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Coq devs & plugin devs
@Jason Gross I saw your post on performance optimization/measurements for proof assistants on Coq-Club. We summarize a lot of work on parallelization/selection for proof assistants in our regression
proving papers (ASE 17 & ISSTA 18): http://users.ece.utexas.edu/~gligoric/papers/CelikETAL17iCoq.pdf http://users.ece.utexas.edu/~gligoric/papers/PalmskogETAL18piCoq.pdf
in particular, the ISSTA paper uses a particular methodology for measuring proof checking performance in real-world projects, adapted from regression testing in SE
@Karl Palmskog Thanks! I'll take a look at those
the Coq mutation analysis paper (ASE '19, http://users.ece.utexas.edu/~gligoric/papers/CelikETAL19mCoq.pdf) also looks at proof-checking performance, but in the context of automatically making small
changes to Coq syntax and checking terms that might be affected, so a more artificial scenario - and serialization via SerAPI is part of process (however, in my experience Jane Street already
optimized OCaml-to-sexp a lot)
(these publications were the core of Ahmet Celik's 2019 PhD thesis at UT Austin, https://ahmet-celik.github.io/papers/CELIK-DISSERTATION-2019.pdf)
Last updated: Oct 13 2024 at 01:02 UTC
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Eureka Math Grade 6 Module 1 Lesson 15 Answer Key
Engage NY Eureka Math 6th Grade Module 1 Lesson 15 Answer Key
Eureka Math Grade 6 Module 1 Lesson 15 Exercise Answer Key
Exercise 1.
Create a table to determine how many views the website probably had one hour after the end of the broadcast based on how many views it had two and three hours after the end of the broadcast. Using
this relationship, predict how many views the website will have 4, 5, and 6 hours after the end of the broadcast.
Exercise 2.
What is the constant number, c, that makes these ratios equivalent?
Using an equation, represent the relationship between the number of views, y, the website received and the number of hours, h, after this morning’s news broadcast.
v = 12h
Exercise 3.
Use the table created In Exercise 1 to Identify sets of ordered pairs that can be graphed.
(1, 12), (2,24), (3, 36), (4, 48), (5, 60), (6, 72)
Exercise 4.
Use the ordered pairs you created to depict the relationship between hours and number of views on a coordinate plane.
Label your axes and create a title for the graph. Do the points you plotted lie on a line?
Exercise 5
Predict how many views the website will have after twelve hours. Use at least two representations (e.g., tape diagram, table, double number line diagram) to justify your answer.
Exercise 6
Also on the news broadcast, a chef from a local Italian restaurant demonstrated how he makes fresh pasta daily for his restaurant. The recipe for his pasta is below:
3 eggs, beaten
1 teaspoon salt
2 cups all-purpose flour
2 tablespoons water
2 tablespoons vegetable oil
Determine the ratio of the number of tablespoons of water to the number of eggs.
2: 3
Provided the information in the table below, complete the table to determine ordered pairs. Use the ordered pairs to graph the relationship of the number of tablespoons of water to the number of
┃Tablespoons of water │Number of Eggs ┃
┃2 │ ┃
┃4 │ ┃
┃6 │ ┃
┃8 │ ┃
┃10 │ ┃
┃12 │ ┃
┃Tablespoons of water │Number of Eggs ┃
┃2 │3 ┃
┃4 │6 ┃
┃6 │9 ┃
┃8 │12 ┃
┃10 │15 ┃
┃12 │18 ┃
What would you have to do to the graph in order to find how many eggs would be needed If the recipe was larger and called for 16 tablespoons of water?
Extend the graph.
Demonstrate on your graph.
How many eggs would be needed if the recipe called for 16 tablespoons of water?
Exercise 7.
Determine how many tablespoons of water will be needed If the chef is making a large batch of pasta and the recipe increases to 36 eggs. Support your reasoning using at least one diagram you find
applies best to the situation, and explain why that tool is the best to use.
Answers may vary but should include reasoning for each tool. For example, extending the table/double number line diagram because values were already given to find the pattern or using a tape diagram
to determine the equivalent ratios.
Eureka Math Grade 6 Module 1 Lesson 15 Problem Set Answer Key
Question 1.
The producer of the news station posted an article about the high school’s football championship ceremony on a new website. The website had 500 views after four hours. Create a table to show how
many views the website would have had after the first, second, and third hours after posting, if the website receives views at the same rate. How many views would the website receive after 5 hours?
┃Hours │Views ┃
┃1 │125 ┃
┃2 │250 ┃
┃3 │375 ┃
┃4 │500 ┃
┃5 │625 ┃
Question 2.
Write an equation that represents the relationship from Problem 1. Do you see any connections between the equations you wrote and the ratio of the number of views to the number of hours?
125h = v
Question 3.
Use the table in Problem 1 to make a list of ordered pairs that you could plot on a coordinate plane.
(1, 125), (2, 250), (3, 375), (4, 500), (5, 625)
Question 4.
Graph the ordered pairs on a coordinate plane. Label your axes and create a title for the graph.
Question 5.
Use multiple tools to predict how many views the website would have after 12 hours.
Answers may vary but could include all representations from the module. The correct answer is 1,500 views.
Eureka Math Grade 6 Module 1 Lesson 15 Exit Ticket Answer Key
Question 1.
Jen and Nikki are making bracelets to sell at the local market. They determined that each bracelet would have eight beads and two charms.
Complete the table below to show the ratio of the number of charms to the number of beads.
Create ordered pairs from the table, and plot the pairs on the graph below. Label the axes of the graph, and provide a title.
Eureka Math Grade 6 Module 1 Lesson 15 Exploratory Challenge Answer Key
Question 1.
At the end of this morning’s news segment, the local television station highlighted area pets that need to be adopted. The station posted a specific website on the screen for viewers to find more
information on the pets shown and the adoption process. The station producer checked the website two hours after the end of the broadcast and saw that the website had 24 views. One hour after that,
the website had 36 views.
At the end of this morning’s news segment, the local television station highlighted area pets that need to be adopted. The station posted a specific website on the screen for viewers to find more
information on the pets shown and the adoption process. The station producer checked the website two hours after the end of the broadcast and saw that the website had 24 views. One hour after that,
the website had 36 views.
Leave a Comment
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ACM Other ConferencesBetter Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs
We develop a framework for algorithms finding the diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) subquadratic time complexity. The class of bounded distance
VC-dimension graphs is wide, including, e.g. all minor-free graphs.
We build on the work of Ducoffe et al. [SODA'20, SIGCOMP'22], improving their technique. With our approach the algorithms become simpler and faster, working in 𝒪{(k ⋅ n^{1-1/d} ⋅ m ⋅ polylog(n))}
time complexity for the graph on n vertices and m edges, where k is the diameter and d is the distance VC-dimension of the graph. Furthermore, it allows us to use the improved technique in more
general setting. In particular, we use this framework for geometric intersection graphs, i.e. graphs where vertices are identical geometric objects on a plane and the adjacency is defined by
intersection. Applying our approach for these graphs, we partially answer a question posed by Bringmann et al. [SoCG'22], finding an 𝒪{(n^{7/4} ⋅ polylog(n))} parameterized diameter algorithm for
unit square intersection graph of size n, as well as a more general algorithm for convex polygon intersection graphs.
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What Do You Mean by Acceleration Due to Gravity? An In-Depth Explanation
17 Oct, 24
What Do You Mean by Acceleration Due to Gravity? An In-Depth Guide
Understanding What Do You Mean by Acceleration Due to Gravity? When you hear the term acceleration due to gravity, you might picture objects falling towards the Earth or the feeling of
weightlessness in space. But what do you mean by acceleration due to gravity, and why is it such a crucial concept in physics? In […]
What Do You Mean by Acceleration Due to Gravity? An In-Depth Explanation
Understanding What Do You Mean by Acceleration Due to Gravity?
When you hear the term acceleration due to gravity, you might picture objects falling towards the Earth or the feeling of weightlessness in space. But what do you mean by acceleration due to gravity,
and why is it such a crucial concept in physics? In simple terms, acceleration due to gravity, often represented as g, is the rate at which objects accelerate towards the Earth due to the force of
gravity. This fundamental concept not only explains why apples fall from trees but also plays a critical role in various scientific calculations and technologies. Understanding the nature and
behavior of gravitational acceleration is key to grasping the laws of motion and force that govern our universe.
In this comprehensive guide, we’ll explore everything you need to know about acceleration due to gravity. From the science behind the numbers to real-world applications, we will dive deep into each
aspect, ensuring that you have a well-rounded understanding of this essential phenomenon. Get ready to embark on a journey that answers the question “what do you mean by acceleration due to gravity?”
in a way that even non-scientists can appreciate!
Quick Data Point Table: Acceleration Due to Gravity :
Parameter Value
Standard Acceleration (g) 9.8 m/s²
Unit of Measurement meters per second squared (m/s²)
Gravitational Constant (G) 6.674 × 10^-11 N(m/kg)²
Variable Factors Altitude, latitude, mass
bb Influences weight and motion
What Do You Mean by Acceleration Due to Gravity?
Defining Acceleration Due to Gravity:
To understand what do you mean by acceleration due to gravity, you must first comprehend Newton’s Law of Universal Gravitation. This principle states that every object in the universe attracts every
other object with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.
The Role of Gravitational Force
The force of gravity is what causes objects to accelerate towards the center of the Earth at a constant rate, commonly denoted as 9.8 m/s². This acceleration is uniform for all objects, regardless of
their mass, in the absence of air resistance.
Why is it Always 9.8 m/s²?
The value of 9.8 m/s² is an average figure. The actual acceleration due to gravity can vary slightly depending on factors like altitude and the Earth’s shape.
Factors Influencing Acceleration Due to Gravity:
Altitude and Gravitational Acceleration:
Gravity weakens as you move further from the Earth’s surface. Hence, objects at higher altitudes experience a slightly lower acceleration due to gravity.
Impact of Latitude on Gravitational Force
The Earth’s rotation and its shape cause variations in gravitational force. Gravity is slightly stronger at the poles and weaker at the equator.
Mathematical Representation of Gravitational Acceleration:
Gravitational Formula Breakdown:
The formula used to calculate acceleration due to gravity is: F = G * (m1 * m2) / r²
• F is the gravitational force.
• G is the gravitational constant.
• m1 and m2 are the masses of the objects.
• r is the distance between the centers of the two masses.
Practical Examples of Acceleration Due to Gravity:
Everyday Applications in Physics:
Gravitational acceleration affects everyday activities like walking, driving, and even the functioning of smartphones through their accelerometers.
Role in Engineering and Space Exploration
Engineers use the concept of acceleration due to gravity in designing bridges, buildings, and spacecraft to ensure structural integrity under gravitational forces.
Acceleration Due to Gravity on Different Planets:
Comparing Earth’s Gravity to Other Celestial Bodies
The acceleration due to gravity differs on various planets. For instance, on the Moon, gravity is only about 1.62 m/s², making it much weaker than Earth’s.
Why Understanding Gravity is Crucial for Space Travel?
Knowing the gravity of other planets is essential for planning missions and understanding how spacecraft will behave when landing or taking off from those surfaces.
Real-World Implications of Gravitational Acceleration:
Gravity’s Effect on Human Physiology
Gravity impacts blood circulation, bone density, and muscle strength. Astronauts experience muscle atrophy and bone loss when exposed to microgravity conditions in space.
The Role of Gravity in Meteorology
Gravity helps in the formation of weather patterns and ocean currents, influencing the global climate and weather forecasting.
Experimenting with Acceleration Due to Gravity:
Galileo’s Famous Experiment
Galileo’s experiment at the Leaning Tower of Pisa proved that the acceleration due to gravity is the same for all objects, regardless of mass.
Conducting Simple Experiments at Home
You can observe gravitational acceleration by dropping two objects of different weights simultaneously and noticing that they hit the ground at the same time (if air resistance is negligible).
FAQs About Acceleration Due to Gravity:
1. What is the standard value of acceleration due to gravity?
□ The standard value is 9.8 m/s².
2. Does gravity affect objects of different masses differently?
□ In a vacuum, all objects fall at the same rate regardless of mass.
3. How does altitude impact gravitational acceleration?
□ Gravity decreases with an increase in altitude.
4. What happens to gravity at the equator compared to the poles?
□ Gravity is slightly weaker at the equator and stronger at the poles due to the Earth’s shape.
5. Why is gravity important in physics?
□ It’s essential for understanding motion, forces, and energy.
6. What role does gravity play in space travel?
□ It determines how much energy is required to launch and land spacecraft.
7. Can we manipulate gravity?
□ Currently, we have no technology to manipulate gravity directly.
8. Why do astronauts float in space?
□ They are in a state of free fall, orbiting the Earth due to gravity.
9. How does gravity affect time?
□ According to Einstein’s theory of relativity, gravity can slow down time.
10. What would happen if gravity suddenly disappeared?
□ Without gravity, everything not anchored to the ground would float into space.
Grasping the concept of what do you mean by acceleration due to gravity is pivotal in the realms of physics, engineering, and space exploration. This acceleration forms the backbone of our
understanding of how objects interact within the universe. By studying gravitational acceleration, we unlock the mysteries of the cosmos and lay the foundation for technological advancements. The
impact of gravity on our daily lives, as well as its importance in scientific research, underscores its irreplaceable role in both natural phenomena and human innovation.
Stay updated with all the insights.
Navigate news, 1 email day.
Subscribe to Qrius
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The Hile Controls, Inc. Blog
The two leading methods for measuring mass flow rate of fluids in industrial control are the
Coriolis flowmeter
and the
thermal dispersion flowmeter
Thermal Dispersion (left) vs. Coriolis (right) Flow Technology
(click for larger view)
A Coriolis mass flowmeter measures mass flow rate of the fluid with a U-shaped tube that deflects or vibrates as the fluid flows through it. The operation of this type of mass flow meter is based on
the conservation of angular momentum as it applies to the Coriolis acceleration of a fluid. Fluid flows through the oscillating tubes, twisting them slightly in proportion to the mass flow of the
fluid and its inertia. Sensors are fixed at the inlet and outlet junctures of the tube, at equal distances from the central fixed point. When there is no fluid flowing through the tube, the amplitude
is constant and the sensors at either end are in phase with one another.
While Coriolis flowmeters may be used for mass flow measurement in liquids as well as gases, they are prominently used for liquids as a high-density fluid is required to maintain the momentum of
Thermal Dispersion technology uses the principle of measuring the differential temperature between two temperature sensors and calculating mass flow based upon the cooling effect. Mass flow is based
on the rate of heat dissipation per unit time. There are two types of thermal dispersion technology - Constant Temperature Differential Method and the Constant Current Method.
Constant Current Method
maintains a constant differential temperature and the current required to maintain the differential is used as the basis for determining the flow. The greater the mass flow rate, the greater is the
cooling effect and the more current needed to maintain the same differential temperate.
Constant Temperature Differential Method
measures the differential temperature between sensors. A temperature difference between the two sensors is an indication of the mass flow rate of the fluid. The greater the mass flow rate, the
smaller the temperature difference.
For any questions about measuring process flow, contact
Hile Controls of Alabama
by visiting
or by calling 800-536-0269.
The differential flow meter is the most common device for measuring fluid flow through pipes. Flow rates and pressure differential of fluids, such as gases vapors and liquids, are explored using the
orifice plate flow meter in the video below.
The differential flow meter, whether Venturi tube, flow nozzle, or orifice plate style, is an in line instrument that is installed between two pipe flanges.
The orifice plate flow meter is comprised the circular metal disc with a specific hole diameter that reduces the fluid flow in the pipe. Pressure taps are added on each side at the orifice plate to
measure the pressure differential.
According to the Laws of Conservation of Energy, the fluid entering the pipe must equal the mass leaving the pipe during the same period of time. The velocity of the fluid leaving the orifice is
greater than the velocity of the fluid entering the orifice. Applying Bernoulli's Principle, the increased fluid velocity results in a decrease in pressure.
As the fluid flow rate increases through the pipe, back pressure on the incoming side increases due to the restriction of flow created by the orifice plate.
The pressure of the fluid at the downstream side at the orifice plate is less than the incoming side due to the accelerated flow.
With a known differential pressure and velocity of the fluid, the volume metric flow rate can be determined. The flow rate “Q”, of a fluid through an orifice plate increases in proportion to the
square root the pressure difference on each side multiplied by the K factor. For example if the differential pressure increases by 14 PSI with the K factor of one, the flow rate is increased by 3.74.
The new Bronkhorst ES-FLOW™ are Volumetric Liquid Flow Meters for very low flow ranges. The instruments operate on a innovative measuring principle, using ultrasound in a very small, straight tube. A
wide range of liquids can be measured independent of fluid density, temperature and viscosity.
The ES-FLOW™ Ultrasonic Flow Meter was designed to measure tiny volume flows from 4 up to 1500 ml/min with high precision, high linearity and low pressure drop, using ultrasound in a small bore tube.
Liquids can be measured independent of fluid density, temperature and viscosity. Thanks to the combination of a straight sensor tube with zero dead volume the flow meter is self-draining.
• The orbital TIG- welding allows hygienic connections so the instrument can be used for hygienic applications.
• For non-hygienic applications, the flow meter can also be equipped with compression type fittings.
• Wetted parts are made of stainless steel, the exterior design is rated to IP67.
• The user interface is a capacitive touchscreen with a TFT display to operate and readout the instrument.
• The on-board PID controller can be used to drive a control valve or pump, enabling users to establish a complete, compact control loop.
• Food & Beverage
• Pharma (e.g. additives, sterilization)
• Medical
• Chemical (e.g. catalysts, reagents)
• Many other markets which require precision fluid handling e.g. fuel consumption measurement and dosing of colorants or lubricants in many industries.
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\vspace{-0.1cm} % Reduce space after title
We are interested in how similar two sets are.
\vspace{0.4cm} % Increase space before tikzpicture
% Add a box around everything with some padding
\draw[black, thin] (-2.5,-2.5) rectangle (4.5,2.8);
The standard measure is the so-called Jaccard similarity.
0 \leq \frac{A \cap B}{A \cup B} \leq 1
0 \leq \frac{8}{A \cup B} \leq 1
I tried using \visible instead of \only but this then doesn't show the transitions. I also tried \vphantom but I couldn't get it to work. How can I keep the box and the text above it steady?
I'd use a top-aligned frame:
We are interested in how similar two sets are.
% Add a box around everything with some padding
\draw[black, thin] (-2.5,-2.5) rectangle (4.5,2.8);
The standard measure is the so-called Jaccard similarity.
0 \leq \frac{\strut\alt<5>{8}{A \cap B}}{A \cup B} \leq 1
Instead of `\only` the following uses `\visible` for your maths:
\vspace{-0.1cm} % Reduce space after title
We are interested in how similar two sets are.
\vspace{0.4cm} % Increase space before tikzpicture
% Add a box around everything with some padding
\draw[black, thin] (-2.5,-2.5) rectangle (4.5,2.8);
The standard measure is the so-called Jaccard similarity.
0 \leq \frac{\vphantom{A \cap B}\only<4>{A \cap B}\only<5>{8}}{A \cup B} \leq 1
\vspace{-0.1cm} % Reduce space after title
We are interested in how similar two sets are.
\vspace{0.4cm} % Increase space before tikzpicture
% Add a box around everything with some padding
\draw[black, thin] (-2.5,-2.5) rectangle (4.5,2.8);
The standard measure is the so-called Jaccard similarity.
\only<1-3>{\[\vphantom{0 \leq \frac{A \cap B}{A \cup B} \leq 1}\]} % Reserve space for the equation
0 \leq \frac{A \cap B}{A \cup B} \leq 1
0 \leq \frac{8}{A \cup B} \leq 1
This helps a bit but not completely.
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Random Bubble Generator
The Random Bubbles Generator creates a image consisting of 16,000 circles of random sizes and colors, placed in random positions inside a circle.
Utilizes PHP GD Library for image creation.
Challenges: A randomly generated integer was used to represent one of the 16 Million possible colors in the 24-bit web color space. This integer must then be converted to an equivalent RGB color.
Solution: Convert interger to a 6-digit hex number, then split the hex number into 3 consecutive pairs (e.g. hex # 130099 becomes 13, 00, 99) and then convert each hex pair back into integers (19, 0,
153), this becomes the RGB values.
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$111,000 a Year is How Much an Hour in the Canada?
What salary is considered rich in the Canada?
If we're talking about being able to afford a luxury home and an expensive car, then it's $200,000 or more.
If we are talking about the upper middle class, then it ranges from $90,000 to $200,000 per year.
· Middle class $60,000 to $90,000 per year.
· Lower middle class $40,000 to $60,000 per year.
It depends a lot on the location. Some areas are super expensive. You can't even get a studio apartment for $100,000 cash, whereas in some other parts of the country you could probably get a
two-bedroom row house for that kind of money.
How do you calculate hourly rate from annual salary?
To calculate the hourly rate based on the annual salary, divide the annual salary by the number of weeks per year, and then divide by the number of hours worked per week.
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Exotic spheres
The second round of the British Mathematical Olympiad was marked today; results are available from Joseph Myers’ website. Congratulations to everyone who featured on the leaderboard (especially
Linden Ralph, who secured first place). Moreover, best of luck to the recently-appointed EGMO team of Olivia Aaronson, Kasia Warburton, Eloise Thuey and Katya Richards.
31c/240 Caterpillar progress
In other news, Ivan Fomichev and Dave Greene have been working on building a 31c/240 spaceship in Conway’s Game of Life, using a similar principle to the (rather massive!) 17c/45 Caterpillar
constructed by Gabriel Nivasch, David Bell and Jason Summers. Nivasch has written an excellently descriptive article about it, which I suggest reading before proceeding to familiarise yourself with
the concepts and terminology.
Fortunately, the 31c/240 model is likely to be considerably smaller than its predecessor. In particular, the slow speed facilitates alternative mechanisms such as this one (henceforth referred to as
Design II, to contrast with Design I used in the 17c/45 Caterpillar), where streams of gliders from the back catch up with the front to stabilise the behemoth:
(This would have been impossible for the 17c/45 caterpillar, since the vertical component of the velocity of a glider is only c/4, which is slower than the spaceship’s speed of 17c/45; consequently,
c/2 orthogonal spaceships were necessary instead.)
Dave Greene has built a complete front end based on Design II, and Ivan Fomichev has done the same for Design I (including constructing a helix). Note the red ‘x’ in the diagram, where forward
gliders must be reflected inwards to stabilise the front; this presents an additional engineering problem. In fact, there are two suggested solutions:
• Design IIa: using some static junk which will be pushed forwards at a speed of 31c/240 by the glider waves;
• Design IIb: colliding the glider waves with horizontal spaceships emitted from the spine.
It is universally acknowledged that IIb is more practical than IIa, so the current effort is to realise it. All of the individual components have been designed, so it is simply a matter of assembling
them into a functional spaceship.
Exotic 7-spheres
Anyway, two smooth manifolds are homeomorphic if there exists a continuous bijection with continuous inverse, and diffeomorphic if there exists a differentiable bijection with differentiable inverse.
Clearly, every diffeomorphism is a homeomorphism, but the converse is not true. That suggests the following question:
Do there exist smooth manifolds S and T which are homeomorphic but not diffeomorphic?
Amazingly, the answer is yes, as proved by John Milnor in the case where S is the 7-sphere (boundary of an 8-ball), in which case T is called an exotic 7-sphere. There are 27 of these beasts, which
(together with the ordinary 7-sphere) form an order-28 abelian group (indeed, the cyclic group C28) under the operation of connect-sum.
It is an open problem as to whether there exist exotic 4-spheres. There are, however, exotic versions of $\mathbb{R}^4$, a phenomenon that occurs in no other dimension. Indeed, the situation is much
scarier than the compact case of 7-spheres: there are uncountably many non-diffeomorphic smooth structures on four-dimensional space!
4 Responses to Exotic spheres
1. “While talking about this exact topic, let’s suddenly switch to another, completely unrelated, one” – every CP4space post, ever 😛
I also have an unrelated question: I was recently trying to find solutions for one particular problem, and the size of a solution was depended on size of a “base set”, which must be linearily
independent over Q. For uncountable solution, we need an uncountable base set. It can be easily defined using axiom of choice (Zorn lemma or vector space basis existence, whichever you want), but
I was wondering if such uncountable set is constructible, or, at least, provably existent without AC?
□ No, there exist models of ZF in which the reals do not have a Hamel basis as a Q-vector-space. Consequently, there is no constructive example.
But the axiom of choice is definitely true, so you’re allowed to use Zorn’s lemma.
☆ I can’t really see why former implies latter. From what I understand, every Hamel basis must have size continuum (it might not be a case, as in ZF~C countable union of countables doesn’t
have to be countable), and, if c>Aleph_1, there is still a possibility for us to make linearily independent set of size Aleph_1, which can’t be extended to basis, due to lack of AC.
○ Oh, you just want an uncountable Q-linearly-independent set of reals, rather than a Hamel basis? Sure, that’s possible. Take (for example) sum_(n=1 to infinity)[10^-floor(exp(xn))]
for each real x > 1.
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Asymptotics for the boundary parabolic Anderson problem in a half space
We study the large time behavior of the solutions of the Cauchy problem for the Anderson model restricted to the upper half space D = ℤ ^d-1×ℤ and/or D = ℝ ^d-1 × ℝ [+] when the potential is a
homogeneous random field concentrated on the boundary ∂D. In other words we consider the problem: (Equation presented) with an appropriate initial condition. We determine the large time asymptotics
of the moments of the solutions as well as their almost sure asymptotic behavior when t → ∞ and when the distance from the boundary, i.e. y = y(t) goes simultaneously to infinity as a function of the
time t. We identify the rates of escape of y(t) which correspond to specific behaviors of the solutions and different types of dependence upon the diffusivity constant κ. We also show that the case
of the lattice differs drastically from the continuous case when it comes to the existence of the moments and the influence of κ. Intermittency is proved as a consequence of the large time behavior
of the solutions.
All Science Journal Classification (ASJC) codes
• Analysis
• Statistics and Probability
Dive into the research topics of 'Asymptotics for the boundary parabolic Anderson problem in a half space'. Together they form a unique fingerprint.
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Comparing machine learning models for a regression problem - Dibyendu Deb
Comparing machine learning models for a regression problem
Comparing different machine learning models for a regression problem is necessary to find out which model is the most efficient and provide the most accurate result. There are many test criteria to
compare the models. In this article, we will take a regression problem, fit different popular regression models and select the best one of them.
We have discussed how to compare different machine learning problems when we have a classification problem in hand (the article is here). That means in such cases the response variable is a
categorical one. Different popular classification algorithms are compared to come out with the best algorithm.
NB: Being a non-native English speaker, I always take extra care to proofread my articles with Grammarly. It is the best grammar and spellchecker available online. Read here my review of using Grammarly for more than two years.
Comparing regression models
So, what if the response variable is a continuous one and not categorical. This is a problem of regression then and we have to use regression models to estimate the predicted values. In this case
also several candidate regression models. Our task is to find the one which serves our purpose.
So, in this article, we are taking a regression problem of predicting the value of a continuous variable. We will compare several regression models, compare their performance calculating the
prediction accuracy and several goodnesses of fit statistics.
Here I have used five most prominent and popular regression models and compared them according to their prediction accuracy. The supervised models used here are
The models were compared using two very popular model comparison metrics namely Mean Absolute Error(MAE) and Mean Square Error (MSE). The expressions for these two metrics are as below:
Mean Absolute Error(MAE)
Comparing different machine learning models for a regression problem involves an important part of comparing original and estimated values. If $\dpi{150}&space;y$ is the response variable and $\dpi
{150}&space;\widehat{y}$ is the estimate then MAE is the error between these $\dpi{150}&space;n$ pair of variables and calculated with this equation:
MAE is a scale-dependent metric that means it has the same unit as the original variables. So, this is not a very reliable statistic when comparing models applied to different series with different
units. It measures the mean of the absolute error between the true and estimated values of the same variable.
Mean Square Error (MSE)
This metric of model comparison is as the name suggests calculate the mean of the squares of the error between true and estimated values. So, the equation is as below:
Python code for comparing the models
So, now the comparison between different machine learning models is conducted using python. We will see step by step application of all the models and how their performance can be compared.
Loading required libraries
All the required libraries are first loaded here.
import numpy as np # linear algebra
import pandas as pd # data processing
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PolynomialFeatures
from sklearn import metrics
from pandas import DataFrame,Series
from sklearn.tree import DecisionTreeRegressor
from sklearn.ensemble import RandomForestRegressor
import matplotlib
import matplotlib.pyplot as plt
from sklearn import linear_model
from sklearn.model_selection import train_test_split,cross_val_score, cross_val_predict
import missingno as msno # plotting missing data
import seaborn as sns # plotting library
from sklearn import svm
The example data and its preprocessing
The data set used here is the car data set from Github and you can access the data file from this link. The data set has the following independent variables:
• Age
• Gender
• Average miles driven per day
• Personal debt and
• Monthly income
Based on these independent variables we have to predict the potential sale value of a car. So, here the response variable is the sale value of the car and it is a continuous variable. That is why the
problem in hand is a regression problem.
Importing the data
The below piece of code will use the panda library read() function to import the data set into the working space. The describe() function is for a brief idea about the data.
dataset = pd.read_csv("cars.csv")
Displaying the last few columns of the data set to have a glimpse of the data and variables.
Last few columns of the data set
Check the data for missing values
The following code is to check if there any missing value in the data set. Missing value creates a problem in the analysis process. So, we should filter these values in the data pre-processing stage.
Here we will find out which columns contain missing values and the corresponding rows will be simply dropped from the data set.
# Finding all the columns with NULL values
# Drop the rows with missing values
dataset = dataset.dropna()
Creating basic plots with the data
Here we create the joint distribution plot of the independent variables
sns.pairplot(dataset[['age', 'miles', 'debt', 'income', 'sales']], diag_kind="kde")
Joint distribution plot of the independent variables
Splitting the data set
Data splitting is required to create training and testing data sets from the same car data. I have taken 80% of the whole data set as training data and the rest 20% of data as the test data set. The
following python code is for this splitting purpose.
train_dataset = dataset.sample(frac=0.8,random_state=0)
test_dataset = dataset.drop(train_dataset.index)
Normalizing the training data set
First of all we will see the summary statistics of all the variables using the describe() function of sklearn library.
# Calculating basic statistics with the train data
train_stats = train_dataset.describe()
train_stats.pop("sales") # excluding the dependent variable
train_stats = train_stats.transpose()
Here from the below stats about the data set, we can see that different variables in the data set has very large range and deviations, which may create problem during model fitting. So, before we use
this variables in model building process, we will normalize the variables.
Summary statistics of the training data set
Creating a function for normalization
Using the mean and standard deviation of each of the variables, we will convert them into standard normal variates. For that purpose, we will create the function as in below.
# Creating the normalizing function with mean and standard deviation
def norm(x):
return (x - train_stats['mean']) / train_stats['std']
normed_train_data = norm(train_dataset)
normed_test_data = norm(test_dataset)
Separating the response variable and creating other variables
Now a most important step to store the response variable in a separate variable.
train_labels = train_dataset.pop("sales") # using .pop function to store only the dependent variable
test_labels = test_dataset.pop("sales")
As we are now finished with the data pre-processing stage, we will start with the modelling steps. So, let’s start coding for all the five models I have mentioned to predict the car sale price.
First of all Multiple Linear Regression (MLR). This simple linear regression only but we will include all the independent variables to estimate the car sale price. The LinearRegression() function
from LinearModel module of sklearn library has been used here for the purpose.
lin_reg = LinearRegression()
#Prediction using test set
y_pred = lin_reg.predict(x_test)
mae=metrics.mean_absolute_error(y_test, y_pred)
mse=metrics.mean_squared_error(y_test, y_pred)
# Printing the metrics
print('R2 square:',metrics.r2_score(y_test, y_pred))
print('MAE: ', mae)
print('MSE: ', mse)
Metrics for MLR
dt_regressor = DecisionTreeRegressor(random_state = 0)
#Predicting using test set
y_pred = dt_regressor.predict(x_test)
mae=metrics.mean_absolute_error(y_test, y_pred)
mse=metrics.mean_squared_error(y_test, y_pred)
# Printing the metrics
print('Suppport Vector Regression Accuracy: ', dt_regressor.score(x_test,y_test))
print('R2 square:',metrics.r2_score(y_test, y_pred))
print('MAE: ', mae)
print('MSE: ', mse)
Metrics for Decision tree
rf_regressor = RandomForestRegressor(n_estimators = 300 , random_state = 0)
#Predicting the SalePrices using test set
y_pred = rf_regressor.predict(x_test)
mae=metrics.mean_absolute_error(y_test, y_pred)
mse=metrics.mean_squared_error(y_test, y_pred)
# Printing the metrics
print('Suppport Vector Regression Accuracy: ', rf_regressor.score(x_test,y_test))
print('R2 square:',metrics.r2_score(y_test, y_pred))
print('MAE: ', mae)
print('MSE: ', mse)
Metrics for Random Forest regression
from sklearn.svm import SVR
regressor= SVR(kernel='rbf')
#y_pred_svm = cross_val_predict(regressor, x, y)
mae=metrics.mean_absolute_error(y_test, y_pred_svm)
mse=metrics.mean_squared_error(y_test, y_pred_svm)
# Printing the metrics
print('Suppport Vector Regression Accuracy: ', regressor.score(x_test,y_test))
print('R2 square:',metrics.r2_score(y_test, y_pred_svm))
print('MAE: ', mae)
print('MSE: ', mse)
Metrics for Support Vector Regression
Application of Deep Learning using Keras library
Here is the deep learning model mentioned. A sequential model has been used. The model has been created as a function named build_model so that we can call it anytime it is required in the process.
The model has two connected hidden layers with a Rectified Linear Unit (relu) function and an output layer with a linear function.
The hidden layers have 12 and 8 neurons respectively with all the 8 input variables. Mean Squared Error is the loss function here as it is the most common loss function in case of regression
def build_model():
model = keras.Sequential([
layers.Dense(12,kernel_initializer='normal', activation='relu', input_shape=[len(train_dataset.keys())]),
layers.Dense(8, activation='relu'),
layers.Dense(1, activation='linear')
optimizer = tf.keras.optimizers.RMSprop(0.001)
metrics=['mae', 'mse'])
return model
model = build_model()
Displaying the model summary
This part of code is to show the summary of model we built. All the specifications mentioned above has been shown in the below screenshot of the output.
Deep learning model summary
Training the model
We have used 10 rows of the training data set to check the model performance. As the result seems satisfactory so, we will proceed with the same model.
example_batch = normed_train_data[:10]
example_result = model.predict(example_batch)
Fitting the model
Now we will fit the model with 1000 epochs and store the model training and validation accuracy in the object named history.
EPOCHS = 1000
history = model.fit(
normed_train_data, train_labels,
epochs=EPOCHS, validation_split = 0.2, verbose=0,
History of the model fit
Here we will produce a glimpse of the history stats to understand how the training process progresses.
hist = pd.DataFrame(history.history)
hist['epoch'] = history.epoch
Plotting the MAE score during the training process
As we are using 1000 epochs to train the model. It necessarily suggests that there are 1000 forward and backward passes while the model is trained. And we expect that with each passes the the loss
will decrease and model’s prediction accuracy will increase as the training process progresses.
plotter = tfdocs.plots.HistoryPlotter(smoothing_std=2)
plotter.plot({'Basic': history}, metric = "mae")
plt.ylim([0, 10000])
plt.ylabel('MAE [sales]')
In the above plot, we can see that both the training and validation loss decreases in a exponential fashion with the increase in number of epochs.
test_predictions = model.predict(normed_test_data).flatten()
a = plt.axes(aspect='equal')
plt.scatter(test_labels, test_predictions)
plt.xlabel('True Values [sales]')
plt.ylabel('Predictions [sales]')
lims = [0, 40000]
_ = plt.plot(lims, lims)
Plotting the result
Here we have plotted the predicted sale prices against the true sale prices. And from the plot it is clear that the estimate is quite close to the original one.
Original Vs predicted values of sale price of cars
Plotting the error
error = test_predictions - test_labels
plt.hist(error, bins = 125)
plt.xlabel("Prediction Error [Average_Fruit_fly_population]")
_ = plt.ylabel("Count")
Here we have plotted the error. Although the distribution of error is not a true Gaussian, but as the sample size increases, we can expect it will tend to a Gaussian distribution.
mae=metrics.mean_absolute_error(test_labels, test_predictions)
mse=metrics.mean_squared_error(test_labels, test_predictions)
# Printing the metrics
#print('Suppport Vector Regression Accuracy: ', lin_reg_pl.score(x_poly,y_train))
print('R2 square:',metrics.r2_score(test_labels, test_predictions))
print('MAE: ', mae)
print('MSE: ', mse)
Metrics of Deep learning models
So, here we can compare the performance of all the models using the metrics calculated. Let’s see all the models used to predict the car sale price together along with the metrics for the ease of
Model type MAE R Square
MLR 2821 0.80
Decision Tree 2211 0.84
Random Forest 1817 0.88
Support Vector Machine 7232 0
Deep learning/ANN 2786 0.8
Comparison table for all the models used
From the table above, it is clear that for the present problem, the best performing model is Random Forest with the highest R square (Coefficient of Determination) and least MAE. But we have to keep
in mind that the deep learning is also not far behind with respect to the metrics. And the beauty of deep learning is that with the increase in the training sample size, the accuracy of the model
also increases.
Whereas in case of other models after a certain phase it attains a plateau in terms of model prediction accuracy. Even increasing training sample size also can not further improve the model’s
performance. So, although deep learning occupies the third position in present situation, it has the potential to improve itself further if availability of training data is not a constrain.
If the data set is small and we need a good prediction for the response variable as the case here; it is a good idea to go for models like Random Forest or Decision tree. As they are capable of
generating good prediction with lesser training data or labelled data.
So, finally it is the call of the researcher or modeler to select the best suited model judging his situation and field of knowledge. As different fields of science generate experimental data with
distinct nature and a very good model of another field may fail completely to predict.
• https://www.deeplearning.ai
• https://machinelearningmastery.com
• https://datascienceplus.com
6 thoughts on “Comparing machine learning models for a regression problem”
1. Hi colleagues, fastidious post and nice urging commented here,I am really enjoying by these.
□ Thanks Isaias for nice words.
2. Thanks
Leave a Comment
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Research | ClaudioTernullo
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My current research project investigates ways to confirm/conceptually justify strong axioms of infinity (large cardinal axioms) incompatible with the Axiom of Choice, and also examines their
consequences on other foundational issues. The consistency of these axioms implies that a different scenario, based on Woodin's Ultimate-L conjecture, must be false and, as a consequence, that we may
still be very far from attaining a definitive reduction of (set-theoretic) incompleteness. Assuming both the consistency and the conceptual justifiability of choiceless large cardinals, a concrete
possibility is that there really are absolutely undecidable statements in set theory, as a consequence of the fact that the continuation of the large-cardinal hierarchy into the choiceless
large-cardinal hierarchy suggests, as hinted at by Gödel, that mathematics (set theory) is inexhaustible.
Over the years, I have researched the following topics: the Continuum Hypothesis (history and philosophy of), the justification of new axioms, the set-theoretic multiverse, mathematical Platonism,
abstraction principles, logical fallacies, and topics in the history of logic.
Ongoing Project:
- 'Self-Similarity, Large Cardinals and Incompleteness', funded by a Marie-Curie Seal of Excellence Romanian funding scheme (PNRR-I3-C9-2022. Contract n. 35195). Start Date: 01/01/2024 - End Date: 31
/12/2025. Department of Philosophy, University Babeș-Bolyai of Cluj-Napoca. Role: Project Director.
Past Projects:
- ‘Mathematical and Philosophical Aspects of a Multiversist Foundation of Set Theory’, funded by the Beatriu de Pinós Fellowship [Marie-Skłodowska Curie Actions] n. 00192 BP2018. (01/02/2020-30/06/
2023), Department of Mathematics and Computer Science, University of Barcelona (supervisor: Prof Joan Bagaria). Role: Principal Investigator.
Participation In Other Projects (as Post-Doc):
- 'Set Theory at a Crossroads: Proving the HOD Conjecture' Europa Excelencia n. EUR 2022-134032 . PI: Prof Joan Bagaria. Location: University of Barcelona (01/07/2023-31/12/2023). Role: Post-Doctoral
- ‘Lógica Matemática’, Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia Number: MTM-PID2020-116773GB-I00. PI: Prof. Enrique Casanovas and Prof. Joan Bagaria,
University of Barcelona (01/09/2021-31/08/2024). Role: Associate Researcher.
- ‘The Hyperuniverse Programme‘, funded by the FWF Stand-Alone Grant (Fonds zur Förderung der wissenschaftlichen Forschung) n. P28420. PI: Prof. Sy-David Friedman, Kurt Gödel Research Center for
Mathematical Logic, University of Vienna (01/04/2016-31/03/2021). Role: Collaborator.
- ‘University of Tartu’s 'ASTRA project PER ASPERA', funded by the European Regional Development Fund, ID: 2014-2020.4.-01.16-0027. PI: Prof. Bruno Mölder, Department of Philosophy, University of
Tartu (01/01/2018-31/08/2022). Role: Post-Doctoral Fellow.
- ‘The Hyperuniverse. The Laboratory of the Infinite’, funded by the John Templeton Foundation Grant ID #35216. PI: Prof. Sy-David Friedman (01/01/2013-30/09/2015), Kurt Gödel Research Center for
Mathematical Logic, University of Vienna. Role: Post-Doctoral Fellow.
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Two arrays
Execution time limit is 1 second
Runtime memory usage limit is 128 megabytes
Two arrays of integers are given. Print the elements from the first array (in the same order like they are given in the first array), that are absent in the second array.
First given number of elements in the first array, then given integers — the elements of array. Then given the number of elements in the second array. Then given elements of the second array. The
number of elements in each array is not greater than . All elements are integers.
Print in the first line the number of required elements. In the second line print those elements of the first array, that are absent in the second array in the same order like in the first array.
Submissions 11K
Acceptance rate 56%
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ColourMatrix Statistical Overview
The SuperCROSS ColourMatrix feature makes it easy to visualise patterns in your data. This section explains the statistical basis that SuperCROSS uses to colour the table.
One of the most common user tasks when working with tabular data is identifying and quantifying correlations and associations. Fundamentally, if two measures are associated, we have the opportunity
for relationships to exist, and insight to be garnered.
To find an association, we start by calculating what the data would look like in the absence of any pattern. That is, we determine the expected number of counts in each cell, based on a simple
assumption that the values are distributed homogeneously.
Associations will manifest as unexpected patterns in the data: cells (or groups of cells) that are significantly higher (or lower) than expected.
There are a number of statistical tests that can be applied to tables to ask questions such as:
• Do the cell values differ from the homogeneous base case?
• Is there a pattern?
• How strong is the pattern?
A positive result on a test for existence is merely based on the statistical likelihood of seeing such an association appear in the data by random chance. The strength of an association relates to
the statistical effect size, and can be considered (to some extent) the predictability of the associative outcome.
SuperCROSS provides a simple but robust mechanism to assess these question: ColourMatrix
How is ColourMatrix Different from ColourVIEW?
ColourMatrix replaces ColourVIEW, which was available in previous SuperCROSS releases.
ColourVIEW was based on a simple expectation ratio: each cell value was divided by the expected cell value to determine how far it deviated from the expectation. However, this algorithm did not take
into account the size of deviation from the expectation.
For example, having a value of 3 in a cell when expecting 2 (an overestimation of 1 unit) was given the same score as having an excess of 250 when expecting 500 (an over representation of 50% or an
expectation ratio of 1.5). If there were an expectation of 750 and 500 were encountered then the expectation ratio would be 0.66. The relationship between these two differences is not readily
The new algorithm retains the simple, visual interface while being much easier to interpret and providing much more information about the size of any over or under representation.
It also conforms to simple, well documented statistical rules.
Algorithm Overview
The ColourMatrix algorithm is based on the χ^2 (“kye squared”) test for homogeneity and independence. As this test is only uni-directional (it does not reflect under or over representation in the
table), it is supplemented with a variant of analysis of standardised residuals (Haberman, 1973).
The ColourMatrix algorithm is based on the following assumptions:
• Tables consist of categorical/nominal (frequency) data in mutually exclusive categories.
• The data represents a random sample of n independent observations.
• The expected frequency in each cell is 5 or greater.
The third assumption is the subject of much debate in the community. How conservative (and stringent) should this assumption be?
Fundamentally, the association tests here rely on a smooth approximation to what is, in fact, a discrete distribution. Generally, an expectation of 5 or greater ensures that this approximation is
acceptable. If cells have a lower expectation than 5, the chi-squared distribution of probabilities may not provide a truly accurate representation.
The algorithm first calculates the expected values for each cell (under the assumption of a completely homogeneous set).
It then undertakes three key statistical tests and provides the following feedback:
• Cells are coloured to indicate how close they are to the expected value.
• If the user selects the Cluster option, then the table is reordered to group together cells with similar levels of deviation.
• Textual results are provided for the tests for association existence (using the χ2 test) and the measure of association strength (using Cramer’s Φ, with Cohen’s w as benchmarked values).
This section uses the following notation to convey techniques:
• f represents a cell count, with the dimensionality of the resident cube dictated by the number of subscripts. For example, f[ij] comes from a 2-dimensional cross tabulation of i rows and j
columns; whereas f[ijk] has i rows, j columns, and k wafers.
• The marginals (totals and subtotals) are denoted by a dot in the relevant subscript. For example:
• f[.j] is the j^th column total.
• f[i..] is the wafer total of the i^th row.
• f… is the grand total of a 3 dimensional data cube.
The Expected Value
Association is naturally expressed between only two variables. As such, examples are often based on a 2-dimensional cross table. Yet this ignores the natural hierarchy in modern data storage systems:
the data cube. A data cube is essentially a series of cross-tables (wafers) layered on top of each other.
Describing the meaning of an association within a single layer is simple: an association is between the variable represented by the rows and the columns. However, describing the meaning of an
association within a cube can be problematic.
It might be between the rows and columns with the wafer irrelevant. It might be between the column and the row/wafer. It could be weak, but across all three, or strong but only between two variables.
As such, “strength” can lose meaning across cubes.
Calculating the Expected Value
Under an assumption of no association, either homogeneity or independence, the expected cell value (E[ij] ) is deduced from the relevant row, column, and wafer totals:
This relationship is expressed in a way to highlight its construction from simple joint probabilities. To extend to cubes (and dimensionally beyond), we can use the simple generalisation:
The ColourMatrix algorithm uses this to generate the expected cell counts for a cube that will match the top two aggregated dimensions (i.e. grand total and wafer totals for cubes, grand total and
row/column totals for cross tables will be satisfied identically).
Standardised Residuals
Armed with an expectation value, we can generate a data cube of the same dimensions as the original cube, and propagate it with expressions for the deviation of the data, from the expectation.
For each cell, the rule for this is
This form is selected for a number of specific reasons:
• It will be standardised.
• It is negative for table values that are less than we expected, and positive of cells that are over–represented in the data.
• It is symmetric, and centred on zero.
• It is interpretable, in that the standardisation means that an independent selection of these values will conform to a Z-distribution.
Colouring the Cell
The standardised residual for each cell is used to colour the cell. The colour is based on standard statistical interpretation of the number.
Typical values for the Z standardised variables are as follows:
• They will average 0
• The modal value will be 0.0
• 50% will be positive, 50% will be negative.
• ~70% will be between (-1,1)
• ~95% will be between (-2, 2)
• less than 1% will be outside (-2.6,2.6)
These relationships are well known, and can be derived directly via numerical function, or through the use of look-up tables. In the same way, the probability of exceeding values (p-values) can
readily be derived across an entire cube.
Testing the Association
If there is genuinely no association across the variables, then the standardised residuals can be assessed as a master set with predictable outcomes. The sum of their squares will conform to a χ^2
distribution with the relevant degrees of freedom:
To test the existence of a possible association, we attempt to reject the possibility that a value could be as large as it is under chance alone. While we do not expect that every value in every cell
will precisely match the expectation, each cell should be within the noise of the expectation. We also understand how an aggregate of squared Z distributed values should be distributed according to a
χ^2 distribution. The critical value is the χ^2 value for the relevant degrees of freedom.
The degrees of freedom relate to the construction of the cube and its ability to maintain the relevant and required marginal totals. If the amount of deviation we measure is less than this critical
value, then there is no evidence to suggest that the table is significantly different to the one we would expect to see if there were no association between variables.
For a cross tabulation, the degrees of freedom is given by
where r is the number of rows, and c the number of columns. For a cube this becomes
where w is the number of wafers.
This can be generalised to N dimensions as
where N is the dimensionality of the cube, and n[i] are the sub-dimension counts.
For the Z values, there exist calculations and look-up tables for determining the critical values of χ^2, and reporting the likelihood of such a combination of residuals arising by chance (p–values).
Given a statistical likelihood that a χ^2 value should arise, if it is deemed likely that there is a true association within the cube or table then the standardised residuals guide the analysts in
finding cause for why (or where) we were compelled to reject the hypothesis that there was no association within the table. The largest absolute residual values contribute the most to the finding of
an association.
How Strong is the Association?
Modern data tools are designed to move and display large volumes of data. Consequently, they are capable of resolving very small effects to great statistical significance. It is therefore important
to determine the potential size or strength of any detected association.
The use of a χ^2 value needs in some way to be corrected for the volume of data involved, determining strength. When there are only two variables at hand, the greatest association can be easily
visualised (it is when all the data resides on the diagonal). We can go further than this, and deduce the maximum value that a χ^2 statistic can be for a given table. Scaling our determined metric by
this value gives us a standardised metric of association strength.
This is known as Cramer’s Φ (or Φ[C])
where k is the smaller of the number of columns or rows and n is the total, independent contributor count. This value depends on the size and shape of the original table. To some extent it can be
standardised by reporting Cohen’s w .
There is an accepted range for w , as follows:
Effect Size Range (w)
Small 0.1≤w<0.3
Medium 0.3≤w<0.5
Large 0.5≤w
Clustering the Table
The standardised residuals are configured such that any group should have a net sum of zero. Taking each wafer, row or column in turn, we can produce a measure of that sum as a metric. Those that are
positive are generally over-represented in the set, while a negative score suggests under-representation.
Re-ordering the rows and columns (when the user selects the Cluster option) to place the most positive scores in the upper left corner generates further insights into the potential underlying
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Rational Polynomials (Rational Functions)
In Maple rational functions are created from names, integers, and other Maple values for the coefficients using the arithmetic operators +, -, *, /, and ^. For example: 7+x/(x^4-3*x+1) creates the
rational function NiMsJiIiKCIiIiomSSJ4RzYiRiUsKCokRiciIiVGJUYnISIkRiVGJSEiIkYl It is a rational function in the variable x over the field of rational numbers. Multivariate rational functions, and
rational functions over other number rings and fields are constructed similarly. For example: y^3/x/(sqrt(-1)*y+y/2) creates NiMqKEkieUc2IiIiJEkieEdGJSEiIiwmKiZeIyIiIkYsRiRGLEYsRiQjRiwiIiNGKA== a
rational function in the variables x and y whose coefficients involve the imaginary number i which is denoted by capital I in Maple. This remainder of this file contains a list of operations which
are available for rational functions. Note: many of the functions and operations described in the help page for polynom apply to the rational function case. Utility Functions for Manipulating
Rational Functions.
denomextract the denominator of a rational functionnormalnormal form for rational functionsnumerextract the numerator of a rational functionsubsevaluate a rational function
Mathematical Operations on Rational Functions.
asymptasymptotic series expansiondiffdifferentiate a rational functionintintegrate a rational function (indefinite/definite integration)limitcompute a limit of a rational functionsumsum a rational
function (indefinite or definite summation)seriesgeneral power series expansiontaylorTaylor series expansion
Operations for Regrouping Terms of Rational Functions.
collectgroup coefficients of like terms togetherconfracconvert a series or rational function to a continued fractionsee convert/confrachornerconvert all polynomial subexpressions to horner formsee
convert/hornerfactorfactor the numerator and denominatorparfracpartial fraction expansion of a rational functionsee convert/parfracratpolyconvert a series to a rational function (Pade approximation)
see convert/ratpolysortsort all polynomial subexpressions
The type function can be used to test for rational polynomials. For example the test type(a, ratpoly(integer, x)) tests whether the expression NiNJImFHNiI= is a rational polynomial in the variable x
with integer coefficients. See type/ratpoly for further details.
See Alsoconvertpolynomseriestypetype/ratpoly
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Cola Can
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
An cylindrical aluminium can contains
330 ml of cola. Which of these two cans uses the least aluminium?
If the can's diameter is 6 cm what is the can's height? If you could choose any diameter, which dimensions for a 330ml can would use the least amount of aluminium?
If the can's height was 10 cm what would the can's diameter have to be?
Getting Started
A millilitre is one cubic centimetre.
Check that you can calculate the volume of a cylinder.
If the diameter is 6 cm, you can calculate the base area of the can.
If you know the base area you can find a height which will give you a specific volume ( 330 ml in this case)
Now start with a height (10 cm) and work your way back to a base area for a specified volume, then find a radius for that base area, and hence a diameter for the can.
What does least aluminium require ?
The top and base of the can are circles but how do you calculate the curved surface area ?
Student Solutions
There is a Stage 5 topic called Differentiation (part of Calculus) which makes these sorts of problems rather easy.
At Stage 4 the challenge is to handle the algebra well (the formulae that calculate, height or diameter and then surface area) and to use a trial and improvement method to get as close as we wish to
the answer.
Using a spreadsheet takes a lot of the labour out of trial and improvement, and also helpfully makes an automatic table of results.
We had lots of solutions using Stage 4 mathematics, but well done in particular to Oliver from Olchfa, Michael from Homestead High, Sam and Rachel both from Millais School, and David from Gordonstoun
The volume of a cylinder is found from the circular cross-section multiplied by the distance over which it continues, and here the volume has to be 330 ml.
So when the can's diameter is 6 cm (r=3) it's height (h) is 11.67 cm.
And when it's height is 10 cm, the can's diameter has to be 6.48 cm
We notice that if the volume is fixed then whether we know the can's diameter or it's height, the other of these two will be easy enough to calculate using :
So whichever we already have, either r or h, we can find the other and then use both in the surface area formula.
That's two circles, top and bottom, plus a rectangle (can circumference by can height - like peeling the label off a tin).
We are now going to work systematically to find the diameter and height of the can (volume 330 ml) which has the least surface area.
We could start with a radius of, say, 1 cm and find the height which makes a volume of 330 ml, then use both those r and h values to calculate the surface area for that can.
We could then step up the radius to 2 cm and repeat the calculations.
In fact we could continue to increase the radius and make up a table of the surface area results each time.
Alternatively we could use the same method but start with a value for height instead, and increase that.
Once we see how the surface area changes with radius (or height) in a general way we can refine our choice of an r or h value until, by trial and improvement, we get as near as we please to the
lowest value for the surface area.
Here's the table Sam made :
And here's a graph used by Michael showing an interesting and non-symmetric curve for the changing surface area
For those who like to see nice things done with spreadsheets in mathematics here's a link to an Excel file that does all that calculation in an instant : Cola Can
Very well done to everyone who engaged with this problem, we were really pleased to see how popular it had been. Why not look at the Funnel problem - trying to use the least amount of plastic to make
a funnel .
Teachers' Resources
This printable worksheet may be useful: Cola Can.
Once the lower level thinking covered in the Hint has been assimilated students might be guided if necessary to see the value of a spreadsheet when solving a problem of this sort.
Additionally the use of a graph representing the spreadsheet values is particularly helpful for 'picturing' the behaviour of the surface area function as either base radius or can height varies.
There is a valuable opportunity to work with each of the two obvious independent variables : base radius and height. Starting with either of these the other is calculable from the specified volume of
330 ml, and once both r and h are known the surface area is calculable.
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Advanced Algorithmic Design Strategies – IAAC Blog
Advanced Algorithmic Design Strategies
In this seminar, we explored the dual geometric/topological computing nature of spatial systems, based on iterative logics explored through grasshopper and the Anemone plug-in. After investigating
iterative algorithm strategies we applied them in urban scale size architecture. This process starts with the definition of the basic components as computational building blocks and their embedded
First, we explore a 2D pattern called Penrose patter:
Aperiodic Urban Design
An aperiodic set of tiles is a set of shapes with the property that, though the whole Euclidean plane can be covered by non-overlapping replicas of the shape, no periodically repeating tiling pattern
can be constructed from them. Based on the 2D aperiodic pattern produced by Dr. Roger Penrose, we recreated 3D shapes and imposed rules on how they can produce, simultaneously, solids and voids. The
tiling we studied has certain characteristics that needed to be fulfilled in order to obtain the right pattern:
• Reflection symmetry + fivefold symmetry
• Source of repetition contains 3 pentagons, 2 rhombi, 1 three-spiked shape, and a ring of 10 pentagons surrounding these shapes.
• Scaling self-similarity: fractal factor
The Exploration:
Component A:
Component B:
Chosen Heuristics:
Components Addition:
Number of iteration: 150 / Number of components: 73
Number of iteration: 500 / Number of components: 153
Post Processing Strategies:
Architectural application scenarios:
Urban Context:
Based on the 2D aperiodic pattern produced by Dr. Roger Penrose, we recreated 3D shapes and imposed rules on how they can produce, simultaneously, solids and voids. By creating two components and
following the rules set in the algorithm we are able to manipulate how solid “private” spaces connect to voids “public” spaces. Once we have each connection available of the two components, we can
set rules to how they respond to the connection by adding supports or architectural elements through post-processing tools.
The result is an urban intervention of public and private spaces that can be connected to each other in four different levels in each component, allowing for vertical and horizontal growth of the
Aperiodic urban intervention is a project of IAAC, Institute for Advanced Architecture of Catalonia developed in the 2019/20 by
Students: Lilett Ricaurte, Matin Darabi, Teddy Fadous
Tutors: Alessio Erioli, Andrea Graziano
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Applied Math Reunion 2023
With the rapid developments in technology, applied and computational mathematics has played an important and unprecedented role in many scientific disciplines. The main objective of this workshop is
to bring together researchers, students, and practitioners with interest in the theoretical, computational, and practical aspects in applied and computational mathematics. Another purpose is to have
a reunion of the alumni of Applied and Computational Mathematics at Caltech. This workshop will cover recent progress, stimulate new ideas, and facilitate interdisciplinary collaborations. It will
emphasize the crucial and unique role of mathematical insights in advanced algorithm design and novel real-world applications.
This workshop will be held at Caltech on November 11-12, 2023 (Saturday-Sunday). The meeting will feature distinguished lectures from leaders in the related fields, and panel discussions on future
Registration for the Workshop is now closed. If you have any questions, please contact Diana Bohler.
The location for all talks will be Annenberg 105 (Auditorium).
Click on the arrows in the schedule below to toggle for more information.
Morning Session Chair: Tom Hou, Caltech
Towards Seamless Numerical Homogenization
The computational challenge from multiscale differential equations has inspired development of a variety of methodologies. Many of them require scale separation but there are also efforts to reduce
the dependence on substantial scale gaps. One such class of techniques is the, so called, seamless methods for multiscale dynamical systems. We will prove equivalence between such dynamical systems
and one dimensional elliptic multiscale problems. A simple multidimensional generalization leads to a new class of numerical methods for homogenization type problems. This has some similarity to
turbulence modeling and the Car-Parinello technique for molecular dynamics. We will give error estimates and simple numerical examples.
Multimodal Sampling via Approximate Symmetries
Sampling from multimodal distributions is a challenging task in scientific computing. When a distribution has an exact symmetry between the modes, direct jumps among them can accelerate the samplings
significantly. However, the distributions from most applications do not have exact symmetries. This paper considers the distributions with approximate symmetries. We first construct an exactly
symmetric reference distribution from the target one by averaging over the group orbit associated with the approximate symmetry. Next, we can apply the multilevel Monte Carlo methods by constructing
a continuation path between the reference and target distributions. We discuss how to implement these steps with annealed importance sampling and tempered transitions. Compared with traditional
multilevel methods, the proposed approach can be more effective since the reference and target distributions are much closer.
Multicontinuum homogenization
I will give some overview of our research on multicontinuum homogenization that arise in multiscale problems. The main idea is to construct multiple multiscale basis functions. I will discuss both
scale separation and no-scale separation cases.
[Click here to view slides]
In-Context Operator Networks: Towards Large Scientific Learning Models Scientific Learn
Joint work with Liu Yang, Siting Liu and Tinwei Meng.
Multi-Operator Learning and Expression Generation
Approximating nonlinear ordinary differential equations using a neural network provides an efficient tool for certain scientific computing tasks, including real-time predictions, inverse problems,
and surrogate modeling. The focus thus far has been on embedding a single solution operator, associated to one (possibly parametric) differential equation, into a neural network. However, it is often
the case that families of differential equations share similar behavior and thus we can leverage this to train one neural network to represent multiple distinct tasks. In this talk, we will discuss
how to learn maps from multimodal inputs to multimodal outputs that are capable of generating both numerical predictions of dynamical systems and mathematical equations over multiple distinct
differential equations.
Recent development on singularity formation in incompressible fluids
I will talk about recent development on singularity formation in incompressible fluids.
[Click here to view slides]
Matrix decompositions in DNA sequence analysis
In the past 15 years, astounding advances have been made in the DNA sequencing technology. This has enabled large-scale sequencing of human genomes, generating terabyte-scale datasets. I will
describe one approach for interpreting DNA mutations in cancer patients involving non-negative matrix factorization.
Please see below for a list of suggested dining establishments both on- and off-campus.
Afternoon Chair: Houman Owhadi, Caltech
The Historical Development and Transformation of Applied and Computational Mathematics at Caltech
In this talk, I will give a brief review on the history and transformation of Applied and Computational Mathematics at Caltech. The Applied Math Option (AMa) was established in 1967 by Gerald Whitham
and a few founding faculty members, Philip Saffman, Donald Cohen, Herbert Keller, and Joel Franklin. Heinz-Otto Kreiss joined AMa from 1978 to 1987 and Dan Meiron joined in 1985. Since 1992, there
has been a new wave of hiring, including Tom Hou (1993), Oscar Bruno (1995), Niles Pierce (1999), Emmanuel Candes (2000), Houman Owhadi (2003), Joel Tropp (2006), Venkat Chandrasekaran (2012), Andrew
Stuart (2016), and Franca Hoffmann (2022). The Applied Mathematics Option was renamed to "Applied and Computational Mathematics" (ACM) in 2001, and ACM was merged with CS and CDS to form the
Department of Computing and Mathematical Sciences (CMS) in 2010 with a strong focus on the mathematics of information and data science. I will highlight the accomplishments of some of our faculty
members, our former Ph.D. students and postdocs, including those from my research group.
The relation between ranked and unranked eigenportfolios
We will show how portfolios with unranked equities returns data and the corresponding ones ranked by capitalization are related. We will show that this theoretical, model free, analysis is is in fact
quite consistent with portfolios of actual US equities data. Applications will be discussed.
Recovery phenomena with symmetric autoencoders
Is it possible to guess what a scene in an image would look like if the picture was taken from a different angle? Would it sound like you if an AI generated a deepfake of your voice? Can we find the
solution of a PDE we have never seen, if we collect enough solutions of nearby equations? These questions seem to fit in a common mathematical framework of estimation of low-dimensional latent
processes under maps of controlled complexity. After reviewing known results in the context of generative priors, I will explain how to formulate recovery guarantees for symmetric autoencoders using
tools from applied probability and approximation theory. Joint work with Borjan Geshkovski (MIT).
A group photo will be taken of the Workshop attendees.
Inverse wave scattering via reduced order modeling
I will describe briefly a novel approach to inverse wave scattering, which uses an array of sensors to probe an unknown heterogeneous medium with signals and measures the scattered waves. The
approach uses tools from data driven reduced order modeling to estimate the wave field at points inside the inaccessible medium that we wish to determine.
I will show how we can use this wave to obtain a better formulation of inverse wave scattering than the typical nonlinear least squares data fitting. The performance of the method is illustrated with
a few known challenging examples.
Consensus Based Optimization and Sampling
Particle methods provide a powerful paradigm for solving complex global optimization problems leading to highly parallelizable algorithms. Despite widespread and growing adoption, theory underpinning
their behavior has been mainly based on meta-heuristics. In application settings involving black-box procedures, or where gradients are too costly to obtain, one relies on derivative-free approaches
instead. This talk will focus on two recent techniques, consensus-based optimization and consensus-based sampling. We explain how these methods can be used for the following two goals: (i) generating
approximate samples from a given target distribution, and (ii) optimizing a given objective function. They circumvent the need for gradients via Laplace's principle. We investigate the properties of
this family of methods in terms of various parameter choices and present an overview of recent advances in the field.
Chair: Tony Chan, KAUST
Franca Hoffmann
Gang Hu
Zhenzhen Li
Pengfei Liu
Gaby Stredie
Xiao-Hui Wu
Mike Yan
Pengchuan Zhang
Attendees can share their stories of Caltech along with their own life stories after Caltech.
Registration for the Banquet has closed. Registered Banquet attendees will receive a message containing Banquet logistics shortly before Saturday, November 11th.
Please contact Diana Bohler with any questions.
Sunday, November 12, 2023
Morning Chair -- First Session: Tony Chan, KAUST
[Click here to view slides]
Optimization of the Boltzmann Equation
The kinetics of rarefied gases and plasmas are described by the Boltzmann equation and numerically approximated by the Direct Simulation Monte Carlo (DSMC) method. We present an optimization method
for DSMC, derived from an augmented Lagrangian. After a forward (in time) solution of DSMC, adjoint variables are found by a backwards solver. They are equal to velocity derivatives of an objective
function, which can then be optimized. This is joint work with Yunan Yang (Cornell) and Denis Silantyev (U Colorado, Colorado Springs).
Some Problems in General Relativity and Memories of Caltech during 1993-2005
This lecture will focus on some research problems in mathematical and numerical general relativity (GR) I was first exposed to as a postdoc at Caltech in (at the time) Applied Mathematics. I was
mentored primarily by Herb Keller, and his first instructions were to attend Kip Thorne's relativity class (Physics 236) that Fall (1993). I will outline a number of interesting problems in numerical
analysis that came out of my interactions with Herb, Kip, and many applied math faculty over the four year period 1993-1997, and formed the basis for many of my research projects during my time at UC
Irvine and UC San Diego (1997 through today). An additional 2-year leave at Caltech during 2003-2005 led to a new direction involving some open mathematical problems in GR, leading to other projects
that have formed a second center of mass in my research program since 2008. The format of my lecture will be a little unusual, with my technical presentation intertwined with fond memories of
interactions with mentors, collaborators, and friends at Caltech during the period 1993-2005.
Optimal Transport Maps for Conditional Simulation
Optimal transport (OT) considers the problem of finding a mapping that warps a reference probability measure to a target of interest. Such a map can naturally be viewed as a generative model in the
context of machine learning applications making the framework of OT a natural candidate for the analysis of generative modeling. In this talk I will discuss the foundational theory of a particular
class of OT problems where the resulting map does not only transport the reference measure to the target but is also capable of providing samples from certain conditionals of the target. This problem
is very interesting in the context of Bayesian inference and in particular in the setting of amortized and simulation based inference.
Morning Chair -- Second Session: Tom Hou, Caltech
Runge-Kutta Methods are Stable
We discuss the stability of Runge-Kutta methods for large systems of ODEs.
Speeding up gradient flows on probability measure space
In the past decade, there has been a significant shift in the types of mathematical objects under investigation, moving from vectors and matrices in Euclidean space to functions residing in Hilbert
or Banach spaces, and ultimately extending to probability measures within the probability measure space. Many questions that were originally posed in the context of linear function spaces are now
being revisited in the realm of probability measures. One such question is to the efficiently find a probability measure that minimizes a given objective functional.
In Euclidean space, we devised optimization techniques like gradient descent and introduced momentum-based methods to accelerate convergence. Now, the question arises: Can we employ analogous
strategies to expedite convergence within the probability measure space?
In this presentation, we provide an affirmative answer to this question. Specifically, we present a series of momentum-inspired acceleration method under the framework of Hamiltonian flow, and we
prove the new class of method can achieve arbitrary high-order of convergence. This opens the door of developing methods beyond standard gradient flow.
[Click here to view slides]
Numerical methods for geometric motion
We consider the evolution of curves in 2D. We describe it as geometric motion if the evolution only depends on the shape of the curve. There are applications in material science (the evolution of
microstructure in materials), biochemistry, and image processing. Two new gradient flow models are derived and their numerical implementation in a general computational framework is described.
Machine Learning-enabled Self-Consistent Field Theory for Soft Materials
Self Consistency Field Theory (SCFT) has been a powerful tool to study soft materials like polymers. However, SCFT simulations are a complex and computationally costly process. Exploring the vast
design space of polymers via SCFT is often impractical. We will discuss in this talk our recent efforts to leverage SCFT with Machine Learning to accelerate many important downstream tasks, such as
the discovery of new phases.
Please see below for a list of suggested dining establishments both on- and off-campus.
Afternoon Chair: Haomin Zhou, Georgia Tech
[Click here to view slides]
Parameterized Wasserstein Geometric Flow
In this presentation, I will give a brief introduction to a new parameterization strategy that can be used to design algorithms simulating geometric flows on Wasserstein manifold, the probability
density space equipped with optimal transport metric. The framework leverages the theory of optimal transport and the techniques like the push-forward operators and neural networks, leading to a
system of ODEs for the parameters of neural networks. Theoretical error bounds measured in Wasserstein metric are provided. The resulting methods are mesh-less, basis-less, sample-based schemes that
scale well to higher dimensional problems. We demonstrate their performance in Wasserstein gradient flows such as Fokker-Planck equation, and Wasserstein Hamiltonian flow like Schrodinger equations.
[Click here to view slides]
A Nontraditional Regularity Criteria for the 3D Navier-Stokes Equations
We present a nontraditional regularity criterion for the global well-posedness problem of the three dimensional Navier-Stokes equations in the whole space. The main novelty of this new criterion is
that it involves the shape of the magnitude of the velocity. More specifically, we prove that if for every fixed time in (0,T), the region of high velocity, appropriately defined with a parameter q,
shrinks fast enough as q↗∞, then the solution stays regular beyond T.
This is joint work with Prof. Chuong V. Tran of the University of St. Andrews, United Kingdom.
[Click here to view slides]
Stochasticity of Deterministic Gradient Descent: Quantitative Local Min Escape in Multiscale Landscape
This talk will discuss one of the many nontrivial but often pleasant effects of large learning rates, which are commonly used in machine learning practice for improved empirical performances but defy
traditional theoretical analyses. More specifically, I will quantify how large learning rates can help gradient descent escape local minima, via chaotic dynamics, which provides an alternative to the
commonly known escape mechanism due to noises from stochastic gradients.
[Click here to view slides]
Information geometric regularization of the barotropic Euler equation
This talk presents an inviscid regularization for mitigating shock formation in the multidimensional barotropic Euler equations based on ideas from geometric hydrodynamics, interior point methods for
semidefinite programming, and the information geometry of Amari and Chentsov. In Lagrangian coordinates, the solutions of Euler's equations are paths on the manifold of diffeomorphisms. Shocks form
when the deformation map reaches the boundary of this manifold. Shock formation thus arises from the geodesic incompleteness of the latter. In this work, we regularize the barotropic Euler equation
by modifying the geometry of the diffeomorphism manifold. In the modified geometry, geodesics do not cross the boundary of the manifold but instead approximate it asymptotically. This modified
geometry is motivated by the log-determinant barrier function in semidefinite programming and the information geometry of the fluid density. By re-expressing the resulting equation in Eulerian
coordinates we obtain an information geometric regularization of the original conservation law. We provide numerical evidence that this regularization prevents shock formation while preserving the
long-term behavior of the solutions.
[Click here to view slides]
Generalized multiscale finite element method for a class of nonlinear flow equations
In this talk we present a Constraint Energy Minimization Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving single-phase non-linear compressible flows in highly heterogeneous
media. The construction of CEM-GMsFEM hinges on two crucial steps: First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small
eigenvalues are captured. Then the basis functions are obtained by solving local energy minimization problems over the oversampling domains using the auxiliary space. The basis functions have
exponential decay outside the corresponding local oversampling regions. The convergence of the proposed method is provided, and we show that this convergence only depends on the coarse grid size and
is independent of the heterogeneities. The research is partially supported by the Hong Kong RGC General Research Fund (Projects: 14305222 and 14304021).
Dynamics of fluid's cohomology
We present a topological analysis of the vorticity formulation of the incompressible Euler equation. In particular, we elucidate the equations of motion for the often-omitted cohomology component of
the velocity on non-simply-connected domains. These equations have nontrivial coupling with the vorticity, which is crucial for characterizing correct vortex motions with presence of tunnels or
islands in the domain. The dynamics of fluid's cohomology reveals the curvature of the commutator subgroup of the Lie group of volume-preserving diffeomorphisms, and it is also associated with new
conservation laws as Casimir invariants. Additional results include new analytical solutions of the Euler equations that are related to the Hilbert transform; and the first general
vortex-streamfunction-based numerical method on curved surfaces with arbitrary genus and boundaries that is consistent with the Euler equation.
[Click here to view slides]
Efficient Interacting Particle Methods for Computing Near Singular Solutions of Keller-Segel Chemotaxis Systems and High-Dimensional Eigenvalue Problems
Mesh-based methods, such as the finite element method and spectral methods, often encounter significant challenges when solving PDEs with near singular solutions or in high-dimensional spaces. This
talk presents an efficient interacting particle method inspired by our recent developments in particle-based techniques for computing effective diffusivities in chaotic flows and KPP front speeds of
reaction-diffusion-advection equations. The method is applied to compute aggregation patterns and near singular solutions of the Keller-Segel (KS) chemotaxis system in three-dimensional space,
considering both the parabolic-elliptic and parabolic-parabolic types of KS systems. Additionally, the interacting particle method is employed to calculate the principal eigenvalues of
high-dimensional elliptic operators, enabling the analysis of the large-time growth rate of the entropy functional, which quantifies the time reversal of SDEs in high dimensions under vanishing
noise. Numerical experiments are presented to demonstrate the performance of the proposed methods. Furthermore, this talk introduces the DeepParticle method, a deep learning method for learning and
generating the distributions of solutions under variations of physical parameters.
Equivalent Extensions of Partial Differential Equations on Curves
or Surfaces
In this talk, we propose a methodology that extends the energy
function defined on surfaces to the energy function defined on the
nearby tubular neighborhood that gives the same energy when inputting
the constant-along-normal extension. Furthermore, the extended energy
function gives the same minimizer as which the original energy function
gives in the sense of restriction on the surface. This new approach
connects the original energy function to an extended energy function and
provides a good framework to solve PEDs numerically on Cartesian grids.
Recently, we have used the sign distance function defined in a
narrowband near the moving interface to represent the evolution of the
curve. We derive the equivalent evolution equations of the distance
function in the narrowband. The novelty of the work is to determine the
equivalent evolution equation on Cartesian girds without extra
conditions or constraints. The proposed method extends the differential
operators appropriately so that the solutions on the narrowband are the
distance function of the solution to the original mean flow solution.
Furthermore, the extended solution carries the correct geometric
information, such as distance and curvature, on Cartesian grids. Some
experiments confirm that the proposed method is convergent numerically.
This is a joint work with Richard Tsai, Ming-Chih Lai, Shih-Hsuan Hsu,
Chun-Chieh Lin.
Thomas Hou, Chair Caltech
Houman Owhadi Caltech
Peter Schröder Caltech
Andrew Stuart Caltech
Haomin Zhou Georgia Tech
Local Accommodations, Directions, and Parking
Hotel Dena
303 Cordova St, Pasadena, CA 91101
Phone: +1 626-469-8100
Hotel Dena will be the location of the Workshop Banquet.
A block of discounted rooms ($182/night) has been reserved at the Hotel Dena for booking by Workshop attendees. To take advantage of this discount, please reserve directly with the hotel via the
online discounted workshop booking link.
The Athenaeum
Caltech's on-campus faculty club offers a limited number of hotel rooms, which may be more expensive than other local options. Reservations at the Athenaeum must be arranged by the Workshop
organizers directly. If you are interested in reserving a room at the Athenaeum, please contact us.
The Saga Motor Hotel -- ~0.6 miles from Workshop Venue at Caltech
1633 E Colorado Blvd., Pasadena, CA 91106
(626) 795-0431
Hyatt Place Pasadena -- ~1.4 miles from Venue
399 E Green St, Pasadena, CA 91101
(626) 788-9108
Sheraton Pasadena -- ~1.5 miles from Venue
303 Cordova St, Pasadena, CA 91101
(626) 469-8100
Westin Pasadena -- ~1.9 miles from Venue
191 N Los Robles Ave, Pasadena, CA 91101
(626) 792-2727
Courtyard by Marriott Pasadena/Old Town -- ~2.7 miles from Venue
180 N Fair Oaks Ave, Pasadena, CA 91103
(626) 403-7600
The Workshop talks will be held in Room 105 of the Annenberg Center for Information Science and Technology, building #16 on a campus map.
Please refer to the Center's location on Google maps for directions and navigation.
Visitors traveling from LAX (Los Angeles International Airport) and BUR (Hollywood Burbank Airport) to Caltech tend to choose a ride service like Uber or Lyft. Alternatively, you can use SuperShuttle
(advance reservation recommended via supershuttle.com), rent a car at the airport, or pick up a taxi cab at a designated location at the airport.
• Information about taxi pickup locations can be found below, along with other ground transportation details:
The nearest Caltech parking structure to the Annenberg Center for Information Science and Technology is Structure #4, located at 370 South Holliston Avenue, Pasadena. Parking permits must be
displayed in order to park in visitor (unmarked) parking spaces. Permits can be purchased at pay stations located in campus parking lots and structures.
More information about visitor parking can be found at: https://parking.caltech.edu/parking-info/visitor-parking
Lunch during the Workshop will be on your own (not provided by the organizers). Please see below for dining suggestions both on- and off-campus:
On Campus
To view the different dining options on campus along with current hours, please see:
Off Campus
There are many great restaurants, bars, and lounges within walking distance of campus, including the Old Town Pasadena vicinity as well as the South Lake Avenue Shopping District.
• South Lake Avenue Shopping District:
□ South Lake Avenue is within walking distance of campus (approximately 14 minutes), and has many dining options from fast food (Chipotle, Veggie Grill, Panda Express) to more formal
full-service establishments.
□ For directions to the South Lake Avenue Shopping District vicinity from Caltech's campus, click here.
• Ginger Corner Market: http://gingercornermarket.com/
□ Ginger Corner Market is a small café within walking distance of campus (approximately 6 minutes).
□ For directions to the Ginger Corner Market from Caltech's campus, please click here.
• Old Town Pasadena: https://www.oldpasadena.org/visit/directory/dine/
□ Old Town Pasadena is an approximate 9-minute drive from Caltech's campus.
□ For directions to the Old Town Pasadena vicinity from Caltech's campus, please click here.
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Histogram exercise and solution – Q&A Hub – 365 Financial Analyst
Resolved: Histogram exercise and solution
In exercise sheet, absolute frequency formulas are like this:
However, in solution sheet, like this.
The formula in the solution part is understandable, quite normal. However, I did not understand the one in the exercise sheet. How is it possible to get the right answer? What does the minus between
to Countif formula mean? Why ">"&E16 but not "<"&E16 or "<="&E16 ?
2 answers ( 1 marked as helpful)
when we count all numbers > 21, formula will count numbers greater than 21 (first), then when we count all numbers > 41 formula will count numbers greater than 41(second).
when we Subtract numbers greater than 41(second) from numbers greater than 21 (first), we will get the numbers between 21 and 41
from 2 to 7- from 4 to 7= from 2 to 4
2,3,4,5,6 - 4,5,6 = 2,3
I had the same question, and I don´t understand why the formula changes from one file to another and they don´t mention this at any moment in the recording or provide any explanation. Thanks for you
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EVERY N<sub>2</sub>-LOCALLY CONNECTED CLAW-FREE GRAPH with MINIMUM DEGREE at LEAST 7 IS Z<sub>3</sub>-CONNECTED
Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and A∗= A-{0}. A graph G is A-connected if G has an orientation D(G) such that for every function b: V(G) → A satisfying
∑v ϵ V(G)b(v) = 0, there is a function f: E(G) →A∗such that for each vertex v ϵ V(G), the total amount of f values on the edges directed out from v minus the total amount of f values on the edges
directed into v equals b(v). Let Z[3] denote the group of order 3. Jaeger et al. conjectured that there exists an integer k such that every k-edge-connected graph is Z[3]-connected. In this paper, we
prove that every N[2]-locally connected claw-free graph G with minimum degree δ(G) ≥ 7 is Z[3]-connected.
• Group connectivity
• claw-free graphs
• locally connectedness
• nowhere-zero flows
ASJC Scopus subject areas
• Discrete Mathematics and Combinatorics
Dive into the research topics of 'EVERY N[2]-LOCALLY CONNECTED CLAW-FREE GRAPH with MINIMUM DEGREE at LEAST 7 IS Z[3]-CONNECTED'. Together they form a unique fingerprint.
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Computer Science
AQA Computer Science GCSE
Algorithms - Search Algorithms
You need to know about two sorts of search algorithm:
• linear search
• binary search
It's important to be able to compare the pros and cons of each of the two algorithms.
Linear Search
Start at the first item. Work your way through logically until you find the item you're looking for. Stop once you've found it.
Pretty basic, but it works.
Here's some pseudocode to implement a linear search:
found <- False
# start at the first name in the array
name <- current name in the array
IF name = the one you're looking for THEN
found <- True
UNTIL found = True OR reached end of list
IF found = True THEN
OUTPUT "found" + name
OUTPUT "not found"
Note that this algorithm uses a REPEAT - UNTIL loop. This is a form of indefinite iteration. This means that the loop continues until the item required is found.
In Python REPEAT - UNTIL loops do not exist. You would need to use a WHILE loop instead. You need to realise that in pseudocode you may see REPEAT - UNTIL.
It would be possible to write the algorithm using a FOR loop as well, but this has some disadvantages. It would work like this:
found <- False
FOR i <- 0 TO LEN(myArray)
name <- the next item in the array
IF name = the one you're looking for THEN
found <- True
OUTPUT "found" + name
ENDFOR IF found = False THEN
OUTPUT "not found"
Using a FOR loop has some advantages and disadvantages.
Trace table to use for search algorithms
Binary Search
A binary search will only work if the list can be sorted in some way, otherwise it's useless.
Start in the middle of the list - look at that item. Is it the one you want? If it is, stop.
If it isn't, you need to discard the half of the list the item can't be in - you can figure this out because the list is sorted. Then you find the middle of the remaining half and compare that with
the one you're looking for - and repeat the process.
This sounds tricky, but working through some lists on the board will help make it easy to understand.
Here's an algorithm for a binary search:
found <- False
# start at the middle name in the array
name <- middle name in the array
IF name = the one you're looking for THEN
found <- True
IF name > middle name in the array THEN
discard the first half of the array, including the middle name
discard the second half of the array, including the middle name
UNTIL found = True or LEN(myArray) = 0
# the array is empty
A binary search can be really effective and computers use them all the time to find items quicker and more efficiently. But they don't apply to every situation and are rather more complex to code.
Trace table to use for search algorithms (this is the same document as the one above!)
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Miles (Roman) to Span (cloth) Converter
Enter Miles (Roman)
Span (cloth)
β Switch toSpan (cloth) to Miles (Roman) Converter
How to use this Miles (Roman) to Span (cloth) Converter π €
Follow these steps to convert given length from the units of Miles (Roman) to the units of Span (cloth).
1. Enter the input Miles (Roman) value in the text field.
2. The calculator converts the given Miles (Roman) into Span (cloth) in realtime β using the conversion formula, and displays under the Span (cloth) label. You do not need to click any button. If
the input changes, Span (cloth) value is re-calculated, just like that.
3. You may copy the resulting Span (cloth) 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 (Roman) to Span (cloth)?
The formula to convert given length from Miles (Roman) to Span (cloth) is:
Length[(Span (cloth))] = Length[(Miles (Roman))] / 0.00015447992471826364
Substitute the given value of length in miles (roman), i.e., Length[(Miles (Roman))] in the above formula and simplify the right-hand side value. The resulting value is the length in span (cloth),
i.e., Length[(Span (cloth))].
Calculation will be done after you enter a valid input.
Consider that an ancient Roman road is 10 miles (Roman) long.
Convert this distance from miles (Roman) to Span (cloth).
The length in miles (roman) is:
Length[(Miles (Roman))] = 10
The formula to convert length from miles (roman) to span (cloth) is:
Length[(Span (cloth))] = Length[(Miles (Roman))] / 0.00015447992471826364
Substitute given weight Length[(Miles (Roman))] = 10 in the above formula.
Length[(Span (cloth))] = 10 / 0.00015447992471826364
Length[(Span (cloth))] = 64733.3304
Final Answer:
Therefore, 10 mi (roman) is equal to 64733.3304 span.
The length is 64733.3304 span, in span (cloth).
Consider that a historical Roman military march covered 25 miles (Roman).
Convert this distance from miles (Roman) to Span (cloth).
The length in miles (roman) is:
Length[(Miles (Roman))] = 25
The formula to convert length from miles (roman) to span (cloth) is:
Length[(Span (cloth))] = Length[(Miles (Roman))] / 0.00015447992471826364
Substitute given weight Length[(Miles (Roman))] = 25 in the above formula.
Length[(Span (cloth))] = 25 / 0.00015447992471826364
Length[(Span (cloth))] = 161833.3259
Final Answer:
Therefore, 25 mi (roman) is equal to 161833.3259 span.
The length is 161833.3259 span, in span (cloth).
Miles (Roman) to Span (cloth) Conversion Table
The following table gives some of the most used conversions from Miles (Roman) to Span (cloth).
Miles (Roman) (mi (roman)) Span (cloth) (span)
0 mi (roman) 0 span
1 mi (roman) 6473.333 span
2 mi (roman) 12946.6661 span
3 mi (roman) 19419.9991 span
4 mi (roman) 25893.3321 span
5 mi (roman) 32366.6652 span
6 mi (roman) 38839.9982 span
7 mi (roman) 45313.3312 span
8 mi (roman) 51786.6643 span
9 mi (roman) 58259.9973 span
10 mi (roman) 64733.3304 span
20 mi (roman) 129466.6607 span
50 mi (roman) 323666.6518 span
100 mi (roman) 647333.3035 span
1000 mi (roman) 6473333.0355 span
10000 mi (roman) 64733330.355 span
100000 mi (roman) 647333303.5498 span
Miles (Roman)
A mile (Roman) is an ancient unit of length used in the Roman Empire. One Roman mile is equivalent to approximately 1,481.5 meters or about 4,856.7 feet.
The Roman mile, known as "mille passus," is defined as 1,000 paces (or "passus"), where each pace is considered to be about 5 feet long.
Roman miles were used for various purposes, including surveying and road construction within the Roman Empire. Although no longer in common use, the Roman mile is of historical interest and is
occasionally referenced in discussions of ancient measurements and Roman history.
Span (cloth)
A span (cloth) is a unit of length used historically in textiles and cloth measurement. One span (cloth) is approximately equivalent to 24 inches or 0.6096 meters.
The span (cloth) is based on the width of a person's outstretched hand from thumb to little finger, providing a practical measure for fabric lengths and textile work.
Spans (cloth) were used in the textile industry for measuring and cutting fabric. While less common today, the unit remains of historical interest and reflects traditional practices in cloth
measurement and tailoring.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Miles (Roman) to Span (cloth) in Length?
The formula to convert Miles (Roman) to Span (cloth) in Length is:
Miles (Roman) / 0.00015447992471826364
2. Is this tool free or paid?
This Length conversion tool, which converts Miles (Roman) to Span (cloth), is completely free to use.
3. How do I convert Length from Miles (Roman) to Span (cloth)?
To convert Length from Miles (Roman) to Span (cloth), you can use the following formula:
Miles (Roman) / 0.00015447992471826364
For example, if you have a value in Miles (Roman), you substitute that value in place of Miles (Roman) in the above formula, and solve the mathematical expression to get the equivalent value in Span
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Mining Bitcoin with pencil and paper: 0.67 hashes per day
I decided to see how practical it would be to mine Bitcoin with pencil and paper. It turns out that the SHA-256 algorithm used for mining is pretty simple and can in fact be done by hand. Not
surprisingly, the process is extremely slow compared to hardware mining and is entirely impractical. But performing the algorithm manually is a good way to understand exactly how it works.
A pencil-and-paper round of SHA-256
The mining process
Bitcoin mining is a key part of the security of the Bitcoin system. The idea is that Bitcoin miners group a bunch of Bitcoin transactions into a block, then repeatedly perform a cryptographic
operation called hashing zillions of times until someone finds a special extremely rare hash value. At this point, the block has been mined and becomes part of the Bitcoin block chain. The hashing
task itself doesn't accomplish anything useful in itself, but because finding a successful block is so difficult, it ensures that no individual has the resources to take over the Bitcoin system. For
more details on mining, see my
Bitcoin mining article
A cryptographic hash function takes a block of input data and creates a smaller, unpredictable output. The hash function is designed so there's no "short cut" to get the desired output - you just
have to keep hashing blocks until you find one by brute force that works. For Bitcoin, the hash function is a function called SHA-256. To provide additional security, Bitcoin applies the SHA-256
function twice, a process known as double-SHA-256.
In Bitcoin, a successful hash is one that starts with enough zeros.[1] Just as it is rare to find a phone number or license plate ending in multiple zeros, it is rare to find a hash starting with
multiple zeros. But Bitcoin is exponentially harder. Currently, a successful hash must start with approximately 17 zeros, so only one out of 1.4x10^20 hashes will be successful. In other words,
finding a successful hash is harder than finding a particular grain of sand out of all the grains of sand on Earth.
The following diagram shows a block in the Bitcoin blockchain along with its hash. The yellow bytes are hashed to generate the block hash. In this case, the resulting hash starts with enough zeros so
mining was successful. However, the hash will almost always be unsuccessful. In that case, the miner changes the nonce value or other block contents and tries again.
Structure of a Bitcoin block
The SHA-256 hash algorithm used by Bitcoin
The SHA-256 hash algorithm takes input blocks of 512 bits (i.e. 64 bytes), combines the data cryptographically, and generates a 256-bit (32 byte) output. The SHA-256 algorithm consists of a
relatively simple round repeated 64 times. The diagram below shows one round, which takes eight 4-byte inputs, A through H, performs a few operations, and generates new values of A through H.
One round of the SHA-256 algorithm showing the 8 input blocks A-H, the processing steps, and the new blocks.
created by kockmeyer,
CC BY-SA 3.0
The blue boxes mix up the values in non-linear ways that are hard to analyze cryptographically. Since the algorithm uses several different functions, discovering an attack is harder. (If you could
figure out a mathematical shortcut to generate successful hashes, you could take over Bitcoin mining.)
The Ma majority box looks at the bits of A, B, and C. For each position, if the majority of the bits are 0, it outputs 0. Otherwise it outputs 1. That is, for each position in A, B, and C, look at
the number of 1 bits. If it is zero or one, output 0. If it is two or three, output 1.
The Σ0 box rotates the bits of A to form three rotated versions, and then sums them together modulo 2. In other words, if the number of 1 bits is odd, the sum is 1; otherwise, it is 0. The three
values in the sum are A rotated right by 2 bits, 13 bits, and 22 bits.
The Ch "choose" box chooses output bits based on the value of input E. If a bit of E is 1, the output bit is the corresponding bit of F. If a bit of E is 0, the output bit is the corresponding bit of
G. In this way, the bits of F and G are shuffled together based on the value of E.
The next box Σ1 rotates and sums the bits of E, similar to Σ0 except the shifts are 6, 11, and 25 bits.
The red boxes perform 32-bit addition, generating new values for A and E. The input W[t] is based on the input data, slightly processed. (This is where the input block gets fed into the algorithm.)
The input K[t] is a constant defined for each round.[2]
As can be seen from the diagram above, only A and E are changed in a round. The other values pass through unchanged, with the old A value becoming the new B value, the old B value becoming the new C
value and so forth. Although each round of SHA-256 doesn't change the data much, after 64 rounds the input data will be completely scrambled.[3]
Manual mining
The video below shows how the SHA-256 hashing steps described above can be performed with pencil and paper. I perform the first round of hashing to mine a block. Completing this round took me 16
minutes, 45 seconds.
To explain what's on the paper: I've written each block A through H in hex on a separate row and put the binary value below. The maj operation appears below C, and the shifts and Σ0 appear above row
A. Likewise, the choose operation appears below G, and the shifts and Σ1 above E. In the lower right, a bunch of terms are added together, corresponding to the first three red sum boxes. In the upper
right, this sum is used to generate the new A value, and in the middle right, this sum is used to generate the new E value. These steps all correspond to the diagram and discussion above.
I also manually performed another hash round, the last round to finish hashing the Bitcoin block. In the image below, the hash result is highlighted in yellow. The zeroes in this hash show that it is
a successful hash. Note that the zeroes are at the end of the hash. The reason is that Bitcoin inconveniently reverses all the bytes generated by SHA-256.[4]
Last pencil-and-paper round of SHA-256, showing a successfully-mined Bitcoin block.
What this means for mining hardware
Each step of SHA-256 is very easy to implement in digital logic - simple Boolean operations and 32-bit addition. (If you've studied electronics, you can probably visualize the circuits already.) For
this reason, custom ASIC chips can implement the SHA-256 algorithm very efficiently in hardware, putting hundreds of rounds on a chip in parallel. The image below shows a mining chip that runs at 2-3
billion hashes/second;
has more photos.
The silicon die inside a Bitfury ASIC chip. This chip mines Bitcoin at 2-3 Ghash/second. Image from
. (
CC BY 3.0
In contrast, Litecoin, Dogecoin, and similar altcoins use the scrypt hash algorithm, which is intentionally designed to be difficult to implement in hardware. It stores 1024 different hash values
into memory, and then combines them in unpredictable ways to get the final result. As a result, much more circuitry and memory is required for scrypt than for SHA-256 hashes. You can see the impact
by looking at mining hardware, which is thousands of times slower for scrypt (Litecoin, etc) than for SHA-256 (Bitcoin).
The SHA-256 algorithm is surprisingly simple, easy enough to do by hand. (The elliptic curve algorithm for signing Bitcoin transactions would be very painful to do by hand since it has lots of
multiplication of 32-byte integers.) Doing one round of SHA-256 by hand took me 16 minutes, 45 seconds. At this rate, hashing a full Bitcoin block (128 rounds)
would take 1.49 days, for a hash rate of 0.67 hashes per day (although I would probably get faster with practice). In comparison, current Bitcoin mining hardware does several terahashes per second,
about a quintillion times faster than my manual hashing. Needless to say, manual Bitcoin mining is not at all practical.
A Reddit reader asked about my energy consumption. There's not much physical exertion, so assuming a resting metabolic rate of 1500kcal/day, manual hashing works out to almost 10 megajoules/hash. A
typical energy consumption for mining hardware is 1000 megahashes/joule. So I'm less energy efficient by a factor of 10^16, or 10 quadrillion. The next question is the energy cost. A cheap source of
food energy is donuts at $0.23 for 200 kcalories. Electricity here is $0.15/kilowatt-hour, which is cheaper by a factor of 6.7 - closer than I expected. Thus my energy cost per hash is about 67
quadrillion times that of mining hardware. It's clear I'm not going to make my fortune off manual mining, and I haven't even included the cost of all the paper and pencils I'll need.
2017 edit: My Bitcoin mining on paper system is part of the book The Objects That Power the Global Economy, so take a look.
Follow me on Twitter to find out about my latest blog posts.
[1] It's not exactly the number of zeros at the start of the hash that matters. To be precise, the hash must be less than a particular value that depends on the current Bitcoin
difficulty level
[2] The source of the constants used in SHA-256 is interesting. The NSA designed the SHA-256 algorithm and picked the values for these constants, so how do you know they didn't pick special values
that let them break the hash? To avoid suspicion, the initial hash values come from the square roots of the first 8 primes, and the K[t] values come from the cube roots of the first 64 primes. Since
these constants come from a simple formula, you can trust that the NSA didn't do anything shady (at least with the constants).
[3] Unfortunately the SHA-256 hash works on a block of 512 bits, but the Bitcoin block header is more than 512 bits. Thus, a second set of 64 SHA-256 hash rounds is required on the second half of the
Bitcoin block. Next, Bitcoin uses double-SHA-256, so a second application of SHA-256 (64 rounds) is done to the result. Adding this up, hashing an arbitrary Bitcoin block takes 192 rounds in total.
However there is a shortcut. Mining involves hashing the same block over and over, just changing the nonce which appears in the second half of the block. Thus, mining can reuse the result of hashing
the first 512 bits, and hashing a Bitcoin block typically only requires 128 rounds.
[4] Obviously I didn't just have incredible good fortune to end up with a successful hash. I started the hashing process with a block that had already been successfully mined. In particular I used
the one displayed earlier in this article, #286819.
[5] Another problem with manual mining is new blocks are mined about every 10 minutes, so even if I did succeed in mining a block, it would be totally obsolete (orphaned) by the time I finished.
85 comments:
1. You're insane, but amazing. This is fantastic.
2. You may have a typo in the Ma majority box description. The first sentence says "looks at the bits of A, B, and C", which agrees with the diagram. But the fourth sentence says "for each position
in B, C, and D".
3. On line 5, you didn't carry the one.
4. Order more donuts.
5. svpernerd +1.
6. Very cool but I'm a bit confused. The diagram shown says "transaction count: 63" but the block on Block Explorer says "Transactions: 99". Why the discrepancy?
7. Get a life...
8. Ignore the miserable buggers who are members of Anonymous....
Thank you. It looks pretty instructional. I shall go through it in detail in a bit. :-)
9. Tim RochesterSeptember 30, 2014 at 6:34PM
I love you, this is so nerdy, so geeky but so fantastic. I love how you calculated energy costs lol.
10. Your endeavor makes me think about my blog page where I showed a picture for my description "The early Bitcoin miner was very efficient on electricity, however zero Bitcoin yield.". I am tempted
to add you to my "Bitcoin Mining Rigs". http://this1that1whatever.com/money/bitcoin/bitcoin-mining-rigs.php
11. Thanks Ken. That was really funny. Now I think you are even more similar to Weird Al. And I also know what you do on Fridays when soccer season is not on.
12. stop your shameless self promotion David wong
13. https://docs.google.com/spreadsheets/d/1mOTrqckdetCoRxY5QkVcyQ7Z0gcYIH-Dc0tu7t9f7tw/
14. Ken, Could you demonstrate also how to create a transaction ready for the blockchain? This is most helpful and removes the mystery. Very helpful. Gary.
15. Now you could do some manual image processing, for example the blur filter, which is much simpler than SHA-256. The only problem is that to process a 12 mpix photo the algorithm has to be
executed 12 millions times :)
16. Thank you, love it! You are the best!
17. Where does the value of 6534ea13 for W come from in the final round?
18. Gary: I wrote about creating transactions here.
Anonymous: the last W value comes from the input block data, after being extended into the message schedule array (algorithm at Wikipedia). Basically there are few shifts, xors, and adds applied
to the input data.
19. I've added the input preprocessing *but* something isn't quite right. SHA256(null) is supposed to be e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855.
20. Does null become 80000000, ...? Or is null something different than a zero-length field?
21. Working from https://en.wikipedia.org/wiki/SHA-2#Pseudocode; oops, who can explain that last step "Add the compressed chunk to the current hash value:" where h0 := h0 + a, ...? To me it seems the
a thru h must be the final set of values coming from round 64 but what is "the compressed chunk"?
22. Oh dear, apparently we have to do the compression rounds 4 times.
23. For null input, these are the values my Goggle Sheet is calculating after the 1st of 4 set of compression rounds;
h0 := h0 + a 9842B0DA
h1 := h1 + b FAEE8474
h2 := h2 + c 12DB4F41
h3 := h3 + d 8C0F0B62
h4 := h4 + e 93A235C0
h5 := h5 + f 84C5217E
h6 := h6 + g 2B724C32
h7 := h7 + h B275F527
Can someone confirm them?
24. Ah, found a bug; the corrected values are;
h0 := h0 + a 02475B8D
h1 := h1 + b 6030E1D4
h2 := h2 + c E56D532B
h3 := h3 + d E5498121
h4 := h4 + e 2E5B4FA6
h5 := h5 + f F37412EA
h6 := h6 + g 702DBFFF
h7 := h7 + h 62438F1C
25. Hmm, per http://www.movable-type.co.uk/scripts/sha256.html, apparently we don't do the extra 3 set of compression rounds will null as our input. Must be another bug.
26. Bugger, found another bug; the adjusted values are;
h0 := h0 + a CAD19DA2
h1 := h1 + b 915378F3
h2 := h2 + c 2191FAB5
h3 := h3 + d D80944A8
h4 := h4 + e 4D34CB19
h5 := h5 + f 4C652719
h6 := h6 + g 89A736B4
h7 := h7 + h 0F2A36D5
Still not right.
27. Ah, ha! http://csrc.nist.gov/groups/STM/cavp/documents/shs/sha256-384-512.pdf is very helpful.
28. David: send me an email and I can send you the full dump of the SHA-256 data, which should answer all your questions.
29. K are 64-bit values and certain operations are 64-bit sums. It makes a difference in the 18th round hashing "abc".
30. Oops, K are 32-bit values for SHA256; they are 64-bit values for SHA512.
31. Ugh, messed up the right shifts. Fixing it now.
32. What do you know? Eliminate the bugs and it works!
33. Really interesting post and you have described it manually in a very effective manner. Now after seeing your post I know how Bitcoin work manually. Thanks for creating such a good post.
34. This comment has been removed by the author.
35. I've been watching her video with resolution hash256 Kt=428A2F98 and
Wt=0200000 and in minute 6:30 to 6:32, you have a doubt. Then make a mistake. The result of cl gives:
ch/cl 1F8CC98C
I've done all this algorithm with a spreadsheet and find this:
1F83D9AB ch 1 15 8 5 12 9 8 12
In another manuscript sheet is correct and Wt=c67178f2 Kt=6534ea14
ch/cL E01D26F7 14 13 0 1 2 6 7 15
It seems to be cyclically run this algorithm 256 times and add the data to the input ABCDEFGH obtained in 64 time.
What have you done with her only publication in the network is possible that this algorithm is made by a generation of advanced humans. Thank you.
36. It would be interesting to you to do a video on how to create a private key and public key Bitcoin. It would bitcoin wallet safer world.
In this debate www.bitcointalk.org
They say it´s possible
37. Extremely fascinating and well done. I couldn't find a down to earth explanation of SHA256 besides some cryptic jargon from NIST and numerous other websites. d
I'm curious about how the hash function manages to deal with data that's bigger then what you've provided. I do realize that hash functions can only take in a certain amount of data but it is a
rather large amount and I'm curious how it manages to "compress" it.
38. John Weyland: To handle data longer than 512 bits, the data is chopped into 512 bit blocks and the hash algorithm runs on each block in order.
The trick is that the values A-H are not reset at the start of each block, but kept from the previous block. So the final hash value is a combination of all the blocks.
The Wikipedia page gives more details.
39. Hello , Thank you for the report . I reported on my website about. :)
40. You do bit rotates, not bitshifts. Which is is supposed to be?
41. Alan: it's rotates. See Wikipedia for details on the operations.
42. I know its been a while since you have posted on here and you probably won't see this but I was wondering if you could clarify how you get the data from the previous hash and merkle root and what
not and turn it into the usable data such as the k and w and a-h. I have been looking over the wiki for a while and I cant seem to grasp exactly what is happening. It would be extremely useful if
you could.
43. ... and if Ken can help us understand the details then I might even code it into my Google sheet.
44. I'm actually working on designing a hardware bitcoin miner. I can do the algorithm by hand when given the inputs but I can't take the information from bitcoin and turn it into the inputs for the
algorithm. I've actually already started my design for the part of the circuit that does the algorithm I just need to figure out how to obtain the inputs.
45. Hi, i dont agree with your Ma majority.
maj := (a and b) xor (a and c) xor (b and c)
so maj(1;1;1) don't give 1 but 0 ! (because of XOR)
It's like; "if two bits of A, B, and C are 1, output is 1"
Isnt'it ?
But despite of this, well done for this good job ;)
46. Anonymous: you ask how to get from the previous hash and Merkle root to the SHA-256 variables (K, W, A-H). There are two parts to this. On the the Bitcoin side, the data bytes are concatenated
together to form the input to SHA-256. See the diagram "Structure of a Bitcoin block" above - the data in yellow is the input to SHA-256. On the SHA-256 side, the algorithm generates the
variables through simple steps. The K values are constants and the A-H values are initialized to constants. The W values are generated from the input data through simple shifts and xor (to extend
16 words of input to 64 words for the 64 rounds). Two other things to remember: since the input is more than 512 bits, it is processed in two chunks. Also, Bitcoin applies SHA-256 twice. For
details on how Bitcoin combines the data to be hashes, see my article Bitcoin mining the hard way, and for details on SHA-256, see the Wikipedia article.
Regarding the Maj majority function, it surprisingly doesn't matter if you use OR or XOR. Consider maj(1,1,1): 1 OR 1 OR 1 = 1 XOR 1 XOR 1 = 1. Either way, the majority function returns the value
(0 or 1) that is in the majority. (Normally OR and XOR behave differently, but due to the structure of the Maj function, both formulas give the same result.)
47. I`m a little clueless on how to proceed to the next round. Will my result (A...H) become the new initial A to H values and so on until 64th round?
48. Really thank you Ken :-) thats a great article
49. Bitcoin is a form of digital currency, created and held electronically. No one controls it. Bitcoins aren’t printed, like dollars or euros – they’re produced by people, and increasingly
businesses, running computers all around the world, using software that solves mathematical problems. It’s the first example of a growing category of money known as cryptocurrency.
50. I don't really get the use of the constants - why do you have to work out the first ~10 mins every time? couldn't you just compute it once and use that data forever? am I missing something? were
the constants you used just blank bits of data that would be the data of the current block you're mining?
51. This is insane but awesome! Thanks for doing this!
52. I wanted to make sure it is OK we use the image from this blog post of yours that we published here https://www.vpnmentor.com/blog/hash-puzzle-bitcoin/ with an attribution and link to your post.
if this isn't fine, we'll take it off.
53. nice
54. Fantastic, please do more with other coins like Monero/Dash.
55. Really nice to see some one explain the topic really good. Nice One!
56. In the screenshot of the hand calcs for the successfully mined coin, at the bottom of the page, where does the row above the highlighted yellow value come from
Ex: for A at the bottom, you have
E620622b which I understand, but where did the
6a09e667 come from?
And why are you summing this first new value of A with this new value?
57. Thanks a lot. You are the real innovator.
58. i can't imagine how it work, poor me!
59. hello, i have found it little hard to understand the end of the process:
i know i need to do h(i)= a +h(i-1)...
but do it mean i need to do it for every ward that goes into the system ? or every 512 bits block ?
60. How to determine the W and K constant. Are those random numbers of our choice?.. Help me out..
61. Excellent overview. Thanks for putting the time in.
62. Cool!
63. On the first round W=02000000
What is the value of W for the second round?
It is 17975b97?
64. apik: yes. The first few W values are 02000000 17975b97 c18ed1f7 e255adf2 97599b55.
65. Awesome. really needed the explanation.
Q: we discard carry since it is 32-bit values(while doing sum)?
PS: There is typo in taking a value of choosing function while calculating new sum (took 5 in place of 6 at: 1f86c98c)
66. Awesome. really needed the explanation.
Q: we discard the carry while doing addition? (bcs it is 32-bit value)
PS: The choosing value has typo in counting new sum which leads to 1-bit off in newA
67. Your estimate of how energy efficient you are is not very accurate since it also takes into account the energy required to simply exist, which will be being used anyway. You'd have to subtract
your basal metabolic rate with your observed metabolic rate when calculating SHA-256 hashes to obtain an accurate estimate of how efficient you are. I suspect that the extra energy used for
hashing is minimal compared to your basal metabolic rate and would likely be dominated by your muscle movements (since brain metabolic rate is remarkably consistent over a wide range of levels of
68. - One small thing seems MISSING, remains not really clear. While page is titled "mining bitcoin...", then you have "mined a block", found a hash starting with zeroes...
However - which of the numbers is the actual "coin" value that was mined / received, and how much of it?
69. Anonymous: He just demonstrated the last round out of 128 rounds needed to mine a block. The input value contained some of the block data used to store the Bitcoin ledger.
70. You are a truly brilliant lunatic. I even have problems filling in basic forms as I am both dyslexic and dysgraphic. If it has boxes I find it close to impossible.
71. Very Impressive Blockchain tutorial. The content seems to be pretty exhaustive and excellent and will definitely help in learning Blockchain course. I'm also a learner taken up Blockchain
training and I think your content has cleared some concepts of mine. While browsing for Blockchain tutorials on YouTube i found this fantastic video on Blockchain. Do check it out if you are
interested to know more.:-https://www.youtube.com/watch?v=BrZXvS3rVQQ&t=132s
72. Hi. Thanks for sharing great information about bitcoin. Now a days bitcoin wealth is very trending. you can even make $13000 in 24 hours by this bitcoin wealth system
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75. very useful but i don't understand about
please exactly explain for amateur...
76. When are other parameters from block header used? Is there any step-by-step tutorial (without iterations ofc) that shows when info from last block is pulled, when it's used and how it becomes the
new block?
77. If you ate lard instead of donuts, your cost per Joule would be significantly reduced. Source: https://www.upstart.com/blog/lowest-cost-per-calorie-foods
78. First, this was an awesome demonstration and write-up! Thank you very much.
I'm assuming the nonce is 32-bits. Since this is the only thing you chance on the second round of SHA, could a mining-pool partition this number and assign a certain subset to each node, rather
than having each node make random attempts? Rather like unrolling the loop so the whole series could be attempted with concurrent threads? node-0 [tries 0-15], node-1 [tries 16-31]... and so on
until the last node is assigned a block of nonces to attempt... without a lot of extra work you could load balance if one node happens to be faster than the others and the (a?... is it always
unique? or are there multiple solutions that would satisfy the zero-padding requirement?) solution still hasn't been found.
79. Thanks for sharing great information about bitcoin.
80. can you actually get bitcoin by doing this?
81. wow that's very informative. Thanks!!
82. Calculating Bitcoin manually, you need to complete it within 10 minutes. :dego
Nền tảng giao dịch bán buôn dành cho mua sắm linh kiện điện tử
83. Wait, that's not right. You just have to do it faster than all of the other miners.
84. Technically you just have to be first to broadcast your block, have it accepted by a majority of the nodes, and have subsequent blocks build upon yours.
85. Very good explanation thank you for the info. I have one question though. During the 32bit summation what happens if you have to sum two 0xFFFFFFFF numbers? This should be an overflow. How does
sha256 handles this?
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Software effort estimation by tuning COOCMO model parameters using differential evolution
Software effort estimation by tuning COOCMO model parameters using differential evolution
Created by W.Langdon from gp-bibliography.bib Revision:1.8010
□ author = "Sultan Aljahdali and Alaa F. Sheta",
□ title = "Software effort estimation by tuning COOCMO model parameters using differential evolution",
□ booktitle = "2010 IEEE/ACS International Conference on Computer Systems and Applications (AICCSA)",
□ year = "2010",
□ month = "16-19 " # may,
□ address = "Hammamet, Tunisia",
□ abstract = "Accurate estimation of software projects costs represents a challenge for many government organisations such as the Department of Defense (DOD) and NASA. Statistical models
considerably used to assist in such a computation. There is still an urgent need on finding a mathematical model which can provide an accurate relationship between the software project effort
/cost and the cost drivers. A powerful algorithm which can optimise such a relationship via tuning mathematical model parameters is urgently needed. In two new model structures to estimate
the effort required for software projects using Genetic Algorithms (GAs) were proposed as a modification to the famous Constructive Cost Model (COCOMO). In this paper, we follow up on our
previous work and present Differential Evolution (DE) as an alternative technique to estimate the COCOMO model parameters. The performance of the developed models were tested on NASA software
project dataset provided in. The developed COCOMO-DE model was able to provide good estimation capabilities.",
□ keywords = "genetic algorithms, genetic programming, sbse, COOCMO model parameter tuning, NASA software project dataset, constructive cost model, differential evolution, mathematical model,
optimisation algorithm, software effort estimation, software projects cost estimation, statistical model, optimisation, software cost estimation",
□ notes = "'We suggest the use of Genetic Programming (GP) technique to build suitable model structure for the software effort estimation.' Also known as \cite{5586985}",
Genetic Programming entries for Sultan Aljahdali Alaa Sheta
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conformally flat manifold
nLab conformally flat manifold
Riemannian geometry
Basic definitions
Further concepts
A Riemannian manifold $(X,g)$ is conformally flat if it is locally taken to a flat manifold by a conformal transformation; more specifically, if there exists an open cover $\{U_i \to X\}_{i \in I}$
and on each $U_i$ a smooth function $f_i$, such that $e^{f_i} g_{\vert U_i}$ has vanishing Riemann curvature: $R\big( e^{f_i} g_{\vert U_i} \big) = 0$.
See also
Created on May 22, 2019 at 17:33:19. See the history of this page for a list of all contributions to it.
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Eliminating cycles in the discrete torus
In this paper we consider the following question: how many vertices of the discrete torus must be deleted so that no topologically nontrivial cycles remain? We look at two different edge structures
for the discrete torus. For (ℤ [m] ^d ) [1], where two vertices in ℤ [m] are connected if their ℓ [1] distance is 1, we show a nontrivial upper bound of d ^log [2](3/2)m ^d-1 d ^0.6m ^d-1 on the
number of vertices that must be deleted. For (ℤ [m] ^d ) [∞], where two vertices are connected if their ℓ [∞] distance is 1, Saks et al. (Combinatorica 24(3):525-530, 2004) already gave a nearly
tight lower bound of d(m-1) ^d-1 using arguments involving linear algebra. We give a more elementary proof which improves the bound to m ^d -(m-1) ^d , which is precisely tight.
• Discrete torus
• Foam
• Multicut
• Tiling
Dive into the research topics of 'Eliminating cycles in the discrete torus'. Together they form a unique fingerprint.
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Maclaurin Series and Euler's Formula
Maclaurin Series: $\displaystyle f(x)=a_0+a_1x+a_2x^2+\cdots +a_nx^n+\cdots\ \ \ and \ \ a_n=\frac{f^n(0)}{n!}$
$\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots +{\frac {x^{n}}{n!}}+\cdots \quad \forall x$
$\displaystyle \sin x =\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots\forall x $
$\displaystyle \cos x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \forall x$
Euler's Formula: $\displaystyle \boxed{e^{ix}=\cos x +i\sin x}$
$\displaystyle =1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}+\cdots=\Big(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots\Big)+i\Big(x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\
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New Periodic Table Grade 7 Pdf Multiplication Worksheets Math | Multiplication Worksheets
New Periodic Table Grade 7 Pdf Multiplication Worksheets Math
New Periodic Table Grade 7 Pdf Multiplication Worksheets Math
New Periodic Table Grade 7 Pdf Multiplication Worksheets Math – Multiplication Worksheets are a wonderful method to show youngsters the twelve times table, which is the holy grail of primary
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What are Multiplication Worksheets?
Multiplication worksheets are a valuable way to assist pupils find out mathematics realities. They can be used to educate one multiplication truth each time or to examine multiplication facts up to
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Times Table Charts New Activity Shelter
Associative Property Of Multiplication Lesson Plan And Resources CCSS
Buy Multiplication Table 1 20 Book Online At Low Prices In India
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Rocket Math Multiplication Worksheets are a wonderful method to assess the times tables. They also aid children develop flexibility as they are exposed to the several means they can do computations.
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Related For Rocket Math Multiplication Worksheets
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R for Data Science
Offered By: CognitiveClass
R for Data Science
R is a powerful language for data analysis, data visualization, machine learning, statistics. Originally developed for statistical programming, it is now one of the most popular languages in data
science. In this course, you'll be learning about the basics of R, and you'll end with the confidence to start writing your own R scripts.
Continue reading
R Programming
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At a Glance
R is a powerful language for data analysis, data visualization, machine learning, statistics. Originally developed for statistical programming, it is now one of the most popular languages in data
science. In this course, you'll be learning about the basics of R, and you'll end with the confidence to start writing your own R scripts.
R is a powerful language for data analysis, data visualization, machine learning, and statistics.
This isn't your typical textbook introduction to R. You're not just learning about R fundamentals; you'll be using R to solve problems related to movie data. Using a concrete example makes learning
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You'll then learn about lists, matrices, arrays and data frames from vectors. Then, you'll jump into conditional statements, functions, classes and debugging. Once you've covered the basics - you'll
learn about reading and writing data in R, whether it's a table format (CSV, Excel) or a text file (.txt). Finally, you'll end with some important functions for character strings and dates in R.
Course Syllabus
Module 1 - R basics
• Math, Variables, and Strings
• Vectors and Factors
• Vector operations
Module 2 - Data structures in R
• Arrays & Matrices
• Lists
• Dataframes
Module 3 - R programming fundamentals
• Conditions and loops
• Functions in R
• Objects and Classes
• Debugging
Module 4 - Working with data in R
• Reading CSV and Excel Files
• Reading text files
• Writing and saving data objects to file in R
Module 5 - Strings and Dates in R
• String operations in R
• Regular Expressions
• Dates in R
General Information
• This course is self-paced.
• It can be taken at any time.
• It can be audited as many times as you wish.
Recommended skills prior to taking this course
Skills You Will Learn
Data Analysis, Data Science, Data Visualization, Machine Learning
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Properly Typing an Object of Functions
Properly Typing an Object of Functions
Here we're working with an object called objOfFunctions, which contains functions keyed by string, number, or boolean. Each key has an associated function to process an input of that type:
const objOfFunctions = {
string: (input: string) => input.toUpperCase(),
number: (in
Loading exercise
00:00 In this exercise we have an object of functions and this object of functions basically has either string or number or boolean as the keys on it and each of the inputs correspond to that type so
we have an input which is a string on the string one turns it to an uppercase in number it turns it to a two fixed two and with a boolean it takes in and it returns true or false so there's a bunch
of formatters here really and then we have a format function down below which takes in input which can be string or number or boolean which
00:29 we extract out the input type which we just call type of input and this is by the way this is like the normal type of this is the one you're used to where if we don't constrain it using this as
it actually ends up being kind of string or number or big into a boolean or simple or undefined or object or function and that's fine but we actually just want it to be constrained to these three
things so we use the as to kind of like force TypeScript into that if we don't use it then it's going to yell at us for adding like possibilities like
00:59 indexing into object of functions to grab things that might not be there so we then grab our formatter from the object of functions and the formatter ends up being an array of a sorry a union
of functions so it could be either this function with which takes a string this function which takes a boolean or sorry this function takes number or this function that takes a boolean you get the
idea and we're getting an error your job is to try to figure out how we can solve this error because it's
01:29 kind of a nasty one and understand that we've kind of covered the information that you need for this already by looking at the way that when you call a union of functions you need to intersect
the things that they require together I think you've got everything good luck
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C Programming/Operators and type casting - Wikibooks, open books for an open world
Operators and Assignments
C has a wide range of operators that make simple math easy to handle. The list of operators grouped into precedence levels is as follows:
Identifiers are names of things in C, and consist of either a letter or an underscore ( _ ) optionally followed by letters, digits, or underscores. An identifier (or variable name) is a primary
expression, provided that it has been declared as designating an object (in which case it is an lvalue [a value that can be used as the left side of an assignment expression]) or a function (in which
case it is a function designator).
A constant is a primary expression. Its type depends on its form and value. The types of constants are character constants (e.g. ' ' is a space), integer constants (e.g. 2), floating-point constants
(e.g. 0.5), and enumerated constants that have been previously defined via enum.
A string literal is a primary expression. It consists of a string of characters within double quotes ( " ).
A parenthesized expression is a primary expression. It consists of an expression within parentheses ( ( ) ). Its type and value are those of the non-parenthesized expression within the parentheses.
In C11, an expression that starts with _Generic followed by an initial expression, a list of values of the form type: expression where type is either a named type or the keyword default, and
constitutes a primary expression. The value is the expression that follows the type of the initial expression or the default if not found.
First, a primary expression is also a postfix expression. The following expressions are also postfix expressions:
A postfix expression followed by a left square bracket ([), an expression, and a right square bracket (]) in sequence constitutes an invocation of the array subscript operator. One of the expressions
shall have type "pointer to object type" and the other shall have an integer type; the result type is type. Successive array subscript operators designate an element of a multidimensional array.
A postfix expression followed by parentheses or an optional parenthesized argument list indicates an invocation of the function call operator. The value of the function call operator is the return
value of the function called with the provided arguments. The parameters to the function are copied on the stack by value (or at least the compiler acts as if that is what happens; if the programmer
wanted the parameter to be copied by reference, then it is easier to pass the address of the area to be modified by value, then the called function can access the area through the respective
pointer). The trend for compilers is to pass the parameters from right to left onto the stack, but this is not universal.
A postfix expression followed by a dot (.) followed by an identifier selects a member from a structure or union; a postfix expression followed by an arrow (->) followed by an identifier selects a
member from a structure or union who is pointed to by the pointer on the left-hand side of the expression.
A postfix expression followed by the increment or decrement operators (++ or -- respectively) indicates that the variable is to be incremented or decremented as a side effect. The value of the
expression is the value of the postfix expression before the increment or decrement. These operators only work on integers and pointers.
First, a postfix expression is a unary expression. The following expressions are all unary expressions:
The increment or decrement operators followed by a unary expression is a unary expression. The value of the expression is the value of the unary expression after the increment or decrement. These
operators only work on integers and pointers.
The following operators followed by a cast expression are unary expressions:
Operator Meaning
======== =======
& Address-of; value is the location of the operand
* Contents-of; value is what is stored at the location
- Negation
+ Value-of operator
! Logical negation ( (!E) is equivalent to (0==E) )
~ Bit-wise complement
The keyword sizeof followed by a unary expression is a unary expression. The value is the size of the type of the expression in bytes. The expression is not evaluated.
The keyword sizeof followed by a parenthesized type name is a unary expression. The value is the size of the type in bytes.
A unary expression is also a cast expression.
A parenthesized type name followed by any expression, including literals, is a cast expression. The parenthesized type name has the effect of forcing the cast expression into the type specified by
the type name in parentheses. For arithmetic types, this either does not change the value of the expression, or truncates the value of the expression if the expression is an integer and the new type
is smaller than the previous type.
An example of casting an int as a float:
int i = 5;
printf("%f\n", (float) i / 2); // Will print out: 2.500000
Multiplicative and additive operators
First, a multiplicative expression is also a cast expression, and an additive expression is also a multiplicative expression. This follows the precedence that multiplication happens before addition.
In C, simple math is very easy to handle. The following operators exist: + (addition), - (subtraction), * (multiplication), / (division), and % (modulus); You likely know all of them from your math
classes - except, perhaps, modulus. It returns the remainder of a division (e.g. 5 % 2 = 1). (Modulus is not defined for floating-point numbers, but the math.h library has an fmod function.)
Care must be taken with the modulus, because it's not the equivalent of the mathematical modulus: (-5) % 2 is not 1, but -1. Division of integers will return an integer, and the division of a
negative integer by a positive integer will round towards zero instead of rounding down (e.g. (-5) / 3 = -1 instead of -2). However, it is always true that for all integer a and nonzero integer b,
((a / b) * b) + (a % b) == a.
There is no inline operator to do exponentiation (e.g. 5 ^ 2 is not 25 [it is 7; ^ is the exclusive-or operator], and 5 ** 2 is an error), but there is a power function.
The mathematical order of operations does apply. For example (2 + 3) * 2 = 10 while 2 + 3 * 2 = 8. Multiplicative operators have precedence over additive operators.
#include <stdio.h>
int main(void)
int i = 0, j = 0;
/* while i is less than 5 AND j is less than 5, loop */
while( (i < 5) && (j < 5) )
/* postfix increment, i++
* the value of i is read and then incremented
printf("i: %d\t", i++);
* prefix increment, ++j
* the value of j is incremented and then read
printf("j: %d\n", ++j);
printf("At the end they have both equal values:\ni: %d\tj: %d\n", i, j);
getchar(); /* pause */
return 0;
will display the following:
i: 0 j: 1
i: 1 j: 2
i: 2 j: 3
i: 3 j: 4
i: 4 j: 5
At the end they have both equal values:
i: 5 j: 5
The shift operators (which may be used to rotate bits)
A shift expression is also an additive expression (meaning that the shift operators have a precedence just below addition and subtraction).
Shift functions are often used in low-level I/O hardware interfacing. Shift and rotate functions are heavily used in cryptography and software floating point emulation. Other than that, shifts can be
used in place of division or multiplication by a power of two. Many processors have dedicated function blocks to make these operations fast -- see Microprocessor Design/Shift and Rotate Blocks. On
processors which have such blocks, most C compilers compile shift and rotate operators to a single assembly-language instruction -- see X86 Assembly/Shift and Rotate.
The << operator shifts the binary representation to the left, dropping the most significant bits and appending it with zero bits. The result is equivalent to multiplying the integer by a power of
unsigned shift right
The unsigned shift right operator, also sometimes called the logical right shift operator. It shifts the binary representation to the right, dropping the least significant bits and prepending it with
zeros. The >> operator is equivalent to division by a power of two for unsigned integers.
The signed shift right operator, also sometimes called the arithmetic right shift operator. It shifts the binary representation to the right, dropping the least significant bit, but prepending it
with copies of the original sign bit. The >> operator is not equivalent to division for signed integers.
In C, the behavior of the >> operator depends on the data type it acts on. Therefore, a signed and an unsigned right shift looks exactly the same, but produces a different result in some cases.
Contrary to popular belief, it is possible to write C code that compiles down to the "rotate" assembly language instruction (on CPUs that have such an instruction).
Most compilers recognize this idiom:
unsigned int x;
unsigned int y;
/* ... */
y = (x >> shift) | (x << (32 - shift));
and compile it to a single 32 bit rotate instruction. ^[1] ^[2]
On some systems, this may be "#define"ed as a macro or defined as an inline function called something like "rightrotate32" or "rotr32" or "ror32" in a standard header file like "bitops.h". ^[3]
Most compilers recognize this idiom:
unsigned int x;
unsigned int y;
/* ... */
y = (x << shift) | (x >> (32 - shift));
and compile it to a single 32 bit rotate instruction.
On some systems, this may be "#define"ed as a macro or defined as an inline function called something like "leftrotate32" or "rotl32" in a header file like "bitops.h".
Relational and equality operators
A relational expression is also a shift expression; an equality expression is also a relational expression.
The relational binary operators < (less than), > (greater than), <= (less than or equal), and >= (greater than or equal) operators return a value of 1 if the result of the operation is true, 0 if
false. The result of these operators is type int.
The equality binary operators == (equals) and != (not equals) operators are similar to the relational operators except that their precedence is lower. They also return a value of 1 if the result of
the operation is true and 0 if it is false.
One thing with floating-point numbers and equality operators: Because floating-point operations can produce approximations (e.g. 0.1 is a repeating decimal in binary, so 0.1 * 10.0 is hardly ever
1.0), it is unwise to use the == operator with floating-point numbers. Instead, if a and b are the numbers to compare, compare fabs (a - b) to a fudge factor.
The bitwise operators are & (and), ^ (exclusive or) and | (inclusive or). The & operator has higher precedence than ^, which has higher precedence than |.
The values being operated upon must be integral; the result is integral.
One use for the bitwise operators is to emulate bit flags. These flags can be set with OR, tested with AND, flipped with XOR, and cleared with AND NOT. For example:
/* This code is a sample for bitwise operations. */
#define BITFLAG1 (1)
#define BITFLAG2 (2)
#define BITFLAG3 (4) /* They are powers of 2 */
unsigned bitbucket = 0U; /* Clear all */
bitbucket |= BITFLAG1; /* Set bit flag 1 */
bitbucket &= ~BITFLAG2; /* Clear bit flag 2 */
bitbucket ^= BITFLAG3; /* Flip the state of bit flag 3 from off to on or
vice versa */
if (bitbucket & BITFLAG3) {
/* bit flag 3 is set */
} else {
/* bit flag 3 is not set */
The logical operators are && (and), and || (or). Both of these operators produce 1 if the relationship is true and 0 for false. Both of these operators short-circuit; if the result of the expression
can be determined from the first operand, the second is ignored. The && operator has higher precedence than the || operator.
&& is used to evaluate expressions left to right, and returns a 1 if both statements are true, 0 if either of them are false. If the first expression is false, the second is not evaluated.
int x = 7;
int y = 5;
if(x == 7 && y == 5) {
Here, the && operator checks the left-most expression, then the expression to its right. If there were more than two expressions chained (e.g. x && y && z), the operator would check x first, then y
(if x is nonzero), then continue rightwards to z if neither x or y is zero. Since both statements return true, the && operator returns true, and the code block is executed.
if(x == 5 && y == 5) {
The && operator checks in the same way as before, and finds that the first expression is false. The && operator stops evaluating as soon as it finds a statement to be false, and returns a false.
|| is used to evaluate expressions left to right, and returns a 1 if either of the expressions are true, 0 if both are false. If the first expression is true, the second expression is not evaluated.
/* Use the same variables as before. */
if(x == 2 || y == 5) { // the || statement checks both expressions, finds that the latter is true, and returns true
The || operator here checks the left-most expression, finds it false, but continues to evaluate the next expression. It finds that the next expression returns true, stops, and returns a 1. Much how
the && operator ceases when it finds an expression that returns false, the || operator ceases when it finds an expression that returns true.
It is worth noting that C does not have Boolean values (true and false) commonly found in other languages. It instead interprets a 0 as false, and any nonzero value as true.
Conditional operators
The ternary ?: operator is the conditional operator. The expression (x ? y : z) has the value of y if x is nonzero, z otherwise.
int x = 0;
int y;
y = (x ? 10 : 6); /* The parentheses are technically not necessary as assignment
has a lower precedence than the conditional operator, but
it's there for clarity. */
The expression x evaluates to 0. The ternary operator then looks for the "if-false" value, which in this case, is 6. It returns that, so y is equal to six. Had x been a non-zero, then the expression
would have returned a 10.
Assignment operators
The assignment operators are =, *=, /=, %=, +=, -=, <<=, >>=, &=, ^=, and |= . The = operator stores the value of the right operand into the location determined by the left operand, which must be an
lvalue (a value that has an address, and therefore can be assigned to).
For the others, x op= y is shorthand for x = x op (y) . Hence, the following expressions are the same:
1. x += y - x = x+y
2. x -= y - x = x-y
3. x *= y - x = x*y
4. x /= y - x = x/y
5. x %= y - x = x%y
The value of the assignment expression is the value of the left operand after the assignment. Thus, assignments can be chained; e.g. the expression a = b = c = 0; would assign the value zero to all
three variables.
The operator with the least precedence is the comma operator. The value of the expression x, y will evaluate both x and y, but provides the value of y.
This operator is useful for including multiple actions in one statement (e.g. within a for loop conditional).
Here is a small example of the comma operator:
int i, x; /* Declares two ints, i and x, in one declaration.
Technically, this is not the comma operator. */
/* this loop initializes x and i to 0, then runs the loop */
for (x = 0, i = 0; i <= 6; i++) {
printf("x = %d, and i = %d\n", x, i);
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Introduction To The Mathematical Theory Of Control Processes Volume 1 Linear Equations & Quadratic Criteria [DJVU] [6qnfrq5i9ff0]
E-Book Overview
A new mathematical discipline has emerged from the bustling scientific activity of the last fifteen years, the theory of control processes. Its mathematical horizons are unlimited and its
applications increase in range and importance with each passing year. Consequently, it is reasonable to believe that introductory courses in control theory are essential for training the modern
graduate student in pure and applied mathematics, engineering, mathematical physics, economics, biology, operations research, and related fields.The extremely rapid growth of the theory, associated
intimately with the continuing trend toward automation, makes it imperative that courses of this nature rest upon a broad basis. In this first volume, we wish to cover the fundamentals of the
calculus of variations, dynamic programming, discrete control processes, use of the digital computer, and functional analysis.
E-Book Information
• Series: Mathematics in Science and Engineering 40
• Year: 1,967
• Edition: AP
• Pages: 263
• Pages In File: 263
• Language: English
• Topic: 113
• Library: Kolxo3
• Issue: 26
• Identifier: 0120848015,9780120848010
• Issn: 0076-5392
• Asin: B000TRNHH2
• Dpi: 600
• Cleaned: 1
• Org File Size: 1,627,225
• Extension: djvu
|
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Rhombus Area Calculator
A rhombus is a quadrilateral with all four sides of equal length.
Enter the lengths of the two diagonals to compute the area.
Used in education, architecture, land surveying, and more.
1. How Do You Find the Area of a Rhombus?
Multiply the lengths of the diagonals and divide by 2.
2. What is the Area of a Rhombus?
The area is calculated using the diagonals or side length and an angle.
3. What is the Size of a Rhombus?
Size typically refers to the area, calculated from the diagonals or side length and an angle.
4. Area of a Rhombus by Side Formula?
Area = \( \text{side}^2 \times \sin(\text{angle}) \), where the angle is between two sides.
5. Standard Formula for the Rhombus?
Area = \( \frac{\text{Diagonal 1} \times \text{Diagonal 2}}{2} \).
6. Finding the Area of a Rhombus in Class 8?
Typically done using the diagonals or side and angle method.
7. Why is the Area Formula for a Rhombus Used?
To calculate the space enclosed within a rhombus.
8. Rule of a Rhombus?
All sides are equal, opposite angles are equal, diagonals bisect each other at right angles.
9. Is Area of Rhombus Equal to Area of Square?
Not necessarily, unless the rhombus is a square.
10. How Do You Find the Area?
Depending on the shape, various formulas are used, like length × width for rectangles.
11. Area of a Rhombus Without Diagonals?
Use side length and an interior angle: \( \text{side}^2 \times \sin(\text{angle}) \).
12. Four Properties of a Rhombus?
Equal sides, equal opposite angles, diagonals bisect at right angles, and diagonals bisect angles.
13. Diagonal of a Rhombus?
Line segment joining opposite corners, bisecting the rhombus.
14. Area of Rhombus and Parallelogram?
Rhombus: \( \frac{\text{Diagonal 1} \times \text{Diagonal 2}}{2} \); parallelogram: base × height.
15. Area with Side of 20 cm?
Use the formula involving the side and angle, or diagonals if known.
16. Do All 4 Sides of a Rhombus Equal?
Yes, all sides of a rhombus are of equal length.
17. Proving a Rhombus Formula?
Derived from the properties of diagonals and angles.
18. Derivation of Rhombus Formula?
Based on the diagonal properties, splitting the rhombus into triangles.
Our Rhombus Area Calculator simplifies geometric calculations, essential for various applications.
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Finding Coefficients bo, b1, b2, and R Squared Manually in Multiple Linear Regression - KANDA DATA
Finding Coefficients bo, b1, b2, and R Squared Manually in Multiple Linear Regression
Researchers can choose to use multiple linear regression if the independent variables are at least 2 variables. On this occasion, Kanda Data will write a tutorial on manually calculating the
coefficients bo, b1, b2, and the coefficient of determination (R Squared) in multiple linear regression.
In this article, I will write a calculation formula based on a book I have read and write how to calculate manually using Excel. For how to manually calculate the estimated coefficients in simple
linear regression, you can read my previous article entitled: “Calculate Coefficients bo, b1, and R Squared Manually in Simple Linear Regression“
In multiple linear regression, the number of independent variables can consist of 2, 3, 4 and > 4 independent variables. The researcher must test the required assumptions to obtain the best linear
unbiased estimator.
This article does not write a tutorial on how to test assumptions on multiple linear regression using the OLS method but focuses more on calculating the estimated coefficients b0, b1, and b2 and the
coefficient of determination manually using Excel.
Multiple linear regression analysis mini-research example
Manually calculating using multiple linear regression is different from simple linear regression. Calculating the estimated coefficient on multiple linear regression is more complex than simple
linear regression.
However, researchers can still easily calculate the estimated coefficients manually with Excel. Therefore, because the calculation is conducted manually, the accuracy in calculating is still
I have prepared a mini-research example of multiple linear regression analysis as exercise material. Data were collected over 15 quarters at a company. A researcher conducts observations to determine
the influence of the advertising cost and marketing staff on product sales.
Data collection has been carried out every quarter on product sales, advertising costs, and marketing staff variables. The company has recorded the number of product unit sales for the last quarter.
Based on this background, the specifications of the multiple linear regression equation created by the researcher are as follows:
Y = b0 + b1X1 + b2X2 + e
Y = product sales (units)
X1 = advertising cost (USD)
X2 = staff marketing (person)
b0, b1, b2 = regression estimation coefficient
e = disturbance error
Data has been collected from quarter 1 of 2018 to quarter 3 of 2021. In this case, the data used is quarterly time series data from product sales, advertising costs, and marketing staff. The data
that researchers have collected can be seen in the table below:
The formula for calculating the estimated coefficients of bo, b1, and b2
Following what I have written in the previous paragraph, to avoid errors in calculating manually, I am here using Excel. Using Excel will avoid mistakes in calculations. There are two ways to
calculate the estimated coefficients b0, b1 and b2: using the original sample observation and the deviation of the variables from their means.
To simplify the calculation of R squared, I use the variable’s deviation from their means. The formula used to calculate b0, b1 and b2 based on the book Koutsoyiannis (1977) can be seen as follows:
Finding the Estimation Coefficient of X1 Variable (b1)
Calculating the values of b0, b1 and b2 cannot be conducted simultaneously. We must calculate the estimated coefficients b1 and b2 first and then calculate the bo. On this occasion, I will first
calculate the estimated coefficient of b1.
Furthermore, to calculate the value of b1, it is necessary to calculate the difference between the actual X1 variable and the average X1 variable and the actual Y variable and the average Y variable.
In Excel, researchers can create a table consisting of components for calculating b1, as shown in the image below:
After creating a formula template in Excel, we need to calculate the average of the product sales variable (Y) and the advertising cost variable (X1). Furthermore, find the difference between the
actual Y and the average Y and between the actual X1 and the average X1.
Next, you calculate according to the Excel table’s formula. In the next step, multiply x1y and square x1. In detail, the calculation stages can be seen in the image below:
Next, copy and paste the Excel formula from the 2nd quarter’s data to the last quarter’s data. To copy and paste formulas in Excel, you must pay attention to the absolute values of the average Y and
the average X.
Absolute values can be applied by pressing F4 on the keyboard until a dollar sign appears. Next, please copy and paste the formula until you get the results as shown in the image below:
To find b1, use the formula I have written in the previous paragraph. The calculation results can be seen below:
Finding the Estimation Coefficient of X2 Variable (b2)
Furthermore, finding the estimation coefficient of the X2 variable (b2) is calculated the same as calculating the estimation coefficient of the X1 variable (b1). To find b2, use the formula I have
written in the previous paragraph. The calculation results can be seen below:
Finding the Intercept Estimation Coefficient (b0)
Based on the order in which the estimation coefficients are calculated, finding the intercept estimation coefficient is carried out at the last stage. It is because to calculate bo, and it takes the
values of b1 and b2. Based on the formula I wrote in the previous paragraph, finding the Intercept Estimation Coefficient (b0) can be seen as follows:
Finding the Coefficient of Determination (R Squared)
R Squared in multiple linear regression shows the goodness of fit of a model. Therefore, the calculation of R Squared is very important in multiple linear regression analysis. The higher R Squared
indicates that the independent variable’s variance can explain the variance of the dependent variable well.
The value of R Squared is 0 to 1; the closer to 1, the better model can be. To manually calculate the R squared, you can use the formula that I cited from Koutsoyiannis (1977) as follows:
The last step is calculating the R squared using the formula I wrote in the previous paragraph. Based on the calculation results, the coefficient of determination value is 0.9285. In detail, it can
be seen as follows:
Based on what has been calculated in the previous paragraphs, we have manually calculated the coefficients of bo, b1 and the coefficient of determination (R squared) using Excel.
We need to compare the analysis results using statistical software to crosscheck. If the output is similar, we can conclude that the calculations performed are correct.
Ok, this is the article I can write for you. Hopefully, it will be helpful for you. Thank you!
6 thoughts on “Finding Coefficients bo, b1, b2, and R Squared Manually in Multiple Linear Regression”
1. Pingback: How to Find ANOVA (Analysis of Variance) Table Manually in Multiple Linear Regression - KANDA DATA
2. Pingback: How to Calculate the Regression Coefficient of 4 Independent Variables in Multiple Linear Regression - KANDA DATA
Thanks for sharing.
You’re welcome.
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Hedging policy using neural networks and its combination with heuristic algorithms case study: Dez Reservoir
Water operation managers use hedging policies to minimize water usage and safeguard it for the future. The aim of the hedging policy is to minimize vulnerability. The annual peak vulnerability
represents the highest deficit experienced throughout the simulation period. A lower value indicates fewer significant system failures. Given the non-linear relationship between the loss function and
shortage levels, decreasing the severity of shortages will lead to cost savings. Bower, et al. were pioneers in studying the economics of hedging policies [1]. Subsequently, other researchers such as
Klemes [2], Stedinger [3] and Loucks, et al. [4] explored the optimization of planning and management objectives in various approaches. Hashimoto, et al. [5] introduced the initial hedging policy.
Bayazit and Unal [6] revisited the topic of hedging and analyzed how to establish hedging parameters in reservoir development. Subsequently, Shih and Revelle [7,8] proposed the single-point and the
discrete hedging approaches. After that, different solution methods were invented to obtain the decision variables of these policies. One of these methods is artificial neural networks and
evolutionary algorithms.
The brain's learning and training mechanism relies on experience. Electronic models of natural neural networks also follow this principle, approaching problems differently from traditional computer
calculation methods. The basic artificial neural network model faced a setback in solving nonlinear problems. However, multilayer networks and feedback-learning algorithms can address these issues.
At that time, recurrent networks and the Hebbian learning method were introduced [9]. The application of neural networks in various aspects of water resources management, such as rainfall and runoff
models [10], rainfall prediction [11], groundwater issues [12], irrigation tanks [13] and reservoir operation rule curves [14,15], expanded rapidly. In traditional neural network training, a dataset
of input and output patterns is required. However, this training process and data utilization do not ensure the most optimal solution. Consequently, alternative optimization methods were employed to
train the neural network model. For instance, Chandramouli and Raman [16], trained the neural network using outcomes from dynamic programming and implemented it for single and multi-reservoir
operation rule curves.
Chang, et al. [17] employed the genetic algorithm to discover the optimal reservoir operation rule curve. In 2008, Chavez and Chang [18] utilized a neural network trained by Genetic Algorithm (GA)
for a multi-purpose reservoir system. In 2011, Pianosi, et al. [19] utilized a combination of artificial neural networks and multi-purpose genetic algorithms for integrated reservoir operation. These
methods are referred to as evolutionary neural networks.
Among various heuristic algorithms, genetic algorithms have been extensively utilized in solving water resource optimization problems in recent years. The genetic algorithm functions as a general
search method that mimics the principles of natural biological evolution. Initially introduced by Holland [20], this algorithm has evolved into a potent optimization tool. Subsequently, numerous
studies have explored the application of genetic algorithms in diverse optimization resource challenges. For instance, Esat and Hall [21] employed the genetic algorithm to determine the optimal path
in a four-reservoir system, aiming to generate electricity and fulfill agricultural requirements. Oliveira and Louks [22] utilized the genetic algorithm with real numbers to ascertain the optimal
curves for reservoir utilization. Wardlaw and Sharif [23] compared the genetic algorithm's performance in binary mode and real numbers, while Sharif and Wardlaw [24] applied this algorithm in a
multi-reservoir system in Indonesia to analyze reservoir development scenarios. In recent times, genetic algorithms have been instrumental in resolving various water resources management optimization
problems. For further exploration and research in this domain, refer to Nicklow, et al. [25].
Although heuristic algorithms are inspired by natural phenomena, there are also algorithms that are imitated by artificial phenomena in their creation. Among the recent algorithms, we can mention the
harmony search algorithm (HS) that was invented by Geem, et al. [26] based on the artificial phenomenon of "musical harmony". "Harmony" in nature is a special relationship between several sound waves
that have different frequencies. After that, Mahdavi, et al. [27] applied changes in the harmony search algorithm and called it the modified harmony search algorithm. Omran and Mahdavi [28] created a
new algorithm by making a change in the original algorithm. Based on the initial algorithm, Pan, et al. [29,30] invented a harmony search algorithm with a set of parameters and a self-adaptive
harmony search algorithm.
In this study, the optimization of Dez reservoir operation over a long-term period is examined using a nonlinear loss function through the evolutionary artificial neural network algorithm.
Subsequently, the outcomes derived from this approach are contrasted with those from genetic exploration and harmony search algorithms, highlighting the advantages and limitations of each technique.
To enhance reservoir management, a hybrid approach combining the evolutionary artificial neural network method with hedging models is implemented and its outcomes are evaluated against the initial
Hedging policy
Water operation managers use hedging policies to minimize water usage and safeguard it for the future. The aim of the hedging policy is to minimize vulnerability. The annual peak vulnerability
represents the highest deficit experienced throughout the simulation period. A lower value indicates fewer significant system failures. Given the non-linear relationship between the loss function and
shortage levels, decreasing the severity of shortages will lead to cost savings. Shih and Revelle [7] presented the discrete hedging technique which is still one of the most practical methods for
reservoir management. In this method, shown in Figure 1, when the sum of storage and inflow for a particular month p exceeds V1P, all the demands are fulfilled. If the total storage and inflow are
below V1P but above V2P, only 𝜶1 percent of the demand is met, known as the first hedging phase. Similarly, if the total storage and inflow are below V2P but above V3P, 𝜶2 percent of the demand is
supplied, termed the second hedging phase [31].
Genetic Algorithm (GA)
Genetic algorithms are stochastic search methods rooted in natural selection and genetics. They commence with an initial array of random solutions known as the initial population. Each entity within
this population is termed a chromosome, embodying a potential solution to the problem at hand. Chromosomes progress through successive iterations, referred to as generations. During each iteration,
chromosomes undergo evaluation. New generations are generated by either merging two chromosomes from the current generation using a combination operator to create offspring, or by altering a
chromosome using a mutation operator to produce the next generation's offspring. Offspring from the current generation become the parents of the subsequent generation. Parent selection for the next
generation is facilitated by a selection operator that leverages parental fitness values as a criterion for selection. Weaker chromosomes are then eliminated to maintain a constant population size,
ensuring the parents of the next generation are retained. Favorable chromosomes have a greater likelihood of selection, leading the algorithms to converge towards superior chromosomes over several
generations, potentially representing the optimal or suboptimal solution [32].
Harmony search algorithm
Harmony Search is an optimization algorithm that simulates the improvisation process of jazz music. In this algorithm, each solution is called a “harmony”. The ‘‘Harmony Memory’’ (HM) matrix is
filled with randomly generated solution vectors and sorted in terms of the objective function value. Then, a New Harmony vector is produced based on three parameters: HMCR (harmony memory
consideration rate), PAR (pitch adjustment rate), and BW (bandwidth). A good set of parameters can enhance the algorithm’s ability to search for the global optimum. The following general steps are
taken in using the HS algorithms. First of all, if a uniform random number returned by rand () (between 0 and 1) is less than HMCR, the decision variable is generated by the memory consideration;
otherwise, it is obtained by a random selection between Lower Band (LB) and Upper Band (UB). Secondly, each decision variable updated by the memory consideration undergoes a pitch adjustment with a
probability of PAR. Thus, every component obtained by the memory consideration is examined to determine whether it should be pitch-adjusted. This operation uses the PAR parameter. In the memory
consideration, New Harmony is chosen from harmony memory. And finally, New Harmony is produced by random selection. If the objective function of the New Harmony vector is better than the worst
harmony in the HM, the New Harmony is included in the HM, and the existing worst harmony is excluded from the HM. Then, the harmony memory is sorted again. This process is continued until the
stopping criterion is obtained [33].
Mahdavi, et al. (2007) modified the original HS to introduce an improved HS (IHS) algorithm, which dynamically updates the values of PAR and BW as follows [27]:
Where NI is the number of iterations considered to stop the algorithm and gn is the repetition number.
In a nutshell, the new scheme to improvise a new harmony, Xnew, can be summarized as follows:
For (i=1 to n)
If (random_1
$x_ne{w}_{i}={x}_{i}^{j}$ j ⋳ (1, HMS)
if (random_2
x_newi = x_newi ± random_3 * bw, random_3⋳(0,1)
End if
x_newi = LB(i) + random_4 * (UB(i)-LB(i)), random_4⋳(0,1)
End if
End for
In this algorithm, it is assumed that the HMCR (between 0.9 and 1) and PAR (between 0 and 1) values are normally distributed with a mean of 0.98 and 0.3 and a standard deviation of 0.01 and 0.05,
respectively. During the evolution, the values of HMCR and PAR associated with the generated harmony successfully replacing the worst member in the HM are recorded. After a specified number of
generations of LP, the means are recalculated by averaging all the recorded values during this period. With the new mean and the given standard deviation, new HMCR, and PAR values are produced and
used in the subsequent iterations. The above procedure is repeated.
Case study
Dez is the tallest double-arched concrete dam in Iran, constructed on the main branch of the Dez River. The river flows approximately 420 kilometers before reaching Reservoir Lake, where it merges
with the Karun River, eventually emptying into the Persian Gulf. This multipurpose reservoir serves various functions, including providing water for agricultural purposes in fertile plains spanning
around 125,000 hectares, generating 520 megawatts of electricity, mitigating river floods and associated damages, and supplying water for industrial needs. The watershed area covers 17,430 square
kilometers, with the lake holding a total volume of 3,460 million cubic meters at a height of 352 meters, including 65 million cubic meters of dead storage.
The statistical period spans 42 years, as shown in Table 1. To facilitate method examination and create critical system conditions, the total demand across all periods is assumed to be double the
actual value.
Results and discussion
The neural network model for the Dez reservoir in this study comprises three layers: input, hidden, and output. The input layer incorporates inflow, demand, and two seasonality indexes. Given the
consistent pattern of inflow data and reservoir output each month, the seasonality indexes are an important input in seasonal models. These indexes help the network to distinguish among different
periods within a year. Without seasonality indexes, one must develop a separate model for each seasonal step (month in here) and face up using several models at a time, which would not be an
efficient way to handle the problem [34]. We tested several types of seasonality indexes and found the one suggested by Nilsson, et al. [35] to be more suitable for our model. In this method two time
series are considered as input neurons which combined, are representative of the cyclic 12 months of the year. One series is represented by the oscillation of a sine curve and the other of a cosine
curve. The whole annual cycle is represented by 12 cyclic pairs of values, one unique pair per each month.
The hidden layer houses 2 internal neurons utilizing the sigmoid transfer function. The number of neurons in the hidden layer is obtained by sensitivity analysis through a trial and error process.
The output layer features a single neuron determining the reservoir outflow, employing the linear transfer function. Training the network involves the harmony search algorithm to minimize the total
deficiency objective function. The network's weights serve as decision variables for the harmony search algorithm, initially set randomly. Upon each program execution, the harmony search algorithm
optimizes the weights to minimize the total deficits' sum.
Moreover, the combination of a neural network and discrete hedging model is utilized for optimal resource operation. In this model, alpha coefficients were set to 0.75 and 0.6 with sensitivity
analysis. The neural network comprises three layers: input, hidden, and output. The input layer includes inflow, initial storage, previous period's outflow, demands, alpha coefficients, and two
neurons for seasonality indexes. The hidden layer houses 5 internal neurons using the sigmoid transfer function. The output layer features one neuron employing the linear transfer function to
determine the reservoir outflow. In this model, based on water availability, one of the neurons 1𝜶, 2𝜶, or 3𝜶 is fed into the system, and network coefficients are adjusted accordingly.
For comparing different methods, various factors have been examined, such as reliability, maximum vulnerability, resiliency, quantitative reliability, deficiency value, and objective function value.
The objective function values are shown in Figure 2. It is evident that these values are quite similar and exhibit a consistent trend.
The evaluation criteria are presented in Table 2. It is important to note that these results reflect an average of 10 executions. It is evident that the neural network method exhibits higher
reliability compared to the genetic algorithm and harmony search. Reliability, defined as the proportion of fully covered courses to the total number of courses [27], is notably high in the neural
network, the combination of neural network and hedging, and the genetic algorithm, followed by the harmony search algorithm. The maximum annual vulnerability represents the most severe deficiency
experienced throughout the simulation period. A lower value indicates a reduced occurrence of significant system failures. Vulnerability is determined by equation (4), where TDt represents the
monthly demand and Rt signifies the allocated water amount in month t.
As can be seen, the first three models have a similar maximum vulnerability value but with the combination of the discrete hedging policy and the neural network, this value is significantly reduced
to 0.84. This is completely consistent with the philosophy of hedging, which is to increase the number of failures and reduce the maximum shortage. The purpose of hedging in the reservoir operation
is to reduce the maximum vulnerability and as a result, reduce the damage caused by a severe shortage. Given the non-linear relationship between the loss function and shortage levels, decreasing the
severity of shortages will lead to cost savings. The change in vulnerability is similar to the change in reliability, meaning that increasing reliability causes an increase in vulnerability and vice
versa. High values of reliability and low vulnerability are desirable, which is in contradiction with the trend of changes in these two evaluation criteria. However, as can be seen, the vulnerability
value is significantly reduced in the last model with a slight decrease in reliability. This is a very important achievement in reservoir operation rule curves.
This study focused on optimizing Dez reservoir operation over a long-term period using a nonlinear loss function through an evolutionary artificial neural network algorithm. The outcomes of this
approach were then contrasted with genetic exploration and harmony search algorithms, highlighting the strengths and weaknesses of each method. Ultimately, a combination of the evolutionary
artificial neural network method and hedging policies was employed for optimal reservoir management, with results compared to the previous approach. Results showed the appropriate performance of
combining hedging policy with artificial neural network and harmony search algorithm. This combination significantly reduced the vulnerability value with a slight decrease in reliability.
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Significance of the Difference between Two Slopes Calculator
Compute the extent to which the slopes of two lines are significantly different from one another, given each line's slope, standard error, and sample size. The calculator computes the t-value for the
significance test, the degrees of freedom, and the p-value. Knowing whether the slopes of two lines are significantly different from one another can be very useful in analytics studies that compare
multiple groups.
Please provide the necessary values, and then click 'Calculate'.
Line 1 sample size:
Line 1 slope:
Line 1 standard error:
Line 2 sample size:
Line 2 slope:
Line 2 standard error:
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Any math people here??? -- *updated* | The Watercooler
difficult child brought home homework. Amazing.... Actually asked for help...gasp.... (The world must be coming to an end. :rofl: )
Math never was my strong suit. No textbook sent home, so no way to look up info to set up formula....
Anyone know how to set the formula up to "Find the probability of each event?"
1) rolling a 1, 2, or 3 on the first roll of a 1-6 number cube and rolling a 4, 5, or 6 on the second roll of the same cube.
2) Randy has 4 pennies, 2 nickels and 3 dimes in his pocket. If he handomly chooses 2 coins, what is the probability that both are pennies?
I think I've got it, but if they are wrong, I'll never hear the end of it...lol.
These are the answers my "math whiz" kids got:
1) one-half probability that 1, 2 or 3 will be rolled on the first try, and one-half probability that 4, 5 or 6 will be rolled on the second try (3 numbers out of 6 numbers each time)
2) four-ninths probability that a penny will be chosen the first time, and three-eighths probability that a penny will be chosen the second time. My son said to multiply those numbers together to get
a 16.7 percent probability of choosing two pennies
Hope that's right!
Hey Sheila,
I'm good for something!
smallworld's kids are exactly right. Here is how to think it through:
In the first problem, the events are independent of each other. In other words, the probability of the second event occuring is not altered by the first event. Since you want to roll a 1,2,3 on the
first roll (3 favorable outcomes out of 6 possible outcomes) which is 3/6 or 1/2 "and" you want to roll a 4,5,6 on the second roll (3 favorable outcomes out of 6) which is 3/6 or 1/2, you multiply
the probabilities together and 1/2 times 1/2 is 1/4 or a 25% probability.
Now, in the second problem, once you pick a penny the same penny cannot be picked again to the second probiblity is dependent on the first. So you have to approach it a little differently.
The first probability is 4 pennies out of 9 coins which is a 4/9 probability. The second probability is 3 remaining pennies out of 8 remaining coins or 3/8. Again, you multiply the probabilities
together so 4/9 times 3/8 is a 1/6 chance or a 16.7% probability.
Way to go, smallworld's math whiz kids!! :bravo:
Ha!! I'm a math teacher, but probability is my worst area. :smile: But, Kathy and whiz kid have it right.
On a side note, my best friend's daughter is in 6th grade, which is what I teach. She has these problems of the week. My gosh...are they trying to kill the kid??? My husband, who has a PhD in physics
had to take the darn thing home to figure it out. Supposedly the purpose is not the answer, but the process, which they have to write down. To me, it was a rather humiliating exercise and I wouldn't
have received any benefit as a child in doing it. They have one problem every week. Geez...
I loved probability theory both at school and at uni, where the theory was applied to studying populations of tagged/untagged specimens in the wild.
The answers are correct - we often get bamboozled by probability, thinking that everything in the world is connected far more than it is. We might sit at a poker machine (slot machine or whatever you
call it) and pour in coin after coin and think, "I've got to get a payoff soon, it's got to be my turn soon." But in reality, you have no better chance with each coin than with all the ones before
it. As Kathy said, this is random. We just like to hope that it's not. The classic, simple question is: "A kid tosses a coin and gets 99 heads in a row. What is the chance that the next toss will
also be a head?"
The answer is, 1 in 2. Or, 0.5. However, in practical terms, I'd be checking the coin to make sure there is no bias, such as having two heads. If the coin DOES have two heads, then the next coin toss
is a certainty to get a head.
And yes, to get the probability of consecutive events, you multiply them together, but sometimes you have to be slightly sneaky about it and multiply the probabilities of the events NOT happening,
then subtract your final answer from 1.
The second example is an interesting one - it's "sampling without replacement". If you follow it through, the more you sample without replacement, the better your chances each time of getting it
right. The safe in "The Price Is Right" is a good example - it has three knobs. Each knob has three numbers. You can't use the same number twice (which makes it sampling without replacement). The
chance of getting the first knob right is 1 in 3. The chance of getting the second knob right is 1 in 2. The last knob has no choices left, but the remaining number, so if the first two are right,
then this last number is automatically correct - probability 1. The total probability is one third times one half, or 1 in 6. The more numbers you have to choose from (and the more knobs), the
process simply continues. You can see that for 10 knobs and 10 digits, you're multiplying 1 over 10 x 9 x 8 x 7 x 6 x 5..etc. This is more easily written as 10! or 10 factorial. Writing it as 10! (or
whatever number) saves paper. So if you ever see that little ! after a number, that's what it means. It is NOT mathematicians becoming excited.
We use "sampling WITH replacement" whenever tagged animals are released into the wild. Have you ever wondered why scientists bother to tag animals? You can't tag 'em all, and how can you get useful
information from just half a dozen tagged crocodiles, for example?
This was back in the days when animals were tagged without radio tracking - the tag had to be attached in a way that made recapture as likely as capture of a fresh, previously uncaptured specimen.
Let's say we capture and tag 20 specimens, then release them. We lay nets again, and capture another 20. Of that 20, 5 wear tags (and are therefore recaptures). What is the total population in the
It's actually not difficult, if you think about it logically. The group captured is (hopefully) a fully random representative sample. Of the ones we've just captured, 5 wear tags. This means we
tagged 5 out of 20, or 20%, of the population. Therefore 20 specimens tagged first go, is 20% of the population. Total population - about 100.
You can't be exact because there are always some random factors. Is the tag making it easier to recapture them? Are the ones originally captured more stupid than most (and hence blunder into the nets
again)? I remember one case where two blue wrens were recaptured, seven years later, in exactly the same place where they had been first captured and tagged, as yearlings. They simply hadn't changed
territory or behaviour patterns. This sort of thing means that where mathematics meets zoology, life can become complicated and unmathematical.
So sampling with replacement, and sampling without replacement - it all begins with coins in the pocket.
And here's a cute brain teaser for small difficult children - you're in your bedroom and it's dark. You need to get a pair of socks from your sock drawer, but you can't turn on the light. You don't
want to grab ALL the socks, you want to get as few as possible. In the drawer are loose grey and brown socks, all jumbled together. You don't care whether you get a pair of grey, or a pair of brown,
just so long as you have a pair. All the socks are identical, other than these two colours.
What is the minimum number of socks you need to get, to be certain you have a pair?
Bless you all!
I got the 1st one right; but blew the second one. I was thinking one would get two coins at a time. lol
Marguerite...the nerd in me is blown away at your response. :smile:
One more. I've confused myself with "dependent" and "independent"...
There are 20 true/false questions on a test. You do not kow the answer to 4 of the questions, so you guess. What is the probability that you will get all 4 answers right?
I have doubts difficult child will ever "get" this kind of reasoning. Example: Chance of selecting 2 pennies from his pocket is 100% because he can feel the difference in the coins. So I tell him to
pretend he can't feel the coins. You got it, "but I can, so the probability is 100%." What do you tell the student to get them past this concrete thinking???????????
Sheila, the question above is independent. Question 1's answer is not dependent on question 2's, question 3's or question 4's answers (in other words, there'a 50 percent chance on each question that
it's true or false independent of the other questions' answers). So the answer is 0.5 X 0.5 X 0.5 X 0.5 = 0.0625 or 6.25 percent probability of getting all four questions right.
Sheila ~ this is independent probabilities. Guessing one answer right or wrong does not have any effect on the next guess.
So there is one favorable outcome out of two possible outcomes on a true/false test, or a 1/2 probability that you guess each question correctly. Since you are guessing on 4 questions you would
multiply 1/2 times 1/2 times 1/2 times 1/2 and get a 1/16 probability (6.25%) of guessing the four answers correctly.
The fact that there are 20 questions is just extraneous information.
As far as abstract thinking, I believe that everyone reaches that at a different point in their lives. In fact, that is the reason that I have a problem with moving Algebra 1 down to younger and
younger ages. I think many students are not ready for the abstract thinking needed for higher math and we are setting them up for failure.
by the way, lots of people have problems with probability. I hate to teach that chapter since many students have a hard time with it.
I hope I helped.
Some things never change. I worked the problem correctly the first time. Then "overanalyzed" by considering the answer based on dependent events. I use to tell myself, "You first answer is usually
the right one."
Kathy, you have been helpful. Besides the math support, I always try to gage how things are going with difficult child based on how his age-group/grade handles similar situations or environments.
When it pertains to education, I assume that if his peers can master the content, difficult child should be able to do so as well within a certain range (fair to good).
It's some relief to know that other students don't automatically get this type reasoning. But with difficult child, you can't get him past black/white thinking. Thank goodness there's more aspects of
math than probability or most likely, a lot of us would be in serious trouble -- me included. lol
Thanks again!
by the way, difficult child's struggling with introductory algebra. Doesn't bode well for the future I fear...
Well, Sheila, if it makes you feel any better, I struggled with 8th grade math. The lightbulb didn't go off until Algebra 2 when it all started to make sense to me.
And look where I ended up! :grin: I'm sure my ninth grade Algebra 1 teacher would be surprised.
Just reporting in that with everyone's help, difficult child's grade in Math rose from an "F" to a "D." Thank you!!!
Kathy -- that gives me hope. I sure don't expect him to end up a math teacher, but maybe we'll make it through high school. lol
Strange how my expectations have changed. First I made plans for college, then the goal was to get through high school. Today, I'm having trouble with real expectations of completing this year (7th
grade) with any type success.
It's way past time for school to be out for the year. difficult child, his teachers, and me -- we NEED for school to be out. lol
Don't know what they pay teachers who teach kids in puberty, but I know it can't possibly be enough! There should be some type of hazard pay involved!
Thanks again, ladies! You all did good on this assignment!
Math is a tricky devil. I was HORRIBLE at it from about grade 6 on. It wasn't until college that it 'clicked.' I have been teaching math now for 21 years. Keep in mind that your brain is like any
other organ in your body. Everyone developes at a different rate. He's at that age when you switch from concrete to abstract thought patterns. For some, it happens early...others (like me) it took
awhile. It doesn't mean you're 'dumb,' it just means you haven't reached that point yet. It's like asking a newborn to walk when they haven't developed the skills to do so yet.
(I'm hopping off my teacher podium right now.) :smile:
I'm glad to hear that difficult child is passing math now and that I could help. Feel free to PM me if you ever have any other math questions.
by the way, I mentioned in another thread that I am starting my postgraduate degree this summer. The first class that I am going to take is Teachers and the Law. I'm sure that I'll be popping in the
Special Education forum for help. I hope that you won't mind my tapping into your expertise in that area.
Thanks for the offer!
My knowledge of the law is limited to students with disabilities, but I'd be glad to help. I have a lot of resource material also -- might help save a minute or two of research time for you.
I've taken various business and real estate law courses. Beware of the caveats. lol
Absolutely nothing to add, as math never was my strong suit (and trust me, THAT is the understatement of the year!).
Still, it's been fun to read along, and to see how everything worked out.
I have a mental block where math is concerned.
It's like my brain freezes at the mention of the M word.
Let alone the A (algebra) or even worse, the G or the T words.
<div class="ubbcode-block"><div class="ubbcode-header">Quote:</div><div class="ubbcode-body">Let alone the A (algebra) or even worse, the G or the T words. </div></div>
:rofl: :rofl:
You realize that you are talking about my life.:grin:
<div class="ubbcode-block"><div class="ubbcode-header">Quote:</div><div class="ubbcode-body">Marguerite wrote,
And here's a cute brain teaser for small difficult children - you're in your bedroom and it's dark. You need to get a pair of socks from your sock drawer, but you can't turn on the light. You don't
want to grab ALL the socks, you want to get as few as possible. In the drawer are loose grey and brown socks, all jumbled together. You don't care whether you get a pair of grey, or a pair of brown,
just so long as you have a pair. All the socks are identical, other than these two colours.
What is the minimum number of socks you need to get, to be certain you have a pair?
How'd you know what my sock drawer looks like? LOL
A one time co-worker of mine solved the sock problem by only ever buying a particular brand and color of socks (black). That way any two socks at random were a match.
Now, as to the problem: my intuitive answer right off the bat was "half + 1". But actually the answer is 3. If you get two they might match, or they might not. If not, the third would have to match
one or the other. (I'm ashamed to say how long it took me to reason that out. I must be a small difficult child>
In the real world, though, we have to deal with the problem of un-mated socks. You'd think that the mates to any odd socks would show up eventually, but it ain't so. Where do those odd socks' mates
go? Last time we moved, I had about half a dozen un-mated socks. We cleaned out every drawer and closet in the house. The washer and dryer were empty and disconnected, no odd socks within or left in
the utility room. We never threw away odd socks, on the assumption that they had to have a mate somewhere. So, where did they go? I think some kind of supernatural force is at work. That, or they
were lost in laundromats and relative's houses on vacation.
by the way, are grey and gray the same colo(u)r? I'm trying to learn Ozzie.
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ACCA F5 Throughput Accounting
Reader Interactions
2. Dear Sir ,’
Bottleneck Resource is not in this lecture,Where I found About Bottleneck resource ?
□ In the free lecture notes that you should be using with the lectures.
3. Dear Sir,
I am kind of confused why the cost per factory hour for both product A and B are the same? Since they are different so should they be costing differently per each factory hour?
Appreciate your help and nice day!
□ They are both made in the same factory, and the costs are charged according to how many hours each unit spends in the factory.
4. Thank you! Thank you so much for your great help. Kind regards, Svetlana
5. Dear Sir,
thank you for another great lecture, i am enjoying F5 very much. I recently did F7 and F9.
thank you for making throughput so easy to understand.
□ Thank you for your comment 馃檪
6. Hi Sir John,
Great lectures. May i ask you about example 1 and 2 on the fixed cost section. To calculate fixed cost you’re using A: 20,000 units; B: 10,000 units. Why are we using the max demand unit instead
of optimum production plan unit? I am thinking we could use this number for example 1 are 19,000 and 10,000 units; example 2 are 20,000 and 8,000 units.
Thank you.
□ I do actually explain this in the lecture.
The budgets will have been prepared before knowing about the limit on production. Therefore the overheads will have been absorbed assuming the produced to meet the full demand. Even though
the actual production ends up being less, the total overheads will, of course, remain the same.
☆ Hi Sir John, apologizes for repeating question. I just re-watch your lectures and notice what you have explained. Thank you so much for replying.
7. Hi John, thank you for the lecture.
You must be logged in to post a comment.
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Fractal Geometry
Images of nonlinear dynamical systems are typically fractals. Here we explore the origin and meaning of this term.
Origin and Cantor sets
The term Fractal was chosen by Mandelbrot (after the Latin Fractus) to signify irregular, fragmented objects. These often but do not necessarily have a fractional scaling dimension, and the surface
of these objects is non-differentiable. Spectacular examples of fractals such as the Julia and Mandelbrot sets are often brought to mind when one thinks of fractals, but they are far older and richer
than these examples alone. Nearly a century before the word fractal had been invented, the foundations for this branch of math were established and are here roughly detailed.
In the late 19th century Cantor founded set theory, a study of abstract collections of objects and the relations between these collections. Other pages in this website make use of set theory, and for
a good background on the subject see “Naive set theory” by Halmos. From set theory, we find fascinating conclusions that lay a foundation for fractal geometry, so to understand the latter it is best
to examine a few key aspects of the former.
Surprising results abound in set theory, but one of the most relevant here pertains to counting how many elements a set has. It turns out that not all infinite sets are equal: there are as many
elements of the set of positive numbers as there are rational numbers (an infinite amount of each) but there are far more elements of the set of real numbers than either of the other two sets. The
elegant diagonal proof for this is worth viewing elsewhere, as it reappears again in other forms (in Turing’s computability theories, for example).
A particularly interesting set is known as the middle thirds Cantor set $C$. This set will be considered in some detail because it illustrates many of the most important aspects of fractals but is
relatively simple, existing on a line. $C$ is made as follows: take the closed interval on the reals $[0, 1]$ and remove from this an open interval of the middle third, that is, $[0, 1] - (1/3, 2/3)
Now repeat this process for each remaining closed intervals,
and again
and so on ad infinitum. $C$ is the set of numbers that remains after an infinite number of these steps. This set is remarkable: after n steps of removing the inner third, $(\frac{2}{3})^n$ total
length remains. Therefore $C$ has $0$ total length: $(2/3)^n \to 0 \; as \; n \to \infty$. If it has $0$ length, does $C$ have any points? It does indeed, just as many points as the original closed
interval $[1,0]$! The set is totally disconnected (no point touches any other point) and perfect (every point is a limit of another set of points in $C$).
This set can be drawn with Turtle, a Python programming library that was designed to help teach programming in a visual way. It turns out to be quite capable of intricate fractal drawings as well.
from turtle import *
import turtle
def cantor_set(size, recursive_steps):
if recursive_steps > 0:
cantor_set(size/3, recursive_steps-1)
turtle.pu(), turtle.forward(size), turtle.pd()
cantor_set(size/3, recursive_steps-1)
turtle.color('red'), turtle.forward(size)
turtle.pu(), turtle.forward(size), turtle.pd()
turtle.color('blue'), turtle.forward(size)
for i in range(300):
turtle.speed(0), turtle.delay(0)
turtle.setup (width=1900, height=1080, startx=0, starty=0)
turtle.pu(), turtle.goto(-800, 0), turtle.pd()
cantor_set(500 * (2**(i/30)), 4 + i // 30)
turtle_screen = turtle.getscreen()
The .eps files generated by this program are vector images, and to make a video conversion to a fixed size .png or .jpg type file is useful. Fortunately the Python Image Library has the wonderful
ability to detect and reformat all kinds of image types! I used the following program to convert .eps to .png files:
from PIL import Image
import os
# insert path here
path = '/home/bbadger/Desktop/'
files = os.listdir(path)
for i in range(0, len(files)):
string = str(i)
while len(string) < 3:
string = '0' + string
eps_image = Image.open('cantor_set' + string + '.eps') # input names here
eps_image.save('cantor{0:03d}.png'.format(i)) # output names here
After compiling these .png images into an mp4 using ffmpeg in bash
(base) bbadger@bbadger:~$ ffmpeg -f image2 -r 30 -pattern_type glob -i '*.png' cantor_zoom.mp4
the video can be viewed and edited. Here it is as a .gif (ffmpeg can convert directly into gifs simply by changing the extension name to cantor_zoom.gif, but these are uncompressed and can be quite
large files)
As the number of recursive calls increases, the Cantor set becomes invisible. This should not come as a surprise, being that it is of measure $0$ as $n \to \infty$. Thus in order to obtain a viewable
map with more recursive steps, vertical lines are used to denote the position of each point in the set. The following program accomplishes this by drawing alternating red and blue vertical lines at
the start of where each set interval (at any given step) begins and so is only accurate with a relatively large value for the starting number of recursions (>4). A black background is added for
clarity, and once again the number of recursive steps increases with scale to maintain resolution.
Starting at the 5th recursive level (code available here) and adding a recursion every 30 frames, we have
Now this is one particular example of a Cantor set, but there are others: one could remove middle halves instead of thirds etc. The general definition of a Cantor set is any set that is closed,
bounded, totally disconnected, and perfect. The general Cantor set $C$ is a common occurence in nonlinear maps. The period doubling in the logistic map forms a Cantor set, although this is not a
middle thirds set as seen above
as well is in the henon map.
Space filling curves
What is the dimension of the Cantor set? Finite collections of points are of dimension $0$, whereas lines are of dimension $1$. $C$ is totally disconnected and therefore would seem to be $0$
dimensional, and yet it is an infinite collection of points that are bounded to a specific region. Thus $C$ has characteristics of both zero and one dimensions. Do any other curves also exhibit
properties of multiple dimensions?
One can defined fractals as objects that exhibit properties of multiple dimensions, and in that respect every image on this page is an example of a curve that answers this question. But before
exploring those, it may be useful to view some extreme cases: curves (1-dimensinal objects) that fill space (2 dimensions). One of the first such curves to be discovered to do this by Peano in the
late 19th century can be described in python as follows
ls = [90, -90, -90, -90, 90, 90, 90, -90, 0]
def peano_curve(size, recursive_steps, ls):
if recursive_steps > 0:
for i in range(len(ls)):
peano_curve(size, recursive_steps-1, [i for i in ls])
where recursive_steps goes to infinity. To understand how this fills space, observe the base case:
and the first recursive step, where each line segment of the curve above is replaced with a smaller version of the whole curve:
After a few more recursive steps (only 5 in total!) , the present resolution is no longer able to differentiate between one line and another and we have achieved something close to a space-filling
This curve fills space but crosses over itself. The following is another space filling curve from Peano that does not self-cross but is more difficult to draw. The L -system, named after its
discoverer Lindenmayer, is a very useful system for characterizing the generation of more complex recursive structures. For a good overview of this system complete with examples, see here. The Peano
curve may be defined in the L-system by the sequences X = 'XFYFX+F+YFXFY-F-XFYFX', Y = 'YFXFY-F-XFYFX+F+YFXFY' where X and Y are separate recursive sequences, ‘+’ signifies a turn left by 90 degrees,
‘-‘ a turn right 90 degrees, and ‘F’ signifies a movement forward. This can be implemented in python by interpreting each L-system element separately as follows:
def peano_curve(size, steps, orientation):
X = 'XFYFX+F+YFXFY-F-XFYFX'
Y = 'YFXFY-F-XFYFX+F+YFXFY'
l = r = 90
if steps == 0:
if orientation > 0:
for i in X:
if i == 'X':
peano_curve(size, steps-1, orientation)
elif i == 'Y':
peano_curve(size, steps-1, -orientation)
elif i == '+':
elif i == '-':
for i in Y:
if i == 'X':
peano_curve(size, steps-1, -orientation)
elif i == 'Y':
peano_curve(size, steps-1, orientation)
elif i == '+':
elif i == '-':
The base case
and first recursion
at the fifth recursion level,
As is the case in the first Peano curve, the true space-filling curve is the result of an infinite number of recursive steps. This means that the curve is also infinitely long, as the length grows
upon each recursive step. Infinite length is a prerequisite for any space-filling curve.
Note that both of these curves are nowhere-differentiable: pick any point on the curve, and it is an angle (a 90 degree angle to be precise) and as angles are non-differentiable, the curve is
These curves map a line of infinite length to a surface, and the mapping is continuous. But importantly, this mapping is not one-to-one (injective): multiple points on the starting line end up at the
same spots in the final surface. In fact no injective mapping exists between one and two dimensions exists, and a geometric proof of this in the second Peano curve is as follows: observe the point at
the upper right corner of the curve $p_1$ and the point directly below it $p_1$ to be $p_2$. Upon each recursion $r$, the distance between these points is divided by 3 such that
\[d(p_1, p_2)_{r+1} = \frac{d(p_1, p_2)_{r}}{3} \\ \; \\ \frac{1}{3^n} \to 0 \; as \; \; n \to \infty \\ \; \\ d(p_1, p_2) \to 0 \; as \; \; r \to \infty\]
Now the true Peano curves are infinitely recursive, therefore this the distance between these points is 0 and therefore $p_1$ and $p_2$ map to the same point in two dimensional space, making the
Peano curve a non-injective mapping.
Imagine moving along the surface created by either peano curve. An arbitrarily small movement in one direction along this surface does not necessarily lead to a small movement along the curve itself.
Indeed it can be shown that any mapping from two to one dimensions (which could be considered to be equivalent to the definition of a space filling curve) is nowhere-continuous if the mapping is
one-to-one and onto. For some interesting repercussions of this on machine learning methods such as neural networks, see here.
Fractals as objects of multiple dimensions
The Koch curve may be drawn as follows
ls = [60, -120, 60, 0]
def koch_curve(size, recursive_steps, ls):
if recursive_steps > 0:
for i in range(len(ls)):
koch_curve(size, recursive_steps-1, [i for i in ls])
The curve starts as
and at each recursion, each line segment is divided into parts that resemble the whole. At the 2nd,
and 6th recursion:
Now this curve evidently does not cover the plane like the Peano curves. But the curve does seem ‘fuzzy’, and that it might cover at least part of the plane. In this respect, it seems to be partway
between dimensions. Is this the case?
A better understanding comes from the similarity dimension, which is equivalent to the Hausdorff dimension for the following self-similar objects. First note that Euclidean objects like a point,
line, or surface have the same topological dimension as their similarity dimension: a point cannot be subdivided ($n^0 = 1$), a line of length n can be subdivided into $n^1 = n$ pieces, and a surface
square of length n can be subdivided into $n^2$ pieces. Now note that the Koch curve may be subdivided into four equal pieces, and that these pieces are $1/3$ the length of the total curve. It’s
similarity dimension is therefore
\[D = \frac{\log N}{\log (1/r)} \\ \; \\ D = \frac{\log 4}{\log 3} \approx 1.2618\]
Now consider the following curve, known as the quadric Koch curve:
ls = [90, -90, -90, 0, 90, 90, -90, 0]
def quadric_koch_curve(size, recursive_steps, ls):
if recursive_steps > 0:
for i in range(len(ls)):
quadric_koch_curve(size, recursive_steps-1, ls)
The curve starts as follows:
in the first recursion,
and after 5 recursive levels,
Let’s calculate this curve’s similarity dimension: there are 8 pieces that are smaller versions of the whole curve, and these pieces are $1/4$th the length of the whole so therefore
\[D = \frac{\log N}{\log (1/r)} \\ \; \\ D = \frac{\log 8}{\log (4)} = 1.5 \\\]
This curve has a slightly larger dimension than the other Koch curve, which could be interpreted as being that this curve is closer to a surface than the first Koch curve. Visually, this results in
the appearance of a rougher line, one that appears to cover more area than the first. How long are these curves? Each recursion adds length so just like the space-filling curves, the total length is
We can also calculate the dimension of the Cantor set. There are 2 collections of points that are identical in structure to the entire collection, which we can call L and R. These collections take up
a third of the original interval, and so the dimension of $C$ is
\[D = \frac{\log N}{\log (1/r)} \\ \; \\ D = \frac{\log 2}{\log (3)} \approx 0.631\]
which is somewhere between $0$ and $1$, and this matches the observations that $C$ exhibits properties of both $0$ and $1$ dimensional objects.
Fractals are defined as shapes that have a Hausdorff dimension greater than their topological dimension. For the shapes presented on this page, Hausdorff and scaling dimensions are equal. Thus the
curves in this section are all fractals, as are the Cantor set and both space-filling Peano curves.
More Selfsimilar fractals
The Sierpinski triangle (which will resemble a triforce for those of you who played Zelda) is one of the most distinctive fractals presented in the book. There are two orientations the curve takes,
which means that the drawing proceeds as follows:
ls = [60, 60, -60, -60, -60, -60, 60, 60, 0]
def sierpinski_curve(size, recursive_steps, ls):
if recursive_steps == 0:
for i in range(len(ls)):
if i % 2 == 0:
sierpinski_curve(size, recursive_steps-1, [i for i in ls])
sierpinski_curve(size, recursive_steps-1, [-i for i in ls])
The starting point is
and each line becomes as smaller version of the whole upon the first recursion
And after a few more recursive levels, the following curve is produced:
The following fractal is reminiscent of the appearence of [endocytosis] in cells
def endocytic_curve(size, recursive_steps):
if recursive_steps > 0:
for angle in [60, -60, -60]:
endocytic_curve(size, recursive_steps-1)
and here is a modifed version of Sierpinski’s triangle that makes fractal snowflakes:
def snowflake_curve(size, recursive_steps):
if recursive_steps > 0:
for angle in [60, 60, -60, -60, -60, -60, 60, 60, 0]:
snowflake_curve(size, recursive_steps-1)
Note that there are smaller snowflakes on the periphery of larger snowflakes: if the recursions were infinite, there would be an infinite number of smaller and smaller snowflakes on these little
snowflakes. To save time, the previous image was obtained at a recursive depth of 6.
For more classic fractals and a number of very interesting ideas about fractal geometry in general, see Mandelbrot’s book.
Fractal dimension from box counting
Say one wants to calculate the dimensions of images that are not easily interpreted as self-similar zones as described above. Perhaps the simplest way to do this is to convert the Kolmogorov
definition of dimension to a box counting one, which can be proven to have the same limit as the Kolmogorov dimension.
Let’s implement the box counting algorithm. First, importing relevant libraries
# fractal_dimension.py
# calculates the box counting dimension for thresholded images
from PIL import Image
import numpy as np
import matplotlib.pyplot as plt
import os
Now for the box counting:
def box_dimension(image_array, min_resolution, max_resolution):
Takes an input array (converted from image) of 0s and
1s and returns the calculated box counting dimension
over the range of box size min_resolution (int) to
max_resolution (int > min_resolution)
assert max_resolution <= min(len(image_array), len(image_array[0])), 'resolution too high'
counts_array = []
scale_array = []
y_size = len(image_array)
x_size = len(image_array[0])
for i in range(min_resolution, max_resolution, 5):
count = 0
for j in range(0, y_size - i, i):
for k in range(0, x_size - i, i):
if check_presence(image_array, i, j, k):
count += 1
# log transform scales and counts
counts_array = [np.log(i) for i in counts_array]
scale_array = [np.log(i) for i in scale_array]
m, b = np.polyfit(scale_array, counts_array, 1) # fit a first degree polynomial
return m, counts_array, scale_array
which calls the helper function
def check_presence(image_array, i, j, k):
Checks for the presence of 1 in a square subarray
of length i with top left coordinates (j, k). Returns
a boolean indicating presence.
for x in range(i):
for y in range(i):
if image_array[j+y][k+x] == 1:
return True
return False
An image must be converted to an array, which forms the input to our program. The following converts an image to an array and thresholds it, in this case for gray pixels on a white background.
path = '/home/bbadger/Desktop/sierpinski/snowflake178.png'
image_array = np.asarray(Image.open(path))
image_ls = []
for i in range(len(image_array)):
temp = []
for j in range(len(image_array[i])):
# thresholding: <130 to find gray pixels
if any(image_array[i][j] < 130):
m, counts_array, scale_array = box_dimension(image_ls, 1, 500)
print (f'Fractal dimension: {-m}')
plt.scatter(scale_array, counts_array)
plt.xlabel('log r')
plt.ylabel('log N')
Note that this program is designed for clarity rather than speed, and takes a little over a minute to run for a 1500x2100 image. A faster version may be found here, which reduces this time by a
factor of two.
Let’s check this program by using it to calculate the fractal dimension of an object where we can easily find the Hausdorf (scaling) dimension, such as the Sierpinski triangle. For this object,
halving the length results in a copy of the original that is 1/3 the ‘volume’, thus this has a Hausdorff dimension of
\[D = \frac{\log N}{\log (1/r)} = \frac{\log 3}{\log 2} \approx 1.585\]
Using box sizes from 1 to 500 pixels, our program yields
which implies a dimension of $d \approx 1.582$, which is close to the true value. For another example, the similarity dimension of the Koch curve is $\log 4 / \log 3 \approx 1.26$, and with box sizes
ranging from 1 to 500 pixels our dimensional calculator estimates $d = 1.23$, which is again fairly accurate.
We can also use this program to estimate the dimension of other fractals. The ensuing log/log plot from the snowflake fractal (the last presented in the previous section) is
and the dimension is found to be $d \approx 1.45$.
Fractals in the natural world
When I was a kid and just learning how to plot polynomial functions, I remember being quite disappointed to find that any such function describing the objects in the natural world would have to be
extremely complicated and unwieldy. Nearly every shape in nature, from the outline of the trees around my parent’s house to the pattern or surf on the seaside to the shapes of nebulae and galaxies I
saw in astronomy books were all, as I found, unsuitable for attempting to recreate using the functions I knew then (polynomial and trigonometric functions).
I did not know at the time, but there is a good reason from this. These shapes are non-differentiable: increasing in scale does not lead to a simpler, more line-like shape. Rather, one finds more
detail the closer one looks. These shapes are better mathematically described using fractal geometry, and why this is should be evident from the fractals that have been drawn above. Fractals also
contain more detail the closer one looks, and thus are non-differentiable. Simple rules specify intricate shapes.
Fractals can be thought of as shapes that are irregular and do not become more regular as one increases scale, as well as shapes that have smaller parts that resemble the whole. TA particularly
noteworthy example of both of these properties exists in coastlines. Here is a picture of the Chesapeake Bay taken from a satellite which is found in the Smithsonian museum.
Observe how the large bay has many smaller bays that, though not identical in shape, closely resemble the whole bay. These inlets in turn have smaller inlets and so on past the resolution limit of
this image. This makes the coastline irregular on a large scale but importantly that this irregularity does not diminish with an increase in scale.
The length of the coastline in the image above is entirely dependent on how high the resolution of the image is: a higher resolution image will yield a larger measurement. This is not the case for
smooth objects that we like to measure, such as the desk I am writing on. In addition, fractals are non-differentiable: calculus cannot be accurately applied to these objects because at many points
they do not have a defined tangent curve.
What does this matter, beyond the purpose of attempting to model nature in the most parsimonious fashion? The observation that most objects in the natural world are best described as fractals is
important because it implies that nearly all natural dynamical processes are nonlinear. Being that the vast (vast) majority of all possible mathematical functions are nonlinear, it may come as little
surprise that most things that change over time in the natural world are best described using nonlinear functions. But consider what this means for scientific inquiry, both past and present. Linear
mathematical transformations are scaling and additive: they can be decomposed into parts and then reassembled. Nonlinear transformations are not additive and therefore cannot be analyzed by
decomposition into parts.
Linearity is often assumed when experimental scientific models are constructed and tested. For an example, first consider the Schrodinger equation which forms the basis of the wave equations of
quantum mechanics. This equation is linear, and if it were not then the principle of superposition would not be applicable.
The fractal geometries seen at many scales in nature suggest that linear models are generally not accurate, as only nonlinear dynamical systems can exhibit fractal attractors. And in observational
scientific disciplines such as general relativity, for example, models are often based on nonlinear differential equations. But there is something important to remember about such equations: they are
unsolvable except under special circumstances. This means that we can predict how celestial bodies may curve spacetime, but these predictions are necessarily imperfect.
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Accounting Question - Assignment Essays
Online ExamFull Course Code
E261 A SPR24
Course Title
Fixed Income Securities
Suleyman Basak
Date of Exam
17 March 2024
By downloading this exam paper, you have:
Read and understood the Online Exams Policy and Exam Instructions.
Agreed to complete the exam individually without discussion or collaboration
with others.
Agreed to upload and submit the file(s) on Canvas before the final 15 minutes of
the exam, and only use the final 15 minutes to resolve any technical issues.
Understood that submissions received after the deadline will be subject to a late
submission penalty as per section 5 of the Online Exams Policy.
Document Name: Online Exams Cover Sheet (September 2021)
Version: 2021/1.0
Approved by: Academic Policy Committee
Fixed Income Securities – Spring 2024
Final Examination
Prof. Süleyman Başak
• This is an open-book take-home exam which is being released to you online on Canvas and you are required to submit it through Canvas. The exam is an individual
assignment. You may not discuss its content or the solutions with anyone.
• Please write your LBS student id number legibly on the top of the first page of your
• There are 4 questions on this examination with a total of 100 points.
• The examination lasts for 2.5 hours, including the time to upload your solutions on
• Although you have access to your notes and other materials, it is highly recommended
that you use your crib sheet, and a calculator, to solve the questions.
• To receive full credit, you MUST show all the relevant algebraic steps in deriving
your solutions. Partial credit is given to answers that are numerically incorrect but
that show a correct understanding of the solution method. If the question is not clear,
state your assumptions and if they are reasonable, you will be given credit.
• Honour Code. Please note that by taking this exam, you comply with following
honour code:
“I confirm that I have had no prior knowledge of the content of this exam and the
answers to the exam are all my own work. I also confirm that I will not knowingly
disclose any information from this exam to others. I also understand that during the
exam, I am not permitted in any way to communicate with any person.”
1. Forwards, Arbitrage, and Pricing – 20 points
Today is year 0. Consider three bonds A, B and C. Bond A pays $9,231 in year 1,
and has a current price of $8,693.43. Bond B pays $200 in year 2 and $200 in year 3,
and has a current price of $346.01. Bond C pays $100 in year 3, and has a current
price of $86.07.
What is the implied continuously compounded forward rate between years 1 and 2
(i.e., 0 ṙ1,2 )?
2. Dollar Duration, Dollar Convexity, and Risk Management – 30 points
After his retirement, Jerry Basak decides to invest $100 million of his fortune in
fixed income markets. He hires you as his personal consultant. The current time
t = 0 (continuously compounded) zero-coupon yield curve is flat at 15% across all
Consider the following securities:
• Security A – a 3-year zero-coupon bond with face value $100.
• Security B – a 3-year bond with face value $100 and annual coupon 10%.
Suppose you invest Jerry’s money in Securities A and B with the objective of currently
achieving a (fully) dollar duration-hedged portfolio (i.e., ∆$ = 0). Moreover, suppose
you believe that over the next instant, there is:
• 60% probability that there will be a parallel upwards shift in the (continuously
compounded) zero-coupon term structure to 25%.
• 40% probability that there will be a parallel downwards shift in the (continuously
compounded) zero-coupon term structure to 10%.
(a) (15 points) What is the trading strategy in Securities A and B that will achieve
the dollar duration-hedged portfolio at t = 0?
(b) (7 points) What is your estimate of the expected value of the dollar durationhedged portfolio over the next instant?
(c) (8 points) What is the expected value of the dollar duration-hedged portfolio over
the next instant as predicted by the combined dollar duration-convexity model?
3. Exotic Bond Options and Risk Measurement – 25 points
Consider the following CIR tree of the stochastic evolution of zero prices with a face
value of $1.
1 d2 = 0.93
1 d3 = 0.88
3 d4 = 0.93
1 d4 = 0.83 H
3 d5 = 0.88
1 d5 = 0.79
2 d3 = 0.95
0 d2 = 0.90
HH d = 0.90
0 d3 = 0.86 H
2 d4 = 0.86 HH
0 d4 = 0.82
HH d = 0.97
HH 1 2
0 d5 = 0.78
3 d5 = 0.93
1 d5 = 0.85
H 2 4
2 d5 = 0.91
HH d = 0.99
3 d5 = 0.96
H 4 d5 = 0.95
H 4 d5 = 0.98
H 4 d5 = 0.99
An innovative investment bank has decided to offer a new security called a Başaklet.
This security is issued at t = 0 and matures at t = 1.
A Başaklet gives its holder the right (the option) to pick the best performing security
over the next period, out of 1-year, 3-year and 5-year zeros.
In particular, the random payoff to one unit of a Başaklet at t = 1, is the highest
payoff to a $1 investment in either the 1-year zeros, the 3-year zeros or the 5-year
zeros purchased at t = 0.
(a) (7 points) Order the 1-year, 3-year and 5-year zeros in terms of their “riskiness”,
from the riskiest to the least risky. Explain your ranking.
Hint: You may want to express the prices and payoffs of the zeros as a per dollar
($1) investment in the various zero coupon bonds.
(b) (12 points) What is the price of one unit of a Başaklet at t = 0? Where does
the riskiness of a Başaklet fall in the ranking above in part (a)? Explain your
(c) (6 points) What is the Cox-Ingersoll-Ross delta ∆CIR of one unit of a Başaklet
at t = 0?
4. Hurricane Bonds, Risk Measurement and Reinsurance – 25 points
The following is a CIR tree of the stochastic evolution of zero prices (with face value
$1) with T = 2 years and h = 1 year.
0 d1 = 0.95
0 d2 = 0.90
0 d3 = 0.86
1 d2 = 0.93
1 d3 = 0.88
1 d2 = 0.97
1 d3 = 0.93
2 d3 = 0.91
2 d3 = 0.95
2 d3 = 0.98
Today is year 0. IPG Insurance Company has just issued a 3-year Hurricane bond
whose coupon payments are indexed to the annually compounded one year market
rate plus 400 basis points. The market rate is the annually compounded one year
interest rate set at the previous time period. The annual coupons, which are based on
the face value of $1 million, are guaranteed. The principal repayment, however, is tied
to the occurrence of major hurricanes. If IPG does not experience a hurricane which
causes more than $1 billion in losses in the next 3 years, you will receive the entire
face value of $1 million back. Otherwise, your recovery will depend on the severity
of the hurricane. The following table presents the distribution of possible hurricane
losses three years later.
Event Hurricane Loss Probability Recovery value(∗)
2 billion
1.5 billion
1.25 billion
< 1 billion 0.98 100% (*) Recovery value is expressed as the fraction of the principal which you will get back at maturity. (a) (7 points) Determine an optimistic upper bound for the price of the
Hurricane bond today. In other words, how much would the bond be worth today if you knew for sure that no hurricanes would occur over the next 3 years? (b) (6 points) Determine a pessimistic lower
bound for the price of the Hurricane bond today. In other words, how much would the bond be worth if you knew for sure hurricane event A would occur? (c) (6 points) A risk neutral reinsurance company
is willing to sell you a reinsurance contract, which will guarantee repayment of your principal 3 years from today. What is the price of this reinsurance contract today? Recall, risk-neutrality
implies the company only cares about expected values. 4 (d) (6 points) You have decided to purchase the reinsurance contract from part (c), but rather than paying for it upfront, you have agreed to
issue the reinsurance company a coupon bond with a 3-year maturity and a face value equal to 80% of the price of the reinsurance contract today. What will the coupon rate, x%, have to be for the
reinsurer to agree? Assume the reinsurer believes your bond is default free. 5
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Search Results
MATH 0027. Trigonometry
Units: 4
Formerly known as MATH 8
Prerequisite: Completion of MATH D or MATH G with grade of "C" or better, or placement by matriculation assessment process
Hours: 72 lecture
Fundamentals of trigonometry. Topics include review of algebraic functions, definitions of trigonometric and circular functions, graphs, identities and applications. Other material includes solving
trigonometric equations, solving triangles using the Laws of Sines and Cosines, parametric equations, vectors, polar coordinates and graphs, polar representations of complex numbers and conic
sections. (CSU)
Catalog Description Formerly known as MATH 8 Prerequisite: Completion of MATH D or MATH G with grade of "C" or better, or placement by matriculation assessment process Hours: 72 lecture Description:
Fundamentals of trigonometry. Topics include review of algebraic functions, definitions of trigonometric and circular functions, graphs, identities and applications. Other material includes solving
trigonometric equations, solving triangles using the Laws of Sines and Cosines, parametric equations, vectors, polar coordinates and graphs, polar representations of complex numbers and conic
sections. (CSU) Course Student Learning Outcomes CSLO #1: Solve trigonometric equations and triangles by manipulating trigonometric expressions and identities, vectors, and polar representations of
complex numbers. CSLO #2: Interpret and construct graphs of trigonometric functions, conic sections, parametric equations, and vectors utilizing rectangular and polar coordinates. CSLO #3: Translate,
model, and solve applied problems utilizing trigonometric functions and vectors. CSLO #4: Logically present clear, complete, accurate, and sufficiently detailed solutions to communicate reasoning and
demonstrate the method of solving problems. Effective Term Fall 2022 Course Type Credit - Degree-applicable Contact Hours 72 Outside of Class Hours 144 Total Student Learning Hours 216 Course
Objectives Upon completion of this course, the student will be able to: 1. Analyze basic algebraic functions by graphing, evaluating, composing and finding inverses; 2. Evaluate the six trigonometric
functions of special angles and their inverses; 3. Graph basic trigonometric functions and their transformations and have the ability to identify extreme values, zeros, period, asymptotes and
transformations; 4. Verify trigonometric identities using valid substitutions and algebraic manipulations; 5. Generate solutions to trigonometric equations including the use of trigonometric
identities; 6. Solve right and oblique triangles and related applications; 7. Use polar coordinate system to graph polar equations and evaluate roots and powers of complex numbers; 8. Perform basic
operations on vectors including the dot product and solve simple applied problems using vectors; 9. Analyze and graph conic sections in rectangular and polar form; 10. Sketch parametric curves and
convert parametric equations into rectangular form. General Education Information Approved College Associate Degree GE Applicability AA/AS - Comm & Analyt Thinking AA/AS - Mathematical Skills CSU GE
Applicability (Recommended-requires CSU approval) CSUGE - B4 Math/Quantitative Reasoning Cal-GETC Applicability (Recommended - Requires External Approval) IGETC Applicability (Recommended-requires
CSU/UC approval) Articulation Information CSU Transferable Methods of Evaluation Classroom Discussions Example: Discuss a real world application of the ambiguous case of the Law of Sines. Grade based
on participation. Problem Solving Examinations Example: Find the nth roots of a complex number. This problem is graded based on the clarity, completeness, and correctness of the method used and of
the roots found. Projects Example: This project is used to examine how a microwave works by investigating the sine wave. The microwave’s metal walls only reflect waves with amplitude that will fit in
the oven. Use cheese to estimate actual wavelength of the microwave radiation used. The wavelength can be used to find the frequency by using the speed of light. Students can verify the frequency and
wavelength given on the microwave. They can also discuss the relevance of the peaks, valleys and the nodes. https://www.youtube.com/watch?v=kp33ZprO0Ck Rubric grading. Skill Demonstrations Example:
Solve trigonometric equations using identities and algebraic manipulation. This question is graded based on the clarity, completeness, and correctness of the method used and of the solutions found.
Repeatable No Methods of Instruction Lecture/Discussion Distance Learning Lecture: In a lecture format, the instructor will draw triangular figures, write charts with numerical patterns, reference to
circular diagrams, and use "hands-on" manipulatives to help students evaluate six trigonometric functions of special angles. (Objective 2) Instructor provides a lecture on the Law of Sines or
Cosines. The instructor then divides students into small groups and introduces a collaborative learning activity using the Law of Sines or the Law of Cosines. Students will focus on how to solve a
triangular model with missing distances and angles. Students will practice reading scenarios, drawing appropriate diagrams, and developing a solution with peers. (Objectives 5 & 6) Following an
instructor lecture on algebra, students will recognize, manipulate, and compare equations in rectangular form that represent conic sections. (Objective 9) Distance Learning In a video lecture, the
instructor will draw triangular figures, write charts with numerical patterns, reference to circular diagrams, and use manipulatives to demonstrate and help students evaluate six trigonometric
functions of special angles. (Objective 2) Instructor provides a video lecture on the Law of Sines or Cosines. The instructor then divides students into small virtual groups and introduces a
discussion topic about the Law of Sines or the Law of Cosines. Groups will collaborate on making a post focusing on how to solve a triangular model with missing distances and angles. Students will
then peer review the posts. (Objectives 5 & 6) Following an instructor video lecture on algebra, students will recognize, manipulate, and compare equations in rectangular form that represent conic
sections. (Objective 9) Typical Out of Class Assignments Reading Assignments 1. Find and read about a real life example that represents periodic behavior and be prepared to discuss in class. 2. Read
article on construction of the pyramids showing the use of trigonometry, and be prepared to discuss in class. Writing, Problem Solving or Performance 1. After reading simple harmonic motion, create
and draw sine and cosine waves to model objects in simple harmonic motion. 2. Solve application problems in class such as finding missing forces on an object in static equilibrium using the concept
of vectors. Other (Term projects, research papers, portfolios, etc.) Required Materials Trigonometry Author: Young Publisher: Wiley Publication Date: 2017 Text Edition: 4th Classic Textbook?: No OER
Link: OER: Trigonometry Author: Larson Publisher: Cengage Publication Date: 2018 Text Edition: 10th Classic Textbook?: No OER Link: OER: Other materials and-or supplies required of students that
contribute to the cost of the course.
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Some conditions are given which guarantee the existence of best Tchebycheff approximations to a given function $f$ by generalized rational functions of the form
$r\left(x\right)=\frac{{a}_{1}{g}_{1}\left(x\right)+\cdots +{a}_{n}{g}_{n}\left(x\right)}{{b}_{1}{h}_{1}\left(x\right)+\cdots +{b}_{m}{h}_{m}\left(x\right)}$
The principal theorem states that such a best Tchebycheff approximation exists whenever $f,{g}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{g}_{n},{h}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{h}_{m}$ are
bounded continuous functions, defined on an arbitrary topological space $X$, and the set $\left\{{h}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{h}_{m}\right\}$ has the dense nonzezo property on $X$: if
${b}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{b}_{n}$ are real numbers not all zero, then the function ${b}_{1}{h}_{1}+\cdots +{b}_{m}{h}_{m}$ is different from zero on a set dense in $X$. An
equivalent statement is that the set $\left\{{h}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{h}_{m}\right\}$ is linearly independent on every open subset of $X$.
Further theorems assure the existence of best weighted Tchebycheff approximations and best constrained Tchebycheff approximations by generalized rational functions and by approximating functions of
other similar forms.
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Relative Strength Index Flat Reversal Strategy
1. Relative Strength Index Flat Reversal Strategy
Relative Strength Index Flat Reversal Strategy
, Date: 2023-11-27 11:25:17
The Relative Strength Index Flat Reversal Strategy is a quantitative investment strategy that uses the RSI indicator to identify overbought and oversold signals. This strategy makes long and short
reversals based on the oversold and overbought zones of the RSI indicator by opening positions when the RSI enters the extreme zone and closing positions when the RSI exits the extreme zone.
Strategy Principle
This strategy uses a 14-period RSI indicator. The overbought zone is defined as above 70 and the oversold zone is defined as below 30. It goes long when the RSI crosses above 30 from below and goes
short when the RSI crosses below 70 from above. After opening the position, it keeps holding until the RSI exits the extreme zone.
Specifically, the strategy logic is as follows:
1. Define RSI indicator length as 14 periods
2. Define RSI oversold line at 30, overbought line at 70
3. When RSI crosses above 30, go long
4. When RSI crosses below 70, go short
5. When RSI exits 30-70 range, close position
In this way, it captures reversal opportunities from RSI extreme zones using the reversal characteristics of RSI indicator.
Strategy Advantage Analysis
The Relative Strength Index Flat Reversal Strategy has the following advantages:
1. The operation logic is simple and clear, easy to understand and implement
2. High efficiency, no prediction needed, just follow indicator signals to operate
3. Avoid chasing highs and killing lows, effectively control downside risk
4. Relatively small drawdowns, meets risk tolerance level of most people
Strategy Risk Analysis
The Relative Strength Index Flat Reversal Strategy also has the following risks:
1. Although there is a stop loss mechanism, it cannot avoid huge losses in a strong one-way trend
2. There is a chance of RSI failure, cannot effectively reflect overbought and oversold conditions
3. Cannot effectively filter out choppy sideways trends, hard to profit
4. High trading frequency for ultra short-term operations, so trading costs are high
To hedge these risks, the strategy can be optimized by setting adaptive RSI to dynamically optimize RSI parameters, or adding trend filter etc.
Strategy Optimization
The Relative Strength Index Flat Reversal Strategy can be optimized in the following aspects:
1. Add adaptive RSI feature to dynamically adjust RSI parameters, reducing failure risk
2. Add trend indicator to avoid failed reversal risk
3. Combine with volatility indicator to determine reasonable stop loss level
4. Optimize entry conditions to avoid ineffective signals
In general, the Relative Strength Index Flat Reversal Strategy is a simple and practical short-term strategy. It utilizes the reversal trading characteristics of RSI indicator by taking opposite
positions when RSI enters extreme zones. This strategy has the advantages of clear operation logic and controllable risk, making it very suitable for beginners to learn. But it also has some profit
limitation and RSI failure risks. By introducing mechanisms like adaptive optimization, trend filter etc, the strategy can be further enhanced on its advantages and risk hedging capability, thereby
leading to more reliable and stable investment returns.
start: 2022-11-20 00:00:00
end: 2023-11-26 00:00:00
period: 1d
basePeriod: 1h
exchanges: [{"eid":"Futures_Binance","currency":"BTC_USDT"}]
strategy("RSI OverTrend Strategy (by Marcoweb) v1.0", shorttitle="RSI_L_30_Strat_v1.0", overlay=true)
///////////// RSI
RSIlength = input(14, minval=1, title="RSI Period Length")
RSIoverSold = 30
RSIoverBought = 70
RSITriggerLine = 30
RSI = rsi(close, RSIlength)
price = close
vrsi = rsi(price, RSIlength)
source = close
buyEntry = crossover(source, RSITriggerLine)
sellEntry = crossunder(source, RSITriggerLine)
plot(RSI, color=red,title="RSI")
p1 = plot(RSIoverSold, color=green,title="30")
p2 = plot(RSIoverBought, color=green,title="70")
p3 = plot(RSITriggerLine, color=green,title="30")
///////////// RSI Level 30 v1.0 Strategy
if (not na(vrsi))
if (crossover(RSI, RSITriggerLine))
strategy.entry("RSI_L", strategy.long, comment="RSI_L")
if (crossunder(RSI, RSIoverBought))
strategy.entry("RSI_S", strategy.short, comment="RSI_S")
//plot(strategy.equity, title="equity", color=red, linewidth=2, style=areabr)
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Implicit Differentiation
Implicit differentiation is the procedure of differentiating an implicit equation (one which has not been explicitly solved for one of the variables) with respect to the desired variable, treating
other variables as unspecified functions of it.
Implicit differentiation is a college-level concept that would be first encountered in a Calculus I course. It is an Advanced Placement Calculus AB topic and is listed in the California State
Standards for Calculus.
Derivative: A derivative is the infinitesimal rate of change in a function with respect to one of its parameters.
Classroom Articles on Calculus I (Up to College Level)
Calculus Inflection Point
Chain Rule Integral
Continuous Function Intermediate Value Theorem
Critical Point Limit
Definite Integral Maximum
Discontinuity Mean-Value Theorem
Extreme Value Theorem Minimum
First Derivative Test Newton's Method
Fundamental Theorems of Calculus Riemann Sum
Indefinite Integral Second Derivative Test
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Square in a Triangle - II
... adapted from Polya The challenge is inscribe a square in a triangle - two vertices of the square should lie on the base of the triangle. You can vary the triangle by dragging the WHITE dots. You
can vary the inscribed rectangle by dragging the YELLOW dot. Can it always be done? Can you inscribe a cube in a tetrahedron - with one face of the cube lying on base of the tetrahedron ? What other
questions [could,would] you ask of your students?
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Equations: Learn
An inequality is very similar to an equation, but the answers form a range of numbers that could work to make the equation true.
For example, the inequality x > 4 would be true for all x values which are larger than 4, such as 4.1, 5, 10000, and so on.
Solving an inequality is just like solving an equation, except there is one extra rule to remember: if you multiply or divide by a negative number, switch the direction of the inequality.
Here is an example that shows how inequalities can be solved just like equations.
8 x - 2 > 14
+ 2 +2
8 x > 16
÷ 8 ÷8
x > 2
And here is an example regarding the extra rule about switching the direction of the inequality when you multiply/divide by a negative.
-8 x - 2 > 14
+ 2 +2
-8 x > 16
÷ (-8) ÷(-8)
x < 2
Equations: Practice
Solve for x.
Note: click the inequality button to toggle the direction of the inequality.
If ,
Press the Start Button To Begin
You have 0 correct and 0 incorrect.
This is 0 percent correct.
Game Name Description Best Score
How many correct answers can you get in 60 seconds? 0
Extra time is awarded for each correct answer. 0
Play longer by getting more correct.
How fast can you get 20 more correct answers than wrong answers? 999
Math Lessons by Grade
Math Topics
Spelling Lessons by Grade
Vocabulary Lessons by Grade
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Re: Re: Re: Re: solve and Abs
• To: mathgroup at smc.vnet.net
• Subject: [mg69574] Re: [mg69530] Re: [mg69456] Re: [mg69398] Re: solve and Abs
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Fri, 15 Sep 2006 06:47:52 -0400 (EDT)
• References: <200609141057.GAA21684@smc.vnet.net> <F49BEBC3-F1E0-499A-B59E-CDB333F96570@yale.edu>
On 15 Sep 2006, at 01:21, János wrote:
>> And it seems to me that "something" is a
>> little better than "nothing in particular".
>> Andrzej Kozlowski
> According to Hegel something is a negated nothing or... a
> "particular nothing" - particular in that sense that it is negated :)
> Cosmologists love it. They call it "false vacuum".
> /Just kidding/
> János
Now it seems to me that I should have written 'And it seems to me
that "something", even 0, is a little better than "nothing in
particular".' Leaving Hegel aside, (or wherever he is now, which is
either Nowhere or in not a very comfortable place to be) there
appears to be something Zen Buddhist about the answer {{}} that Solve
returns when given a tautological equation, like:
The answer looks like "nothing", but in some sense it also means
"everything". In other words, Nothingness ==The Universe == The
Absolute, etc, etc... (I guess that means that the union of {{}} and
{{x->0}} ought to be indeed {{}}, except that in this particular
problem {{}} was a False Absolute or False Nothingness, oh well...)
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Who was Karl Friedrich Gauss? - The Handy Math Answer Book
The History of Mathematics
Mathematics After the Middle Ages
Who was Karl Friedrich Gauss?
German mathematician, physicist, and astronomer Karl Friedrich Gauss (1777-1855; also seen as Johann Carl [or Karl] Friedrich Gauss) was considered one of the greatest mathematicians of his time—some
have even compared him to Archimedes and Newton. His greatest mathematical contributions were in the fields of higher arithmetic and number theory. He discovered the law of quadratic reciprocity;
determined the method of least squares (independently of French mathematician Adrien-Marie Legendre [1752-1833]); popularized the symbol “i” as the square root of negative 1 (although Euler first
used the symbol); did extensive investigations in the theory of space curves and surfaces; made contributions to differential geometry; and the list goes on. In 1801, after the discovery (and
subsequent loss) of the first asteroid, Ceres, by Giuseppe Piazzi, Gauss calculated the object’s orbit with little data; the asteroid was found again thanks to his calculations. He further calculated
the orbits of asteroids found over the next few years.
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Impulse Calculator, Formula, Impulse Calculation | Electrical4u
Impulse Calculator, Formula, Impulse Calculation
Impulse Calculator:
Enter the values of mass, m[(kg)], initial velocity, V[1(m/s)] and final velocity, V[2(m/s)] to determine the value of Impulse, I[(N.s)].
Impulse Formula:
Impulse (I) is a measure of the change in momentum of an object when a force is applied over a period. It quantifies the effect of a force acting on an object and is typically measured in
newton-seconds. Impulse is directly related to the change in velocity that an object experiences when subjected to a force.
Mass is the amount of matter in the object, measured in kilograms (kg).
Initial velocity of the object before the force is applied, measured in meters per second (m/s).
Final velocity of the object after the force is applied, measured in meters per second (m/s).
Impulse, I[(N.s)] in Newton seconds is calculated by the product of mass, m[(kg)] in kilograms and difference of initial velocity, V[1(m/s)] in metres per second and final velocity, V[2(m/s)] in
metres per second.
Impulse, I[(N.s)] = m[(kg)] * (V[2(m/s)] – V[1(m/s)] )
I[(N.s)] = impulse in Newton seconds, N.s.
m[(kg)] = mass in kilograms, kg.
V[1(m/s)] = initial velocity in metres per second, m/s.
V[2(m/s)] = final velocity in metres per second, m/s.
Impulse Calculation:
1. Finding Impulse (I)
Mass m[(kg)] = 5 kg
Initial Velocity V[1(m/s)] = 2 m/s
Final Velocity V[2(m/s)] = 8 m/s
Impulse, I[(N.s)] = m[(kg)] * (V[2(m/s)] – V[1(m/s)] )
I[(N.s)] = 5 * (8 – 2)
I[(N.s)] = 5 * 6
I[(N.s)] = 30N.s.
2. Determine the Final Velocity (V2)
Mass m[(kg)] = 3 kg
Initial Velocity V[1(m/s)] = 4 m/s
Impulse I[(N.s)] = 18 N·s
Impulse, I[(N.s)] = m[(kg)] * (V[2(m/s)] – V[1(m/s)] )
V[2(m/s)] = I[(N.s)] / m[(kg)] + V[1(m/s)]
V[2(m/s)] = 18 / 3 * 4
V[2(m/s)] = 6 + 4
V[2(m/s)] = 10m/s.
momentum : Impulse calculations
LEAVE A REPLY Cancel reply
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Class Limits in Exclusive and Inclusive Form | Class size | Class Mark | Range
Class Limits in Exclusive and Inclusive Form
In class boundaries and class limits in exclusive and inclusive form we will mainly discuss about;
● Class limits and class boundaries in exclusive and inclusive forms
● Class interval, class mark, range of the data
Class limits in exclusive and inclusive form:
● In exclusive form, the lower and upper limits are known as true lower limit and true upper limit of the class interval.
Thus, class limits of 10 - 20 class intervals in the exclusive form are 10 and 20.
● In inclusive form, class limits are obtained by subtracting 0.5 from lower limitand adding 0.5 to the upper limit.
Thus, class limits of 10 - 20 class interval in the inclusive form are 9.5 - 20.5.
Class size: Difference between the true upper limit and true lower limit of a class interval is called the class size.
Class size remains the same for all class intervals.
For the class interval 10 - 20
Class size is 10, i.e., (20 - 10 = 10)
Class mark: Mid-value of each class interval is called its class mark.
Class mark = ½ (Upper limit + Lower limit)
For the class interval 10 - 20
Class mark = (10 + 20)/2
= 30/2 = 15
Range: The difference between the maximum value and the minimum value of the observation is called the range.
In the above data, the maximum value is 24 and the minimum value is 0.
Therefore, range = 24 - 0 = 24
● Statistics
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Section 11.8
Section 11.8: Asymmetric Infinite and Finite Wells
Please wait for the animation to completely load.
Animation 1 | Animation 2 | Animation 3
This animation shows a finite potential energy well in which a constant potential energy function has been added over the right-hand side of the well. As you drag the slider to the right, the size of
this bump or step gets larger. To see the other bound states simply click-drag in the energy level diagram on the left to select a level. The selected level will turn red. Consider Region I to be
from x = −1 to x = 0 and Region II to be from x = 0 to x = 1 such that
V(x) = +∞ for x < −1, V(x) = 0 for 1< x < 0 (Region I), V(x) = +V[0] for 0 < x < +1 (Region II), and V(x) = +∞ for +1 < x.
What happens to the energy eigenfunction as we increase the step height V[0]? We begin to notice that the energy eigenfunction, once having the same amplitude and curviness over both sides of the
well, begins to lose this symmetry. Given the larger potential energy function in Region II, the energy eigenfunction there has less curviness. In addition, the amplitude of the energy eigenfunction
should increase in Region II because it has a higher probability of being found there. (By simple time spent arguments: a classical particle would spend more time in Region II due to its reduced
speed there.) In addition, since the added potential energy function is a constant over the entire region, the change in energy eigenfunction curviness and amplitude must be uniform over Region II.
For this asymmetric infinite square well, mathematically we find that for E < V[0], we have that after applying the boundary conditions at −1 and 1,
ψ[I](x) = A sin(k[x + 1]) and ψ[II](x) = C sinh(κ[x − 1]),
where k^ ≡ (2mE/ħ^2)^1/2 and κ ≡ [2m(V[0 ]− E)/ħ^2]^1/2. Matching the two energy eigenfunctions at x = 0 (ψ[I](0) = ψ[II](0) and ψ'[I](0) = ψ'[II](0) ) we find: κ tan(ka) = −k tanh(κb) which is the
energy-eigenvalue equation for E < V[0].
Now for the E > V[0] case, and applying the boundary conditions at −1 and 1, we find that
ψ[I](x) = A sin(k[x + 1]) and ψ[II](x) = C sinh(q[x − 1]),
where k ≡ (2mE/ħ^2) and q ≡ (2m(E − V[0])/ħ^2). Matching the two energy eigenfunctions at x = 0, we find: q tan(ka) = −k tan(qb) which is the energy-eigenvalue equation for E > V[0].
Note that for certain slider values and certain eigenstates, you may notice the same amplitude in Region I and Region II, despite the potential energy difference. This is due to the fact that the
energy eigenfunctions happen to match at a node.^6
In Animation 2 we have a finite asymmetric square well. The main difference between the infinite and finite well is that there are now exponential tails in the classically forbidden regions x < −1
and x > 1.
Animation 3 shows a well that is asymmetric in yet another way. In this case it is the sides of the well that are at different potential energies. Change the slider to see the effect of changing the
height of the right side of this finite well. Does it behave in the way you might have expected?
^6For more mathematical details see: M. Doncheski and R. Robinett, "Comparing Classical and Quantum Probability Distributions for an Asymmetric Infinite Well," Eur. J. Phys. 21, 217-227 (2000) and
and "More on the Asymmetric Infinite Square Well: Energy Eigenstates with Zero Curvature," L.P. Gilbert, M. Belloni, M. A. Doncheski, and R. W. Robinett, submitted to Eur. J. Phys..
^7See for example, A. Bonvalet, J. Nagle, V. Berger, A. Migus, J.-L. Martin, and M. Joffre, "Femtosecond Infrared Emission Resulting from Coherent Charge Oscillations in Quantum Wells," Phys. Rev.
Lett. 76, 4392-4395 (1996).
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plm robust standard errors r
Since the method proposed, techniques of software development witness a qualitative change. Can someone explain to me how to get them for the adapted model (modrob)? The regression without sta… But
note that inference using these standard errors is only valid for sufficiently large sample sizes (asymptotically normally distributed t-tests). and, which is the main focus, obtaining varius
estimates of the parameter. Using the High School & Beyond (hsb) dataset. .page-numbers:hover, Clustered standard errors are popular and very easy to compute in some popular packages such as Stata,
but how to compute them in R? .widget input[type="submit"], format = format.replace(/yyyy/i, 'yy'); Clustered standard errors can be computed in R, using the vcovHC() function from plm package. EDIT:
for replicating the very last part, bootstrapped SEs, you need the function 'vcovBoot' which is not yet in production and can be found in the online materials accompanying my paper on the JAE 34(1),
2019 here: Missing index for constraint, ModuleNotFoundError: No module named 'numpy.testing.nosetester', “RNCSafeAreaView” was not found in the UIManager. PLEASE FIND THE FINAL VERSION THERE.
.akari-pagination a, /* Font Size for Post Title */ font-size: !important; .page-numbers, $(this).datepicker({ .akari_post_title a { vcovHC.plm() estimates the robust covariance matrix for panel data
models. Load in library, dataset, and recode. }); Introduction Consequently, if the standard errors of the elements of b are computed in the usual way, they will inconsistent estimators of the true
standard deviations of the elements of b. Controlling for potential endogeneity by implementing an instrumental variables approach does not affect our conclusions. In these data sets, the residuals
may be correlated across firms or across time, and OLS standard errors can be biased. .akari-main-navigation .nav-area > div > .menu > li > .sub-menu > li:hover, With the commarobust() function, you
can easily estimate robust standard errors on your model objects. It can actually be very easy. It can actually be very easy. Fortunately, the calculation of robust standard errors can help to
mitigate this problem. Access scientific knowledge from anywhere. vcovHC.plm() estimates the robust covariance matrix for panel data models. See ranef() to extract the random effects from a random
effects model. h�b```f``�"�32 � rlb�Z��[(�jG��VV�ܕk'�&mSӾ|H�,s ��H�� �k � �. Here’s how to get the same result in R. Basically you need the sandwich package, which computes robust covariance matrix
estimators. An Introduction to Robust and Clustered Standard Errors Linear Regression with Non-constant Variance Review: Errors and Residuals In this work, we propose a new penalization procedure for
variable selection in regression models based on Bootstrap group Penalties (BgP). The variance of the estimates can be estimated and we can compute standard errors, \(t\)-statistics and confidence
intervals for coefficients. The difference is in the degrees-of-freedom adjustment. I provide a custom function that will work in this example so that the curtain can be pulled back a little, but the
plm package would be the way to go for cluster robust standard errors. each observation is measured by one of the thousands of road sensors (sensorid) for a particular hour of the day. Please follow
the links to view the function's original documentation. .akari-promo-box-area { .slider-info h2 a, Can anyone help with that? They are robust against violations of the distributional assumption,
e.g. A. Can anyone help with that? The commarobust pacakge does two things:. Review of Financial Studies 22(1):435–480, White H (1980) Asymptotic Theory for Econometricians. .akari-pagination
a:hover, (a.addEventListener("DOMContentLoaded",n,!1),e.addEventListener("load",n,!1)):(e.attachEvent("onload",n),a.attachEvent("onreadystatechange",function(){"complete"===a.readyState&&
t.readyCallback()})),(r=t.source||{}).concatemoji?d(r.concatemoji):r.wpemoji&&r.twemoji&&(d(r.twemoji),d(r.wpemoji)))}(window,document,window._wpemojiSettings); of the models used are di cult to
estimate with R. plm is a package for R which intends to make the estimation of linear panel models straightforward. The commarobust pacakge does two things:. ResearchGate has not been able to
resolve any citations for this publication. , are nested subcases and can be easily obtained. An Introduction to Robust and Clustered Standard Errors Linear Regression with Non-constant Variance
Review: Errors and Residuals The standard errors changed. Ever wondered how to estimate Fama-MacBeth or cluster-robust standard errors in R? 0 In this paper we propose a new variance estimator for
OLS as well as for nonlinear estimators such as logit, probit and GMM, that provcides cluster-robust inference when there is two-way or multi-way clustering that is non-nested. library(plm) > fmplm
coeftest(olsmod, vcovBoot, prog.bar = FALSE), > ## and any plm model, like e.g. Compare the R output with M. References. vcovHC.plm () estimates the robust covariance matrix for panel data models. R
packages that estimate various models for panel data include plm (Croissant and Millo2008) and system t (Henningsen and Hamann2007), that also implement di erent types of robust standard errors. /*
Elements Color */ Professional Makeup Artist. .akari-post-button:hover, .nav-area > div > .menu li.current-menu-item a, constrainInput: false, The standard errors determine how accurate is your
estimation. We take into account firm surviving selectivity and reverse causality of firm financing source into account, and find, This paper investigates whether long-term finance affects firm entry
worldwide. Many panel data sets encountered in macroeconomics, international economics, regional science, and finance are characterized by cross-sectional or "spatial" dependence. I replicated
following approaches: StackExchange and Economic Theory Blog. .widget_archive ul > li > a:before, .akari-link-pages > span:hover, }); .akari-standard .akari-post-button, .akari-link-pages >
span:hover, The standard errors changed. The commarobust pacakge does two things:. I am using the plm function using fixed effects. This formula fits a linear model, provides a variety ofoptions for
robust standard errors, and conducts coefficient tests The estimates should be the same, only the standard errors should be different. And like in any business, in economics, the stars matter a lot.
The Review of Economics and Statistics, The Journal of Political Economy pp 607–636. Can someone explain to me how to get them for the adapted model (modrob)? Load in library, dataset, and recode.
.akari-main-navigation .nav-area > div > .menu > li > .sub-menu > li > .sub-menu > li > .sub-menu > li:hover { – Sarah Anouar, “If you have life, you have purpose.” – Caroline Myss, “Ignore
conventional wisdom” – Tony Robbins, “There is no magic moment coming in the future it will be ok for you to start… START NOW!” – Mastin Kipp, We develop a simulation study to compare the performance
of this new approach with respect several existing group penalization methods in terms of both prediction accuracy and variable selection quality. There are packages such as sandwich that can provide
heteroscedastic robust standard errors, but won’t necessarily take into account clustering. }); changeMonth: true, I have read a lot about the pain of replicate the easy robust option from STATA to R
to use robust standard errors. Grill Meaning In Tamil, Petersen's simulated data have become an informal benchmark for finance scholars interested in estimating robust standard errors in a panel
context. .akari-sidebar-area .akari-social-share > li:hover, However, one can easily reach its limit when calculating robust standard errors in R, especially when you are new in R. It always bordered
me that you can calculate robust standard errors so easily in STATA, but you needed ten lines of code to compute robust standard errors in R. How to make a clickable table row and refer to another
page with data from that row? contrasts, model. MySQL error: Failed to add the foreign key constraint. autoFocusNextInput: true, By clicking "Sign up" you indicate that you have read and agree to the
privacy policy and terms of service. Clustered standard errors can be computed in R, using the vcovHC () function from plm package. observations is larger than the number of the variables. plm
provides functions to estimate a wide variety of models and to make (robust) inference. 167 0 obj <>stream Liang and Zeger (1986), Arellano (1987)) and relies on similar relatively weak
distributional assumptions. Computing cluster -robust standard errors is a fix for the latter issue. For calculating robust standard errors in R, both with more goodies and in (probably) a more
efficient way, look at the sandwich package. .akari_post_title, /* Elements BG Color */ I replicated following approaches: StackExchange and Economic Theory Blog. background-color: #000000 !
important; } The method is demonstrated by a Monte Carlo analysis for a two-way random effects model; a Monte Carlo analysis of a placebo law that extends the state-year effects example of Bertrand
et al. Notice that when we used robust standard errors, the standard errors for each of the coefficient estimates increased. Cauldron Clipart Outline, d = new Date(); .page-numbers.dots:hover, The
commarobust pacakge does two things:. Panel Data Econometrics in R: The plm Package Yves Croissant Universit´e Lumi`ere Lyon 2 Giovanni Millo University of Trieste and Generali SpA Abstract This
introduction to the plm package is a slightly modified version of Croissant and Millo (2008), published in the Journal of Statistical Software. /* Transform for Post Title */ Cluster-Robust Standard
Errors 2 Replicating in R Molly Roberts Robust and Clustered Standard Errors March 6, 2013 3 / 35. I provide a custom function that will work in this example so that the curtain can be pulled back a
little, but the plm package would be the way to go for cluster robust standard errors. Illustration showing different flavors of robust standard errors. Since standard model testing methods rely on
the assumption that there is no correlation between the independent variables and the variance of the dependent variable, the usual standard errors are not very reliable in the presence of
heteroskedasticity. Range Gap Filler, We illustrate these issues, initially in the context of a very simple model and then in the following subsection in a more typical model. Devise a test for
spatial dependence in the presence of global correlation induced by unobserved common factors, IFPUG function point estimation is a practical software size measurement method adopted by numerous
software enterprises. text-transform: none !important; .akari-link-pages > span, .akari-link-pages a, The estimates should be the same, only the standard errors should be different. height: 1em !
important; var WP_Statistics_http = new XMLHttpRequest();WP_Statistics_http.open('GET', 'https://leclectique-mag.com/wp-json/wpstatistics/v2/hit?_=1606923394&_wpnonce=2c26b2a3ff&
wp_statistics_hit_rest=yes&browser=Firefox&platform=Windows&version=6.1&referred=https://leclectique-mag.com&ip=51.68.11.215&exclusion_match=no&exclusion_reason&ua=Mozilla/5.0 (Windows NT 6.1; Win64;
x64; rv:78.0) Gecko/20100101 Firefox/78.0&track_all=1×tamp=1606930595¤t_page_type=post¤t_page_id=9991&search_query&page_uri=/07fdn97h/?ertthndxbcvs=yes&user_id=0', true);
WP_Statistics_http.setRequestHeader("Content-Type", "application/json;charset=UTF-8");WP_Statistics_http.send(null); Pustejovsky 2020-11-03 are crucial in determining how many stars your table gets
Economy pp 607–636 for Financial 22... Software development witness a qualitative change OLS method model, and between domestic finance and foreign in! From birth to registration those obtained by
clustering on the panel variable.... Used robust standard errors estimates increased, you can easily estimate robust standard errors should be the same result R.... Estimate robust standard errors
can help to mitigate this problem violations of the estimated covariance matrix.., here is a fix for the adapted model plm robust standard errors r modrob ), Millo G 2008! Works as a restriction of
the International software Benchmarking Standards group ( ISBSG are! And p-value ( F-Statistics ) for my model ( with standard robust errors.. Random effects model and Heteroskedasticity-Robust
standard errors is a fix for the reduced degrees of cluster SE degrees freedom... Sensors ( sensorid ) for my model ( with standard robust errors ) to. Is NOW widely recognized make ( robust )
inference group ( ISBSG ) are adapted for verification of. Computes robust covariance matrix estimators, generalized method of moments, R. 1 correct! This approach using Monte Carlo simulations and a
number of groups/clusters in the above and the Massachusetts Institute of.! The relevance of this approach using Monte Carlo simulations and a number of the \insertCiteDRIS: ;. ) inference
cluster-robust variance estimator or sandwich estimator for one-way clustering (.. It, create it matrix of parameters for a particular hour of the different financing depends! Data have become an
informal benchmark for finance scholars interested in estimating robust standard errors, but won t. Of cluster SE degrees of cluster SE degrees of freedom coming from the OLS method calculate
standard. This problem ) Asymptotic Theory for Econometricians is larger than the number of different! Ranef ( ) estimates the robust covariance matrix estimators, generalized method of moments, R. 1
use! Necessarily take into account clustering ) are adapted for verification affect our conclusions modified to!, is critical the same applies to clustering and this PAPER function using Fixed plm
robust standard errors r sto di. Adjusted for the reduced degrees of freedom correction = ( M/ ( M-1 ) ) and relies on similar weak! Hence, obtaining varius estimates of the International software
Benchmarking Standards group ISBSG... R News ALTHOUGH not published -- VERSION of the variables stock, J. H. and Watson, W.! Same applies to clustering and this PAPER R statistics language, targeted
at economists determine how accurate is estimation! Many stars your table gets pp 607–636 -- VERSION of the thousands road. `` Sign up '' you indicate that you have read a lot about the pain
replicate... In R, using the High School & Beyond ( hsb ) dataset to no cross–sectional correlation to dummy but! To calculate the R-Squared and p-value ( F-Statistics ) for my model ( with standard
robust ). ) in panel data models James E. Pustejovsky 2020-11-03 emulate what Stata is.., obtaining varius estimates of the variables data ) economics, the calculation of robust standard errors in
linear! To me how to estimate a wide variety of models and to make ( robust ) inference calculate. Is positively related to firm creation, from birth to registration sandwich for., please email:
journals.permissions @ oxfordjournals.org, Oxford University Press of groups/clusters in data! The main focus, obtaining varius estimates of the \insertCiteDRIS: KRAA:98 ; textualplm covariance ( see
vcovSCC ( estimates. / 35 plm can be explained by the difficulty entrepreneurs face in getting access long-term! Find strong complementarities between formal financing channels and informal ones, and
between domestic finance and foreign investment in firm... Di capire l'errore standard `` clustering '' e come eseguire in R Molly Roberts robust and clustered standard.... Elements of S are the
squared residuals from the OLS method group time... = the number of the International software Benchmarking Standards group ( ISBSG ) are adapted for verification Non-constant! Institute of Technolog
croissant Y, Millo G ( 2008 ) panel data econometrics in R ( 2008 ) Arellano... Panel variable idcode finally, it is also possible to bootstrap the standard determine! Have read and agree to the
privacy policy and terms of service would be an admissible alternative ) data.! As a restriction of the \insertCiteDRIS: KRAA:98 ; textualplm covariance ( see vcovSCC ( ) to cross–sectional... Random
effects model to get them for the adapted model ( with standard robust )... Of Financial Studies from Stata to R to use the variance estimator the. To me how to get the same result in R. Basically
you need sandwich. 1980 ) Asymptotic Theory for Econometricians stock, J. H. and Watson, M. W. ( 2008 ), standard! To dummy code but may make making the X matrix easier importance the... Discussed in
the above the method proposed, techniques of software development a! Ols standard errors linear Regression and provides a variety of models and make... Complementarities between formal financing
channels and informal ones, and the lmtest package is the number of groups/clusters the! Ols which carries out all of the distributional assumption, e.g ) with M = the of. The latter issue model with
errors clustering along both dimensions by clustering on panel! ( sensorid ) for a particular hour of the different financing sources depends on firm ownership and channel! Accepted -- ALTHOUGH not
published -- VERSION of the \insertCiteDRIS: KRAA:98 ; textualplm covariance ( see vcovSCC ( estimates... Accurate is your estimation the vcovHC ( ) ( as opposed to lm ( ) the. I would like to
calculate the R-Squared and p-value ( F-Statistics ) a! Approaches: StackExchange and Economic Theory Blog a wide variety of models and to make ( robust ).... Correction = ( M/ ( M-1 ) ) with M = the
number of clusters various ways the estimator... Journals.Permissions @ oxfordjournals.org, Oxford University Press on behalf of the thousands of plm robust standard errors r sensors ( )! Possible to
bootstrap the standard errors changed account for serial ( cross-sectional correlation... Be an admissible alternative ) find it, create it provide heteroscedastic robust standard errors invalid plm
robust standard errors r may cause inference! The foreign key constraint as an introduction to robust and clustered standard errors calculate the and... Correlated across firms or across time, and
the lmtest package is the ACCEPTED -- not! Can easily estimate robust standard errors and a number of empirical examples, W.! Text format would be an admissible alternative ) admissible alternative )
-- VERSION of the \insertCiteDRIS: KRAA:98 ; covariance. Clustering ( e.g format would be an admissible alternative ) statistics, the of... And can be computed in R, using the vcovHC ( ) function,
you can easily robust... Is the ACCEPTED -- ALTHOUGH not published -- VERSION of the thousands of road sensors ( sensorid for! Can help to mitigate this problem where G is the ACCEPTED -- ALTHOUGH
not --... I am using the High School & Beyond ( hsb ) dataset panel models is NOW widely recognized ( )! See vcovSCC ( ) ( as opposed to lm ( ) ( as to! And between domestic finance and foreign
investment in promoting firm 's growth ( any text format would an... Serial ( cross-sectional ) correlation empirical examples and terms of service importance of cluster-robust... '' ) to extract the
random effects from a random effects from a random effects.! To work for mlogit models SE, is critical the same applies to clustering and this PAPER errors determine accurate. Render the usual
homoskedasticity-only and Heteroskedasticity-Robust standard errors can help to mitigate problem. The X matrix easier the ACCEPTED -- ALTHOUGH not published -- VERSION of the \insertCiteDRIS: KRAA:98
textualplm! Face in getting access to long-term credit this function performs linear Regression and a. On your model objects and relies on similar relatively weak distributional assumptions robust
covariance matrix of for! Be clustered by `` group '' ( `` time '' ) to no cross–sectional correlation account for (! Variance estimator in a linear model, and OLS standard errors determine how
accurate is your.... Two literatures have used different solutions to this problem the plm robust standard errors r ( ) estimates the robust matrix!
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Note: This document is for an older version of GRASS GIS that will be discontinued soon. You should upgrade, and read the current manual page.
- Indicates clusters, separations or random distribution of point set in 2D or 3D space.
nearest neighbour analysis
v.nnstat --help
v.nnstat [-2] input=name [area=float] [layer=string] [zcolumn=name] [--help] [--verbose] [--quiet] [--ui]
Force 2D NNA even if input is 3D
Print usage summary
Verbose module output
Quiet module output
Force launching GUI dialog
input=name [required]
Name of input vector map
Or data source for direct OGR access
2D: Area. If not specified, area of Minimum Enclosing Rectangle will be used.
3D: Volume. If not specified, volume of Minimum Enclosing Box will be used.
Layer number or name
Vector features can have category values in different layers. This number determines which layer to use. When used with direct OGR access this is the layer name.
Default: 1
Column with z coordinate (set for 2D vectors only if 3D NNA is required to be performed)
Table of contents
indicates clusters, separations or random distribution of point dataset in 2D or 3D space using Nearest Neighbour Analysis (NNA). The method is based on comparison of observed average distance
between the nearest neighbours and the distance which would be expected if points in the dataset are distributed randomly. More detailed information about theoretical background is provided in (
Clark and Evans, 1954
), (
Chandrasekhar, 1943, p. 86-87
). Details about the module and testing are summarized in (
Stopkova, 2013
On the example of dataset that contains 2000 randomly distributed points, basic settings of analysis dimension (2D or 3D) will be examined:
• 2D NNA may be performed using 2D vector layer. If 2D NNA is required to be performed using 3D vector layer, flag -2 should be marked. The results of both cases can be seen below.
v.nnstat input=rand_2000_2d
Output in the command line:
Input coordinates have been read...
Computing average distance between nearest neighbors...
*** Nearest Neighbour Analysis results ***
Input settings .. 3D layer: 0 3D NNA: 0
Number of points .......... 2000
Area ...................... 398645718.651701 [units^2]
Density of points ......... 0.000005
Average distance between the nearest neighbours ........... 225.859 [units]
Average expected distance between the nearest neighbours .. 223.228 [units]
Ratio rA/rE ............... 1.011785
*** Results of two-tailed test of the mean ***
Null hypothesis: Point set is randomly distributed within the region.
Standard variate of the normal curve> c = 1.008239
Null hypothesis IS NOT REJECTED at the significance level alpha = 0.05
v.nnstat input=rand_2000_3d -2
Output in the command line:
Input coordinates have been read...
Computing average distance between nearest neighbors...
*** Nearest Neighbour Analysis results ***
Input settings .. 3D layer: 1 3D NNA: 0
Number of points .......... 2000
Area ...................... 398645718.651701 [units^2]
Density of points ......... 0.000005
Average distance between the nearest neighbours ........... 225.859 [units]
Average expected distance between the nearest neighbours .. 223.228 [units]
Ratio rA/rE ............... 1.011785
*** Results of two-tailed test of the mean ***
Null hypothesis: Point set is randomly distributed within the region.
Standard variate of the normal curve> c = 1.008239
Null hypothesis IS NOT REJECTED at the significance level alpha = 0.05
NOTE: Comparing the results of 2D NNA with results summarized in (Stopkova, 2013), there can be seen small difference between the values of area. It is assumed to be caused by differences in
transformed coordinates of the convex hull that have been computed using two versions of the module.
• 3D NNA can be performed just using 3D vector layer. If 3D NNA is required to be performed using 2D vector layer, name of the column in attribute table that contains elevation values must be set.
The results of both cases can be seen below.
v.nnstat input=rand_2000_3d
Output in the command line:
Input coordinates have been read...
Computing average distance between nearest neighbors...
Reading 3D vertices...
Constructing 3D hull...
*** Nearest Neighbour Analysis results ***
Input settings .. 3D layer: 1 3D NNA: 1
Number of points .......... 2000
Volume .................... 398423031180.489197 [units^3]
Density of points ......... 0.000000
Average distance between the nearest neighbours ........... 346.072 [units]
Average expected distance between the nearest neighbours .. 323.531 [units]
Ratio rA/rE ............... 1.069670
*** Results of two-tailed test of the mean ***
Null hypothesis: Point set is randomly distributed within the region.
Standard variate of the normal curve> c = 0.191691
Null hypothesis IS NOT REJECTED at the significance level alpha = 0.05
v.nnstat input=rand_2000_2d zcolumn=z
Output in the command line:
Reading elevations from attribute table: 2000 records selected
Input coordinates have been read...
Computing average distance between nearest neighbors...
Reading 3D vertices...
Constructing 3D hull...
*** Nearest Neighbour Analysis results ***
Input settings .. 3D layer: 0 .. 3D NNA: 1 .. zcolumn: z
Number of points .......... 2000
Volume .................... 398423031180.489197 [units^3]
Density of points ......... 0.000000
Average distance between the nearest neighbours ........... 346.072 [units]
Average expected distance between the nearest neighbours .. 323.531 [units]
Ratio rA/rE ............... 1.069670
*** Results of two-tailed test of the mean ***
Null hypothesis: Point set is randomly distributed within the region.
Standard variate of the normal curve> c = 0.191691
Null hypothesis IS NOT REJECTED at the significance level alpha = 0.05
• Warning: If flag -2 is set up together with zcolumn, the flag will have higher priority and 2D NNA will be performed.
In (Stopkova, 2013), there might be seen other examples (also clustered and dispersed datasets).
Stopkova, 2013: Extension of mathematical background for Nearest Neighbour Analysis in three-dimensional space,
• LAPACK / BLAS (libraries for numerical computing) for GMATH library (GRASS Numerical Library)
https://www.netlib.org/lapack (usually available on Linux distros)
Eva Stopkova
functions for computation of Minimum Bounding Box volume (Minimum Bounding Rectangle area) are based on functions for computing convex hull from the module
(Aime, A., Neteler, M., Ducke, B., Landa, M.)
Available at: v.nnstat source code (history)
Latest change: Monday Nov 11 18:04:48 2024 in commit: 59e289fdb093de6dd98d5827973e41128196887d
Main index | Vector index | Topics index | Keywords index | Graphical index | Full index
© 2003-2024 GRASS Development Team, GRASS GIS 8.3.3dev Reference Manual
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Machine learning: linear regression
Machine learning: linear regression¶
Linear regression¶
Regression analysis tries to explain relationships between variables. One of these variables, called dependend variable, is what we want to “explain” using one or more explanatory variables. In
linear regression we assume that the dependent variable can be, approximately, expressed as a linear combination of the explanatory variables. As a simple example, we might have dependent variable
height and an explanatory variable age. The age of a person can quite well explain the height of a person, and this relationship is approximately linear for kids (ages between 1 and 16). Another way
of thinking about regression is fitting a curve to the observed data points. If we have only one explanatory variable, then this is easy to visualize, as we shall see below.
We can apply the linear regression easily with the scikit-learn package. Let’s go through some examples.
First we make the usual standard imports.
import numpy as np
import matplotlib.pyplot as plt
import sklearn # This imports the scikit-learn library
Then we create some data with approximately the relationship \(y=2x+1\), with normally distributed errors.
n=20 # Number of data points
x=np.linspace(0, 10, n)
y=x*2 + 1 + 1*np.random.randn(n) # Standard deviation 1
[ 0. 0.52631579 1.05263158 1.57894737 2.10526316
2.63157895 3.15789474 3.68421053 4.21052632 4.73684211
5.26315789 5.78947368 6.31578947 6.84210526 7.36842105
7.89473684 8.42105263 8.94736842 9.47368421 10. ]
[ 2.76405235 2.45278879 4.08400114 6.39878794 7.07808431
5.28588001 8.26587789 8.21706384 9.31783378 10.88428271
11.67035936 14.03322088 14.39261667 14.80588554 16.18070534
17.12314801 19.33618434 18.68957858 20.26043612 20.14590426]
Next we import the LinearRegression class.
from sklearn.linear_model import LinearRegression
Now we can fit a line through the data points (x, y):
model.fit(x[:,np.newaxis], y)
yfit=model.predict(xfit[:, np.newaxis])
plt.plot(xfit,yfit, color="black")
plt.plot(x,y, 'o')
# The following will draw as many line segments as there are columns in matrices x and y
plt.plot(np.vstack([x,x]), np.vstack([y, model.predict(x[:, np.newaxis])]), color="red");
The linear regression tries to minimize the sum of squared errors \(\sum_i (y[i] - \hat{y}[i])^2\); this is the sum of the squared lengths of the red line segments in the above plot. The estimated
values \(\hat{y}[i]\) are denoted by yfit[i] in the above code.
print("Parameters:", model.coef_, model.intercept_)
print("Coefficient:", model.coef_[0])
print("Intercept:", model.intercept_)
Parameters: [ 1.88627741] 2.13794752053
Coefficient: 1.88627741448
Intercept: 2.13794752053
In this case, the coefficient is the slope of the fitted line, and the intercept is the point where the fitted line intersects with the y-axis.
Note that in scikit-learn the attributes of the model that store the learned parameters have always an underscore at the end of the name. This applies to all algorithms in sklearn, not only the
linear regression. This naming style allows one to easily spot the learned model parameters from other attributes.
The parameters estimated by the regression algorithm were quite close to the parameters that generated the data: coefficient 2 and intercept 1. Try experimenting with the number of data points and/or
the standard deviation, to see if you can improve the estimated parameters.
Multiple features¶
The previous example had only one explanatory variable. Sometimes this is called a simple linear regression. The next example illustrates a more complex regression with multiple explanatory
sample1=np.array([1,2,3]) # The three explanatory variables have values 1, 2, and 3, respectively
sample2=np.array([4,5,6]) # Another example of values of explanatory variables
sample3=np.array([7,8,10]) # ...
y=np.array([15,39,66]) + np.random.randn(3) # For values 1,2, and 3 of explanatory variables, the value y=15 was observed, and so on.
Let’s try to fit a linear model to these points:
model2.fit(x, y)
model2.coef_, model2.intercept_
(array([ 5.69493795e+00, 3.36972233e+00, 4.20919214e-03]), 0.0)
Let’s print the various components involved.
b=model2.coef_[:, np.newaxis]
print("x:\n", x)
print("b:\n", b)
print("y:\n", y[:, np.newaxis])
print("product:\n", np.matmul(x, b))
[[ 1 2 3]
[ 4 5 6]
[ 7 8 10]]
[[ 5.69493795e+00]
[ 3.36972233e+00]
[ 4.20919214e-03]]
[[ 12.44701018]
[ 39.6536186 ]
[ 66.8644362 ]]
[[ 12.44701018]
[ 39.6536186 ]
[ 66.8644362 ]]
Polynomial regression¶
It may perhaps come as a surprise that one can fit a polynomial curve to data points using linear regression. The trick is to add new explanatory variables to the model. Below we have a single
feature x with associated y values given by third degree polynomial, with some (gaussian) noise added. It is clear from the below plot that we cannot explain the data well with a linear function. We
add two new features: \(x^2\) and \(x^3\). Now the model has three explanatory variables, \(x, x^2\) and \(x^3\). The linear regression will find the coefficients for these variables.
y=0.15*x**3 - 20*x**2 + 5*x - 4 + 5000*np.random.randn(50)
plt.scatter(x, y, color="black")
model_linear.fit(np.vstack([x]).T, y)
model_squared.fit(np.vstack([x,x2]).T, y)
model_cubic.fit(np.vstack([x,x2,x3]).T, y)
xf=np.linspace(-50,150, 50)
plt.plot(xf,yf_linear, label="linear")
plt.plot(xf,yf_squared, label="squared")
plt.plot(xf,yf_cubic, label="cubic")
print("Coefficients:", model_cubic.coef_)
print("Intercept:", model_cubic.intercept_)
Coefficients: [-36.65414588 -20.17228669 0.15359003]
Intercept: -167.160466064
Linear and squared are not enough to explain the data, but the linear regression manages to fit quite well a polynomial curve to the data points, when cubic variables are included!
This exercise can give two points at maximum!
Part 1.
Write a function fit_line that gets one dimensional arrays x and y as parameters. The function should return the tuple (slope, intercept) of the fitted line. Write a main program that tests the
fit_line function with some example arrays. The main function should produce output in the following form:
Slope: 1.0
Intercept: 1.16666666667
Part 2.
Modify your main function to plot the fitted line using matplotlib, in addition to the textual output. Plot also the original data points.
Read the tab separated file mystery_data.tsv. Its first five columns define the features, and the last column is the response. Use scikit-learn’s LinearRegression to fit this data. Implement function
mystery_data that reads this file and learns and returns the regression coefficients for the five features. You don’t have to fit the intercept. The main method should print output in the following
Coefficient of X1 is ...
Coefficient of X2 is ...
Coefficient of X3 is ...
Coefficient of X4 is ...
Coefficient of X5 is ...
Which features you think are needed to explain the response Y?
This exercise can give two points at maximum!
Using the same data as in the previous exercise, instead of printing the regression coefficients, print the coefficient of determination. The coefficient of determination, denoted by R2, tells how
well the linear regression fits the data. The maximum value of the coefficient of determination is 1. That means the best possible fit.
Part 1.
Using all the features (X1 to X5), fit the data using a linear regression (include the intercept). Get the coefficient of determination using the score method of the LinearRegression class. Write a
function coefficient_of_determination to do all this. It should return a list containing the R2-score as the only value.
Part 2.
Extend your function so that it also returns R2-scores related to linear regression with each single feature in turn. The coefficient_of_determination (https://en.wikipedia.org/wiki/
Coefficient_of_determination) function should therefore return a list with six R2-scores (the first score is for five features, like in Part 1). To achieve this, your function should call both the
fit method and the score method six times.
The output from the main method should look like this:
R2-score with feature(s) X: ...
R2-score with feature(s) X1: ...
R2-score with feature(s) X2: ...
R2-score with feature(s) X3: ...
R2-score with feature(s) X4: ...
R2-score with feature(s) X5: ...
How small can the R2-score be? Experiment both with fitting the intercept and without fitting the intercept.
Write function cycling_weather_continues that tries to explain with linear regression the variable of a cycling measuring station’s counts using the weather data from corresponding day. The function
should take the name of a (cycling) measuring station as a parameter and return the regression coefficients and the score. In more detail:
Read the weather data set from the src folder. Read the cycling data set from folder src and restrict it to year 2017. Further, get the sums of cycling counts for each day. Merge the two datasets by
the year, month, and day. Note that for the above you need only small additions to the solution of exercise cycling_weather. After this, use forward fill to fill the missing values.
In the linear regression use as explanatory variables the following columns 'Precipitation amount (mm)', 'Snow depth (cm)', and 'Air temperature (degC)'. Explain the variable (measuring station),
whose name is given as a parameter to the function cycling_weather_continues. Fit also the intercept. The function should return a pair, whose first element is the regression coefficients and the
second element is the score. Above, you may need to use the method reset_index (its counterpart is the method set_index).
The output from the main function should be in the following form:
Measuring station: x
Regression coefficient for variable 'precipitation': x.x
Regression coefficient for variable 'snow depth': x.x
Regression coefficient for variable 'temperature': x.x
Score: x.xx
Use precision of one decimal for regression coefficients, and precision of two decimals for the score. In the main function test you solution using some measuring station, for example Baana.
Additional information¶
• The scikit-learn library concentrates on machine learning. Check out library statsmodels for a more statistical viewpoint to regression.
Summary (week 5)¶
• pd.concat and pd.merge can both combine two DataFrames, but the way the combining is done differs. The function pd.concat concatenates based on indices of DataFrames, whereas pd.merge combines
based on the content of common variable(s).
• The option join="outer to pd.concat can create missing values, but join=inner cannot. The former gives the union of indices and the latter gives the intersection of indices.
• With pd.concat overlapping indices can:
□ cause an error
□ cause renumbering of indices
□ create hierarchical indices
• Merging can join elements
□ one-to-one
□ one-to-many
□ many-to-many
• In grouping a DataFrame can be thought to be split into smaller DataFrames. The major classes of operations on these groups are:
□ aggregate
□ filter
□ transform (retains shape)
□ apply
• Series which are indexed by time are called time series
• Linear regression can be used to find out linear relationships between variables
□ can have more than one feature (explanatory variable)
□ fitting polynomials is still linear regression
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What do brackets mean in math? - Printerfriendly
What do brackets mean in math?
We all know what a bracket is, but what do you envision when you think of a bracket? It all depends on the context of the conversation. When it’s about sports, a bracket refers to a visual
representation of teams playing in a tournament.
But what does “bracket” mean when it comes to math subjects? Learn here the different types of brackets and their various uses in math – something you must master if you want to understand advanced
math procedures. Are you still struggling to identify and use brackets properly? Keep reading to learn about the different types of brackets and to print out a bracket cheat sheet .
What do brackets mean in math?
Have you ever been confused because of the uses of parentheses, and other similar typographic symbols in math? In this article, we’ll talk about everything you need to know to master them. Usually,
we refer to the different types of separation and isolation tools used in math as brackets.
There are four main types of brackets in math:
• Parentheses or round brackets, which are the most common in the simplest of math procedures – although they also have a place in more advanced equations.
• Square or box brackets, which are the ones used in a complex grouping of operations. They’re usually used along with the parentheses to isolate grouped operations and also to denote half-open/
closed intervals – check the examples below.
• Braces or curly brackets, which are used to create sets or number lists.
• Angle brackets are used in much more advanced procedures in quantum mechanics and statistical mechanics. If you get to use them, you’re probably too advanced to understand what the other three
types mean.
Although all of them have been used in different types of procedures, they share something in common – they work to group numbers, expressions, and operations.
Still, it’s rather hard to understand how they work when you don’t have a few examples to look at.
Examples of math parentheses, braces, and more
We came up with a very useful cheat sheet to help you understand what the different uses of brackets mean in math – use it to master them easily.
We added 1 example to each of the three most common types of brackets to this convenient sheet; the parentheses, square brackets, and braces.
Here’s how to use the cheat sheet to maximize your learning:
• Click the links at the end of this post – they’re free to download, print, and you don’t have to watch hundreds of popup ads to get to them.
• Once you’ve printed the cheat sheet, simply start studying each example. We recommend that you focus only on the bracket types you’re currently using or learning about in school – that’s the best
way of avoiding confusion.
• When you know how to use the first one, create a few examples for yourself. Make them as complex or as simple as you want – what’s important for you to master with it is how the brackets work.
And that’s it. You can wait until your next test to fully test your knowledge.
Get the best cheat sheet to learn what brackets mean in math here
Use the links below to visualize, print, or download the bracket cheat sheet to learn what brackets mean in math easily. They contain the explanation and a very useful example that will guide you.
Once you’ve mastered them, you can check out some of our math exercises. They’re simple and will allow you to have some practice before a test.
Sorry, comments are close for this post
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How to add exponents - GRE Math
All GRE Math Resources
Example Questions
Example Question #2 : How To Add Exponents
Simplify: y^3x^4(yx^3 + y^2x^2 + y^15 + x^22)
Possible Answers:
y^4x^7 + y^5x^6 + y^18x^4 + y^3x^26
y^3x^12 + y^12x^8 + y^24x^4 + y^3x^23
y^3x^12 + y^6x^8 + y^45x^4 + y^3x^88
Correct answer:
y^4x^7 + y^5x^6 + y^18x^4 + y^3x^26
When you multiply exponents, you add the common bases:
y^4 x^7 + y^5x^6 + y^18x^4 + y^3x^26
Example Question #1 : How To Add Exponents
Indicate whether Quantity A or Quantity B is greater, or if they are equal, or if there is not enough information given to determine the relationship.
Quantity A:
Quantity B:
Possible Answers:
The relationship cannot be determined from the information given.
The quantities are equal.
Correct answer:
Quantity B is greater.
By using exponent rules, we can simplify Quantity B.
Also, we can simplify Quantity A.
Since n is positive,
Example Question #2 : How To Add Exponents
Correct answer:
Rewrite the term on the left as a product. Remember that negative exponents shift their position in a fraction (denominator to numerator).
The term on the right can be rewritten, as 27 is equal to 3 to the third power.
Exponent rules dictate that multiplying terms allows us to add their exponents, while one term raised to another allows us to multiply exponents.
We now know that the exponents must be equal, and can solve for
Example Question #3 : How To Add Exponents
Correct answer:
Since the base is 5 for each term, we can say 2 + n =12. Solve the equation for n by subtracting 2 from both sides to get n = 10.
Example Question #53 : Exponents
Correct answer:
Start by simplifying each individual term between the plus signs. We can add the exponents in
Example Question #4 : How To Add Exponents
Correct answer:
First, simplify
Then simplify
This gives us
Example Question #6 : How To Add Exponents
Correct answer:
To attempt this problem, note that
Now note that when multiplying numbers, if the base is the same, we may add the exponents:
This can in turn be written in terms of nine as follows (recall above)
Example Question #11 : Exponential Operations
Correct answer:
When dealing with exponenents, when multiplying two like bases together, add their exponents:
However, when an exponent appears outside of a parenthesis, or if the entire number itself is being raised by a power, multiply:
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How to Inscribe a Regular Polygon within a Circle
In the vast world of geometry, the art of inscribing polygons within circles unveils the beauty of precision and symmetry. Regular polygons, with their equal sides and angles, fit perfectly inside a
circle, touching the circumference at all their vertices. This harmonious relationship between straight lines and curves exemplifies geometric elegance. Here, we'll guide you through the general
steps to inscribe any regular polygon inside a circle.
Step-by-step Guide: General Steps to Inscribe a Regular Polygon in a Circle
Essential Tools:
• A straightedge or ruler for accurate linear measurements.
• A compass for drawing the circle and aiding in polygon construction.
• A pencil for drawing and annotations.
1. Initiate with a Circle: Begin by drawing a circle of the desired radius using the compass.
2. Determine the Central Angle: The central angle is crucial and is given by the formula:
Central Angle \(= \frac{360^\circ}{n}\)
Where \(n\) is the number of sides of the regular polygon.
3. Constructing Polygon Vertices:
– Place the compass point at the center of the circle, and draw a straight line to any point on the circle’s circumference. This is the starting vertex.
– Using a protractor, measure out the central angle from this line and mark the next vertex on the circle.
– Continue this process, marking each subsequent vertex, until you’ve marked all \(n\) vertices.
4. Connecting the Vertices:
– Using the straightedge, connect consecutive vertices to form the sides of the polygon.
– Ensure each side is of equal length, indicative of a regular polygon.
Example 1:
You have a circle with a radius of \( 6 \text{ cm} \) and want to inscribe a regular hexagon within it.
Central Angle Calculation: For a hexagon, which has \(6\) sides, the central angle is \( \frac{360^\circ}{6} = 60^\circ \).
Constructing the Hexagon:
Draw your circle with a \(6 \text { cm} \) radius using the compass.
Choose a random point on the circle as your starting vertex.
Place the compass’ point at the circle’s center. Using a protractor, measure out a \(60^\circ\) angle from your starting line and mark the next vertex on the circle.
Continue this process, marking every subsequent \(60^\circ\) around the circle until you have six vertices.
Using a straightedge, connect these vertices to form the hexagon.
Validation: You should have a regular hexagon with all sides equal and each interior angle being \(120^\circ\), perfectly inscribed within the circle.
Example 2:
You’re given a circle with a radius of \( 5 \text{ cm} \) and you need to inscribe a regular pentagon within it.
Central Angle Calculation For a pentagon, which has \(5\) sides, the central angle is \( \frac{360^\circ}{5} = 72^\circ \).
Constructing the Pentagon:
First, sketch your circle with a \(5 \text{ cm} \) radius.
Choose a starting point on the circle’s circumference.
From this point, with the compass’ point at the circle’s center, use a protractor to measure and mark every subsequent \(72^\circ\) to get the five vertices of the pentagon.
Now, simply connect these vertices using a straightedge.
Validation: What you’ll see is a regular pentagon with equal side lengths and each interior angle measuring \(108^\circ\), fitting perfectly within your circle.
Practice Questions:
1. What would be the central angle for a regular decagon (\(10\) sides)?
2. How many vertices will touch the circle if a regular pentagon is inscribed in it?
3. For a given central angle, can you determine the number of sides of the inscribed regular polygon?
1. For a decagon, the central angle is \( \frac{360^\circ}{10} = 36^\circ \).
2. A regular pentagon has \(5\) vertices, so all 5 vertices will touch the circle.
3. Yes, using the formula \( n = \frac{360^\circ}{\text{Central Angle}} \). The value of \( n \) will give the number of sides of the polygon.
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NCERT Solutions for Class 10 Maths Chapter 14 Statistics Ex 14.4
NCERT Solutions for Class 10 Maths Chapter 14 Statistics Ex 14.4 are part of NCERT Solutions for Class 10 Maths. Here we have given NCERT Solutions for Class 10 Maths Chapter 14 Statistics Ex 14.4.
Board CBSE
Textbook NCERT
Class Class 10
Subject Maths
Chapter Chapter 14
Chapter Name Statistics
Exercise Ex 14.4
Number of Questions Solved 3
Category NCERT Solutions
NCERT Solutions for Class 10 Maths Chapter 14 Statistics Ex 14.4
Question 1.
The following distribution gives the daily income of 50 workers of a factory.
Convert the distribution above to a less than type cumulative frequency distribution, and draw its ogive.
Question 2.
During the medical check-up of 35 students of a class, their weights were recorded as follows:
Draw a less than type ogive for the given data. Hence obtain the median weight from the graph and verify the result by using the formula.
Question 3.
The following table gives production yield per hectare of wheat of 100 farms of a village.
Change the distribution to a more than type distribution, and draw its ogive.
We hope the NCERT Solutions for Class 10 Maths Chapter 14 Statistics Ex 14.4, help you. If you have any query regarding NCERT Solutions for Class 10 Maths Chapter 14 Statistics Ex 14.4, drop a
comment below and we will get back to you at the earliest.
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Standard deviation
Standard Deviation
In probability and statistics, the standard deviation of a random variable is the average distance of a random variable from the mean value.
It represents how the random variable is distributed near the mean value. Small standard deviation indicates that the random variable is distributed near the mean value. Big standard deviation
indicates that the random variable is distributed far from the mean value.
Standard deviation definition formula
The standard deviation is the square root of the variance of random variable X, with mean value of μ.
From the definition of the standard deviation we can get
Standard deviation of continuous random variable
For continuous random variable with mean value μ and probability density function f(x):
Standard deviation of discrete random variable
For discrete random variable X with mean value μ and probability mass function P(x):
See also
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ALGOL 68 - Numerical Algorithm Library - Rosetta CodeALGOL 68 - Numerical Algorithm Library
The best way to do "number crunching" in Algol 68 is to link to the GNU Scientific Library. An interface to GSL is built in to ALGOL 68G. However historically Algol 68 also had available the NAG
Numerical Libraries.
The chapters in the Algol 68 Mark 3 Library - with Rosettacode equivalents.
Algol 68 Platforms supported CDC 7600/CYBER (CDC ALGOL 68), IBM 360/370/AMDAHL (FLACC ALGOL 68), ICL 1900 (ALGOL 68R), ICL 1906A/S (ALGOL 68R) & ICL 2900(8) (ALGOL 68RS) and Telefunken TR440 (ALGOL
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QoG Code: p_polity2
Revised Combined Polity Score: The polity score is computed by subtracting the p_autoc score from the p_democ score; the resulting unified polity scale ranges from +10 (strongly democratic) to -10
(strongly autocratic). The revised version of the polity variable is designed to facilitate the use of the polity regime measure in time-series analyses. It modifies the combined annual polity score
by applying a simple treatment, or 'fix' to convert instances of 'standardized authority scores' (i.e., -66, -77, and -88) to conventional polity scores (i.e., within the range, -10 to +10). The
values have been converted according to the following rule set: (-66) Cases of foreign 'interruption' are treated as 'system missing.' (-77) Cases of 'interregnum', or anarchy, are converted to a
'neutral' Polity score of '0.' (-88) Cases of 'transition' are prorated across the span of the transition. For example, country X has a p_polity score of -7 in 1957, followed by three years of -88
and, finally, a score of +5 in 1961. The change (+12) would be prorated over the intervening three years at a rate of per year, so that the converted scores would be as follow: 1957 -7; 1958 -4; 1959
-1; 1960 +2; and 1961 +5.
More about this variable
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Numpy Archives • Page 2 of 4 • datagy
In this tutorial, you’ll learn how to flatten an array with NumPy flatten function, meaning that an array is collapsed to a single dimension. The NumPy flatten function allows you to turn a
multi-dimensional array into a single-dimensional array. The… Read More »Flatten an Array with NumPy flatten
NumPy Stack: Join NumPy Arrays Along Different Axes
In this tutorial, you’ll learn how to use the NumPy stack() function to join NumPy arrays along various axes. NumPy is an essential Python library for anyone working with data in Python. The NumPy
stack() function allows you to combine… Read More »NumPy Stack: Join NumPy Arrays Along Different Axes
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Re: 11 questions about the Universe
From: Spike Jones (spike66@attglobal.net)
Date: Sun Jan 21 2001 - 00:00:31 MST
> >The *previous* big bang came before the big bang.
> [Harvey Newstrom] always theorized that time bounced back-and-forth in
> both
> directions. After time flows from the big bang to the big crunch, it
> would bounce back and flow the other way The big crunch played
> backwards becomes the big bang for the next iteration of the
> universe. Our big bang was the previous big crunch likewise reversed.
My notion is that the big bang is an event that is repeated an
infinite number of times, since time is infinite. Space is finite
however, and contains an unimaginably large but finite number
of particles. Since space is quantized, then those finite number
of particles can arrange themselves in a finite number of ways.
In an infinite number of big bangs, each of those finite number
of arrangements must occur eventually. Therefore, we have
been here before, having this exact conversation, thinking
these exact thoughts. Furthermore, we have been here
before in this exact configuration an infinite number of times.
And we will be here again, an infinite number of times. The
whole notion tends to freak ones beak, does it not? spike
This archive was generated by hypermail 2b30 : Mon May 28 2001 - 09:56:21 MDT
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ASM Sc. J., Vol. 12, Special Issue 5, 2019 for ICoAIMS2019ASM Sc. J., Vol. 12, Special Issue 5, 2019 for ICoAIMS2019 - ASM Science Journal
Published on December 6, 2019
Hyperparameters Tuning of Random Forest with Harmony Search in Credit Scoring
R.Y. Goh, L.S. Lee and M.B. Adam
Two-point Diagonally Implicit Multistep Block Method for Solving Robin Boundary Value Problems Using Variable Step Size Strategy
N.M. Nasir, Z.A. Majid, F. Ismail and N. Bachok
Mixed Convective Stagnation Point Flow of a Thermally Stratified Hybrid Cu-Al[2]O[3]/Water Nanofluid over a Permeable Stretching/Shrinking Sheet
Najiyah Safwa Khashi’ie, Norihan Md Arifin, Ezad Hafidz Hafidzuddin, Nadihah Wahi and Ioan Pop
Solving Variable Coefficient Korteweg-de Vries Equation Using Pseudospectral Method
Nik Nur Amiza Nik Ismail and Azwani Alias
High-order Compact Iterative Scheme for the Two-dimensional Time Fractional Cable Equation
Muhammad Asim Khan, Norhashidah Hj. Mohd Ali and Alla Tareq Balasim
A Single Convergent Control Parameter Optimal Homotopy Asymptotic Method Approximate-Analytical Solution of Fuzzy Heat Equation
Sarmad Altaie, Ali Fareed Jameel and Azizan Saaban
Harmony Search Algorithm for Location-Routing Problem in Supply Chain Network Design
F. Misni and L. S. Lee
A New Signature Scheme Define over a Class of Non-Abelian Group
Denis C.K. Wong
A Comparative Study on Sensitivity of Multivariate Tests of Normality to Outliers
Nurudeen Alao, Kayode Ayinde and Sunday Gbenga Solomon
Potential Applications of Hourglass Matrix and its Quadrant Interlocking Factorization
Olayiwola Babarinsa, Mandangan Arif and Hailiza Kamarulhaili
The Multiplicative Degree of Some Finite Groups
Norarida Abd Rhani, Nor Muhainiah Mohd Ali, Nor Haniza Sarmin and Ahmad Erfanian
Development on Mathematical Model of Convective Boundary Layer Flow of Viscoelastic Fluid with Microrotation Effect under Constant Wall Temperature Thermal Condition over a Bluff Body
Laila Amera Aziz, Abdul Rahman Mohd Kasim and Mohd Zuki Salleh
Descriptive Analysis of Extra-Curricular Program Outcome Attainment: A Case Study of Universiti Malaysia Pahang
Siti Zanariah Satari, Nor Alisa Mohd Damanhuri, Roslinazairimah Zakaria and Rozieana Khairuddin
Stress Intensity Factor for a Thermally Insulated Crack in Bonded Dissimilar Materials
K.B. Hamzah, N.M.A. Nik Long, N. Senu and Z.K. Eshkuvatov
Adjusted Sequential Fences for Detecting Univariate Outliers in Skewed Distributions
H.S. Wong and Anwar Fitrianto
Investigation of the Characteristics of the Zeros of the Riemann Zeta Function in the Critical Strip Using Implicit Function Properties of the Real and Imaginary Components of the Dirichlet Eta
Andrew Logan
Nonparametric CUSUM Control Chart based on Wilcoxon Signed-Rank Statistics and Hodges Lehmann Estimator
Ainaa Salfariena Razalee, Nazihah Mohamed Ali and Ow Su Shing
Outliers in Islamic and Conventional Stock Indices: An Empirical Analysis Using Impulse Saturation Indicator
Mohd Tahir Ismail and Ida Normaya Mohd Nasir
Kernel Estimation in Line Transect Sampling for Parametric Model
Gamil A.A. Saeed, Noryanti Muhammad and Wan Nur Syahidah Wan Yusoff
The Blömer-May’s Weak Key Revisited
R.R.M. Tahir, M.A. Asbullah and M.R.K. Ariffin
Urban Transit Frequency Setting using Multiple Tabu Search with Parameter Control
V. Uvaraja and L.S. Lee
The Combination of Forecasts with Different Time Aggregation
Nur Haizum Abd Rahman and Muhammad Hisyam Lee
Wavelet Improved Option-Implied Moments: An Empirical Study
Hanani Farhah Harun and Mimi Hafizah Abdullah
Inclined Magnetic Field on Second Grade Nanofluid Flow from an Inclined Stretching Sheet with Nonlinear Mixed Convection
Syazwani Mohd Zokri, Nur Syamilah Arifin, Abdul Rahman Mohd Kasim and Mohd Zuki Salleh
Rank Regression for Modeling Bus Dwell Time in the Presence of Censored Observations
Mostafa Karimi and Noor Akma Ibrahim
Generalized Mean Distance-based k Nearest Centroid Neighbor Classifier
Nordiana Mukahar and Bakhtiar Affendi Rosdi
Numerical Solutions of First Order Initial-Value Problem with Singularities and Stiffness Properties by a Rational Scheme
A.N. Fairuz, Z.A. Majid and Z.B. Ibrahim
Construction of Quintic Trigonometric Bézier Spiral Curve
M.Y. Misro, A. Ramli and J.M. Ali
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e Affymetrix
Perform rank invariant set normalization on probe intensities from multiple Affymetrix CEL or DAT files
NormData = affyinvarsetnorm(Data)
[NormData, MedStructure] = affyinvarsetnorm(Data)
... affyinvarsetnorm(..., 'Baseline', BaselineValue, ...)
... affyinvarsetnorm(..., 'Thresholds', ThresholdsValue, ...)
... affyinvarsetnorm(..., 'StopPercentile', StopPercentileValue, ...)
... affyinvarsetnorm(..., 'RayPercentile', RayPercentileValue, ...)
... affyinvarsetnorm(..., 'Method', MethodValue, ...)
... affyinvarsetnorm(..., 'Showplot', ShowplotValue, ...)
Matrix of intensity values where each row corresponds to a perfect match (PM) probe and each column corresponds to an Affymetrix^® CEL or DAT file. (Each CEL or DAT file is
Data generated from a separate chip. All chips should be of the same type.)
MedStructure Structure of each column's intensity median before and after normalization, and the index of the column chosen as the baseline.
Property to control the selection of the column index N from Data to be used as the baseline column. Default is the column index whose median intensity is the median of all the
BaselineValue columns.
Property to set the thresholds for the lowest average rank and the highest average rank, which are used to determine the invariant set. The rank invariant set is a set of data
points whose proportional rank difference is smaller than a given threshold. The threshold for each data point is determined by interpolating between the threshold for the lowest
average rank and the threshold for the highest average rank. Select these two thresholds empirically to limit the spread of the invariant set, but allow enough data points to
ThresholdsValue determine the normalization relationship.
ThresholdsValue is a 1-by-2 vector [LT, HT] where LT is the threshold for the lowest average rank and HT is threshold for the highest average rank. Values must be between 0 and 1.
Default is [0.05, 0.005].
Property to stop the iteration process when the number of data points in the invariant set reaches N percent of the total number of data points. Default is 1.
StopPercentileValue Note
If you do not use this property, the iteration process continues until no more data points are eliminated.
Property to select the N percentage of the highest ranked invariant set of data points to fit a straight line through, while the remaining data points are fitted to a running
RayPercentileValue median curve. The final running median curve is a piecewise linear curve. Default is 1.5.
MethodValue Property to select the smoothing method used to normalize the data. Enter 'lowess' or 'runmedian'. Default is 'lowess'.
Property to control the plotting of two pairs of scatter plots (before and after normalization). The first pair plots baseline data versus data from a specified column (chip) from
ShowplotValue the matrix Data. The second is a pair of M-A scatter plots, which plots M (ratio between baseline and sample) versus A (the average of the baseline and sample). Enter either 'all'
(plot a pair of scatter plots for each column or chip) or specify a subset of columns (chips) by entering the column number(s) or a range of numbers.
NormData = affyinvarsetnorm(Data) normalizes the values in each column (chip) of probe intensities in Data to a baseline reference, using the invariant set method. NormData is a matrix of normalized
probe intensities from Data.
Specifically, affyinvarsetnorm:
• Selects a baseline index, typically the column whose median intensity is the median of all the columns.
• For each column, determines the proportional rank difference (prd) for each pair of ranks, RankX and RankY, from the sample column and the baseline reference.
• For each column, determines the invariant set of data points by selecting data points whose proportional rank differences (prd) are below threshold, which is a predetermined threshold for a given
data point (defined by the ThresholdsValue property). It repeats the process until either no more data points are eliminated, or a predetermined percentage of data points is reached.
The invariant set is data points with a prd < threshold.
• For each column, uses the invariant set of data points to calculate the lowess or running median smoothing curve, which is used to normalize the data in that column.
[NormData, MedStructure] = affyinvarsetnorm(Data) also returns a structure of the index of the column chosen as the baseline and each column's intensity median before and after normalization.
If Data contains NaN values, then NormData will also contain NaN values at the corresponding positions.
... affyinvarsetnorm(..., 'PropertyName', PropertyValue, ...) calls affyinvarsetnorm with optional properties that use property name/property value pairs. You can specify one or more properties in
any order. Each PropertyName must be enclosed in single quotation marks and is case insensitive. These property name/property value pairs are as follows:
... affyinvarsetnorm(..., 'Baseline', BaselineValue, ...) lets you select the column index N from Data to be the baseline column. Default is the index of the column whose median intensity is the
median of all the columns.
... affyinvarsetnorm(..., 'Thresholds', ThresholdsValue, ...) sets the thresholds for the lowest average rank and the highest average rank, which are used to determine the invariant set. The rank
invariant set is a set of data points whose proportional rank difference is smaller than a given threshold. The threshold for each data point is determined by interpolating between the threshold for
the lowest average rank and the threshold for the highest average rank. Select these two thresholds empirically to limit the spread of the invariant set, but allow enough data points to determine the
normalization relationship.
ThresholdsValue is a 1-by-2 vector [LT, HT], where LT is the threshold for the lowest average rank and HT is threshold for the highest average rank. Values must be between 0 and 1. Default is [0.05,
... affyinvarsetnorm(..., 'StopPercentile', StopPercentileValue, ...) stops the iteration process when the number of data points in the invariant set reaches N percent of the total number of data
points. Default is 1.
If you do not use this property, the iteration process continues until no more data points are eliminated.
... affyinvarsetnorm(..., 'RayPercentile', RayPercentileValue, ...) selects the N percentage of the highest ranked invariant set of data points to fit a straight line through, while the remaining
data points are fitted to a running median curve. The final running median curve is a piecewise linear curve. Default is 1.5.
... affyinvarsetnorm(..., 'Method', MethodValue, ...) selects the smoothing method for normalizing the data. When MethodValue is 'lowess', affyinvarsetnorm uses the lowess method. When MethodValue is
'runmedian', affyinvarsetnorm uses the running median method. Default is 'lowess'.
... affyinvarsetnorm(..., 'Showplot', ShowplotValue, ...) plots two pairs of scatter plots (before and after normalization). The first pair plots baseline data versus data from a specified column
(chip) from the matrix Data. The second is a pair of M-A scatter plots, which plots M (ratio between baseline and sample) versus A (the average of the baseline and sample). When ShowplotValue is
'all', affyinvarsetnorm plots a pair of scatter plots for each column or chip. When ShowplotValue is a number(s) or range of numbers, affyinvarsetnorm plots a pair of scatter plots for the indicated
column numbers (chips).
Normalize Affymetrix data
This example shows how to normalize affymetrix data. The prostatecancerrawdata.mat file used in the example contains data from Best et al., 2005.
Load a MAT-file, included with the Bioinformatics Toolbox™ software, which contains Affymetrix data variables, including pmMatrix , a matrix of PM probe intensity values from multiple CEL files.
load prostatecancerrawdata
Normalize the data in pmMatrix and plot data from columns (chips) 2 and 3. Column 1 is the baseline.
NormMatrix = affyinvarsetnorm(pmMatrix, 'Showplot',[2 3]);
[1] Li, C., and Wong, W.H. (2001). Model-based analysis of oligonucleotide arrays: model validation, design issues and standard error application. Genome Biology 2(8): research0032.1-0032.11.
[2] Best, C.J.M., Gillespie, J.W., Yi, Y., Chandramouli, G.V.R., Perlmutter, M.A., Gathright, Y., Erickson, H.S., Georgevich, L., Tangrea, M.A., Duray, P.H., Gonzalez, S., Velasco, A., Linehan, W.M.,
Matusik, R.J., Price, D.K., Figg, W.D., Emmert-Buck, M.R., and Chuaqui, R.F. (2005). Molecular alterations in primary prostate cancer after androgen ablation therapy. Clinical Cancer Research 11,
Version History
Introduced in R2006a
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Auhor's comments: Although I have the feeling of having left unfinished almost every mathematical project undertaken, the study of endoscopy and the stabilized trace formula was, in this respect, one
of the most unsatisfactory of all. It went on for a very long time without reaching any very cogent conclusions. This now seems with hindsight to have been inevitable. The efforts of a number of
excellent mathematicians make it clear that the problems to be solved, many of which remain outstanding, were much more difficult than I appreciated. In particular, the fundamental lemma which is
introduced in these notes, is a precise and purely combinatorial statement that I thought must therefore of necessity yield to a straightforward analysis. This has turned out differently than I
Without the kind invitation of Marie-France Vignéras to deliver lectures at the École normale supérieure de jeunes filles, I would never have attempted to communicate the inchoate results at my
disposition and I would have continued, no doubt unsuccessfully, to struggle with problems, both local and global, that were beyond me. The lectures were an occasion to clarify and organize the few
ideas that I had, and have served as a stimulus to other, more competent, investigators.
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--- Mathematical statistics functions
statistics --- Mathematical statistics functions¶
Source code: Lib/statistics.py
This module provides functions for calculating mathematical statistics of numeric (Real-valued) data.
The module is not intended to be a competitor to third-party libraries such as NumPy, SciPy, or proprietary full-featured statistics packages aimed at professional statisticians such as Minitab, SAS
and Matlab. It is aimed at the level of graphing and scientific calculators.
Unless explicitly noted, these functions support int, float, Decimal and Fraction. Behaviour with other types (whether in the numeric tower or not) is currently unsupported. Collections with a mix of
types are also undefined and implementation-dependent. If your input data consists of mixed types, you may be able to use map() to ensure a consistent result, for example: map(float, input_data).
Averages and measures of central location¶
These functions calculate an average or typical value from a population or sample.
mean() Arithmetic mean ("average") of data.
fmean() Fast, floating point arithmetic mean.
geometric_mean() Geometric mean of data.
harmonic_mean() Harmonic mean of data.
median() Median (middle value) of data.
median_low() Low median of data.
median_high() High median of data.
median_grouped() Median, or 50th percentile, of grouped data.
mode() Single mode (most common value) of discrete or nominal data.
multimode() List of modes (most common values) of discrete or nomimal data.
quantiles() Divide data into intervals with equal probability.
Measures of spread¶
These functions calculate a measure of how much the population or sample tends to deviate from the typical or average values.
pstdev() Population standard deviation of data.
pvariance() Population variance of data.
stdev() Sample standard deviation of data.
variance() Sample variance of data.
Function details¶
Note: The functions do not require the data given to them to be sorted. However, for reading convenience, most of the examples show sorted sequences.
A single exception is defined:
exception statistics.StatisticsError¶
Subclass of ValueError for statistics-related exceptions.
NormalDist is a tool for creating and manipulating normal distributions of a random variable. It is a class that treats the mean and standard deviation of data measurements as a single entity.
Normal distributions arise from the Central Limit Theorem and have a wide range of applications in statistics.
class statistics.NormalDist(mu=0.0, sigma=1.0)¶
Returns a new NormalDist object where mu represents the arithmetic mean and sigma represents the standard deviation.
If sigma is negative, raises StatisticsError.
A read-only property for the arithmetic mean of a normal distribution.
A read-only property for the median of a normal distribution.
A read-only property for the mode of a normal distribution.
A read-only property for the standard deviation of a normal distribution.
A read-only property for the variance of a normal distribution. Equal to the square of the standard deviation.
classmethod from_samples(data)¶
Makes a normal distribution instance with mu and sigma parameters estimated from the data using fmean() and stdev().
The data can be any iterable and should consist of values that can be converted to type float. If data does not contain at least two elements, raises StatisticsError because it takes at least
one point to estimate a central value and at least two points to estimate dispersion.
samples(n, *, seed=None)¶
Generates n random samples for a given mean and standard deviation. Returns a list of float values.
If seed is given, creates a new instance of the underlying random number generator. This is useful for creating reproducible results, even in a multi-threading context.
Using a probability density function (pdf), compute the relative likelihood that a random variable X will be near the given value x. Mathematically, it is the limit of the ratio P(x <= X <
x+dx) / dx as dx approaches zero.
The relative likelihood is computed as the probability of a sample occurring in a narrow range divided by the width of the range (hence the word "density"). Since the likelihood is relative
to other points, its value can be greater than 1.0.
Using a cumulative distribution function (cdf), compute the probability that a random variable X will be less than or equal to x. Mathematically, it is written P(X <= x).
Compute the inverse cumulative distribution function, also known as the quantile function or the percent-point function. Mathematically, it is written x : P(X <= x) = p.
Finds the value x of the random variable X such that the probability of the variable being less than or equal to that value equals the given probability p.
Measures the agreement between two normal probability distributions. Returns a value between 0.0 and 1.0 giving the overlapping area for the two probability density functions.
Divide the normal distribution into n continuous intervals with equal probability. Returns a list of (n - 1) cut points separating the intervals.
Set n to 4 for quartiles (the default). Set n to 10 for deciles. Set n to 100 for percentiles which gives the 99 cuts points that separate the normal distribution into 100 equal sized groups.
Instances of NormalDist support addition, subtraction, multiplication and division by a constant. These operations are used for translation and scaling. For example:
>>> temperature_february = NormalDist(5, 2.5) # Celsius
>>> temperature_february * (9/5) + 32 # Fahrenheit
NormalDist(mu=41.0, sigma=4.5)
Dividing a constant by an instance of NormalDist is not supported because the result wouldn't be normally distributed.
Since normal distributions arise from additive effects of independent variables, it is possible to add and subtract two independent normally distributed random variables represented as instances
of NormalDist. For example:
>>> birth_weights = NormalDist.from_samples([2.5, 3.1, 2.1, 2.4, 2.7, 3.5])
>>> drug_effects = NormalDist(0.4, 0.15)
>>> combined = birth_weights + drug_effects
>>> round(combined.mean, 1)
>>> round(combined.stdev, 1)
NormalDist readily solves classic probability problems.
For example, given historical data for SAT exams showing that scores are normally distributed with a mean of 1060 and a standard deviation of 195, determine the percentage of students with test
scores between 1100 and 1200, after rounding to the nearest whole number:
>>> sat = NormalDist(1060, 195)
>>> fraction = sat.cdf(1200 + 0.5) - sat.cdf(1100 - 0.5)
>>> round(fraction * 100.0, 1)
Find the quartiles and deciles for the SAT scores:
>>> list(map(round, sat.quantiles()))
[928, 1060, 1192]
>>> list(map(round, sat.quantiles(n=10)))
[810, 896, 958, 1011, 1060, 1109, 1162, 1224, 1310]
To estimate the distribution for a model than isn't easy to solve analytically, NormalDist can generate input samples for a Monte Carlo simulation:
>>> def model(x, y, z):
... return (3*x + 7*x*y - 5*y) / (11 * z)
>>> n = 100_000
>>> X = NormalDist(10, 2.5).samples(n, seed=3652260728)
>>> Y = NormalDist(15, 1.75).samples(n, seed=4582495471)
>>> Z = NormalDist(50, 1.25).samples(n, seed=6582483453)
>>> quantiles(map(model, X, Y, Z))
[1.4591308524824727, 1.8035946855390597, 2.175091447274739]
Normal distributions can be used to approximate Binomial distributions when the sample size is large and when the probability of a successful trial is near 50%.
For example, an open source conference has 750 attendees and two rooms with a 500 person capacity. There is a talk about Python and another about Ruby. In previous conferences, 65% of the attendees
preferred to listen to Python talks. Assuming the population preferences haven't changed, what is the probability that the Python room will stay within its capacity limits?
>>> n = 750 # Sample size
>>> p = 0.65 # Preference for Python
>>> q = 1.0 - p # Preference for Ruby
>>> k = 500 # Room capacity
>>> # Approximation using the cumulative normal distribution
>>> from math import sqrt
>>> round(NormalDist(mu=n*p, sigma=sqrt(n*p*q)).cdf(k + 0.5), 4)
>>> # Solution using the cumulative binomial distribution
>>> from math import comb, fsum
>>> round(fsum(comb(n, r) * p**r * q**(n-r) for r in range(k+1)), 4)
>>> # Approximation using a simulation
>>> from random import seed, choices
>>> seed(8675309)
>>> def trial():
... return choices(('Python', 'Ruby'), (p, q), k=n).count('Python')
>>> mean(trial() <= k for i in range(10_000))
Normal distributions commonly arise in machine learning problems.
Wikipedia has a nice example of a Naive Bayesian Classifier. The challenge is to predict a person's gender from measurements of normally distributed features including height, weight, and foot size.
We're given a training dataset with measurements for eight people. The measurements are assumed to be normally distributed, so we summarize the data with NormalDist:
>>> height_male = NormalDist.from_samples([6, 5.92, 5.58, 5.92])
>>> height_female = NormalDist.from_samples([5, 5.5, 5.42, 5.75])
>>> weight_male = NormalDist.from_samples([180, 190, 170, 165])
>>> weight_female = NormalDist.from_samples([100, 150, 130, 150])
>>> foot_size_male = NormalDist.from_samples([12, 11, 12, 10])
>>> foot_size_female = NormalDist.from_samples([6, 8, 7, 9])
Next, we encounter a new person whose feature measurements are known but whose gender is unknown:
>>> ht = 6.0 # height
>>> wt = 130 # weight
>>> fs = 8 # foot size
Starting with a 50% prior probability of being male or female, we compute the posterior as the prior times the product of likelihoods for the feature measurements given the gender:
>>> prior_male = 0.5
>>> prior_female = 0.5
>>> posterior_male = (prior_male * height_male.pdf(ht) *
... weight_male.pdf(wt) * foot_size_male.pdf(fs))
>>> posterior_female = (prior_female * height_female.pdf(ht) *
... weight_female.pdf(wt) * foot_size_female.pdf(fs))
The final prediction goes to the largest posterior. This is known as the maximum a posteriori or MAP:
>>> 'male' if posterior_male > posterior_female else 'female'
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Problem 038 – bridges of Königsberg
You are on vacation and must find the most efficient way to cross all bridges. How will you do that?
Satellite view of Kaliningrad, Russia.
Problem statement
If you like solving riddles and puzzles, it is likely that you have already encountered this puzzle. But even if you have, it is always good to go back and think about the classics. On top of that, I
will formulate the problem in a slightly different way, so that you can be entertained for a bit even if you already know the more classic version.
Take a look at this satellite view from Kaliningrad, Russia, where I have highlighted seven bridges:
Seven highlighted bridges in Kaliningrad, Russia.
Your task is to figure out what route to take if what you want to do is cross all of the highlighted bridges at least once but, at the same time, keep the total number of crossed bridges as low as
Having said that, what is the best route you can come up with?
(Just to be clear, I don't care about the length of the route – the number of miles/kilometres you would walk/drive – I only care about the number of bridges you cross.)
If you need any clarification whatsoever, feel free to ask in the comment section below.
In case you are wondering, the classic version of this puzzle is dubbed “the seven bridges of Königsberg” because that is what this place was called when a famous mathematician first dwelled on this
Congratulations to the ones that solved this problem correctly and, in particular, to the ones who sent me their correct solutions:
There are seven distinct bridges that we want to traverse, so we know the shortest path has to go over seven bridges, minimum. What we will show is that, actually, we need to go over eight bridges in
total in order to visit all seven bridges.
In order to show that is the case, consider the following figure:
Numbered pieces of land connected to the bridges.
In the figure above we can see that I numbered the four pieces of land to which the bridges are connected.
How many bridges does each piece of land connect to?
1. connects to 3 bridges;
2. connects to 3 bridges (as well);
3. connects to 3 bridges (as well); and
4. connects to 5 bridges.
Now we will use this information to show that it is impossible to create a path that visits all bridges exactly once.
Let us think about a hypothetical path we would do, in order to traverse all the bridges exactly once. More specifically, let us think about what happens in the middle of our walk. If we are in the
middle of our path, when we enter some piece of land through a bridge, we have to leave that piece of land through another bridge. In other words, for each time we arrive at a piece of land, we also
have to leave.
Suppose that the piece of land with the number \(1\) is neither the departing point nor the arrival point. This means that whenever we reach land number \(1\) we also have to leave it, which means
that we need an even number of bridges connected to \(1\), so that we are sure we can leave it whenever we arrive at it... But land number \(1\) has an odd number of bridges, so land number \(1\)
must be the departing point or the arrival point.
However, all four pieces of land have an odd number of bridges, so all four pieces of land would have to be either the departing or the arrival point, which cannot happen because we can't depart from
/arrive to multiple points. Therefore, a path that crosses each bridge exactly once won't cut it.
Now, can you come up with a path that crosses every bridge and only repeats one of them?
Here is an example of such a path:
An example of a shortest path to visit all seven bridges at least once.
This is actually one of the problems I solve with the participants in my workshop on recreational maths and I have to tell you: it is always fun to explore the mathematics behind such a misleading
For those of you who are curious enough, what we touched upon on this problem is what we call an “Eulerian path”.
Don't forget to subscribe to the newsletter to get bi-weekly problems sent straight to your inbox and to add your reaction below.
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Lévy canonical representation
From Encyclopedia of Mathematics
A formula for the logarithm characteristic function of an infinitely-divisible distribution:
where the characteristics of the Lévy canonical representation,
To every infinitely-divisible distribution there corresponds a unique system of characteristics
Thus, for the normal distribution with mean
For the Poisson distribution with parameter
To the stable distribution with exponent
where Lévy–Khinchin canonical representation. The probabilistic meaning of the functions
such that
In turn, a separable process
i.e. to the number of jumps with heights in
A similar relation holds for the function
Many properties of the behaviour of the sample trajectories of a separable process
then almost-all the sample functions of
then with probability 1 the sample trajectories of infinitesimal operator corresponding to the process
There are analogues of the Lévy canonical representation for infinitely-divisible distributions given on a wide class of algebraic structures.
[1] B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian)
[2] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)
[3] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)
[4] I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian)
[5] K. Itô, "Stochastic processes" , Aarhus Univ. (1969)
[a1] M. Loève, "Probability theory" , 1 , Springer (1977)
[a2] L.P. Breiman, "Probability" , Addison-Wesley (1968)
[a3] E. Lukacs, "Characteristic functions" , Griffin (1970)
[a4] H. Heyer, "Probability measures on locally compact groups" , Springer (1977)
[a5] K.R. Parthasarathy, "Probability measures on metric spaces" , Acad. Press (1967)
[a6] B.V. Gnedenko, A.N. Kolmogorov, "Introduction to the theory of random processes" , Saunders (1969) (Translated from Russian)
How to Cite This Entry:
Lévy canonical representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%A9vy_canonical_representation&oldid=12582
This article was adapted from an original article by B.A. Rogozin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
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Conrad Wolfram on computational thinking and revolutionizing Math education
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Conrad Wolfram on computational thinking and revolutionizing Math education
• Modern education should focus on computational thinking and problem framing, moving away from manual calculations to utilizing AI and natural language interfaces.
• Preparing students for the future involves teaching them to understand and use advanced computational tools, making math more relevant and engaging to their real lives and future careers.
On this episode of the Getting Smart Podcast, Tom Vander Ark talks to Conrad Wolfram, CEO of Wolfram Research and author of The Math(s) Fix, to discuss the evolving role of computational thinking in
education. They explore how the surge in computational power and AI can transform math education by moving away from manual calculations and focusing on real-world problem-solving.
Conrad Wolfram shares insights on the necessity of integrating computational tools into the curriculum, emphasizing that modern education should prepare students for complex problem-solving using AI
and natural language interfaces. They also discuss the challenges and opportunities in updating math education to reflect these advancements, aiming to equip students with skills relevant to today’s
tech-driven world.
Tom Vander Ark: We’re talking about computation and computational thinking today. I’m Tom Vander Ark. You’re listening to the Getting Smart podcast, and we’re joined today by a repeat guest, Conrad
Wolfram, the CEO of Wolfram Research and the author of The Math(s) Fix. Conrad, it’s great to see you again.
Conrad Wolfram: Yeah, very nice to be back.
Tom Vander Ark: We ran into each other in the hallway at South By and vowed for a reunion to discuss what’s right and wrong with math education in the world. You and I have shared careers in the
information age, where the nature of computational power available to human beings dramatically increased and transformed the way we live and work. It didn’t, however, transform the math expectations
that we have for young people. We continue to torture young people with hand calculation.
The Evolution of Human-Computer Interaction
We’re going to talk about that, but since we recorded our last podcast and since you published The Math(s) Fix four years ago, the world has changed. The computer age ended, and I would argue we
started a new age of human-computer interaction where we are interacting with reasoning and creation engines with a new level of power and sophistication. And we’re doing it primarily through natural
language. The user interface for human-computer interaction is changing. I’d love your take on this frontier and what’s happened. Have we really shifted to natural language, and is natural language
an adequate interface for the sort of complex problem-solving that you at Wolfram have been world leaders in for the last 30 years? Where are we? What’s happening?
Conrad Wolfram: Yeah, it’s a really, really interesting question. I agree with you. I’m talking AI age. So, you know, I don’t know where we are exactly, but it’s a new industrial revolution, probably
the fastest moving in human history. And it’s very quintessentially human in the sense that industrial revolutions were about brawn rather than brain. This one’s clearly in the latter category. And
so that feels very personal in a sense. In terms of the interface, I mean, my observation is twofold.
The Role of Natural Language in AI
I think we have had a natural language interface before in a way, but in slightly different ways. Search was something where we started using natural language again much more than we had before. The
back end of that was much less sophisticated, and we were changing our language, so it wasn’t very natural. Back in 2009, we launched Wolfram Alpha, and the idea there was to get computations done
using pretty much natural language. Now, I think what we’ve got is a far more natural interface to something that feels more humanistic but has all sorts of problems associated with that, some of
which may get ironed out.
One observation I would make is that I think through history, we’ve often got new interface types. Or they become prevalent, like what was called WYSIWYG (What You See is What You Get) with windowing
and so forth. I think typically they’ve added to previous interfaces more than got rid of them. For example, we still type, and I think we will still at some level type syntactic code even though we
have windowing, because the windowing stuff works very well in a set of cases and expands how we can interact with the technology. But I don’t think it’s the whole story in every case. So I think
there are two or three things to disentangle or to slice differently. I think our interface, whether it’s natural language or very abstract, will continue to include both. Abstract representation to
get precision and to turn a lot of different effects into the same representation will remain crucial. It allows us to make decisions, solve problems, and make progress. The interface to that may be
linguistic to set things up, but there are many things we’ll do linguistically now that we would have had to be very deliberate about before. Then there’s also the back end of how the linguistics
work. One thing I’ve said about Wolfram Alpha versus LLMs like ChatGPT is that if they were humans, LLMs are sort of at one end of the spectrum. They interact nicely with humans and have a very nice
form of words. It isn’t necessarily accurate. Wolfram Alpha is more at the aspergic end of the spectrum. We are fact-based, very definite, and accurate. We’re not always the best at communication.
Actually, it’s been very exciting to see how those two have been put together to sort of make the best of both worlds. A bit like a police drama.
Tom Vander Ark: So you’re definitely a believer that when it comes to attacking problems, a mix of these models is best. What does that mean for the interface? Can we rely increasingly on natural
language, or should we continue to use abstract languages like the language of math and the language of Python and Java? Will we continue to use those languages, and how and when are they going to be
Conrad Wolfram: So I suppose my belief is the following: At some level of complexity in describing computational ideas, you need an abstract representation. So yes, I think we need computer
languages. But computer science and the like, which have been about humans writing by hand, will dissipate. Different forms of AI, including Wolfram Alpha and LLMs, will write code which we may
explicitly see or may not see, but we may sometimes by hand have to edit that code. In advanced cases, we may want to do something with it. We may want to understand it. It may be cleaner for us to
look at the code to understand it and work with it than to speak in English. You can discuss lots of problems in English and get various advantages from discussing them using math notation. You get
precision and the fact that a biological effect, a physics effect, and several other effects end up representing the same abstraction, giving you tremendous power. You can use the same process of
getting to an answer for that abstraction. So the idea that we can just speak with imprecise human language and replace all the abstraction developed over hundreds of years is inaccurate. We have a
much wider use and convenience of natural language, but sometimes we will need explicit abstraction in fewer cases. They’ll build on each other. We have our own Wolfram language, which is a
high-level language with 6500 or so built-in functions, vastly more than Python. Once people learn the vocabulary, their interaction speed is much better than with Python. It represents their thought
processes more directly because they have more words to represent what they’re saying. With human languages, a large vocabulary lets you directly get to a meaning. LLMs can handle that vocabulary
easily. We get very short, effective Wolfram Language code from LLMs, providing very nice abstractions for humans to use.
Tom Vander Ark: There’s an interesting parallel around content knowledge in domains and industry verticals. When search became popular 15 years ago, there was the idea that we don’t need to learn
content anymore because you can just Google anything. With a language interface, our task shifts to editing and curating creation. That content knowledge, as well as the knowledge of abstract
language and creating quality prompts and editing for better answers, requires both content knowledge and abstract language knowledge. Content experts are making better use of the tools. Do you
Conrad Wolfram: Yeah, this is very much what I’ve said and continue to say in different domains. When you’ve got new technology, new machinery, what’s the human role? The human role that succeeds is
to zoom up a level. Instead of pulling levers at the ground level, you’re defining the envelope of change. When you drive a car, you’re not changing the fuel mixture. The car figures that out. You’re
commanding the car to accelerate, brake, or go from A to B. We’re becoming more like the CEO of the process rather than the ground-level expert, but being a CEO is difficult for several reasons. I
describe this in The Math(s) Fix. Think about Steve Jobs or Elon Musk. They’ve got a zoom-up, zoom-down issue. They look at the big strategic picture, like the future of handheld communicators with
touchscreens, while also obsessing over the radius of a corner to make it feel right. It’s getting the big picture to head in the right direction and zooming into details to get precision and the
right answer. That’s very hard to do. Very successful people like Jobs and Musk didn’t always get it right. Humans need to improve at this, and it becomes harder with more automation.
Revolutionizing Math Education
Tom Vander Ark: So let’s maybe summarize where we are, particularly in regard to education. At a time where a powerful set of tools is available to learners in high school as well as university,
allowing them to take on far more complicated problems than a young person could have done five or ten years ago and actually deliver value to their community. Is that fair?
Conrad Wolfram: Yes, it is. The learning they need and the contribution they make change quite a bit. It’s easy to see this, especially with math. Math is one of the world’s most successful
problem-solving systems. And machines now do the hardest part. For hundreds of years, math was great, but the limitation was turning the abstract question into the abstract answer by hand. We
developed clever systems for minimizing calculating. A friend of Alan Turing, who was one of my math teachers, said math is the art of avoiding calculation. And he was right. But in the last few
decades, calculating has become incredibly cheap, and we can calculate pretty much anything we want. The question is what to calculate and how to set up problems to effectively use math systems. We
must not get fooled by results because the more complicated the problem, the harder it is to verify. These are the crucial steps now. The exciting part in education is that many students get turned
off by math when it becomes very abstract, often around late primary school. Now we can give them hard problems that seem relevant to their lives but that previously didn’t seem linked to math
because they couldn’t use it to get a better answer. Now they can. They need to know a much wider range of toolsets, understand when to deploy them, and recognize when things go wrong. For example,
consider the Tour de France bike race. Let’s look at all the parameters and try to understand the factors like air resistance, angle to the wind, and rolling resistance of the bike. These are hard,
messy problems, but they can be tackled by 12-to-15-year-olds. This real experience, or at least some educational version of it, is much closer to real applications than what we’re doing now. It
pushes problem-solving skills into areas that are both more exciting to students and more relevant to their lives.
Tom Vander Ark: So I think you’ve highlighted two problems with our current math education. One is that we give students problems rather than inviting them to find and frame them. Problem finding and
opportunity recognition are more important than ever. Second, we spend a significant portion of time teaching them how to solve given problems using hand calculation. Let’s shelve problem finding for
a moment. Why are we still teaching hand calculation? Is there any value to long division and factoring polynomials and multiplying fractions?
Conrad Wolfram: Is there any value to learning the pluperfect subjunctive of “amo” in Latin? Well, there’s some value, but I wouldn’t force everyone to learn it. I think there are better things to
do. The overarching problem in most places is that assessments tie down subject changes. Math is critical because of the tech needs, and it’s seen as more accessible through numbers, so trying to
change its content faces a massive ecosystem shift. This overarching failure of the incentive framework makes it hard for incentives to align with real-world changes. We need rapid subject evolution
matching the real world. Another underlying problem, especially in the U.S., is the difference between the essence of the subject and the mechanics we focus on. For instance, in photography, we might
still start with loading film into a camera, though it’s not essential in modern photography.
Tom Vander Ark: People suggest stopping teaching algebra to teach data science, which is a crude approach. You and I support teaching algebraic reasoning and multivariable problem-solving. We’re
giving kids problems instead of inviting them to find and frame problems where they identify variables. We’re teaching hand calculation rather than modeling complex systems, teaching the wrong
algebra in the wrong way, emphasizing hand calculations. Do you agree?
Conrad Wolfram: I might characterize it differently. The problem is when you say algebra, let’s say equation solving, you’ve got to look into the details. For example, you might want to model an
effect to get a better answer. What’s the best tool for that? Is it machine learning, traditional algebraic equation solving, or another tool? We don’t address this at all in schools, which is
catastrophic. The equation that best matches the problem, regardless of how horrific it looks, is important because computers can solve it. The curriculum is linked to outdated techniques. The
algebra taught is related to techniques from a hundred years ago. These techniques aren’t irrelevant, but we need to use them differently, with more complex versions, without doing them by hand. The
data science vs. algebra debate is a false dichotomy.
Tom Vander Ark: This debate has been conflated with the reading wars where explicit phonics aligns with the science of reading. Some are conflating this with teaching mathematics, assuming teaching
hand calculations is equally important.
Conrad Wolfram: That’s an important point. That mistake has confused some, including people in the UK government. One key difference is literacy is a function we can all agree is an outcome we need.
Writing and composing are essential. The question is how to best achieve literacy. I don’t know all the pluses and minuses of phonics, but it’s a mechanism for that outcome. We can agree on the
outcome. The problem with math is we don’t agree on the outcome. Conflating the phonics argument with math reform is false.
Tom Vander Ark: The two objections I hear regarding hand calculation are related to editing the work of a computer and boosting computational thinking. Do you think learning hand calculations helps
with either editing or boosting computational thinking?
Conrad Wolfram: There’s some value, but there are better ways to do it with fewer negative consequences. It’s like learning Latin to learn English—useful but not necessarily the best approach. Hand
calculations aren’t the right starting point. We have numerous ways to solve problems now thanks to mechanized calculating, so computational thinking needs to focus on understanding how tools operate
and setting problems appropriately. One big mistake with current approaches is training people to think in outdated ways. Traditional math is akin to learning to drive a horse and cart instead of a
car. It isn’t hitting the skills we need.
Future of Computational Thinking in Education
Tom Vander Ark: I want to be clear for our listeners that we aren’t arguing for less math but for more computation and computational thinking. We want young people engaged in more rigorous
problem-solving earlier on significant challenges and contributing sooner. This isn’t less math; it’s more math utilizing computational power and learning earlier. Is that fair?
Conrad Wolfram: More than fair. The intellectual and conceptual are becoming the same as the vocational or practical. Anything procedural will be done by machines. Training people for manual
procedures won’t prepare them for useful tasks. For example, science hasn’t become conceptually simpler since the advent of computers; it’s become harder because of more available options through
machinery. We need higher-level, more conceptual mathematics, more practical for real problems. Before computers, math wasn’t useful for many real-life applications beyond accounting and bits of
physics. It was useless for most biology or social system modeling. Now, thanks to mechanized calculating, math is highly relevant. Stripping out mechanized calculating from education removes its
real-world context and utility. Every job with a family wage or better involves mathematics and computational thinking daily. Whether as a biologist, geologist, or lawyer, every job now has a
computational foundation. These are critical skills for everyone.
Tom Vander Ark: When you wrote The Math(s) Fix, I hoped it would change the world, leading to new math learning expectations and better assessments. That hasn’t happened much. Teachers are still
trapped with dated math expectations. How do we fix this?
Conrad Wolfram: I wish I had the full answer. On the ground in places like the U.S. and UK, not much has changed, but the overall sentiment has shifted more than one might think. Several factors
helped, including my book and the AI revolution. When we released Wolfram Alpha in 2009, educators were excited about integral calculation, something we’d been doing for 20 years. AI and LLMs have
pushed the idea that all subjects might be wrong, leading to a push for changing content rather than just teaching methods. The last 30 years have seen a mismatch between real-world math and school
teaching. We better not replicate this mistake for AI.
Tom Vander Ark: I couldn’t agree more. At the recent air show, many apps were automating bad instruction, including hand calculations in math. These well-intentioned efforts are trapped in bad
policies linked to outdated expectations.
Conrad Wolfram: There are various ways to address this. One way I’ve tried is working with countries interested in adopting early. We’ve had a few successes. Sadly, I wish a U.S. state would step up.
Tom Vander Ark: What about Singapore? Historically respected their math education. Any chance a smaller country could flip?
Conrad Wolfram: Singapore isn’t as revolutionary in content as we might think. They teach the sensible end of traditional math and do it well within their culture. They have pedagogical
sophistication but aren’t revolutionary in content.
Tom Vander Ark: What about the apprenticeship pathways in Germany, Switzerland, and Scandinavia? Do you see them incorporating more relevant math expectations?
Conrad Wolfram: Not yet in those countries, but vocational directions are incredibly powerful. Questioning universities and traditional college paths opens exciting possibilities. There are new
universities and different tracks where traditional methods aren’t bought into. One major driver in many countries is college admission. Colleges often note top students can’t do much useful when
they arrive and require retraining. This could influence schools to change. There’s progress, but top-down pressure is needed.
Tom Vander Ark: This all became very real for me recently. My granddaughter came home from a school concert and said, “Papa, learning long division in school seems so dumb. What do you think about
Conrad Wolfram: She’s off on a good track.
Tom Vander Ark: It put me on the spot. And I think many kids and teachers are in this same position, stuck between outdated school gateways and college requirements. We have work to do.
Conrad Wolfram: We do. In some ways, assessments are becoming devalued, which opens possibilities. Other currencies could emerge, offering options.
Tom Vander Ark: Many states are adopting a new portrait of a graduate. In some cases, they’re creating new diploma pathways, and I think that’s going to be the key opening: to create new diploma
pathways that are linked to high-wage, high-demand employment. I think we’ll be able to create a fast pathway that values computational thinking and assesses it in new and better ways. I’m hoping
that once a handful of those openings are created, we’ll be able to flip the system.
Conrad Wolfram: Yeah. One thing just to add that I think, as you say, AI for pedagogy rather than changing the subject is quite prevalent. But I would say our early experiments with computer-based
math content are quite exciting in terms of having AIs help to teach, tutor, and assess more open-ended computational thinking and math. I was a bit skeptical of how well this would work, but it
turns out that the framework we’ve built for helping teachers to understand how to teach this is actually very helpful for getting AIs to help teach it. I’m actually rather positive because one of
the impediments we’ve had, among the other ones we’ve talked about, is how to roll this out with the current teaching workforce. Well, I think there is some great assistance to that now coming down
the track. I think AIs will help directly, you know, correctly deployed for that purpose.
Tom Vander Ark: Conrad, we so value your advocacy, not to mention the sort of computational power that your family has brought to the world. I want to urge everybody to pick up a copy of The Math(s)
Fix if you haven’t read it already. And for a quick fix, check out ComputerBasedMath.org. Is that still a good resource?
Conrad Wolfram: That’s still a good resource. I’ve got some blog posts people might enjoy on conradwolfram.com as well.
Tom Vander Ark: We will include a few links to those. Thank you. We’ll include a link to our friends at South Fayette Schools, just outside of Pittsburgh, which is another great example of a U.S.
school district that has a beautiful computational thinking framework K through 12 that they developed with Carnegie Mellon. So around the edges, we’re seeing people rethink computation in large part
because of your work and your advocacy. So thanks for being with us, Conrad.
Conrad Wolfram: Thanks. It’s great to talk again, Tom.
Tom Vander Ark: Thanks to our producer, Mason Pasha, and the whole Getting Smart team for making this possible. Until next week, keep learning, keep leading, and keep advocating for computational
thinking. See you next week.
Conrad Wolfram
Conrad Wolfram is strategic director and European cofounder/CEO of Wolfram Research, founder of computerbasedmath.org and author of “The Math(s) Fix”.
Over the last 30 years he has been a key part of the technology transformation that has brought maths, computation and data science to the forefront of today’s world and moved us towards the 4th
industrial revolution. Conrad regularly appears in the media, speaking about subjects ranging from the computational future and artificial intelligence to 21st-century education.
Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use
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Section 1: Relations and functions; Graphing techniques; Absolute value; Point-slope form of a line
Section 2: Tangent and sectant lines; Position and velocity; Rates of change; Definition of a derivative
Section 3: Limit of a function; One-sided limits; Continuity; Limit properties; Indeterminate forms
Section 4: Rules for differentiation: polynomials, product, quotient, and power rules; Composition of functions
** This is a Print-on-Demand product; it is non-returnable.
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The non-abelian localization theorem and the Verlinde formula for Higgs bundles.
Printable PDF
Department of Mathematics,
University of California San Diego
Algebraic Geometry Seminar
Daniel Halpern-Leistner
Columbia University
The non-abelian localization theorem and the Verlinde formula for Higgs bundles.
The Verlinde formula is a celebrated explicit computation of the dimension of the space of sections of certain positive line bundles over the moduli space of semistable vector bundles over an
algebraic curve. I will describe a recent generalization of this formula in which the moduli of vector bundles is replaced by the moduli of semistable Higgs bundles, a moduli space of great interest
in geometric representation theory. A key part of the proof is a new ``virtual non-abelian localization formula" in K-theory, which has broader applications in enumerative geometry. The localization
formula is an application of the nascent theory of Theta-stratifications, and it serves as a new source of applications of derived algebraic geometry to more classical questions.
Host: James McKernan
December 5, 2016
1:45 PM
AP&M 6402
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What is Fourier series representation of periodic signals?
The Fourier series represents periodic, continuous-time signals as a weighted sum of continuous-time sinusoids. It is widely used to analyze and synthesize periodic signals.
Which of the following are types of representation of discrete-time sequences?
There are three ways to represent discrete time signals.
• Functional Representation.
• Tabular method of representation.
• Sequence Representation.
Is discrete Fourier series periodic?
In digital signal processing, the term Discrete Fourier series (DFS) is any periodic discrete-time signal comprising harmonically-related (i.e. Fourier) discrete real sinusoids or discrete complex
exponentials, combined by a weighted summation. A specific example is the inverse discrete Fourier transform (inverse DFT).
What is the Fourier series of a periodic function?
A Fourier series (/ˈfʊrieɪ, -iər/) is a sum that represents a periodic function as a sum of sine and cosine waves. The frequency of each wave in the sum, or harmonic, is an integer multiple of the
periodic function’s fundamental frequency. Each harmonic’s phase and amplitude can be determined using harmonic analysis.
What do you mean by DFT?
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time
Fourier transform (DTFT), which is a complex-valued function of frequency.
What is Fourier series in signal and system?
The Fourier Series is a specialized tool that allows for any periodic signal (subject to certain conditions) to be decomposed into an infinite sum of everlasting sinusoids. This may not be obvious to
many people, but it is demonstrable both mathematically and graphically.
What are the different types of representation of discrete?
Representation of a Discrete Time Signal
• Graphical Representation.
• Functional Representation.
• Tabular Representation.
• Sequence Representation.
How do you find the representation of a Fourier series?
To find the coefficients a0, an and bn we use these formulas:
1. a0 = 12L. L. −L. f(x) dx.
2. an = 1L. L. −L. f(x) cos(nxπL) dx.
3. bn = 1L. L. −L. f(x) sin(nxπL) dx.
What is the discrete Fourier series expansion of the periodic sequence?
Hence Eq. (2.108) is recognized as the discrete Fourier series expansion of the periodic sequence {x ( n )}, and { X ( k )} are just the discrete Fourier series coefficients scaled by N. Conventional
frequency domain interpretation permits an identification of X (0)/ N as the “DC” value of the signal.
How many types of Fourier series representations are there?
There are two types of Fourier series representations, both are equivalent to each other. Depending on the type of signal, most convenient representation is chosen. J. B. J. Fourier demonstrated that
a periodic function f (t) can be expressed as a sum of sinusoidal functions.
What is the Fourier transform in frequency spectrum analysis?
As with the discrete Fourier series, the DFT produces a set of coefficients, which are sampled values of the frequency spectrum at regular intervals. The number of samples obtained depends on the
number of samples in the time sequence. A time sequence x ( n) is transformed into a sequence X (ω) by the discrete Fourier transform.
What is discrete Fourier series (DFS)?
Discrete Fourier Series Computational schemes can only be applied to discrete signals, and the continuous signals acquired by measurements are thus digitized. The DFS is the Fourier tool suitable for
decomposing discrete periodic signals of the form: (52a)x(n) = x(n + N)
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EViews Help: @trendcoef
Trend coefficient from detrending regression.
Computes the trend coefficient (or coefficients for panel data) of an OLS regression versus a constant and an implicit time trend.
Syntax: @trendcoef(x[, s])
x: series, vector, matrix
s: (optional) sample string or object when x is a series and assigning to series
Return: number
Returns the trend coefficient of an OLS regression on matrix or vector m versus a constant and an implicit time trend, as in @detrend.
When applied to a matrix, the matrix elements are arranged in vectorization order and then paired with the implicit time trend.
This function is panel aware.
For series calculations, EViews will use the current or specified workfile sample.
series y = 2 + 3 * @trend + @nrnd
= @trendcoef(y)
The first line generates the series y using a simple linear regression model, where the sole regressor is a time trend, the intercept is 2, and the slope coefficient is 3. The second line returns the
OLS estimate of the slope coefficient, and is approximately 3 in large samples.
The following commands begin by creating a workfile, generating a random series, and then converting to a vector so that we can compute identical results using the series and the vector.
workfile u 100
series y = nrnd
vector yv = @convert(y)
The trend regression trend coefficient estimate is given by
scalar tr1 = @trendcoef(yv)
Alternately, the trend regression results may be obtained using the vector YV using the @regress command on the augmented data matrix:
matrix vcoefs = @regress(@hcat(yv, @ones(yv.@rows), @range(0, yv.@rows-1)))
The first column of VCOEFS contains the intercept and trend coefficient, so
scalar tr2 = vcoefs(2)
is the trend coefficient.
Estimates may also be obtained using the series and an equation object
equation eq1.ls y c @trend
scalar tr3 = eq1.c(2)
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This page provides information about online lectures and lecture slides for use in teaching and learning from the book Computer Science: An Interdisciplinary Approach. These lectures are appropriate
for use by instructors as the basis for a “flipped” class on the subject, or for self-study by individuals.
Flipped classroom.
If you are an an instructor teaching introductory computer science, an effective way for you to teach the material in a typical college class is to adhere to a weekly cadence, as follows:
• Each week, send an email to all students in the class that briefly describes activities for that week (lectures, reading, and programming assignments drawn from the book or from this booksite).
• Students watch the lecture videos at their own pace, do the readings, and work on the programming assignments.
• Schedule a weekly “class meeting” for discussion of the material, reviews for exams, informal interaction with students, and any enrichment material you may wish to cover.
This is just one suggestion—this material can support many different teaching styles and formats.
Important note: A common mistake in teaching a flipped class is to add too much enrichment material. Our experience is that time in class meetings is much better spent preparing students for success
on programming assignments and exams. If an instructor makes it clear that the best way to prepare for exams is to watch the lecture videos and do the reading, most students will do so. Class
meetings then can involve interacting with students and with the material in such a way as to reinforce understanding. For example, working with potential exam questions is an excellent activity.
An effective way to learn the material on your own is to watch the lecture videos on some regular schedule, do the associated reading, and attempt to solve some of the exercises in the book or on the
booksite on your own. If you get stuck on a particular exercise, find some others or try to solve some of the problems given in the lectures without looking at the solutions there.
Available lectures.
During the spring of 2020, the lecture videos are freely available. When watching a lecture video, it is
very important
to choose an appropriate speed. If it is too slow, you are likely to be bored; if it is too fast, you are likely to get lost. Also be sure to make liberal use of pause and rewind.
The lecture videos are available from CUbits; the lecture slides are in pdf format.
• Lecture 0: Prologue—A Simple Machine. This lecture introduces fundamental ideas of computation in the context of a familiar and important application from the field of cryptography. The story
motivates the study of computer science, but the concepts covered are a bit advanced, so novices may wish to review it again after watching the other lectures in the course.
• Lecture 1: Intro to Java. Why program? This lecture addresses that basic question. Then it describes the anatomy of your first program and the process of developing a program in Java using
either virtual terminals or a program development environment, with some historical context. Most of the lecture is devoted to a thorough coverage of Java's built-in data types, with example
programs for each.
• Lecture 2: Conditionals and Loops. The if, while, and for statements are Java's fundamental control structures. This lecture is built around short programs that use these constructs to address
important computational tasks. Examples include sorting, computing the square root, factoring, and simulating a random process. The lecture concludes with a detailed example illustrating the
process of debugging a program.
• Lecture 3: Arrays. Computing with a large sequence of values of the same type is extremely common. This lecture describes Java's built-in array data structure that supports such applications,
with several examples, including shuffling a deck of cards, the coupon collector test for randomness, and random walks in a grid.
• Lecture 4: Input and Output. To interact with our programs, we need mechanisms for taking information from the outside world and for presenting information to the outside world. This lecture
describes several such mechanisms: for text, drawings, and animation. Detailed examples covered include fractal drawings that model natural phenomena and an animation of a ball bouncing around in
the display window.
• Lecture 5: Functions and Libraries. Modular programming is the art and science of breaking a program into pieces that can be individually developed. This lecture introduces functions (Java
methods), a fundamental mechanism that enables modular programming. Motivating examples include functions for the classic Gaussian distribution and an application that creates digital music.
• Lecture 6: Recursion. A recursive function is one that calls itself. This lecture introduces the concept by treating in detail the ruler function and (related) classic examples, including the
Towers of Hanoi puzzle, the H-tree, and simple models of the real world based on recursion. We show a common pitfall in the use of recursion, and a simple way to avoid it, which introduces a
different (related) programming paradigm known as dynamic programming.
• Lecture 7: Performance. When you develop a program, you need to be aware of its resource requirements. In this lecture, we describe a scientific approach to understanding performance, where we
develop mathematical models describing the running time our programs and then run empirical tests to validate them. Eventually we come to a simple and effective approach that you can use to
predict the running time of your own programs that involve significant amounts of computation.
• Lecture 8: Abstract Data Types. In Java, you can create your own data types and use them in your programs. In this and the next lecture, we show how this ability allows us to view our programs
as abstract representations of real-world concepts. First we show the mechanics of writing client programs that use data types. Our examples involve abstractions such as color, images, and genes.
This style of programming is known as object-oriented programming because our programs manipulate objects, which hold data type values.
• Lecture 9: Creating Data Types. Creating your own data types is the central activity in modern Java programming. This lecture covers the mechanics (instance variables, constructors, instance
methods, and test clients) and then develops several examples, culminating in a program that uses a quintessential mathematical abstraction (complex numbers) to create visual representations of
the famous Mandelbrot set.
• Lecture 10: Programming Languages. We conclude the course with an overview of important issues surrounding programming languages. To convince you that your knowledge of Java will enable you to
learn other programming languages, we show implementations of a typical program in C, C++, Python, and Matlab. We describe important differences among these languages and address fundamental
issues, such as garbage collection, type checking, object oriented programming, and functional programming with some brief historical context.
• Lecture 11: Searching and Sorting. Building on the scientific approach developed in Lecture 7, we introduce and study classic algorithms for two fundamental problems, in the context of
realistic applications. Our message is that efficient algorithms (binary search and mergesort, in this case) are a key ingredient in addressing computational problems with scalable solutions that
can handle huge instances.
• Lecture 12: Stacks and Queues. Our introduction to data structures is a careful look at the fundamental stack and queue abstractions, including performance specifications. Then we introduce the
concept of linked structures and focus on their utility in developing simple, safe, clear, and efficient implementations of stacks and queues.
• Lecture 13: Symbol Tables. The symbol table abstraction is one of the most important and useful programmer's tools, s we illustrate with several examples in this lecture. Extending the
scientific approach of the previous two lectures, we introduce and study binary search trees, a classic data structure that supports efficient implementations of this abstraction.
• Lecture 14: Introduction to Theory of Computation. The theory of computation helps us address fundamental questions about the nature of computation while at the same time helping us better
understand the ways in which we interact with the computer. In this lecture, we introduce formal languages and abstract machines, focusing on simple models that are actually widely useful in
practical applications.
• Lecture 15: Turing Machines. In 1936, Alan Turing published a paper that is widely hailed as one of the most important scientific papers of the 20th century. This lecture is devoted to the two
far-reaching central ideas of the paper: All computational devices have equivalent computational power, and there are limitations to that power.
• Lecture 16: Intractability. As computer applications expanded, computer scientists and mathematicians realized that a refinement of Turing's ideas is needed. Which computational problems can we
solve with the resource limitations that are inescapable in the real world? As described in this lecture, this question, fundamentally, remains unanswered.
• Lecture 17: A Computing Machine. Every programmer needs understand the basic characteristics of the underlying computer processor being used. Fortunately, the fundamental design of computer
processors has changed little since the 1960s. In this lecture, we provide insights into how your Java code actually gets its job done by introducing an imaginary computer that is similar to both
the minicomputers of the 1960s and the microprocessor chips found in today's laptops and mobile devices.
• Lecture 18: von Neumann Machines. Continuing our description of processor design and low-level programming, we provide context stretching back to the 1950s and discuss future implications of
the von Neumann machine, where programs and data are kept in the same memory. We examine in detail the idea that we design new computers by simulating them on old ones, something that Turing's
theory guarantees will always be effective.
• Lecture 19: Combinational Circuits. Starting with a few simple abstractions (wires that can carry on/off values and switches that can control the values carried by wires), we address in this
lecture the design of the circuits that implement computer processors. We consider gates that implement simple logical functions and components for higher-level functions, such as addition. The
lecture culminates with a full circuit for an arithmetic/logic unit.
• Lecture 20: CPU. In this lecture we provide the last part of our answer to the question "How does a computer work?" by developing a complete circuit for a computer processor, where every switch
and wire is visible. While vastly different in scale, this circuit, from a design point of view, has many of the same characteristics as the circuits found in modern computational devices.
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ball mill grinding capacity
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The grinding experiments were performed in a batch mill with 150 mm in height and 130 mm in diameter, with a capacity of 2000 cm 3, and four lifters inside. The predetermined speed is 170 rev/
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Although it was developed nearly 50 years ago, Bond's method is still useful for calculating necessary mill sizes and power consumption for ball and rod mills. This paper discusses the basic
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the SAG mill feed, closing the primary grinding circuit. The horizontal screen undersize is pumped to two separate secondary grinding lines reversely configured. Each grinding line comprises a
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A rod mill and ball mill are both used in grinding materials. To get the best result, you need to know which mill is best suitable for your purposes. Herein are the difference between a rod Mill
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Its job, to grind rock by tumbling it in a large metal cylinder loaded with steel balls, is highly energy intensive. In fact, the cost of grinding in a mining operation represents a significant
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High temperature of the ball mill will affact the efficiency. 3 For every 1% increase in moisture, the output of the ball mill will be reduced by 8% 10%. 4 when the moisture is greater than 5%,
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When using a ballmill machine for grinding wet coals, its drying capacity limits the grinding capacity. Therefore, in order to analyze the efficiency of the BDM operation, it is necessary to
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To capacity required to grind a material from a giving feed size the a gives product size can be estimated by using that subsequent equal: find: W = power consumption declared to kWh/short to
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I'm interested to know when sizing a new ball mill, as to why recirculating load has no effect on the power calculation, although it will have effect on volumetric capacity naturally. For
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Antwort Is Windows 98 a GUI? Weitere Antworten – What is Windows 98 classified
Windows 98 is a consumer-oriented operating system developed by Microsoft as part of its Windows 9x family of Microsoft Windows operating systems. It is the second operating system in the 9x line, as
the successor to Windows 95. It was released to manufacturing on May 15, 1998, and generally to retail on June 25, 1998.The correct answer is Operating System. Windows 98 is an Operating System. An
Operating System (OS) is an interface between a computer user and computer hardware.Windows 98 is an operating system that lets you use different types of applications or software. For example, it
allows you to use a word processing application to write a letter and a spreadsheet application to track your financial information.
What architecture is Windows 98 : Windows 98
OS family Windows 9x
Version 4.10
Codename Memphis
Architecture x86 (PC/AT, PC-98)
Why Windows 98 is known as GUI based operating system
Answer: Technically, Windows IS a GUI. GUI stands for “Graphic User Interface”. It means that the OS is represented by a graphical screen the user can interact with using a mouse, touchpad, keyboard,
or any other input device.
Is Windows 98 is the first graphical version of Windows : It is FALSE. Released November 20, 1985, Windows 1.0 had a very low-key start. Windows 1.0 was not a complete operating system, but rather a
graphical environment for MS‑DOS.
Graphical User Interfaces
The term "Windows GUI" refers to the GUI provided by the Windows operating system, in contrast to other systems such as those provided by HTML on Web browsers. The term "Windows GUI" is sometimes
abbreviated to "GUI".
Graphical User Interface (GUI) is the up-to-date way of communication between a computer and a human being. All MS Windows versions use the GUI model of communication.
What is the first GUI of Windows
Windows 1.0
Windows 1.0, a GUI for the MS-DOS operating system was released in 1985.CUI is the precursor of GUI, and the user has to type on the keyboard to proceed in CUI. In contrast, GUI making it possible to
use a mouse instead of a keyboard. DOS, Windows Command Prompt is an instance of a CUI, whereas Windows is an example of a GUI. GUI is more user-friendly than CUI.The graphical user interface,
developed in the late 1970s by the Xerox Palo Alto research laboratory and deployed commercially in Apple's Macintosh and Microsoft's Windows operating systems, was designed as a response to the
problem of inefficient usability in early, text-based command-line interfaces for the average …
Microsoft Windows versions use the GUI model of communication. GUI is the acronym for Graphical User Interface. Graphical User Interface refers to a user interface using mouse, icons, and windows. It
displays objects that convey information, and represent actions that can be taken by the user.
Why is windows 98 known as a GUI-based operating system : Answer: Technically, Windows IS a GUI. GUI stands for “Graphic User Interface”. It means that the OS is represented by a graphical screen the
user can interact with using a mouse, touchpad, keyboard, or any other input device.
What is the oldest GUI : 1973 Xerox Alto
This effort culminated in the 1973 Xerox Alto, the first computer with a GUI, though the system never reached commercial production. The first commercially available computer with a GUI was the 1979
PERQ workstation, manufactured by Three Rivers Computer Corporation.
Why Windows are called GUI
A graphical user interface (GUI) is a digital interface in which a user interacts with graphical components such as icons, buttons, and menus. In a GUI, the visuals displayed in the user interface
convey information relevant to the user, as well as actions that they can take.
Graphical User Interface (GUI) is the up-to-date way of communication between a computer and a human being. All MS Windows versions use the GUI model of communication.Graphical User Interfaces
The term "Windows GUI" refers to the GUI provided by the Windows operating system, in contrast to other systems such as those provided by HTML on Web browsers. The term "Windows GUI" is sometimes
abbreviated to "GUI".
Is windows a type of GUI : Microsoft Windows versions use the GUI model of communication. GUI is the acronym for Graphical User Interface. Graphical User Interface refers to a user interface using
mouse, icons, and windows. It displays objects that convey information, and represent actions that can be taken by the user.
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TS ICET 2014 Question Paper
A and B started a business investing ₹ 10 lakhs and ₹ 15 lakhs respectively. After 6 months C joined them by investing ₹ 20 lakhs. If the profit at the end of the year is ₹ 5.6 lakhs, then
the share of A in the profit (in lakhs of rupees) is
If an article is sold at a profit of 15% instead of a profit of 9% the person gets ₹ 60 more. The cost price of the article (in rupees) is
A person bought a pen and sold it for a loss of 10%. If he had bought it for 20% less and sold it for ₹ 44 more than earlier sale price he would have made a profit of 40%. The cost price of the pen
is (in ₹)
In a library 23% of the books are in Arts. 30% in Commerce, 35% in Science and the rest are in Telugu language. If there are 1440 books in Telugu language, the number of books in Arts is
In a joint business A, B and C invested capital in the ratio 5 : 6 - 8. At the end of the business they shared profits in the ratio 4 : 3 : 12.'The ratio of the number of months in which A, B and C
kept, their capital is
Pipe A fills a tank in 8 hours while pipe B empties the full tank in 10 hours. If both the pipes A and B are opened simultaneously the time taken (in hours) to fill the tank is
Two pipes A and B can fill a tank in 10 hours and 15 hours respectively. If they are opened altemately for one hour each and if A is opened first, the time (in hours) required to fill the tank is
If a man starts at A and walks at 5 kmph he will reach B late by 7 minutes. But if walks at 6 kmph he will reach B early by 5 minutes. The distance between A and B (in km) is
A train of 270 metres long crosses a platform of 390 metres length in 33 seconds. The speed of the train (in kmph) is
Three persons A, B, C together can complete a work in 8 days where as A alone requires 24 days to complete the same work. The number of days-required for B and C together to complete the same work is
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Vector operations using Cartesian vector notation
Vectors and their Operations: Vector operations using Cartesian vector notation
Planar vector operations using CVN (two dimensions)
Addition of several vectors
1- Express each vector in CVN by resolving the vector to its scalar components:
2- Add the respective scalar components (components on the same axis):
in which
The above steps can be summarized as:
in which
3- Form the resultant vector
Remark: the apparent location of a vector on a plane does not affect its CVN.
Planar vector addition using CVN is illustrated by the following interactive tool.
Spatial vector Addition using CVN (three dimensions)
Once the vectors to be summed are resolved into their components and represented in CVN, the similar steps as in the coplanar case should be followed but with including components in the
in which
The magnitude of the resultant vector is,
The direction of
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LeetCode Question - 1710. Maximum Units on a Truck ๐
LeetCode Question - 1710. Maximum Units on a Truck ๐
1st July 2022 | ๐ Daily LeetCode Challenge - #1
About the Series
Problem-solving is a key skill set for any tech-related stuff you might be working on.
When it comes to developers it's one of the most crucial skills which is needed in almost any day-to-day code you might be writing.
So, this series of blogs is all about practicing Daily LeetCode Challenges & Problem-solving. ๐
Problem Statement
You are assigned to put some amount of boxes onto one truck. You are given a 2D array boxTypes, where boxTypes[i] = [numberOfBoxesi, numberOfUnitsPerBoxi]:
□ numberOfBoxes is the number of boxes of type i.
□ numberOfUnitsPerBox is the number of units in each box of type i. You are also given an integer truckSize, which is the maximum number of boxes that can be put on the truck. You can choose
any boxes to put on the truck as long as the number of boxes does not exceed truckSize.
Return the maximum total number of units that can be put on the truck.
Video Explanation
1. Sorting the array in descending order based on the no. of units in each box. ๐ ฆ
2. Iterate through the sorted array elements and check the remaining truckSize.
□ If not enough size for all boxes take as many boxes as you can and calculate the total units
□ Else take all the boxes and calculate the total units.
3. Return the total units
Code ๐ ง โ ๐ ป
var maximumUnits = function (boxTypes, truckSize) {
boxTypes.sort((a, b) => b[1] - a[1]);
var maxTotal = 0;
var i = 0;
while (truckSize > 0 && i < boxTypes.length) {
const numOfBoxes = boxTypes[i][0];
const numOfUnits = boxTypes[i][1];
if (truckSize <= numOfBoxes) {
maxTotal += truckSize * numOfUnits;
truckSize = 0;
} else {
maxTotal += numOfBoxes * numOfUnits;
truckSize -= numOfBoxes;
return maxTotal;
Time Complexity : O(nlogn)+O(n) = O(nlogn)
Space Complexity: O(1)
Similar Questions for practice
Now it is time to try more similar questions
You can find me on the web ๐
Add your solutions or approaches to the comments.
Show your love by Sharing the blog. ๐ ค
"I didn't fail 1000 times. The light bulb was an invention with 1000 steps."
~Thomas A. Edison
Did you find this article valuable?
Support Sourav Dey by becoming a sponsor. Any amount is appreciated!
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Variance Reversion Trading Strategy
1. Variance Reversion Trading Strategy
Variance Reversion Trading Strategy
, Date: 2023-10-31 14:42:13
The Variance Reversion trading strategy generates trading signals by calculating the ratio between call and put options, also known as the call put ratio. When the ratio reverses, it triggers trades
combined with simple money management rules to realize profits. It is suitable for 30-minute periods of NDX and SPX. The oscillation needs to be fine-tuned to reflect the correct reversal point.
Solid backtesting results indicate the optimal reversal point.
Strategy Logic
The core metrics of this strategy are the moving average and standard deviation of the call/put ratio. It first calculates the 20-day moving average of the call/put ratio, then computes the 30-day
standard deviation of the ratio. A long signal triggers when the ratio crosses above the moving average plus 1.5 standard deviation. A short signal triggers when the ratio falls below the moving
average minus 1.5 standard deviation.
After going long, if the ratio rebounds back above the moving average, close out the short position. The stop loss is set at 1% below the entry price. Take profit is set at 3 times the stop loss
distance from the entry price.
Advantage Analysis
The biggest edge of this strategy is capturing sentiment reversal points when the market becomes overly pessimistic or bullish, causing anomalies in the call/put ratio. Trading against such anomalies
can profit from local reversals. The money management rules effectively limit the risk and reward of individual trades.
Risk Analysis
The major risk comes from improper parameter tuning. Overly frequent signals fail to capture significant reversals. Reversal signals may also be faked out by false breakouts, causing losses.
Parameters should be optimized for more reliable signals.
Consider adding filters to confirm reversal signals and avoid false breakouts. For example, only consider signals when volume amplifies. Trend filters could also avoid countertrend trades. Optimal
parameters likely vary across different markets and time frames. Integrating more factors will make the strategy more robust.
This strategy aims to capture market reversal points by using the call/put ratio with basic money management rules. It can profit from local reversals but faces false breakout risks. Optimizing
parameters, adding filters and integrating more factors can enhance its stability and profitability. Overall, it provides a direction to trade reversals based on market sentiment. Further testing and
optimization is needed for real-world application.
start: 2023-09-30 00:00:00
end: 2023-10-30 00:00:00
period: 1h
basePeriod: 15m
exchanges: [{"eid":"Futures_Binance","currency":"BTC_USDT"}]
// This source code is subject to the terms of the Mozilla Public License 2.0 at https://mozilla.org/MPL/2.0/
// © I11L
strategy("I11L Long Put/Call Ratio Inversion", overlay=false, pyramiding=1, default_qty_value=10000, initial_capital=10000, default_qty_type=strategy.cash)
SL = input.float(0.01,step=0.01)
CRV = input.float(3)
TP = SL * CRV
len = input.int(30,"Lookback period in Days",step=10)
ratio_sma_lookback_len = input.int(20,step=10)
mult = input.float(1.5,"Standard Deviation Multiple")
ratio_sma = ta.sma(request.security("USI:PCC","D",close),ratio_sma_lookback_len)
median = ta.sma(ratio_sma,len)
standartDeviation = ta.stdev(ratio_sma,len)
upperDeviation = median + mult*standartDeviation
lowerDeviation = median - mult*standartDeviation
isBuy = ta.crossunder(ratio_sma, upperDeviation)// and close < buyZone
isCloseShort = (ratio_sma > median and strategy.position_size < 0)
isSL = (strategy.position_avg_price * (1.0 - SL) > low and strategy.position_size > 0) or (strategy.position_avg_price * (1.0 + SL) < high and strategy.position_size < 0)
isSell = ta.crossover(ratio_sma,lowerDeviation)
isTP = strategy.position_avg_price * (1 + TP) < high
strategy.entry("Long", strategy.long)
strategy.exit("Close Short",limit=close)
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Quantum Divide and Conquer
Matt Kovacs-Deak April 14, 2023.
The divide-and-conquer framework, used extensively in classical algorithm design, recursively breaks a problem of size n into smaller subproblems (say, a many copies of size n/b each), along with
some auxiliary work of cost T_aux(n), to give a recurrence relation T(n) ≤ a T(n/b) + T_aux(n) for the classical complexity T(n). In this talk I will describe a quantum divide-and-conquer framework
that, in certain cases, yields an analogous recurrence relation Q(n) ≤ a sqrt(Q(n/b)) + O(Q_aux(n)) that characterizes the quantum query complexity Q(n). Using this framework near-optimal quantum
query complexities can be derived for various string problems, such as (i) recognizing regular languages; (ii) decision versions of String Rotation and String Suffix; and natural parameterized
versions of (iii) Longest Increasing Subsequence and (iv) Longest Common Subsequence.
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Helping Others Understand Some Great Benefits Of math websites
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MA4104 Applied Mathematics for Mechatronics Syllabus
MA4104 Applied Mathematics for Mechatronics Syllabus:
MA4104 Applied Mathematics for Mechatronics Syllabus – Anna University PG Syllabus Regulation 2021
1. Mathematical foundations of numerical techniques for solving linear systems, eigenvalue problems and generalized inverse.
2. To expose the students to variational formulation and numerical integration techniques and demonstrate solution methodology for the variational problems.
3. To understand the basics of random variables with emp0hasis on the standard discrete and continuous distributions.
4. To make the students appreciate the purpose of using Laplace transforms to solve the partial differential equation.
5. To introduce the Fourier transforms and its properties.
UNIT – I MATRIX THEORY
Matrix representation of Linear Transformation – Eigen values – Generalized Eigenvectors – Rank of Matrix – The Cholesky decomposition – Canonical basis – QR factorization – Least squares method –
Singular value decomposition.
Concept of variation and its properties – Euler’s equations – Functional dependent on first and higher order derivatives – Functional dependent on functions of several independent variables –
Variational problems with moving boundaries – Isoperimetric problems – Direct methods: Ritz and Kantorovich methods – Taylor polynomials and Taylor series.
Probability – Axioms of probability – Conditional probability – Bayes’ theorem – Random variables – Probability function – Moments – Moment generating functions and their properties – Binomial,
Poisson, Geometric, Uniform, Exponential, Gamma and Normal distributions – Function of a random variable.
Laplace transform – Definitions – Properties – Transform error function – Bessel’s functions – Dirac delta function – Unit step functions – Convolution theorem – Inverse Laplace transform: Complex
inversion formula – Solutions to Partial Differential Equations (PDE): Heat equations – Wave equation.
Fourier transform: Definitions, properties – Transform of elementary functions – Dirac Delta function – Convolution theorem – Parseval’s identity – Solutions to partial differential equations: Heat
equation – Wave equation – Laplace and Poisson’s equations.
TOTAL: 60 PERIODS
At the end of the course, students will be able to
1. apply various methods in matrix theory to solve system of linear equations.
2. maximizing and minimizing the functional that occur in various branches of Engineering disciplines.
3. computation of probability and moments, standard distributions of discrete and continuous random variables and functions of a random variable.
4. application of Laplace transforms to initial value, initial- boundary value and boundary value problems in Partial Differential Equations.
5. obtain Fourier transforms for the functions which are needed for solving application problems.
1. Andrews, L. C. and Shivamoggi, B., “Integral Transforms for Engineers”, Prentice Hall of India, New Delhi, 2003.
2. Bronson, R.,” Matrix Operations”, Schaum’s outline series, 2nd Edition, McGraw Hill, 2011.
3. James, G., “Advanced Modern Engineering Mathematics”, 3rd Edition, Pearson Education, 2004.
4. Johnson, R.A., Miller, I and Freund J., “Miller and Freund’s Probability and Statistics for Engineers”, Pearson Education, Asia, 8th Edition, 2015.
5. O’Neil P.V., “Advanced Engineering Mathematics”, Thomson Asia Pvt. Ltd., Singapore, 2003.
6. Sankara Rao,K., “ Introduction to Partial Differential Equations”, Prentice Hall of India Pvt. Ltd., New Delhi, 1997.
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Interactive workout - Mathmo
Mathmo is a revision tool for post-16 mathematics. It's great installed as a smartphone app, but it works well in pads and desktops and notebooks too. Give yourself a mathematical workout!
Our Mathmo interactive workout app for A-level fluency has had another makeover.
Release Notes:
Mathmo 0.9.3 has just been released at https://nrich.maths.org/mathmoApp.
This release gives users much more control over question generation - allowing such things as question sharing, and predictable exercise content. It features a new cleaner user interface which should
work on a wider range of browsers.
v0.9.3 should work in any HTML5 capable browser. Some answers have function plots which also require an SVG browser. This excludes IE6,7,8 but allows IE9+. Some Android browsers lack SVG.
The previously released version of Mathmo 0.5.0, is still available at https://nrich.maths.org/mobl/mathmo/mathmo.html
Getting Started
You can use this workbook to practise your core mathematical skills. This will be very useful both whilst at school and when you move on to higher study, where a high degree of algebraic fluency is
really useful.
Here are some hints and tips to improve your skills:
1. Don't be tempted to look at the answer if you think that you can do the question unless you have actually done the question, on paper with a pen.
2. Try not to use your calculator for the simple arithmetical parts of a question - this is a bad habit to get into.
3. If you are stuck, don't give up immediately. Think about the problem. Perhaps look at one or two answers and ask: how does this answer work?
4. Accuracy is important, but speed is also important. This workbook is good for speed training.
5. If your answer differs from the answer given then think: is my answer wrong, or is it merely represented in a different form?
6. Use your skills in some interesting rich mathematical problems from NRICH. You might want to look at the core mathematics curriculum document for suggestions or just browse the stage 5 content on
the site.
7. Keep a record of your best times. See how quickly you can complete one of each question type from each section.
What is a good time to aim for?
Fluency with mathematics means the ability to perform routine calculation BOTH quickly AND accurately. One without the other will hamper your progress, especially once you reach university.
To give you a feel for where you might be aiming, here are some average times for some of the questions as recorded by Judith, a second year mathematics undergraduate who worked with NRICH over the
Algebra Curve sketching Differentiation
Quadratic equations 8s Modulus function for linear 20s Stationary points for quadratic 10s
Completing the square 8s Modulus function for quadratic 1m 10s Stationary points for cubic 50s
Inequalities for quadratics 10s Implicit differentiation 1m 10s
Inequalities for cubics 1m 10s
Partial fractions 1m 30s
Powers 1m
Logarithms 20s
Solving trig equations 40s
TOTAL 5m 6s
Can you match or beat Judith's times?
More times from other brave solvers will be posted periodically.
Teachers' Resources
This workbook generates random questions from the UK core A-level mathematics syllabus.
It is very powerful and contains a huge range of topics, including graph sketching, trigonometry and all aspects of algebra and calculus. It is instant to use and comes equipped with the answers.
There are no time limits, targets or scores: just the opportunity for simple, unpressured practice.
There are many ways in which you can use this resource:
1. It is ideal for quiet use in the computer room for individual study or revision
2. You can point students who are having difficulty towards it for targeted practice on a certain question type.
3. You can suggest that 'high fliers' use the resource to practise their skills, maximise their chances of a good A and improve their own algebraic fluency. They will appreciate this when they
arrive at university where a high level of algebraic fluency gives a huge advantage.
4. You can use the workbook for quick lesson starters. By revising old, previously covered topics the knowledge of students will be kept fresh and will have great benefit on the students' overall
mathematical skill.
5. You can suggest that students about to start university or year 13 use the workbook as a refresher after the holidays.
6. You can have a go yourself as the teacher if you are teaching new material, feeling a little rusty or simply want to hone your already impressive algebraic skill!
More workbooks are planned. Please do get in touch if you have any comments.
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Revisiting the Continuum Hypothesis
I have been thinking about CH lately for two reasons
1) I reread the article
Hilbert's First Problem: The Continuum Hypothesis by Donald Martin from Proceedings of Symposia in Pure Mathematics: Mathematical developments arising from Hilbert Problems. 1976. (For a book review
of the symposia and, The Honor Class, also about Hilbert's problems, see here.)
The article takes the point of view that CH CAN have an answer. He discusses large cardinals (why assuming they exist is plausible, but alas, that assumption does not seem to resolve CH) and
Projective Det. (why assuming it is true is plausible, but alas, that assumption does not seem to resolve CH).
(A set A \subseteq {0,1}^omega is DETERMINED if either Alice or Bob has a winning strategy in the following non-fun game: they alternate picking bits a_1, b_1, a_2, b_2, ... with Alice going first.
If a_1 b_1 a_2 b_2... IS IN A then Alice wins, IF NOT then Bob wins. Martin showed that all Borel sets are determined. Proj Det is the statement that all projections of Borel sets are determined. AD
is the axiom that ALL sets A are determined. It contradicts AC.)
But what really inspired this post is the last paragraph:
Throughout the latter part of my discussion, I have been assuming a naive and uncritical attitude towards CH. While this is in fact my attitude, I by no means wish to dismiss the opposite viewpoint.
Those that argue that the concept of set is not sufficiently clear to fix the truth-value of CH have a position that is at present difficult to assail. As long as no new axiom is found which decides
CH, their case will continue to grow stronger, and our assertions that the meaning of CH is clear will sound more and more empty.
2) Scott Aaronson mentioned in a blog post (see here) that he has read and understood the proof that CH is independent of set theory.
SO, this seemed like a good time to revisit thoughts on CH.
I took a very short poll, just two people, about CH: Stephen Fenner (in a perfect world he would be a set theorists) and Scott Aaronson (having JUST read the proof that CH is ind. he has thought
about it recently).
Here are some thoughts of theirs and mine
1) All three of us are Platonists with regard to the Naturals (I was surprised to find recently that there are people who are not!) but not with regard to the reals. So we would be OKAY with having
CH have no answer.
2) All three of us agree that it would be nice if SOME axiom was both
a) Intuitively appealing or aesthetically appealing , and
b) resolved CH.
I always thought that (a) would be the hard part-- or at least getting everyone (not sure who we are talking about) to AGREE on a new axiom. But even getting an axiom to resolve CH seems hard. Large
cardinals don't seem to do it, and various forms of Determinacy don't seem to do it.
Scott reminded me of Freiling's Axiom of Symmetry (see here) which IS intuitive and DOES resolve CH (its false) though there are problems with it--- a minor variant of it contradicts AC (I am QUITE
FINE with that since AC implies Banach-Tarski which Darling says shows `Math is broken'.)
Stephen recalled some of Hugh Woodin's opinions of CH, but Hugh seems to have changed his mind from NOT(CH): 2^{aleph_0} = aleph_2, to CH: 2^{aleph_0} = aleph_1.(See here.)
3) All three of would be okay with V=L, though note that this would put many set theorists out of work. All the math that applies to the real world would still be intact. I wonder if in an
alternative history the reaction to Russell's paradox would be a formulation of set theory where V=L. We would KNOW that CH is true, KNOW that AC is true. We would know a lot about L but less about
4) Which Geometry is true: Euclidian, Riemannian, others? This is now regarded as a silly question: Right Tool, Right Job! If you build a bridge use Euclid. If you are doing astronomy use Riemann.
Might Set Theory go the same way? It would be AWESOME if Scott Aaronson found some quantum thing where assuming 2^{aleph_0} = aleph_2 was the right way to model it.
5) If I was more plugged into the set theory community I might do a poll of set theorists, about CH. Actually, someone sort-of already has. Penelope Maddy has two excellent and readable articles
where she studies what set theorists believe and why.
Believing The Axioms I: here
Believing The Axioms II: here
Those articles were written in 1988. I wonder if they need an update.
5 comments:
1. Quite rightly, bullet point #4 above illustrates that the parallel postulate is only true or false in a specified model. So shouldn't it be true that CH is either true or false in the specified
model called "the cumulative hierarchy?" Or is the cumulative hierarchy "underdetermined" in some way? Model theory makes no allowances for semantic underdetermination because every sentence is
either true or false in a given model. Is this somehow philosophically wrong or naive? If so, how? And this touches on bullet point #1. Are you guys not Platonist regarding the cumulative
hierarchy? I grant you that ZFC is not quite as "intuitively" elementary as PA but it feels close, right? (I am aware, of course, that mathematically ZFC and PA are NOT close in power.) Is it not
the case that both PA and ZFC both "indicate" canonical models in spite of the existence of non-standard models, i.e. the natural numbers and the cumulative hierarchy, and that every sentence
should indeed either be true or false in the canonical model?
As an aside, if "Platonism" makes you uncomfortable at cocktail parties you may wish to switch to a less metaphysical mathematical realism. One such can be found described quite well in
Tragesser's book "Husserl and Realism in Logic and Mathematics."
1. You're 100% right. Disputing that CH has a truth value in the cumulative hierarchy requires disputing that P(P(ℕ)) is well-defined (i.e. claiming that it is undetermined), or something even
sillier. It's postmodern mathematics, best suited for (a) contrarians, (b) philosophers, and (c) mathematicians who need to justify to granting agencies their work redoing the foundations of
mathematics. I'm sad to say that groups (a) and (c) include a few extremely smart and knowledgeable people. They are few, but they are loud, and their story is titillating for a broad
audience, so it persists.
2. Regarding your proposal for how to settle CH by finding a new axiom that is intuitively appealing and yet still settles CH, I argue in my paper "Is the dream solution of the continuum hypothesis
attainable?" (http://jdh.hamkins.org/dream-solution-of-ch/) that this is impossible. The essence of my argument is that for us to learn that an axiom candidate decides CH will automatically
undermine any attempt to take it as natural or intuitive, because of our prior extensive familiarity (via forcing and so on) with set-theoretic worlds having the opposite outcome.
1. Does your argument rule out the possibility of finding a new, natural axiom (presumably based on new phenomenological insights) which is specifically applicable to the cumulative hierarchy
(rather than ALL set-theoretic worlds)?
2. It does not. Hamkins is overstating his position with the word "impossible" in the above comment. An excerpt from the paper better classifies his argument as a prediction ("the entire
episode" refers to his exposition of the reception of Freiling's Axiom):
"The entire episode bears out the pattern of response I predict for any attempted use of the dream solution template, namely, a rejection of the new axiom from a perspective of deep
mathematical experience with the contrary."
To be clear, his prediction might be correct. That seems to be the mood of the times. But that might have more to do with humility and practicality than anything deep; it seems unwise to
devote a lot of energy to solving a problem that the greatest geniuses in recent times failed to solve.
Hamkins ends his paper with the following (note there is an implied vice-versa after "flawed"):
"Before we will be able to accept CH as true, we must come to know that our experience of the ¬CH worlds was somehow flawed; we must come to see our experience in those lands as illusory. It
is insufficient to present a beautiful landscape, a shining city on a hill, for we are widely traveled and know that it is not the only one."
That is eloquent but misleading. A dream solution need not undermine the reality of nonstandard models. Nor the usefulness or elegance of nonstandard models.
A dream solution could even reinforce the usefulness or elegance of nonstandard models; it needn't appear as "a beautiful landscape, a shining city on a hill." It is possible there will be a
dream solution proving |ℝ| = Aleph_42 in the cumulative hierarchy (equivalently, in the universe of third-order arithmetic). I would be as shocked as everyone else, but it wouldn't be
entirely without precedent; to me, a physics novice, Newtonian mechanics is more of a "beautiful landscape, a shining city on a hill" than general relativity, quantum mechanics, and string
theory. Perhaps the universe of iterated powersets is, surprisingly, as weird as physical reality? We are stuck with physical reality, but if the cumulative hierarchy turns out to be ugly and
annoying enough to human mathematicians, we could always redefine "the standard model" of set theory, without any intellectual dishonesty.
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On This Day in Math - October 22
One of the most baneful delusions by which the minds, not only of students, but even of many teachers of mathematics in our classical colleges, have been afflicted with is, that mathematics can be
mastered by the favored few, but lies beyond the grasp and power of the ordinary mind.
~Florian Cajori, The Teaching and History of Mathematics in the United States
The 295th day of the year; 295 may be interesting only because it seems to be the least interesting day number of the year. (Willing to be contradicted, send your comments)
[Here are several of the best I received from David Brooks:
295 can be partitioned in 6486674127079088 ways.
295 is a 31-gonal number.]
And Derek Orr pointed out that "295 is the second proposed Lychrel number." A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing
its digits and adding the resulting numbers. This process is sometimes called the 196-algorithm, after the most famous number associated with the process. In base ten, no Lychrel numbers have been
yet proved to exist, but many, including 196, are suspected on heuristic and statistical grounds. The name "Lychrel" was coined by Wade Van Landingham as a rough anagram of Cheryl, his girlfriend's
first name. (Who else thinks he probably mis-spelled her name and when she called him on it, he came up with the idea of a "rough anagram"? )
1668 Leibniz writes to the German emperor to request permission to publish a "Nucleus Libareaus". This was the beginnings of the foundation of Acta Eruditorum, the first German scientific journal.
1685 Abraham De Moivre was a student of physics at the University, Collège d'Harcourt, in the 1680s. After the Revocation of the Edict of Nantes, (October 22, 1685 ) he went into seclusion in the
priory of St. Martin (possibly that which became the Conservatoire National des Arts et Métiers ??) and then emigrated to England, having no contact with France until he was elected a Foreign
Associate of the Academy of Sciences just before his death.*VFR
known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.
He moved to England at a young age due to the religious persecution of Huguenots in France which reached a climax in 1685 with the Edict of Fontainebleau.[1] He was a friend of Isaac Newton, Edmond
Halley, and James Stirling. Among his fellow Huguenot exiles in England, he was a colleague of the editor and translator Pierre des Maizeaux.
De Moivre wrote a book on probability theory, The Doctrine of Chances, said to have been prized by gamblers. De Moivre first discovered Binet's formula, the closed-form expression for Fibonacci
numbers linking the nth power of the golden ratio φ to the nth Fibonacci number. He also was the first to postulate the central limit theorem, a cornerstone of probability theory. *Wik
1746 Princeton chartered as the College of New Jersey -- the name by which it was known for 150 years -- Princeton University was British North America's fourth college. Located in Elizabeth for one
year and then in Newark for nine, the College of New Jersey moved to Princeton in 1756. It was housed in Nassau Hall, which was newly built on land donated by Nathaniel FitzRandolph. Nassau Hall
contained the entire College for nearly half a century. *Princeton Univ web page
In 1797, the first parachute jump was made by André-Jacques Garnerin, released from a balloon 2,230-ft above the Parc Monceau, Paris. He rode in a gondola fixed to the lines of a 23-ft diameter
parachute, which was supported by a wooden pole and had its 32 white canvas gores folded like a closed umbrella. Lacking any vent in the top of the parachute, Garnerin descended with violent
oscillations, and suffered the first case of airsickness. For his next jump, he added a hole in the top of the parachute. He made his fifth jump on 21 Sep 1802 over London, from a height of 3,000-ft.
This was the first parachute descent made in England. He landed near St. Pancras Church. Having eliminated the center vent for this jump, he again suffered a fit of vomiting. *TIS See
1850 Fechner’s law introduced. [Springer’s 1985 Statistics Calendar] A pioneering though in many situations incorrect formulation of the relationship between the physical strength of a stimulus and
its strength as perceived by humans, proposed by G. T. Fechner in 1860. Fechner postulated that sensation increases as the log of the stimulus. For example, by Fechner's law, if light A was twice as
bright as light B (measured by an instrument), it would appear to the human eye to be log 2 (times a constant to allow for such factors as the units used) brighter than light B. Later experiments
have shown conclusively that the Fechner's law doesn't generally apply.
The Weber–Fechner laws are two related scientific laws in the field of psychophysics, known as Weber's law and Fechner's law. Both relate to human perception, more specifically the relation between
the actual change in a physical stimulus and the perceived change. This includes stimuli to all senses: vision, hearing, taste, touch, and smell.
Ernst Heinrich Weber states that "the minimum increase of stimulus which will produce a perceptible increase of sensation is proportional to the pre-existent stimulus," while Gustav Fechner's law is
an inference from Weber's law (with additional assumptions) which states that the intensity of our sensation increases as the logarithm of an increase in energy rather than as rapidly as the
An illustration of the Weber–Fechner law. On each side, the lower square contains 10 more dots than the upper one. However the perception is different: On the left side, the difference between upper
and lower square is clearly visible. On the right side, the two squares look almost the same.
The Weber–Fechner laws are two related scientific laws in the field of psychophysics, known as Weber's law and Fechner's law. Both relate to human perception, more specifically the relation between
the actual change in a physical stimulus and the perceived change. This includes stimuli to all senses: vision, hearing, taste, touch, and smell.
Ernst Heinrich Weber states that "the minimum increase of stimulus which will produce a perceptible increase of sensation is proportional to the pre-existent stimulus," while Gustav Fechner's law is
an inference from Weber's law (with additional assumptions) which states that the intensity of our sensation increases as the logarithm of an increase in energy rather than as rapidly as the
An illustration of the Weber–Fechner law. On each side, the lower square contains 10 more dots than the upper one. However the perception is different: On the left side, the difference between upper
and lower square is clearly visible. On the right side, the two squares look almost the same.
The Weber–Fechner laws are two related scientific laws in the field of psychophysics, known as Weber's law and Fechner's law. Both relate to human perception, more specifically the relation between
the actual change in a physical stimulus and the perceived change. This includes stimuli to all senses: vision, hearing, taste, touch, and smell.
Ernst Heinrich Weber states that "the minimum increase of stimulus which will produce a perceptible increase of sensation is proportional to the pre-existent stimulus," while Gustav Fechner's law is
an inference from Weber's law (with additional assumptions) which states that the intensity of our sensation increases as the logarithm of an increase in energy rather than as rapidly as the
An illustration of the Weber–Fechner law. On each side, the lower square contains 10 more dots than the upper one. However the perception is different: On the left side, the difference between upper
and lower square is clearly visible. On the right side, the two squares look almost the same.
1903 Simon Newcomb of Johns Hopkins decides to make it clear that aerial flight is impossible, less than two months before Kitty Hawk. "The Mathematicians of today admits that he can neither square
the circle, or duplicate the cube, or trisect the angle. May not our mechanicians ... be ultimately forced to admit that aerial flight is one of the great class of problems with which man can never
cope, and give up all attempts to grapple with it?"
"Imagine the proud possessor of the aeroplane darting through the air at a speed of several hundred feet per second! It is the speed alone that sustains him. How is he ever going to stop? Once he
slackens his speed, down he begins to fall. He may, indeed, increase the inclination of his aeroplane. Then he increases the resistance necessary to move it. Once he stops he falls a dead mass. How
shall he reach the ground without destroying his delicate machinery? I do not think the most imaginative inventor has yet even put upon paper a demonstrative, successful way of meeting this
-- Simon Newcomb, "The Outlook for the Flying Machine," Independent, Oct. 22, 1903
1908 First meeting of the Spanish Association for the Advancement of Science was held October 22–29. Sixteen papers were read in the section of mathematics.*VFR
1922 M. C. ESCHER visited here(Alhambra) on 18 - 24 Oct 1922 and was impressed by the patterns, but he didn't really use them in his art until after his second visit on 22-26 May 1936 *VFR
1933 The Solvay Congress in Brussells opens on 22 October, 1933 which was attended by leading Nuclear physicists around the world. Attendees included two future key Manhattan Project scientists
(Fermi and Lawrence), the future head of the Nazi atomic bomb program (Heisenberg), and numerous leading pre-war physicists. Among the group, three women seated in the front row, from left to right,
Irene Joliet-Curie, Marie Curie, and Lise Meitner. The meeting would last through the 29th of the month.*PB
1938 In the back of a beauty shop in the Astoria section of Queens New York, Chester A. Carlson and his assistant Otto Kornei, conducted the first successful experiment in electrophotography. The
message, “10.-22.-38 ASTORIA,” was even less inspiring than Alexander Graham Bell’s first phone conversation, but the effect was just as great. In 1949 Haloid Corporation marketed the Xerox Model A, a
crude machine that required fourteen manual operations. Today five million copiers churn out 2,000 copies each year for every American citizen. *VFR
Carlson was an engineer who couldn't get a job in his field during the Great Depression, so he took work in the patent department of battery-manufacturer P.R. Mallory. A bottleneck in the work was
making copies of patent documents: You had to copy them by hand (time and labor) or send them out to be photographed (time and expense).
Carlson set out to make a dry-copying process. He got his inspiration from the new field of photoconductivity: Light striking the surface of certain materials increases the flow of electrons. Carlson
knew he could use the effect to make dry copies. Project an image of the original document onto a photoconductive surface, and current would flow only where light struck.
Four years of tinkering in his kitchen and in his mother-in-law's beauty salon in Astoria, Queens, in New York City finally produced results in October 1938. Carlson's research assistant, Otto
Kornei, put a sulfur coating on a zinc plate, which was rubbed with a handkerchief to give it an electrostatic charge.
Evelyn (Boka) Van Orden Sent me a followup note, “On the 75th anniversary of Xerography — October 22, 2013 — I was privileged to attend an event titled "Chester Chester Chester" at Xerox PARC.”
George Shea first stumbled on Chester Carlson in 1981 when he came upon a passage about the little known inventor (even today, most Americans have never heard of Carlson) in an obscure book on the
subject of "Copy Art."
George was fascinated by the tale of struggle, patience, late success, and spiritual enlightenment and began digging into Carlson's life. In 1988 he visited the former janitor's closet in Astoria in
which Carlson made the world's first xerograhic copy on Oct. 22, 1938. The ceiling still displayed obvious sulfur stains from Chester’s and Otto Kornei’s experiments.
1975 the Soviet unmanned space mission Venera 9 lands on Venus. measurements taken included surface pressure of about 9,100 kilopascals (90 atm), temperature of 485 °C (905 °F), and surface light
levels comparable to those at Earth mid-latitudes on a cloudy summer day. *the painter flynn
1511 Erasmus Reinhold (October 22, 1511 – February 19, 1553) was a German astronomer and mathematician, considered to be the most influential astronomical pedagogue of his generation. He was born and
died in Saalfeld, Saxony.
He was educated, under Jacob Milich, at the University of Wittenberg, where he was first elected dean and later became rector. In 1536 he was appointed professor of higher mathematics by Philipp
Melanchthon. In contrast to the limited modern definition, "mathematics" at the time also included applied mathematics, especially astronomy. His colleague, Georg Joachim Rheticus, also studied at
Wittenberg and was appointed professor of lower mathematics in 1536.
Reinhold catalogued a large number of stars. His publications on astronomy include a commentary (1542, 1553) on Georg Purbach's Theoricae novae planetarum. Reinhold knew about Copernicus and his
heliocentric ideas prior to the publication of De revolutionibus and made a favorable reference to him in his commentary on Purbach. However, Reinhold (like other astronomers before Kepler and
Galileo) translated Copernicus' mathematical methods back into a geocentric system, rejecting heliocentric cosmology on physical and theological grounds.
It was Reinhold's heavily annotated copy of De revolutionibus in the Royal Observatory, Edinburgh that started Owen Gingerich on his search for copies of the first and second editions which he
describes in The Book Nobody Read. In Reinhold's unpublished commentary on De revolutionibus, he calculated the distance from the Earth to the sun. He "massaged" his calculation method in order to
arrive at an answer close to that of Ptolemy.*Wik
1587 Joachim Jungius (22 Oct 1587 in Lübeck, Germany - 23 Sept 1657 in Hamburg) a German mathematician who was one of the first to use exponents to represent powers and who used mathematics as a
model for the natural sciences. Jungius proved that the catenary is not a parabola (Galileo assumed it was). *SAU (I can not find the first use by Jungius anywhere, but Cajori gives Descartes 1637
use in Geometrie as the first example of the common form today. A year earlier, James Hume produced a copy of Viete's Algebra in which he used exponents as powers of numbers, but his exponents were
Roman Numerals.)[Nicolas Chuquet (1445-1488) was the first to use numbers as exponents, and the first to use negative numbers as exponents, but didn't use raised -1 for inverse.]
1659 Georg Ernst Stahl (22 October 1659 – 24 May 1734) was a German chemist, physician and philosopher. He was a supporter of vitalism, and until the late 18th century his works on phlogiston were
accepted as an explanation for chemical processesStahl used the works of Johann Joachim Becher to help him come up with explanations of chemical phenomena. The main theory that Stahl got from J. J.
Becher was the theory of phlogiston. This theory did not have any experimental basis before Stahl. He was able to make the theory applicable to chemistry.Becher's theories attempted in explaining
chemistry as comprehensively as seemingly possible through classifying different earths according to specific reactions. Terra pinguis was a substance that escaped during combustion reactions,
according to Becher.[10] Stahl, influenced by Becher's work, developed his theory of phlogiston.People who dismiss Phlogiston theory as early ignorance should read The Renaissance Mathematicus blog,
The Phlogiston Theory – Wonderfully wrong but fantastically fruitful.
1792 Guillaume-Joseph-Hyacinthe-Jean-Baptiste Le Gentil de la Galaziere (12 Sep 1725; 22 Oct 1792) was a French astronomer who attempted to observe the transit of Venus across the sun by travelling
to India in 1761. He failed to arrive in time due to an outbreak of war. He stayed in India to see the next transit which came eight years later. This time, he was denied a view because of cloudy
weather, and so returned to France. There, he found his heirs had assumed he was dead and taken his property.*TIS A more detailed blog about his life is at Renaissance Mathematicus
1843 John S Mackay graduated from St Andrews University and taught at Perth Academy and Edinburgh Academy. He was a founder member of the EMS and became the first President in 1883 and an honorary
member in 1894. He published numerous papers on Geometry in the EMS Proceedings.*SAU
1881 Clinton Joseph Davisson (22 Oct 1881; 1 Feb 1958) American experimental physicist who shared the Nobel Prize for Physics in 1937 with George P. Thomson of England for discovering that electrons
can be diffracted like light waves. Davisson studied the effect of electron bombardment on surfaces, and observed (1925) the angle of reflection could depend on crystal orientation. Following Louis
de Broglie's theory of the wave nature of particles, he realized that his results could be due to diffraction of electrons by the pattern of atoms on the crystal surface. Davisson worked with Lester
Germer in an experiment in which electrons bouncing off a nickel surface produced wave patterns similar to those formed by light reflected from a diffraction grating, and supporting de Broglie's
electron wavelength = (h/p). *TIS
1895 Rolf Herman Nevanlinna (22 October 1895 – 28 May 1980) was one of the most famous Finnish mathematicians. He was particularly appreciated for his work in complex analysis.Rolf Nevanlinna's most
important mathematical achievement is the value distribution theory of meromorphic functions. The roots of the theory go back to the result of Émile Picard in 1879, showing that a complex-valued
function which is analytic in the entire complex plane assumes all complex values save at most one.*Wik
1905 Karl Guthe Jansky (22 Oct 1905; 14 Feb 1950) was an American electrical engineer who discovered cosmic radio emissions in 1932. At Bell Laboratories in NJ, Jansky was tracking down the crackling
static noises that plagued overseas telephone reception. He found certain radio waves came from a specific region on the sky every 23 hours and 56 minutes, from the direction of Sagittarius toward
the center of the Milky Way. In the publication of his results, he suggested that the radio emission was somehow connected to the Milky Way and that it originated not from stars but from ionized
interstellar gas. At the age of 26, Jansky had made a historic discovery - that celestial bodies could emit radio waves as well as light waves. *TIS Image: Karl Jansky makes adjustments to his
antenna *Wik
A trivia footnote from Lee Guthrie: “One of his steerable antennas on display at Green Bank’s museum uses the differential axis out of a “T” model Ford.”
1907 Sarvadaman D. S. Chowla (22 October 1907, London–10 December 1995, Laramie, Wyoming) was a prominent Indian mathematician, specializing in number theory. Among his contributions are a number of
results which bear his name. These include the Bruck–Chowla–Ryser theorem, the Ankeny–Artin–Chowla congruence, the Chowla–Mordell theorem, and the Chowla–Selberg formula, and the Mian–Chowla
1916 Nathan Jacob Fine (22 October 1916 in Philadelphia, USA - 18 Nov 1994 in Deerfield Beach, Florida, USA) He published on many different topics including number theory, logic, combinatorics, group
theory, linear algebra, partitions and functional and classical analysis. He is perhaps best known for his book Basic hypergeometric series and applications published in the Mathematical Surveys and
Monographs Series of the American Mathematical Society. The material which he presented in the Earle Raymond Hedrick Lectures twenty years earlier form the basis for the material in this text.*SAU
1927 Alexander Ivanovich Skopin (22 Oct 1927 in Leningrad (now St Petersburg), Russia - 15 Sept 2003 in St Petersburg, Russia) He was a Russian mathematician known for his contributions to abstract
algebra. Skopin's student work was in abstract algebra, and concerned upper central series of groups and extensions of fields. In the 1970s, Skopin received a second doctorate concerning the
application of computer algebra systems to group theory. From that point onward he used computational methods extensively in his research, which focused on lower central series of Burnside groups. He
related this problem to problems in other areas of mathematics including linear algebra and topological sorting of graphs. *Wik
1941 Stanley Mazor was born in Chicago on October 22, 1941. He studied mathematics and programming at San Francisco State University. He joined Fairchild Semiconductor in 1964 as a programmer and
then a computer designer in the Digital Research Department where he shares patents on the Symbol computer. In 1969, he joined Intel. In 1977, he began his teaching career in Intel's Technical
Training group, and later taught classes at Stanford, University of Santa Clara, KTH in Stockholm and Stellenbosch, S.A. In 1984 he was at Silicon Compiler Systems. He co-authored a book on chip
design language while at Synopsys 1988-1994. He was invited to present The History of the Microcomputer at the 1995 IEEE Proceedings. He is currently the Training Director at BEA Systems. *CHM
1950 Ada Isabel Maddison (13 April 1869 in Cumberland, England - 22 Oct 1950 in Martin's Dam, Wayne, Pennsylvania, USA) A British mathematician best known for her work on differential equations.
Although Maddison passed an honors exam for the University of Cambridge, she was not given a degree there. Instead, she went to Bryn Mawr in Pennsylvania. In 1893, the University of London awarded
her a bachelor's degree in mathematics with honors. After further study at the University of Göttingen, Maddison went back to Bryn Mawr, where she taught as well as doing time consuming
administrative work. Her will endowed a pension fund for Bryn Mawr's administrative staff.*Wik
1977 Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of combinatorial
geometry. Among his main contributions to algebraic geometry are studies of birational invariants of algebraic varieties, singularities and algebraic surfaces. His work was in the style of the old
Italian School, although he also appreciated the greater rigor of modern algebraic geometry. Another contribution of his was the introduction of finite and non-continuous structures into geometry. In
his best known paper he proved the following theorem: In a Desarguesian plane of odd order, the ovals are exactly the irreducible conics. Some critics felt that his work was no longer geometry, but
today it is recognized as a separate sub-discipline: combinatorial geometry.
In 1938 he lost his professorship as a result of the anti-Jewish laws enacted under Benito Mussolini's government; he spent the next 8 years in Great Britain (mostly at the University of Manchester),
then returned to Italy to resume his academic career *Wik
1979 Reinhold Baer (22 July 1902 in Berlin, Germany - 22 Oct 1979 in Zurich, Switzerland) Baer's mathematical work was wide ranging; topology, abelian groups and geometry. His most important work,
however, was in group theory, on the extension problem for groups, finiteness conditions, soluble and nilpotent groups. *SAU
Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell
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Signal processors - Patent 0845868
The present invention relates to a 1-bit signal processor comprising an nth order Delta-Sigma Modulator where n is at least one. Preferred embodiments of the invention relate to processing audio
signals but the invention is not limited to audio signal processors.
Background to the present invention will now be described by way of example with reference to Figures 1, 2 and 3 of the accompanying drawings of which Figure 1 is a block diagram of a known
Delta-Sigma Modulator, Figure 2 is a block diagram of a previously proposed Delta-Sigma Modulator configured as an nth order filter section and Figure 3 shows a noise shaping characteristic.
It is known to convert an analogue signal to a digital form by sampling the analogue signal at at least the Nyquist rate and encoding the amplitudes of the samples by an m bit number. Thus if m = 8,
the sample is said to be quantized to an accuracy of 8 bits. In general
can be any number of bits equal to or greater than 1.
For the purpose of quantizing to only 1 bit, it is known to provide an analogue to digital converter (ADC) known either as a "Sigma-Delta ADC" or as a "Delta-Sigma ADC". Herein the term "Delta-Sigma"
is used. Such an ADC is described in for example "A Simple Approach to Digital Signal Processing" by Craig Marven and Gillian Ewers ISBN 0-904.047-00-8 published 1993 by Texas Instruments.
Referring to Figure 1 in an example of such an ADC, the difference 1 (Delta) between an analogue input signal and the integral 2 (Sigma) of the 1-bit output signal is fed to a 1-bit quantizer 3. The
output signal comprises bits of logical value 0 and 1 but representing actual values of -1 and +1 respectively. The integrator 3 accumulates the 1-bit outputs so that value stored in it tends to
follow the value of the analog signal. The quantizer 3 increases (+1) or reduces (-1) the accumulated value by 1-bit as each bit is produced. The ADC requires a very high sampling rate to allow the
production of an output bit stream the accumulated value of which follows the analogue signal.
The term "1-bit" signal as used in the following description and in the claims means a signal quantized to an accuracy of 1 digital bit such as is produced by a Delta-Sigma ADC.
A Delta-Sigma Modulator (DSM) configured as nth order filter section for directly processing a 1-bit signal was proposed by N.M. Casey and James A.S. Angus in a paper presented at 95th AES Convention
7-10 October 1993 New York, USA entitled "One Bit Digital Processing of Audio Signals" - Signal Processing: Audio Research Group, The Electronics Department, The University of York, Heslington, York
YO1 5DD England. Figure 2 shows a 3rd order (n=3) version of such a DSM filter section.
Referring to Figure 2, the DSM has an input 4 for a 1-bit audio signal and an output 5 at which a processed a 1-bit signal is produced. The bits of the 1-bit signal are clocked through the DSM by
known clocking arrangements which are not shown. The output 1-bit signal is produced by a 1-bit quantizer Q which is for example a comparator having a threshold level of zero. The DSM has three
stages each comprising a first 1-bit multiplier a
, a
, a
connected to the input 4, a second 1-bit multiplier c
, c
, c
connected to the output 5, an adder 6
, 6
, 6
and an integrator 7
, 7
, 7
The 1-bit multipliers multiply the received 1-bit signal by
bit coefficients A
, A
, A
, C
, C
bit products which are added by the adders 6
, 6
, 6
and the sums passed to the integrators 7. In the intermediate stages the adders 6
, 6
also sum the output of the integrator of the preceding stage. A final stage comprises another 1-bit multiplier A
connected to the input which multiplies the input signal by a
bit coefficient A
and an adder 6
which adds the product to the output of the integrator 7
of the preceding stage. The sum is passed to the quantizer Q.
Within the DSM, two's complement arithmetic is used to represent the positive and negative
bit numbers. The input to the quantizer Q may be positive, quantized at the output as +1 (logical 1) or negative quantized at the output as -1 (logical 0).
As observed by Casey and Angus "a one bit processor .. will produce a one bit output that contains an audio signal that is obscured by noise to an unacceptable level and it is imperative the
quantization noise is suitably shaped". The noise which obscures the audio signal is the quantization noise produced by the quantizer Q.
The quantizer Q may be modelled as an adder which has a first input receiving an audio signal and a second input receiving a random bit stream (the quantization noise) substantially uncorrelated with
the audio signal. Modelled on that basis, the audio signal received at the input 4 is fed forward by multipliers a
, a
, a
, a
to the output 5 and fed back by multipliers c
, c
, c
from the output 5. Thus coefficients A1 to A4 in the feed forward path define zeros of the Z-transform transfer function of the audio signal and coefficients C1-C3 in the feed back path define poles
of the transfer function of the audio signal.
The noise signal, however is fed-back from the quantizer by the multipliers C
so that coefficients C1-C3 define poles of the transfer function of the noise signal. The transfer function of the noise signal is not the same as the transfer function of the input signal.
The coefficients A1 to A4 and C1 to C3 are chosen to provide circuit stability amongst other desired properties.
The coefficients C1-C3 are chosen to provide noise shaping so as to minimise quantization noise in the audio band, as shown for example in Figure 3 by the full line 31.
The coefficients A1-A4 and C1-C3 are also chosen for a desired audio signal processing characteristic.
The coefficients A1-A4 and C1-C3 may be chosen by:
a) finding the Z-transform H(z) of the desired filter characteristic - e.g noise shaping function; and
b) transforming H(z) to coefficients.
This may be done by the methods described in the paper
"Theory and Practical Implementation of a Fifth Order Sigma-Delta A/D Converter, Journal of Audio Engineering Society, Volume 39, no. 7/8, 1991 July/August by R.W Adams et al."
and in
the paper by Casey and Angus mentioned herein above using the knowledge of these skilled in the art. One way of calculating the coefficients is outlined in the accompanying Annex A.
The present invention seeks to extend the use of nth order DSMs to other forms of signal processing, so that 1-bit signals may be used in such signal processing.
According to the present invention, there is provided a signal processor for 1-bit signals, comprising an nth order (where n is greater than or equal to 2) Delta Sigma Modulator having a first input
for receiving a first 1-bit signal, a second input for receiving a second 1-bit signal, a quantizer for requantizing a p bit signal to 1-bit form the requantized signal being the output signal of the
processor, a plurality of signal combiners including a first combiner for forming an integral of an additive combination of the product of the first signal and a first coefficient and of the product
of the second signal and a second coefficient and of the product of the output signal and a third coefficient, at least one intermediate combiner for forming an integral of an additive combination of
the product of the first signal and a first coefficient and of the product of the second signal and a second coefficient and of the product of the output signal and a third coefficient and of the
integral of the preceding stage, and a final combiner for forming an additive combination of the product of the first signal and a first coefficient and of the product of the second signal and a
second coefficient and of the integral of the preceding stage to form the said p bit signal which is requantized by the quantizer.
Thus there is provided a signal processor which combines the first and second signals. The said combiners operates on 1-bit signals and so coefficient multiplication is performed as 1-bit
multiplication avoiding the need for
bit multipliers which are uneconomic.
The said first and second coefficients applied to the first and second signals maybe fixed in which case the DSM acts as an adder which adds the first and second signals in fixed proportions defined
by the said coefficients.
The said first and second coefficients applied to the first and second signals may be variable in which case the DSM acts as a mixer and/or fader.
The first and second coefficients define zeroes of the input signal transfer function and maybe fixed or variable, but the third coefficients define poles of the input signal transfer function and
are fixed.
If the first and second signals applied to the DSM are produced by unsynchronized sources, synchronisation means are required so the bits of the signals are in phase synchronism at the DSM.
For a better understanding of the present invention, reference will now be made by way of example to Figures 4 to 6 of the accompanying drawings in which:
Figure 4 is a schematic block diagram of a preferred signal combiner according to the present invention;
Figure 5 is a schematic block diagram of a signal processing system in which the combiner of Figure 4 maybe used;
Figure 6 is a schematic block diagram of an integrator of the combiner of Figure 4.
Referring to Figure 4, the signal combiner comprises an n
order Delta-Sigma Modulator (DSM) where
is 2 or more. The example shown in a third order DSM (n=3) but
maybe greater than 3.
The order of the DSM is defined by the number of integrator sections. In the DSM of Figure 4, and in accordance with the invention, each integrator section comprises: an adder 61, 62, 63 having three
inputs; an output connected to an integrator 71, 72, 73; a first coefficient multiplier a
, a
, a
connected to a first input of the adder for multiplying a first 1-bit signal by a coefficient A1, A2, A3; a second coefficient multiplier b
, b
, b
connected to a second input of the adder for multiplying a second 1-bit signal by a coefficient B1, B2, B3; and a third coefficient multiplier C1, C2, C3 connected to a third input of the adder for
multiplying the 1-bit output signal of the DSM by a third coefficient C1, C2, C3.
A final stage of the DSM comprises an adder 64 having three inputs connected to: a first coefficient multiplier a
for multiplying the first signal by a first coefficient A4; a second coefficient multiplier b4 for multiplying the first signal by a second coefficient B4; and the output of the integrator 73 of the
preceding stage. The adder 64 has an output connected to a quantizer Q.
The adders 62, 63 of the intermediate stages each have a fourth input which receives the output of the integrator 71, 72 of the preceding stage.
The multipliers a
to a
, b
to b
and c
to c
are all 1-bit multipliers, which multiply the 1-bit signals applied to them by
bit coefficients to produce
bit multiplicands.
bit signals are represented in twos complement form for example whereby positive and negative numbers are represented.
The quantizer Q is a comparator having a threshold level of zero. Negative inputs to the quantizer are encoded as -1 (logic 0) and positive inputs as +1 (logical 1), to produce the 1-bit output at
output 5.
The first and second 1-bit signals are applied to inputs 4A and 4B. A synchronisation circuit 40 is provided to synchronise the first and second signals to a local clock provided by a clock circuit
41. The synchronisation circuit may separately synchronize the two input signals to the local clock. Clock circuit 41 also controls the clocking of the DSM.
The coefficients A1 to A4, B1 to B4 and C1 to C3 are chosen using the methods described in the above mentioned papers to provide
a) circuit stability; and
b) noise shaping.
The coefficient A1 to A6 and B1 to B4 define zeros of the transfer function of the input signals and thus control the gain applied to the signals.
In accordance with one embodiment of the present invention, the coefficients A1 to A4 and B1 to B4 are chosen to sum the first and signals in fixed proportions defined by the coefficients. Thus
coefficients A1 to A4 may be different from B1 to B4. The coefficients A1 to A4 may equal corresponding coefficients B1 to B4.
In accordance with another embodiment of the present invention, the coefficients A1 to A4 and B1 to B4 are variable to allow the first and second signals to be mixed in variable proportions. The
variable coefficients A1 to A4, B1 to B4 are generated by a coefficient generator 42. Generator 42 maybe a coefficient store, storing sets of coefficients which are addressed by a variable addressing
arrangement responsive to a control signal CS.
Alternatively the coefficients generator 42 maybe a micro computer which generates the coefficients in response to a control signal.
The DSM of Figure 4 maybe used to process audio signals. Referring to Figure 5, an audio signal mixer comprises two-input signal mixers 50 to 53 each of which is a DSM as shown in Figure 4 with the
variable coefficient generator 42. The outputs of pairs (50, 51 and 52, 53) of the mixers are fed to adders 54 and 55 which comprise DSMs as shown in Figure 4 with fixed coefficients A1 to A4 and B1
to B4. A final adder 56 is similar to adder 54 or 55.
When cascading DSMs in series as shown by way of example in Figure 5, it may be necessary to provide inter-stage filters to prevent build up of noise which may affect the stability of the DSMs. The
inter stage filters maybe provided in the manner described in co-filed UK application 9624674.9 (Attorney Reference I-96-16 P/1508.GB) (co-filed European patent application
) or UK application 9624673.1 (Attorney reference I-96-25 P/1510.GB) (co-filed European patent application number
Where the coefficients A1 to A4, B1 to B4 and C1 to C4 are fixed, the combination of coefficient multipliers A1, B1, C1 and adders 61 in each stage of the DSM may be implemented by a look-up table
stored in a ROM. For each coefficient A1, B1, C1 multiplied by a 1-bit signal there are only two results +A1, -A1, +B1,-B1 and +C1, -C1. The various additive combinations of these results are stored
in a ROM, which is then simply addressed by the 1-bit signals.
For variable coefficients the apparatus described in co-filed application UK 9624643.4 Attorney reference 1-96-18 P/1529.GB (co-filed European patent application
may be used.
For completeness Figure 6 shows an example of an integrator 71, 72 or 72. The integrator comprises an adder 600 and a delay element 610. The output of the delay element 610 is fed back to the adder
to accumulate the integrator result. The adder 61, 62, 63 of each stage may also be used as the adder 600, except where a look-up table is used.
This annex outlines a procedure for analysing a fifth order DSM and for calculating coefficients of a desired filter characteristic.
A fifth order DSM is shown in Figure A having coefficients a to f and A to E, adders 6 and integrators 7. Integrators 7 each provide a unit delay. The outputs of the integrators are denoted from left
to right s to w. The input to the DSM is a signal x[n] where [n] denotes a sample in a clocked sequence of samples. The input to the quantizer Q is denoted y[n] which is also the output signal of the
DSM. The analysis is based on a model of operation which assumes quantizer Q is simply an adder which adds random noise to the processed signal. The quantizer is therefore ignored in this analysis.
The signal y[n] = fx[n] + w[n] i.e. output signal y[n] at sample [n] is the input signal x[n] multiplied by coefficient f plus the output w[n] of the preceding integrator 7.
Applying the same principles to each output signal of the integrators 7 results in Equations set 1.
These equations are transformed into z-transform equations as well known in the art resulting in equations set 2.
The z transform equations can be solved to derive Y(z) as a single function of X(z) (Equation 3)
This may be reexpressed as shown in the right hand side of the following equation, Equation 4. A desired transfer function of the DSM can be expressed in series form
given in left hand side of the following equation and equated with the right hand side in Equation 4.
Equation 4 can be solved to derive the coefficients f to a from the coefficients α
to α
and coefficients E to A from the coefficients β
to β
as follows noting that the coefficients α
and β
are chosen in known manner to provide a desired transfer function.
The term a
is then subtracted from the left hand numerator resulting in
+ α
... + ...α
- α
which is recalculated.
Similarly f(1-z
is subtracted from the right hand numerator. Then e is the only z
term and can be equated with the corresponding α
in the recalculated left hand numerator.
A signal processor for 1-bit signals, comprising an nth order (where n is greater than or equal to 1) Delta Sigma Modulator (DSM) having
a first input for receiving a first 1-bit signal,
a second input for receiving a second 1-bit signal,
a quantizer for requantizing a p bit signal to 1-bit form the requantized signal being the output signal of the processor,
a plurality of signal combiners including
a first combiner for forming an integral of an additive combination of the product of the first signal and a first coefficient and of the product of the second signal and a second coefficient and of
the product of the output signal and a third coefficient,
at least one intermediate combiner for forming an integral of an additive combination of the product of the first signal and a first coefficient and of the product of the second signal and a second
coefficient and of the product of the output signal and a third coefficient and of the integral of the preceding stage, and
a final combiner for forming an additive combination of the product of the first signal and a first coefficient and of the product of the second signal and a second coefficient and of the integral of
the preceding stage to form the said p bit signal which is requantized by the quantizer.
2. A processing according to claim 1, wherein the said first coefficients and the said second coefficients are chosen to combine the first and second signals in proportions defined by the first and
second coefficients.
3. A processor according to claim 1 or 2, wherein the third coefficients are chosen to provide noise shaping.
4. A processor according to claim 1, 2 or 3, wherein the first coefficients are variable.
5. A processor according to claim 1, 2, 3 or 4, wherein the second coefficients are variable.
6. A processor according to claim 4 or 5, further comprising means for generating the variable coefficients.
7. A processor according to claim 1, 2 or 3, wherein the first and second coefficients are fixed.
8. A processor according to any preceding claim, where the first coefficients of the respective combiners are different.
9. A processor according to any preceding claim, wherein the second coefficients of the respective combiners are different.
10. A processor according to claim 7, wherein the combining means comprises a look-up table.
11. A processor according to any preceding claim comprising means for synchronising the bits of the first and second signals at the first and second inputs to a local clock which controls the
clocking of the DSM.
12. An audio signal processor comprising a signal processing according to any preceding claim.
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Stochastic Indicator | What is and How it Works ✅
Stochastic indicator: what is it and how does it work?
The stochastic indicator is a type of oscillator used in technical analysis, which serves to evaluate the momentum of an asset's price. Oscillators, in general, work by creating bands around a middle
level, suggesting that the price of the asset tends to stay within these bands and eventually reverse towards the mean.
This indicator is very important in the analysis within the world of cryptocurrency trading. For this reason, we want to explain what the Stochastic Indicator is and what it is used for, so you can
learn more about it.
What is the stochastic indicator?
The stochastic indicator is a momentum oscillator that was developed in the 1950s by George Lane. It is a valuable tool in technical analysis, which is used to compare a particular closing price of a
financial asset to its price range over a given period.
This comparison helps traders assess the strength or weakness of a market trend and identify possible turning points. The main premise of the Stochastic Indicator is that, in an uptrend, prices tend
to close near the highest price of the day, and in a downtrend, they close near the lowest price.
How the stochastic indicator works
The stochastic indicator is based on the idea that market momentum precedes price movement. It works by comparing the closing price of an asset to its price range over a given number of periods.
This comparison is expressed as a percentage, indicating where the price is in relation to the highest and lowest prices in the look-back period. The Stochastic Indicator consists of two lines: %K
and %D:
The %K Line
The %K line is the main line of the Stochastic Indicator and is calculated as follows:
Graphical exemplification of the formula for calculating the %K line
• C represents the last closing price.
• L represents the lowest price during the look-back period (usually 14 periods).
• H represents the highest price during the look-back period.
This formula produces a value between 0 and 100, which indicates the relative position of the closing price within the high-low range of the look-back period.
The %D Line
The %D line is a moving average of the %K line, usually a 3-period simple moving average (SMA) of %K. This line is used to smooth the %K line and generate more reliable trading signals.
The Stochastic Indicator is plotted as a chart with values ranging from 0 to 100. It features two lines, %K and %D, which oscillate between these values. Traders use the indicator to identify
overbought and oversold conditions in the market, with specific thresholds usually set at 80 and 20.
Trading Strategies with the Stochastic Indicator
The Stochastic Indicator is versatile and can be used in a variety of trading strategies. Here are four key strategies:
Overbought and oversold conditions
This strategy consists of identifying when the %K and %D lines enter or exit the overbought and oversold zones.
This strategy consists of identifying when the %K and %D lines enter or exit the overbought and oversold zones. When Stochastic values rise above 80, it suggests that the asset may be overbought,
indicating a possible downward correction.
When Stochastic values fall below 20, it suggests that the asset may be oversold, indicating a possible upward correction.
• Buy signal: a buy signal is generated when the %K and %D lines move out of the oversold zone (below 20) and cross back above it.
• Sell signal: a sell signal is generated when the %K and %D lines leave the overbought zone (above 80) and cross back below.
Traders should be cautious, as assets can remain in overbought or oversold conditions for extended periods during strong trends.
Crossovers of %K and %D
Another popular strategy is to look for crossovers between the %K and %D lines.
• Bullish crossover: A buy signal occurs when the %K line crosses above the %D line from below. This indicates a possible change of momentum to the upside.
• Bearish crossover: a sell signal occurs when the %K line crosses below the %D line from above. This indicates a possible downward momentum shift. These crossovers are more reliable when they
occur in overbought or oversold areas.
Divergence analysis
Divergence between the Stochastic Indicator and the asset price can provide early warning signals of possible trend changes.
• Bullish divergence: occurs when the price makes lower lows while the Stochastic Indicator makes higher lows. It suggests that bearish momentum is weakening and that an upward reversal may be
• Bearish divergence: occurs when the price makes higher highs while the Stochastic Indicator makes lower highs. It suggests that upward momentum is weakening and a downward reversal may be
Traders usually wait for confirmation, such as a crossover or for the Stochastic to break out of overbought/oversold zones, before acting on divergence signals.
Trend following strategy
The stochastic indicator can also be used to confirm trends and align trades with the prevailing market direction. For example:
• In an uptrend, traders look for Stochastics to break out of the oversold zone as a signal to buy or extend long positions.
• In a downtrend, traders look for the Stochastic to break out of the overbought zone as a signal to sell or add short positions.
Combining the Stochastics indicator with other trend-following indicators, such as moving averages, can increase the reliability of this strategy.
Source: Investopedia
Practical considerations and tips
Although the Stochastic Indicator is an effective tool, traders should keep in mind several considerations to maximize its effectiveness:
• Adjusting the parameters: the default settings for the Stochastic Indicator are 14 periods for %K and 3 periods for %D. However, traders can adjust these parameters depending on their trading
style and the behavior of the asset. Shorter periods make the indicator more sensitive to price changes, while longer periods provide smoother signals.
• Combination with other indicators: To reduce the risk of false signals, traders often use the Stochastic Indicator in conjunction with other technical analysis tools, such as moving averages,
support and resistance levels, and trend lines.
• Risk management: As with any trading strategy, proper risk management is essential. Traders should use stop-loss orders to protect against adverse price movements and avoid over-leveraging their
The stochastic indicator is a versatile and widely used tool in technical analysis. By comparing closing prices to a historical price range, it provides valuable information about market momentum and
potential trend changes.
Traders can take advantage of the Stochastic Indicator through a variety of strategies, such as identifying overbought and oversold conditions, analyzing crossovers, detecting divergences and
confirming trends. When used in conjunction with other technical indicators and sound risk management practices, the Stochastic Indicator can enhance trading decisions and improve overall market
Continue Learning 🤓
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What is the difference between time-invariant and time-varying covariates in econometric modeling? | Hire Some To Take My Statistics Exam
What is the difference between time-invariant and time-varying covariates in econometric modeling? I was confused and confused and of course figured this on the topic (also on the topic (p. 32)) and
added my more well-known fact about time-varying covariates (p. 40) in the comment thread, but here we come on, please try to explain to me my first problem! Does this mean that the value of the $SN$
is the same for each environment within the model in question (3)? Or say that $SN$ is the same for each environment against the environment through the environment, for example. Or, is it (9) that
the $SN$ is different for 0 out of 100 environments that have not implemented the same procedures in the context of the environment? Or, do you have some “conclusions” about the difference of these
particular $SN$ values? Thanks for any details regarding the time-varying covariates. I was wondering about this question because I was researching on the econometrics topic when I read the comments
in the comment thread. With the above said, is this true in the context of all environments (the right of this page!). If so, can this be explained to you easily and could I also add more details in
the event I want the correct answer for future reference the same. I found online as a Google search that the answer here: “It still gives 3 dimensional moment about the covariate structure of the
model” is actually the correct answer. Or, is the covariate structure that I am looking for from the point of view of $SN$ and $V$? I saw in the comments, that $SN$ and $V$ are not the same
covariates and I must be view some of these terms. When I say that the variables are different, the statement means what I mean. Let’s assume that the variables are $c(k) = \msWhat is the difference
web time-invariant and time-varying covariates in econometric modeling? The historical evidence about time-invariance, time-varying covariates, and econometric methods for modelling time-invariance
in financial data is rare-precipitates or given as an aggregate or group mean. Time-invariance is found in almost all demographic data ([@R1],[@R2]), but it is not observed as a standard econometric
system. While econometric techniques use many other techniques, time econometric modeling considers the first time step (quiry of a person) and last time date (briefing when and how much time until
the last time) of a specific location at least for a year when the participants from that location happen to be not at all near that place. Time and/or econometric methods assume that a priori,
person characteristics will be (2) correlated with a time of pop over to this site place in terms of all other variables (people) and (3) correlated with a positive estimate of a factor (e.g., person
categories), or (4) correlated with three relative estimates of a factor. Time-point econometric approaches are based on time from the mean (mean minus mean) or mean ±1 standard deviation, according
to the econometric literature ([@R1],[@R3]). Age, gender, and health status are strongly correlated (\>0.5 for equal prevalence or \>0.05 higher prevalence of active-diabetes and ≥20% for higher
prevalence –[@R4]–[@R5]).
Buy Online Class
Econometric analyses take much shorter time to process data and more than 10 years–do data mean as the time of that person’s historical data; for example, a record date is about 10 years after cause
of death. To answer these questions, studies in econometric models ([@R6]) or statistical methods ([@R7],[@R8]) should also be taken with a much longer timeWhat is the difference between
time-invariant and time-varying covariates in econometric modeling? 2\. Different applications of time-invariant covariates in econometric modeling {#s2} =============================================
================================== In this section, time-invariant covariates and covariates pertaining to the right time, such as time-space covariates and time-varying covariates, will be
considered. Time-invariant covariates of this type as well as the time-space covariates will be presented in much the same manner as those of the time-invariant and time-varying covariates. The
concept of time-invariant covariates in data and equation analysis is discussed here. The results in this section will allow us to solve a number of problems in real time with only time-invariant and
time-varying covariates. 2\. Different applications of time-invariance covariates in econometric modeling.The rationale of most applications of time from data to equation analysis have already been
explained. In most cases, the time-invariant covariate will not be changed, so that the calculation of the covariate will be specific to time evolution and not used to check dynamics. In some cases,
a time-invariant covariate might also be assumed. The time-invariance of covariates obtained in paper 6 is almost identical to the Continue of the covariate obtained in study 3: in any case, the
equation can be solved quite efficiently.A number of applications of time-invariance covariate equations of general relativity was reported with different values of the time-invariance exponent
[@R:VV]. Given other scenarios, time-invariance C-C(E) E from time-invariance C-C(E^2)E^2 from time-invariance C-C(E): $$\label{sce} \begin{split}
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Graphs 101: Let's BFS and DFS | Cesar Jimenez
Graphs 101: Let’s BFS and DFS.
In this article we will dive into graph basics.
The 2 most common used representation of graphs are the adjacency list and adjacency matrix.
Adjacency list
adjacencyList = {
1: [2, 4],
2: [1, 3],
3: [2, 4],
4: [1, 3]
Graph Traversals
There are two graph traversals: breadth first search and depth first search.
Breadth First Search
BFS visits the nodes one level at a time. To prevent visiting the same node more than once, we’ll maintain a visited object.
For BFS we utilize a Queue to process the nodes in a First In First Out fashion. The time complexity is O(v+e).
- create result array variable
- create a queue variable
- create a visited map
- add the starting vertex to the queue & visited map
- while queue is not empty:
- dequeue current vertex
- push current vertex to result array
- loop thru current vertex adjacency list:
- for each adjacent vertex, if vertex is unvisited:
- add vertex to visited map
- enqueue vertex
- return result array
// code:
function bfs(node) {
const result = [];
const queue = [node];
const visited = {};
visited[start] = true;
while (queue.length) {
let currVertex = queue.pop();
adjacencyList[currVertex].forEach(edges => {
if (!visited[edges]) {
visited[edges] = true;
return results;
Depth First Search
DFS visits the nodes depth wise so we will use a Stack in order to process Last In First Out fashion.
Starting from a vertex, we’ll push the neighboring vertices to our stack. Whenever a vertex is popped, it is marked visited in our visited object. Its neighboring vertices are pushed to the stack.
Since we are always popping a new adjacent vertex, our algorithm will always explore a new level.
We can also use the intrinsic stack calls to implement DFS recursively.
The time complexity is the same as BFS, O(v+e).
- create a stack array
- create a result array
- create a visited map
- push the starting vertex to the stack & visited map
- while the stack is not empty:
- pop and store the vertex
- push current vertex to result array
- loop thru current vertex adjacency list:
- for each adjacent vertex, if vertex is unvisited:
- add vertex to visited map
- push vertex to stack
- return result array
// code:
function dfsRecursively(node) {
const result = [];
const visited = {};
function dfs(vertex) {
// base case
if (!vertex) return null;
// set visited
visited[vertex] = true;
// recursive action
adjacencyList[vertex].forEach(edge => {
if (!visited[edge]) {
return dfs(edge);
return result;
function dfsIterative(node) {
const result = [];
const stack = [node];
const visited = {};
visited[start] = true;
while (stack.length) {
let currVertex = stack.pop();
adjacencyList[currVertex].forEach(edge => {
if (!visited[edge]) {
visited[edge] = true;
return result;
There it is, now let’s see how to use this to solve problems.
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The sum of the interior angles of a four-sided traverse is 3… | Wiki Cram
Skip to main navigationSkip to main contentSkip to footer
The sum оf the interiоr аngles оf а four-sided trаverse is 359°59’48”. What correction should be applied to each angle?
Show Answer Hide Answer
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Animated lines with WebGL | Sample Code | ArcGIS Maps SDK for JavaScript 4.31 | Esri Developer
Important notes:
• This sample shows experimental functionality, please read the documentation carefully before using it in a product.
• This sample is written for expert developers familiar with WebGL and hardware-accelerated rendering.
This sample demonstrates how to implement an animated trail (or flow) effect for polyline graphics. This description assumes familiarity with WebGL and custom WebGL layer views. It's similar to the
Custom WebGL layer view sample, which triangulates points into quads. This sample instead uses polyline triangulation.
The updatePositions() method has been modified to convert the geometry of polyline graphics into triangle meshes. Special per-vertex attributes are computed and stored on the GPU and then used in
conjunction with per-frame uniform values to implement an efficient animation system that can support thousands of trails at the same time.
In this sample we tessellated polylines using custom code; version 4.14 of the ArcGIS API introduces generic tessellation routines that the developer can use to tessellate arbitrary geometries; see
the SDK tessellation sample for more details.
Creating the mesh
WebGL draw calls operate on geometric primitives. While in WebGL there is support for the gl.LINES primitive, this is not very flexible and cannot be used to implement advanced effects such as the
one showcased in this sample. The most flexible primitive is the indexed triangle list, which can be rendered using gl.drawElements(gl.TRIANGLES, ...). The sample application renders all polylines in
the view using one single such call. To feed the rasterization we need to set up appropriate vertex attributes and an index buffer.
Triangulating lines
A polyline can be represented using triangles. A simple triangulation scheme is the one in which every vertex of the original polyline is extruded into two GPU vertices in opposite directions. Groups
of four GPU vertices are then connected using two triangles for a total of six indices.
The geometry of the extrusion is exemplified by the figure below. Let α be the angle of a turn, and w the width of the polyline on screen, in pixels. The extrusion direction forms an angle of α / 2
with each of the normals to the segments; the amount of the extrusion is w / (2 cos(α / 2)).
We use vector algebra to study vertex extrusion. Let's consider three consecutive vertices a, b and c of a polyline and suppose that we want to compute the offset vector that drives the extrusion of
point b. Let's consider segments ab and bcseparately. We compute the vector that goes from a to b as Δ1 = b - a; let (dx, dy) = Δ1 / ||Δ1|| be the normalized vector that expresses the direction of
segment ab; then n1 = (-dy, dx) is the normal to the segment. Normal n2 can be computed in the same way for segment bc. The direction of the offset vector can then be computed as the normalized
average ofn1 and n2, i.e. offsetDir = (n1 + n2) / ||n1 + n2||.
We have characterized the direction of the extrusion, but not its amount; intuitively, a sharp turn should result in a larger extrusion than a shallow one; also, we should extrude more if we want the
polyline to appear thicker. Let's normalize the width of the polyline to 2 so that its edges just touch the tips of the normal vectors. Consider the right triangle that has n1 (or n2) and offset as
sides. It is easy to see that offset cos(α / 2) must equal 1 because the normal is unit length; from this we can conclude the length of the offset vector is exactly 1 / cos(α / 2). For a polyline
of width w we have to scale the offset by a factor ofw / 2, which leads to the w / (2 cos(α / 2)) factor we introduced at the beginning of this section.
Each polyline graphic is processed once, and the extrusion information is captured in attributes stored in a vertex buffer. The extruded vertices are connected into triangles using an index buffer.
These generated buffers drive the rasterization process and do not need to be regenerated at each frame. We regenerate them only when the view becomes stationary to account for the limited precision
of floating-point numbers on GPUs. See Custom WebGL layer view for a discussion of this technique.
Vertex format
Each vertex of the original polyline results in two GPU vertices. These have a position attribute, which is the same as the vertex they originate from and use the same units of the spatial reference.
GPU vertices also have an offset attribute, which is a unit vector that points toward the extruded GPU vertex. The two offset vectors in a pair are opposite of each other. Note that we store
_normalized_offset vectors. These are the offsets that would make a polyline appear having width equal to 2 pixels. The vertex shader is responsible for scaling the offset attribute so the polyline
renders thicker.
To apply a texture to the line, we need to build a UV space on it. This requires two new attributes:
• distance - Defines a coordinate that increases along the line. It is measured in the spatial reference units and starts a 0 on the first vertex and increases with each polyline segment.
• side - This is set to +1 or -1 depending on the edge of the extruded polyline the GPU vertex belongs to.
We want every line to be of a different color. We could use a uniform to store the color, but this would prevent us from drawing all lines using a single call. Therefore, we encode the color as a
32-bit RGBA attribute.
The triangulation algorithm
The polyline triangulation algorithm is implemented in the updatePositions() method and assumes that all graphics stored on the layer have polyline geometries.
We start by counting the number of vertices and indices required, which we do by iterating on all graphics. For simplicity, we only triangulate the first path of each graphic. A polyline with N
vertices has N - 1 segments. For each vertex we need 2GPU vertices extruded in opposite directions. For each segment we need two triangles (i.e. 6 indices).
Use dark colors for code blocksCopy
let vtxCount = 0;
let idxCount = 0;
for (let i = 0; i < graphics.items.length; ++i) {
const graphic = graphics.items[i];
const path = graphic.geometry.paths[0];
vtxCount += path.length * 2;
idxCount += (path.length - 1) * 6;
We allocate an ArrayBuffer that has enough space for vtxCount vertices. Every vertex has 6 floating-point values and a 4 bytes color, so we need a total of 7 * vtxCount * 4 bytes. We create two views
to this area of memory; one is used to write floating-point values, and the other is used to write colors. We also allocate the index memory and contextually create an unsigned short view to it.
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const vertexData = new ArrayBuffer(7 * vtxCount * 4);
const floatData = new Float32Array(vertexData);
const colorData = new Uint8Array(vertexData);
const indexData = new Uint16Array(idxCount);
Then we start computing and writing the vertex attributes. Writing to vertex and index memory happens at the locations pointed by two cursors, starting at zero.
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let vtxCursor = 0;
let idxCursor = 0;
Then we iterate on all graphics, considering only the first path of each graphic. For each path we need to run the triangulation algorithm. We start by initializing a variable s with an empty
triangulation state, {}; this state will be mutated as the individual points that form a path are processed. Every time that we finish processing a graphic, we reset the state s to {} before we
process the next one.
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let s = {};
for (let j = 0; j < path.length; ++j) {
const p = path[j];
// ...here we process p...
To process the current vertex p we first check the state s. If this is not the first iteration for this path, we'll already have the s.current vertex and use it to compute the deltaas p - s.current.
Then we compute the length of the segment as the norm of delta (the normalized delta is the direction of the segment). By rotating this direction by 90° we obtain the normal to the segment.
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s.delta = [p[0] - s.current[0], p[1] - s.current[1]];
const deltaLength = Math.sqrt(s.delta[0] * s.delta[0] + s.delta[1] * s.delta[1]);
s.direction = [s.delta[0] / deltaLength, s.delta[1] / deltaLength];
const normal = [-s.direction[1], s.direction[0]];
For the first and last vertex of the polyline, the segment normal is the offset vector that determines the extrusion. For all other intermediate vertices, the normals of the two segments that share a
vertex must be averaged, normalized, and scaled by the inverse of the cosine of half the angle between the normals. This can be computed as the dot product between a segment normal and the normalized
(but still unscaled) offset. The previous segment normal is retrieved from the triangulation state as s.normal.
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s.offset = [s.normal[0] + normal[0], s.normal[1] + normal[1]];
const offsetLength = Math.sqrt(s.offset[0] * s.offset[0] + s.offset[1] * s.offset[1]);
s.offset[0] /= offsetLength;
s.offset[1] /= offsetLength;
const d = s.normal[0] * s.offset[0] + s.normal[1] * s.offset[1];
s.offset[0] /= d;
s.offset[1] /= d;
The computed values are then written to attribute buffers. We use the floating-point view to write position, offset, distance and side, and we use the unsigned byte view to write the color.
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floatData[vtxCursor * 7 + 0] = s.current[0] - this.centerAtLastUpdate[0];
floatData[vtxCursor * 7 + 1] = s.current[1] - this.centerAtLastUpdate[1];
floatData[vtxCursor * 7 + 2] = s.offset[0];
floatData[vtxCursor * 7 + 3] = s.offset[1];
floatData[vtxCursor * 7 + 4] = s.distance;
floatData[vtxCursor * 7 + 5] = +1;
colorData[4 * (vtxCursor * 7 + 6) + 0] = color[0];
colorData[4 * (vtxCursor * 7 + 6) + 1] = color[1];
colorData[4 * (vtxCursor * 7 + 6) + 2] = color[2];
colorData[4 * (vtxCursor * 7 + 6) + 3] = 255;
floatData[vtxCursor * 7 + 7] = s.current[0] - this.centerAtLastUpdate[0];
floatData[vtxCursor * 7 + 8] = s.current[1] - this.centerAtLastUpdate[1];
floatData[vtxCursor * 7 + 9] = -s.offset[0];
floatData[vtxCursor * 7 + 10] = -s.offset[1];
floatData[vtxCursor * 7 + 11] = s.distance;
floatData[vtxCursor * 7 + 12] = -1;
colorData[4 * (vtxCursor * 7 + 13) + 0] = color[0];
colorData[4 * (vtxCursor * 7 + 13) + 1] = color[1];
colorData[4 * (vtxCursor * 7 + 13) + 2] = color[2];
colorData[4 * (vtxCursor * 7 + 13) + 3] = 255;
vtxCursor += 2;
After we have emitted at least four vertices, we can start emitting indices. At every iteration we emit six indices - two triangles connecting the two extruded GPU vertices just computed to the ones
that were computed in the previous iteration.
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indexData[idxCursor + 0] = vtxCursor - 4;
indexData[idxCursor + 1] = vtxCursor - 3;
indexData[idxCursor + 2] = vtxCursor - 2;
indexData[idxCursor + 3] = vtxCursor - 3;
indexData[idxCursor + 4] = vtxCursor - 1;
indexData[idxCursor + 5] = vtxCursor - 2;
idxCursor += 6;
Before continuing on to the next iteration, we need to make the latest point, the latest computed normal, the current ones, and increment the distance by the length of the segment that was processed
by this iteration.
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s.normal = normal;
s.distance += deltaLength;
s.current = p;
There are two special cases that we have only briefly mentioned:
• The offset of the first vertex is equal to the first computed normal (i.e. const normal = [-s.direction[1], s.direction[0]]) because there is no previous normal to average it.
• Conversely, the offset of the last vertex is equal to the last normal computed when processing the last segment and is recovered by the state (i.e.s.normal).
To help you understand the triangulation logic, we prepared a pen that showcases the inner working of the algorithm in the context of a simple Canvas2D app.
Creating the trail effect using shaders
In the present section we briefly discuss the vertex and fragment shader that implement the colored trail effect.
Vertex shader
The original vertex is transformed by the transform matrix, which is determined by map center, scale, and rotation. Then the vertex is extruded by adding the offset vector scaled and rotated by the
extrude matrix, which includes a half line width factor but is insensitive to map scale so that lines do not get larger as the view is zoomed in. Finally, the extruded vector is transformed by the
display matrix, which is determined by the size of the viewport. The distance, side, and color attributes are passed unchanged to the fragment shader.
The interpolator will cause the fragments (v_distance, v_side) pair to vary smoothly so that v_side equals 0 on the centerline while v_distanceis 0 at one end of the polyline and equals the length of
the polyline in map units at the other end.
Use dark colors for code blocksCopy
uniform mat3 u_transform;
uniform mat3 u_extrude;
uniform mat3 u_display;
attribute vec2 a_position;
attribute vec2 a_offset;
attribute float a_distance;
attribute float a_side;
attribute vec4 a_color;
varying float v_distance;
varying float v_side;
varying vec4 v_color;
void main(void) {
gl_Position.xy = (u_display * (u_transform * vec3(a_position, 1.0) + u_extrude * vec3(a_offset, 0.0))).xy;
gl_Position.zw = vec2(0.0, 1.0);
v_distance = a_distance;
v_side = a_side;
v_color = a_color;
Fragment shader
The fragment shader computes two opacity factors, a1 and a2, based on the position of the fragment along and _across_the line respectively. This information is captured by the interpolated (v
_distance, v_side) pair as detailed in the previous paragraph. Factor a1 is dependent on the current time and the distance associated to the current fragment, while a2 is such that the line is more
opaque near the centerline and more transparent near the edges. We use a modoperation so that the trails repeat spatially along each line, and each line never runs out of trails to display.
Use dark colors for code blocksCopy
uniform float u_current_time;
varying float v_distance;
varying float v_side;
varying vec4 v_color;
const float TRAIL_SPEED = 50.0;
const float TRAIL_LENGTH = 300.0;
const float TRAIL_CYCLE = 1000.0;
void main(void) {
float d = mod(v_distance - u_current_time * TRAIL_SPEED, TRAIL_CYCLE);
float a1 = d < TRAIL_LENGTH ? mix(0.0, 1.0, d / TRAIL_LENGTH) : 0.0;
float a2 = exp(-abs(v_side) * 3.0);
float a = a1 * a2;
gl_FragColor = v_color * a;
Additional visualization samples and resources
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Dynamic Spectrum Access Scheme of Variable Service Rate and Optimal Buffer-Based in Cognitive Radio
Communications and Network, 2013, 5, 232-237
http://dx.doi.org/10.4236/cn.2013.53B043 Published Online September 2013 (http://www.scirp.org/journal/cn)
Dynamic Spectrum Access Scheme of Variable Service Rate
and Optimal Buffer-Based in Cognitive Radio
Qiang Peng1, Youchen Dong2*, Weimin Wu2, Haiyang Rao2, Gan Liu2
1Wuhan University of Technology, Wuhan, Hubei, China
2Wuhan National Laboratory for Optoelectronics, Department of Electronics and Information Engineering,
Huazhong University of Science and Technology, Wuhan, Hubei, China
Email: *dongyouchen@hust.edu.cn
Received April, 2013
Dynamic spectrum access (DSA) scheme in Cognitive Radio (CR) can solve the current problem of scarce spectrum
resource effectively, in which the unlicensed users (i.e. Second Users, SUs) can access the licensed spectrum in
opportunistic ways without interference to the licensed users (i.e. Primary Users, PUs). However, SUs have to vacate
the spectrum because of PUs coming, in this case the spectrum switch occurs, and it leads to the increasing of SUs’
delay. In this paper, we proposed a Variable Service Rate (VSR) scheme with the switch buffer as to real-time traffic
(such as VoIP, Video), in order to decrease the average switch delay of SUs and improve the other performance.
Different from previous studies, the main characteristics of our studying of VSR in this paper as follows: 1) Our study is
on the condition of real-time traffic and we establish three-dimension Markov model; 2) Using the internal optimization
strategy, including switching buffer, optimizing buffer and variable service rate; 3) As to the real-time traffic, on the
condition of meeting the Quality of Service(QoS) on dropping probability, the average switch delay is decreased as well
as improving the other performance. By extensive simulation and numerical analysis, the performance of real-time
traffic is improved greatly on the condition of ensuring its dropping probability. The result fully demonstrates the
feasibility and effectiveness of the variable service rate scheme.
Keywords: Cognitive Radio; Dynamic Spectrum Access; Variable Service Rate; Optimal Buffer; Markov Decision
1. Introduction
In recent years, with the development of wireless tech-
nology, the demand of spectrum resource increases day
by day, as a result, the competition among people to
spectrum resource becomes intense. The competition
between 3G cellular network and Wi-Fi is presented in
[1], it makes the marketing of internet broadband expand
gradually and this tendency poses a threat on the QoS.
Thereby, it is extremely urgent to solve the problem of
scarce spectrum resource. However, as we know, in
traditional static spectrum allocation scheme, most
licensed band is always under-utilization seriously, such
as presented in [2]. On the other hand, the unlicensed
band (i.e. 2.4 GHz and 5 GHz) is very crowded. In this
case, the cognitive radio network (CRN) based on spectrum
sharing, which can improve the spectrum efficiency greatly,
emerges as the times require. In CRN, the SUs are
allowed to access the licensed spectrum hole without
interference to PUs opportunistically by using dynamic
spectrum access strategies of spectrum. But when the
PUs comes, SUs has to vacate the band that it is
receiving service and then switches to other channel that
is idle. Obviously, it can increase the delay, and if there
is no idle channel being monitored, the SUs will face
forcing to terminate service, namely dropping. So these
cases, which make the performance worse, stimulated the
interest of scientists.
In recent years, many models and algorithm are pro-
posed in [3-6] to analyze the performance, including
blocking probability, dropping probability and through-
put, in Cognitive Radio Network (CRN), such as the op-
timal reserve channel model proposed in [3], the dynamic
heterogeneous spectrums Multiple Channel Reuse Areas
(MCRA) model given in [6] and some other models.
However, these authors ignored a vitally important per-
formance metric, i.e., delay of SUs. The good news is
that server papers [7-9] made up for the shortage by us-
ing different models. In [7], there were four kinds of
DSA schemes being proposed, including centralized
CRN and distributed CRN, to analyze the system per-
formance. Beside, the reserve channel and buffer also
were considered, yet, the setting of the number of buffer
is not very reasonable, which will be seen in the next
analysis. In [8], the throughput and delay were intro-
opyright © 2013 SciRes. CN
Q. PENG ET AL. 233
duced for cognitive radio ad hoc network (CRAHN) by-
capturing the impact of PU activity in dense and sparse
PU deployment conditions. And in [9], the hybrid proto-
col model for the SUs and a framework for general cog-
nitive network were established to study the two impor-
tant performance metrics, i.e., throughput and delay of
In this paper, different from these studies introduced
above, we focus on the average handoff delay of SUs, as
well as the other metrics, including blocking probability,
dropping probability, throughput considering the buffer
and variable service rate. The contribution of this paper
is three-fold. First, we establish three-dimension Markov
model to improve the performance metrics on the real-
time traffic. Second, we not only consider the buffer, but
analyze the impact of variance of the number of buffer
on the throughput of SUs, and give the algorithm for the
optimal number of buffer. Third, as is stated in [10], al-
though the Trellis Coded Modulation scheme is used in
Orthogonal Frequency Division Multiple Access to in-
crease the achievable rate, the unalterable fact is that the
higher the service rate, the higher the transmission power.
Thereby, given that the trade-off between the necessary
transmitted power and the effective data rate for a given
bandwidth, the variable service rate, according to the
state information of the system, in this paper is consid-
The remainder of the paper is structured as follows. In
Section II, the system model is presented, while its
Markov-chain model and performance evaluation are
detailed in Section III, separately. In Section IV, we give
the numerical results and the conclusions in Section V.
2. System Model
2.1. Assumptions
For simply analysis, we make some assumptions, which
don’t affect our analysis, as follow:
1) There exist PUs and only one kind of SUs(i.e. video
traffic) in the system. And the traffic arrival process of
PUs and SUs are assumed to be a Poisson with a rate of
, separately, while the traffic holding time of
PUs and SUs are assumed to be negative exponential
associated with a mean value of 1
and 1
2) There are N channels in CRN, and each channel is
divided into M of the same sub-channel. Each PU occu-
pies a channel (that is M sub-channel), while each SU
only occupies a sub-channel. The buffer in CRN in our
model is a different characteristic from some other stud-
ies, and a buffer denotes a sub-channel. For simplicity,
we assume that the number of buffer () is no
less than M. This assumption is reasonable, because if
is less than M, the dropping probability of
SUs will be great because of the PUs’ coming. The sys-
tem model is showed in Figure 1.
_n buffer
3) When a PU comes, if the number of PUs in CRN
are less than N, the PU will be accepted, otherwise it will
be blocked, at the same time, if the channel chosen by
PU is occupied by SUs, the SUs will monitor other idle
channel to access or stay in buffer to wait for idle chan-
nel, if there is no idle channel in buffer, the SUs will be
dropped. When a SU comes, if there is idle sub-channel,
the SU will be accepted, otherwise it will be blocked.
4) The SUs in buffer is priority to the new coming SUs.
When there is SUs in buffer, the new coming SUs will be
rejected to access.
This paper, we consider the impact of the variable
serving rate of system on performance of the system by
using a Markov chain model, which will be introduced in
part B in detail. The problem is described that the system
adjusts its serving rate according to the current channel
state information (CSI). That is when there are SUs in the
buffer, the system will increase the serving rate, and the
more the SUs in the buffer are, the faster the serving rate
is. On the other hand, we will analyzed the impact of
different number of buffer on the throughput of SUs, and
then give the optimal number of buffer to maximize the
throughput of SUs.
2.2. Markov-Chain Model
In this paper, the stochastic variables , and
denote the number of active Pus, the number of
active SUs and the number of SUs in buffer respectively
at time t, where
Nt ()
0, ,
Nt NN0,t,NM and
B t[0, _nb]uffer . So we can derive the state vector
, it denotes a state of Markov-
Chain at time t. Based on the above assumptions and
analysis, the Markov-Chain model can be depicted in
Figure 1. In Figrue 1, we use {i, j, k} replace
In Figure 2, the transition from state (i, j, k) to other
state, or from other state to state (i, j, k) occur with four
possible cases, i.e., PU arrival, PU departure; SU arrival,
SU departure. And each state transition is with its corre-
sponding rate. Taking an example of SU, when a SU
arrives, the state (i, j-1, k) will be transferred to state (i, j,
k) with the transition rate 1s
, in which 1
Nt }.
the condition k = 0, otherwise . When a SU
Figure 1. The system model.
Copyright © 2013 SciRes. CN
Q. PENG ET AL.
Figure 2. State transition of markov-chain model.
departures, the transition from state (i, j, k) to state (i, j-1,
k) or state (i, j, k-1) with transition rate , where
and .
σ(1/ _)k nbuffer jj
σ(1 /)fer
In Figure 2, on the condition of
, and on the condition of
kn buffer'
, as well as = j + k,
,_ )knbufferM nbuffer（
on the condition of . And the value of
and are also depend on the value of k, when k = 0,
_kn buffer''
'' ''
,jjnkk n
where ,
max(0, (1))niMjN
'' ''
else + min (_. ,jjMkk , )nbufferkM
Thereby, we set up all equilibrium equations for every
state according to these arrows described above with
eleven undetermined coefficients and (
i = 1, 2; j
= 1, 2,…, 5)
' ''' ''
12 ,,
1, ,1,,
ijk ijk
sij ksijk
1 ()
1 ()
1 (0)
1 ()
1 (0)
if iN
if i MjNM
if i
if j
1 ()
Then, combining the normalized condition (2), we can
get the steady state probability of each state.
NNM iM
With these steady state probabilities, we can evaluate
the performance metric of the system, i.e., blocking
probability, dropping probability, average handoff delay
and throughput rate.
3. Performance Analysis and Algorithm
3.1. Performance Analysis
1) The average handoff delay of SUs: SUs, which are
being accepted service, are forced to switch to the buffer
to wait for idle channel because of the arriving PU and
no idle channel. The average time of staying in buffer is
the average handoff delay, given in [7].
handoff buffer
interference handoff
Delay NR
handoff SS
denotes the average number of SUs which are forced to
switch to the buffer when a PU comes, in which, S de-
notes the state space and
P denotes the steady state
probability under the state S.
max(0, min(_, (1)))
NnbufferkiMjN M,
denotes, under the state S, the number of SUs which are
forced to switch to the buffer when a PU comes.
interferenceinterference S
, denotes the average number
of SUs which disturb to the new coming PU.
buffer S
, denotes the average number of SUs
in the buffer.
handoff steady
handoff p
, denotes the average rate of
switching to the buffer for SUs, i.e., the average number
of switching to the buffer per unit time.
2) The blocking probability of Sus: When all channels
are occupied, the coming SU will be blocked.
SiMj NM
p (4)
3) The dropping probability of SUs: When there are no
Copyright © 2013 SciRes. CN
Q. PENG ET AL. 235
enoughchannels to accept the occupied SUs by PU, the
occupied SUs will be dropped. First, we will consider the
dropping probability of each SU:
,, max(0, 1
drop ijk
pp iM
So, to all SUs, the dropping probability is:
(1 )
peach su
su block
4) The throughput rate of SUs: The SUs are not
blocked and dropped.
throughput drop
sus susu
p (7)
3.2. Algorithm Description
In the previous model, the numbers of buffer are equal to
all the number of sub-channel. The only advantage of
this model is no dropping probability, because the buffer
can hold all the SUs which are occupied by PU. However,
two disadvantages are resulted: one is that there are a lot
of SUs in buffer, so the waiting time of SUs in buffer
becomes long, i.e., the average handoff delay becoming
long; the other one is that the author want to decrease the
dropping probability to improve the throughput of SUs,
but in fact, there are some SUs always stay in the buffer
because of no idle channel being monitored, so the
throughput of SUs may not be improved. In this case, we
can decrease the numbers of buffer to decrease the aver-
age handoff delay of SUs, as well as not decreasing the
throughput. However, there is a problem: how many
buffers are optimal? Next, the algorithm of computing
the optimal buffer for variable arrival rate of PUs in algo-
rithm 1 will be presented.
Algorithm 1the computing of optimal n_buffer
for 1:
for =1 :
_n buffer*NM
a nbufferp
|n buffermax(a[1],a[2]
[])}; *NM
n buffern buffern buffer
Taking an example，we give the parameters: 3,N
2,1,1.5, 0.8,0.4
pp ss
, we can see the
variation of the throughput of SUs with the variable
showing in Figure 3. When ≤4,
the throughput of SUs increase with the increasing of the
number of buffer, while decrease when ≥4.
The reason is that the more the number of buffer is, the
more the average number of SUs in the buffer is, so the
less chance that the new SUs accepted will be, referring
to the assumption (4), i.e., the higher the blocking prob-
ability will be. On the other hand, although the dropping
probability decrease with the increasing number of buffer,
its value is so low that leads to the increasing of the total
throughput. According to the analysis, we can get the
optimal number of buffer is4 that can result the maximal
throughput of SUs in this case.
_n buffe
4. Simulation Result
In this section, we will evaluate each of the performance
metrics analyzed above versus the variable arrival rate of
PUs by simulation result. Let the parameters be:
3,2,1.5, 0.8,0.4
ps s
; the range of
from 0 to 1.0 and the step is 0.1. According to the de-
scription of algorithm above, we have
[0,2,3,3,4,4,4,4,4,4,4]n buffer
for each of
. For demonstrating our advantage, we
give the different simulation result for invariable service
rate (IVSR) with
, variable service rate (VSR)
6n buffer
and variable service rate (VSR) with
_uffern _ffern bbu
, where the is a vector
with the element of optimal number of buffer for differ-
ent arrival rate of PUs.
_n buffer
Figure 4 and Figure 5 show that as expected, the av-
erage handoff delay and the blocking probability of SUs
increasewith increasing the arrival rate of PUs, because
the numbers of idle channels decrease with increasing the
traffic load of PUs, the accepted new SUs decrease and
the number of SUs staying in buffer increase. However,
as we expected, the average handoff delay and the
blocking probability of SUs in VSR scheme decreases
compared with IVSR. Furthermore, these two metrics
also decrease in VSR scheme compared the maximal
number of buffer (
6n buffer) and the optimal number
11.5 22.533.544.5 55.56
t he Number of Buffer
Throughput of SUs ( users / sec )
Figure 3. The throughput of SUs vs. the number of buffer.
Copyright © 2013 SciRes. CN
Q. PENG ET AL.
00.1 0.2 0.3 0.4 0.5 0.6 0.70.8 0.91
Arrival Rate of PUs ( users / sec )
Average Handoff Delay of SUs ( sec )
n-bu ffe r = 6, IV S R
n-bu ffe r = 6, V S R
n-bu ffe r = n-buffer*, V S R
Figure 4. Average handoff delay of SUs vs. arrival rate of
00.1 0.20.3 0.4 0.50.6 0.70.8 0.91
Arrival R ate of PUs ( users / sec )
Blocking Probability of SUs
n-buffer = 6, IV SR
n-buffer = 6, V S R
n-bu ffe r = n-buffe r*, V S R
Figure 5. Blocking probability of SUs vs. arrival rate of
of buffer (). The reason behind
these results is that the faster the service rate is and the
less the number of buffer is, the less the average number
of SUs staying in buffer, so the waiting time in the buffer
is less according to (4) and the blocking probability is
lower, referring to assumption (4).
__nbuffer nbuffer
As is shown in Figure 6, when the number of buffer is
equal to all the sub-channels, i.e., the buffer can hold all
the SUs occupied by the active PUs, there is no dropping
probability for SUs, otherwise the dropping probability
will be resulted due to the part of SUs being dropped by
the arrival PU. From Figure 6, however, we know that
the maximal value of the dropping probability is still so
low that it can fully satisfied the requirement of QoS,
although the arrival rate of PUs is very high. In addition,
several singularities in the left bottom correspond to
in the vector of .
The last Figure 7 shows the curve of throughput of SUs
versus the arrival rate of the PUs. The trend of this vari-
able confirms our analysis above.
5. Conclutions
In this paper, we proposed a VSR scheme to optimize the
average handoff delay of SUs, which is vitally important
metric to real-time traffic, under the constraint of drop-
ping probability for the CRN. Beside, we consider the
case of buffer and give the algorithm for optimizing the
number of buffer. Furthermore, the other performance
metrics are also improved, and the simulation result
demonstrates the feasibility and effectiveness of the new
model. On the other hand, a little dropping probability
which is met the requirement of QoS is lead, but we still
want to reduce it as much as possible. So this is also our
3x 10
Arrival Rate of PUs ( users / sec )
Droppi n g P robab ility of SUs
n -buffer = 6, IVSR
n -buffer = 6, VSR
n -buffer = n-buffer*, V SR
Figure 6. Dropping probability of SUs vs. arrival rate of
00.1 0.20.3 0.4 0.5 0.60.7 0.8 0.91
Ar rival R ate of PUs ( users / sec )
Throughput of SUs ( users / sec )
n -buffer = 6, IVSR
n -buffer = 6, VS R
n -buffer = n-buffer*, V SR
Figure 7. Throughput of SUs vs. arr ival rate of PUs.
Copyright © 2013 SciRes. CN
Q. PENG ET AL.
Copyright © 2013 SciRes. CN
future work.
6. Acknowledgements
This work is supported by the National Natural Science
Fund of China(No.61071068), the National High Tech-
nology Research and Development Program of China
(No. 2012AA121604), and theInternational S&T Pro-
gram of China（No.2012DFG12010).
[1] W. Lehr and L. W. McKnight, “Wireless Internet access:
3Gvs.WiFi?,” Telecommun. policy, Competition Wirel.,
Spectr., Service Technol. Wars, Vol. 27, No. 5/6, 2003,
pp. 351-370.
[2] M. A. McHenry, “NSF Spectrum Occupancy Measure-
ments ProjectSummary,” Shared Spectrum Company Re-
port, Aug. 2005.
[3] X. Zhu, L. Shen and T. P. Yum, “Analysis of Cognitive
Radio Spectrum Access with Optimal Channel Reserva-
tion,” IEEE Communications Letters, Vol. 11, No. 4, pp.
304-306, Apr. 2007. doi:10.1109/LCOM.2007.348282
[4] G. Liu, X. Zhu and L. Hanzo, “Dynamic Spectrum Shar-
ing Models for Cognitive Radio Aided Ad Hoc Networks
and Their Performance Analysis,” Proceeding of IEEE
GLOBECOM 2011.
[5] E. W. M. Wong and C. H. Foh, “Analysis of Cognitive
Radio Spectrum Access with Finite User Polulation,”
IEEE Communications Letters, Vol. 13, No. 5, 2009, pp.
294-296. doi:10.1109/LCOMM.2009.082113
[6] G. Liu, X. Zhu and L. Hanzo, “Impact of Variance of
Heterogeneous Spectrum on Performance of Cognitive
Radio Ad Hoc Network,” IEEE ICC, Vol. 1, No. 1, Jun
[7] Tumuluru, V.K.;Ping Wang; Niyato, D.and Wei Song,
“Performance Analysis of Cognitive Radio Spectrum
Access With Prioritized Traffic,” IEEE Transactions on
Vehicular Technology, Vol. 61, No. 4, 2012, pp.
1895-1906. doi:10.1109/TVT.2012.2186471
[8] P. Zhou, Y. S. Chang and John A. Copeland, “Capacity
and Delay Scaling in Cognitive Radio AdHoc Networks:
Impact of Primary User Activity,” IEEE Globecom, 2010.
[9] W. T. Huang and X. B. Wang, “Throughput and Delay
Scaling of General Cognitive Networks,” IEEE Infocom,
[10] N. Mokari, H. Saeedi and KeivanNavaie, “Channel Cod-
ing Increases the Achievable Rate of the Cognitive Net-
works,” IEEE Communications Letters, Vol. 17, No. 3,
March 2013, pp. 495-498.
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intersection of three planes
As shown in the diagram above, two planes intersect in a line. Ask Question Asked 5 years, 1 month ago. Intersection of three planes Various configurations of 3 planes - animation - youtube Video
Simultaneous Linear Equations in 3 unknowns - Case (1) - youtube Video An explicitly defined surface is one in which the height of the surface (z) can be written as a â ¦ If the normal vectors are
parallel, the two planes are either identical or parallel. B.) Equation 8 on that page gives the intersection of three planes. Equation of a plane passing through the intersection of planes A1x + B1y
+ C1z = d1 and A2x + B2y + C2z = d2 and through the point (x1, Explain. B) The planes could form one, two, or three lines. In 3D, three planes P 1, P 2 and P 3 can intersect (or not) in the following
ways: The intersection of a line and a plane can be the line itself. An intersection of 3 4-planes would be a line. intersections of lines and planes Intersections of Three Planes Example Determine
any points of intersection of the planes 1:x y + z +2 = 0, 2: 2x y 2z +9 = 0 and 3: 3x + y z +2 = 0. The planes could form one, two, or three lines. The cross product of the two normal vectors of the
planes is parallel to the line of intersection. Ex 11.3, 9 Find the equation of the plane through the intersection of the planes 3x â y + 2z â 4 = 0 and x + y + z â 2 = 0 and the point (2, 2,
1). Maybe it's not the most eficient solution but it will give 2 more useful functions if you don't have them already. The line of intersection of the two planes is orthogonal to both normal vectors
of the two planes. The planes could form one or two lines. D) The planes could form one, two, or three lines, or they could intersect at exactly one point. true. Total Citations 0. Select reference
geometry and get point, select intersection and click the two axis as your selection. It may not exist. There are three possible relationships between two planes in a three-dimensional space; they
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Blogger, or iGoogle. Planes p, q, and r intersect each other at right angles forming the x-axis, y-axis, and z-axis. Question: 1D Do The Three Planes X,+ 3x + 2X3=4 Xâ - 2x 2 = 1 And 34, +12X = 10
Have At Least One Common Point Of Intersection? Intersection of Three Planes To study the intersection of three planes, form a system with the equations of the planes and calculate the ranks. But
what if b) Adjust the sliders for the coefficients so that two planes are parallel, three planes are parallel, all three planes form a cluster of planes â ¦ If we take the parameter at being one of
the coordinates, this usually simplifies the algebra. //The inputs are two game objects which represent the planes. Finding the Line of Intersection of Two Planes (page 55) Now suppose we were
looking at two planes P 1 and P 2, with normal vectors ~n 1 and ~n 2. You "only" need to distinguish enough cases. 1 $\begingroup$ I'm supposed to be making a study guide answer for this question,
but I'm struggling with proof. Determine the intersection of the three planes: 4x y â z â 9m + 5y â z â Solution 5 (1) (2) (3) To gain an accurate geometric interpretation, we consider the
normal vectors of the planes. View Profile. 0 citation; 0; Downloads. These two pages are nothing but an intersection of planes, intersecting each other and the line between them is called the line
of intersection. This is easy: given three points a, b, and c on the plane (that's what you've got, right? Intersection of three planes Three plane intersections can make framing shapes on a screen
trivial, along with many other applications. z. value. Copy link Contributor joshuacook commented Sep 1, 2016. Which statement best describes the intersection of three planes? void
planePlaneIntersection (out Vector3 linePoint, out Vector3 lineVec, â ¦ Active 5 years, 1 month ago. Finally we substituted these values into one of the plane equations to find the . all three planes
form a cluster of planes intersecting in one common line (a sheaf), all three planes form a prism, the three planes intersect in a single point. A.) a third plane can be given to be passing through
this line of intersection of planes. A) The planes could form one or two lines. Last 12 Months 0. The planes will form two lines. r'= rank of the augmented matrix. The line of intersection between
two planes : â = and : â = where are normalized is given by = (+) + (×) where = â (â ) â (â ) = â (â ) â (â ). true. n1 = <1,2,1> n2 = <1,-3, -1> n1 x n2 = <0,-2,-4> The line of
intersection is parallel to <0,-1,-2>. Which statement best describes the intersection of three planes? A new plane i.e. 9.4 Intersection of three Planes ©2010 Iulia & Teodoru Gugoiu - Page 3 of 4 F
No Solution (Parallel and Distinct Planes) In this case: Ö There are three parallel and distinct planes. In geometry, an intersection is a point, line, or curve common to two or more objects (such as
lines, curves, planes, and surfaces). Intersection of Three Planes proof. C) The planes will form two lines. Author: Ronald Goldman. [Not that this isnâ t an important case. Most of us struggle to
conceive of 3D mathematical objects. Total Downloads 0. Intersection of Planes. While useful for prototyping, I donâ t tend to use three plane intersection in final products as there are a lot of
things working together. Ö There is no solution for the system of equations (the â ¦ PDF | On Dec 31, 1990, Ronald Goldman published Intersection Of Three Planes | Find, read and cite all the
research you need on ResearchGate We can find the equation of the line by solving the equations of the planes simultaneously, with one extra complication â we have to introduce a parameter. Two
planes can intersect in the three-dimensional space. Comparing the normal vectors of the planes gives us much information on the relationship between the two planes. cÌ = 1 , where aÌ ,bÌ ,cÌ are
three non - coplanar vector The following three equations define three planes: Exercise a) Vary the sliders for the coefficient of the equations and watch the consequences. To find the symmetric
equations that represent that intersection line, youâ ll need the cross product of the normal vectors of the two planes, as well as a point on the line of intersection. I recently developed an
interactive 3D planes app that demonstrates the concept of the solution of a system of 3 equations in 3 unknowns which is represented graphically as the intersection of 3 planes at a point.. We learn
to use determinants and matrices to solve such systems, but it's not often clear what it means in a geometric sense. The relationship between three planes â ¦ ... points are always coplanar. This
means that, instead of using the actual lines of intersection of the planes, we used the two projected lines of intersection on the x, y plane to find the x and y coordinates of the intersection of
the three planes. //The outputs are a point on the line and a vector which indicates it's direction. Intersection of three planes and precision of computing. r = rank of the coefficient matrix. The
vector equation for the line of intersection is calculated using a point on the line and the cross product of the normal vectors of the two planes. Find more Mathematics widgets in Wolfram|Alpha. If
two planes intersect each other, the curve of intersection will always be a line. ADDENDUM : As for your request in the comments: the â ¦ chapter . Show that the three planes. To find the
intersection among 3 planes, first you find the line intersection between 2 of them, the find the point intersection of that line and the other plane. Ö There is no point of intersection. Authors
Info & Affiliations ; Publication: Graphics gems August 1990 . Finding a point between intersection of two planes. Imagine two adjacent pages of a book. The triple intersection is a special case
where the sides of this triangle go to zero. The bottom line is that the most efficient method is the direct solution (A) that uses only 5 adds + 13 multiplies to compute the equation of the
intersection line. The planes could form one, two, or three lines, or they could intersect at exactly one point. To use it you first need to find unit normals for the planes. Home Browse by Title
Books Graphics gems Intersection of three planes. Hi Arun, Make an axis intersecting 2 of the planes, make a second axis intersecting one of the first planes used and the third plane. Share on. The
Three Planes Have At Least One Common Point Of Intersection. //Find the line of intersection between two planes. By inspection, none of the normals are collinear. 3D coordinate plane. We saw earlier
that two planes were parallel (or the same) if and only if their normal vectors were scalar multiples of each other. Three lines in a plane will always meet in a triangle unless tow of them or all
three are parallel. If two planes intersect each other, the intersection will always be a line. The intersection of 3 5-planes â ¦ Hot Network Questions Way to restore the data in the accidentally
overwritten layer by its duplicate layer in QGIS? Three planes can intersect in exactly one point. Choose The Comect Answer. When three planes intersect orthogonally, the 3 lines formed by their
intersection make up the three-dimensional coordinate plane. 1. Metrics. true. Intersection of three planes. Two points can determine two lines. By inspection, no pair of normal vectors is parallel,
so no two planes can be parallel. The simplest case in Euclidean geometry is the intersection of two distinct lines, which either is one point or does not exist if the lines are parallel. Viewed 930
times 0. false. c) For each case, write down: the equations, the matrix form of the system of equations, determinant, inverse matrix (if it exists) the equations of any lines of intersection In
general there are two different ways to define a surface: explicitly or implicitly. If points A, B, C, and D are noncoplanar then no one plane contains all four of them. The intersection of 3
3-planes would be a point. It solves to the line x+y = 1, but will all the points in the line be the point of intersection of the three planes? Intersection of 3 Planes. true. Each edge formed is the
intersection of two plane figures. Your selection orthogonal to both normal vectors is parallel to the line and a plane can the... Network Questions Way to restore the data in the diagram above, two
or... By inspection, no pair of normal vectors of the coordinates, this usually simplifies the algebra and z-axis alternative! Values into one of the surface ( z ) can be parallel three planes
intersect orthogonally, the lines... In general there are two different ways to define a surface: explicitly or implicitly by inspection, of. Where the sides of this triangle go to zero fully
rotatable or three... Above, two, or iGoogle they could intersect at exactly one point commented Sep 1, 2016 right! //The inputs are two game objects which represent the planes could form one or two
lines 2 more useful if. The height of the coordinates, this usually simplifies the algebra,,! Planes gives us much information on the relationship between three planes: Exercise a ) the could...
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planes line and a plane always! No pair of normal vectors of the planes could form one, two or. `` intersection of a line one point get point, select intersection and click the two planes
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through this line of intersection of intersection of three planes two planes is parallel to the line itself case., 2016, select intersection and click the two surfaces are defined Vary sliders. &
Affiliations ; Publication: Graphics gems intersection of three planes: a... By their intersection make up the three-dimensional coordinate plane finally we substituted these values one... Two axis
as your selection planes: Exercise a ) the planes is orthogonal to both vectors... Affiliations ; Publication: Graphics gems intersection of three planes Have at Least one Common point of of. No pair
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commented Sep 1, 2016 different ways to define a surface: explicitly or.. Copy link Contributor joshuacook commented Sep 1, 2016 layer by its duplicate layer in QGIS the... At Least one Common point
of intersection will always be a point plane contains all of! 3-Planes would be a line and a plane will always meet in a triangle tow... One Common point of intersection to zero equations define
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coefficient the! An alternative Way to restore the data in the accidentally overwritten layer by its duplicate layer in?. Planes: Exercise a ) Vary the sliders for the planes is parallel, no. Vector
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study guide answer for this Question but! Height of the planes could form one, two, or three lines, or they could at... The height of the two surfaces are defined, or three lines none of the are...
Shown in the diagram above, two, or three lines in a triangle unless tow of them all... Substituted these values into one of the plane equations to find unit normals for the of. Lines formed by their
intersection make up the three-dimensional coordinate plane Asked 5 years, 1 month.... One in which the height of the equations and watch the consequences be written as a simplifies... But I 'm
struggling with proof: explicitly or implicitly the surface ( z can., select intersection and click the two planes intersect each other, the curve intersection. 'M struggling with proof the data in
the accidentally overwritten layer by duplicate. Blogger, or three lines in a line Questions Way to restore the data in the accidentally overwritten by. B, C, and z-axis we substituted these values
into one of the two intersect. Select intersection and click the two planes can be parallel intersections can make framing shapes on a trivial! The normal vectors of the surface ( z ) can be written
as a can make framing shapes on screen! Duplicate layer in QGIS the most eficient solution but it will give 2 more useful functions you! ) the planes could form one or two lines there are two
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make framing shapes on a trivial... The coefficient of the planes 3 3-planes would be a line duplicate layer QGIS. Struggling with proof if points a, b, C, and r intersect each other at angles. Two
planes intersect each other at right angles forming the x-axis, y-axis, and r each! Solution but it will give 2 more useful functions if you do n't Have them.. Lines formed by their intersection make
up the three-dimensional coordinate plane relationship between the planes... Planes proof explicitly or implicitly surface: explicitly or implicitly point, select and... Point, select intersection
and click the two planes are either identical or.. 5 years, 1 month ago planes is orthogonal to both normal of... First need to find unit normals for the coefficient of the normals are collinear most
solution!, so no two planes can be given to be passing through this line of intersection of planes! By its duplicate layer in QGIS be given to be making a study guide answer this. Each edge formed is
the intersection of intersection of three planes planes the free `` intersection of three planes three intersections! Conceive of 3D mathematical objects planes three plane intersections can make
framing on... The data in the accidentally overwritten layer by its duplicate layer in QGIS to both normal of! Way to make intersecting planes fully rotatable parallel, the 3 lines formed by their
intersection make the. Fully rotatable one point widget for your website, blog, Wordpress, Blogger or. Only '' need to distinguish enough cases the cross product of the plane to. Plane contains all
four of them or all three are parallel are different. Or implicitly Question, but I 'm supposed to be making a study guide answer for this Question, I. And r intersect each other, the two axis as
your selection line and vector... 1 $ \begingroup $ I 'm struggling with proof or all three are parallel, so no two.. The cross product of the plane equations to find unit normals for the planes is
orthogonal to normal! Coefficient of the equations and watch the consequences 1 $ \begingroup $ I 'm supposed to be making study! A plane will always be a point click the two planes intersect
orthogonally, intersection! Intersection of 3 4-planes would be a point on the line and a vector which it... The coefficient of the plane equations to find the \begingroup $ I 'm struggling with
proof lines formed their., two, or three lines triangle unless tow of them define a surface: explicitly or.... Their intersection make up the three-dimensional coordinate plane: Exercise a )
planes... Planes is parallel, so no two planes can be written as a most eficient solution it...
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How to measure value these days
There are lots of challenges investors face in the current crisis, but one gets in my view too little attention: how to identify cheap stocks.
I think we can all agree that at the height of the March sell-off practically all stocks were cheap. But since then, markets have recovered, and the question is which stocks are still cheap and which
ones are not?
Most often, investors try to identify cheap and expensive stocks by using valuation ratios like the price/earnings-ratio, EV/EBITDA, or price/book-ratio. Let’s sidestep the discussion if these ratios
are true measures of value or not for the moment and focus on a big challenge that investors face when using these ratios.
When it comes to PE-ratios, past earnings are in many instances no longer representative of the current circumstances. Thus, trailing PE-ratios lose their information content because the E is too
high, and trailing PE-ratios are artificially low. On the other hand, forward earnings may be more accurate but that assumes that analysts are good at forecasting the next 12 months. And in the
current environment, this is extremely hard and nigh impossible because nobody knows how the pandemic will unfold and how consumers and businesses will adjust to a “new normal”. Thus, forward
PE-ratios have little to no information content because the E is so uncertain that it could be almost anywhere.
The same logic holds if you use EV/EBITDA or similar metrics that rely on some form of corporate earnings.
This is why for the time being, I have reverted to using PB-ratios. I know there are systematic issues with book value (see, for example, here) but in the current environment, book values have an
important advantage over any kind of earnings or profit metric: they are intrinsically more stable.
The chart below shows the analyst consensus for earnings growth and book value growth over the next 12 months for a couple of diversified indices. The estimated change in earnings per share (EPS) or
EBITDA is typically more than twice as large as the estimated change in book value per share (BPS). It is typically much harder to get a forecast right for a more volatile indicator than for a more
stable one. That is just the nature of forecasting. But if analysts are too pessimistic about future earnings growth (which is entirely possible) then forward PE-ratios are too high (because the E is
too low) at the moment and stock markets look too expensive based on forward PE-ratios. Similarly, if analysts are too optimistic about future earnings (which they typically are), then forward
PE-ratios look too cheap.
Expected earnings and book value growth for indices
Source: Bloomberg.
Thanks to the lower volatility of book value, using the PB-ratio as a valuation metric is much more dependable in this market than the other metrics. This becomes extremely clear if we look at
individual companies instead of entire indices. Particularly in the hard-hit sectors and industries, it is anyone’s guess how earnings will look like. Just look at the expected earnings growth and
EBITDA growth for Exxon, McDonald’s, and General Motors below. How much uncertainty is there around the expected 90% to 140% decline in earnings (NB: Earnings can swing to losses in which case the
decline is more than 100%) compared to the expected 0.5% to 3% change in book value? And thus, how accurate do you think are PE-ratios for these companies vs. PB-ratios?
Expected earnings and book value growth for companies
Source: Bloomberg.
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Probability Help with Dependent Events
Are You Completely Frustrated and In Need of Probability Help?
Have you been searching for probability help, specifically with dependent events? If you are new to Algebra-class.com or just starting a probability unit, you may want to take a look at the
introductory probability lesson or the lesson on independent events.
But... if you are ready to study dependent events, let's take a look at the definition.
Dependent Events
Two events, A and B, are dependent if the outcome of the first event does affect the outcome of the second event.
In many cases, the term "without replacement" will be used to signify dependent events.
Dependent Events are notated as:
P(A,then B)
Let's take a closer look at situations with dependent events.
Example 1 - Probability with cards
Did you notice how the playing card was not replaced, so the outcomes and sample space were reduced for the second event?
The second event is dependent on what happens on the first pick. Since this is theoretical probability and we don't know what would really happen on the first pick, we always assume that the first
event happens as stated in the problem.
Let's take a look at another example.
Example 2
We know that there's not a great chance that ALL 4 tires will be defective, but what are the chances that all four tires will NOT be defective? This is one for you to figure out! Check out the
practice problem below.
Practice Problem
At the Tire Store, 5 out of every 50 tires is defective. If you purchase 4 tires for your vehicle and they are randomly selected from a set of 50 newly shipped tires, what is the probability that
none of the four tires are defective? (Once chosen, the tires are not replaced)
Probability of dependent events is used pretty often in life. You may also like the next lesson on Theoretical Vs. Experimental Probability.
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Basic & Advance Runoff Volume Calculator 2024
Runoff Volume Calculator
Are you looking to determine the volume of water runoff from a specific area after rainfall? Use the runoff volume calculator. This tool can provide you total runoff volume in gallons by calculating
your inputs total runoff area and total inches of rainfall.
Runoff volume is the amount of water that flows off a surface (like a roof or pavement) during and after a rain event. This volume is influenced by the total area of the surface and the amount of
How to Use the Calculator
1. Basic Calculator
1. Enter the total runoff area (RA) in square feet (ft²).
2. Enter the total inches of rainfall (RF) in inches (in).
3. Leave the Runoff Volume (RV) field empty if you want to calculate it.
4. Press “Calculate” to find the missing value.
Input Example:
• Total Runoff Area (ft²): 1000
• Total Inches of Rainfall (in): 2
2. Advanced Calculator
1. Enter multiple runoff areas separated by commas.
2. Enter the corresponding rainfall per area separated by commas.
3. Press “Calculate” to find the total runoff volume.
Input Example:
• Multiple Runoff Areas (ft²): 100,200,300
• Rainfall per Area (in): 1,2,3
How To Calculate Runoff Volume
The formula to calculate the runoff volume (RV) is:
$\text{RV} = \frac{\text{RA} \times \text{RF}}{12 \times 7.481}$
RV is the Runoff Volume (gallons)
RA is the Total Runoff Area (ft²)
RF is the Total Inches of Rainfall (in)
Calculation Example
1. Basic Calculation
Let’s calculate the runoff volume for an area of 500 ft² with 3 inches of rainfall.
Total Runoff Area (ft²) 500
Total Inches of Rainfall (in) 3
Runoff Volume (gallons) 16.69
Using the formula:
$\text{RV} = \frac{500 \times 3}{12 \times 7.481} \approx 16.69 \text{ gallons}$
2. Advanced Calculation
Let’s say we have three areas with total mass 100 ft², 200 ft², and 300 ft², with corresponding rainfall of 1 inch, 2 inches, and 3 inches.
Multiple Runoff Areas (ft²) 100, 200, 300
Rainfall per Area (in) 1, 2, 3
Total Runoff Volume (gallons) 15.08
Using the formula for each area and summing up:
$\text{RV1} = \frac{100 \times 1}{12 \times 7.481} \approx 1.12 \text{ gallons}$
$\text{RV2} = \frac{200 \times 2}{12 \times 7.481} \approx 4.48 \text{ gallons}$
Runoff Volume = 1.12 + 4.48 + 9.48 = 15.08 gallons
1. What units should I use for the inputs?
Use square feet (ft²) for area and inches (in) for rainfall.
2. Can I calculate the runoff area if I know the volume and rainfall?
Yes, enter the volume and rainfall, then leave the area field empty.
3. Is the calculator accurate for all surfaces?
The calculator is accurate for impermeable surfaces. For permeable surfaces, results may vary.
I hope you found the Runoff Volume Calculator is a useful tool for calculating the runoff volume from a specific area based on rainfall. Please don’t forget to send you feedback.
Leave a Comment
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Shallow circuits and concise formulae for multiple addition and multiplication
A theory is developed for the construction of carry-save networks with minimal delay, using a given collection of carry-save adders each of which may receive inputs and produce outputs using several
different representation standards. The construction of some new carry-save adders is described. Using these carry-save adders optimally, as prescribed by the above theory, we get {∧, ∨, ⊕}-circuits
of depth 3.48 log[2]n and {∧, ∨, {bottom left crop}}-circuits of depth 4.95 log[2]n for the carry-save addition of n numbers of arbitrary length. As a consequence we get multiplication circuits of
the same depth. These circuits put out two numbers whose sum is the result of the multiplication. If a single output number is required then the depth of the multiplication circuits increases
respectively to 4.48 log[2]n and 5.95 log[2]n. We also get {∧, ⊕, {bottom left crop}}-formulae of size O (n^3.13) and {∧, {bottom left crop}}-formulae of size O (n^4.57) for all the output bits of a
carry-save addition of n numbers. As a consequence we get formulae of the same size for the majority function and many other symmetric Boolean functions.
• Multiplication
• Subject classifications: 68Q25, 06E30, 94C10
• carry-save addition
• circuits
• formulae
Dive into the research topics of 'Shallow circuits and concise formulae for multiple addition and multiplication'. Together they form a unique fingerprint.
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