filling layers are carried out. The up-sampling layer can
retain the image features extracted from the earlier convolutional layer to the maximum extent, which plays a good role in the subsequent feature extraction.
Through the padding layer and the convolutional layer with a nonlinear activation function, the size of the output matrix is completely consistent with the original input signature image, which is
enhanced for the subsequent data. The final output result of the data enhancement module is obtained by multiplying the data enhancement weight matrix with the input matrix of the original input
signature image. In this way, the feature data on the image can be enlarged and the handwriting features can be further extracted. The symbol ⊗ represents element multiplication. Through the
processing of the module, the generated image $x ˜$ becomes a new image after data enhancement, which has the same size as the original signature x.
3.5. Loss Function
The loss function can evaluate the performance of the model, which is the most critical part of machine learning, especially in the training phase. Suppose elements combination {(
= 1, …, N} is a sample of the training dataset, and
are the
-th verification sample and the genuine verification, respectively. The value
∈ {0, 1} is (
), where
= 0 indicates the verification sample
is forged compared to
, and where
= 1 indicates genuine. With the training dataset, this study aims to optimize the network parameters consisting of two parts. Both the original sample and the image enhanced sample are involved in
the training process, the loss function of the model consists of two losses: the loss of the original sample to contrast and the loss of new sample comparison after data enhancement. For the original
handwritten signature (
xi, si, yi
are the output of the features from the original samples.
P(F(x[i]), F(s[i]))
is the signature verification probability predicted. To deal with the extreme imbalance between positive and negative offline signatures, this study adopts the Focal loss function [
] to express the loss of verification results. As a binary classification problem, the loss
$F L O ( x i , s i )$
result can be calculated:
$F L O ( x i , s i ) = { − α ( 1 − y ^ ) γ log ( y ^ ) , y = 1 − ( 1 − α ) y ^ γ log ( 1 − y ^ ) , y = 0 y ^ = P ( ( F ( x i ) , F ( s i ) )$
Based on the input handwriting signatures
, the image enhanced signatures are defined as
$x ˜ i$
$s ˜ i$
. In handwriting identification studies, (
$x ˜ i$
$s ˜ i$
) should have the same predictive value as the original signature image (
), for the data enhancement of the signature, the image does not change the writing style of the signature. Therefore, the loss function
$F L E ( x ˜ i , s ˜ i )$
for (
$x ˜ i$
$s ˜ i$
) after image enhancement should be defined as follows
$F L E ( x ˜ i , s ˜ i ) = { − α ( 1 − y ^ ) γ log ( y ^ ) , y = 1 − ( 1 − α ) y ^ γ log ( 1 − y ^ ) , y = 0 y ^ = P ( ( F ( x ˜ i ) , F ( s ˜ i ) )$
Hence, the final loss is the total loss of training samples by combining the losses of the two parts, which can be calculated as follows
$L o s s = ∑ i N { F L O ( x i , s i ) + λ ⋅ F L E ( x ˜ i , s ˜ i ) }$
where λ is a hyperparameter whose function is to balance the weight of two parts of the loss and is an empirical value.
3.6. Algorithm Design
Here, we discuss about the training process of the proposed algorithm, which is demonstrated in Algorithm 1.
Algorithm 1: Training Process of the Proposed Algorithm
Require: set up the batch size m, the maximum number of epoch k, the learning rate LR, and the penalty factors $λ$
Require: Initialize the weights of the networks $θ$.
forepoch number = 1 : kdo
Randomly select m images from the training image dataset: $x i$
Select m corresponding genuine images from the preprocessed dataset: $s i$
Calculate the eigenvector and the Loss according to the network weights $θ$.
Update the weights of the networks $θ$.
$∇ θ 1 k ∑ i = 1 k [ F L O ( x i , s i ) + λ ⋅ F L E ( x ˜ i , s ˜ i ) ]$
End for
4. Empirical Studies
4.1. General Settings
This study tested the proposed model on four challenging offline handwritten signature datasets: the Chinese signature dataset, CEDAR signature datasets, BHSig-Hindi signature datasets, and
BHSig-Bengali signature datasets, which come from four different languages, respectively: Chinese, English, Hindi, and Bengali. Taking the CEDAR dataset [
] as an example, this dataset contains a total of 55 English signatures, and 24 forged handwritten signatures and 24 genuine signatures were written for each person, so there are C
= 276 pairs of positive signatures. Combining the genuine and forged signatures of each person, there are 24 × 24 = 576 pairs of negative signatures. According to the characteristics of machine
learning, 50 signature samples were randomly selected in this study to train model parameters, and the remaining 5 signature samples were used as validation samples. BHSig260 signature Dataset [
] was divided into the BHSIG-Bengali signature set and BHSIG-Hindi signature set, which were trained and verified independently, respectively. The BHSIG-Bengali signature dataset contains a total of
100 handwritten signature images with Bengali signatures. Each person has 30 forged handwritten signatures and 24 genuine signatures, so there are C
= 276 pairs of positive signatures. Combining the genuine and forged signatures of each person, there are 30 × 24 = 720 pairs of negative signatures. Bhsig—Hindi Signature Dataset is another subset
of BHSig260, which contains a total of 160 handwritten signature images of people signing in Hindi. Similar to Bengali signatures, each person has 30 forged handwritten signatures and 24 real
signatures, so there are C
= 276 pairs of genuine signatures. Combining the genuine and forged signatures of each person, there are 30 × 24 = 720 pairs of negative signatures. The Chinese signature dataset contains 500 groups
of signature data signed in Chinese, and each group of signature data contains one check sample signature and three genuine signatures. The check sample signatures need to be used to verify whether
they are genuine handwriting or not, and the other three sample signatures are genuine handwriting that has been confirmed to be written by oneself. The signatures of each dataset come from different
scenarios and different sampling times.
Chinese Handwritten Signature Dataset: Since the previous Chinese handwriting signatures were imitated in the laboratory and the data amount was small, there was no suitable Chinese handwriting
signature dataset. Therefore, we collected a multi-source Chinese handwriting signature dataset with a large period and strong practical significance. This data set includes both positive and
negative signatures sample., which are from the National Forensic Center of Southwest University of Political Science and Law between 2009 and 2020. As it is a real case, these signatures all come
from real-life signatures such as credit card consumption signatures, personal file signatures, signatures in document contracts. In a real setting, to ensure the consistency of the data set sample
data, the datasets collected an examination of each signature handwriting and three certified true signature handwriting, and formed a set of data signature handwriting. There are altogether 500 sets
of such signature handwriting data, including 220 sets of negative signature handwriting data and 280 sets of positive signature handwriting data. The handwriting data of each signature applying for
handwriting identification are highly similar. All the signed signatures were scanned into images at 300DPI. The Chinese offline signature dataset consists of 500 names and 2000 signature images.
This dataset has the characteristics of multi-source, real, and large scale. First, all the signatures are from real cases, which is often challenging. Secondly, the Chinese data set belongs to a
relatively large-scale data set, in which the handwriting period of decades. Third, real signatures are collected at different times and in different scenarios, and the signatures of the same person
may be significantly different. All of these characteristics make this data set very valuable and challenging.
To further understand the sample information of signature images in each dataset, some sample signatures are shown in
Table 1
. In each dataset, the data set details are shown in
Table 2
Evaluation Metrics:
In this study, a group of positive samples is composed of two genuine signatures written by the same person, and the corresponding recognition decision label y = 1. The evaluation metrics are based
on the prediction of the sample pairs in all validation sets and the statistical analysis of the predicted results. Three evaluation indicators were used to evaluate and compare the proposed method
with other methods: false acceptance rate (FAR), false rejection rate (FRR), and accuracy (ACC). The false acceptance rate is defined as the ratio of the number of false acceptances divided by the
number of negative signature samples. The false rejection rate is defined as the ratio of the number of false rejections divided by the number of positive signature samples. Lower FRR or FAR and
higher ACC mean better performance. They are calculated as follows:
$FAR = F a l s e p o s i t i v e F a l s e p o s i t i v e + T r u e n e g a t i v e × 100 % FRR = F a l s e n e g a t i v e F a l s e n e g a t i v e + T r u e p o s i t i v e × 100 % ACC = T r u e
p o s i t i v e + T r u e n e g a t i v e T r u e p o s i t i v e + T r u e n e g a t i v e + F a l s e p o s i t i v e + F a l s e n e g a t i v e × 100 %$
This study compares our proposed model with eight involved state-of-the-art models, including five writer-independent methods models (SigNet [
], Surroundness [
], Chain code [
], Ensemble Learning [
], Morphology [
], and DeepHSV [
]) and three writer-dependent models (Chain code [
], Texture Feature [
] and Fusion of HTF [
Table 3
describes the main descriptions of the correlation models.
4.2. Comparison with State-of-the-Art Models
This study was independent of the writer, and we labeled writer-independent (WI) in the table of experimental results, such as [
]. It also lists the results of methods that rely on the writer himself and are labeled writer-dependent (WD), such as [
]. The writer-independent approaches train just one model for all test writers, and writer-dependent approaches train a single-minded model for each writer. The writer-dependent model generally has
better performance than the writer-dependent model but requires training in everyone’s signature sample, which is impractical and cannot be generalized to unobserved people. This study is based on
writer-independent and only one model parameter set is trained in the same data set.
The performance of the proposed model was compared with that of the four state-of-the-models, and detailed comparisons are given in
Table 4
Table 5
below. In these experiments, To make the networks learn as many characteristic attributes of signatures as possible, we attempt to remove noise and keep foreground information about the signature
itself. Firstly, the OTSU algorithm [
] was used to separate foreground and background regions, and the batch normalization was utilized to normalize signature images. Second, the background pixel value is converted to 255 and the
original pixel value of signature strokes is retained. The running environment of the model is based on Pytorch 1.3.1 framework, using the NVIDIA 2080Ti GPU graphics card. In this study, the
stochastic gradient descent optimization method was adopted, the basic learning rate was 1 × 10
, and the batch size was set as 32. The λ in Equation (5) is set as the empirical value 2.5. The proposed model is compared with the baseline model, the traditional Siamese neural network method, and
the classical cross-entropy loss function method. As shown in
Table 4
Table 5
, our model achieves better performance than state-of-the-art models.
4.3. Chinese Signature Dataset
This study is the first to identify handwriting in a real case. The research results of this study will be helpful to judicial identification and have important research value. In the Chinese offline
signature handwriting data set, the handwriting information collected is from real handwriting identification cases from 2008 to 2020. In addition, due to a large number of Chinese fonts, similar
appearance, ease of confusion, and the randomness of Chinese signatures, the results of handwriting identification will be affected to a certain extent. In addition, different from the multi-arc
features of Latin letters, the focus of Chinese handwriting identification is also different. The main characteristics of Chinese signatures are the special and stable parts such as stroke crossing,
connection, and collocation. We fully believe that it will contribute to the field of Chinese offline handwritten signature verification and related research. As shown in
Table 6
, compared with other current methods, this study plays a certain role in handwriting prediction, which is a significant attempt in the field of Chinese signature identification.
4.4. Process Visualization
Figure 6
shows the feature extraction process of signature image features. It can be seen from the figure that at the beginning of training, neural network learning features mainly focus on texture features
of handwritten pictures, as shown in
Figure 6
b–d. With the deepening of training, the features learned are gradually abstract, which can be understood as handwriting style features.
From the above conclusions, we can draw the following important implications:
• Compared with previous methods, this model has better prediction performance. On the CEDAR signature dataset, the FRR, FAR, and ACC of the proposed method reach 6.78%, 4.20%, and 95.66%,
respectively, which are superior to the existing comparison methods under all evaluation indicators. On the BHSIG-Bengali and BHSIG-Hindi signature datasets, our model achieves ACC of 90.64% and
88.98%, respectively, which is superior to other models. These results show that our method is superior to other comparison methods. In addition, our writer-independent approach still performs
better than the writer-dependent approach.
• The data enhancement method adopted in this study is only related to the original input signature image. The original input signature image is processed by a series of neural networks to generate
a data enhancement weight matrix. Finally, the degree of image data enhancement is adjusted by adjusting the proportion of the weight matrix, which improves the accuracy of experimental results,
and the proposed model has strong robustness.
• The focal Loss function is very effective for solving the problem of unbalanced positive and negative data.
• The proposed model also has good performance in Chinese signature datasets, and this conclusion will be helpful for further research on offline Chinese signature verification.
5. Conclusions
Aiming at solving the problem of offline handwritten signature verification, this study proposes a two-stage Siamese neural network model to extract the writers’ writing style. Based on the
end-to-end image enhancement learning method and Focal Loss function, the proposed model can effectively solve the problem of imbalance of positive and negative samples, achieve good performance on
challenging datasets with three different languages, and also work well in Chinese offline handwritten signature dataset. To evaluate the proposed model, we conduct extensive experiments on four
challenging handwritten signature datasets with different languages. The results demonstrate that the proposed model achieves better performance than the state-of-the-art models. Future work will
focus on the study of Chinese handwriting signatures and improving the accuracy of Chinese handwriting identification.
Author Contributions
Conceptualization, W.X.; methodology, W.X.; software, W.X.; validation, W.X.; formal analysis, Y.D.; investigation, W.X.; resources, Y.D.; data curation, W.X.; writing-original draft preparation,
W.X.; writing-review and editing, Y.D.; visualization, W.X.; supervision, Y.D.; project administration, Y.D.; funding acquisition, W.X. All authors have read and agreed to the published version of
the manuscript.
This work was supported by the Science and Technology Project of Chongqing Education Commission, No. KJQN202100304 and the Key Cooperation Project of Chongqing Municipal Education Commission, No.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
For the collection of Chinese handwriting datasets, we thank the National Forensic Center of Southwest University of Political Science and Law.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 6. (a) Original Signature images, (b–d), feature extraction in CNN process, (e) Signatures after data enhancement.
Genuine Forgery
Datasets CEDAR BHSig-B BHSig-H CHINESE
languages English Bengali Hindi Chinese
People 55 100 160 500
Signatures 2640 5400 8640 2000
Total sample 46,860 99,600 159,360 1500
Positive: negative 276:576 276:720 276:720 840:660
Model Description
SigNet The writer independent Siamese network model proposed in 2017 [37] and is often applied to signature verification.
Surroundness A signature feature extraction model based on envelopment was proposed in 2012 [38].
Chain code In 2013 [39], a model based on the histogram features of chain codes was proposed and enhanced by Laplacian Gaussian filter.
Eensemble Learning Deep learning model proposed in 2019 [40], which improves an integration model for offline writer independent signature verification.
Morphology Feature analysis technology based on multi-layer perceptron was proposed in 2010 [41].
Texture Feature a texture-oriented signature verification method was proposed in 2016 [42]. It has good performance for Indian scripts.
Fusion of HTF A Signature verification model proposed in 2019 [6]. It adopts discrete wavelet and local quantized patterns features
DeepHSV A neural network model proposed in 2019 [30], which improves the network with a two-channel CNN network
Method Type FRR FAR ACC
Morphology WI 12.39 11.23 88.19
Surroundness WI 8.33 8.33 91.67
Chain code WD 9.36 7.84 92.16
Ensemble Learning WI 8.48 7.88 92.00
ISNN + CrossEntropy WI 9.38 7.68 92.55
SNN + Focal Loss WI 8.92 6.94 93.47
Our method WI 6.78 4.20 95.66
BHSig-Bengali BHSig-Hindi
Method Type FRR FAR ACC FRR FAR ACC
SigNet WI 13.89 13.89 86.11 15.36 15.36 84.64
Texture Feature WD 33.82 33.82 66.18 24.47 24.47 75.53
Fusion of HTF WD 18.42 23.10 79.24 11.46 10.36 79.89
DeepHSV WI 11.92 11.92 88.08 13.34 13.34 86.66
ISNN + CrossEntropy WI 18.64 12.86 86.66 15.63 15.49 84.54
SNN + Focal Loss WI 16.87 9.43 87.69 13.38 10.91 84.79
Our method WI 14.25 6.41 90.64 12.29 9.6 88.98
Method Type FRR FAR Acc
SigNet WI 42.36 42.36 57.64
DeepHSV WI 41.87 41.87 58.13
SNN + CrossEntropy WI 38.98 35.77 64.79
ISNN + CrossEntropy WI 33.66 31.24 68.88
SNN + Focal Loss WI 36.74 30.92 65.88
ISNN + Focal Loss WI 32.18 30.59 70.31
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Xiao, W.; Ding, Y. A Two-Stage Siamese Network Model for Offline Handwritten Signature Verification. Symmetry 2022, 14, 1216. https://doi.org/10.3390/sym14061216
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Xiao, Wanghui, and Yuting Ding. 2022. "A Two-Stage Siamese Network Model for Offline Handwritten Signature Verification" Symmetry 14, no. 6: 1216. https://doi.org/10.3390/sym14061216
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Binary Jumbled Pattern Matching on Trees and Tree-Like Structures
Binary jumbled pattern matching asks to preprocess a binary string $$S$$S in order to answer queries $$(i,j)$$(i,j) which ask for a substring of $$S$$S that is of length $$i$$i and has exactly $$j$$j
1-bits. This problem naturally generalizes to vertex-labeled trees and graphs by replacing “substring” with “connected subgraph”. In this paper, we give an $$O(n^2 / \log ^2 n)$$O(n2/log2n)-time
solution for trees, matching the currently best bound for (the simpler problem of) strings. We also give an $${O}({g^{2 / 3} n^{4 / 3}/(\log n)^{4/3}})$$O(g2/3n4/3/(logn)4/3)-time solution for
strings that are compressed by a context-free grammar of size $$g$$g in Chomsky normal form. This solution improves the known bounds when the string is compressible under many popular compression
schemes. Finally, we prove that on graphs the problem is fixed-parameter tractable with respect to the treewidth $$w$$w of the graph, even for a constant number of different vertex-labels, thus
improving the previous best $$n^{O(w)}$$nO(w) algorithm.
Bibliographical note
Publisher Copyright:
© 2014, Springer Science+Business Media New York.
• Grammar compression
• Graph motifs
• Pattern matching
• Permutation pattern matching
• Tree pattern matching
ASJC Scopus subject areas
• General Computer Science
• Computer Science Applications
• Applied Mathematics
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Can others disable my system?
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Home | Hippocampus's Garden
This post attempts to take a deeper look at F1 score. Do you know that, for calibrated classifiers, the optimal threshold is half the max F1? How come? Here it's explained.
This post steps forward to multiple linear regression. The method of least squares is revisited --with linear algebra.
This post summarizes the basics of simple linear regression --method of least squares and coefficient of determination.
Is the sample correlation coefficient an unbiased estimator? No! This post visualizes how large its bias is and shows how to fix it.
The correlation coefficient is a familiar statistic, but there are several variations whose differences should be noted. This post recaps the definitions of these common measures.
When you sample from a finite population without replacement, beware the finite population correction. The samples are not independent of each other.
What is unbiased sample variance? Why divide by n-1? With a little programming with Python, it's easier to understand.
How come ROC-AUC is equal to the probability of a positive sample ranked higher than negative ones? This post provides an answer with a fun example. | {"url":"https://hippocampus-garden.com/tags/stats/","timestamp":"2024-11-11T09:55:59Z","content_type":"text/html","content_length":"239587","record_id":"<urn:uuid:1e59eaa6-bdcb-4d91-8a19-91dc67d86a1b>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00883.warc.gz"} |
Centrifugal Force Calculator
What is Centrifugal Force?
Centrifugal force is an apparent force that is felt by an object moving in a circular path that acts outwardly away from the center of rotation. It’s not a real force in the classical sense, but
rather a result of inertia—the tendency of an object to resist any change in its state of rest or uniform motion.
When an object moves in a circle, it constantly changes direction, requiring a force directed towards the center of the circle to cause this change. This force is known as centripetal force.
Centrifugal force, on the other hand, is the sensation of an outward force felt by the object in motion. It’s an inertial force that appears to act on all objects when viewed in a rotating frame of
Understanding Through an Example:
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the inward pull (centripetal force) that keeps you moving in a circle.
Misconceptions About Centrifugal Force:
It’s crucial to note that centrifugal force is often misunderstood. It’s not a force that ‘acts’ on an object in the same way gravity or electromagnetic forces do. Instead, it’s a perceived force due
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Centrifugal Force Equation
\( F = m \cdot r \cdot \omega^2 \)
• F – Centrifugal Force,
• m – Mass of the object,
• r – Radius of the circular path,
• ω – Angular velocity.
Centrifugal Force is a force that acts outward on a body moving around a center, arising from the body’s inertia. It is crucial in various engineering and physical applications such as automotive
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Real-World Applications of Centrifugal Force
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Centrifugal Force Calculator: Interactive Exploration
Understanding centrifugal force is greatly aided by interactive tools. Our Centrifugal Force Calculator allows users to input values for mass, radius, and angular velocity to instantly calculate the
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FAQs: Addressing Common Questions
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Is it possible to find assistance with Integral Calculus integration exam accommodations for students with disabilities? | Hire Someone To Do Calculus Exam For Me
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6 Corrected exercises on transport problems - Complex systems and AI
6 Corrected exercises on transport problems
A business needs to transport supplies from factories to construction sites. The three factories have a supply capacity of 300, 300, 100 respectively. And the three yards require 200, 200, 300
Shipping costs are shown in the following chart:
Figure out how to distribute supplies.
Wheat is harvested in the Midwest and stored in grain elevators in three different cities: Kansas City, Omaha and Des Moines. These grain elevators supply three flour mills located in Chicago, St.
Louis and Cincinnati. Grain is shipped to the mills in railroad cars, each car holding a ton of wheat. Each grain elevator is able to supply the following number of tons (i.e. railroad cars) of wheat
to the mills on a monthly basis.
Each mill demands the following number of tons of wheat per month.
The cost of transporting a ton of wheat from each grain elevator (source) to each mill (destination) differs depending on the distance and the rail system. These costs are shown in the following
table. For example, the cost to ship a ton of wheat from the grain elevator in Omaha to the mill in Chicago is 7 $.
The problem is to determine the number of tons of wheat to be transported from each grain elevator to each mill on a monthly basis in order to minimize the total transport cost. Is there an
alternative solution? If yes which ?
From the linear program below, formulate the transport problem and solve it.
Is there an alternative solution? If yes which ?
Take the same problem as Exercise 2. Change the demand in Cincinnati to 350 tons. Solve.
Solve the following LP problem:
The Atlantic Coast Conference is hosting four basketball games on a particular night. The conference office wants to assign four teams of officials to the four games in order to minimize the total
distance covered by the officials. The distances in miles of each team of officials at each match venue are shown in the table below: | {"url":"https://complex-systems-ai.com/en/theorie-des-graphes/corrected-exercises-on-transportation-problems/","timestamp":"2024-11-05T06:04:44Z","content_type":"text/html","content_length":"157706","record_id":"<urn:uuid:f353504a-4189-47b0-9316-9a9ea67c200c>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00311.warc.gz"} |
How to Calculate a Yield to Maturity Loan
••• calculator chart image by max blain from Fotolia.com
The yield on a variable-price loan or bond is calculated using the yield to maturity equation. This equation uses the current market price, the time to maturity of the bond, the payments and the face
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appropriately price the bonds in their inventory.
Subtract the face value (F) of the bond from the current market price (P). For example, if F is $100 and P is $90, then P - F = -$10.
Divide this value by the number of years to maturity (n), as in (F-P)/n. If n = 5, then (F-P)/n = -$2.
Add the interest payment (C) to this value, as in C +(F-P)/n. If C is $5, then C +(F-P)/n = $3.
Divide the combined amount from Step 3 by the price plus face value divided by 2, as in (C +(F-P)/n) / ((F+P)/2). That is, 3 divided by 95 ($100 plus $90 divided by 2) equals .0315789.
The final value from Step 4, multiplied by 100 to get a percentage, is the yield to maturity. Yield to maturity = (C +(F-P)/n) / ((F+P)/2). In the example, the yield to maturity equals 3.158 percent. | {"url":"https://pocketsense.com/calculate-yield-maturity-loan-6879718.html","timestamp":"2024-11-05T19:43:47Z","content_type":"text/html","content_length":"178345","record_id":"<urn:uuid:ebc207b7-5c2e-4eac-af30-1ec30b3e1c50>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00683.warc.gz"} |
Planck Length
There are many indications that in quantum gravity there might exist a minimal observable distance on the order of the Planck length. The emergence of a minimal length is usually considered a
dynamical phenomenon, related to the fact that at the Planck scale there are violent fluctuations of the metric and even topology changes, as in Wheeler space-time foam/ from Quantum groups, gravity,
and the generalized uncertainty principle by Michele Maggiore
if one wants to probe an event in the length scale of Planck length with a photon, by the uncertainty principle, the particle has to have roughly Planck energy. Now according to general relativity, a
photon on such energy scales causes a gravitational collapse and therefore it does not yield any information of the event. The gravitational collapse is caused by the fact that the Schwarzschild
radius of a particle with Planck energy is approximately equal to the Planck length. Consequently, due to the uncertainty principle and the Schwarzschild radius, the very measurement of an event in
this length scale creates a black hole and no information about this event will emerge. The region with Planck-length of radius, therefore becomes in a sense noncontinuous, i.e. experimentally not
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | {"url":"https://physicstravelguide.com/advanced_notions/planck_length","timestamp":"2024-11-11T23:31:20Z","content_type":"text/html","content_length":"78497","record_id":"<urn:uuid:849989b0-0524-40e1-b0e1-fcec4c242735>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00767.warc.gz"} |
Gödel's Theorems from the History to the Demonstrations
One of the greatest discoveries of all times explained in a bunch of simple words
This is a collection of three threads published on Twitter, about Goedel's Theorems.
Today I want to start a potentially little series of threads about one of my biggest passions: Gödel's theorems. One of the greatest discoveries of all time that I'm sure will blow your mind! I'll
structure the series as follows.
First: The history that preceded the theorems
Then: The theorems. Explanation and some common misconceptions
And finally: Shallow overview of the demonstrations.
Let's start with the history that preceded Gödel's results. It is pretty much the history of math so let's see if we can comprise it in a thread.
The history
The study of math as science started in ancient Greece. Although math was applied earlier in other ancient civilizations, philosophers like Tales were the first that studied mathematical abstractions
like shapes without asking for the practical purpose of that study. Math started to have its very own questions that only could be answered inside Math itself.
Math became a science. Pythagoreans were a philosophy group that defended that the essence of the universe was numbers.
But their theory was based on the hypothesis that all numbers could be expressed as a fraction of two integer numbers. It seemed an acceptable supposition until...
A member of the Pythagorean school discovered some numbers that cannot be expressed that way.
For example sqrt(2) and sqrt(5). Those are presumably the first known irrational numbers. So all the Pythagorean theory was reduced to ashes. That was the first big crisis of math.
What was wrong with the Pythagorean theory? They assumed as true a proposition that is false inside that theory. You can prove some numbers cannot be expressed as a fraction using just the theorems,
elements, and operations of the theory.
Greeks realized they needed to change the way math was developed so far. And then Euclid wrote one of the most important books of the history of science: "The Elements". Almost all the geometry we
learned until high school was written by Euclid 2500 years earlier. But the best part was the method Euclid used to formulate his geometry.
"The Elements" showed the first example of an axiomatic theory. An axiomatic theory is built on top of very simple propositions that are assumed as true (axioms). Every other proposition needs to be
demonstrated from the axioms by following a set of rules that state how we can go from proposition A to proposition B.
The process of going from the set of axioms to some proposition A is called demonstration. When we demonstrate proposition A we say that A is a theorem in our theory. Let's see how history continues.
Euclid built his geometry on top of five axioms. The first four of them seemed pretty simple but the fifth was trickier. It's well known that Euclid himself tried to demonstrate the fifth axiom from
the other four.
More than 2000 years after the first publication of "The Elements", mathematicians were still figuring out how to remove the fifth axiom by demonstrating it. The result of those studies unveiled an
astonishing fact: Euclid just defined one of the many possible geometries🤯. If you change the fifth axiom a little bit, you can end up with a perfectly defined (although very crazy) geometry
As a side note, the General Theory of Relativity demonstrated that the geometry of our universe is non-Euclidean (it's one of the new crazy ones).
But this was a big problem! Mathematicians thought the fifth axiom could be demonstrated someday and they built the entire Math building on top of the robust and unique Euclidian geometry. This meant
the second deep crisis of Math. We need to rebuild the whole thing again!
But what do we mean by building the entire Math on top of something?
It is defining some axioms in a way that any mathematical proposition can be either proved or refuted by a demonstration process. Many of the greatest mathematicians of all time worked hard on that
problem. And then, in 1931, a 25 yo man destroyed that intention. Kurt Gödel proved that such a system was impossible to build. He proved that there are true propositions that cannot be proved in
some theories. He proved some things cannot be proved! 🤯🤯🤯
The Theorems
First things first. Let's talk about some important concepts.
We saw what an axiomatic theory is. Well, there are two properties that you'd like to have if you were an axiomatic theory:
Consistency and Completeness
Consistency: A consistent theory is one in which a proposition can be either true or false but not both. In other words, a theory without contradictions. Inconsistent theories are useless because you
can prove anything from them... Yeah, anything. There's a funny story of Bertrand Russell proving that if 2+2=5 then he was the Pope😆
Completeness: A complete theory is one in which all the true propositions are provable inside the theory. The doctoral thesis of Gödel was the demonstration of the completeness of the first-order
logic (with 23 yo).
Now we can continue with the history.
So, mathematicians were trying to build math on top of other ground different from geometry. They picked number theory (arithmetic, natural numbers) as the new foundations. The main reason: it was
axiomatized some years before. To give you an idea of the magnitude of Gödel discovery I'm going to mention some of the mathematicians trying to rebuild math:
• David Hilbert💪
• Bertrand Russel💥
• Ackerman❗
• John Von Neuman🔥😱🤯💫
They were trying to prove that the number theory was both Consistent and Complete. That way Math would be safe. The entire Math would be contradictions free and everything could be proven. It seemed
to be a matter of time before the proof arrived. Actually, some sub-theories of arithmetic were proven to be both consistent and complete. Gödel himself was working on that but he realized this:
Theorem 1: About incompleteness.
For any axiomatic theory that includes a certain part of arithmetic, if it is consistent then, it is incomplete
This means that all theories that include the number theory, contain true propositions that we'll never be able to prove inside that theory!
All the work of some of the greatest mathematicians of all times was in vain. Jon Von Neuman never worked again in Logic.
But for those who have some hope in their hearts. I remind you that there are two theorems.
Theorem 2: About consistency.
For any consistent theory that contains a certain part of arithmetic the consistency of the theory is not provable.
Precisely one of those true but not provable propositions is the consistency of the theory itself! So, 0 out of 2. No consistency and no completeness. Math can't be built that way. We have to live
with that. There are true propositions out there we'll never prove😔. End of story.
Now, let's talk about some misconceptions generated from the theorems. First I'd like you to note that both theorems say "with a certain amount of arithmetic".
We will be talking about that amount in the next section. For now, just suppose a theory containing the arithmetic.
Misconception number one:
Gödel said: for any sufficiently complex theory if it is consistent, then it is incomplete
❌ There is this idea of anything more complex than number theory has the conditions to apply Gödel's incompleteness theorem. Real numbers is a complete theory. And real numbers are at least as
complex as natural numbers. It is not about complexity. It is about how natural numbers are defined. That definition has the "poison".
Misconception number two:
The truth is unreachable for scientists
❌ Ok, some true propositions can't be proven in some theories. But maybe there are other alternative theories. Furthermore, experiments and observations are other methods to discover the truth about
our universe.
Misconception number three:
There is no philosophic system that can explain the universe
❌ The explanation of the universe doesn't have to do with natural numbers necessarily. And Gödel's theorems don't apply when there is no arithmetic in the theory.
Of course, there are lots more misconceptions about Gödel's results. But I'll stop here🥵. What bout some demonstrations?
Sketch of the demonstrations
Let's try to understand how Gödel was able to prove that there are not provable propositions and let's do it as smoothly as we can🙄.
By the end, we'll have proved one of the most mind-blowing results ever.
From now we denote with T an axiomatic theory that contains the number theory (theory of natural numbers).
The first Gödel's theorem states that: If T is consistent, then it is incomplete.
It means that there are true propositions in T that can't be proven.
Think about this proposition: "This proposition is not a theorem of T", or equivalently, "This proposition is not provable in T".
If the proposition was true, then it was a true but unprovable proposition in T. Problem solved. End of the thread. WAIT ⛔️
Well, there is a little problem with that last proposition. It talks about the theory T, but it is not a proposition in T. There's a difference between talking about something and being part of it.
Let's see how Goedel did it.
He created a code for every proposition and proof in T, in a way that every one of those propositions and proofs had its own and unique natural number that identifies it. That way, we can talk about
the theory from the language of numbers. That's what we call Gödel numeration. But remember that T contains the number theory. And that's the fact Gödel took advantage of.
Expressing propositions with numbers is a way to talk about T from within T itself. But how?
By saying: "N is not the code of any theorem in T", we are talking about T. But being the code of a theorem in T is an arithmetic property, and T contains the arithmetic.
So now we can state the proposition: "The code of this proposition is not the code of any theorem in T".
But that previous proposition can't still be formulated in T. That's a non-valid syntax. A proposition can't talk about itself in that way.
We need to achieve that in a more subtle way.
For that, we'll use the method proposed by Quine that is called "quinning". Let's see the following statement:
"yields a proposition with property P when appended to its own quotation." yields a proposition with property P when appended to its own quotation.
We can substitute the quotation for any other sentence. But when using the same sentence the statement starts to talk about itself!
Gödel proposed another method but it is trickier.
Let's denote the previous sentence with th letter G. So, if G is true then G has the property P and if G has the property P, then G is true. Let's do the last twist!
Let's make P = "its code is not the code of any theorem in T". Now, if G is false, then it is not provable, and that makes P true, and that makes G also true. So, T would be inconsistent. If T is
consistent, then G is true, then P is also true, and G is not provable!
We did it!
G is true but not provable in T!!!
What about the second theorem?
"The consistency of T is not provable in T"
That's a "direct" result from the previous demonstration!
If T is consistent, then G is true. And that's a demonstration of G, and G is not provable!
What we know is that if T is consistent then G is true but not provable. But (and precisely because of the previous sentence), the consistency of T is not provable either.
And that's it. We proved the Goedel theorems!
Of course, these demonstrations are not too accurate but they comprise the main ideas behind those mind-blowing results.
So, this is the end of the series😢. Gödel's theorems started a revolution. A revolution that resulted in the birth of Computer Science. But that's another history. Maybe I'll write my own version of
that history in the form of threads in the future😉.
Did you find this article valuable?
Support Jose Jorge Rodriguez by becoming a sponsor. Any amount is appreciated! | {"url":"https://blog.jayjaydev.com/godels-theorems-from-the-history-to-the-demonstrations","timestamp":"2024-11-09T07:56:57Z","content_type":"text/html","content_length":"197770","record_id":"<urn:uuid:48dbcfc0-e539-4d4b-964c-a62c21e22e23>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00470.warc.gz"} |
Topological gauge theories and group cohomology
We show that three dimensional Chern-Simons gauge theories with a compact gauge group G (not necessarily connected or simply connected) can be classified by the integer cohomology group H^4(BG, Z).
In a similar way, possible Wess-Zumino interactions of such a group G are classified by H^3(G, Z). The relation between three dimensional Chern-Simons gauge theory and two dimensional sigma models
involves a certain natural map from H^4(BG, Z) to H^3(G, Z). We generalize this correspondence to topological "spin" theories, which are defined on three manifolds with spin structure, and are
related to what might be called Z[2] graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the
special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.
All Science Journal Classification (ASJC) codes
• Statistical and Nonlinear Physics
• Mathematical Physics
Dive into the research topics of 'Topological gauge theories and group cohomology'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/topological-gauge-theories-and-group-cohomology","timestamp":"2024-11-13T22:10:45Z","content_type":"text/html","content_length":"48411","record_id":"<urn:uuid:b8c2948b-795d-4d03-b6fd-1e216c8249e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00451.warc.gz"} |
Maths - Factor Theorem
If $f(x)$ is a polynomial and $f(p) = 0$, what must be true?
$(x - p)$ is a factor of $f(x)$.
If $f(x)$ is a polynomial and $(x-p)$ is a factor of $f(x), what must bbe true?
\[f(p) = 0\]
If $f(x)$ is a polynomial and $f\left(\frac{b}{a}\right) = r$, what must be true?
Dividing $f(x)$ by $(ax - b)$ has remaineder $r$.
If $f(x)$ is a polynomial and dividing by $(ax - b)$ has remainder $r$, what must be true?
\[f\left(\frac{b}{a}\right) = r\]
Related posts | {"url":"https://ollybritton.com/notes/a-level/maths/topics/factor-theorem/","timestamp":"2024-11-05T10:25:51Z","content_type":"text/html","content_length":"503528","record_id":"<urn:uuid:f7582ab3-2ebe-4f35-82b8-565b6fb3512c>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00497.warc.gz"} |
Line-of-Sight Networks
Random geometric graphs have been one of the fundamental models for reasoning about wireless networks: one places n points at random in a region of the plane (typically a square or circle), and then
connects pairs of points by an edge if they are within a fixed distance of one another. In addition to giving rise to a range of basic theoretical questions, this class of random graphs has been a
central analytical tool in the wireless networking community. For many of the primary applications of wireless networks, however, the underlying environment has a large number of obstacles, and
communication can only take place among nodes when they are close in space and when they have line-of-sight access to one another — consider, for example, urban settings or large indoor environments.
In such domains, the standard model of random geometric graphs is not a good approximation of the true constraints, since it is not designed to capture the line-of-sight restrictions. Here we propose
a random-graph model component, as well an approximation question, in which we seek to connect a set of given nodes in such an environment by adding a small set of additional “relay” nodes. both
range limitations and line-of-sight constraints, and we prove asymptotically tight results for k-connectivity. Specifically, we consider points placed randomly on a grid (or torus), such that each
node can see up to a fixed distance along the row and column it belongs to. (We think of the rows and columns as “streets” and“avenues” among a regularly spaced array of obstructions.) Further, we
show that when the probability of node placement is a constant factor larger than the threshold for connectivity, near-shortest paths between pairs of nodes can be found, with high probability, by an
algorithm using only local information. In addition to analyzing connectivity and k-connectivity, we also study the emergence of a giant
Publisher Statement
All Rights Reserved | {"url":"https://kilthub.cmu.edu/articles/journal_contribution/Line-of-Sight_Networks/6706628/1","timestamp":"2024-11-10T02:51:04Z","content_type":"text/html","content_length":"130173","record_id":"<urn:uuid:f1152e88-1446-42f0-92ac-e626275fe5bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00592.warc.gz"} |
Miles (US survey) to Perch Converter
Enter Miles (US survey)
β Switch toPerch to Miles (US survey) Converter
How to use this Miles (US survey) to Perch Converter π €
Follow these steps to convert given length from the units of Miles (US survey) to the units of Perch.
1. Enter the input Miles (US survey) value in the text field.
2. The calculator converts the given Miles (US survey) into Perch in realtime β using the conversion formula, and displays under the Perch label. You do not need to click any button. If the input
changes, Perch value is re-calculated, just like that.
3. You may copy the resulting Perch value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Miles (US survey) to Perch?
The formula to convert given length from Miles (US survey) to Perch is:
Length[(Perch)] = Length[(Miles (US survey))] / 0.003124993501330624
Substitute the given value of length in miles (us survey), i.e., Length[(Miles (US survey))] in the above formula and simplify the right-hand side value. The resulting value is the length in perch,
i.e., Length[(Perch)].
Calculation will be done after you enter a valid input.
Consider that a cross-country road trip covers 2,000 miles (US survey).
Convert this distance from miles (US survey) to Perch.
The length in miles (us survey) is:
Length[(Miles (US survey))] = 2000
The formula to convert length from miles (us survey) to perch is:
Length[(Perch)] = Length[(Miles (US survey))] / 0.003124993501330624
Substitute given weight Length[(Miles (US survey))] = 2000 in the above formula.
Length[(Perch)] = 2000 / 0.003124993501330624
Length[(Perch)] = 640001.3309
Final Answer:
Therefore, 2000 mi is equal to 640001.3309 perch.
The length is 640001.3309 perch, in perch.
Consider that a long-distance bike ride spans 100 miles (US survey).
Convert this distance from miles (US survey) to Perch.
The length in miles (us survey) is:
Length[(Miles (US survey))] = 100
The formula to convert length from miles (us survey) to perch is:
Length[(Perch)] = Length[(Miles (US survey))] / 0.003124993501330624
Substitute given weight Length[(Miles (US survey))] = 100 in the above formula.
Length[(Perch)] = 100 / 0.003124993501330624
Length[(Perch)] = 32000.0665
Final Answer:
Therefore, 100 mi is equal to 32000.0665 perch.
The length is 32000.0665 perch, in perch.
Miles (US survey) to Perch Conversion Table
The following table gives some of the most used conversions from Miles (US survey) to Perch.
Miles (US survey) (mi) Perch (perch)
0 mi 0 perch
1 mi 320.0007 perch
2 mi 640.0013 perch
3 mi 960.002 perch
4 mi 1280.0027 perch
5 mi 1600.0033 perch
6 mi 1920.004 perch
7 mi 2240.0047 perch
8 mi 2560.0053 perch
9 mi 2880.006 perch
10 mi 3200.0067 perch
20 mi 6400.0133 perch
50 mi 16000.0333 perch
100 mi 32000.0665 perch
1000 mi 320000.6655 perch
10000 mi 3200006.6547 perch
100000 mi 32000066.5465 perch
Miles (US survey)
A mile (US survey) is a unit of length used primarily in land surveying in the United States. One US survey mile is defined as exactly 5,280 feet or approximately 1,609.347 meters.
The US survey mile is used in land measurement and mapping, differing slightly from the international mile, which is defined as exactly 1,609.344 meters.
US survey miles are utilized primarily in the United States for property measurement, land surveying, and other applications requiring precise distance measurement. This unit ensures consistency and
accuracy in surveying and mapping activities.
A perch is a unit of length used primarily in land measurement and surveying. One perch is equivalent to 16.5 feet or approximately 5.0292 meters.
The perch is defined as 16.5 feet, which is the same length as a rod or a pole, and is used in various practical applications such as land measurement and construction.
Perches are used in land surveying, property measurement, and agricultural contexts. The unit provides a convenient measurement for shorter distances and has historical significance in land
measurement practices.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Miles (US survey) to Perch in Length?
The formula to convert Miles (US survey) to Perch in Length is:
Miles (US survey) / 0.003124993501330624
2. Is this tool free or paid?
This Length conversion tool, which converts Miles (US survey) to Perch, is completely free to use.
3. How do I convert Length from Miles (US survey) to Perch?
To convert Length from Miles (US survey) to Perch, you can use the following formula:
Miles (US survey) / 0.003124993501330624
For example, if you have a value in Miles (US survey), you substitute that value in place of Miles (US survey) in the above formula, and solve the mathematical expression to get the equivalent value
in Perch. | {"url":"https://convertonline.org/unit/?convert=miles_us_survey-perch","timestamp":"2024-11-10T12:54:28Z","content_type":"text/html","content_length":"91569","record_id":"<urn:uuid:3f2a0474-872c-4c6a-824f-c9e853c08a6a>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00462.warc.gz"} |
VRWorks - 360 Video - Multiband Blending
Multiband Blending
ROIs and Laplacian Image Generation
The first step of our multiband blending implementation is the computation of the Region of Interest (ROI) in the output buffer corresponding to each input camera feed. The next step is to project
each input into the corresponding ROI. This computation takes the camera parameters and the desired resolution of the output into account. Thereafter, the Laplacian pyramid is generated for each of
the projected inputs.
The projected frames are finally blended at each level using masks and the final output is synthesized.
Mask Generation
Masks determine the path that the seams will follow. The masks are compute at the base level and a Gaussian pyramid of this mask is generated to blend at each level. The width of the region to be
blended at each level increases at subsequent down-sampled levels.
Number of Levels
All the pyramids used have the same number of levels. The current implementation computes the number of levels from the output buffer resolution such that at the lowest level the smallest surface
dimension is no less than 16 pixels (capped at 8 levels).
Multiband blending is very sensitive to the type of filter used, both for downsampling and upsampling. The repeated upsampling and downsampling required can cause minor artifacts in the smaller
levels to be greatly amplified as the pyramid is synthesized.
CUDA Streams and Multi-GPU Scaling
Our Multiband implementation maps very well to CUDA Streams and it is a good candidate for multi-GPU scaling. Most of the processing is performed on a per-camera basis and only the last blending +
synthesizing stages require inputs from each of the camera pipelines.
Once the ROIs are generated, the first steps are to project the images into the base level of their image pyramids (purple), generate Gaussian and Laplacian pyramids, blend and synthesize. Note that
with this approach there is no need for synchronization until the blending stage, which means the CUDA streams can be executed on different GPUs. | {"url":"https://developer.nvidia.com/vrworks/vrworks-360video/multiband-blending","timestamp":"2024-11-09T10:49:27Z","content_type":"text/html","content_length":"41573","record_id":"<urn:uuid:9d6f21d5-5795-4750-8b7e-a0f07080ce42>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00784.warc.gz"} |
What is the shape of a pentagon?What is the shape of a pentagon?
Sorry, you do not have permission to ask a question, You must login to ask a question.
What is the shape of a pentagon?
A pentagon is a two-dimensional geometric shape that has five straight sides and five angles. Each angle of a regular pentagon (all sides and angles are equal) measures 108 degrees, and the sum
of all angles of a pentagon is 540 degrees. The shape of a pentagon can be described as a closed polygon wRead more
A pentagon is a two-dimensional geometric shape that has five straight sides and five angles. Each angle of a regular pentagon (all sides and angles are equal) measures 108 degrees, and the sum
of all angles of a pentagon is 540 degrees.
The shape of a pentagon can be described as a closed polygon with five sides and five angles, where each side connects two consecutive vertices or corners of the shape. A regular pentagon has
rotational symmetry of order 5, which means that it looks the same after a rotation of 72 degrees, 144 degrees, 216 degrees, or 288 degrees.
See less
A pentagon is a polygon with five sides and five angles. It can have a variety of shapes depending on the length and orientation of its sides and angles. However, a regular pentagon, which is a
polygon with five sides of equal length and five interior angles of equal measure (108 degrees), has a disRead more
A pentagon is a polygon with five sides and five angles. It can have a variety of shapes depending on the length and orientation of its sides and angles. However, a regular pentagon, which is a
polygon with five sides of equal length and five interior angles of equal measure (108 degrees), has a distinctive shape that looks like a five-pointed star. It has five straight sides that
connect to form five interior angles, each of which measures 108 degrees.
See less
You must login to add an answer. | {"url":"https://expertcivil.com/question/what-is-the-shape-of-a-pentagon/","timestamp":"2024-11-05T12:10:00Z","content_type":"text/html","content_length":"273182","record_id":"<urn:uuid:fdcd4113-ea22-4c14-a65e-6f5cb96ae94c>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00177.warc.gz"} |
Fall 2024 Department Colloquium
The talks will typically take place on Tuesdays at 4:00-5:00pm in Adel Room 164. Please contact Nandor Sieben if you would like to give a talk or have a question about the colloquium.
Tuesday 8/27 at 4:00-4:50
Short organizational meeting
Tuesday 9/3 at 4:00-4:50
Speaker: Shannon Guerrero Title: Teacher-authored culturally responsive mathematics curriculum: Lessons from the Diné Institute for Navajo Nation Educators
Abstract: This presentation shares research conducted with a long-term professional development program for teachers in Navajo-serving schools, with a focus on the mathematics curriculum developed by
teachers in this program. We address the research question: To what extent and in what ways do teachers in DINÉ math seminars develop curriculum units that evidence culturally responsive principles
and mathematics education best practices? We analyze teacher-authored mathematics curriculum from two years of this program and share key lessons learned in supporting teachers from
Indigenous-serving schools to engage culturally responsive principles in their math teaching and learning.
Tuesday 9/10 at 4:00-4:50
Speaker: Jeffrey Covington Title: Curve fitting and Gaussian process regression
Abstract: Gaussian process regression, also known as Kriging, is a powerful curve fitting and statistical regression tool which has grown in popularity over the past few years. I’ve found it to be a
powerful tool in my own research in spatial and dynamical modeling. This talk will introduce (kernel-based) Gaussian processes and their applications, relate them to curve fitting through Gaussian
process regression, and explore the power and pitfalls of the methods.
Tuesday 9/17 at 4:00-4:50
Speaker: Michele Torielli Title: Hyperplane arrangements and signed graphs
Abstract: In this talk, we will recall the notion of hyperplane arrangement and discuss when an arrangement is free. We will then discuss their connection with simple graphs and finally, generalize
this connection to the notion of signed graphs.
Tuesday 9/24 at 4:00-4:50
Speaker: Roy St. Laurent Title: Diversions: My Favorite Logic Puzzles from Nikoli
Abstract: This is not a research talk. I will introduce three or four of my favorite Nikoli puzzles, with an opportunity for you to solve examples of them yourself. Paper copies will be provided,
just bring a pencil (or pen).
Nikoli is a Japanese publisher that specializes in culture-independent puzzles – especially pencil/paper logic puzzles played on a rectangular grid. I have been hooked on Nikoli puzzles since being
introduced to them by now-retired Mathematics professor Dr. Steve Wilson. The English names of some of my favorites are Akari, Fillomino, Hashiwokakero, Heyawake, Hitori, Kakuro, Nurikabe,
Slitherlink, Yajilin.
While this isn’t a research talk, a couple of interesting questions come to mind that will be raised during the talk.
Tuesday 10/1 at 4:00-4:50
Site visit week. No colloquium.
Tuesday 10/8 at 4:00-4:50
No colloquium.
Tuesday 10/15 at 4:00-4:50
Speaker: Peter Vadasz Title: Magnetostrophic Flow and Electromagnetic Columns in Magneto-Fluid Dynamics and short updates on my previous presentations of Quantum Mechanics and Newtonian Gravitational
Abstract: An analogy between magneto-fluid dynamics (MFD/MHD) and geostrophic flow in a rotating frame of reference including the existence of electromagnetic columns identical to Taylor-Proudman
columns is identified and demonstrated theoretically. The latter occurs in the limit of large values of a dimensionless group representing the magnetic field number. Such conditions are shown to be
easily satisfied in reality. Consequently, the electromagnetic fluid flow subject to these conditions is two dimensional and the streamlines are being shown to be identical to the pressure lines in
complete analogy to rotating geostrophic flows. An experimental setup is suggested to confirm the theoretical results experimentally.
Tuesday 10/22 at 4:00-4:50
Speaker: Rachel A Neville Title: A Fractal Dimension for Measures via Persistent Homology
Abstract: A fractal is a geometric object that displays self-similarity on all scales. There are common examples, such as the Sierpinsky Triangle or the Koch snowflake. The “fractal dimension”
quantifies the complexity of the fractal. While fractal dimensions are most classically defined for a space, there are a variety of fractal dimension definitions for a measure, including the
Hausdorff or packing dimension of a measure. In this talk, I will give a brief introduction to fractal dimension and describe how persistent homology can be used in order to define a family of
fractal dimensions. I will end with a discussion of some work done with John Leland on a persistence based distribution test.
Tuesday 10/29 at 4:00-4:50
Speaker: Sam Harris Title: The Max 3-Cut problem for graphs
Abstract: A famous problem in graph theory is the Max Cut problem: given an undirected graph G, determine whether there is a partition of the vertices of G into two subsets so that every edge of G is
“cut” by the partition. While this problem is NP-hard, the non-commutative version is surprisingly solvable in polynomial time, and even can be used to approximate the (classical) maximum cut of a
graph. In this talk, we’ll look at some of the history of both problems, and also look at recent work on the Max 3-Cut problem and its noncommutative variants.
Tuesday 11/5 at 4:00-4:50
Speaker: Dana Ernst Title: Pattern-avoiding Cayley permutations via combinatorial species
Abstract: Any permutation of $n$ may be written in one-line notation as a sequence of entries representing the result of applying the permutation to the identity $12\cdots n$. If $p$ and $q$ are two
permutations, then $p$ is said to contain $q$ as a pattern if some subsequence of the entries of $p$ has the same relative order as all of the entries of $q$. If $p$ does not contain a pattern $q$,
then $p$ is said to avoid $q$. One of the first notable results in the field of permutation patterns was obtained by MacMahon in 1915 when he proved that the ubiquitous Catalan numbers count the
123-avoiding permutations. The study of permutation patterns began receiving focused attention following Knuth’s introduction of stack-sorting in 1968. Knuth proved that a permutation can be sorted
by a stack if and only if it avoids the pattern 231 and that the Catalan numbers also enumerate the stack-sortable permutations. In this talk, we study pattern avoidance in the context of Cayley
permutations, which were introduced by Mor and Fraenkel in 1983. A Cayley permutation is a finite sequence of positive integers that include at least one copy of each integer between one and its
maximum value. When possible we will take a combinatorial species-first approach to enumerating Cayley permutations that avoid patterns of length two, pairs of patterns of length two, patterns of
length three, and pairs of patterns of length three with the goal of providing species, exponential generating series, and counting formulas. We also include several conjectures and open problems.
Tuesday 11/12 at 4:00-4:50
Speaker: Jim Swift Title:
Tuesday 11/19 at 4:00-4:50
Speaker: Minah Kim Title:
Tuesday 11/26 at 4:00-4:50
Speaker: Adeolu Taiwo Title:
Spring 25
Tuesday 1/21 at 4:00-4:50
Speaker: Annie and Gina Title:
Tuesday 3/8 at 4:00-4:50
Speaker: Andrew Schultz (Wellesley College) Title:
Tuesday 4/1 at 4:00-4:50
Speaker: Joe Polman (CU Boulder) CSTL STEM Education speaker series Title: | {"url":"https://naumathstat.github.io/seminars/colloquium/","timestamp":"2024-11-04T14:52:19Z","content_type":"application/xhtml+xml","content_length":"13279","record_id":"<urn:uuid:0f95b50e-73c4-4c49-bf95-3e00d4e52ef7>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00546.warc.gz"} |
33 Fantastic Times Tables Games For Kids - Teaching Expertise
Multiplication can be a challenging step in your kiddos’ math journey. However, with these captivating math games, multiplication will become a walk in the park! This fantastic collection of awesome
resources is guaranteed to have your learners begging for more times tables fun. Whether it’s card games, online activities, or even crafting – you’re sure to find the perfect fit to engage your
budding mathematicians in a multi-faceted learning experience!
1. Rock, Paper, Times Tables
Here’s an activity that adds an entertaining spin to the classic game of Rock, Paper, Scissors! Invite your learners to partner up and on the third lowering of their fists have them each reveal a
number. Then, challenge them to multiply the numbers to see who can call out the correct answer first.
Learn More: Childhood 101
2. Times Tables Matching Game & Book
This interactive game is perfect for hands-on learning! Simply invite your kiddos to read through this colorful times table book to brush up on their skills. Then, allow them to use the cards
provided to match their answers with the corresponding sum.
Learn More: YouTube
3. Multiplication Bingo
Multiplication Bingo is sure to be a hit in your next math class! Start by presenting your students with multiplication sums and then prompt them to calculate the sum, before placing a token on the
correct answer displayed on their Bingo card.
Learn More: Amazon
4. Multiplication Flash Cards
These flash cards are a wonderful resource that your little brainiacs can use to polish their multiplication skills! Invite them to draw cards from the deck and then challenge them to solve each
problem as quickly as possible.
Learn More: Amazon
5. Online Times Tables Practice
Fine-tune your kids’ times tables with a plethora of online games! Simply set aside time during math class where they can take a break and play these educational games. It’s a fantastic way to
solidify their knowledge and develop a strong mathematical foundation.
Learn More: Coko Games
6. Times Tales
Treat your littles to a visually rich learning experience! Invite them to eagerly listen and observe as you have them watch these captivating math videos. It’s a great way to diversify their learning
experience as you bring this digital teacher in for the day.
Learn More: Time Tales
7. Dice Games for Multiplication Mastery
Here’s another way that your kiddos can use dice in their math lessons! Challenge them to a game of multiplication checkers as they use dice to formulate their sum. Then, prompt them to multiply the
numbers together before placing a token on the correct answer.
Learn More: Walmart
8. Printable Multiplication Board Games
This adorable multiplication game is guaranteed to have your learners begging for one more go! To play this game, invite them to roll some dice and then have them move their game piece according to
the number they roll. Once they’ve landed their piece, prompt them to correctly solve the problem – if they’re correct they’ll stay on that spot, but if they get the answer wrong they’ll have to move
back to their previous position.
Learn More: Math Easily
9. Multiplication Spinner Game
Combine crafts and math for a memorable learning experience! Simply prompt your kids to color in a number on their number grid. Then, have them spin their spinners and multiply the two numbers once
the spinner stops.
Learn More: The Mailbox
10. Pirate Quest
Take your little brainiacs on a swashbuckling adventure with this captivating activity! Prompt them to roll a pair of dice and then have them use their numbers as coordinates to pinpoint the correct
square. Once they’ve found their square, challenge them to add up their coordinates and multiply the answer by the value in its allocated block.
Learn More: Childhood 101
11. Scoop it! Times Tables Game
This treat of a game provides a sweet twist to math lessons. Task your learners with cutting out these colorful ice cream scoops and then prompt them to place their scoops in the correct outline to
form their times table treat!
Learn More: Childhood 101
12. Times Tables Songs
Invite your little ones to a math-inspired, musical interlude! Encourage them to listen carefully as you play catchy times table songs for them to sing along to.
Learn More: YouTube
13. Make Math Stick
Make your math lessons stick with this ingenious activity! Simply start by formulating a secret message by assigning numbers to letters. Then, let your kids get code cracking as they solve
multiplication sums to unlock your secret message.
Learn More: No Time for Flashcards
14. Multiplication Flowers
Invite your littles to create magical multiplication flowers in this creative activity! Simply provide them with these easy-to-use worksheets and encourage them to follow the petals as they fill in
the answers to each multiplication problem.
Learn More: Etsy
15. Multiplication Patterns and Need for Speed
Put a spin on mundane math lessons with a multiples-inspired game of Speed! Challenge your kiddos with playing each variation of this classic game by first having them pair up. If they choose the “5
Speed” they’ll have to place down cards that are multiples of 5 as fast as they can to see who can empty their deck first.
Learn More: YouTube
16. Times Tables Magic
Teach your little ones multiplication through literature. Invite them to get comfortable as you read them times table tales – the perfect way to conclude your multiplication lessons!
Learn More: Amazon
17. Multiplication Splat!
This fun-filled game is a great way to develop your learners’ fast-paced thinking. Simply start by laying cards down that contain multiplication problems. Then, call out an answer and prompt your
kiddos to flip over the corresponding problem card as fast as they can before shouting “Splat!”.
Learn More: Lakeshore Learning
18. Spiral Multiplication
Gather your students as you take them on a whirlwind of a learning experience! Simply provide them with these topsy-turvy multiplication worksheets and task them with answering each question.
Learn More: Dad Worksheets
19. Multiplication Race Track
It’s a sprint to the finish in this Multiplication Race Track game. Challenge your students with solving multiplication problems, where each correct answer moves their cars closer to the checkered
Learn More: Toy Theater
20. Gone Fishing Multiplication
This engaging game is perfect for hands-on learning. Start by laying out stapled paper fish numbers in front of your kiddos. Then, invite them to solve various multiplication sums before prompting
them to catch the correct fish using a magnetic fishing rod.
Learn More: TES
21. Times Tables Puzzle Pieces
Here’s a fun-filled puzzle game with a mathematical twist! Simply begin by cutting out these multiplication table templates according to their colors. You’ll then task your kids with putting the
puzzle back together so that each row and column matches the product in the middle.
Learn More: This Reading Mama
22. Multiplication Fact Flip
Spark your littles’ competitive spirit with these times table flip cards. Encourage them to solve each sum displayed on the card before flipping over the end piece to see whether they’ve calculated
the correct number. Have them play on their own or add an element of friendly fun by challenging them to see who can solve each card the fastest!
Learn More: Inspired Elementary
23. Space Mission Multiplication
This cool game is perfect for treating your learners to some tech time. Invite them to embark on a space mission as they solve multiplication sums to propel their spaceship forward. It’s an
adventurous approach to mastering their multiplication skills!
Learn More: ABC Ya
24. Muffin Tin Math
Invite your little ones to an interactive math experience! Simply provide them with Muffin Tin Math supplies before placing multiplication sums in each muffin hole. Then, engage them in solving the
sums by having them place the equivalent number of muffin toys in each hole.
Learn More: YouTube
25. Multiplication Fact Jenga
Add a mathematical twist to the classic Game of Jenga! Task your kiddos with setting up a Jenga tower with blocks that’ve been labeled with multiplication sums. Allow them to then take turns to pull
out a block before prompting them to solve its sum. If they provide the correct answer they’ll keep their block, but if they provide an incorrect answer they’ll have to place their block on top of
the tower.
Learn More: Multiplication
26. Multiplication Obstacle Course
Reinforce your littles’ multiplication knowledge in this dynamic game! Simply set up an obstacle course in your classroom where each obstacle has a corresponding multiplication problem. Kick off the
fun by encouraging them to solve each problem before they proceed to the next obstacle.
Learn More: Teachers Pay Teachers
27. Sticky Note Multiplication Hunt
Send your kids on a quest to uncover hidden mathematical knowledge! Begin by hiding sticky notes with multiplication problems all over your classroom. Then, challenge them to find each note before
solving its displayed problem.
Learn More: STEM Camp
28. Roll and Multiply Challenge
Multiply your learners’ knowledge with this interactive game! Task them with rolling two dice and then prompt them to multiply the two numbers to calculate the correct answer.
Learn More: Childhood 101
29. Multiplication Memory Grid
Combine your students’ memory and math skills in one fun-filled game! Simply start by placing cards face down, where one side has multiplication problems and the other side has the answers. Encourage
them to flip over two cards at a time to try and match each sum with its corresponding answer.
Learn More: 3 Dinosaurs
30. Times Tables Shootout
Here’s another engaging activity that your little ones can enjoy on their digital devices. In this game, they’ll have to solve multiplication sums and then shoot down the correct answer. It’s a great
way to incorporate technology into their learning experience!
Learn More: Times Tables
31. Multiplication Beat the Clock
Put your kiddos’ knowledge to the test in this fast-paced game. Start by handing out multiplication worksheets before starting a timer and challenging them to solve each equation as fast as they can!
Learn More: Twinkl
32. Times Tables Hopscotch
Take learning outdoors in this vibrant activity! Simply draw a hopscotch grid on the ground, but instead of numbers, write down multiplication problems in each square. Then, prompt your kids to hop
and skip through the maze as they call out each correct answer as they go.
Learn More: Coffee Cups and Crayons
33. Multiplication Mystery Bags
Spice up your little learners’ math lessons with these sensory-rich mystery bags! To do this, you’ll fill bags with objects such as marbles or coins, and then label each bag with a multiplication
problem. Invite your learners to then solve the problem and check their answer by counting the items in the bag.
Learn More: Teachers Pay Teachers | {"url":"https://www.teachingexpertise.com/math/times-tables-games/","timestamp":"2024-11-11T20:04:08Z","content_type":"text/html","content_length":"88452","record_id":"<urn:uuid:8005e7c1-ab9c-4c29-a5be-0476120f67c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00571.warc.gz"} |
R Bar Chart Multiple Series 2024 - Multiplication Chart Printable
R Bar Chart Multiple Series
R Bar Chart Multiple Series – You can create a Multiplication Graph Nightclub by marking the columns. The remaining column must say “1” and signify the amount increased by one. About the right hand
part in the dinner table, brand the columns as “2, 8, 6 and 4 and 9”. R Bar Chart Multiple Series.
Tips to discover the 9 times multiplication desk
Discovering the 9 periods multiplication desk is just not easy. There are several ways to memorize it, but counting down is one of the easiest. In this particular technique, you set the hands around
the desk and amount your hands one at a time from a single to ten. Fold your 7th finger so that you can begin to see the ones and tens onto it. Then matter the amount of hands on the left and proper
of your folded away finger.
When discovering the kitchen table, young children can be intimidated by bigger figures. The reason being incorporating bigger phone numbers repeatedly becomes a task. However, you can exploit the
hidden patterns to make learning the nine times table easy. One of the ways would be to write the 9 periods kitchen table over a cheat page, study it deafening, or process composing it lower
regularly. This process can certainly make the dinner table more unique.
Habits to look for on a multiplication graph or chart
Multiplication graph bars are ideal for memorizing multiplication specifics. You can get the item of two numbers by looking at the rows and columns from the multiplication graph. As an example, a
line which is all twos as well as a row that’s all eights should meet up with at 56. Patterns to consider with a multiplication chart bar are exactly like those who work in a multiplication desk.
A pattern to search for on a multiplication graph or chart may be the distributive property. This property might be noticed in every columns. For example, a product or service x two is equivalent to
5 various (occasions) c. This same residence is applicable to any column; the sum of two columns means value of one other column. As a result, an odd variety periods a much number is an even number.
A similar is applicable to these products of two odd figures.
Setting up a multiplication chart from memory
Building a multiplication graph or chart from memory space might help little ones discover the diverse amounts inside the times dining tables. This simple exercising will allow your child to remember
the figures and see how you can multiply them, that helps them later on once they find out more challenging math concepts. For any exciting and good way to commit to memory the figures, you are able
to organize coloured buttons in order that each one corresponds to particular instances dinner table number. Make sure to tag every single row “1” and “” to help you quickly identify which variety is
available initially.
As soon as children have learned the multiplication chart club from storage, they ought to commit them selves on the process. This is why it is far better to utilize a worksheet rather than a classic
laptop to practice. Colorful and computer animated figure templates can entice the feelings of your kids. Before they move on to the next step, let them color every correct answer. Then, exhibit the
graph or chart within their examine area or bedrooms to serve as a reminder.
Utilizing a multiplication chart in everyday life
A multiplication graph demonstrates how to multiply amounts, a person to 10. Furthermore, it shows the product of two phone numbers. It can be beneficial in everyday life, like when dividing funds or
collecting details on people. The subsequent are among the ways you can use a multiplication chart. Utilize them to assist your child comprehend the strategy. We have now pointed out just some of the
most frequent purposes of multiplication tables.
Use a multiplication chart to help you your kids learn to decrease fractions. The key is to follow the denominator and numerator left. Using this method, they may realize that a small fraction like 4
/6 could be lowered to a fraction of 2/3. Multiplication charts are particularly helpful for young children mainly because they enable them to recognize amount styles. You can find Cost-free computer
models of multiplication chart pubs on the internet.
Gallery of R Bar Chart Multiple Series
Highcharts Multiple Series In Highcharter R Stacked Barchart Stack
Multiple Bar Charts In R Data Tricks
Microsoft Excel Custom Error Bars On Bar Chart With Multiple Series
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Constructing small-bias sets from algebraic-geometric codes
We give an explicit construction of an ∈-biased set over k bits of size O (k/∈^2log(1/∈))^5/4. This improves upon previous explicit constructions when ∈ is roughly (ignoring logarithmic factors) in
the range [k^-1.5, k^-0.5]. The construction builds on an algebraic-geometric code. However, unlike previous constructions we use low-degree divisors whose degree is significantly smaller than the
genus. Studying the limits of our technique, we arrive at a hypothesis that if true implies the existence of ∈-biased sets with parameters nearly matching the lower bound, and in particular giving
binary error correcting codes beating the Gilbert-Varshamov bound.
Original language English
Title of host publication Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009
Pages 191-197
Number of pages 7
State Published - 2009
Event 50th Annual Symposium on Foundations of Computer Science, FOCS 2009 - Atlanta, GA, United States
Duration: 25 Oct 2009 → 27 Oct 2009
Publication series
Name Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print) 0272-5428
Conference 50th Annual Symposium on Foundations of Computer Science, FOCS 2009
Country/Territory United States
City Atlanta, GA
Period 25/10/09 → 27/10/09
• Algebraic-geometric codes
• Small-bias sets
Dive into the research topics of 'Constructing small-bias sets from algebraic-geometric codes'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/constructing-small-bias-sets-from-algebraic-geometric-codes","timestamp":"2024-11-15T04:23:32Z","content_type":"text/html","content_length":"49966","record_id":"<urn:uuid:86ce262c-8c93-4e61-9eb5-efe0f8a8ca52>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00367.warc.gz"} |
Using Three-Peak Analysis for Projected Exclusion Limits in Vector Dark Matter Detection | HackerNoon
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University.
Table of Links
2 Calculating the Stochastic Wave Vector Dark Matter Signal
3 Statistical Analysis and 3.1 Signal Likelihood
4 Application to Accelerometer Studies
4.1 Recasting Generalised Limits onto B − L Dark Matter
6 Conclusions, Acknowledgments, and References
A Equipartition between Longitudinal and Transverse Modes
B Derivation of Marginal Likelihood with Stochastic Field Amplitude
D The Case of the Gradient of a Scalar
3 Statistical Analysis
We now consider the projected exclusion limits that a generic experiment would be able to set using our three-peak analysis. To do this, we use a series of likelihood-ratio tests.
3.1 Signal Likelihood
For our likelihood, we follow a hybrid frequentist-Bayesian approach, defining a marginalized likelihood in which all nuisance parameters are integrated out. In our case, these are the random
Rayleigh parameters, α, and random uniform DM phases, φ. Such a hybrid approach has already been used in the context of ultralight bosonic dark matter [42, 79]. Our work differs from Ref. [79] since
they focused on an axion-like signal as opposed to that from vector DM. It goes beyond Ref. [42] since they did not consider the peaks arising from the rotation of the Earth in their analysis.
The full likelihood over all frequency space is then given by the product of the likelihoods in each frequency bin,
where pi represents the excess power density in the ith frequency bin, p is the full data vector, and the product runs over all Nbins frequency bins. Ultimately, since our signal only manifests in
three bins, it suffices for us to consider only those bins that could potentially contain a signal, and we may ignore all other bins. We can express the likelihood in this way because each bin is
statistically uncorrelated, as we show in Appendix C. This is in contrast with the analysis performed in Ref. [45], where a similar study was conducted in the case of the gradient of a scalar in the
time domain. There, a complicated covariance matrix had to be computed to account for correlations in the signal at different times. In Fourier space, these covariances disappear. The power of
performing this analysis in the frequency domain is thus not only that the signal is contained within a small number of bins, but also that these bins are statistically independent, which allows us
to treat the statistics in a significantly simpler way.
Crucially, once the latitude of the experiment, ϕ, is fixed, the likelihood depends on the product of all experimental variables via the dimensionless parameter β. This means that we can set a more
holistic limit that is independent of the specifics of an experiment. Once the form of A (which depends on both the experiment and the DM model), the observation time Tobs, and the noise profile σ
are known, the ensuing limit on β can be recast to one on the model parameters of interest. This makes our analysis, both the results and overall logic, as generally useful as possible. | {"url":"https://hackernoon.com/using-three-peak-analysis-for-projected-exclusion-limits-in-vector-dark-matter-detection","timestamp":"2024-11-07T14:02:31Z","content_type":"text/html","content_length":"223136","record_id":"<urn:uuid:19cda652-362e-4961-b7e1-db711ec1e4ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00522.warc.gz"} |
Writing in Geology to Reduce Math Anxiety
Students in GLY 4866: Computational Geology at the University of South Florida learn to solve mathematical problems in a geologic context. Quantitative literacy – a fundamental set of skills and
habits of mind – is essential to geologists in any of a variety of occupations, and this course helps prepare students for those careers.
Based on suggestions from prior interviews with alumni, we introduced a detailed reading and writing assignment into the course for fall 2016. Students submitted a general written statement about how
the felt about math, and then read The Math Instinct: Why you're a Mathematical Genius (along with Lobsters, Fish, Cats, and Dogs) by Keith Devlin over the course of the semester. After each of the
13 chapters, students gave a short written weekly response, and at the conclusion of the semester they submitted a 1-2 page paper outlining their feelings about the book and how their attitude toward
math had changed over the semester (if applicable).
Although the assignment was given without the intent of publication, after grade submission for the semester, a post hoc record review was approved by the USF IRB due to the minimal risk to students
once the data was de-identified.
Analysis by grounded theory coding indicates significant gains in student attitudes toward their own math confidence, with similar reductions in stated math anxiety. While the relatively small sample
size (n=28) and ad hoc nature of the study preclude making generalizations beyond this semester, results are very promising. Similar assignments are planned for future semesters, and future studies
include more formal attitude surveys and post-class interviews.
Conference Earth Educators Rendezvous Oral Presentation Abstracts
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Theory of Sound Field
Code Completion Credits Range
XP02TZP ZK 4 2P
Course guarantor:
The aim of this course is deeper understanding the fundamentals of physical acoustics. The continuity equation, Euler and Navier-Stokes equations and the energy equation are derived from the
prime laws of fluid dynamics. These equations are utilized for derivation of a linear wave equation under the acoustical approximation; its special solutions are discussed. General solutions of
the wave equation and Helmholtz equation are formulated using the integrals of Kirchhoff-Helmholtz and Rayleigh. Using these integrals, some problems of acoustic radiation and diffraction are
studied. Problem of the acoustic field description is further developed using the methods of Fourier acoustics.
Foundations of physics, foundations of vector analysis, Founations of Fourier transform.
Syllabus of lectures:
1.Recapitulation: differential operators, Gauss law, 1D wave equation, method of characteristics, d'Alembert solution of wave equation.
2.3D wave equation, planar, spherical and cylindrical wave.
3.Acoustic particle, Lagrange and Euler description of fluid motion, material derivative, continuity equation.
4.Euler and Navier-Stokes equation, viscosity, rotational and irrotational field, velocity potential.
5.Energy equation, equation of state.
6.Acoustic approximation of the fluid-dynamics equations, wave equation for acoustic pressure and velocity potential.
7.Acoustic intensity, acoustic energy density, planar wave, specific acoustic impedance, representation using phasors.
8.Acoustic field generated by a pulsating sphere, radiated power, simple and volume source.
9.Homogeneous and inhomogeneous Helmholtz equation, free-field Green's function.
10.Helmholtz-Kirchhoff integral, application for a volume source, Sommerfeld radiation condition.
11.Rayleigh integral, far-field approximation, far-field of a circular piston, directivity.
12.Acoustic field at the axis of a circular piston, near-field, transition to far-field, Rayleigh distance.
13.Fourier transform of transient sound field, circular aperture diffraction.
14.Fourier acoustics: description of sound radiation, evanescent wave, acoustical holography.
Syllabus of tutorials:
Study Objective:
Study materials:
1.D. T. Blackstock, Fundamentals of Physical Acoustics, Wiley-Interscience, 2000.
2.P. M. Morse, K. Uno Ingard, Theoretical Acoustics, Princeton University Press, 1987.
3.E. G. Williams, Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography, Academic Press, 1999.
4.J. W. Goodman, Introduction to Fourier Optics, Roberts and Company Publishers, 2004.
5.D. J. Griffiths, Introduction to Electrodynamics, Addison Wesley, 1999.
Further information:
Time-table for winter semester 2024/2025:
Červenka M.
Wed 14:30–16:00
(lecture parallel1)
Time-table for summer semester 2024/2025:
Time-table is not available yet
The course is a part of the following study plans: | {"url":"https://bilakniha.cvut.cz/en/predmet11508304.html","timestamp":"2024-11-08T06:16:57Z","content_type":"text/html","content_length":"23554","record_id":"<urn:uuid:b4e7bfbb-20f1-4013-ba16-069dc8f69ee7>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00427.warc.gz"} |
Learn To Use XLOOKUP [2 Full Examples!]
It’s time we introduce you to the new member of the modern function family of Excel – the XLOOKUP function.
XLOOKUP is the successor to the conventional VLOOKUP, INDEX & MATCH, LOOKUP, and the HLOOKUP function.
You must have heard the meme floating on the internet these days that says.
There are two types of people in the Excel world; the type that masters VLOOKUP & XLOOKUP. And the type that hovers about the ones who master VLOOKUP and XLOOKUP!
Which type are you? If you have been ‘type two’ all this time, this article is sure to pave your way to ‘type one’!
XLOOKUP Explained
Through XLOOKUP, Excel has offered a one-in-all solution to all the VLOOKUP problems of Excel users.
It allows you to look up data, both horizontally and vertically, to the above and the left.
Users can define multiple criteria and can seek a whole row or column of data as the return value instead of a single value only.
This powerful successor of the VLOOKUP function is all that Excel users have been pleading for the last three decades.
We teach using XLOOKUP on our MS Excel intermediate course given how easy it is to use.
The syntax of the XLOOKUP function looks as below.
= XLOOKUP (lookup_value, lookup_array, return_array, [if_not_found], [match_mode], [search_mode])
A little too long to decipher? Let’s break it down into individual arguments to make better sense of it.
• Lookup_Value – the first argument represents the value to look for in a given dataset.
• Lookup_array – the second argument refers to the data range where the value is to be looked for.
• Return_Array – the third argument refers to the array/range from where the value that is to be returned.
• If_not_found – the fourth argument is an optional one that refers to the value to be returned if the desired value is not found. If the lookup value doesn’t exist in the data set and the
‘if_not_found’ argument is omitted, Excel returns the #N/A error.
• Match_Mode – the fifth argument is also optional. It refers to the match type to be performed by Excel:
□ Exact Match: Under this option, Excel looks out for an exact match of the lookup value and returns the #N/A error if not found. To set the match mode to the exact match, set the fifth
argument to ‘0’.
□ Exact or next smaller: Under this option, Excel looks out for an exact match of the lookup value. If an exact match is not found, Excel returns the next smaller value to the lookup value. Set
up ‘-1’ as the fifth argument to put the match mode to ‘Exact or next smaller.
□ Exact or next larger: Under this option, Excel looks out for exact matches of the lookup value. If an exact match is not found, Excel returns the next larger value to the lookup value. Set up
‘1’ as the fifth argument to put the match mode to ‘Exact or next smaller’.
□ Wildcard character: Setting up the fifth argument to ‘2’ puts the match mode to the wildcard character match.
If the fifth argument is omitted, Excel, by default, sets it to 0; the exact match mode.
• Search_Mode – the sixth and the last argument to the XLOOKUP formula is an optional one. It guides the direction of search and can be set to four modes:
□ Search First to Last: Set this argument to ‘0’ or leave it omitted to perform the search for the lookup value from first to last.
□ Search Last to First: Set this argument to ‘-1’ to perform the search for the lookup value from last to the first.
□ Binary Search on Ascending Data: Setting this argument to ‘2’ performs a binary search on ascendingly sorted data.
□ Binary Search on Descending Data: Setting this argument to ‘-2’ performs a binary search on data sorted in descending order.
Return Value
The XLOOKUP function returns the value from the return array that matches the lookup value and the supporting specified criteria.
Functions Library
To find the XLOOKUP function from the Functions Library, go as follows.
Formulas > Functions Library > Lookup & Reference > XLOOKUP
Version of Excel
The XLOOKUP function is an advanced function of Excel that is only available to the users of Office 365. Users of the previous Excel versions from 2010 to 2019 will not be able to access the XLOOKUP
Pro Tip!
Is there some way out you can use the modern-day XLOOKUP function without being an Office 365 user? Yes, through Excel For Web!
Log on to your Microsoft OneDrive account and launch Excel online to have access to the latest functions of Excel.
OneDrive is a great tool, you can even use it to recover Unsaved Files in Excel.
How is XLOOKUP of use?
It took Microsoft around three decades to come up with a function as wholesome and robust as the XLOOKUP.
This function is a successor of the VLOOKUP function and is primarily designed to offer a solution to all the problem areas where VLOOKUP and other functions failed.
Why would you want to use XLOOKUP? Consider the data below.
The data consists of employee names alongside the number of items sold by each of them.
To the right side of the image, is a key that grades the performance of employees based on the number of units sold by them.
If you want to grade the performance of all the employees as ‘Good’, ‘Average’ or ‘Bad’ – could this be done with a single function?
Yes, the XLOOKUP function. See below.
Seems like magic? There are many more ways how XLOOKUP can ease your job.
• Use XLOOKUP to fetch a value, array, or range from a given data set.
• Use XLOOKUP to fill grids based on a given criterion in the blink of an eye.
• Use XLOOKUP to create an Excel dashboard by selecting only the data that you need.
XLOOKUP V/s. VLOOKUP
What differentiates XLOOKUP from its predecessor, the VLOOKUP? Below are the four main areas where XLOOKUP outstands the contemporary VLOOKUP function.
1. Vertical and Horizontal Lookup
The VLOOKUP function is designed to perform a vertical lookup only. Take a quick look at the example below to find the difference between a horizontal and a vertical lookup.
With VLOOKUP, you can find the any sector’s revenue for a particular state (say State B’s fiscal sector revenue) by performing VLOOKUP.
However, you cannot find any state’s revenue for a particular sector.
This is because it requires a horizontal lookup of values which the VLOOKUP function fails to offer.
Using the XLOOKUP function, you can look up the data both ways.
This is essential a way to compare your columns, but lets you do it with more than two!
2. Search Mode & Match Mode
The XLOOKUP function offers two optional arguments that help users to define the match mode.
For instance, do you want Excel to return an exact match from the lookup array or an approximate match (higher or smaller values)?
Similarly, search mode allows users to tell if the lookup value needs to be searched for, from the left of the lookup array or the right.
VLOOKUP fails to offer both the above-said features, making it rigid to use.
3. If Not Found Argument
The very common #N/A error of VLOOKUP comes to the screen when Excel fails to find an exact match of what you’re looking for.
While XLOOKUP does no different, it allows users to replace the nasty looking #N/A with any value/dialogue of their choice. You might even choose to leave it vacant.
4. Lookup Array and Return Array can be separately identified
Take a quick look at the data below:
The above data is not in a row but in two stacked tables.
XLOOKUP can search for a value from such scattered data if the dimensions are compatible. See below.
However, VLOOKUP cannot handle such datasets.
For VLOOKUP to work on such data, the data must be arranged together in a row as below.
XLOOKUP Example 1 – Basic Exact Match
It’s time we delve into examples that demonstrate the uses of the XLOOKUP function.
The first example in this article covers the basic function of the XLOOKUP function – seeking an exact match.
The image below contains data for the employees of an organization.
For each employee, the available details include the first name and the employee ID; however, the last name is missing.
Another dataset, as shown below, includes the employee IDs and the respective last names of employees.
Now what, copy-pasting?
Do note that the sequence of employee IDs in the second table do not match the sequence in the first table.
Copy-pasting the correct last name against each first name might take you ages.
Using the XLOOKUP function under the exact match mode, bringing these two lists together is only about a minute.
Begin writing the XLOOKUP function as follows.
= XLOOKUP (B2, $H$2:$H$10, $I$2:$I$10)
• The first argument is set to B2, which contains the lookup value – the employee ID against which the last name is required.
• The second argument includes a reference to the lookup_array where the lookup_value is to be looked for. Column H contains the employee ID against the last names and is selected as the
• The third argument consists of the return_array, from where the value is to be returned. We want the last name of employees which are situated in Column I.
• The fourth argument of ‘if not found’ is omitted.
• The fifth argument is omitted as we want an exact match of the employee ID. Excel would have set it to ‘0’ by default.
• The sixth argument is omitted as we want Excel to perform the search from first to last. Upon being omitted, Excel by default sets it to ‘0’.
And you’re good to go. Press ‘Enter’ to see the following results.
Drag and drop to yield the same results for the remaining employees.
XLOOKUP Example 2 – Multiple Values
One main factor that distinguishes XLOOKUP from VLOOKUP is its ability to return not only a single value, but a range of values.
This feature of the XLOOKUP function was much demanded and is of great use. Earlier, to have multiple values returned from a lookup range, multiple VLOOKUP functions were to be combined.
But the very advanced XLOOKUP function can do this in a single formula. See below how.
Let’s expand the example stipulated above to include further detail about employees, as shown below.
The above data now also constitutes the date of joining and leaving of each employee alongside their designation.
Can we fetch the last name, joining and leaving date, and designation of each employee to the first table all at once? With the XLOOKUP function, yes.
Compose the XLOOKUP function as follows:
= XLOOKUP (B2, $H$2:$H$10, $I$2:$L$10)
While everything remains the same as the above formula, what has changed?
We have only changed the third argument, return_array from I2:I10 to I2:L10. The range I2:L10 includes four columns.
For each employee code (the lookup value) Excel would then return the value of all the corresponding columns from Column I to Column L.
Time to see this in action? See Below.
For each employee code, the XLOOKUP function returns multiple cell values from all the specified corresponding columns.
Don’t stop only there. Drag and drop the above function to the entire list to have your data sorted in only a second.
Point to Ponder: How is this different from the VLOOKUP function?
Can VLOOKUP not perform the above operation? The precise answer to this is, that a single VLOOKUP cannot. To fetch more than one value like the above example, you need to operate multiple VLOOKUP
For instance, to fetch all the details for Employee Code BX12360, write the VLOOKUP function as follows:
Pay close attention to the last argument that specifies the ‘Column’ number of the table from where we want the return value. Here it is set up as ‘2’.
Excel would therefore return the value from the second column (Column I) of the specified table array of H2:L10, which contains Employee last names only.
What about the other values? The VLOOKUP functions need to be set up for each of them again and again with the last argument changing to the return value column number.
With the XLOOKUP, you can target as many return value columns as desired – that’s when you know how badly XLOOKUP was needed.
XLOOKUP Troubleshooting
There are a variety of errors the XLOOKUP function might pose while you continue to twitch it in different ways. Once you know what each error has to say, resolving it shouldn’t take you a great deal
of effort.
#REF Error
The very annoying reference error is set forth by the XLOOKUP function when it is being operated in two or more workbooks at the same time.
This might be the case when the lookup array and the return array reside in one workbook, and the XLOOKUP is employed in another workbook.
That’s no big deal as Excel can handle that effortlessly.
However, for this to be done, both the workbooks must be simultaneously launched in the background.
If either of the workbooks is shut, Excel would end up returning the #REF error.
Simply open both the workbooks to get rid of the said error.
If the error still persists, use the formula auditing tools to help see why!
#VALUE Error
The #VALUE error posed by Excel is an indication that the lookup array and the return array specified by you are not compatible in terms of dimensions.
For example, if your lookup array has 7 rows but your return array is only 5 rows long, Excel would give back a #VALUE Error. See below:
Excel fails to return the PCT Code for Kiwi because of incompatible dimensions of both arrays. Change the return array to make it parallel to the lookup array to see the #VALUE error vanish away.
The return array is changed from B2:B6 to B2:B8 – compatible with the lookup array of A2:A8.
Absolute References
This is not a problem with Excel but a problem with the drag-and-drop function. See the example below.
Here we have set the formula for Cell C2 by setting the Marks key as the lookup array and the remarks key as the return array.
The formula runs perfectly well for the first cell. However, drag and drop it down to the entire list, and the results would distort as follows.
The #N/A error tells the lookup value is not found.
This is because when dragged and dropped, Excel automatically updates the cell references, and the lookup array and the return array have changed.
Turn the lookup array and the return array into absolute references by navigating the cursor to each cell reference in the formula bar and pressing the F4 Key. See how the results change.
In the fast-paced world of today, smart and swift functions like the XLOOKUP are a necessity. Going through the above article and understanding the functionality of each example therein can help you
master the XLOOKUP function with sheer ease. | {"url":"https://www.acuitytraining.co.uk/news-tips/master-the-xlookup-function-in-excel/","timestamp":"2024-11-10T08:21:35Z","content_type":"text/html","content_length":"385835","record_id":"<urn:uuid:3224a88a-6b16-4863-b06c-79ded93c2dc0>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00479.warc.gz"} |
Matrix Nullity Calculator - Calculator Wow
Matrix Nullity Calculator
In the vast landscape of linear algebra, matrices serve as the foundation for myriad mathematical operations. The Matrix Nullity Calculator steps onto the stage, offering a key to unlocking the
intricacies of nullity – a concept pivotal in understanding the structure and properties of matrices. This article embarks on a journey to explore the significance of the Matrix Nullity Calculator,
its application, and answers to common queries that arise in the realm of linear algebra.
The Importance of Matrix Nullity
Nullity, in the context of matrices, unveils essential insights into their structure. The Matrix Nullity Calculator becomes a compass, guiding mathematicians and scientists through the intricate
relationships within a matrix. Understanding nullity is crucial for comprehending the dimensions of the null space, providing valuable information about the solutions to linear equations and the
overall behavior of the matrix.
How to Use the Matrix Nullity Calculator
Despite its seemingly abstract nature, the Matrix Nullity Calculator is remarkably user-friendly. Input the total number of columns and the rank of the matrix into the designated fields. A simple
click on the ‘Calculate Nullity’ button reveals the nullity of the matrix – a numerical representation that contributes to a deeper understanding of its mathematical properties. This tool empowers
students, researchers, and professionals to explore the realms of linear algebra with precision and clarity.
10 FAQs and Answers
1. What is nullity in linear algebra?
• Answer: Nullity represents the dimension of the null space of a matrix. It provides insights into the linear relationships and dependencies within the columns of the matrix.
2. Why is nullity important in matrix theory?
• Answer: Nullity is crucial for understanding the rank-nullity theorem, which establishes a relationship between the rank, nullity, and the total number of columns in a matrix.
3. Can the Matrix Nullity Calculator handle rectangular matrices?
• Answer: Yes, the calculator is versatile and applicable to both square and rectangular matrices, accommodating diverse dimensions.
4. How is nullity related to the solutions of linear equations?
• Answer: Nullity is directly connected to the number of free variables in the solutions of a system of linear equations, providing information about the system’s consistency and solutions.
5. Does nullity change if columns are multiplied by a scalar?
• Answer: Multiplying columns by a scalar does not affect the nullity. Nullity remains unchanged as long as the linear relationships within the matrix columns remain the same.
6. Can nullity be greater than the number of columns?
• Answer: No, nullity cannot exceed the total number of columns. It is bounded by the dimensions of the matrix and provides valuable insights into its linear independence.
7. Is nullity related to the invertibility of a matrix?
• Answer: Yes, nullity is inversely related to the invertibility of a matrix. A matrix is invertible if and only if its nullity is zero.
8. How does the Matrix Nullity Calculator handle non-square matrices?
• Answer: The calculator accommodates non-square matrices seamlessly, providing accurate nullity calculations based on the given dimensions.
9. Is nullity affected by row operations on a matrix?
• Answer: Nullity remains unaffected by elementary row operations. It is a property intrinsic to the matrix structure and not altered by row transformations.
10. Can nullity be negative?
• Answer: No, nullity is a non-negative integer representing the dimension of the null space. It cannot be negative.
As we conclude our expedition into the realm of linear algebra, the Matrix Nullity Calculator stands as a beacon, illuminating the path to a deeper understanding of matrices and their inherent
structures. Nullity, once an abstract concept, transforms into a tangible value, offering profound insights into the linear relationships within matrices. In a world driven by mathematical precision,
this calculator becomes an invaluable tool for students, educators, and researchers alike, unraveling the complexities and unveiling the elegance of linear algebra. The Matrix Nullity Calculator
invites us to explore the beauty of mathematical abstraction and empowers us to decipher the language of matrices with newfound clarity and confidence. | {"url":"https://calculatorwow.com/matrix-nullity-calculator/","timestamp":"2024-11-13T08:31:06Z","content_type":"text/html","content_length":"65654","record_id":"<urn:uuid:865d9563-0e00-440e-b0e3-5d711f7d8b89>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00134.warc.gz"} |
Barbie has taught math and worked in student support at community colleges and other postsecondary schools for twenty years. Most of her students were math-insecure. Barbie is also a parent and
regular school volunteer, using her math teaching skills alongside local teachers to support and bolster the students and families as much as she can.
Some posts contain affiliate links. If you purchase through these links, I may earn a small commission at no additional cost to you. You can read more about how I choose affiliates and products at my
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Resources - Introductory Calculus Courses
For students in quantitative course, Smith offers a variety of resources, free of charge.
• The Spinelli Center for Quantitative Learning is staffed every week day with tutors for a variety of subjects including mathematics. There are lots of tables and chairs for you to work at, and if
you find you need a little boost when working on a problem, the Q-tutors have many areas of expertise, from algebra to calculus and beyond.
• The Calculus Counselor also works out of the Spinelli Center, and he holds both drop-in hours and individual appointments available here.
• The Math TAs hold drop-in tutoring hours in the evening, Sunday – Thursday from 7-9pm in the Math Forum, Burton Hall 3rd floor. These tutors work closely with the faculty teaching the
introductory courses and are prepared to handle questions from all the introductory courses. Certain tutors are even designated for MTH 153: Discrete Mathematics and MTH211: Linear Algebra and
MTH212: Multivariable Calculus on specific nights of the week.
• The Department of Mathematics & Statistics also runs Calculus Training Groups (CTG), pairing interested students with a peer mentor to meet for one hour once a week to spend extra time on MTH111:
Calculus 1 and MTH112: Calculus 2. Participants are expected to commit to regularly scheduled meeting times and occasional review sessions and extra workshops on study skills. | {"url":"https://www.science.smith.edu/calculus/resources/","timestamp":"2024-11-11T21:07:32Z","content_type":"text/html","content_length":"39060","record_id":"<urn:uuid:7b35121a-b303-4edb-b7e3-58649c816143>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00106.warc.gz"} |
How To Prepare For A Pre-Algebra Exam In High School And Calculus And Algebra In College - Hire Someone To Take My Exam | Pay Me To DO Your Online Examination
Pay Someone to Do University Exam to Improve Your Score is a common practice amongst students. Pre-algebra and Calculus in 30 days give students with the basic math skills and confidence they need to
excel on the Calculus and Algebra exam, giving them a solid foundation for all other mathematics subjects.
If you have a poor grade and have little or no prior knowledge of the subject, a good grasp of basic concepts will help you understand concepts better and gain an edge over your peers. This can also
help you learn new techniques and concepts that can help you get the most out of your exams. Whether you pay someone to do the exam for you or not, however, there are some things you can do to
improve your score.
The first thing you can do to help yourself is to practice your skills and knowledge of pre-algebra and calculus by doing a little research online. You can find a number of sample exam questions on
websites which offer free calculators to try and practice your skills. It is best to choose a site that offers multiple choice, short answer, and full range tests. Try to choose tests that test many
different concepts.
You may also want to start taking college level courses. This can give you a good base of knowledge for the Calculus and Algebra exam. It should help you learn more advanced concepts such as graph
theory and discrete math more easily.
In addition to taking pre-algebra, you should also consider taking Calculus and Algebra at the same time. By doing this, you will be able to cover both subjects in one semester, thus allowing you to
have one less test to take. If you take the calculus first, you will probably find yourself with more questions.
If you are not quite sure whether you should take Calculus or Algebra before you take the college-level course, consider trying it together and see which is easier for you to follow and understand.
Taking Calculus first can help you make up your mind but if Algebra will give you more challenges and questions.
Pre-algebra and Calculus at the same time can also help you work out which subjects are easier to master. {, as well as which ones are harder. When you have one test to study for, it will be much
easier to focus on the harder subjects. and work your way through the difficult topics faster. If the subject topics you are unsure about are easier than the others, then the quicker you can work
through them the better.
There are a number of resources available to help you prepare for the pre-algebra and calculus exams in high school. If you want to do your own research online, you can use books and magazines, but
you can also consider taking a Calculus refresher course or enrolling in a college course so that you can refresh your knowledge with extra practice tests.
Once you get a feel for the topics you will need to know about Calculus and Algebra, you can then start applying it to a real-life situation. For example, if you are taking the college-level course
and have a couple of minutes to spare on the exam, try to take a Calculus or Algebra quiz that requires a simple problem and a direct answer from the professor.
Instead of just guessing how to solve the problem, try to solve it in a step by step manner and use a practice problem that you can use when taking the exam. Doing this will help you to memorize the
concepts of Calculus and Algebra.
After you finish your exam, make sure you take a review quiz to assess your results. You may have questions that are not directly related to the subject matter. and you should try to think of answers
that are easy and apply to the real-world situations you will be faced with during the exam.
You can even consider taking the pre-algebra and Calculus courses in high school and then taking the college-level course in college and then taking the exam again in college and taking the quiz once
more to refresh your knowledge. Taking the pre-calculus exams as an elective course in college can also help you learn new concepts that you could not know before. Once you finish the pre-algebra and
Calculus requirements in high school and the Calculus requirements in college, you will have a strong foundation on which to build.
How To Prepare For A Pre-Algebra Exam In High School And Calculus And Algebra In College | {"url":"https://hireforexamination.com/how-to-prepare-for-a-pre-algebra-exam-in-high-school-and-calculus-and-algebra-in-college","timestamp":"2024-11-03T09:36:46Z","content_type":"text/html","content_length":"88294","record_id":"<urn:uuid:35f4df00-7c9e-41a0-8594-a5fc4d962b4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00241.warc.gz"} |
Jonathan Payne (University of Sheffield (PhD)): Publications - PhilPeople
• 640
Neo-Fregeans such as Bob Hale and Crispin Wright seek a foundation of mathematics based on abstraction principles. These are sentences involving a relation called the abstraction relation. It is
usually assumed that abstraction relations must be equivalence relations, so reflexive, symmetric and transitive. In this article I argue that abstraction relations need not be reflexive. I
furthermore give an application of non-reflexive abstraction relations to restricted abstraction principles
• 578
Neo-Fregean approaches to set theory, following Frege, have it that sets are the extensions of concepts, where concepts are the values of second-order variables. The idea is that, given a
second-order entity $X$, there may be an object $\varepsilon X$, which is the extension of X. Other writers have also claimed a similar relationship between second-order logic and set theory,
where sets arise from pluralities. This paper considers two interpretations of second-order logic—as being either extens…
Read more
• 518
Harold Hodes in [1] introduces an extension of first-order modal logic featuring a backtracking operator, and provides a possible worlds semantics, according to which the operator is a kind of
device for ‘world travel’; he does not provide a proof theory. In this paper, I provide a natural deduction system for modal logic featuring this operator, and argue that the system can be
motivated in terms of a reading of the backtracking operator whereby it serves to indicate modal scope. I prove soundn…
Read more
• 23
As the title of this book suggests, the main focal point is the so-called Yablo Paradox,11First formulated by Stephen Yablo. an infinitary, apparently non-circular paradox involving truth, w... | {"url":"https://philpeople.org/profiles/jonathan-payne/publications?order=viewings","timestamp":"2024-11-11T05:09:41Z","content_type":"text/html","content_length":"35564","record_id":"<urn:uuid:4c75271d-1044-4a02-8d8b-aa1208ac85fd>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00411.warc.gz"} |
Understanding radio polarimetry
Up: Abstract
Astron. Astrophys. Suppl. Ser. 143, 515-534
Understanding radio polarimetry
IV. The full-coherency analogue of scalar self-calibration: Self-alignment, dynamic range and polarimetric fidelity
J.P. Hamaker
Send offprint request: J.P. Hamaker,
e-mail: hamaker@nfra.nl
Netherlands Foundation for Research in Astronomy, Postbus 2, 7990 AA Dwingeloo, The Netherlands
Received September 23, 1998; accepted January 13, 2000
Paper II of this series studied the calibration process in mostly qualitative terms. In developing the underlying mathematics this paper completes that analysis and extends it in several directions.
It exploits the analogy between scalar and matrix algebras to reformulate the self-calibration method in terms of
In the same way that scalar selfcal leaves the brightness scale undefined, matrix selfcal gives rise to a more complicated indeterminacy. The calibration is far from complete: self-alignment
describes more properly what is actually achieved. The true brightness is misrepresented in the image obtained by an unknown brightness-scale factor (as in scalar selfcal) and an undefined
poldistortion of the Stokes brightness. The latter is the product of a polrotation of the polvector (Q,U,V) and a polconversion between unpolarized and polarized brightness. The relation of these
concepts to conventional "quasi-scalar'' calibration methods is demonstrated.
Like scalar selfcal, matrix self-alignment is shown to suppress spatial scattering of brightness in the image, which is a condition for attaining high dynamic range. Poldistortion of the brightness
is an in-place transformation, but must be controlled in order to obtain polarimetric fidelity. The theory is applied to reinterpret the quasi-scalar methods of polarimetry including those of Paper
II, and to prove two major new assertions: (a.) An instrument calibrated on an unpolarized calibrator measures the degree of polarization correctly regardless of poldistortion; (b.) Under the usual a
priori assumptions, a heterogeneous instrument (i.e. one with unequal feeds) can be completely calibrated without requiring a phase-difference measurement.
Key words: instrumentation: interferometers -- instrumentation: polarimeters -- methods: analytical -- methods: observational -- techniques: interferometric -- techniques: polarimetric
Up: Abstract
Copyright The European Southern Observatory (ESO) | {"url":"https://aas.aanda.org/articles/aas/full/2000/09/h1201/h1201.html","timestamp":"2024-11-10T18:37:27Z","content_type":"text/html","content_length":"7344","record_id":"<urn:uuid:39c86ae4-0790-4a46-8120-2e1cb2723f6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00192.warc.gz"} |
How do I find the integral
intx/(x-6)dx ? | HIX Tutor
How do I find the integral #intx/(x-6)dx# ?
Answer 1
#int x/{x-6}dx=x+6ln|x-6|+C#
Let us look at some details.
#int x/{x-6}dx#
by the sunstitution #u=x-6#, #Rightarrow x=u+6# #Rightarrow dx=du#
#=int {u+6}/u du#
by splitting the integrand,
#=int (1+6/u) du#
by putting #u=x-6# back in,
by letting #C=C_1-6#,
I hope that this was helpful.
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Answer 2
To integrate ( \int \frac{x}{x-6} , dx ), we perform polynomial long division to simplify the integrand.
Step 1: Perform polynomial long division.
[ \begin{array}{c|c} x & 1 \ \hline x-6 & x \end{array} ]
Step 2: Multiply the divisor ( x - 6 ) by ( 1 ) to obtain ( x - 6 ).
Step 3: Subtract ( x - 6 ) from ( x ) to get ( 6 ).
Step 4: Since the degree of the remainder (which is ( 6 )) is greater than or equal to the degree of the divisor ( x - 6 ), we rewrite ( \frac{x}{x-6} ) as ( 1 + \frac{6}{x-6} ).
Now, we can integrate ( 1 + \frac{6}{x-6} ) separately.
[ \int \frac{x}{x-6} , dx = \int \left(1 + \frac{6}{x-6}\right) , dx ]
Integrating ( 1 ) with respect to ( x ) gives ( x ), and integrating ( \frac{6}{x-6} ) with respect to ( x ) gives ( 6\ln|x-6| + C ).
[ \int \frac{x}{x-6} , dx = x + 6\ln|x-6| + C ]
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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ball mill motor
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CGPA To Percentage Calculator
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CGPA To Percentage Calculator Online Tool
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• Candidates have to select their respective grading scales.
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CGPA to Percentage Calculator for Engineering:
In engineering, your CGPA is determined by the range of grades you receive. For instance, if you receive an A in a subject, your grade point would be 9 and your marks will fall between 90 and 95.
Credit points are used in the calculation of the CGPA.
With the aid of an illustration, let's learn how to compute CGPA in engineering:
Consider that your institution gives each of physics and mathematics a total of 4 credit points. Imagine the grade points and the range of grades are as follows:
• 90-10 is a 10 and it is O
• 80-89 is a 9 and it is A
• 70-79 is an 8 and it is B
... and so on. Imagine that your math score is 96 and your physics score is 86. As a result, you will receive a 9 for physics and a 10 for mathematics.
CGPA to Percentage Calculator Anna University:
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Formula for CGPA to Percentage Calculator Anna University: CGPA*10.
The CGPA to Percentage Calculator out of 10 is used for the calculation at Anna University.
Letter grade and Grade points for Anna University:
Here we have tabulated the Letter grade and Grade points for Anna University.
│ Letter grade and Grade points for Anna University │
│ Letter Grade │ Grade points │
│O (Outstanding) │10 │
│A+ (Excellent) │9 │
│A (Very Good) │8 │
│B+ (Good) │7 │
│B (Average) │6 │
│RA (Reappearance Registration) │0 │
│SA (Shortage of Attendance) │0 │
│W (Withdrawal) │0 │
CGPA to Percentage Calculator VTU
CGPA Full Form
The VTU uses a grading system in which marks are converted to grades. The semester grade point average (SGPA) and cumulative grade point average are then announced for each semester's results (CGPA).
CGPA to Percentage Calculator VTU Conversion Formula: [CGPA-0.75]*10
CGPA to Percentage Calculator CBSE:
Every year, thousands of applicants take the CBSE test. Students experience tension and worry after exams because of the outcomes. In 2011, CBSE stopped issuing grades according to the CGPA grading
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some students may feel the need to double-check.
One of the most widely used and trustworthy methods of CGPA to Percentage Calculator CBSE is:
For average CGPA:
A sum of all the Grade Points (GP) of each subject must be divided by 5.
For example, suppose an examinee receives the following scores on each subject: Subject 1: 8, Subject 2: 9, Subject 3: 9, Subject 4: 9.5, and Subject 5: 7.
The total GPs will therefore be 8+9+9+9+9.5+7=42.5.
By dividing 42.5 by 5, we obtain 8.5, which represents the total CGPA.
CGPA to Percentage Calculator Out of 10:
Most universities and school board marks have been calculated out of 10. Also, if aspirants are applying for any government exam, they need to calculate their percentage of marks into CGPA. For that
here we have given the CGPA to Percentage Calculator out of 10 for your reference.
Percentage To CGPA Calculator Online Tool
Here we have added the details of the percentage to the CGPA calculator steps for the candidate's reference. The candidate should enter the percentage. Follow the steps and use the percentage to CGPA
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Candidates can follow the steps provided below using the percentage to CGPA Converter.
• Candidates have to select their respective grading scales.
• Candidates should enter their respective percentages obtained in the Input Score tab.
• Then, click the Calculate button.
• The CGPA will be displayed below.
Check the respective CGPA obtained by you by using the free percentage to CGPA calculator online tool. So, follow the steps mentioned above to convert the percentage to CGPA easily and accurately.
Download Last 6 Months Current Affairs
CGPA To Percentage Calculator Online Tool - FAQs
Here we have given common FAQs regarding CGPA to Percentage Calculator Online Tool for your reference.
Q. What is the full form of CGPA?
A. The full form of CGPA is Cumulative Grade Point Average.
Q. How can I convert the CGPA to percentage accurately?
A. The CGPA can be converted into percentages based on certain formulas. Here we have added the CGPA to percentage Conversion Online tool for easy access for the candidates.
General Awareness Smart Analysis
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matrix inverse of $I + A$ where $A$ is skew-symmetric
I am looking for a formula or result for $$(I + A)^{-1}$$ where $I$ is the identity matrix and $A$ is skew-symmetric ($A^T = -A$). I have spent a lot of time looking online and through various
sources but can't find anything.
1 answer
I found a solution for the case I am considering, which is when $A$ is a $3{\times}3$ cross product matrix of vector $a \in \mathbb{R}^3$: $$A = \begin{bmatrix} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\
-a_2 & a_1 & 0 \end{bmatrix}, ~~~~~ a = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} .$$ Using the explicit equation for the inverse of a $3{\times}3$ matrix, the inverse can be derived as: $$( I
+ A )^{-1} = \frac{1}{1+\|a\|^2} ((1 + \|a\|^2)I - A + A^2) .$$ I'm not sure if this can be extended to more general cases.
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Tarski–Grothendieck set theory
Tarski–Grothendieck set theory (TG, named after mathematicians
non-conservative extension of
Zermelo–Fraenkel set theory
(ZFC) and is distinguished from other axiomatic set theories by the inclusion of
Tarski's axiom
, which states that for each set there is a
Grothendieck universe
it belongs to (see below). Tarski's axiom implies the existence of
inaccessible cardinals
, providing a richer
than ZFC. For example, adding this axiom supports
category theory
formal verification of proofs
Tarski–Grothendieck set theory starts with conventional Zermelo–Fraenkel set theory and then adds “Tarski's axiom”. We will use the axioms, definitions, and notation of Mizar to describe it. Mizar's
basic objects and processes are fully formal; they are described informally below. First, let us assume that:
TG includes the following axioms, which are conventional because they are also part of
It is Tarski's axiom that distinguishes TG from other axiomatic set theories. Tarski's axiom also implies the axioms of
More formally:
${\displaystyle \forall x\exists y[x\in y\land \forall z\in y(z\subseteq y\land {\mathcal {P}}(z)\subseteq y\land {\mathcal {P}}(z)\in y)\land \forall z\in {\mathcal {P}}(y)(eg (z\approx y)\to z\
in y)]}$
where “${\displaystyle {\mathcal {P}}(x)}$” denotes the power class of x and “${\displaystyle \approx }$” denotes equinumerosity. What Tarski's axiom states (in the vernacular) is that for each set $
{\displaystyle x}$ there is a Grothendieck universe it belongs to.
That ${\displaystyle y}$ looks much like a “universal set” for ${\displaystyle x}$ – it not only has as members the powerset of ${\displaystyle x}$, and all subsets of ${\displaystyle x}$, it also
has the powerset of that powerset and so on – its members are closed under the operations of taking powerset or taking a subset. It's like a “universal set” except that of course it is not a member
of itself and is not a set of all sets. That's the guaranteed Grothendieck universe it belongs to. And then any such ${\displaystyle y}$ is itself a member of an even larger “almost universal set”
and so on. It's one of the strong cardinality axioms guaranteeing vastly more sets than one normally assumes to exist.
Implementation in the Mizar system
The Mizar language, underlying the implementation of TG and providing its logical syntax, is typed and the types are assumed to be non-empty. Hence, the theory is implicitly taken to be non-empty.
The existence axioms, e.g. the existence of the unordered pair, is also implemented indirectly by the definition of term constructors.
The system includes equality, the membership predicate and the following standard definitions:
Implementation in Metamath
The Metamath system supports arbitrary higher-order logics, but it is typically used with the "set.mm" definitions of axioms. The ax-groth axiom adds Tarski's axiom, which in Metamath is defined as
⊢ ∃y(x ∈ y ∧ ∀z ∈ y (∀w(w ⊆ z → w ∈ y) ∧ ∃w ∈ y ∀v(v ⊆ z → v ∈ w)) ∧ ∀z(z ⊆ y → (z ≈ y ∨ z ∈ y)))
See also
External links | {"url":"https://findatwiki.com/Tarski%E2%80%93Grothendieck_set_theory","timestamp":"2024-11-08T17:11:33Z","content_type":"text/html","content_length":"141243","record_id":"<urn:uuid:bff740ab-b0a5-4c1a-8fd3-d60684778a84>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00326.warc.gz"} |
Calculus/Centre of mass - Wikibooks, open books for an open world
← Work Calculus Kinematics →
Centre of mass
r → G = ∑ k = 1 n m k r k → ∑ k = 1 n m k {\displaystyle {\vec {r}}_{G}={\frac {\displaystyle \sum \limits _{k=1}^{n}m_{k}{\vec {r_{k}}}}{\displaystyle \sum \limits _{k=1}^{n}m_{k}}}}
This page or section of Calculus is a stub.
You can help Wikibooks by expanding it. | {"url":"https://en.m.wikibooks.org/wiki/Calculus/Centre_of_mass","timestamp":"2024-11-04T14:45:40Z","content_type":"text/html","content_length":"25437","record_id":"<urn:uuid:17f6607f-b609-4963-b9c2-61ae931dccba>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00515.warc.gz"} |
Backing up HP28s user data via IR port
01-09-2023, 02:26 AM
Post: #1
voltaage Posts: 28
Junior Member Joined: Oct 2021
Backing up HP28s user data via IR port
Hello and Happy New 2023.
I finally upgraded myself from a long friend of mine, my HP 28S, with the mighty 50g. I've been using the little clamshell and it's been a blast. I love the environment, the keys, the display... the
of using it.
But, as far as temptation goes, the desire of exploring the huge xxxRPL library written for the 48 series and up, with the consquent frustration of keying in all the code just to painstaikingly check
of errors (seriously, it was far easier with numeric codes) pushed me to get this one. Plus, it's been a long time since I wanted this (1 decade +) but never bother because of some fear it would be
awkward to have the alpha keyboard and the functions in all one space.
So as you might have read the title, I want to backup some data that I have on the I/O limited 28S. I've
read this thread
where it mentions how to build an IR Arduino device to capture the printer output. But that in just a glance requires the use of a special HP transceiver module that is no longer sold (and it's not
cheap to buy).
One of the options that I'm thinking of is listing the program on the redeye, scanning, OCR'ing it and upload into the 50g. But that imply some editing to make sure it's been been transferred
properly. Especially special RPL symbols in programs. Probably it works I need to see.
Is there a more automatic way of doing this job? Like when you dump image roms in the 50g?
Bear in mind that I'm new to the use of the 50g, so I beg your pardon if the answer is too obvious but my inexperience is blindfolding me.
01-09-2023, 12:01 PM
(This post was last modified: 01-09-2023 12:02 PM by robjordan.)
Post: #2
robjordan Posts: 22
Junior Member Joined: Nov 2022
RE: Backing up HP28s user data via IR port
By coincidence I've also been going down this path, in my case to back up equations from the HP-17bII. I read Martin Hepperle's documents that you referenced, and I was lucky enough to get one of the
C4103a HP IR receivers for £5 from ebay. In fact, oddly enough there were two relatively cheap ones on ebay in the UK the same day. So I would suggest to keep tracking ebay in case you find a cheap
There are other ways... this fellow on
was successful in building an IR interface to a 28S using Vishay TSOP4133 IR receiver, even though there are real difficulties in receiving the HP IR protocol using this type of integrating IR
So I have built a device similar to Martin's. I used a different processor (an ATTiny), and wrote my own software inspired by Martin's approach (because - you know - fun!). I connect my device to a
USB port and run a serial terminal on my PC (I run Linux), and when I print an equation on the calculator, the characters appear on the serial terminal, and I can cut and paste them into a document
to save the equation in case the calculator loses its memory.
I guess in your case, you could use the text you cut from the serial terminal to import into the HP50G.
My code is
, and I'm attaching a couple of photos. It needs some tweaks to support 28S (I have one, so I may progress with this shortly).
35, 12C, 17BII, 27S, 28S, 32SII, 48G, 49G, 50G, 35S.
01-09-2023, 03:23 PM
Post: #3
rprosperi Posts: 6,625
Super Moderator Joined: Dec 2013
RE: Backing up HP28s user data via IR port
(01-09-2023 02:26 AM)voltaage Wrote: So as you might have read the title, I want to backup some data that I have on the I/O limited 28S. I've read this thread where it mentions how to build an
IR Arduino device to capture the printer output. But that in just a glance requires the use of a special HP transceiver module that is no longer sold (and it's not cheap to buy).
These units are readily available on eBay, starting at $14 w/shipping included (though possibly more in Germany?). Seems pretty cheap to me... These have been used by numerous members here and have
been found to reliable and consistent (which not all old IR devices can clam). Just an FYI.
--Bob Prosperi
01-09-2023, 05:35 PM
Post: #4
EugeneNine Posts: 331
Senior Member Joined: Feb 2017
RE: Backing up HP28s user data via IR port
Doesn't the 28 transfer via kermit like the 48?
Even so when you print from the 48 its characters so capturing a print you wouldn't need to OCR.
01-09-2023, 07:24 PM
Post: #5
C.Ret Posts: 291
Member Joined: Dec 2013
RE: Backing up HP28s user data via IR port
(01-09-2023 02:26 AM)voltaage Wrote: Is there a more automatic way of doing this job?
Why not just use an IR printer to print all your programs and data in your directories and subdirectories?
The VARS command (from MEMORY menu) used in conjunction with PRVAR (from PRINT menu) produces clear, detailed, and labeled printouts of the contents of any user's directory.
01-09-2023, 10:14 PM
Post: #6
Jake Schwartz Posts: 360
Senior Member Joined: Dec 2013
RE: Backing up HP28s user data via IR port
(01-09-2023 05:35 PM)EugeneNine Wrote: Doesn't the 28 transfer via kermit like the 48?
Even so when you print from the 48 its characters so capturing a print you wouldn't need to OCR.
The 28 has no I/O ports.
01-09-2023, 11:57 PM
Post: #7
EugeneNine Posts: 331
Senior Member Joined: Feb 2017
RE: Backing up HP28s user data via IR port
(01-09-2023 10:14 PM)Jake Schwartz Wrote:
(01-09-2023 05:35 PM)EugeneNine Wrote: Doesn't the 28 transfer via kermit like the 48?
Even so when you print from the 48 its characters so capturing a print you wouldn't need to OCR.
The 28 has no I/O ports.
via IR.
The 48 will speak kermit wired or IR
01-10-2023, 02:41 AM
Post: #8
DavidM Posts: 998
Senior Member Joined: Dec 2013
RE: Backing up HP28s user data via IR port
(01-09-2023 11:57 PM)EugeneNine Wrote: The 48 will speak kermit wired or IR
I don't believe the 28 has bi-directional IR capability, and likewise doesn't have any documented support for communications (other than some printing commands, of course).
It may be possible to run the
command on a
to receive the results of a print command issued on the 28. I don't believe there's a version of INPRT for the 49/50 platforms, though.
Does anyone have both a 28S and a 48sx/gx that they could try this on? The description for INPRT includes this comment:
Quote:"For the HP 28, HP 18C, and HP 19B: Just line up the calculator's LED between the E and T of HEWLETT on the HP 48's Hewlett-Packard logo."
So it appears that it may work to at least get your programs in text form.
01-12-2023, 12:13 AM
Post: #9
FLISZT Posts: 78
Member Joined: Nov 2022
RE: Backing up HP28s user data via IR port
As DavidM said, the hp-28 does not have bi-directional IR capability and, furthermore, the infrared signals of the hp-28 are different from those used by the hp-50.
So, unfortunately, my hp-28s can't send data to one of my hp-50g.
Two hp-50 can "exchange" (not full-duplex!) data via the IF port or via a specific and (I think) very rare wire, if one owns one.
It must be possible for two hp-48 as well, at least of the same serie: S/SX ↔ S/SX or G/GX ↔ G/GX.
But S/SX ↔ G/GX… does it work?
There was a thread in the forum years ago, if I remember correctly, that talked about all this (IF and hp-28-48-etc)... but I didn't find it!
Sanyo CZ-0124 ⋅ TI-57 ⋅ HP-15C ⋅ Canon X-07 + XP-140 Monitor Card ⋅ HP-41CX ⋅ HP-28S ⋅ HP-50G ⋅ HP-50G
01-12-2023, 12:25 AM
Post: #10
EugeneNine Posts: 331
Senior Member Joined: Feb 2017
RE: Backing up HP28s user data via IR port
so its been a few years but I would send IR from my 48sx to various pc's over serial. The trick is most bought IR adapters have a small mcu in them which 'speaks' IRDA and the HP's don't. You have to
build your own by just connecting an IR tx/rx pair to the serial port. Its been many years but I'd use one of the IR pair sets Radio Shack sold and a rs232 level shifter like the max232.
Now a days I'm using a USB-rs232 but going from usb 5v to +/-12v for rs232 and then back to 5v for the ir pair is overkill, it would be easier to use one of the USB-serial @5v or 3.3v like what are
used for programming small mcu' like Arduino and such. I should still have some parts lying around to test with.
01-12-2023, 06:53 PM
Post: #11
FLISZT Posts: 78
Member Joined: Nov 2022
RE: Backing up HP28s user data via IR port
According to this
in the Old HP Forum, a device, that allowed data transmission (via infrared) from a hp-28 to a PC, did exist.
I wish I had known that at the time!
Sanyo CZ-0124 ⋅ TI-57 ⋅ HP-15C ⋅ Canon X-07 + XP-140 Monitor Card ⋅ HP-41CX ⋅ HP-28S ⋅ HP-50G ⋅ HP-50G
01-12-2023, 07:29 PM
Post: #12
FLISZT Posts: 78
Member Joined: Nov 2022
RE: Backing up HP28s user data via IR port Bruno
Sanyo CZ-0124 ⋅ TI-57 ⋅ HP-15C ⋅ Canon X-07 + XP-140 Monitor Card ⋅ HP-41CX ⋅ HP-28S ⋅ HP-50G ⋅ HP-50G
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Integer Divisibility Review
Integer Divisibility Review
We will now review some of the recent material regarding integer divisibility.
• On the The Well-Ordering Principle of the Natural Numbers that the Well-Order Principle for the set of natural numbers $\mathbb{N}$ states the following important fact: Every NONEMPTY subset of
the natural numbers has a least element. That is, if $A \subseteq \mathbb{N}$ and $A \neq \emptyset$ then there exists an element $x \in A$ such that $x \leq y$ for all $y \in A$.
• On the Integer Divisibility we said that if $a, b \in \mathbb{Z}$ then $b$ is Divisible by $a$ written $a \mid b$ if there exists an integer $q \in \mathbb{Z}$ such that:
$$aq = b$$
• We then proved some basic properties regarding divisibility which are summarized in the following table:
(a) If $a \mid b$ and $b \mid c$ then $a \mid c$.
(b) If $a \mid b$ and $a \mid c$ then for all $x, y \in \mathbb{Z}$, $a \mid (bx + cy)$.
(c) If $a \mid b$ and $a \mid c$ then $a \mid (b + c)$ and $a \mid (b - c)$.
• On The Division Algorithm page we looked at a very important algorithm known as the division algorithm. It states that if $a, b \in \mathbb{Z}$ and $b > 0$ then there exists unique integers $q, r
\in \mathbb{Z}$ with $0 \leq r < b$ such that:
\quad a = bq + r
• For example, consider $a = 12$ and $b = 5$. Then $12 = 5(2) + 2$ where $q = 2$ and $r = 2$.
• On The Greatest Common Division Between Integers page we said that if $a, b \in \mathbb{Z}$ then the Greatest Common Divisor of $a$ and $b$ denoted $\gcd (a, b)$ is the integer $d \in \mathbb{Z}$
if $d \mid a$ and $d \mid b$ ($d$ is a common divisor of $a$ and $b$) and if $c $ ]] is any divisor of [[$ a$ and $b$ then $c \leq d$ ($d$ is the greatest of such common divisors).
• We said that the integers $a$ and $b$ are Relatively Prime if $\gcd (a, b) = 1$. | {"url":"http://mathonline.wikidot.com/integer-divisibility-review","timestamp":"2024-11-13T18:22:50Z","content_type":"application/xhtml+xml","content_length":"17074","record_id":"<urn:uuid:96196b01-9932-408e-89c9-8cefee298ec8>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00090.warc.gz"} |
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Samacheer Kalvi 6th Maths Solutions Term 3 Chapter 2 Integers Intext Questions
You can Download Samacheer Kalvi 6th Maths Book Solutions Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations.
Tamilnadu Samacheer Kalvi 6th Maths Solutions Term 3 Chapter 2 Integers Intext Questions
(Try These Text Book Page No. 26)
Question 1.
Read the following numbers orally.
(i) + 24
(ii) -13
(iii) -9
(iv) 8
(i) + 24 is read as ‘ plus 24’ or ‘positive 24’
(ii) -13 is read as ‘minus 13’ or ‘negative 13’
(iii) -9 is read as ‘minus 9’ or ‘ negative 9’
(iv) 8 is read as ‘plus 8’ or ‘positive 8’ or simply ‘8’
Question 2.
Draw a number line and mark the following integers.
(i) 0
(ii) -6
(iii) 5
(iv) -8
Question 3.
Are all natural numbers integers?
Yes, all natural numbers are positive integers
Question 4.
Which part of the integers are not whole numbers?
The numbers 0, 1, 2, 3, are whole numbers. Therefore negative part of the integers are not whole numbers.
Question 5.
How many units should you move to the left of 3 to reach -4.
From 3, we should move 7 units to the left to reach -4.
(Activity Textbook Page No. 27)
Question 1.
Ask your parents / grandparents about the depth at which the various types of vegetables (seeds) should be planted, for their better and efficient growth. For the same, draw a number line indicating
the depth of various vegetable seeds. (Draw the planting chart!).
Activity to be done by the students themselves
(Try These Text Book Page No. 29)
Question 1.
Find the opposite of the following numbers :
(i) 55
(ii) -300
(iii) +5080
(iv) -2500
(v) 0
For every positive integer, there is a corresponding negative integer as its opposite number and vice versa
Question 2.
Represent the following situations as integers.
i) A loss of ₹ 2000
ii) 2018 CE (AD)
iii) Fishes found at 60 m below the sea level
iv) 18°C below 0°C
v) Gaining 13 points
vi) A jet plane at a height of 2500 m
(i) Since loss is negative, a loss of t 2000 is denoted as – ₹ 2000
(ii) Here CE (AD) can be considered as a positive number. Hence CE (AD) 2018 can be represented as + 2018.
(iii) Below sea level is represented as -60 m.
(iv) 18°C below 0°C is represented as – 18°C
(v) Gaining 13 points can be represented as + 13
(vi) Height of 2500 m is represented as + 2500 m.
Question 3.
Suppose in a building, there are 2 basement floors. If the ground floor is denoted as zero, how can we represent the basement floors?
If the ground floor is denoted as 0, basement floors are represented as -1 and -2.
(Try This Textbook Page No. 31)
Question 1.
For two numbers, say 3 and 5, we know that 5 > 3. Will there be a change in the inequality if both the numbers have negative sign before them?
We know that 5 > 3
But when we change the sign of both the numbers we get – 5 and – 3.
Here -5 is 2 units to the left of-3 and so – 5 < 3.
∴ The inequality changes when the signs of both the numbers are changed.
(Activity Textbook Page No. 31)
Question 1.
Take two cards from a deck of playing cards and identify, which is greater between them, assuming that the Joker card represents zero, black cards represent positive integers, red cards represent
negative integers and the cards A, J, Q and K represent 1,11,12 and 13 respectively.
Activity to be done by the students themselves
(Try These Textbook Page No. 31)
Question 1.
Is -15 < -26? Why?
Given statement is wrong.
We will draw the number line and plot the numbers on it.
So -26 < -15
Question 2.
Which is smaller -3 or -5? Why?
Comparing the numbers from the number line.
– 5 is 2 units to the left of -3 and so -5 is smaller than -3. i.e. – 5 < – 3
Question 3.
Which is greater 7 or -4? Why?
7 is a positive integer and lies to the right of 0.-4 is a negative integer and lies to the left of 0
∴ 7 > -4
Question 4.
Which is the greatest negative integer?
– 1 is the greatest negative integer.
Question 5.
Which is the smallest positive integer?
+ 1 is the smallest positive integer.
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Imaginary Numbers - Calculating | Value of i in Math
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Imaginary Numbers
A Greek mathematician called Hero of Alexandria first invented imaginary numbers. Later in 1572, an Italian mathematician Gerolamo Cardano developed the rules for multiplying imaginary numbers. These
numbers are helpful in finding the square roots of negative numbers.
Let us learn more about imaginary numbers along with their definition, examples, and geometrical interpretation. Also, let us see how to do calculations on these numbers.
1. What are Imaginary Numbers?
2. Geometrical Interpretation of Imaginary Numbers
3. i in Math
4. Calculating Imaginary Numbers
5. Tips and Tricks on Imaginary Numbers
6. FAQs on Imaginary Numbers
What are Imaginary Numbers?
Imaginary numbers are numbers that result in a negative number when squared. They are also defined as the square root of negative numbers. An imaginary number is the product of a non-zero real number
and the imaginary unit "i" (which is also known as "iota"), where i = √(-1) (or) i^2 = -1.
Let's try squaring some real numbers:
(−2)^2 = −2×−2 = 4
5^2 = 5×5 = 25
(1.3)^2 = 1.3×1.3 = 1.69
Is any of the results a negative number? No. It means the square of any real number is always positive. Then what number results in a negative square? It is an imaginary number. In math, we come
across the square root of negative numbers many times, especially in the case of solving quadratic equations using the quadratic formula. In such cases, the usage of imaginary numbers is mandatory.
Here are some imaginary numbers examples:
• √(-4) = √(-1) · √4 = i (2) = 2i
• √(-3) = √(-1) · √3 = i √3
In the above examples, 2i and i √3 are imaginary numbers. We can see that each of these numbers is a product of a non-zero real number and i. Thus, we can derive a rule for imaginary numbers which
These are a part of complex numbers, where a complex number is the sum of a real number and an imaginary number. i.e., a complex number is of the form a + ib, where both 'a' and 'b' are real numbers
and here bi is an imaginary number.
Geometrical Interpretation of Imaginary Numbers
We usually represent a complex number a+bi by a point (a, b) in the Argand plane. For example, a complex number 1-3i represents the point (1, -3) on the Argand plane. So an imaginary number bi (which
can be written as 0 + bi) represents a point (0, b) on the plane and hence it is a point on the vertical axis (imaginary axis). Thus, the imaginary numbers always lie on the vertical axis of an
Argand plane. Here are a few examples.
i in Math
What exactly is the term i in the complex number x+iyx+iy corresponding to the point P(x, y)? Clearly, "i" cannot be a real number, because if i was real, then x+iyx+iy would also be real, and would
lie on the real axis itself. Thus, if i was real, then x+iy would only represent points on the real axis, rather than points in a plane.
If we want to represent points in a plane using numbers, i must be a non-real number, in the sense that it must not lie in the Real set. We interpret i in math as follows: "i is one unit in the
direction perpendicular to the real axis". In the above figure, we can see that the point (0, 1) is nothing but "i".
Thus, in the complex number x+iy, the term iy becomes a non-real number: it represents y times i, that is, y units in the i direction. Hence x+iy represents a point that can be reached by moving x
units in the Real direction and y units in the i direction (from the origin):
Value of i
It turns out that arithmetically, i^2=−1. In other words, the value of i is the square root of −1. For now, don’t ask how! As we said earlier, the various different parts of the puzzle that is
Complex Numbers will fall into place as you delve deeper into this subject.
At this point, just keep in mind that:
• i is a non-real number (it lies outside the Real set).
• i represents one unit perpendicular to the Real direction.
• yi represents y units perpendicular to the Real direction
• x+iy represents the point (x, y)
• i is the square root of −1, or, i^2=−1.
When we say that i is a non-real number, we do not mean that i does not exist or is a figment of our imagination. We mean to say that i is non-real in the sense that it does not like in the Real set.
However, it is a perfectly valid mathematical entity. i is also known by the name "iota".
Calculating Imaginary Numbers
We can do all calculations on imaginary numbers also just like how we do with real numbers. Let us see how we do them.
Adding/Subtracting Imaginary Numbers
Adding or subtracting imaginary numbers is just like how we combine the like terms in algebra. For example:
• 2i + 3i = 5i
• 2i - 3i = -i
Multiplying Imaginary Numbers
We multiply the imaginary numbers just like how we multiply the terms in algebra. Here, we may have to use the rule of exponents a^m × a^n = a^m+n. But here, we have to take care of the fact that i^2
= -1. Here are some examples.
• 2i × 3i = 6i^2 = 6(-1) = -6
• 3i^2 × -5i^3 = -15i^5 = -15 (i^2)^2 i = -15 (-1)^2 i = -15i
We may feel that simplifying the powers of iota is difficult. Here are some rules that make the process of finding powers of "i" easier while multiplying complex numbers.
• i^4k = 1
• i^4k+1 = 1
• i^4k+2 = -1
• i^4k+3 = -i
where k is a whole number. These rules state that "i" raised to a number is equal to the "i" raised to the number which is the remainder obtained by dividing the original number by 4. For example:
Any power of i is equal to one of 1, i, -i, and -1 after simplification. We can understand this from the figure below.
These rules really made calculating the product of imaginary numbers very easy.
Dividing Imaginary Numbers
While dividing imaginary rules, we use the rule of exponents a^m / a^n = a^m-n. In the result after division, we usually do not keep "i" in the denominator. If we get so, then we use the rule 1/i =
-i (this is because 1/i = 1/i · i/i = i/i^2 = i/(-1) = -i). Here are some examples:
• 4i/2i = 2
• 3i^3 / 4i^6 = 3/(4i^3) = 3/(4(i^2)(i)) = -3/(4i) = (-3/4)(-i) = 3i/4
Tips and Tricks on Imaginary Numbers
• When the minus sign comes out of the square root, it becomes i because the value of i in math is, i = √(-1).
• \(\sqrt[n]{x}\) can be simplified using the imaginary numbers where x is negative and n is even number.
• When "i" is the denominator (or in case of negative powers of i), use the rule 1/i= -i.
• i^n = i^r, where 'r' is the remainder obtained by dividing n by 4. Then we can apply one of the rules: i^2 = -1, i^3 = -i, i^4 = 1.
Examples on Imaginary Numbers
1. Example 1: Determine the value of (3i)^2.
(3i)^2 = (3i)(3i)
= 9i^2
= 9(-1)
= -9
The third step is because by the definition of imaginary numbers i = √(-1) and this means that i^2 = -1.
Answer: (3i)^2 = -9.
2. Example 2: Express the roots of the quadratic equation x^2 + x + 1 = 0 in terms of imaginary numbers.
Comparing the given quadratic equation with ax^2 + bx + c = 0, we get a = 1, b = 1, and c = 1. Substituting these in the quadratic formula:
x = (-b ± √(b^2 - 4ac) ) / (4a)
= (-1 ± √(1^2 - 4 · 1 · 1) ) / (4·1)
= (-1 ± √(1 - 4) ) / 4
= (-1 ± √(-3) ) / 4
= (-1 ± i√3 ) / 4
Answer: The roots are (-1 + i√3 ) / 4 and (-1 - i√3 ) / 4.
3. Example 3: Find the values of (a) i^101 and (b) i^-199.
To simplify this, we use the rule of imaginary numbers which says i^n = i^r, where n is the remainder that is obtained by dividing n by 4.
(a) i^101 = i^1 = i
(b) i^-199 = 1/i^199
= 1/i^3
= 1/(-i) (because i^3 = i^2 (i)= (-1)(i) = -i)
= -1/i
= - (-i) (because 1/i = -i)
= i
Answer: (a) i^101 = i (b) i^-199 = i.
View Answer >
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Practice Questions on Imaginary Numbers
Check Answer >
FAQs on Imaginary Numbers
What is Imaginary Numbers Definition?
An imaginary number is a number that is the product of a non-zero real number and the iota "i". Here, i = √(-1) or i^2 = -1. These numbers are helpful to find the square root of negative numbers.
Some examples of imaginary numbers are -4i, 6i, i, etc.
What is the Value of i in Math?
"i" in math is known as an imaginary unit. Its value is √-1. It is used to calculate the square roots of negative numbers. It is also a part of complex numbers.
What is the Process of Simplifying Imaginary Numbers?
To add/subtract imaginary numbers, we just add/subtract their coefficients. For example, -2i + 8i - 4i = i. We multiply/divide imaginary numbers just like how we do with terms in algebra. But we use
the facts i^2 = -1 and 1/i = -i while multiplying/dividing imaginary numbers.
What is the Difference Between Imaginary Numbers and Complex Numbers?
An imaginary number is of the form bi whereas a complex number is of the form a+bi where both a and b are real numbers. All imaginary numbers are complex numbers (as 'a' may be 0) but all complex
numbers may not be imaginary numbers.
What is the Difference Between Imaginary Numbers and Real Numbers?
A real number can be a natural number, a whole number, an integer, a rational number, or an irrational number. But an imaginary number is the product of a real number and "i" where i = √(-1). We use
imaginary numbers to evaluate the square root of negative numbers. For example, √(-9) = √(-1) . √9 = i (3) = 3i.
What are Imaginary Numbers Rules?
Here are some rules that are helpful in calculating imaginary numbers.
• i = √(-1)
• i^2 = -1
• i^3 = -i
• i^4 = 1
• i^n = i^r, where r is the remainder obtained by dividing n by 4.
• 1/i = -i and this rule is helpful for calculating the negative powers of i.
How to Write a Real Number in Terms of i in Math?
Any real number x can be written as x + i0. In other words, a real number can also be thought of as a complex number in which the non-real part is 0. Thus, every real number is a complex number. Put
differently, the Real set is a subset of the Complex set. This should be geometrically obvious as well: the Real Line is a subset of the Complex Plane.
What are the Applications of Imaginary Numbers?
• Imaginary numbers are used to simplify the square roots of negative numbers.
• They are used in solving quadratic equations.
• They are used in the formation of complex numbers.
• They are used in physics and in advanced calculus.
Is 0 an Imaginary Number?
Though 0 can be written as 0i, it is not an imaginary number as it is not associated with the square root of any negative number. That is why we clearly mention in the imaginary number definition
that it is of the form bi where 'b' is a non-zero real number.
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Kitchen Backsplash Cost Calculator – Estimate Your Budget
This tool helps you estimate the total cost of installing a kitchen backsplash based on your specific measurements and preferences.
This kitchen backsplash cost calculator allows you to estimate the total cost of installing a backsplash in your kitchen. Here’s how you can use it:
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6. Click on the “Calculate” button to see the total estimated cost which will be displayed in the “Result” field.
The calculator computes the area in square feet and then calculates the cost of tiles and installation separately based on the per square foot costs you enter. It then sums these two costs and
applies any discount before displaying the final estimated cost.
• The calculator assumes a rectangular backsplash area. For other shapes, please consult a professional.
• Prices can vary widely depending on the region and specific vendor.
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Computing the Lipschitz constant needed for fast scene recovery from CASSI measurements
Computing the Lipschitz constant needed for fast scene recovery from CASSI measurements
Niels Chr. Overgaard · Anders Holst
# 289
] [
Paper PDF
Tue 1 Oct 1:30 a.m. PDT — 3:30 a.m. PDT
The linear inverse problem associated with the standard model for hyperspectral image recovery from CASSI measurements is considered. This is formulated as the minimization of an objective function
which is the sum of a total variation regularizer and a least squares loss function. Standard first-order iterative minimization algorithms, such as ISTA, FISTA and TwIST, require as input the value
of the Lipschitz constant for the gradient of the loss function, or at least a good upper bound on this value, in order to select appropriate step lengths. For the loss term considered here, this
Lipschitz constant equals the square of the largest singular value of the measurement map. In applications, this singular value is usually computed directly as the largest eigenvalue of a huge square
matrix. This computation can sometimes be a bottleneck in an otherwise optimized algorithm. In this paper we effectively eliminate this bottleneck for CASSI reconstructions by showing how the
Lipschitz constant can be calculated from a square matrix whose size is easily three orders of magnitudes smaller than in the direct approach.
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Another Approach to Proof of the Twin Prime Conjecture
The Hotchkiss proof of the Twin Prime Conjecture presents a comprehensive argument demonstrating the impossibility of a largest twin prime pair. Beginning with the assumption of the existence of such
a pair and its implications for fundamental theorems like the Prime Number Theorem and Euclid’s Theorem, the proof methodically deconstructs the notion of a largest twin prime pair. Utilizing the
Hotchkiss Prime Theorem alongside established principles in number theory, the proof establishes the independence and mutual exclusivity of primes in defined sets, highlighting the interaction
between these sets and composite values. By demonstrating that any assumed largest twin prime pair leads to a contradiction with the concept of infinitely many primes and the inclusive nature of the
defined sets, the proof concludes that there will always be another twin prime pair beyond any assumed largest pair. Overall, the Hotchkiss proof provides a rigorous and compelling argument for the
validity of the Twin Prime Conjecture.
Optimized Primes
Assume for contradiction that there exists a largest twin prime pair, denoted as (p, q), where p and q are both prime numbers and there are no twin prime pairs beyond this pair.
Existence of a Largest Twin Prime Pair:
• Let (p, q) be the largest twin prime pair, where p and q are consecutive primes such that q = p + 2. By assumption, there are no twin prime pairs beyond (p, q).
Consequence for the Prime Number Theorem:
• The prime number theorem states that the number of primes less than or equal to a given number x is approximately x / ln(x). If there were a largest twin prime pair (p, q), it would imply that
there is a finite limit to the number of primes, contrary to the prime number theorem, which asserts that the number of primes is unbounded. Therefore, the existence of a largest twin prime pair
contradicts the prime number theorem.
Consequence for Euclid’s Theorem:
• Euclid’s theorem states that there are infinitely many primes. If there were a largest twin prime pair (p, q), it would imply that there is a largest prime, namely q, and thus a finite number of
primes. This contradicts Euclid’s theorem, which guarantees the existence of infinitely many primes.
Overall Contradiction:
• The prime number theorem and Euclid’s theorem are both well-established and widely accepted in mathematics. The assumption of a largest twin prime pair leads to a contradiction with these
fundamental theorems. Since the prime number theorem and Euclid’s theorem have been rigorously proven and are integral to number theory, the existence of a largest twin prime pair cannot hold
• Therefore, we conclude that the existence of a largest twin prime pair would invalidate both the prime number theorem and Euclid’s theorem, leading to a contradiction. Consequently, the Twin
Prime Conjecture, which asserts the existence of infinitely many twin primes, must be true.
In order to prove this, we will apply the Hotchkiss Prime Theorem.
1. “Hotchkiss Prime Theorem”:
Let A = {6x + 5 | x ∈ ℤ} be the set of all numbers of the form 6x + 5, and B = {6y + 7 | y ∈ ℤ} be the set of all numbers of the form 6y + 7. Let AA, AB, and BB represent the sets of products:
AA = {(6x + 5)(6y + 5) | x, y ∈ ℤ}
AB = {(6x + 5)(6y + 7) | x, y ∈ ℤ}
BB = {(6x + 7)(6y + 7) | x, y ∈ ℤ} Then, any number that is an element of A or B but not an element of AA, AB, or BB is a prime number.
Assumption: Assume there exists a number k that is:
• Composite (not prime).
• An element of either set A or B (i.e., it’s of the form 6x + 5 or 6y + 7).
• Not an element of AA, AB, or BB.
Case 1: k is of the form 6x + 5 (k ∈ A)
Since k is composite, it has at least two factors, say a and b, where a > 1 and b > 1. Since k is odd, both a and b must be odd. Considering the possible forms of odd numbers in relation to multiples
of 6, we have the following subcases:
• Subcase 1.1: a = (6x + 1) and b = (6y + 1)k = a * b = (6x + 1)(6y + 1) = 36xy + 6x + 6y + 1, which is an element of AA.
• Subcase 1.2: a = (6x + 1) and b = (6y + 5)k = a * b = (6x + 1)(6y + 5) = 36xy + 36x + 5, which is an element of AB.
• Subcase 1.3: a = (6x + 5) and b = (6y + 5)k = a * b = (6x + 5)(6y + 5) = 36xy + 60x + 25, which is an element of AA.
• Subcase 1.4: a = (6x + 5) and b = (6y + 1)k = a * b = (6x + 5)(6y + 1) = 36xy + 30x + 5, which is an element of AB.
Case 2: k is of the form 6y + 7 (k ∈ B)
This case follows a similar logic to Case 1. We analyze the possible forms of factors a and b(both must be odd) and arrive at similar contradictions:
• Subcase 2.1: a = (6x + 1) and b = (6y + 1)k = a * b = (6x + 1)(6y + 1) = 36xy + 6x + 6y + 1, which is an element of BB.
• Subcase 2.2: a = (6x + 1) and b = (6y + 7)k = a * b = (6x + 1)(6y + 7) = 36xy + 42x + 7, which is an element of AB.
• Subcase 2.3: a = (6x + 7) and b = (6y + 7)k = a * b = (6x + 7)(6y + 7) = 36xy + 84x + 49, which is an element of BB.
• Subcase 2.4: a = (6x + 7) and b = (6y + 1)k = a * b = (6x + 7)(6y + 1) = 36xy + 42y + 7, which is an element of AB.
Contradiction: In all subcases, we’ve shown that if k is a composite number of the form 6x+ 5 or 6y + 7, it must be an element of AA, AB, or BB. This contradicts our initial assumption that k is not
an element of those sets.
Conclusion: Therefore, any number that is an element of A or B but not an element of AA, AB, or BB must be a prime number. This completes the proof.
2. Theorem: There are infinitely many prime numbers in sets A and B.
Euclid’s Theorem:
Euclid’s theorem states that there are infinitely many prime numbers. This means there is no largest prime number, and the set of prime numbers is infinite.
Characterization of Sets A and B:
Sets A and B are defined as follows:
• Set A: {6k + 5 | k ∈ Z} (All integers of the form 6k + 5)
• Set B: {6k + 7 | k ∈ Z} (All integers of the form 6k + 7)
Infinite Primes in Sets A and B:
Consider the primes in sets A and B. These primes are of the form 6k±1 for some integer k.
Since there are infinitely many prime numbers, and every prime greater than 3 can be expressed in the form 6k±1, there are infinitely many prime numbers in sets A and B.
Conclusion: Because there are infinitely many primes in sets A and B, the theorem is proven to be true.
This proof establishes the connection between the infinitude of prime numbers and the presence of infinitely many prime numbers in sets A and B. It clarifies the theorem and provides a logical
argument supported by Euclid’s theorem and the characterization of sets A and B.
3. Theorem: Identification of Twin Prime Pairs by Hotchkiss Sets A, B, AA, AB, and BB
• Set A: {6k + 5 | k ∈ Z} (All integers of the form 6k + 5)
• Set B: {6k + 7 | k ∈ Z} (All integers of the form 6k + 7)
• Set AA: {a × a | a ∈ A} (All products of two elements in A)
• Set AB: {a × b | a ∈ A, b ∈ B} (All products of one element in A and one in B)
• Set BB: {b × b | b ∈ B} (All products of two elements in B)
Theorem: Two prime numbers, p and q, form a twin prime pair if and only if:
• Both p and q are in sets A and B respectively (or vice versa), and
• Neither p nor q are in sets AA, AB, or BB, and
• p and q differ by 2 (i.e., p = q ± 2).
Part 1: If two prime numbers p and q form a twin prime pair, then they meet the conditions of the theorem.
• Condition 1: If p and q form a twin prime pair, then they must differ by 2. If p is in set A (p = 6k + 5), then q must be in set B (q = 6k + 7) or vice versa.
• Condition 2: Since p and q are prime numbers, they cannot be factored into two smaller integers. Therefore, neither p nor q can be formed by the product of two elements from sets A and B. Thus,
they are not in sets AA, AB, or BB.
• Condition 3: This is a direct consequence of the definition of twin primes.
Part 2: If two prime numbers p and q meet the conditions of the theorem, then they form a twin prime pair.
• Condition 1: Since p and q are in sets A and B respectively (or vice versa), they are both prime numbers.
• Condition 2: Since neither p nor q is in AA, AB, or BB, they cannot be factored into two smaller integers.
• Condition 3: Since p and q differ by 2, they fulfill the definition of a twin prime pair.
Conclusion: We have shown that two prime numbers p and q form a twin prime pair if and only if they meet the conditions of the theorem. Therefore, the sets A, B, AA, AB, and BB, along with the
requirement that the prime numbers differ by 2, can effectively identify twin prime pairs.
4. Theorem: All pairs of twin primes greater than (3,5) are contained in sets A and B.
Characterization of Twin Primes:
Twin primes are pairs of prime numbers that differ by 2. For any twin prime pair (p,q), we have q=p+2.
Form of Twin Primes:
Twin primes are of the form 6k±1 (except for 3 and 5). This means, for some integer k, p and q can be expressed as 6k−1 and 6k+1, respectively.
Prime Number Representation:
All prime numbers greater than 3 can be expressed as either 6k+1 or 6k+56, where k is a non-negative integer. This corresponds to sets A and B, respectively.
Twin Primes as A or B:
Since twin primes are of the form 6k±1, they must belong to either set A or set B.
• For 6k−1, this corresponds to set A.
• For 6k+1, this corresponds to set B.
Exclusion from AA, AB, BB:
By the definition of twin primes, p and q cannot be products of elements from sets A and B. Therefore, they cannot be in sets AA, AB, or BB.
All pairs of twin primes, being of the form 6k±1 and not being products of elements from sets A and B, are indeed contained within sets A and B.
Thus, we have proven that all pairs of twin primes are contained within sets A and B, as per the given theorem.
5. Theorem: Independence and Mutual Exclusivity of Primes in Sets A and B
Let A = {6x + 5 | x ∈ ℤ} and B = {6y + 7 | y ∈ ℤ} be defined as sets containing numbers of the form 6x + 5 and 6y + 7, respectively. Let AA, AB, and BB represent the composite sets formed by the
products within sets A and B as follows:
AA = {(6x + 5)(6x’ + 5) | x, x’ ∈ ℤ}
AB = {(6x + 5)(6y + 7) | x, y ∈ ℤ}
BB = {(6y + 7)(6y’ + 7) | y, y’ ∈ ℤ}
Then, the following theorem holds:
• Independence of Primes in Sets A and B: The primes in sets A and B are independent variables, meaning that the occurrence of a prime in one set does not affect the likelihood of finding a prime
in the other set.
• Mutual Exclusivity of Primes in Sets A and B: The primes in sets A and B are mutually exclusive sets, indicating that a number cannot simultaneously belong to both sets A and B.
• Interactions in Composite Sets AA, AB, and BB: The composite values within sets AA, AB, and BB represent interactions between elements of sets A and B. These composite values constitute
interactions that invalidate the possibility of a prime number existing within sets A or B due to their composite nature.
Independence of Primes in Sets A and B:
• The primes in sets A and B are of the form 6x + 5 and 6y + 7, respectively, where x and y are integers.
• The occurrence of a prime in set A does not affect the form of primes in set B, and vice versa. Therefore, the primes in sets A and B are independent variables.
Mutual Exclusivity of Primes in Sets A and B:
• By definition, a number of the form 6x + 5 cannot simultaneously be of the form 6y + 7, and vice versa.
• Therefore, a prime number belonging to set A cannot belong to set B, and vice versa. This establishes mutual exclusivity.
Interactions in Composite Sets AA, AB, and BB:
• The composite values within sets AA, AB, and BB represent the products of elements from sets A and B.
• These composite values result from interactions between elements of sets A and B.
• Since these interactions produce composite values, any number in sets AA, AB, or BB cannot be a prime number.
• Thus, the presence of composite values within sets AA, AB, and BB invalidates the possibility of primes existing within sets A or B.
Conclusion: The theorem demonstrates the independence and mutual exclusivity of primes in sets A and B, while also highlighting the interactions within composite sets AA, AB, and BB that preclude the
existence of primes within sets A or B.
6. Theorem: Let’s assume that there exists a largest twin prime pair (p, q) such that p and q are both elements of sets A and B, respectively, and there are no twin prime pairs beyond this largest
Let (p, q) be the largest twin prime pair, where p=6k−1 and q=6k+1 for some integer k.
By assumption, there are no twin prime pairs beyond (p, q).
Consider the sets A and B:
• Set A contains all numbers of the form 6x+5, which includes primes greater than 3.
• Set B contains all numbers of the form 6y+7, which also includes primes greater than 3.
Since sets A and B include all primes greater than 3, and there are infinitely many primes according to Euclid’s Theorem, there must be infinitely many values of k such that the pairs (6k – 1, 6k +
1) form twin primes.
Therefore, if we assume there is a largest twin prime pair (p, q), there would always be another twin prime pair (6k−1,6k+1) for some k beyond this largest pair.
This contradicts our initial assumption that there exists a largest twin prime pair.
Conclusion: The assumption that there exists a largest twin prime pair leads to a contradiction with the concept of infinitely many primes and the comprehensive inclusion of primes greater than 3 in
sets A and B. Therefore, there will always be another twin prime pair beyond any assumed largest pair. | {"url":"https://n01r.com/another-approach-to-proof-of-the-twin-prime-conjecture/","timestamp":"2024-11-06T18:40:40Z","content_type":"text/html","content_length":"125982","record_id":"<urn:uuid:d0494fce-9199-4711-8687-0b54131b6bef>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00411.warc.gz"} |
COMPREHENSIVE crossword clue - All synonyms & answers
COMPREHENSIVE Crossword Clue & Answer
'COMPREHENSIVE' is a 13 letter Word starting with C and ending with E
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Top answers for COMPREHENSIVE crossword clue from newspapers
Definition of Comprehensive
• broad in scope; "a comprehensive survey of world affairs"; including all or everything; "comprehensive coverage"; "a comprehensive history of the revolution"; "a comprehensive survey"; "a
comprehensive education"
COMPREHENSIVE Crossword puzzle solutions
53 Solutions - 8 Top suggestions & 45 further suggestions. We have 53 solutions for the frequently searched for crossword lexicon term COMPREHENSIVE. Furthermore and additionally we have 45 Further
solutions for this paraphrase.
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Find for us the 54nth solution for COMPREHENSIVE and send it to our e-mail (crossword-at-the-crossword-solver com) with the subject "New solution suggestion for COMPREHENSIVE". Do you have an
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Solution GENERAL is our most searched for solution by our visitors. Solution GENERAL is 7 letters long. We have 3 further solutions of the same word length.
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We have 53 solutions to the crossword puzzle COMPREHENSIVE. The longest solution is INCORPORATING with 13 letters and the shortest solution is FAT with 3 letters.
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With help from our search you can look for words of a certain length. Our intelligent search sorts between the most frequent solutions and the most searched for questions. You can completely free of
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The lenght of the solutions is between 3 and 13 letters. In total we have solutions for 11 word lengths.
More clues you might be interested in | {"url":"https://www.the-crossword-solver.com/word/comprehensive","timestamp":"2024-11-05T00:50:13Z","content_type":"text/html","content_length":"287336","record_id":"<urn:uuid:b0119228-fcc5-489f-b127-b004b0856d3d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00750.warc.gz"} |
: The water ionization constant
K[w]: The water ionization constant
Important note: all constants refered to: K[c], K[w], K[a], and K[b] are temperature-dependent. All discussions are assumed to be at 25 °C, i.e. standard temperature.
The following equation describes the reaction of water with itself (called autoprotolysis):
H[2]O + H[2]O ⇌ H[3]O^+ + OH¯
The equilibrium constant for this reaction is written as follows:
[H[3]O^+] [OH¯]
K[c] = –––––––––––
[H[2]O] [H[2]O]
However, in pure liquid water, [H[2]O] is itself a constant value. To demonstrate this, consider 1000 mL of water with a density of 1.00 g/mL. This 1.00 liter (1000 mL) would weigh 1000 grams. This
mass divided by the molecular weight of water (18.0152 g/mol) gives 55.5 moles. The "molarity" of this water would then be 55.5 mol / 1.00 liter or 55.5 M.
The solutions studied in introductory chemistry are so dilute that the "concentration" of water is unaffected. So 55.5 molar can be considered to be a constant if the solution is dilute enough.
Cross-multiplying the above equation gives:
K[c] [H[2]O] [H[2]O] = [H[3]O^+] [OH¯]
Since the term K[c] [H[2]O] [H[2]O] is a constant, let it be symbolized by K[w], giving:
K[w] = [H[3]O^+] [OH¯]
This constant, K[w], is called the water autoprotolysis constant or water autoionization constant. (Sometimes the prefix auto is dropped, as was done in the title of this section.) It can be
determined by experiment and has the value 1.011 x 10¯^14 at 25 °C. Generally, a value of 1.00 x 10¯^14 is used.
From the chemical equation just above, it can be seen that, for every H[3]O^+ produced, one OH¯ is also produced. Another way to say is that the molar ratio of H[3]O^+ to OH¯ in the water is 1 to 1.
This means that the same amount of each ion is present in pure water. In other words, in pure water, [H[3]O^+] = [OH¯].
Therefore the values of [H[3]O^+] and [OH¯] can be determined by taking the square root of K[w]. Hence, both [H[3]O^+] and [OH¯] equal 1.00 x 10¯^7 M in pure water. This leads to several important
results in the acid base world.
Result #1: The pH of pure water is 7
By definition, pH = −log [H[3]O^+]
The pH of pure water then equals −log 10¯^7, which is 7.
Result #2: If the pH or the pOH is known, the other can be found.
Take the negative logarithm of each side of the K[w] equation as follows:
−log K[w] = −log [H[3]O^+] + −log [OH¯]
−log 1.00 x 10¯^14 = −log [H[3]O^+] + −log [OH¯]
Note the use of the add sign on the right side of the equation. The result is ususally written as:
pK[w] = pH + pOH = 14
This is an extremely important equation. Learn it well.
Result #3: If the [H[3]O^+] or the [OH¯] is known, the other can be found.
Simply divide K[w] by the known value to get the other.
Suppose [H[3]O^+] is known, then:
[OH¯] = K[w] / [H[3]O^+]
Suppose [OH¯] is known, then:
[H[3]O^+] = K[w] / [OH¯]
Result #4: In a solution, if one variable ( [H[3]O^+] or [OH¯] ) changes value (either up or down), the other variable will change in the opposite direction.
The change in values will still preserve this fundamental equality:
K[w] = [H[3]O^+] [OH¯]
Suppose [H[3]O^+] became larger, therefore the [OH¯] becomes smaller.
Suppose [OH¯] became larger, therefore the [H[3]O^+] becomes smaller.
This change happens automatically and cannot be stopped. | {"url":"https://web.chemteam.info/AcidBase/Kw.html","timestamp":"2024-11-04T07:03:41Z","content_type":"text/html","content_length":"5214","record_id":"<urn:uuid:ffb36e27-0ac2-4b6b-b684-544051b0716a>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00639.warc.gz"} |
Make Your Own Math Manipulatives Kit FREE | Orison Orchards
Math Manipulatives; Make Your Own DIY Kit FREE
Over the past four decades, studies conducted in several different countries across multiple grade levels indicate that math manipulatives substantially improve mathematical understanding and
ability. Math manipulatives help children to understand abstract concepts by equating them to something concrete, using the manipulatives.
Young children think of their world in concrete ways. They collect facts about objects they experience in daily life and organize those facts as they classify objects. Concrete thinking is focused
upon gathering pieces of information about tangible objects.
Abstract thinking typically doesn’t develop until children are older. As children develop the skill of abstract thinking, they learn to deal with concepts they haven’t directly experienced, learning
to draw their own conclusions from previous experiences.
Math manipulatives have improved mathematical understanding so greatly that several curriculum developers have begun selling math curricula entirely based on math manipulatives. Learn more about them
in my Homeschool Math Curriculum Comparison Guide.
Math manipulatives are so important that every homeschool family should have a set. If you can’t afford $150 or more to buy a ready-made set of math manipulatives, or if you just don’t want to pay
that much, you can certainly create your own with things you have around the house (or a quick trip to the dollar store) and your printer.
I store all of our math manipulatives in a simple plastic file box from the organization aisle at Walmart. I love that everything is in one place and I can easily remove it to a high shelf, away from
my kids, when we’re not using it. Not that that always works. Somehow they search out (and lose or destroy) everything, but this still helps.
15 Math Manipulatives Your Kit Should Include:
1. Craft Sticks and Rubber Bands
You can find huge packages of craft sticks and rubber band at the dollar store. Craft sticks are easy to group, which makes them perfect for learning place value.
You’ll want at least 350 craft sticks and 20 or so rubber bands, so you can bundle the craft sticks into ten groups of 10, two groups of 100, and have some left over as ones. I like to teach place
value by having my child trade me ten ones for a bundle of ten, or ten bundles of ten for a bundle of 100, so they fully understand the concepts of “borrowing” and “carrying”.
That number will allow you to work on two-digit addition and subtraction, and demonstrate 3-digit, though only having 2 bundles of 100 limits you to three-digit numbers starting with 1 or 2. But it
demonstrates the concept in a concrete, easily understood way.
Math concepts to teach with craft sticks:
• Place value
• Borrowing and carrying
• Addition and Subtraction
• Multiplication and Division
2. Counters
A pile of small, countable items will help your child recognize that written numbers represent real items (number recognition and correspondence). They can also help your child learn to count by
rote. Those are some of the earliest learned concepts.
Dried beans, legos, wooden blocks, pennies, colored beads or anything you have around the house will work. It’s best, though, if they are identical, so your child is just working on one concept and
not confused by trying to sort colors or shapes.
I keep a small container of coins (to use as both counters and to learn about money) and a couple of small bags of assorted counters in our math manipulative bin. Occasionally, for added interest and
fun, I add edible counters, like m&m’s, jelly beans, marshmallows or cheerios to our kit. You wouldn’t want to use edible counters all the times, since they’re consumable, but they provide a fun
Later, your child can use these to master ordinal numbers (first, second, third…), addition and subtraction, compare numbers (more, less and equivalence) and eventually group them for multiplication,
division and fractions.
This set of printable Rainbow Bear Counters with fun math games and activities includes bear counters in six colors, plus 18 fun math games to play with the bear counters. Use coupon code ‘Math25’ to
download your rainbow bear counters and fun math games for only $3.
Math concepts to teach with counters:
• Counting
• Number recognition
• Number correspondence
• Ordinal numbers
• More, less and equivalence
• The concept of ten (use counters with ten frames)
• Addition and subtraction
• Multiplication and division
3. Dice are fun math manipulatives
You’ll need multiple dice for all the math games you’ll be playing. You can either rob your game closet, or you can pick up a package of these at the dollar store. I bought these, 6 dice for $1, at
my local dollar store.
Dice are also a fun way to work on subitization, or the ability to “see” numbers at a glance. They can be used all by themselves to drill math facts or practice mental math in a fun and interesting
Math games to play with dice:
Pig: You’ll need a pair of dice, paper and pencil for scoring. The goal is to be the first player to reach 100. The first player rolls both dice, calculates the sum, then rolls again if he or she
wants to. The next sum is added to the first. The player can roll as many times as he chooses, but if he rolls a one on either die, he scores 0 for that round and it’s the next players turn. If he
rolls 1’s on both dice, his entire total is scrapped and he starts over again at 0.
Multiplication Tetris: You’ll need a pair of dice, a sheet of graph paper and pencil per player, and a box of crayons. Players take turns rolling both dice and drawing arrays on their graph paper,
according to the numbers rolled, then color it. For example, if 4 and 3 are rolled, the player draws a 4 x 3 rectangle on the graph paper. Players should draw rectangles strategically so they fit
together like in Tetris. The first player with a full paper wins.
4. Blocks
Blocks can be used to teach multiple mathematical concepts. Blocks that are all one shape or color make it simpler to learn a single concept at a time.
Blocks can be used to teach patterns, they can be used as counters, and they can help your child to learn to compare sizes such as taller, bigger, or shorter. You can use any type of blocks to build
towers, comparing heights, walls or snakes to compare length.
Concepts to teach with blocks:
• Comparison (Build towers or walls and compare the sizes: tall, taller, tallest, long, longer, longest).
• Pattern (if you have different colors or shapes)
• Counting
• Place Value
5. Pattern Blocks
Pattern blocks are different than building blocks. They’re comprised of 6 shapes in 6 colors (green triangles, orange squares, blue parallelograms, tan rhombuses, red trapezoids, and yellow
Disregarding the thickness, pattern blocks are two dimensional shapes. They are designed so the sides are all the same length except for the trapezoid, which has 1 side that is twice as long, making
it easy to nest the shapes together to build figures.
Pattern blocks help students develop spatial reasoning. As students become more familiar with composition and decomposition of shapes, they recognize patterns more easily. This feature makes it
possible for the shapes to nest together and provides for a wide range of explorations
Jessica Brown has printable pattern blocks and tons of fun patterns for using them. When I use paper pattern blocks I don’t laminate them because that makes them hard to nest together. I prefer to
just print them on cardstock.
Concepts to teach with pattern blocks:
• Patterns
• Geometric designs
• Spatial Relations
• Area and other geometry concepts
6. Deck of cards for math games
Cards with numbers on the face are such a fun way drill math facts without it seeming like drill! Here are several fun games to play with cards that will teach mathematical concepts from number
recognition to drilling math facts.
Memory: Lay multiples of each number out, face down, on the table and scramble them around. Players take turns choosing two cards at a time. Players keep matching pairs and take an additional turn
following each match. Alternately, you could match cards that add up to ten, so 7 and 3 match or 6 and 4 match.
War: The deck is divided evenly between two players, with cards face down on the table. Each player should have a stack of 26 cards face down in front of him. Each player turns up a card at the same
time and the player with the higher card takes both cards and places them on the bottom of his stack, face down.
In the event that the cards are the same rank, you have ‘War’. Each player turns up one card face down and one card face up. The player with the higher cards takes both piles. If the turned-up cards
are again the same rank, each player places another card face down and turns another card face up. The player with the higher card takes all of the cards, and so on until one player has all of the
Ten: Play this math card game alone or as a team. Lay out a deck cards on the table, face up (number cards only — leave out face cards). Players take turns removing sets of cards that add up to 10,
trying to accumulate as many cards as possible. So 1+3+2+4 is preferable to 5+5. The player with the most cards wins. Alternately, you could decide that fewer cards are preferable. Or you could
decide to only choose pairs.
Number Bingo: Lay out a 4 x 4 array of cards, face up (number cards only — leave out face cards) in front of each player. This is is ‘bingo card’. Remaining cards are placed face down in front of the
caller. The caller flips over a card. All players with that number on their board turns the card face down. Play continues until one player has a row flipped over horizontally, vertically, or
diagonally and calls “Bingo!”
Twenty-one: This is traditionally a gambling game (aka Blackjack), but we don’t bet. This is a great way to practice arithmetic skills. The objective is to get to 21 or as close as possible. Number
cards are worth their face value, jacks, kings and queens are worth 10. Aces can be either 1 or 11 — the holder of the card gets to choose. To start, the dealer deals two cards from his face-down
deck to each player. After all cards are dealt, all players may look at their two cards and add up their total. In order, players may request a “hit” (another card from the dealer). You may hit any
number of times in one turn. Do this until you are satisfied with your card total or bust (over 21). Once you’re satisfied with your card total, you “stay”. The winner is the person closest to 21
without going over, or busting.
Number Scrabble: A dealer deals each player 10 cards (numbers only), then places one card face up in the center and the remaining cards face down in a pile on the edge of the table. Players take
turns placing cards in sequential order, building off the card already in the center. For example, if the card in the center is a 3, a player could place a 2 or a 4 below, above or the the right or
left. The next player would build on that sequence or start another in a different direction. Players can place more than one card at a time. If the center card is 3, and a player has a 2, a 4 and a
5, he could place them all at once. All cards that are next to each other must be sequential.
Solitaire for number sequencing.
7. Clock
You might have an old analog clock around the house you could use. It doesn’t need to be anything special, just a clock with moveable hands and numbers (not Roman numerals).
I’ve used this inexpensive clock in our homeschool for years. It’s geared on the back so the hands move together, helping kids see the way hands move together typically. But that isn’t necessary.
You can make your own clock that will do in a pinch, using the free printable at the bottom of the page and a brad.
Games for learning to tell time:
Daily schedule: Create a visual schedule of your daily activities, with pictures of getting dressed at 6:30 am, pictures of breakfast at 7 am, pictures of practicing (instruments or whatever) at 8
am, pictures of school at 9 am, etc…
Stop the clock: We play this with at least two children, but you could play it non-competitively with one child. Each child has a clock with moveable hands (free printable at the bottom of this
page). I yell out a time, or display the time on a digital device. Players have to make their clock hands match the specified time. I start with o’clock times, move to half past, quarter of, quarter
after, then to more specific times.
8. Money
Coins are essential for teaching children to count money. You might have a toy cash register that came with plastic coins and fake bills, or a game with money you can play with. Or you could just use
real money. That’s what I do, because my kiddos kept losing the play money and I hate spending real money on fake money.
I just use the loose change I find in our pockets at the end of the day. It gets collected for awhile in a jar and then added to our manipulative kit, along with bills of all denominations, which I
am more careful about.
Coins can also be used for:
• Use pennies as counters
• Use nickels, dimes, or quarters to practice counting by 5s, 10s, or 25s.
• Use pennies and dimes to teach place value.
• Use pennies and dimes to teach adding and subtracting 2-digit numbers, with or without regrouping.
• Use pennies and dimes to stand for tenths and hundredths to teach decimals.
Pennies, dimes and 1 dollar bills (1s, 10s and 100s) are great for teaching place value. So are 1 dollar bills, 10 dollar bills and 100 dollar bills from your Monopoly set.
9. Ruler and other measuring tools
You probably already have rulers, yard sticks, measuring tapes, measuring spoons, measuring cups, quart/gallon pitchers and a kitchen scale around your house. If not, you can pick them up at the
dollar store.
I actually don’t keep my kitchen tools in my math manipulative kit because I only have one set of them. Plus, they’d take up too much room, we only use them for math infrequently, and we homeschool
at the kitchen table anyway (so they’re very accessible).
Concepts you can teach with measuring tools:
• length
• size
• weight
• metric system
• unit conversion
10. Hundreds Chart
A hundreds chart is just a 10 x 10 array of all the numbers from 1 to 100 in order. It’s useful to have one with the numbers filled in and a blank one, laminated, for your child to practice filling
Dry erase markers can be difficult for little hands to manage, so I actually prefer to print several and not laminate them for my preK and kinders to practice with. Older, more dexterous kids do fine
with the laminated, reusable version.
You’ll find two hundreds charts in the free printable at the bottom of this post, one blank and one with the numbers filled in.
Games to play with your hundreds chart:
Missing Digits: I’ll use a black dry-erase marker to black out multiple squares on the laminated hundreds chart with the numbers filled in. My child has to figure out the missing numbers then wipe
off the marker to check his guess. Black out more numbers for more experienced children, or just write in a few numbers on the empty chart.
Hundreds Bingo: Give your child a couple of rolls of smarties or a handful of m&m’s or other small treat. Call out numbers and have him cover the corresponding number on his chart. Work to cover a
whole row or go for blackout.
Mystery Numbers: You choose a number and give the other players clues, such as, “The mystery number has two digits.” “The mystery number is greater than 60.” ” They mystery number is even.” until the
players guess your mystery number correctly.
Twenty Questions: You choose a number. Players may ask up to 20 yes/no questions to try to guess the number.
Don’t eat Pete!: Choose one player to be the guesser. He leaves the room. The remaining players choose a number to be Pete. Cover all 100 numbers with a smartie or other small candy, make sure
everyone knows which is Pete, and invite the guesser back in. The guesser starts guessing numbers randomly. As he says the number out loud, he removes and eats the candy (or places it in his pile to
save for later). When he finally reaches Pete, everyone shouts, “Don’t eat Pete!” and his turn is done. Another guesser is chosen and everything is repeated.
11. Ten Frames math manipulatives
A ten frame is just a 2 x 5 array of squares, used to visually represent ten in a simple way. Because our number system is based on ten, having a thorough understanding of ten and combinations that
make ten or can be derived from ten is critical.
Concepts taught with ten frames:
Ten frames are excellent ways to present sums and differences up to ten in a visual way. You can model addition with two different colors of counters, one per space. For example, six green counters
and four red counters makes ten.
You can also model subtraction by filling the spaces with a single color of counters, then removing some. Stack two of the ten frames to create a 4 x 5 array when your child is ready to add and
subtract into the teens.
Ten frames are also helpful when it comes to teaching multiplication and division. Model multiplication by grouping equal quantities of counters in each square. For example, 10 groups with 4 counters
each is 40.
Division is just the opposite. “If I have 40 counters and divide them equally between the squares, how many will be in each square?” It’s a great way to teach remainders as well. And you don’t have
to use all ten squares.
The free printable at the bottom of this post includes a couple of ten frames.
12. Fraction Strips and Circles
Both of these are excellent ways to visually represent division and fractions. Fractions can be hard to grasp until kids can actually hold the pieces in their hands and see that 1/4 is smaller than 1
The free printable (watch for the big, blue button) math manipulatives at the bottom of this post include a set of fraction strips and a set of fraction circles from whole to 1/16.
Concepts taught with fraction strips and circles:
• Whole and part
• Multiplication of fractions
• Division of fractions
• Fraction equivalents
• Addition and subtraction of fractions
13. Base Ten Set
Base Ten Blocks provide hands-on ways to learn number concepts such as place value, regrouping and decimals. They physically represent mathematical concepts so students can develop a deeper
understanding. The blue button at the bottom of this post will take you to a free printable that includes a base ten set.
Concepts to teach with a base ten set:
• Adding and Subtracting
• Multiplication and Division
• Number sense
• Place Value
• Regrouping
• Counting
• Decimals
14. Picture Books make great visual math manipulatives
I constantly trade picture books in an out of my math manipulative kit, depending on the concepts we’re working on at the time. We visit the library weekly, so I just look ahead in my children’s math
books and try to find a couple of math picture books to go along with upcoming concepts each week.
Picture books teach shapes, numbers, counting, prepositions, comparison words and so much more. Picture books are a fun way to learn concepts because they’re so visual.
15. A Chalkboard or Dry Erase Board
This is not a necessity, but if you have one around the house, this will make learning to write numbers more fun.
First you can just write a single number on the board and have your child count out the corresponding number of counters. Next, you can use it to write addition facts, placing the corresponding
number of counters above each number.
My kids like the 9 x 12 size so they can work independently. They love having their own tools. I also appreciate that the smaller size boards fit into our math manipulative kit. I buy the 2-sided
kind with a blank side for math and a lined side for writing.
Now That’s a (practically free!) Math Manipulatives Kit
There you have it! A very simple math manipulative kit that is every bit as effective as the expensive, commercial kits. You probably have most of the items you’ll need around your house already,
except for the printables, which you may download free below.
Math manipulatives don’t need to be complicated!
Are you looking for a complete, gameified, FUN math curriculum?
Kids don’t hate math, they hate feeling stuck. We Play Math eliminates frustration through gamification. Short, animated videos teach concepts and games and puzzles provide practice. Our math arcade
facilitates instant, automatic recall of critical math facts.
Sign up here for your 25-page FREE Math Manipulatives printable, which includes:
• Ten Frames
• Base Ten Set
• Practice Clock
• Hundreds Chart (1 blank and 1 filled in)
• Fraction Strips
• Fraction Circles
Pin this DIY Math Manipulatives kit for later!
8 Comments
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On the abundance of primordial helium
We have used recent observations of helium-4, nitrogen, and oxygen from some four dozen, low-metallicity, extragalactic H II regions to define mean N vs. O, ^4He vs. N, and ^4He vs. O relations which
are extrapolated to zero metallicity to determine the primordial ^4He mass fraction Y[P]. The data and various subsets of the data, selected on the basis of nitrogen and oxygen, are all consistent
with Y^P = 0.232 ± 0.003. For the 2 σ (statistical) upper bound we find Y[P]^2 σ ≤ 0.238. Estimating a 2% systematic uncertainty (σ[syst] = ± 0.005) leads to a maximum upper bound to the primordial
helium mass fraction: Y[P]^MAX = Y[P]^2 σ + σ[syst] ≤ 0.243. We compare these upper bounds to Y[P] with recent calculations of the predicted yield from big bang nucleosynthesis to derive upper bounds
to the nucleon-to-photon ratio η(η[10] ≡ 10^10[η]) and the number of equivalent light (≲10 MeV) neutrino species. For Y[p] ≤ 0.238 (0.243), we find η[10] ≤ 2.5(3.9) and N[v] ≤ 2.7(3.1). If indeed Y
[P] ≤ 0.238, then BBN predicts enhanced production of deuterium and helium-3 which may be in conflict with the primordial abundances inferred from model-dependent (chemical evolution) extrapolations
of solar system and interstellar observations. Better chemical evolution models and more data - especially D-absorption in the QSO Ly-α clouds - will be crucial to resolve this potential crisis for
BBN. The larger upper bound, Y[P] ≤ 0.243, is completely consistent with BBN which, now, bounds the universal density of nucleons ( for Hubble parameter 40 ≤ H[o] ≤ 100 km s^-1 Mpc^-1 and cosmic
background radiation temperature T = 2.726 ± 0.010) to lie in the range 0.01 ≤ Ω[BBN] ≤ 0.09 (for H[o] = 50 h[50] km s^-1 Mpc^-1, 0.04 ≤ Ω[BBN] h[50]^2 ≤ 0.06).
• Cosmology: theory
• Nuclear reactions, nucleosynthesis, abundances
Dive into the research topics of 'On the abundance of primordial helium'. Together they form a unique fingerprint. | {"url":"https://experts.umn.edu/en/publications/on-the-abundance-of-primordial-helium","timestamp":"2024-11-05T19:31:06Z","content_type":"text/html","content_length":"54236","record_id":"<urn:uuid:648dccf0-d42b-4f57-a03a-ab9ba8bce219>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00194.warc.gz"} |
Lesson: Equally likely outcomes | KS3 Maths | Oak National Academy
Equally likely outcomes
I can state when outcomes will be equally likely.
I can state when outcomes will be equally likely.
Lesson details
Key learning points
1. A set of possible outcomes can be stated.
2. Outcomes can be evaluated to decide how likely they are.
3. Outcomes with the same likelihood can be identified.
Common misconception
Rolling a 6 on a dice is less likely to happen than rolling other numbers. Or ticket 001 is less likely to be drawn in a raffle than other tickets.
The random nature of picking a raffle or rolling a dice means all outcomes are equally likely as each other.
• Outcome - An outcome is a result of a trial.
• Likelihood - Likelihood describes the chance that a particular event will occur.
• Trial - A trial is a single predefined test.
• Sample space - A sample space is all the possible outcomes of a trial. A sample space diagram is a systematic way of producing a sample space.
Ask pupils to design a their own single-trial game where the likelihoods of winning and losing are equal.
Teacher tip
6 Questions
Select all the numbers that are multiples of 3.
Select all the numbers that are factors of 3.
What fraction of the circle is shaded?
Correct answer: $${3}\over{10}$$
In which diagrams is half of the circle shaded?
Correct Answer: An image in a quiz
Correct Answer: An image in a quiz
Jun wants to shade in $${1}\over{3}$$ of this whole circle. Jun should shade in sectors of the circle.
Correct Answer: 4, four, four sectors, 4 sectors
4, four, four sectors, 4 sectors
Match each letter to the fraction of the circle that is labelled with that letter.
Correct Answer:a,$${1}\over{4}$$
Correct Answer:b,$${1}\over{3}$$
Correct Answer:c,$${5}\over{12}$$
6 Questions
Match each key words with its definition.
Correct Answer:trial,a single predefined test
trial -
a single predefined test
Correct Answer:outcome,a result of a trial
outcome -
a result of a trial
Correct Answer:sample space,all the possible outcomes of a trial
sample space -
all the possible outcomes of a trial
Correct Answer:likelihood,describes the chance that a particular event will occur
likelihood -
describes the chance that a particular event will occur
Complete the following sample space for the outcomes of the spinner. ξ = {A, I, L, R, }.
In which spinners are the outcomes 'win' and 'lose' equally likely?
Correct Answer: An image in a quiz
Correct Answer: An image in a quiz
Which outcomes are equally likely in this spinner?
Correct answer: A and B
A and B
In which spinners are all the outcomes equally likely?
Correct Answer: An image in a quiz
Correct Answer: An image in a quiz
A regular six-sided dice is rolled once. In which sample space are the outcomes equally likely?
ξ = {factor of 6, not a factor of 6}
ξ = {multiple of 3, not a multiple of 3}
Correct answer: ξ = {odd, even}
ξ = {odd, even}
ξ = {square number, not a square number} | {"url":"https://www.thenational.academy/teachers/programmes/maths-secondary-ks3/units/probability-possible-outcomes/lessons/equally-likely-outcomes","timestamp":"2024-11-14T07:20:42Z","content_type":"text/html","content_length":"302863","record_id":"<urn:uuid:f5136fa5-f67c-41c0-b30d-f010f2611752>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00126.warc.gz"} |
Lesson 4: How Far is it from Typical?
Lesson 4: How Far is it from Typical?
Students will understand that the mean of the absolute deviations (MAD) is a way to assess the degree of variation in the data from the mean and adjusts for differences in the number of points in the
data set (n). The MAD measures the total distance between all the data values from the mean and divides it by the number of observations in the data set.
1. Masking tape (or painter’s tape) – approximately 4-5 feet long – one for each student team
2. How Far Apart? handout (LMR_2.6_How Far Apart) – will be used again in Lesson 17
3. Exported, printed, and reproduced class’s Personality Color survey data
measures of variability (or spread) deviation mean of absolute deviations (MAD)
Essential Concepts:
Essential Concepts:
MAD measures the variability in a sample of data - the larger the value, the greater the variability. More precisely, the MAD is the typical distance of observations from the mean. There are other
measures of spread as well, notably the standard deviation and the interquartile range (IQR).
1. Remind students that they learned about 2 different measures of center during the previous 2 lessons: the mean and the median. Have the students recall when it is appropriate to use each value
based on the shape of the distribution.
1. Mean – use with symmetric distributions.
2. Median – use with skewed distributions or when there are outliers.
2. Inform the students that, during today’s lesson, they will learn about measures of variability – also known as measures of spread. These values show us how much the quantitative data varies from
the center of a distribution. Similar to measures of center, we will use two different measures of spread: (1) the mean of absolute deviations (MAD), and (2) the interquartile range (IQR).
Note: IQR will be discussed in detail during Lesson 5.
3. Introduce the term deviation. Using Think, Pair, Share, ask students what they think this word means and how it could relate to variability. A deviation is the act of departing from an
established course or accepted standard. Common synonyms include departure, detour, difference, digression, divergence, fluctuation, inconsistency, modification, shift, etc.
4. On the classroom floor next to each student team, place a 4-5 foot long piece of masking tape (or painter’s tape). Then, propose the following scenario:
Your team has been invited to guest star at the circus! You have been asked to perform as part of the tightrope act – a routine that requires tremendous focus and balance to walk across a tightly
pulled rope that is suspended high in the air. In order to practice your balancing skills, the circus has provided your team with a line of tape that will represent the tightrope.
5. Have the students consider the piece of tape (aka the rope) to be the “typical” path they must take to finish the circus act. Since they do not want to fall from the suspended tightrope while
performing at the actual circus, they will need to practice walking directly on the middle of the line at all times. If they deviate from the line, they will no longer be walking the “typical”
path, and will likely fall.
6. Each team should select one student to be their starting performer.
7. In teams of 4, one student is the performer, two are measuring the distance of the deviation (one on each side of the tape), and one is the recorder.
8. Place a ruler perpendicular to the “rope” and measure the distance, in centimeters, from the path to the center of the back of their heel as the student walks and attempts to balance across the
9. The performer will walk the tightrope by looking straight up to the sky – first they look to place a foot on the line, then walk naturally while looking up to the sky, and repeating one step at a
time for 4 steps, measuring after each step. Any time the performer missteps, this is considered a variation from the typical value. You can have students take turns so everyone gets a chance to
balance, walk, and to measure, depending on time in your class.
10. Now that the students have an idea about what it means to deviate from something they consider “typical,” they can start looking at distributions to see how data points vary from their typical
11. Inform students that they were observing deviations from typical while calculating actual differences between the rope and the performer’s steps. When data are quantified with numbers, we can
then calculate how far away each value is from the center.
12. One such calculation that is popular among data scientists is the mean of absolute deviations (MAD). Ask students to consider the components of the MAD in math terms, and brainstorm what the MAD
value might represent.
mean – an average
absolute – in mathematics, we talk about absolute value, the positive difference between 2 numerical values
deviation – as discussed earlier in the lesson, deviation represents how much things vary
13. Using the 3 components in Step 12, explain that the MAD measures the absolute distance of each data point from the mean, and then finds the average of all those distances.
14. Display the formula for the MAD distribution for the whole class to see.
15. Discuss what each symbol in the formula means and how we use it to perform the calculation. x[i] represents each individual data point, x̄ represents the mean value, and n represents the total
number of observations. The symbol Σ represents the summation – this tells us to add up all the absolute distances from each point to the mean.
16. To practice using this formula with actual data, students will calculate and compare the MAD values for 2 distributions.
17. Distribute the How Far Apart? handout (LMR_2.6), which contains 2 of the dot plots - plots (a) and (c) from the Where is the Middle? handout (LMR_2.5) used in Lesson 3. As before, the dot plots
depict the number of candies eaten by a group of 17 high school students on different days of the week. The means are also given.
The calculations for each plot are shown below for the teacher’s reference.
MAD for plot (a)
MAD for plot (c) \begin{align*}&space;MAD&space;&=&space;\frac{3|0-2.53|+0|1-2.53|+4|2-2.53|+5|3-2.53|+5|4-2.53|}{17}&space;\\&space;&=&space;\frac{3(2.53)+0(1.53)+4
18. Students may work in pairs to complete the handout. After all student pairs have come to an agreement on their answers, pose the following questions to the class as a whole:
1. Which MAD value did you think would be larger based only on the look/shape of the distributions? Why? Since plot (c) is skewed to the left, it probably has a larger MAD because more points
will be further away from the mean than in plot (a).
2. Which MAD value was actually larger when you calculated it? The MAD value for plot (c) was larger (1.1418 > 0.8253).
3. Did your prediction match the actual calculated values, or were you surprised by the results? Yes. The distribution with the wider spread (more variability) had the larger MAD value.
19. To continue exploring with the class’s Personality Color survey data, student teams should calculate the MAD value for their Blue scores. Does the MAD value seem reasonable based on the dot plot
they created during Lesson 2?
Class Scribes:
One team of students will give a brief talk to discuss what they think the 3 most important topics of the day were.
Homework & Next Day
Students should calculate the MAD values for each of the other 3 personality color scores and compare the values of the 4 color scores.
Declutter your Environment Pane
Unit 2 utilizes new datasets so it's a good idea to declutter your Environment Pane.
Refer students to this video or demonstrate it on your own Environment.
LAB 2A: All About Distributions
Complete Lab 2A prior to Lesson 5. | {"url":"https://curriculum.idsucla.org/unit2/lesson4/","timestamp":"2024-11-11T06:26:47Z","content_type":"text/html","content_length":"81186","record_id":"<urn:uuid:ff418dc1-4a0b-4a2f-8372-2c72f18cfc57>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00885.warc.gz"} |
Re: [AMBER] MMPBSA Entropies: N-mode and Quasi Harmonic calculations
From: Jason Swails <jason.swails.gmail.com> Date: Tue, 17 Jul 2012 08:58:46 -0400
They are two completely different methods for calculating vibrational
entropies. One relies more or less on PCA vectors to describe vibrations
(with the corresponding eigenvalues as the vibrational frequencies). The
other relies on a normal mode approximation whereby the Hessian is
diagonalized to obtain frequencies.
To illustrate how these methods could give drastically different entropies,
imagine a large free energy well whose fine-grained structure has many
local minima (the best example I could find in 3 seconds is here:
). As
long as the barriers between the minima are relatively low, you will get
vibrational frequencies that look like they come from the vibrational
frequencies derived from the large, single free energy well. Nmode
entropies, on the other hand, will appear as though they come directly from
whatever local free energy minimum they were minimized to (so a local
minima, rather than the global well). I think these would give quite
different answers, and would help explain why quasi-harmonic calculations
take so many frames to converge.
Also, the solvation method you used for the normal mode analysis is almost
certainly different than what you used to propagate the dynamics. Ergo,
even if your system stayed in the same free energy well in a single
simulation, the shape of that well would be different between vacuum/igb=1
(the only two options for nmode) and that of the original simulation
(likely explicit solvent). As a result, you should expect different
If you want to get the *same* answers, run a simulation in implicit solvent
(igb=1) at very low temperatures (20-50K), and do your quasi-harmonic calc
with those frames. The low temperature should force you to stay in a
single well. Then, perform a normal mode calculation on a minimized
structure (making sure to adjust the temperature in both the ptraj code and
nab/sff code to get the right values, or calculate the entropy of each mode
by hand given the frequencies). I think I recall a colleague of mine
running this exact test and finding they converged to the same answer (but
it was a long time ago and the details are long buried :))
On Tue, Jul 17, 2012 at 7:38 AM, George Tzotzos <gtzotzos.me.com> wrote:
> I would be grateful if anyone could provide an explanation regarding the > huge differences in total Delta S given by the two methods. > > Below are the results obtained for exactly the same
trajectories > > Solvated complex topology file: 2wc6_bom_solv.prmtop > |Complex topology file: 2wc6_bom.prmtop > |Receptor topology file: 2wc6.prmtop > |Ligand topology file: bom.prmtop > |Initial
mdcrd(s): prod_10ns.mdcrd > | prod_2ns.mdcrd > | prod_4ns.mdcrd > | prod_6ns.mdcrd > | prod_8ns.mdcrd > | > |Best guess for receptor mask: ":1-141" > |Best guess for ligand mask: ":142" > |Ligand
residue name is "BOM" > | > |Calculations performed using 1000 frames. > |NMODE calculations performed using 20 frames. > > DELTA S total= -21.7086 +/- 1.5570 > > ENTROPY RESULTS (QUASI-HARMONIC
APPROXIMATION) CALCULATED WITH PTRAJ: > > DELTA S: -12.5924 -10.2513 -56.3566 -79.2006 > > > Many thanks in advance > > George > > > _______________________________________________ > AMBER mailing
list > AMBER.ambermd.org > http://lists.ambermd.org/mailman/listinfo/amber >
Jason M. Swails
Quantum Theory Project,
University of Florida
Ph.D. Candidate
AMBER mailing list
Received on Tue Jul 17 2012 - 06:00:02 PDT | {"url":"http://archive.ambermd.org/201207/0319.html","timestamp":"2024-11-08T02:31:53Z","content_type":"application/xhtml+xml","content_length":"12558","record_id":"<urn:uuid:342e0c25-2955-4a19-bf3a-16a88d686f7c>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00449.warc.gz"} |
Using the MATLAB Unit Testing Infrastructure for Grading Assignments
Steven Lord, Andy Campbell, and David Hruska are members of the Quality Engineering group at MathWorks who are guest blogging today to introduce a new feature in R2013a, the MATLAB unit testing
infrastructure. There are several submissions on the MATLAB Central File Exchange related to unit testing of MATLAB code. Blogger Steve Eddins wrote one highly rated example back in 2009. In release
R2013a, MathWorks included in MATLAB itself a MATLAB implementation of the industry-standard xUnit testing framework.
If you're not a software developer, you may be wondering if this feature will be of any use to you. In this post, we will describe one way someone who may not consider themselves a software developer
may be able to take advantage of this framework using the example of a professor grading students' homework submissions. That's not to say that the developers in the audience should move on to the
next post; you can use these tools to test your own code just like a professor can use them to test code written by his or her students.
There is a great deal of functionality in this feature that we will not show here. For more information we refer you to the MATLAB Unit Testing Framework documentation.
In order to use this feature, you should be aware of how to define simple MATLAB classes in classdef files, how to define a class that inherits from another, and how to specify attributes for methods
and properties of those classes. The object-oriented programming documentation describes these capabilities.
Problem Statement
As a professor in an introductory programming class, you want your students to write a program to compute Fibonacci numbers. The exact problem statement you give the students is:
Create a function "fib" that accepts a nonnegative integer n and returns
the nth Fibonacci number. The Fibonacci numbers are generated by this
F(0) = 1
F(1) = 1
F(n) = F(n-1) + F(n-2) for integer n > 1
Your function should throw an error if n is not a nonnegative integer.
Basic Unit Test
The most basic MATLAB unit test is a MATLAB classdef class file that inherits from the matlab.unittest.TestCase class. Throughout the rest of this post we will add additional pieces to this basic
framework to increase the capability of this test and will change its name to reflect its increased functionality.
dbtype basicTest.m
1 classdef basicTest < matlab.unittest.TestCase
3 end
test = basicTest
test =
basicTest with no properties.
Running a Test
To run the test, we can simply pass test to the run function. There are more advanced ways that make it easier to run a group of tests, but for our purposes (checking one student's answer at a time)
this will be sufficient. When you move to checking multiple students' answers at a time, you can use run inside a for loop.
Since basicTest doesn't actually validate the output from the student's function, it doesn't take very long to execute.
results = run(test)
results =
0x0 TestResult array with properties:
0 Passed, 0 Failed, 0 Incomplete.
0 seconds testing time.
Let's say that a student named Thomas submitted a function fib.m as his solution to this assignment. Thomas's code is stored in a sub-folder named thomas. To set up our test to check Thomas's answer,
we add the folder holding his code to the path.
dbtype fib.m
1 function y = fib(n)
2 if n <= 1
3 y = 1;
4 else
5 y = fib(n-1)+fib(n-2);
6 end
Test that F(0) Equals 1
The basicTest is a valid test class, and we can run it, but it doesn't actually perform any validation of the student's test file. The methods that will perform that validation need to be written in
a methods block that has the attribute Test specified.
The matlab.unittest.TestCase class includes qualification methods that you can use to test various qualities of the results returned by the student files. The qualification method that you will
likely use most frequently is the verifyEqual method, which passes if the two values you pass into it are equal and reports a test failure if they are not.
The documentation for the matlab.unittest.TestCase class lists many other qualification methods that you can use to perform other types of validation, including testing the data type and size of the
results; matching a string result to an expected string; testing that a given section of code throws a specific errors or issues a specific warning; and many more.
This simple test builds upon generalTest by adding a test method that checks that the student's function returns the value 1 when called with the input 0.
dbtype simpleTest.m
1 classdef simpleTest < matlab.unittest.TestCase
2 methods(Test)
3 function fibonacciOfZeroShouldBeOne(testCase)
4 % Evaluate the student's function for n = 0
5 result = fib(0);
6 testCase.verifyEqual(result, 1);
7 end
8 end
9 end
Thomas's solution to the assignment satisfies this basic check. We can use the results returned from run to display the percentage of the tests that pass.
results = run(simpleTest)
percentPassed = 100 * nnz([results.Passed]) / numel(results);
disp([num2str(percentPassed), '% Passed.']);
Running simpleTest
Done simpleTest
results =
TestResult with properties:
Name: 'simpleTest/fibonacciOfZeroShouldBeOne'
Passed: 1
Failed: 0
Incomplete: 0
Duration: 0.0112
1 Passed, 0 Failed, 0 Incomplete.
0.011168 seconds testing time.
100% Passed.
Test that F(pi) Throws an Error
Now that we have a basic positive test in place we can add in a test that checks the behavior of the student's function when passed a non-integer value (like n = pi) as input. The assignment stated
that when called with a non-integer value, the student's function should error. Since the assignment doesn't require a specific error to be thrown, the test passes as long as fib(pi) throws any
dbtype errorCaseTest.m
1 classdef errorCaseTest < matlab.unittest.TestCase
2 methods(Test)
3 function fibonacciOfZeroShouldBeOne(testCase)
4 % Evaluate the student's function for n = 0
5 result = fib(0);
6 testCase.verifyEqual(result, 1);
7 end
8 function fibonacciOfNonintegerShouldError(testCase)
9 testCase.verifyError(@()fib(pi), ?MException);
10 end
11 end
12 end
Thomas forgot to include a check for a non-integer valued input in his function, so our test should indicate that by reporting a failure.
results = run(errorCaseTest)
percentPassed = 100 * nnz([results.Passed]) / numel(results);
disp([num2str(percentPassed), '% Passed.']);
Running errorCaseTest
Verification failed in errorCaseTest/fibonacciOfNonintegerShouldError.
Framework Diagnostic:
verifyError failed.
--> The function did not throw any exception.
Expected Exception Type:
Evaluated Function:
Stack Information:
In C:\Program Files\MATLAB\R2013a\toolbox\matlab\testframework\+matlab\+unittest\+qualifications\Verifiable.m (Verifiable.verifyError) at 637
In H:\Documents\LOREN\MyJob\Art of MATLAB\errorCaseTest.m (errorCaseTest.fibonacciOfNonintegerShouldError) at 9
Done errorCaseTest
Failure Summary:
Name Failed Incomplete Reason(s)
errorCaseTest/fibonacciOfNonintegerShouldError X Failed by verification.
results =
1x2 TestResult array with properties:
1 Passed, 1 Failed, 0 Incomplete.
0.026224 seconds testing time.
50% Passed.
Another student, Benjamin, checked for a non-integer value in his code as you can see on line 2.
dbtype fib.m
1 function y = fib(n)
2 if (n ~= round(n)) || n < 0
3 error('N is not an integer!');
4 elseif n == 0 || n == 1
5 y = 1;
6 else
7 y = fib(n-1)+fib(n-2);
8 end
Benjamin's code passed both the test implemented in the fibonacciOfZeroShouldBeOne method (which we copied into errorCaseTest from simpleTest) and the new test case implemented in the
fibonacciOfNonintegerShouldError method.
results = run(errorCaseTest)
percentPassed = 100 * nnz([results.Passed]) / numel(results);
disp([num2str(percentPassed), '% Passed.']);
Running errorCaseTest
Done errorCaseTest
results =
1x2 TestResult array with properties:
2 Passed, 0 Failed, 0 Incomplete.
0.010132 seconds testing time.
100% Passed.
Basic Test for Students, Advanced Tests for Instructor
The problem statement given earlier in this post is a plain text description of the homework assignment we assigned to the students. We can also state the problem for the students in code (if they're
using release R2013a or later) by giving them a test file they can run just like simpleTest or errorCaseTest. They can directly use this "requirement test" to ensure their functions satisfy the
requirements of the assignment.
dbtype studentTest.m
1 classdef studentTest < matlab.unittest.TestCase
2 methods(Test)
3 function fibonacciOfZeroShouldBeOne(testCase)
4 % Evaluate the student's function for n = 0
5 result = fib(0);
6 testCase.verifyEqual(result, 1);
7 end
8 function fibonacciOfNonintegerShouldError(testCase)
9 testCase.verifyError(@()fib(pi), ?MException);
10 end
11 end
12 end
In order for the student's code to pass the assignment, it will need to pass the test cases given in the studentTest unit test. However, we don't want to use studentTest as the only check of the
student's code. If we did, the student could write their function to cover only the test cases in the student test file.
We could solve this problem by having two separate test files, one containing the student test cases and one containing additional test cases the instructor uses in the grading process. Can we avoid
having to run both test files manually or duplicating the code from the student test cases in the instructor test? Yes!
To do so, we write an instructor test file to incorporate, through inheritance, the student test file. We can then add additional test cases to the instructor test file. When we run this test it
should run three test cases; two inherited from studentTest, fibonacciOfZeroShouldBeOne and fibonacciOfNonintegerShouldError, and one from instructorTest itself, fibonacciOf5.
dbtype instructorTest.m
1 classdef instructorTest < studentTest
2 % Because the student test file is a matlab.unittest.TestCase and
3 % instructorTest inherits from it, instructorTest is also a
4 % matlab.unittest.TestCase.
6 methods(Test)
7 function fibonacciOf5(testCase)
8 % Evaluate the student's function for n = 5
9 result = fib(5);
10 testCase.verifyEqual(result, 8, 'Fibonacci(5) should be 8');
11 end
12 end
13 end
Let's look at Eric's test file that passes the studentTestFile test, but in which he completely forgot to implement the F(n) = F(n-1)+F(n-2) recursion step.
dbtype fib.m
1 function y = fib(n)
2 if (n ~= round(n)) || n < 0
3 error('N is not an integer!');
4 end
5 y = 1;
It should pass the student unit test.
results = run(studentTest);
percentPassed = 100 * nnz([results.Passed]) / numel(results);
disp([num2str(percentPassed), '% Passed.']);
Running studentTest
Done studentTest
100% Passed.
It does NOT pass the instructor unit test because it fails one of the test cases.
results = run(instructorTest)
percentPassed = 100 * nnz([results.Passed]) / numel(results);
disp([num2str(percentPassed), '% Passed.']);
Running instructorTest
Verification failed in instructorTest/fibonacciOf5.
Test Diagnostic:
Fibonacci(5) should be 8
Framework Diagnostic:
verifyEqual failed.
--> NumericComparator failed.
--> The values are not equal using "isequaln".
Actual Value:
Expected Value:
Stack Information:
In C:\Program Files\MATLAB\R2013a\toolbox\matlab\testframework\+matlab\+unittest\+qualifications\Verifiable.m (Verifiable.verifyEqual) at 411
In H:\Documents\LOREN\MyJob\Art of MATLAB\instructorTest.m (instructorTest.fibonacciOf5) at 10
Done instructorTest
Failure Summary:
Name Failed Incomplete Reason(s)
instructorTest/fibonacciOf5 X Failed by verification.
results =
1x3 TestResult array with properties:
2 Passed, 1 Failed, 0 Incomplete.
0.028906 seconds testing time.
66.6667% Passed.
Benjamin, whose code we tested above, wrote a correct solution to the homework problem.
results = run(instructorTest)
percentPassed = 100 * nnz([results.Passed]) / numel(results);
disp([num2str(percentPassed), '% Passed.']);
Running instructorTest
Done instructorTest
results =
1x3 TestResult array with properties:
3 Passed, 0 Failed, 0 Incomplete.
0.015946 seconds testing time.
100% Passed.
In this post, we showed you the basics of using the new MATLAB unit testing infrastructure using homework grading as a use case.
We checked that the student's code worked (by returning the correct answer) for one valid value and worked (by throwing an error) for one invalid value. We also showed how you can use this
infrastructure to provide an aid/check for the students that you can also use as part of your grading.
We hope this brief introduction to the unit testing framework has shown you how you can make use of this feature even if you don't consider yourself a software developer. Let us know in the comments
for this post how you might use this new functionality. Or, if you've already tried using matlab.unittest, let us know about your experiences here.
Published with MATLAB® R2013a
To leave a comment, please click here to sign in to your MathWorks Account or create a new one. | {"url":"https://blogs.mathworks.com/loren/2013/03/14/using-the-matlab-unit-testing-infrastructure-for-grading-assignments/?s_tid=blogs_rc_2&from=en","timestamp":"2024-11-07T05:40:00Z","content_type":"text/html","content_length":"192893","record_id":"<urn:uuid:9c5b3a5d-b3fd-4f86-90b0-8e86dd472e34>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00449.warc.gz"} |
Q9: Answers – Paper 2 June 19 – AQA GCSE Maths Higher
Helpful Links
Mark Scheme
In a sport, injury time is added time played at the end of a match.
The table shows the injury time, t (minutes) played in 380 matches.
a) Circle the two words that describe the data.
b) Which class interval contains the median?
You must show your working.
c) What percentage of the matches had more than 6 minutes of injury time?
Click here for a printable version of this question!
In a sport, injury time is added time played at the end of a match.
The table shows the injury time,
(minutes) played in 380 matches.
a) Circle the two words that describe the data.
[1 mark]
b) Which class interval contains the median?
You must show your working.
[2 marks]
c) What percentage of the matches had more than 6 minutes of injury time?
[2 marks] | {"url":"https://www.elevise.co.uk/kd9a.html","timestamp":"2024-11-03T09:51:08Z","content_type":"text/html","content_length":"96964","record_id":"<urn:uuid:580b7303-1156-40c2-a8f1-329d6f2329f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00695.warc.gz"} |
Tutoring Business Calculus, Linear Equations in Business Situations
We spent some time looking at linear equations in business situations. Profit, cost, revenue, and demand. They work like similar problems, just with business concepts incorporated. They tend to use
letters in place of x and y, but they play the same roles – independent and dependent variables. Be careful to not switch the order of numbers.
When getting a vertex, it’s usually helpful to have the equation in the standard order so the coefficients also are in the order you expect.
When graphing a parabola, one approach that can be useful (if possible) is to get the x-intercepts by factoring and get the vertex using the standard form. | {"url":"https://nealien.com/tutoring-business-calculus-linear-equations-in-business-situations/","timestamp":"2024-11-14T11:20:08Z","content_type":"text/html","content_length":"27155","record_id":"<urn:uuid:ff2048d9-5b19-4a0e-99ff-cd20cf09db25>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00762.warc.gz"} |
5.4 – Using MCMC To Interpret Single-Cell Experiments (Dr. Huy Vo)
Title: Tutorial — MCMC Methods for Quantification of Parameter Uncertainties
Lecturer: Dr. Huy Vo
Lecturer Website: https://www.engr.colostate.edu/~munsky/
Lecturer Email: huy.vo@colostate.edu
Learning Objectives:
☆ Learn how to run an MCMC sampling algorithm to quantify model uncertainty given experimental data
Dr. Huy Vo is a postdoctoral researcher in the Munsky Group at Colorado State University. He earned a Ph.D. degree in Mathematics at the University of Alabama in 2017. His current research focuses on
developing new computational tools to design informative single-cell experiments that account for both intrinsic noise and measurement uncertainty. Other interests: parameter estimation, uncertainty
quantification, model reduction for stochastic gene expression models, software development.
Title: Basics of Probability Distributions and Statistics for Single-Cell Data
Abstract: abc abc abc
Suggested Reading or Key Publications:
Links to Relevant Software:
□ A Gene is ON at some time 𝑡=0. It can turn OFF at a stochastic rate of 5/min*. It can also create one mRNA at a time at a stochastic rate of 20/min. What is the distribution of mRNA created
before the gene turns OFF?
□ A Gene is ON at some time 𝑡=0. It later turns OFF at exactly 𝑡=1/5 min. It can also create one mRNA at a time at a stochastic rate of 20/min. What is the distribution of mRNA created before
the gene turns OFF?
□ Why are the two random variables above different? Which is more variable?
□ Consider 2 genes that are both ON and both can turn OFF with a stochastic rate of 5/min.
☆ What is the distribution of time until the first of these genes turn OFF?
☆ What is the distribution of time until BOTH of these genes turn OFF?
□ A Gene is ON at some time t=0. It can turn OFF at a stochastic rate of 5/min. What is the probability that it is still ON at a time t = 1 min?
□ What is the Fano Factor of a Poisson random variable? Of an exponential random variable? How do these depend on the mean of the random variables?
□ What is the Coefficient of Variation (std/mean) of a Poisson random variable? Of an exponential random variable? How do these depend on the mean of the random variable?
□ Consider two independent normal distributed random variables both with mean of 2 and a standard deviation of 1. What is the distribution of the sum of pairs of these two random variables?
What is its mean? What is its standard deviation?
□ Consider two identical (non-independent) normal distributed random variables both with mean of 2 and a standard deviation of 1. What is the distribution of the sum of pairs of these two
random variables? What is its mean? What is its standard deviation?
□ When will the Central Limit Theorem fail to work? | {"url":"https://q-bio.org/wp/uqbio_21_lecture_5p4_vo/","timestamp":"2024-11-09T13:32:28Z","content_type":"text/html","content_length":"53423","record_id":"<urn:uuid:b27a9283-eb98-4218-8a1a-7f9b0c623dae>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00531.warc.gz"} |
Rust Operators : The Four Pillars of Empire
As we know, Operators are Key Pillars of every programming language, Instead of talking about other things, Let’s talk about only Rust Operators the four pillars of empire. Sounds weird? I know
because you are in TechAlgoSpotlight. We only do weird things 😅
Basically, operators are nothing. They used to do various types of operations in values or maybe variables. So, Rust also came with this type of common functionality in four parts. Arithmetic
Operators, Logical Operators, Comparison Operators and Compound Assignment Operators.
Are you ready to see each of them? Let’s get started with wasting a lot of time. 😉
!Rust Operators Okay TechAlgoSpotlight
1. Rust with Arithmetic Operators
As we know everywhere arithmetic operators are only used for addition, subtraction, multiplication and division. Rust did the same things without any fear.
Let’s think about two variables, the First X and the Second Y. Let me show you something in table form. I know, you know these things, But as a responsible admin, I have to show you, How many types
of arithmetic operators are available in Rust.
Operators Using What We Can Do
+ (Addition) X + Y
- (Subtraction) X – Y
* (Multiplication) X * Y
/ (Division) X / Y
% (Remainder) X % Y
Rust with Addition, Subtraction and Multiplication Operators
We will see the Division operator after some time. The longer you wait, The more you get. 😂
fn main() {
let a = 18;
let b = 2;
// add two variables using + operator
let x = a + b;
println!("{} + {} = {}", a, b, x);
// subtract two variables using - operator
let y = a - b;
println!("{} - {} = {}", a, b, y);
// multiply two variables using * operator
let z = a * b;
println!("{} * {} = {}", a, b, z);
Output (Result)
18 + 2 = 20
18 - 2 = 16
18 * 2 = 36
Have seen magic? Rust just did with giving us the output. Let’s see some more magical things with Division Operators.
Rust with Division Operator
fn main() {
let dividend = 21;
let divisor = 8;
// arithmetic division using / operator with integers
let division = dividend / divisor;
println!("{} / {} = {}", dividend, divisor, division);
As per the code, you can understand. We did just a simple division with / Operators. But wait here is the twist come.
Regular calculation gives us 2.625. But in Rust, When we use division operators for operation. Instead of a floating value, they give us an Int value (2). I don’t know why Rust acts like this But
maybe internally they like JavaSciprt.
Anyway, What if we want float values with a division operator? So, silently Rust introduced a trick to achieve floating values by passing floating integers.
Spotlight: Floating Integer is like x = 2 works as an integer but x = 2.0 works as an floating integer.
Let’s take a small example while closing our eyes.
fn main() {
let dividend = 21.0;
let divisor = 8.0;
// arithmetic division using / operator with floating point values
let division = dividend / divisor;
println!("{} / {} = {}", dividend, divisor, division);
As we discussed in Spotlight. Floating Integer is nothing but a copycat of Float.
By assigning dividend and divisor as a float value. We got a float value, Simple. Now, we have to learn the remainder. So let’s see how remainder operators work.
Rust with Remainder Operator
fn main() {
let dividend = 21;
let divisor = 8;
// arithmetic remainder using % operator
let remainder = dividend % divisor;
println!("{} % {} = {}", dividend, divisor, remainder);
As we don’t know the remainder, So, let’s understand the code first. Instead of using other arithmetic operators, They used %. It works like the operator name. Just give us reminders. Nothing Else.
I don’t think so any other arithmetic operators remaining. 🤔 Let’s see Assignment Operators with a cup of tea.
2. Rust with Assignment Operator
Assignment operator used for assigning values. That’s it!!! 🤣
Here we are using = for assigning 007 to bobbyDeol. Maybe in the future bobbyDeol become 007. But as syntax vice, it’s 007. Assignment operators are like I am giving you something, Hold it until you
Do you have questions like, What if we have to change the value of a variable? Let’s see how we can do by putting the small keyword mut (Mutable).
As we discussed bobbyDeol example of assigning 007. Sowe can not change. But after using mut keyword, We can change bobbyDeol to 700.
Now mut keyword is introduced and I would like to tell you more about this. let me share one example with you.
Rust Compound Assignment Operators
The simplest definition of this operator is we can do some operations with shorter syntax. Let me show you How we can do in Rust.
let mut x = 1;
// compound assignment operators
x += 3;
Rust has some fancy terms. Looks like, they use almost JavaScript Syntax. 😆 Let me share some of this fancy operators list with you.
Operator Example Equivalent To
+= (addition assignment) a += b a = a + b
-= (subtraction assignment) a -= b a = a – b
*= (multiplication assignment) a *= b a = a * b
/= (division assignment) a /= b a = a / b
%= (remainder assignment) a %= b a = a % b
Let’s wrap this operator with our cup of tea. Because you need coffee to understand the next operator.
3. Rust Comparison Operators
The heading says everything. Explanation is not required here.
This is called > (greater than). This operator is only required when you have to compare something. After comparison, the Output will be either true or false. You can this operator as a relational
• TRUE : If Condition Satisfied.
• FALSE : If Condition Not Satisfied.
Let’s list out some of the comparison operators
Operator Example Description
> (Greater than) a > b true if a is greater than b
< (Less than) a < b true if a is less than b
>= (Greater than or equal to) a >= b true if a is greater than or equal to b
<= (Less than or equal to) a <= b true if a is less than or equal to b
== (Equal to) a == b true if a is equal to b
!= (Not equal to) a != b true if a is not equal to b
Oh, We forgot the example of comparison operators. Let me share 😅
Rust Comparison Operators Example
fn main() {
let a = 7;
let b = 3;
// use of comparison operators
let c = a > b;
let d = a < b;
let e = a == b;
println!("{} >= {} is {}", a, b, c);
println!("{} <= {} is {}", a, b, d);
println!("{} == {} is {}", a, b, e);
7 > 3 is true
7 < 3 is false
7 == 3 is false
As we know if the condition is satisfied then TRUE, Otherwise FALSE. That’s all about comparison operators. Actually, I don’t know much about this operator because I don’t like it. Simple 🫠🫠
I think we have to conclude with our last operator. The logic killer operator. It’s called Logical Operator.
4. Rust Logical Operators
Find a second coffee, Because this operator required energy to understand. As you know words say everything. But the tricky part is, It’s not about logical things. It’s all about logical plus
comparison. Let me show you how this works.
It’s like comparing two logic. && This is an AND operator. AND Operator returns TRUE If both conditions are satisfied.
I know only three logical conditions 😅, Let me share them one by one quickly via table
• Operator: && (Logical AND)
□ Example: exp1 && exp2
□ Description: returns true if both exp1 and exp2 are true
□ Description: returns true if any one of the expressions is true
• Operator: ! (Logical NOT)
□ Example: !exp
□ Description: returns true if the expression is false and returns false, if it is true
Rust Logical Operators Example
fn main() {
let a = true;
let b = false;
// logical AND operation
let c = a && b;
// logical OR operation
let d = a || b;
// logical NOT operation
let e = !a;
println!("{} && {} = {}", a, b, c);
println!("{} || {} = {}", a, b, d);
println!("!{} = {}", a, e);
true && false = false
true || false = true
!true = false
Basically, && Operater Works If both condition is TRUE, ** ** Operartor works if one or more condition is TRUE and ! Operartor used for when we don’t want to satisify condition forcefully.
That’s It!! 🙌🙌
Today we show four pillars of Operators.
• Arithmetic Operators
• Compound Assignment Operators
• Logical Operators
• Comparison Operators (Relational Operators)
We show the each of the operators examples with 2 cup of Coffee and 1 Cup of Tea. Why i am telling you because reading is easy but make content more readble and understandable is little difficult.🥲
Insted of sharing my childhood troma let’s wrap up now!!. We will see Rust conditions in upcomming articles. | {"url":"https://techalgospotlight.com/rust-operators-the-four-pillars-of-empire/","timestamp":"2024-11-04T17:10:13Z","content_type":"text/html","content_length":"296840","record_id":"<urn:uuid:f2d38d60-247c-42b3-95a3-e1bbd54e956c>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00734.warc.gz"} |
Innovation: Maths Clocks and Time Management
A maths watch is a timepiece that uses mathematical expressions and equations to display the current time. Unlike traditional analog or digital clocks, maths clocks offer a creative and
unconventional way to represent time. Maths clocks replace traditional hour and minute hands or numerical numbers with mathematical symbols, equations, or numerical representations. These symbols and
equations act as visual indicators of time, requiring users to interpret and calculate current hour and minute values.
The maths clock challenges the familiar format of timing and encourages individuals to engage their mathematical thinking skills. They provide a new way of perceiving time and offer a visual and
mental exercise that can be both intriguing and educational. Maths clocks come in a variety of designs, from binary clocks that use binary codes to represent time to base-12 clocks that divide the
day into 12 units instead of 24. Some maths clocks have equations, formulas or mathematical symbols to display time. Each design brings its own unique style and adds to the overall aesthetic appeal
of the maths watch.
Today’s society is characterized by rapid technological advances, increased connectivity, and fast-paced lifestyles. People often find themselves juggling multiple responsibilities, deadlines and
commitments at once. With so much to do and limited time available, effective time management becomes important. Without proper time management, individuals can feel stressed, overwhelmed and
constantly behind on work.
Managing time efficiently has numerous benefits. It allows individuals to get more done in less time, reducing stress levels, increasing productivity and improving overall well-being. Effective time
management enables individuals to prioritize work, make better decisions and maintain a healthy work-life balance. Effective time management is closely tied to achieving personal and professional
goals. When time is managed effectively, individuals have the opportunity to focus on important tasks, invest in self-improvement, and make progress toward their desired outcomes. It improves
productivity, efficiency, and effectiveness in various aspects of life.
Effective time management not only influences professional success, but also contributes to a higher quality of life. It allows individuals to allocate time for hobbies, leisure activities,
relationships, self-care, and personal growth. Managing time effectively allows people to live more fulfilling and balanced lives. In today’s fast-paced world, where time is a precious resource,
mastering time management is essential. It enables individuals to control their schedules, make the most of their time, and achieve a sense of accomplishment and satisfaction both personally and
The Basics of Maths Clocks
Maths Clocks introduces a unique way of displaying time by replacing conventional hour and minute hands or numerical numbers with mathematical expressions and equations. Maths clocks use different
mathematical symbols to represent different elements of time. For example, instead of using traditional hour hands, maths clocks can use equations or mathematical symbols such as π (pi) or √2 to
indicate the current hour.
To read the time on a maths clock, the displayed equations or expressions should be interpreted. Each equation represents a specific time value, and users must decipher the equation to determine the
corresponding hour or minute. Types of mathematical expressions or equations commonly used in maths clocks include simple equations like “2 + 3 = 5” to represent 2:35, or more complex equations
involving logarithms, trigonometric functions, or geometric patterns.
Reading time on a maths clock often requires mental computation to accurately decode the equations or expressions. Users must think mathematically to identify hour and minute values based on given
mathematical representations. A maths clock not only serves as a timing device but also provides opportunities to learn and practice mathematical skills. They encourage users to apply their
mathematical knowledge and improve their mental math abilities. The maths clock brings a sense of intellectual stimulation and aesthetic appeal to timing. By combining mathematics with the
representation of time, they offer a visually appealing and engaging way to view and interact with the concept of time.
Maths clocks come in different designs and variations, each using a distinct mathematical approach to representing time.
Binary clocks: Binary clocks use a binary number system to display time. Instead of a traditional hour and minute hands, they use rows or numbers of LED lights to represent the binary values of the
hours and minutes. Each row corresponds to a specific power of the two, and the lit LEDs indicate the “on” bits, which are then converted to decimal values for reading the time.
Base-12 Clock: The Base-12 clock divides the day into 12 units instead of the conventional 24 hours. These watches use the duodecimal system, where each hour represents a different symbol or number.
Some base-12 clocks may also include non-standard minute intervals, deviating further from the standard timekeeping system.
Equation-based clocks: Equation-based clocks use mathematical equations or formulas to represent time. Instead of displaying numeric numbers, these watches use mathematical expressions involving
variables, constants and arithmetic operations. To determine the current time, the user must solve the equations.
Other Variations: Beyond binary, base-12, and equation-based clocks, there are additional maths clock variants with their unique approach to time presentation. For example, there are watches that use
geometric patterns or shapes to indicate time or watches that incorporate advanced mathematical concepts like fractals or mathematical series.
Each type of maths watch offers its own aesthetic appeal and functional features. Some maths watch prioritize simplicity and readability, while others focus on intricate design and artistic elements.
Users can choose a maths clock that matches their mathematical interests, personal preferences and the level of complexity they desire.
The existence of different types of maths clocks demonstrates the creativity and versatility of combining timekeeping with math. It allows individuals to select maths watches that resonate with their
mathematical curiosity, enhance their time management skills and complements their personal style.
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2021 Fall AMC 12B Problems/Problem 9
Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$?
$\textbf{(A)} \: 9\pi \qquad\textbf{(B)} \: 12\pi \qquad\textbf{(C)} \: 18\pi \qquad\textbf{(D)} \: 24\pi \qquad\textbf{(E)} \: 27\pi$
Solution 1 (Cosine Rule)
Construct the circle that passes through $A$, $O$, and $C$, centered at $X$.
Also notice that $\overline{OA}$ and $\overline{OC}$ are the angle bisectors of angle $\angle BAC$ and $\angle BCA$ respectively. We then deduce $\angle AOC=120^\circ$.
Consider another point $M$ on Circle $X$ opposite to point $O$.
As $AOCM$ is an inscribed quadrilateral of Circle $X$, $\angle AMC=180^\circ-120^\circ=60^\circ$.
Afterward, deduce that $\angle AXC=2·\angle AMC=120^\circ$.
By the Cosine Rule, we have the equation: (where $r$ is the radius of circle $X$)
The area is therefore $\pi r^2 = \boxed{\textbf{(B)}\ 12\pi}$.
~Wilhelm Z
Solution 2
We have $\angle AOC = 120^\circ$.
Denote by $R$ the circumradius of $\triangle AOC$. In $\triangle AOC$, the law of sines implies $\[ 2 R = \frac{AC}{\sin \angle AOC} = 4 \sqrt{3} . \]$
Hence, the area of the circumcircle of $\triangle AOC$ is $\[ \pi R^2 = 12 \pi . \]$
Therefore, the answer is $\boxed{\textbf{(B) }12 \pi}$.
~Steven Chen (www.professorchenedu.com)
Solution 3
As in the previous solution, construct the circle that passes through $A$, $O$, and $C$, centered at $X$. Let $Y$ be the intersection of $\overline{OX}$ and $\overline{AB}$.
Note that since $\overline{OA}$ is the angle bisector of $\angle BAC$ that $\angle OAC=30^\circ$. Also by symmetry, $\overline{OX}$$\perp$$\overline{AB}$ and $AY = 3$. Thus $\tan(30^\circ) = \frac
{OY}{3}$ so $OY = \sqrt{3}$.
Let $r$ be the radius of circle $X$, and note that $AX = OX = r$. So $\triangle AYX$ is a right triangle with legs of length $3$ and $r - \sqrt{3}$ and hypotenuse $r$. By Pythagoras, $3^2 + (r - \
sqrt{3})^2 = r^2$. So $r = 2\sqrt{3}$.
Thus the area is $\pi r^2 = \boxed{\textbf{(B)}\ 12\pi}$.
Solution 5 (SIMPLE)
The semiperimeter is $\frac{6+6+6}{2}=9$ units. The area of the triangle is $9\sqrt{3}$ units squared. By the formula that says that the area of the triangle is its semiperimeter times its inradius,
the inradius $r=\sqrt{3}$. As $\angle{AOC}=120^\circ$, we can form an altitude from point $O$ to side $AC$ at point $M$, forming two 30-60-90 triangles. As $CM=MA=3$, we can solve for $OC=2\sqrt{3}$.
Now, the area of the circle is just $\pi*(2*\sqrt{3})^2 = 12\pi$. Select $\boxed{B}$.
~hastapasta, bob4108
Solution 6 (Ptolemy)
Call the diameter of the circle $d$. If we extend points $A$ and $C$ to meet at a point on the circle and call it $E$, then $\bigtriangleup OAE=\bigtriangleup OCE$ . Note that both triangles are
right, since their hypotenuse is the diameter of the circle. Therefore, $CE=AE=\sqrt{d^2-12}$. We know this since $OC=OA=OB$ and $OC$ is the hypotenuse of a $30-60-90$ right triangle, with the longer
leg being $\frac{6}{2}=3$ so $OC=2\sqrt{3}$. Applying Ptolemy's Theorem on cyclic quadrilateral $OCEA$, we get $2({\sqrt{d^2-12}})\cdot{2\sqrt{3}}=6d$. Squaring and solving we get $d^2=48 \
Longrightarrow (2r)^2=48$ so $r^2=12$. Therefore, the area of the circle is $\boxed{12\pi}$
Video Solution (Just 3 min!)
~Education, the Study of Everything
Video Solution by TheBeautyofMath
See Also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. | {"url":"https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_12B_Problems/Problem_9","timestamp":"2024-11-06T18:30:22Z","content_type":"text/html","content_length":"59827","record_id":"<urn:uuid:b81fd7d0-1793-4a0c-9181-608c0e6ac98f>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00735.warc.gz"} |
Math, Grade 7, Proportional Relationships, Identifying Proportional Relationships
A Park Ticket
A Park Ticket
Discuss the following with your classmates.
• At an amusement park in California, the cost of admission is $75.
• This table shows the costs of tickets for different numbers of adults.
• You can see from the table that the cost of the tickets is always 75 times the number of adults. You can express this relationship using the ratio 75:1.
• When the ratio between two varying quantities remains constant, the relationship between the two quantities is called a proportional relationship.
• The relationship between the number of adults and the cost of tickets is constant; thus, the relationship is a proportional relationship. | {"url":"https://oercommons.org/courseware/lesson/2956/student/?section=10","timestamp":"2024-11-08T23:58:15Z","content_type":"text/html","content_length":"47198","record_id":"<urn:uuid:83d7ff95-6f99-4525-9f29-31ce32313fb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00336.warc.gz"} |
Modulation and Transmission
Peter Dordal, Loyola University CS Department
How do we transmit sound (or any other analog or digital signal) using electromagnetic waves? We modulate the EM wave (the carrier) in order to encode the signal. The result no longer is a single
frequency; it is a band of frequencies. This spreading of the single-frequency carrier to a band of frequencies is fundamental.
If we are sending the signal a long way (over transmission lines), then we may very well also need to modulate some form of carrier. In the digital context, this is sometimes known as encoding.
Suppose we want to transmit a square wave, alternating from -1 to +1. We have a band-width of 4 Mhz. How does the data rate ("bandwidth" in the digital sense) compare to the band-width in the
spectrum sense? The simple modulation here is an attempt to generate a square wave out of sine waves.
Case 1: we use sin(2π ft) + (1/3)sin(2π (3f)t) + (1/5)sin(2π (5f)t) (that is, three terms of the Fourier series), where f = 1 Mhz.
Look at this with fourier.xls: does it look squarish?
The frequencies are 1 Mhz, 3 Mhz and 5 Mhz. The band-width is 5-1 = 4 Mhz.
The data rate is 2 Mbps (sort of; we are sending 1 M 1-bits and 1 M 0-bits per second)
Note dependence on notion of what waveform is "good enough"
Case 2: If we double all the frequencies to 2 MHz, 6 MHz, 10 MHz, we get band-width 8MHz, data rate 4Mbps. This is the same as above with f=2 Mhz.
Case 3: We decide in the second case that we do not need the 10 MHz component, due to a more accurate receiver. The base frequency is f = 2MHz, frequencies 2MHz, 6MHz, band-width is now 6MHz - 2MHz =
4MHz, data rate: 4Mbps
Look at this with fourier.xls.
Note that we're really not carrying data in a meaningful sense; we can't send an arbitrary sequence of 0's and 1's this way. However, that's done mostly to simplify things.
Note also the implicit dependence on bandwidth of the fact that we're decomposing into sinusoidal waves.
Voice transmission: frequency band-width ~ 3-4kHz (eg 300 Hz to 3300 Hz)
64Kbps encoding (8 bits sampled 8000 times a second)
Modems have the reverse job: given the band-width of ~3 kHz, they have to send at 56 kbps!
Note: we're looking at DISCRETE frequency spectrum (periodic signals). CONTINUOUS frequency spectrum also makes mathematical sense, but is kind of technical
Note that frequency-domain notion depends on fundamental theorem of Fourier analysis that every periodic function can be expressed as sum of sines & cosines (all with period an integral multiple of
Band-width of voice: <=4 kHz
This is quite a different meaning of band-width from the digital usage, where a 64kbps channel needs a bandwidth of 64kbps.
But if we wanted to encode the digitized voice back into an analog voice band-width, we'd have to encode 16 bits per cycle (Hertz), which is a little tricky.
Amplitude modulation & band-width
Note that AM modulation (ALL modulation, in fact) requires a "band-width"; ie range of frequencies. This will be very important for cellular.
AM:amplitude = [1+data(t)]*sin(2π ft)
f is "carrier" high frequency; eg 100,000
If data(t) = sin(2π gt), g a much lower frequency (eg 1000)
Then sin(2π ft)*sin(2π gt) = 0.5 cos(2π (f-g)t) - 0.5 cos(2π (f+g)t)
band of frequencies: (f-g) to (f+g)
band-width: 2g
Example: beats+modulation.xls, beats+modulation.ods
The sincfilter.ods spreadsheet, demonstrating filtering.
Note that most of the low-pass filtering is done within the first full cycle.
Discussion of the graph
analog transmission: needs amplifiers to overcome attenuation
digital transmission: store-and-forward switches?
Switches do signal regeneration, not amplification; noise is NOT added. BUT: we need them a lot more often.
data may need some form of encoding: analog may use something like AM modulation, or equalization.
Digital encoding: NRZ is basic (1=on, 0=off), but isn't good in real life.
Analog data:
Analog signal: commonly some form of modulation on a different frequency
Digital signal: something like PCM sampling
Digital data:
Analog signal: this is what modems generate
Digital signal: we need some form of encoding.
Data: the original data format
Signal: the signal actually transmitted
Transmission: how we handle that signal on the wire.
analog v data v (encoding | transmission)
Note analog data / analog signal is an odd case.
See http://intronetworks.cs.luc.edu/current/html/links.html#encoding-and-framing as a reference on encoding of (short-haul) digital signals on a wire.
Signals v Transmission:
Normally these should match. Note special case of analog signal / digital transmission, which is taken to mean that the analog signal encodes a digital signal in a way that the repeater can decode
and re-encode.
Transmission impairments
• Attenuation
□ high-frequency attenuation: signal distortion
• Delay distortion
• Noise
□ crosstalk (noise from other signals)
□ impulse noise (eg noise from electrical appliances)
1. need strength for reception
Ethernet problem with detecting collisions
2. need strength above noise level
3. attenuation increases with frequency, leading to distortion
"loading coils" in voice telephony: cut off frequencies outside of a desired range; tremendous out-of-band attenuation
attenuation measured in dB; dB per unit distance
Here is a graph from Stallings showing the relative attenuation of voice frequencies:
Note use of equalization to make up for high-frequency loss
Brief review of decibels: logarithmic scale of relative power:
db = 10 log[10] (P/P[baseline])
3 dB = 2Γ log10(2) = .30103
5 dB = 3Γ
7 dB = 5Γ (5Γ = 10Γ /2Γ ; 7 = 10-3)
10 dB = 10Γ
20 dB = 100Γ
Attenuation problems 1 & 2 above can be addressed with amplification.
3rd problem introduces fundamental distortion; digitization solves this but analog equalization can work too.
Digital: high-frequency attenuation => signal degradation
Attenuation: leads to distortion of relative frequency strengths
Delay Distortion (like differential frequency attenuation, but different): different (sine) frequencies travel at different speeds.
Again, this leads to digital signal degradation, and some audio distortion.
Thermal noise:
N (watts) = kTB, B=band-width, T=temp, k=Boltzmann's constant (small!). Thermal noise is relevant for satellite transmission, but other sources of noise are usually more important for terrestrial
Note that thermal noise is proportional to the analog bandwidth; ie, it affects all frequencies identically.
It is often more convenient to use logarithms (base 10):
Noise in dBW = log(k) + log(T) + log(B)
log(k) = -228.6 dBW
Suppose the temperature is 300K (27 degrees C) and the band-width is 20 MHz. Then the thermal noise is
-228.6 + 10 log 300 + 10 log 20,000,000 = -228.6 + 24.8 + 73 = -133.5 dBW
dBW is the difference (in decibels) from a 1 Watt reference signal.
Intermodulation noise
This is the noise created by two signals interacting, in the same medium
Brief discussion on why it isn't universal.
Intermodulation noise requires some nonlinear interaction between the signals!
A linear combination of frequencies f1 and f2 (ie just transmitting them side-by-side in space) does not produce energy at f1+f2.
Noise created by two signals interacting, on adjacent wires
Impulse noise
• Fridge story
• Bad for data
• Most significant noise for ordinary wiring
Interference (a form of impulse noise, from sharers of your frequency range)
Somebody else is using your frequency. Perhaps to make microwave popcorn. (Or perhaps you are simply driving around in the country listening to the radio.)
Other sources of noise:
• poor connectors
• cosmic rays / sunspots (form of impulse noise)
• signal reflections from connectors/taps
Channel capacity
Nyquist's Theorem: the signal rate is the rate of sending data symbols. Nyquist's theorem states that
maximum binary signal rate = 2 B
Where B is the width of the frequency band (that is, the "band-width").
This can be hard to realize in practice.
Signal rate v data rate: if we use binary signaling (binary encoding), then this means
max data rate = 2 B (binary version)
We might also send symbols (signal elements) each encoding L bits, in which case the data rate is LΓ signal_rate. One way to do this is to use multi-level encoding, using M=2^L distinct signal values
(eg distinct amplitudes, etc). In this case we have
max data rate = 2 B Γ log[2](M) (multi-level version)
For binary signals, M=2 and log[2](M)=1, so we just get the binary-version formula. Log[2](M) is the number of bits needed to encode M, that is, the number of bits per symbol.
Signal rate is sometimes called "modulation rate". It is traditionally measured in baud. Note that for a 56k "baud" modem, it's the data rate that is 56kbps; the signaling rate is 8000/sec.
Compare Nyquist to the Sampling Theorem, which says that if a sine wave has frequency B, then it can be exactly reproduced if it is sampled at a rate of 2B. (Note: the sampling theorem allows for
exact reproduction only if the sampled values are exact. In real life, the sampled values are digitized, and thus "rounded off"; this is called quantizing error.)
Basis of Nyquist's theorem: fundamental mathematics applied to individual sine waves.
The data rate is sometimes called "bandwidth" in non-analog settings.
The Nyquist limit does not take noise into account.
Note that if we are talking about a single sin(x), then analog band-width = 0! sin(x) does not carry any useful information.
Example 1: M=8, log[2](M) = 3. Max data rate is 6B.
With M levels, we can carry log[2](M) bits where we used to only carry 1 bit.
Why can't we just increase the bits per signal indefinitely, using multi-level encoding with more and more levels?
Answer: noise.
The Shannon-Hartley Theorem uses the noise level to give an upper bound on the communications throughput.
If S is the signal power and N is the noise power, then the signal-to-noise ratio, SNR, is S/N. This is often measured in dB, though in the formula below we want the actual ratio (25 dB = 300Γ )
Shannon-Hartley claim:
C β € B log[2](SNR + 1)
where B = band-width, C = maximum channel capacity
Example: 3000Hz voice bandwidth, S/N = 30 dB, or a ratio of 1000.
C = 3000*log[2](1000) = 3000*10 = 30kbps
Note that increasing signal strength does tend to increase noise as well. Also, increasing band-width increases noise more or less in proportion. So: increasing B does lead to more thermal noise, and
thus by Nyquist's formula SNR will decrease.
Here's a quick attempt to justify the Shannon-Hartley formula, borrowed from www.dsplog.com/2008/06/15/shannon-gaussian-channel-capacity-equation. It essentially derives Hartley's original formula.
Let us start with the assumption that S[v] is the maximum signal voltage, and N[v] is the range of noise voltage; noise itself ranges from β N[v]/2 to +N[v]/2. N[v] is much smaller than S[v]. We'd
like to choose an M so the M different voltages 0, S[v]/(M-1), 2S[v]/M, ..., (M-1)S[v]/(M-1) are all distinct. This means that the step between two adjacent voltages is at least as large as N[v], as
the upper voltage can have N[v]/2 subtracted while the lower voltage can have N[v]/2 added. This means S[v]/(M-1) = N[v], or
M = S[v]/N[v]+1 = (Sv+N[v])/N[v]
The number of bits we can send with M levels is log[2](M) = log[2](S[v]/N[v]+1).
We're using voltages here; we really want to use power, which, all else being equal, is proportional to the square of the voltage. Let S = S[v]^2 and N = N[v]^2, so (S/N)^1/2 = S[v]/N[v]. We now
have, ignoring the "+1" because S/N is large,
log[2]M = (1/2) log[2](S/N)
If B is the bandwidth then Nyquist's theorem says in effect that the maximum symbol rate is 2B. This means that our data rate is 2B Γ (1/2) log[2](S/N) = B log[2](S/N).
We've said nothing about the idea that noise is statistically distributed following the Gaussian distribution. But this is a first step.
Let us equate the Shannon and Nyquist formulas for C:
C = 2B log[2](M) β € B log[2](SNR+1)
M^2 β € SNR+1
Suppose we take SNR = 251Γ ; from the above we can infer that we can have at most M=16 signal levels.
56kbps modem: C=56kbps, B=3100Hz. C/B = 18
18 = log[2](1 + SNR); SNR ~ 2^18 = 260,000 = 54 dB
Nyquist and 56Kbps modem: B=4kHz; 128 = 2^7 levels
Shannon and 28/56Kbps modems
noise is proportional to bandwidth; let N[0] = noise power per Hertz.
E[b] = energy per bit of signal (eg wattage of signal Γ time-length of bit; this decreases with increased signaling rate (shorter time)) or lower average signal energy.
Ratio is E[b]/N[0]
Note that this is a dimensionless quantity, though as a ratio of energy levels it is often expressed (logarithmically) in dB.
bit error rate decreases as this increases; significant for optical fiber designers
We will often assume N[0] is all thermal noise, equated to kB, but the notion makes sense when there is other noise too.
Here is figure 5.4 from Stallings involving BER v E[b]/N[0]:
Transmission Media
EM spectrum:
1000 voice
10^6 AM radio
10^7 shortwave
10^8 FM radio, television
10^9 Wi-Fi, microwave ovens
10^10 microwaves
10^12 Low end of infrared
10^14 Red end of visible light
Attenuation of various media (from Stallings)
Twisted pair (loading coils) 0.2 dB/km at 1kHz
Twisted pair (voice-grade) 0.7 dB/km at 1kHz, 25 dB/km at 1 MHz
Cat-3 12 dB/km at 1 MHz
Coax 2dB/km at 1 MHz
Coax (fat) 7 dB/km at 10 MHz
Fiber 0.2 - 0.5 dB/km
consider attenuation & interference for the following.
Note: attenuation measured in dB/km! What are the implications of this!
At 16 mHz, attenuation per tenth of a km:
13 dB (cat 3)
8 dB (cat 5) (80 dB/km)
Why is it TWISTED??
summary: Coax has less attenuation, much less crosstalk/interference, but is $$
fiber modes
• step-index multimode: reflection off fiber surface
• graded-index multimode: light is refracted away from surface due to changes in refractive index
• single-mode: single light ray down the center of the fiber
light source: lasers or LEDs (the latter is cheaper)
ADSL issues:
Stallings table 4.2: Cat-3 twisted pair has an attenuation of 2.6dB/100m!! (at 1 mHz)
Over the maximum run of 5 km, this works out to an incredible 90dB loss! And residential phone lines are not twisted-pair.
384Kbps: 17,000 feet
1.5mbps: 12,000 feet
ADSL must deal with tremendous signal attenuation!
Thermal noise becomes very serious!
Satellite note: I used to have satellite internet.
My transmitter was 2 watts. This reached 23,000 miles.
The central problem with satellite phone (and internet) links is delay. The round-trip distance (up-down-up-down) is 4x23,000 miles, for a minimum propagation delay of 495 ms. For Internet, there was
usually an additional 500 ms of queuing delay.
Frequencies: < 1.0 gHz: noisy
> 10 gHz: atmospheric attenuation
Wi-fi uses the so-called "ISM" band, at around 2.4 gHz
4.3: propagation
High-frequency is line-of-sight, but low frequency (<= ~ 1 mHz) bends
In between is "sky-wave" or ionospheric skip (2-30mHz)
Once upon a time, AT&T had chains of microwave towers, 40-50 miles apart. They would relay phone calls. They're obsolete now, replaced by fiber. The concrete tower in the picture below, built in
1950, is the original phone microwave-relay tower; the newer steel tower arrived in 1957. (Further information about this particular site can be found at long-lines.net/places-routes/LeeIL;
information about the entire system can be found at long-lines.net.) The single-story base building is huge; it was built to house vacuum-tube electronics and early transistor technology. Nowadays
the electronics fit within the base of each antenna.
Suppose you could have 100 mHz of band width (eg 2.5-2.6 gHz). At 4 kHz per call, that works out to 25,000 calls. That many calls, at 64kbps each, requires a 1.6-gbit fiber line. In the SONET
hierarchy, that just below OC-36/STS-36/STM-12. Single fiber lines of up to STM-1024 (160 Gbps; almost 100 times the bandwidth) are standard, and are usually installed in multiples.
Is it cheaper to bury 50 miles of cable, or build one tower?
Now suppose you decide (before construction begins) that you need 10Γ more capacity. What then?
Attenuation, inverse-square v exponential
water vapor: peak attenuation at 22gHz (a 2.4gHz microwave is not "tuned" to water)
rain: scattering
oxygen: peak absorption at 60 gHz
cell phones: 824-849mhz
pcs: 1.9ghz
Attenuation along a wire (coax, twisted pair, or anything else) is exponential, while wireless attenuation is proportional to the square of the distance, meaning that in the long run wire attenuation
becomes much more significant than wireless.
Every time you double the distance with wireless, the signal strength goes down by a factor of 4, which is a 6 dB loss. Suppose a cable has a loss of 3 dB per 100 m (factor of 2). Suppose the wired
signal is 10 db ahead at 100 m. We get the following
│distance │wired │wireless │
│100 m │0 db │-10 db │
│200 m │-3 db │-16 db │
│400 m │-9 db │-22 db │
│800 m │-21 db│-28 db │
│1600 m │-45 db│-34 db │
Starlight (a form of wireless) is detectable at distances of 100's of light-years.
Techniques for Modulation and Encoding
5.1 digital data/digital signal
NRZ is the "simple" encoding. But on short-haul links it has problems with clock drift. It has additional problems with long-haul encodings.
See also http://intronetworks.cs.luc.edu/current/html/links.html#encoding-and-framing about the encoding of short-haul digital signals on a wire.
data rate v modulation rate: these are often not the same
(ethernet: data rate 10Mbps, modulation rate 20Mbaud)
phone modems: data rate 56kbps, modulation rate 7kbaud
RZ, NRZ
analog band width: avoid needing waveforms that are too square
DC component (long distances don't like this)
NRZ flavors
inversion (NRZ-I) v levels (NRZ-L)
differential coding (inversion) may be easier to detect than comparison to reference level
Also, NRZ-I guarantees that long runs of 1's are self-clocked
DC component: non-issue with short (LAN) lines, larger issue with long lines
losing count / clocking (note that NRZ-I avoids this for 1's)
• no DC component
• no long runs of 0 (or any constant voltage level)
• no reduction in data rate through insertion of extra bits
bipolar (bipolar-AMI): 1's are alternating +/-; 0's are 0
Fixes DC problem! Still 0-clocking problem
Note that bipolar involves three levels: 0, -1, and +1.
biphase: (bi = signal + clock)
Example: Manchester (10mbps ethernet)
10mbps bit rate
20mbps baud rate (modulation rate)
bipolar-8-zeros (B8ZS)
This is what is used on most North American T1 lines (I'm not sure about T3, but probably there too)
1-bits are still alternating +/-; 0-bits are 0 mostly.
If a bytes is 0, that is, all the bits are 0s (0000 0000), we replace it with 000A B0BA, where A = sign of previous pulse and B=-A.
This sequence has two code violations. The receiver detects these code violations & replaces the byte with 0x00.
Note the lack of a DC component
Example: decoding a signal
Bipolar-HDB3: 4-bit version of B8ZS
│ 4-bit data │ 5-bit code │
│ 0000 │ 11110 │
│ 0001 │ 01001 │
│ 0010 │ 10100 │
│ 0011 │ 10101 │
│ ... │ │
│ 1100 │ 11010 │
│ 1101 │ 11011 │
│ 1110 │ 11100 │
│ 1111 │ 11101 │
│ IDLE │ 11111 │
│ DEAD │ 00000 │
│ HALT │ 00100 │
4b/5b involves binary levels, unlike bipolar. It does entail a 20% reduction in the data rate.
It is used in 100-mbit Ethernet
Fig 5.3 (8th, 9th edition): spectral density of encodings. Spectral density refers to the band-width that the signal needs.
Lowest to highest:
1. biphase (Manchester, etc)
2. AMI,
3. B8ZS
Latter is narrower because it guarantees more transitions
=> more consistent frequency
Fig 5.4: theoretical bit error rate
biphase is 3 dB better than AMI: not sure why. This means that, for the same bit error rate, biphase can use half the power per bit.
HDLC Bit Stuffing
The HDLC protocol sends frames back-to-back on a serial line; frames are separated by the special bit-pattern 01111110 = 0x7E. This is, however, an ordinary byte; we need to make sure that it does
not appear as data. To do that, the bit stuffing technique is used: as the sender sends bits, it inserts an extra 0-bit after every 5 data bits. Thus the pattern 01111110 in data would be sent as
011111010. Here is a longer example:
data: 0111101111101111110
sent as: 011110111110011111010
The receiver then monitors for a run of 5 1-bits; if the next bit is 0 then it is removed (it is a stuffed bit); if it is a 1 then it must be part of the start/stop symbol 01111110.
Some consequences:
1. We have guaranteed a maximum run of 6 1-bits; if we interchange 0's and 1's and use NRZ-I, bit-stuffing has solved the clocking problem for us.
2. The transmitted size of an HDLC data unit depends on the particular data, because the presence of stuffed bits depends on the particular data. This will ruin any exact synchronization we had
counted on; for example, we cannot use HDLC bit-stuffing to encode voice bytes in a DS0 line because the extra stuffed bits will throw off the 64000-bps rate.
3. The data sent, and the 01111110 start/stop symbol, may no longer align on any byte boundaries in the underlying transmission bitstream.
see also http://intronetworks.cs.luc.edu/current/html/links.html#framing.
Analog data / Digital signal
(Stallings 5.3)
sampling theorem: need to sample at twice the max frequency, but not more
basic idea of PCM: we sample at regular intervals (eg 1/8000 sec), digitize the sample amplitude, and send that.
PCM stands for Pulse Code Modulation; it replaced an earlier analog strategy called PAM: Pulse Amplitude Modulation. In PAM, the signal was sampled and then a brief carrier pulse of that amplitude
was sent. This is a little like AM, except the pulses could be short, and time-division-multiplexed (below) with other voice channels. The C in PCM means that the analog signal was replaced by a
"code" representing its amplitude. This is all meant to explain why digital sampling, which is what PCM is, gets the word "modulation" in its name, which is really not applicable.
In the early days, one sampler (PCM encoder) could sample multiple analog input lines, saving money on electronics.
sampling error v quantization error
nonlinear encoding versus "companding" (compression/expansion)
The voice-grade encoding used in the US is known as π -law (mu-law) encoding; π is a constant used in the scaling formula, set equal to 255. We define F(x) as follows, for -1<=x<=1 (sgn(x) = +1
for x>0 and -1 for x<0):
F(x) = sgn(x)*log(1+π *|x|) / log(1+π ),
Note that for -1<=x<=1 we also have -1<=F(x)<=1. If x is the signal level, on a -1β €xβ €1 scale, then F(x) is what we actually transmit. More precisely, we transmit 128*F(x), rounded off to the
nearest 8-bit integer. The use of F(x) has the effect of nonlinear scaling, meaning that for x close to 0 there are still a wide range of levels. Consider the following few values:
F(1)=1, F(-1)=-1, F(0)=0
F(0.5)= .876, Γ 128 = 112
F(0.1)= .591 Γ 128 = 76
F(0.01)= .228 Γ 128 = 29
F(0.001)= .041 Γ 128 = 5
These last values mean that faint signals (eg, x = 0.001) still get transmitted with reasonably limited quantizing roundoff. A signal around x = 0.01 can get rounded off by at most 1/2Γ 29 β 0.016;
a signal around x = 0.001 gets rounded off by at most 1/2Γ 5 = 0.1. With linear scaling, a signal level of 0.01 (relative to the maximum) would encode as 1 (0.01 Γ 128 = 1.28 β 1), and anything
fainter would round off to 0.
This is often called companding, for compression/expanding; note that it is done at the per-sample level. If everyone's voice energy ranged, say, from 100% down to a floor of 20% of the maximum,
companding wouldn't add anything. But voice energy in fact has a much wider dynamic range.
Music, of course, has an even larger dynamic range, but musical encoding almost always uses 16 bits for sampling, meaning that plain linear encoding can accurately capture a range from 100% down to
at least 0.1%. 8-bit π -law companding is often considered to be about as accurate, from the ear's perspective, as 12 or 13-bit linear encoding.
Demo of what happens if you play a π -law-encoded file without the necessary expansion: faint signals (including hiss and static) get greatly amplified. To get sox to accept this, rename
cantdo.ulaw to cantdo.raw and then:
play -r 8000 -b 8 -c 1 -e signed-integer cantdo.raw
A-law encoding: slightly different formula, used in Europe.
By comparison to companding, compression may involve taking advantage of similarities in a sequence of samples. MP3 is a form of true compression, though it is not used in telephony (because it is
hard to do in real time). G.729 is a high-performance form of true compression frequently used in voice.
delta modulation: This involves encoding the data as a sequence of bits, +1 or -1. The signal level moves up by 1 or down by 1, respectively. This limits how closely the encoded signal can track the
actual signal. I have no idea if this is actually used. It has a bias against higher frequencies, which is ok for voice but not data
advantage: one bit! However, higher sampling rates are often necessary.
Performance of digital voice encoding:
voice starts out as a 4kHz band-width.
7-bit sampling at 8kHz gets 56kbps, needs 28kHz analog band-width (by Nyquist)
(Well, that assumes binary encoding....)
BUT: we get
• digital repeaters instead of analog amplifiers
• digital reliability
• no cumulative noise
• can use TDM instead of FDM
• digital switching
voice: often analog=>digital, then encoded as analog signal on the transmission lines!
Analog data / Analog signal
(Stallings 5.4)
Why modulate at all? The primary reasons are
• to be able to support multiple non-interfering channels (FDM, or Frequency-Division Multiplexing)
• to be able to take advantage of higher-frequency transmission characteristics (you can't broadcast voice frequencies!)
AM and FM radio is the classic example. Cellular telephony would be analog data / digital signal
The simplest is AM.
AM band-width usage is worth noting
new frequencies at carrier +/- signal are generated because of nonlinear interaction (the modulation process itself).
Single Side Band (SSB): slightly more complex to generate and receive, but:
• half the band-width
• no energy at the carrier frequency (this is "wasted" energy)
Sound files: beats.wav v modulate.wav
Latter has nonlinearities
(1+sin(sx)) sin(fx) = sin(fx) + sin(sx)sin(fx)
= sin(fx) + 0.5 cos((f-s))x) - 0.5 cos((f+s)x)
reconsider "intermodulation noise". This is nonlinear interactions between signals, which is exactly what modulation here is all about.
Angle Modulation (FM and PM)
FM is Frequency Modulation; PM is Phase Modulation. These can be hard to tell apart, visually.
Let m(t) = modulation signal (eg voice or mustic).
The (transmitted) signal is then
A cos (2π ft + π (t))
FM: k*m(t) = π '(t) (that is, π (t) = β «m(t)dt). If m(t) = c is constant for an interval, then π (t) = kct = k[1]t; that is, we have the transmitted signal as
A cos (2π ft + kct) = A cos (2π (f+kc/2π ) t),
a signal with the fixed (higher) frequency f+kc/2π .
(We are assuming m(t) is a constant level, not a constant frequency)
PM: k*m(t) = π (t). m(t) = const => π (t) = const. We shift phase for the duration of the constant interval, but the base frequency changes only when m(t) is changing.
See modulation.xls.
Somewhat surprisingly, FM and PM often sound very similar. One reason for this is that the derivative (and so the antiderivative) of a sine wave is also a sine wave. There's distortion in terms of
frequency, but most voice frequencies are in a narrow range.
Picture: consider a signal m(t) = 0 0 1 1 1 1 1 1 0 0 0 0
FM,PM both need more band-width than AM
AM: band-width = 2B, B=band-width of orig signal
FM,PM: band-width = 2(Ξ²+1)B, where again B = band-width of original signal. This is Carson's Rule.
For PM, Ξ² = n[p]A[max], A_max = max value of m(t) and np is the "phase modulation index", a quantity proportional to k in the PM rule k*m(t) = π (t).
For FM, Ξ² = π «F/B, π «F = peak frequency difference. A value of Ξ²=2, for example, would mean that in encoding an audio signal with band-width 4 KHz, the modulated signal varied in frequency by a
total range of 8 KHz. Having Ξ² low reduces the band-width requirement, but also increases noise. Also note that in our Ξ²=2, the total band-width needed for the modulated signal wold be 24 KHz.
Digital data / Analog signal
Stallings 5.2
modems, long lines & fiber
(even long copper lines tend to work better with analog signals)
ASK: AM modulation using something like the NRZ signal as the input. It is a "naive" encoding, though used for fiber
FSK: FM modulation. 1-bits are transmitted by brief pulses at frequency f[1] (that is, A cos(2π f[1]t)), while 0-bits are transmitted by brief pulses at another frequency f[2]. The bit-time must be
long enough that the two frequencies f[1] and f[2] are easily distinguished!
On optical fiber, FSK is represented by color shift.
PSK: easier to implement (electrically) than FSK. 0-bits might be sent as A cos(2π ft), while for 1-bits the waveform might change to A cos(2π ft + π )
Superficially, ASK appears to have zero analog band-width, but this is not really the case!
ASK: 1 bit /hertz => 4000 bps max over voice line
1 bit/ 2Hz, 2400 Hz carrier => 1200 bps.
FSK analog band-width = high_freq - low_freq
MFSK: we use multiple frequencies to send several bits at once. If we have four frequencies to use (eg f-3d, f-d, f+d and f+3d, where f is the "carrier"), then one frequency encodes two bits. We
might even label the frequencies with the bits encoded: f[00], f[01], f[10], f[11].
BFSK v MFSK: fig 5.9 for MFSK.
BFSK: fig 5.8: old modems, full-duplex
One direction of the signal might use the frequency band 600-1600 Hz, and the other direction might use the band 1800-2800 Hz.
MFSK: the trouble is, it takes time to recognize a frequency (several cycles at least!)
FSK is supposedly more "noise-resistant" than ASK, but fig 5.4 shows the same graph of Eb/N[0] v BER for the two. (PSK is shown 3 dB lower (better) in the graph)
BPSK: decoding starts to get very nonintuitive!
DPSK: differential, like differential NRZ
QPSK: 4 phase choices, encoding 00, 01, 10, 11
9600bps modem: really 2400 baud; 4 bits per signal element (12 phase angles, four of which have two amplitude values, total 16 distinct values per signal, or 4 bits)
Nyquist limit applies to modulation rate: noise reduces it.
56Kbps modems: use PCM directly.
Station gets data 7 bits at a time, every 1/8 ms, and sets the output level to one of 128 values.
If there is too much noise for the receiver to distinguish all those values, then use just every other value: 64 values, conveying 6 bits, for 48kbps. Or 32 values (using every fourth level), for 5*8
= 40 kbps.
Quadrature Amplitude Modulation, QAM
This involves two separate signals, sent 90Β° out of phase and each amplitude-modulated (ASK) separately. Because the two carriers are 90Β° out of phase (eg sin(ft) and cos(ft)), the combined signal
can be accurately decoded.
We will ignore the QAM details.
An Example
The following example is due to Oona RΓ€isΓ€nen, via her blog at windytan.com/2014/02/mystery-signal-from-helicopter.html. We make use of audacity and the SoX program.
We start with a police helicopter video at youtube.com/watch?v=TCKRe4jJ0Qk. What is that buzzing noise? The engine? Step 0 is to save the sound track as an mp3 file (police_chase.mp3), using, say,
Next, using Audacity, convert the mp3 file to .wav format. While we're here, note the distinctive appearance of the left channel.
The next step is to extract the left channel using
sox police_chase.wav -c 1 left.wav remix 1
Zoom in on the left.wav file. It appears to be a mixture of higher-frequency and lower-frequency sine waves. The high-frequency wavelength is about .45 ms, making the frequency ~2200 Hz; the
lower-frequency wavelength is about .85 ms, making the frequency ~1200 Hz. These numbers turn out to match the Bell 202 modulation scheme, which uses FSK: data is sent at a rate of 1200 bps, with
1-bits encoded as a single wavelength at 1200 Hz and 0-bits encoded as 1.83 wavelengths (1/1200 sec) at 2200 Hz. Bell 202 modulation is still used to transmit CallerID data to analog landline phones.
How do we demodulate the signal? One approach is to apply lowpass and highpass filters about the midpoint, 1700 Hz, and compare the outputs:
sox left.wav hi.wav sinc 1700
sox left.wav lo.wav sinc -1700
Combine the two channels:
sox --combine merge lo.wav hi.wav both.wav remix 1 2
and look at the two signals side-by-side. To demodulate, we'd need to do the following:
• find the envelope of the sine wave
• figure out at what points we want to be doing the sampling. Theoretically this is once every 1/1200 sec, but we have to figure out how to resynchronize clocks occasionally to account for clock
An easier way to demodulate is to use minimodem:
minimodem --receive 1200 -f left.wav | tr '\200-\377\r' '\000-\177\n'
The "tr" (translate) command unsets the high-order bit to get 7-bit ascii, with \n replacing \r.
This gives us latitude and longitude coordinates that match up fairly well with the path of the helicopter! Consider the first data point:
N390386 W0943420
This appears to be in "decimal minutes" format:
39 3.86, -94 34.20
Rules for entering latitude and longitude into maps.google.com are at support.google.com/maps/answer/18539.
The video itself starts at 39Β°03'51.6"N 94Β°34'12.0"W, at the corner of Volker Blvd and Troost Ave; this is almost two miles due south of the coordinates above.
The Swope Parkway Tennis Courts are at 39Β°02'28.9"N 94Β°33'43.7"W. In the video we pass these at T=17 sec; the helicopter is looking south and the car is heading east, about to head under I-71.
The video has 1428 seconds and the telemetry data has 5706 lines, for just about exactly 4 lines (2 position records) a second. At T=3:30 (210 seconds) the car is at the intersection of I-71 and 39th
street. That's line 840 of the file, where the coordinates are N390269 W0943368. If we plot that on google maps, as 39 2.69, -94 33.68, we get I-71 and 45st street; the helicopter is now "only" six
blocks behind.
(Note that the telemetry sound fluctuates twice a second; that is, once per record! At 1200 bits/sec, we can send 150 bytes/sec. The actual position records are ~48 bytes long, with null bytes added
to take up the slack.
Brief note on synchronous v asynchronous transmission
Sender and receiver clocks MUST resynchronize at times; otherwise, the clock drift will eventually result in missed or added bits.
Asynchronous: resynchronise before/after data, eg with a "stop bit" before and after each byte. This is common approach with serial lines, eg to modems.
Synchronous: send data in blocks too big to wait to resynchronize at the end, but embed synchronization in the data (with NRZ-I, for example, we usually resynchronize on each 1-bit).
Manchester (a form of synchronous): we interleave clock transitions with data transitions.
More efficient techniques make sure there are enough 1's scattered in the data itself to allow synchronization without added transitions. Example: 4b/5b: every 5 bits has at least 2 transitions (2
Brief note on PACKETs as a form of multiplexing
The IP model, with relatively large (20 byte for IP) headers that contain full delivery information, is an approach allowing a large and heterogeneous network. But simpler models exist.
The fundamental idea of packets, though, is that each packet has some kind of destination address attached to it. Note that this may not happen on some point-to-point links where the receiver is
unambiguous, though what "flow" the packet is part of may still need to be specified.
HDLC packet format: omit
Voice channels
The basic unit of telephony infrastructure is the voice channel, either a 4 KHz analog channel or a 64 kbps DS0 line. A channel here is the line between two adjacent switching centers; we might also
call them channel segments. An end-to-end voice path is a sequence of channels. To complete a call, we do two things:
1. reserve an end-to-end sequence of voice channels for the call
2. at each switch along the way, arrange for the output of a channel to be forwarded (switched) to the next channel in the path.
Channels are either end-user lines or are trunk channels; the latter are channels from one switching center to the next. Within the system, channels are identified by their Circuit Identification
Code. It is the job of Signaling System 7 (in particular, the ISDN User Part, or ISUP, of SS7, to handle the two steps above). The spelling "signalling" is common in this context. SS7 also involves
conveying information such as caller-ID and billing information.
Note that VoIP does not involve anything like channels; we just send packets until a link is saturated. The channel-based system amounts to a hard bandwidth reservation (with hard delay bounds!) for
every call.
The channel is the logical descendant of the physical circuit. At one point, the phone system needed one wire per call. Channels allow the concept of multiplexing: running multiple channels over a
single cable. We'll now look at three ways of doing this:
• L-carrier
• DS (T-carrier) lines
• SONET
More on the signaling and switching processes below
FDM (Frequency Division Multiplexing)
AM radio is sort of the archetypal example. This is a fundamentally analog technique, though we can use FDM and digital data (eg ASK or FSK).
ATT "L-carrier" FDM
voice example
4kHz slots; 3.1kHz actual bandwidth (300 Hz - 3400 Hz). AM SSB (upper sideband) modulation onto a carrier frequency f transforms this band into the band [f, f+4kHz], of the same width. Note that
without SSB, we'd need double the width; FM would also use much more bandwidth than the original 4kHz.
ATT group/supergroup hierarchy: Table 8.1
│ name │ composition │ # channels │
│ Group │ │ 12 │
│ Supergroup │ 5 groups │ 5 Γ 12 = 60 │
│ Mastergroup │ 10 supergroups │ 10 Γ 60 = 600 │
│ Jumbogroup │ 6 mastergoups │ 6 Γ 600 = 3600 │
│ Mastergroup Multiplex │ N mastergroups │ N Γ 600 │
L-carrier: used up through early 1970s
Why bundle calls into a hierarchy of groups? So you can multiplex whole trunks onto one another, without demuxing individual calls. Peeling out a single call is relatively expensive, particularly if
we want to replace that slot with a new call. For one thing, additional noise is introduced.
Even the repeated modulation into larger and larger groups introduces noise.
Brief comparison of Stallings Fig 5-8 (below) and Fig 8-5 (above). Both show side-by-side bands, interfering minimally. The first is of two bands in the voicerange (1 kHz and 2 kHz respectively),
representing a modem sending in opposite directions. The second is of multiple 4 kHz voice bandsAM-modulated (using SSB) onto carriers of 60 kHz, 64 kHz, 68 kHz, .... | {"url":"https://pld.cs.luc.edu/courses/346/sum15/mnotes/modulation_transmission.html","timestamp":"2024-11-01T23:08:21Z","content_type":"text/html","content_length":"64443","record_id":"<urn:uuid:edc76afd-5e2f-4f4b-b531-438adcc73dca>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00215.warc.gz"} |
How many similarity theorems are there?
How many similarity theorems are there?
3 theorems
In today’s geometry lesson, you’re going to learn about the triangle similarity theorems, SSS (side-side-side) and SAS (side-angle-side). In total, there are 3 theorems for proving triangle
similarity: AA Theorem. SAS Theorem.
What are the rules for similar triangles?
Two triangles are similar if they meet one of the following criteria. : Two pairs of corresponding angles are equal. : Three pairs of corresponding sides are proportional. : Two pairs of
corresponding sides are proportional and the corresponding angles between them are equal.
What are the 3 similarity theorems?
These three theorems, known as Angle – Angle (AA), Side – Angle – Side (SAS), and Side – Side – Side (SSS), are foolproof methods for determining similarity in triangles.
Are theorems always true?
A theorem is a statement having a proof in such a system. Once we have adopted a given proof system that is sound, and the axioms are all necessarily true, then the theorems will also all be
necessarily true.
What are the 3 triangle similarity theorems?
How do you tell if two triangles are similar?
Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion . In other words, similar triangles are the same shape, but not
necessarily the same size. The triangles are congruent if, in addition to this, their corresponding sides are of equal length.
How do you prove two triangles are similar?
If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. We know this because if two angle pairs are the same, then the third pair must also be
equal. When the three angle pairs are all equal, the three pairs of sides must also be in proportion.
What theorem can you use to prove triangles are similar?
Two triangles can be proved similar by the angle-angle theorem which states: if two triangles have two congruent angles, then those triangles are similar. This theorem is also called the
angle-angle-angle (AAA) theorem because if two angles of the triangle are congruent, the third angle must also be congruent.
How do you determine if a triangle is similar?
There are three ways to find if two triangles are similar: AA, SAS and SSS: AA stands for “angle, angle” and means that the triangles have two of their angles equal. If two triangles have two of
their angles equal, the triangles are similar.
How do you prove SSS theorem in similar triangles?
Part 3 of 4: Using the Side-Side-Side Theorem Define the Side-Side-Side (SSS) Theorem for similarity. Two triangles would be considered similar if the three sides of both triangles are of the same
proportion. Measure the sides of each triangle. Using a ruler, measure all three sides of each triangle. Calculate the proportions between the sides of each triangle.
How do you solve similar triangles?
You can solve certain similar triangle problems using the Side-Splitter Theorem. This theorem states that if a line is parallel to a side of a triangle and it intersects the other two sides, it
divides those sides proportionally. See the below figure. | {"url":"https://www.handlebar-online.com/usefull-tips/how-many-similarity-theorems-are-there/","timestamp":"2024-11-04T08:23:42Z","content_type":"text/html","content_length":"44040","record_id":"<urn:uuid:68fd7816-adba-4f00-958d-334a20354457>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00105.warc.gz"} |
An interview with Brian Russell
Brian Russell holds a B.Sc. from the University of Saskatchewan, a M.Sc. from Durham University, U.K., and a Ph.D. from the University of Calgary, all in geophysics. He worked for Chevron, Teknica
and Veritas before co-founding Hampson- Russell Software with Dan Hampson in 1987, a company that develops interactive seismic analysis software for the oil industry. Hampson-Russell is now a
subsidiary of CGG, where Brian is Vice President, Software and a CGG Fellow. His research interests include rock physics, seismic inversion and seismic attribute analysis. Brian is a Past-President
of both the Society of Exploration Geophysicists (SEG) and the Canadian SEG (CSEG) and has received Honorary Membership from both societies, as well as the Cecil Green Enterprise Award from SEG
(jointly with Dan Hampson). Brian is a director on the CSEG Foundation Board, Chairman of the Board of the Pacific Institute for the Mathematical Sciences (PIMS) and also an Adjunct Professor in the
Department of Geoscience at the University of Calgary. He is registered as a Professional Geophysicist (P.Geoph.) in the Province of Alberta.
Doing mathematics is like exercising. Do a little bit every day and you stay in shape, either intellectually (in the case of math) or physically (in the case of exercising). Neglect it and your
muscles (intellectual or physical) fade away.
Geophysics is a hard science. By that I mean that it is a science based on ‘hard’ facts, but also that it can be difficult. We all struggled through tough math and physics classes at university to
get our degrees. But once we were in the working world, especially if we became seismic interpreters, we tended to leave the details to the specialists. Indeed, picking up a copy of Geophysics and
trying to read every article is a daunting task. And I do not expect that every exploration geophysicist should be able to understand the latest implementation of Green’s functions in anisotropic
depth imaging. However, I do think that an appreciation of some of the fundamental applied mathematical ideas in our profession can go a long way towards enhancing your enjoyment and appreciation of
your day-to-day job.
Two examples
Let me illustrate my point with two equations. Let us start with:
d = Gm
where d is our data, a set of n geophysical observations, m is our model, a set of k model parameters, and G is a linearized relationship that relates the observations to the parameters. This
ubiquitous equation can be found in every area of geophysics, from seismology through potential fields to electromagnetic theory. The simplicity of the way I have written the equation hides the fact
that d is usually written as an n-dimensional vector, m as a k-dimensional vector, and G as an n row by k column matrix.
The way that you react to these equations tells me a lot about you as a geophysicist.
Solving the equation is a little more difficult. Since n is usually greater than k, the solution can be written:
m = (G^TG+mI)^-1G^Td = C^-1h
where C is the autocorrelation matrix found by multiplying the G matrix by its transpose G^T (and adding a little pre-whitening by multiplying the value λ by I, the k by k identity matrix), and h is
the zero-lag cross-correlation vector, found by multiplying the transpose of the G matrix by the data vector. Again, this equation, sometimes called the Normal Equation, is ubiquitous in geophysics.
It is the basis of deconvolution, AVO attribute analysis, post- and pre-stack inversion, refraction and reflection statics, and so on. So, what lesson should we take away from these equations?
My advice
The way that you react to these equations tells me a lot about you as a geophysicist. If you are thinking: ‘what’s the big deal, I use those types of equations every day,’ you probably don’t need my
advice. If you are thinking: ‘yes, I saw those equations once in a class, but haven’t thought about them for years,’ perhaps I can inspire you to look at them again. On the other hand, if you are
thinking: ‘why would I ever need to use those boring-looking equations,’ you are a tougher challenge! I would recommend starting with these equations and really trying to understand them (perhaps you
will need to dust off your linear algebra, and I recommend the book by Gilbert Strang). Then, pick up a copy of Geophysics, or any geophysics textbook, and see how many of the equations can be
expressed in the same way. Or, take some quantitative industry training courses and see what the mathematics is really telling you about your data.
I guarantee it will be good for you!
Brian, by saying ‘don’t neglect your math’, I guess you are essentially saying math is a tool we use to learn geophysics and so we as geophysicists should not shy away from it. On the other extreme,
a heavy dose of math may put people off and there is a huge need to be able to explain difficult math concepts in simple terms. Could you please elaborate on this?
This is an interesting question because although we know that the underlying principles of geophysics are mathematically based, and although we spend a lot of time working through these equations at
university, many of us stop using any mathematics at all when we join the industry and essentially just look at the data. This is good up to a point, but what I am saying in this article that you
should continue to keep up with the mathematical/physical principles. In the article I use generalized linear inversion and give everyone a “homework” assignment to see how often it is used in
seismic data analysis. I could have used a simpler expression like the NMO equation or the AVO equation. I feel it is important to know these key equations and how their application affects your data
and its interpretation. But how much math is too much? I agree that the journal Geophysics has become too specialized for the average interpreter to read, and I would not expect most of our members
to slog through the details. I agree that we need more readable descriptions and, luckily, there is a new journal coming out from AAPG and SEG called Interpretation, which will fill this gap.
In your article, ‘See the big picture’, you say ‘An integrated project in any area of geophysics involves data acquisition, modeling, analysis and interpretation…, in the 21st century no one person
can be a specialist in even a sub-set of these different areas’. In my mind, for the areas of geophysics you mention, any geophysicist is still well-versed in most of these areas, if not all. Would
you agree?
I agree that every exploration geophysicist should have a basic understanding of every aspect of the exploration process, from acquisition through processing to interpretation. But each area has
become so complex that, unlike forty years ago, nobody can be on top of all the details. For example, as this year’s SEG, there were literally dozens of papers on new marine acquisition design
techniques, so it is very hard for someone who does not specialize in marine acquisition to master it all. Thus, while every geophysicist should understand the principles of marine acquisition, we
should not expect that every geophysicist could put together the specs for the latest multi-azimuth survey in the Gulf of Mexico. We have to work in teams and rely on the expertise of other members
of the team. That was my point here. I think we are in agreement on the fundamentals.
A little later in the same article, you mention about ‘collaboration’ with engineers and geologists to understand how they perceive many of the terms we deal with such as anisotropy, or a seismic
response of shale overlaying a carbonate reef. Yes, this will help us understand how they view our problems. I think what is more important is how we can help them quantify their analysis, e.g. by
using seismic geomechanics, etc. This is what I think is interdisciplinary co-operation and a dire need in unconventional resource characterization. How would you react to that?
I think my answer here is similar to the last one. Although we should attack this problem with our own specialist tools, such as azimuthal AVO, we should make an effort to understand the language of
the engineers, and to communicate our results to them in their language. For example, it is fairly easy for us to transform our measurements from P and S-wave velocity and density to Young’s modulus
and Poisson’s ratio. In doing so, we make a step towards “speaking” the same language as the engineer.
Embarking on a lifetime of learning is what has been done by many individuals in the recent past. However, in the present scenario, its need has increased several fold. I think the exposure to
inter-disciplinary ideas within asset teams in oil and gas companies is a good way to learn these, as well as with readily accessible literature and continuing education courses. Is there is way to
somehow quicken this process? How about the organization of inter-disciplinary workshops such as the SEG/AAPG/SPE URTeC held in Denver last August?
I am all in favour of this and, although I did not get to the workshop you mention, I have been to several inter-disciplinary workshops. However, what I always observe at such workshops is that there
is never a totally equal mix of representatives from all the disciplines and that there is usually one main organizer whose discipline was overwhelmingly represented. (For example, I would love to
see the attendance list for the Denver conference and see if the mix was equal thirds among geophysicists, geologists and engineers.) Having said this, we definitely need to carry on trying to
improve the communication among our disciplines, and this is a good way.
You mention as an example how one could begin to understand anisotropy in terms of Thomsen’s parameters, what they mean and how these details are relevant to the particular task being done. Let me
take it from here and ask you this: for shale resource plays, while shales have intrinsic anisotropy (which is a case of VTI), the presence of kerogen in shale source rocks enhances this anisotropy
(as enunciated by Vernik, 1994) so that it is a case of strong anisotropy now. Do you think Thomsen’s analysis of weak-anisotropy would be adequate in characterizing shale source rocks? If not, what
type of analysis is required to handle strong anisotropy?
I think that Thomsen’s theory of weak anisotropy is adequate considering all the other factors that have to be taken into account. However, his theory for VTI anisotropy has to be extended to other
forms of anisotropy such as HTI, TTI, and orthorhombic, as shown by researchers such as Rüger, Tsvankin and Gretchka in several papers. Also, when modeling the data we do not need to use the Thomsen
approximation but can use the full form of the stiffness or compliance matrix.
As an extension of the previous question, if there are natural fractures in the shale source rocks (which are formed as the kerogen matures) it becomes a case of HTI. So, the situation is complicated
in that strong VTI anisotropy clubbed with HTI, is now a case of orthorhombic anisotropy. How could one handle such a characterization? Is it being attempted? I may mention that this needs the
appropriate mathematical tools to carry out the analysis.
As you can see, my previous answer anticipated this question. (I didn’t read ahead, really!). Yes, my colleague Dr. Jon Downton is working on performing orthorhombic modeling and also orthorhombic
analysis using AVAz data, and I know that other researchers are doing the same, such as Professor Boris Gurevich at Curtin University in Perth. For a more detailed explanation, please look up papers
by these individuals.
In shale resource rocks, pre-stack impedance inversion is usually carried out and the spatial variation of the derived attributes is examined to locate the sweet spots. The lowering of P-impedance
for example is usually taken to imply large porosity. This implication may be flawed, as such a lowering could be due to large porosity, could be due to softer mineralogy or small pore-aspect ratio.
One or more of these properties could be influencing the P-impedance. How does one resolve this problem?
I agree that just a pre-stack inversion in shale plays may not be enough. As discussed by Goodway et al. (2010) in The Leading Edge (TLE), the use of pre-stack inversion (in their case, where the
parameters have been converted to Lambda-Rho and Mu-Rho) gives good discrimination between ductile and non-ductile shales in the isotropic case. However, to evaluate parameters like closure stress,
especially in the anisotropic case, the authors recommend using either shear-wave splitting measurements or AVAz. My colleagues at Hampson-Russell in Houston also wrote a TLE article (Sena et al.,
2011) in which they show that a number of attributes besides those from pre-stack inversion (including AVAz attributes) need to be utilized when analyzing resource shale plays. In their case they
used a multi-attribute transform to combine the attributes and predict sweet spots for drilling. | {"url":"https://www.csegrecorder.com/interviews/view/interview-with-brian-russell","timestamp":"2024-11-10T02:57:49Z","content_type":"text/html","content_length":"31407","record_id":"<urn:uuid:6f8d37d1-882d-47d1-8062-0bfe373e6f8c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00406.warc.gz"} |
Lesson 2
Function Notation
2.1: Back to the Post! (10 minutes)
The goal of this warm-up is to motivate the need for a notation that can be used to communicate about functions.
Students analyze three graphs from an earlier lesson, interpret various points on the graphs, and use their analyses to answer questions about the situations. This work requires students to make
careful connections between points on the graphs, pairs of input and output values, and verbal descriptions of the functions. Students find that, unless each feature and the function being referenced
is clearly articulated, which could be tedious to do, what they wish to communicate about the functions may be ambiguous or unclear.
When answering the last two questions, students are likely to find the prompts lacking in specificity and to probe: “for which day?” Suggest that they answer based on their interpretation of the
Then, look for students who assume that the questions refer to one particular function and those who assume they refer to all three functions (and consequently answer them for each function). Ask
them to share their interpretations during the whole-class discussion.
Student Facing
Here are the graphs of some situations you saw before. Each graph represents the distance of a dog from a post as a function of time since the dog owner left to purchase something from a store.
Distance is measured in feet and time is measured in seconds.
1. Use the given graphs to answer these questions about each of the three days:
1. How far away was the dog from the post 60 seconds after the owner left?
2. How far away was the dog from the post when the owner left?
3. The owner returned 160 seconds after he left. How far away was the dog from the post at that time?
4. How many seconds passed before the dog reached the farthest point it could reach from the post?
2. Consider the statement, “The dog was 2 feet away from the post after 80 seconds.” Do you agree with the statement?
3. What was the distance of the dog from the post 100 seconds after the owner left?
Activity Synthesis
Invite students to share their response to the first set of questions.
To help illustrate that it could be tedious to refer to a specific part of a function fully and precisely, ask each question completely for each of the three days. (For instance, “How far away was
the dog from the post 60 seconds after the owner left on Day 1? How far away was the dog from the post 60 seconds after the owner left on Day 2?”) If students offer a numerical value (for instance,
“1.5 feet”) without stating what question it answers or to what quantity it corresponds to, ask them to clarify.
Next, select previously identified students to share their responses to the last two questions. Regardless of whether students chose to answer them for a particular day or for all three days, point
out that the answers depend on the day. When the day (or the function) is not specified, it is unclear what information is sought.
Explain that sometimes we need to be pretty specific when talking about functions. But to be specific could require many words and become burdensome. Tell students that they will learn about a way to
describe functions clearly and succinctly.
2.2: A Handy Notation (15 minutes)
In this activity, students learn that function notation can be used as a handy shorthand for communicating about functions and specific parts or features of a function. They interpret statements that
are written in this notation and use the notation to refer to points on a graph or to represent simple verbal statements about a function.
Explain to students that one way to talk about functions precisely and without wordy descriptions is by naming the functions and using function notation.
• Suppose we give a name to each function that relates the dog’s distance from the post and the time since the dog owner left: function \(f\) for Day 1, function \(g\) for Day 2, function \(h\) for
Day 3. The input of each function is time in seconds, \(t\).
• To represent “the distance of the dog from the post 60 seconds after the owner left,” we can simply write: \(f(60)\). To express the same quantity for the second and third day, we can write \(g
(60)\) and \(h(60)\).
Ask students to refer to the three graphs from the warm-up to answer the questions.
Student Facing
Let’s name the functions that relate the dog’s distance from the post and the time since its owner left: function \(f\) for Day 1, function \(g\) for Day 2, function \(h\) for Day 3. The input of
each function is time in seconds, \(t\).
1. Use function notation to complete the table.
│ │day 1│day 2│day 3│
│a. distance from post 60 seconds after the owner left │ │ │ │
│b. distance from post when the owner left │ │ │ │
│c. distance from post 150 seconds after the owner left │ │ │ │
2. Describe what each expression represents in this context:
1. \(f(15)\)
2. \(g(48)\)
3. \(h(t)\)
3. The equation \(g(120) = 4\) can be interpreted to mean: “On Day 2, 120 seconds after the dog owner left, the dog was 4 feet from the post.”
What does each equation mean in this situation?
1. \(h(40) = 4.6\)
2. \(f(t) = 5\)
3. \(g(t) = d\)
Anticipated Misconceptions
Students may ignore the function name and attend only to the input value. For instance, they may say “\(f(60)\) means that 60 seconds have passed.” Explain that the input value of 60 or \(t=60\) does
represents that 60 seconds have passed, but the expression \(f(60)\) represents the output value of the function. In this case, it means the dog’s distance from the post, on Day 1, 60 seconds after
its owner left.
Activity Synthesis
Invite students to share their responses. As students begin to share, they may be unsure as to how to express the notation orally. Explain that the expression \(f(60)\) is read "\(f\) of 60," \(g
(150)\) is read "\(g\) of 150," and \(h(t)\) is read "\(h\) of \(t\)."
To make sure students see the structure of this new notation, consider displaying it and annotating each part, as shown here.
Clarify that:
• The notation \(f(x)\) is read “\(f\) of \(x\).” It tells us that \(f\) is the name of the function, \(x\) is the input of the function, and \(f(x)\) is the output or the value of the function
when the input is \(x\).
• The statement \(g(t) = d\) is read: “\(g\) of \(t\) is equal to \(d\).” It tells us that \(g\) is the name of the function and \(t\) is the input. It also tells us that \(g(t)\) is the output or
the value of the function at \(t\), and \(g(t)\) has the same value as \(d\).
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their description of what each expression represents in context, present an incorrect response and explanation. For
example, “\(f(15)\) is the input value. It tells us that 15 seconds have passed.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with
a partner, monitor for students who clarify that \(f(15)\) represents an output value, specifically the distance of the dog from the post 15 seconds after the owner left on day 1. This helps students
evaluate, and improve upon, the written mathematical arguments of others, as they interpret function notation in context.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding
of function notation. Provide students with a physical copy of the annotated function notation.
Supports accessibility for: Conceptual processing; Language
2.3: Birthdays (10 minutes)
This activity reinforces students’ understanding about what makes a relationship between two variables a function, namely, that it gives a unique output for each input. It also prompts students to
use function notation to express a functional relationship that does not involve numerical values for its input and output.
Student Facing
Rule \(B\) takes a person’s name as its input, and gives their birthday as the output.
│ input │ output │
│Abraham Lincoln │February 12 │
│ │ │
│ │ │
│ │ │
Rule \(P\) takes a date as its input and gives a person with that birthday as the output.
│ input │ output │
│August 26 │Katherine Johnson │
│ │ │
│ │ │
│ │ │
1. Complete each table with three more examples of input-output pairs.
2. If you use your name as the input to \(B\), how many outputs are possible? Explain how you know.
3. If you use your birthday as the input to \(P\), how many outputs are possible? Explain how you know.
4. Only one of the two relationships is a function. The other is not a function. Which one is which? Explain how you know.
5. For the relationship that is a function, write two input-output pairs from the table using function notation.
Student Facing
Are you ready for more?
1. Write a rule that describes these input-output pairs:
2. Here are some input-output pairs with the same inputs but different outputs:
What rule could define function \(v\)?
Anticipated Misconceptions
If a student wonders what happens to a person born on February 29, tell them that the output of the function is the original birth date, not the annual birthday.
Activity Synthesis
Discuss with students:
• “Why is \(B\) a function, but \(P\) isn’t?” (Each input for \(B\) has a unique output, while inputs for \(P\) may have several outputs. For example, March 14 is the birthday of Albert Einstein,
Stephen Curry, Billy Crystal, Simone Biles and many other people. February 12 is the birthday of Abraham Lincoln and Charles Darwin.)
• “Would it be acceptable to express relationship \(P\) using function notation, for instance,: \(P(\text{August 26})= \text{Katherine Johnson}\)? Why or why not?” (No, because this notation is
reserved for functions.)
Some students might wonder if \(B\) is still a function if multiple people have the same name. For instance, there might be a few people named Katherine Johnson, and if we enter “Katherine Johnson”
as the input for \(B\), we would likely get different birthdays for the output.
Acknowledge that this is true, and that \(B\) would only be a function if it assumes that no two people have the same full name, or if another identifier could be used to tell apart people with the
same first name and last name (for instance, if a middle name or initial is also used, or if a number is added to each Katherine Johnson to distinguish them from one another).
Conversing: MLR2 Collect and Display. Listen for and collect vocabulary, gestures, and diagrams students use to describe what makes a relationship between two variables a function. Capture student
language that reflects a variety of ways to describe the characteristics of and differences between a relationship that is a function and a relationship that is not a function. Amplify the words
“unique,” “input,” and “output.” Write the students’ words on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed.
This will help students read and use mathematical language during their partner and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Lesson Synthesis
Refer back to the bagel shop activity from the opening lesson. Invite students to consider some statements about the function they saw in that situation. Display the following for all to see:
The best price for bagels, in dollars, is a function of the number of bagels bought, \(n\).
Arrange students in groups of 2. Ask partners to take turns reading and interpreting the statements in function notation. Each person should:
• Read the statement aloud to their partner.
• Identify the input, the output, and the function in the statement.
• Explain the meaning of the entire statement using a complete sentence.
If students say that the first two statements have no outputs, clarify that both \(b(2)\) and \(b(6)\) represent outputs, even though the value of each is not stated.
2.4: Cool-down - A Growing Puppy (5 minutes)
Student Facing
Here are graphs of two functions, each representing the cost of riding in a taxi from two companies—Friendly Rides and Great Cabs.
For each taxi, the cost of a ride is a function of the distance traveled. The input is distance in miles, and the output is cost in dollars.
• The point \((2,5.70)\) on one graph tells us the cost of riding a Friendly Rides taxi for 2 miles.
• The point \((2, 4.25)\) on the other graph tells us the cost of riding a Great Cabs taxi for 2 miles.
We can convey the same information much more efficiently by naming each function and using function notation to specify the input and the output.
• Let’s name the function for Friendly Rides function \(f\).
• Let's name the function for Great Cabs function \(g\).
• To refer to the cost of riding each taxi for 2 miles, we can write: \(f(2)\) and \(g(2)\).
• To say that a 2-mile trip with Friendly Rides will cost \$5.70, we can write \(f(2)=5.70\).
• To say that a 2-mile trip with Great Cabs will cost \$4.25, we can write \(g(2)=4.25\).
In general, function notation has this form:
It is read “\(f\) of \(x\)” and can be interpreted to mean: \(f(x)\) is the output of a function \(f\) when \(x\) is the input.
The function notation is a concise way to refer to a function and describe its input and output, which can be very useful. Throughout this unit and the course, we will use function notation to talk
about functions. | {"url":"https://curriculum.illustrativemathematics.org/HS/teachers/1/4/2/index.html","timestamp":"2024-11-03T07:01:08Z","content_type":"text/html","content_length":"125813","record_id":"<urn:uuid:d061e7cd-cf58-412d-8481-b5e2aae7bf26>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00705.warc.gz"} |
Yield | Data Encyclopedia
Staking Yield Metrics
Annual real and nominal percentage rates and yields for active stakers. A staker is an account that has staked native assets to participate in network consensus.
A real staking rate is adjusted to the network's inflation rate.
APR metrics are calculated without taking compounding into account. APY metrics take compounding into account for chains where this is available. Otherwise the two should be identical.
Hourly APR Metrics are annualized based on a 24h rolling period
Availability for Assets
Validator Yield Metrics
Annual real and nominal percentage rates and yields for active validators. A validator is a staker account that has staked native assets and validates blocks in the network's consensus algorithm.
A real staking rate is adjusted to the network's inflation rate.
APR metrics are calculated without taking compounding into account. APY metrics take compounding into account for chains where this is available. Otherwise the two should be identical.
Hourly APR Metrics are annualized based on a 24h rolling period
Availability for Assets
Delegator Yield Metrics
Annual real and nominal percentage rates and yields for active validators. A delegator is a staker account that has entrusted its staked native assets to a validator to participate in network
A real staking rate is adjusted to the network's inflation rate.
APR metrics are calculated without taking compounding into account. APY metrics take compounding into account for chains where this is available. Otherwise the two should be identical.
Hourly APR Metrics are annualized based on a 24h rolling period
Availability for Assets | {"url":"https://docs.coinmetrics.io/network-data/network-data-overview/staking/yield","timestamp":"2024-11-08T04:27:42Z","content_type":"text/html","content_length":"659022","record_id":"<urn:uuid:22195f24-bda5-4004-80c7-c9d20346cfda>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00011.warc.gz"} |
Design of Experiments Online Course - DOE Training | GoSkills
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What you will learn
In this Design of Experiments online course, you will learn the Design of Experiments or DOE. This design technique, which can be applied in several different methods, takes the results from a few
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In addition to covering experimental design approaches and methodologies, you will learn to use popular statistical analysis application Minitab to conduct your DOE and apply your results.
The design of experiments technique is incredibly powerful when working with new products, new technologies, or when migrating an existing technology into a new application. It is also very helpful
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Who this Design of Experiments online course is for
This course can be taken as part of the GoSkills Lean Six Sigma Black Belt training program, to prepare for certification with IASSC. It is also a good stand-alone course to improve proficiency and
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This course will be from the standpoint of helping you to make wise decisions about your product and process design and management, not conducting mathematical proofs or solving complex matrix
• 24 practical tutorials with videos, reference guides, exercises and quizzes.
• Designed to prepare you in part for the IASSC Black Belt exam. To prepare in full, you should take all eight courses in our Lean Six Sigma Black Belt program.
• Identify when and why to do a DoE, and recognize the steps of the process.
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Once enrolled, our friendly support team and tutors are here to help with any course related inquiries.
Skill level: Intermediate
Certificate: Yes
Lessons: 26
Accredited by: CPD
Pre-requisites: None
Video duration: 2h 25m
Estimated study time: 13h for all materials
Experimental Design Approaches Free Lesson
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Experiments and Design
The design process typically relies on experiments to create and analyze data that is used when making design decisions. This data is invaluable to the design team as they strive to create a superior
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Trial and Error
The simplest experimental design approach is trial and error. If subject matters experts are generating the trial design, this can be successful. However, if the trial fails, this approach can lead
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One Factor At A Time
The OFAAT method is often considered the best scientific method for creating a plan of experiments. It is very controlled, and the design performance often grows in capability over time. But it is
also the most timing consuming and expensive approach when conducting a set of experiments.
Full Factorial Design of Experiments
A full factorial DOE conducts a set of experiments with carefully controlled configurations of the independent or control factors in the design. The results are statistically analyzed to create a
design space equation that can be used to optimize the design. It is faster and cheaper than OFAAT, but longer and more costly than a lucky guess with Trial and Error.
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Fractional Factorial Design of Experiments
A fractional factorial DOE conducts only a fraction of the experiments done with the full factorial DOE. It then statistically analyzes the results to fine tune the design and normally does a second
optimizing study. Even though there are typically several sets of experiments, the total is still less than the number conducted with a full factorial study and much less than OFAAT.
Theory of Design of Experiments
This lesson provides a high level description of the DOE process that applies to any type of DOE. It also answers the questions of when to do a DOE and why to do a DOE.
DOE Studies
This lesson explains the preparation needed to initiate a DOE study of any type. It includes a discussion on setting the DOE objective and has a checklist of questions that will need to be answered
either before the study starts or early in the study design.
Full Factorial Design of Experiments
Full Factorial DOE Methodology
This lesson describes the eight steps to be followed when conducting a full factorial DOE.
Factor Selection
This lesson explains the different types of factors that are involved in the DOE study design including control factors and response factors. The characteristics that should be used when selecting
each type of factor are discussed.
Full Factorial DOE Study Design
This lesson explains how to design the study so that the statistical analysis can be performed. The preparation of the test sample configurations is explained. The use of design features of
replication, center points and blocking are also addressed.
Conducting the Study
This lesson addresses how to execute and control each of the experimental runs in the study. It also explains the importance of the measurement system that is used.
DOE Functional Equation
The statistical analysis of the full factorial DOE results in the determination of the coefficients for a design space equation that relates all the control factors to the response factors. This
equation includes interaction effects between control factors. This equation can then be used by designers to solve for the best overall system performance.
DOE in Minitab
Minitab is the statistical analysis software application that is most often used with Lean Six Sigma projects. Minitab has a Wizard that guides you through the setup and design of a Design of
Experiments study. This lesson demonstrates how to use that Wizard.
Fractional Factorial Design of Experiments Free Lesson
Fractional Factorial Pros and Cons
This lesson compares the difference between the full factorial approach and fractional factorial approaches. It explains the pros and cons of using a fractional factorial methodology.
Fractional Factorial DOE Methodology
This lesson describes the nine steps to be followed when using one of the fractional factorial DOE methods. The emphasis is one how the steps differ from the full factorial DOE methodology.
Try it!
Confounding Effects
This lesson explains the importance of designing a fractional factorial DOE study using a set of experiments that is balanced and orthogonal. Otherwise the runs can become confounded and that will
invalidate the statistical analysis of the results.
Factor Selection
This lesson builds on the previous factor selection lesson. However, now it addresses how the factor selection process changes as a fractional factorial DOE progresses through two or three levels of
Plackett-Burman DOE
The Plackett-Burman DOE is a special case fractional factorial DOE. It is used as a screening study when there are a large number of control factors. This lesson explains when to use Plackett-Burman
DOE and how to design this type of study.
Taguchi DOE
The Taguchi DOE is a special case fractional factorial DOE. It is used primarily for analyzing manufacturing processes. The Taguchi DOE separates the control factors into two categories and analyzes
them with different DOE approaches. This lesson explains the characteristics of this type of study.
Applying DOE Results
DOE Analysis in Minitab
This lesson reviews the different types of graphical and tabular results for a DOE study that are generated by Minitab. Each of these types of results provides a different perspective on the analysis
of the design that is being studied.
DOE Factorial Plots
One of the most common techniques for analyzing the results of a DOE study in Minitab is to review the factor plots. These will provide insight into the optimal settings for control factors. The
interactive plots will also highlight the settings associated with local maximum or minimum performance levels.
DOE in Design Creation
The DOE results can be used by design teams to make wise design decisions. This lesson will address how to use the DOE results in predicting system performance, designing system controls and
establishing tolerances on system control and response factors.
Path of Steepest Ascent/Descent
Some DOE analyses will indicate that the optimal performance of the system would occur when control factors are set beyond the bounds of the study. When this occurs, it is best to shift the study to
the likely region of optimal performance and then determine the best control factor settings. Following the path of steepest ascent or descent will ensure that the new analysis is conducted in a
region with maximum or minimum performance.
DOE in Design Optimization
The DOE results can be used by design teams to improve and optimize an existing design based upon new needs or uses. The structure of the DOE study, particularly the fractional factorial DOE
methodologies, allows the design team to easily establish optimal performance in a variety of settings.
DOE in Problem Solving
The DOE results can be used by problem solving teams, such as Lean Six Sigma project teams, to identify which factors provide the major contribution to the problem or problem performance. It can also
be used to explain the expected benefit from implementing different types of solutions.
DOE Keys to Success
This final lesson reviews the key principles that must be followed when conducting a DOE study. It highlights the benefit of each and the dangers if the principle is not properly applied. | {"url":"https://www.goskills.com/Course/Design-Experiments","timestamp":"2024-11-10T01:21:16Z","content_type":"text/html","content_length":"179516","record_id":"<urn:uuid:101a0ffd-0776-47d2-8b2a-3aa66fb4fd28>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00780.warc.gz"} |
Christian Schweizer
Parity-dependent state transfer for direct entanglement generation
As quantum information technologies advance they face challenges in scaling and connectivity. In particular, two necessities remain independent of the technological implementation:
the need for connectivity between distant qubits and the need for efficient generation of entanglement. Perfect State Transfer is a technique which realises the time optimal transfer of a quantum
state between distant nodes of qubit lattices with only nearest-neighbour couplings, hence providing an important tool to improve device connectivity. Crucially, the transfer protocol results in
effective parity-dependent non-local interactions, extending its utility to the efficient generation of entangled states. Here, we experimentally demonstrate Perfect State Transfer and the generation
of multi-qubit entanglement on a chain of superconducting qubits. The system consists of six fixed-frequency transmon qubits connected by tunable couplers, where the couplings are controlled via
parametric drives. By simultaneously activating all couplings and engineering their individual amplitudes and frequencies, we implement Perfect State Transfer on up to six qubits and observe the
respective single-excitation dynamics for different initial states. We then apply the protocol in the presence of multiple excitations and verify its parity-dependent property, where the number of
excitations within the chain controls the phase of the transferred state. Finally, we utilise this property to prepare a multi-qubit Greenberger-Horne-Zeilinger state using only a single transfer
operation, demonstrating its application for efficient entanglement generation.
Efficient decoupling of a non-linear qubit mode from its environment
To control and measure the state of a quantum system it must necessarily be coupled to external degrees of freedom. This inevitably leads to spontaneous emission via the Purcell effect,
photon-induced dephasing from measurement back-action, and errors caused by unwanted interactions with nearby quantum systems. To tackle this fundamental challenge, we make use of the design
flexibility of superconducting quantum circuits to form a multi-mode element — an artificial molecule — with symmetry-protected modes. The proposed circuit consists of three superconducting islands
coupled to a central island via Josephson junctions. It exhibits two essential non-linear modes, one of which is flux-insensitive and used as the protected qubit mode. The second mode is flux-tunable
and serves via a cross-Kerr type coupling as a mediator to control the dispersive coupling of the qubit mode to the readout resonator. We demonstrate the Purcell protection of the qubit mode by
measuring relaxation times that are independent of the mediated dispersive coupling. We show that the coherence of the qubit is not limited by photon-induced dephasing when detuning the mediator mode
from the readout resonator and thereby reducing the dispersive coupling. The resulting highly protected qubit with tunable interactions may serve as a basic building block of a scalable quantum
processor architecture, in which qubit decoherence is strongly suppressed.
Effective non-local parity-dependent couplings in qubit chains
For the efficient implementation of quantum algorithms, practical ways to generate many-body entanglement are a basic requirement. Specifically, coupling multiple qubit pairs at once
can be advantageous and can lead to multi-qubit operations useful in the construction of hardware-tailored algorithms. Here we harness the simultaneous coupling of qubits on a chain and engineer a
set of non-local parity-dependent quantum operations suitable for a wide range of applications. The resulting effective long-range couplings directly implement a parametrizable Trotter-step for
Jordan-Wigner fermions and can be used for simulations of quantum dynamics, efficient state generation in variational quantum eigensolvers, parity measurements for error-correction schemes, and the
generation of efficient multi-qubit gates. Moreover, we present numerical simulations of the gate operation in a superconducting quantum circuit architecture, which show a high gate fidelity of >
99.9% for realistic experimental parameters.
Z2 lattice gauge theories and Kitaev’s toric code: A scheme for analog quantum simulation
Kitaev’s toric code is an exactly solvable model with Z2-topological order, which has potential applications in quantum computation and error correction. However, a direct experimental
realization remains an open challenge. Here, we propose a building block for Z2 lattice gauge theories coupled to dynamical matter and demonstrate how it allows for an implementation of the
toric-code ground state and its topological excitations. This is achieved by introducing separate matter excitations on individual plaquettes, whose motion induce the required plaquette terms. The
proposed building block is realized in the second-order coupling regime and is well suited for implementations with superconducting qubits. Furthermore, we propose a pathway to prepare topologically
non-trivial initial states during which a large gap on the order of the underlying coupling strength is present. This is verified by both analytical arguments and numerical studies. Moreover, we
outline experimental signatures of the ground-state wavefunction and introduce a minimal braiding protocol. Detecting a π-phase shift between Ramsey fringes in this protocol reveals the anyonic
excitations of the toric-code Hamiltonian in a system with only three triangular plaquettes. Our work paves the way for realizing non-Abelian anyons in analog quantum simulators. | {"url":"https://circuitqed.net/publications_author/christian-schweizer/","timestamp":"2024-11-12T20:10:01Z","content_type":"text/html","content_length":"28997","record_id":"<urn:uuid:071ac5af-4a81-47b0-a586-e2bc3a5a9d45>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00484.warc.gz"} |