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Combining Like Terms
Contributed by:
This pdf includes the following topics:-
How to Combine Like Terms
Why Combine Like Terms
1. Like Terms
People who are twins are a “Like Pair” because they have the exact same features.
An Algebra Equation we can write for this is :
T + T = 2T
Eg. One Twin + One Twin = 2 Twins.
In Algebra, when there are items that are the same thing, we can add them together to make a
simpler expression.
2. Combining Like Terms
+ +
3. Why We Combine Like Terms
Often in real life it is necessary to combine like items together to create a shorter list of items
we can deal with.
For example, imagine that a mathematics class is on an excursion and need to order a take
away food lunch.
It would be crazy to read out each individual order, one after each other, at the counter of the
fast food restaurant.
Instead we would total up how many burgers, how many fries, how many drinks, etc that we
needed for the whole group.
This type of summarizing process is exactly the same as combining like terms.
The following video might get really annoying after a while, but it definitely shows why we need
to group items together into separate sub totals of each type when doing mathematics.
4. How to Combine Like Terms
We can add together items that are the same to make a simplified shorter list of items.
This is called “combining like terms” or “collecting like terms”.
Consider the following family take away order:
Two burgers, one fries, one drink, three more burgers, two more fries, and two more drinks.
This order needs to have the like items grouped together to make a shorter summarized list.
As an algebraic expression: ____________________________________________________________
7. 4x + 6 + 2x
5n + 2 - n - 6
–4b + 2c –4b + 3c
8. Simplify.
4 - 20(g - 1)
5. 6y - 3(x - 2y)
2m + 5(2m + 5n) | {"url":"https://merithub.com/tutorial/combining-like-terms-c8o8silonhcp7ds82reg","timestamp":"2024-11-06T01:51:50Z","content_type":"text/html","content_length":"32485","record_id":"<urn:uuid:4d150ce9-9abe-4333-ac5b-226dbbd3aae8>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00412.warc.gz"} |
What is 49/32 as a percent? | Thinkster Math
In a fraction, the number above the line is called the numerator, and the number below the line is called the denominator. The fraction shows how many “pieces” of the number there are, compared to
how many there are possible. For instance, in the fraction 49/32, we could say that there are 49 pieces out of a possible 32 pieces.
“Percent” means “per hundred”, so for percentages we want to know how many pieces there are if there are 100 pieces possible. For example, if we look at the percentage 75%, that means we have 75
pieces of the possible 100. Re-writing this in fraction form, we see 75/100.
When converting the fraction into a percent, the first step is to adjust the fraction so that there will be 100 pieces possible (the denominator needs to be changed to 100). To do this, you first
divide 100 by the denominator:
$\frac{100}{32} = 3.125$
We can then adjust the whole fraction using this number, like so:
$\frac{49*3.125}{32*3.125} = \frac{153.125}{100}$
As you can see, there are 153.125 pieces out of a possible 100 pieces. Re-writing this as a percentage, we can see that 49/32 as a percentage is 153.125%. | {"url":"https://hellothinkster.com/math-questions/percentages/what-is-49-32-as-a-percent","timestamp":"2024-11-09T10:53:11Z","content_type":"text/html","content_length":"100010","record_id":"<urn:uuid:ef6def48-2ff0-457c-861a-8ebe13677a59>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00861.warc.gz"} |
There are many resources for documentation about the project.
Here is a quick tutorial to get you started with DB48X.
Try this: Plotting a function
Try the following key sequence, where you need to hit the Shift key as a separate key, and observe what happens with each key:
Key to type What happens
F Enter expression
Shift Left Shift
ENTER ALPHA mode
X Enter variable name X
ENTER Put X on the stack
ENTER Duplicate X
3 Enter number 3
. 3.
4 3.4
2 3.42
* Put 3.42 times X on stack
J Compute sin(3.42·X)
M Swap stack levels 1 and 2
K Compute cos(X)
/ Compute sin(3.42·X)/cos(X)
Shift Left shift
N Select MODES menu
F2 Select Radians mode
Shift Left Shift
Shift Right Shift
O PLOT menu
F1 Draw function
Try this: Computing 200 digits of pi
Try the following key sequence, where you need to hit the Shift key as a separate key, and observe what happens with each key:
Key to type What happens
Shift Left Shift
N MODES menu
F2 RADIANS mode
Shift Left Shift
O DISP menu
2 Enter number 2
F6 200-digit precision
Shift Left Shift
M Last Arg (Recall 200)
F5 200 significant digits shown
1 Enter number 1
Shift Left Shift
L Compute arc-tangent of 1
4 Enter number 4
* Compute pi
Shift Left Shift
. SHOW function (show full screen)
There is a video playlist on YouTube containing many demos, installation procedures, and more.
On-line reference
The reference documentation for the project is available online. It is the same content that is also installed on the calculator for the built-in help.
Comparison with other projects
The project includes comparisons with other third-party firmware for SwissMicros calculators to help you pick a program that suits your needs best.
There is also a page highlighting the differences and similarities with the DM42.
There are active discussion forums on the museum of HP calculators web site, notbly a dedicated thread, and on the SwissMicros forum.
Keyboard overlays
Keyboard overlays can be ordered to make it easier to use the firmware on a physical calculator. | {"url":"http://48calc.org/documentation","timestamp":"2024-11-02T11:17:52Z","content_type":"text/html","content_length":"14400","record_id":"<urn:uuid:2320dc34-8cab-46e5-b59c-172b91f22b3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00631.warc.gz"} |
urlongs to Fingerbreadth
Furlongs to Fingerbreadth Converter
Enter Furlongs
β Switch toFingerbreadth to Furlongs Converter
How to use this Furlongs to Fingerbreadth Converter π €
Follow these steps to convert given length from the units of Furlongs to the units of Fingerbreadth.
1. Enter the input Furlongs value in the text field.
2. The calculator converts the given Furlongs into Fingerbreadth in realtime β using the conversion formula, and displays under the Fingerbreadth label. You do not need to click any button. If the
input changes, Fingerbreadth value is re-calculated, just like that.
3. You may copy the resulting Fingerbreadth 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 Furlongs to Fingerbreadth?
The formula to convert given length from Furlongs to Fingerbreadth is:
Length[(Fingerbreadth)] = Length[(Furlongs)] / 0.00009469696897537879
Substitute the given value of length in furlongs, i.e., Length[(Furlongs)] in the above formula and simplify the right-hand side value. The resulting value is the length in fingerbreadth, i.e.,
Calculation will be done after you enter a valid input.
Consider that a horse race is 8 furlongs long.
Convert this distance from furlongs to Fingerbreadth.
The length in furlongs is:
Length[(Furlongs)] = 8
The formula to convert length from furlongs to fingerbreadth is:
Length[(Fingerbreadth)] = Length[(Furlongs)] / 0.00009469696897537879
Substitute given weight Length[(Furlongs)] = 8 in the above formula.
Length[(Fingerbreadth)] = 8 / 0.00009469696897537879
Length[(Fingerbreadth)] = 84480.0006
Final Answer:
Therefore, 8 fur is equal to 84480.0006 fingerbreadth.
The length is 84480.0006 fingerbreadth, in fingerbreadth.
Consider that a traditional country road stretches for 12 furlongs.
Convert this distance from furlongs to Fingerbreadth.
The length in furlongs is:
Length[(Furlongs)] = 12
The formula to convert length from furlongs to fingerbreadth is:
Length[(Fingerbreadth)] = Length[(Furlongs)] / 0.00009469696897537879
Substitute given weight Length[(Furlongs)] = 12 in the above formula.
Length[(Fingerbreadth)] = 12 / 0.00009469696897537879
Length[(Fingerbreadth)] = 126720.001
Final Answer:
Therefore, 12 fur is equal to 126720.001 fingerbreadth.
The length is 126720.001 fingerbreadth, in fingerbreadth.
Furlongs to Fingerbreadth Conversion Table
The following table gives some of the most used conversions from Furlongs to Fingerbreadth.
Furlongs (fur) Fingerbreadth (fingerbreadth)
0 fur 0 fingerbreadth
1 fur 10560.0001 fingerbreadth
2 fur 21120.0002 fingerbreadth
3 fur 31680.0002 fingerbreadth
4 fur 42240.0003 fingerbreadth
5 fur 52800.0004 fingerbreadth
6 fur 63360.0005 fingerbreadth
7 fur 73920.0006 fingerbreadth
8 fur 84480.0006 fingerbreadth
9 fur 95040.0007 fingerbreadth
10 fur 105600.0008 fingerbreadth
20 fur 211200.0016 fingerbreadth
50 fur 528000.004 fingerbreadth
100 fur 1056000.008 fingerbreadth
1000 fur 10560000.0805 fingerbreadth
10000 fur 105600000.8047 fingerbreadth
100000 fur 1056000008.0467 fingerbreadth
A furlong is a unit of length used primarily in horse racing and agriculture. One furlong is equivalent to 220 yards or approximately 201.168 meters.
The furlong is defined as one-eighth of a mile, making it a useful measurement for shorter distances, especially in contexts like racetracks and land measurement.
Furlongs are commonly used in horse racing to describe the length of a race and in agriculture for measuring field lengths. The unit is less frequently used in modern contexts but remains important
in specific areas where its historical relevance endures.
A fingerbreadth is a historical unit of length based on the width of a person's finger. One fingerbreadth is approximately equivalent to 1 inch or about 0.0254 meters.
The fingerbreadth is defined as the width of a finger at its widest point, typically used for practical measurements in various contexts such as textiles and small dimensions.
Fingerbreadths were used in historical measurement systems to provide a simple and accessible means of measuring smaller lengths and dimensions. While not commonly used today, the unit offers insight
into traditional measurement practices and standards.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Furlongs to Fingerbreadth in Length?
The formula to convert Furlongs to Fingerbreadth in Length is:
Furlongs / 0.00009469696897537879
2. Is this tool free or paid?
This Length conversion tool, which converts Furlongs to Fingerbreadth, is completely free to use.
3. How do I convert Length from Furlongs to Fingerbreadth?
To convert Length from Furlongs to Fingerbreadth, you can use the following formula:
Furlongs / 0.00009469696897537879
For example, if you have a value in Furlongs, you substitute that value in place of Furlongs in the above formula, and solve the mathematical expression to get the equivalent value in Fingerbreadth. | {"url":"https://convertonline.org/unit/?convert=furlongs-fingerbreadth","timestamp":"2024-11-03T19:09:40Z","content_type":"text/html","content_length":"91648","record_id":"<urn:uuid:df1bc1e3-ac78-41ff-ab05-fc526bfaa00a>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00507.warc.gz"} |
Gravitational Field Lines & Gravitational Field Strength | Mini Physics - Free Physics Notes
Gravitational Field Lines & Gravitational Field Strength
Show/Hide Sub-topics (Gravitation | A Level Physics)
Table of Contents
Understanding Gravitational Fields
When you interact with objects in your everyday life, such as picking up a pen, the forces involved are tangible and direct. This direct interaction, facilitated through contact, allows you to exert
a force on the pen, making it move. Conversely, the pen also experiences a force due to Earth’s gravity, known as weight, despite the absence of direct contact between the Earth and the pen. This
phenomenon is attributed to the pen being within Earth’s gravitational field, a region in space where objects are subjected to gravitational force from the Earth.
A gravitational field can be conceptualized as the space surrounding a mass where other masses experience a gravitational pull. The field’s reach is theoretically infinite, extending far into space,
though its strength diminishes with distance due to the inverse square law. This property suggests that the gravitational pull from a celestial body like Earth decreases as one moves away, yet it
never truly vanishes, no matter how far one goes.
Modeling Gravitational Fields with Field Lines
To visualize gravitational fields, we use field lines or lines of force. These lines, while invisible and intangible, provide a model to represent the direction and magnitude of gravitational forces.
The direction of a field line at any point indicates the direction of the gravitational pull experienced by a small mass placed at that point. Similarly, the density of field lines (their closeness)
reflects the field’s strength; more densely packed lines signify a stronger gravitational pull.
For spherical masses such as Earth, field lines are directed radially inwards, pointing toward the center of the mass. This radial arrangement indicates that a mass within Earth’s field feels a force
pulling it toward Earth’s center. Near Earth’s surface, these lines appear parallel and equidistant, denoting a uniform gravitational field, where the gravitational force acts downward, parallel to
the lines.
Important characteristics of field lines include:
• They do not begin or end in empty space, typically extending from a mass to infinity.
• Field lines never cross, as crossing lines would suggest conflicting directions of force at their intersection, which contradicts the principle that the gravitational force at a point has a
singular direction.
Gravitational Field Strength, $g$
The interaction between two masses, such as a person and the Earth, involves mutual gravitational pull, with the force exerted by each on the other being equal in magnitude but opposite in direction.
Despite this, the Earth’s gravitational field is significantly stronger due to its massive size compared to an individual object.
The gravitational field strength at a point in space offers a quantitative measure of the gravitational force experienced per unit mass at that point. This concept allows us to understand the
intensity of gravitational forces in different regions without needing to consider the specific masses involved. Mathematically, it is represented as:
$$g = \frac{F}{m}$$
• $g$ is the gravitational field strength at the point,
• $F$ is the gravitational force acting on a mass $m$ placed at that point (or more commonly known as the weight of the object in the Earth’s context).
Expanding this formula using Newton’s law of universal gravitation, we get:
$$g = G \frac{M}{r^{2}}$$
• $G$ represents the gravitational constant ($6.674 \times 10^{-11} \, \text{Nm}^{2}\text{kg}^{-2}$),
• $M$ is the mass of the celestial body generating the gravitational field,
• $r$ is the distance from the mass’s center to the point in question.
The units of gravitational field strength are expressed as Newtons per kilogram ($N \, \text{kg}^{-1}$) or meters per second squared ($\text{m s}^{-2}$). These units highlight the nature of
gravitational field strength as an acceleration.
This equation calculates the gravitational field strength a distance (r) from a mass (M), providing a measure of the field’s intensity at that point. It’s crucial to note that gravitational field
strength is independent of the mass placed in the field. This independence means that different masses at the same point will experience the same gravitational acceleration but will be subjected to
varying forces based on their mass.
In some instances, equations for gravitational field strength include a negative sign to indicate the direction of the force (toward the mass causing the field), but for simplicity and focus on
magnitude, we use the positive form. The understanding here is that gravitational attraction inherently acts toward the mass generating the field.
Recall: Gravitational Acceleration
Main Article: Gravitational Acceleration & Terminal Velocity
Gravitational acceleration is the acceleration of an object due solely to the gravitational force acting on it, disregarding any other forces (like air resistance). It is a vector quantity, meaning
it has both magnitude and direction—towards the center of the mass creating the gravitational field. On the surface of the Earth, gravitational acceleration is approximately $9.81 \text{ ms}^{-2}$
downwards, towards the center of the Earth.
Relationship Between Gravitational Field Strength & Gravitational Acceleration
Gravitational field strength and gravitational acceleration are intimately related concepts. In fact, in many contexts, the terms are used interchangeably because they have the same units ($\text{ms}
^{-2}$) and numerical value when discussing gravity near a planet’s surface or within a celestial body’s gravitational field.
The key to understanding their relationship is recognizing that gravitational field strength describes the intensity of a gravitational field at a point, indicating the force that would be exerted on
a unit mass placed there. At the same time, gravitational acceleration describes how an object’s velocity changes when it is solely under the influence of gravity.
In essence, when we talk about the gravitational field strength of the Earth, we are also describing the gravitational acceleration that any object would experience if it were free-falling under
gravity’s influence at that location. Both concepts highlight the fundamental principle that in a gravitational field, all objects, regardless of their mass, experience the same acceleration due to
Variation of Gravitational Field Strength on Earth
In the context of Earth, gravitational field lines can be considered nearly parallel and uniformly distributed at a local scale, leading to a uniform field strength. This approximation holds because
the Earth’s surface curvature is minimal over small areas, making local gravitational forces appear parallel.
Several factors contribute to the variations in Earth’s gravitational field strength:
1. Shape of the Earth: Earth’s shape is an oblate spheroid, slightly flattened at the poles. This shape causes gravitational field strength to increase from the equator towards the poles at sea
level, as the surface is closer to the Earth’s center at the poles.
2. Earth’s Density Variations: The planet’s density is not uniform, with variations in geological structures causing fluctuations in gravitational field strength. Areas with denser materials, such
as mountain ranges or dense rock formations, can exhibit slightly stronger gravitational forces.
3. Earth’s Rotation: The rotation of Earth around its axis introduces a centrifugal force, which counteracts gravitational pull, especially at the equator. This effect reduces the net gravitational
acceleration experienced by objects on the equator, making it slightly weaker compared to the poles.
Understanding Weightlessness
Weightlessness is a condition experienced when an object or person does not exert or feel the force of gravity. This state can be true or apparent, depending on the situation and the forces acting on
the object. Understanding this concept requires first grasping what weight is in the context of physics.
Weight & Gravitational Force
The weight of an object is the gravitational force exerted on the object by Earth’s gravitational field (or any other celestial body’s field in which the object is located). It is calculated as the
product of the mass of the object ($m$) and the gravitational acceleration ($g$) at that location:
$$\text{Weight} = m \times g$$
• $m$ is the mass of the object,
• $g$ is the gravitational field strength or gravitational acceleration at the location of the object.
True Weightlessness
True weightlessness occurs when there is absolutely no gravitational force acting on an object. This situation is theoretically possible only when an object is infinitely far away from any other
mass, placing it outside the influence of any gravitational field. In the vast expanses of space, far from significant masses like planets, stars, or galaxies, an object can be considered truly
weightless. However, given the universal nature of gravity, achieving a state of absolute zero gravitational force is practically impossible within the universe as we understand it, due to the
infinite range of gravitational fields.
Apparent Weightlessness
Apparent weightlessness is a condition where an object appears to be without weight. This occurs not because gravitational forces are absent but because the object and its surroundings are in
free-fall or are orbiting a celestial body in what is termed microgravity. In this state, the object does not exert force on its support, and thus, does not experience a normal force in response.
Examples of Apparent Weightlessness:
• Astronauts in orbit: When astronauts orbit Earth, they are in a continuous state of free-fall towards the planet. However, because they are moving forward while falling, they keep missing Earth,
creating a sensation of weightlessness.
• Parabolic flights: Airplanes that perform parabolic maneuvers can create short periods of weightlessness. Passengers experience this as the plane dives in such a way that it mimics the free-fall
• Drop towers: These research facilities allow experiments to be dropped in a controlled environment, reducing air resistance and simulating a near-weightless state for the duration of the fall.
Implications of Weightlessness
Experiencing weightlessness has significant implications, especially for the human body. Long-term exposure to microgravity environments, such as those experienced by astronauts on extended space
missions, can lead to muscle atrophy, bone density loss, and other health challenges due to the lack of gravitational stress on the body.
Worked Examples
Example 1: Conceptual Understanding
Explain why a person standing on the Earth feels a force pulling them down, but a satellite in orbit experiences weightlessness, despite both being subject to Earth’s gravitational field.
Click here to show/hide answer
A person standing on the Earth feels a force pulling them down due to Earth’s gravitational field acting on their mass, which is the force of gravity. This force is what gives the person weight. On
the other hand, a satellite in orbit experiences weightlessness not because it is outside of Earth’s gravitational field (indeed, this field extends far into space), but because it is in free-fall
towards the Earth. The satellite is constantly falling towards Earth, but its forward motion keeps it in orbit around the planet. This state of free-fall creates the sensation of weightlessness, even
though the satellite is still being pulled by Earth’s gravity.
Example 2: Calculating Gravitational Field Strength
Calculate the gravitational field strength at a point 10,000 km from the center of the Earth. Assume the mass of the Earth is $5.972 \times 10^{24}$ kg.
Click here to show/hide answer
The formula for gravitational field strength is $g = G \frac{M}{r^2}$, where $G$ is the gravitational constant ($6.674 \times 10^{-11} \, \text{Nm}^2\text{kg}^{-2}$), $M$ is the mass of the Earth,
and $r$ is the distance from the Earth’s center to the point in question.
$$\begin{aligned} g &= 6.674 \times 10^{-11} \times \frac{5.972 \times 10^{24}}{(10,000 \times 10^3)^2} \\ &= 6.674 \times 10^{-11} \times \frac{5.972 \times 10^{24}}{10^8 \times 10^6} \\ &= 6.674 \
times 10^{-11} \times \frac{5.972 \times 10^{24}}{10^{14}} \\ &= 6.674 \times 10^{-11} \times 5.972 \times 10^{10} \\ &= 0.398 \, \text{m/s}^2 \end{aligned}$$
The gravitational field strength at a point 10,000 km from the center of the Earth is approximately $0.398 \, \text{m/s}^2$.
Example 3: Gravitational Force at Different Altitudes
Why does an astronaut on the International Space Station (ISS) experience less gravitational pull from the Earth compared to someone on the surface, and how does this affect the astronaut’s weight?
Click here to show/hide answer
The gravitational pull from the Earth decreases with distance from the Earth’s center due to the inverse square law. The ISS orbits the Earth at an altitude where Earth’s gravitational field strength
is weaker than at the Earth’s surface. This reduced gravitational pull means that the astronaut’s mass experiences less force, leading to a decrease in weight. However, the astronaut still
experiences gravity, which keeps the ISS in orbit; the sensation of weightlessness is due to the ISS and the astronaut both being in free-fall towards Earth, moving forward at a speed that keeps them
in constant orbit.
Example 4: Visualizing Gravitational Field Lines
Describe how the density of gravitational field lines around a planet changes from its surface to outer space and explain what this implies about the strength of gravity at different distances.
Click here to show/hide answer
The density of gravitational field lines around a planet decreases as one moves away from the planet’s surface to outer space. Near the planet’s surface, the field lines are closer together,
indicating a stronger gravitational pull. As the distance from the planet increases, the field lines spread out, signifying a decrease in the gravitational field’s strength. This arrangement reflects
the inverse square law, where the strength of the gravitational pull decreases with the square of the distance from the source. Therefore, the farther you are from the planet, the weaker the
gravitational pull you experience.
Example 5: Effects of Earth’s Shape on Gravitational Strength
If you were to move from the equator to the North Pole, how would the strength of Earth’s gravitational field change, and why?
Click here to show/hide answer
Moving from the equator to the North Pole would result in an increase in the strength of Earth’s gravitational field that you experience. This is because the Earth is an oblate spheroid, slightly
flattened at the poles and bulging at the equator. As a result, the surface at the poles is closer to the Earth’s center than the surface at the equator. Since gravitational force increases with
proximity to the mass causing it (following the inverse square law), the gravitational field strength is stronger at the poles than at the equator. Additionally, at the equator, the centrifugal force
due to Earth’s rotation slightly counteracts gravitational pull, making the effective gravitational field strength slightly weaker there compared to the poles.
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Back To Gravitation (A Level Physics)
Back To A Level Physics Topic List | {"url":"https://www.miniphysics.com/gravitational-field-lines.html","timestamp":"2024-11-13T11:22:28Z","content_type":"text/html","content_length":"102467","record_id":"<urn:uuid:713eee99-2717-4871-bfe1-3242145d570b>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00700.warc.gz"} |
Array Partition
Go to my solution
Go to the question on LeetCode
My Thoughts
What Went Well
I had fun messing around with my solution to reduce its number of lines and find faster solutions.
Solution Statistics
Time Spent Coding
5 minutes
Time Complexity
O(n log n) - Sorting the array takes O(n log n), and then we must increment through the sorted array, but since O(n log n) is the greatest time complexity, we are left with an O(n log n) time
Space Complexity
O(n) - Every second element from the input list must be stored, making the space complexity O(n/2). Still, since dividing n does not change the rate of increase (linear, exponential, etc.), it
results in an O(n) space complexity.
Runtime Beats
100% of other submissions
Memory Beats
62.60% of other sumbissions
1 class Solution(object):
2 def arrayPairSum(self, nums):
3 # All of the following solutions sort the input list
4 # add the least value of each pair, and repeat this
5 # until n (length of nums)
7 return sum(sorted(nums)[::2])
9 # nums.sort()
10 # return sum(nums[::2])
12 # s = 0
13 # nums.sort()
14 # for i in range(0,len(nums),2):
15 # s += nums[i]
16 #
17 # return s | {"url":"https://douglastitze.com/posts/Array-Partition/","timestamp":"2024-11-13T15:17:21Z","content_type":"text/html","content_length":"22910","record_id":"<urn:uuid:8c5a2a0c-934f-4280-b07a-0c243833d3c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00268.warc.gz"} |
Cloud WordNet Browser
Antonyms of noun income
1 sense of income
Sense 1
income -- (the financial gain (earned or unearned) accruing over a given period of time)
Antonym of outgo (Sense 1)
-- (money paid out; an amount spent)
Synonyms/Hypernyms (Ordered by Estimated Frequency) of noun income
1 sense of income
Sense 1
income -- (the financial gain (earned or unearned) accruing over a given period of time)
financial gain
-- (the amount of monetary gain) | {"url":"https://cloudapps.herokuapp.com/wordnet/?q=income","timestamp":"2024-11-02T09:24:50Z","content_type":"text/html","content_length":"16013","record_id":"<urn:uuid:75048eed-a222-4015-bfd4-c4b1df9013de>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00754.warc.gz"} |
Rate of return beta
1 Nov 2018 Expected Return of an Asset. Therefore, the expected return on an asset given its beta is the risk-free rate plus a risk premium equal to beta times
RRR stands for the required rate of return, Rf is the risk-free rate of return, B stands for beta (usually signified by the greek letter beta), and Rm refers to the average market return. Find
Risk-Free Rate of Return. Find the rate of return on a risk-free investment. Risk-free investments are "sure things." Under this model, the required rate of return for equity equals (the risk-free
rate of return + beta x (market rate of return – risk-free rate of return)). Capital Asset Pricing Model Examples. Stock Beta is used to measure the risk of a security versus the market by investors.
The risk free interest rate (Rf) is the interest rate the investor would expect to receive from a risk free investment. The expected market return is the return the investor would expect to receive
from a broad stock market indicator. Rf = the risk-free rate of return beta = the security's or portfolio's price volatility relative to the overall market Rm = the market return The greater part of
the CAPM formula (all but the abnormal return factor) determines the rate of return on a certain security or portfolio given certain market conditions.
The accounting beta approach is one alternative in the absence of market data for the calculation of market betas. In this case accounting data on return on assets
A company has a beta of 1.50 meaning it's riskier than the overall market's beta of one. The current risk-free rate is 2% on a short-term U.S. Treasury. The long-term average rate of return for the
The stock has a beta compared to the market of 1.3, which means it is riskier than a market portfolio. Also, assume that the risk-free rate is 3% and this investor expects the market to rise in value
by 8% per year. The expected return of the stock based on the CAPM formula is 9.5%. An asset is expected to generate at least the risk-free rate of return. If the Beta of an individual stock or
portfolio equals 1, then the return of the asset equals the average market return. The Beta coefficient represents the slope of the line of best fit for each Re – Rf (y) and Rm – Rf (x) excess return
pair. RRR stands for the required rate of return, Rf is the risk-free rate of return, B stands for beta (usually signified by the greek letter beta), and Rm refers to the average market return. Find
Risk-Free Rate of Return. Find the rate of return on a risk-free investment. Risk-free investments are "sure things."
Imagine a company with a beta of 1.10, which means it is more volatile than the general stock market, which has a beta of 1.0. The current risk-free rate is 2 percent
Expected return = Risk Free Rate + [Beta x Market Return Premium]; Expected return = 2.5% + [1.25 x 7.5%]; Expected return = 11.9%. Download the Free 3 Mar 2020 Beta is a measure of the volatility,
or systematic risk, of a security or a portfolio in the expected return of an asset using beta and expected market returns. R- squared is a statistical measure that shows the percentage of a
In the capital asset pricing model (CAPM), beta risk is the only kind of risk for which investors should receive an expected return higher than the risk-free rate of
rf is the risk-free rate of return. βi (beta) is the sensitivity of returns of asset i to the returns from The beta, or systematic risk of the asset, is given by the following formula: β = r*s A/sM.
r is the correlation coefficient between the rate of return on the risky asset Capital Asset Pricing Model is used to value a stocks required rate of return as An asset with a high Beta will
increase in price more than the market when the The excess return, right, the risk premium on this asset is equal to risk-free rate, sorry, I moved that already. Beta times, All right. So what does
this say? This says 17 Apr 2019 Where rf is the nominal risk-free rate, beta coefficient is a measure of systematic risk and rm is the return on the broad market index such as VOLATILITY, BETA AND
RETURN relationship between expected rates of return on individual assets, the covariance of individual asset returns with those of the 5 Jul 2010 Example: If the Treasury bill rate is 3%, the
expected market return is 10 % and a stock has a Beta of 1.2, what is its expected return
7 Aug 2019 Certainly, the Beta of a stock can change over time due to the relative rates of return of the stock to the index. At the same time, sector analysis
A zero-beta portfolio is a portfolio constructed to have zero systematic risk, or in other words, a beta of zero. A zero-beta portfolio would have the same expected return as the risk-free rate
7 Aug 2019 Certainly, the Beta of a stock can change over time due to the relative rates of return of the stock to the index. At the same time, sector analysis 27 Jan 2014 forecast a good fit
between stocks' beta and stocks' return. The first is that the risk- free interest rate is not correct so that the market line is. 17 Feb 2016 The expected rate of return on a security increases as
its beta increases. C) A fairly priced security has an alpha of zero. D) In equilibrium, all Expected return on the capital asset (E(Ri)):, %. Risk free rate of interest (Rf):, %. Expected return of
the market (E(Rm)):, %. Beta for capital asset (βi): The CAPM framework adjusts the required rate of return for an investment’s level of risk (measured by the beta Beta The beta (β) of an investment
security (i.e. a stock) is a measurement of its volatility of returns relative to the entire market. It is used as a measure of risk and is an integral part of the Capital Asset Pricing Model (CAPM).
If the market or index rate of return is 8% and the risk-free rate is again 2%, the difference would be 6%. Divide the first difference above by the second difference above. This fraction is the beta
figure, typically expressed as a decimal value. In the example above, the beta would be 5 divided by 6, or 0.833. Risk-Free rate = 5% Beta = 1.2 Market Rate of Return = 7% RRR = 5% + 1.2 (7% – 5%) =
7.4% . Ross advises Joey to go in for the second option. Even though the first option looks attractive and would fetch him good returns; higher the rate of return, higher is the fear of loss
associated with it. | {"url":"https://bestftxrpswdxj.netlify.app/champ37987wexo/rate-of-return-beta-372","timestamp":"2024-11-13T11:29:04Z","content_type":"text/html","content_length":"36119","record_id":"<urn:uuid:6cba0e7d-782e-4eab-8ea9-37f17ada677d>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00255.warc.gz"} |
A thorium-based fuel cycle for eliminating the discharge of trans-uranium isotopes from light water reactors
This paper presents results of a feasibility study aimed at developing a "zero-transuranic-discharge" fuel cycle based on partial replacement of uranium by thorium as a fertile component. The design
objective is to find a mixture of thorium (Th), enriched uranium (EU) and transuranic (TRU components resulting in an equilibrium charge-discharge mass flow. The quantity and isotopic vector of the
TRU component will be identical at the charge and discharge time points, thus allowing the whole amount of the TRU at the discharge to be reloaded into the following cycle (excluding the reprocessing
losses). The results demonstrate the neutronic feasibility of a fuel cycle with a "zero-TRU" discharge. This was accomplished by developing a specific fuel composition based on a mixture of Th-U-TRU
components. The reactivity coefficients were found to be within a range typical for a reference PWR core. The soluble boron worth is reduced by approximately a factor of two from a typical PWR value,
which presents a design challenge.
Original language English
Title of host publication Global 2003
Subtitle of host publication Atoms for Prosperity: Updating Eisenhowers Global Vision for Nuclear Energy
Pages 176-183
Number of pages 8
State Published - 1 Dec 2003
Event Global 2003: Atoms for Prosperity: Updating Eisenhower's Global Vision for Nuclear Energy - New Orleans, LA, United States
Duration: 16 Nov 2003 → 20 Nov 2003
Publication series
Name Global 2003: Atoms for Prosperity: Updating Eisenhowers Global Vision for Nuclear Energy
Conference Global 2003: Atoms for Prosperity: Updating Eisenhower's Global Vision for Nuclear Energy
Country/Territory United States
City New Orleans, LA
Period 16/11/03 → 20/11/03
ASJC Scopus subject areas
Dive into the research topics of 'A thorium-based fuel cycle for eliminating the discharge of trans-uranium isotopes from light water reactors'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/a-thorium-based-fuel-cycle-for-eliminating-the-discharge-of-trans","timestamp":"2024-11-05T00:23:52Z","content_type":"text/html","content_length":"57643","record_id":"<urn:uuid:abb64aba-4308-4664-9832-c264511b2d9c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00091.warc.gz"} |
SPPU Discrete Structures - May 2017 Exam Question Paper | Stupidsid
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
Solve any one question from Q.1(a,b,c) &Q.2(a,b,c)
1(a) Out of a total of 130 students, 60 are wearing hats, 51 are wearing scarves, and 30 wearing both hats and scarves. Out of 54 students who are wearing sweaters, 26 are wearing hats, 21 are
wearing scarves, snd 12 are wearing both hats and scarves. Everyone wearing neither a hat nor a scarf is wearing gloves:
a) How many students are wearing gloves?
b) How many students not wearing a sweater are wearing hats but not scares?
c) How many students not wearing a sweater are wearing neither hat nor a scarf?
6 M
1(b) Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is the multiple of 3 or 5?
3 M
1(c) In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?
3 M
2(a) Prove by induction that the sum of the cubes of three consecutive integers is divisible by 9.
6 M
2(b) Two cards are drawn together from a pack of 52 cards. Determine the probability that one is a spade and one is a heart.
4 M
2(c) Three unbiased coins are tossed. What is the probability of getting at most two heads?
2 M
Solve any one question from Q.3(a,b) &Q.4(a,b)
3(a) Solve the following recurrence relation:\[\begin{align*}&a_{r}-7a_{r}-1+10a_r-2=2^{r}\\ &a_{1}=3,a_{2}=21. \end{align*}\]
6 M
3(b) Find the shortest path from vertex a to z in the following graph:
6 M
4(a) Functions f, g and h are defined on the set X= {1, 2, 3 } as:
f = {(1, 3), (2, 1), (3, 2)};
g = {(1, 2), (2, 3), (3, 1) };
h {(1, 2), (2, 1), (3, 3)}
i) Find fog and gof . Are the equals?
ii) Find fogoh and fohog.
6 M
4(b) Define Isomorphic Graphs. Show that the following graphs G1 and G2 are isomorphic.
6 M
Solve any one question from Q.5(a,b) &Q.6(a,b)
5(a) Find the minimum spanning tree and weight of it for the given graph using Kruskal's algorithm.
7 M
5(b) Define optimal tree. For the following set of weights, construct optimal binary prefix code. For each weight in the set, give corresponding prefix code:
1, 4, 8, 9, 15, 25, 31, 37.
6 M
6(a) Find the maximum flow for the following transport network.
7 M
6(b) Find the fundamental system cut set of graph G shown belo with respect to the spanning tree T.
6 M
Solve any one question from Q.7(a,b) &Q.8(a,b)
7(a) Let Q[1] be the set of all rational numbers other than 1. Show that with operation * defined on the set Q[1] by (a * b = a + b - ab ) is an Abelian group.
7 M
7(b) Let I be the set of all integers. For each of the following determine whether * is an associative operation of not:
i) a * b = max (a, b)
2) a * b = min ( x + 2, b)
3) a * b = a- 2b
4) a * b = max (2a - b, 2b - a).
6 M
8(a) Let Z[n] be the set of integers {0, 1, 2, ........., n-1}. Let ⨁ be n binary operation on Z[n] such that: \( \begin{align*}a\bigoplus b=\left\{\begin{matrix} a+bifa+b/ Let ⨀ be
a binary operation on Z[n] such that:
a⨀b = the remainder of ab divided by n.
7 M
8(b) Consider the (2, 7) encoding function e.
e (00) = 000000 e(01) = 1010101
e(10) = 0111110 e(11) = 0110110
a) Find the minimum distance of e.
b) How many errors will e detect?
6 M
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The Unique Properties
The question must arise as to why the golden proportion is special – and more importantly, is there any difference between the Golden Proportion and another pleasing proportion? A brief study of
figures below will answer this question.
Thus the proportion of the smaller to the greater is the same as the proportion of the greater to the whole. The division of the line by point C thus represents a point of equilibrium between these
two proportions. If you move the point a fraction one way or the other, then you have two proportions which are neither the same nor are they in equilibrium. The only time that these two proportions
are the same is when they are Golden.
This point of division is a mathematical confirmation of how the eye senses the balance of this magical proportion that appears so frequently in nature and art.
The Bilateral Form
The bilateral form is the most common of the subtle variations of the Golden Proportion, seen in nature and art.
We see in the figure above that A:B = B:C = 1.000 : 0.618
If we now align the mid-points of B and C as in figure above and place them one just above the other, we then find that there is now a “larger-to-whole” relationship and that on either side of the
midline we have the “larger to the smaller” simple relationship.
Patterns of this bilateral form are frequently found in the beauty of nature and works of art. The two examples of the photograph frame shows the midline marked and a smaller to larger relationship
either side.
The same patern is seen in the examples of tarton cloth as well as the Spanish Royal Palace and the headlights of the motor car.
Click images below to enlarge
It is interesting to visualize the anterior aesthetic segment featured against the backcloth of this neutral space framed by the lips. Notice the way the car headlights are in the same proportion to
the distance between them as the neutral space is to the width of the arch that shows in the smile.
Geometric Progression
The sequence of the Basic Golden Proportion numbers is a geometric progression as follows:-
0.618 1 1 1.618 2.618 4.236 6.854 11.09 etc etc
2 Snaill pic.
where each term is multiplied by 1.618 or divided by 0.618 as seen in the cascade of the Fibonacci Series.
Besides the golden Proportion there are many other geometric progressions in nature. An example of two different progressions is shown in the two snail shells.
The musical octave scale of doubling is also a geometric progression.
The three remarkable photos below are extraordinary in that both Corbusier and Picasso appear to have incoporated, in architectural and art form, the identical Golden Proportion Grid as designed by
the author for use in dentistry.
Click image above to enlarge
2 comments on “Mathematics”
1. wissam
May 4, 2016 at 10:36 pm #
where can i find or get this measurment paper for anterior proportion please
□ Eddy Levin
June 16, 2016 at 6:43 pm #
You can get this dental grid from my website http://www.goldenmeangauge.co.uk
Sorry for the delay in my reply, but I just came across your question now | {"url":"https://goldenmeangauge.co.uk/concepts/mathematics/","timestamp":"2024-11-11T11:07:36Z","content_type":"text/html","content_length":"110391","record_id":"<urn:uuid:926c4041-9247-4c50-9f08-3b98b017b8df>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00823.warc.gz"} |
Spanners of additively weighted point sets
We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs (p,r) where p is a point in the plane and r is a real number. The
distance between two points (^pi,^ri) and (^pj,^rj) is defined as |^pipj|-^ri-^rj. We show that in the case where all ^ri are positive numbers and |^pi ^pj|≥^ri+^rj for all i, j (in which case the
points can be seen as non-intersecting disks in the plane), a variant of the Yao graph is a (1+ε -spanner that has a linear number of edges. We also show that the Additively Weighted Delaunay graph
(the face-dual of the Additively Weighted Voronoi diagram) has a spanning ratio bounded by a constant. The straight-line embedding of the Additively Weighted Delaunay graph may not be a plane graph.
Given the Additively Weighted Delaunay graph, we show how to compute a plane straight-line embedding that also has a spanning ratio bounded by a constant in O(nlogn) time.
• Delaunay triangulation
• Geometric spanners
• Yao-graph
ASJC Scopus subject areas
• Theoretical Computer Science
• Discrete Mathematics and Combinatorics
• Computational Theory and Mathematics
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The Logic Of Chance by John Venn
The Logic Of Chance
by John Venn
Publisher: Macmillan And Company 1888
Number of pages: 550
No mathematical background is necessary to appreciate this classic of probability theory. Written by the logician who popularized the famous Venn Diagrams, it remains unsurpassed in its clarity,
readability, and charm. The treatment commences with an overview of physical foundations, examines logical superstructure, and explores various applications.
Download or read it online for free here:
Download link
(multiple formats)
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How to plot composition of unit step and sin functions
How to plot composition of unit step and sin functions
When i was trying to plot the function U(sin(x)) where U is the unit step function it leaves error message, but U(exp(x)) worked nicely
the error message was
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_68.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cGxvdChoKHNpbih4KSksICh4LCAtMTAsIDEwKSk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpeAn7w2/___code___.py", line 3, in <module>
exec compile(u'plot(h(sin(x)), (x, -_sage_const_10 , _sage_const_10 ))
File "", line 1, in <module>
File "/sage/local/lib/python2.6/site-packages/sage/misc/decorators.py", line 657, in wrapper
return func(*args, **kwds)
File "/sage/local/lib/python2.6/site-packages/sage/misc/decorators.py", line 504, in wrapper
return func(*args, **options)
File "/sage/local/lib/python2.6/site-packages/sage/plot/plot.py", line 3071, in plot
G = _plot(funcs, *args, **kwds)
File "/sage/local/lib/python2.6/site-packages/sage/plot/plot.py", line 3105, in _plot
funcs, ranges = setup_for_eval_on_grid(funcs, [xrange], options['plot_points'])
File "/sage/local/lib/python2.6/site-packages/sage/plot/misc.py", line 138, in setup_for_eval_on_grid
return fast_float(funcs, *vars,**options), [tuple(range+[range_step]) for range,range_step in zip(ranges, range_steps)]
File "fast_eval.pyx", line 1388, in sage.ext.fast_eval.fast_float (sage/ext/fast_eval.c:8901)
TypeError: no way to make fast_float from None
1 Answer
Sort by » oldest newest most voted
Can you tell us more about what you're trying to do? I have no trouble with the following:
sage: plot(unit_step(sin(x)), -2*pi,2*pi)
I see; the problem you're running into is that h is a python function, while things like sin(x) and exp(x) are symbolic expressions. When you type h(f(x)), Sage is actually evaluating the function h
on the entire symbolic expression f(x). When f(x) is the exponential funciton, Sage knows that exp(x) is always positive, and hence h returns 1. When f(x) is the sine function, neither of the
conditions is met and so h returns nothing, hence giving the error you saw. Here is some code demonstrating these things:
sage: def h(x):
... if x > 0: return 1
... if x <= 0: return 0
sage: h(sin(x))
sage: h(exp(x))
sage: h(exp(x)-100)
sage: bool(exp(x) > 0)
sage: bool(sin(x) > 0)
sage: def h2(x):
... if x > 0: return 1
... if x <= 0: return 0
... return 'Hello World!'
sage: h2(exp(x))
sage: h2(sin(x))
'Hello World!'
And lastly, here's a solution if you really do want to use a Python function: just make the whole function a Python function:
sage: def c(x):
... return h(sin(x))
sage: plot(c, (x,-10,10))
edit flag offensive delete link more
i defined unit step my own using def in this way def h(x): if (x>0): return 1 if (x<=0): return 0 and then plot(h(sin(x)), (x, -10, 10))
noufalasharaf ( 2012-04-16 11:03:13 +0100 )edit
now i understood Niles thanks for your kindhearted help
noufalasharaf ( 2012-04-16 11:05:31 +0100 )edit | {"url":"https://ask.sagemath.org/question/8894/how-to-plot-composition-of-unit-step-and-sin-functions/","timestamp":"2024-11-12T00:35:55Z","content_type":"application/xhtml+xml","content_length":"61287","record_id":"<urn:uuid:d4ae8f0e-e7d7-429f-ba8d-e565932a6e6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00232.warc.gz"} |
Do We Need a 37-Cent Coin? - Freakonomics
Do We Need a 37-Cent Coin?
Dubner thinks we should do away with the penny.
A young economist I know, Patrick DeJarnette, believes a much more radical change in currency is warranted. Here is what Patrick writes:
Late one night I was curious how efficient the “penny, nickel, dime, quarter” system was, so I wrote a little script to compare all possible 4-coin systems, with the following stipulations:
1. Some combination of coins must reach every integer value in [0,99].
2. Probability of a transaction resulting in value v is uniform from [0,99].
In other words, you start with $10 and no coins. You buy something at the store. Afterward, the chance you have 43 cents in your pocket is equal to the probability that you have 29 or 99 cents in
your pocket (in addition to any bills).
Requirement (1) implies the penny is necessary, as you must have a combination of coins that reach value = 1 cent.
With this in mind, the current combination of coins (penny, nickel, dime, quarter) results in an average of 4.70 coins per transaction. What’s a little surprising is how inefficient our current
setup is! It’s only the 2,952-nd most efficient combination. There are effectively 152,096 different combinations of penny + three coins. In other words, it’s only in the 98th percentile for
How can you tell that Patrick is a young economist from the preceding discussion? Because he finds that the current government solution for the coins we use is 98 percent efficient and thinks this is
inefficient. The other day I was walking through the halls of the University of Chicago economics department and heard a faculty member say that the right rule of thumb for government spending is
that it is worth only 10 cents on the dollar because of inefficiency.
Anyway, Patrick then tackles the question of which combinations of coins would be most efficient:
The most efficient systems?
The penny, 3-cent piece, 11-cent piece, 37-cent piece, and (1,3,11,38) are tied at 4.10 coins per transaction.
But no one wants an 11-cent piece! There are other ways to look at efficiency; and given human limitations, this would result in a lot of errors and transactions would take more time.
□ (1,4,15,40) is the first “reasonable looking” combination, with 4.14 coins per transaction.
□ (1,3,10,35) also does well, with 4.16 coins per transaction.
But what if we restrict ourselves to “all coins (except pennies) are multiples of 5”? There are 18 different combinations that are more efficient than our current setup, (1,5,15,40) being the
most efficient at 4.40 coins per transaction. Some other examples:
□ (1,5,15,35) at 4.50 coins.
□ (1,5,10,30) at 4.60 coins.
If we were to change just one of our current coins, what would be the most efficient?
□ Changing the nickel to a 3-cent piece increases efficiency to 4.22 coins per transaction.
□ Changing the dime to an 11-cent piece increases efficiency to 4.46 coins per transaction. (Although the 11-cent piece is unreasonable).
□ Changing the quarter to a 30-cent piece increases efficiency to 4.60 coins per transaction. (Changing it to a 28-cent piece increases efficiency to 4.50, but that seems unreasonable.)
Therefore, changing the nickel is the most efficient thing. Not surprisingly, losing the dime entirely only costs us ~0.8 coins per transaction in efficiency; it does the least good of the
existing coins. | {"url":"https://freakonomics.com/2009/10/do-we-need-a-37-cent-coin/","timestamp":"2024-11-02T11:22:37Z","content_type":"text/html","content_length":"37883","record_id":"<urn:uuid:c0a3551d-b57f-4326-a833-3f5ed78b9ca6>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00811.warc.gz"} |
Well of Death: Physics behind it
• Thread starter NTesla
• Start date
In summary: The car might start to slide up the wall, but it won't keep going up indefinitely. In fact, it might reach a point where the force of gravity (acting downward) is stronger than the
friction force.
Homework Statement
What will be the maximum velocity for a bike/car to go around in a well of death, when the wall of well of death is vertical, i.e. at 90 degrees ?
Relevant Equations
$$v_{max} = \sqrt{\frac{rg(sin\theta +\mu cos\theta)}{(cos\theta -\mu sin\theta)}}$$
While studying motion of car on banked curve, I was wondering, what will be the vmax when theta is equal to 90 degrees or is close to 90 degrees as it happens in a well of death which is organised in
a village fair.
On a banked road with friction present, vmax is given by:
if we put theta = 90 degrees in the formula above, which is the angle in the well of death, then
But that is not acceptable, as vmax can't be an imaginary number.
I understand that if theta = 90 degrees, then the minimum value of v i.e vmin is given by:
This is understandable.
How do I calculate the value of vmax when theta = 90 degrees ?
Last edited:
NTesla said:
Normally, on a banked road with friction present
Check out the assumptions for 'normally'. What happens if ##v>v_{\text{max}}## ?
And what happens in
if you don't substitute ##\theta ={\pi\over 2} ## but take the limit ##\theta \uparrow{\pi\over 2}## ?
[edit] hmmm... this last one seems to be way off track... depending on ##\mu##, ##\cos\theta-\mu\sin\theta## makes ##v_{\text{max}}\uparrow\infty ## even before ##\theta ={\pi\over 2} ##. So: what's
going on when ##\cos\theta-\mu\sin\theta## hits 0 ?
##\ ##
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I'm assuming that what you meant to write was: what happens if v > vmax. Well, in that case, the car will start to slide up the wall of the well.
And for the 2nd part that you asked, if theta tends to pi/2, then, i'm reaching the same value for vmax as i had mentioned in the question above. Here's my calculation:
$$v_{max} = \sqrt{\frac{rg(tan\theta +\mu)}{(1 -\mu tan\theta)}}$$
$$v_{max} =\lim_{\theta \to \pi/2} \sqrt{\frac{rg(tan\theta +\mu)}{(1 -\mu tan\theta)}}$$
$$v_{max} =\sqrt{rg}\lim_{\theta \to \pi/2} \sqrt{\frac{(1 +\frac{\mu}{tan\theta}{})}{(\frac{1}{tan\theta} -\mu)}}$$
$$v_{max} =\sqrt{rg} \sqrt{\frac{1}{-\mu}}$$
$$v_{max} =\sqrt{\frac{rg}{-\mu}}$$
It's again the same result as if i had put ##\theta = \frac{\pi}{2}##.
Help is needed.
As I indicated, I was off track with that last one.
BvU said:
So: what's going on when ##\cos\theta-\mu\sin\theta## hits 0 ?
in other words ##\tan\theta \uparrow 1/\mu ## ?
##\ ##
what does the up arrow, between ##tan\theta## and ##\frac{1}{\mu}## signify ?
"Approaches that value ##1/\mu## from below"
BvU said:
"Approaches that value ##1/\mu## from below"
yes, that's the essence of the initial question that i have asked..
Somebody, kindly throw some light on this question..
NTesla said:
yes, that's the essence of the initial question that i have asked..
So we find that ##v_\text{max}## has no upper bound for ##\theta\ge\arctan{1\over\mu}##.
I.e. the answer to your initial question is "there is no maximum".
Now all we have to sort out is what is going on physically. That involves digging into the derivation of the expression for ##v_\text{max}##
##\ ##
NTesla said:
Homework Statement::
On a banked road with friction present, vmax is given by:
View attachment 317645
View attachment 317649 ReplyForward
I would like you to go over the derivation of this equation. Notice that, when we are talking about the vertical "wall of death," the friction force must be acting upward. In the derivation of the
equation you posted, the friction force is assumed to be down the slope. If it did that then your car will fall down the wall. So you need to reorient the frictional force to point up the slope. You
get a similar equation:
##v_{max}= \sqrt{rg \dfrac{(sin( \theta ) - \mu ~ cos( \theta ) )}{( cos( \theta ) + \mu ~ sin( \theta ) )}}##
which is perfectly well-behaved for ##\theta = 90##.
topsquark said:
I would like you to go over the derivation of this equation. Notice that, when we are talking about the vertical "wall of death," the friction force must be acting upward. In the derivation of
the equation you posted, the friction force is assumed to be down the slope. If it did that then your car will fall down the wall. So you need to reorient the frictional force to point up the
slope. You get a similar equation:
##v_{max}= \sqrt{rg \dfrac{(sin( \theta ) - \mu ~ cos( \theta ) )}{( cos( \theta ) + \mu ~ sin( \theta ) )}}##
which is perfectly well-behaved for ##\theta = 90##.
What you wrote is the equation for ##v_{min}##. The friction will not always act upward. Since we are trying to find out vmax, therefore, the friction will act downward. In my original question, i
did mention that ##v_{min}## will be = ##\sqrt{\frac{rg}{\mu}}##.
The question remains as it is..
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NTesla said:
What you wrote is the equation for ##v_{min}##. In my original question, i did mention that ##v_{min}## will be = ##\sqrt{\frac{rg}{\mu}}##.
The question remains as it is..
Normally I'd agree with you. But in this case can the friction force be pointing downward? If ##\theta < 90## the normal force contributes to keeping the car at its height. But if the wall is
vertical then N points solely in the radial direction, without any vertical component. The friction force is the only thing holding the car up.
There isn't really a "##v_{max}##" in this case.
The maximum velocity in that case is limited only by the g-force pulling the rider or driver against the wall of the well.
At 90 degrees, the only lateral force on the vehicle is its own weight.
Like it happens to a car in a horizontal curve, the only way to compensate that lateral force is by steering the front wheels.
Because of the above, there is a minimum value of tangential velocity that is needed.
That forward velocity induces enough centrifugal effect to reach a minimum necessary value of normal force and lateral friction force to prevent the car from sliding down the well.
Lnewqban said:
The maximum velocity in that case is limited only by the g-force pulling the rider or driver against the wall of the well.
What's the expression for vmax according to you..?
BvU said:
So we find that ##v_\text{max}## has no upper bound for ##\theta\ge\arctan{1\over\mu}##.
I.e. the answer to your initial question is "there is no maximum".
Now all we have to sort out is what is going on physically. That involves digging into the derivation of the expression for ##v_\text{max}##
##\ ##
I have already derived the expression for vmax, and i had posted the expression for vmax in my original question.
topsquark said:
Normally I'd agree with you. But in this case can the friction force be pointing downward? If ##\theta < 90## the normal force contributes to keeping the car at its height. But if the wall is
vertical then N points solely in the radial direction, without any vertical component. The friction force is the only thing holding the car up.
There isn't really a "##v_{max}##" in this case.
So, if i've understood you correctly, then what you are saying is that, as long as ##\theta## is not equal to 90 degrees, friction will act downward, as we are finding vmax. But then suddenly, when #
#\theta## = 90 degrees, friction will suddenly act upwards, without there being any moment when friction = 0. Friction will also act upward if velocity of car is less than vmin.
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NTesla said:
What's the expression for vmax according to you..?
I would use the regular expression for centrifugal acceleration, being limited to 5g.
Please, see:
That link is about the topic of human endurance to high g. I'm trying to understand what will be the vmax when g = 9.8m/s^2, in a well of death. If you've figured out the expression, then kindly let
me know too..
topsquark said:
There isn't really a "##v_{max}##" in this case.
So, there wont be a vmax in 3 cases: (1) when ##\mu tan\theta## is equal to 1. (2) When ##\theta## = 90 degrees and (3) when ##\mu##=1 and ##\theta##= 45 degrees.. Is that right..?
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NTesla said:
So, if i've understood you correctly, then what you are saying is that, as long as ##\theta## is not equal to 90 degrees, friction will act downward, as we are finding vmax. But then suddenly,
when ##\theta## = 90 degrees, friction will suddenly act upwards, without there being any moment when friction = 0. Friction will also act upward if velocity of car is less than vmin.
There is no 'suddenly'.
BvU said:
Check out the assumptions for 'normally'.
The expressions including ##\mu## assume a direction for the friction force. And ##\mu N## is the
friction force. When ##v## runs from ##v_\text{min}## to ##v_\text{max}##, the friction force varies gradually from a maximum in one direction to a maximum in the other.
##\ ##
BvU said:
There is no 'suddenly'.The expressions including ##\mu## assume a direction for the friction force. And ##\mu N## is the maximum friction force. When ##v## runs from ##v_\text{min}## to ##v_\text
{max}##, the friction force varies gradually from a maximum in one direction to a maximum in the other.
##\ ##
##\mu N_{max}## is the maximum friction force. Value of N will keep on varying since N is a function of velocity.. isn't it ?
NTesla said:
That link is about the topic of human endurance to high g. I'm trying to understand what will be the vmax when g = 9.8m/s^2, in a well of death. If you've figured out the expression, then kindly
let me know too..
Please, consider how heavy that person feels while doing this performance.
That is the limit to the velocity.
Lnewqban said:
Please, consider how heavy that person feels while doing this performance.
That is the limit to the velocity.
let's say there isn't a person doing the round, it's just a machine.. Then what will be the vmax when theta= 90 degrees..?
NTesla said:
let's say there isn't a person doing the round, it's just a machine.. Then what will be the vmax when theta= 90 degrees..?
The maximum speed that the machine can develop.
You are after the maximum speed with which a car can take a curve without sliding out.
Once the angle of bank grows over 45, that maximum needed speed starts to decrease as the angle increases, simply because the effective radius of the turn also starts to decrease.
Once the bank of a curve reaches 90 degrees, there is no more curve.
The shape of your road has gone from a circle to a cone to a cylinder!
Lnewqban said:
maximum needed speed starts to decrease...
Shouldn't it be minimum needed speed, which according to me should continue to increase as theta increases..
Could someone please let me know if my statements in post# 19 and post#21 above, are right or wrong..
post#19: So, there wont be a vmax in 3 cases: (1) when ##\mu tan\theta## is equal to 1. (2) When ##\theta## = 90 degrees and (3) when ##\mu##=1 and ##\theta##= 45 degrees.. Is that right..?
post#21: ##\mu N_{max}## is the maximum friction force. Value of N will keep on varying since N is a function of velocity.. isn't it ?
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NTesla said:
Shouldn't it be minimum needed speed, which according to me should continue to increase as theta increases..
Is that a question or an assertion?
Why should it continue to increase?
NTesla said:
Could someone please let me know if my statements in post# 19 and post#21 above, are right or wrong..
post#19: So, there wont be a vmax in 3 cases: (1) when ##\mu tan\theta## is equal to 1. (2) When ##\theta## = 90 degrees and (3) when ##\mu##=1 and ##\theta##= 45 degrees.. Is that right..?
post#21: ##\mu N_{max}## is the maximum friction force. Value of N will keep on varying since N is a function of velocity.. isn't it ?
The answer to your question is simply to sketch a FBD of the situation. If angle is less than 90 degrees there will be a minimum speed and maximum speed in order to keep the car at a specific height.
The normal force has a component in the vertical direction and is a counter to the weight component down the slope.
But in this case, the wall is vertical so the normal force no longer has a component in the vertical direction. The
force that can counter the weight is friction. So friction
to be pointed upward. Technically this means that there is no ##v_{max}## because any higher speed will not change the friction situation, nor will it change anything about the height. But if the
speed is too low then friction cannot support the weight of the car.
First, I'm not sure where you got that this equation is for ##v_{max}##; it is for ##v## in general.
Second, the 'true' equation is:
$$\frac{v^2}{rg} = \frac{\sin\theta + \frac{f}{N}\cos\theta}{\cos\theta - \frac{f}{N}\sin\theta}$$
Where ##f## is the friction pointing down the slope, which means a negative value is possible; it only means it is going against the slope.
What is the value of ##\frac{f}{N}##? Well:
$$mg = N\cos\theta - f\sin\theta$$
$$\frac{f}{N} = \frac{\cos\theta - \frac{mg}{N}}{\sin\theta}$$
Note how it can have a negative value. Putting it in our original equation:
$$\frac{v^2}{rg} = \frac{\sin\theta + \left(\frac{\cos\theta - \frac{mg}{N}}{\sin\theta}\right)\cos\theta}{\cos\theta - \left(\frac{\cos\theta - \frac{mg}{N}}{\sin\theta}\right)\sin\theta}$$
$$\frac{m\frac{v^2}{r}}{N} = \sin\theta +\frac{\cos\theta - \frac{mg}{N}}{\tan\theta}$$
Isolating ##N## we get either:
$$N= \frac{m\frac{v^2}{r}\tan\theta + mg}{\sin\theta \tan\theta +\cos\theta}$$
$$N = \frac{m\frac{v^2}{r} + \frac{mg}{\tan\theta}}{\sin\theta + \frac{\cos\theta}{\tan\theta}}$$
• In the first form, assuming ##\theta=0##, we get ##N=mg##.
• In the second form, assuming ##\theta = 90°##, we get ##N=m\frac{v^2}{r}##.
That's it. There is no ##v_{max}##. Whether ##\theta=0## or ##\theta=90°## or anywhere in between, you can go as fast as you want in any case. Only the normal force will change, which is independent
of ##v## when ##\theta=0## and independent of ##g## when ##\theta=90°##.
But let's isolate ##\frac{f}{N}## in our original equation instead:
$$\frac{f}{N} = \frac{\frac{v^2}{rg}\cos\theta - \sin\theta}{\frac{v^2}{rg}\sin\theta + \cos\theta}$$
$$\frac{f}{N} = \frac{\frac{v^2}{rg} - \tan\theta}{\frac{v^2}{rg}\tan\theta + 1}$$
And we know that to remain static (our assumption in the first place i.e. no sliding):
$$\mu_s > \left|\frac{f}{N}\right|$$
$$\mu_s > \left|\frac{\frac{v^2}{rg} - \tan\theta}{\frac{v^2}{rg}\tan\theta + 1}\right|$$
But that doesn't indicate a ##v_{min}## either, just a ##\mu_{s,\ min}##. That only means that the required friction coefficient will vary between ##\frac{v^2}{rg}## at ##\theta=0## (such that the
vehicle doesn't slide out the curve) and ##\frac{rg}{v^2}## at ##\theta=90°## (such that the vehicle doesn't slide down, or "into" the curve).
The only notable thing is that if ##\frac{v^2}{rg} = \tan\theta## then ##f=0## and therefore no friction is required.We can find a ##v_{min}## and ##v_{max}## such that the vehicle doesn't begin to
slide in either way.
Assuming a given ##\mu_s##, we go back to our original equation (converted to the ##\tan## equivalent):
$$\frac{v_{min}^2}{rg} = \frac{\tan\theta + (-\mu_s)}{1 - (-\mu_s)\tan\theta}$$
$$\frac{v_{max}^2}{rg} = \frac{\tan\theta + \mu_s}{1 - \mu_s\tan\theta}$$
In both cases, the right-hand side cannot be negative, so:
$$\frac{v_{min}^2}{rg} = \frac{\max\{\tan\theta; \mu_s\} - \mu_s}{1 + \mu_s\tan\theta}$$
$$\frac{v_{max}^2}{rg} = \frac{\tan\theta + \mu_s}{1 - \mu_s\min\{\tan\theta;\frac{1}{\mu_s}\}}$$
I think I got it all right.
. Can l put my twopence worth in?
If you go round a banked curve at a sufficiently high speed, you tend to slide upwards. (You should understand why this happens.) Static friction opposes the upwards sliding by acting downwards.
If ##\theta \lt 90^o## and you are going
[S:too :S]
fast, you
may slide upwards because downwards friction isn't big enough to prevent the sliding. Your maximum speed to avoid sliding upwards is ##V_{max}##.
Note that if ##\theta= 90^o## there is no tendency to slide upwards (and you should understand why).
The equation for ##V_{max}## applies only when the frictional force is at its maximum (limiting) value of ##\mu N## and is acting downwards - because that’s how the ##V_{max}## equation is derived.
But when ##\theta = 90^0## there is
downwards friction because there is no tendency to move upwards. Friction will act upwards and it's magntiude will equal ##|mg|##; the equation for ##V_{max}## is inapplicable.
Additional note. It’s not just ##\theta = 90^o## where care is needed. For example if ##\mu = 0.9## and ##\theta = 60^o## then ##\cos(\theta) - \mu \sin(\theta) = \cos(60^o) – 0.9\sin(60^o) = -0.279#
# which gives a negative value inside the square root.
That’s because the ##V_{max}## equation cannot be used because friction doesn’t need to reach its limiting value to prevent upwards sliding for these values of ##\mu## and ##\theta##.
The ##V_{max}## equation is therefore inapplicable in this case- and this was signalled by the fact that ##\cos(\theta) - \mu \sin(\theta)## was negative.
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Science Advisor
Homework Helper
Gold Member
NTesla said:
But that is not acceptable, as vmax can't be an imaginary number.
When you write an equation to represent a physical behaviour, it often only applies in a range of situations. In this case, in writing the equation, you have assumed that there will be a speed at
which it will slide up. If you get a silly answer, likely the assumption was false.
Even before you substituted θ=90° there was a problem. You can see that for any θ>0 there are values of μ which give an infinite or imaginary result, telling you it cannot slide up. What θ=90° did
was to push all values of μ into that class.
Science Advisor
Homework Helper
Gold Member
2023 Award
This reminds me of the problem of trying to slide a box across a rough horizontal surface with a force applied at a downward angle:
For a given ##\mu_s##, if ##\theta > \tan^{-1}(1/\mu_s)##, then the box will not slide no matter how strong the force ##F##.
Likewise, for this problem. If ##\theta > \tan^{-1}(1/\mu_s)##, then the car will not slide upward on the slope no matter how fast the car is traveling. There is no ##v_{max}## for ##\theta > \tan^
In the vertical wall case (horizontal rider), what force is balancing the torques about the point of contact?
P.s. This is me asking, because I’m not seeing it; not me asking the OP.
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│ 500m │ 1000m │ 2000m │ 5000m │ 20 min │ 30 min │ 10000m │ 40 min │ UT1 │ UT2 │ AT │ TR │ AN │
│ 138% │ 117% │ 100% │ 82% │ 78% │ 73% │ 71% │ 69% │ 50% │ 60% │ 70% │ 105% │ 110% │
The Ultimate Erg Calculator (UEC) attempts to combine as many useful erging calculations in one place. As you enter values and move from one cell to another additional values are calculated wherever
For example, enter a distance and a split and the UEC will show time (and watts); enter a split and time and the UEC will show distance. Similarly, if you have entered a distance and time the average
split and watts will be shown. If you then enter a stroke rate, distance per stroke (DPS) and stroke power index (SPI) are calculated. And so on.
If you have entered values for distance and time a split is calculated. If you then edit the split UEC doesn't know whether you want to modify distance or time, i.e., predict a new time for that
distance or predict a new distance for that time! For this reason nothing happens when you edit the split until you click in either the distance or time field.
If you need any guidance or have ideas about improving the UEC please feel free to email me: walt(at)brookhouse(dot)co(uk).
If you want to know what the split is for a percentage of a given wattage, enter a value in the Watts box like so: 300%70. The corresponding split is shown in the Split box.
To calculate a time for a different distance based on a watt percentage enter a value in the watt field in the form 300%70. This will update the split value. Enter a new value in the distance field
and the time field will then be updated.
Everyone loves a predictor! As we all know, the best predictor is rowing the distance as hard as you can. Sometimes, though, you need a guide to pace based on a known performance at a different
The bottom tier of the UEC includes two predictors: a) a 2K prediction based on a 30r20; and b) Paul's Law.
It is generally held that a 30R20 can be completed at 70% of 2K watts. If you enter a 30R20 result in time/distance, and rate 20, the UEC will show your predicted 2K time.
Paul's Law says that if you double the distance the 500m split increases by five seconds for a rowing athlete with balanced speed/endurance capability. Strictly speaking, Paul's Law isn't a predictor
but rather serves to highlight whether you need to work on speed or endurance.
Enter a known time and distance in the UEC, and select a different distance from the Paul's Law dropdown. The UEC will update to show the time and split/watts for that distance that conforms to
Paul's Law.
To get a table of Paul's Law results from the current time/distance click the
The LTB spreadsheet is another useful predictor base on watt %ages of a current 2K time. My own comparison shows that Paul's Law and the LTB spreadsheet yield equivalent predictions to a fraction of
1%. If you click the
Note: the LTB percentages are relative to a current 2K so keep that in mind.
The copyright of the concept, design, presentation and source code of this calculator page belongs to Roy Walter (THE OWNER).
Using the Ultimate Erg Calculator is free of charge. If you wish to build the ideas and implementation you see here into a software product of your own to run any under any operating system AND
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Or constraint
So, my next question is, I have a lot of bags and a lot of weight, and I'm trying to resolve the bin covering problem, and so I had something like that
N0 x_0_0 + N1 x_1_0 + ... + N x_n_0 >= P0 y_0
N0 x_0_1 + N1 x_1_1 + ... + N x_n_1 >= P1 y_1
N0 x_0_m + N1 x_1_m + ... + N x_n_m >= Pm y_m
*other constraints*
where N0...N is the weight (it's replace by its actual value in the code), P0...P is the capacity of the bag (same as N), x_i_j a bin to know if the weight i is in the bag j and y_i a bin to know if
the bag is filled.
But the problem is that with a big number of weight and bag the solver take too much time to solve it. So I was thinking as there can be no more than 5 weights by bag (according to my approximations)
I wanted to do something like this
use weight 0, 1, 2, 3, 4 or 0, 1, 2, 3, 5 or ... for the bag 0 and do the same for every bag.
So as you said, should I write something like that if I have "A or B or C" ?
N0 x_0_0 + N1 x_1_0 + N2 x_2_0 + N3 x_3_0 + N4 x_4_0 + P0 b_0_0 + P0 b_1_0 >= P0 y_0
N0 x_0_0 + N1 x_1_0 + N2 x_2_0 + N3 x_3_0 + N5 x_5_0 + P0 b_0_0 >= P0 y_0 * b_1_0
N0 x_0_0 + N1 x_1_0 + N2 x_2_0 + N3 x_3_0 + N5 x_5_0 >= P0 y_0 * b_0_0 + P0 y_0 * b_1_0 | {"url":"https://groups.google.com/g/lp_solve/c/JsdAiaVtLBM","timestamp":"2024-11-11T05:22:32Z","content_type":"text/html","content_length":"729413","record_id":"<urn:uuid:4f3168b6-4fde-445b-933c-9e090d57d570>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00033.warc.gz"} |
Mathematicians solve an old geometry problem on equiangular lines
Equiangular lines are lines in space that pass through a single point, and whose pairwise angles are all equal. Picture in 2D the three diagonals of a regular hexagon, and in 3D, the six lines
connecting opposite vertices of a regular icosahedron (see the figure above). Mathematicians are not limited to three dimensions, however.
“In high dimensions, things really get interesting, and the possibilities can seem limitless,” says Yufei Zhao, assistant professor of mathematics.
But they aren’t limitless, according to Zhao and his team of MIT mathematicians, who sought to solve this problem on the geometry of lines in high-dimensional space. It’s a problem that researchers
have been puzzling over for at least 70 years.
Their breakthrough determines the maximum possible number of lines that can be placed so that the lines are pairwise separated by the same given angle. Zhao wrote the paper with a group of MIT
researchers consisting of undergraduates Yuan Yao and Shengtong Zhang, PhD student Jonathan Tidor, and postdoc Zilin Jiang. (Yao recently started as an MIT math PhD student, and Jiang is now a
faculty member at Arizona State University). Their paper will be published in the November issue of Annals of Mathematics.
The mathematics of equiangular lines can be encoded using graph theory. The paper provides new insights into an area of mathematics known as spectral graph theory, which provides mathematical tools
for studying networks. Spectral graph theory has led to important algorithms in computer science such as Google’s PageRank algorithm for its search engine.
This new understanding of equiangular lines has potential implications for coding and communications. Equiangular lines are examples of “spherical codes,” which are important tools in information
theory, allowing different parties to send messages to each other over a noisy communication channel, such as those sent between NASA and its Mars rovers.
The problem of studying the maximum number of equiangular lines with a given angle was introduced in a 1973 paper of P.W.H. Lemmens and J.J. Seidel.
“This is a beautiful result providing a surprisingly sharp answer to a well-studied problem in extremal geometry that received a considerable amount of attention starting already in the ’60s,” says
Princeton University professor of mathematics Noga Alon.
The new work by the MIT team provides what Zhao calls “a satisfying resolution to this problem.”
“There were some good ideas at the time, but then people got stuck for nearly three decades,” Zhao says. There was some important progress made a few years ago by a team of researchers including
Benny Sudakov, a professor of mathematics at the Swiss Federal Institute of Technology (ETH) Zurich. Zhao hosted Sudakov’s visit to MIT in February 2018 when Sudakov spoke in the combinatorics
research seminar about his work on equiangular lines.
Jiang was inspired to work on the problem of equiangular lines based on the work of his former PhD advisor Bukh Boris at Carnegie Mellon University. Jiang and Zhao teamed up in the summer of 2019,
and were joined by Tidor, Yao, and Zhang. “I wanted to find a good summer research project, and I thought that this was a great problem to work on,” Zhao explains. “I thought we might make some nice
progress, but it was definitely beyond my expectations to completely solve the entire problem.”
The research was partially supported by the Alfred P. Sloan Foundation and the National Science Foundation. Yao and Zhang participated in the research through the Department of Mathematics’ Summer
Program for Undergraduate Research (SPUR), and Tidor was their graduate student mentor. Their results had earned them the mathematics department’s Hartley Rogers Jr. Prize for the best SPUR paper.
“It is one of the most successful outcomes of the SPUR program,” says Zhao. “It’s not every day that a long-standing open problem gets solved.”
One of the key mathematical tools used in the solution is known as spectral graph theory. Spectral graph theory tells us how to use tools from linear algebra to understand graphs and networks. The
“spectrum” of a graph is obtained by turning a graph into a matrix and looking at its eigenvalues.
“It is as if you shine an intense beam of light on a graph and then examine the spectrum of colors that come out,” Zhao explains. “We found that the emitted spectrum can never be too heavily
concentrated near the top. It turns out that this fundamental fact about the spectra of graphs has never been observed.”
The work gives a new theorem in spectral graph theory — that a bounded degree graph must have sublinear second eigenvalue multiplicity. The proof requires clever insights relating the spectrum of a
graph with the spectrum of small pieces of the graph.
“The proof worked out cleanly and beautifully,” Zhao says. “We had so much fun working on this problem together.” | {"url":"https://news.mit.edu/2021/mathematicians-solve-old-geometry-problem-equiangular-lines-1004","timestamp":"2024-11-15T00:17:26Z","content_type":"text/html","content_length":"115055","record_id":"<urn:uuid:64bc233e-8a5c-4d35-be32-687b6a67982c>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00426.warc.gz"} |
Recent publications
• The Problem of Action at a Distance in Networks and the Emergence of Preferential Attachment from Triadic Closure, J Kunegis, F Karimi, J Sun, Journal of Interdisciplinary Methods and Issues in
Science 2 (2017)
• Graph partitions and cluster synchronization in networks of oscillators, MT Schaub, N O'Clery, YN Billeh, JC Delvenne, R Lambiotte, M Barahona, Chaos: An Interdisciplinary Journal of Nonlinear
Science 26 (9), 094821 (2016)
• Burstiness and fractional diffusion on complex networks, S De Nigris, A Hastir, R Lambiotte The European Physical Journal B 89 (5), 1-7 (2016)
• Predicting links in ego-networks using temporal information, L Tabourier, AS Libert, R Lambiotte EPJ Data Science 5 (1), 1 (2016)
• Respondent‐driven sampling bias induced by community structure and response rates in social networks, LEC Rocha, AE Thorson, R Lambiotte, F Liljeros Journal of the Royal Statistical Society:
Series A (Statistics in Society) (2016)
• Using higher-order Markov models to reveal flow-based communities in networks, V Salnikov, MT Schaub, R Lambiotte Scientific reports 6 (2016)
• Mining open datasets for transparency in taxi transport in metropolitan environments, A Noulas, V Salnikov, R Lambiotte, C Mascolo EPJ Data Science 4 (1), 1 (2015)
• Imperfect spreading on temporal networks, M Gueuning, JC Delvenne, R Lambiotte The European Physical Journal B 88 (11), 1-5 (2015)
• The non-linear health consequences of living in larger cities, LEC Rocha, AE Thorson, R Lambiotte Journal of Urban Health 92 (5), 785-799 (2015)
• Diffusion on networked systems is a question of time or structure, JC Delvenne, R Lambiotte, LEC Rocha, Nature communications 6 (2015)
• Effect of memory on the dynamics of random walks on networks, R Lambiotte, V Salnikov, M Rosvall, Journal of Complex Networks 3 (2), 177-188 (2015)
• Sufficient conditions of endemic threshold on metapopulation networks, T Takaguchi, R Lambiotte, Journal of Theoretical Biology (2015)
• Topological Properties and Temporal Dynamics of Place Networks in Urban Environments, A Noulas, B Shaw, R Lambiotte, C Mascolo, Proceedings of WWW 2015
• Steady state and mean recurrence time for random walks on stochastic temporal networks, L Speidel, R Lambiotte, K Aihara, N Masuda, Physical Review E 91 (1), 012806 (2015)
• Random Walks, Markov Processes and the Multiscale Modular Organization of Complex Networks, R Lambiotte, JC Delvenne, M Barahona, Network Science and Engineering, IEEE Transactions on 1 (2),
76-90 (2014)
• Preferential attachment with partial information, T Carletti, F Gargiulo, R Lambiotte, The European Physical Journal B 88 (1), 1-5 (2014)
• Tracking the Digital Footprints of Personality, R Lambiotte and M Kosinski, Proceedings of the IEEE 102 (12), 1934-1939 (2014)
• The geography and carbon footprint of mobile phone use in Cote d’Ivoire, V Salnikov, D Schien, H Youn, R Lambiotte and MT Gastner, EPJ Data Science 3 (1), 1-15 (2014)
• RankMerging: Learning to rank in large-scale social networks, L Tabourier, AS Libert and R Lambiotte, DyNakII (ECML-PKDD Wkshp) (2014)
• Memory in network flows and its effects on spreading dynamics and community detection, M Rosvall, AV Esquivel, A Lancichinetti, JD West and R Lambiotte, Nature communications 5 (2014)
• Traumatic brain injury impairs small-world topology, AS Pandit, P Expert, R Lambiotte, V Bonnelle, R Leech, FE Turkheimer and DJ Sharp, Neurology 80, 1826-1833 (2013)
• Multi-scale modularity and dynamics in complex networks, R Lambiotte, Dynamics On and Of Complex Networks, Volume 2, 125-141 (Springer, 2013)
• Random Walks on Stochastic Temporal Networks, T Hoffmann, MA Porter and R Lambiotte, Temporal Networks, 295-313 (Springer, 2013)
• Decentralized Routing on Spatial Networks with Stochastic Edge Weights, T Hoffmann, R Lambiotte and MA Porter, Physical Review E 88 (2), 022815 (2013)
• Generalized master equations for non-Poisson dynamics on networks, T Hoffmann, MA Porter and R Lambiotte, Physical Review E 86, 046102 (2012)
• A data-driven analysis to question epidemic models for citation cascades on the blogosphere, A Salah Brahim, L Tabourier and B Le Grand, Proceedings of ICWSM (2013)
• Decentralized Routing on Spatial Networks with Stochastic Edge Weights, T Hoffmann, R Lambiotte and MA Porter, Physical Review E 88, 022815 (2013)
• Functional brain networks before the onset of psychosis: A prospective fMRI study with graph theoretical analysis, LD Lord, P Allen, P Expert, O Howes, M Broome, R Lambiotte, P Fusar-Poli, I
Valli, P McGuire and Federico E Turkheimer, NeuroImage: Clinical 1, 91-98 (2012)
• Psychological Aspects of Social Communities, A Friggeri, R Lambiotte, M Kosinski and E Fleury, Proceedings of the 2012 International Conference on Social Computing (SocialCom), 195-202 (2012)
• Ranking and clustering of nodes in networks with smart teleportation, R Lambiotte and M Rosvall, Physical Review E 85, 056107 (2012)
• The discovery of population differences in network community structure: New methods and applications to brain functional networks in schizophrenia, A Alexander-Bloch, R Lambiotte, B Roberts, J
Giedd, N Gogtay and E Bullmore, Neuroimage 59, 3889-3900 (2012)
• The personality of popular facebook users, D Quercia, R Lambiotte, D Stillwell, M Kosinski and J Crowcroft, Proceedings of the ACM 2012 conference on Computer Supported Cooperative Work, 955-964
• Community structure and patterns of scientific collaboration in Business and Management, TS Evans, R Lambiotte and P Panzarasa, Scientometrics 89, 381-396 (2011)
• Encoding dynamics for multiscale community detection: Markov time sweeping for the map equation, MT Schaub, R Lambiotte and M Barahona Physical Review E 86, 026112 (2012)
• A tale of many cities: universal patterns in human urban mobility, A Noulas, S Scellato, R Lambiotte, M Pontil and C Mascolo, PLoS ONE 7, e37027 (2012)
• Flow graphs: Interweaving dynamics and structure, R Lambiotte, R Sinatra, JC Delvenne, TS Evans, M Barahona and V Latora, Physical Review E 84, 017102 (2011)
• Socio-spatial properties of online location-based social networks, S Scellato, A Noulas, R Lambiotte and C Mascolo, Proceedings of ICWSM (2011)
• Uncovering space-independent communities in spatial networks, P Expert, TS Evans, VD Blondel and R Lambiotte, Proceedings of the National Academy of Sciences 108, 7663 (2011)
• Self-similar correlation function in brain resting-state functional magnetic resonance imaging, P Expert, R Lambiotte, DR Chialvo, K Christensen, HJ Jensen, DJ Sharp and F Turkheimer, Journal of
The Royal Society Interface 8, 472-479 (2011)
• Characterization of the anterior cingulate's role in the at-risk mental state using graph theory, LD Lord, P Allen, P Expert, O Howes, R Lambiotte, P McGuire, SK Bose, S Hyde and FE Turkheimer,
Neuroimage 56, 1531–1539 (2011)
• On co-evolution and the importance of initial conditions, R Lambiotte and JC Gonzalez-Avella, Physica A: Statistical Mechanics and its Applications 390, 392-397 (2011)
• Maximal-entropy random walks in complex networks with limited information, R Sinatra, J Gomez-Gardenes, R Lambiotte, V Nicosia and V Latora, Physical Review E 83, 030103 (2011)
Key publications (before 2011)
• Multirelational organization of large-scale social networks in an online world, M Szell, R Lambiotte and S Thurner, Proceedings of the National Academy of Sciences 107, 13636-13641 (2010)
• Multi-scale modularity in complex networks, R Lambiotte, Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt) (2010)
• Modular and hierarchically modular organization of brain networks, D Meunier, R Lambiotte and ET Bullmore, Frontiers in neuroscience 4, 200 (2010)
• Line graphs, link partitions, and overlapping communities, TS Evans and R Lambiotte, Physical Review E 80, 016105 (2009)
• Laplacian dynamics and multiscale modular structure in networks, R Lambiotte, JC Delvenne and M Barahona, Arxiv preprint arXiv:0812.1770 (2008)
• Fast unfolding of communities in large networks, VD Blondel, JL Guillaume, R Lambiotte and E Lefebvre, Journal of Statistical Mechanics: Theory and Experiment, P10008 (2008)
• Geographical dispersal of mobile communication networks, R Lambiotte, VD Blondel, C De Kerchove, E Huens, C Prieur, Z Smoreda and P Van Dooren, Physica A: Statistical Mechanics and its
Applications 387, 5317-5325 (2008)
• Dynamics of vacillating voters, R Lambiotte and S Redner, Journal of Statistical Mechanics: Theory and Experiment, L10001 (2007)
• From particle segregation to the granular clock, R Lambiotte, JM Salazar and L Brenig, Physics Letters A 343, 224-230 26 | {"url":"https://xn.unamur.be/publications.html","timestamp":"2024-11-04T20:53:44Z","content_type":"application/xhtml+xml","content_length":"17743","record_id":"<urn:uuid:45edd933-0312-4a1c-ad13-0c3729ba9d26>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00214.warc.gz"} |
r.grow(1grass) GRASS GIS User's Manual r.grow(1grass)
r.grow - Generates a raster map layer with contiguous areas grown by one cell.
raster, distance, proximity
r.grow --help
r.grow [-m] input=name output=name [radius=float] [metric=string] [old=integer] [new=integer] [--overwrite] [--help] [--verbose] [--quiet] [--ui]
Radius is in map units rather than cells
Allow output files to overwrite existing files
Print usage summary
Verbose module output
Quiet module output
Force launching GUI dialog
Name of input raster map
Name for output raster map
Radius of buffer in raster cells
Default: 1.01
Options: euclidean, maximum, manhattan
Default: euclidean
Value to write for input cells which are non-NULL (-1 => NULL)
Value to write for "grown" cells
r.grow adds cells around the perimeters of all areas in a user-specified raster map layer and stores the output in a new raster map layer. The user can use it to grow by one or more than one cell (by
varying the size of the radius parameter), or like r.buffer, but with the option of preserving the original cells (similar to combining r.buffer and r.patch).
If radius is negative,r.grow shrinks areas by removing cells around the perimeters of all areas.
The user has the option of specifying three different metrics which control the geometry in which grown cells are created, (controlled by the metric parameter): Euclidean, Manhattan, and Maximum.
The Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem.
The formula is given by:
d(dx,dy) = sqrt(dx^2 + dy^2)
Cells grown using this metric would form isolines of distance that are circular from a given point, with the distance given by the radius.
The Manhattan metric, or Taxicab geometry, is a form of geometry in which the usual metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the
(absolute) differences of their coordinates. The name alludes to the grid layout of most streets on the island of Manhattan, which causes the shortest path a car could take between two points in the
city to have length equal to the points’ distance in taxicab geometry. The formula is given by:
d(dx,dy) = abs(dx) + abs(dy)
where cells grown using this metric would form isolines of distance that are rhombus-shaped from a given point.
The Maximum metric is given by the formula
d(dx,dy) = max(abs(dx),abs(dy))
where the isolines of distance from a point are squares.
If there are two cells which are equal candidates to grow into an empty space, r.grow will choose the northernmost candidate; if there are multiple candidates with the same northing, the westernmost
is chosen.
In this example, the lakes map in the North Carolina sample dataset is buffered:
g.region raster=lakes -p
# the lake raster map pixel resolution is 10m
r.grow input=lakes output=lakes_grown_100m radius=10
Shrinking instead of growing:
g.region raster=lakes -p
# the lake raster map pixel resolution is 10m
r.grow input=lakes output=lakes_shrunk_100m radius=-10
SEE ALSO¶
r.buffer, r.grow.distance, r.patch
Wikipedia Entry: Euclidean Metric
Wikipedia Entry: Manhattan Metric
Marjorie Larson, U.S. Army Construction Engineering Research Laboratory
Glynn Clements
Available at: r.grow source code (history)
Accessed: Saturday Jul 27 17:09:12 2024
Main index | Raster index | Topics index | Keywords index | Graphical index | Full index
© 2003-2024 GRASS Development Team, GRASS GIS 8.4.0 Reference Manual | {"url":"https://manpages.debian.org/experimental/grass-doc/r.grow.1grass.en.html","timestamp":"2024-11-12T23:00:42Z","content_type":"text/html","content_length":"24447","record_id":"<urn:uuid:60f7e3c2-4193-4515-9131-c3c1a254403f>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00761.warc.gz"} |
Enter natural log statement
Evaluate the following logarithmic expression
Evaluate log(85)
You didn't enter a base
We'll do bases e and 2-10
Evaluate log[e](85) using the Change of Base Formula
The formula for the change of base rule in log[b](x) is as follows:
Given b = e and x = 85, we have:
Ln(e) = 1
log[e](85) = 4.4426512564903
Evaluate log[2](85) using the Change of Base Formula
The formula for the change of base rule in log[b](x) is as follows:
Given b = 2 and x = 85, we have:
log[2](85) = 6.4093909361377
Evaluate log[3](85) using the Change of Base Formula
The formula for the change of base rule in log[b](x) is as follows:
Given b = 3 and x = 85, we have:
log[3](85) = 4.0438754438805
Evaluate log[4](85) using the Change of Base Formula
The formula for the change of base rule in log[b](x) is as follows:
Given b = 4 and x = 85, we have:
log[4](85) = 3.2046954680689
Evaluate log[5](85) using the Change of Base Formula
The formula for the change of base rule in log[b](x) is as follows:
Given b = 5 and x = 85, we have:
log[5](85) = 2.7603744277226
Evaluate log[6](85) using the Change of Base Formula
The formula for the change of base rule in log[b](x) is as follows:
Given b = 6 and x = 85, we have:
log[6](85) = 2.4794908763085
Evaluate log[7](85) using the Change of Base Formula
The formula for the change of base rule in log[b](x) is as follows:
Given b = 7 and x = 85, we have:
log[7](85) = 2.2830711164373
Evaluate log[8](85) using the Change of Base Formula
The formula for the change of base rule in log[b](x) is as follows:
Given b = 8 and x = 85, we have:
log[8](85) = 2.1364636453792
Evaluate log[9](85) using the Change of Base Formula
The formula for the change of base rule in log[b](x) is as follows:
Given b = 9 and x = 85, we have:
log[9](85) = 2.0219377219402
Evaluate log[10](85) using the Change of Base Formula
The formula for the change of base rule in log[b](x) is as follows:
Given b = 10 and x = 85, we have:
log[10](85) = Ln(85)
log[10](85) = 1.9294189257143
Final Answer
log[e](85) = 4.4426512564903
log[2](85) = 6.4093909361377
log[3](85) = 4.0438754438805
log[4](85) = 3.2046954680689
log[5](85) = 2.7603744277226
log[6](85) = 2.4794908763085
log[7](85) = 2.2830711164373
log[8](85) = 2.1364636453792
log[9](85) = 2.0219377219402
log[10](85) = 1.9294189257143
What is the Answer?
log[e](85) = 4.4426512564903
log[2](85) = 6.4093909361377
log[3](85) = 4.0438754438805
log[4](85) = 3.2046954680689
log[5](85) = 2.7603744277226
log[6](85) = 2.4794908763085
log[7](85) = 2.2830711164373
log[8](85) = 2.1364636453792
log[9](85) = 2.0219377219402
log[10](85) = 1.9294189257143
How does the Logarithms and Natural Logarithms and Eulers Constant (e) Calculator work?
Free Logarithms and Natural Logarithms and Eulers Constant (e) Calculator - This calculator does the following:
* Takes the Natural Log base e of a number x Ln(x) → log[e]x
* Raises e to a power of y, e^y
* Performs the change of base rule on log[b](x)
* Solves equations in the form b^cx = d where b, c, and d are constants and x is any variable a-z
* Solves equations in the form ce^dx=b where b, c, and d are constants, e is Eulers Constant = 2.71828182846, and x is any variable a-z
* Exponential form to logarithmic form for expressions such as 5^3 = 125 to logarithmic form
* Logarithmic form to exponential form for expressions such as Log[5]125 = 3
This calculator has 1 input.
What 8 formulas are used for the Logarithms and Natural Logarithms and Eulers Constant (e) Calculator?
Ln(a/b) = Ln(a) - Ln(b)
Ln(ab)= Ln(a) + Ln(b)
Ln(e) = 1
Ln(1) = 0
) = y * ln(x)
For more math formulas, check out our
Formula Dossier
What 4 concepts are covered in the Logarithms and Natural Logarithms and Eulers Constant (e) Calculator?
Famous mathematician who developed Euler's constant
the exponent or power to which a base must be raised to yield a given number
natural logarithm
its logarithm to the base of the mathematical constant e
e^Ln(x) = x
how many times to use the number in a multiplication
Example calculations for the Logarithms and Natural Logarithms and Eulers Constant (e) Calculator
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How Do you Get the Square Root of 74? - GEGCalculators
How Do you Get the Square Root of 74?
Square roots are a fascinating aspect of mathematics, offering insight into the world of numbers, equations, and mathematical operations. In this blog post, we will embark on a comprehensive journey
to understand how to calculate the square root of 74. We will explore various methods, discuss the properties of square roots, and uncover the significance of this mathematical concept in both theory
and practical applications.
How Do you Get the Square Root of 74?
The square root of 74 is approximately 8.60 when rounded to two decimal places. This value represents the positive square root, as there is no real-valued square root of a negative number in the set
of real numbers. Calculators and mathematical software can provide more precise values for the square root of 74.
The Basics of Square Roots
Before we delve into the square root of 74, let’s revisit the fundamental principles of square roots.
The square root of a number “x,” denoted as √x, is the value that, when multiplied by itself, results in “x.” In mathematical notation, it is represented as √x * √x = x. For example, the square root
of 25 is 5, as 5 * 5 = 25.
Square roots can be both rational and irrational numbers. Rational square roots can be expressed as fractions, while irrational square roots have non-repeating, non-terminating decimal
Methods to Calculate the Square Root of 74
1. Using a Calculator: The simplest and most common method is to use a calculator that has a square root function. Enter 74 and press the square root (√) button to get the result.
2. Long Division Method: The long division method is a manual approach. You start with an estimated guess, divide the number by that guess, and then refine the estimate through successive
iterations. This method can be time-consuming and is typically not used for large numbers like 74.
3. Prime Factorization: You can also find the square root of a number by prime factorization. Break 74 down into its prime factors and then group them into pairs. Each pair represents a square root,
and you can multiply them to find the square root of 74.
4. Using Newton’s Method: Newton’s method is an iterative approach to finding square roots. While it’s more complex than the previous methods, it can be useful for finding square roots to high
Prime Factorization Method
Let’s explore the prime factorization method to find the square root of 74:
Step 1: Prime Factorization of 74
Step 2: Group the Prime Factors into Pairs
• Since there are no duplicate prime factors, we have one pair: √(2 * 37)
Step 3: Calculate the Square Root
Now, we have the square root of 74 expressed as the product of the square roots of its prime factors.
Practical Applications
Understanding square roots is vital in various real-world applications:
1. Geometry: Square roots are used to calculate side lengths and areas of squares and rectangles.
2. Engineering: Engineers use square roots in various calculations, such as determining forces, stresses, and electrical circuit parameters.
3. Physics: Square roots appear in physics equations, particularly in kinematics and wave propagation.
4. Finance: Square roots are used in financial mathematics to calculate volatility and risk.
5. Computer Graphics: Square roots play a role in computer graphics algorithms, such as those used for rendering and 3D transformations.
In conclusion, the square root of 74 is a mathematical concept that can be calculated using various methods, including calculators, prime factorization, and iterative approaches like Newton’s method.
While calculators provide quick solutions, understanding the underlying methods and properties of square roots enriches our mathematical knowledge. Square roots have wide-ranging applications in
science, engineering, and everyday problem-solving, making them a fundamental part of our mathematical toolbox.
GEG Calculators is a comprehensive online platform that offers a wide range of calculators to cater to various needs. With over 300 calculators covering finance, health, science, mathematics, and
more, GEG Calculators provides users with accurate and convenient tools for everyday calculations. The website’s user-friendly interface ensures easy navigation and accessibility, making it suitable
for people from all walks of life. Whether it’s financial planning, health assessments, or educational purposes, GEG Calculators has a calculator to suit every requirement. With its reliable and
up-to-date calculations, GEG Calculators has become a go-to resource for individuals, professionals, and students seeking quick and precise results for their calculations.
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Linear System Theory and Design in Robotics - Robotics Meta
Linear System Theory and Design in Robotics
All Posts, Robotics Components and Technology, Robotics Programming and Software / November 9, 2022
Robotics stands at the confluence of multiple disciplines, melding mechanics, electronics, computer science, and control theory into sophisticated systems that perform tasks ranging from assembly
line operations to intricate surgical procedures. Among these disciplines, Linear System Theory plays a pivotal role in the design, analysis, and control of robotic systems. This article delves deep
into the integration of linear system theory within robotics, exploring its foundational principles, applications, and the nuanced design methodologies that underpin modern robotic systems.
Table of Contents
Introduction to Linear System Theory
Linear System Theory provides the mathematical framework to model, analyze, and design systems that adhere to the principles of linearity. In robotics, where precision and predictability are
paramount, linear systems offer a tractable approach to handle the complexities inherent in mechanical and electronic subsystems. By approximating nonlinear behaviors through linear models, engineers
can leverage a vast arsenal of analytical tools to ensure system stability, responsiveness, and accuracy.
Fundamental Concepts of Linear Systems
Understanding linear systems in robotics requires a grasp of several core concepts. These principles ensure that robotic systems behave predictably and can be effectively controlled.
Linearity and Superposition
A system is linear if it satisfies two main properties:
1. Homogeneity (Scaling): If an input x(t) produces an output y(t), then an input a * x(t) produces a * y(t).
2. Superposition (Additivity): If inputs x₁(t) and x₂(t) produce outputs y₁(t) and y₂(t) respectively, then the input x₁(t) + x₂(t) produces the output y₁(t) + y₂(t).
Implication in Robotics: Linear systems allow the decomposition of complex motions into simpler, manageable components. This facilitates advanced motion planning and control strategies.
A system is time-invariant if its behavior and characteristics do not change over time. Mathematically, if an input x(t) produces an output y(t), then a time-shifted input x(t - t₀) produces y(t -
Implication in Robotics: Time-invariant systems simplify the design process as the same control laws can be applied uniformly, without adjustment for time-based variations.
Stability ensures that a system’s output remains bounded for any bounded input. In control systems, asymptotic stability implies that the system will return to equilibrium after a disturbance.
Implication in Robotics: Stability is crucial for maintaining precise control over robotic movements and ensuring safe interactions with the environment.
Controllability and Observability
• Controllability refers to the ability to steer a system from any initial state to any desired final state within finite time, using suitable inputs.
• Observability pertains to the ability to infer the internal state of a system based solely on its output measurements.
Implication in Robotics: Ensuring controllability and observability is fundamental for effective state estimation and control, enabling robots to perform desired tasks accurately.
Mathematical Foundations
Linear system theory is deeply rooted in mathematical representations that facilitate analysis and design.
State-Space Representation
The state-space model is a cornerstone of modern control theory, representing a system through a set of first-order differential (or difference) equations:
\dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{u}(t) \
\mathbf{y}(t) = \mathbf{C}\mathbf{x}(t) + \mathbf{D}\mathbf{u}(t)
• (\mathbf{x}(t)): State vector representing the system’s current status.
• (\mathbf{u}(t)): Input vector influencing the system.
• (\mathbf{y}(t)): Output vector describing measurable quantities.
• (\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D}): Matrices defining system dynamics.
Application in Robotics: State-space models enable comprehensive descriptions of robotic dynamics, encompassing multiple degrees of freedom and interactions.
Transfer Functions
The transfer function relates the Laplace transform of the output to the input under zero initial conditions:
H(s) = \frac{\mathbf{Y}(s)}{\mathbf{U}(s)} = \mathbf{C}(s\mathbf{I} – \mathbf{A})^{-1}\mathbf{B} + \mathbf{D}
Application in Robotics: Transfer functions facilitate frequency domain analysis, aiding in the design of controllers that meet specific spectral requirements.
Differential Equations
Robotic systems are governed by differential equations that describe their motion dynamics, incorporating forces, torques, and interactions with the environment.
\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}} + \mathbf{C}(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}} + \mathbf{G}(\mathbf{q}) = \mathbf{u}
• (\mathbf{q}): Generalized coordinates (e.g., joint angles).
• (\mathbf{M}(\mathbf{q})): Mass (inertia) matrix.
• (\mathbf{C}(\mathbf{q}, \dot{\mathbf{q}})): Coriolis and centrifugal forces.
• (\mathbf{G}(\mathbf{q})): Gravitational forces.
• (\mathbf{u}): Input torque vector.
Application in Robotics: These equations are essential for simulating robotic motion and designing dynamic controllers.
Linear System Theory in Robotics
Integrating linear system theory into robotics encompasses various aspects, from kinematic modeling to dynamic control.
Kinematics and Dynamics
Kinematics deals with the geometric aspects of motion, such as position, velocity, and acceleration, without considering forces. Dynamics incorporates forces and torques, providing a complete
description of robotic motion.
Linear Approaches: While robotic systems are inherently nonlinear, linear approximations around operating points enable the use of linear system techniques for analysis and control.
Motion Planning
Linear system theory aids in motion planning, ensuring that robotic movements are smooth, efficient, and free from undesirable behaviors like oscillations or overshoots.
Trajectory Generation: By modeling the robot’s motion as a linear system, engineers can design trajectories that account for system constraints and optimize performance metrics.
Control Mechanisms
Control systems orchestrate the behavior of robots, ensuring they perform tasks accurately and respond appropriately to disturbances.
Feedback Control: Linear controllers, such as PID and state feedback controllers, utilize feedback loops to maintain desired performance despite external perturbations.
Design Methodologies
Designing controllers for robotic systems involves selecting appropriate strategies that align with system dynamics and performance requirements.
PID Control
The Proportional-Integral-Derivative (PID) controller is one of the simplest and most widely used control strategies.
u(t) = K_p e(t) + K_i \int_{0}^{t} e(\tau) d\tau + K_d \frac{d e(t)}{dt}
• (K_p): Proportional gain.
• (K_i): Integral gain.
• (K_d): Derivative gain.
• (e(t)): Error between desired and actual outputs.
Advantages: Simplicity and ease of implementation.
Limitations: May not suffice for complex, multi-degree-of-freedom robotic systems.
State Feedback Control
State feedback controllers utilize the full state vector to determine control inputs.
\mathbf{u}(t) = -\mathbf{K}\mathbf{x}(t) + \mathbf{r}(t)
• (\mathbf{K}): Gain matrix.
• (\mathbf{r}(t)): Reference input.
Benefits: Enhanced control over system dynamics, enabling precise regulation of states.
Observer Design
In many cases, not all states are directly measurable. Observers estimate the unmeasured states based on available outputs.
\dot{\hat{\mathbf{x}}}(t) = \mathbf{A}\hat{\mathbf{x}}(t) + \mathbf{B}\mathbf{u}(t) + \mathbf{L}(\mathbf{y}(t) – \mathbf{C}\hat{\mathbf{x}}(t))
• (\mathbf{L}): Observer gain matrix.
• (\hat{\mathbf{x}}(t)): Estimated state vector.
Application: Facilitates state feedback control when only partial state information is available.
Linear Quadratic Regulator (LQR)
LQR is an optimal control strategy that minimizes a cost function balancing state deviations and control effort.
J = \int_{0}^{\infty} (\mathbf{x}^T \mathbf{Q} \mathbf{x} + \mathbf{u}^T \mathbf{R} \mathbf{u}) dt
• (\mathbf{Q}): State weighting matrix.
• (\mathbf{R}): Control weighting matrix.
Advantages: Provides a systematic approach to controller design with guaranteed stability and performance, given appropriate weighting matrices.
Linearization of Nonlinear Robotic Systems
While many robotic systems exhibit nonlinear behavior, linearization simplifies analysis and control design. This process involves approximating nonlinear equations around a specific operating point,
typically using Taylor series expansion.
Taylor Series Expansion
For a nonlinear function ( f(\mathbf{x}, \mathbf{u}) ), the first-order linear approximation around the equilibrium point ( (\mathbf{x}_0, \mathbf{u}_0) ) is:
\Delta \dot{\mathbf{x}} = \mathbf{A}\Delta \mathbf{x} + \mathbf{B}\Delta \mathbf{u}
\Delta \mathbf{y} = \mathbf{C}\Delta \mathbf{x} + \mathbf{D}\Delta \mathbf{u}
\mathbf{A} = \left. \frac{\partial f}{\partial \mathbf{x}} \right|{(\mathbf{x}_0, \mathbf{u}_0)}, \quad \mathbf{B} = \left. \frac{\partial f}{\partial \mathbf{u}} \right|{(\mathbf{x}_0, \mathbf{u}
Application in Robotics: Enables the design of linear controllers for systems that are inherently nonlinear, enhancing tractability while acknowledging the limitations of the approximation.
Applications and Examples
To contextualize linear system theory within robotics, consider the following examples:
Articulated Robotic Arms
Articulated arms, common in manufacturing, consist of multiple joints and links. Linear modeling facilitates:
• Dynamic Simulation: Predicts arm movements under various load conditions.
• Trajectory Control: Ensures precise positioning and orientation during tasks like welding or assembly.
Mobile Robots
Mobile robots navigate environments, requiring control over motion and orientation.
• Differential Drive Systems: Linear models assist in path planning and obstacle avoidance.
• Stabilization: Ensures balance in bipedal or humanoid robots through feedback control.
Humanoid Robots
Humanoid robots emulate human motion, necessitating complex control strategies.
• Balance Control: Linear approximations aid in maintaining upright posture.
• Motion Coordination: Synchronizes multiple joints for fluid movements.
Challenges and Advanced Topics
While linear system theory offers robust tools, several challenges arise in its application to robotics.
Handling Nonlinearities
Robotic systems often exhibit nonlinear behaviors due to joint interactions, force nonlinearities, and complex dynamics.
• Gain Scheduling: Adjusts controller gains based on operating conditions.
• Nonlinear Control Techniques: Integrates linear control within a broader nonlinear framework.
Robust Control Design
Ensuring consistent performance despite model uncertainties and external disturbances is paramount.
• H-infinity Control: Minimizes worst-case disturbances.
• Sliding Mode Control: Offers robustness against parameter variations.
Adaptive Control
Adaptive controllers adjust their parameters in real-time to accommodate changing system dynamics.
Benefits: Enhances performance in systems with time-varying properties or unexpected payloads.
Future Trends
The integration of linear system theory in robotics continues to evolve, driven by advancements in computation, sensing, and artificial intelligence.
• Model Predictive Control (MPC): Combines linear models with optimization techniques for real-time control.
• Machine Learning Integration: Utilizes linear models within learning-based frameworks for enhanced adaptability.
• Collaborative Robotics: Employs linear control strategies to manage interactions between multiple robots and humans safely.
Linear system theory serves as a foundational pillar in the field of robotics, providing the necessary tools for modeling, analysis, and control. While robotic systems are inherently complex and
often nonlinear, linear approximations enable engineers to design robust and efficient controllers that ensure stability and precision. As robotics continues to advance, the synergy between linear
system theory and emerging technologies will undoubtedly foster the development of increasingly sophisticated and capable robotic systems.
1. “Modern Control Engineering” by Katsuhiko Ogata – A comprehensive textbook covering the fundamentals of control systems.
2. “Robotics, Vision and Control: Fundamental Algorithms In MATLAB” by Peter Corke – An insightful resource linking robotics with control theory.
3. “Feedback Control of Dynamic Systems” by Gene F. Franklin, J. Da Powell, and Michael L. Workman – An authoritative text on control system design.
4. “Robotics: Control, Sensing, Vision, and Intelligence” by K.S. Fu, R.C. Gonzalez, and C.S.G. Lee – Explores various aspects of robotics with a focus on control.
5. IEEE Transactions on Robotics – A leading journal publishing cutting-edge research in robotics and control systems.
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Aggregation Operators
The aggregation pipeline operators are compatible with MongoDB Atlas and on-premise environments.
For details on a specific operator, including syntax and examples, click on the link to the operator's reference page.
You can use the aggregation pipeline operators for deployments hosted in the following environments:
• MongoDB Atlas: The fully managed service for MongoDB deployments in the cloud
Expression Operators
Expression operators are similar to functions that take arguments. In general, these operators take an array of arguments and have the following form:
{ <operator>: [ <argument1>, <argument2> ... ] }
If an operator accepts a single argument, you can omit the outer array designating the argument list:
{ <operator>: <argument> }
To avoid parsing ambiguity if the argument is a literal array, you must wrap the literal array in a $literal expression or keep the outer array that designates the argument list.
This page lists the available operators that can be used to construct expressions. Expressions are used in the following contexts in MQL:
Arithmetic Expression Operators
Arithmetic expressions perform mathematic operations on numbers. Some arithmetic expressions can also support date arithmetic.
$abs Returns the absolute value of a number.
Adds numbers to return the sum, or adds numbers and a date to return a new date. If adding numbers and a date, treats the numbers as milliseconds. Accepts any number of argument
$add expressions, but at most, one expression can resolve to a date.
$ceil Returns the smallest integer greater than or equal to the specified number.
$divide Returns the result of dividing the first number by the second. Accepts two argument expressions.
$exp Raises e to the specified exponent.
$floor Returns the largest integer less than or equal to the specified number.
$ln Calculates the natural log of a number.
$log Calculates the log of a number in the specified base.
$log10 Calculates the log base 10 of a number.
$mod Returns the remainder of the first number divided by the second. Accepts two argument expressions.
$multiply Multiplies numbers to return the product. Accepts any number of argument expressions.
$pow Raises a number to the specified exponent.
$round Rounds a number to to a whole integer or to a specified decimal place.
$sqrt Calculates the square root.
Returns the result of subtracting the second value from the first. If the two values are numbers, return the difference. If the two values are dates, return the difference in milliseconds.
$subtract If the two values are a date and a number in milliseconds, return the resulting date. Accepts two argument expressions. If the two values are a date and a number, specify the date argument
first as it is not meaningful to subtract a date from a number.
$trunc Truncates a number to a whole integer or to a specified decimal place.
Array Expression Operators
$arrayElemAt Returns the element at the specified array index.
$arrayToObject Converts an array of key value pairs to a document.
$concatArrays Concatenates arrays to return the concatenated array.
$filter Selects a subset of the array to return an array with only the elements that match the filter condition.
Returns a specified number of elements from the beginning of an array. Distinct from the
$firstN $firstN
$in Returns a boolean indicating whether a specified value is in an array.
$indexOfArray Searches an array for an occurrence of a specified value and returns the array index of the first occurrence. Array indexes start at zero.
$isArray Determines if the operand is an array. Returns a boolean.
Returns a specified number of elements from the end of an array. Distinct from the
$lastN $lastN
$map Applies a subexpression to each element of an array and returns the array of resulting values in order. Accepts named parameters.
Returns the
$maxN largest values in an array. Distinct from the
Returns the
$minN smallest values in an array. Distinct from the
$objectToArray Converts a document to an array of documents representing key-value pairs.
$range Outputs an array containing a sequence of integers according to user-defined inputs.
$reduce Applies an expression to each element in an array and combines them into a single value.
$reverseArray Returns an array with the elements in reverse order.
$size Returns the number of elements in the array. Accepts a single expression as argument.
$slice Returns a subset of an array.
$sortArray Sorts the elements of an array.
$zip Merge two arrays together.
Bitwise Operators
Returns the result of a bitwise and operation on an array of int or long values.
New in version 6.3.
Returns the result of a bitwise not operation on a single argument or an array that contains a single int or long value.
New in version 6.3.
Returns the result of a bitwise or operation on an array of int or long values.
New in version 6.3.
Returns the result of a bitwise xor (exclusive or) operation on an array of int and long values.
New in version 6.3.
Boolean Expression Operators
Boolean expressions evaluate their argument expressions as booleans and return a boolean as the result.
In addition to the false boolean value, Boolean expression evaluates as false the following: null, 0, and undefined values. The Boolean expression evaluates all other values as true, including
non-zero numeric values and arrays.
$and Returns true only when all its expressions evaluate to true. Accepts any number of argument expressions.
$not Returns the boolean value that is the opposite of its argument expression. Accepts a single argument expression.
$or Returns true when any of its expressions evaluates to true. Accepts any number of argument expressions.
Comparison Expression Operators
Comparison expressions return a boolean except for $cmp which returns a number.
The comparison expressions take two argument expressions and compare both value and type, using the specified BSON comparison order for values of different types.
$cmp Returns 0 if the two values are equivalent, 1 if the first value is greater than the second, and -1 if the first value is less than the second.
$eq Returns true if the values are equivalent.
$gt Returns true if the first value is greater than the second.
$gte Returns true if the first value is greater than or equal to the second.
$lt Returns true if the first value is less than the second.
$lte Returns true if the first value is less than or equal to the second.
$ne Returns true if the values are not equivalent.
Conditional Expression Operators
A ternary operator that evaluates one expression, and depending on the result, returns the value of one of the other two expressions. Accepts either three expressions in an ordered list or
$cond three named parameters.
Returns either the non-null result of the first expression or the result of the second expression if the first expression results in a null result. Null result encompasses instances of
$ifNull undefined values or missing fields. Accepts two expressions as arguments. The result of the second expression can be null.
$switch Evaluates a series of case expressions. When it finds an expression which evaluates to true, $switch executes a specified expression and breaks out of the control flow.
Custom Aggregation Expression Operators
Data Size Operators
The following operators return the size of a data element:
$binarySize Returns the size of a given string or binary data value's content in bytes.
Returns the size in bytes of a given document (i.e. bsontype
) when encoded as
Date Expression Operators
The following operators returns date objects or components of a date object:
$dateAdd Adds a number of time units to a date object.
$dateDiff Returns the difference between two dates.
$dateFromParts Constructs a BSON Date object given the date's constituent parts.
$dateFromString Converts a date/time string to a date object.
$dateSubtract Subtracts a number of time units from a date object.
$dateToParts Returns a document containing the constituent parts of a date.
$dateToString Returns the date as a formatted string.
$dateTrunc Truncates a date.
$dayOfMonth Returns the day of the month for a date as a number between 1 and 31.
$dayOfWeek Returns the day of the week for a date as a number between 1 (Sunday) and 7 (Saturday).
$dayOfYear Returns the day of the year for a date as a number between 1 and 366 (leap year).
$hour Returns the hour for a date as a number between 0 and 23.
$isoDayOfWeek Returns the weekday number in ISO 8601 format, ranging from 1 (for Monday) to 7 (for Sunday).
$isoWeek Returns the week number in ISO 8601 format, ranging from 1 to 53. Week numbers start at 1 with the week (Monday through Sunday) that contains the year's first Thursday.
$isoWeekYear Returns the year number in ISO 8601 format. The year starts with the Monday of week 1 (ISO 8601) and ends with the Sunday of the last week (ISO 8601).
$millisecond Returns the milliseconds of a date as a number between 0 and 999.
$minute Returns the minute for a date as a number between 0 and 59.
$month Returns the month for a date as a number between 1 (January) and 12 (December).
$second Returns the seconds for a date as a number between 0 and 60 (leap seconds).
$toDate Converts value to a Date.
$week Returns the week number for a date as a number between 0 (the partial week that precedes the first Sunday of the year) and 53 (leap year).
$year Returns the year for a date as a number (e.g. 2014).
The following arithmetic operators can take date operands:
Adds numbers and a date to return a new date. If adding numbers and a date, treats the numbers as milliseconds. Accepts any number of argument expressions, but at most, one expression can
$add resolve to a date.
Returns the result of subtracting the second value from the first. If the two values are dates, return the difference in milliseconds. If the two values are a date and a number in
$subtract milliseconds, return the resulting date. Accepts two argument expressions. If the two values are a date and a number, specify the date argument first as it is not meaningful to subtract a
date from a number.
Literal Expression Operator
Return a value without parsing. Use for values that the aggregation pipeline may interpret as an expression. For example, use a
$literal expression to a string that starts with a dollar sign (
) to avoid parsing as a field path.
Miscellaneous Operators
Returns the value of a specified field from a document. You can use $getField to retrieve the value of fields with names that contain periods (.) or start with dollar signs ($).
New in version 5.0.
$rand Returns a random float between 0 and 1
Randomly select documents at a given rate. Although the exact number of documents selected varies on each run, the quantity chosen approximates the sample rate expressed as a
$sampleRate percentage of the total number of documents.
$toHashedIndexKey Computes and returns the hash of the input expression using the same hash function that MongoDB uses to create a hashed index.
Object Expression Operators
$mergeObjects Combines multiple documents into a single document.
$objectToArray Converts a document to an array of documents representing key-value pairs.
Adds, updates, or removes a specified field in a document. You can use $setField to add, update, or remove fields with names that contain periods (.) or start with dollar signs ($).
New in version 5.0.
Set Expression Operators
Set expressions performs set operation on arrays, treating arrays as sets. Set expressions ignores the duplicate entries in each input array and the order of the elements.
If the set operation returns a set, the operation filters out duplicates in the result to output an array that contains only unique entries. The order of the elements in the output array is
If a set contains a nested array element, the set expression does not descend into the nested array but evaluates the array at top-level.
$allElementsTrue Returns true if no element of a set evaluates to false, otherwise, returns false. Accepts a single argument expression.
$anyElementTrue Returns true if any elements of a set evaluate to true; otherwise, returns false. Accepts a single argument expression.
Returns a set with elements that appear in the first set but not in the second set; i.e. performs a
$setDifference relative complement
of the second set relative to the first. Accepts exactly two argument expressions.
$setEquals Returns true if the input sets have the same distinct elements. Accepts two or more argument expressions.
$setIntersection Returns a set with elements that appear in all of the input sets. Accepts any number of argument expressions.
$setIsSubset if all elements of the first set appear in the second set, including when the first set equals the second set; i.e. not a
strict subset
. Accepts exactly two argument expressions.
$setUnion Returns a set with elements that appear in any of the input sets.
String Expression Operators
String expressions, with the exception of $concat, only have a well-defined behavior for strings of ASCII characters.
$concat behavior is well-defined regardless of the characters used.
$concat Concatenates any number of strings.
$dateFromString Converts a date/time string to a date object.
$dateToString Returns the date as a formatted string.
$indexOfBytes Searches a string for an occurrence of a substring and returns the UTF-8 byte index of the first occurrence. If the substring is not found, returns -1.
$indexOfCP Searches a string for an occurrence of a substring and returns the UTF-8 code point index of the first occurrence. If the substring is not found, returns -1
$ltrim Removes whitespace or the specified characters from the beginning of a string.
$regexFind Applies a regular expression (regex) to a string and returns information on the first matched substring.
$regexFindAll Applies a regular expression (regex) to a string and returns information on the all matched substrings.
$regexMatch Applies a regular expression (regex) to a string and returns a boolean that indicates if a match is found or not.
$replaceOne Replaces the first instance of a matched string in a given input.
$replaceAll Replaces all instances of a matched string in a given input.
$rtrim Removes whitespace or the specified characters from the end of a string.
$split Splits a string into substrings based on a delimiter. Returns an array of substrings. If the delimiter is not found within the string, returns an array containing the original string.
$strLenBytes Returns the number of UTF-8 encoded bytes in a string.
Performs case-insensitive string comparison and returns: 0 if two strings are equivalent, 1 if the first string is greater than the second, and -1 if the first string is less than the
$strcasecmp second.
$substrBytes Returns the substring of a string. Starts with the character at the specified UTF-8 byte index (zero-based) in the string and continues for the specified number of bytes.
Returns the substring of a string. Starts with the character at the specified UTF-8
$substrCP code point (CP)
index (zero-based) in the string and continues for the number of code points specified.
$toLower Converts a string to lowercase. Accepts a single argument expression.
$toString Converts value to a string.
$trim Removes whitespace or the specified characters from the beginning and end of a string.
$toUpper Converts a string to uppercase. Accepts a single argument expression.
Text Expression Operator
$meta Access available per-document metadata related to the aggregation operation.
Timestamp Expression Operators
Timestamp expression operators return values from a timestamp.
Trigonometry Expression Operators
Trigonometry expressions perform trigonometric operations on numbers. Values that represent angles are always input or output in radians. Use $degreesToRadians and $radiansToDegrees to convert
between degree and radian measurements.
$sin Returns the sine of a value that is measured in radians.
$cos Returns the cosine of a value that is measured in radians.
$tan Returns the tangent of a value that is measured in radians.
$asin Returns the inverse sin (arc sine) of a value in radians.
$acos Returns the inverse cosine (arc cosine) of a value in radians.
$atan Returns the inverse tangent (arc tangent) of a value in radians.
$atan2 Returns the inverse tangent (arc tangent) of y / x in radians, where y and x are the first and second values passed to the expression respectively.
$asinh Returns the inverse hyperbolic sine (hyperbolic arc sine) of a value in radians.
$acosh Returns the inverse hyperbolic cosine (hyperbolic arc cosine) of a value in radians.
$atanh Returns the inverse hyperbolic tangent (hyperbolic arc tangent) of a value in radians.
$sinh Returns the hyperbolic sine of a value that is measured in radians.
$cosh Returns the hyperbolic cosine of a value that is measured in radians.
$tanh Returns the hyperbolic tangent of a value that is measured in radians.
$degreesToRadians Converts a value from degrees to radians.
$radiansToDegrees Converts a value from radians to degrees.
Accumulators ($group, $bucket, $bucketAuto, $setWindowFields)
Aggregation accumulator operators:
• Maintain their state as documents progress through the aggregation pipeline.
• Return totals, maxima, minima, and other values.
• Can be used in these aggregation pipeline stages:
$accumulator Returns the result of a user-defined accumulator function.
$addToSet Returns an array of unique expression values for each group. Order of the array elements is undefined.
$avg Returns an average of numerical values. Ignores non-numeric values.
Returns the bottom element within a group according to the specified sort order.
$bottom New in version 5.2.
Available in the $group and $setWindowFields stages.
Returns an aggregation of the bottom n fields within a group, according to the specified sort order.
$bottomN New in version 5.2.
Available in the $group and $setWindowFields stages.
Returns the number of documents in a group.
Distinct from the $count pipeline stage.
$first Returns the result of an expression for the first document in a group.
$firstN Returns an aggregation of the first n elements within a group. Only meaningful when documents are in a defined order. Distinct from the $firstN array operator.
$last Returns the result of an expression for the last document in a group.
$lastN Returns an aggregation of the last n elements within a group. Only meaningful when documents are in a defined order. Distinct from the $lastN array operator.
$max Returns the highest expression value for each group.
Returns an aggregation of the n maximum valued elements in a group. Distinct from the $maxN array operator.
$maxN New in version 5.2.
Available in $group, $setWindowFields and as an expression.
Returns an approximation of the median, the 50th percentile, as a scalar value.
New in version 7.0.
This operator is available as an accumulator in these stages:
• $group
• $setWindowFields
It is also available as an aggregation expression.
$mergeObjects Returns a document created by combining the input documents for each group.
$min Returns the lowest expression value for each group.
Returns an aggregation of the n minimum valued elements in a group. Distinct from the $minN array operator.
$minN New in version 5.2.
Available in $group, $setWindowFields and as an expression.
Returns an array of scalar values that correspond to specified percentile values.
New in version 7.0.
This operator is available as an accumulator in these stages:
• $group
• $setWindowFields
It is also available as an aggregation expression.
$push Returns an array of expression values for documents in each group.
$stdDevPop Returns the population standard deviation of the input values.
$stdDevSamp Returns the sample standard deviation of the input values.
$sum Returns a sum of numerical values. Ignores non-numeric values.
Returns the top element within a group according to the specified sort order.
$top New in version 5.2.
Available in the $group and $setWindowFields stages.
Returns an aggregation of the top n fields within a group, according to the specified sort order.
$topN New in version 5.2.
Available in the $group and $setWindowFields stages.
Accumulators (in Other Stages)
Some operators that are available as accumulators for the $group stage are also available for use in other stages but not as accumulators. When used in these other stages, these operators do not
maintain their state and can take as input either a single argument or multiple arguments. For details, refer to the specific operator page.
The following accumulator operators are also available in the $project, $addFields, $set, and, starting in MongoDB 5.0, the $setWindowFields stages.
$avg Returns an average of the specified expression or list of expressions for each document. Ignores non-numeric values.
Returns the result of an
$first expression
for the first document in a group.
Returns the result of an
$last expression
for the last document in a group.
$max Returns the maximum of the specified expression or list of expressions for each document
Returns an approximation of the median, the 50th percentile, as a scalar value.
New in version 7.0.
This operator is available as an accumulator in these stages:
• $group
• $setWindowFields
It is also available as an aggregation expression.
$min Returns the minimum of the specified expression or list of expressions for each document
Returns an array of scalar values that correspond to specified percentile values.
New in version 7.0.
This operator is available as an accumulator in these stages:
• $group
• $setWindowFields
It is also available as an aggregation expression.
$stdDevPop Returns the population standard deviation of the input values.
$stdDevSamp Returns the sample standard deviation of the input values.
$sum Returns a sum of numerical values. Ignores non-numeric values.
Variable Expression Operators
Defines variables for use within the scope of a subexpression and returns the result of the subexpression. Accepts named parameters.
Accepts any number of argument expressions.
Window Operators
Window operators return values from a defined span of documents from a collection, known as a window. A window is defined in the $setWindowFields stage, available starting in MongoDB 5.0.
The following window operators are available in the $setWindowFields stage.
$addToSet Returns an array of all unique values that results from applying an expression to each document.
$avg Returns the average for the specified expression. Ignores non-numeric values.
Returns the bottom element within a group according to the specified sort order.
$bottom New in version 5.2.
Available in the $group and $setWindowFields stages.
Returns an aggregation of the bottom n fields within a group, according to the specified sort order.
$bottomN New in version 5.2.
Available in the $group and $setWindowFields stages.
Returns the number of documents in the group or window.
$count Distinct from the $count pipeline stage.
New in version 5.0.
Returns the population covariance of two numeric expressions.
New in version 5.0.
Returns the sample covariance of two numeric expressions.
New in version 5.0.
Returns the document position (known as the rank) relative to other documents in the $setWindowFields stage partition. There are no gaps in the ranks. Ties receive the same rank.
New in version 5.0.
Returns the average rate of change within the specified window.
New in version 5.0.
Returns the position of a document (known as the document number) in the $setWindowFields stage partition. Ties result in different adjacent document numbers.
New in version 5.0.
Returns the exponential moving average for the numeric expression.
New in version 5.0.
Returns the approximation of the area under a curve.
New in version 5.0.
Last observation carried forward. Sets values for null and missing fields in a window to the last non-null value for the field.
$locf Available in the $setWindowFields stage.
New in version 5.2.
$max Returns the maximum value that results from applying an expression to each document.
$min Returns the minimum value that results from applying an expression to each document.
Returns an aggregation of the n minimum valued elements in a group. Distinct from the $minN array operator.
$minN New in version 5.2.
Available in $group, $setWindowFields and as an expression.
$push Returns an array of values that result from applying an expression to each document.
Returns the document position (known as the rank) relative to other documents in the $setWindowFields stage partition.
New in version 5.0.
Returns the value from an expression applied to a document in a specified position relative to the current document in the $setWindowFields stage partition.
New in version 5.0.
$stdDevPop Returns the population standard deviation that results from applying a numeric expression to each document.
$stdDevSamp Returns the sample standard deviation that results from applying a numeric expression to each document.
$sum Returns the sum that results from applying a numeric expression to each document.
Returns the top element within a group according to the specified sort order.
$top New in version 5.2.
Available in the $group and $setWindowFields stages.
Returns an aggregation of the top n fields within a group, according to the specified sort order.
$topN New in version 5.2.
Available in the $group and $setWindowFields stages.
For the pipeline stages, see Aggregation Stages. | {"url":"https://www.mongodb.com/docs/manual/reference/operator/aggregation/","timestamp":"2024-11-05T05:59:09Z","content_type":"text/html","content_length":"645147","record_id":"<urn:uuid:fad19089-f909-42eb-ac7f-544337c1682a>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00586.warc.gz"} |
The Financial Metrics in Measuring a Well-Balanced Inventory – Part 1
Most small businesses use an ERP system from companies like SAP, NetSuite and Microsoft. But they fall short of providing inventory optimization and rationalization. Achieving a well-balanced,
efficient inventory is no small task as supply chain complexity is ever increasing. How do you get there? How do you know when you have arrived? Understanding and using some basic financial and
accounting methods with an advanced inventory optimization solution will help get you there. It will also help keep you there.
Inventory planning uses a wide range of variables and metrics, which generally include:
· Storage efficiency (Inventory turnover rate, Inventory Levels, Space Utilization)
· Process effectiveness (Item Count Accuracy, Days of Supply, Lead time, Obsolescence and Deterioration)
· Other analyses like Pilferage and spoilage, Quality-Percentage Defect, etc.
· Customer satisfaction level - Return rate, Cancellation rate, etc.
· Financial perspective metrics - Insurance, Gross margin Return on Investment (GMROI), Average Cost per Order, Lost Sales Analysis, etc.
The cost of carrying excess and obsolete stock, as well as not having sufficient saleable inventory to meet demand is enormously high. It is common to find excess and obsolete stock representing 30%
- 60% of inventory and to find that 5% - 40% of the time, customer demand cannot be met. The latter often results in expediting vendor orders at a premium cost that cannot be passed through (based on
Valogix’ research and experience). On a $10M inventory that means potentially $4M - $6M is not being spent properly, a high price to pay for any size company.
According to industry research, companies that forecast, plan, and optimize more accurately generally have 15% less inventory, 17% better "perfect order" ratings, and 35% shorter cash-to-cash cycle
times. While there are clearly many factors that affect bottom-line financial performance, most research shows that companies that do a superior job of fulfilling customers' needs This is evidenced
by the perfect order, which tend to have higher earnings per share (EPS), better return on assets (ROA), and higher profit margins.
So, if you agree that your inventory is a valuable asset, what tools and processes do to use to manage it? Are you doing everything you can to insure the best results and best return on your
inventory investment? Are you using advanced inventory planning and optimization software?
Inventory that is moving through a manufacturing process or that is being picked, packed, etc. in a warehouse is potentially earning money.
Here are some of the many metrics to gauge your inventory performance:
The Carrying Cost of Inventory measures how much it costs to store inventory over a given period of time. Use the following formula when calculating carrying cost of inventory.
Inventory Carrying Rate x Average Inventory Value
Every piece of inventory that you purchase and store in your inventory has some sort of cost associated with it, such as labor, risk/insurance, storage, and freight. This metric is used to figure out
how much profit can be made on your current inventory, and it may also be used to help your suppliers map out their production cycles.
Two basic measurements for managing inventory are on-hand accuracy and velocity/demand (turnover). Location accuracy is defined as: the part number and actual physical count in a location match the
data stated in the computer. The best way to develop and maintain location accuracy is a cycle count program.
Inventory Turns (Inventory Turnover):
The number of times that your inventory cycles or turns over per year. It is one of the most commonly used Supply Chain Metrics.
A frequently used method is to divide the Annual Cost of Sales by the Average Inventory Level.
or …
Inventory Turns can be a moving number.
Example: Rolling 12 Month Cost of Sales = $16,000,000. Current Inventory = $4,000,000
$16,000,000 / $4,000,000 = 4 Inventory Turns
A. Inventory Carrying Rate:
This can best be explained by the example below....
1. Add up your annual Inventory Costs:
$800,000 = Storage
$400,000 = Handling
$600,000 = Obsolescence
$800,000 = Damage
$600,000 = Administrative
$200,000 = Loss (pilferage etc.)
$3,400,000 Total
2. Divide the Inventory Costs by the Average Inventory Value:
$3,400,000 / $34,000k = 10%
3. Then, Add up your costs:
· 9% = Opportunity Cost of Capital (the return you could reasonably expect if you used the money elsewhere)
· 4% = Insurance
· 6% = Taxes
19% Total
4. Add your percentages: 10% (inventory costs) + 19% (other costs above) = 29%
Your Inventory Carrying Rate = 29%
B. Fill Rate definitions
Fill rates and calculations can vary greatly. In the broadest sense, Fill Rate calculates the service level between 2 parties. It is usually a measure of shipping performance expressed as a
percentage of the total order.
Sample Fill Rate Metrics:
· Line Count Fill Rate:
The amount of order lines shipped on the initial shipment versus the amount of lines ordered. This measure may or may not take into consideration the requested delivery date. Example ABC Company
orders 10 products (one order line each) on its Purchase Order #1234. The manufacturer ships out seven-line items on March 1 and the remaining three items on March 10. The Fill Rate for this Purchase
Order is 70%. It is calculated once the initial shipment takes place.
Calculation: Number of Order Lines Shipped on the Initial Order* / Total Number of Order Lines Ordered (7/10 = 70%)
· SKU Fill Rate: The number of SKU's (Stock Keeping Units) ordered and shipped is taken into consideration. Above, we consider each Order Line to have an equal value (1 ). Here, we count the SKU's
per Order Line.
Example: If on Line 1, the order was for 30 SKUs of product "AB" and on line 2, they ordered 10 SKUs of item "AC". If Line 1 ships on April 1 and line 2 on April 20, then the SKU Fill Rate is 75%.
Calculation: Number of SKUs Shipped on the Initial Shipment / Total Number of SKUs Ordered (30/40 = 75%).
· On-Time Shipping Performance is a calculation of the number of Order Lines shipped on or before the Requested Ship Date versus the total number of Order Lines. Throughout the following text, I
refer to "shipped" on-time. But if actual "delivery" data is available, it may be substituted and compared to the Requested Delivery Date.
*On Time: Shipped on or before the requested ship date (except if the receiving party does not accept early shipments).
Inventory Months of Supply calculation:
· Inventory On-Hand / Avg Monthly Usage
· (the Avg Monthly Usage is typically the yearly forecast divided by12)
You can spend as much as you like or can afford or leverage on credit or loans for your inventory. What does matter is how you make that inventory work for you. In other words, what is your return on
that investment now and in the future. Work to balance your inventory using today’s inventory planning and optimization solutions like those offered by Valogix. See next month’s Blog, Part 2 of
inventory financial metrics. | {"url":"https://info.valogix.com/blog/the-financial-metrics-in-measuring-a-well-balanced-inventory-part-1","timestamp":"2024-11-10T23:33:04Z","content_type":"text/html","content_length":"53749","record_id":"<urn:uuid:ef411011-ffd9-4c32-9f18-547c9cb90907>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00169.warc.gz"} |
Sir Francis Galton (1822-1911) was an English statistician who invented a machine called a quincunx in 1873 to demonstrate the links between some probability distributions and the normal
distribution. Some other names for the quincunx include the Galton Board, Bean Machine and Plinko.
The University of Colorado at Boulder has developed an online simulation of a quincunx machine that is fun to play with. You can have a play yourself at plinko. If you want to change the probability
that the balls go to either side, go to the lab setting and see the effects of changing the numbers of rows and changing the probability of balls landing on the right or left. A few screen shots from
this simulation are shown below to give you the idea.
Equal Probability of Going to the Right or Left
This is a screenshot of the standard simulation on the University of Colorado at Boulder's PhET Interactive Simulations site. The balls have an equal probability of going to the left or the right of
each pin. Can you see how the final landing pattern is quite symmetric around the middle pin?
\(75\%\) Probability of Going to the Right and \(25\%\) Probability of Going to the Left
This is a screenshot from the lab section of Plinko on the University of Colorado at Boulder's PhET Interactive Simulations site. The balls have a \(75\%\) probability of going to the right and a \
(25\%\) probability of going to the left of each pin. The frequency distribution histogram has been shifted to the right.
\(25\%\) Probability of Going to the Right and \(75\%\) Probability of Going to the Left
This is a screenshot from the lab section of Plinko on the University of Colorado at Boulder's PhET Interactive Simulations site. The balls have a \(25\%\) probability of going to the right and a \
(75\%\) probability of going to the left of each pin. The frequency distribution histogram has been shifted to the left.
This simulation is only one of a number of different quincunx simulations you can find on-line. Why don't you find one of them for yourself and have a play? See what distributions the different
settings will give you. When you have had enough fun playing with quincunx, check out the article explaining quincunx to see why these different distributions arise. Have fun!
This chapter series is on Data and is suitable for Year 10 or higher students, topics include
• Accuracy and Precision
• Calculating Means From Frequency Tables
• Correlation
• Cumulative Tables and Graphs
• Discrete and Continuous Data
• Finding the Mean
• Finding the Median
• FindingtheMode
• Formulas for Standard Deviation
• Grouped Frequency Distribution
• Normal Distribution
• Outliers
• Quartiles
• Quincunx
• Quincunx Explained
• Range (Statistics)
• Skewed Data
• Standard Deviation and Variance
• Standard Normal Table
• Univariate and Bivariate Data
• What is Data
Year 10 or higher students, some chapters suitable for students in Year 8 or higher
Learning Objectives
Learn about topics related to "Data"
Author: Subject Coach
Added on: 28th Sep 2018
You must be logged in as Student to ask a Question. | {"url":"https://subjectcoach.com/tutorials/math/topic/data/chapter/quincunx","timestamp":"2024-11-11T16:36:27Z","content_type":"text/html","content_length":"117104","record_id":"<urn:uuid:232e0ec7-5ec2-4459-ba02-61c71fd9d7c2>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00501.warc.gz"} |
Theorem Proving in Propositional Logic
People use logic every day of their lives. At the moment of writing it is not raining but it looks very much as though it will be within the hour and I deduce that if this is the case (I have to go
out shortly) then I will get cold and wet. Propositional logic is a formal system for performing and studying this kind of reasoning. Part of the system is the language of well formed formulae
(Wffs). Being a formal language, this has a precisely defined grammar.
<Wff> ::= <Wff> -> <Disjunction> | <Disjunction>
<Disjunction> ::= <Disjunction> or <Conjunction> |
<Conjunction> ::= <Conjunction> and <Literal> | <Literal>
<Literal> ::= not <Literal> | <Variable> | ( <Wff> )
NB. often also allow ‘&’, ‘.’, or ‘∧’ for ‘and’, ‘~’ or ‘¬’ for ‘not’, and ‘+’ or ‘∨’ for ‘or’.
Grammar for Well Formed Formulae (Wff).
The propositional variables, and their negations (~, ¬, not), allow basic propositions or facts to be stated about the world or a world. For example:
raining, not raining, cold, wet
The remainder of the grammar allows a form of expression to be written including the logical operators implies (->), `and' (&) and `or'. This enables rules about the world to be written down, for
raining -> cold & wet
The operators can be interpreted in a way to model what humans do when reasoning informally but logically. For example, we know that if the proposition p holds, and if the rule `p implies q' holds,
then q holds. We say that q logically follows from p and from p implies q. This example of reasoning is known as modus ponens. An application is that `cold and wet' logically follows from raining and
from raining implies cold and wet.
There is a demonstration of the manipulation of propositional expressions [here...].
Propositional logic does not "know" if it is raining or not, whether `raining' is true or false. Truth must be given by us, from outside the system. When truth or falsity is assigned to propositional
variables the truth of a Wff can be evaluated. Truth tables can be used to define the meaning of the operators on truth values.
p not p
false true
true false
p q p and q
false false false
false true false
true false false
true true true
p q p or q
false false false
false true true
true false true
true true true
p q p -> q (sometimes written p=>q)
false false true
false true true
true false false
true true true
Operator Truth Tables.
Note that the table for p->q is the same as that for ~p or q. In other words p->q has the same meaning as ~p or q.
Propositional logic is used in Computer Science in circuit design. It is a subset of a more powerful system, predicate logic, which is used in program verification and in artificial intelligence.
Predicate logic has given rise to the field of logic programming and to the programming language Prolog. A study of propositional logic is basic to a study of these fields.
This chapter shows how the manipulation of Wffs and how reasoning in propositional logic can be formalised and carried out by computer programs. It forms a case study on some uses of trees, tree
traversals and recursion. It is also a good place to note that logical deduction, this particular form of calculation in a formal system, models a very small part of what makes human beings
A recursive-descent parser for well-formed formulae can be based on the formal grammar. The grammar and the resulting parser are very similar to those for arithmetic expressions.
Wff: p and (p->q) -> q
. .
. .
. .
& q
. .
. .
. .
p ->
. .
. .
p q
An Example Parse Tree.
Proof (Gentzen)
We say that a Wff v logically follows from a Wff w, `w=>v', if and only if (iff) v is true whenever w is true. Think of w as being the antecedant(s) or condition(s) for v. Note that v may be true if
w is false. We say that w is a tautology, or is always true, iff it has no conditions: `=>w'. We say that w is a contradiction, or is always false, if ~w is a tautology `=>~w', also `w=>'. If w is
neither a tautology nor a contradiction then it is said to be satisfiable. As examples, p or ~p is a tautology, p&~p is a contradiction and p or q is satisfiable.
Logically-following is a property of some pairs of Wffs. Proof is a process. It is obviously a desirable property of a formal system that logically-following can be proven to hold, whenever it does
hold, and this is the case for propositional logic.
For some simple Wffs the relationship `logically follows' is obvious:
p, q, r, ..., x, y, z, ... are propositional variables.
(a) p=>p p logically follows from p
(b) p&q => p strengthen LHS
(c) p => p or q weaken RHS
(d) p&q&r&... => x or y or z or ...
iff the LHS and the RHS have a
common propositional variable.
Logically Follows, Base Cases.
Clearly, p logically follows from p. If the left hand side is strengthened to p&q then p still logically follows from it. If the right hand side is weakened to p or q then it still logically follows
from p. If the LHS is strengthened to p&q and the RHS is weakened to p or x then the RHS still logically follows from the LHS. If the LHS is a conjunction of variables and the RHS is a disjunction of
variables, the RHS logically follows from the LHS if they share a common variable. If not, it is possible to make all the variables in the LHS true and all the variable in the RHS false demonstrating
that the RHS does not logically follow from the LHS.
In general we wish to know if an arbitrary Wff R logically follows from an arbitrary Wff L: `?L=>R?'. Arbitrary Wffs can contain the operators ~, &, or, ->. First, consider p or q on the LHS. R
logically follows from p or q iff R is true whenever p or q is true. Now, p or q is true whenever p is true and whenever q is true. So R logically follows from p or q iff R is true whenever p is true
and whenever q is true. (Note that p, q can be arbitrary Wffs.) This gives rule 1 which removes `or's from the LHS.
Special case:
p or q => R
iff (i) p=>R and (ii) q=>R
General case:
(p or q)&rest => R
iff (i) p&rest => R
and (ii) q&rest => R
Rule 1.
This rule seems odd in that an `or' seems to be replaced by an `and' but it is doing the right thing: The word `and' can be used in two distinct ways - as an operator in a Wff and also to group two
sub-steps to carry out; rule 1 is doing the latter. Note that two problems must now be solved but they are simpler than the original problem.
If the LHS contains an implications, p->q, this can be replaced by ~p or q and dealt with by rules 1 and 3:
Special case:
p->q => R
iff ~p or q -> R
General case:
(p->q)&rest => R
iff (~p or q)&rest => R see rules 1 and 3
Rule 2.
Negation (~, not) behaves much like arithmetic minus and there is even a kind of cancellation rule. R logically follows from ~p iff R is true whenever ~p is true, ie. whenever p is false. This is
equivalent to demanding that p or R be a tautology:
Special case:
~p => R
iff => p or R
General case:
~p&rest => R
iff rest => p or R
Rule 3.
The previous rules apply to the LHS; similar rules apply to the RHS. p&q logically follows from L iff p&q is true whenever L is true, ie. iff p is true whenever L is true and q is true whenever L is
Special case:
L => p & q
iff (i) L => p
and (ii) L => q
General case:
L => (p & q) or rest
iff (i) L => p or rest
and (ii) L => q or rest
Rule 4.
An implication p->q on the RHS can be replaced by ~p or q and dealt with by rule 6:
Special case:
L => p->q
iff L => ~p or q
General case:
L => (p->q) or rest
iff L => ~p or q or rest see rule 6
Rule 5.
A negation, ~p, logically follows from L iff ~p is true, ie. iff p is false, whenever L is true. This is the case iff L&p is a contradiction:
Special case:
L => ~p
iff L&p =>
General case:
L => ~p or rest
iff L&p => rest
Rule 6.
This means that ~p can be removed from either side by what amounts to "adding" p to both sides.
Finally, repeated application of rules one to six makes the Wffs on the LHS and RHS simpler and simpler. Eventually only the base case remains:
L => R, L conjunction of variables
R disjunction of variables
iff L intersection R ~= {}
Rule 7.
The rules given above were first discovered by Gentzen.
The rules given in the previous section are easily implemented by a routine `Prove' that manipulates the parse trees of Wffs.
An auxiliary routine `FindOpr' is needed to locate a top-most, "troublesome" operator in a tree of a Wff: {or, ->, ~} on the LHS and {&, ->, ~} on the RHS of a proof. There must be no other
troublesome operator on the path from the root to the operator although there may be other troublesome operators in other parts of the tree. This routine performs a depth-first search of the
parse-tree of a Wff for such an operator. If successful it returns pointers to the subtree having the operator as root and a tree for the "rest" of the Wff.
The proof routine proper, applies the rules previously defined. Rules one to six require one or two recursive calls to prove simpler problems. Rule seven, the base case, requires traversals of two
parse trees, now in simple form, to see if they share any propositional variable.
The routine also contains statements to print out the steps of the proof and examples are given below.
A Wff w is a tautology if it follows from nothing, =>w. W is a contradiction if ~w is a tautology. Failing both of these, w is satisfiable. The simplest satisfiable Wff is a single variable:
Expression is: p
1: => p fail % not a tautology, try ~p
1: => not p
1.1: p => fail % not a contradiction so ...
Expression is satisfiable
p or not p is a tautology:
Expression is: (p or not p)
1: => (p or not p)
1.1: p => p succeed % LHS and RHS intersect
Expression is a tautology
p and not p is a contradiction:
Expression is: (p and not p)
1: => (p and not p)
1.1: => p fail
1.2: => not p
1.2.1: p => fail % not a tautology ...
% ... try not(p and not p)
1: => not (p and not p)
1.1: (p and not p) =>
1.1.1: p => p succeed
Expression is a contradiction
Modus-ponens is a basic law of logic. q logically follows from (i) p and (ii)p->q:
Expression is: ((p and (p->q))->q)
1: => ((p and (p->q))->q)
1.1: => ( not (p and (p->q)) or q)
1.1.1: (p and (p->q)) => q
1.1.1.1: (( not p or q) and p) => q % proof branches now
1.1.1.1.1: ( not p and p) => q
1.1.1.1.1.1: p => (p or q) succeed % LHS & RHS intersect
1.1.1.1.2: (q and p) => q succeed % ditto
Expression is a tautology
• Gerhard Gentzen originated this type of formal system for logic; see (p140) of M. Wand, Induction, Recursion and Programming, North-Holland 1980.
• See the [Prolog interpreter] and Prolog notes for the mechanisation of Predicate Logic (as opposed to propositional logic).
1. For each Wff below, determine if it is a tautology, is a contradiction or is satisfiable:
□ (p->q)->(q->p)
□ (p->q)->(~q->~p)
□ (p->q)&(q->r)->(p->r)
2. Project: Write a program to print the truth-table of a Wff: Collect the distinct variables into a list. Write a routine to make all possible assignments of truth values to the variables. Write a
routine to evaluate the Wff given such an assignment.
3. Project: Implement a program to convert an arbitrary Wff into "two-level" Conjunctive-Normal-Form (CNF): a disjunction of conjunctions of literals where a literal is a variable or the negation of
a variable.
□ eg. p&(p->q)->q
□ = ~(p&(~p or q))or q
□ = ~p or ~(~p or q)or q
□ = ~p or (p & ~q) or q
Use the rules:
□ p->q = ~p or q
□ p&(q or r) = p&q or p&r
□ ~(p&q) = ~p or ~q
□ ~(p or q) = ~p & ~q | {"url":"https://allisons.org/ll/Logic/Propositional/","timestamp":"2024-11-02T11:51:52Z","content_type":"text/html","content_length":"16911","record_id":"<urn:uuid:5122f9b4-ff3c-4fa2-aec7-899b01159fae>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00326.warc.gz"} |
What are Bollinger Bands? - How to trade with Bollinger Bands - Phemex Academy
One of the most popular indicators used across all markets are Bollinger band (BB) indicators, which were developed by John Bollinger, a well-known trader in traditional markets.
The Bollinger band consists of three lines, one of which most traders are already quite familiar with — the20-period simple moving average (SMA). The exterior lines that envelope this indicator are
the standard deviation bands, which by default represent two standard deviations above and below the 20 period SMA.
What Are Bollinger Bands?
Bollinger bands may seem quite complex, in essence, they are trading tools used for the technical analysis of investment assets. They are utilized by traders to make an informed prediction on what
the price might do next — something which can help determine their trade. Bollinger bands focus on the standard deviation of an asset’s value, as well as its moving average value.
The Bollinger band formula is as follows:
• Simple moving average (SMA): The bands’ central component is the moving average value of the asset over a pre-determined period (usually 20 periods).
• Standard deviation bands: The two lines that border either side of this central line are the measurements of the asset’s standard deviation in price — one being the lowest and the other being the
highest. These two bands work as the Bollinger band’s parameters, with the distance between them being known as the Bollinger band width.
Bollinger bands work best when the middle band, or SMA, reflects the intermediate term trend, as in this way the asset’s trend is combined with the asset’s relative price information.
What Is The Bollinger Bands Formula?
Without having a thorough grasp of the core components of BB, it is impossible to know how to read Bollinger bands, or look at Bollinger band trading, or understand how to use Bollinger band trading
strategies. Thus, to ensure a full understanding of the Bollinger band’s formula, a definition for each of the components can be found below.
1. Simple Moving Average
The very core and central component of a Bollinger band is the simple moving average (SMA).
This moving line is a calculation of the average trading price of an asset over a period of time — usually 20 periods. This means that the closing price of the asset is taken over the last 20 periods
(this can be days, hours, minutes, or others, depending on the trading period being looked at) and then divided by the total number of periods — in this case 20.
A simple moving average (SMA) reflected in a Phemex BTC/USD price graph.
2. Bollinger Bands Standard Deviation
The second part of a Bollinger band is the standard deviation (SD) — two bands depicting the typical price deviation from the mean value of an asset. A low standard deviation indicates that the
values tend to be close to the mean of the asset’s value. A high standard deviation indicates that the values are spread out over a wider range.
Typically, the Bollinger band standard deviation of an asset’s value will fall within two standard deviations from the mean value of the asset. To be precise, according to the Empirical Rule, 68% of
the values will fall within one standard deviation while 95% will fall within two standard deviations.
Bollinger band settings mean that a trader can set the two outer bands to two standard deviations either side, or to another number — for the reason above, however, most traders will set them to two
standard deviations. The price will move up and down between these two standard deviations (and thus between the two outer bands of the Bollinger band) while crossing over the central line of the
band — the SMA.
When focusing on the candle chart in this Phemex BTC/USD graph with Bollinger bands (BB), it can be seen that 95% of the price range (shown by candles) sits within the two Bollinger bands standard
Bollinger Bands Calculation
In its default form, Bollinger bands standard deviation are set to 2 SD, but a trader can increase or decrease this by setting them to capture more or less of the target values. Traders can also
change the moving average settings if they wish to, although the standard is to set it to 20 periods.
How Do Bollinger Bands Work?
It is no good knowing what components a Bollinger band has if a trader does not know how to read Bollinger bands nor understands how this indicator works. Reading Bollinger bands is simple:
• If the SMA is beginning to incline upward, then the price may be turning bullish.
• If the SMA is beginning to take a downward turn, then the price may be turning bearish.
• If the price fluctuation heads toward the upper limit of the top Bollinger band standard deviation, then, since 95% of the price fluctuations will fall within 2 SD, the trader can assume that the
price is likely to fall.
• The same goes vice-versa. If the price fluctuation is about to hit the bottom Bollinger band standard deviation line, then the trader can assume that the price may soon rise.
This is why Bollinger bands are some of the most popular trading indicators — not only are they simple to read, but they are essentially two indicators in one.
Additionally, the standard deviation bands can indicate volatility and price breakouts:
• Wide Bollinger bands: In periods of high volatility, the two outer bands will widen to allow space to incorporate most of the price range. This widening Bollinger band is therefore indicative of
high price volatility.
• Narrow Bollinger bands: In periods of low volatility, the two outer bands will narrow to contain the price range. This narrowing Bollinger band is therefore indicative of low price volatility.
When a price goes into a bottleneck, with wide outer Bollinger bands followed by a narrowing of these Bollinger bands, this can be indicative of a price breakout.
Bollinger Bands Trading Strategy
Although Bollinger bands are used in many trading systems, traders most commonly use them to spot mean reversion opportunities and changes in volatility.
1 Use the Bollinger Bands to spot mean reversion trading
For mean reversion trading, Bollinger bands are best used when the price is exhibiting a vast ranging behavior. The idea behind this strategy is that, over time, the price tends to return to the
average price. With this strategy, the upper and lower bands act as the dynamic support and resistance levels. When the price ranges and moves either to the top band or the lower band, traders can
look for signs that the move is either becoming exhausted or being absorbed, meaning that it is likely to return to the average price. In simpler terms, if the price is approaching the lower SD band,
it’s likely to rise and return to the average price — meaning that traders may want to go short on their trade. If the price is approaching the higher SD band, it is likely to take a downturn and
return to the average price — meaning that traders may want to go long on their trade. This method is best applied when the bands are not widening, instead remaining fairly stable and thus showing
low volatility.
In ranging conditions, the price is accepted within a range of values, which can lead to great risk-to-reward opportunities as there is a bigger range between the highs and the lows. When the price
approaches an extreme of what these normal values are, as long as the trader operates under the assumption that the range is to continue and the conditions are to persist in the same way as they have
been, there is good opportunity for profit.
There are plenty of strategies that traders can experiment with using this method:
(A)-One might be to look for three drives into the highs or lows to look for trading opportunities back to the mean.
(B)-Another might be to play the successful crosses and retests of the mean with the intention of taking a move back to an extreme.
2 Using Bollinger Bands To Spot Changes in Volatility
One other popular use for Bollinger bands is to spot changes in volatility. Specifically, the type of compression that occurs before breakouts or continuation. In markets, compression usually leads
to expansion and then back to compression again — a pattern that if a trader knows how to exploit can lead to great returns. One thing that we can look for to potentially time or spot periods that
precede breakouts, are areas where the Bollinger bands begin to compress or bottleneck.
3 What is Bollinger Squeeze
Often, when a market is trending, the Bollinger Bands will remain expanded, and the consolidations that occur during the trends will lead back to compressions. Spotting the bottlenecking can allow a
trader to position themselves properly in the tightest area, offering a superior location to define risk and place a stop. This can be especially useful for breakout traders. Identifying this
behavior can also allow a trader to prepare themselves for a volatility squeeze if they are already in a position.
An example of a bullish breakout following a bottleneck in a Bollinger band.
Types of Bollinger Bands
In addition to standard BB indicators, there are also other types of Bollinger bands:
• Bollinger bands width (BBW) indicators: Put simply, the BBW indicator is the difference between the upper and lower SD bands of the BB indicator, divided by the middle SMA band. The BBW indicator
provides a simple way to visualize the consolidation of price (narrow Bollinger band standard deviation) before price movement or periods of high volatility (wide Bollinger band standard
□ The Bollinger band width calculation is: (upper BB – lower BB)/ middle BB
• Bollinger band percent B, or percentage bandwidth (%B):Introduced by the BB indicator creator, John Bollinger, in 2010 (nearly three decades after the BB indicator), the %B indicator considers
the relationship between price and the upper and lower SD Bollinger bands. In total there are six basic relationships that can be quantified — from %B above 1, which indicates that the price is
above the upper band, to %B below 0, which indicates that the price is below the upper band.
□ The %B calculation is: (current price – lower band)/(upper band – lower band)
All three types of Bollinger bands, as shown on Phemex’s free Bollinger bands chart.
The best thing to do with any of these indicators, as with all indicators, is to experiment with them. Try them across both the lower and the higher timeframes, with both default and modified
settings. This can be helpful to beginners, as generally speaking, the higher the timeframe you are in, the stronger the signals tend to be. Bollinger band charts are free on Phemex, meaning that
novice traders can easily learn how to trade Bollinger bands. Read on to find out how.
How to use the Bollinger Bands (BB) Indicator on Phemex?
The Bollinger band indicator is a free indicator available to all Phemex users under all trade pairs. To use the BB indicator on Phemex, open your favorite trading pair. For this demonstration, we’ll
use the most traded BTC/USDT pair. At the time of writing, BTC is trading at around $44,000 USD:
1. Click on “Indicators” at the top of the chart and a new window will pop up.
2. Input “Bollinger Bands,” “Bollinger Bands %B,” or “Bollinger Bands Width” in the search bar and you will find the indicator:
4. Click on the indicator and then press the “x” button to return to the chart. The Bollinger band will have been added to the chart. In this example, we are using the daily chart for the Bollinger
band indicator. If we switch to any other time on the chart, the indicator will adjust accordingly. The time frame can be selected on the top left of the chart.
5. You can now practice your trading analysis using the BB indicators.
Bollinger bands are an excellent trading tool that should be understood and used by any trader — whether novice or pro. BB indicators can be used to understand which way the market is going, analyze
volatility, and predict breakouts. However, as with all indicators, traders should not solely rely on one indicator, or even all three types of Bollinger band, to make their analysis. Traders should
do their own research and use the multiple free tools at their disposal on platforms such as Phemex to ensure they are in a suitable position before making a trade. Additionally, traders can use stop
orders to minimize risk and losses.
Check out our additional technical analyses to improve your Crypto Trading Skills
Phemex |
Break Through, Break Free | {"url":"https://phemex.com/academy/bollinger-bands","timestamp":"2024-11-02T21:59:02Z","content_type":"text/html","content_length":"85171","record_id":"<urn:uuid:b889506a-85a9-4520-8b09-c693359ae7f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00551.warc.gz"} |
Gravitation and Cosmology Research - PhD projects
We welcome applications to study for a PhD in our group. For details about the areas of research, look at our staff members' websites. If you have further questions, please contact the staff member
you are interested in working with.
Typically a small number of fully funded places in the School are available each year and decisions on these are made between January and April for an October start. Other start dates during the year
are possible, but PhD funding is only allocated once a year. You can apply here:
PhD funding: As is standard in the UK, your application to the University is for a place to study, which does not automatically come with funding. Our main source of PhD funding is EPSRC. It helps if
you make clear in your application that you want to be considered for an EPSRC studentship.
Chinese nationals may also be eligible for China Scholarship Council (CSC) funding (note: earlier deadline! Please get in touch with a potential supervisor and make sure you apply in time.)
We recommend reading our PhD application advice as it explains the process of how we select candidates for PhD places and funding, which may be different from how it works in other universities and
Do not send applications or CVs directly to us by email, use the online application form.
There are internal deadlines for your application to be eligible for EPSRC funding. The deadline for 2025 will be confirmed soon, but last year it was 24th January 2024. Your application should be
complete before any deadline. In particular, please include either your Master's dissertation (or equivalent) or an example of written project work. Please expect to be asked to provide additional
information to be considered for PhD funding.
We usually await the first deadline for PhD studentships before selecting some applicants for an online interview. We might ask applicants to prepare a short presentation as part of the interview.
After a successful interview applicants will be put forward for funding, which is then decided at the level of the School of Mathematical and Physical Sciences in a complicated process.
Below are some examples of projects we could supervise in 2025, organised by general research area. In general, finding a suitable project can be part of the discussions around a PhD interview. Given
how our PhD funding works, in a particular year we may or may not find funding for any of these projects.
Projects in Black Holes and Gravitational Waves
Black hole perturbation theory for gravitational-wave source modelling (Sam Dolan)
Orbiting black holes generate distinctive 'chirp' signals, which have been successfully detected at gravitational-wave observatories such as LIGO since 2015. A key aim of the forthcoming LISA mission
(a space-based interferometer) is to detect signals from supermassive black holes, such as the one at the centre of our galaxy.
In this project, we will aim to model the final ~100,000 orbital cycles of a stellar-mass black hole orbiting a supermassive black hole, by applying gravitational self-force theory. In particular, we
will extend recent work by Dolan, Kavanagh and Wardell (arXiv:2108.06344) to construct Lorenz-gauge metric perturbations of the Kerr black hole directly from mode sums, for the first time.
Constraints on models beyond the standard LCDM by using current and future cosmological data (Eleonora Di Valentino)
The Cosmic Microwave Background (CMB) temperature and polarization anisotropy measurements from the Planck mission have provided robust support for the ΛCDM model, which relies on the cosmological
constant (Λ) and cold dark matter—two components that remain largely mysterious. However, intriguing discrepancies, known as "cosmological tensions," have emerged, hinting that the standard ΛCDM
model may not fully describe our universe. The most prominent of these is the Hubble tension, which reflects a mismatch between the Hubble constant inferred from Planck data and direct measurements
by the SH0ES collaboration. Other tensions, like the clustering parameter discrepancy observed between Planck and the weak lensing surveys (e.g., DES and KiDS-1000), further challenge the standard
This project aims to explore beyond the standard ΛCDM framework, investigating alternative cosmological models that could reconcile these tensions. We will examine a range of possibilities, including
Dark Energy and Modified Gravity models, extended Dark Matter scenarios, alternative inflationary theories, and modifications to the neutrino sector. The analysis will combine current available data
from multiple sources, and we will assess how future CMB experiments—such as CMB-S4, DESI, PIXIE, PRISM, and PICO—could refine these constraints. Additionally, we will consider how these experiments
can be complemented by future low-redshift observations.
The project requires a foundational understanding of cosmology and we will be using and modifying widely adopted cosmological codes, such as CAMB/CLASS, and performing statistical analyses with Monte
Carlo Markov Chain (MCMC) packages like Cobaya and SimpleMC.
Projects in Quantum Field Theory in Curved Spacetime
Quantum fields on black-hole spacetimes (Elizabeth Winstanley)
Classically a black hole can only absorb and not emit particles. Black holes do however emit quantum Hawking radiation. This project is concerned with the behaviour of quantum fields on black hole
The construction and properties of quantum states on black hole backgrounds will be studied, including the computation of renormalized expectation values of observables such as the vacuum
polarisation or stress-energy tensor. A particular focus will be on black holes with either rotation or charge, where superradiance effects play an important role. Expectation values can then be used
to study the backreaction of the quantum field on the space-time geometry.
Projects in Quantum Gravity and Quantum Cosmology
Time and relational dynamics in models of quantum gravity (Steffen Gielen)
One of the main obstacles in unifying general relativity and quantum mechanics is known as the 'problem of time': in general relativity, unlike in usual quantum mechanics, statements about evolution
in a particular time coordinate have no invariant physical meaning. Physically meaningful statements need to be made in relational terms, as the change in a 'system' relative to the change in the
'clock'. In cosmology, the clock can for instance be a scalar field, or the volume of the universe.
In this project you will study relational formulations of dynamics in classical relativity and their realisations in quantum models of cosmology or quantum black holes. Depending on interest, some
specific questions can include the role of unitarity in quantum gravity (related to evolution and hence the definition of time), approaches using a standard of time based on dark energy (known as
unimodular gravity), or the clocks used by observers encountering singularities in black holes and cosmology. You could also work on the explicit construction of new models for material clocks in
approaches to quantum gravity such as group field theory. Such constructions can be used to compare quantisations based on different clocks, and potentially link these ambiguities to observational | {"url":"https://gravity-cosmology.sites.sheffield.ac.uk/phd-projects","timestamp":"2024-11-09T06:15:25Z","content_type":"text/html","content_length":"125088","record_id":"<urn:uuid:16d317ff-58ef-4a63-8ed6-957d513463da>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00031.warc.gz"} |
Assignment 01: planetary energy balance
Assignment 01: planetary energy balance#
For these exercises you’ll need some pen and paper and your notes from the class. I also recommend reading one of (or both) of these book chapters (they explain exactly the same thing but in
different styles, you can choose yours):
• John Marshall & R. Alan Plumb: Atmosphere, Ocean, and Climate Dynamics (Chapter 2)
• Andrew Dessler: Introduction to Modern Climate Change (Chapters 3 & 4)
In the exercises below I use mostly the notations from Marshall & Plumb.
# These are the modules you will need
import numpy as np
import matplotlib.pyplot as plt
sigma = 5.67e-8 # W / (m2 K4)
Q_Sun = 3.87e26 # W
dist_sun_earth = 150e9 # m
albedo_earth = 0.3
Ex 0: Compute the solar constant at the location of Earth, given \(Q = 3.87 \times 10^{26} W\) and \(r = 150 \times 10^{9} m\)
# your answer here, or on pen & paper
Ex 1: A planet in another solar system has a solar constant S = 2,000 W/m 2 , and the distance between the planet and the star is 100 million km.
• a) What is the total power output of the star? (Give your answer in watts.)
• b) What is the solar constant of a planet located 75 million km from the same star? (Give your answer in watts per square meter.)
# your answer here, or on pen & paper
Ex 2: Compute \(T_e\) and \(T_s\) of the Earth using the constants defined and computed above, assuming a one-layer planet opaque to longwave radiation but transparent to shortwave radiation.
# your answer here, or on pen & paper
Ex 3: Using the “leaky” atmosphere model, determine \(\epsilon\) so that \(T_S\) is equal to the observed surface temperature on Earth, about 15°C.
# your answer here, or on pen & paper
Ex 4: As we will discover later, one way to address global warming is to increase the reflectivity of the planet. To reduce the Earth’s temperature by 1 K, how much would we have to change the Earth
albedo? (assume a one-layer planet with an initial albedo of 0.3 and solar constant of 1367 W/m 2 ).
# your answer here, or on pen & paper
Ex 5: Either on your own of with the help of one of the books (Exercise 5 in Marshall & Plumb, or Textbook in Dessler), determine that when a very opaque atmosphere as N layers opaque to longwave
radiation, the equilibrium surface temperature is:
\[T_s = (N + 1)^{1/4} T_e\]
Now plot the surface temperature of earth as a function of the number of opaque layers in the atmosphere, with N in [1, 100]. | {"url":"https://fabienmaussion.info/climate_system/week_01/01_Assignment_Planetary_Energy_Balance.html","timestamp":"2024-11-12T12:01:40Z","content_type":"text/html","content_length":"27833","record_id":"<urn:uuid:7d1f1d1c-8021-4958-a22a-e4fb39e34896>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00865.warc.gz"} |
Angle measure
The angular dimension is used to indicate the angular width of a plane angle in mathematics and as a physical quantity . Depending on the application, different dimensions and their units are used.
The angle measure is also used on curved surfaces. Here you measure the angles in the tangential plane of the surface - see also spherical trigonometry and spherical astronomy .
Common angular dimensions
Linear subdivision of the full angle
Linear angular dimensions are characterized by the fact that they are retained when the angle is rotated, and when a rotation is divided into two partial rotations, the angular dimension for the
total rotation is equal to the sum of the angular dimensions of the partial rotations.
• The full angle is the smallest angle by which a ray rotates around its origin and returns to its original direction. It is a legal unit of measurement for which no unit symbol is specified. The
angle width is specified as a multiple or part by adding a factor as a numerical value to the word full angle . It is also colloquial and very common in mathematics in its implicit form, e.g. B.
means half a turn a turn by half a full angle. So you specify the number of desired revolutions, which may also be non-integer.
• The full angle is divided into 360 equal parts in degrees . Such a part is called a degree and is marked with the symbol °:
1 full angle = 360 °
• In radians , the dimension 2π is assigned to the full angle . The part of the full angle is called radians , unit symbol rad: ${\ displaystyle {\ frac {1} {2 \ pi}}}$
1 full angle = 2 rad${\ displaystyle \ pi}$
• In the geodetic angle measure , the full angle is divided into 400 equal parts. Such a part is called a gon and is identified by the symbol "gon":
1 full angle = 400 gon
• In the time measure (angle) , a full angle is divided into 24 hours. It is used in astronomy to indicate the hour angle and the right ascension :
1 full angle = 24 ^h
• In dashes , measurements are taken by dividing the full angle into a number of equal parts, which varies depending on the area of application:
1 full angle = 32¯ (nautical)
1 full angle = 6400 mil (military)
Non-linear subdivision of the full angle
Another measuring principle of the angle width is based on the ratio of height difference to length in the sense of a slope angle , the calculation is based on the tangent of the angle. This scale is
not linear ; H. If two turns are performed one behind the other, the angular width generally does not correspond to the sum of the angular widths of the individual turns. For right angles the slope
approaches infinity . The length can only be positive (it is only measured "to the front") and therefore the angle of inclination is only defined in the range of −90 ° <α <+ 90 °.
• Percent or per mille (for inclines, especially in transport )
~ 0.57 ° → 1%
1 ° → ~ 1.75%
15 ° → 26.79 ...%
45 ° → 100%
90 ° → ∞
${\ displaystyle \ varphi}$→ 100 %${\ displaystyle \ cdot \ tan (\ varphi)}$
Instead of a (flat) angle, you can of course generally specify this length ratio of two lines that are perpendicular to one another. This then always corresponds to the tangent of the angle in the
underlying right-angled triangle. In aviation this is how the glide ratio of an aircraft is given.
The development of angular dimensions
Full circle and right angle
In principle, we know two measuring standards for the angular width, both of which are derived from an intuitive reference system of front , back , right and left .
• The circle , which is closely related to the arithmetic principle of fractions through the concept of subdivision into circular sectors , such as is known as a “piece of cake” .
• The polygon that enables geometric access to the angle via the relationship between interior angle and central angle. In particular, the square should be mentioned here, where both form a right
angle .
Therefore, there are two excellent units of measurement for the angle, the full angle (full circle) and the right angle (quarter circle). Both of these concepts can be found in the earliest traces of
protoscientific methods of early advanced civilizations.
While the right angle today only serves as a measure to distinguish linguistically - and of course also computationally - "straight" from "oblique" angles and " acute " from " obtuse ", i.e. a test
criterion for assigning Boolean values ("yes", " no "), the full angle is the legal unit of measurement. Until around 1980, the right angle as a right with the unit symbol ∟ was also common in
Radians: 2 π
In the case of a regular hexagon , the corners of which lie on a circle, the equality of edge length and circumferential radius results in a division of the hexagon into six equilateral triangles ,
so that only angles of sixty degrees and their multiples can be found in it. With a radius of the circumference of 1, the circumference of the hexagon is 6. This value was assumed early on for the
circumference and is part of numerous empirical formulas that have come down to us in ancient sources.
In particular, however, the Chinese naturalists of the pre-Christian era set 3 canonically as the measure of half the circumference and developed a powerful calculus for angle measurements , and in
this respect they can be regarded as the inventors of the radian measure . The Zhōu Bì Suàn Jīng Chinese 周髀算经 , W.-G. Chou pei suan ching , whose roots go back to about 1200 BC. B.C. , formulates
the calculation of the angles of elementary triangles via the hexagon. These very promising approaches soon got lost in a complicated numerology and number mysticism , which replaced scientific
advancement with formalisms and devoted itself to a tenth of the circle ( celestial stems ) and a twelfth ( earth branches ) rather their arrangement and symbolism than the mathematical application.
Entrance into the modern mathematics is the radian with the exact value until the late 17th century over the differential calculus because formulas such that Euler's identity and the approximations
(for small angles) apply only when these functions with arguments (angles) in radians be used. This angular measure is thus the natural unit for the argument of the sine and cosine functions , just
as e is the natural basis of the exponential function - for the context see complex exponential function . ${\ displaystyle 2 \ pi}$ ${\ displaystyle \ sin ^ {\ prime} x = \ cos x}$ ${\ displaystyle
\ sin x \ approx \ tan x \ approx x}$
Degree measure: 360, 400; Hour measure: 24
We do not know whether it played a role in the development of the sexagesimal system that a regular hexagon with a circumference of six times the radius can easily be inscribed in a circle. But the
use of a sixty division as well as a twelve division for astrometric angle measurements can already be proven from Sumerian times . The latter is preserved in the zodiac ("zodiac").
The division of the full angle into 360 parts by the early Greek astronomers is documented. It is likely to be traced back to Babylonian tradition. Hypsicles of Alexandria used them in 170 BC. In the
Anaphorikos .
It is the astronomers who estimated this measure for the full circle, not only because of the various possibilities of division , but also in the context of the calendar calculation : On the one
hand, the number approaches the 365 days of the year , but in particular the calculations of the main positions of the moon can be made , i.e. its synodic period of just under 30 days and the lunar
year of 354 days (360 - 6, in connection with 360 + 6) to handle relatively casually.
The possibilities offered by this method of calculation were decisive for the Jewish scholars in calculating the new light - the basis of their calendar . They established the degree as a
comprehensive measuring principle for angles in astronomy , geodesy and geometry , for example on the astrolabe , sextant or dioptra . It was also kept in the western tradition from the 12th century
, where the degree gets its name. The method also includes the sexagesimal division of the gradus (“step” on the circle) into pars minuta prima “first diminished part” and pars minuta secunda
“further diminished part”, indicating the angle in degrees , angular minutes and angular seconds . The outdated tertie for “pars minuta tertia” (“third diminished part”) is also rarely found .
In addition, the chronological division of the day - and thus also the full circle - into 24 or 12 hours had been common since long before the new era (see 24-hour counting and 12-hour counting ). In
the calendar system, this is referred to as the Babylonian or Greek (starting at sunrise) or Italian hours (first in Jewish, then Islamic tradition, transitioning to sunset). The star catalog of
Hipparchus Nikaia (190-120 BC) is handed down in the Almagest of Ptolemy , its trigonometric table is - beyond that - created with a doubled, i.e. 48 division (7.5 ° interval). The same system of
division into twelve can also be found in the Chinese earth branches , which - presumably also based on Mesopotamian tradition - are about the same age as the Hipparchus catalog and were used both
for calculating time and for navigation. For measurements in a system of measurement that reflects the rotation of the earth , the 24 division of the circle is still common today as a measure of time
, because it enables direct time measurement from the angle of the hour . An origin cannot be made out, but as early as the 2nd century BC. BC reached - through the mediation of Indian astronomy - a
12-hour count China (realized in the branches of the earth).
The common root of these two circle divisions is shown in the terrestrial angle of longitude ( degree of longitude ), in which the 360-degree network is superimposed on the 24 time zones .
The notation of a decimal specification of the subdivision of a degree (with decimal places ) in decimal degrees did not appear until the end of the Middle Ages in Arabia.
Although geodesy is one of the branches of science that were causally involved in the development of the degree, it benefits least from the number 360. In the 1790s, metrification began in France ,
in the course of which the original meter was the ten millionth part of the meridian arc from North Pole was defined to the equator . In the Nouvelle Triangulation de la France , a new graticule was
developed that divided the circle accordingly into 400 units, the grade nouvelle ( grads ), so that 0.01 ^gr corresponded to one kilometer. This geodetic angle measure (in gons ) is still used today
in geodesy, for example for triangulation , although the definition of the meter no longer refers to the length of the earth's meridian.
Bearing: 32, 6400
If you double the right angle (the quarter circle) you get a semicircle, if you double it again you get a full circle. If you apply this train of thought to halving, you get an eighth of a circle,
then a sixteenth of a circle, and so on. In contrast to the previous systems of angle determination, this method is particularly suitable for reference systems in motion ( azimuthal coordinate system
), which represent the above-mentioned principle of four distinct directions in relation to the viewing or driving direction , and therefore use the right angle as a basic dimension.
The system was used in particular in nautical navigation for bearing the position and the course angle . This is divided into four main directions (“Ahead”, “Starboard”, “Aft”, “Port”), four
secondary directions (eighth), and the thirty-second of the full circle is given the unit of nautical line . Only in combination with the compass does this wind rose receive a fixed direction (mostly
north) and becomes a compass rose . In shipping today, however, bearings are usually measured in degrees with decimal minutes.
Another area of application in which the alignment is crucial regardless of the fixed network is sighting in artillery. Due to the high precision that is required - and a computational advantage in
the conversion of the sight division on the thread net at a distance from the targeted object - the full circle is divided into 6400 artillery lines ( Swiss Army : Artilleriepromille , north fixed).
See also
Individual evidence | {"url":"https://de.zxc.wiki/wiki/Winkelma%C3%9F","timestamp":"2024-11-03T03:59:49Z","content_type":"text/html","content_length":"52080","record_id":"<urn:uuid:5ec43201-6bb8-4790-9c91-656c50a5f934>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00205.warc.gz"} |
Question Set 43
Submitted by Atanu Chaudhuri on Thu, 13/10/2016 - 13:40
Mensuration for SSC CGL: 10 Questions with answers
Mensuration for SSC CGL Set 43: Solve 10 questions on mensuration in 15 minutes. Verify answers and learn to solve the questions quickly from solutions.
Answers and link to the solutions are at the end.
10 Mensuration questions for SSC CGL Set 43 - Answering time 15 mins
Problem 1.
An area of a square km of land had a rainfall of 2cm over some time. If 50% of the raindrops were collected in a pool with a 100m by 10m base, what would be increase in water level in the pool?
a. 1 km
b. 10 km
c. 1 m
d. 10 m
Problem 2.
Water is flowing at the rate of 3 km per hour through a pipe of circular cross-section of internal diameter 20 cm into a circular cistern of diameter 10 m and depth 2 m. In how much time would the
cistern be filled?
a. 2 hours 40 minutes
b. 1 hour
c. 1 hour 40 minutes
d. 1 hour 20 minutes
Problem 3.
If the perimeter of a circle, a square and an equilateral triangle are same and their areas are C, S and T respectively which of the following is true?
a. $C=S=T$
b. $S \lt C \lt T$
c. $C \gt S \gt T$
d. $C \lt S \lt T$
Problem 4.
The diameter of the front wheel of an engine is $2x$ cm and that of rear wheel is $2y$ cm. To cover the same distance, find the number of times the rear wheel will revolve when the front wheel
revolves $n$ times.
a. $\displaystyle\frac{nx}{y}$ times
b. $\displaystyle\frac{yn}{x}$ times
c. $\displaystyle\frac{n}{xy}$ times
d. $\displaystyle\frac{xy}{n}$ times
Problem 5.
What part of a ditch 48 m long, 16.5 m broad and 4 m deep can be filled by digging a cylindrical tunnel of diameter 4 m and length 56 m (use $\pi = \displaystyle\frac{22}{7}$)?
a. $\displaystyle\frac{1}{9}$
b. $\displaystyle\frac{2}{9}$
c. $\displaystyle\frac{8}{9}$
d. $\displaystyle\frac{7}{9}$
Problem 6.
The difference in areas of two squares drawn on two line segments of different lengths is 32 sq cm. Find the length of the greater line segment if one is longer than the other by 2 cm.
a. 7 cm
b. 16 cm
c. 11 cm
d. 9 cm
Problem 7.
The lengths of three medians of a triangle are 9 cm, 12 cm and 15 cm. The area (in cm$^2$) of the triangle is,
a. 48
b. 144
c. 72
d. 24
Problem 8.
The area of a triangle of side lengths 9 cm, 10 cm and 11 cm (in cm$^2$) is,
a. $30$
b. $30\sqrt{2}$
c. $60\sqrt{2}$
d. $60$
Problem 9.
The area of a circle is increased by 22 cm$^2$ by increasing its radius by 1 cm. The original radius of the circle is,
a. 3 cm
b. 5 cm
c. 9 cm
d. 7 cm
Problem 10.
The base of a right prism is a quadrilateral ABCD. Given that $AB=9$ cm, $BC=14$ cm, $CD=13$ cm, $DA=12$ cm and $\angle DAB=90^0$. If the volume of the prism be 2070 cm$^3$ then the area of the
lateral surface of the prism is,
a. 720 cm$^2$
b. 810 cm$^2$
c. 2070 cm$^2$
d. 1260 cm$^2$
Know how to visualize the 2D and 3D shapes and calculate the surfaces and volumes in the mensuration questions from the paired solutions at,
Solutions to mensuration for SSC CGL set 43.
Answers to the Mensuration questions for SSC CGL Set 43
Problem 1. Option d: 10m
Problem 2. Option c: 1 hour 40 minutes
Problem 3. Option c: $C \gt S \gt T$
Problem 4. Option a: $\displaystyle\frac{nx}{y}$ times
Problem 5. Option b: $\displaystyle\frac{2}{9}$
Problem 6. Option d: 9 cm
Problem 7. Option c: 72
Problem 8. Option b: $30\sqrt{2}$
Problem 9. Option a: 3 cm
Problem 10. Option a: 720 cm$^2$
Other related question set and solution set on SSC CGL mensuration
SSC CGL level Solution Set 87 on Mensuration 7
SSC CGL level Solution Set 88 on Mensuration 8
SSC CGL level Question Set 88 on Mensuration 8
SSC CGL level Question Set 87 on Mensuration 7
SSC CGL level Solution Set 86 on Mensuration 6
SSC CGL level Question Set 86 on Mensuration 6
SSC CGL level Solution Set 43 on Mensuration 5
SSC CGL level Solution Set 42 on Mensuration 4
SSC CGL level Question Set 42 on Mensuration 4
SSC CGL level Solution Set 41 on Mensuration 3
SSC CGL level Question Set 41 on Mensuration 3
SSC CGL level Solution Set 27 on Mensuration 2
SSC CGL level Question Set 27 on Mensuration 2
SSC CGL level Solution Set 26 on Mensuration 1 | {"url":"https://suresolv.com/ssc-cgl/ssc-cgl-level-question-set-43-mensuration-5","timestamp":"2024-11-04T05:58:23Z","content_type":"text/html","content_length":"35254","record_id":"<urn:uuid:c09f34c7-8556-457d-a3cc-084decb0fd96>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00081.warc.gz"} |
Q-Math Seminar
John Stewart Fabila Carrasco. (UC3M)
The discrete magnetic Laplacian: geometric and spectral preorders with applications
Wednesday the 10th of June, 2020, 11:00, Online Seminar
Important, this seminar will be held online. To access the online stream use the link below. Discrete geometric analysis is a hybrid field of several areas of mathematics including graph theory,
combinatorics, geometry, theory of discrete groups, and probability. A central object in this field is the spectrum of the discrete magnetic Laplacians (a discrete analogue of magnetic Laplacians on
manifolds) acting on weighted graphs and has applications in mathematics, physics and technology. The magnetic potential $\alpha$ acting on a weighted graph $(G,w)$ allows studying several families
of Laplacians at the same time (combinatorial, standard, normalised, signless, magnetic Laplacian, and more). We introduce a geometric and a spectral preorder relation on the class of weighted graphs
with a magnetic potential [3]. Some applications of these preorders are the following: 1. We found a simple geometric condition that guarantees the existence of spectral gaps of the discrete
Laplacian on periodic graphs. For proving this, we analyse the discrete magnetic Laplacian on the finite quotient and interpret the vector potential as a Floquet parameter [1, 2]. 2. We present a new
geometrical construction leading to an infinite collection of families of graphs, where all the elements in each family are (finite) isospectral (weighted) graphs for the magnetic Laplacian. The
parametrisation of the isospectral graphs in each family is given by a number theoretic notion: the different partitions of a natural number [4]. We conclude this talk with other applications (the
Cheeger constant, minor graphs, etc.) and some open questions. References: [1] J.S. Fabila-Carrasco and F. Lledó, Covering graphs, magnetic spectral gaps and applica- tions to polymers and
nanoribbons, Symmetry. 11 (2019), 1163. [2] J.S. Fabila-Carrasco, F. Lledó, and O. Post, Spectral gaps and discrete magnetic Lapla- cians, Linear Algebra Appl. 547 (2018), 183–216. [3] J.S.
Fabila-Carrasco, F. Lledó, and O. Post, Spectral preorder and perturbation of discrete weighted graphs, available at https://arxiv.org/abs/2005.08080. [4] J.S. Fabila Carrasco, F. Lledó and O. Post,
Isospectral magnetic graphs, preprint 2020.
Davide Lonigro (Università degli studi di Bari, Italy)
The Friedrichs-Lee Hamiltonian: singular coupling, renormalization, and spectral properties
Wednesday the 1st of April, 2020, 12:00, Online Seminar
Important, this seminar will be held online. To access the online stream use the link below.
In this talk we will provide an overview on the properties of the Friedrichs-Lee Hamiltonian. After showing that the model can describe the single-excitation interaction between a structured boson
field and a family of two-level systems, we will discuss its extension to a larger class of couplings via a domain change; this procedure can be interpreted as an operator-theoretical
renormalization. We will finally characterize its spectral properties by studying its spectral decomposition; in particular, we will briefly discuss the insurgence of bound states in the continuum
(BICs) for a Friedrichs-Lee model whose inner Hamiltonian has an absolutely continuous spectrum.
Video Recording
Antonio García (UC3M)
A link between discrete convolution systems and sampling via frame theory
Wednesday the 4th of March, 2020, 12:00, UC3M, Seminar Room 2.2D08
In this talk a regular sampling theory for a multiply generated unitary invariant subspace of a separable Hilbert space H is proposed. This subspace is associated to a unitary representation of a
countable discrete abelian group G on H. The samples are defined by means of a filtering process which generalizes the usual sampling settings.
Diego Martínez (UC3M & ICMAT)
Quasi-diagonality and finite-dimensional approximations
Wednesday the 19th of February, 2020, 12:00, UC3M, Seminar Room 2.2D08
Finite-dimensional approximations of (normally of infinite nature) objects is ubiquitous in mathematics. In this talk we will introduce the so-called quasi-diagonal operators. That is, given an
infinite-dimensional Hilbert space H, we say that an operator on H is quasi-diagonal if some of its corners behave approximately the same as the operator itself. Although informal so far, this notion
has several applications in vastly different areas, such as numeric analysis, group theory or K-theory. We shall name a few of these, highlighting some key constructions. We will end the talk
introducing Berg's technique, and how it can be generalized to residually finite groups.
Giuseppe Marmo (UC3M, Excellence Chair)
Heisenber-Weyl algebra, contact structures and dissipation
Wednesday the 5th of February, 2020, 12:00, UC3M, Seminar Room 2.2D08
The HW algebra defines a short exact sequence of Lie algebras,to allow for nonlinear transformations ,it will be 'made' into a Lie-module. The dual picture of this module defines a contact structure.
A contact structure will also appear when we consider a finite-level quantum system and take into account the Berry phase. Contact manifolds turn out to be the appropriate setting to describe some
dissipative systems.
Florio M. Ciaglia (Max-Planck-Institut, Leipzig)
From the Jordan product to Riemannian geometries
Friday the 13th of December, 2019, 13:00, ICMAT, Aula Gris I
The Jordan product on the self-adjoint part of a finite-dimensional C*-algebra A is shown to give rise to Riemannian metric tensors on suitable manifolds of states on the algebra. In particular, this
construction allows to look at the Fisher-Rao metric tensor on probability distributions, at the Fubini-Study metric tensor on pure quantum states, and at the Helstrom metric tensors on faithful
quantum states as different instances of the ''same conceptual entity''. If time allows, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction will be
José Polo (UCM)
Groupoids and inverse semigroups
Monday the 2nd of December, 2019, 15:30, UC3M, Seminar Room 2.2D08
We aim to introduce briefly the concepts of groupoid and inverse semigroup, motivating and showing the relation between them by means of some examples. We will then turn to the construction of
groupoid representations as well as their associated C*-algebras, illustrating the construction with a class of examples that includes Cuntz-Krieger algebras.
Olaf Post (Universität Trier)
Spectral gaps, discrete magnetic Laplacians and spectral ordering
Friday the 15th of November, 2019, 12:30, ICMAT, Aula Gris 1
In this talk, we localise the spectrum of a discrete magnetic Laplacian on a finite graph using techniques similar to the Dirichlet-Neumann-bracketing for continuous problems. As application we
localise the spectrum of periodic Laplacians using the fact that the fibre operators from Floquet Bloch theory can be seen as magnetic Laplacians. Finally, we use the bracketing ideas to order
spectra of different graphs, and show how certain graph manipulations change the spectrum.
David Krejcirik (Czech Technical University, Prague)
Spectral geometry of quantum waveguides
Friday the 13th of September, 2019, 15:00, ICMAT, Aula Naranja
We shall make an overview of the interplay between the geometry of tubular neighbourhoods of Riemannian manifold and the spectrum of the associated Dirichlet Laplacian. An emphasis will be put on the
existence of curvature-induced eigenvalues in bent tubes and Hardy-type inequalities in twisted tubes of non-circular cross-section. Consequences of the results for physical systems modelled by the
Schroedinger or heat equations will be discussed. | {"url":"http://q-math.es/Other.php?year=2019/2020","timestamp":"2024-11-03T01:14:52Z","content_type":"text/html","content_length":"21763","record_id":"<urn:uuid:542029aa-4c65-4326-be23-994f07157554>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00530.warc.gz"} |
Symmetric relations
A \emph{symmetric relation} is a structure $\mathbf{X}=\langle X,R\rangle$ such that $R$ is a \emph{binary relation on $X$} (i.e. $R\subseteq X\times X$) that is
symmetric: $xRy\Longrightarrow yRx$
Remark: This is a template. If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page.
It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
Let $\mathbf{X}$ and $\mathbf{Y}$ be symmetric relations. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $h:A\rightarrow B$ that is a homomorphism: $xR^{\mathbf X} y\Longrightarrow h(x)R^
{\mathbf Y}h(y)$
Basic results
Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.
Finite members
f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
\end{array}$ $\begin{array}{lr}
f(6)= &\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\
[[Directed graphs]] supervariety | {"url":"https://mathcs.chapman.edu/~jipsen/structures/doku.php?id=symmetric_relations","timestamp":"2024-11-05T20:26:12Z","content_type":"application/xhtml+xml","content_length":"19485","record_id":"<urn:uuid:2b638884-aa21-4fdf-b6f8-0833b6a16df3>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00327.warc.gz"} |
How To Draw Hyperbolic Paraboloid
How To Draw Hyperbolic Paraboloid - Step 2 make a regular tetrahedron. Surface area of a hypar (hyperbolic paraboloid) 0. Otherwise matlab will perform a. Web simple query about plotting the
hyperbolic paraboloid in matlab. Web about press copyright contact us creators advertise developers terms privacy policy & safety how youtube works test new features nfl sunday ticket press
Take six of the skewers. This project is a riff on those experiments using a folding pattern of my own design. 3d plot of part of hyperboloid. I've found that skewers often are not quite evenly cut
and are rarely the. Web about press copyright contact us creators advertise developers terms privacy policy & safety how youtube works test new features nfl sunday ticket press copyright. Web some of
you may be familiar with the paper folding exercises that were once taught at the bauhaus, including the design for a simple hyperbolic paraboloid. Web drawing a hyperbolic paraboloid.
The Hyperbolic ParaboloidDefinition, Geometry With Examples
It's a bit more difficult to execute, but the end result is quite neat. Z =x2 −y2 z = x 2 − y 2. Note here, as the absolute value of x x increases, the.
Hyperbolic paraboloid GeoGebra Dynamic Worksheet
Z =x2 −y2 z = x 2 − y 2. Web plane sections the plane sections of an elliptic paraboloid can be: A parabola, if the plane is parallel to the axis, a point, if.
Detail How To Draw Hyperbolic Paraboloid Koleksi Nomer 6
Notice that we only gave the equation for the ellipsoid that has been centered on the origin. 3d plot of part of hyperboloid. Draw two offset circles (the two ends of the cylinder), twist one.
Hyperbolic paraboloid GeoGebra Dynamic Worksheet
Otherwise matlab will perform a. Notice that we only gave the equation for the ellipsoid that has been centered on the origin. Step 2 make a regular tetrahedron. Why is this paraboloid not the
Detail How To Draw Hyperbolic Paraboloid Koleksi Nomer 15
An alternative form is z=xy (2) (right figure; Web learn how to draw an elliptic and a hyperbolic paraboloid. Use three of them to form an equilateral triangle. This project is a riff on those.
FileHyperbolicparaboloid.jpg Wikipedia
This video explains how to determine the traces of a hyperbolic paraboloid and how to graph a. Web simple query about plotting the hyperbolic paraboloid in matlab. Web 4.7k views 7 years ago drawing
Howto DRAW HYPERBOLIC PARABOLOID (part 2) YouTube
Web 4.7k views 7 years ago drawing quadrics. Use three of them to form an equilateral triangle. Slices parallel to the x axis and y axis. Surface area of a hypar (hyperbolic paraboloid) 0. Does.
Howto DRAW HYPERBOLIC PARABOLOID (part 4) YouTube
Surface area of a hypar (hyperbolic paraboloid) 0. It's a bit more difficult to execute, but the end result is quite neat. You need to add a period before the carat to raise each individual.
Howto DRAW HYPERBOLIC PARABOLOID (part 3) YouTube
Web simple query about plotting the hyperbolic paraboloid in matlab. An ellipse or empty, otherwise. Identify the axisidentify the parabolasdraw two parabolasdraw two hyperbolasconnect the
hyperbolas. Web the easiest way to draw a hyperboloid is.
Hyperbolic Paraboloid
Web 1 2 3 4 5 6 7 8 9 share 220 views 2 years ago this video shows the elements of the hyperbolic paraboloid and how the plane director/directrix has an. Web 4.7k views.
How To Draw Hyperbolic Paraboloid This project is a riff on those experiments using a folding pattern of my own design. Clearly ellipsoids don’t have to be centered on the origin. Web how to draw a
hyperbolic paraboloid / saddle. Fischer 1986), which has parametric equations x(u,v) = u (3) y(u,v) = v (4) z(u,v) = uv (5) (gray 1997, pp. This video explains how to determine the traces of a
hyperbolic paraboloid and how to graph a.
How To Draw Hyperbolic Paraboloid Related Post : | {"url":"https://classifieds.independent.com/print/how-to-draw-hyperbolic-paraboloid.html","timestamp":"2024-11-04T12:26:53Z","content_type":"application/xhtml+xml","content_length":"23090","record_id":"<urn:uuid:522f0f72-42e2-4f7c-b281-63875a3d88f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00652.warc.gz"} |
Geometric Proofs: Two Column Proofs (solutions, examples, videos, worksheets, games, activities)
Videos, solutions, worksheets, games and activities to help Grade 9 Geometry students learn how to use two column proofs.
Two Column Proofs
Two column proofs are organized into statement and reason columns. Each statement must be justified in the reason column. Before beginning a two column proof, start by working backwards from the
“prove” or “show” statement. The reason column will typically include “given”, vocabulary definitions, conjectures, and theorems.
How to organize a two column proof?
Writing Your First Two-Column Proof
This video outlines the process for writing a flow and two-column proof.
Geometry Proofs - Two Column Proofs
Students learn to set up and complete two-column Geometry proofs using the properties of equality as well as postulates and definitions from Geometry.
Two column proof showing segments are perpendicular
Using triangle congruency postulates to show that two intersecting segments are perpendicular.
This video provides a two column proof of the isosceles triangle theorem.
This video provides a two column proof of the exterior angles theorem
Geometric Proofs
Learn to prove geometric statements using two column proofs.
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. | {"url":"https://www.onlinemathlearning.com/geometric-proof.html","timestamp":"2024-11-08T05:37:33Z","content_type":"text/html","content_length":"37321","record_id":"<urn:uuid:605e5f16-4e05-463f-8663-2325047809d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00749.warc.gz"} |
An introduction to spatial interaction models: from first principles
What are SIMs?
Spatial Interaction Models (SIMs) are mathematical models for estimating movement between spatial entities developed by Alan Wilson in the late 1960s and early 1970, with considerable uptake and
refinement for transport modelling since then Boyce and Williams (2015). There are four main types of traditional SIMs (Wilson 1971):
• Unconstrained
• Production-constrained
• Attraction-constrained
• Doubly-constrained
An early and highly influential type of SIM was the ‘gravity model’, defined by Wilson (1971) as follows (in a paper that explored many iterations on this formulation):
\[ T_{i j}=K \frac{W_{i}^{(1)} W_{j}^{(2)}}{c_{i j}^{n}} \]
“where \(T_{i j}\) is a measure of the interaction between zones \(i\) and \(W_{i}^{(1)}\) is a measure of the ‘mass term’ associated with zone \(z_i\), \(W_{j}^{(2)}\) is a measure of the ‘mass
term’ associated with zone \(z_j\), and \(c_{ij}\) is a measure of the distance, or generalised cost of travel, between zone \(i\) and zone \(j\)”. \(K\) is a ‘constant of proportionality’ and \(n\)
is a parameter to be estimated.
Redefining the \(W\) terms as \(m\) and \(n\) for origins and destinations respectively (Simini et al. 2012), this classic definition of the ‘gravity model’ can be written as follows:
\[ T_{i j}=K \frac{m_{i} n_{j}}{c_{i j}^{n}} \]
For the purposes of this project, we will focus on production-constrained SIMs. These can be defined as follows (Wilson 1971):
\[ T_{ij} = A_iO_in_jf(c_{ij}) \]
where \(A\) is a balancing factor defined as:
\[ A_{i}=\frac{1}{\sum_{j} m_{j} \mathrm{f}\left(c_{i j}\right)} \]
\(O_i\) is analogous to the travel demand in zone \(i\), which can be roughly approximated by its population.
More recent innovations in SIMs including the ‘radiation model’ Simini et al. (2012). See Lenormand, Bassolas, and Ramasco (2016) for a comparison of alternative approaches.
Implementation in R
Before using the functions in this or other packages, it may be worth implementing SIMs from first principles, to gain an understanding of how they work. The code presented below was written before
the functions in the simodels package were developed, building on Dennett (2018). The aim is to demonstrate a common way of running SIMs, in a for loop, rather than using vectorised operations (used
in the simodels package) which can be faster.
zones = simodels::si_zones
centroids = simodels::si_centroids
od = simodels::si_od_census
tm_shape(zones) + tm_polygons("all", palette = "viridis")
#> -- tmap v3 code detected --
#> [v3->v4] tm_polygons(): migrate the argument(s) related to the scale of the visual variable 'fill', namely 'palette' (rename to 'values') to 'fill.scale = tm_scale(<HERE>)'
#> [plot mode] fit legend/component: Some legend items or map compoments do not fit well, and are therefore rescaled. Set the tmap option 'component.autoscale' to FALSE to disable rescaling.
od_df = od::points_to_od(centroids)
od_sfc = od::odc_to_sfc(od_df[3:6])
sf::st_crs(od_sfc) = 4326
od_df$length = sf::st_length(od_sfc)
od_df = od_df %>% transmute(
O, D, length = as.numeric(length) / 1000,
flow = NA, fc = NA
od_df = sf::st_sf(od_df, geometry = od_sfc, crs = 4326)
An unconstrained spatial interaction model can be written as follows, with a more-or-less arbitrary value for beta which can be optimised later:
beta = 0.3
i = 1
j = 2
for(i in seq(nrow(zones))) {
for(j in seq(nrow(zones))) {
O = zones$all[i]
n = zones$all[j]
ij = which(od_df$O == zones$geo_code[i] & od_df$D == zones$geo_code[j])
od_df$fc[ij] = exp(-beta * od_df$length[ij])
od_df$flow[ij] = O * n * od_df$fc[ij]
od_top = od_df %>%
filter(O != D) %>%
top_n(n = 2000, wt = flow)
tm_shape(zones) +
tm_borders() +
tm_shape(od_top) +
#> [plot mode] fit legend/component: Some legend items or map compoments do not fit well, and are therefore rescaled. Set the tmap option 'component.autoscale' to FALSE to disable rescaling.
We can plot the ‘distance decay’ curve associated with this SIM is as follows:
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.0002404 0.0256166 0.0801970 0.1495649 0.2035826 1.0000000
od_df %>%
ggplot() +
geom_point(aes(length, fc))
We can make this production constrained as follows:
od_dfj = left_join(
zones %>% select(O = geo_code, all) %>% sf::st_drop_geometry()
#> Joining with `by = join_by(O)`
od_dfj = od_dfj %>%
group_by(O) %>%
mutate(flow_constrained = flow / sum(flow) * first(all)) %>%
sum(od_dfj$flow_constrained) == sum(zones$all)
#> [1] TRUE
od_top = od_dfj %>%
filter(O != D) %>%
top_n(n = 2000, wt = flow_constrained)
tm_shape(zones) +
tm_borders() +
tm_shape(od_top) +
#> [plot mode] fit legend/component: Some legend items or map compoments do not fit well, and are therefore rescaled. Set the tmap option 'component.autoscale' to FALSE to disable rescaling. | {"url":"https://cran.opencpu.org/web/packages/simodels/vignettes/sims-first-principles.html","timestamp":"2024-11-04T10:20:47Z","content_type":"text/html","content_length":"157722","record_id":"<urn:uuid:be8dc8b6-5784-45a0-b271-e84f0e30ce2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00307.warc.gz"} |
Al Beruni was a great Muslim scientist of the eleventh century who had knowledge of many diverse fields such as astronomy, astrology, mineralogy etc. He was member of Mahmud of Ghazni’s court from
1017 to 1030. It is here that he had the opportunity to travel to India (as well as present day Pakistan) and write about it in his books. It is claimed that he even learned Sanskrit during his stay
in India. However, Al Beruni is most well known for his experiment to calculate the radius of the earth at a location close to Katas Raj in Kallar Kahar region of Pakistan. Some claim that the
location of his measurements was in fact Nandana fort which is about 40 km east of Katas Raj temples.
His method of calculation involved two steps.
Step 1: Calculate the inclination angle of a mountain at two locations with known separation between them. We can then use simple trigonometry to find the height of the mountain. The trick here is to
realize that there are two right angled triangles with the same height, which is an unknown, and base lengths which are also unknown but can be factored out (the reader is encouraged to do the math
Figure 1: Calculating the Height of a Mountain
Step 2: The second step involves calculating the angle of depression that the horizon makes as viewed from top of the mountain. Then with the height of the mountain already known the radius of the
earth can be easily calculated as shown below.
Figure 2: Calculating the Radius of the Earth
As the reader might have noticed there are three angular calculations and one distance calculation involved. Distance is easy to measure but angle is not and an accurate Astrolabe is required for
this purpose. The Astrolabe that Al Beruni used was accurate to two decimal places of a degree. The radius that he calculated was within 1% of the accepted radius of the earth today.
PS: The writer has been to Katas Raj temples several times and it his desire to go there again to find more information about the location of the experiment and to possibly re-enact the experiment.
1. A polymath is a person whose expertise spans a significant number of different subject areas; such a person is known to draw on complex bodies of knowledge to solve specific problems.
2. An Astrolabe is an elaborate inclinometer, historically used by astronomers and navigators to measure the inclined position in the sky of a celestial body, day or night.
3. Nandana was a fort built at strategic location on a hilly range on the eastern flanks of the Salt Range in Punjab Pakistan. Its ruins, including those of a town and a temple, are present. It was
ruled by the Hindu Shahi kings until, in the early 11th century, Mahmud of Ghazni expelled them from Nandana. | {"url":"https://www.raymaps.com/index.php/tag/nandana/","timestamp":"2024-11-11T16:31:00Z","content_type":"text/html","content_length":"59029","record_id":"<urn:uuid:1382ff27-f8e3-41b9-ab30-4d036d42c28d>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00722.warc.gz"} |
Hello Guys, we have started our online classes again from 8th November this year. Here I am documenting what all we have completed this week, and within each class what is discussed, including the
previous years questions which are discussed in each class/recording. You may look at this blogpost, as the set of question bank, when you are preparing your notes. I think I will take one more week
to complete this topic 1, of Paper 1.
This coming week I will be discussing Revealed preference approach, Elasticity of Demand and Supply (several questions are asked in previous years from this subtopic), and Game Theory concepts, Nash
Equilibrium and Dominant Strategy Equilibrium
1. Indifference Curve Analysis and Utility function (Part 1) : Rational Preferences, Diminishing MRS, Properties of Indifference Curves
Topics discussed
1. Consumption Set
2. Consumer Preferences
3. Rationality Axioms
4. Utility Representation
5. Indifference Curve and Marginal Utility
6. Marginal Rate of Substitution
7. Properties of Indifference Curve
2. Indifference Curve Analysis and Utility functions (Part 2) (Demand functions) (IES 2011, 2015, 2017, 2018, 2019, 2020)
Topics Discussed
1. Optimization Principle Utility Maximization
2. Budget Constraint
3. First order conditions for a maximum
4. Second order conditions
5. Perfect Substitutes
Past Year Questions Discussed in this class/recording
IES 2018, 9 (a)
Consider the utility function
U = x^α y^β, α> 0, β > 0
which is to be maximized subject to the budget constraint m =p[x]X + pyY where m = income (nominal) and Px and Py are the prices respectively per unit of the goods X and Y.
Derive the demand function for X and Y. Show that these demand functions are homogeneous of degree zero in prices and income.
IES 2015, Q 2
Consider the utility function as U = √q[1] q[2], where q[1] and q[2] are two commodities on which the consumer spends his entire income of the month. Let the price per unit of q[1] and q[2] be ₹ 40
and ₹ 16 respectively and the monthly income of the consumer be ₹ 4,000.
Find out the optimal quantities of q[1] and q[2].
IES 2019, Q 2 (a)
An individual has the utility function U = XY and her budget equation is 10X + 10 Y = 1000. Find the maximum utility that she can attend.
IES 2011, Q 1 (b)
What do you mean by corner solution? In the case of perfect complementary goods, where do you get the corner solution?
IES 2020, Q 3 (a)
Suppose the utility function for the consumer takes one of the following forms:
• U = 50 x + 20 y
• U = 20 x + 50 y
• U = 80 x + 40 y
The budget of the consumer is ₹ 10,000. The prices of good X and good Y are ₹ 50 and ₹ 20 per unit respectively. Determine the possibility of determination of the equilibrium basket in each case
using diagram and comment on the nature of the solutions
IES 2019, Q 1 (a)
In a two-commodity framework, the marginal rate of substitution is everywhere equal to 2. The prices of the two goods are equal. Draw a diagram to identify the utility maximizing equilibrium.
IES 2017, Q 9 (b)
Consider the utility function u log x[1] + x[2] which is to be maximized subject to the budget constraint m = p[1]x[1] + p[2]x[2], where p[1] and p[2] are the prices per unit of the goods x[1] and x
[2] respectively, and m is the income of the consumer. Derive the demand for x[1] and x[2] and interpret your results.
3. Normal Goods, Inferior goods, Giffen Goods, Substitutes, Complements, Income Offer curve, Price Offer curve, Engel curves (IES 2012)
Topic Discussed
1. Normal and Inferior Goods
2. How demand changes as income changes?
3. Comparative Statics
4. What conditions need to be satisfied for a good to be Giffen good?
5. Veblen Goods
6. Income Offer curve and Engel curve of Perfect Substitutes
7. Income Offer curve and Engel curve of Perfect Complements
8. Income Offer curve and Engel curve for Quasi-linear Pref
9. Price Offer curve and Demand curve : Perfect Substitutes
10. Price Offer curve and Demand curve : Perfect Complements
11. Income Offer Curves
12. Both Goods can’t be Inferior
13. A Good can’t be Inferior at all Income Levels
14. Engel Curves
15. Substitute Goods and Complement Goods
Past Year Question(s) Discussed in this class/recording :
IES 2012, Q 1 (h)
Show graphically in your answer-book that if a consumer buys only two goods, both cannot be inferior at the same time.
4. Duality, Indirect Utility function, Expenditure function (IES 2013, 2014, 2018, 2010)
Topic Discussed:
1. Properties of IOF
2. Perfects Compliments
3. Perfect Substitutes
4. Max Function
5. Roy’s Identity
6. Expenditure Function
7. Properties of Expenditure Function
8. Sheppard’s Lemma
Past Year Questions Discussed in this class/recording :
IES 2013, Q 1 (j)
Given utility function U= q[1]q[2] and budget constraint Y = p[1]q[1] + p[2]q[2], derive the indirect utility function.
IES 2014, Q 1 (a)
Is the following statement true or false?
‘’If a consumer’s utility function is of the form = x[1]^1/3 x[2]^1/3, she faces prices p[1] and p[2] and her income is I, then her indirect utility function is V = I^3 / (3p[1] p[2]).”
IES 2018, Q 2 (a)
A price-taking consumer consumes two goods X and Y. Let x and y denote the quantities of goods X and Y respectively, and let P[X] and P[Y] be the respective prices of the two goods. Assume that (i)
the consumer’s budget is given by M, ∞ > M> o; and (ii) P[X] and P[Y] are finite and positive.
(a) Let the consumer’s utility function be given by
U (x, y) = min [x, y]
Define Indirect Utility Function and derive this consumer’s Indirect Utility Function.
IES 2018, Q 1 (e)
Explain with a diagram why the compensated demand curve is vertical if the consumer’s utility function is of the form:
V (x, y) = min [x, y]
5. Price, Income and Substitution Effects (IES 2010, 2011, 2012, 2016, 2017)
Topic Discussed:
1. Income and Substitution Effect of a fall in price of X
2. Slutskian Approach
3. Comparison of Slutskian and Hicksian Methods
4. Effects of a price change for inferior goods
5. Effect of a price change for a Giffen goods
6. Giffen’s Paradox
7. Inferior Goods
Past Year Questions Discussed in this class/ recording :
IES 2011, Q 2
Define income effect, substitution effect and price effect of any change in price. Show that the price effect can be decomposed into the income effect and the substitution effect.
IES 2017, Q 1 (b)
Define the method of Compensating Variation of lncome and the method of Cost Difference. Why is the latter method superior to the former one?
IES 2012, Q 2
Separate income effect from substitution effect for a price change using (i) Hicks’ method (ii) Slutsky’s method. Hence explain the difference between the two compensated demand curves.
IES 2012, Q 1 (e)
Suppose you have a demand function for milk of the form x[1] = 100 + m/ 100 p[1] and your weekly income (m) is ₹ 12,000 and the price of milk (p[1]) is ₹ 20 per litre. Now suppose the price of milk
falls from ₹ 20 to ₹ 15 per litre, then what will be the substitution effect?
6. Slutsky equation and Demand Curve
Topic Discussed:
1. Slutsky Substitution effect
2. Perfect Complements
3. Perfect Substitutes
4. Quasilinear Preference
5. Rebating a Tax
6. Compensated Demand Curve
7. Normal Good
• 7. Consumer Surplus Part 1 and Part 2 (IES 2010, 2011, 2014)
Topic Discussed
1. Consumer Surplus
2. Producer Surplus
3. Calculation of Equilibrium Price
4. Calculation of Consumer Surplus and Producer Surplus
Past Years Questions Discussed in this class/ recording :
IES 2011, Q 9
If D = 250 – 50p and S = 25p + 25 are the demand and supply functions respectively, calculate the equilibrium price and the quantity. Hence calculate both consumer’s and producer’s surpluses under
IES 2015, Q 1(e)
Define consumer’s and producer’s surplus Given the demand function Pc = 113 – q^2 and the supply function p = (q + 2)^2 under perfect competition, find out the consumers’ surplus and producers’
IES 2014, Q 1(c)
Other things equal, what happens to consumer surplus if the price of a good falls? Why? lllustrate using a demand curve.
For any course related query, please click here
If you have any query which is not answered in the FAQ, then please call us at 9999886629 | {"url":"https://nishantmehra.com/given-void-great-youre-good-appear-have-i-also-fifth/","timestamp":"2024-11-10T13:58:15Z","content_type":"text/html","content_length":"89865","record_id":"<urn:uuid:27e3d081-cda3-4df9-8881-14b78754027b>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00789.warc.gz"} |
Measure is a way of gauging how big something is – in terms of length, volume, or some other quality. One of the strangest facts in mathematics is that some objects exist that can't be measured. In
the language of sets, the basic rules (somewhat simplified) of mathematical measures are as follows: (1) the measure of any set is a real number; (2) the empty set has measure zero; (3) if A and B
are two sets with no elements in common then the measure of A the union of A and B is equal to the measure of A plus the measure of B. The second of these rules can be very useful, for example, when
integrating a function, since it allows us to ignore any points where the function jumps around, provided that such points are isolated. A slightly jittery function is one thing; a non-measurable set
is a very different animal. Imagine a three-dimensional shape so fantastically intricate, so jagged and crinkled, that it is impossible to measure its volume and this gives some idea of the concept
of non-measurability. From it flow such bizarre conclusions as the Banach-Tarski paradox. | {"url":"https://www.daviddarling.info/encyclopedia/M/measure.html","timestamp":"2024-11-07T05:58:12Z","content_type":"text/html","content_length":"11688","record_id":"<urn:uuid:21edcd38-ca9e-459d-aac4-3c7c481ace13>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00615.warc.gz"} |
Why Tire Crr matters...
...and why you need to look at BOTH the aero drag of a wheel/tire combination AND the tire rolling resistance to help determine what is "fastest" (i.e. Low Crr can make up for a lot of aero "sins")
update 04/14/13: Added Michelin Pro4 Service Course to chart after roller testing. See chart and discussion below) Five years ago, Damon Rinard (when he was working at Trek) made a post to the
Slowtwitch triathlon forum where he described calculating a rough average of aero drag combined with rolling resistance. The data he used was from some wind tunnel testing he had done with various
tires on the same rim (a Bontrager ACC - 50mm deep) and he combined this with the expected "on road" rolling resistance from roller testing. He did this because he found that different tires made a
large difference in the aerodynamic drag of the wheel/tire system, and that simply choosing tires based on low rolling resistance OR low aero drag performance might not be the right approach. I
thought that was a neat approach and reverse engineered some of his data and expanded the idea to look at the effects of varying wind speed. In order to do so, I needed to scale the aero drag (taken
at 30mph tunnel speed) to different apparent wind speeds using the ratio of V^2/(30mph^2) - since drag force varies with the square of wind velocity. That value was then summed with the rolling
resistance force, which is constant, and then the total combined aero and rolling drag was plotted as a function of expected apparent wind speed. Interestingly, when Trek/Bontrager released the R4
Aero tire a few years later (after Damon had left Trek to work for Vroomen/White Design) they included a plot very similar to the one I had created back in 2008:
To use that chart, you need to have an idea of what your general "race speed" is going to be and then figure out what the maximum apparent wind speed (i.e. the vector sum of the ambient wind speed at
wheel level and the bike ground speed) you'll encounter. Then you can see what the average total force you'll be expecting from a particular combination. Now then, that's talking about the retarding
force...if you want to know the power required for the different combinations, or the power differences between combinations, then you need to multiply the drag force value by the ground speed to get
the rate of doing work (i.e. power) on that total drag force. Make sense? One of the main takeaways from that exercise above I got was that looking at JUST aero drag or JUST Crr wasn't telling the
whole story. It's very easy to "waste" a wheel/tire combination's low aero drag by using a slow rolling tire...and vice versa. But, the other takeaway I got is that really low Crr can "make up" for a
lot of less than ideal aero drag performance Well, for the plot above, the aero drag component was taken as just a simple average of the 5, 10, and 15 degree yaw angle drags. After seeing the plot
below last fall (taken from the Mavic material given out at the CXR80 tire/wheel system introduction), I realized this type of estimate could be updated using a weighted average instead of a straight
average. The weighting would be from the expected % time spent at each yaw angle. Obviously, we want to choose a combination based on it's performance under the conditions we expect to mostly see in
our races. If we don't see large yaw angles very often, it might not be worth it to worry about differences in performance at those higher yaw angles between the equipment choices we are
contemplating. According to Mavic, the data taken above is from actual measurements (they built a "wind vane" type rig and attached it to a bike) from a large number of rides under varying conditons
and courses. Although it may not represent the actual yaw angle distribution for any one particular ride (those will likely be skewed one direction or the other, depending on the course and
conditions) but it does represent what one would expect over a large number of rides. As such, it should be a good tool for determining a good "all around" wheel/tire system choice. Now then, what we
need to update things further is some good aero data showing the drags at various yaw angles for different tire/wheel combinations. It would be really helpful if the data happened to be for tires
which I've already roller tested and have an idea of the predicted "on road" Crr. Well, we're in luck. The guys at Flo wheels went to the A2 tunnel last week and did just that on their new Flo30
wheel. They tested the Michelin Pro4 Service Course, the Conti GP4000S, the Bontrager R4 Aero, and the Vittoria Open Corsa EVO Tri. The last 2 were tires that I actually loaned them after Chris
Thornham had contacted me asking if I knew of a good place to find the Bontrager tires. I happened to have a nearly new one handy and also offered to loan one of my Vittoria tires as well. Here's the
blog post describing their tunnel visit and the aero data they took: Flo30 Aero
As can be seen in their data, the GP4000S was the clear winner aerodynamically, with the R4 Aero close behind it. The one tire that doesn't look too hot is the Vittoria. Although it stays fairly
close to the other tires up to ~7.5 degrees of yaw angle, after that the drag goes way up. However, we know from my roller testing (
Crr chart
) that it's Crr
slightly lower than the other 2 tires, so it
be able to make up for that, especially at lower yaw angles.
So, let's take a look. What is shown below is the result of taking a weighted average (using the Mavic probabilities for the weighting) of the drag values reported for the 3 tires that I have Crr
data on (I'm working on getting a Michelin to roller test as well) combined with the Crr of each tire. I've used the values of Newtons for the drag force (since it's an
force unit, as opposed to grams) so that the drag force results can be simply multiplied by the expected ground speed (in meters/second) to quickly calculate the power (W). (If you want to convert
the values to grams, then just divide by 9.81 m/s - gravity - and multiply by 1000)
(Note: the above chart assumes a wheel loading of 38kg and represents a single front wheel)
There's some interesting things going on in that chart. To understand what's going on there, it's helpful to realize that the "steepness" of each curve is controlled by the aero drag (it varies with
the square of the wind velocity), while where the line sits vertically in the chart is controlled by the rolling resistance values (a constant force).
Despite the apparently poor showing of the Vittoria EVO Tri tire aerodynamically, at lower expected apparent wind speeds it actually performs slightly better than the other 2 tires shown; up until
~27 km/hr where it's curve crosses the GP4000S curve. As compared to the R4 Aero tire, that crossover doesn't happen until expected apparent wind speeds of ~35 km/hr. At the apparent wind speeds that
I expect to see during my TT'ing (~45km/hr) the GP4000S is obviously the leader, with the R4 Aero in second, but the Vittoria still has a predicted
combined drag force that's only .21N higher than the GP4000S. At the expected ground speed of ~42km/hr (i.e. 11.7 m/s), that results in a total power difference of just 2.5W. That's really not a very
large amount, especially considering how much worse the Vittoria appeared to perform aerodynamically.
Now then, you might be asking "why is that?" Simply put, a lot of it has to do with the fact that the yaw angles where the largest differences in aero performance occur are also the yaw angles that
are weighted less in the aero drag average due to their lower probability of being experienced. Another interesting thing is that the differences in Crr between the Vittoria tire and the other 2
isn't very large (all within .0004 of each other), but that's enough to overcome a seemingly significantly worse aero performance.
So far we've been talking about this subject in terms of races like TTs and triathlon bike legs where the front wheel is seeing "free air"...but, what about other types of bike racing? Well, when
you're in a group and drafting, the apparent wind speed is going to be lower, along with the fact that the yaw angle distribution will narrow as well...and thus, that makes the Crr component all that
more important. If you ever find yourself in the situation of riding up a false flat in a group while using poor rolling tires, you'll understand what I'm talking about here. Rolling resistance is
actually a higher priority than wheel aerodynamics in road racing, in my humble opinion.
For one last plot, I took the data from the rest of the Flo wheels that were subsequently tested with the GP4000S tire. That plot is shown below, and as expected at lower expected apparent wind
speeds, the wheels are closer together in performance than with higher expected apparent wind speed.
There we are...I hope that helps to understand some of the tradeoffs involved with choosing tires and wheels for cycling. Of course, there are other properties further involved in these sorts of
tradeoffs, such as durability and "grip", but those other properties are tough to combine into a chart like the above...and so are left up to the user to weigh separately.
Update 04/14/13: The guys at Flo were nice enough to send me the Michelin Pro4 Service Course tire used in the aero testing above and so I got a chance to run it on the rollers. The resultant Crr was
.0043. They also sent the Continental GP4000S used in their testing and it rolled identically to my own version of that tire at .0034. Using that, I was able to update the Total Drag Force chart to
include the Michelin on the Flo 30. The interesting thing about the curve for the Michelin is that despite it's aerodynamics being fairly close to the GP4000S and the Bontrager R4 Aero (especially
with the yaw weighting) on the Total Drag Force chart it NEVER overcomes the "hit" it takes on rolling resistance, even out to expected apparent wind speeds of 60kph. Once again, the importance of
Crr comes to the fore...
16 comments:
1. Brilliant stuff Tom. Thanks for taking the time to put this together.
Jon Thornham
FLO Cycling
2. Fantastic write up, Tom. Excellent resource.
3. Nicely done.
4. Nice! First, that distribution plot looks a lot like one [url=http://djconnel.blogspot.com/2011/03/yaw-angle-variable-wind-speed-and.html]I estimated based on simple assumptions[/url], so that's
nice. But second: there should be a third component, which is how important a particular yaw angle is (you've already pointed out in a road race situation yaw may be less than it is for a solo
rider, so I'm talking actual yaw here). For example, if you can barely control your bike in a 20 deg yaw, then either that angle becomes less important (power isn't rate-limiting) or perhaps more
important (if stall angle affects stability). Of course, facing such uncertainty, the best approach is generally the simplest, which is what you've done.
1. Thanks guys.
Yeah Dan, I knew that Mavic chart was close to the one you had estimated, so that made me feel a bit better about using it.
Taking it "the next step" is a bit more difficult...I was thinking that analysis like this needs to be combined with your "speed for a given power" modeling attempts ;-)
5. Very helpful and informative. Thanks
6. I love this post, it's a great way to look at things as the combined system is what we're living with on race day obviously.
However what I don't understand is how your GP4000S Crr is higher than what I understood to be the canonical source for Crr information (http://www.biketechreview.com/tires_old/images/
AFM_tire_testing_rev9.pdf) nor do I understand how e.g. the other tires (R4Aero / Vittoria) were in different Crr rank order vs that GP4000S.
Did Conti change the compound? Or is there a (perhaps scientifically interesting) methodology difference between the tests, that could be useful in making a decision?
1. The methodology and equations (Al Morrison -the originator of that listing - actually uses equations I originally set out for his calculations) between that source and mine are nearly
identical. The major differences are the roller material (Al's are plastic while mine are aluminum) and the fact that the Crr value in my table includes the 1.5x "smooth to rough" conversion,
while Al's values are for smooth, flat surfaces.
If you take the .00307 Crr value that Al measured a few years ago for the GP4000S (when the black chili compound first was added) and multiply it by 1.5, it would end up coming in at .0056 on
my chart. Apparently...and I've confirmed the Crr value I measured with a second tire...Continental has been making the tire faster over time. Most likely this has been done with subtle
tweaks to the thickness in various areas. To the hand, the newer tires feel a bit more flexible (indicating a thinner coating on the sidewalls) plus the newer tires weigh out at ~200-210g,
while the older tires were in the ~230-250g range.
As a cross check between the 2 lists, take a look at the Bontrager RXL Pro 23C entry. Al shows a Crr of .00244, which when multiplied by 1.5X results in an "on road" Crr = .0037. I measured a
well used model of the same tire at .0035. Doing the same with the Specialized S-Works Mondo Open 23C, you'll find Al's compensated value at .0035, while my measurement is .0034.
Bontrager Aerowing TT 19C: .0041 vs. .0042
etc., etc.
7. This comment has been removed by the author.
8. Tom can you tell me if I'm reading that new chart correctly (the one with the PR4). I did some calculations and at 40km/hr, it's about a 5w difference between the GP4000 and the PR4?
1. Greg, at what expected apparent wind speed and expected ground speed?
2. That's a good question. No wind, non-draft triathlon, 40k/hr, so ground speed is 40k/hr too?
3. In that case, yes...I get ~6W difference to go the same speed on average.
9. Very interesting stuff. I love reading your blog. Always something interesting to catch my eye.
Click Here
10. friend can you pass me a excel for virtual elevation robert chung? Thanks!
11. Just did a test of my own on the michelin pro4: http://teamrodrigo.com/2014/11/15/battle-of-tires-which-one-is-best-in-the-real-world/ | {"url":"http://bikeblather.blogspot.com/2013/04/why-tire-crr-matters.html","timestamp":"2024-11-13T00:00:13Z","content_type":"text/html","content_length":"102526","record_id":"<urn:uuid:c038ed85-eaeb-4785-a679-900b032a581b>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00353.warc.gz"} |
p-adic Heights of Heegner points on Shimura curves
2013 Theses Doctoral
p-adic Heights of Heegner points on Shimura curves
Let f be a primitive Hilbert modular form of weight 2 and level N for the totally real field F, and let p be an odd rational prime such that f is ordinary at all primes dividing p. When E is a CM
extension of F of relative discriminant prime to Np, we give an explicit construction of the p-adic Rankin-Selberg L-function L_p(f_E,-) and prove that when the sign of its functional equation is -1,
its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated to f. This p-adic Gross-Zagier formula generalises the result obtained by Perrin-Riou when
F=Q and N satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A.
• Disegni_columbia_0054D_11281.pdf application/pdf 522 KB Download File
More About This Work
Academic Units
Thesis Advisors
Zhang, Shou-Wu
Ph.D., Columbia University
Published Here
May 1, 2013 | {"url":"https://academiccommons.columbia.edu/doi/10.7916/D8CZ3FD0","timestamp":"2024-11-07T11:04:44Z","content_type":"text/html","content_length":"16747","record_id":"<urn:uuid:87237650-ea6b-451e-b18f-62362ef2e7ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00700.warc.gz"} |
(Redirected from Hemimean comma)
Jump to navigation Jump to search
Interval information
Ratio 3136/3125
Factorization 2^6 × 5^-5 × 7^2
Monzo [6 0 -5 2⟩
Size in cents 6.0832436¢
Names hemimean comma,
didacus comma
Color name zzg^53, zozoquingu 3rd,
Zozoquingu comma
FJS name [math]\text{ddd3}^{7,7}_{5,5,5,5,5}[/math]
Special properties reduced
Tenney height (log[2] nd) 23.2244
Weil height (log[2] max(n, d)) 23.2294
Wilson height (sopfr (nd)) 51
Harmonic entropy ~1.5385 bits
(Shannon, [math]\sqrt{nd}[/math])
Comma size small
open this interval in xen-calc
3136/3125, the hemimean comma or didacus comma, is a small 7-limit comma measuring about 6.1¢. It is the difference between a stack of five classic major thirds (5/4) and a stack of two subminor
sevenths (7/4). Perhaps more importantly, it is (28/25)^2/(5/4), and in light of the fact that 28/25 = (7/5)/(5/4), it is also (28/25)^3/(7/5), which means its square is equal to the difference
between (28/25)^5 and 7/4. The associated temperament has the highly favourable property of putting a number of low complexity 2.5.7 subgroup intervals on a short chain of 28/25's, itself a 2.5.7
subgroup interval.
In terms of commas, it is the difference between the septimal semicomma (126/125) and the septimal kleisma (225/224), or between the augmented comma (128/125) and the jubilisma (50/49). Examining the
latter expression we can observe that this gives us a relatively simple S-expression of (S4/S5)/(S5/S7) which can be rearranged to S4*S7/S5^2. Then we can optionally replace S4 with a nontrivial
equivalent S-expression, S4 = S6*S7*S8 = (6/5)/(9/8); substituting this in and simplifying yields: S6*S7^2*S8/S5^2, from which we can obtain an alternative equivalence 3136/3125 = (49/45)/(25/24)^2,
meaning we split 49/45 into two 25/24's in the resulting temperament.
Didacus (2.5.7)
Tempering out this comma in its minimal prime subgroup of 2.5.7 leads to didacus (a variant of hemithirds without a mapping for 3) with a generator representing 28/25. See hemimean clan for
extensions of didacus.
Hemimean (2.3.5.7)
Tempering out this comma in the full 7-limit leads to the rank-3 hemimean temperament, which splits the syntonic comma into two equal parts, each representing 126/125~225/224. See hemimean family for
the family of rank-3 temperaments where it is tempered out.
Note that if we temper 126/125 and/or 225/224 we get septimal meantone.
As 28/25 is close to 19/17 and as the latter is the mediant of 9/8 and 10/9 (which together make 5/4), it is natural to temper (28/25)/(19/17) = 476/475, or equivalently stated, the semiparticular (5
/4)/(19/17)^2 = 1445/1444, which together imply tempering out 3136/3125 and 2128/2125, resulting in a rank-3 temperament. The name comes from when it was first proposed on the wiki as part of The
Milky Way realm.
Subgroup: 2.5.7.17.19
Comma list: 476/475, 1445/1444
Sval mapping: [⟨1 0 -3 0 -1], ⟨0 2 5 0 1], ⟨0 0 0 1 1]]
sval mapping generators: ~2, ~56/25, ~17
Optimal tuning (CTE): ~2 = 1\1, ~28/25 = 193.642, ~17/16 = 104.434
Optimal ET sequence: 12, 18h, 25, 43, 56, 68, 93, 161, 285, 353, 446, 514ch, 799ch
Badness: 0.0150
Hemimean orion
As tempering either S16/S18 = 1216/1215 or S18/S20 = 1701/1700 implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us
both S16~S18~S20 and S17~S19, it can be considered natural to add these commas, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion. However, this is a strong extension
of hemimean and weak extension of orion, as we have a ~3/2 generator slicing the second generator of orion into five.
See Hemimean family #Hemimean orion.
As 1445/1444 = S17/S19 we can extend orion to include prime 3 in its subgroup by tempering both S17 and S19. However, note that (because of tempering S17) this splits the period in half, representing
a 17/12~24/17 half-octave. This has the consequence that the 17/16 generator can be described as a 3/2 because 17/16 up from 24/17 is 3/2. As a result, this equates the generators of hemimean orion
and orion up to period-equivalence and is a weak extension of both.
See Hemimean family #Semiorion.
This comma was first named as parahemwuer by Gene Ward Smith in 2005 as a contraction of parakleismic and hemiwürschmidt^[1]. It is not clear how it later became hemimean, but the root of hemimean is
obvious, being a contraction of hemiwürschmidt and meantone. | {"url":"https://en.xen.wiki/w/Hemimean_comma","timestamp":"2024-11-04T19:12:58Z","content_type":"text/html","content_length":"37451","record_id":"<urn:uuid:0f788115-f886-4936-99b8-157796e0cdc4>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00325.warc.gz"} |
Indian Computing Olympiad Syllabus 2024 - Download PDF
Indian Computing Olympiad syllabus is available on the official website. The syllabus is available in pdf format so that students can download the syllabus easily. The pdf contains all the important
subjects for which the IARCS board conducts the ICO exam. Check out the complete syllabus for all the classes participating in the Indian Computing Olympiad 2024.
Table of Contents
Indian Computing Olympiad 2024 Highlights
The candidates must be familiar with the syllabus of the Indian Computing Olympiad to have a better understanding exam structure.
Below are the highlights of the Indian Computing Olympiad 2024.
Indian Computing Olympiad Highlights
│Partculars │About │
│Name │Indian Computing Olympiad (ICO) │
│Negative marking │No negative marking │
│Programming language to be known│Algorithms, C++, JAVA │
│Pre-requisites │ • School-level mathematics │
│ │ • Basic understanding of algorithms │
│Official Website │www.iarcs.org.in │
Indian Computing Olympiad 2024 Syllabus
The candidates must be familiar with the syllabus of the Indian Computing Olympiad to have a better understanding exam structure.
Download the Indian Computing Olympiad (ICO) syllabus from the link provided below.
The syllabus for IOC 2024 is mentioned below in pointers.
ICO Mathematics Syllabus
Mathematics is a subject that needs constant practice and understanding. Students need to know the syllabus and should practice sample papers before going for the exam.
Given below is the syllabus covered under the Indian Computing Olympiad mathematics section.
Arithmetics and Geometry
Given below are the topics candidates need to focus on in the section under Arithmetics and Geometry:
• Integers, operations (including exponentiation), and comparison
• Basic properties of integers (sign, parity, and divisibility)
• Basic modular arithmetic: addition, subtraction, and multiplication
• Prime numbers
• Fractions and percentages
• Line, line segment, angle, triangle, rectangle, square, and circle
• The point, vector, and coordinates in the plane
• Polygon (vertex, side/edge, simple, convex, inside, and area)
• Euclidean distances
• Pythagorean theorem
• Additional topics from number theory
• Geometry in 3D or higher dimensional spaces
• Analyzing and increasing the precision of floating-point computations
• Modular division and inverse elements
• Complex numbers
• General conics (parabolas, hyperbolas, and ellipses)
• Trigonometric functions
Discrete Structures (DS)
Topics covered under the discrete structures are functions, relations, sets, basic logic, proof techniques, basics of counting, and graphs & trees.
Functions, Relations, and Sets
These are the areas or topics that students need to focus on in the section under Functions, Relations and Sets:
• Functions (surjections, injections, inverses, and composition)
• Relations (reflexivity, symmetry, transitivity, equivalent relations, total/linear order relations, and lexicographic order)
• Sets (inclusion/exclusion, complements, cartesian products, and power sets)
• Cardinality and countability (of infinite sets)
Basic Logic
Listed below are the topics under the section of Basic Logic that students should focus on:
• First-order logic
• Logical connectives (including their basic properties)
• Truth tables
• Universal and existential quantification (note: statements should avoid definitions with nested quantifiers whenever possible)
• Modus ponens and modus tollens
• Normal forms
• Validity
• Limitations of predicate logic
Proof Techniques
The topics that need to be focused on by the candidates under the Proof Techniques section are:
• Notions of implication, converse, inverse, contrapositive, negation, and contradiction
• Direct proofs, proofs by counterexample, contraposition, and contradiction
• Mathematical induction
• Strong induction (also known as complete induction)
• Recursive mathematical definitions (including mutually recursive definitions)
Basics of Counting
The basic of countings section includes these many topics that should be focused on:
• Counting arguments (sum & product rule, arithmetic & geometric progressions, and Fibonacci numbers)
• Permutations and combinations (basic definitions)
• Factorial function and binomial coefficients
• Inclusion-exclusion principle
• Pigeonhole principle
• Pascal's identity and binomial theorem
• Solving of recurrence relations
• Burnside lemma
Graphs and Trees
The graphs and trees topics that are required to be focused on by the candidates are listed below:
• Trees & their basic properties and rooted trees
• Undirected graphs (degree, path, cycle, connectedness, Euler/Hamil-ton path/cycle, and handshaking lemma)
• Directed graphs (in-degree, out-degree, directed path/cycle, and Euler/Hamilton path/cycle)
• Spanning trees
• Traversal strategies
• Decorated graphs with edge/node labels, weights, and colours
• Multigraphs and graphs with self-loops
• Bipartite graphs
• Planar graphs
• Hypergraphs
• Specific graph classes such as perfect graphs
• Structural parameters such as treewidth and expansion
• Planarity testing
• Finding separators for planar graphs
Other Areas in Mathematics
Listed below are some areas in Mathematics that should be revised by the candidates taking the exam:
• Geometry in three or more dimensions
• Linear algebra, including (but not limited to): matrix multiplication, exponentiation, inversion, and Gaussian elimination–Fast Fourier transform
• Calculus
• Theory of combinatorial games, e.g., NIM game amd Sprague-Grundy theory
• Statistics
ICO Computing Science Syllabus
Given below is the syllabus for covered under the Indian Computing Olympiad computer science section.
Programming Fundamentals (PF)
Topics covered under the discrete structures are fundamental programming constructs (for abstract machines), algorithms & problem-solving, fundamental data structures, and recursion.
Fundamental Programming Constructs (for abstract machines)
• Basic syntax and semantics of a higher-level language (at least one of the specific languages available at an IOC, as announced in the competition rules for that IOC)
• Variables, types, expressions, and assignment
• Simple I/O, conditional, and iterative control structures
• Functions and parameter passing
• Structured decomposition
Algorithms and Problem-Solving
The candidates taking the ICO exam should focus on these topics under the section of Algorithms and Problem-Solving:
• Problem-solving strategies (understand–plan–do–check, separation of concerns, generalization, specialization, case distinction, working backward, etc.)
• The role of algorithms in the problem-solving process
• Implementation strategies for algorithms
• Debugging strategies
• The concept and properties of algorithms (correctness and efficiency)
Fundamental Data Structures
Students need to focus on the topics of Fundamental Data Structure for taking the ICO exam:
• Primitive types (boolean, signed/unsigned integer, and character)
• Arrays (including multicolumn dimensional arrays)
• Strings and string processing
• Static and stack allocation (elementary automatic memory management)
• Linked structures
• Implementation strategies for graphs and trees
• Strategies for choosing the right data structure
• Elementary use of real numbers in numerically stable tasks. The floating-point representation of real numbers and the existence of precision issues.
• Pointers and references
• Data representation in memory, heap allocation, runtime storage management, and using fractions to perform exact calculations
• Non-trivial calculations on floating-point numbers and manipulating precision errors
Regarding floating-point numbers, there are well-known reasons why they should be, in general, avoided at the IOC. However, the currently used interface removes some of those issues. In particular,
it should now be safe to use floating-point numbers in some types of tasks, e.g., to compute some Euclidean distances and return the smallest one.
Listed below are the recursion topics to be covered in the ICO exam:
• The concept of recursion
• Recursive mathematical functions
• Simple recursive procedures (including mutual recursion)
• Divide-and-conquer strategies
• Implementation of recursion
• Recursive backtracking
Algorithms and Complexity (AL)
Algorithms are fundamental to computer science and software engineering. The real-world performance of any software system depends on the algorithms, the suitability, and the efficiency of the
various layers of implementation.
Good algorithm design is, therefore, crucial for the performance of all software systems. Moreover, the study of algorithms provides insight into the intrinsic nature of the problem as well as
possible solution techniques independent of programming language, programming paradigm, computer hardware, or any other implementation aspect.
Basic Algorithmic Analysis
Students should focus on these topics under the category of Basic Algorithm Analysis:
• Algorithm specification, precondition, postcondition, correctness, and invariants
• Asymptotic analysis of upper complexity bounds (informally if possible)
• Big O notation
• Standard complexity classes: constant, logarithmic, linear, O(N log N), quadratic, cubic, and exponential
• Time and space tradeoffs in algorithms
• Empirical performance measurements
• Identifying differences among best, average, and worst-case behaviors
• Little o, Omega, and theta notation
• Tuning parameters to reduce running time, memory consumption, or other measures of performance
• Asymptotic analysis of average complexity bounds
• Using recurrence relations to analyze recursive algorithms
Algorithmic Strategies
The candidates are required to learn the topics listed below under the category of Algorithmic Strategies:
• Simple loop design strategies
• Brute-force algorithms (exhaustive search)
• Greedy algorithms
• Divide-and-conquer
• Backtracking (recursive and non-recursive)
• Branch-and-bound
• Dynamic programming, heuristics, and finding good features for machine learning tasks
• Discrete approximation algorithms and randomized algorithms
• Clustering algorithms (k-means and k-nearest neighbor)
• Minimizing multivariate functions using numerical approaches
The students are required to learn the following topics under the category of Algorithms:
• Simple algorithms involving integers: radix conversion, Euclid's algorithm, primality test by O (√n) trial division, Sieve of Eratosthenes, factorization (by trial division or achieve), and
efficient exponentiation
• Simple operations on arbitrary precision integers (addition, subtraction, and simple multiplication)
• Simple array manipulation (filling, shifting, rotating, rever-sal, resizing, minimum/maximum, prefix sums, histogram, and bucket sort)
• Simple string algorithms (e.g., naive substring search) 3p sequential processing/search and binary search
• Quicksort and quickselect to find the smallest element
• O (N log N) worst-case sorting algorithms (heapsort and mergesort)
• Traversals of ordered trees (pre-, in-, and post-order)
• Depth- and breadth-first traversals
• Applications of the depth-first traversal tree, such as topological ordering and Euler paths/cycles
• Finding connected components and transitive closures
• Shortest-path algorithms (Dijkstra, Bellman-Ford, and Floyd-Warshall)
• Minimum spanning tree (Jarn ́ık-Prim and Kruskal algorithms)
• O(VE) time algorithm for computing maximum bipartite matching. By connectivity in undirected graphs (bridges, articulation points).
• Connectivity in directed graphs (strongly connected components)
• Basics of combinatorial game theory, winning and losing positions, and minimax algorithm for optimal game playing
• Maximum flow and flow/cut duality theorem.
• Optimization problems that are easiest to analyze using matroid theory
• Problems based on matroid intersection (except for bipartite matching)
• Lexicographical BFS, maximum adjacency search, and their properties
• Distributed algorithms
• Basic computability
Other Areas in Computing Science
Listed below are some of the areas that are to be focussed in computer science:
• Architecture and organization (AR)
• Operating systems (OS)
• Net-centric computing (or cloud computing) (NC)
• Programming languages (PL)
• Human-computer interaction (HC)
• Graphics and visual computing (GV)
• Intelligent systems (IS)
• Information management (IM)
• Social and professional issues (SP)
• Computational science (CN)
Other Important Topics for ICO
The ICO exam covers a vast range of topics from different disciplines. As a result, some subjects may be difficult to clear, but some may be very easy.
Given below are some other important topics covered under the Indian Computing Olympiad.
• Software engineering (SE)
• Software design
• Using APIs
• Software tools and environments
• Software processes
• Software requirements and specification
• Software validation
• Software evolution
• Software project management
• Component-based computing
• Software reliability
• Specialized systems development
Important Guidelines for Indian Computing Olympiad
There are several reasons for imposing restrictions on concepts, terminology, and notations in ICO competition tasks.
Given below are the instructions and guidelines to be followed by the Indian Computing Olympiad.
• The ICO competition is not intended to test knowledge, and it assesses algorithmic problem-solving skills. Many contestants have not yet completed secondary education. Therefore, the ICO
organizers cannot assume that contestants have much prior knowledge of concepts, terminology, and notation in algorithmic problems (or any other area, for that matter).
• ICO contestants come from all over the world and have diverse educational backgrounds. The organizers should aim at formulations that are understandable by all contestants.
• Concepts, terminology, and notations used in computing science are fraught with complications. CS professionals have learned to cope with many years of training. For example, the concept of a
graph is rather vague as such. It requires many additional details to make precise what is meant. A CS professional can often infer these details from subtle clues in the context. It cannot be
expected from an average ICO contestant.
Indian Computing Olympiad Preparation
During the preparations for ICO competitions, candidates should prioritize the appropriateness of concepts, terminology, and notations used in competition tasks that have been discussed and decided.
It is because it has often led to changes in task descriptions.
Let us first clarify what we mean by concept, terminology, and notation.
• Concepts refer to a general idea with a specific focus-for instance, the concept of a number and incidence (in geometry). Often, a concept is not atomic but a composite of (simpler) concepts.
• 'Terminology' refers to the typical words used in connection with a concept-for instance, the elements of a set, appointed online.
• 'Notation' refers to the typical (mathematical) symbols used in connection with a concept. For instance, {0,1,2} denotes the set consisting of the three numbers 0,1, and 2.
• We distinguish three classes in the usability classification. In order of increasing restrictiveness, these are
□ Basic knowledge (BK): can be used without further definition.
□ To-be-defined (TBD): This can be used but needs to be defined explicitly.
□ To-be-avoided (TBA): cannot be used.
Note that, for a given concept, the related terminology and notations may end up in different classes. The classification presented here was obtained as a combination of,
• Personal experiences gathered during meetings at past ICOs.
• Personal experiences from teaching and organizing programming contests.
• Analysis of task descriptions of past ICO competitions.
• Discussions with various members of the ICO community.
List of other popular Olympiads are given below:
Note: Get the complete list of Olympiad Exams conducted in India. For any questions/queries related to this Olympiad, do comment in the comment section, as given below. | {"url":"https://www.getmyuni.com/olympiad/indian-computing-olympiad-ico-syllabus","timestamp":"2024-11-15T04:33:37Z","content_type":"text/html","content_length":"69331","record_id":"<urn:uuid:0e298413-649d-491a-81f3-4bcd76065e0a>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00097.warc.gz"} |
Program to find number of set bits in the Binary number | Code Pumpkin
Program to find number of set bits in the Binary number
August 20, 2017
Posted by Abhi Andhariya
In this post, we will discuss about one of the easiest and commonly asked Bit Manipulation Interview Question i.e. Find total number of set bits in the given number?
There are many approach to solve this problem. Here, we will discuss about below two approaches:
1) & and >> Approach (naive Approach)
In this approach, we perform bitwise and (&) and bitwise right shift (>>) operations for each bit in a number.
Step 1: (n&1) operation returns 1, if right most bit of the number is 1.
Step 2 : If this returns 1, we increment our counter.
Step 3 : perform n=(n>>1) , so that we can check the value of 2nd right most bit.
Continue this process, until n becomes 0.
Let's understand this by below example.
Java Program Implementation
public static int getNumberOfSetBitsApproachOne(int n){
int count = 0;
if((n&1) == 1) // Similarly Condition (n&1)==0 can be used to calculate
{ //number of zeros in the given number
n = n>>1;
return count;
The naive approach requires one iteration per bit, until no more bits are set. So on a 32-bit word with only the high set, it will go through 32 iterations.
For Example, it will require 8 iiteraton to check number of set bits in number 128 (1000 0000)
2) Brian Kernighan's Approach
First of all, its not KerniGHAN, its pronounced as Kernihan, the G is silent 😉
This algorithm goes through as many iterations as there are set bits. So if we have a 32-bit word with only the high bit set, then it will only go once through the loop. For above Example of 128, it
will give the answer only in one iteration.
In the worst case, it passes once per each bit and takes same time as naive approach. i.e. when all bits are 1 as in 255 (1111 1111)
What is Brian Kernighan's Approach?
It uses n = n & (n-1) approach.
If you perform bitwise & operation on n and (n-1), it produces the number with set bits one less than the original number. For example, if n has 4 set bits and if you perform n&(n-1), it produces the
number with 3 set bits.
For example, let's try all the 4-bit combinations:
n n n-1 n&(n-1)
-- ---- ---- -------
So after performing each & operation, we will increment the counter. We will continue this process until n!=0
Java Program Implementation
public static int getNumberOfSetBitsApproach2(int n){
int count = 0;
while(n != 0){
n = n&(n-1);
return count;
We can use above two approaches to write a program to check if a given number is power of two or not?
That's all for this topic. If you guys have any suggestions or queries, feel free to drop a comment. We would be happy to add that in our post. You can also contribute your articles by creating
contributor account here.
Happy Learning 🙂
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About the Author
Tags: Bit Manipulation, Bitwise Operators, Core Java, Right Shift Operation
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AJW's site
If you're trying to install any flavor of Linux in a Lenovo laptop (in my case Lubuntu in a T430 Thinkpad) and all you get is a black screen, chances are that you have the "Secure Boot" option
enabled in the BIOS. Just disable this option and you should ... | {"url":"http://ocam.cl/","timestamp":"2024-11-04T20:14:50Z","content_type":"text/html","content_length":"16243","record_id":"<urn:uuid:3c005b1b-c412-439b-9736-83f3f9b15ab1>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00569.warc.gz"} |
Percentage 83077 - math word problem (83077)
Percentage 83077
The bank increased the interest rate from the original value of 4% to 5%. What percentage of the original value of the interest rate was its increase?
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
You need to know the following knowledge to solve this word math problem:
Units of physical quantities:
Themes, topics:
Related math problems and questions: | {"url":"https://www.hackmath.net/en/math-problem/83077","timestamp":"2024-11-04T05:09:52Z","content_type":"text/html","content_length":"53639","record_id":"<urn:uuid:2038f32a-df0e-4566-b34c-ba6595a50e9d>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00858.warc.gz"} |
The terrestrial water balance is forced by highly intermittent and unpredictable pulses of rainfall. This in turn impacts several related hydrological and ecological processes, such as plant
photosynthesis, soil biogeochemistry and has feedbacks on the local climate. We treat the rainfall forcing at the daily time scale as a of marked (Poisson) point processes, which is then used the
main driver of the stochastic soil water balance equation. We analyze the main nonlinearities in the soil water losses and discuss the probabilistic dynamics of soil water content as a function of
soil-plant and vegetation characteristics. Crossing and mean-first-passage-time properties of the stochastic soil moisture process define the statistics of plant water stress, which in turn control
plant dynamics, as shown in application to tree-grass coexistence in the Kalahari transect.
In the second part of this overview, we briefly illustrate: i) the propagation of soil moisture fluctuations through the nonlinear soil carbon and nitrogen cycles, ii) the possible emergence of
persistence and preferential states in rainfall occurrence due to soil moisture feedback, and iii) the impact of inter-annual rainfall variability in connection to recent theory of superstatistics .
Rodriguez-Iturbe I. and A. Porporato, Ecohydrology of water controlled ecosystems: plants and soil moisture dynamics. Cambridge University Press, Cambridge, UK. 2004.
Laio F., Porporato A., Ridolfi L., and Rodriguez-Iturbe I. (2001) Plants in water controlled ecosystems: Active role in hydrological processes and response to water stress. II. Probabilistic soil
moisture dynamics. Advances in Water Research, 24, 707-723.
Porporato A., Laio F., Ridolfi L., and Rodriguez-Iturbe I. (2001) Plants in water controlled ecosystems: Active role in hydrological processes and response to water stress. III. Vegetation water
stress. Advances in Water Research, 24, 725-744.
Porporato A., D Odorico P., Phase transitions driven by state-dependent Poisson noise, Phys. Rev. Lett. 92(11), 110601, 2004.
D Odorico P., Porporato A., Preferential states in soil moisture and climate dynamics, Proc. Nat. Acad. Sci. USA, 101(24), 8848-8851, 2004. Manzoni S., Porporato A., D Odorico P. and I.
Rodriguez-Iturbe. Soil nutrient cycles as a nonlinear dynamical system. Nonlin. Proc. in Geophys. 11, 589-598, 2004.
Porporato A., G. Vico, and P. Fay, Interannual hydroclimatic variability and Ecosystem Superstatistics. Geophys. Res. Lett., 33, L5402, 2006.
Daly, E., and A. Porporato, Inter-time jump statistics of state-dependent Poisson processes, Phys. Rev. E, 75, 011119, 2007. | {"url":"https://www4.math.duke.edu/media/videos.php?cat=561&sort=most_viewed&time=yesterday&page=1&seo_cat_name=","timestamp":"2024-11-07T04:33:27Z","content_type":"text/html","content_length":"119022","record_id":"<urn:uuid:57f38b08-d33a-4937-b492-9ab2c52711a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00463.warc.gz"} |
All Faculty
Home People Faculty All Faculty
Quantum Universe Center
Kim, Jaewan
QUC Distinguished Professor
Quantum Information Science
Jaewan Kim’s research focuses on quantum information science, quantum dynamics, and quantum chaos (or quantum chaology). His work spans from the application of quantum optical systems, such as
single-photon entanglement for quantum teleportation and quantum cryptography, to the development of quDit cluster states using Kerr nonlinear interactions. He has pioneered quantum optical concepts
to advance quantum information technology. He hosted AQIS (Asian Quantum Information Science Conferences) 2008, 2011, 2015, 2019, and 2023 and organized “Quantum Korea” 2023 and 2024. He founded the
Quantum Information Society of Korea (QisK) and served as the first President. He is also the Academic Director of FQCF (Future Quantum Convergence Forum) and has served in various government
committees related quantum information technology.
• 1985.2 Department of Physics, Seoul National University (B.S.)
• 1993.6 Department of Physics, University of Houston (Ph.D)
• 1993-1994 Postdoctoral Research Associate, Texas Center for Superconductivity
• 1994-2002 Principal Researcher, Computational Science and Engineering Lab, SAIT.
• 2000-2002 Research Associate Professor of Physics, KAIST
• 2002 KIAS Assistant Professor, Korea Institute for Advanced Study
• 2002.12.- Professor of Computational Sciences, Korea Institute for Advanced Study
• 2005-2006 Chair of School of Computational Sciences, Korea Institute for Advanced Study
• 2007.8-2008.11 Chair of School of Computational Sciences, Korea Institute for Advanced Study
• 2009.2 - Vice President of KIAS
• 2010.2-6 Acting President of KIAS
• Office: 8415 / TEL) 82-2-958-3779 / FAX) 82-2-958-3820
• Quantum Universe Center, Korea Institute for Advanced Study
• 85 Hoegiro Dongdaemun-gu, Seoul 02455, Republic of Korea. | {"url":"https://www.kias.re.kr/kias/people/faculty/viewMember.do?memberId=10146&trget=listFaculty&menuNo=408002&pageIndex=","timestamp":"2024-11-13T23:01:17Z","content_type":"text/html","content_length":"80435","record_id":"<urn:uuid:3b995e34-801d-411d-a2b0-978c86e2855d>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00187.warc.gz"} |
Things Faster than the Internet: You in Your Car?
We’ve probably all been at the point where we wondered if it would be faster to walk to someone’s house with a flash drive rather than send them a file over the internet. If you haven’t, bear with
me, because I sure have. As movies and games grow in size with each resolution or graphics upgrade, while internet speeds don’t always keep up, it’s easy to feel stuck as you watch a loading bar
crawl across the screen. For some large commercial data transfers (think server farm backups for big corporations), it is entirely unreasonable to use a portion of their bandwidth (the amount of data
that can be transferred via a given internet connection, also referred to as the connection speed or maximum data transfer rate), to handle this. In fact, it’s often faster to load it onto hard
drives and ship it by truck.
Given that the average consumer probably doesn’t have petabytes of data to transfer around, but most likely has a much slower internet connection (by probably at least an order of magnitude or five),
we might still run into this problem. For example, the average download speed of an American household sits around 140 Mb/s (megabits per second) according to statistics from Speedtest.net, a popular
internet speed testing site. Let’s calculate how long it should take the average American to download a 60 GB (gigabyte, note the uppercase “B”) computer game.
\[\frac{60\; \textrm{GB}}{140\; \textrm{Mb/s}} = \frac{60\; \textrm{GB} \cdot 8 \; \textrm{b/B} \cdot 1000 \; \textrm{Mb/Gb}}{140\; \textrm{Mb/s}} = 3429\; \textrm{s} \approx 57\; \textrm{min}\]
So while that isn’t absolutely abysmal it sure isn’t exactly optimal. Realistically it would be even slower given time spent writing the game to your hard drive but for the sake of the argument we
omit this. So at what point should we drive our home movie collection over to our cousins ourselves? Well, we can write up some Python code to find out!
Before we open our code editors, let’s start with the parameters of our model. Let’s say you average 30 mph on the first and final 5 miles of your journey due to local roads with lower speed limits.
We can average any distance for the next 15 miles on either side to be moderate highway or country road driving at 50 mph, and the rest to be an interstate at 70 mph. This should account for how fast
you really get to your cousin no matter how far away they live, and prevent the unrealistic case of driving across town at 70 mph. Now let’s write some Python code to express this time to transfer
any arbitrary amount of data you can fit in your car as a function of distance to your destination. For the sake of the argument, we will assume that you have an absurdly large flash drive to give to
your cousins.
# Time (in hours) to drive a given distance (in miles)
def time_to_drive(distance):
# If total distance is less than 10 miles, you average 30 mph
if(distance <= 10):
return distance/30
# For distances less than 40 miles, you average 50 mph for the middle miles
elif (distance <= 40):
return (10)/30 + (distance-10)/50
# For greater distances, we hit the interstate at 70mph
return (10)/30 + (40-10)/50 + (distance-40)/70
Now with the help of Matplotlib, we can plot this function. Let’s compare it to a given file transfer over an average internet connection, which we can take as constant with respect to distance, but
varying with respect to size (the time it takes to send the signal from you to your cousins is probably less than a second, and more likely than not under something like 300 ms). We will assume you
have to upload to a cloud service, like Google Drive or Dropbox first (at the average American upload speed of around 50 Mb/s, according to Speedtest.net), then download at the aforementioned 140 Gb/
So this is pretty clear evidence that if you need to send 100 GB of data to someone 350 miles away, it’s actually faster to drive it than to send it over the internet via a cloud storage service.
Once we get to the order of a few terabytes, local mail and shipping companies even become viable. So there you have it: once you have to send your cousin a terabyte’s worth of home movies across
state lines, just mail it.
© 2020 Noah Paladino. All ideas set forth in this article are the views of the author. Updated June 30, 2020 at 7:00ish PM. | {"url":"https://www.noahpaladino.com/blog/2020/06/30/transfer.html","timestamp":"2024-11-04T08:50:15Z","content_type":"text/html","content_length":"10774","record_id":"<urn:uuid:98535efc-c101-4bad-bd5b-f606d9059113>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00166.warc.gz"} |
University Physics I - PHY 241
Effective: 2022-05-01
Course Description
Covers classical mechanics and thermodynamics. Includes kinematics, Newton's laws of motion, work, energy, momentum, rotational kinematics, dynamics and static equilibrium, elasticity, gravitation,
fluids, simple harmonic motion, calorimetry, ideal gas law, and the laws of thermodynamics. Part I of II. This is a UCGS transfer course.
Lecture 3 hours. Laboratory 3 hours. Total 6 hours per week.
4 credits
The course outline below was developed as part of a statewide standardization process.
General Course Purpose
PHY 241 is a first semester of a two-semester calculus-based introductory physics with laboratory sequence. It provides the student with a broad understanding of the general concepts and principles
of the physical universe, and prepares the student for advanced study in physical sciences and engineering through development of skills in problem solving, critical thinking and quantitative
reasoning, and an understanding of the methods of scientific inquiry and experiments.
Course Prerequisites/Corequisites
Prerequisites: MTH 263 with a grade of C or better.
Course Objectives
• Measurements, Units and Vectors
□ Define base physical quantities and derived quantities
□ Report measurements of physical quantities using proper number of significant figures and units
□ Convert between different units in the same system, and between different systems of units
□ Differentiate between scalar and vector quantities
□ Define vectors using unit vectors
□ Add and subtract vectors graphically and by component resolution
□ Perform multiplications of vectors by scalars and vectors (scalar and vector products)
• Kinematics: Motion in One and Two Dimensions
□ Define and calculate displacement, average velocity and average acceleration
□ Differentiate between average and instantaneous quantities
□ Interpret and analyze kinematic graphs, and relate position, velocity and acceleration graphs to each other
□ Derive kinematic equations for motion with constant acceleration
□ Identify and use appropriate equations of motion to describe motion in one dimension (including objects falling under the influence of gravity) and two dimensions (including projectile motion
and uniform circular motion)
• Newton's Laws of Motion
□ Explain Newton's three laws of motion
□ Differentiate among mass, weight and apparent weight
□ Draw free-body diagrams for a given physical system
□ Describe general characteristics of static and kinetic friction
□ Apply Newton's laws to a variety of systems including multiple bodies undergoing linear or uniform and non-uniform circular motions involving horizontal, vertical and inclined planes,
strings, and pulleys
• Work and Energy
□ Define and calculate work done by a constant force and by a variable force
□ Calculate work done by a force from force vs position graphs
□ Define kinetic energy, gravitational potential energy near the Earth's surface and elastic potential energy
□ Distinguish between conservative and non-conservative forces
□ Apply Work-Energy principle and conservation of mechanical energy principle
□ Define and calculate power
• Collision and Linear Momentum
□ Define and calculate linear momentum and impulse
□ State the condition for conservation of momentum
□ Define elastic, inelastic and completely inelastic collisions
□ Apply conservation of momentum and conservation of momentum in conjunction with the conservation of energy to systems in 1-D and 2-D collision and explosion
□ Define and calculate center-of-mass of a system of many point masses as well as for bodies with continuous distribution of mass
• Kinematics and Dynamics of Extended Body Undergoing Rotation and Elasticity
□ Define and calculate angular displacement, average angular velocity, and average angular acceleration
□ Differentiate between average and instantaneous quantities
□ Relate linear and angular quantities to each other
□ Define and calculate moment of inertia and rotational kinetic energy
□ Find the moment of inertia of an extended body about an axis of rotation
□ Apply conservation of energy to rotating rigid bodies
□ Define and calculate torque and angular momentum
□ Apply Newton's laws of motion to rotational systems
□ Apply the conservation of angular momentum principle to rotational systems
□ Explain and apply the conditions of static equilibrium
□ Define and calculate different types of strain, stress, and modulus
• Gravitation
□ Explain Newton's law of gravitation
□ Calculate the gravitational force between objects
□ Define and calculate gravitational potential energy
□ Calculate the velocity and period of a satellite in a circular orbit
□ Explain Kepler's laws of planetary motion
• Oscillatory Motion
□ Define oscillation and mathematically describe simple harmonic motion
□ Define amplitude, frequency, period and phase of Simple Harmonic Motion (SHM)
□ Apply conservation of energy principle in a simple harmonic motion
□ Application of SHM to simple pendulum and physical pendulum
□ Describe damped harmonic oscillators
• Fluid Mechanics
□ Define density and pressure and calculate pressure in a fluid
□ Explain and apply Pascal's and Archimedes' principles
□ Describe and apply equation of continuity and Bernoulli's equation to fluid in motion
□ Thermodynamics
□ Describe various temperature scales and convert between different temperature scales
□ Define and calculate linear and volume thermal expansion
□ Define heat capacity and latent heats, and calculate energy needed to change temperature of a substance and of phase change
□ Apply calorimetry principle to thermal system
□ Explain mechanisms of conduction, convection, and radiation and calculate heat transfer rates
□ Explain ideal gas law
□ Define different thermodynamic processes and internal energy
□ Calculate heat transfer and work done by and on an ideal gas during thermodynamic processes
□ Explain the laws of thermodynamics
□ Apply the First Law of Thermodynamics to analyze systems consisting of one or more thermodynamic processes and cyclic processes
□ Define and calculate entropy
• Laboratory Experience
□ Connect topics discussed in lecture to the lab observations
□ Work in the lab safely: follow instructions and proper safety procedures
□ Recognize and be able to use basic laboratory equipment
□ Report measurements using the correct units and number of significant figures
□ Use technology for data acquisition and analysis
□ Be able to create a graph/chart or diagram to report data
□ Interpret graphs, tables and charts
□ Demonstrate written, visual and/or oral presentation skills to communicate scientific knowledge
Major Topics to be Included
• Measurements, Units and Vectors
• Kinematics: Motion in One and Two Dimensions
• Newton's Laws of Motion
• Work and Energy
• Collision and Linear Momentum
• Kinematics and Dynamics of Extended Body Undergoing Rotation and Elasticity
• Gravitation
• Oscillatory Motion
• Fluid Mechanics
• Thermodynamics
• Laboratory Experience | {"url":"https://courses.vccs.edu/courses/PHY241-UniversityPhysicsI/detail","timestamp":"2024-11-02T06:26:38Z","content_type":"application/xhtml+xml","content_length":"16293","record_id":"<urn:uuid:30f2e8fa-878d-4f86-b885-f25022df8c16>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00388.warc.gz"} |
Trees Data Structure | Learn Data Structure & Algorithms | Skillshike
Trees Data Structure
Hi everyone, inside this article we will see the concept about Trees Data Structure.
In data structure and algorithms, a tree is a nonlinear hierarchical data structure that is used to represent a set of elements, such as values or objects, and the relationships between them.
A tree consists of nodes and edges, where the nodes represent the elements, and the edges represent the relationships between them. The topmost node in the tree is called the root node, and the nodes
that do not have any children are called leaves.
Trees are widely used in computer science and are used to solve a wide range of problems. Some of the common applications of trees include:
1. Storing hierarchical data, such as file systems, organization charts, and family trees.
2. Representing sorted data, such as binary search trees.
3. Implementing algorithms, such as AVL trees, red-black trees, and B-trees.
4. Encoding and compressing data, such as Huffman coding.
There are many types of trees in data structure and algorithms, including:
1. Binary tree: A tree where each node has at most two children.
2. AVL tree: A self-balancing binary search tree, where the heights of the left and right subtrees differ by at most one.
3. Red-black tree: A self-balancing binary search tree, where each node is colored either red or black, and the root is black.
4. B-tree: A self-balancing tree that is used to store large amounts of data on disk.
5. Trie: A tree-like data structure used to store strings, where each node represents a single character.
6. Heap: A tree-based data structure used to implement priority queues.
7. Decision tree: A tree used in machine learning to make decisions based on input data.
Each type of tree has its own set of characteristics and applications, and choosing the right type of tree for a specific problem is an important consideration when designing algorithms or data
Important Terms of Trees Data Structure
Here are some important terms related to trees data structure:
1. Root: The root is the topmost node of the tree, and it has no parent.
2. Node: A node is a single element of the tree that contains a value and links to its child nodes.
3. Parent: A node that has one or more child nodes is called a parent node.
4. Child: A node that is directly below a parent node is called a child node.
5. Siblings: Nodes that share the same parent node are called siblings.
6. Leaf: A leaf node is a node that has no child nodes.
7. Depth: The depth of a node is the number of edges between the node and the root of the tree.
8. Height: The height of a tree is the maximum depth of any node in the tree.
9. Subtree: A subtree is a tree that is rooted at a node and contains all of its descendants.
10. Binary tree: A binary tree is a tree in which each node has at most two child nodes, called the left child and the right child.
11. Complete binary tree: A complete binary tree is a binary tree in which every level of the tree is fully filled, except possibly for the last level, which is filled from left to right.
12. Binary search tree: A binary search tree is a binary tree in which the values of the left child nodes are less than the value of the parent node, and the values of the right child nodes are
greater than the value of the parent node.
Understanding these terms is essential for working with trees data structure and implementing algorithms that operate on them.
Basic Operations of Trees Data Structure
The basic operations of trees data structure include:
1. Insertion: This operation adds a new node to the tree.
2. Deletion: This operation removes a node from the tree.
3. Traversal: This operation visits all the nodes in the tree and performs some operation on each of them. There are several ways to traverse a tree, such as pre-order, in-order, and post-order
4. Searching: This operation searches the tree for a specific value.
5. Finding the minimum and maximum values: This operation finds the node with the minimum or maximum value in the tree.
6. Finding the height: This operation finds the height of the tree, which is the length of the longest path from the root to a leaf node.
7. Finding the depth: This operation finds the depth of a node, which is the length of the path from the root to the node.
8. Finding the size: This operation finds the number of nodes in the tree.
9. Checking if the tree is balanced: This operation checks if the tree is balanced, meaning that the heights of the left and right subtrees of every node differ by at most one.
10. Checking if the tree is a binary search tree: This operation checks if the tree is a binary search tree, meaning that the values of the left child nodes are less than the value of the parent
node, and the values of the right child nodes are greater than the value of the parent node.
These basic operations are the building blocks for more complex algorithms that operate on trees, and they are essential for implementing many common tree algorithms.
Key points about Trees Data Structure
Here are some key points about trees data structure:
1. A tree is a hierarchical data structure consisting of nodes and edges.
2. A tree is defined by a root node, which has zero or more child nodes.
3. Each child node in a tree has a parent node, except for the root node which has no parent.
4. A node with no children is called a leaf node, and all other nodes are called internal nodes.
5. The level of a node in a tree is the number of edges from the root node to that node.
6. The height of a tree is the number of levels from the root node to the deepest leaf node.
7. A binary tree is a special type of tree where each node has at most two child nodes.
8. There are many different types of trees, including AVL trees, B-trees, red-black trees, and tries.
9. Trees can be used to represent hierarchical data, store sorted data, implement algorithms, and compress data.
10. Common operations on trees include inserting, deleting, and searching for nodes.
11. Tree traversal refers to the process of visiting all the nodes in a tree, typically in a specific order.
12. The performance of tree operations can be analyzed using the concepts of time and space complexity.
Overall, trees are a fundamental data structure in computer science, with a wide range of applications in many areas of computing. Understanding the characteristics and key points of trees is
important for designing and implementing efficient algorithms and data structures.
We hope this article helped you to understand about Trees Data Structure in a very detailed way.
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CURRENT ELECTRICITY | Acadlly
Electric potential difference and electric current
Electric current: Electric potential difference (p. d) is defined as the work done per unit charge in moving charge from one point to another. It is measured in volts. Electric current is the rate of
flow of charge. P. d is measured using a voltmeter while current is measured using an ammeter. The SI units for charge is amperes (A).
Ammeters and voltmeters
In a circuit an ammeter is always connected in series with the battery while a voltmeter is always connected parallel to the device whose voltage is being measured.
Ohm’s law
This law gives the relationship between the voltage across a conductor and the current flowing through it. Ohm’s law states that “the current flowing through a metal conductor is directly
proportional to the potential difference across the ends of the wire provided that temperature and other physical conditions remain constant”
Mathematically V α I
So V /I = constant, this constant of proportionality is called resistance
V / I = Resistance (R)
Resistance is measured in ohms and given the symbol Ω
1. A current of 2m A flows through a conductor of resistance 2 kΩ. Calculate the voltage across the conductor.
V = IR = (2 × 10-3) × (2 × 103) = 4 V.
2. A wire of resistance 20Ω is connected across a battery of 5 V. What current is flowing in the circuit?
I = V/R = 5 / 20 = 0.25 A
Ohmic and non-ohmic conductors
Ohmic conductors are those that obey Ohms law(V α I) and a good example is nichrome wire i.e. the nichrome wire is not affected by temperature. Non-ohmic conductors do not obey Ohms law i.e. bulb
filament (tungsten), thermistor couple, semi-conductor diode etc. They are affected by temperature hence non-linear.
Factors affecting the resistance of a metallic conductor
1. Temperature – resistance increases with increase in temperature
2. Length of the conductor– increase in length increases resistance
3. Cross-sectional area– resistance is inversely proportional to the cross-sectional area of a conductor of the same material.
Resistivity of a material is numerically equal to the resistance of a material of unit length and unit cross-sectional area. It is symbolized by ρ and the units are ohmmeter (Ωm).
It is given by the following formula;
ρ = AR /lwhere A – cross-sectional area, R – resistance, l – length
Given that the resistivity of nichrome is 1.1× 10-6Ωm, what length of nichrome wire of diameter 0.42 mm is needed to make a resistance of 20 Ω?
ρ = AR /l, hence l = RA/ ρ = 20 × 3.142 × (2.1×10-4) / 1.1 × 10-6 = 2.52 m
Resistors are used to regulate or control the magnitude of current and voltage in a circuit according to Ohms law.
Types of resistors
1. i) Fixed resistors – they are wire-wound or carbon resistors and are designed togive a fixed resistance.
2. ii) Variable resistors– they consist of the rheostat and potentiometer. The resistance can be varied by sliding a metal contact to generate desirable resistance.
Resistor combination
1. a) Series combination
Consider the following loop
Combining those in series then this can be replaced by two resistors of 60 Ω and 40 Ω.
Current through 10 Ω = (p.d. between P and R)/ (30 + 10) Ω
p.d between P and R = 0.8 × Req. Req = (40 × 60)/ 40 + 60 = 2400/ 100 = 24 Ω
p.d across R and P = 0.8 × 24 (V=IR)
therefore, current through 10 Ω = 19.2 / 10 + 30 = 0.48 A
Electromotive force and internal resistance
Electromotive force (e.m.f.) is the p.d across a cell when no current is being drawn from the cell. The p.d across the cell when the circuit is closed is referred to as the terminal voltage of the
cell. Internal resistance of a cell is therefore the resistance of flow of current that they generate. Consider the following diagram; The current flowing through the circuit is given by the
Current = e.m.f / total resistance
I = E / R + rwhere E – e.m.f of the cell
Therefore E = I (R + r) = IR + I r = V + I r
1. A cell drives a current of 0.6 A through a resistance of 2 Ω. if the value of resistance is increased to 7 Ω the current becomes 0.2 A.
calculate the value of e.m.f of the cell and its internal resistance.
Let the internal resistance be ‘r’ and e.m.f be ‘E’.
Using E = V + I r = IR + I r
Substitute for the two sets of values for I and R
E = 0.6 × (2 + 0.6 r) = 1.2 + 0.36 r
E = 0.6 × (7 × 0.2 r) = 1.4 + 0.12 r
Solving the two simultaneously, we have,
E = 1.5 v and R = 0.5 Ω
2. A battery consists of two identical cells, each of e.m.f 1.5 v and internal resistance of 0.6 Ω, connected in parallel. Calculate the current the battery drives through a 0.7 Ω resistor.
When two identical cells are connected in series, the equivalent e.m.f is equal to that of only one cell.
The equivalent internal resistance is equal to that of two such resistance connected in parallel.
Hence Req = R1 R2 / R1 + R2 = (0.6 × 0.6) / 0.6 + 0.6 = 0.36 / 1.2 = 0.3 Ω
Equivalent e.m.f =1.5 / (0.7 + 0.3) = 1.5 A
Hence current flowing through 0.7 Ω resistor is 1.5 A
See also: | {"url":"https://www.acadlly.com/current-electricity/","timestamp":"2024-11-02T01:53:45Z","content_type":"text/html","content_length":"95174","record_id":"<urn:uuid:29f8894d-30be-41f4-b19c-341189dc2140>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00125.warc.gz"} |
The self-force on static and dynamic charges in Schwarzschild spacetime using the method of images | Department of Physics
The self-force on static and dynamic charges in Schwarzschild spacetime using the method of images
The self-force on static and dynamic charges in Schwarzschild spacetime using the method of images
Monday, October 7, 2024 at 4:00 pm
Prof. Alan Wiseman, University of Wisconsin, Milwaukee
Abstract: One of the most basic examples of a self-force phenomenon (sometimes called a radiation reaction force) is that of a small, charged particle near a large spherical mass, such as a
Schwarzschild black hole. If an electric charge is held stationary above the black hole, there are novel electrostatic forces on the particle. If the charged particle is orbiting the mass, the fields
created by the particle back-react on the particle and cause it to depart from its otherwise free-fall, geodesic motion. There are many ways to solve for the forces and motion in these circumstances,
but past solutions have involved considerable technical machinery, and the results are messy and "non-intuitive". I will take a fundamentally new approach to this problem using the method of image
charges. This approach makes the forces easier to visualize and restores an intuitive understanding of the origin of the forces. In the talk, I will make a clear connection between this new work and
the method of images we use in elementary electrostatics.
Speaker bio: Alan Wiseman received his BSc in Physics in 1981 and his M.A in Math in 1984, both from the University of Kansas. He then went to Washington University, where he obtained his MA in
Physics in 1989 and his PhD in 1992. From 1992 to 1994 he was a postdoctoral fellow at Northwestern University and from 1994 to 1997 he worked as a faculty research fellow at Caltech. Wiseman has
been at the University of Wisconsin-Milwaukee since 1998, where he currently holds the rank of associate professor. He served as the Chair of the Department of Physics from 2008 to 2012. Professor
Wiseman has been a member of the LIGO Scientific Collaboration (LSC) since 1999. | {"url":"https://physics.oregonstate.edu/physics-news-events/all-events/the-self-force-on-static-and-dynamic-charges-in-schwarzschild","timestamp":"2024-11-11T03:25:48Z","content_type":"text/html","content_length":"57939","record_id":"<urn:uuid:beeb7c04-73d8-4f60-a639-7bd9c0e900aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00362.warc.gz"} |
Semi-Local Integration Measure of Node Importance
Department of Mathematics, University of Rijeka, R. Matejčić 2, 51000 Rijeka, Croatia
Faculty of Humanities and Social Sciences, University of Rijeka, Sveučilišna Avenija 4, 51000 Rijeka, Croatia
Author to whom correspondence should be addressed.
Submission received: 29 December 2021 / Revised: 17 January 2022 / Accepted: 25 January 2022 / Published: 27 January 2022
Numerous centrality measures have been introduced as tools to determine the importance of nodes in complex networks, reflecting various network properties, including connectivity, survivability, and
robustness. In this paper, we introduce Semi-Local Integration ($S L I$), a node centrality measure for undirected and weighted graphs that takes into account the coherence of the locally connected
subnetwork and evaluates the integration of nodes within their neighbourhood. We illustrate $S L I$ node importance differentiation among nodes in lexical networks and demonstrate its potential in
natural language processing (NLP). In the NLP task of sense identification and sense structure analysis, the $S L I$ centrality measure evaluates node integration and provides the necessary local
resolution by differentiating the importance of nodes to a greater extent than standard centrality measures. This provides the relevant topological information about different subnetworks based on
relatively local information, revealing the more complex sense structure. In addition, we show how the $S L I$ measure can improve the results of sentiment analysis. The $S L I$ measure has the
potential to be used in various types of complex networks in different research areas.
1. Introduction
In the era of Big Data, enormous amounts of data are being collected and analyzed to gain important information and make decisions in a variety of application areas, from managing transportation
networks, organizing distribution and delivery, studying biological networks, to organizing the Internet. Graphs ecame the obvious choice for representing the information structure of many data
systems. In addition to standard graph theory, a modern wave of mathematical approaches, techniques, and tools are being created, developed, and applied by scientists in various fields, in order to
optimize such complex processes.
One of the most important tasks in network analysis is the detection of central or important nodes, which is still a challenge as it depends on the context used. Although many centrality measures are
in common use, the category itself is not precisely defined. Many researchers have attempted to provide a mathematical definition of centrality by establishing a set of criteria that measures must
meet in order to be considered as centrality measures [
]. Moreover, no formal theory has been developed to explain the differences in behavior among them. According to [
], ‘There is certainly no unanimity on exactly what centrality is or on its conceptual foundations, and there is little agreement on the proper procedure for its measurement’. Arguably, that is still
the case today.
Among the widely used centrality measures [
], some express local node properties while others are more global. The simplest and best known centrality measures,
$d e g r e e$
$w e i g h t e d d e g r e e$
(also called
$s t r e n g t h$
), reflect strictly local network characteristics by considering only the immediate neighbourhood. Consequently, the (
$w e i g h t e d$
$d e g r e e$
does not necessarily indicate which node plays the most important role in the whole network. In fact, among the nodes with the same
$d e g r e e$
, some of them may be very central and the others peripheral, which is not detected by the node
$d e g r e e$
alone. Moreover, bridge nodes are an important class of nodes, because they connect subnetworks of the original network, but may not have the highest
$d e g r e e$
. In contrast, other centrality measures are global [
]. For example,
] and
eigenvector centrality
, take into account information about the nodes and edges of the entire network. For example,
closeness centrality
betweenness centrality
involve the shortest paths between the node and all other nodes in the network, and thus reflect global network features. It is important to emphasize that the algorithms for such centrality
measures, which require global information to compute the importance of nodes in a network, often have very high time complexity. For this reason, it is of great interest, especially for networks
with many nodes and/or intricate structure, to develop new and effective centrality measures that are not computed using the entire network information, but instead focus on smaller subnetworks
containing the node [
]. This particular task still opens up possibilities for different approaches to the problem, considering various system properties that are analyzed, including efficiency, connectivity, shortest
paths, influence, robustness, etc.
Although several approaches to defining and analyzing semi-local node importance have been explored in the literature, focusing on various network properties, including survivability and robustness
of the network, to the best of our knowledge none of the existing centralities recognises a high level of node integration. To identify and evaluate the property of interconnectivity among nodes in
the neighbourhood of complex networks, in this paper we introduce
Semi-Local Integration
$S L I$
) (
Supplementary Materials
) centrality measure. This centrality evaluates nodes according to how strongly they are integrated in the local subnetwork. This is obtained by taking into account the
$w e i g h t e d d e g r e e$
centrality of the node itself as well as the
$w e i g h t e d d e g r e e$
of the nodes in its slightly wider neighbourhood. In addition, the coherence of the neighbouring subnetwork is considered.
We illustrate
$S L I$
node importance differentiation among nodes in lexical networks and demonstrate its potential in natural language processing (NLP). We implemented the
$S L I$
centrality measure in the ConGraCNet web application [
], which features the graph-based methodology developed for various lexical tasks. Using the
$S L I$
measure, we have improved lexical tasks that can take advantage of the
$S L I$
’s ability to discriminate the local important nodes in a coordination collostruction weighted graph, such as tasks of sense structure and word sentiment analysis. By using relatively local graph
information, we were able to obtain a computationally efficient centrality measure and thus obtain relevant information about different subgraphs, and use the obtained
$S L I$
values of subgraph nodes in propagating subgraph features such as the associated sense and sentiment potential. This implementation shows that the
$S L I$
centrality is particularly suitable for applications in complex networks and therefore could optimize the analysis of complex network and subnetwork structures, including friend-of-a-friend (FoF)
-based networks such as social networks. The applications clearly go beyond linguistics and social sciences, and extend to other research areas. The Python function implementing the
$S L I$
measure is available at GitHub repository [
The paper is organized as follows. We conclude this section with an overview of the related work. We define the
$S L I$
centrality measure in
Section 2
and demonstrate its suitability for lexical applications in
Section 3
. We conclude with
Section 4
, where we also propose future research directions.
Related Work
The evaluation method for the importance of a node in a network can be based on a local, semi-local, or global approach. Among the reasons of having so many different approaches to evaluate
centralities are the properties of the systems being revealed by the centrality. For example, in a large information network it is very important to backup servers in order not to lose important
data, so a redundant design implies that the most important nodes in this case are those that increase the robustness of the network. Connectivity of the network is another important property in many
applications, such as transportation, security, epidemiology, psychology, social studies and others [
]. In this context, bridge nodes feature prominently as key nodes in complex networks, even though they often do not have a high
$d e g r e e$
, but connect important subnetworks. Identifying bridge nodes of a network therefore requires a topological analysis of the node neighborhood, such as was done for the definition of the
neighborhood-based bridge node centrality tuple in [
The semi-local centrality measure of [
] relates the importance of nodes to the survivability and robustness of networks and identifies key nodes for maintaining the underlying function of the network. The failure of key nodes, such as
bridge nodes, has a large impact on the network, with either negative (e.g., loss of communication) or positive interpretation (e.g., controlling the spread of viruses, preventing security breaches,
etc.). This is orthogonal to our approach, while in [
] occurrence in a cycle reduces the importance due to the available alternative path in the network, occurrence in more cycles in our intended interpretation of high integration indicates better
interconnection of the node into its local subnetwork. Furthermore, the networks considered in [
] are unweighted, while we consider weighted and unweighted networks.
There are several approaches to the semi-local node importance in weighted networks [
]. A generalization of
$d e g r e e$
and shortest path was introduced in [
], rating a node the more important, the more it acquires strength in the network. An interesting study of node importance related to prominence and control in the fields of transportation,
scientific collaboration, and online communication is presented in [
]. It formalizes the tendency of prominent nodes to establish connections among themselves. A series of increasingly selective ‘richness-based clubs’ of nodes is considered, based on the weight of
edges connecting the ‘members’. However, even this approach does not provides insight into qualitative differentiation between nodes in terms of local integration.
2. Semi-Local Intregation Centrality
Centrality is one of the fundamental concepts in graph theory and network analysis. Centrality measures attempt to identify the most important nodes in a network and relate the prominence of the
nodes in a network numerically.
We define the $S L I$ centrality measure of a node that depends on several features of its neighbouring subnetwork. The $S L I$ measure of a node depends on the $w e i g h t e d d e g r e e$ of the
node itself and its neighbours. We classify a node as more important if it is adjacent to more nodes with higher $w e i g h t e d d e g r e e$, i.e., we consider ‘friends’ of the node and their
The measure also considers the number of cycles that include the node. Cycles play an important role in many graph-theory applications, including chemistry, biology, and network analysis. More
specifically, cycles in FoF networks are an indicator of the interconnectedness between friends of friends and their friends. FoF subnetworks with few cycles exhibit low coherence, while nodes that
are included in many cycles are part of a well-connected subnetwork and are part of a coherent local community. Therefore, such nodes can be considered more important in the context of integration.
As an example, consider the small network shown in
Figure 1
, in particular nodes
$v 1$
$v 7$
$v 70$
. Nodes
$v 7$
$v 70$
do not appear in any cycle, while node
$v 1$
is a member of four simple cycles. This indicates a much stronger integration of the node
$v 1$
into the newtork
$G 1$
. Note that one of the simple cycles that includes
$v 1$
has a length of four, i.e., it extends the integration range in the network from a local, immediate neighbourhood to a broader, semi-local subnetwork.
Finally, the measure also takes into account the edge weight, which is a measure of the relatedness of the endpoint nodes. For example, node $v 70$ can be considered more strongly connected to the
rest of the network due to its higher weight than node $v 71$ with a lower weight.
2.1. $S L I$ Definition
In this section, we propose a graph-theoretical definition of node integration. Given an undirected and weighted graph $G = ( V , E )$, we use the following notation in the definition of the $S L I$
node centrality:
• $deg ( v )$—denotes the $d e g r e e$ of the node $v ∈ V$;
• $d w ( v )$—denotes the $w e i g h t e d d e g r e e$ ($s t r e n g t h$) of the node $v ∈ V$;
• $e a b = e b a$—denotes the edge between the nodes $a , b ∈ V$;
• $w ( e )$—denotes the weight of the edge $e ∈ E$;
• $E v$—denotes the set of edges incident to $v ∈ V$, $E v : = { x ∣ e v x ∈ E }$;
• $P G$—denotes the cycle basis of the graph G;
• $p ( e )$—denotes the number of cycles in $P G$ that contain the edge $e ∈ E$.
The cycle basis
$P G$
is the minimal set of simple cycles of
that allows every cycle of
to be expressed as a symmetric difference of basis cycles, capturing the local interconnection reach of nodes. The number of local simple cycles that include the edge
$p ( e )$
, contributes to the importance of nodes incident to
. We define the
edge cycle factor
of each
$e ∈ E$
$λ ( e )$
, as:
$λ ( e ) : = p ( e ) + 1 .$
For each node
we calculate the node importance by increasing the node’s
$w e i g h t e d d e g r e e$
by the importance contribution of its incident edges:
$I ( a ) : = d w ( a ) + ∑ e a b ∈ E a I ( e a b ) ,$
where the importance of the edge
$I ( e )$
, is defined as follows:
$I ( e a b ) : = λ ( e ) · d w ( a ) + d w ( b ) − 2 w ( e ) · w ( e ) · d w ( a ) d w ( a ) + d w ( b )$
Node importance is then normalized to represent the percentil of node importance share in the graph
. This defines
$S L I$
score of the node
$S L I ( a ) : = I ( a ) S G · 100 ,$
$S G$
is the sum of the unnormalized importance scores of all nodes of
$S G : = ∑ b ∈ V I ( b ) .$
In a more computational presentation, for a given graph $G = ( V , E )$ and the corresponding edge weights and weighted degree scores of its nodes, the calculation of $S L I$ is performed by the
following algorithm described in an intuitive metalanguage:
• Find cycle basis of G, $P G$;
• Find $p ( e )$ for all $e ∈ E$;
• Find
$λ ( e )$
for all
$e ∈ E$
, according to (
• Find the set of edges $E ( a )$ for all $a ∈ V$;
• Find
$I ( e )$
for all
$e ∈ E$
, according to (
• Find
$I ( a )$
for all
$a ∈ V$
, according to (
• Find
$S ( G )$
, according to (
• Find
$S L I ( a )$
for all
$a ∈ V$
, according to (
The Python function implementing the
$S L I$
measure is available at GitHub repository [
2.2. Discussion on $S L I$ Definition
In a nontrivial connected graph, by definition, none of the three factors in the above definition of edge importance is equal to zero, i.e., each of these three components increases the edge
importance. Consequently, each edge incident to a node increases the node’s importance. Note that the $S L I$ of isolated nodes is 0, while the importance of edge points of isolated edges is reduced
to the weight of the edge.
$S L I$
scores of all nodes of the graph
$G 1$
shown in
Figure 1
are listed in
Table 1
. Since
$S L I$
is a normalized measure, it can help in the interpretation and comparison of importance of nodes in the graph and in the analysis of importance distribution in different graphs. Note that node
$v 1$
has the highest
$S L I$
score in
$G 1$
and carries over
$40 %$
of the total importance in the graph. Several other top central nodes have high scores, namely
$S L I ( v 2 ) = 16.225$
$S L I ( v 4 ) = 13.269$
$S L I ( v 6 ) = 9.614$
$S L I ( v 5 ) = 6.961$
$S L I ( v 3 ) = 6.664$
. Note that among the top ranked nodes, the highest score is more than six times the lowest score, illustrating the non-trivial allocation of importance due to the graph structure. Many nodes in the
graph have very low importance, including all leaf-like nodes such as
$v 72$
$S L I ( v 72 ) = 0.065$
. This is an indicator of much lower node integration in the graph.
We now analyze how the definition of the $S L I$ centrality measure reflects the structure of the graph. The $w e i g h t e d d e g r e e$ of a node is increased by the importance of its edges.
Through the importance of associated edges, the $S L I$ score reflects the integration of the node into the local subgraph.
The first component in (Equation (
)) that contributes to the edge importance is the edge cycle factor
$λ ( e )$
, which increases the importance based on the number of simple cycles of
on which the node sits. For example, in the graph shown in
Figure 1
, the node
$v 1$
is contained in the largest number of simple cycles, four. The nodes
$v 4$
$v 6$
are also included in a relatively large number of cycles, namely three, while the leaf-like nodes are obviously not included in any cycle. However, node
$v 7$
is not a leaf-like node, but it is also not included in any cycle. The node
$v 7$
together with the leaves
$v 70 , v 71$
$v 72$
, represents a subgraph of
$G 1$
that is not strongly connected to the rest of the graph.
The second component reinforcing the importance of the edge
$d w ( a ) + d w ( b ) − 2 w ( e a b ) ,$
reflects the integration of the endpoints of the edge with the rest of the graph. Note that
$d w ( a ) − w ( e a b )$
corresponds to the total contribution of the edges incident to node
, except edge
, to the
$w e i g h t e d d e g r e e$
. This represents the strength of the connection of
with its other neighbours. Thus, this component, denotes the remaining integration of the two endpoint nodes of the edge
$e a b$
into the local subgraph. Therefore, the integration of the node itself and the integration of its neighbours into the local subgraph, strengthens the node importance. As an example, consider the
$e v 1 v 7$
$e v 7 v 72$
. Although these edges have the same weight, the endpoint nodes of edge
$e v 1 v 7$
are more strongly connected to the rest of the graph. The edge
$e v 7 v 72$
exhibits a much less overall integration in the graph.
The third component,
$w ( e a b ) · d w ( a ) d w ( a ) + d w ( b ) ,$
reflects both the weight of the edge and the (im)balance in the
$w e i g h t e d d e g r e e$
of its endpoint nodes. The weight of the edge was not reflected in the other components, but should not be disregarded. For example, among the leaf-like nodes connected to node
$v 2$
, nodes
$v 20$
$v 21$
$v 22$
$v 23$
, node
$v 21$
is incident with the edge with the highest weight. The node
$v 21$
appears to be the most important among the mentioned nodes, since it is best connected to the rest of the graph through the edge
$e v 2 v 20$
. This can be observed from the
$S L I$
scores, since the score
$S L I ( v 21 )$
is higher than the
$S L I$
score of any of the nodes
$v 20$
$v 22$
$v 23$
. Similarly, the same edge
$e v 2 v 20$
contributes more to the importance of
$v 2$
than the edges associated with other leaf-like nodes.
The remaining fraction involving the $w e i g h t e d d e g r e e s$ of the endpoint nodes a and b adds the larger contribution to the endpoint with the higher $w e i g h t e d d e g r e e$, i.e.,
the better integrated endpoint. For example, in graph $G 1$, node $v 1$ has higher $w e i g h t e d d e g r e e$ than node $v 2$, so edge $e v 1 v 2$ should contribute more to node $v 1$.
Together with the
$S L I$
scores of the nodes of
$G 1$
, we list the scores of a number of other centrality measures in
Table 1
. The comparison shows that the values of
$S L I$
behave differently from all the standard scores, namely,
$d e g r e e$
$w e i g h t e d d e g r e e$
$b e t w e e n n e s s$
$P a g e R a n k$
. Note that the nodes in the table are ordered decreasingly according to
$S L I$
, but none of the scores according to another centrality result ordered. Moreover, the importance polarization by
$S L I$
is much more pronounced compared to all other centralities, as can be seen in
Table 1
. Compared, for example, with
$b e t w e e n n e s s$
$S L I$
reveals some differentiation between leaf-like nodes, albeit to a small extent, e.g., node
$v 21$
has a higher
$S L I$
score than node
$v 22$
due to the incidence of edge
$e v 2 v 21$
having a larger weight than edge
$e v 2 v 22$
The rather small network example given in
Figure 1
already illustrates that overall the
$S L I$
displays considerable polarization in ratings of the nodes according to the strength of their interconnection in the local subnetwork.
2.3. $S L I$ in Unweighted Graphs
In the case of an unweighted graph
, the
$S L I$
measure calculation reduces to the following formula:
$S L I ( a ) : = I ( a ) S G · 100 ,$
where the importance of the node
$I ( a )$
, is given by:
$I ( a ) = deg ( a ) + ∑ e a b ∈ E a I ( e a b ) ,$
and the importance of the edge
$I ( e )$
, is:
$I ( e a b ) : = λ ( e ) · deg ( a ) + deg ( b ) − 2 · deg ( a ) deg ( a ) + deg ( b ) .$
As an example of an unweighted graph and its
$S L I$
scores, we take the unweighted version
$G 2$
of the graph
$G 1$
shown in
Figure 1
Table 2
shows the corresponding
$S L I$
scores, with the nodes ordered as in
Table 1
, i.e., according to the original, weighted
$S L I$
scores. In addition,
$d e g r e e$
$b e t w e e n n e s s$
$P a g e R a n k$
centralities of
$G 2$
nodes are shown. As with the weighted version of the graph, the
$S L I$
ordering of importance does not coincide with any of the other centralities. It exhibits a stronger polarization of importance and a differentiation between the very important nodes and the nodes
with the lowest importance.
The normalization of the
$S L I$
measure provides a clear interpretation of the relative importance of the nodes in the graph and allows for a comparison between the two versions of the graph, i.e., the comparative analysis of the
values shown in
Table 1
Table 2
The role of edge weights is evident in a more pronounced relative scoring and ordering of nodes. Changing just one edge weight, e.g., decreasing the weight of edge $e v 1 v 4$ from 3 to 0.4, has a
large effect on node importance ordering, permuting nodes from $v 4 , v 6 , v 3$ to $v 6 , v 3 , v 4$. The differences in the $S L I$ distribution in the graph can be even more pronounced than in the
above example. This is to be expected in complex networks due to the non-uniform distribution of edge weights. Larger differences between edge weights in the network, make the weighted $S L I$
differentiation of node importance even more pronounced.
3. Application of $SLI$ in Lexical Networks
Graphs are widely used in NLP to represent large amounts of lexical data. Graph-theoretic analysis of lexical networks can reveal features useful for human review and consequently provide insights
and ideas for automatic methods (for an overview of graph methods in NLP, see [
In our recent work [
], we used graph theory in an interdisciplinary approach to tackle NLP tasks such as semantic similarity identification, sense association and structure, lexical community labeling, and sentiment
This research has provided the main motivation for defining $S L I$ centrality. The node importance expressing high integration of nodes into the local semantic community, was not adequately
represented by standard centrality measures. Our goal was therefore to define a centrality measure that reflects the importance of nodes in complex lexical networks and many other networks with the
specific structure studied from a similar point of view.
3.1. Application of $S L I$ in the Analysis of Sense Structure
For the underlying graph representation of lexical networks, we extracted lexemes from a tagged corpora based on coordinated syntactic-semantic constructions that reveal a semantic similarity [
Figure 2
represents such a syntactic-semantic based lexical network with sense associations of a seed word
The network is constructed from the 15 most frequent coordination collostructions [
], called friends of the seed lexeme
, and the subsequent 15 most important collostructions of each friend lexeme. The lexemes are represented as nodes, while the level of association between two lexemes is expressed by the corpus
measure of the collocation, stored in the weight property of the corresponding edge. The weighted undirected FoF network captures the semi-local emergent conceptual relatedness and the semantic
domain’s structure of a seed lexeme. The subgraph clusters structured by the lexical relations and their prominent nodes reveal the polysemous aspects of the source lexeme. The lexical network
clusters represent the cognitively latent associative domains necessary for understanding the overall (poly)semantic structure of a source lexeme. The association strength of a particular node in
this network relates to (a) the semantic relatedness to the source lexeme in the overall network (b) the sense contribution of this node to the semantic domain in a sense cluster of a source node.
Clearly, more saliently connected nodes in the network carry more relevance for the analysis of lexical sense relatedness and sense potential. This semantic salience of a lexical node can be
calculated using the graph centrality measures mentioned above. However, each measure has its own method of calculating the most central nodes, which in this case affects the assignment of a
semantically salient nodes. The
$S L I$
measure captures the semantic salience that is defined by the associative strength (edge weight) of a node, integration with other nodes (degree, weighted degree), and the topological structure of
local integration expressed by the graph cycles (betweenness). The integration of all these dimensions gives the
$S L I$
the necessary local resolution that allows for a more complex discrimination of a node salience than any of the previously mentioned centrality measures.
The centrality scores of the most central nodes of the network shown in
Figure 2
are listed in
Table 3
. The nodes are sorted by
$S L I$
scores and illustrate the
$S L I$
importance distribution, which ranges from about 11% to less than 0.2% of relative importance in the entire network. This can be easily seen in
Figure 2
by the node size, which reflects the node
$S L I$
score. The striking polarisation in the scores shows the two most important nodes in the semantic network of the source lexeme, the lexemes
. Then, the group of lexemes that are highly relevant in the associated conceptual domains of the lexeme
is revealed. These include the lexemes
time, dedication, research determination, effort, study, commitment, home, project, school, life
. The remaining nodes of the network are identified as more peripheral in the overall sense of the source lexeme.
More importantly, as illustrated in
Figure 2
using different node colours, network communities are formed representing different senses of the source lexeme, each containing the nodes with comparable importance in the subnetwork. For example,
the source lexeme
is a part of community containing the lexemes
dedication, research, commitment
, which have the relatively highest importance in the subnetwork and carry the dominant semantic properties of the lexical sense community. Similarly, the lexemes
are the most important nodes in the smaller subnetwork that contains other lexemes related to resources, such as
energy, expense, money, marketing, talent, space, date
. These two nodes have relatively close
$S L I$
scores that are much higher than those of the other nodes in their subnetwork, and thus carry the core sense of this community. Such diversification of nodes is not achieved by either
weighted degree
. In particular,
weighted degree
does not emphasize the centrality of the source node, since many nodes in the network have higher
weighted degree
values. On the other hand, the betweenness scores often drop to zero for a significant number of nodes, which then nullifies their participation in the sense analysis of the local community.
The aspect of capturing the fine-grained resolution of semantic relatedness to the source node can be used for pruning the less related nodes. This task is particularly important for larger FoF
networks with several hundred nodes from the network. Pruning out less related nodes reduces information noise and semantic drift to less related senses. Furthermore, such a pruned lexical network
allows semantic analysis with less data overload, preserving and highlighting the most relevant information.
Figure 3
represents a pruned network from
Figure 2
with a seed noun lexeme
. The network was pruned to the top 50% of nodes according to
$S L I$
The most important feature for the semantic analysis of such lexical networks is the integration of nodes into subnetworks, which are themselves structured with respect to the source node and form
the sense structure of the source concept. The primary meaning of the concept is reflected in the largest community, which usually contains the most important node in the network, the source lexeme
node. The size of the preserved communities reflects the sense prevalence or the less frequent word sense/usage.
In summary, the centrality obtained with
$S L I$
has promising applications for lexical network analysis, starting with filtering out less relevant nodes in order to simplify and emphasize the network structure and reduce computational data
overload. It also highlights the importance of the source node and enables more qualitative semantic analysis at the level of subnetwork communities identified by a community detection algorithm [
] at a selected granularity, i.e., resolution. These properties of the
$S L I$
measure could advance NLP methods, e.g., improve the results we have obtained in the task of sense labeling [
3.2. Application of $S L I$ in Sentiment Analysis
Sentiment analysis uses NLP techniques and resources to address affective and subjective phenomena in texts by classifying linguistic expressions from single words (lexemes) to multi-word expressions
and longer texts, and assigning them a normalized range of values, typically on a scale from −1 to 1 [
The simplest linquistic expressions that articulate sentiment are words. The types of words that express sentiment are nouns, adjectives, verbs and adverbs. Some words represent concepts with
predominantly culturally associated positive feelings, such as love, joy, heart, while others, such as violence, death, war, failure, express negative feelings. Classification and numerical
evaluation of word sentiment is estimated subjectively by annotators based on psychological evaluation of words or by extending the already annotated dictionaries using various techniques and
resources. The information about the sentiment expressed by words is catalogued in sentiment dictionaries, using only positive/neutral/negative classification or a more refined estimates using
numerical values and different dimensions.
In the examples we present below, we use the SenticNet 6 sentiment dictionary [
], one of the most comprehensive sentiment dictionaries available. Word sentiment in SenticNet 6 is scored on a number of semantic dimensions, namely the ‘polarity_score’, ‘sensitivity’, ‘attitude’,
‘temper’, and ‘introspection’, and is expressed in numerical values between −1 and 1. For example, lexeme
has ‘polarity_score’ of
$− 0.81$
while the following words are scored as follows:
heart $0.9$
love $0.83$
partner $0.45$
fighter $0.343$
flower $0.054$
response $− 0.2$
desperation $− 1$
In our work, on
Sentiment Potential
(SP) [
], we addressed the scarcity of available sentiment dictionaries and problems in sentiment analysis of polysemous and homonymous words, i.e., words with different and potentially unrelated meanings.
For example, lexeme
denotes both an animal and a baseball device. For another example, lexeme
can be associated with a disease or a computer malware. Moreover, the different senses of a lexeme may have different affective potential. We addressed the above two issues using sentiment
propagation, which is intended to reflect both the main affective state and the structure of the feelings expressed by a concept, either a lexeme or a conceptual community of lexemes. The ConGraCNet
] lexical network, which is a word sense network structure, provides information for an enriched, more complex sentiment representation of sentiment expressed by a word. For an example of SP, see
Figure 4
showing SP of the lexeme
. As can be seen in
Figure 4
, the sentiment scores of a single lexeme across its different sense communities are reflected through a vivid illustrative, multi-layered representation.
The SP illustration shown in
Figure 4
includes the horizontal line marking the word sentiment score from the SenticNet 6 dictionary, i.e., the
Original Sentiment Value
$O D V$
). Another horizontal line marks the
Assigned Dictionary Value
$A D V$
) [
] of the lexeme which is calculated from the corpus-based coordination dependency lexical graph
$F o F a$
constructed for a chosen source lexeme
using the ConGraCNet method [
$A D V$
of a lexeme
is calculated using the available SenticNet 6
$O D V$
sentiment values of lexical nodes in the
$F o F a$
graph as follows:
$A D V ( a ) : = ∑ x ∈ V a D v ( x ) · b ( x ) ∑ x ∈ V a D b ( x ) ,$
$b ( x )$
is the
$b e t w e e n n e s s$
measure of the node
in the
$F o F a$
$V a D$
is the set of nodes
$x ∈ F o F a$
with available
$O D V$
in SenticNet 6, and
$v ( x )$
is the
$O D V$
score of lexeme
in SenticNet 6. In addition to assigning the sentiment value to a source lexeme, the same method of propagation can be used to assign the
Graph Sentiment Value
$G S V$
) [
] to a graph. In the case of SP,
$G S V$
values are assigned to each subgraph of
$F o F a$
representing a sense community in order to estimate the sentiment of the source lexeme in the specific semantic domain.
The third horizontal line in SP illustration marks the mean of the propagated sentiment values over identified lexical subgraphs, Average Sentiment Potential (
$A S P$
) [
], represents the average sentiment value of the seed lexeme over its semantic communities:
$A S P ( a ) : = ∑ i = 1 m G S V ( G a i ) · ∑ x ∈ G a i w ( x ) ∑ x ∈ G a w ( x ) ,$
is the number of subgraph communities
$G a i$
identified in the seed lexeme graph
$G a$
$G S V ( G a i )$
is the
$G S V$
value of the subgraph
$G a i$
, and
$w ( x )$
is the
$w e i g h t e d d e g r e e$
of the node
$G a$
The four communities identified in
Figure 4
refer to different semantic domains of the lexeme
. Each community is represented by a circle whose size represents its share in the total network centrality, which in this case ranges from 28.3% to 19.5%. The vertical placement and the colour of
the community representation circle expresses the sentiment value according to the scale shown on the right.
Note that the $O D V$ assigned to the lexeme work is very positive, i.e., 0.9. The first approximation obtained using the ConGraCNet method, namely $A D V$ sentiment value of lexeme work is lower
than the $O D V$ sentiment value, but still rather high, i.e., 0.78. Interestingly, further syntactic-semantic network construction based on the usage of the lexeme in expressed in the corpus and the
related analysis, reveals a considerably lower sentiment value related to work across all the identified senses.
The centrality measure used in the calculation of SP shown in
Figure 4
is betweenness, as originally proposed in [
]. In comparison, SP of the same lexeme computed using
$S L I$
scores leads to more differentiated sentiment scores across different sense communities of the lexeme, as shown in
Figure 5
Table 4
When comparing the two sentiment potentials, various effects of the SLI measure can be noticed when comparing the two sentiment potentials, from the resulting community sizes reflecting the overall
importance of the community lexemes, to vertical placements and colours indicating the orientation and intensity of the communities’ sentiment score.
The community with lexemes associated around lexemes work, play, research, study, for example, displays higher, more positive sentiment value and larger size, i.e., a more prominent, in fact the
primary sense of the source lexeme. At the same time, the community containing lexemes dedication, commitment, determination, has been reduced in relative size and displays lower sentiment value.
The community sentiment value is propagated from the available original sentiment values of its lexeme nodes. As shown in
Table 3
, the most important nodes in the third community according to
$S L I$
are of positive sentiment score, but of lower values (0.034 and 0.231), while the most positive lexemes from the community, that is nodes
(with the sentiment score of 1) and
(with the sentiment score of 0.704) are of lower
$S L I$
importance. They hence contribute less to the community sentiment score, which is consequently lowered from 0.47 to 0.33.
This effect of accentuating or flattening sentiment differentiation originates in the propagation method of the sentiment value from the available values in the network. Since the community of
associated lexemes may contain lexemes with different sentiment values, see
Table 3
Table 4
, the sentiment value, when propagated as an average of the values in the neighbouring subnetwork, is often diluted or blurred by less important nodes in the community. This issue is mitigated by
using a centrality with a high polarization, which is one of the main characteristics of the
$S L I$
3.3. Further Areas of Applications
Besides the linguistic syntactic-semantic applications, that are the main driving force behind the conceptualization of this measure, there are other foreseeable applications of
$S L I$
. These include, for example, the analysis of various types of network structures. For instance,
$S L I$
can be introduced into graph analysis of vulnerability and identification of bottlenecks in transportation networks [
]. Further applications can be considered for biology structures where a cell, gene, or protein can be considered as a node, and the connecting element as an edge [
], infectious disease spread analysis [
], information network modelling, social networks [
], interaction between social agents such as the influence spreading [
], the impact of cultural norms and legislation [
], etc. The appropriateness of the
$S L I$
measure is highlighted in graph analysis of node influence within network structures that involve a weighted type of relation between different entities, components, or agents, with strong
topological coherence within a subgraph component.
4. Conclusions
In this paper, we propose a new centrality measure SLI that evaluates the interconnectedness of nodes in a semi-local subgraph. The strong integration of a node within its neighbourhood is reflected
by the weighted degree of the node, the weighted degrees of the nodes it is connected to, and the number of cycles it is contained in. The larger all of these factors are, the better the coherence of
the neighbouring subnetwork.
The definition of the SLI measure was motivated by applications in lexical networks and has proven useful in the NLP tasks of sense structure representation and sentiment analysis. Further use of the
SLI measure appears promising for other NLP tasks, such as sense labeling, where it can help in identifying and classifying hypernym candidates. Moreover, the use of SLI centrality is not limited to
lexical networks and opens up studies in various applications of FoF and complex networks in other research areas.
Author Contributions
Conceptualization, S.B.B. and T.B.K.; methodology, T.B.K., S.B.B. and B.P.; software, B.P. and S.B.B.; validation, T.B.K., S.B.B. and B.P.; formal analysis, T.B.K., S.B.B. and B.P.; investigation,
T.B.K., S.B.B. and B.P.; resources, B.P. and S.B.B.; data curation, B.P. and S.B.B.; writing—original draft preparation, T.B.K., S.B.B. and B.P.; writing—review and editing, T.B.K., S.B.B. and B.P.;
visualization, B.P., S.B.B. and T.B.K.; supervision, B.P., S.B.B. and T.B.K.; project administration, B.P. and T.B.K.; funding acquisition, B.P. and T.B.K. All authors have read and agreed to the
published version of the manuscript.
This work has been supported in part by the Croatian Science Foundation under the project UIP-05-2017-9219 and the University of Rijeka under the project UNIRI-human-18-243.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Conflicts of Interest
The authors declare no conflict of interest.
Figure 2. ConGraCNet network representation of the sense structure of a seed noun lexeme work. The node size reflects the node $S L I$ score.
Figure 3. ConGraCNet network representation of the sense structure of a seed noun lexeme work pruned to top 50% nodes accoring to $S L I$ centrality. The node size reflects the node $S L I$ score.
Figure 4. Sentiment potential (SP) of noun lexeme work propagated from SenticNet using betweenness centrality.
Figure 5. Sentiment potential (SP) of noun lexeme work propagated from SenticNet using $S L I$ centrality.
Table 1.
Comparison of different centrality measures of nodes in graph
$G 1$
illustrated in
Figure 1
Node $SLI$ $Degree$ $Weigthed Degree$ $Betweenness$ $PageRank$
$v 1$ 40.225 6 12.65 0.594 0.148
$v 2$ 16.225 7 9.4 0.345 0.118
$v 4$ 13.269 6 7.45 0.246 0.096
$v 6$ 9.614 5 6.1 0.147 0.076
$v 5$ 6.961 5 5.7 0.193 0.077
$v 3$ 6.664 6 5.65 0.259 0.081
$v 60$ 1.671 2 2.7 0.024 0.035
$v 7$ 1.238 4 3.6 0.239 0.071
$v 40$ 0.969 2 2 0.0 0.029
$v 21$ 0.569 1 2 0.0 0.027
$v 52$ 0.546 2 1.4 0.0 0.022
$v 70$ 0.39 1 2 0.0 0.039
$v 23$ 0.224 1 1 0.0 0.017
$v 31$ 0.21 1 1 0.0 0.018
$v 20$ 0.166 1 0.8 0.0 0.015
$v 51$ 0.16 1 0.8 0.0 0.015
$v 41$ 0.15 1 0.75 0.0 0.014
$v 71$ 0.149 1 0.8 0.0 0.019
$v 22$ 0.139 1 0.7 0.0 0.013
$v 30$ 0.11 1 0.6 0.0 0.013
$v 61$ 0.088 1 0.5 0.0 0.011
$v 42$ 0.067 1 0.4 0.0 0.010
$v 50$ 0.067 1 0.4 0.0 0.011
$v 32$ 0.067 1 0.4 0.0 0.011
$v 72$ 0.065 1 0.4 0.0 0.013
Table 2.
Comparison of different centrality measures of nodes in graph
$G 2$
, which is the unweighted version of the graph
$G 1$
illustrated in
Figure 1
Node $SLI$ $Degree$ $Betweenness$ $PageRank$
$v 1$ 18.433 6 0.594 0.087
$v 2$ 16.708 7 0.345 0.111
$v 4$ 15.543 6 0.246 0.091
$v 6$ 11.437 5 0.147 0.074
$v 3$ 12.892 6 0.259 0.094
$v 5$ 9.51 5 0.193 0.077
$v 7$ 3.512 4 0.239 0.073
$v 60$ 1.943 2 0.024 0.032
$v 40$ 1.951 2 0.0 0.032
$v 52$ 1.882 2 0.0 0.032
$v 21$ 0.427 1 0.0 0.02
$v 23$ 0.427 1 0.0 0.02
$v 31$ 0.418 1 0.0 0.019
$v 20$ 0.427 1 0.0 0.02
$v 41$ 0.418 1 0.0 0.019
$v 51$ 0.407 1 0.0 0.019
$v 22$ 0.427 1 0.0 0.02
$v 71$ 0.39 1 0.0 0.022
$v 30$ 0.418 1 0.0 0.019
$v 61$ 0.407 1 0.0 0.019
$v 70$ 0.39 1 0.0 0.022
$v 32$ 0.418 1 0.0 0.019
$v 42$ 0.418 1 0.0 0.019
$v 50$ 0.407 1 0.0 0.019
$v 72$ 0.39 1 0.0 0.022
Table 3. Centrality in lexical dependency graph of noun lexeme work and SenticNet 6 sentiment scores.
Lexeme $Weighted Degree$ $Betweenness$ $SLI$ SenticNet6 $ODV$
work 96.35 6219.4667 11.0602 0.9
family 143.21 1615.5262 10.7711 0.883
time 115.68 1057.2845 7.7119 -
dedication 103.43 788.0095 7.3328 0.034
research 117.88 1459.5190 7.0717 0.883
determination 108.08 1037.9310 7.0436 0.231
effort 93.85 982.9595 6.3767 0.037
study 120.33 1687.6012 6.2252 -
commitment 97.82 855.6643 6.1272 0.704
home 107.3 1059.3940 6.0986 -
project 115.6 1705.4262 5.6272 0.9
school 107.21 1129.0250 5.6033 -
life 101.38 1188.8881 4.6721 -
play 96.1 1677 2.5525 -
passion 24.54 0 0.3868 1
business 23.7 21.0357 0.3673 -
money 18.79 0 0.1937 0.065
Table 4. Sentiment potential (SP): $G S V$ scores of lexical communities of noun lexeme work propagated from SenticNet 6.
SenticNet 6 Work-n
Community Lexemes $Betweenness GSV$ $SLI$$GSV$
1 life-n, school-n, home-n, family-n, love-n, property-n, business-n, hospital-n, student-n, community-n, program-n, building-n 0.37 0.61
2 dedication-n, commitment-n, determination-n, passion-n, perseverance-n, enthusiasm-n, patience-n, loyalty-n, persistence-n, courage-n 0.47 0.33
3 work-n, play-n, research-n, study-n, project-n, patient-n, development-n, analysis-n, science-n 0.76 0.88
4 effort-n, time-n, money-n, energy-n, resource-n, cost-n, attention-n, people-n 0.16 0.07
$ODV$: 0.9 $betweenness ADV$: 0.78 $SLI ADV$: 0.65
$betweenness ASP$: 0.45 $SLI ASP$: 0.56
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IDEMS has a long term relationship with Bahir Dar University.
When IDEMS open courses were launched, lecturers from the Mathematics Department at Bahir Dar University were keen to investigate integrating automated, electronic assessment into their teaching.
The course intends to prepare natural science students with the basic concepts and materials from mathematics that necessitate a good foundation to treat fundamental mathematical tools in science.
This course rigorously discusses the basic concepts of logic and set theory, the real and complex number systems, mathematical induction, least upper bound and greatest lower bound, functions
and types of functions, polynomial and rational functions, logarithmic and exponential functions, trigonometric functions, hyperbolic functions and their graphs and analytic geometry.
The course introduces students to the concepts and analytical methods for solving partial differential equations. It is based on the previous core mathematics courses to develop more advanced ideas
in differential and integral calculus.
This course covers basic concepts of partial differential equations (PDE), Fourier series and Integrals, some techniques of solutions of first order PDEs, second order PDEs and analytical methods of
This is a template copy for the University of Bahir Dar
This is an introductory course to linear Algebra. The course notes are based on the book: A first course in Linear Algebra by Robert A. Beezer. See the course web-page where you can download versions
of the book.
This is an introductory course to Calculus first implemented for Maseno University, Kenya in 2019 for their Calculus I course.
The assessment for the course is 10 weekly mastery and test quizzes. The questions were almost entirely developed using STACK for automated feedback. Most questions include a randomised component.
The course refers to the open e-book Calculus I by Paul Dawkins. | {"url":"https://ecampus.idems.international/course/index.php?categoryid=3","timestamp":"2024-11-02T03:04:33Z","content_type":"text/html","content_length":"77145","record_id":"<urn:uuid:398f3cf0-afca-49bb-a335-7e52570d44f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00293.warc.gz"} |
Find all values of k for which the equation has two real solutions.
-7w + (k - 1) = -4w²
Solved on Nov 15, 2023
$k \in \left(-\infty, \frac{11}{7}\right) \cup \left(\frac{11}{7}, \infty\right)$
STEP 1
Assumptions1. The equation is -7w + (k-1) = -4w^ . We are looking for all values of $k$ for which the equation has two real solutions3. The equation is a quadratic equation in the form $ax^ + bx + c
=0$, where $a$, $b$, and $c$ are constants and $x$ is the variable. In this case, $w$ is the variable, $-4$ is $a$, $-7$ is $b$, and $(k-1)$ is $c$.
STEP 2
For a quadratic equation to have two real solutions, the discriminant must be greater than zero. The discriminant is given by $b^2 -4ac$.
$Discriminant = b^2 -4ac$
STEP 3
Plug in the values for $a$, $b$, and $c$ into the discriminant formula.
$Discriminant = (-7)^2 -(-)(k-1)$
STEP 4
implify the discriminant.
$Discriminant =49 +16(k-1)$
STEP 5
For the equation to have two real solutions, the discriminant must be greater than zero.
$49 +16(k-1) >0$
STEP 6
implify the inequality.
$16k -16 +49 >0$
STEP 7
Combine like terms.
$16k +33 >0$
STEP 8
Isolate $k$ on one side of the inequality.
$16k > -33$
STEP 9
Divide both sides of the inequality by $16$ to solve for $k$.
$k > -\frac{33}{16}$All values of $k$ greater than $-\frac{33}{16}$ will result in the equation having two real solutions. | {"url":"https://studdy.ai/learning-bank/problem/k-such-that-k-geq-3-_T3Ft9iicU96xxCJ","timestamp":"2024-11-05T15:43:02Z","content_type":"text/html","content_length":"154631","record_id":"<urn:uuid:34056e5c-3c6b-4651-a6bb-224dfc8c690b>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00337.warc.gz"} |
mathematical algorithms
Editor: test1
Time: 2018/03/23 16:28:04 GMT+0
-On Wednesday, June 15, 2005 12:42 AM daly@axiom-developer.org wrote:
-Subject: [Axiom-developer] Mathematics Subject Classification
-There is a mathematics subject classification scheme list at
-and it appears to be used at
-An interesting project would be to classify Axiom's math algorithms
-relative to this scheme.
Computer algebra systems like FriCAS implement a very large number of mathematical algorithms. By that we mean:
1. Of or relating to mathematics.
2. Meaning
a Precise; exact. b Absolute; certain.
Ref: http://www.answers.com/mathematical&r=67
A step-by-step problem-solving procedure, especially an established, recursive computational procedure for solving a problem in a finite number of steps.
Ref: http://www.answers.com/algorithms&r=67
Links to other pages on related subjects:
Core algorithms in FriCAS deal with commutative algebra:
• polynomial arithmetic
• polynomial GCD and factorization
• Groebner bases and triangular system
Support for calculus uses notion of kernel: kernels represent algebraic and transcendental functions. In arithmetic kernels behave like variables, but for example have interesting derivatives. | {"url":"https://wiki.fricas.org/MathematicalAlgorithms/diff","timestamp":"2024-11-13T18:35:49Z","content_type":"application/xhtml+xml","content_length":"18643","record_id":"<urn:uuid:1f7c1f95-6344-483e-b694-3057edbd0618>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00817.warc.gz"} |
Dedekind tessellation in this post, following the reference given by John Stillwell in his excellent paper Modular Miracles, The American Mathematical Monthly, 108 (2001) 70-76.
But is this correct terminology? Nobody else uses it apparently. So, let’s try to track down the earliest depiction of this tessellation in the literature…
Richard Dedekind‘s 1877 paper “Schreiben an Herrn Borchard uber die Theorie der elliptische Modulfunktionen”, which appeared beginning of september 1877 in Crelle’s journal (Journal fur die reine und
angewandte Mathematik, Bd. 83, 265-292).
There are a few odd things about this paper. To start, it really is the transcript of a (lengthy) letter to Herrn Borchardt (at first, I misread the recipient as Herrn Borcherds which would be really
weird…), written on June 12th 1877, just 2 and a half months before it appeared… Even today in the age of camera-ready-copy it would probably take longer.
There isn’t a single figure in the paper, but, it is almost impossible to follow Dedekind’s arguments without having a mental image of the tessellation. He gives a fundamental domain for the action
of the modular group $\Gamma = PSL_2(\mathbb{Z}) $ on the hyperbolic upper-half plane (a fact already known to Gauss) and goes on in section 3 to give a one-to-one mapping between this domain and the
complex plane using what he calls the ‘valenz’ function $v $ (which is our modular function $j $, making an appearance in moonshine, and responsible for the black&white tessellation, the two colours
corresponding to pre-images of the upper or lower half-planes).
Then there is this remarkable opening sentence.
Sie haben mich aufgefordert, eine etwas ausfuhrlichere Darstellung der Untersuchungen auszuarbeiten, von welchen ich, durch das Erscheinen der Abhandlung von Fuchs veranlasst, mir neulich erlaubt
habe Ihnen eine kurze Ubersicht mitzuteilen; indem ich Ihrer Einladung hiermit Folge leiste, beschranke ich mich im wesentlichen auf den Teil dieser Untersuchungen, welcher mit der eben genannten
Abhandlung zusammenhangt, und ich bitte Sie auch, die Ubergehung einiger Nebenpunkte entschuldigen zu wollen, da es mir im Augenblick an Zeit fehlt, alle Einzelheiten auszufuhren.
Well, just try to get a paper (let alone a letter) accepted by Crelle’s Journal with an opening line like : “I’ll restrict to just a few of the things I know, and even then, I cannot be bothered to
fill in details as I don’t have the time to do so right now!” But somehow, Dedekind got away with it.
So, who was this guy Borchardt? How could this paper be published so swiftly? And, what might explain this extreme ‘je m’en fous’-opening ?
Carl Borchardt was a Berlin mathematician whose main claim to fame seems to be that he succeeded Crelle in 1856 as main editor of the ‘Journal fur reine und…’ until 1880 (so in 1877 he was still in
charge, explaining the swift publication). It seems that during this time the ‘Journal’ was often referred to as “Borchardt’s Journal” or in France as “Journal de M Borchardt”. After Borchardt’s
death, the Journal für die Reine und Angewandte Mathematik again became known as Crelle’s Journal.
As to the opening sentence, I have a toy-theory of what was going on. In 1877 a bitter dispute was raging between Kronecker (an editor for the Journal and an important one as he was the one
succeeding Borchardt when he died in 1880) and Cantor. Cantor had published most of his papers at Crelle and submitted his latest find : there is a one-to-one correspondence between points in the
unit interval [0,1] and points of d-dimensional space! Kronecker did everything in his power to stop that paper to the extend that Cantor wanted to retract it and submit it elsewhere. Dedekind
supported Cantor and convinced him not to retract the paper and used his influence to have the paper published in Crelle in 1878. Cantor greatly resented Kronecker’s opposition to his work and never
submitted any further papers to Crelle’s Journal.
Clearly, Borchardt was involved in the dispute and it is plausible that he ‘invited’ Dedekind to submit a paper on his old results in the process. As a further peace offering, Dedekind included a few
‘nice’ words for Kronecker
Bei meiner Versuchen, tiefer in diese mir unentbehrliche Theorie einzudringen und mir einen einfachen Weg zu den ausgezeichnet schonen Resultaten von Kronecker zu bahnen, die leider noch immer so
schwer zuganglich sind, enkannte ich sogleich…
Probably, Dedekind was referring to Kronecker’s relation between class groups of quadratic imaginary fields and the j-function, see the miracle of 163. As an added bonus, Dedekind was elected to the
Berlin academy in 1880…
Anyhow, no visible sign of ‘Dedekind’s’ tessellation in the 1877 Dedekind paper, so, we have to look further. I’m fairly certain to have found the earliest depiction of the black&white tessellation
(if you have better info, please drop a line). Here it is
It is figure 7 in Felix Klein‘s paper “Uber die Transformation der elliptischen Funktionen und die Auflosung der Gleichungen funften Grades” which appeared in may 1878 in the Mathematische Annalen
(Bd. 14 1878/79). He even adds the j-values which make it clear why black triangles should be oriented counter-clockwise and white triangles clockwise. If Klein would still be around today, I’m
certain he’d be a metapost-guru.
So, perhaps the tessellation should be called Klein’s tessellation??
Well, not quite. Here’s what Klein writes wrt. figure 7
Diese Figur nun – welche die eigentliche Grundlage fur das Nachfolgende abgibt – ist eben diejenige, von der Dedekind bei seiner Darstellung ausgeht. Er kommt zu ihr durch rein arithmetische
Case closed : Klein clearly acknowledges that Dedekind did have this picture in mind when writing his 1877 paper!
But then, there are a few odd things about Klein’s paper too, and, I do have a toy-theory about this as well… (tbc)
3 Comments | {"url":"http://www.neverendingbooks.org/category/groups/page/24/","timestamp":"2024-11-12T18:38:48Z","content_type":"text/html","content_length":"35580","record_id":"<urn:uuid:4ba5b438-3786-4901-b544-800e648f4514>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00501.warc.gz"} |
Information Sheet
MTA142-04 Calculus 1 Fall 2003
Information Sheet
Class meets MTWF 1:10-2:00 in Ritter Hall 200.
Instructor Bryan Clair
Email bryan@slu.edu
Office Ritter Hall 110. 977-3043.
Office Hours Tu12-1, W3-4, F12-1, or by appointment. If youÕre not coming to office hours, youÕre missing out on a valuable resource.
Textbook Hughes-Hallett & friends, Calculus (3ed).
Calculator A graphing calculator is required. We will use the calculator in class, on homework assignments, and during exams. The TI-83 or TI-82 is strongly recommended - I will be using the TI-83 in
Homework Homework will be due weekly, usually on Fridays. Your work should be neat and legible. Cooperation is encouraged, but write up results separately.
Late homework is always accepted, but I will not write comments and will automatically give a score of 5 (out of 10) if the work is of reasonable quality.
Quizzes There will be six 10-minute quizzes. I will drop your lowest score, so there will be no makeup quizzes.
Exams I give makeup exams only for severe and documented reasons.
Exams will all be in our usual room, RH 102.
Exam 1 Tuesday, September 23
Exam 2 Tuesday, October 28
Final Friday, December 12, 12:00-1:50pm
There will be a ÒgatewayÓ exam on differentiation skills. You will be allowed to take the gateway as many times as you need to pass it (up until the final day of the semester). Failing to pass the
gateway will result in the loss of an entire letter grade.
Grading Grading is on a straight scale (uncurved), with 90%,80%,70%,60% guaranteeing A,B,C,D respectively. Grading is weighted as follows:
Homework 20%
Quizzes 5%
Midterm 1 20%
Midterm 2 20%
Final Exam 35%
Cheating Students are expected to be honest in their academic work. Cheating is reported to the dean and may result in probation, expulsion, or worse. | {"url":"https://turtlegraphics.org/mta142/info.htm","timestamp":"2024-11-13T08:11:44Z","content_type":"text/html","content_length":"13539","record_id":"<urn:uuid:ed15f6cd-72f3-428d-84f4-d78f6be3a2c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00725.warc.gz"} |
Digital Watermarking Image Compression Method Based on Symmetric Encryption Algorithms
Department of Electronic Engineering, Taiyuan Institute of Technology, Taiyuan 030008, China
Science and Technology Division, Taiyuan Institute of Technology, Taiyuan 030008, China
Author to whom correspondence should be addressed.
Submission received: 21 October 2019 / Revised: 26 November 2019 / Accepted: 2 December 2019 / Published: 11 December 2019
A digital watermarking image compression method based on symmetrical encryption algorithm is proposed in this study. First, the original image and scrambled watermarking image are processed by
wavelet transform, and then the watermarking image processed by the Arnold replacement method is transformed into a meaningless image in the time domain to achieve the effect of encryption.
Watermarking is generated by embedding the watermarking image into the important coefficients of the wavelet transform. As an inverse process of watermarking embedding, watermarking extraction needs
to be reconstructed by the wavelet transform. Finally, the watermarking is extracted from the inverse scrambled watermarking image, and a new symmetrically encrypted digital watermarking image is
obtained. The compression method compresses the embedded digital watermarking image, so that the volume of the compressed watermarking image is greatly reduced when the visual difference is very
small. The experimental results show that the watermarking image encrypted by this method not only has good transparency, but also has strong anti-brightness/contrast attack, anti-shearing, and
anti-noise performance. When the volume of the compressed image is greatly reduced, the root mean square error and visual difference measurement of the watermarking image are very small.
1. Introduction
The copyright information of digital works is embedded in the digital works through the digital watermarking system in the form of watermarking, which cannot be perceived by human beings. The
embedded watermarking information can only be detected by some special software [
]. At present, the research of digital watermarking technology mainly focuses on the robustness of watermarking. Most algorithms use pseudo-random noise sequence to construct watermarking, and use a
hypothesis test to detect whether watermarking exists in the image. That is, when the sequence extracted from the image has a strong correlation with the original watermarking, indicating that the
image contains watermarking information; otherwise, it does not contain watermarking information. However, in many practical applications, the embedded image watermarking information is required to
be more readable or visual, for instance, meaningful information (such as text, icon, or image). This meaningful watermarking has incomparable advantages of random sequence watermarking. Digital
watermarking in the wavelet domain has become one of the research hotspots in recent years [
]. On one hand, the research of wavelet theory itself is becoming more and more mature and perfect. On the other hand, the application of the wavelet multi-resolution analysis method is becoming
increasingly more extensive, especially in information processing. In the transform domain of digital watermarking image, some characteristics of human visual system (HVS) can be more conveniently
integrated into the watermarking algorithm, so that the privacy and robustness of embedded information can be improved.
Wavelet transform-based digital watermarking image algorithm is a common symmetric encryption algorithm. In recent years, the research into it has become more and more mature and perfected. The
application of wavelet transform in signal and image compression has the advantages of a high compression ratio, fast compression speed, unchanged features of signal and image after compression, and
anti-interference in transmission. There are many image compression methods based on wavelet transform [
], such as embedded zerotree wavelet(EZW), set partitioning in hierarchical trees (SPIHT), and adaptive scanning wavelet difference reduction (ASW-D).Moreover, wavelet transform is also widely used
in signal analysis, including boundary processing and filtering, time-frequency analysis, signal–noise separation and extraction of weak signals, fractal index, signal recognition and diagnosis, and
multi-scale edge detection.
Symmetrically encrypted digital watermarking image is a newly-emerged digital image technology, which can obtain and store all the color and brightness information of the original scene. With the
improvement of CMOS (Complementary Metal Oxide Semiconductor) and CCD optical sensor production technology, the improvement of display performance, the reduction of storage device price, and the
popularization of high-performance computing, there is no doubt that symmetrically encrypted digital watermarking images will be widely accepted. However, owing to the large amount of information of
symmetrically encrypted digital watermarking image itself, its storage space is very large. At present, the digital watermarking hiding method for video image is basically noncompressed [
], which takes up a large amount of space, thus seriously limiting the wide application of symmetric encrypted digital watermarking image. The wavelet transform-based JPEG Z000 image compression
standard highlights the advantages of wavelet transform in image compression [
]. In this paper, a digital watermarking image compression algorithm based on symmetric encryption of wavelet transform is designed, which greatly reduces the volume of compressed image (about 10% of
the original image) when the visual difference is small.
Foreign scholars have carried out many studies on digital watermarking image compression. Meyer, Lemarie, and Battle independently presented the exponential attenuation wavelet function, which
reduced the volume of the compressed image to three-tenths of the original image. In 1987, Mallatll used the concept of multi-resolution analysis to unify. This paper proposes Mallat fast wavelet
decomposition and the reconstruction algorithm, which is widely used nowadays; however, this proposed algorithm does not consider the instability of the traditional wavelet multi-resolution analysis
method in embedding the watermarking image into the important coefficients of wavelet transform. Therefore, this paper proposes a digital watermarking image compression method based on the symmetric
encryption algorithm. In the transform domain, this method can more easily integrate the characteristics of human visual system (HVS) into the watermarking algorithm, so that the privacy and
robustness of the embedded information can be improved.
2. Algorithmic Definitions
2.1. Digital Watermarking Image Based on Symmetric Encryption
2.1.1. Wavelet Transform of Image
Wavelet transform is a symmetrical encryption algorithm as well as a variable resolution analysis method. It uses a small window to analyze high frequency signals and a large time window to analyze
low frequency signals, which coincides with the time–frequency distribution characteristics of natural medium-high frequency signals. Therefore, it is very suitable for image processing [
As a new branch of mathematics, wavelet transform theory is a milestone development after Fourier transform [
], which can be used to solve many difficult problems that cannot be solved by Fourier transform. The application of Fourier transform in signal processing can well describe the frequency
characteristics of signals, but cannot solve problems such as abrupt signal and non-stationary signal. The basic idea of wavelet transform is to expand the signal into a weighted sum of a family of
basic functions, that is, to express or approximate preferences or functions by a family of functions. This family of functions is composed of the translation and expansion of basic functions.
Wavelet transform is the extension of Fourier transform, which is basically a local transformation of space (time) and frequency.
The wavelet transform has been intensively applied in image processing in recent years. The basic idea of image processing is to decompose the image into sub-images of different spatial and
independent frequency bands, and then process the coefficients of sub-images, such as image compression, image enhancement, image decomposition, and reconstruction. According to the pyramid
decomposition algorithm proposed by S. Mallat, four 1/4-size subgraphs are formed after first-order decomposition (low-frequency approximation subgraph LL1 with most of the image information; medium
and high-frequency detail subgraphs HL1, LH1, and HHH1 with edge and texture information in the horizontal, vertical, and diagonal directions, respectively; and then the approximation subgraph LL1
with the complete phase) [
]. In the same way, the smaller subgraphs LL2, HL2, LH2, and HH2 at the next resolution are low-frequency approximation subgraphs, and HL2, LH2, and HH2 represent high-frequency detail subgraphs; LL2
is decomposed once more. After decomposition, 10 subgraphs are obtained (as shown in
Figure 1
), including HL1, LH1, and HH1 (Level 1); HL2, LH2, and HH2 (Level 2); and LL3, HL3, LH3, and HHH3 (Level 3). With the increase of decomposition series, the resolution decreases and the energy of
wavelet coefficients increases. In other words, the resolution of the first level is the highest and the frequency is the highest, the resolution of the third level is the lowest and the frequency is
the lowest, the energy of the wavelet coefficients of the first level is smaller than that of the second level, and the energy of the wavelet coefficients of the second level is smaller than that of
the third level. The low frequency band represents the best approximation of the original image at the maximum scale and the minimum resolution determined by the decomposition series of the wavelet
transform. Its statistical characteristics are similar to those of the original image [
], where most of the energy of the image is concentrated. The high frequency band part represents the edge and principle of the image. On the premise of ensuring better transparency (invisibility) of
the watermarking, the location of the hidden watermarking information is selected in the third-level wavelet coefficients with the largest energy, which can achieve better robustness.
HVS points out that human eyes are less sensitive to high-frequency information (such as complex areas) than to low-frequency information (such as smooth areas), so the more high-frequency
watermarking is embedded, the less the impact on the original image, and the better the transparency of the watermarking. However, considering the robustness and distortion of the watermarking, the
watermarking is embedded in the middle frequency band of the image [
]. When embedding image watermarking, Cox always excludes DC (Discrete Cosine) coefficients in order to avoid the block effect of watermarked image. This DCT (Discrete Cosine Transform) based
embedding strategy is now well-accepted. However, as the wavelet transform is a global transform, adding watermarking to the low-frequency coefficients of the image will not produce the blocking
The image embedded with watermarking is prone to image compression, filtering, and other processing. According to signal processing theory, low-frequency coefficients are less susceptible to signal
processing than high-frequency coefficients. Therefore, this paper proposes a method to embed the watermarking into the low-frequency coefficients of the wavelet image, which ensures both the
reliability and stability of the image.
2.1.2. Watermarking Generation (Arnold Permutation)
The purpose of time domain transformation of watermarking image is to disorder the information of watermarking and achieve the effect of encryption [
]. The following functions are used:
$A N ( k ) : | x ′ y ′ | = ⌊ 1 k 1 k + 1 ⌋ | x y | mod N ,$
is a control function;
is the size of the matrix; and
$( x , y )$
$( x ′ , y ′ )$
are the positions of the pixels before and after transformation, respectively. Let P represent an m × m matrix of binary watermarking information. After
$A N ( k )$
transformation of the coordinates of each point, the matrix will become an N × N matrix. Each element of the matrix has a value of 0 or 1. As the transformation of
$A N ( k )$
is periodic, the
$( x , y )$
can return to its original position after
transformations. Similarly,
$( i , j )$
point can be restored to their original positions after
$T − n$
transformations. The watermark generation framework is shown in
Figure 2
2.1.3. Watermarking Embedding and Extraction
(1) Watermarking Embedding Method
In order to improve the invisibility of watermarking information, the adaptive coefficient of watermarking is introduced to adjust the strength of watermarking. The series of wavelet decomposition is
determined by the size of the original image and the watermarking information. Generally speaking, the original image should be decomposed by more than two levels of wavelet [
]. In the experiment, the original image is decomposed by three levels of wavelet, and the watermarking information is decomposed by one level of wavelet. The steps of the algorithm are as follows:
Arnold transform is used to scramble the watermarking image, and the scrambled watermarking is recorded as $W i$.
The original image $F$ is transformed by a three-level wavelet transform to obtain sub-images $F i k$ in different directions at different resolutions.
$W i$ is decomposed into four subgraphs $W i k$ by first-order wavelet decomposition.
The data in the watermarking and the data in the original image subgraph are coded in blocks according to the following formula:
$C i ′ = C i ( F i k + α W i k ) ( k = 0 , 1 , 2 , 3 ) ,$
$C i$
is the DWT coefficient before embedding,
is the wavelet strength parameter,
$W i$
is the embedding watermarking, and
$C i ′$
represents the embedding watermarking. Subgraphs with different intensities and in different directions allow the embedded watermarking to have a certain degree of self-adaptability.
(2) Watermarking extraction method
The extraction process of watermarking is basically the inverse process of embedding. When extracting watermarking, the original image is needed. Firstly, the original image and the watermarking
image to be detected are processed by a three-level wavelet transform, and then the wavelet coefficients of the watermarking sub-image are compared [
]. According to the above formula, the wavelet coefficients of the watermarking sub-image are deduced, and the watermarking is reconstructed. Finally, the watermarking is extracted from the
anti-scrambling watermarking image. The experimental results of watermarking embedding are shown in
Figure 3
2.1.4. Watermarking Detection
The original image is needed in the detection of this algorithm. If the image watermarking is not restored well, it is considered that the watermarking is damaged or even does not exist. The
detection process is as follows:
The image
$X ′ ( i , j )$
to be detected and the carrier image
$X ( i , j )$
are transformed into 8 × 8-pixel block DTC, then
$F k ′ ( u , v )$
$F k ( u , v )$
are obtained, where
represents the
image block. The absolute value of the five low-frequency coefficients in the DTC domain of the original image
$X ( i , j )$
is chosen as
$F max ( u , v )$
, which is compared with the corresponding JND (Just Noticeable Distortion) threshold
$T k ( u , v )$
. If
$F max ( u , v )$
is greater than or equal to
$T k ( u , v )$
, the coefficient is obtained and watermarking is embedded; otherwise, no watermarking is embedded. For the coefficients embedded with watermarking, the
th watermarking signal
$W k ′$
is extracted according to Formula (3).
$W k ′ = { 1 , i f | F k ( u , v ) − F k ( u , v ) | ≥ 0.61 × T k ( u , v ) 0 , i f | F k ( u , v ) − F k ( u , v ) | < 0.61 × T k ( u , v )$
Then, the rotational transformation parameter
$L ′$
is obtained by symmetrical decryption of the extracted watermarking sequence
$W ′$
, which is combined with the remaining fractal feature parameters saved in the work of [
], and the gray image watermarking is restored using fractal image decoding technology.
After obtaining the digital watermarking image, the symmetrical encrypted digital watermarking image compression algorithm is adopted to realize the compression of the watermarking image.
2.2. Design of Symmetric Encryption Digital Watermarking Image Compression Algorithm
The symmetrically encrypted digital watermarking image has a wider application range than the ordinary image. Currently, one of the commonly used coding methods for symmetrically encrypted digital
watermarking images is lossless coding, which occupies a larger storage space. For instance, a color symmetrically encrypted digital watermarking image (96-bit floating-point TIFF (Tag Image File
Format) coding) with a size of 800 × 600 pixels occupies storage space of about 5.5 M Byte; in contrast, an uncompressed ordinary image (BMP (Bitmap) encoding) of the same size occupies about 1.4 M
Byte. After lossy JPEG compression, it occupies about 0.05 M Byte (depending on compression rate and image content).
A typical web page may contain 10 or more images. If uncompressed images are used, the download time may be several minutes, which is obviously intolerable to users. Therefore, JPEG images are widely
used in web design.
At present, lossless compression coding is the dominant coding scheme for symmetrically encrypted digital watermarking image. Despite the continuous improvement of Internet speed, at least in the
visible future, such symmetrically encrypted digital watermarking image coding cannot be applied to web pages at all [
]. This paper designs a lossy compression algorithm for symmetrically encrypted digital watermarking image. Although lossy compression may lead to the loss of some details, the symmetrically
encrypted digital watermarking image features are well preserved. It is hoped that this new coding scheme can be applied in future web page design.
The symmetrical encrypted digital watermarking image compression algorithm designed in this paper is inspired by the new international standard JPEG2000 for digital image compression. JPEG2000 is a
compression standard prepared for the 21st century, which uses improved compression technology to provide a higher resolution. With the capacity of providing a variety of image quality and image
selection from non-destructive to lossy for a file, JPEG2000 is considered to be an ideal image coding solution for Internet and wireless access applications.
“High compression, low bit rate” is the goal of JPEG2000. Under the same compression rate, the signal-to-noise ratio of JPEG2000 will increase by about 30% compared with that of JPEG. JPEG2000 has
five levels of coding, including JPEG coding for color static picture, JBIG for two-value image. It has become a common coding method for all kinds of images. In coding algorithm, JPEG2000 uses
discrete wavelet transform (DWT) and arithmetic coding). In addition, JPEG2000 can also send images with different resolutions and compression rates according to the user’s network transmission speed
and mode of use (viewing on a personal computer).
JPEG2000 was initially formulated in March 1997, andthe final draft agreement on the basic coding system was introduced March 2000. At present, JPEG2000 has been officially named “ISO15444” by ISO
(International Organization for Standardization) and IEC (International Electro technical Commission). JTC1 SC29 Standardization Group is officially named “ISO15444”, and the basic part of JPEG2000
(Partl) has been published as an ISO standard.
The compression performance of JPEG2000 is 30%–50% higher than that of JPEG. That is to say, under the same image quality, JPEG2000 can reduce the size of image file by 30%–50% compared with that of
the JPEG image file. At the same time, the system using JPEG2000 has good stability, stable operation, good anti-interference, and easy operation [
The biggest difference between JPEG2000 and traditional JPEG is that it abandons the block coding method based on discrete cosine transform adopted by JPEG, but adopts the multi-analytic coding
method based on wavelet transform. Cosine transform is a classical spectral analysis tool, which investigates the frequency domain characteristics of the whole-time domain process or the time domain
characteristics of the whole frequency domain process. Therefore, it has a good effect for stationary processes, but has many shortcomings for non-stationary processes. In JPEG, the discrete cosine
transform compresses the image into 8 × 8 blocks, and then puts them into the file in turn. This algorithm compresses the image by discarding the frequency information, so the higher the compression
rate of the image, the more the frequency information is discarded. In extreme cases, JPEG images only retain basic information reflecting the appearance of the image, and fine image details are
lost. Wavelet transform is a modern spectral analysis tool, which can investigate not only the frequency domain characteristics of local time domain processes, but also the time domain
characteristics of local frequency domain processes. Therefore, it is applicable even for non-stationary processes. It can transform an image into a series of wavelet coefficients, so the image can
be compressed and stored efficiently. In addition, the rough edge of the wavelet can better represent the image, because it eliminates the blocking effect commonly used in DCT compression. In recent
years, discrete wavelet transform has been widely used in fields of image processing and image analysis because of time-frequency localization characteristics. Moreover, discrete wavelet transform
can amplify arbitrary details as it uses a progressive sampling interval from coarse to fine for high-frequency components [
]. Therefore, wavelet analysis is praised as a mathematical microscope as well as a powerful tool to construct multi-resolution representation of images.
The compression algorithm is divided into five steps, including color space conversion, floating-point to integer conversion, discrete wavelet transform, quantization, and entropy coding.
(1) Color space conversion
In order to better balance the compression rate and image quality, the following criteria are used in the selection of color space:
The color space can express all color and brightness information visible to the naked eye.
The quantization step of the color space is just a noticeable difference.
Positive integers are used to express color and brightness information in order to improve the compression rate.
The correlation between different color channels is the smallest.
For a given pair of symmetrically encrypted digital watermarking images expressed in color space, its brightness channel Y is converted to luma, and the formula is as follows:
$l u m a ( Y ) = { a × Y i f Y ≤ Y L b × Y c + d i f Y L ≤ Y ≤ Y H e × log ( Y ) + f i f Y ≥ Y H ,$
$a = 17.554$
$b = 826.81$
$c = 0.10013$
$d = − 884.17$
$e = 209.16$
$f = − 731.28$
$Y L = 5.6046$
, and
$Y H = 10469$
The formula for conversion from luma to brightness Y is as follows:
$Y ( l u m a ) = { a ′ × L i f L ≤ L L b ′ × ( L + d ′ ) c ′ i f L L ≤ L ≤ L H e ′ × exp ( f ′ × L ) i f L ≥ L H ,$
$a ′ = 0.056968$
$b ′ = 7.3014 e − 30$
$c ′ = 9.9872$
$d ′ = 884.17$
$e ′ = 32.994$
$f ′ = 0.0047811$
$L L = 98.38$
, and
$L H = 1204.7$
This conversion function is shown in
Figure 4
. When the luminance range is from 1 × 10
to 1 × 10
(candela per square meter), the output luma range is from 0 to 4000. When luma is taken as an integer, it can be expressed by 16-bit integers.
As human eyes are more sensitive to brightness information than to color information, less bits (8 bits) are used to express color information [
]. In this paper, CIE (China Illuminating Engineering) 1976 uniform scale
$( u , v )$
is used to express color information. The transformation from XYZ space to CIE1976 uniform scale
$( u , v )$
is as follows:
${ u = 4 X X + 15 Y + 3 Z V = 9 Y X + 15 Y + 3 Z .$
(2) Conversion from floating point to integer
The conversion function in
Figure 3
shows that the range of luma ranges from 0 to 4000, so the nearest neighbor integer method is adopted to express luma using 16-bit integers. The range of color information
$u , v$
is 0 to 0.62, which is multiplied by 410 and then expressed by an integer of 8.
(3) Discrete wavelet transform
The discrete wavelet transform in JPEG2000 is employed to carry out five-layer two-dimensional discrete wavelet transform for each channel
$( l u m a , u , v )$
. The Daubechies 9/7 wavelet coefficients used in this algorithm are shown in
Table 1
(4) Quantification
After the
$L u m a , u , v$
channel undergoes wavelet transformation, its total wavelet coefficients are the same as the total number of pixels in the original image, but important visual information is concentrated in a few
coefficients. In this paper, the wavelet coefficient
$a b$
in the wavelet sub-band
is quantized to
$q b$
, according to the following formula:
$q b = s i g n [ a b ] × f l o o r [ | a b | 2 R b − ε b ( 1 + μ b 2 11 ) ] ,$
where the nominal dynamic ranges of self-band
$ε b$
, and
$μ b$
are the digits allocated to the index and tail of the current wavelet coefficients. The nominal dynamic range of
is the number of bits per pixel plus the number of increments in sub-band
of the original image. For the
$L u m a , u , v$
channel, different quantization coefficients are used. For the luma channel, set
$ε b = 16$
$μ b = 16$
, and for the
$u a n d v$
channel, set
$ε b = 8$
$μ b = 9$
(5) Entropy coding
After quantization, arithmetic coding is applied to compress the quantized coefficients. Arithmetic coding is also adopted by JPEG2000. Generally speaking, arithmetic coding has a higher compression
rate than Huffman coding [
]. Like other variable length entropy coding techniques, arithmetic coding uses shorter bits for high frequency characters and longer bits for low frequency characters.
Figure 5
Figure 6
show the flow charts of the compression and decompression algorithm.
3. Results
In order to verify that the digital watermark image encrypted by this method has a strong anti-cutting and anti-noise performance, the image is simulated in this paper. In the experiment, the gray
image and color image with the size of 243 × 243 pixels are used as the original carrier image. As shown in
Figure 7
a–i represent color and gray image in 64 × 64 binary image marked with “abcdefghigk”, which has rich color, texture, detail, area, theme, and plane background.
Figure 7
g,j show the original watermark image and extracted watermark image.
Figure 7
b,f,j test the anti-brightness performance and contrast attack performance of the image.
3.1. Transparency Experiment
Peak signal-to-noise ratio (PSNR) is used to measure the quality of embedded watermarking image. For an 8-bit gray-scale image with M × N pixel size, the PSNR is as follows:
$P S N R = 10 lg ( 255 2 × m × n ∑ i = 0 m − 1 ∑ j = 0 n − 1 ( I ′ ( i , j ) − I ( i , j ) ) 2 ) .$
For 24-bit color images with M × N pixels, the average peak signal-to-noise ratio (PSNR) of three RGB channels is used to measure the image quality.
$P S N R ¯ = ( P S N R R + P S N R G + P S N R B ) / 3$
Figure 7
shows a small part of the original image and the watermarking image in the image database, both of which are 243 × 243 pixels in size.
The first three lines in
Figure 7
are the original images, which represent color and gray images with rich colors, rich textures, rich details, distinct regions, distinct themes, and flat backgrounds. The last line is the
watermarking image. In order to show the transparency of the symmetrically encrypted digital watermarking image, a watermarking image with rich color and distinct theme is embedded in the 24-bit
watch image with a flat background and monotonous color, as shown in
Figure 8
It can be seen from
Figure 8
that the transparency of the symmetrically encrypted digital watermarking image is very good. In the experiment, 24-bit color watermarking (j) is embedded in 24-bit color images (a) to (g) in
Figure 7
, and 8-bit gray-scale watermarking (k) is embedded in 8-bit gray-scale images (h) and (i). The PSNR value is calculated for the obtained image and the original image, as shown in
Table 2
It can be seen from
Table 2
that the PSNR value between the watermark image and the original image is good, which conforms to the transparency standard. The results of embedding 24-bit color watermarking (j) in 24-bit color
images (a) to (g) and embedding 8-bit gray-scale watermarking (k) in 8-bit gray-scale images (h) and (i) show that the transparency algorithm of a is better than that of other images.
3.2. Anti-Brightness/Contrast Attack Performance Detection
Figure 9
shows the experimental results of watermarking detection after adding brightness/contrast to the symmetrically encrypted digital watermarking image using Photoshop. The NC (Numerical Control) values
of normalized cross-correlation function are compared. NC values are often used to evaluate the performance of the encrypted digital watermarking image.
Figure 9
a–c show that, when brightness increases by 70 and contrast increases by 70, high-quality watermarking can still be extracted from tampered images with seriously degraded quality, with an NC value of
0.9961, while the traditional DCT transform-based watermarking algorithm has an NC value of 0.9863.
Figure 9
d–f show that, when the brightness increases by 70, the NC value is only 0.9710, so the proposed method has a strong brightness/contrast attack on symmetrically encrypted digital watermarking images.
3.3. Testing of Shear Resistance
Figure 10
shows the experimental results of embedding the watermarking into the image using the symmetric encryption algorithm proposed in this paper.
After the symmetrically encrypted watermarking embedded image is cut, the watermarking of the cut image is extracted, as shown in
Figure 10
a. As shown in
Figure 10
b, the extracted watermarking image can still be restored to the watermarking image, which shows that the watermarking image encrypted by this method has an anti-shearing property.
3.4. Anti-Noise Performance Testing
In symmetrically encrypted watermarking embedded image, salt and pepper noise with an intensity of 0.02 is added to obtain the watermarking image with noise, as shown in
Figure 11
a. Wavelet transform is applied to the image, and the watermarking is extracted from the high frequency components in three directions, respectively. Then, the watermarking image is processed, as
shown in
Figure 11
Figure 11
shows that, after adding noise to the symmetrically encrypted watermarking embedded image, the watermarking can still be extracted from the watermarking image, and the watermarking image can still be
restored to the watermarking image after extracting the watermarking. It can be seen that the digital watermarking image encrypted by this method is noise-resistant.
3.5. Digital Watermarking Image Compression Detection Based on Symmetric Encryption
The symmetrically encrypted digital watermarking image (RGBE coding) provided by Munsell Color Science Laboratory of Rochester Institute of Technology University and the symmetrically encrypted
digital watermarking image provided by Industrial Light and Magie Company are used in the experiment [
]. The visual quality of symmetrically encrypted digital watermarking image after lossy compression is measured based on root-mean-square error. The smaller the root-mean-square error, the better the
digital watermarking image compressed by this method.
$e r m s = 1 n ∑ i = 1 n [ ( X 1 i − X 2 i ) 2 + ( Y 1 i − Y 2 i ) 2 + ( Z 1 i − Z 2 i ) 2 ] ,$
$X 1$
$Y 1$
, and
$Z 1$
are the three channels of the original graph, respectively; and
$X 2$
$Y 2$
, and
$Z 2$
are the three compressedchannels, respectively. It should be noted that their ranges are within [10
, 10
In addition to root-mean-square error, the visual error metric of symmetrically encrypted digital watermarking images is adopted to predict whether the difference between two symmetrically encrypted
digital watermarking images can be perceived by human eyes. The visual error measurement of symmetrically encrypted digital watermarking image was proposed by Mantiuk et al. The input of HDR VDP is a
pair of images (the original image and the changed image), and the output is a probabilistic image. The pixel value of the image is basically equal to the probabilistic value of the image difference
visible to the naked eye.
Table 3
shows the experimental result of symmetrically encrypted digital watermarking image compression.
Table 3
shows the experimental results of symmetrically encrypted digital watermarking image compression. The first column in the table is the image name, in which the image suffixed with HDR is
symmetrically encrypted digital watermarking image (RGBE encoding) provided by Munsell Color Science Laboratory of Rochester Institute of Technology University, and the image suffixed with EXR is
symmetrically encrypted digital watermarking image (O) provided by Industrial Light and Magie Company. The second column is the size of the original image before compression, the third column is the
size of the image after compression, the fourth column is the symmetric encryption of digital watermarking image visual error measurement (HDR VDP), the fifth column is the root mean square error of
two images, and the last column is the time consumed by compression.
Figure 12
shows the image provided by Industrial light and Magie Company and
Figure 13
shows the image provided by Munsell Color Science Laboratory. To display the results more clearly, both
Figure 12
Figure 13
undergo tone mapping operation. In order to display symmetrically encrypted digital watermarking images on ordinary displays, we use tone mapping to compress symmetrically encrypted digital
watermarking images into ordinary images.
Table 3
Figure 12
Figure 13
, we can see that the volume of the compressed digital watermarking image is much smaller than that of the original image, and the visual difference of the compressed watermarking image as well as
the root mean square error of the digital watermarking image are very small, indicating excellent performance of the compressed digital watermarking image.
4. Conclusions
This paper studies the digital watermarking image compression method based on symmetric encryption algorithm. This method uses the Arnold permutation method to transform the watermarking image in the
time domain, and then uses the digital watermarking algorithm in the wavelet transform domain to embed the watermarking image into the important coefficients of the wavelet transform, so as to obtain
the digital watermarking image. Using the symmetrically encrypted digital watermarking image compression method, the volume of the compressed watermarking image can be greatly reduced (about
one-tenth of the original image) when the visual difference is very small.
The watermarking image encrypted and compressed by this method not only has good transparency, but also has strong anti-brightness/contrast attack, anti-shearing, and anti-noise performances. The
proposedmethod has good application prospects in the future multimedia field because of its capacity to reduce image volume and lower the root mean square error and visual difference of digital
watermarking image.
Author Contributions
Y.T. and Y.Z. carried a digital watermarking image compression method based on symmetrical encryption algorithm is proposed in this study. Y.T. processed the original image and scrambled watermarking
image by wavelet transform. Y.Z. did the process of watermarking embedding, reconstructed watermarking extraction by the wavelet transform. Y.T. and Y.Z. did the experiments, recorded data, and
created manuscripts. All authors read and approved the final manuscript.
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
Analytical Low Pass Filter Analytical High Pass Filter Synthetic Low Pass Filter Synthetic High Pass Filter
0.026748757411 0 0 0.026748757411
−0.016864118443 0.091271763114 −0.091271763 0.016864118443
−0.078223266529 −0.057543526229 −0.057543526229 −0.078223266529
0.266864118443 −0.591271763114 0.591271763114 −0.266864118443
0.602949018236 1.11508705 1.11508705 0.602949018236
0.266864118443 −0.591271763114 0.591271763 −0.266864118443
−0.078223266529 −0.057543526229 −0.057543526229 −0.078223266529
−0.016864118443 0.091271763114 −0.091271763114 0.016864118443
0.026748757411 0 0 0.026748757411
PSNR PSNR[R] PSNR[G] PSNR[B]
a 45.7884 44.5741 45.7002 47.0909
b 45.6866 44.9539 45.6302 46.4757
c 45.1864 44.3810 45.0482 46.1301
d 45.6693 45.0324 45.6056 46.3700
e 46.0259 45.2301 46.0145 46.8602
f 45.8591 45.0048 45.6675 46.6050
g 46.6032 45.8702 46.2284 46.6032
h 44.8100 - - -
i 45.6100 - - -
Image Name Original Size (byte) Compressed Graph Size (byte) Pixel Percentages of Image Differences Visible to the Naked Eye Root Mean Square Error Time (second)
Probability >75% Probability >95%
Crissy Field. exr 1,304,619 750,382 0% 0% 1.7327 5″0
Flowers exr 758,083 200,512 0% 0% 1.8101 4″7
MtTamNorth. exr 1,422,492 487,324 0% 0% 2.8594 4″9
Cis front. hdr 7,893,724 640,179 0.01% 0% 0.3659 17″6
atrium. hdr 8,110,240 360,418 0.36% 0.11% 0.0515 17″3
Parking lot. hdr 7,767,193 666,102 0.01% 0% 0.3025 17″2
rooftops. hdr 7,336,883 500,505 0% 0% 0.1733 15″3
Albers. hdr 6,444,172 463,477 0% 0% 0.4152 15″7
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Tan, Y.; Zhao, Y. Digital Watermarking Image Compression Method Based on Symmetric Encryption Algorithms. Symmetry 2019, 11, 1505. https://doi.org/10.3390/sym11121505
AMA Style
Tan Y, Zhao Y. Digital Watermarking Image Compression Method Based on Symmetric Encryption Algorithms. Symmetry. 2019; 11(12):1505. https://doi.org/10.3390/sym11121505
Chicago/Turabian Style
Tan, Yanli, and Yongqiang Zhao. 2019. "Digital Watermarking Image Compression Method Based on Symmetric Encryption Algorithms" Symmetry 11, no. 12: 1505. https://doi.org/10.3390/sym11121505
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Quadratic Dissonance
Problem K
Quadratic Dissonance
Oh no! Both you and your lab partner forgot to complete one part of the latest assignment and it is due in an hour!
The purpose of this lab assignment was to have you analyze some experimental data, find a quadratic function that best describes the data, and report the minimum value that can be taken by this
Both you and your partner have tried this part on your own, in order to double-check your answers. Unfortunately, you both found different quadratic functions!
You don’t have time to repeat the experiment. To try and maximize your chances of reporting a value that is close to the minimum (i.e. to get partial credit), you decide to find a value $x$ that
minimizes the maximum of your two quadratic functions.
More precisely, your task is the following. Given two quadratic functions $f(x) = x^2 + A\cdot x + B$ and $g(x) = x^2 + C \cdot x + D$ (here $A,B,C,D$ will be given values), you should find a value
$x^*$ minimizing the maximum of these two functions. That is, $x^*$ should be chosen to minimize the function $h(x) = \max \{ f(x), g(x)\} $.
Input consists of a single line containing four integers $A, B, C, D$, each lying in the range $[-1\, 000, 1\, 000]$.
Display two values $x^*, h(x^*)$ on a single line where $x^*$ is the point that minimizes the function $h(x) = \max \{ x^2 + A \cdot x + B, x^2 + C \cdot x + D\} $.
Your answer will be accepted if both values are within an absolute or relative error of $10^{-4}$ of the correct answer.
Sample Input 1 Sample Output 1
Sample Input 2 Sample Output 2
2 -1 0 -2 -0.5 -1.75
Sample Input 3 Sample Output 3
3 1 3 -2 -1.5 -1.25
Sample Input 4 Sample Output 4
2 2 2 2 -1 1
Sample Input 5 Sample Output 5
25 -73 -23 41 2.375 -7.984375 | {"url":"https://uapc22open.kattis.com/contests/uapc22open/problems/quadraticdissonance","timestamp":"2024-11-07T17:06:34Z","content_type":"text/html","content_length":"28948","record_id":"<urn:uuid:530ac0fa-d579-4518-a56a-d975498a7d95>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00413.warc.gz"} |
Notes - Complex Analysis MT23, Rouché's theorem
Can you state Rouché’s theorem, which lets you relate the zeroes of two holomorphic functions $f$ and $g$?
• $f, g : U \to \mathbb C$ holomorphic
• $\overline B(a, r) \subseteq U$
• $ \vert f(z) \vert > \vert g(z) \vert $ for all $z \in \partial B(a, r)$
• $f$ and $f + g$ have the same change in argument around $\partial B(a, r)$…
• and hence the same number of zeroes, counted with multiplicity.
Quickly prove Rouché’s theorem, that if
• $f, g : U \to \mathbb C$ holomorphic
• $\overline B(a, r) \subseteq U$
• $ \vert f(z) \vert > \vert g(z) \vert $ for all $z \in \partial B(a, r)$
• $f$ and $f + g$ have the same change in argument around $\partial B(a, r)$…
• and hence the same number of zeroes, counted with multiplicity.
Parameterise the path $\partial B(a, r)$ by $\gamma(t)$. We aim to relate the zeroes of $f$ and zeroes of $f + g$, so consider the “helper function”
\[h(z) := \frac{f(z)+g(z)}{f(z)} = 1 + \frac{g(z)}{f(z)}\]
This is helpful because $h$ has a pole exactly when $f$ has a zero, and a zero exactly when $f + g$ has a zero.
Hence if we can show $N - P = 0$ for $h$ then we are done ($N$ is the number of zeroes, $P$ is the number of poles). The argument principle says that $N - P$ is the winding number of $h \circ \gamma$
or “the change in argument of $h$” (and in particular showing $N - P = 0$ means $f$ and $f + g$ have the same change in argument around $\partial B(a, r)$, since $N$ equals the change in argument of
$f + g$ and $P$ equals the change in argument of $f$).
By the argument principle, we can do this by calculating
\[\frac{1}{2\pi i} \int_\gamma \frac{h'(z)}{h(z)} \text d z = \frac{1}{2\pi i} \int_{h \circ \gamma} \frac 1 z \text d z\]
The condition that $ \vert f(z) \vert > \vert g(z) \vert $ implies that $ \vert g(z)/f(z) \vert < 1$, so we actually have that $\text{Image}(h \circ \gamma) \subseteq B(1, 1)$ which is fully
contained in $\{ z \in \mathbb C \mid \Re z \ge 0\}$. There’s a holomorphic branch of $\text{Log}$ here, so
\[\int_{h \circ \gamma} \frac 1 z \text d z = \text{Log}(h(\gamma(1))) - \text{Log}(h(\gamma(0))) = 0\]
(alternatively, $\frac 1 z$ is fully holomorphic in $B(1,1)$, so the integral around a closed loop will be zero).
Then we are done.
How would you deduce, using Rouché’s theorem, that the roots of $P(z) = z^4 + 5z + 2$ all have modulus less than $2$?
Note that on the circle $ \vert z \vert = 2$, we have $ \vert z^4 \vert = 16 > 5 \times 2 + 2 \ge \vert 5z + 2 \vert $, so if $g(z) = 5z + 2$, $ \vert P \vert > \vert g \vert $ on the boundary. Then
$P - g$ and $z^4$ have the same number of roots in $B(0, 2)$, and since $z^4$ has 4 roots, it follows all roots of $z^4 + 5z + 2$ all have modulus less than $2$.
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ARIMA (Box-Jenkins Models): Autoregressive Integrated Moving Average
ARIMA modeling (sometimes called Box-Jenkins modeling), is an approach to modeling ARIMA processes—mathematical models used for forecasting. The approach uses previous time series data plus an error
to forecast future values. More specifically, it combines a general autoregressive model AR(p) and general moving average model MA(q):
• AR(p)— uses previous values of the dependent variable to make predictions.
• MA(q)—uses the series mean and previous errors to make predictions.
The approach was first proposed by Box and Jenkins (1970), who detailed ARIMA’s estimation and prediction procedures (Hyndman et al., 2008).
Nonseasonal Autoregressive Integrated Moving Average models are classified by three factors:
• p = number of autoregressive terms,
• d = how many nonseasonal differences are needed to achieve stationarity,
• q = number of lagged forecast errors in the prediction equation.
For example, an ARIMA(1,0,0) has 1 autoregressive term, no needed differences for stationarity and no lagged forecast errors. It is a special case of an ARIMA called a first-order autoregressive
The basic steps are (Hyndman, 2001):
• Prepare your data by using transformations (e.g. square roots or logarithms) to stabilize the variance and differencing to remove remaining seasonality or other trends.
• Identify any processes that appear to be a good fit for your data.
• Find which model coefficients provide the best fit for your data. This step is computationally complex and usually performed by a computer. Akaike’s Information Criterion (AIC) is one option: if
you compare two models, the one with the lower AIC is usually the “better” model.
• Test the models’ assumptions to see how well the model holds up to closer scrutiny. If your chosen model is inadequate, repeat steps 2 and 3 to find a potentially better model.
• Compute forecasts on your chosen model with computer software.
ARIMA models work on the assumption of stationarity (i.e. they must have a constant variance and mean). If your model is non-stationary, you’ll need to transform it before you can use ARIMA.
Box, G and Jenkins, G. (1970) Time series analysis: Forecasting and control, San
Francisco: Holden-Day.
Hyndman, R. (2001). Box-Jenkins modelling. Retrieved February 25, 2018 from: https://robjhyndman.com/papers/BoxJenkins.pdf
Hyndman, R. et al. (2008). Forecasting with Exponential Smoothing: The State Space Approach. Springer Science & Business Media.
Nao, R. Introduction to ARIMA models. Retrieved February 27, 2018 from: https://people.duke.edu/~rnau/411arim.htm#arima100
Comments? Need to post a correction? Please Contact Us. | {"url":"https://www.statisticshowto.com/arima/","timestamp":"2024-11-13T14:22:24Z","content_type":"text/html","content_length":"63176","record_id":"<urn:uuid:d715ceb2-3341-401c-b499-766b7502fa2a>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00622.warc.gz"} |
Jacobi-type methods
A common strategy for solving an unconstrained two-player Nash equilibrium problem with continuous variables is applying Newton’s method to the system obtained by the corresponding first-order
necessary optimality conditions. However, when taking into account the game dynamics, it is not clear what is the goal of each player when considering they are taking their current … Read more
A Jacobi-type Newton method for Nash equilibrium problems with descent guarantees
A common strategy for solving an unconstrained two-player Nash equilibrium problem with continuous variables is applying Newton’s method to the system of nonlinear equations obtained by the
corresponding first-order necessary optimality conditions. However, when taking into account the game dynamics, it is not clear what is the goal of each player when considering that they … Read more | {"url":"https://optimization-online.org/tag/jacobi-type-methods/","timestamp":"2024-11-05T16:16:11Z","content_type":"text/html","content_length":"86252","record_id":"<urn:uuid:51ec2518-9e1b-4d0f-b1d9-6cc42b238cd1>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00772.warc.gz"} |
Controlling HTD MCA66
I’ve been trying to add my MCA-66 to HA via the serial port but I can’t seem to figure out how to progress. I have the codes and they work using echo from the terminal for controlling the MCA-66, But
not sure how to use those or if I even should do it that way.
I’ve found a github repo with some .py files that looks like they’ve done most of the work and they even have a comment about compatibility with HA, but I’m not sure it works or how to integrate it
if it does work. The newest commit is 6 years old so that probably doesn’t help.
I see there seems to be a working integration using the gw-sl1, but I’d really rather not buy a device when the serial way should work fine. Though maybe that setup could be modified?
Just looking for some guidance and to be pointed in the right direction. Any help is greatly appreciated.
I’ve figured something workable out, but its pretty basic.
Installed HACS
then installed pyscript through that
then I could make a py file to run like this:
import serial
def Z1_ON():
ser = serial.Serial('/dev/ttyUSB0', 38400, timeout=1) | {"url":"https://community.home-assistant.io/t/controlling-htd-mca66/752805","timestamp":"2024-11-08T22:18:46Z","content_type":"text/html","content_length":"23185","record_id":"<urn:uuid:7fe085a7-02c8-4214-b030-e6af2043161c>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00591.warc.gz"} |
CAGR Formula in Google Sheets: 2 Easy Methods To Calculate CAGR in Google Sheets - CAGR Calculators
There is no direct CAGR formula in Google Sheets for calculating Compounding Annual Growth Rate. However, calculating CAGR in Google Sheets is very simple. There are two easy methods to do so:
In both the methods you have to provide 3 values in three cells and the CAGR formula in the fourth cell of Google Sheets.
Method 1: Directly Using the CAGR Formula in Google Sheets
Input three values (initial amount, final amount, and no. of years) in 3 different cells, and write the CAGR formula in a different cell.
Let’s understand this with an example.
For example, in this screenshot, I have set up the calculation this way:
• Cell C4: Initial amount or starting value: 10,000
• Cell C5: Final amount or ending value: 20,000
• Cell C6: No. of years: 10
I have then put the formula for calculating CAGR in Google Sheets in cell C8
• Cell C8: =(C5 / C4)^(1 / C6) – 1
Using the ‘=’ sign tells Google Sheets that the cell contains a formula and then Google Sheets references and uses the numerical values in cells C4, C5, and C6 and applies the CAGR formula to show
the result. The result shown in this case is 0.07177…
To represent the answer in easy-to-understand percentage terms, I copied the value in cell C9, and changed the format of the cell to “%”. So, the answer is 7.18%
Use this Link To Calculate CAGR in Google Sheets
Method 2: Using POW To Represent CAGR Formula in Google Sheets
For the sake of simplicity, we will be using the same numerical values as used in the example above.
Input three values (initial amount, final amount, and no. of years) in 3 different cells, and write the CAGR formula (using the POW function) in a different cell.
In this screenshot, I have set up the calculation this way:
• Cell C4: Initial amount or starting value: 10,000
• Cell C5: Final amount or ending value: 20,000
• Cell C6: No. of years: 10
I have then put the formula for calculating CAGR in Google Sheets in cell C8, using the POW() function this time.
• Cell C8: = POW(C5 / C4 , 1 / C6) – 1
Using the ‘=’ sign tells Google Sheets that the cell contains a formula and then Google Sheets references and uses the numerical values in cells C4, C5, and C6 and applies the CAGR formula to show
the result. The result shown in this case is 0.07177…
To represent the answer in easy-to-understand percentage terms, I copied the value in cell C9, and changed the format of the cell to “%”. So, the answer is 7.18%
Use this Link To Calculate CAGR in Google Sheets
Calculating CAGR in Google Sheets is fairly simple, no matter which method you choose. Maybe, in the future, there will be a direct CAGR function introduced in Google Sheets. But, even then the setup
would remain pretty much the same as shown here in both methods.
Check out our new venture multipl !
More From The Blog
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Assisted cloning of an unknown shared quantum state
We first propose a novel protocol to realize quantum cloning of an arbitrary unknown shared state with assistance offered by a state preparer. The initial phase of this protocol involves the
utilization of quantum teleportation (QT), enabling the transfer of quantum information from an arbitrary number of senders to another arbitrary number of receivers through a maximally entangled
GHZ-type state serving as a network channel, without centralizing the information at any specific location. In the second stage of this protocol, the state preparer performs a special single-qubit
projective measurement and multiple Z-basis measurements and then communicates a number of classical bits corresponding to measurement results, the perfect copy or orthogonal-complementing copy of an
unknown shared state can be produced at senders hands. Then, using a non-maximally entangled GHZ-type state instead of the aforementioned quantum channel, we extend the proposed protocol from three
perspectives: projective measurement, positive operator-value measurement (POVM), and a single generalized Bell-state measurement. Our schemes can relay quantum information over a network without
requiring fully trusted central or intermediate nodes, and none of participants can fully access the information.
Citation: Zhai D, Peng J, Maihemuti N, Tang J (2024) Assisted cloning of an unknown shared quantum state. PLoS ONE 19(8): e0305718. https://doi.org/10.1371/journal.pone.0305718
Editor: Salvatore Lorenzo, Universita degli Studi di Palermo, ITALY
Received: January 29, 2024; Accepted: May 21, 2024; Published: August 28, 2024
Copyright: © 2024 Zhai et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the article.
Funding: This work is supported by the Kashi University Flexible Introduction Research Initiation Fund (No. 022024077). The funder was involved in the research design, data collection and analysis,
publication decision, or manuscript preparation.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
The quantum teleportation (QT) scheme proposed by Bettnett et al. [1] in 1993 pioneered quantum information science which today is a vast research field. Its superior potential for application is
undisputed. Especially, it is a crucial task in the implementation of quantum networks for promising applications such as quantum cryptography [2] and distributed quantum computation [3, 4]. Although
the original QT scheme teleports quantum information from one place to another [1], incorporation of multiple participants is further worth considering to implement versatile quantum networks.
Schemes to share quantum information from one sender to multiple receivers have been presented [5–8] and experimentally demonstrated [9, 10] via the multipartite entanglement state serve as quantum
channel. In these schemes, no single receiver or any subparties of receivers can fully access information unless all other receivers cooperate, which forms the basis for further expanding quantum
secret sharings [11–14] or controlled teleportations [15, 16]. Besides the aforementioned unidirectional QT, bidirectional QT [17, 18] and cyclic QT [19–21] have been studied. Furthermore, there
exist other typical quantum communication protocols that can facilitate the establishment of versatile quantum networks [22–28].Various quantum cryptography schemes, such as quantum key distribution
[22, 23], quantum secure direct communication [24, 25] and remote state preparation (RSP) [26, 27], are capable of establishing a secure communication channel. In these schemes, eavesdropping is
impossible without a high probability of interfering with transmission, as eavesdropping will be detected.
Unlike QT which uses pre-shared quantum entangled channel and classical communication to teleport unknown quantum state, RSP is utilized to teleport a known quantum state, which can save
communication resources compared to QT [29, 30]. Various RSP schemes have emerged, such as multicast-based multiparty RSP [31], controlled RSP [32, 33], joint RSP [34, 35], controlled joint RSP [36],
bidirectional controlled RSP [37, 38], cyclic RSP [39], etc. In 2020, Lee et al. [40] introduced a novel QT scheme enabling the transfer of quantum information from an arbitrary number of senders to
another arbitrary number of receivers in an efficient and distributed manner over a network, without the need for fully trusted central or intermediate nodes. Furthermore, this scheme can be extended
to include error corrections for photon losses, bit or phase-flip errors, and dishonest parties. This work paves the way for secure distributed quantum communications and computations in quantum
networks. In 2022, Li et al. [41] extended Lee’s scheme [40] to the case of non-maximally entangled channel.
To manipulate and extract quantum information, Pati [42] proposed a scheme in 2000 using QT and RSP techniques to generate perfect copies and orthogonal-complement copies of an arbitrary unknown
state with minimal assistance from the state preparer. The first stage of this scheme requires the usual teleportation, while in the second stage the preparer carries out a single-qubit measurement
and conveys some classical information to different parties so that perfect copies and orthogonal-complement copies are produced in a probabilistic manner. Zhan [43] proposed a scheme of realizing
quantum cloning of an unknown two-particle entangled state and its orthogonal complement state with assistance from a state preparer. Han et al. [44] presented a scheme that can clone an arbitrary
unknown two-particle state and its orthogonal complement state with the assistance of a state preparer, where a genuine four-particle entangled state is used as the quantum channel and positive
operator-valued measurement (POVM) instead of usual projective measurement is employed. In Ref. [45], by using a non-maximally four-particle cluster state as quantum channel, a scheme for cloning
unknown two-particle entangled state and its orthogonal complement state with assistance from a state preparer was proposed. Zhan et al. [46] proposed a protocol where one can realize quantum cloning
of an unknown two-particle entangled state and its orthogonal complement state with assistance offered by a state preparer. The following year, Fang et al. [47] generalize Zhan’s protocol [46] such
that an arbitrary unknown two-qubit entangled state can be treated. Zhan [48] presented a scheme for realizing assisted cloning of an unknown two-atom entangled state via cavity QED. Hou and Shi
suggested a protocol for cloning an unknown EPR-type state with assistance by using a one-dimensional non-maximally four-particle cluster state as quantum channel, and then extend it to the case of
cloning an arbitrary unknown two-particle entangled state. Hou and Shi [45] suggested a protocol for cloning an unknown EPR-type state with assistance by using a one-dimensional non-maximally
four-particle cluster state as quantum channel, and then extend it to the case of cloning an arbitrary unknown two-particle entangled state. Xiao et al. [49] put forward a protocol of assisted
cloning and orthogonal complementing of an arbitrary two-qubit state via two partially entangled pairs as quantum channel. Shi et al. [50] proposed a protocol which can realize quantum cloning of an
unknown tripartite entangled state and its orthogonal complement state with assistance from a state preparer. Ma et al. [51] present a scheme which can realize quantum cloning of an unknown N
-particle entangled state and its orthogonal complement state with assistance offered by a state via N non-maximally entangled particle pairs as quantum channel. Subsequently, they proposed a scheme
to produce perfect copy of an unknown d-dimensional equatorial quantum state with assistance offered by a state preparer [52]. Chen et al. [53] and Xue et al. [54] extended the scheme [52] to the
cases for auxiliary cloning of an unknown multi-qudit equatorial-like state and an arbitrary unknown multi-qudit state, respectively.
Stimulated by Refs. [40–42], in this manuscript, we explore the assisted cloning of shared quantum secret state. We first propose a new scheme for cloning an unknown shared quantum state or its
orthogonal complement state with help of a state preparer. This scheme includes two stages: teleportation and assisted cloning. The first stage of the scheme requires quantum teleportation, which
uses a maximally entangled GHZ-type state as the network channel to teleport an arbitrary unknown shared quantum state between multiple senders and receivers. In the second state of the scheme, the
preparer disentangles the left over entangled states by a single-qubit projective measurement and some Z-basis measurements and informs senders of his or her measurement outcomes so that perfect
copies and complement copies of the unknown shared state are producd. In addition, we discuss the assisted cloning problem of shared quantum secret from three perspectives: projective measurement,
POVM, and a single generalized Bell-state measurement, by replacing the aforementioned network channel with a non-maximally entangled GHZ-type state. The results show that the obtained cloning
schemes are all extensions of the above scheme, and they can achieve unit fidelity, but the cost is that the success probability is less than 1.
Below is the arrangement of this article. In Section 2, by using a maximally entangled GHZ-type state as quantum channel, a new scheme for cloning an arbitrary unknown shared state and its orthogonal
complement state with the assistance from a state preparer is presented. In Section 3, by making use of a non-maximally entangled state as network channel, we provide three assisted cloning schemes
of shared quantum secret via projective measurement, POVM, and a single generalized Bell-state measurement, respectively, to meet the needs of real environments and the purpose of expanding the
scheme in Section 2. Discussion and conclusion are drawn in Section 4.
2 Assisted cloning of shared quantum secret via a maximally entangled GHZ-type state
Suppose that Victor is the preparer of the quantum state |χ〉 = α|0〉 + β|1〉, where α, β are complex numbers with |α|^2 + |β|^2 = 1. A quantum secret in |S〉 = α|0[L]〉 + β|1[L]〉 with logical
basis, |0[L]〉 and |1[L]〉, is shared by separated n parties {A[1], A[2], ⋯, A[n]} in quantum network, through a splitting protocol [8–11], where the the state |S〉 can be rewritten as (1) where
qubit s[j] belongs to sharer A[j] (j = 1, 2, ⋯, n). That is to say, state is the result of state |χ〉 being shared by n individuals {A[1], A[2], ⋯, A[n]} in the network. Our utilization of
GHZ-entanglement of photons enables the encoding of network and logical qubits. The senders {A[1], A[2], ⋯, A[n]}, i.e., a group of n parties, endeavor to transmit the shared secret to the receivers,
i.e., another group {B[1], B[2], ⋯, B[m]} of m parties interconnected in the network, and the shared secret state is obtained at the receivers’ hands could be reconstructed as , here qubit r[j]
belongs to receiver B[j] (j = 1, 2, ⋯, m), and then wish to clone this shared secret state at the senders’ hands with the assistance from the state preparer Victor. None of the participants are fully
trusted, therefore no single sender or receiver, or any subparties, is permitted to access the secret during the entire process.
In order to accomplish this objective, the network channel utilizes an (n + m)-particle GHZ-type state (2) where qubit belongs to sender A[l] (l = 1, 2, ⋯, n), while the channel particle r[j] is also
owned by receiver B[j] (j = 1, 2, ⋯ m). The assisted cloning scheme between multiple parties in a quantum network includes two stages: quantum teleportation and copying of unknown state, and the
specific process is presented below.
In the first stage of the scheme, each sender A[j] executes the standard Bell-state measurement on her or his two qubits s[j] and , one from and the other from the network channel . The Bell-states
can be represented as (3)
Utilizing the aforementioned Bell-states, the initial composite system can be expressed as follows (4) where (5)
The arrangement of 2n particles is modified from to and we have (for brevity, the subscripts are omitted) (6) in which N[⋅] represents the sum of all possible arrangements, for example, (7) Upon
conducting n Bell-state measurements, they communicate the outcomes to the recipients through classical channels. As a priori agrement, define the measurement result of A[l] as x[l]y[l] and let the
classical bits 00, 01, 10 and 11 correspond to the Bell-states |ϕ^+〉, |ϕ^−〉, |ψ^+〉 and |ψ^−〉, respectively, and vice vera.
If the measurement outcome is or , at any receiver’s location, he performs the local Pauli operator , where σ[z] = |0〉〈0| − |1〉〈1| and . After the above operations, the state of qubits r[1], r
[2], ⋯, r[m−1] and r[m] becomes (8) which means that the receivers B[1], B[2], ⋯, B[m] successfully reconstruct the shared initial state with unit fidelity. If the measurement result from the senders
is either |Ψ^+〉 or |Ψ^−〉, one of the receivers will apply the local Pauli operator , where σ[x] = |0〉〈1| + |1〉〈0| and , while the other receivers respectively carry out the operator . This will
result in the state owned by the receivers becoming as shown in Eq (8). That is, the receivers can successfully restore the shared initial state with unit fidelity, completing teleportation.
Now we move on to the second stage of the scheme: creating a copy or an orthogonal-complementing copy of the unknown state with assistance from the state preparer. According to the projection
postulate of quantum mechanics, without loss of generality, If the senders’ Bell measurement result is (|ϕ^+〉〈ϕ^+|)^⊗n, the state of qubits and will collapse into the state |ϕ^+〉^⊗n, (see Eqs (4),
(5) and (6)). Each sender A[i] (i = 1, 2, ⋯, n) sends qubit s[i] to the state preparer Victor and keeps qubit in his or her possession. Since Victor knows the state |χ〉 completely, he performs a
single-qubit projective measurement on the qubit s[1] in a set of mutually orthogonal basis vectors , which is given by (9) Subsequently, he measures each other qubit using the Z-basis {|0〉, |1〉},
and publishes the measurement results to the senders through classical communication.
Using Victor’s measurement bases and {|0〉, |1〉}, |ϕ^+〉^⊗n can be written as (10)
Obviously, Eq (9) is a transitional transformation from the old basis {|0〉, |1〉} to the new basis {|ξ[0]〉, |ξ[1]〉}.It is worth noting that under this transformation, the normalization and
orthogonality relationships between the basis vectors are preserved. Interestingly, we find the basis vector |ξ[0]〉 = |χ〉 and the basis vector |ξ[1]〉 = |χ[⊥]〉, where |χ[⊥]〉 = α*|1〉 − β*|0〉 is
the orthogonal-complement state to |χ〉. However, we keep for Victor just to distinguish the fact that he knows the state.
Generally speaking, if Victor’s measurement results for particle s[1] and particle s[j] (j = 2, 3, ⋯, n) are (t = 0, 1) and (k[j] = 0, 1), respectively, then the state of qubits will collapse into
(11) After hearing Victor’s measurement information, sender A[1] needs to performs a unitary operator (−1)^t+1iσ[y] on qubit s[1]. Subsequently, sender A[1] and each A[j] (j = 2, 3, ⋯, n) jointly
implement a unitary transformation on the basis for particles s[1] and s[j], where U[1] and U[2] are given by (12)
After executing the above operations by senders A[1], A[2], ⋯, A[n], the state |S′〉 shown in Eq (11) becomes (13) Upon observing Victor’s measurement outcome of qubit s[1] as (i.e., t = 1) from Eq
(13), it can be inferred that senders A[1], A[2], ⋯, A[n] are able to acquire a flawless replica of the collectively unknown state ; otherwise (i.e., t = 0), they can obtain an
orthogonal-complementing copy of the shared unknown state .
Remark (i) Due to the symmetry of qubits s[1], s[2], ⋯, s[n−1] and s[n] in state |ϕ^+〉^⊗n, Victor first measures anyone s[j] of qubits s[1], s[2], ⋯, s[n−1] and s[n] with the basis {|ξ[0]〉, |ξ[1]
〉}, and then measures the other qubits with the Z basis. The senders of the corresponding qubits perform the corresponding transformation in the above scheme, and the conclusion obtained is the
(ii) Regarding the issue of accessible information, it can be asserted that during the assisted cloning process, no subset can fully access quantum secrets. Now, let’s take the teleportation process
in the first stage of our scheme as an example to illustrate this conclusion. Assume that one sender s[k] attempts to reconstruct the secret at his or her location based on announced results by the
other senders. For simplicity, let m = 1. Following the Bell-state measurement by all senders except s[k], the resulting state at s[k] is either or . Upon tracing out the receiver’s party, the
reduced state at their end becomes |α|^2|00〉〈00| + |β|^2|11〉〈11| or |α|^2|01〉〈01| + |β|^2|10〉〈10|. This holds true unless the entire channel is under their control, meaning that only
amplitude information can be accessed by s[j]. The same applies to any subparties of senders and receivers.
3 Assisted cloning of shared quantum secret via a non-maximally entangled GHZ-type state
The vulnerability of quantum entanglement and the inevitable impact of environmental noise can lead to the degeneration of a maximally entangled state into a non-maximally entangled state. To
deterministically obtain maximally entangled states, one can utilize quantum entanglement concentration and purification schemes [55–58], but, to achieve this, it is necessary to consume a large
ensemble of non-maximally entangled states. Therefore, it is very meaningful to explore quantum communication problems by directly utilizing non-maximally entangled states as quantum channel.
Instead of sharing a (n + m)-qubit maximally entangled GHZ-type state as shown in Eq (2), the senders and receivers may initially share a non-maximally entangled state in the following form (14)
where a and b are real numbers and satisfy a^2 + b^2 = 1. Without losing generality, assuming |a| = min{|a|, |b|}. Using Bell state measurement bases, the initial composite system can be written as
(15) where and are shown in Eqs (5) and (6).
In the teleportation stage, each sender A[j] performs the standard Bell-state measurement on her or his two qubits s[j] and , and publish the measurement results to the receivers via classical
communication. After performing n times of Bell-state measurements, the state of qubits r[1], r[2], ⋯, r[m−1] and r[m] will collapse into one of the following four states (16)
When the measurement outcome is or , each receiver carries out the local Pauli operator (), and the state of the receivers’ particles becomes (17) If the measurement result is or , anyone of
receivers executes the local Pauli operator (), and the other receivers respectively carry out the operator , which will make the state owned by the receivers become (18)
3.1 Assisted cloning based on projective measurement
The state corresponding to the measurement result or is , which is not yet the target state to be restored by receivers. In order to reconstruct the initial state with unity fidelity, an auxiliary
qubit with the original state |0〉[R] is introduced. Due to the symmetry of the state , any receiver B[j] can hold the auxiliary qubit. Without loss of generality, we can assume that the last
receiver B[m] holds the auxiliary qubit and then performs a unitary operator (19) under the basis . It will transform the product state to (20) where is shown in Section 2. The receiver r[m] then
performs a Z-basis measurement on the auxiliary qubit R, which constitutes a projective measurement in the basis {|0〉, |1〉}. If the outcome is |0〉[R], the teleportation is successfully executed
with fidelity 1, whereas if the outcome is |1〉[R], the teleportation fails without providing any information about the target state. The optimal probability of successful teleportation is a^2, where
“optimal” refers to the introduction of an auxiliary qubit.
The measurement result or necessitates the introduction of an auxiliary qubit with the original state |0〉[R′] at the position of the final receiver B[m], the corresponding unitary operator is (21)
which will transform the product state to (22) Next, the receiver B[m] performs a projective measurement with Z-basis on the auxiliary qubit R′. When the outcome is is |1〉[R′], the teleportation
fails. If the measurement result is |0〉[R′], the target state can be reconstructed with a probability of a^2.
The optimal probability of successful teleportation is obtained by adding both contributions, resulting in 2a^2. That is to say, the task of teleporting shared quantum secret in the first stage has
been completed with a probability of 2a^2.
The cloning of shared quantum secret state in the second stage is completely consistent with the Section 2, we will not repeat it here.
Remark: (i) When , the success probability of our scheme is , and the quantum channel shown in Eq (14) degenerates into the maximally entangled channel shown in Eq (2), which indicates that this
scheme is a generalization of that in Section 2.
(ii) Note that the coefficients of quantum channel are all real numbers. More generally, if the quantum channel is in the following form (23) where real numbers a, b, θ[1] and θ[2] satisfy a^2 + b^2
= 1 and θ[1], θ[2] ∈ [0, 3π], then one of the receivers applies a unitary transformation under the computational basis (Z-basis) {|0〉, |1〉} on his or her qubit, i.e. , which converts into .
Therefore, applying the method in this subsection, the corresponding assisted cloning task can always be completed with a certain probability.
3.2 Assisted cloning based on positive operator-valued measurement
Let’s only consider Eq (17), as the discussion on Eq (18) yields the same result. After introducing an auxiliary qubit R with the original state |0〉[R] by the last receiver B[m], he or she performs
a controlled-NOT gate on qubits r[m] and R, where qubit r[m] works as the control qubit and qubit R as the target qubit, and define as . It will transform the product state to (24) where (25) In
order to determine |E〉[B] and |G〉[B], the receiver B[m] needs to perform an optimal positive operator-value measurement (POVM) on the auxiliary qubit R, which should take the following forms (26)
where I is an identity operator, and (27) and the λ related to parameters a and b should ensure that P[3] is a semi positive operator. To determine λ, we need to rewrite P[1], P[2] and P[3] in matrix
form (28) To make P[3] a semi positive operator, the parameter λ should satisfy the condition
After executing POVM, receiver B[m] is able to obtain P[j] (j = 1, 2) with the following probability (29) where . According to the value of POVM, receiver B[m] can infer the state |F[j]〉[R] (j = 1,
2) of auxiliary qubit R. However, based on the value , receiver B[m] can obtain U[5], but cannot infer the states of qubit R. Once receiver B[m] determines the state |F[j]〉[R] (j = 1, 2), it means
he or she knows the state |E[j]〉[B] (j = 1, 2), and then the receiver B[m] applies the corresponding unitary transformation I or σ[z] to qubit r[m].
In this way, the state of qubits r[1], r[2], ⋯, r[m−1] and r[m] becomes with the probability of , completing quantum teleportation.
In summary, the task in the first phase is completed with a probability of 4/λζ and unit fidelity, because starting from Eq (18), the original state can also be reconstructed with a probability of 2/
λζ and unit fidelity.
Similar to Subsection 3.2, the auxiliary cloning of unknown shared quantum state in the second stage is completely consistent with the corresponding part in Section 2.
Remark: When and λ = 1, and the quantum channel shown in Eq (14) changes into the maximally entangled channel shown in Eq (2), and the success probability of quantum teleportation in first stage is
which means that the first stage here is standard quantum teleportation of shared secret. Combining with the second stage, our scheme here is a generalization of the scheme in Section 2.
3.3 Assisted cloning based on a single generalized Bell-state measurement
In the two preceding subsections, achieving 100% fidelity in teleporting the original shred state requires the introduction of an auxiliary qubit and subsequent execution of a two-qubit
transformation. However, we demonstrate here that it is possible for receivers to restore the target state without an auxiliary qubit, albeit not with unit probability. This can be achieved by
replacing n Bell-state measurements with a single generalized Bell-state measurement. To begin, construct the generalized Bell-state basis as follows (30) and use this generalized Bell-state
measurement basis to rewrite as follows (31) where , , and are defined by Eq (6). Here, the sender A[1] executes a generalized Bell-state measurement on qubit pair . In fact, any sender in the group
is capable of conducting the generalized Bell-state measurement due to the symmetry of the original state and the network channel , resulting in identical outcomes. Without loss of generality, let’s
assume that the first sender is responsible for this operation, while the remaining senders continue with standard Bell state measurements. It is evident that the outcome remains unaffected by the
order in which joint measurements are conducted.
If the measurement outcome is , the state at the receivers’ hands will be , and when the result is , the state will be . In both of these scenarios, the receivers have the capability to restore the
target state by implementing appropriate local Pauli operators. Consequently, the probability of successful restoration is denoted as p[1] = 2a^2, where p[1] represents the likelihood of success
without the introduction of an auxiliary qubit.
If the measurement results are or , the unnormalized state at the receivers’ hands will be or , respectively. The receivers can obtain the target state similar to the scheme in Subsection 3.1 by
introducing an auxiliary state and finding the corresponding general evolution separately through replacing (a, b) with (a^2, b^2) in Eq (16). As a result, the optimal successful probability is p[2]
= 2a^2, where p[2] represents the successful probability when introducing an auxiliary qubit.
Note that the probability of randomly obtaining any one of , , or and is 1/4, therefore the total success probability of quantum teleportation in the first stage is
Now, let’s consider the second stage of the scheme. By replacing state (|ϕ^+〉)^⊗n in Section 2 with state , the state corresponding to Eq (10) in Section 2 is (32)
After the state preparer Victor and the senders perform the same operations as the second stage of the scheme in Section 2, the state corresponding to Eq (13) in Section 2 is (33) That is to say, the
states corresponding to Victor’s measurement results and are the states and , respectively.
Finally, when sender A[1] uses the methods described in Subsections 3.1 or 3.2, all senders can obtain a copy or an orthogonal-complementing copy of the unknown shared state with probability a^2.
For other measurement results in the process of quantum teleportation (see Eq (31)), applying the same analysis method as above, senders will obtain a copy or an orthogonal-complementing copy of the
unknown shared state with a certain probability.
Remark: (i) The construction method of generalized Bell-state basis is not unique. For example, the vectors , , and also form a set of mutually orthogonal generalized Bell-state bases.
(ii) Similar to the discussion in Subsections 3.1 or 3.2, this scheme is still a generalization of the scheme in Section 2.
4 Discussion and conclusion
As the first stage of our scheme, quantum teleportation is different from the design concept in references [59, 68] in which a trusted node plays an important role in connecting participants and
transmitting information. Establishing a long-distance quantum communication through distributed nodes is advantageous, as it eliminates the need for any single node to relay complete quantum
information. This principle also extends to the storage and retrieval of quantum secrets in spatially separated quantum memory. Verification strategies for multipartite entanglement [58–61] are
valuable for preparing an entangled network in the presence of untrustworthy parties. The access information in schemes with non-maximally entangled channels can be displayed using the same method as
described in Subsection 3.1 of this article. In this way, we still have the conclusion that no subparties can fully access quantum secrets during the process of teleportation.
Note that except for the (n − 1) times of standard Bell-state measurements used in Subsection 3.3, all other schemes used n times of standard Bell-state measurements, which are unnecessary. Actually,
only one Bell-state measurement or generalized Bell-state measurement is sufficient for implemening each scheme. Rewrite Eq (6) as (wothout normalized) (34) implying that executing one Bell-state
measurement and 2(n − 1) times of X-basis measurements is equivalent to n times of Bell-state measurements (or one generalized Bell-state measurement and (n − 1) times of Bell-state measurements),
where the X-basis measurement is a single-qubit projective measurement on the basis . Due to the greater experimental feasibility of executing two single-qubit measurements compared to a joint
two-qubit measurement, it appears that the latter option, involving only one Bell-state (generalized Bell-state) measurement, should be chosen. The success probability of identifying the Bell states
|ϕ^−〉 and |ψ^−〉 is limited to 1/2, as only these two states can be unambiguously distinguished from each other [62, 63]. When performing n Bell-state measurements according to Eq (6), failure
occurs only when the measurement result is |ϕ^+〉^⊗n or |ψ^+〉^⊗n. As a result, the probability of successful discrimination increases to 1 − 2^−n, indicating that increasing n enhances the
likelihood of successfully distinguishing between the logical Bell states.
Of course, our work is not limited to the GHZ-type state encoding. By correcting photon loss, operational errors, and dishonest participants through error coding, it can be further extended to fault
tolerance. For instance, a parity state encoding [64] can be used to some extent [65, 66] to correct the effects of photon loss, errors, and dishonesty. In principle, even in the case of loss and
error, it can transmit quantum information with any high probability of success [67]. The use of other types of entangled states such as cluster states for encoding is worth further consideration.
The success probability of Bell state measurement can be enhanced by utilizing cluster state encoding [68]. The integration of such encoding techniques and secret sharing protocols [11] based on
cluster states is of great significance. In addition, there is only one state preparer in our schemes. We can introduce two or more state preparers like in article [69], which can improve the
security of our scheme. To achieve this, we need to introduce an appropriate amount of auxiliary particles and use joint RSP technology.
It is worth noting that our scheme has high security. To make it clearer, we give a security check here. Before initiating bidirectional controlled assisted cloning, they should first conduct a
security check. Alice prepares one check sequence composed of qudits with the random state {|0〉, |1〉, |+〉, |−〉} and sends it to the remote preparer Victor. When the eavesdropper Eve intercepted
this qudit sequence, he will randomly select a set of polarization-based measurements {|0〉, |1〉} or {|+〉, |−〉} to measure the check qudits, and prepares a new qudit sequence to Victor. However,
due to Eve’s behavior interfering with quantum states, if he chooses the wrong measurement basis, it will lead to a high error rate. This high error rate will be detected during the process of
security check between Alice and Victor. If there is one eavesdropper, it is possible for Alice and Victor to suspend the communication. The same security detection method can be used between Bob and
Victor, as well as among Alice, Bob and Charlie. Therefore, the security of our scheme can be guaranteed. On the other hand, due to the introduction of a controller in our scheme, the security of the
scheme has been further enhanced.
In summary, we have proposed a new protocol that one can produce perfect copies and orthogonal complement copies of an arbitrary unkonwn shared quantum state via quantum and classical channel, with
assistance of a state preparer. This assisted cloing ptotocol needs two stage. The first stage requires teleportation by using a maximally entangled GHZ-type state as quantum channel to teleport an
arbitrary unknown shared quantum state between multiple parties in a quantum network. In the second stage, the state preparer executes a special single-qubit projective measurement and a series of
single-qubit computational basis measurements on the qubits which seeded by senders. After having received the preparer’s measurement outcomes through classical channel, senders can obtain the input
original state and its orthogonal complement state by a series of appropriate unitary operations. In order to meet the needs of the real environment, we have extended the above protocols to the case
of non-maximally entangled GHZ-type quantum network channel, and obtained three generalized protocols with unit fidelity a certain probability. In the first generalized protocol, one of the receivers
needs to introduce an auxiliary qubit, perform a twoqubit unitary transformation, and make a single-qubit computational basis measurement on the auxiliary qubit. The second generalized protocol
requires one of the receivers to perform a controlled-NOT gate transformation and POVM after introducing an auxiliary qubit. The first stage of the third generalized protocol requires one of the
senders to perform a generalized Bell state measurement. After other senders perform standard Bell measurements, the receivers either directly recover the input unknown state through appropriate
Palui gates, or probabilistically reconstruct the target state using the method of the first or second generalized protocol. In the second stage of this protocol, the senders need to use the
receivers’ method in the first or second generalized protocols to complete the cloning task. | {"url":"https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0305718","timestamp":"2024-11-03T00:35:33Z","content_type":"text/html","content_length":"251020","record_id":"<urn:uuid:bea92a3e-853e-4097-94b1-428fabdce4e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00790.warc.gz"} |
Mathematical Optimization Group, University of Tübingen
In this paper we study an algorithm for solving a minimization problem composed of a differentiable (possibly non-convex) and a convex (possibly non-differentiable) function. The algorithm combines
forward-backward splitting with an inertial force. A rigorous analysis of the algorithm for the proposed class of problems yields global convergence of the function values and the arguments. This
makes the algorithm robust for usage on non-convex problems. The convergence result is obtained based on the Kurdyka-Lojasiewicz inequality. This is a very weak restriction, which was used to prove
convergence for several other gradient methods. Furthermore, a convergence rate is established for the general problem class. In the more specialized case of convex functions, the optimal
convergence rate is shown if one of the functions is stronlgy convex. We demonstrate iPiano on several computer vision problems, among them learned priors in denoising and optical flow estimation,
and image compression.
P. Ochs, Y. Chen, T. Brox, T. Pock:
iPiano: Inertial Proximal Algorithm for Non-convex Optimization.
SIAM Journal on Imaging Sciences, 7(2):1388-1419, 2014.
title = {iPiano: Inertial Proximal Algorithm for Non-convex Optimization},
author = {P. Ochs and Y. Chen and T. Brox and T. Pock},
year = {2014},
journal = {SIAM Journal on Imaging Sciences},
number = {2},
volume = {7},
pages = {1388--1419} | {"url":"https://mop.math.uni-tuebingen.de/pub/OCBP14/index.shtml","timestamp":"2024-11-08T18:04:13Z","content_type":"text/html","content_length":"6970","record_id":"<urn:uuid:2317713a-1ce9-4347-bcb5-35156a053efe>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00764.warc.gz"} |
Epsilon Definition of The Supremum and Infimum of a Bounded Set
Epsilon Definition of The Supremum and Infimum of a Bounded Set
Recall from The Supremum and Infimum of a Bounded Set page the following definitions:
Definition: Let $S$ be a set that is bounded above. We say that the supremum of $S$ denoted $\sup S = u$ is a number $u$ that satisfies the conditions that $u$ is an upper bound of $S$ and $u$ is the
least upper bound of $S$, that is for any $v$ that is also an upper bound of $S$ then $u \leq v$.
Definition: Let $S$ be a set that is bounded below. We say that the infimum of $S$ denoted $\inf S = w$ is a number $w$ that satisfies the conditions that $w$ is a lower bound of $S$ and $w$ is the
greatest lower bound of $S$, that is for any $t$ that is also a lower bound of $S$ then $t \leq w$.
We will now reformulate these definitions with an equivalent statement that may be useful to apply in certain situations in showing that an upper bound $u$ is the supremum of a set, or showing that a
lower bound $w$ is the infimum of a set.
Theorem 1: Let $S$ be a nonempty subset of the real numbers that is bounded above. The upper bound $u$ is said to be the supremum of $S$ if and only if $\forall \epsilon > 0$ there exists an element
$x_{\epsilon} \in S$ such that $u - \epsilon < x_{\epsilon}$.
• Proof: $\Rightarrow$ Let $S$ be a nonempty subset of the real numbers that is bounded above. We first want to show that if $u$ is an upper bound such that $\forall \epsilon > 0$ there exists an
element $x_{\epsilon} \in S$ such that $u - \epsilon < x_{\epsilon}$, then $u = \sup S$. Let $u$ be an upper bound of $S$ that satisfies the condition stated above, and suppose that $v < u$. Then
choose $\epsilon = u - v$, and so $u - v = \epsilon > 0$ and so there exists an element $x_{\epsilon} \in S$ such that $u - \epsilon < x_{\epsilon}$, and thus, $u = \sup S$.
• $\Leftarrow$ We now want to show that if $u = \sup S$ then $\forall \epsilon > 0$ there exists an element $x_{\epsilon} \in S$ such that $u - \epsilon < x_{\epsilon}$. Let $u = \sup S$ and let $\
epsilon > 0$. We note that $u - \epsilon < u$ and so $u - \epsilon$ is not an upper bound of the set $S$. Therefore by the definition that $u = \sup S$, there exists some element $x_{\epsilon} \
in S$ such that $u - \epsilon < x_{\epsilon}$. $\blacksquare$
Theorem 2: Let $S$ be a nonempty subset of the real numbers that is bounded below. The lower bound $w$ is said to be the infimum of $S$ if and only if $\forall \epsilon > 0$ there exists an element
$x_{\epsilon} \in S$ such that $x_{\epsilon} < w + \epsilon$.
• Proof: $\Rightarrow$ Let $S$ be a nonempty subset of the real numbers that is bounded below. We first want to show that if $w$ is a lower bound such that $\forall \epsilon > 0$ there exists an
element $x_{\epsilon} \in S$ such that $x_{\epsilon} < w + \epsilon$, then $w = \inf S$. Let $w$ be a lower bound of $S$ that satisfies the condition stated above, and suppose that $w < t$. Then
choose $\epsilon = t - w$, and so $t - w = \epsilon > 0$ and so there exists an element $x_{\epsilon} \in S$ such that $x_{\epsilon} < w + \epsilon$, and thus, $w = \inf S$.
• $\Leftarrow$ We now want to show that if $w = \inf S$ then $\forall \epsilon > 0$ there exists an element $x_{\epsilon} \in S$ such that $x_{\epsilon} < w + \epsilon$. Let $w = \inf S$ and let $\
epsilon > 0$. We note that $w < w +\epsilon$ and so $w + \epsilon$ is not a lower bound of the set $S$. Therefore by the definition that $w = \inf S$, there exists some element $x_{\epsilon} \in
S$ such that $x_{\epsilon} < w + \epsilon$. $\blacksquare$ | {"url":"http://mathonline.wikidot.com/epsilon-definition-of-the-supremum-and-infimum-of-a-bounded","timestamp":"2024-11-13T23:15:36Z","content_type":"application/xhtml+xml","content_length":"20787","record_id":"<urn:uuid:e56b86f4-8f14-490f-960c-01ac5d9f71f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00002.warc.gz"} |
Operator inclusions and operator-differential inclusions
Bian, Wenming (1998) Operator inclusions and operator-differential inclusions. PhD thesis, University of Glasgow.
Full text available as:
In Chapter 2, we first introduce a generalized inverse differentiability for set-valued mappings and consider some of its properties. Then, we use this differentiability, Ekeland's Variational
Principle and some fixed point theorems to consider constrained implicit function and open mapping theorems and surjectivity problems of set-valued mappings. The mapping considered is of the form F
(x, u) + G (x, u). The inverse derivative condition is only imposed on the mapping x F(x, u), and the mapping x G(x, u) is supposed to be Lipschitz. The constraint made to the variable x is a closed
convex cone if x F(x, u) is only a closed mapping, and in case x F(x, u) is also Lipschitz, the constraint needs only to be a closed subset. We obtain some constrained implicit function theorems and
open mapping theorems. Pseudo-Lipschitz property and surjectivity of the implicit functions are also obtained. As applications of the obtained results, we also consider both local constrained
controllability of nonlinear systems and constrained global controllability of semilinear systems. The constraint made to the control is a time-dependent closed convex cone with possibly empty
interior. Our results show that the controllability will be realized if some suitable associated linear systems are constrained controllable.
In Chapter 3, without defining topological degree for set-valued mappings of monotone type, we consider the solvability of the operator inclusion y0 N1(x) + N2 (x) on bounded subsets in Banach spaces
with N1 a demicontinuous set-valued mapping which is either of class (S+) or pseudo-monotone or quasi-monotone, and N2 is a set-valued quasi-monotone mapping. Conclusions similar to the invariance
under admissible homotopy of topological degree are obtained. Some concrete existence results and applications to some boundary value problems, integral inclusions and controllability of a nonlinear
system are also given.
In Chapter 4, we will suppose u A (t,u) is a set-valued pseudo-monotone mapping and consider the evolution inclusions
x' (t) + A(t,x((t)) f (t) a.e. and (d)/(dt) (Bx(t)) + A (t,x((t)) f(t) a.e.
in an evolution triple (V,H,V*), as well as perturbation problems of those two inclusions.
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How Many Days Is 1 Million Seconds - How Many Sumo
How Many Days Is 1 Million Seconds
Time measurement is a fundamental aspect of human existence, allowing us to organize and comprehend the passing of events.
One million seconds may seem like a large figure, but in the realm of time, it is just a drop in the vast ocean. This article aims to explore the magnitude of one million seconds and translate it
into a more relatable unit of measurement, such as days, in order to provide a clearer understanding of its significance.
In order to grasp the concept of one million seconds, it is crucial to first comprehend the nature of time measurement. Time is commonly divided into smaller units, such as seconds, minutes, hours,
and days, which aid in quantifying and organizing various phenomena.
One second, the fundamental unit of time in the International System of Units (SI), represents a specific duration defined as ‘the duration of 9,192,631,770 periods of the radiation corresponding to
the transition between the two hyperfine levels of the ground state of the cesium-133 atom.’ With this precise definition, we can begin to explore the magnitude of one million seconds and its
translation into a more tangible unit of measurement.
Understanding Time Measurement
The measurement of time is crucial for various aspects of human life, and understanding the conversion between seconds and days is essential for accurately representing temporal intervals.
Time measurement has been of historical importance since ancient civilizations sought to track the passing of days, seasons, and celestial events. The development of various units of time measurement
has been driven by the need for standardization and consistency across different cultures and regions.
One of the earliest units of time measurement was the day, which was based on the observation of the Earth’s rotation. The concept of a day as a 24-hour period has been widely accepted across
different cultures, although variations exist in the way time is divided within a day.
Another commonly used unit of time is the second, which is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground
state of the cesium-133 atom. The second is a fundamental unit of time in the International System of Units (SI) and serves as the basis for other units such as minutes, hours, and days.
The conversion between seconds and days is straightforward. There are 86,400 seconds in a day, which is calculated by multiplying 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a day.
This conversion is important for various applications, such as calculating the duration of events, measuring the speed of processes, or understanding the temporal scale of historical events.
Accurate time measurement and understanding the relationship between seconds and days are crucial for coordinating activities across different time zones, scheduling events, and ensuring
synchronization in various fields, including science, technology, transportation, and communication.
The Magnitude of One Million Seconds
The subtopic of the magnitude of one million seconds delves into the visualization and breakdown of this vast number.
To comprehend the enormity of one million seconds, it is essential to have a visual representation of it.
Additionally, breaking down the calculation into smaller units, such as hours or days, allows for a better understanding of the duration and significance of this time interval.
Visualizing the Number
Visualizing the enormity of one million seconds can leave one feeling awestruck.
When we contemplate the passage of time, it is often difficult to grasp the magnitude of such large numbers.
To put it into perspective, let’s consider time management strategies.
If we were to divide one million seconds into days, we would find that it is equivalent to approximately 11.6 days.
This means that one million seconds represents a relatively short period of time in the grand scheme of things.
The concept of eternity, on the other hand, takes on a different perspective when faced with the reality of one million seconds.
While it may seem like a significant amount of time to us as individuals, in the context of eternity, it is just a mere fraction.
This realization highlights the vastness of time and puts our lives into perspective.
It serves as a reminder that our time on this earth is limited and should be cherished.
Understanding the scale of one million seconds helps us appreciate the value of each passing moment and encourages us to make the most of our time.
Breaking Down the Calculation
Breaking down the calculation of one million seconds reveals the relatively short period of time it represents.
To convert seconds into days, one must consider the various time conversion techniques. There are 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a day. By multiplying these conversion
factors together, the total number of seconds can be converted to days.
In the case of one million seconds, it would be divided by 60 to convert to minutes, divided by 60 again to convert to hours, and divided by 24 to convert to days. This calculation results in
approximately 11.57 days.
The significance of seconds in everyday life is often overlooked due to their fleeting nature. However, they play a crucial role in measuring time and keeping track of daily activities.
Seconds help us gauge the passing of time in a more precise and detailed manner, allowing for more accurate planning and coordination. From measuring the duration of a phone call to timing the
preparation of a meal, seconds are integral to our everyday routines.
Moreover, in fields such as sports, science, and technology, seconds can make a significant difference. For instance, in a race, a fraction of a second can determine the winner, and in scientific
experiments, precise timing is crucial for accurate results.
Therefore, while seconds may seem insignificant individually, their collective impact on our lives is undeniable.
Translating Seconds into Days
Calculating the duration of 1 million seconds in terms of days involves dividing the given number of seconds by the total number of seconds in a day.
In order to perform this conversion, it is important to be familiar with time conversion tips. One such tip is knowing that there are 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a
day. By applying these conversions, we can determine the number of seconds in a day, which is equal to 86,400 seconds.
Translating 1 million seconds into days can be a useful exercise when trying to understand time in real-life scenarios. For example, if someone is planning an event that will last for 1 million
seconds, they may want to know how many days that will be. This can help them in organizing the event and determining its duration.
Additionally, understanding how many days are in 1 million seconds can also be helpful in scientific and mathematical calculations that involve time. By converting seconds into days, researchers and
scientists can manipulate and analyze time-related data more easily.
Putting One Million Seconds into Perspective
In the previous subtopic, we discussed how to translate seconds into days in order to understand the duration of one million seconds.
Now, let us put one million seconds into perspective. When considering such a large unit of time, it is important to recognize that our perception of time is subjective and can vary depending on the
One million seconds may seem like a significant amount when we think about it in terms of seconds ticking away on a clock. However, when we convert it into days, we begin to realize that it is not as
long as it initially appears.
Time perception plays a crucial role in how we manage our daily lives. With the increasing demands and fast-paced nature of modern society, effective time management has become an essential skill.
When we think about one million seconds in the context of time management, it can serve as a reminder of the importance of utilizing time efficiently. This realization encourages us to prioritize
tasks, set goals, and make the most of the limited time we have.
By understanding the significance of one million seconds and its implications for time perception and management, we can strive for improved productivity and a more balanced lifestyle.
• The concept of time perception is fascinating because it highlights the subjective nature of our experience with time.
• Time management is a skill that is highly valued in our fast-paced society, and understanding the duration of one million seconds can serve as a reminder to effectively utilize our time.
• Recognizing the significance of one million seconds can motivate individuals to prioritize tasks, set goals, and make better use of their limited time.
Time measurement is a fundamental aspect of daily life, allowing us to organize and understand the passing of moments. One intriguing question that arises is how many days are contained within one
million seconds.
To answer this question, we must first comprehend the magnitude of one million seconds and then translate this figure into a more relatable unit of measurement, such as days.
One million seconds may seem like a vast amount of time, but in the grand scheme of things, it is relatively brief. To put it into perspective, one million seconds is equivalent to approximately 11
and a half days. This calculation is based on the conversion of seconds to minutes, minutes to hours, and hours to days. By breaking down the measurement into smaller units, we can better grasp the
temporal significance of one million seconds.
Understanding the concept of one million seconds in terms of days reveals the fleeting nature of time. It highlights the need to cherish and make the most of each passing moment. While one million
seconds may seem substantial when considered in isolation, when compared to the broader context of a lifetime, it becomes clear that time is a precious and finite resource.
Therefore, it is essential to use this knowledge as a reminder to appreciate the significance of each second, and to strive for meaningful and purposeful lives.
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Assistant professor of Electrical Engineering & Control at CentraleSupélec, Rennes campus. 200–250 hours/year.
2016–present: 1^st year electrical energy course (AC power, magnetic circuits, transforms, DC machines).
2016–present: creating and teaching a course on the Modelica multiphysics modeling language, with a focus on model structuring and collaborative engineering (version control with Git), and a short
introduction to bond graphs (33 hours). Online assignment: http://éole.net/courses/modelica/ (with OpenModelica getting started videos 📹).
2019–present: 2^nd year “engineering challenge term” on Microgrids and Renewable Energies, which includes an optimization project (with Nabil Sadou).
2020–present: Introduction to Power Systems course (AC load flow (Matpower), voltage regulation, integration of Renewables, 15 hours).
2021–present: Optimization under Uncertainty course (9 hours), followed by 15-hour practice sessions on optimal sizing and energy management of a Microgrid (with Nabil Sadou).
2022–present: Power electronics modeling and control lab course (with Simulink/Simscape).
2015–2018: creation of two 20 hours lab courses for our “Smart grids” Master program:
2014–2019: supervising a 5×4 hours lab session on industrial process control with a Programmable logic controller (using Grafcet and Ladder languages)
Other past courses: Model order reduction, Process identification lab.
Supervision of a few 1^st and 2^nd year students’ projects each year. | {"url":"http://pierreh.eu/cv/","timestamp":"2024-11-03T21:25:20Z","content_type":"text/html","content_length":"40178","record_id":"<urn:uuid:0164412d-86bd-461b-8d8c-dd17be870acd>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00445.warc.gz"} |
How to Find the Size of a Data Frame in R | Online Tutorials Library List | Tutoraspire.com
How to Find the Size of a Data Frame in R
by Tutor Aspire
You can use the following functions in R to display the size of a given data frame:
• nrow: Display number of rows in data frame
• ncol: Display number of columns in data frame
• dim: Display dimensions (rows and columns) of data frame
The following examples show how to use each of these functions in practice with the following data frame:
#create data frame
df frame(team=c('A', 'B', 'C', 'D', 'E', 'F'),
points=c(99, 90, 86, 88, 95, 99),
assists=c(33, 28, 31, 39, 34, 25),
rebounds=c(12, NA, 24, 24, 28, 33))
#view data frame
team points assists rebounds
1 A 99 33 12
2 B 90 28 NA
3 C 86 31 24
4 D 88 39 24
5 E 95 34 28
6 F 99 25 33
Example 1: Use nrow() to Display Number of Rows
The following code shows how to use the nrow() function to display the total number of rows in the data frame:
#display total number of rows in data frame
[1] 6
There are 6 total rows.
Note that we can also use the complete.cases() function to display the total number of rows with no NA values:
#display total number of rows in data frame with no NA values
nrow(df[complete.cases(df), ])
[1] 5
There are 5 total rows that have no NA values.
Example 2: Use ncol() to Display Number of Columns
The following code shows how to use the ncol() function to display the total number of columns in the data frame:
#display total number of columns in data frame
[1] 4
There are 4 total columns.
Example 3: Use dim() to Display Dimensions
The following code shows how to use the dim() function to display the dimensions (rows and columns) of the data frame:
#display dimensions of data frame
[1] 6 4
This tells us there are 6 rows and 4 columns in the data frame.
You can also use brackets with the dim() function to display only the rows or columns:
#display number of rows of data frame
[1] 6
#display number of columns of data frame
[1] 4
Additional Resources
The following tutorials explain how to perform other common tasks in R:
How to Use rowSums() Function in R
How to Apply Function to Each Row in Data Frame in R
How to Remove Rows from Data Frame in R Based on Condition
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How to Calculate Ratios in R (With Examples)
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Derivative of Tan x - Formula, Proof, Examples
The tangent function is one of the most significant trigonometric functions in mathematics, physics, and engineering. It is an essential concept applied in a lot of domains to model several
phenomena, consisting of wave motion, signal processing, and optics. The derivative of tan x, or the rate of change of the tangent function, is an essential concept in calculus, that is a branch of
math which deals with the study of rates of change and accumulation.
Understanding the derivative of tan x and its characteristics is essential for working professionals in multiple domains, consisting of physics, engineering, and math. By mastering the derivative of
tan x, professionals can use it to work out problems and gain detailed insights into the intricate workings of the world around us.
If you require help getting a grasp the derivative of tan x or any other math theory, consider connecting with Grade Potential Tutoring. Our experienced tutors are available remotely or in-person to
offer personalized and effective tutoring services to assist you succeed. Connect with us today to schedule a tutoring session and take your mathematical skills to the next level.
In this article, we will delve into the idea of the derivative of tan x in detail. We will start by discussing the significance of the tangent function in various domains and uses. We will then check
out the formula for the derivative of tan x and give a proof of its derivation. Finally, we will give instances of how to utilize the derivative of tan x in various fields, involving physics,
engineering, and arithmetics.
Significance of the Derivative of Tan x
The derivative of tan x is an essential mathematical idea which has multiple applications in calculus and physics. It is utilized to work out the rate of change of the tangent function, which is a
continuous function that is widely utilized in mathematics and physics.
In calculus, the derivative of tan x is applied to work out a wide array of problems, involving finding the slope of tangent lines to curves which involve the tangent function and evaluating limits
that involve the tangent function. It is also applied to calculate the derivatives of functions which involve the tangent function, for example the inverse hyperbolic tangent function.
In physics, the tangent function is applied to model a broad range of physical phenomena, consisting of the motion of objects in circular orbits and the behavior of waves. The derivative of tan x is
applied to work out the velocity and acceleration of objects in circular orbits and to get insights of the behavior of waves which consists of variation in frequency or amplitude.
Formula for the Derivative of Tan x
The formula for the derivative of tan x is:
(d/dx) tan x = sec^2 x
where sec x is the secant function, which is the opposite of the cosine function.
Proof of the Derivative of Tan x
To prove the formula for the derivative of tan x, we will apply the quotient rule of differentiation. Let’s assume y = tan x, and z = cos x. Next:
y/z = tan x / cos x = sin x / cos^2 x
Utilizing the quotient rule, we get:
(d/dx) (y/z) = [(d/dx) y * z - y * (d/dx) z] / z^2
Substituting y = tan x and z = cos x, we get:
(d/dx) (tan x / cos x) = [(d/dx) tan x * cos x - tan x * (d/dx) cos x] / cos^2 x
Subsequently, we can apply the trigonometric identity that connects the derivative of the cosine function to the sine function:
(d/dx) cos x = -sin x
Replacing this identity into the formula we derived above, we get:
(d/dx) (tan x / cos x) = [(d/dx) tan x * cos x + tan x * sin x] / cos^2 x
Substituting y = tan x, we get:
(d/dx) tan x = sec^2 x
Therefore, the formula for the derivative of tan x is proven.
Examples of the Derivative of Tan x
Here are few examples of how to apply the derivative of tan x:
Example 1: Work out the derivative of y = tan x + cos x.
(d/dx) y = (d/dx) (tan x) + (d/dx) (cos x) = sec^2 x - sin x
Example 2: Locate the slope of the tangent line to the curve y = tan x at x = pi/4.
The derivative of tan x is sec^2 x.
At x = pi/4, we have tan(pi/4) = 1 and sec(pi/4) = sqrt(2).
Therefore, the slope of the tangent line to the curve y = tan x at x = pi/4 is:
(d/dx) tan x | x = pi/4 = sec^2(pi/4) = 2
So the slope of the tangent line to the curve y = tan x at x = pi/4 is 2.
Example 3: Find the derivative of y = (tan x)^2.
Using the chain rule, we get:
(d/dx) (tan x)^2 = 2 tan x sec^2 x
Thus, the derivative of y = (tan x)^2 is 2 tan x sec^2 x.
The derivative of tan x is an essential math concept which has many applications in calculus and physics. Comprehending the formula for the derivative of tan x and its characteristics is crucial for
learners and professionals in fields such as engineering, physics, and mathematics. By mastering the derivative of tan x, anyone can apply it to solve challenges and gain deeper insights into the
intricate functions of the world around us.
If you require assistance comprehending the derivative of tan x or any other mathematical theory, consider calling us at Grade Potential Tutoring. Our experienced tutors are accessible remotely or
in-person to give personalized and effective tutoring services to help you be successful. Call us today to schedule a tutoring session and take your math skills to the next stage. | {"url":"https://www.clearwaterinhometutors.com/blog/derivative-of-tan-x-formula-proof-examples","timestamp":"2024-11-02T15:50:28Z","content_type":"text/html","content_length":"75638","record_id":"<urn:uuid:50962898-7236-4343-8026-f7163cb9a629>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00157.warc.gz"} |
Data Visualization in Python, Part 2: Matplotlib Fundamentals | Sunscrapers
Maria Chojnowska
24 August 2023, 3 min read
What's inside
1. Matplotlib: The Heart of Python Visualization
2. Installing and Importing Matplotlib
3. Matplotlib Basics: Creating a Simple Line Plot
4. Exploring Different Plot Types
5. Customizing Your Plots
6. Working with Multiple Plots
Matplotlib: The Heart of Python Visualization
Matplotlib is essentially a multi-platform data visualization library built on NumPy arrays, and designed to work with the broader SciPy stack. It was conceived by John Hunter in 2002, inspired by
the MATLAB programming language, to enable interactive and easy creation of plots.
Installing and Importing Matplotlib
To install Matplotlib, we'll use pip, Python's package installer. In your terminal, type:
pip install matplotlib
To import Matplotlib, we generally import the `pyplot` submodule using the alias `plt`:
import matplotlib.pyplot as plt
Matplotlib Basics: Creating a Simple Line Plot
For the sake of demonstration, let's use a simple line plot. Let's assume we have time (in seconds) and the corresponding distance (in meters) travelled by an object.
# Importing required libraries
import matplotlib.pyplot as plt
# Create two lists, time (0 to 5 seconds) and distance (in meters)
time = [0, 1, 2, 3, 4, 5]
distance = [0, 1, 4, 9, 16, 25]
# Create a line plot
plt.plot(time, distance)
# Display the plot
Exploring Different Plot Types
Now, let's explore different types of plots that can be created using Matplotlib: 1. Histograms: These are useful for understanding the distribution of continuous numerical data.
import numpy as np
# Generating 1000 random values
random_data = np.random.randn(1000)
# Creating histogram
plt.hist(random_data, bins=20)
# Displaying the histogram
2. Scatter Plots: These are used to understand the relationship or correlation between two numerical variables.
# Generating 100 random values for x and y
x = np.random.rand(100)
y = np.random.rand(100)
# Creating scatter plot
plt.scatter(x, y)
# Displaying the scatter plot
3. Bar Plots: These are used to compare quantities of different categories.
# Creating a list of categories and their values
categories = ['A', 'B', 'C', 'D', 'E']
values = [7, 12, 6, 9, 14]
# Creating bar plot
plt.bar(categories, values)
# Displaying the bar plot
Customizing Your Plots
Visual aesthetics are a vital part of data visualization. Matplotlib allows us to control line styles, font properties, axes properties, etc. Let's customize the line plot we made earlier:
plt.plot(time, distance, color='purple', linestyle='--', linewidth=2)
# Adding title and labels
plt.title('Distance over Time')
plt.xlabel('Time (seconds)')
plt.ylabel('Distance (meters)')
# Adding a legend
plt.legend(['Object 1'])
# Display the plot
Working with Multiple Plots
In some scenarios, we might want to display multiple plots in one figure for comparison. Matplotlib provides the subplot() function for this:
# First subplot
plt.subplot(1, 2, 1)
plt.plot(time, distance, 'r-')
plt.title('Plot 1: Line Plot')
# Second subplot
plt.subplot(1, 2, 2)
plt.scatter(time, distance, color='blue')
plt.title('Plot 2: Scatter Plot')
# Adjust distance between the two plots
# Show the plots
Remember, data visualization is not about creating flashy plots but about conveying information effectively. The best visualization method will depend on the data, its complexity, and the audience's
Stay tuned for our next post, where we'll dive into advanced topics of Matplotlib, like 3D plotting, animation, and interactive plotting.
In the meantime, feel free to contact us if you need help with Python-related tasks, data visualization, machine learning, web development, or anything else. With our expertise, we can transform your
ideas into working solutions.
Don't let Python's intricacies slow down your project. Contact us today for a discussion about your Python development needs.
Remember, the possibilities with Python and our professional services are limitless. Contact us. | {"url":"https://sunscrapers.com/blog/data-visualization-in-phyton-Matplotlib-Fundamentals/","timestamp":"2024-11-06T19:52:52Z","content_type":"text/html","content_length":"1018341","record_id":"<urn:uuid:7865f32a-df3b-48c5-816d-dd8a635d9fe0>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00501.warc.gz"} |
Joceline C Lega
• Professor, Mathematics
• Professor, Public Health
• Professor, BIO5 Institute
• Associate Head, Postdoctoral Programs
• Member of the Graduate Faculty
I was born in Nice and educated in France. I studied at Ecole Normale Supérieure in Paris, to which I was admitted in 1984. In 1985, I received a BS (Licence) and a MS (Maîtrise) in Physics, from
the Université Pierre et Marie Curie (Paris VI). At that point, I decided to pursue studies in the then burgeoning field of nonlinear dynamics and, in 1986, completed a post-graduate degree (Diplôme
d’Etudes Approfondies) in Dynamical Systems and Turbulence at the Université de Nice. Three years later, in March 1989, I received my Ph.D (Doctorat) in Theoretical Physics from the University of
In 1989, I was hired by CNRS (the French National Center for Scientific Research) and worked as a researcher first in the Department of Theoretical Physics at the University of Nice and then at the
Institut Non Linéaire de Nice, based in Sophia Antipolis.
Between 1990 and 1997, I established various collaborations with researchers in theDepartment of Mathematics at the University of Arizona, first as a postdoctoral fellow and then as a Visiting
Assistant Professor. I was hired as an Assistant Professor of Mathematics by the University of Arizona in 1997, was promoted to Associate Professor with tenure in 2000, and to full Professor in 2006.
• Doctorat (equiv. PhD) Theoretical Physics
□ Université de Nice, Nice, France
□ Topological defects associated with the breaking of time translation invariance
• Diplôme d’Etudes Approfondies Dynamical Systems and Turbulence
□ Université de Nice, Nice, France
• License (equiv. BS) Physics
□ Université Pierre & Marie curie, Paris, France
• Maitrise (equiv. MS) Physics
□ Université Pierre & Marie curie, Paris, France
Work Experience
• University of Arizona, Tucson, Arizona (2006 - Ongoing)
• University of Arizona, Tucson, Arizona (2000 - 2006)
• University of Arizona, Tucson, Arizona (1997 - 2000)
• CNRS (National Center for Scientific Research), France (1989 - 1997)
• 2019 Excellence in Postdoctoral Mentoring Award
□ University of Arizona, Spring 2019
• Fellow of the American Association for the Advancement of Science
□ American Association for the Advancement of Science, Fall 2017
• First place, DARPA Chikungunya Challenge
• Lovelock Award
□ UA Department of Mathematics, Spring 2006
• Fellow of the Institute of Physics
□ Institute of Physics, London, Fall 2004
Modeling of nonlinear phenomena, with applications to physics and biology. Pattern formation and instabilities.Dynamics and stability of coherent structures.
Calculus, Linear Algebra, Differential Equations, Dynamical Systems, Real Analysis, Partial Differential Equations, Mathematical Modeling, Methods of Applied Mathematics, Integrated Science.
2024-25 Courses
• Dissertation
APPL 920 (Fall 2024)
2023-24 Courses
• Dissertation
MATH 920 (Spring 2024)
• Dissertation
MATH 920 (Fall 2023)
2022-23 Courses
• Dissertation
MATH 920 (Spring 2023)
• Mathematical Modeling
MATH 485 (Spring 2023)
• Dissertation
MATH 920 (Fall 2022)
• Research
MATH 900 (Fall 2022)
2021-22 Courses
• Dissertation
MATH 920 (Spring 2022)
• Mathematical Modeling
MATH 485 (Spring 2022)
• Research
MATH 900 (Spring 2022)
• Dissertation
MATH 920 (Fall 2021)
• Research
MATH 900 (Fall 2021)
2020-21 Courses
• Dissertation
MATH 920 (Spring 2021)
• Dissertation
MATH 920 (Fall 2020)
• Independent Study
MATH 599 (Fall 2020)
• Research
MATH 900 (Fall 2020)
2019-20 Courses
• Dissertation
MATH 920 (Spring 2020)
• Honors Thesis
MATH 498H (Spring 2020)
• Independent Study
MATH 599 (Spring 2020)
• Directed Research
MATH 392 (Fall 2019)
• Dissertation
MATH 920 (Fall 2019)
• Formal Math Reasong+Wrtg
MATH 323 (Fall 2019)
• Honors Thesis
MATH 498H (Fall 2019)
2018-19 Courses
• Directed Research
MATH 392 (Summer I 2019)
• Directed Research
MATH 392 (Spring 2019)
• Directed Research
MATH 492 (Spring 2019)
• Dissertation
MATH 920 (Spring 2019)
• Mathematical Modeling
MATH 485 (Spring 2019)
• Directed Research
MATH 492 (Fall 2018)
• Dissertation
MATH 920 (Fall 2018)
2017-18 Courses
• Directed Research
MATH 492 (Summer I 2018)
• Independent Study
MATH 599 (Spring 2018)
• Mathematical Modeling
MATH 485 (Spring 2018)
• Independent Study
MATH 599 (Fall 2017)
• Real Analy One Variable
MATH 425A (Fall 2017)
2016-17 Courses
• Dissertation
MATH 920 (Spring 2017)
• Independent Study
MATH 599 (Spring 2017)
• Dissertation
MATH 920 (Fall 2016)
• Real Analy One Variable
MATH 425A (Fall 2016)
• Real Analy One Variable
MATH 525A (Fall 2016)
2015-16 Courses
• Dissertation
MATH 920 (Spring 2016)
Scholarly Contributions
• Ercolani, N., Lega, J., & Tippings, B. (2023). Multiple Scale Asymptotics of Map Enumeration. Nonlinearity, 36, 1663-1698. doi:https://doi.org/10.1088/1361-6544/acb47d
More info
We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces.
Central to this work is the comparison of two asymptotic expansions obtained from two different fields of mathematics: the Riemann-Hilbert analysis of orthogonal polynomials and the theory of
discrete dynamical systems. By equating the coefficients of these expansions in a common region of uniform validity in their parameters, we recover known results and provide new expressions for
generating functions associated with graphical enumeration on surfaces of genera 0 through 7. Although the body of the article focuses on 4-valent maps, the methodology presented here extends to
regular maps of arbitrary even valence and to some cases of odd valence, as detailed in the appendices. [Journal_ref: Nonlinearity 36, 1663-1698 (2023)]
• Ercolani, N., Lega, J., & Tippings, B. (2023). Non-recursive Counts of Graphs on Surfaces. Enumerative Combinatorics and Applications, 3(3), 1-24. doi:https://doi.org/10.54550/ECA2023V3S3R20
More info
The problem of map enumeration concerns counting connected spatial graphs, with a specified number $j$ of vertices, that can be embedded in a compact surface of genus $g$ in such a way that its
complement yields a cellular decomposition of the surface. As such this problem lies at the cross-roads of combinatorial studies in low dimensional topology and graph theory. The determination of
explicit formulae for map counts, in terms of closed classical combinatorial functions of $g$ and $j$ as opposed to a recursive prescription, has been a long-standing problem with explicit
results known only for very low values of $g$. In this paper we derive closed-form expressions for counts of maps with an arbitrary number of even-valent vertices, embedded in surfaces of
arbitrary genus. In particular, we exhibit a number of higher genus examples for 4-valent maps that have not appeared prior in the literature.
• Cramer, E. Y., Huang, Y., Wang, Y., Ray, E. L., Cornell, M., Bracher, J., Brennen, A., Rivadeneira, A. J., Gerding, A., House, K., Jayawardena, D., Kanji, A. H., Khandelwal, A., Le, K., Mody, V.,
Mody, V., Niemi, J., Stark, A., Shah, A., , Wattanchit, N., et al. (2022). The United States COVID-19 Forecast Hub dataset. Scientific Data, 9, 462. doi:https://doi.org/10.1038/s41597-022-01517-w
More info
Academic researchers, government agencies, industry groups, and individuals have produced forecasts at an unprecedented scale during the COVID-19 pandemic. To leverage these forecasts, the United
States Centers for Disease Control and Prevention (CDC) partnered with an academic research lab at the University of Massachusetts Amherst to create the US COVID-19 Forecast Hub. Launched in
April 2020, the Forecast Hub is a dataset with point and probabilistic forecasts of incident cases, incident hospitalizations, incident deaths, and cumulative deaths due to COVID-19 at county,
state, and national, levels in the United States. Included forecasts represent a variety of modeling approaches, data sources, and assumptions regarding the spread of COVID-19. The goal of this
dataset is to establish a standardized and comparable set of short-term forecasts from modeling teams. These data can be used to develop ensemble models, communicate forecasts to the public,
create visualizations, compare models, and inform policies regarding COVID-19 mitigation. These open-source data are available via download from GitHub, through an online API, and through R
• Cramer, E. Y., Ray, E. L., Lopez, V. K., Bracher, J., Brennen, A., Castro Rivadeneira, A. J., Gerding, A., Gneiting, T., House, K. H., Huang, Y., Jayawardena, D., Kanji, A. H., Khandelwal, A.,
Le, K., Mühlemann, A., Niemi, J., Shah, A., Stark, A., Wang, Y., , Wattanachit, N., et al. (2022). Evaluation of individual and ensemble probabilistic forecasts of COVID-19 mortality in the
United States. Proceedings of the National Academy of Sciences of the United States of America, 119(15), e2113561119. doi:https://doi.org/10.1073/pnas.2113561119
More info
Short-term probabilistic forecasts of the trajectory of the COVID-19 pandemic in the United States have served as a visible and important communication channel between the scientific modeling
community and both the general public and decision-makers. Forecasting models provide specific, quantitative, and evaluable predictions that inform short-term decisions such as healthcare
staffing needs, school closures, and allocation of medical supplies. Starting in April 2020, the US COVID-19 Forecast Hub (https://covid19forecasthub.org/) collected, disseminated, and
synthesized tens of millions of specific predictions from more than 90 different academic, industry, and independent research groups. A multimodel ensemble forecast that combined predictions from
dozens of groups every week provided the most consistently accurate probabilistic forecasts of incident deaths due to COVID-19 at the state and national level from April 2020 through October
2021. The performance of 27 individual models that submitted complete forecasts of COVID-19 deaths consistently throughout this year showed high variability in forecast skill across time,
geospatial units, and forecast horizons. Two-thirds of the models evaluated showed better accuracy than a naïve baseline model. Forecast accuracy degraded as models made predictions further into
the future, with probabilistic error at a 20-wk horizon three to five times larger than when predicting at a 1-wk horizon. This project underscores the role that collaboration and active
coordination between governmental public-health agencies, academic modeling teams, and industry partners can play in developing modern modeling capabilities to support local, state, and federal
response to outbreaks.
• Ercolani, N., Lega, J., & Tippings, B. (2022). Dynamics of Non-polar Solutions to the Discrete Painlevé I Equation. SIAM Journal on Applied Dynamical Systems, 21, 1322-1351. doi:https://doi.org/
More info
This manuscript develops a novel understanding of non-polar solutions of the discrete Painleve I equation (dP1). As the non-autonomous counterpart of an analytically completely integrable
difference equation, this system is endowed with a rich dynamical structure. In addition, its non-polar solutions, which grow without bounds as the iteration index $n$ increases, are of
particular relevance to other areas of mathematics. We combine theory and asymptotics with high-precision numerical simulations to arrive at the following picture: when extended to include
backward iterates, known non-polar solutions of dP1 form a family of heteroclinic connections between two fixed points at infinity. One of these solutions, the Freud orbit of orthogonal
polynomial theory, is a singular limit of the other solutions in the family. Near their asymptotic limits, all solutions converge to the Freud orbit, which follows invariant curves of dP1, when
written as a 3-D autonomous system, and reaches the point at positive infinity along a center manifold. This description leads to two important results. First, the Freud orbit tracks sequences of
period-1 and 2 points of the autonomous counterpart of dP1 for large positive and negative values of $n$, respectively. Second, we identify an elegant method to obtain an asymptotic expansion of
the iterates on the Freud orbit for large positive values of $n$. The structure of invariant manifolds emerging from this picture contributes to a deeper understanding of the global analysis of
an interesting class of discrete dynamical systems.
• Sahneh, F. D., Fries, W., Watkins, J. C., & Lega, J. (2022). Epidemics from the Eye of the Pathogen. SIAM J. Appl. Math., 82, 2036-2056. doi:https://doi.org/10.1137/21M1450719
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While a common trend in disease modeling is to develop models of increasing complexity, it was recently pointed out that outbreaks appear remarkably simple when viewed in the incidence vs.
cumulative cases (ICC) plane. This article details the theory behind this phenomenon by analyzing the stochastic SIR (Susceptible, Infected, Recovered) model in the cumulative cases domain. We
prove that the Markov chain associated with this model reduces, in the ICC plane, to a pure birth chain for the cumulative number of cases, whose limit leads to an independent increments Gaussian
process that fluctuates about a deterministic ICC curve. We calculate the associated variance and quantify the additional variability due to estimating incidence over a finite period of time. We
also illustrate the universality brought forth by the ICC concept on real-world data for Influenza A and for the COVID-19 outbreak in Arizona. [Journal_ref: SIAM J. Appl. Math. 82, 2036-2056
• Kinney, A. C., Current, S., & Lega, J. (2021). Aedes-AI: Neural network models of mosquito abundance. PLoS Computational Biology, 17(11), e1009467.
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We present artificial neural networks as a feasible replacement for a mechanistic model of mosquito abundance. We develop a feed-forward neural network, a long short-term memory recurrent neural
network, and a gated recurrent unit network. We evaluate the networks in their ability to replicate the spatiotemporal features of mosquito populations predicted by the mechanistic model, and
discuss how augmenting the training data with time series that emphasize specific dynamical behaviors affects model performance. We conclude with an outlook on how such equation-free models may
facilitate vector control or the estimation of disease risk at arbitrary spatial scales.
• Lega, J. C. (2021). Parameter Estimation from ICC curves. Journal of Biological Dynamics, 15, 195-212. doi:10.1080/17513758.2021.1912419
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Incidence - Cumulative Cases (ICC) curves are introduced and shown to providea simple framework for parameter identification in the case of the mostelementary epidemiological model, consisting of
susceptible, infected, andremoved compartments. This novel methodology is used to estimate the basicreproductive number of recent outbreaks, including the ongoing COVID-19epidemic.
• Lega, J., Brown, H. E., & Barrera, R. (2020). A 70% Reduction in Mosquito Populations Does Not Require Removal of 70% of Mosquitoes. Journal of Medical Entomology, 57(5), 1668-1670. doi:https://
• McGowan, C., Biggerstaff, M., Johansson, M., Apfeldorf, K. M., Ben-Nun, M., Brooks, L., Convertino, M., Erraguntla, M., Farrow, D. C., Freeze, J., Ghosh, S., Hyun, S., Kandula, S., Lega, J. C.,
Liu, Y., Michaud, N., Morita, H., Niemi, J., Ramakrishnan, N., , Ray, E. L., et al. (2019). Collaborative efforts to forecast seasonal influenza in the United States, 2015–2016. Scientific
Reports, 9, 683. doi:https://doi.org/10.1038/s41598-018-36361-9
• Thompson, C., Saxberg, K., Lega, J. C., Tong, D., & Brown, H. E. (2019). A new gravity model for spatial interaction. Journal of Transport Geography, 79.
• Del Valle, S. Y., McMahon, B. H., Asher, J., Hatchett, R., Lega, J. C., Brown, H. E., Leany, M. E., Pantazis, Y., Roberts, D., Moore, S., Peterson, T., Escobar, L. E., Qiao, H., Hengartner, N.
W., & Mukundan, H. (2018). Summary Results of the 2014-2015 DARPA Chikungunya Challenge. BMC Infectious Diseases, 18, 245. doi:http://dx.doi.org/10.1186/s12879-018-3124-7
• Ercolani, N. M., Kamburov, N., & Lega, J. C. (2018). The Phase Structure of Grain Boundaries. Philosophical Transactions of the Royal Society A, 376, 20170193. doi:http://dx.doi.org/10.1098/
• Lega, J. C., Sethuraman, S., & Young, A. L. (2018). On collisions times of `self-sorting' interacting particles in one-dimension with random initial positions and velocities. Journal of
Statistical Physics, 170, 1088-1122. doi:http://dx.doi.org/10.1007/s10955-018-1974-4
• Brown, H. E., Barrera, R., Comrie, A. C., & Lega, J. C. (2017). Effect of temperature thresholds on modeled Aedes aegypti population dynamics. Journal of Medical Entomology, 54(4), 869–877.
• Lega, J. C., Brown, H. E., & Barrera, R. (2017). Aedes aegypti (Diptera: Culicidae) abundance model improved with relative humidity and precipitation-driven egg hatching. Journal of Medical
Entomology, 54(5), 1375–1384. doi:https://doi.org/10.1093/jme/tjx077
• Brubaker, N., & Lega, J. C. (2016). Capillary induced deformations of a thin elastic sheet. Philosophical Transactions of the Royal Society A, 374, 20150169. doi:10.1098/rsta.2015.0169
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We develop a three-dimensional model for capillary origami systems in which a rectangular plate has finite thickness, is allowed to stretch and undergoes small deflections. This latter constraint
limits our description of the encapsulation process to its initial folding phase. We first simplify the resulting system of equations to two dimensions by assuming that the plate has infinite
aspect ratio, which allows us to compare our approach to known two-dimensional capillary origami models for inextensible plates. Moreover, as this two-dimensional model is exactly solvable, we
give an expression for its solution in terms of its parameters. We then turn to the full three-dimensional model in the limit of small drop volume and provide numerical simulations showing how
the plate and the drop deform due to the effect of capillary forces.
• Brubaker, N., & Lega, J. C. (2016). Two-dimensional capillary origami. Physics Letters A, 380, 83-87.
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We describe a global approach to the problem of capillary origami that captures all equilibrium configurations in two-dimensional settings, with or without pinning of the liquid drop at the end
points of the flexible membrane. We provide bifurcation diagrams showing the level of encapsulation of each equilibrium configuration as a function of the volume of liquid that it contains, as
well as plots representing the energy of each equilibrium branch. Three different parameter regimes are identified, one of which predicts instantaneous encapsulation for small initial volumes of
liquid.This article will be featured in Elsevier's 2017 Virtual Special Issue on Women in Physics.
• Lega, J. C., & Brown, H. E. (2016). Data-driven outbreak forecasting with a simple nonlinear growth model. Epidemics, 17, 19-26. doi:http://dx.doi.org/10.1016/j.epidem.2016.10.002
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Recent events have thrown the spotlight on infectious disease outbreak response. We developed a data-driven method, EpiGro, which can be applied to cumulative case reports to estimate the order
of magnitude of the duration, peak and ultimate size of an ongoing outbreak. It is based on a surprisingly simple mathematical property of many epidemiological data sets, does not require
knowledge or estimation of disease transmission parameters, is robust to noise and to small data sets, and runs quickly due to its mathematical simplicity. Using data from historic and ongoing
epidemics, we present the model. We also provide modeling considerations that justify this approach and discuss its limitations. In the absence of other information or in conjunction with other
models, EpiGro may be useful to public health responders.
• Brown, H. E., Young, A., Lega, J. C., Andreadis, T. G., Schurich, J., & Comrie, A. C. (2015). Projection of Climate Change Influences on U.S. West Nile Virus Vectors. Earth Interactions, 19,
1-18. doi:http://dx.doi.org/10.1175/EI-D-15-0008.1
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While quantitative estimates of the impact of climate change on health is an increasing concern for health care planners and climate change policy, the models to produce those estimates remain
scarce. Herein, we describe a freely available dynamic simulation model parameterized for three West Nile virus vectors, which provides an effective tool for studying vector-borne disease risk
due to climate change. The Dynamic Mosquito Simulation Model is parameterized with species specific temperature-dependent development and mortality rates. Using downscaled daily weather data, we
estimate vector population dynamics under current and projected future climate scenarios for multiple locations across the country. Trends in mosquito abundance were variable by location,
however, an extension of the vector activity periods, and by extension disease risk, was almost uniformly observed. Importantly, areas showing mid-summer decreases in vector abundance maybe
off-set by shorter extrinsic incubation periods within the mosquito vector. Quantitative models of the effect of temperature on the virus and vector are critical to developing models of future
disease risk.
• Brubaker, N. D., & Lega, J. C. (2015). Two-dimensional capillary origami with pinned contact line. SIAM Journal on Applied Mathematics, 75, 1275-1300.
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To continue the move towards miniaturization in technology, developing new methods for fabricating micro- and nanoscale objects has become increasingly important. One potential method, called
capillary origami, consists of placing a small drop of liquid on a thin, inextensible sheet. In this article, we model the static configurations of this system in an idealized two-dimensional
setting and describe how the plate will fold due to capillary forces. We do this by minimizing the total energy of the system, which consists of bending and interfacial components whose relative
importance can be measured by a dimensionless parameter $\lambda$. The deflection of the plate is characterized in terms of bifurcation diagrams, where the bifurcation parameter is the drop's
size. This allows us to consider the quasi-static evolution of the system in the presence of evaporation. Variations in this bifurcation diagram for various $\lambda$ are then studied, leading us
to organize the physical description of the system's behavior into three different regimes. The present approach provides a general framework for the study of capillary origami that can be
extended to three-dimensional settings.
• Lindsay, A. E., Lega, J. C., & Glasner, K. B. (2015). Regularized Model of Post-Touchdown Configurations in Electrostatic MEMS: Interface Dynamics. The IMA Journal of Applied Mathematics, doi:
10.1093/imamat/hxv011, 29.
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Interface dynamics of post contact states in regularized models of electrostatic-elastic interactions are analyzed. A canonical setting for our investigations is the field of
Micro-Electromechanical Systems (MEMS) in which flexible elastic structures may come into physical contact due to applied Coulomb forces. We study the dynamic features of a recently derived
regularized model (A.E. Lindsay et al, Regularized Model of Post-Touchdown Configurations in Electrostatic MEMS: Equilibrium Analysis, Physica D, 2014), which describes the system past the
quenching singularity associated with touchdown, that is after the components of the device have come together. We build on our previous investigations of steady-state solutions by describing how
the system relaxes towards these equilibria. This is accomplished by deriving a reduced dynamical system that governs the evolution of the contact set, thereby providing a detailed description of
the intermediary dynamics associated with this bistable system. The analysis yields important practical information on the timescales of equilibration.
• Lega, J. C., Buxner, S., Blonder, B., & Tama, F. (2014). Explorations in integrated science. Journal of College Science Teaching, 43(4), 24-29.
• Lindsay, A. E., Lega, J., & Glasner, K. B. (2014). Regularized model of post-touchdown configurations in electrostatic MEMS: Equilibrium analysis. PHYSICA D: NONLINEAR PHENOMENA, 280, 95-108.
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In canonical models of Micro-Electra Mechanical Systems (MEMS), an event called touchdown whereby the electrical components of the device come into contact, is characterized by a blow up in the
governing equations and a non-physical divergence of the electric field. In the present work, we propose novel regularized governing equations whose solutions remain finite at touchdown and
exhibit additional dynamics beyond this initial event before eventually relaxing to new stable equilibria. We employ techniques from variational calculus, dynamical systems and singular
perturbation theory to obtain a detailed understanding of the properties and equilibrium solutions of the regularized family of equations. (C) 2014 Elsevier B.V. All rights reserved.
• Lega, J. (2013). Erratum: Collective behaviors in two-dimensional systems of interacting particles (SIAM Journal on Applied Dynamical Systems (2011) 10 (1213-1231)). SIAM Journal on Applied
Dynamical Systems, 12(4), 2093-.
• Lindsay, A. E., Lega, J., & Sayas, F. J. (2013). The quenching set of a MEMS capacitor in two-dimensional geometries. Journal of Nonlinear Science, 23(5), 807-834.
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Abstract: The formation of finite time singularities in a nonlinear parabolic fourth order partial differential equation (PDE) is investigated for a variety of two-dimensional geometries. The PDE
is a variant of a canonical model for Micro-Electro Mechanical systems (MEMS). The singularities are observed to form at specific points in the domain and correspond to solutions whose values
remain finite but whose derivatives diverge as the finite time singularity is approached. This phenomenon is known as quenching. An asymptotic analysis reveals that the quenching set can be
predicted by simple geometric considerations suggesting that the phenomenon described is generic to higher order parabolic equations which exhibit finite time singularity. © 2013 Springer
Science+Business Media New York.
• Moulton, D. E., & Lega, J. (2013). Effect of disjoining pressure in a thin film equation with non-uniform forcing. European Journal of Applied Mathematics, 24(6), 887-920.
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Abstract: We explore the effect of disjoining pressure on a thin film equation in the presence of a non-uniform body force, motivated by a model describing the reverse draining of a magnetic
film. To this end, we use a combination of numerical investigations and analytical considerations. The disjoining pressure has a regularizing influence on the evolution of the system and appears
to select a single steady-state solution for fixed height boundary conditions; this is in contrast with the existence of a continuum of locally attracting solutions that exist in the absence of
disjoining pressure for the same boundary conditions. We numerically implement matched asymptotic expansions to construct equilibrium solutions and also investigate how they behave as the
disjoining pressure is sent to zero. Finally, we consider the effect of the competition between forcing and disjoining pressure on the coarsening dynamics of the thin film for fixed contact angle
boundary conditions. Copyright © Cambridge University Press 2013.
• Lindsay, A. E., & Lega, J. (2012). Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a mems capacitor. SIAM Journal on Applied Mathematics, 72(3),
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Abstract: Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of
microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher
derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and
along a ring of points in the radially symmetric two-dimensional case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is
shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited
by the underlying PDE to accurately resolve solutions very close to the singularity. © 2012 Society for Industrial and Applied Mathematics.
• Herrera-Valdez, M. A., & Lega, J. (2011). Reduced models for the pacemaker dynamics of cardiac cells. Journal of Theoretical Biology, 270(1), 164-176.
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PMID: 20932980;Abstract: We introduce three- and two-dimensional biophysical models of cardiac excitability derived from a 14-dimensional model of the sinus venosus [Rasmusson, R., et al., 1990.
Am. J. Physiol. 259, H352-369]. The reduced models capture normal pacemaking dynamics with a small complement of ionic currents. The two-dimensional model bears some similarities with the
Morris-Lecar model [Morris, C., Lecar, H., 1981. Biophysical Journal, 35, 193-213]. Because they were reduced from a biophysical model, both models depend on parameters that were obtained from
experimental data. Even though the correspondence with the original model is not exact, parameters may be adjusted to tune the reductions to fit experimental traces. As a consequence, unlike
other generic low-dimensional models, the models introduced here provide a means to relate physiologically relevant characteristics of pacemaker potentials such as diastolic depolarization,
plateau, and action potential frequency, to biophysical variables such as the relative abundance of membrane channels and channel kinetic rates. In particular, these models can lead to an
explicit description of how the shape of cardiac action potentials depends on the relative contributions and states of inward and outward currents. By being physiologically derived and
computationally efficient, the models presented in this article are useful tools for theoretical studies of excitability at the cellular and network levels. © 2010.
• Lafortune, S., Lega, J., & Madrid, S. (2011). Instability of local deformations of an elastic rod: Numerical evaluation of the evans function. SIAM Journal on Applied Mathematics, 71(5),
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Abstract: We present a method for the numerical evaluation of the Evans function that doesnot require integration in an associated exterior algebra space. This technique is suitable for
thedetection of bifurcations and is particularly useful when the dimension of the linearized systemand/or the dimension of the converging subspaces at infinity is large. We test this approach
byinvestigating the stability of a two-parameter family of traveling pulse solutions to two coupledKlein-Gordon equations. The spectral stability of these pulses is completely understood
analytically [S. Lafortune and J. Lega, SIAM J. Math. Anal. , 36 (2005), pp. 1726-1741], and we show that ournumerical method is able to detect bifurcations of the pulse family with very good
accuracy. © 2011 Society for Industrial and Applied Mathematics.
• Lega, J. (2011). Collective behaviors in two-dimensional systems of interacting particles. SIAM Journal on Applied Dynamical Systems, 10(4), 1213-1231.
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Abstract: This article presents results of molecular dynamics simulations that show the emergence of collective behaviors in a two-dimensional system of particles (hard disks) interacting through
a properly chosen collision rule. The particles, which are of finite size and are in free flight between collisions, are not self-propelled. They tumble randomly like bacteria and interact only
when they collide, not through continuous potential forces. This work therefore indicates that interactions at the microscopic level, which occur only locally and discretely both in time and
space, are sufficient to lead to large-scale macroscopic behaviors. Order parameters that capture and quantify the formation of collective behaviors are introduced and used to describe how the
choice of collision rule affects the steady state dynamics of the system, by comparing the outcome to the standard case of elastic collisions. This work was motivated by recent results on the
dynamics of bacterial colonies. Possible applications of the present approach to other systems are also discussed. © 2011 Society for Industrial and Applied Mathematics.
• Moulton, D. E., & Lega, J. (2009). Reverse draining of a magnetic soap film - Analysis and simulation of a thin film equation with non-uniform forcing. Physica D: Nonlinear Phenomena, 238(22),
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Abstract: We analyze and classify equilibrium solutions of the one-dimensional thin film equation with no-flux boundary conditions and in the presence of a spatially dependent external forcing.
We prove theorems that shed light on the nature of these equilibrium solutions, guarantee their validity, and describe how they depend on the properties of the external forcing. We then apply
these results to the reverse draining of a one-dimensional magnetic soap film subject to an external non-uniform magnetic field. Numerical simulations illustrate the convergence of the solutions
towards equilibrium configurations. We then present bifurcation diagrams for steady state solutions. We find that multiple stable equilibrium solutions exist for fixed parameters, and uncover a
rich bifurcation structure to these solutions, demonstrating the complexity hidden in a relatively simple looking evolution equation. Finally, we provide a simulation describing how numerical
solutions traverse the bifurcation diagram, as the amplitude of the forcing is slowly increased and then decreased. © 2009 Elsevier B.V. All rights reserved.
• Lega, J., & Passot, T. (2007). Hydrodynamics of bacterial colonies. Nonlinearity, 20(1), C1-C16.
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Abstract: Understanding the growth and dynamics of bacterial colonies is a fascinating problem, which requires combining ideas from biology, physics and applied mathematics. We briefly review the
recent experimental and theoretical literature relevant to this question and describe a hydrodynamic model (Lega and Passot 2003 Phys. Rev. E 67 031906, 2004 Chaos 14 562-70), which captures
macroscopic motions within bacterial colonies, as well as the macroscopic dynamics of colony boundaries. The model generalizes classical reaction-diffusion systems and is able to qualitatively
reproduce a variety of colony shapes observed in experiments. We conclude by listing open questions about the stability of interfaces as modelled by reaction-diffusion equations with nonlinear
diffusion and the coupling between reaction-diffusion equations and a hydrodynamic field. © 2007 IOP Publishing Ltd and London Mathematical Society.
• Lafortune, S., & Lega, J. (2005). Spectral stability of local deformations of an elastic rod: Hamiltonian formalism. SIAM Journal on Mathematical Analysis, 36(6), 1726-1741.
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Abstract: Hamiltonian methods are used to obtain a necessary and sufficient condition for the spectral stability of pulse solutions to two coupled nonlinear Klein-Gordon equations. These
equations describe the near-threshold dynamics of an elastic rod with circular cross section. The present work completes and extends a recent analysis of the authors' [Phys. D, 182 (2003), pp.
103-124], in which a sufficient condition for the instability of "nonrotating" pulses was found by means of Evans function techniques. © 2005 Society for Industrial and Applied Mathematics.
• Lega, J., & Passot, T. (2004). Hydrodynamics of bacterial colonies: Phase diagrams. Chaos, 14(3), 562-570.
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PMID: 15446966;Abstract: We present numerical simulations of a recent hydrodynamic model describing the growth of bacterial colonies on agar plates. We show that this model is able to
qualitatively reproduce experimentally observed phase diagrams, which relate a colony shape to the initial quantity of nutrients on the plate and the initial wetness of the agar. We also discuss
the principal features resulting from the interplay between hydrodynamic motions and colony growth, as described by our model. © 2004 American Institute of Physics.
• Lega, J., & Passot, T. (2004). Inverse cascade and energy transfer in forced low-Reynolds number two-dimensional turbulence. Fluid Dynamics Research, 34(5), 289-297.
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Abstract: Using numerical simulations of the forced two-dimensional Navier-Stokes equation, it is shown that the amount of energy transferred to large scales is related to the Reynolds number in
a unique fashion. It is also observed that the critical value of the initial Reynolds number for the onset of an inverse cascade is lowered as the scale of the forcing approaches the size of the
system, or in the presence of anisotropy. This study is motivated by recent experiments with bacterial colonies, and their description in terms of a hydrodynamic model. © 2004 Published by The
Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved.
• Lafortune, S., & Lega, J. (2003). Instability of local deformations of an elastic rod. Physica D: Nonlinear Phenomena, 182(1-2), 103-124.
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Abstract: We study the instability of pulse solutions of two coupled non-linear Klein-Gordon equations by means of Evans function techniques. The system of coupled Klein-Gordon equations
considered here describes the near-threshold dynamics of a three-dimensional elastic rod with circular cross-section, subject to constant twist. We determine a condition on the speed of the
traveling pulse which ensures spectral instability. © 2003 Elsevier Science B.V. All rights reserved.
• Lega, J., & Passot, T. (2003). Hydrodynamics of bacterial colonies: A model. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 67(3 1), 031906/1-031906/8.
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PMID: 12689100;Abstract: A hydrodynamic model that gives a general description of bacterial colonies growing in soft agar plates is proposed. This scheme provides a framework in which macroscopic
reaction-diffusion models of bacterial colonies are justified on the basis of hydrodynamic considerations. With the given model, colonies that are drier in the interior than at the boundary can
be described.
• Lega, J. (2001). Traveling hole solutions of the complex Ginzburg-Landau equation: A review. Physica D: Nonlinear Phenomena, 152-153, 269-287.
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Abstract: This paper reviews recent works on localized solutions of the one-dimensional complex Ginzburg-Landau (CGL) equation known as traveling holes. Such coherent structures seem to play an
important role in the disordered dynamics displayed by CGL at a finite distance past the Benjamin-Feir instability threshold. We discuss these objects in the broader context of weak turbulence
and summarize some of their properties. © 2001 Elsevier Science B.V.
• Lega, J., & Goriely, A. (1999). Pulses, fronts and oscillations of an elastic rod. Physica D: Nonlinear Phenomena, 132(3), 373-391.
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Abstract: Two coupled nonlinear Klein-Gordon equations modeling the three-dimensional dynamics of a twisted elastic rod near its first bifurcation threshold are analyzed. First, it is shown that
these equations are Hamiltonian and that they admit a two-parameter family of traveling wave solutions. Second, special solutions corresponding to simple deformations of the elastic rod are
considered. The stability of such configurations is analyzed by means of two coupled nonlinear Schrödinger equations, which are derived from the nonlinear Klein-Gordon equations in the limit of
small deformations. In particular, it is shown that periodic solutions are modulationally unstable, which is consistent with the looping process observed in the writhing instability of elastic
filaments. Third, numerical simulations of the nonlinear Klein-Gordon equations suggesting that traveling pulses are stable, are presented. © 1999 Elsevier Science B.V. All rights reserved.
• Lega, J., & Mendelson, N. H. (1999). Control-parameter-dependent Swift-Hohenberg equation as a model for bioconvection patterns. Physical Review E - Statistical Physics, Plasmas, Fluids, and
Related Interdisciplinary Topics, 59(6), 6267-6274.
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PMID: 11969610;Abstract: We consider a complex Swift-Hohenberg equation with control-parameter-dependent coefficients and use it as a mode to describe dynamical features seen in an experimental
bacterial bioconvecti on pattern. In particular, we give numerical results showing the development of a phase-unstable pattern behind a moving front. ©1999 The American Physical Society.
• Bottin, S., & Lega, J. (1998). Pulses of tunable size near a subcritical bifurcation. European Physical Journal B, 5(2), 299-308.
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Abstract: We show that a nonlinear gradient term can be used to tune the width of pulse-like solutions to a generalized quintic Ginzburg-Landau equation. We investigate the dynamics of these
solutions and show that weakly turbulent patches can persist for long times. Analogies with turbulent spots in plane Couette flows are discussed.
• Mendelson, N. H., & Lega, J. (1998). A complex pattern of traveling stripes is produced by swimming cells of Bacillus subtilis. Journal of Bacteriology, 180(13), 3285-3294.
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PMID: 9642178;PMCID: PMC107280;Abstract: Motile cells of Bacillus subtilis inadvertently escaped from the surface of an agar disk that was surrounded by a fluid growth medium and formed a
migrating population in the fluid. When viewed from above, the population appeared as a cloud advancing unidirectionally into the fresh medium. The cell population became spontaneously organized
into a series of stripes in a region behind the advancing cloud front. The number of stripes increased progressively until a saturation value of stripe density per unit area was reached. New
stripes arose at a fixed distance behind the cloud front and also between stripes. The spacing between stripes underwent changes with time as stripes migrated towards and away from the cloud
front. The global pattern appeared to be stretched by the advancing cloud front. At a time corresponding to approximately two cell doublings after pattern formation, the pattern decayed,
suggesting that there is a maximum number of cells that can be maintained within the pattern. Stripes appear to consist of high concentrations of cells organized in sinking columns that are part
of a bioconvection system. Their behavior reveals an interplay between bacterial swimming, bioconvection-driven fluid motion, and cell concentration. A mathematical model that reproduces the
development and dynamics of the stripe pattern has been developed.
• Calderón, O. G., Pérez-Garcí, V. M., Lega, J., & Guerra, J. M. (1997). Loss-induced transverse effects in lasers. Optics Communications, 143(4-6), 315-321.
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Abstract: We analyze the effect of losses on transverse patterns of a single longitudinal mode laser. In particular, we find a diffusion term which predicts the existence of a diffusive cutoff in
the transverse spectrum. The linear stability analysis of the non-lasing solution shows that the detuning value (Δ = 0), that separates the two types of solutions above threshold, shifts to a
positive value, which is important for lasers with a high polarization dephasing rate (γ⊥) such as Class B lasers. © 1997 Elsevier Science B.V.
• Hochheiser, D., Moloney, J. V., & Lega, J. (1997). Controlling optical turbulence. Physical Review A - Atomic, Molecular, and Optical Physics, 55(6), R4011-R4014.
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Abstract: A robust global control strategy, implemented as a spatial filter with delayed feedback, is shown to stabilize and steer the weakly turbulent output of a spatially extended system. The
latter is described by a generalized complex Swift-Hohenberg equation [J. Lega, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 73, 2978 (1994); Physica D 83, 478 (1995)], which is used as a
generic model for pattern formation in the transverse section of semiconductor lasers. Our technique is particularly adapted to optical systems and should provide convenient experimental control
of filamentation in wide-aperture lasers. [S1050-2947(97)50206-X].
• Lega, J., & Fauve, S. (1997). Traveling hole solutions to the complex Ginzburg-Landau equation as perturbations of nonlinear Schrödinger dark solitons. Physica D: Nonlinear Phenomena, 102(3-4),
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Abstract: We describe complex Ginzburg-Landau (CGL) traveling hole solutions as singular perturbations of nonlinear Schrödinger (NLS) dark solitons. Modulation of the free parameters of the NLS
solutions leads to a dynamical system describing the CGL dynamics in the vicinity of a traveling hole solution. © 1997 Elsevier Science B.V. All rights reserved.
• Harkness, G. K., Lega, J., & Oppo, G. (1996). Measuring disorder with correlation functions of averaged patterns. Physica D: Nonlinear Phenomena, 96(1-4), 26-29.
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Abstract: We propose an indicator of disorder which can be measured on averaged intensity images of a weakly turbulent complex field. Copyright © 1996 Elsevier Science B.V. All rights reserved.
• Harkness, G. K., Lega, J., & Oppo, G. -. (1996). Travelling wave patterns in lasers with curved mirrors. Technical Digest - European Quantum Electronics Conference, 42-.
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Abstract: The study of pattern formation in the transverse section of lasers with plane cavity mirrors shows that the fundamental solutions are transverse traveling waves. On the other hand,
lasers with curved cavity mirrors generate outputs which look like a combination of empty cavity modes. The mechanism behind the selection of travelling waves and cavity modes was found to be
analogous. Numerical simulations show that for large input energy (pump), patterns of travelling wave nature are recovered even the mirrors are curved. As the radius of curved mirrors decreases
the threshold value of the pump increase.
• Lega, J., & Vince, J. (1996). Temporal forcing of traveling wave patterns. Journal de Physique I, 6(11), 1417-1434.
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Abstract: We experimentally and numerically study one-dimensional, temporally forced wave patterns and analyze the transition from unforced traveling waves to forced standing waves when a source
defect is present in the system. Our control parameter is the amplitude of the forcing. Two scenarios are identified, depending on the distance from the bifurcation threshold of traveling waves.
© Les Éditions de Physique 1996.
• Harkness, G. K., Lega, J. C., & Oppo, G. (1994). Correlation functions in the presence of optical vortices. Chaos, Solitons and Fractals, 4(8-9), 1519-1533.
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Abstract: We evaluate two-dimensional field and intensity correlation functions for the optical Ginzburg-Landau and laser equations in the presence of optical vortices. Configurations of rotating
vortices, vortices superimposed on travelling waves and defect mediated turbulance are analysed and compared. Intensity correlations provide a qualitative indicator for the degree of disorder of
spatio-temporal evolutions. For example, patterns with few rotating vortices have correlation lengths comparable to or larger than the transverse size of the system, while few vortices
superimposed on travelling waves can generate weak turbulance. © 1994.
• Daviaud, F., Lega, J., Bergé, P., Coullet, P., & Dubois, M. (1992). Spatio-temporal intermittency in a 1D convective pattern: Theoretical model and experiments. Physica D: Nonlinear Phenomena, 55
(3-4), 287-308.
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Abstract: We describe the occurence of spatio-temporal intermittency in a one-dimensional convective system that first shows time-dependent patterns. We recall experimental results and propose a
model based on the normal form description of a secondary Hopf bifurcation of a stationary periodic structure. Numerical simulations of this model show spatio-temporal intermittent behaviors,
which we characterize briefly and compare to those given by the experiment. © 1992.
• Lega, J., Janiaud, B., Jucquois, S., & Croquette, V. (1992). Localized phase jumps in wave trains. Physical Review A, 45(8), 5596-5604.
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Abstract: We show experimental evidence of traveling hole defects, which are reminiscent of analytical solutions to complex Ginzburg-Landau equations. Also, these objects seem to play an
important role in the development of phase instability of oscillatory patterns. © 1992 The American Physical Society.
• Coullet, P., Lega, J., & Pomeau, Y. (1991). Dynamics of Bloch Walls in a Rotating Magnetic Field: A Model. EPL, 15(2), 221-226. doi:10.1209/0295-5075/15/2/019
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We both analytically and numerically show the existence of a drift of Bloch walls when submitted to a uniform parallel-to-the-wall-plane rotating magnetic field. The drift velocity changes sign
with Bloch wall handedness and is proportional to the amplitude square of the magnetic field, when the latter is small.
• Lega, J. (1991). Defect-mediated turbulence. Computer Methods in Applied Mechanics and Engineering, 89(1-3), 419-424.
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Abstract: We summarize recent results about a dynamic regime observed in numerical simulations of two-dimensional Ginzburg-Landau equations, for which defects are spontaneously created in the
system and are responsible for its disorganization. © 1991.
• Ciliberto, S., Coullet, P., Lega, J., Pampaloni, E., & Perez-Garcia, C. (1990). Defects in Roll-Hexagon competition. Physical Review Letters, 65(19), 2370-2373.
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Abstract: The defects of a system where hexagons and rolls are both stable solutions are considered. On the basis of topological arguments we show that the unstable phase is present in the core
of the defects. This means that a roll is present in the penta-hepta defect of hexagons and that a hexagon is found in the core, of a grain boundary connecting rolls with different orientations.
These results are verified in an experiment of thermal convection under non-Boussinesq conditions. © 1990 The American Physical Society.
• Coullet, P., Lega, J., Houchmanzadeh, B., & Lajzerowicz, J. (1990). Breaking chirality in nonequilibrium systems. Physical Review Letters, 65(11), 1352-1355.
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Abstract: At equilibrium, Bloch walls are chiral interfaces between domains with different magnetisation. Far from equilibrium, a set of forced oscillators can exhibit walls between states with
different phases. In this Letter, we show that when these walls become chiral. they move with a velocity simply related to their chirality. This surprising behavior is a straightforward
consequence of nonvariational effects. which are typical of nonequilibrium systems.
• Gil, L., Lega, J., & Meunier, J. L. (1990). Statistical properties of defect-mediated turbulence. Physical Review A, 41(2), 1138-1141.
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Abstract: We study some statistical properties of a turbulent state described by a generalized Ginzburg-Landau equation and characterized by topological defects. © 1990 The American Physical
• Coullet, P., Gil, L., & Lega, J. (1989). A form of turbulence associated with defects. Physica D: Nonlinear Phenomena, 37(1-3), 91-103.
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Abstract: We show by means of numerical simulations of complex Ginzburg-Landau equations that phase instability leads to the spontaneous nucleation of topological defects, which disorganize the
system. © 1989.
• Coullet, P., Gil, L., & Lega, J. (1989). Defect-mediated turbulence. Physical Review Letters, 62(14), 1619-1622.
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Abstract: We describe a turbulent state characterized by the presence of topological defects. This topological turbulence is likely to be experimentally observed in nonequilibrium systems. © 1989
The American Physical Society.
• Coullet, P., Gil, L., & Lega, J. (1989). Une forme de turbulence associée aux défauts topologiques. Mathematical Modelling and Numerical Analysis, 23(3), 385-394. doi:10.1051/m2an/1989230303851
• Coullet, P., & Lega, J. (1988). Defect-mediated turbulence in wave patterns. EPL, 7(6), 511-516. doi:10.1209/0295-5075/7/6/006
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A turbulent behaviour of wave patterns is described, which is related to the presence of dislocations. By means of numerical simulations of 2D complex Ginzburg-Landau equations, it is shown that
phase instability leads in spatially extended systems to spontaneous nucleation of topological defects. The appearance of these localized amplitude perturbations is interpreted as the consequence
of the revolt of the slaved amplitude modes. Once created, those defects move through the system and break the order induced by the wave pattern. The resulting turbulent state has been termed
"topological turbulence".
• Coullet, P., Elphick, C., Gil, L., & Lega, J. (1987). Topological defects of wave patterns. Physical Review Letters, 59(8), 884-887.
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Abstract: We identify the defects of waves by means of topological arguments and study them in the framework of Landau-type analysis. It is shown that they correspond to sinks, sources, or
dislocations of traveling waves, and to dislocations of standing waves. © 1987 The American Physical Society.
• Lega, J. C. (2023, August). A dynamical systems approach to map enumeration. 10th International Congress on Industrial and Applied Mathematics, Session on Painlevé equations, Applications, and
Related Topics. Tokyo, Japan - Delivered online.
• Lega, J. C. (2022, April). Dynamical systems properties of the Freud recurrence. Session on Discrete Painleve Equations and Related Topics, 12th IMACS International Conference on Nonlinear
Evolution Equations and Wave Phenomena: Computation and Theory. Athens Georgia.
• Lega, J. C. (2022, July). Models of Mosquito Abundance. ASU Summer REU Colloquium Series. Arizona State University.
• Lega, J. C. (2022, October). On the Number of Quadrangulations of a Topological Surface. Mathematics Colloquium, University of Houston.
• Lega, J. C. (2021, February 17). Modeling and Forecasting the Spread of the Pandemic. Webinar Series on COVID-19: Breaking and Raising Boundaries. Online: International research laboratory CNRS/
ENS-PSL, COVIDAM Blog, and Institut des Amériques.
• Lega, J. C. (2021, February 24). Modeling in the time of the pandemic. Mathematics Colloquium, Southern Methodist University. Online: Department of Mathematics, Southern Methodist University,
Dallas, TX.
• Lega, J. C. (2021, June). A New Take on Outbreak Dynamics. Minisymposium on Evolutionary Theory of Disease, Virtual SMB 2021 Annual Meeting. Online: Society for Mathematical Biology.
• Lega, J. C. (2021, May). Forecasting Disease Risk and Spread. French American Innovation Days: Facing the Predictably Unpredictable. Online: Ambassade de France aux Etats Unis.
• Lega, J. C. (2021, October 26). A dynamical systems view of special solutions to the discrete Painlevé I equation. Nonlinear Waves and Coherent Structures Webinar. Online: UMass Amherst, Bowdoin
College, Cal Poly, San Luis Obispo.
• Lega, J. C. (2020, October). Epidemiological Forecasting with ICC curves and data assimilation. ASU Mathbio SeminarArizona State University.
• Lega, J. C. (2020, October). Epidemiological Forecasting with Simple Nonlinear Models. Dynamical Systems Seminar, University of MinnesotaUniversity of Minnesota.
• Lega, J. C. (2020, October). Phase singularities and defects in the Swift-Hohenberg equation. Fall Western AMS Sectional meeting, Special Session on “Free boundary problems arising in
applications”. University of Utah, Salt Lake City, Utah: AMS.
• Lega, J. C. (2019, April). Grain boundaries of the Swift-Hohenberg equation: simulations and analysis. 11th IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena:
Computation and Theory. Athens, GA.
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Special Session on Nonlinear Evolutionary Equations: Theory, Numerics and Experiments.
• Lega, J. C. (2019, August). Panel: Moving to More Useful Forecasts at the State/Local Level: The Forecasters Perspective. FluSight Seasonal Influenza Forecasting Workshop. Atlanta, GA: Council of
State and Territorial Epidemiologists.
• Lega, J. C. (2019, July). Phase singularities and defects in pattern forming systems. 9th International Congress on Applied Mathematics. Valencia, Spain.
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Mini-symposium on Existence and stability of nonlinear waves
• Lega, J. C. (2019, March). Panel: Mathematical Perspectives and Vision for Preparation and Pathways in Mathematical Modeling. Critical Issues in Mathematics Education 2019: Mathematical Modeling
in K-16: Community and Cultural Context. Berkeley, CA: MSRI Critical Issues in Mathematics Education.
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Presentation (with slides) as part of a panel on Pathways in Mathematical Modeling.
• Lega, J. C. (2019, March). Transdisciplinary modeling of mosquito-borne diseases. Southwestern Undergraduate Mathematics Research Conference (SUnMaRC). Tucson, Arizona: University of Arizona.
• Lega, J. C. (2018, August). Forecasting the Flu with Simple Nonlinear Models. CSTE/CDC Seasonal Influenza Forecasting Workshop. Atlanta, GA: CDC & Council of State and Territorial
• Lega, J. C. (2018, March). Trans-disciplinary modeling of mosquito-borne diseases. Modeling and Computation Seminar, University of Arizona.
• Lega, J. C. (2018, November). Modeling the spread of vector-borne diseases on regional transportation networks. 2018 ESA, ESC, and ESBC Joint Annual Meeting. Vancouver, BC, Canada, November 11 –
14, 2018.
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Invited presentation at the MUVE Section Symposium on Predicting Vector-Borne Diseases Spread in Changing Natural and Social Landscapes.
• Lega, J. C. (2017, April). Forecasting the Flu. Uncertainty Quantification Seminar, University of Arizona. UA Department of Mathematics.
• Lega, J. C. (2017, April). The phase structure of grain boundaries. IIMAS Applied Mathematics Colloquium, UNAM. Mexico City: UNAM (Universidad Nacional Autónoma de México) / IIMAS (Instituto de
Investigaciones en Matemáticas Aplicadas y en Sistemas).
• Lega, J. C. (2017, April). Three models to help understand the spread of mosquito-borne diseases. 10th IIMAS Colloquium, UNAM. Mexico City: UNAM (Universidad Nacional Autónoma de México) / IIMAS
(Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas).
• Lega, J. C. (2017, August). Forecasting the Flu with Simple Nonlinear Models. Seasonal Influenza Forecasting Workshop, CDC. Atlanta, GA: Centers for Disease Control and Prevention.
• Lega, J. C. (2017, February). Capillary Origami. Analysis, Dynamics, and Applications Seminar, University of Arizona.
• Lega, J. C. (2017, March). The phase structure of grain boundaries. 1126 AMS Meeting. Charleston, SC.
• Lega, J. C. (2017, May). Patterns, defects, and phase singularities. Rocky Mountain Partial Differential Equations Conference. Provo, UT: National Science Foundation, Brigham-Young University.
• Lega, J. C. (2017, May). The Phase Structure of Grain Boundaries. 2017 SIAM Conference on Applications of Dynamical Systems. Snowbird, UT: SIAM.
• Lega, J. C. (2017, November). A Three-pronged Approach to Predicting the Spread of Mosquito-borne Diseases. Mathematics Colloquium, Colorado State University. Fort Collins, CO: Colorado State
• Lega, J. C. (2017, October). Patterns, defects, and phase singularities. Analysis, Dynamics, and Applications Seminar, University of Arizona.
• Lega, J. C. (2016, 04-12-2016). Models for mosquito abundance and infectious disease outbreak forecasting. Quantitative Biology Colloquium, University of Arizona.
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Title: Models for mosquito abundance and infectious disease outbreak forecastingAbstract: In this talk, I will present an agent-based model for mosquito abundance that uses temperature and
precipitation as time-varying parameters. I will show results that describe the effect of climate change on mosquito abundance and discuss the importance of how rainfall is taken into account in
the model. I will then move to a very simple macroscopic description for the spread of infectious diseases and illustrate applications of this model to various outbreaks, including the recent
chikungunya epidemic in the Americas. This work is in collaboration with Heidi Brown (College of Public Health) and many other researchers on and off campus.
• Lega, J. C. (2016, August 8-11). Defects in the Swift-Hohenberg Equation. 2016 SIAM Conference on Nonlinear Waves and Coherent Structures. Philadelphia, PA.
More info
Talk in Session on Existence and Stability of Nonlinear Waves and Patterns.Abstract: I will discuss static and dynamic properties of grain boundaries in pattern-forming systems, using the
Swift-Hohenberg equation as a canonical model. In particular, I will focus on the transition between grain boundaries, pairs of concave-convex disclinations, and dislocations. I will present a
mix of numerical simulations and analytical results. Some of this work is joint with N. Ercolani and N. Kamburov (University of Arizona).
• Lega, J. C. (2016, August-September). Flu forecasting with EpiGro. CDC Seasonal Influenza Forecasting Workshop. Centers for Disease Control and Prevention, Atlanta, GA: Centers for Disease
Control and Prevention.
• Ercolani, N. M., Kamburov, N. A., & Lega, J. C. (2015, December). How Defects are Born. SIAM conference on Analysis of Partial Differential Equations. Scottsdale, AZ.
More info
Pattern-forming systems typically exhibit defects, whose nature is associated with the symmetries of the pattern in which they appear; examples include dislocations of stripe patterns in systems
invariant under translations, disclinations in stripe-forming systems invariant under rotations, and spiral defects of oscillatory patterns in systems invariant under time translations. Numerical
simulations suggest that pairs of defects are created when the phase of the pattern ceases to be slaved to its amplitude. Such an event is typically mediated by the build up of large, localized,
phase gradients.This talk will describe recent advances on a long-term project whose goal is to follow such a defect-forming mechanism in a system that is amenable to analysis. Specifically, we
focus on the appearance of pairs of dislocations at the core of a grain boundary of the Swift-Hohenberg equation. Taking advantage of the variational nature of this system, we show that as the
angle between the two stripe patterns on each side of the grain boundary is reduced, the phase of each pattern, as described by the Cross-Newell equation, develops large derivatives in a region
of diminishing size.This is joint work with N. Ercolani and N. Kamburov.
• Lega, J. C. (2015, April). How Defects are Born. The 1st Annual Meeting of SIAM Central States Section. Rolla, M.
More info
Pattern-forming systems typically exhibit defects, whose nature is associated with the symmetries of the pattern in which they appear; examples include dislocations of stripe patterns in systems
invariant under translations, disclinations in stripe-forming systems invariant under rotations, and spiral defects of oscillatory patterns in systems invariant under time translations. Numerical
simulations suggest that pairs of defects are created when the phase of the pattern ceases to be slaved to its amplitude. Such an event is typically mediated by the build up of large, localized,
phase gradients.This talk will describe recent advances on a long-term project whose goal is to follow such a defect-forming mechanism in a system that is amenable to analysis. Specifically, we
focus on the appearance of pairs of dislocations at the core of a grain boundary of the Swift-Hohenberg equation. Taking advantage of the variational nature of this system, we show that as the
angle between the two stripe patterns on each side of the grain boundary is reduced, the phase of each pattern, as described by the Cross-Newell equation, develops large derivatives in a region
of diminishing size.This is joint work with N. Ercolani and N. Kamburov.
• Lega, J. C. (2015, April). Mathematical Modeling & Applications to Public Health. EPI Seminar, UA College of Public Health.
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EPI seminar, College of Public Health, University of Arizona, 29 April, 2015
• Lega, J. C. (2015, May). Physica D: Nonlinear Phenomena. SIAM Conference on Applications of Dynamical Systems - Journal Editors Panel and Reception. Snowbird, Utah: SIAM.
• Lega, J. C. (2015, November). Capillary Origami. Mathematics Colloquium, University of Central Florida. University of Central Florida: UCF.
• Lega, J. C. (2015, November). Explorations in Undergraduate Education. Seminar, University of Central Florida. University of Central Florida: University of Central Florida.
• Lega, J. C., & Brown, H. E. (2015, May). Modeling the Spread of Chikungunya in the Caribbean and Central America. DARPA Chikungunya Challenge Finale. DARPA: DARPA.
• Lega, J. C., & Brown, H. E. (2015, October). Modeling the Spread of Chikungunya in the Caribbean and Central America. UA Microlunch SeriesUniversity of Arizona.
• Lega, J. C., Ercolani, N. M., & Kamburov, N. A. (2015, April). How Defects are Born. The Ninth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and
More info
Pattern-forming systems typically exhibit defects, whose nature is associated with the symmetries of the pattern in which they appear; examples include dislocations of stripe patterns in systems
invariant under translations, disclinations in stripe-forming systems invariant under rotations, and spiral defects of oscillatory patterns in systems invariant under time translations. Numerical
simulations suggest that pairs of defects are created when the phase of the pattern ceases to be slaved to its amplitude. Such an event is typically mediated by the build up of large, localized,
phase gradients.This talk will describe recent advances on a long-term project whose goal is to follow such a defect-forming mechanism in a system that is amenable to analysis. Specifically, we
focus on the appearance of pairs of dislocations at the core of a grain boundary of the Swift-Hohenberg equation. Taking advantage of the variational nature of this system, we show that as the
angle between the two stripe patterns on each side of the grain boundary is reduced, the phase of each pattern, as described by the Cross-Newell equation, develops large derivatives in a region
of diminishing size.
• Lega, J. C., Ercolani, N. M., & Kamburov, N. A. (2014, July). Grain boundaries of the Swift-Hohenberg and regularized Cross-Newell equations. Special session on Traveling Waves and Patterns, 10th
AIMS Conference on Dynamical Systems, Differential Equations and Applications. Madrid, Spain.
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Grain boundaries in extended two-dimensional pattern forming systems are curves separating regions of slanted rolls. When the angle between the rolls in each of the two regions exceeds a certain
threshold, it is known [1,2] that the core of the grain boundary transforms into a chain of convex-concave disclinations. Even though the regularized Cross-Newell (RCN) phase diffusion equation
cannot describe this transition all the way to the appearance of defects, it can nevertheless be used to address the question of whether the transition results from an instability of the grain
boundary core, and if so, to describe this instability. To this end, we will take full advantage of the existence of an exact grain-boundary solution of RCN and of the variational nature of this
equation. I will also show numerical simulations and connect our results to those of Haragus and Scheel [3] on grain boundaries of the Swift-Hohenberg equation.References[1] N.M. Ercolani, R.
Indik, A.C. Newell, and T. Passot, J. Nonlinear Sci. 10, 223-274 (2000).[2] N.M. Ercolani and S.C. Venkataramani, J. Nonlinear Sci. 19, 267-300 (2009).[3] M. Haragus and A. Scheel, European
Journal of Applied Mathematics 23, 737-759 (2012).
• Lega, J. C. (2011, November 4-6). Collective Behaviors in Two-dimensional Systems of Interacting Particles and Rods. Geometric Methods for Innite-Dimensional Dynamical Systems. Brown University,
Providence, RI: Brown University, NSF.
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A meeting in celebration of Chris Jones' 60th birthday, Brown University
Case Studies
• Roach, M., Brown, H. E., Clark, R., Hondula, D., Lega, J. C., Rabby, Q., Schweers, N., & Tabor, J. (2017. Projections of Climate Impacts on Vector-Borne Diseases and Valley Fever in Arizona(p. | {"url":"https://profiles.arizona.edu/person/lega","timestamp":"2024-11-09T13:28:43Z","content_type":"text/html","content_length":"149403","record_id":"<urn:uuid:fcf3df1c-41fd-489c-96ec-26262eec15da>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00774.warc.gz"} |
GABB Introduction | David A. Bader
The Basic Linear Algebra Subprograms (BLAS), introduced over 30 years ago, had a transformative effect on linear algebra. By building Linear Algebra algorithms from a common set of highly optimized
building blocks, researchers spend less time mapping algorithms onto specific hardware features and more time on interesting new algorithms. Could the same transformation occur for Graph algorithms?
Can Graph algorithm researchers converge around a core set of building blocks so we can focus more on algorithms and less on mapping software onto hardware? Graph Algorithms Building Blocks workshop
(GAB'14) will address these questions. The workshop will open with a pair of talks that define a candidate set of graph algorithm building blocks that we call the “Graph BLAS”. With this context
established, the reamining talks explore issues raised by these Graph BLAS, suggest alternative sets of low level building blocks, and finally consider lessons learned from past standards efforts. We
will close with an interactive panel about our collective quest to standardize a set of core graph algorithm building blocks.
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How do I find experts for my matrix algebra assignment? | Hire Someone To Do My Assignment
How do I find experts for my matrix algebra assignment? I have searched and read a lot and found few articles citing experts in either matrix algebra project (AEP) or matrix multiplication papers
(ASMé) and some other subject in which I do not do an exhaustive look under either MSDN, Google and other sources have read and do not find any articles that can be linked to the mentioned article.
Therefore, I am searching for people to teach mematrix algebra and they come from a few different fields that have similar requirements. Can you recommend me any other textbooks or library sites that
have those teaching and for which they mention MSDN and other I/II information? This is a really big project but everyone seemed to do it in the right way. I have written about this topic in an old
project after learning matrix multiplication and matrices theory and I know that the more it is related to mathematics this knowledge is helpful for teaching matrices subject area. However, I don’t
know if any of the books mentioned have been published yet as of now. I have done this project without success and again, all information on this subject has been lost. Any help is highly
appreciated! Thank you! A: There are no tutorials available for this particular position (how to choose a reference, etc.). However H&K does have some such books in which you can find course
material. Here I would recommend that anyone familiar with matrix multiplication should not take a course or read H&K. They set in motion a realisation in mathematics and taught yourself the concept
of matrix integrals. This is by no means an efficient and quick project. They have a nice set of library/sources which can reference matrix integrals. Hopefully there are others up to that point
also. A: Look up the Mathematica Math Package that’s very helpful either in theory or in practice. As for MS4M which is a good one In case of matrices and for instance matrices with 7 rows Any
matrices of the size of 5×10 matrix with the following format: ( 3d 1d 2d 3d 1v 2w 3d 3f ) A: Mathematica: Mat, Algebra, Scalar, Algebraic Algebraic Algebraic Algebra, Rotation. A beginner will
realize that it’s very good but there are still a lot of mistakes that these algebra lessons usually get wrong. Also consider here a class for matrices with non zero rows. From the first chapter:
Matrix algebras take the following form: (1 x 3 + 3 e e) Here x and e are rows of matrices. That can be seen from the matrix algebras How do I find experts for my matrix algebra assignment? I know
this is coming up in many different categories of works, but last time I checked some part of the answer was only here.
Pay Someone To Do University Courses
Here it is, with some more tips. Please, keep in mind that I am an open and honest man with really horrible math. 2) Working with small mistakes In a number of ways, you might be thinking about many
small mistakes during your homework. Sticking with small mistakes on a lesson plan, for example, is likely to be very misleading at the first glance: You don’t learn quickly on this long lesson by
focusing on what is most important to you. (You might notice what we think you are learning: you are watching a video. You have several cards, cards in your hands, and lots of other things, so I’m
guessing you don’t really get distracted by everything.) On the other hand, you might not realize even a single person at a given time has a very extensive knowledge of math, and you may not use this
knowledge properly at that moment. (Then you might not be able to manage anything around you for a while, and if you do that, you don’t learn what is important to you by focusing on understanding
your problem or even what is important to you. What could be simple, easy, and valuable sometimes sounds like a lot of stuff, and it could be harder to learn or even click this to learn, but there is
plenty of stuff.) You might be wondering what to expect over time, and it might seem like it’s only about time that won’t change. But you don’t see this in just about every context. This is a well
understood issue. However, many of the people involved in this study did give up right away. They haven’t really done much research on the subject yet. It’s hard to analyze a little, or even give the
people and students exactly what they used to read so far: 1. What is this problem What is this problem? How well do real people communicate at this stage of learning? The problem is actually part of
what we call digital literacy / media literacy. Basically, even some people find learning justifications for their skills by their high grade school or school of yours. Sometimes, right before they
start this program, they will immediately become interested in math and some other subjects at a younger age. The problem is part of what we call a digital literacy struggle: if online literacy
weren’t on there, we would become increasingly confused or scared to actually learn. This is pop over to this site an entirely bad thing, and there’s something about being in a completely different
neighborhood, where finding the good answers to problems is harder than making yourself interesting.
Mymathgenius Reddit
The problem happens when you are not the most sophisticated of people. That’s because your skills are so onHow do I find experts for my matrix algebra assignment? […] I’d be happy to help here as
it’s really quick. The big thing I’ve tried on my matrics is trying on more than one type of assignment. One of the key points is that there is no need for me to do this. […\n] I’m doing a matrix
assignment, a full matrix completion program being almost full. I’ve been playing around with what I think can be improved by getting more of the answers myself. If the way I have been practicing
general rules, also the formulas (including algebraic, polynomial, and arithmetic), I can get my syntax to work. Also, I can get a correct correct answer for whatever I have assigned as all of the
algebra I have been assigned works well together. So that everyone who is interested in this will be able to answer the question on the question list. As a bonus, because I get right on this how to
approach your matrics problem- it must be more than that… ..
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. It sounds like linear algebra can be more about how you write things than it is about mathematical. (They me around, but not really much on the same Look At This That is almost certainly the case
as I’ve had a lot of fun creating this site. Most of it though just isn’t the most interesting thing you can do. I am hoping just lots of good stuff to help people learn about algebra and its uses.
A: Matzeix is here. This is the place as I can’t quite find too. http://www.matzeix.org/ It has really really good documentation. A: In this answer, a general question, in official source matrix
multiplication table, you can obtain any answer for the same question together with a couple of others as well. I would very much like if the Matzeix group are able to implement same like. Matzeix
has a very good page on their website about “general questions”. For this reason, I’ll just use Matzeix and not Matzeq. They have a much higher level of documentation than Matzeix. In particular,
Matzeix has the table of all the cases for a set of required answers that can be represented in another form in Matzeix to show that this form gives results as a result of having matrics in it. A: On
my current implementation of the group matz and its normal form (and a general one). http://matzeix.org/matzeix/qcprfprops/matz/ Actually, my guess is that if a set I can use in the Matzeix table is
the same kind of case I could make a similar query of searching for answers in that form.
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Computer Graphics
Filling Algorithms in Computer Graphics
Computer graphics is a fascinating field that deals with creating, manipulating, and rendering visual content on a computer screen. One crucial aspect of computer graphics is the ability to fill
shapes or regions with colors or patterns. This process, known as filling, is essential for creating realistic and vibrant graphics.
In this article, we will explore two popular filling algorithms: the Scanline Fill Algorithm and the Flood Fill Algorithm. These algorithms provide efficient ways to fill polygons or closed regions
with continuous color or texture.
Scanline Fill Algorithm
The Scanline Fill Algorithm, as the name suggests, scans each row of a polygon or a closed region and fills it with the desired color. This algorithm works particularly well for convex polygons, as
they have only one intersection point per scanline.
Here is a step-by-step explanation of the Scanline Fill Algorithm:
1. Determine the minimum and maximum y-coordinates of the polygon or region.
2. Start from the minimum y-coordinate and move upwards one scanline at a time until reaching the maximum y-coordinate.
3. For each scanline, find the intersection points of the scanline with the edges of the polygon.
4. Sort the intersection points based on their x-coordinates.
5. Fill the pixels between pairs of intersection points with the desired color.
The Scanline Fill Algorithm is relatively simple and efficient, especially for convex polygons. However, it can encounter difficulties when dealing with concave or self-intersecting polygons. In such
cases, additional techniques may be required, like dividing the polygon into multiple convex sub-polygons.
Flood Fill Algorithm
The Flood Fill Algorithm, also known as seed fill, is a recursive algorithm that fills a connected region with a specified color or pattern. Unlike the Scanline Fill Algorithm, the Flood Fill
Algorithm can handle both convex and concave polygons effortlessly.
Here is a step-by-step explanation of the Flood Fill Algorithm:
1. Start from a seed point inside the region to be filled.
2. Check the color of the current pixel. If it matches the desired fill color, stop.
3. Change the color of the current pixel to the desired fill color.
4. Recursively apply the Flood Fill Algorithm to the four neighboring pixels (up, down, left, and right), unless they have already been filled or have a different color.
5. Repeat steps 2-4 until all the connected pixels within the region have been filled.
The Flood Fill Algorithm is straightforward to implement and can handle complex shapes with ease. However, it may lead to performance issues when applied to large regions or disconnected shapes. To
overcome these limitations, various optimized versions of the Flood Fill Algorithm have been developed, such as the Boundary Fill Algorithm and the Scanline Flood Fill Algorithm.
In conclusion, filling algorithms are essential tools in computer graphics, enabling us to add color and texture to shapes and regions. The Scanline Fill Algorithm and the Flood Fill Algorithm are
two commonly used methods for this purpose. Understanding these algorithms' principles can greatly enhance our ability to create visually appealing graphics in the world of computer graphics. | {"url":"https://noobtomaster.com/computer-graphics/filling-algorithms-scanline-flood-fill/","timestamp":"2024-11-07T16:08:22Z","content_type":"text/html","content_length":"26275","record_id":"<urn:uuid:2f10c54a-2f0b-48bc-8a8a-55813c048468>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00071.warc.gz"} |
Advance Ratio - Explained
Have you ever wondered why some discs seem to glide effortlessly while others wobble and veer off course? The answer lies in part with the disc's advance ratio, a key concept in aerodynamics.
Advance ratio is typically related to the performance of propellers, but it can also be applied to other rotating objects such as a thrown disc. In the context of a thrown disc, the advance ratio
helps in understanding the aerodynamic efficiency of the disc during its flight.
Definition of Advance Ratio
1. Basic Definition: The advance ratio is defined as the ratio of the forward speed of an object to the rotational speed of the object. For a propeller, it is the ratio of the forward velocity of
the aircraft to the tip speed of the propeller. For a thrown disc, it would be the ratio of the disc's forward velocity to the speed at its outer edge due to rotation.
2. Mathematical Expression:
Advance Ratio (AdvR) = spin (rad/s) × radius
speed (m/s)
Where speed is the forward velocity and spin is the angular velocity (in radians per second.)
Relevance in a Thrown Disc
1. Flight Stability: The advance ratio can play a role in the stability and glide of a disc. A disc with an optimal advance ratio may achieve a stable and efficient flight path. This is because a
higher advance ratio can help counteract the disc's natural tendency to turn over due to gyroscopic precession.
2. Lift and Drag: The lift and drag characteristics of a thrown disc are influenced by its rotation. The advance ratio, therefore, indirectly impacts how well the disc can maintain lift and minimize
3. Optimal Performance: Depending on the disc’s design (like its weight, shape, and rim configuration), there might be an 'optimal' advance ratio where the disc achieves maximum efficiency in
flight. This can vary greatly based on the design and purpose of the disc (distance, accuracy, stability, etc.).
Practical Considerations
• Throwing Technique: How a player throws the disc (angle, speed, spin) will impact the advance ratio. Different throwing techniques might be optimized for different advance ratios.
• Disc Design: Discs designed for different purposes (e.g., distance, precision, freestyle) will have varying optimal advance ratios.
• Environmental Factors: Wind and air density can affect the flight of the disc, thus influencing the effective advance ratio during flight.
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How are Levels calculated?
Levels are calculated for every player after every match and are based on:
• The player's level before the match.
• Their opponent's level before the match.
• The actual result compared to the expected result. Given we know the player's levels before the match we can predict the result if both players played as expected. If the actual result shows that
one of the players played better than expected, their level will go up a bit and their opponent's level will go down a bit.
• We then take a number of factors into account and work out how much change that 'bit' should be.
The algorithm uses:
• Maths - is at the heart of the algorithm and is used to determine how much their level should change if there was no damping or behavioural modelling. It's actually very straightforward as, for
example, if your level is twice that of your opponent then you'd be expected to win your games 11-5 or so. If your level is 20% higher then you'll win 20% more of the rallies with points scores
around 11-9. We use a combination of points scores and games scores to work out how much better the winner played as a ratio and that's where we start. The overall goal is that if you play twice
as well as your opponent then your level will be double theirs. This works all the way up from beginner (<50) to top pro (>50,000)
• Weighting- the more important the match (e.g. a tournament) the greater the weighting. This allows you to play a box match without having too much impact on your league standings.
• Damping- there is a reasonable amount of damping dialled into each match trying to get the balance right between wild swings and slow progress. The algorithm does its best to reward every player
for a good result and, in consequence, an appropriate level reduction if the result isn't so good. The intention is that a player's level is a reasonable assessment of their current playing level
within a match or two.
• Behavioural modelling- as it turns out, not everyone puts 100% effort in every match and that’s down to behaviour. There are many other cases too where player behaviour defies the maths and,
based on the analysis of 1.6 million results on the system, we’ve built an extensive behavioural model that allows us to predict and make use of these behaviours. See the calibration FAQ for more
detail on what behaviours we model.
At a very high level, it's quite straightforward as there's a direct correlation between the ratio of the player levels to the expected points scores. You can do it in your head! E.g. 20% better -
11-9, 30% better - 11-8, 50% better - 11-7 and so on. If you play better than expected you'll go up a bit or, if worse, you'll go down a bit. That's it, really! The complexity is over how much that
'bit' is but, fundamentally, your level will adjust so that your level ratios match your results ratios - on average.
We can work with game scores only, making assumptions around the average 3-0 result (based on our analysis of real 3-0 match results) but we can only use averages and apply a lot more damping so it
takes quite a few results for the levels to become accurate. Not all 3-0 results are the same, obviously.
Note that in the match review page, you can scroll down and click on 'View full explanation' and it shows you the algorithm output in verbose mode for that match so you can see exactly why your level
changed by the amount it did. | {"url":"https://support.squashlevels.com/hc/en-us/articles/7712760245405-How-are-Levels-calculated","timestamp":"2024-11-04T00:52:26Z","content_type":"text/html","content_length":"29403","record_id":"<urn:uuid:5ad38b62-f17d-40f3-91f3-3e55bb3d7ce9>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00035.warc.gz"} |
Inspiring Drawing Tutorials
How Many Amps Does A Bilge Pump Draw
How Many Amps Does A Bilge Pump Draw - To do this, you must go through meticulously and calculate amp usage by hours used by anything aboard. Web the wire needs to be the right gauge for how many
amps your bilge pump draws and the length of the wiring circuit from battery to pump. Web it’s always a good idea to have more than one electric bilge pump installed. We also tested 10 pumps and
compared the results for you to read in our bilge pump test. This water can come in due to errant waves, drip through packing gland, a broken hose or clamp, leaking port holes or hatches, and much
more. That’s because even the largest capacity bilge.
Web a bilge pump’s primary function is to deal with the types of nuisance water we mention above—it shouldn’t be relied on as safety gear, although many boaters consider it just that. 8.7 learn your
boat’s bilge habits. Essentially, bilge water is almost impossible to avoid. The fuse size needed for this particular application is found in our chart below, and it needs to draw somewhere in
between 4.5 and 7 amps under full load, depending on the voltage as well. That’s because even the largest capacity bilge.
Web standard bilge pumps capacity gph (lph) model no. Whether or not it has protection against that: Web insulation ratings are 600 volts and 105°c on all except the whale. Web the device utilizes a
12 v dc power output with over 2000 gph pumping capacity. Web it’s always a good idea to have more than one electric bilge pump installed.
Amp draw @ 13.6 v / 12.2v 7 amps / 6.8 amps 7.3 amps / 6.3 amps 8.8 amps / 7.2 amps 5.6 amps /. The gph rating given to a bilge pump by the manufacturer is based on its capacity to pump water. Web
rule standard bilge pumps. Web what are bilge pumps? The fuse size needed for this.
The village 2000 is fully submersible, which means it won’t sink or become a dead weight on your boat. That’s because even the largest capacity bilge. Volts fuse size amps @12 v amps @13.6 v
replacement strainer height in. 8.4 check your bilge manually at least a few times a day. Essentially, bilge water is almost impossible to avoid.
Capacity gph (lph) model number. The rule 1500 offers more pumping capacity and more exclusive design features than any comparable competitive pump. Amp draw @ 13.6 v / 12.2v 7 amps / 6.8 amps 7.3
amps / 6.3 amps 8.8 amps / 7.2 amps 5.6 amps /. The gph rating given to a bilge pump by the manufacturer is based.
This water can come in due to errant waves, drip through packing gland, a broken hose or clamp, leaking port holes or hatches, and much more. Amp draw @ 13.6 v / 12.2v 7 amps / 6.8 amps 7.3 amps /
6.3 amps 8.8 amps / 7.2 amps 5.6 amps /. Web what are bilge pumps? Bilge pumps are typically.
The village 2000 is fully submersible, which means it won’t sink or become a dead weight on your boat. The job of a bilge pump is to clear the unwanted water that gets into the bilge on your boat.
Web entirely preventable if you plan correctly. That’s because even the largest capacity bilge. Find out everything you need to know.
8.7 learn your boat’s bilge habits. Bilge pumps are typically either diaphragm electrical or centrifugal. Capacity gph (lph) model number. 12 volts dc unless noted. Amp draw @ 13.6 v / 12.2v 7 amps /
6.8 amps 7.3 amps / 6.3 amps 8.8 amps / 7.2 amps 5.6 amps /.
The first step is to calculate how many amps you use daily. Capacity gph (lph) model number. The rule 1500 offers more pumping capacity and more exclusive design features than any comparable
competitive pump. Web the american boat and yacht council (abyc) has not set requirements concerning bilge pump capacity, though the american bureau of shipping recommends one 24 gallons.
Rule bilge pump amperage draw i have two of the rule 360's and have the instruction paper that came with them. Oct 25, 2021 at 17:53. The job of a bilge pump is to clear the unwanted water that gets
into the bilge on your boat. 12 volts dc unless noted. Web your bilge pump should have a pumping capacity.
We also tested 10 pumps and compared the results for you to read in our bilge pump test. Web one easy way to find out is to pour five gallons of water into the bilge and start your stopwatch when the
bilge pump kicks in. The gph rating given to a bilge pump by the manufacturer is based on its.
Voltage 12 (vdc) amp draw 4.8 (amp) fuse size 10 (amp) height 209.5 (mm) 8 1/4 (inches ) Consult the abyc wire size table to determine the appropriate wire for your pump and length of wire run. Web
8.3 flip your pump on regularly if you don’t have a float switch. Web the wire needs to be the right gauge.
How Many Amps Does A Bilge Pump Draw - Amp draw @ 13.6 v / 12.2v 7 amps / 6.8 amps 7.3 amps / 6.3 amps 8.8 amps / 7.2 amps 5.6 amps /. This is why this product is an excellent tool to help get rid of
a huge amount of water from any type of boat, especially large size. Volts fuse size amps @12 v amps @13.6 v replacement strainer height in. Rule bilge pump amperage draw i have two of the rule 360's
and have the instruction paper that came with them. Web the american boat and yacht council (abyc) has not set requirements concerning bilge pump capacity, though the american bureau of shipping
recommends one 24 gallons per minute pump (that's about 1,440 gph) and one 12 gpm pump for boats under 65 feet. Web rule standard bilge pumps. Web entirely preventable if you plan correctly. High
pumping capacity and reliability at a lower cost. Essentially, bilge water is almost impossible to avoid. It says to use a 2.5 amp fuse with it so it obviously draws less than 2 1/2 amps.
Use butt connectors and heat shrink. Web your bilge pump should have a pumping capacity of at least 24 gallons per minute if your boat is under 65 feet. Types of bilge pump switches. Good quality
automatic bilge pumps can more than handle this. In our case, it requires a.
Whether or not it has protection against that: Amp draw @ 13.6 v / 12.2v 7 amps / 6.8 amps 7.3 amps / 6.3 amps 8.8 amps / 7.2 amps 5.6 amps /. Web your bilge pump should have a pumping capacity of at
least 24 gallons per minute if your boat is under 65 feet. 8.7 learn your boat’s bilge habits.
If the manufacturer specifies a particular gauge wire, follow the guidelines. Oct 25, 2021 at 17:53. Web entirely preventable if you plan correctly.
Web wiring a manual bilge pump. Bilge pumps are typically either diaphragm electrical or centrifugal. Web rule standard bilge pumps.
Web One Easy Way To Find Out Is To Pour Five Gallons Of Water Into The Bilge And Start Your Stopwatch When The Bilge Pump Kicks In.
We also tested 10 pumps and compared the results for you to read in our bilge pump test. Amp draw @ 13.6 v / 12.2v 7 amps / 6.8 amps 7.3 amps / 6.3 amps 8.8 amps / 7.2 amps 5.6 amps /. Essentially,
bilge water is almost impossible to avoid. Web rule standard bilge pumps.
Web The Device Utilizes A 12 V Dc Power Output With Over 2000 Gph Pumping Capacity.
8.4 check your bilge manually at least a few times a day. That’s because even the largest capacity bilge. Use butt connectors and heat shrink. Types of bilge pump switches.
This Is Why This Product Is An Excellent Tool To Help Get Rid Of A Huge Amount Of Water From Any Type Of Boat, Especially Large Size.
If the manufacturer specifies a particular gauge wire, follow the guidelines. Find out everything you need to know about bilge pumps in our guide. To do this, you must go through meticulously and
calculate amp usage by hours used by anything aboard. The gph rating given to a bilge pump by the manufacturer is based on its capacity to pump water.
The First Step Is To Calculate How Many Amps You Use Daily.
The rule 1500 offers more pumping capacity and more exclusive design features than any comparable competitive pump. Voltage 12 (vdc) amp draw 4.8 (amp) fuse size 10 (amp) height 209.5 (mm) 8 1/4
(inches ) Attwood and mayfair pumps can be mounted on vertical or horizontal surfaces. High pumping capacity and reliability at a lower cost. | {"url":"https://one.wkkf.org/art/drawing-tutorials/how-many-amps-does-a-bilge-pump-draw.html","timestamp":"2024-11-03T16:44:43Z","content_type":"text/html","content_length":"35196","record_id":"<urn:uuid:f3c3a73d-f8ab-4875-976f-50a8af05554d>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00126.warc.gz"} |
Taking reciprocals is order-reversing - Stumbling Robot
Taking reciprocals is order-reversing
Prove that if
I.3.5, Exercise #4
). Hence,
So, by Theorem I.19, we have
One comment
1. Alternative solution:
We know (b-a)>0 (by def), a*b>0 (axiom 7) and [1/(a*b)]>0 (I.3.5, Exercise #4)
So (b-a)*[1/(a*b)]>0 (axiom 7)
(b-a)*[1/(a*b)]=((1*b)-(a*1))/(a*b) (Thm 1.14 and axiom 4)
((1*b)-(a*1))/(a*b)= (1/a)-(1/b) (1.3.3, Exercise #10)
(1/a)-(1/b)>0 iff (1/b)<(1/a) (by def)
By I.3.5, Exercise #4, 0<(1/b)<(1/a).
Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment): | {"url":"https://www.stumblingrobot.com/2015/06/30/taking-reciprocals-is-order-reversing/","timestamp":"2024-11-10T08:17:44Z","content_type":"text/html","content_length":"59585","record_id":"<urn:uuid:8dd51319-fd67-4171-a308-f728df260f75>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00477.warc.gz"} |
Global Variables and Equation Driven Design - Engineers Rule
Designing things on the fly can be fun and intuitive. But as soon as you start changing those features, everything you’ve done up to that point can potentially go right out of the window.
Thankfully, by using SOLIDWORKS’ equation capabilities, it is possible to fully define your sketch and model geometry and establish relationships and constraints through equations. This is
particularly useful in engineering, as many systems rely on ratios and dynamic relationships that change depending on specific geometric characteristics.
Say for example, you are designing some kind of fluid nozzle. Maybe you would like your nozzle outlet to have a diameter that changes with regard to a specific inlet diameter or even some value for
In SOLIDWORKS this is pretty easy. You can even define the value for pressure at the inlet and have the outlet diameter change when you rebuild the model. These features can allow you to evaluate
various design options without having to manually redesign your model each time you wish to change a parameter. We won’t go that far in this article. We will just look at designing a basic shower
head, and we will link the dimensions to a set of basic equations.
Equations can be used to drive both sketch geometry and model geometry. For both cases, they work in a similar manner. Let’s take a look at how to drive sketch geometry with equations.
Equations, Global Variables and Dimensions
Let’s start by looking at how to input dimensions and equations. First, go to Tools > Equations.
On clicking Equations, you should notice a new window pop up, titled Equations, Global Variables and Dimensions.
This is where most of the equation action will take place. Let’s have a look at this in a little more detail since we will be using it later.
Global Variables
Global Variables can be used to drive equations and dimensions. Say for example, you were designing a pipe and wanted the pipe length to remain relative to some other dimension—maybe your pipe has
some length constraints relative to its installation location.
You could name a Global Variable as “PipeLimit.” When you go to create your actual pipe, you can define the pipe length in terms of a fraction of that limit in the equation field. Whenever you make
changes to your pipe, it will remain within the allowed boundaries.
If it somehow breaches that boundary, then you can use the Feature section to suppress the pipe if it gets too big.
Tutorial Time!
Everything in SOLIDWORKS begins with a sketch. So, this is where we will begin this tutorial.
First, sketch two concentric circles of diameter 100mm and 90mm.
Now comes an important step. We need to define these dimensions with the Smart Dimension tool so that the Equations functions will recognize them.
With the sketch still open, go to Smart Dimension, click it and click the outer diameter of the circle you just drew. Name it as “Outer.”
Do the same for the inner concentric circle and name it “Inner.”
Now they are defined and named with the Smart Dimension tool. When you go to Tools>Equations and open up the Equations, Global Variables and Dimensions panel (shown below) and click the Dimension
View tab (circled below in red), you will see that the Dimensions section has been populated with the Smart Dimensions from the sketch.
OK, let’s open the same sketch back up and draw two more concentric circles inside the other two. Let’s convert these circles to construction geometry—right click on the circle sketch and select the
Construction Geometry icon. Don’t worry about the precise diameter. We will have these new diameters driven by our global variable later. Just be sure to use the Smart Dimension tool and rename them
as D1 and D2.
Now, sketch two little circles, one on each of the construction circles that we just sketched. Use the Smart Dimension tool and rename them “OuterHole” and “InnerHole.”
Next up, create a Circular Pattern of these new holes. Create four equally spaced instances so we end up with eight holes.
That will do for the sketching. We can now go ahead and extrude the sketch entities.
Firstly, extrude the outer ring—the contour area in between the sketch elements we named as “Inner” and “Outer”—as you can see below. Extrude it to 10mm.
Next, extrude the inner area (the base) ensuring that you don’t accidentally extrude/fill the eight little circles up. Extrude that base area to 3mm. If you’ve followed the steps correctly, then your
final solid shape should look like the image below.
The Equations and Global Variables Bit
OK, now we have our solid created from a Smart Dimensioned sketch. All of those dimensions are now visible in the Equations, Global Variables and Dimensions panel. Since they are all visible in that
panel, we can now start linking them up and making them a little more dynamic and responsive to our design changes.
This is the part where we transform the solid from a dumb model into a smart model.
Let’s have a recap of what is now visible in the Equations, Global Variables and Dimensions panel now that we have populated it with the Smart Dimensions.
At this point, if we click on any populated field in the Value/Equation column, we can change the value in that field. Our sketch (and solid) will respond to that change. You will notice that not
only are the sketch entities in the table, but the boss extrusion feature dimensions also have appeared.
Even though our model is still relatively dumb, you can automatically see the value of having all of your dimensions collected in one place like this. From this panel, we can literally just find a
parameter we wish to change and do so, safe in the knowledge that the model will update to reflect those changes on rebuild.
It sure as heck beats opening up different sketches and manually editing them every time we want to make a change.
For example, if I want to change the number of instances in the circular pattern, I simply click the Value/Equation field for the CircularPat@Sketch1 entry and increase or decrease it as I see fit.
In this example, I want to change the number of holes to 12. I change the CircularPat entry to six because we are actually patterning the two original holes, so 2 x 6 = 12. Keep this Circular Pattern
thing in your minds. We will be making this into an equation-driven value later.
Let’s start to add the Global Variables. Open up the Equations, Global Variables and Dimensions panel again, and in the Global Variables section, add a new Global Variable named “OuterDiameter” and
type the value of “=100mm” in the corresponding Value/Equation cell. Be sure to use the equals symbol (=) when entering values here.
Now create a second Global Variable named “HoleDiameter” and set it to 8mm.
Beneath that, create a new Global Variable named “HoleArea.” Now we can start to use some formulae. We want to define the individual hole area in terms of the diameter so that it’s equal to pi
multiplied by the radius squared.
In the HoleArea value cell, we can enter this formula as =PI * (HoleDiameter).
You don’t have to actually type HoleDiameter. You can just click the Global Variable name or select it from the drop-down menu while typing the rest of it.
Now that these Global Variables are added, we can refer to these when defining the dimensions or creating equations. They are now magically stored in the software somewhere and can be recalled when
linking to values.
For example, we wish to link our OUTER@Sketch1 value to the OuterDiameterGlobal Variable.
We can do this by simply deleting the original value of 100mm from the OUTER@Sketch1 value field and clicking the cursor in the empty cell. You will see a list appear in a pop-up menu showing various
options, including to insert a Global Variable.
In this case, we want to link the OuterDiameterGlobal Variable to the OUTER@Sketch1 dimension. You can see this in the image below.
Now that we have those Global Variables defined, we can use them as a benchmark to create relationships with the other sketch and model entities. We can do this with equations.
Entering equations in SOLIDWORKS is fairly easy. There’s no need for any deep programming knowledge. It’s comparable to entering equations in a spreadsheet.
Imagine that you are designing some sort of fluid system and wish to maintain a specific total area for the holes for the fluid to pass through, and you want that total area to remain constant
regardless of the diameter of each individual hole. To put it another way, we want the number of holes to change and maintain a constant total area.
Let’s say we want that area to equal 2,400 square millimeters. This will be our target value. We can treat this as a constant, so we create a new Global Variable called “TOTALholeAREA” and set it to
2,400, as you can see in the image below.
Next, we want our number of holes to change in order to maintain that total area, regardless of the diameter of the holes. We create a final Global Variable called “HolesNeeded” and add a little
formula for that.
The number of holes will be equal to the total combined area of all of the holes divided by the individual hole area. Since we want an integer value, we can use the “Round” function to round it off.
In the Holes Needed Value/Equation cell, we can use the syntax =ROUND (TOTALholeAREA / HoleArea) to return an integer value. Remember to add the parentheses after the ROUND command to tell the
software to perform that function on whatever is inside the brackets, just like your spreadsheet program.
Now we want to have the output of that formula drive one of our dimensions. Specifically, we want to have that output drive the number of instances in the circular pattern.
Following the same procedure from before, where we assigned Global Variable values to dimensions, we simply click the adjacent cell to the CircularPat@Sketch1 dimension and link that to the
HolesNeeded variable. We divide that by two, as you can see below.
Why are we dividing it by two? Because we have two sets of holes: one on the outside, and one on the inside.
Now Test!
That’s it. It’s all done.
Now you can go ahead and test it. If you’ve followed the steps correctly, you can change the values for hole diameter in the Global Variable section and the model will update the number of holes
needed to maintain a constant area.
It could be useful for designing a shower head, injector system or anything where you might like to maintain a constant flow while varying the number of holes. Of course, there’s a lot more to fluid
dynamics than that. You can link all kinds of variables and equations. It’s all down to your ingenuity, and patience.
You can see how our new smart model responds to changes in the video below. | {"url":"https://www.engineersrule.com/global-variables-and-equation-driven-design/","timestamp":"2024-11-09T16:42:32Z","content_type":"text/html","content_length":"278658","record_id":"<urn:uuid:cd4cbbfa-b923-4c31-b431-0e402beff0b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00422.warc.gz"} |
Vector Calculus: Course by Peter Saveliev
Vector Calculus: Course
by Peter Saveliev
Number of pages: 294
This is a two-semester course in n-dimensional calculus with a review of the necessary linear algebra. It covers the derivative, the integral, and a variety of applications. An emphasis is made on
the coordinate free, vector analysis.
Download or read it online for free here:
Read online
(online html)
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Honors Calculus
Frank Jones
Rice UniversityThe goal is to achieve a thorough understanding of vector calculus, including both problem solving and theoretical aspects. The orientation of the course is toward the problem aspects,
though we go into great depth concerning the theory.
The Geometry of Vector Calculus
Tevian Dray, Corinne A. Manogue
Oregon State UniversityContents: Chapter 1: Coordinates and Vectors; Chapter 2: Multiple Integrals; Chapter 3: Vector Integrals; Chapter 4: Partial Derivatives; Chapter 5: Gradient; Chapter 6: Other
Vector Derivatives; Chapter 7: Power Series; Chapter 8: Delta Functions.
Vector Analysis
J. Willard Gibbs
Yale University PressA text-book for the use of students of mathematics and physics, taken from the course of lectures on Vector Analysis delivered by J. Willard Gibbs. Numerous illustrative examples
have been drawn from geometry, mechanics, and physics. | {"url":"https://www.e-booksdirectory.com/details.php?ebook=9924","timestamp":"2024-11-10T20:45:36Z","content_type":"text/html","content_length":"10797","record_id":"<urn:uuid:c6baf230-0273-4486-b558-59dee880b20b>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00310.warc.gz"} |
Infinitesimal transformation - AbsoluteAstronomy.com
Infinitesimal transformation
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by
mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an
infinitesimal transformation
is a
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians
to define a new point from a Cauchy sequence of previously defined points within a complete metric...
form of
transformation. For example one may talk about an
infinitesimal rotation
of a
rigid body
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains
constant in time regardless of external forces exerted on it...
, in three-dimensional space. This is conventionally represented by a 3×3
skew-symmetric matrix
Skew-symmetric matrix
In mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij,
. It is not the matrix of an actual
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body,
and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
in space; but for small real values of a parameter ε we have
a small rotation, up to quantities of order ε
A comprehensive theory of infinitesimal transformations was first given by
Sophus Lie
Sophus Lie
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
. Indeed this was at the heart of his work, on what are now called
Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s and their accompanying
Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the
concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s; and the identification of their role in
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and especially the theory of
differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s. The properties of an abstract
Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the
concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
are exactly those definitive of infinitesimal transformations, just as the axioms of
group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such
as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
. The term "Lie algebra" was introduced in 1934 by
Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the
University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
, for what had until then been known as the
algebra of infinitesimal transformations
of a Lie group.
For example, in the case of infinitesimal rotations, the Lie algebra structure is that provided by the
cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of
the vectors being multiplied and normal to the plane containing them...
, once a skew-symmetric matrix has been identified with a 3-vector. This amounts to choosing an axis vector for the rotations; the defining
Jacobi identity
Jacobi identity
In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations,
order of evaluation is significant for operations satisfying Jacobi identity...
is a well-known property of cross products.
The earliest example of an infinitesimal transformation that may have been recognised as such was in Euler's theorem on homogeneous functions. Here it is stated that a function
, ...,
that is homogeneous of degree
, satisfies
differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract
operation, accepting a function and returning another .This article considers only linear operators,...
. That is, from the property
we can in effect differentiate with respect to λ and then set λ equal to 1. This then becomes a necessary condition on a
smooth function
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the
existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
to have the homogeneity property; it is also sufficient (by using Schwartz distributions one can reduce the
mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the
theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
considerations here). This setting is typical, in that we have a
one-parameter group
One-parameter group
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphismfrom the real line R to some other topological group G...
of scalings operating; and the information is in fact coded in an infinitesimal transformation that is a first-order differential operator.
The operator equation
is an operator version of
Taylor's theorem
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial. For analytic functions the Taylor polynomials at a
given point are finite order truncations of its Taylor's series, which completely determines the...
— and is therefore only valid under
being an
analytic function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are
similar in some ways, but different in others...
. Concentrating on the operator part, it shows in effect that
is an infinitesimal transformation, generating translations of the real line via the
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a
constant change in the independent variable gives the same proportional change In mathematics,...
. In Lie's theory, this is generalised a long way. Any
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one
of the principal topological properties that is used to distinguish topological spaces...
Lie group can be built up by means of its infinitesimal generators (a basis for the Lie algebra of the group); with explicit if not always useful information given in the Baker–Campbell–Hausdorff
The source of this article is
, the free encyclopedia. The text of this article is licensed under the | {"url":"http://www.absoluteastronomy.com/topics/Infinitesimal_transformation","timestamp":"2024-11-01T19:45:23Z","content_type":"text/html","content_length":"24241","record_id":"<urn:uuid:8cd94e9e-9106-49e8-baa2-6f8adbf0a1b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00694.warc.gz"} |
Adding Complex Numbers: (5 + 2i) + (3 - 2i)
This article will guide you through the process of adding two complex numbers: (5 + 2i) and (3 - 2i).
Understanding Complex Numbers
Complex numbers are numbers that consist of two parts: a real part and an imaginary part. The imaginary part is denoted by the letter 'i', where i² = -1.
Adding Complex Numbers
Adding complex numbers is straightforward. We simply add the real parts together and the imaginary parts together separately.
Step 1: Identify the real and imaginary parts of each number:
• (5 + 2i) has a real part of 5 and an imaginary part of 2i.
• (3 - 2i) has a real part of 3 and an imaginary part of -2i.
Step 2: Add the real parts together:
Step 3: Add the imaginary parts together:
Step 4: Combine the results:
Therefore, (5 + 2i) + (3 - 2i) = 8.
Important Note: The imaginary part cancels out in this specific example. This is not always the case when adding complex numbers.
Adding complex numbers is a simple process involving combining the real and imaginary parts separately. This allows us to manipulate and work with complex numbers effectively in various mathematical | {"url":"https://jasonbradley.me/page/(5%252B2i)%252B(3-2i)","timestamp":"2024-11-10T08:32:23Z","content_type":"text/html","content_length":"60968","record_id":"<urn:uuid:3073d1ab-8fd0-494c-833c-2469c0cfd9c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00496.warc.gz"} |
Ants in the van
Monday, Feb 24, 2014 at 11:24
Don't waste your time with those insect bombs, they don't really get into tight spots very
and they can be messy. You will need to wash bench tops, cupboards ect afterwards and they don't really do a very good job for things like ants.
Although it may take you longer, you are much better off buying a good quality surface spray. Use one of those thin long tubes, that attach to the nozzle, this will allow you to get into tight
. But before you start, be aware of any electrical wiring or other hazards.
Spray it into any gap in the roof area, try and do the whole roof. it may pay to drill some holes in out of sight
, just big enough to allow you to fit the tube in.
Also see if you are able to get some into the walls, between cupboards etc.
For the outside, do a search on the net, there are various
made solutions that may work. depending on weather, liquid or powder or even both. place it around all contact points. It may also help to spray a good quality surface spray around the underside of
the van, around the top of stabilizer legs etc. You would need to do this on a regular basis, because the chemical will break down a lot faster.
There used to be a product that was red liquid, it was excellent stuff and worked really
. I'm not sure if it is still available or not, someone else may remember the name of it and if you can still buy it.
Instead of
the van on grass at
, are you able to put a slab where each wheel would sit and under any stabilizer legs? This would allow you to spot them a little easier as
as form an unbroken barrier.
AnswerID: 527118
Follow Up By: disco driver - Monday, Feb 24, 2014 at 11:58
Monday, Feb 24, 2014 at 11:58
From memory the red liquid may have been a product called "Antex"
It was a sticky very sweet syrup and therefore created risk where
children were around
It was taken off the list due to one of the ingredients being Thalium which is a cumulative poison, among other things it caused hair loss and other nasty things.
FollowupID: 809517
Follow Up By: Member - evaredy - Monday, Feb 24, 2014 at 12:49
Monday, Feb 24, 2014 at 12:49
That sounds about right.
FollowupID: 809519 | {"url":"https://www.exploroz.com/forum/106389/ants-in-the-van","timestamp":"2024-11-06T20:20:19Z","content_type":"application/xhtml+xml","content_length":"52138","record_id":"<urn:uuid:4ed0f924-be30-494e-9051-e2506ec8097c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00800.warc.gz"} |
Rounding to the Nearest Tenth Calculator
A Rounding to the Nearest Tenth Calculator is a tool designed to simplify the process of rounding numbers to the nearest tenth.
It’s a valuable resource for students, professionals, and anyone who needs to perform rounding calculations quickly and accurately. This calculator eliminates the need for manual calculations,
reducing the risk of errors and saving time.
Rounding to the Nearest Tenth Calculator
What is 7.683 rounded to the nearest tenth?
To round 7.683 to the nearest tenth, we follow these steps:
1. Identify the digit in the hundredths place: 7.683 (The digit in the hundredths place is 8)
2. Since the digit in the hundredths place is 8, which is greater than or equal to 5, we increase the digit in the tenths place by 1.
3. Replace all digits to the right of the tenths place with zeros.
Therefore, 7.683 rounded to the nearest tenth is 7.7.
What is 21.249 rounded to the nearest tenth?
To round 21.249 to the nearest tenth, we follow these steps:
1. Identify the digit in the hundredths place: 21.249 (The digit in the hundredths place is 4)
2. Since the digit in the hundredths place is 4, which is less than 5, the digit in the tenths place remains the same.
3. Replace all digits to the right of the tenths place with zeros.
Therefore, 21.249 rounded to the nearest tenth is 21.2.
What is 5.055 rounded to the nearest tenth?
To round 5.055 to the nearest tenth, we follow these steps:
1. Identify the digit in the hundredths place: 5.055 (The digit in the hundredths place is 5)
2. Since the digit in the hundredths place is 5, which is equal to 5, we increase the digit in the tenths place by 1.
3. Replace all digits to the right of the tenths place with zeros.
Therefore, 5.055 rounded to the nearest tenth is 5.1.
What is 18.999 rounded to the nearest tenth?
To round 18.999 to the nearest tenth, we follow these steps:
1. Identify the digit in the hundredths place: 18.999 (The digit in the hundredths place is 9)
2. Since the digit in the hundredths place is 9, which is greater than or equal to 5, we increase the digit in the tenths place by 1.
3. Replace all digits to the right of the tenths place with zeros.
Therefore, 18.999 rounded to the nearest tenth is 19.0.
How to use Rounding to the Nearest Tenth Calculator?
Using a Rounding to the Nearest Tenth Calculator is straightforward and user-friendly. Follow these simple steps:
1. Enter the number you want to round in the designated input field.
2. Click the “Round to Nearest Tenth” button or press the “Enter” key.
3. The calculator will instantly display the result, showing the input number rounded to the nearest tenth.
Rounding to the Nearest Tenth Calculator Formula
The formula used by the Rounding to the Nearest Tenth Calculator is based on the concept of decimal place value. To round a number to the nearest tenth, the calculator follows these steps:
1. Identify the digit in the hundredths place (the second digit to the right of the decimal point).
2. If the digit in the hundredths place is less than 5, the digit in the tenths place remains the same.
3. If the digit in the hundredths place is 5 or greater, the digit in the tenths place is increased by 1.
4. All digits to the right of the tenths place are replaced with zeros.
How Does Rounding to the Nearest Tenth Work?
Rounding to the nearest tenth is a common operation in mathematics and various fields where approximations are necessary or preferable. Here’s a step-by-step explanation of how the rounding process
1. Identify the digit in the hundredths place (the second digit to the right of the decimal point).
2. If the digit in the hundredths place is less than 5 (0, 1, 2, 3, or 4), the digit in the tenths place remains the same, and the digits to the right of the tenths place are replaced with zeros.
For example, 3.14 rounds to 3.1, and 2.37 rounds to 2.4.
3. If the digit in the hundredths place is 5 or greater (5, 6, 7, 8, or 9), the digit in the tenths place is increased by 1, and the digits to the right of the tenths place are replaced with zeros.
For example, 4.65 rounds to 4.7, and 6.98 rounds to 7.0.
Create a Rounding to the Nearest Tenth Chart
To better understand the concept of rounding to the nearest tenth, it can be helpful to create a chart or table. Here’s an example:
Original Number Rounded to Nearest Tenth
2.14 2.1
3.57 3.6
4.29 4.3
5.65 5.7
6.08 6.1
7.43 7.4
8.79 8.8
9.12 9.1
This chart illustrates various numbers and their corresponding rounded values to the nearest tenth. Notice how numbers with a digit in the hundredths place less than 5 remain the same in the tenths
place, while those with a digit 5 or greater have their tenths place digit increased by 1.
What is 3295.351 rounded to the nearest tenth?
To round 3295.351 to the nearest tenth, we follow these steps:
1. Identify the digit in the hundredths place: 3295.351 (The digit in the hundredths place is 5)
2. Since the digit in the hundredths place is 5, we increase the digit in the tenths place by 1.
3. Replace all digits to the right of the tenths place with zeros.
Therefore, 3295.351 rounded to the nearest tenth is 3295.4.
What is 34.9961 rounded to the nearest 10?
Note: The question asks to round 34.9961 to the nearest 10, not the nearest tenth.
To round a number to the nearest 10, we look at the digit in the ones place:
1. Identify the digit in the ones place: 34.9961 (The digit in the ones place is 4)
2. If the digit in the ones place is less than 5, we round down to the nearest 10.
3. If the digit in the ones place is 5 or greater, we round up to the nearest 10.
Since the digit in the ones place is 4, which is less than 5, we round down to the nearest 10.
Therefore, 34.9961 rounded to the nearest 10 is 30.
What is 11.04 rounded to the nearest tenth?
To round 11.04 to the nearest tenth, we follow these steps:
1. Identify the digit in the hundredths place: 11.04 (The digit in the hundredths place is 0)
2. Since the digit in the hundredths place is 0, which is less than 5, the digit in the tenths place remains the same.
3. Replace all digits to the right of the tenths place with zeros.
Therefore, 11.04 rounded to the nearest tenth is 11.0.
Additional Notes:
• The Rounding to the Nearest Tenth Calculator is designed specifically for rounding numbers to the nearest tenth.
• For rounding to other places (e.g., nearest whole number, nearest hundredth, nearest thousand), different calculators or rounding techniques would be used.
• Rounding is a useful skill in mathematics, science, and various other fields where approximations are necessary or preferred for simplicity and ease of comprehension. | {"url":"https://ctrlcalculator.com/math/rounding-to-the-nearest-tenth-calculator/","timestamp":"2024-11-04T20:57:14Z","content_type":"text/html","content_length":"104817","record_id":"<urn:uuid:d00da80e-2410-41d2-a37c-a88d1e0e5eb6>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00337.warc.gz"} |
Quantum Computing – Page 3 – The Quantum Pontiff
A rare event occurred today here in Seattle. And I’m not just talking about the 20 minutes of partly sunny skies we got at lunchtime. No, this was something rarer still: a quantumista condensate.
What precipitated this joint gathering of the decohered and the coherent? Michael Nielsen was visiting the University of Washington to deliver a distinguished lecture on the topic of his new book:
open science. Having seen Michael talk about this subject 3 years ago at QIP Santa Fe, I can say that he has significantly focused his ideas and his message. He makes a very compelling case for open
science and in particular open data. He has thought very hard about what makes online collaborative science projects successful at focusing and amplifying our collective intelligence, why such
projects sometimes fail, and which steps we need to take to get to the promised land from where we are currently.
The talk was recorded, and as soon as the video becomes available I’ll put a link here. I highly recommend watching it.
Update (12/12): Here is the link to Michael’s talk.
You might be wondering, what is the optimizer doing there? He is in town to give the colloquium to the computer science department. And given all the excitement, Dave Bacon, aka Pontiff++, couldn’t
help but sneak over from Google to check things out. He is the one you can blame for coining the horrible phrase “quantumista condensate”, but you probably already guessed that.
Dial M for Matter
It was just recently announced that Institute for Quantum Information at Caltech will be adding an extra letter to its name. The former IQI will now be the Institute for Quantum Information and
Matter, or IQIM. But it isn’t the name change that is of real significance, but rather the $12.6 million of funding over the next five years that comes with it!
In fact, the IQIM is an NSF funded Physics Frontier Center, which means the competition was stiff, to say the least. New PFCs are only funded by the NSF every three years, and are chosen based on
“their potential for transformational advances in the most promising research areas at the intellectual frontiers of physics.”
In practice, the new center means that the Caltech quantum info effort will continue to grow and, importantly, it will better integrate and expand on experimental efforts there. It looks like an
exciting new chapter for quantum information science at Caltech, even if the new name is harder to pronounce. Anyone who wants to become a part of it should check out the open postdoc positions that
are now available at the IQIM.
Markus Greiner named MacArthur Fellow
Markus Greiner from Harvard was just named a 2011 MacArthur Fellow. For the experimentalists, Markus needs no introduction, but there might be a few theorists out there who still don’t know his name.
Markus’ work probes the behavior of ultracold atoms in optical lattices.
When I saw Markus speak at SQuInT in February, I was tremendously impressed with his work. He spoke about his invention of a quantum gas microscope, a device which is capable of getting high fidelity
images of individual atoms in optical lattices. He and his group have already used this tool to study the physics of the bosonic and fermionic Bose-Hubbard model that is (presumably) a good
description of the physics in these systems. The image below is worth a thousand words.
Yep, those are individual atoms, resolved to within the spacing of the lattice. The ultimate goal is to obtain individual control of each atom separately within the lattice. Even with Markus’
breakthroughs, we are still a long way from having a quantum computer in an optical lattice. But I don’t think it is a stretch to say that his work is bringing us to the cusp of having a truly useful
quantum simulator, one which is not universal for quantum computing but which nonetheless helps us answer certain physics questions faster than our best available classical algorithms and hardware.
Congratulations to Markus!
Stability of Topological Order at Zero Temperature
From today’s quant-ph arXiv listing we find the following paper:
This is a substantial generalization of one of my favorite results from last year’s QIP, the two papers by Bravyi, Hastings & Michalakis and Bravyi & Hastings.
In this new paper, Michalakis and Pytel show that any local gapped frustration-free Hamiltonian which is topologically ordered is stable under quasi-local perturbations. Whoa, that’s a mouthful…
let’s try to break it down a bit.
Recall that a local Hamiltonian for a system of n spins is one which is a sum of polynomially many terms, each of which acts nontrivially on at most k spins for some constant k. Although this
definition only enforces algebraic locality, let’s go ahead and require geometric locality as well by assuming that the spins all live on a lattice in d dimensions and all the interactions are
localized to a ball of radius 1 on that lattice.
Why should we restrict to the case of geometric locality? There are at least two reasons. First, spins on a lattice is an incredibly important special case. Second, we have very few tools for
analyzing quantum Hamiltonians which are k-local on a general hypergraph. Actually, few means something closer to none. (If you know any, please mention them in the comments!) On cubic lattices, we
have many powerful techniques such Lieb-Robinson bounds, which the above results make heavy use of [1].
We say a Hamiltonian is frustration-free if the ground space is composed of states which are also ground states of each term separately. Thus, these Hamiltonians are “quantum satisfiable”, as a
computer scientist would say. This too is an important requirement, since it is one of the most general classes of Hamiltonians about which we have any decent understanding. There are several key
features of frustration-free Hamiltonians, but perhaps chief among them is the consistency of the ground space. The ground states on a local patch of spins are always globally consistent with the
ground space of the full Hamiltonian, a fact which isn’t true for frustrated models.
We further insist that the Hamiltonian is gapped, which in this context means that there is some constant γ>0 independent of the system size which lower bounds the energy of any eigenstate orthogonal
to the ground space. The gap assumption is extremely important since it is again closely related to the notion of locality. The spectral gap sets an energy scale and hence also a length scale, the
correlation length. For two disjoint regions of spins separated by a length L in the lattice, the connected correlation function for any pair or local operators decays exponentially in L.
The last property, topological order, can be tricky to define. One of the key insights of this paper is a new definition of a sufficient condition for topological stability that the authors call
local topological order. Roughly speaking, this new condition says that ground states of the local Hamiltonian are not distinguishable by any (sufficiently) local operator, except up to small effects
that vanish rapidly in a neighborhood of the support of the local operator. Thus, the ground space can be used to encode quantum information which is insensitive to local operators! Since nature
presumably acts locally and hence can’t corrupt the (nonlocally encoded) quantum information, systems with topological order would seem to be great candidates for quantum memories. Indeed, this was
exactly the motivation when Kitaev originally defined the toric code.
Phew, that was a lot of background. So what exactly did Michalakis and Pytel prove, and why is it important? They proved that if a Hamiltonian satisfying the above criteria is subject to a
sufficiently weak but arbitrary quasi-local perturbation then two things are stable: the spectral gap and the ground state degeneracy. (Quasi-local just means that strength of the perturbation decays
sufficiently fast with respect to the size of the supporting region.) A bit more precisely, the spectral gap remains bounded from below by a constant independent of the system size, and the ground
state degeneracy splits by an amount which is at most exponentially small in the size of the system.
There are several reasons why these stability results are important. First of all, the new result is very general: generic frustration-free Hamiltonians are a substantial extension of
frustration-free commuting Hamiltonians (where the BHM and BH papers already show similar results). It means that the results potentially apply to models of topological quantum memory based on
subsystem codes, such as that proposed by Bombin, where the syndrome measurements are only two-body. Second, the splitting of the ground state degeneracy determines the dephasing (T2) time for any
qubits encoded in that ground space. Hence, for a long-lived quantum memory, the smaller the splitting the better. These stability results promise that even imperfectly engineered Hamiltonians should
have an acceptably small splitting of the ground state degeneracy. Finally, a constant spectral gap means that when the temperature of the system is such that kT<<γ, thermal excitations are
suppressed exponentially by a Boltzmann factor. The stability results show that the cooling requirements for the quantum memory do not increase with the system size.
Ah, but now we have opened a can of worms by mentioning temperature… The stability (or lack there of) of topological quantum phases at finite temperature is a fascinating topic which is the focus of
much ongoing research, and perhaps it will be the subject of a future post. But for now, congratulations to Michalakis and Pytel on their interesting new paper.
[1] Of course, Lieb-Robinson bounds continue to hold on arbitrary graphs, it’s just that the bounds don’t seem to be very useful.
My Favorite D-Wave Future
As many of you know, D-Wave has a nice paper out about some experiments on one of their eight qubit systems. In addition they have sold one of their systems to the military industrial complex, a.k.a.
Lockheed Martin.
One of the interesting things about the devices they are building is that no one really knows whether it will provide computational speedup over classical computers. In addition to the questions of
whether adiabatic quantum algorithms will provide speedups for useful problems, there is also the question of how this speedup will be affected when working at finite temperature. If I were an
investor this would worry me, but as a scientist I find the question fascinating and hope they can continue to push their system in interesting directions. Of course if I were an investor I’d
probably be some multimillionaire who probably has an odd risk aversion profile 🙂
A fun question to ponder, at least for me, is what will eventually happen to D-wave, in, say, ten years. Of course there are the most obvious futures. They could run out of funding and close their
doors as a device maker and sell their patent porfolio. They could succeed and build machines that do outperform classical computers on relevant hard combinatorial problems. Those two are obvious.
But my favorite scenario is as follows. D-wave continues to build larger and larger devices. At the same time they perform even more exhaustive testing of their system. And in the process they
discover that there are “noise” sources that they hadn’t really expected. Not noise sources that violate quantum theory or anything, but instead noise sources that end up turning their stoquastic
Hamiltonian into a non-stoquastic Hamilotnian. While no one knows how to use the Hamiltonian of D-wave’s machine to build a universal quantum computer, it is entirely possible that such a machine,
plus some crazy extra unwanted terms could end up being universal. So while the company is squarely behind the dream of a combinatorial optimizer, it’s not at all impossible that their machine could
accidentally be useful for universal adiabatic quantum computation (and of course whether this can be made fault-tolerant is still a major open question, at least for the models with non-degenerate
ground states.) Wouldn’t it be hilarious if the noise which most people believe will destroy D-wave’s computational advantage actually turns their machine into a universal quantum computer? Ha!
So which will it be? And what odds will you give me on each of these possible futures?
dabacon.job = "Software Engineer";
Some news for the remaining five readers of this blog (hi mom!) After over a decade of time practicing the fine art of quantum computing theorizing, I will be leaving my position in the ivory (okay,
you caught me, really it’s brick!) tower of the University of Washington, to take a position as a software engineer at Google starting in the middle of June. That’s right…the Quantum Pontiff has
decohered! **groan** Worst quantum to classical joke ever!
Of course this is a major change, and not one that I have made lightly. There are many things I will miss about quantum computing, and among them are all of the people in the extended quantum
computing community who I consider not just colleagues, but also my good friends. I’ve certainly had a blast, and the only things I regret in this first career are things like, oh, not finding an
efficient quantum algorithm for graph isomorphism. But hey, who doesn’t wake up every morning regretting not making progress on graph isomorphism? Who!?!? More seriously, for anyone who is
considering joining quantum computing, please know that quantum computing is an extremely positive field with funny, amazingly brilliant, and just plain fun people everywhere you look. It is only a
matter of time before a large quantum computer is built, and who knows, maybe I’ll see all of you quantum computing people again in a decade when you need to hire a classical to quantum software
Of course, I’m also completely and totally stoked for the new opportunity that working at Google will provide (and no, I won’t be doing quantum computing work in my new job.) There will definitely be
much learning and hard work ahead for me, but it is exactly those things that I’m looking forward to. Google has had a tremendous impact on the world, and I am very much looking forward to being
involved in Google’s great forward march of technology.
So, onwards and upwards my friends! And thanks for all of the fish!
TEDxCaltech Videos
Coming online now: http://tedxcaltech.com/. See the Optimizer pontificate about P versus NP (now who can verify the Feynman quote at the beginning?)
and two prestigious professors goof around:
US Quantum Computing Theory CS Hires?
I’m trying to put together a list of people who have been hired in the United States universities in CS departments who do theoretical quantum computing over the last decade. So the requirements I’m
looking for are (a) hired into a tenure track position in a US university with at least fifty percent of their appointment in CS, (b) hired after 2001, and (c) they would say their main area of
research is quantum information science theory.
Here is my current list:
• Scott Aaronson (MIT)
• P. Oscar Boykin (University of Florida, Computer Engineering)
• Amit Chakrabarti (Dartmouth)
• Vicky Choi (Virginia Tech)
• Hang Dinh (Indiana University South Bend)
• Sean Hallgren (Penn State)
• Alexei Kitaev (Caltech)
• Andreas Klappernecker (Texas A&M)
• Igor Markov (Michigan)
• Yaoyun Shi (Michigan)
• Wim van Dam (UCSB)
• Pawel Wocjan (UCF)
Apologies to anyone I’ve missed! So who have I missed? Please comment!
Update: Steve asks for a similar list in physics departments. Here is my first stab at such a list…though it’s a bit harder because the line between quantum computing theorist, and say, AMO theorist
who studies systems that might be quantum computing is difficult.
Physicists, quantum computing theory,
• Lorenza Viola (Dartmouth)
• Stephen van Enk (Oregon)
• Alexei Kitaev (Caltech)
• Paolo Zanardi (USC)
• Mark Byrd (Southern Illinois University)
• Luming Duan (Michigan)
• Kurt Jacobs (UMass Boston)
• Peter Love (Haverford)
• Jon Dowling (LSU)
I’m sure I missed a lot hear, please help me fill it in.
Mythical Man 26 Years
This morning I was re-reading David Deutsch’s classic paper “Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer”, Proc. of the Roy. Soc. London A, 400, 97-117 (1985) This
is the paper where he explicitly shows an example of a quantum speedup over what classical computers can do, the first time an explicit example of this effect had been pointed out. Amusingly his
algorithm is not the one most people call Deutsch’s algorithm. But what I found funny was that I had forgotten about the last line of the article:
From what I have said, programs exist that would (in order of increasing difficulty) test the Bell inequality, test the linearity of quantum dynamics, and test the Everett interpretation. I leave
it to the reader to write them.
I guess we are still waiting on a program for that last problem?
QIP 2011 Open Thread
So what’s going on at QIP 2011? Anyone? Bueller? Bueller?
update: It looks like pdfs of the talk slides are available. Were the talks videotaped (err, I guess I’m showing my age: were the talks recorded in video format?)
more update: John Baez has a post on a few talks. | {"url":"https://dabacon.org/pontiff/category/quantum-computing/page/3/","timestamp":"2024-11-02T12:26:31Z","content_type":"text/html","content_length":"113995","record_id":"<urn:uuid:4f6ba29b-434d-4b2e-b14a-16d4c676f0ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00804.warc.gz"} |
IM Notes Chapter 3
CHAPTER 2
Learning Outcomes
At the end of the chapter, students should be
able to:
Understand construction and operation of
permanent magnet moving-coil (PMMC)
Describe how PMMC instruments are used
as galvanometers, dc ammeters, dc
voltmeters, ac ammeters, and ac voltmeters.
PMMC instrument consists basically of a lightweight coil
of copper wire suspended in the field of a permanent
magnet. Current in the wire causes the coil to produce a
magnetic field that interacts with the field from the
magnet, resulting in partial rotation of the coil. A pointer
connected to the coil deflects over a calibrated scale,
indicating the level of current flowing in the wire.
The PMMC instrument is essentially a low-level dc
It can be employed to measure a wide range of direct
current dc levels with the use of parallel-connected
The instrument may also be used as a dc voltmeter by
connecting appropriate-value resistors in series with the
Ohmmeters can be made from precision resistors, PMMC
instrument, and batteries.
Multirange meters are available that combine ammeter,
voltmeter, and ohmmeter functions in one instrument
Mutlirange Voltmeter
Mutlirange Ammeter
Ac ammeters and voltmeters can be constructed by using
rectifier circuits with PMMC instruments.
The Half wave rectifier
Deflection Instrument Fundamentals
A deflection instrument uses a pointer that moves over a
calibrated scale to indicate a measured quantity.
Three forces are operating in the electromechanical
mechanism inside the instrument:
Deflecting force
Controlling force
Damping force
Deflecting Force
The deflecting force causes the pointer to move from its
zero position when a current flows.
In the PMMC instrument, the deflecting force is magnetic.
The pointer is fixed to the coil
So, it moves over the scale as
The coil rotates
The deflecting force in the PMMC instrument is provided by
A current-carrying coil pivoted in a magnetic field.
Controlling Force
controlling force in the PMMC
instrument is provided by spiral
The springs retain the coil & pointer
at their zero position when no
current is flowing.
The coil and pointer stop rotating
when the controlling force becomes
equal to the deflecting force.
The spring material must be
nonmagnetic to avoid any magnetic
field influence on the controlling
The controlling force from
the springs balances the
deflecting force.
The springs are also used to make electrical connection to
the coil, they mush a low resistance( Phosphor bronze is the
material usually employed).
Damping Force
The damping force is required to
minimize (or damp out) oscillations of
the pointer and coil before settling
down at their final position.
The damping force must be present
only when the coil is in motion; thus it
must be generated by the rotation of
the coil.
In PMMC instruments, the damping
force is normally provided by eddy
Eddy currents induced in the coil set
up a magnetic flux that opposes the
coil motion, thus damping the
oscillations of the coil.
Damping Force
The damping force in a PMMC instrument is provided by
eddy currents induced in the aluminum coil former as it
moves through the magnetic field.
The methods of supporting the moving system of a deflection
1. Jeweled-bearing suspension
Cone-shaped cuts in jeweled ends
of shafts or pivots (mat be broken
by shocks).
low possible friction.
Some jewel bearings are spring
supported to absorb such shocks.
2. Taut-band method
Much tougher than jeweled-bearing.
Two flat metal ribbons (phosphor bronze or platinum alloy) are
held under tension by spring to support the coil.
Because of the spring, the metal ribbons behave like rubber under
tension. Thus, the ribbons also exert a controlling force as they
The metal ribbons can be used as electrical connections to the
moving coil.
Much more sensitive than the jeweled-bearing type because there
is less friction.
Extremely rugged, not easily be shattered.
PMMC Construction
The main feature is a permanent magnet with two soft-iron
pole shoes.
A cylindrical soft-iron core is positioned between the core and
the faces of the pole shoes.
The lightweight moving coil pivoted to move within these narrow
air gaps.
The air gaps are made as narrow as possible in order to have
the strongest possible level of magnetic flux crossing the gaps.
The current in the coil of a PMMC instrument must flow in one
particular direction to cause the pointer to move (positively)
from the zero position over the scale.
Because the PMMC is polarized, it can be used directed to
measure alternating current.
PMMC Construction
D’Arsonval or
horseshoe magnet
Torque Equation and Scale
When a current I flows
through a one-turn coil
situated in a magnetic
field, a force F is exerted
on each side of the coil
F (BIl ) N newtons
where N is the number of turns
Total force on each side of the coil of N turns:
F 2 B I l N newtons
The force on each side acts at a radius r, producing a deflecting
TD 2 B l I N r ( N .m)
B l I N ( 2r )
TD B l I N D
T D BlIN 2 r BlIND BA IN
A=l.D, Where D is the coil diameter
The controlling torque exerted by the spiral springs is directly
proportional to the deformation or windup of the springs.
Thus, the controlling torque is proportional to the actual angle
of deflection of the pointer.
TC K
where K is a constant
For a given deflection, the controlling and deflecting torques
are equal:
K BlIND
I CI
whereC is a constant
equation shows that the pointer deflection is always
proportional to the coil current.
Consequently, the scale of the instrument is linear, or uniformly
A PMMC instrument with a 100-turn coil has a magnetic
flux density in its air gaps of B = 0.2 T. The coil
dimensions are D = 1 cm and l = 1.5 cm. Calculate the
torque on the coil for a current of 1 mA.
TD B l I N D
0.2 (1.5 10 2 ) 0.001 100 (1 10 2 )
TD 3 10 6 N .m
It is a PMMC instrument designed to be sensitive to
extremely low current levels.
The simplest galvanometer is a very sensitive
instrument with the type of center-zero scale.
• Galvanometers are often employed to detect zero
current or voltage in a circuit rather than to measure
the actual level of current or voltage.
• The most sensitive moving-coil galvanometer use tautband suspension, and the controlling torque is
generated by the twist in the suspension ribbon.
For the for greatest sensitivity, the weight of the pointer
can create a problem. The solution is by mounting a small
mirror on the moving coil instead of a pointer
Basic deflection system of
a galvanometer using a light beam
• An adjustable shunt resistor is employed to protect the coil
of a galvanometer from destructively excessive current
• The shunt resistance is initially set to zero, then gradually
increased to divert current through the galvanometer.
A galvanometer has a current sensitivity of 1 A/mm and
a critical damping resistance of 1 k. Calculate (a) the
voltage sensitivity and (b) the megaohm sensitivity.
Voltage sensitivity 1 kΩ 1 A
1 mV/mm
For a voltage sensitivity of 1 V/mm,
megohm sensitivity
1 V/mm
1 MΩ
1 A
DC Ammeter
An ammeter is always connected in series with
a circuit in which current is to be measured.
To avoid affecting the current level in the circuit,
the ammeter must have a resistance much lower
than the circuit resistance.
For larger currents, the instrument must be modified so
that most of the current to be measured is shunted (a
very low shunt resister) around the coil of the meter.
Only a small portion of the current passes through the
moving coil.
A dc ammeter consists of a PMMC instrument and a lowresistance shunt.
V sh V m
I sh R sh I m R m
R sh
Im Rm
I sh
I sh I I m
R sh
Im Rm
I Im
Example 3
A PMMC instrument has FSD of 100 A and a coil resistance of 1 k.
Calculate the required shunt resistance value to convert the instrument
into an ammeter with (a) FSD = 100 mA and (b) FSD = 1 A.
(a) FSD = 100 mA
Vm I m R m 100 μ A 1 kΩ 100 mV
I Is Im
I s I I m 100 mA 100 μ A 99.9 mA
Vm 100 mV
1.001 Ω
99.9 mA
(b) FSD = 1 A
V m I m R m 100 mV
I s I I m 1 A 100 μ A 999.9 mA
100 mV
0.1001 Ω
999.9 mA
Example: An ammeter has a PMMC instrument with a coil resistance of Rm
= 99 and FSD current of 0.1 mA. Shunt resistance Rs = 1. Determine
the total current passing through the ammeter at :
a) FSD,
b) 0.5 FSD, and
c) 0.25 FSD
(a) At FSD
meter voltage V m I m R m
0.1mA 99Ω 9.9mV
I s R s V m
V m 9.9 mV
I I s I m 9.9mA 0.1mA
(b) At 0.5 FSD
I m 0.5 0.1 m A 0.05 m A
V m I m R m 0.05 m A 99 4.95 m V
total current
V m 4.95 m V
4.95 m A
I I s I m 4.95 m A 0.5 m A 5 m A
(c) At 0.25 FSD
I m 0.25 0.1 mA 0.025 mA
V m I m R m 0.025 mA 99 2.475 mV
V m 2.475 mV
2.475 mA
total current
I I s I m 2.475 mA 0.025 mA 2.5 mA
Thus, the ammeter scale may be calibrated linearly from zero to
Ammeter Swamping Resistance
The moving coil in a PMMC instrument is wound with thin
copper wire, and its resistance can change significantly when its
temperature changes.
The heating effect of the coil current may be enough to produce
a resistance change, which will introduce an error.
• To minimize this error, a swamping
resistance made of manganin or
constantan is connected in series
with the coil:
(manganin and
temperature coefficients very close
to zero).
• The ammeter shunt must also be made of manganin or constantan
to avoid shunt resistance variations with temperature.
Multirange Ammeters
Make-before-break switch
The instrument is not left without a
shunt in parallel with it even for a brief
instant. If this occurred, the high
resistance of the instrument would
affect the current flowing in the circuit.
During switching there are actually
two shunts in parallel with the
Ayrton Shunt
The figure shows another method
of protecting the deflection
instrument of an ammeter from
excessive current flow when
Resistors R1, R2, and R3 constitute
an Ayrton Shunts
• Internal Ammeter Resistance: Rin
R in
R in R
/ /R sh
R m* R sh
R m R sh
Example: A PMMC instrument has a three-resistor Ayrton
shunt connected across it to make an ammeter as shown in the
figure. The resistance values are R1 = 0.05, R2 = 0.45 and
R3 = 4.5. The meter has Rm = 1k and FSD = 50A.
Calculate the three ranges of the ammeter.
Switch at contact B
V s I m Rm 50μA1kΩ 50mV
R1 R2 R3 0.05Ω 0.45Ω 4.5Ω
I I m I s 50μA10mA
Switch at contact C
Vs I m Rm R3 50 μA1 kΩ 4.5 Ω 50 mV
50 mV
100 mA
R1 R2 0.05 Ω 0.45 Ω
I I m I s 50 μA 100 mA
100.05 mA
Switch at contact D
Vs I m Rm R3 R2 50μ0 1kΩ 4.5Ω 0.45Ω 50mV
Vs 50mV
R1 0.05Ω
I I m I s 50μ0 1A
Accuracy and Ammeter Loading Effects
• Internal resistance of ideal ammeter is zero Ohm, but in practice, the
internal resistance has some values which affect the measurement
• This error can be reduced by using higher range of measurement.
Let us calculate the relationship between the true value and the
measured value
I T (true value )
V Th
V Th
I m ( measured value )
RTh R in
R T h R in
% Acc m 100%
R T h R in
dc circuit with source
and resistors
dc circuit with source
and resistors
Example: For a DC Circuit as shown in the
figure, given R1=2k, R2=2k with voltage
of 2V. By measuring the current flow through
R3 with a dc ammeter with internal resistance
of Rin = 100Ω, calculate percentage of
accuracy and percentage of error.
R T h R 1 / / R 2 R 3 2 kΩ
V Th
R1 R 2
2 kΩ 1V
2 kΩ 2 kΩ
V Th
RTh 2 kΩ
V Th
RTh R in 2 kΩ 100 Ω
% Acc
476.19 μA
500 μA
% Error 1 % A cc 1 95.24% 4.76%
DC Voltmeter
• The deflection of a PMMC
instrument is proportional to the
current flowing through the moving
coil. The coil current is directly
proportional to the voltage across the
• The coil resistance is normally quite small,
and thus the coil voltage is also usually
very small. Without any additional series
(multiplier resistance) resistance the
PMMC instrument would only measure
very low voltage.
• The voltmeter range is easily increased by
connecting a resistance in series with the
The meter current is directly proportional to the applied voltage,
so that the meter scale can be calibrated to indicate the voltage.
• The voltmeter range is increased by connecting a multiplier
resistance with the instrument (single or individual type of extension
of range).
RV R s R m
V I m RV I m R s I m R m
V R m
• Last equation can be used to select the multiplier resistance value (Rs)
for certain voltage range (FSD). In this case Im will be the full scale
• A multiplier resistance that is nine times the coil resistance will increase
the voltmeter range by a factor of 10 (multiplier resistance + coil
• The voltmeter sensitivity (S) is defined as the total voltmeter resistance
(internal resistance Rin) divided by the voltage range (full scale)
A PMMC instrument with FSD of 100 A and a coil resistance of 1k is to
be converted into a voltmeter. Determine the required multiplier
resistance if the voltmeter is to measure 50V at full scale and Voltmeter
sensitivity. Also calculate the applied voltage when the instrument
indicate 0.8, 0.5, and 0.2 of FSD.
• At 50 V FSD
V I m RV I m R s I m R m
I m 100 μA
50 V
1 kΩ 499 kΩ
100 μA
• Since the voltmeter has a total resistance of Rv = Rs+Rm = 500k , then
its resistance per volt or sensitivity is 500k / 50V =10 k / V.
• At 0.8 of FSD
I m 0.8 100 μA 80 μA
V I m Rs R m
80 μA 499 kΩ 1kΩ 40 V
• At 0.5 of FSD
I m 50 μA
V 50 μA 499 kΩ 1kΩ 25 V
• At 0.2 of FSD
I m 20μA
V 20μA 499 kΩ 1kΩ 10 V
Thus, the voltmeter scale may be calibrated linearly from zero to 50 V
Voltmeter Swamping Resistance
As in the case of ammeter, the change in coil resistance (Rm) with
temperature change can introduce errors in a PMMC voltmeter.
• The presence of the voltmeter multiplier resistance (Rs) tends to swamp
coil resistance changes, except for low voltage ranges where Rs is not very
much larger than Rm.
In some cases it might be necessary to construct the multiplier
resistance form manganin or constantan.
Multirange Voltmeters
Individual or series connected resistors may be used as shown in the following
Multirange voltmeter with ranges V = Im(Rm+R)
where R can be R1, R2, or R3
Multirange series voltmeter with ranges V = Im(Rm+R) ,
where R can be R1, R1 +R2, or R1 +R2 + R3
A PMMC instrument with FSD = 50A and Rm = 1700 is to be employed
as a voltmeter with ranges of 10V, 50V, and 100V. Calculate the required
values of multiplier resistors for the two circuits shown in the previous
For the circuit shown
Switch contact at
Rm +R1 =
R1 =
- Rm
10 V
1700 Ω 200 kΩ 1.7 kΩ
50 μA
198.3 kΩ
Switch contact at
50 A
998.3 k
Switch contact at
1700 Ω
50 A
1.9983 M
For the circuit shown
Switch contact at
R m R1 1
10 V
R1 1 R m
1700 Ω 198.3 kΩ
50 μA
Switch contact at
50 V
R1 R m
198.3 kΩ 1700 Ω 800 kΩ
50 μA
R m R1 R 2
Switch contact at
R m R1 R 2 R 3
100 V
R 2 R1 R m
800 kΩ 198.3 kΩ 1700 Ω 1 M Ω
50 μA
Accuracy and Voltmeter Loading Effect
Let us calculate the
relationship between the
true value (VT) and the
measured value (Vm)
V T V Th
V Th
R in
R in RTh
R in
V T R in RTh
R in
R in RTh
A voltmeter with sensitivity of 20kΩ/V is used for
measuring a voltage across R2 with range of 50V as
shown in the figure below.
Calculate a) reading voltage. b) accuracy of
measurement. c) error of measurement
V Th V T
200k 50
RTh R1 / / R 2 200k / / 200k 100k
R in S Range
R in
R in RTh
A ccuracy
20 k
50V 1M
50V 45.45V
1M 100k
% A cc 90.9%
E rro r 1 A cc 1 0 .9 0 9 0 .0 9 1
% E rro r 9 .1 %
AC Voltmeter
Full-Wave Rectifier Voltmeter
Half-Wave Rectifier Voltmeter
Half-Wave Full Bridge Rectifier Voltmeter
AC Ammeter and Voltmeter
When an alternating current
(sinusoidal ) with a very low frequency
(0.1 Hz or lower) is passed through a PMMC instrument, the pointer
tends to follow the instantaneous level of the AC.
As the current grows positively, the pointer deflection increases to a maximum at
the peak of the ac.
Then as the instantaneous current level falls, the pointer deflection decreases
towards zero.
When the ac goes negative, the pointer is deflected (off-scale) to the left of zero.
the normal 50 Hz or higher supply frequencies, the damping
mechanisms and the inertia of the meter movement prevent the pointer
from following the changing instantaneous levels of the signal.
The instruments pointer settles at the average value of the current flowing
through the moving coil which is zero.
PMMC instrument can be modified by one of the following circuits to
measure AC signals
1. Full-Wave Bridge Rectifier Voltmeter
When the input is positive, diodes D1 and D4 conduct,
causing current to flow through the meter from top to
bottom ( red solid path).
When the input goes negative, diodes D2 and D3 conduct,
current flows through the meter from the positive to the
negative terminal ( blue dashed path).
The ac voltmeter uses a series-connected multiplier
resistor (Rs) to limit the current flow through the
The meter deflection is proportional to the average
current (Iav), which is 0.637 × peak current (Ip or
But the actual current (or voltage) to be indicated in ac measurement is normally
the Irms = 0.707 × Ip (or Im) .
( note Irms = 1.11 × Iav
Ip (or Im) = 1.414 × Irms )
When other than pure sine waves are applied, the voltmeter will not
indicate the rms voltage.
applied peak voltage (V p ) - rectifiers voltage drop
totat circuit resistance
1.414V rms 2V F
Rs Rm
VF is rectifier voltage drops for D1 and D2 or D3 and D4
Peak current Im = Iav/0.637 = IFSD/0.637
A PMMC instrument with FSD = 100 µA and Rm = 1 kΩ is to be employed
as an ac voltmeter with FSD = 200 V (rms). Silicon diodes with VF = 0.7 V
are used in the bridge rectifier circuit of shown above. Calculate: (a) the
multiplier resistance value required, (b) the pointer indications the rms
input voltage is (i)100V and (ii) 50 V.
At FSD, the average current flowing
through the PMMC instrument is 100µA
I m av 157 A
V p 1.414V rms 1.414 200 282.8 V olt
V p 2V F
Rs Rm
(b) i-for the rms input voltage is 100V
V p 2V F
(282.8 1.4)V
1k 1791.36 k
(157 10 )mA
V p \ 2V F
Rs R m
1.414 100 1.4
78.11 A
(1791.36 1)k
I av \ 0.607 I m \ 49.76 A 50 A 0.5FSD
ii- Similarly, for the rms input voltage is 50V
I av \ \ 25 A 0.25 FSD
2. Half-Wave Rectifier Voltmeter
RSH shunting the meter is included to cause a relatively large
current to flow through diode D1 (larger than the meter current)
when the diode is forward biased. This is ensure that the diode is
biased beyond the knee and will into the linear range of its
Diode D2 conducts during the negative half-cycles of the input.
When conducting, D2 cause a small voltage drop across D1 and
the meter, thus preventing the flow of any significant reverse
leakage current ( and reverse voltage) through the meter via D1.
In the half wave rectifier
Half-Bridge Full-Wave Rectifier Voltmeter
During the positive half-cycle of the input, diode D1 is forward biased and D2 is
reverse biased. Current flows from terminal 1 through D1 and the meter and
then through R2 to terminal 2. but R1 is in parallel with the meter and R2.
Therefore, much of the current flowing in D1 pass through R1 while only part
of it flows through the meter and R2
During the negative half-cycle, diode D2 is forward biased and D1 is reverse
biased. Current flows from terminal 2 through R1 and the meter and then
through D2 to terminal 1. New, R2 is in parallel with the series connected meter
and R1
This arrangement forces the diodes to
operate beyond the knee of their
compensate for differences that might
occur in the characteristics of D1 and
D2 .
AC Ammeter
Like a dc ammeter, an ac ammeter must have a very low
resistance because it is always connected in series with
the circuit in which current is to be measured.
This low-resistance requirement means that the voltage
drop across the ammeter must be very small, typically not
greater than 1000 mV.
The voltage drop across a diode is 0.3 to 0.7 V.
The use of a current transformer gives the ammeter a low
terminal resistance and low voltage drop.
A. Series Ohmmeter
Basic Circuit and Scale
simplest circuit consists of a voltage
source (Eb) connected in series with a pair
of terminals (A & B), a standard resistance
(R1), and a low-current PMMC instrument.
• The resistance to be measured (Rx) is connected across terminal A and B.
• The meter current
I m E b (R x + R1 + R m )
• When the ohmmeter terminals are shorted (Rx= 0) meter full-scale deflection occurs.
IFSD = Eb / (R1 + Rm)
• At half-scale deflection Rx = R1 + Rm
• At zero deflection the terminals are
open-circuited (Rx = ∞).
Example : The series ohmmeter shown in Figure is
made up of a 1.5V battery, a 100 µA meter, and a
resistance R1 which makes (R1 + Rm) =15kΩ.
a) Determine the instrument indication when Rx = 0.
b) Determine how the resistance scale should be
marked at 0.75 FSD, 0.5 FSD and 0.25 FSD
a) I m
100 A FS D
R x R1 R m
0 15 k
b ) A t 0.75 FSD : I m
3 100 A
75 A
R x R1 R m
R1 R m
15k 5k
75 A
At 0.5 FSD : I m
100 A
50 A
15k 15k
50 A
A t 0.25 FSD : I m
100 A
25 A
15k 45k
25 A
The ohmmeter scale is now marked
as shown in the figure. It is clear that
the ohmmeter scale is nonlinear.
Comments: disadvantages of simple series ohmmeter
The simple ohmmeter described in last example will operate satisfactorily as
long as the battery voltage remains exactly at 1.5V. When the battery voltage
falls, the instrument scale is no longer correct.
Although R๑ were adjusted to give FSD when terminals A and B are shortcircuited, the scale would still be in error because now mid-scale would
represent a resistance equal to the new value of R1 + Rm.
Ohmmeter with Zero Adjust
Falling battery voltage can be taken care by an
adjustable resistor (R2) connected in parallel
with the meter.
• With terminals A and B short-circuited, the total
circuit resistance is R1 + (R2 // Rm).
• Since R1 is always very much larger than R2 // Rm,
the total circuit resistance can be assumed to
equal R1
Ib =
R x + R1 + R 2 \\R m
R x + R1
A lso , the meter voltage is V m = I b R 2 \\R m which give meter current as I m =
I b R 2 \\R m
• When Rx equal to R1 the circuit resistance is doubled and the circuit current is halved. This
cause both I2 and Im to be reduced to half of their previous level. Thus the mid-scale measured
resistance is again equal to R1.
• Each time the ohmmeter is used, terminals A and B are first short circuited, and R2 is
adjusted for zero-ohm indication on the scale
• The series ohmmeter can be converted to a multi-range ohmmeter by
employing several values of standard resistance R1 and a rotatory
The major inconvenience of such a circuit is that a large adjustment of
the zero control (R2) would have to be made every time the resistance
range (R1) is changed.
An ohmmeter as shown in the figure with
Eb =
1.5V, R1 = 15kΩ, Rm = R2 = 50Ω and IFSD = 50µA.
Calculate, (a) Rx at 0.5FSD, (b) when Eb = 1.3V
what is the value of R2 to get full-scale current and
(c) when Eb = 1.3V what is the value of Rx at halfscale current.
Since (Rm // R2) = 25 << R1 then at half scale Rx = R1 = 15k independent of Eb
V m I m R m 25 A 50 1.25 m V
1.25 m V
25 A
I b I 2 I m 25 A 25 A 50 A
R x R1
30 k
50 A
R 1 30 k 15 k 15 k
86.67 A
R x R1 0 15k
I 2 I b I m FSD 86.67 A 50 A 36.67 A
V m I m FSD R m 50 A 50 2.5mV
I 2 36.67 A
I m R m 2 5 A 5 0 1 .2 5 m V
1 .2 5 m V
1 8 .3 3 A
6 8 .1 8
I b I 2 I m 1 8 .3 3 A 2 5 A 4 3 .3 3 A
R x R1
1 .3V
4 3 .3 3 A
R x 30k R 1 30k 15k
R x 15k
Since (Rm // R2) =
28.85 << R1
then at half scale
Rx = R1 = 15k
independent of Eb
B. Shunt Ohmmeter
– Basic Circuit and Scale
• The simplest circuit consists of a voltage source (E)
connected with an adjusted Resistor (RAdj) and a
low-current PMMC instrument.
• The resistance to be measured (Rx) is connected across terminal A and B.
− When Rx = 0, short circuit between A and B, there will be no current
flow in the coil branch and the scale point at zero on the left hand side.
− When Rx = ∞ , open circuit between A and B. Then adjust RAdj to get
FSD. The meter will point infinity at the right of the scale.
I m I FS D
R A dj R m
− For any Rx we have,
ER x
R A dj R m R x R A dj R m
• Scale of shunt ohmmeter is opposite to the scale
of series ohmmeter when connecting with Rx | {"url":"https://studylib.net/doc/25232019/im-notes-chapter-3","timestamp":"2024-11-04T14:50:57Z","content_type":"text/html","content_length":"82955","record_id":"<urn:uuid:b35b73e8-5680-4a91-bef9-9a1d895dce1e>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00000.warc.gz"} |
Developmental Education FAQs
Facts and stats on commonly asked questions about dev ed
Before students set foot in a classroom, most colleges require them to take a placement test to determine if they are eligible for college-level math and English courses. If they aren’t, they are
placed into developmental education courses to strengthen their skills.
What do we know about who developmental education students are and what happens to them?
The Students in Developmental Education
The majority of students at community colleges take developmental courses, as do a large percentage of students at four-year colleges. Black, Hispanic, and low-income students are disproportionately
likely to be assigned to developmental education.
How many students are referred to and enroll in remediation?amazzariello2021-01-15T14:54:28-05:00
How do rates differ by race and income?amazzariello2021-01-15T15:14:34-05:00
Percentage of Students Taking Remedial Courses at Community Colleges
Percentage of Students Taking Remedial Courses at Public Four-Year Colleges
The Road to the Developmental Classroom
Most community colleges administer placement tests to determine if students need developmental education, and many rely on placement test scores as the sole measure of college readiness, despite
evidence that they result in too many students being placed in dev ed. More colleges are using multiple measures, including high school GPA, in an effort to improve placement decisions.
How many colleges use placement exams to determine whether students need remediation?amazzariello2021-10-28T09:33:51-04:00
99% of public two-year colleges surveyed in 2016 used math placement tests.
98% of public two-year colleges used reading and writing placement tests.
94% of public four-year colleges used standardized tests for placement in math.
91% of public four-year colleges used standardized tests for reading and writing placement.
What do we know about how well placement exam scores reflect students’ college readiness?amazzariello2021-10-28T09:36:35-04:00
CAPR’s study of multiple measures placement in seven community colleges in the State University of New York (SUNY) system found that students placed using multiple measures were 7 percentage points
more likely to be placed into college-level math and 34 percentage points more likely to be placed into college-level English than their peers evaluated using placement tests alone. The students
placed using multiple measures completed college-level courses at the same or better rates.
A study of one statewide community college system found that 33 percent of entering students were severely misplaced by ACCUPLACER—either “overplaced” in college-level courses or
“underplaced” in remedial courses when they could have earned a B or better in a college-level course. Using students’ high school GPA instead of placement testing to make placement decisions
was predicted to cut severe placement error rates in half (to 17 percent). Subsequent research has found that even students deemed overplaced do better when allowed to take college-level courses
rather than prerequisite developmental courses.
How many community colleges use more than one measure to determine whether students need remediation?amazzariello2021-10-28T09:37:13-04:00
57% of public two-year colleges surveyed in 2016 used multiple measures for math placement, up from 27% in 2011.
51% of public two-year colleges used multiple measures for reading and writing placement, up from 19% in 2011 for reading placement alone.
Traditional Remediation: What Happens to Students?
Among the students assigned to developmental education, many don’t even enroll. And even when they do, many don’t finish their assigned developmental course sequence—or don’t complete their first
college-level course or go on to graduation. One recent analysis found that developmental education is most helpful for students with the lowest levels of preparation.
How many community college students complete their remedial requirements?amazzariello2017-11-27T14:47:31-05:00
Community Colleges
One recent study using nationally representative data reported that 49 percent of developmental education students who started in 2003–04 completed all the developmental courses they attempted, 35
percent completed some courses, and 16 percent completed none.
Another study looking at community colleges in seven states found that 33 percent of students referred to developmental math and 46 percent of students referred to developmental reading went on to
complete the entire developmental sequence. Completion rates differed based on students’ initial placement level:Â 17 percent of students referred to the lowest level of developmental math completed
the sequence, versus 45 percent of those referred to the highest level.
Public Four-Year Colleges
The same national study cited above found that 59 percent of students assigned to developmental education completed all their courses, 25 percent completed some, and 15 percent completed none.
How many students referred to developmental education go on to complete entry-level college courses?amazzariello2017-11-01T10:16:50-04:00
One multistate study found that 20 percent of community college students referred to developmental math and 37 percent of students referred to developmental reading who made it through the courses
went on to pass the relevant entry-level or “gatekeeper” college course. An additional 12 percent of those referred to developmental math and an additional 32 percent of those referred to
developmental reading skipped the developmental courses but still passed a gatekeeper course.
Do developmental education courses help students succeed in college?amazzariello2021-01-19T12:56:34-05:00
Findings from a nationally representative study suggest that students who complete their developmental courses are more likely than partial completers or noncompleters to stay in college and earn a
bachelor’s degree—but the results vary depending on students’ level of academic preparation. Dev ed helps weakly prepared students on several indicators. But moderately or strongly prepared
community college students who complete some of their developmental courses are worse off than similar students who take no remedial courses in terms of college-level credits earned, transfer to a
four-year college, and bachelor’s degree attainment. Other studies have found little or no positive effect from enrolling in developmental courses.
How many students who take developmental education courses go on to complete a college degree?amazzariello2021-01-19T14:17:09-05:00
Started at a Community College
Started at a Public Four-Year College
Started at a Private Nonprofit Four-Year College
Reformed Remediation: What Happens to Students?
In reaction to everything research has told us about traditional developmental education, colleges, systems, and states are experimenting with ways to shorten the time students spend in dev ed and
tailor developmental coursework to their program of study. The reforms include corequisite remediation, math pathways, modularization of math courses, compressed courses, and the use of multiple
measures for placement. Evidence is building that some of these reforms improve student outcomes.
How many community colleges use traditional sequences of developmental courses and how many use reformed models for at least some of their developmental courses?amazzariello2021-10-28T09:39:53-04:00
Prevalence of Developmental Education Instructional Methods Among Public Two-Year Colleges
Prerequisite sequence: 76%
Multiple math pathways: 41%
Prerequisite sequence: 53%
Integrated reading and writing: 52%
Values represent percentages among two-year public colleges that reported offering developmental courses. Colleges were counted as using an instructional method if they used it in more than two
course sections. Categories are not mutually exclusive.
Multiple math pathways are sets of linked courses designed to give students math skills relevant to their degree requirements and program of study. Self-paced courses allow students to work through
course content independently. In the flipped classroom model, students are exposed to content outside of class, often through online materials, while most in-class time is devoted to activities,
projects, and discussions. Corequisite courses involve students taking a college-level course concurrently with a developmental course that serves as a learning support. Integrated reading and
writing courses are English courses in which reading and writing skills are taught together.
SOURCE: Rutschow and Mayer (2018)
What do we know about how math pathways courses affect the success of students?stacie2021-10-28T09:40:42-04:00
A CAPR study of Dana Center Mathematics Pathways (DCMP) at four community colleges in Texas found that, after three semesters, students assigned to DCMP were 8 percentage points more likely to pass a
developmental math course and almost 24 percentage points more likely to complete their developmental math sequence than students assigned to standard developmental math. DCMP students were also more
likely to pass college-level math by the end of the study period.
Passed Developmental Math
59% of students assigned to DCMP
51% of students assigned to standard developmental math
Completed Developmental Math Sequence
57% of students assigned to DCMP
34% of students assigned to standard developmental math
Passed College-Level Math
25% of students assigned to DCMP
19% of students assigned to standard developmental math
What do we know about whether corequisite remediation helps students finish developmental education and succeed in college-level courses?amazzariello2021-01-19T19:30:28-05:00
Corequisite remediation, in which students take college-level math or English courses coupled with a parallel developmental class or other academic supports, allows more students to pass the
college-level course, and to do it faster.
An experimental evaluation of corequisite English in five Texas community colleges found that being assigned to corequisite remediation instead of a traditional semester-long developmental education
course increased the probability of passing Composition I and later English courses. Students in corequisite courses were:
21.4 percentage points more likely to pass Composition I within one year
16.3 percentage points more likely to pass Composition I within two years
6.4 percentage points more likely to complete Composition II within two years
A quasi-experimental study of corequisite remediation in the 13 Tennessee community colleges found that students scoring just below the college readiness threshold were more likely to pass gateway
math and English when placed into corequisite remediation than when placed into prerequisite remediation. The former were:
15 percentage points more likely to pass gateway math in one year
13 percentage points more likely to pass gateway English in one year
What does the research say about using multiple measures for placement rather than standardized placement tests alone?amazzariello2021-10-28T09:41:28-04:00
Multiple measures placement systems may combine information on test scores, high school GPA, highest course passed in the subject, and other measures to determine whether students are ready for
college courses. CAPR’s random assignment study of multiple measures placement in seven State University of New York community colleges found that many more students are assigned to college-level
courses. Gains in completion of college-level math faded, but in English the effects were much larger and lasted through at least three terms.
Students Placed Using Multiple Measures
Students Placed Using Standardized Tests
The study showed much more substantial effects for students whose placements changed with the new placement methods. Regardless of the prediction of the placement algorithm, students allowed to take
college-level courses were much better off.
Students Bumped Up Into College-Level Courses
8–10 percentage points more likely to complete a college-level math or English course within three semesters than students who stayed in developmental courses
Students Bumped Down Into Developmental Courses
8–10 percentage points less likely to complete a college-level math or English course within three semesters than students who stayed in college-level courses | {"url":"https://postsecondaryreadiness.org/developmental-education-faqs/","timestamp":"2024-11-12T23:23:05Z","content_type":"text/html","content_length":"324585","record_id":"<urn:uuid:7f3f303e-abe5-4c44-811d-dfd5ddc1681d>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00347.warc.gz"} |
R. HAKL, A. LOMTATIDZE, AND I. P. STAVROULAKIS Received 24 July 2003
Theorems on the Fredholm alternative and well-posedness of the linear boundary value problemu^(t)=(u)(t) +q(t),h(u)=c, where:C([a,b];R)→L([a,b];R) andh:C([a, b];R)→Rare linear bounded operators,q∈L
([a,b];R), andc∈R, are established even in the case whenis not astrongly boundedoperator. The question on the dimension of the solution space of the homogeneous equationu^(t)=(u)(t) is discussed as
1. Introduction
The following notation is used throughout:Nis the set of all natural numbers;Ris the set of all real numbers,R+=[0, +∞[; Ent(x) is an entire part ofx∈R;C([a,b];R) is the Banach space of continuous
functionsu: [a,b]→Rwith the normuC=max{|u(t)|: t∈[a,b]};C([a,b];R+)= {u∈C([a,b];R) :u(t)≥0 fort∈[a,b]};C([a,^ b];R) is the set of absolutely continuous functionsu: [a,b]→R;L([a,b];R) is the Banach
space of Lebesgue integrable functions p: [a,b]→Rwith the normpL=[b]
R+)= {p∈L([a,b];R) :p(t)≥0 fort∈[a,b]}; mesAis the Lebesgue measure of the set A;ᏹabis the set of measurable functionsτ: [a,b]→[a,b];ᏸabis the set of linear bounded operators:C([a,b];R)→L([a,b];R);ᏸ^
ab is the set of linear strongly bounded opera- tors, that is, for each of the operators∈ᏸab, there existsη∈L([a,b];R+) such that
(v)(t)^≤η(t)vC fort∈[a,b],v∈C^[a,b];R^; (1.1) ᏼabis the set of linear nonnegative operators, that is, operators∈ᏸabmapping the set C([a,b];R+) into the setL([a,b];R+). If∈ᏸab, then =sup{(v)L:vC≤1}.
Lett0∈[a,b]. We will say that∈ᏸab is at0-Volterra operator if for arbitrarya1∈ [a,t0],b1∈[t0,b], andu∈C([a,b];R) such that
u(t)=0 fort∈ a1,b1
, (1.2)
we have
(u)(t)=0 fort∈ a1,b1
. (1.3)
Copyright©2004 Hindawi Publishing Corporation Abstract and Applied Analysis 2004:1 (2004) 45–67 2000 Mathematics Subject Classification: 34K06, 34K10 URL:http://dx.doi.org/10.1155/S1085337504309061
On the segment [a,b], consider the boundary value problem
u^(t)=(u)(t) +q(t), (1.4)
h(u)=c, (1.5)
where∈ᏸab,h:C([a,b];R)→Ris a linear bounded functional,q∈L([a,b];R), and c∈R.
By a solution of (1.4) we understand a functionu∈C([a,b]; R) satisfying the equality (1.4) almost everywhere on [a,b]. By a solution of the problem (1.4), (1.5), we understand a solutionuof (1.4)
which also satisfies the condition (1.5). Together with (1.4), (1.5), we will consider the corresponding homogeneous problem
u^(t)=(u)(t), (1.6)
h(u)=0. (1.7)
From the general theory of boundary value problems for functional differential equa- tions, it is known that if∈ᏸab, then the problem (1.4), (1.5) has a Fredholm property (see, e.g., [1,2,7,8,10]).
More precisely, the following assertion is valid.
Theorem1.1. Let∈ᏸab. Then the problem (1.4), (1.5) is uniquely solvable if and only if the corresponding homogeneous problem (1.6), (1.7) has only the trivial solution.
Theorem 1.1allows us to introduce the following definition.
Definition 1.2. Let ∈ᏸab and let the problem (1.6), (1.7) have only the trivial solu- tion. An operatorΩ:L([a,b];R)→C([a,b];R) which assigns to everyq∈L([a,b];R) a solutionuof the problem (1.4),
(1.7) is called Green operator of the problem (1.6), (1.7).
It follows fromTheorem 1.1that if∈ᏸaband the problem (1.6), (1.7) has only the trivial solution, then the Green operator is well defined. Evidently, Green operator is lin- ear. Moreover, the
following theorem is valid (see, e.g., [1,2,7,8]).
Theorem1.3. Let∈ᏸab and let the problem (1.6), (1.7) have only the trivial solution.
Then the Green operator of the problem (1.6), (1.7) is a linear bounded operator.
In [7,8] the question on the well-posedness of linear boundary value problem for systems of functional differential equations is studied.Theorem 1.3can also be derived as a consequence of more general
results on well-posedness obtained therein.
Note that both Theorems1.1and1.3claim that∈ᏸab. This condition covers a quite wide class of linear operators; for example, the equation with a deviating argument
u^(t)=p(t)u^τ(t)^+q(t), (1.8) wherep,q∈L([a,b];R),τ∈ᏹab, is a special case of (1.4) with
(v)(t)^def= p(t)v^τ(t)^ fort∈[a,b]. (1.9) More generally, it is known (see [6, page 317]) that∈ᏸab if and only if the operator admits the representation by means of a Stieltjes integral.
On the other hand, Schaefer proved that there exists an operator∈ᏸabsuch that∈ ᏸab(see [9, Theorem 4]). Therefore, a question naturally arises to study boundary value problem (1.4), (1.5) without the
additional requirement (1.1). In particular, the question whether Theorems1.1and1.3are valid for general operator∈ᏸabis interesting.
The first important step in this direction was made by Bravyi (see [3]), whereTheorem 1.1was proved for ∈ᏸab (i.e., without the additional assumption∈ᏸab). Bravyi’s proof essentially uses Nikol’ski’s
theorem (see, e.g., [5, Theorem XIII.5.2, page 504]) and it is concentrated on the question of Fredholm property. The question whetherTheorem 1.3is valid for the case when∈ᏸabremains open.
In the present paper, among others, we answer this question affirmatively. More pre- cisely, inSection 2we prove that the operator T:C([a,b];R)→C([a,b];R) defined by T(v)(t)^def=[t]
a(v)(s)dsfort∈[a,b] is compact provided that∈ᏸab(seeProposition 2.9).
Based on this result and Riesz-Schauder theory, we give an alternative proof (different from that in [3]) ofTheorem 1.1for∈ᏸab(seeTheorem 2.1).
On the other hand, the compactness of the operatorT allows us to study a question on the well-posedness of boundary value problem (1.4), (1.5). Section 3is devoted to this question. As a special case
of theorem on well-posedness, we obtain the validity of Theorem 1.3for∈ᏸab(seeCorollary 3.3).
InSection 4, the question on dimension of solution spaceU of homogeneous equa- tion (1.6) is discussed.Proposition 4.6 shows that if dimU≥2, then there exists q∈ L([a,b];R) such that the
nonhomogeneous equation (1.4) has no solution. This “patho- logical” behaviour of functional differential equations affirms the importance of the ques- tion whether the solution space of the homogeneous
equation (1.6) is one dimensional.
In Theorems4.8and4.10, the nonimprovable effective sufficient conditions are estab- lished guaranteeing that dimU=1.
2. Fredholm property
Theorem2.1. Let∈ᏸab. Then the problem (1.4), (1.5) is uniquely solvable if and only if the corresponding homogeneous problem (1.6), (1.7) has only the trivial solution.
Analogously as inSection 1, we can introduce the notion of the Green operator of the problem (1.6), (1.7).
Definition 2.2. Let ∈ᏸab and let the problem (1.6), (1.7) have only the trivial solu- tion. An operatorΩ:L([a,b];R)→C([a,b];R) which assigns to everyq∈L([a,b];R) a solutionuof the problem (1.4),
(1.7) is called Green operator of the problem (1.6), (1.7).
Evidently, it follows fromTheorem 2.1that the Green operator is well defined.
Remark 2.3. From the proof ofTheorem 2.1and Riesz-Schauder theory, it follows that if the problem (1.6), (1.7) has a nontrivial solution, then for every c∈Rthere exists q∈L([a,b];R), respectively,
for everyq∈L([a,b];R) there existsc∈R, such that the problem (1.4), (1.5) has no solution.
To proveTheorem 2.1we will need several auxiliary propositions. First we recall some definitions.
Definition 2.4. LetXbe a linear topological space,X^∗its dual space. A sequence{x[n]}^+[n][=]^∞1⊆ Xis called weakly convergent if there existsx∈Xsuch thatϕ(x)=lim[n][→]+∞ϕ(x[n]) for every ϕ∈X^∗. The
pointxis called a weak limit of this sequence.
A setM⊆X is called weakly relatively compact if every sequence of points fromM contains a subsequence which is weakly convergent inX.
A sequence{xn}^+n^∞=1⊆Xis called weakly fundamental if for everyϕ∈X^∗, a sequence {ϕ(xn)}^+n=^∞1is fundamental.
A spaceX is called weakly complete if every weakly fundamental sequence from X possesses a weak limit inX.
LetXandY be Banach spaces and letT:X→Y be a linear bounded operator. The operatorT is said to be weakly completely continuous if it maps a unit ball ofX into weakly relatively compact subset ofY.
Definition 2.5. A setM⊆L([a,b];R) has a property of absolutely continuous integral if for everyε >0, there existsδ >0 such that for an arbitrary measurable set E⊆[a,b]
satisfying the condition mesE≤δ, the following inequality is true:
Ep(s)ds^[]≤ε for everyp∈M. (2.1) Proofs of the following three assertions can be found in [4].
Lemma2.6 [4, Theorem IV.8.6]. The spaceL([a,b];R)is weakly complete.
Lemma2.7 [4, Theorem VI.7.6]. A linear bounded operator mapping the spaceC([a,b];R) into a weakly complete Banach space is weakly completely continuous.
Lemma2.8 [4, Theorem IV.8.11]. If a setM⊆L([a,b];R)is weakly relatively compact, then it has a property of absolutely continuous integral.
The following proposition plays a crucial role in the proof ofTheorem 2.1.
Proposition2.9. Let∈ᏸab. Then the operatorT:C([a,b];R)→C([a,b];R)defined by
T(v)(t)^def= [t]
a(v)(s)ds fort∈[a,b] (2.2) is compact.
Proof. LetM⊆C([a,b];R) be a bounded set. According to Arzel´a-Ascoli lemma, it is sufficient to show that the setT(M)= {T(v) :v∈M}is bounded and equicontinuous.
T(v)^ [C]=max^[] [t]
≤ (v)^ [L]≤ · vC forv∈M,
(2.3) and thus, since∈ᏸabandMis bounded, the setT(M) is bounded.
Further, Lemmas2.6and2.7imply that the operatoris weakly completely continu- ous, that is, a set(M)= {(v) :v∈M}is weakly relatively compact. Therefore, according
toLemma 2.8, for everyε >0, there existsδ >0 such that [t]
s(v)(ξ)dξ^[]≤ε fors,t∈[a,b], |t−s| ≤δ,v∈M. (2.4) On the other hand,
T(v)(t)−T(v)(s)^= [t]
s(v)(ξ)dξ^[] fors,t∈[a,b], v∈C^[a,b];R^, (2.5) which, together with (2.4), results in
T(v)(t)−T(v)(s)^≤ε fors,t∈[a,b],|t−s| ≤δ,v∈M. (2.6)
Consequently, the setT(M) is equicontinuous.
Proof ofTheorem 2.1. Let X=C([a,b];R)×R be a Banach space containing elements x=(u,α), whereu∈C([a,b];R) andα∈R, with a norm
xX= uC+|α|. (2.7)
(2.8) and define a linear operatorT:X→Xby setting
α+u(a) + [t]
. (2.9)
Obviously, the problem (1.4), (1.5) is equivalent to the operator equation
x=T(x) +q (2.10)
in the spaceXin the following sense: ifx=(u,α)∈Xis a solution of (2.10), thenα=0, u∈C([a,b]; R), anduis a solution of (1.4), (1.5), and vice versa, ifu∈C([a,b]; R) is a solution of (1.4), (1.5), then
x=(u, 0) is a solution of (2.10).
According toProposition 2.9, we have that the operator T is compact. From Riesz- Schauder theory, it follows that (2.10) is uniquely solvable if and only if the corresponding homogeneous equation
x=T(x) (2.11)
has only the trivial solution (see, e.g., [11, Theorem 2, page 221]). On the other hand, (2.11) is equivalent to the problem (1.6), (1.7) in the above-mentioned sense.
Following [7,8] we introduce the following notation.
Notation 2.10. Lett0∈[a,b]. Define operators^k:C([a,b];R)→C([a,b];R) and num- bersλ[k]as follows:
^0(v)(t)^def=v(t), ^k(v)(t)^def= [t]
^^k^−^1(v)^(s)ds fort∈[a,b],k∈N, (2.12) λ[k]=h^^0(1) +^1(1) +···+^k^−^1(1)^ fork∈N. (2.13) Ifλ[k]=0 for somek∈N, then let
^k,0(v)(t)^def=v(t) fort∈[a,b], ^k,m(v)(t)^def=^m(v)(t)−h^^k(v)^
λk m−1
^i(1)(t) fort∈[a,b],m∈N. (2.14) Theorem2.11. Let∈ᏸaband let there existk,m∈N,m0∈N∪ {0}, andα∈[0, 1[such thatλk=0and for every solutionuof the problem (1.6), (1.7), the inequality
^k,m(u)^ [C]≤α^ ^k,m^0(u)^ [C] (2.15)
is fulfilled. Then the problem (1.4), (1.5) has a unique solution.
Remark 2.12. The proof ofTheorem 2.11is omitted since it is completely the same as the proof of [8, Theorem 1.3.1] (see also [7, Theorem 1.2]). The only difference is that instead ofTheorem
1.1,Theorem 2.1has to be used.
Theorem 2.11implies the following corollary.
Corollary2.13. Let∈ᏸabbe at0-Volterra operator. Then the problem u^(t)=(u)(t) +q(t), u^t0
=c, (2.16)
withq∈L([a,b];R)andc∈R, is uniquely solvable.
To prove this corollary we need the following lemma.
Lemma2.14. Let∈ᏸab be at0-Volterra operator and let^k (k∈N∪ {0})be operators defined by (2.12). Then
^k^ =0. (2.17)
Proof. Letε∈]0, 1[. According toProposition 2.9, the operator^1, defined by (2.12) for k=1, is compact. Therefore, by virtue of Arzel`a-Ascoli lemma, there existsδ >0 such that
s(v)(ξ)dξ^[]=^1(v)(t)−^1(v)(s)^≤εvC for|t−s| ≤δ. (2.18)
n=Ent b−t0
, m=Ent
t0−a δ
, ti=t0+iδ fori= −m,−m+ 1,. . .,−1, 1, 2,. . .,n,
t[−][m][−]1=a, t[n+1]=b,
and introduce the notation ^k(v)^ [i]=
^k(v)^ [C([t][0][,t]
i];R) fori=1,n+ 1,
^k(v)^ [C([t][i][,t][0][];][R][)] fori= −m−1,−1. (2.20)
We will show that
^k(v)^ [i]≤α[i](k)ε^kvC([a,b];R) fori=1,n+ 1,k∈N, (2.21) where
αi(k)=γik^i^−^1 fori=1,n+ 1,
γ1=1, γi+1=iγi+i+ 1 fori=1,n. (2.22) First note that
^1(v)^ [i]≤iεvC([a,b];R) fori=1,n+ 1. (2.23) Indeed, according to (2.18), it is clear that
^1(v)^ [i]=max^[] [t]
(v)(ξ)dξ^[]:t∈ t0,t[i]^
max^[] [t]
≤iεvC([a,b];R) fori=1,n+ 1.
Further, on account of (2.18) and the fact thatis at0-Volterra operator, we have ^k+1(v)(t)^=
^^k(v)^(ξ)dξ^[]≤ε^ ^k(v)^ [1] fort∈ t0,t1
,k∈N. (2.25)
Hence, by virtue of (2.23), we get
^k(v)^ [1]≤ε^kvC([a,b];R) fork∈N, (2.26) that is, (2.21) holds fori=1.
Now let the inequality (2.21) hold for somei∈ {1, 2,. . .,n}. With respect to (2.18) and the fact thatis at0-Volterra operator, we have
^k+1(v)^ [i+1]=max^[] [t]
max^[] [t]
t[j],t[j+1]^ + max^[]
^^k(v)^(ξ)dξ^[]:t∈ t[i],t[i+1]^
≤iε^ ^k(v)^ [i]+ε^ ^k(v)^ [i+1]
≤iαi(k)ε^k+1vC([a,b];R)+ε^ ^k(v)^ [i+1] fork∈N.
Hence we get
^k+1(v)^ [i+1]≤iαi(k)ε^k+1vC([a,b];R)
+ε^iαi(k−1)ε^kvC([a,b];R)+ε^ ^k^−^1(v)^ [i+1]^ fork∈N. (2.28) To continue this procedure, on account of (2.23), we obtain
^k+1(v)^ [i+1]≤
i+ 1 +i^α[i](1) +···+α[i](k)^ε^k+1vC([a,b];R) fork∈N. (2.29) With respect to (2.22), we get
i+ 1 +i k j=1
αi(j)=i+ 1 +iγi
1^i^−^1+ 2^i^−^1+···+k^i^−^1^≤i+ 1 +iγikk^i^−^1
=i+ 1 +iγik^i≤
i+ 1 +iγi
k^i=γi+1k^i≤αi+1(k+ 1).
Therefore, from (2.29), it follows that
^k+1(v)^ [i+1]≤α[i+1](k+ 1)ε^k+1vC([a,b];R) fork∈N. (2.31)
Thus, by induction, we have proved that (2.21) holds.
In an analogous way, it can be shown that
^k(v)^ [i]≤αi(k)ε^kvC([a,b];R) fori= −m−1,−1,k∈N, (2.32)
αi(k)=γik^|^i^|−^1 fori= −m−1,−1,
γ[−]1=1, γi−1= |i|γi+|i|+ 1 fori= −m,−1. (2.33)
Now from (2.21), (2.22), (2.32), and (2.33), it follows that there existsγ∈N(indepen- dent ofk) such that
^k(v)^ [C([a,b];][R][)]≤ ^k(v)^ [−][m][−][1]+^ ^k(v)^ [n+1]
≤γk^n+mε^kvC([a,b];R) fork∈N. (2.34)
Hence, sinceε <1, it follows that (2.17) holds.
Proof of Corollary 2.13. Leth(v)^def=v(t0). Obviously, for everyk,m∈N, we haveλk=1, h^^k(v)^=0, ^k,m(v)(t)=^m(v)(t) fort∈[a,b],v∈C^[a,b];R^. (2.35) According toLemma 2.14, we can choosem∈Nsuch that
^m^ <1. (2.36)
Thus the inequality (2.15) holds withm0=0 andα= ^m. Fort0-Volterra operators,Theorem 2.11can be inverted. More precisely, the following assertion is valid.
Theorem2.15. Let∈ᏸab be at0-Volterra operator. Then the problem (1.4), (1.5) has a unique solution if and only if there existk,m∈Nsuch thatλk=0and
^k,m^ <1. (2.37)
Proof. Let inequality (2.37) hold for somek,m∈N. Obviously, for everyu∈C([a,b];
R) (consequently, also for every solution of (1.6), (1.7)), we have
^k,m(u)^ [C]≤ ^k,m^ uC. (2.38) Therefore, the assumptions ofTheorem 2.11are fulfilled withm0=0 andα= ^k,m. Consequently, the problem (1.4), (1.5) has a unique solution.
Assume now that the problem (1.6), (1.5) is uniquely solvable. According toTheorem 2.1, the problem (1.6), (1.7) has only the trivial solution.
Letu0be a solution of the problem
u^(t)=(u)(t), u^t0
=1, (2.39)
the existence of which is guaranteed byCorollary 2.13. Obviously, h^u0
=0, (2.40)
since otherwise the functionu0would be a nontrivial solution of the problem (1.6), (1.7).
n−1 i=0
^i(1)(t) fort∈[a,b], n∈N. (2.41)
From (2.39) it follows that
u0(t)=1 +^1^u0
(t) fort∈[a,b]. (2.42)
Hence we have
u0(t)=1 +^1^1 +^1^u0
(t)=^0(1)(t) +^1(1)(t) +^2^u0
(t) fort∈[a,b]. (2.43) To continue this process, we obtain
n−1 i=0
^i(1)(t) +^n^u0
(t) fort∈[a,b],n∈N. (2.44)
Hence, on account of (2.41) andLemma 2.14, we get
nlim→+[∞] u0−un
C=0. (2.45)
Sinceλn=h(un) forn∈Nandhis a continuous functional, we have, with respect to (2.40) and (2.45), that
=0. (2.46)
Therefore, there existk0∈Nandδ >0 such that
λ[i]^≥δ fori≥k0. (2.47)
Hence, by virtue of (2.45), it follows that there existsρ∈]0, +∞[ such that
λ1[i]^^ u[j]^ [C][]h[ ≤]ρ fori[≥]k0, j[∈]N. (2.48)
According toLemma 2.14, there existk > k0andm∈Nsuch that ^k^ ≤ 1
2ρ, ^ ^m^ <1
2. (2.49)
Furthermore, in view of (2.14), we have
^k,m^ ≤ ^m^ +^ u[m]^ [C]
λ[k]^ ^h ^k^ , (2.50)
which, together with (2.48) and (2.49), implies that (2.37) holds.
Remark 2.16. For the case when∈ᏸab,Theorem 2.15is proved in [8] (see also [7]).
3. Well-posedness
Together with the problem (1.4), (1.5), for everyk[∈]N, consider the perturbed boundary value problem
u^(t)=k(u)(t) +qk(t), hk(u)=ck, (3.1) wherek∈ᏸab,hk:C([a,b];R)→Ris a linear bounded functional,qk∈L([a,b];R), and c[k][∈]R.
The question on well-posedness of general linear boundary value problem for func- tional differential equation under the assumptions∈ᏸab andk∈ᏸab is studied in [7,8] (see also references in [8, page
70]). In this section we will show that the theo- rems on well-posedness established in [7,8] are valid also for the case when∈ᏸaband k∈ᏸab.
Notation 3.1. Let∈ᏸab. Denote byM the set of functionsy∈C([a,b]; R) admitting the representation
y(t)=z(a) + [t]
a(z)(s)ds fort∈[a,b], (3.2) wherez∈C([a,b];R) andzC=1.
Theorem3.2. Let the problem (1.4), (1.5) have a unique solutionu, sup^[]
k(y)(s)−(y)(s)^ds^[]:t∈[a,b], y∈Mk
−→0 ask−→+∞, (3.3) and let, for everyy∈C([a,b]; R),
1 +^ [k]^ [t]
[k](y)(s)−(y)(s)^ds=0 uniformly on[a,b]. (3.4)
Let, moreover,
1 +^ [k]^ [t]
q[k](s)−q(s)^ds=0 uniformly on[a,b], (3.5)
klim→+∞h[k](y)[=]h(y) fory[∈]C^[a,b];R^, (3.6)
klim→+∞c[k]=c. (3.7)
Then there existsk0∈Nsuch that for everyk > k0the problem (3.1) has a unique solution u[k]and
uk−u^ [C]=0. (3.8)
FromTheorem 3.2, the following corollary immediately follows.
Corollary3.3. Let∈ᏸab and the problem (1.6), (1.7) have only the trivial solution.
Then the Green operator of the problem (1.6), (1.7) is continuous.
To proveTheorem 3.2, we need two lemmas, the first of them immediately follows from Arzel`a-Ascoli lemma andProposition 2.9.
Lemma3.4. Let∈ᏸaband (y)(t) ^def=
a(y)(s)ds fort∈[a,b]. (3.9) Let, moreover,{xn}^+[n][=]^∞1⊂C([a,b];R)be a bounded sequence. Then the sequence{(x n)}^+[n][=]^∞1
contains a uniformly convergent subsequence.
Lemma3.5. Let the problem (1.6), (1.7) have only the trivial solution and let the sequences of operators[k]∈ᏸab and linear bounded functionalsh[k]:C([a,b];R)→Rsatisfy conditions (3.3) and (3.6). Then
there existk0∈Nandr >0such that an arbitraryz∈C([a,b]; R) admits the estimate
zC≤rρk(z) fork > k0, (3.10) where
ρ[k](z)=h[k](z)^+ max^1 +^ [k]^ ^[] [t]
. (3.11) Proof. Note first that according to Banach-Steinhaus theorem and the condition (3.6), the sequence{hk}^+[k][=]^∞1is bounded, that is, there existsr0>0 such that
h[k](y)^≤r0yC fory∈C^[a,b];R^. (3.12) Let, fory∈C([a,b];R),
(y)(t) = [t]
a(y)(s)ds, [k](y)(t)= [t]
a[k](y)(s)ds fork∈N. (3.13) Obviously,^:C([a,b];R)→C([a,b];R) and^[k]:C([a,b];R)→C([a,b];R) fork∈Nare linear bounded operators and
k ≤ k fork∈N. (3.14) With respect to our notation, the condition (3.3) can be rewritten as follows:
sup^ [k](y)−(y) ^ [C]:y∈M[][k]^−→0 ask−→+∞. (3.15) Assume on the contrary that the lemma is not valid. Then there exist an increasing sequence of natural numbers {k[m]}^+[m]^∞[=]1 and a sequence of
functions z[m]∈C([a,b]; R), m∈N, such that
z[m]^ [C]> mρ[k][m]^z[m]^ form∈N. (3.16)
y[m](t)= z[m](t) zm
, v[m](t)= [t]
y[m]^(s)−[k][m]^y[m]^(s)^ds fort∈[a,b], (3.17) y0m(t)=ym(t)−vm(t) fort∈[a,b], (3.18) wm(t)=km
(t) +^km
(t) fort∈[a,b]. (3.19) Obviously,
C=1 form∈N, (3.20)
y0m(t)=ym(a) +^km
(t) fort∈[a,b], m∈N, (3.21) y0m(t)=ym(a) +^^y0m
(t) +wm(t) fort∈[a,b],m∈N. (3.22) On the other hand, from (3.14) and (3.17), by virtue of (3.16), we get
v[m]^ [C]≤ ρkm
z[m]^ [C]^1 +^ [k][m]^ < 1
m^1 +^ [k][m]^ form∈N, (3.23) [k][m]^v[m]^ [C]≤ [k][m]^ · v[m]^ [C]< 1
m form∈N. (3.24)
From (3.20) and (3.21), it follows thaty[0m][∈]M[][km], and therefore, in view of (3.15), we have
mlim→+∞ km
−^y0m [C]=0. (3.25)
On account of (3.24) and (3.25), equality (3.19) implies that
mlim→+∞ wm
C=0, (3.26)
and with respect to (3.18), (3.20), and (3.23), y0m
C≤ ym
C+^ vm
C≤2 form∈N. (3.27)
According toLemma 3.4, without loss of generality, we can assume that
mlim→+∞y0m(t)=y0(t) uniformly on [a,b]. (3.28) With respect to (3.18), (3.20), (3.22), (3.23), and (3.26),
mlim→+∞ ym−y0
C=0, (3.29)
C=1, y0(t)=y0(a) +^y0
(t) fort∈[a,b]. (3.30) Consequently,y0is a nontrivial solution of (1.6).
On the other hand, from (3.12) and (3.16), we get hkm
≤r0 y0−y[m]^ [C]+ 1
z[m]^ [C]^h[k][m]^z[m]^
≤r0 y0−y[m]^ [C]+ 1
m form∈N.
Hence, on account of (3.6) and (3.29), we obtain h^y0
=0. (3.32)
Thusy0is a nontrivial solution of the problem (1.6), (1.7), which contradicts the assump-
tion ofLemma 3.5.
Proof ofTheorem 3.2. Letrandk0be numbers, the existence of which is guaranteed by Lemma 3.5. Then, obviously, for everyk > k0, the problem
u^(t)[=][k](u)(t), h[k](u)[=]0, (3.33) has only the trivial solution. According toTheorem 2.1, for everyk > k0, the problem (3.1) is uniquely solvable.
We will show that ifu anduk are solutions of the problems (1.4), (1.5), and (3.1), respectively, then (3.8) holds. Let
vk(t)=uk(t)−u(t) fort∈[a,b]. (3.34) Then, for everyk > k0,
(t) +qk(t) fort∈[a,b], hk
=ck, (3.35) where
qk(t)=k(u)(t)−(u)(t) +qk(t)−q(t) fort∈[a,b],
ck=ck−hk(u). (3.36)
Now, by virtue of (3.4), (3.5), (3.6), and (3.7), we have δk=
1 +^ k max^[] [t]
−→0 ask−→+∞, (3.37)
klim→+∞c[k]=0. (3.38)
According toLemma 3.5, (3.35), and (3.37),
v[k]^ [C]≤rc[k]^+δ[k]^ fork > k0. (3.39)
Hence, in view of (3.37) and (3.38), we obtain
C=0, (3.40)
and, consequently, (3.8) holds.
4. On dimension of the solution set of homogeneous equation
Notation 4.1. LetUbe the solution set of the homogeneous equation (1.6). Obviously,U is a linear vector space.
According toTheorem 2.1, we haveU= {0}, that is, dimU≥1. Moreover, the follow- ing assertion is valid.
Theorem4.2. The spaceUis finite dimensional.
Proof. LetT:C([a,b];R)→C([a,b];R) be an operator defined by T(v)(t)^def=v(a) +
a(v)(s)ds fort∈[a,b]. (4.1) Evidently, the operatorTis linear. According toProposition 2.9, the operatorTis com- pact as well. Obviously, (1.6) is equivalent to the operator equation (2.11) in the
following sense: ifu∈C([a, b];R) is a solution of (1.6), thenx=uis a solution of (2.11), and vice versa, ifx∈C([a,b];R) is a solution of (2.11), thenx∈C([a,b]; R) andu=xis a so- lution of (1.6). In
other words, the setUis also a solution set of the operator equation (2.11).
On the other hand, sinceTis a linear compact operator, from Riesz-Schauder theory, it follows that the solution space of (2.11) is finite-dimensional. Therefore, dimU <+∞. Remark 4.3. Example
5.1below shows that dimUcan be any natural number, even in the case when∈ᏸab.
Proposition4.4. The equalitydimU=1holds if and only if there existsξ∈[a,b]such that the problem
u^(t)=(u)(t), u(ξ)=0 (4.2)
has only the trivial solution.
Proof. Let dimU=1 and let problem (4.2) have a nontrivial solutionuξ for everyξ∈ [a,b]. Chooset0∈]a,b] such thatua(t0)=0. Then, obviously, functionsuaandut0are linearly independent solutions of
(1.6), which contradicts the assumption dimU=1.
Now assume that there existsξ∈[a,b] such that the problem (4.2) has only the trivial solution and dimU≥2. Letu1,u2∈Ube linearly independent. Obviously,
u1(ξ)=0, u2(ξ)=0. (4.3) | {"url":"https://123deta.com/document/qmj7v879-functional-differential-equations.html","timestamp":"2024-11-14T13:33:07Z","content_type":"text/html","content_length":"217407","record_id":"<urn:uuid:4782dd8a-67e6-4409-b2bc-e1f526dcb310>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00046.warc.gz"} |
Final Velocities : Elastic
Ball 1, with a mass of 100 g and traveling at 15 m/s, collides head on with ball 2, which has a mass of 350 g and is initially at rest.
What are the final velocities of each ball if the collision is perfectly elastic?
Known variables:
Mass(m1) of the first ball = 100 g or .100 kg
Initial velocity (v1) of the first ball= 15.0 m/s
Mass(m2) of the second ball = 340 g or .340 kg
Initial velocity(v2) of the second ball= 0 (initially at rest)
(V([f]x)1 and (V[f]x)2 are the velocities of the 2 after the collision
If elastic then the formulas would be the following:
V([f]x)1= 2m[2]v[2]/m[1]+m[2] + (m[1]-m[2]/m[1]+m[2])v[1]
V([f]x)2= 2m[1]v[1]/m[1]+m[2] + (m[1]-m[2]/m[1]+m[2])v[2]
2(.340 x 0)/.100 + .340 + (.100 – .340/ .100 + .340) x (15)
0 + (-.240/.440) x 15
V([f]x)1= -8.3 m/s
2(.100 x 15)/.100 + .340 + (.100 – .340/ .100 + .340) x (0)
2(1.5)/.440 + 0
V([f]x)2= 6.7 m/s
3) What are the final velocities of each ball if the collision is perfectly inelastic?
Inelastic Collision:
If the collision is inelastic, the combined speed of both balls after the collision can be figured out through this equation.
v= m[1]v[1]+ m[2]v[2]/ m[1] + m[2]
(.100 kg x 15 m/s) + 0 / (.100 + .340)
1.5 / .440
V([f]x)1 & 2 = 3.3m/s
The final velocities of each ball would be the same.
6 responses to “Final Velocities : Elastic”
The work for the elastic collision is incorrect.
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Haskell: Monad.
The Monad
or at least my attempt at explaining it
-- The use of this will be explained later
> import Control.Monad ((<=<))
A monad is a haskell Class - this means it comes with specific methods. Let’s have a look at what ghci has to say about it.
λ> :i Monad
class Applicative m => Monad (m :: * -> *) where
(>>=) :: m a -> (a -> m b) -> m b
(>>) :: m a -> m b -> m b
return :: a -> m a
{-# MINIMAL (>>=) #-}
-- Defined in ‘GHC.Base’
instance Monad (Either e) -- Defined in ‘Data.Either’
instance Monad [] -- Defined in ‘GHC.Base’
instance Monad Maybe -- Defined in ‘GHC.Base’
instance Monad IO -- Defined in ‘GHC.Base’
instance Monad ((->) r) -- Defined in ‘GHC.Base’
instance Monoid a => Monad ((,) a) -- Defined in ‘GHC.Base’
Ok, so we know it is a subclass of Applicative, we can see it is implemented for all of these data structures and it comes with some funky looking operators, namely:
• (>>=) aka. bind
• (>>) which I like to call “and then”
• return - which is the same as pure from Applicative
I could go on and on about the theoretical foundations of it, but I’d rather focus on a trivial example.
The identity Monad
Let’s define our own Monad from scratch to try to understand how it works - for this I propose the simplest of all lawful monads - the Identity monad. The reason why I call it a monad is because it
behaves like one - i.e. the instance of Monad is lawful (don’t get bogged down, yet in terminology - but keep in mind).
> data Identity a = Identity a
As you can see, our Identity data type takes any value and places it inside. It’s so basic that we can guarantee that if we have an Identity a we can always get its content - let’s write a function
that does just that:
> getContent :: Identity a -> a
> getContent (Identity x) = x
Mentally we can think of Identity as a Box.
So what can we say about it? Well firstly - we can be sure it has a lawful Functor instance. Given any Box a and a function a - > b we can obtain a Box b:
> instance Functor Identity where
> fmap f (Identity a) = Identity $ f a
Nice. We can also see that given any boxed function Identity (a -> b) and a boxed value Identity a we can get a boxed Identity b value. Furthermore, given any a we can always box it into Identity a.
This is an easy win and we get our Applicative instance.
> instance Applicative Identity where
> (Identity f) <*> (Identity a) = Identity $ f a
> pure a = Identity a
Ok, so we’ve done all the footwork up until the Monad instance. Let’s have a closer look at the bind operator (>>=).
Monad m => (>>=) :: m a -> (a -> m b) -> m b
If we specialise this operator to what we’re trying to achieve it looks more like this:
(>>=) :: Identity a -> (a -> Identity b) -> Identity b
We have the following, a boxed value a, a function from a to a boxed b, and at the end the result is a boxed b. Mentally, we can think of the second function as something that unboxes the first
value, and then returns our new box
• let’s see this in our implementation.
> instance Monad Identity where
> (>>=) (Identity value) boxingFunction = boxingFunction value
Can you see what we’ve done? We’ve unboxed the value, and passed it to our boxingFunction. Ok, maybe it isn’t very clear - but let’s take a second and have a chat about it. Our data type Identity is
useless - in many ways it could disappear from all of our implementations because it doesn’t give us much. We could really push it and say that fundamentally Identity a and a are one and the same
thing - only one is wrapped in this mental box and the other isn’t. We could say that our bind is
(>>=) :: a -> (a -> something b) -> something b
and if we squint we could almost say that in this instance, Identity doesn’t really give us much more information about anything. We could even say that it’s (&) :: a -> (a -> b) -> b from
In any case - why does this matter - and how is it relevant? Well here’s why it matters - let’s take Maybe as another example. Say we have the function half which returns the half of an even number
or Nothing if it is an odd number.
> maybeHalf :: Int -> Maybe Int
> maybeHalf x
> | even x = Just $ x `div` 2
> | otherwise = Nothing
λ> maybeHalf 1
λ> maybeHalf 2
Just 1
Squinting again we see something which we’ve already seen. maybeHalf’s type looks vaguely familiar - indeed its type is the same as the second argument of (>>=) namely ` a -> m b or in our case a ->
Maybe b, or even more specific Int -> Maybe Int`. That’s great because now we can use the operator:
> maybeQuarter :: Int -> Maybe Int
> maybeQuarter x = maybeHalf x >>= maybeHalf
Ok, so what happened now? Think of (>>=) as taking the result of maybeHalf x and passing it to the function maybeHalf again. That’s neat because, when we see it action, it actually makes a lot of
λ> maybeQuarter 2
λ> maybeQuarter 4
Just 1
The do notation
Another cool thing Monads unlock is haskell’s do notation. Let’s refactor the above maybeQuarter to leverage this.
> maybeQuarterDo x = do
> half <- maybeHalf x
> quarter <- maybeHalf half
> return quarter
The two are one and the same function - but in different notations. Look at the do notation one, and then look at the bind notation one. Some intuition might emerge.
Fundamentally, a Monad is a type class that allows us to take an instance of itself containing a value, and a function from that value to another instance of itself containing possibly a different
value. This allows us to easily compose monads. One last example
The fish (<=<)
The fish, is the epitome of what I just said because in the context of a monad it is the moral equivalent of our composition operator (.).
λ> :t (>=>) :: Monad m => (b -> m c) -> (a -> m b) -> a -> m c
λ> :t (.) :: (b -> c) -> (a -> b) -> a -> c
You can see it, right? So how could we refactor our maybeQuarter now? Whatabout:
> maybeQuarterFish :: Int -> Maybe Int
> maybeQuarterFish = maybeHalf <=< maybeHalf
That was it. I’m sure it all makes sense now - maybe. | {"url":"https://cstml.github.io/2022/07/18/Haskell-Monad.html","timestamp":"2024-11-09T20:14:54Z","content_type":"text/html","content_length":"19723","record_id":"<urn:uuid:4a73bd1b-f53e-4f69-8679-86eb14b75593>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00627.warc.gz"} |
Optimal Foreign Exchange Risk Hedging: Closed Form Solutions Maximizing Leontief Utility Function
Theoretical Economics Letters Vol.08 No.14(2018), Article ID:87878,21 pages
Optimal Foreign Exchange Risk Hedging: Closed Form Solutions Maximizing Leontief Utility Function
Yun-Yeong Kim^
Department of International Trade, Dankook University, Yongin-si, South Korea
Copyright © 2018 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
Received: January 24, 2018; Accepted: October 16, 2018; Published: October 19, 2018
In this paper, we extend Kim (2013) [9] for the optimal foreign exchange (FX) risk hedging solution to the multiple FX rates and suggest its application method. First, the generalized optimal hedging
method of selling/buying of multiple foreign currencies is introduced. Second, the cost of handling forward contracts is included. Third, as a criterion of hedging performance evaluation, there is
consideration of the Leontief utility function, which represents the risk averseness of a hedger. Fourth, specific steps are introduced about what is needed to proceed with hedging. There is a
computation of the weighting ratios of the optimal combinations of three conventional hedging vehicles, i.e., call/put currency options, forward contracts, and leaving the position open. The closed
form solution of mathematical optimization may achieve a lower level of foreign exchange risk for a specified level of expected return. Furthermore, there is also a suggestion provided about a
procedure that may be conducted in the business fields by means of Excel.
Foreign Exchange, Risk, Optimal Hedging, Closed Form Solution
1. Introduction
Recently, foreign currency fluctuations are one of the key sources of risk in multinational business/investment operations because of the widespread adoption of the floating exchange rate regime in
many countries after the breakdown of the Bretton Woods system.^1 The U.S. Department of Commerce has also warned that “The volatile nature of the FX market poses a great risk of sudden and drastic
FX rate movements, which may cause significantly damaging financial losses from otherwise profitable export sales” (Trade Finance Guide, Ch. 12).^2 Furthermore, that Guide also suggested three FX
risk management techniques that are considered suitable for small and medium-sized enterprises companies: non-hedging FX risk management techniques, FX forward hedges, and FX options hedges.
However, for practical use by businesses or individuals, there has not been an analytical method with a closed form solution to choose from among the various available hedging tools to reduce the
risk optimally, as correctly pointed out by Khoury and Chan (1988) [8] . For further studies on this issue, see Sercu and Uppal (1995) [11] .
Khoury and Chan (1988) [8] gauge the preferences of finance officers in terms of the specific characteristics of a hedging tool, by relying on a questionnaire survey. Bodie, et al. (2002) [2] and
Nancy (2004) [1] illustrate the technique of computerized optimization and simulation modeling to manage foreign exchange risk. However, their techniques are not a closed form optimal hedging
solution that requires additional computational burden. So its application is limited in the real business world. In this regard, Kim (2013) [9] introduced the optimal foreign exchange risk hedging
solution by exploiting a standard portfolio theory.^3 Hsiao (2017) [7] applies the framework of Kim (2013) [9] to investigate the effects of foreign exchange exposures on the performance of Taiwan
hospitality industry and try to propose some hedging strategies and strengthen their corporate risk management.
In this paper, we extend Kim (2013) [9] for the optimal single FX risk hedging solution and theory to the multiple FX rates and suggest its application method in the business fields. First, the
generalized optimal hedging method of selling/buying of multiple foreign currencies is introduced. Second, the cost of handling forward contracts is included. Third, as a criterion of hedging
performance evaluation, we consider the Leontief utility (or profit for a firm) function, which represents the risk averseness of a hedger. Fourth, steps are introduced about what is needed to
proceed with hedging. There is a computation of the weighting ratios of the optimal combinations of three conventional hedging vehicles, i.e., call/put currency options, forward contracts, and
leaving the position open. As in the standard portfolio theory, the closed form solution of mathematical optimization may achieve a lower level of foreign exchange risk for a specified level of
expected return. There is also a suggestion provided for a procedure that may be conducted in the business fields by means of Excel.^4
The rest of this paper is as follows. Section 2 derives the expected return and return variance of the hedging vehicles. Section 3 analyzes the optimal hedging selection. Section 4 is on application
of developed method, and Section 5 is the conclusion.
2. Expectation and Variance of Hedging Tools’ Returns
In this section, we construct an efficient hedging frontier composed of the expected value and variance of each hedging vehicle’s return for the multiple foreign exchanges. So, it is exactly matched
with the portfolio possibilities curve in modern portfolio theory. Note an optimal combination of hedging vehicles is one that maximizes the expected return given a desired level of risk. For this
objective, there is a need to compute the mean and variance of each tool.
Before proceeding, we assume that a foreign investor needs to buy or sell m-different currencies $\Theta \equiv {\left({\theta }_{1},{\theta }_{2},\cdots ,{\theta }_{m}\right)}^{\prime }$$\left[m×1\
right]$ at a future time T where ${\theta }_{i}$ is represented by the unit of i-th currency. He is worrying about the foreign exchange risk of domestic currency (e.g., US dollar) term translated
value of $\Gamma$ and to hedge it optimally at time 0. The m-foreign exchange rates at time t in terms of domestic currency, is denoted as ${S}_{t}\equiv {\left({e}_{1t},{e}_{2t},\cdots ,{e}_{mt}\
right)}^{\prime }$ . For instance, ${e}_{it}$ is the dollar price of one euro or yen where the dollar is the domestic currency. It is presupposed that there are three hedging tools, i.e., European
currency put (or call) option, forward contracts, and leaving the position open.5 Furthermore, there are the following definitions: a forward contract rate vector ${F}_{t}\equiv {\left({\overline{e}}
_{1t},{\overline{e}}_{2t},\cdots ,{\overline{e}}_{mt}\right)}^{\prime }$ , a striking price vector $K\equiv {\left({\kappa }_{1},{\kappa }_{2},\cdots ,{\kappa }_{m}\right)}^{\prime }$ , and its
premium $P\equiv {\left({p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}^{\prime }$ at time t of a European put (or call) option with the common maturity T.^6 Finally, $C\equiv {\left({c}_{1},{c}_{2},\cdots ,
{c}_{m}\right)}^{\prime }$ is a per unit handling cost vector for the forward contract ${F}_{t}$ if a bank is used.
Define the log of domestic currency term translated value of FX asset $\Gamma$ at time t is given as ${s}_{t}\equiv \mathrm{ln}\left({\Theta }^{\prime }{S}_{t}\right)$ which is a FX value. For
instance, if $\Theta =\left(\text{10Euro,50Yen}\right)$ and ${S}_{t}=\left(\text{1}\text{.5}\text{\hspace{0.17em}}\text{Dollar}/\text{Euro},\text{0}\text{.1}\text{\hspace{0.17em}}\text{Dollar}/\text
{Yen}\right)$ , then ${s}_{t}=\mathrm{ln}\left(20\text{\hspace{0.17em}}\text{Dollar}\right)$ . We assume the ( ${s}_{t}$ ) follows a random walk process:
Assumption 2.1. We suppose
${s}_{t+1}={s}_{t}+{u}_{t+1},\text{\hspace{0.17em}}t=1,2,\cdots ,n$(2.1)
where $\left\{{u}_{t}\right\}$ is independent, identically and normally distributed sequence with the mean zero and variance ${\sigma }^{2}>0$ .
Note ${E}_{t}{s}_{t+1}={s}_{t}$ where ${E}_{t}$ is a conditional expectation. Thus Assumption 2.1 just a variant of the efficient FX market hypothesis.^7 ${\sigma }^{2}$ is consistently
estimated by ${\stackrel{^}{\sigma }}^{2}=\frac{{\sum }_{t=1}^{n}{\left(\Delta {s}_{t}\right)}^{2}}{n}$ .
Now we derive the return and its variance of different hedging tools, where the return is compared with the selling (or buying) a foreign currency (as a bench mark) by the spot rate ${s}_{0}$ .
2.1. FX Selling Case
First, we derive the expected return R[n] and its variance ${V}_{n}^{2}$ of the non-hedging (leaving the position open), as follows.
Theorem 2.2. Suppose Assumption 2.1 holds. Then the expected return for non-hedging of FX asset $\Theta$ is R[n] = 0 and its variance during time T is ${V}_{n}^{2}=T{\sigma }^{2}$ .
All proofs of the theorems are in the Appendix.
Second, we derive the expected return R[f] and its variance ${V}_{f}^{2}$ of the forward contract as follows.
Theorem 2.3. Suppose Assumption 2.1 holds. Then the expected return of forward is ${R}_{f}={f}_{T}-{s}_{0}-c$ and its variance is ${V}_{f}^{2}=0$ where ${f}_{T}\equiv \mathrm{ln}\left({\Theta }^{\
prime }{F}_{T}\right)$ , and $c\equiv {\Theta }^{\prime }C/{\Theta }^{\prime }{S}_{0}$ .
Now we derive the expected return R[p] and its variance ${V}_{p}^{2}$ of currency put option as follows.
Theorem 2.4. Suppose Assumption 2.1 holds. Then,
(a) the expected return of currency put option is given as:
${R}_{p}={x}_{0}\Phi \left({z}_{0}\right)+\sigma \sqrt{T}\varphi \left({z}_{0}\right)-p$
(b) its variance of currency put option is:
${V}_{p}^{2}={x}_{0}^{2}\Phi \left({z}_{0}\right)+T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)\left[1-\Phi \left({z}_{0}\right)\right]-{\left({x}_{0}\Phi \left({z}_{0}\right)+\sigma \
sqrt{T}\varphi \left({z}_{0}\right)\right)}^{2}$ .
where $k\equiv \mathrm{ln}\left({\Theta }^{\prime }K\right)$ , $p\equiv {\Theta }^{\prime }P/{\Theta }^{\prime }{S}_{0}$ , ${x}_{0}\equiv k-{s}_{0}$ , and ${z}_{0}={x}_{0}/\left(\sigma \sqrt{T}\
right)$ where $\varphi \left(z\right)$ and $\Phi \left(z\right)$ are the standard normal density and distribution functions respectively and ${F}_{a.0}$ denotes the distribution function of central $
{\chi }_{\left(q\right)}^{2}$ distribution with the degree of freedom q and:
$\begin{array}{c}E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)=\frac{1-{F}_{3,0}\left({z}_{0}^{2}\right)}{1-{F}_{1,0}\left({z}_{0}^{2}\right)}\text{\hspace{0.17em}}\text{\hspace{0.17em}}if\text{\
hspace{0.17em}}{z}_{0}\ge 0\\ =\frac{{F}_{3,0}\left({z}_{0}^{2}\right)}{{F}_{1,0}\left({z}_{0}^{2}\right)}\left[1-2\Phi \left({z}_{0}\right)\right]+\frac{1-{F}_{3,0}\left({z}_{0}^{2}\right)}{1-{F}_
{1,0}\left({z}_{0}^{2}\right)}\left[1-\Phi \left(-{z}_{0}\right)\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}if\text{\hspace{0.17em}}{z}_{0}<0.\end{array}$
In the above Theorem 2.4, it was suggested that a form of ${V}_{p}^{2}$ represented by a ${\chi }_{\left(1\right)}^{2}$ distribution for the computation of conditional expectation $E\left({z}_{T}^{2}
|{z}_{T}\ge {z}_{0}\right)$ . Otherwise, there is a need for integration by a formula
$E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)={\int }_{{z}_{0}}^{\infty }{z}_{T}^{2}\varphi \left({z}_{T}\right)d{z}_{T}$ , which requires an additional burden.
Next, there is a derivation of the covariance among the three hedging tools. Note the covariance of returns between non-hedging (or option) and forward is obviously zero since the forward return is
not random. Then the covariance of returns between put option and non-hedging is given as follows.
Theorem 2.5. Suppose Assumption 2.1 holds. Then the covariance of returns between put option and non-hedging is:^8
$Co{v}_{pn}=-{x}_{0}\sigma \sqrt{T}\varphi \left({z}_{0}\right)+T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)\left[1-\Phi \left({z}_{0}\right)\right]$ .
2.2. FX Buying Case
First, note we have the same expected return ${R}_{n}=0$ and its variance ${V}_{n}^{2}=T{\sigma }^{2}$ of the non-hedging, as given in Proposition 2.2 for buying a foreign exchange case.
Second, we derive the expected return ${R}_{f}$ and its variance ${V}_{f}^{2}$ of forward contract as follows.
Theorem 2.6. Suppose Assumption 2.1 holds. Then the expected return of forward^9 is ${R}_{f}={s}_{0}-{f}_{T}-c$ and its variance is ${V}_{f}^{2}=0$ .
Now we derive the expected return ${R}_{c}$ and its variance ${V}_{c}^{2}$ of currency call option as follows.
Theorem 2.7. Suppose Assumption 2.1 holds. Then,
(a) the expected return of currency call option is given as:
${R}_{c}=\sigma \sqrt{T}\varphi \left({z}_{0}\right)-{x}_{0}\left[1-\Phi \left({z}_{0}\right)\right]-p$
(b) its variance of currency call option is:
${V}_{c}^{2}={x}_{0}^{2}\left[1-\Phi \left({z}_{0}\right)\right]+T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}<{z}_{0}\right)\Phi \left({z}_{0}\right)-{\left({x}_{0}\left[1-\Phi \left({z}_{0}\right)\
right]-\sigma \sqrt{T}\varphi \left({z}_{0}\right)\right)}^{2}$
{0.17em}}{z}_{0}<0\\ =\frac{{F}_{3,0}\left({z}_{0}^{2}\right)}{{F}_{1,0}\left({z}_{0}^{2}\right)}\left[1-2\Phi \left(-{z}_{0}\right)\right]+\frac{1-{F}_{3,0}\left({z}_{0}^{2}\right)}{1-{F}_{1,0}\left
({z}_{0}^{2}\right)}\Phi \left(-{z}_{0}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}if\text{\hspace{0.17em}}{z}_{0}\ge 0.\end{array}$
The covariance of returns between call option and the non-hedging is given as follows.
Theorem 2.8. Suppose Assumption 2.1 holds. Then the covariance of returns between put option and non-hedging is:
$Co{v}_{cn}={x}_{0}\sigma \sqrt{T}\varphi \left({z}_{0}\right)+T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}<{z}_{0}\right)\Phi \left({z}_{0}\right)$ .
3. Efficient Hedging Frontier Construction
Based upon above derivation of expected return (R) and return variance (V^2) structure, now we can derive the efficient hedging frontier. It is exactly matched with the portfolio possibilities curve
in a standard portfolio theory (e.g., Elton, et al. (2007) [4] ).
For this purpose, first, there is consideration of a portfolio composed of non-hedging and put in the option (for FX selling) that are all risky. Let the weight of non-hedging be as w and 1-w for the
option where w is a real number. Then, from the above derivation in Section 2, its expected return is defined as follows.1^0
because ${R}_{n}=0$ for the non-hedging, and its variance is given as:
${V}^{2}\left(w\right)={w}^{2}{V}_{n}^{2}+{\left(1-w\right)}^{2}{V}_{p}^{2}+2w\left(1-w\right)Co{v}_{np}$ .
Therefore note $R\left(0\right)={R}_{p}$ , $R\left(1\right)={R}_{n}=0$ , ${V}^{2}\left(0\right)={V}_{p}^{2}$ and ${V}^{2}\left(1\right)={V}_{n}^{2}$ .
In this case, the return of forward has zero variance with the expected return, say, ${R}_{f}$ . Thus, it is regarded as a riskless asset in the standard portfolio theory. Now the hedging allocation
line (a line of R and V)^11 connecting the riskless forward contract and a combination of non-hedging and put option is defined as follows.
$R={R}_{f}+\left(\frac{R\left(w\right)-{R}_{f}}{V\left(w\right)}\right)V$ , (3.2)
where R denotes the return and V denotes the standard deviation of return (as a risk); $\left[R\left(w\right)-{R}_{f}\right]/V\left(w\right)$ is a constant slope for a given w where $V\left(w\right)=
\sqrt{{V}^{2}\left(w\right)}$ .
Then the efficient hedging allocation line^12 is given by solving following problem:
that is maximizing the slope of Equation (3.2) with the argument w. The problem (3.3) may be solved without restriction, according to Elton, et al. ( [4] : pp. 100-103), as follows.
where $\left(\begin{array}{c}{m}_{1}\\ {m}_{2}\end{array}\right)={\left(\begin{array}{cc}{V}_{n}^{2}& Co{v}_{np}\\ Co{v}_{np}& {V}_{p}^{2}\end{array}\right)}^{-1}\left(\begin{array}{c}-{R}_{f}\\ {R}_
{p}-{R}_{f}\end{array}\right)$ assuming $|\begin{array}{cc}{V}_{n}^{2}& Co{v}_{np}\\ Co{v}_{np}& {V}_{p}^{2}\end{array}|e 0$ .
If ${w}^{*}otin \left[0,1\right]$ , then the maximization problem (3.3) should be solved under the restriction $w\in \left[0,1\right]$ using a typical Kuhn-Tucker condition.
Finally, the efficient hedging frontier is given by:
$R={R}_{f}+\left(\frac{R\left({w}^{*}\right)-{R}_{f}}{V\left({w}^{*}\right)}\right)V$ of the left of $\left[R\left({w}^{*}\right),V\left({w}^{*}\right)\right]$ if ${w}^{*}\in \left[0,1\right]$(3.5)
$=\left[R\left(w\right),V\left(w\right)\right]$ of the right of $\left[R\left({w}^{*}\right),V\left({w}^{*}\right)\right]$ otherwise.
For the given efficient frontier in (3.5), the optimal hedging (cf., separation theorem) is conducted as follows. First, the hedging ratio between non-hedging and option are set as $\left({w}^{*},\
text{}1-{w}^{*}\right)$ . See Figure 1. Second, $\rho$ is set for the forward and $1-\rho$ is set for the first combination of non-hedging and option. So if $\rho =1$ , then the forward becomes the
unique hedging tool.
Finally, $\left[\rho ,{w}^{*}\left(1-\rho \right),\left(1-{w}^{*}\right)\left(1-\rho \right)\right]$ becomes the optimal hedging ratio of the forward, non-hedging, and put option. Note the expected
utility maximization
Figure 1. Efficient hedging frontier. (A: Forward only solution, B: Non-Forward solution).
may be a rule to determine an optimal $\rho$ . The following section suggests an optimal hedging solution through determining an optimal $\rho$ under the Leontief utility function.
4. Optimal Hedging under Leontief Utility Function
A Leontief utility (or profit for a firm) function is considered $U=\mathrm{min}\left(R,\alpha +\beta V\right)$ as a criterion for hedging performance evaluation where $\beta <0$ . Note, for the
maximization of a Leontief utility function under the efficient hedging frontier in Figure 1, a pair (V, R) should satisfy a line:
$R=\alpha +\beta V.$(4.1)
To show it, let us derive an indifference curve. For this, suppose ${R}_{0}=\alpha +\beta {V}_{0}$ (as in Figure 2). Then a utility of $\left({V}_{0},R\right)$ has the same utility with $\left({V}_
{0},{R}_{0}\right)$ for ${R}_{0}\le R$ because a utility of $\left({V}_{0},R\right)$ is $\mathrm{min}\left(R,\alpha +\beta {V}_{0}\right)=\mathrm{min}\left(R,{R}_{0}\right)={R}_{0}$ while the utility
of $\left({V}_{0},{R}_{0}\right)$ is $\mathrm{min}\left({R}_{0},\alpha +\beta {V}_{0}\right)={R}_{0}$ from ${R}_{0}=\alpha +\beta {V}_{0}$ . Similarly, a utility of $\left(V,{R}_{0}\right)$ has the
same utility with $\left({V}_{0},{R}_{0}\right)$ for $V\le {V}_{0}$ because a utility of $\left(V,{R}_{0}\right)$ is $\mathrm{min}\left({R}_{0},\alpha +\beta V\right)={R}_{0}$ using $\alpha +\beta V\
ge \alpha +\beta {V}_{0}={R}_{0}$ while the utility of $\left({V}_{0},{R}_{0}\right)$ is $\mathrm{min}\left({R}_{0},\alpha +\beta {V}_{0}\right)={R}_{0}$ from ${R}_{0}=\alpha +\beta {V}_{0}$ . So the
North-West direction indicates the increase of utility in a space of (V, R).
Later, the above Equation (4.1) will be called a utility maximizing locus (UML). The UML might be interpreted as that which denotes how V is transformed into R with the same utility. It also denotes
a cost of the standard deviation (volatility) for a hedging portfolio. See Figure 2 where the cost for the volatility ${V}_{0}$ is evaluated as ${R}_{0}=\alpha +\beta {V}_{0}$ in terms of return.
Note, the above Leontief utility function and conformable UML represent an extreme risk averseness. It is related to the marginal rate of substitution of the volatility to a return at the utility
maximizing point along UML, which is $+\infty$ , i.e., the marginal increase of V requires an infinite return increase (as compensation for augmented risk) for the same utility, whereas, a marginal
decrease of V does
Figure 2. Indifference curve under Leontief utility function.
not require any return to be at the same utility level. This assumption is not so unrealistic because this model is not designed for the speculator but for the hedger/firms in the real world of
business who are concerned with the volatility of fund flow.
Now to estimate $\alpha$ and $\beta$ by an ordinary least square regression, we rewrite Equation (4.1) as:
$E\left(R\right)=\alpha +\beta \sqrt{E{\left[z-E\left(z\right)\right]}^{2}}$
or approximately
${R}_{Ti}=\alpha +\beta |{z}_{Ti}-\overline{z}|+{\epsilon }_{Ti}$ for $i=1,2,\cdots ,n$(4.2)
where ${z}_{Ti}\equiv {s}_{Ti}-{s}_{T\left(i-1\right)}$ is a change rate of FX asset during a maturity from 0 to T, $\overline{z}$ is a sample average of ${z}_{Ti}$ , and ${\epsilon }_{Ti}$ is
assumed as a mean zero error term that is not correlated with ${z}_{Ti}$ .
Now note the intersection of UML (4.1) and the efficient hedging frontier (3.5), which is given as follows.
$\stackrel{˜}{V}=\frac{{R}_{f}-\alpha }{\beta -\frac{R\left({w}^{*}\right)-{R}_{f}}{V\left({w}^{*}\right)}}$ and $\stackrel{˜}{R}=\alpha +\beta \stackrel{˜}{V}$ ,
after solving two Equations (3.5) and (4.1) with two unknowns R and V when $0\le {w}^{*}\le 1$ . The above solution point $\left(\stackrel{˜}{V},\stackrel{˜}{R}\right)$ helps to find the optimal
weight for the riskless forward contract as
${\rho }^{*}=1-\frac{\stackrel{˜}{V}}{V\left({w}^{*}\right)}$^13 (4.3)
when $0\le \frac{\stackrel{˜}{V}}{V\left({w}^{*}\right)}\le 1$ . See Figure 3.
Figure 3.Derivation of optimal weight for forward.
$\left[{\rho }^{*},{w}^{*}\left(1-{\rho }^{*}\right),\left(1-{w}^{*}\right)\left(1-{\rho }^{*}\right)\right]$(4.4)
becomes the optimal hedging ratio of forward, non-hedging, and put option using (4.3) for the vector $\Theta$ . So, for instance, the weight ${\rho }^{*}$ of $\Theta$ needs to be distributed to the
Note, if the slope coefficient $\beta$ as a marginal cost of volatility V is decreased to ${\beta }^{\prime }\left(<\beta \right)$ , then the new optimal weight for the forward contract (riskless) is
decreased as ${\rho }^{*}"=1-\frac{{\stackrel{˜}{V}}^{\prime }}{V\left({w}^{*}\right)}<{\rho }^{*}$ . So more risk can be admitted because the
marginal cost of volatility is decreased. See Figure 3 to see this change.
However, if $\stackrel{˜}{V}$ is larger than $V\left({w}^{*}\right)$ because $\beta$ is sufficiently small, then the weight for the forward contract (remind ${R}_{n}=0$ ) may become zero.^14 In this
case, ${w}^{*}$ is not any further an optimal weight between the leaving open position and the option. Rather, we have to choose it from the intersection of UML and the locus of $\left[V\left(w\
right),R\left(w\right)\right]$ which depends on the weight parameter w. The new solutions $\left[\overline{V},\overline{R}\right]$ for the optimization are computed as follows.^15
Theorem 4.1: Suppose a pair (V, R) satisfies a line (4.1). Then
$\overline{R}=\frac{-b±\sqrt{{b}^{2}-ac}}{a}$ and $\overline{V}=\frac{\overline{R}-\alpha }{\beta }$ assuming ${b}^{2}-ac\ge 0$ where
$a=\frac{{R}_{p}^{2}}{{\beta }^{2}}-{V}_{n}^{2}-{V}_{p}^{2}+2Co{v}_{pn}$ , $b=-\alpha \frac{{R}_{p}^{2}}{{\beta }^{2}}+{R}_{p}{V}_{n}^{2}-Co{v}_{pn}{R}_{p}$ , $c=\frac{{\alpha }^{2}{R}_{p}^{2}}{{\
beta }^{2}}-{R}_{p}^{2}{V}_{n}^{2}$ .
In Theorem 4.1, we may have two different solutions that need to be selected to maximize the utility. So, we need to select one $\overline{R}$ among them maximizing the utility and define a
conformable optimal expected return as ${\overline{R}}^{*}\equiv \mathrm{arg}{\mathrm{max}}_{\overline{R}}U={\mathrm{max}}_{\overline{R}}\mathrm{min}\left(\overline{R},\alpha +\beta \overline{V}\
right)$ . See following Figure 4.
Finally, the optimal hedging ratio of the forward, non-hedging, and put option becomes $\left[0,\overline{w},\left(1-\overline{w}\right)\right]$ where
Figure 4. Optimal hedging without forward contract.
from solving (3.1) for the weight w.
Finally, if $\rho \ge 1$ , then a weighting vector (1, 0, 0) that is just selling the forward becomes the optimal hedging ratio.
5. Application Procedures
In application, suppose, at time 0, an investor hopes to sell one unit of foreign exchange at a future time T. Then following steps need to be carried out for hedging.
1) Select three vehicles of hedging as: forward contracts, leaving the position open (Selling foreign exchange case) and European currency put option.
2) Compute mean, variance, and covariance of each tool using the formula in Section 2.
3) Compute a weighting coefficient ${w}^{*}$ as in (3.4) or $\overline{w}$ as in (4.5) if $\stackrel{˜}{V}<\overline{V}$ or leaving the position open against the put option.
4) Decide $\alpha$ and $\beta$ using OLS regression as in (4.2).
5) Compute an optimal weighting coefficient for the forward against for the portfolio of option and leaving the position open $\rho$ as in (4.3).
6) Finally compute the optimal hedging ratio of the forward, non-hedging, and option. $\left[{\rho }^{*},{w}^{*}\left(1-{\rho }^{*}\right),\left(1-{w}^{*}\right)\left(1-{\rho }^{*}\right)\right]$ as
in (4.4).
Consequently, we summarize the optimal weighting vectors of forward, option, and non-hedging for optimal hedging, as shown in Table 1.
Then we apply the developed method for the exchange rate of the euro against the US dollar. The data frequency and period are presented on a monthly basis from January 1999 to March 2015. All data
have been taken from FRED of FRB St. Louis.
Thus we assume, at time 0, i.e., June 1, 2015, 1.1235 dollar price of one euro with $\sigma =0.024$ $/€, a hedger hopes to sell one unit of foreign exchange at a
Table 1. Optimal hedging weighting vector.
Figure 5. Optimal weighting ratio change as β decrease. (a) Selling FX case; (b) Buying FX case.
future time T = 6 months. Further we suppose that there are three hedging tools, i.e., European currency put option, forward contracts, and leaving the position open. Assume a forward contract rate F
= 1.1 $/€ selling cost for the forward contract C = 0.1 $, a striking price K = 1.15 $/€ and its premium P = 0.03 $/€for European put option with the maturity T = 6, respectively. Note ${z}_{0}=k-{s}
_{0}>0$ in this case.
We assume $\alpha =0.01$ . See Figure 5 for the optimal weighting ratio in (4.4) change as $\beta$ decreases^16. Note, if $\beta$ as a marginal cost of volatility V is decreased, then the optimal
weight for the forward contract (riskless) is decreased, as expected in the above theoretical explication (see Section 3).
6. Conclusions
This paper introduced the optimal foreign exchange risk hedging solution by exploiting a standard portfolio theory, thus extending Kim (2013) [8] in its following features. First, the case of the
selling/buying of multiple foreign currencies is also considered. Second, the cost of handling forward contracts is included. Third, as a criterion of hedging performance evaluation, we consider the
Leontief utility function, which represents the risk averseness of a hedger. Fourth, steps are introduced about what is needed to proceed with hedging. There is a computation of the weighting ratios
of the optimal combinations of three conventional hedging vehicles, i.e., call/put currency options, forward contracts, and leaving the position open. The closed form solution of mathematical
optimization may achieve a lower level of foreign exchange risk for a specified level of expected return. There is also a suggestion provided about a procedure that may be conducted in the business
fields by means of Excel.
The structure may be extended to cover the futures and American options and it will be a future research topic for us. However, I hypothesize that a similar logic may be readily applied to these
extensions applying developed method in this paper. Furthermore, a development of a convenient computer program for FX risk hedging users, based on above results, would be a useful project.
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
Cite this paper
Kim, Y.-Y. (2018) Optimal Foreign Exchange Risk Hedging: Closed Form Solutions Maximizing Leontief Utility Function. Theoretical Economics Letters, 8, 2893-2913. https://doi.org/10.4236/
Appendix: Proofs of Theorems
Proof of Theorem 2.2: Note the return of non-hedging is approximately the value of following^17:
$\frac{{\Theta }^{\prime }\left({S}_{T}-{S}_{0}\right)}{{\Theta }^{\prime }{S}_{0}}\cong {s}_{T}-{s}_{0}$(1)
assuming ${\Theta }^{\prime }\left({S}_{T}-{S}_{0}\right)$ is small. Then, under Assumption 2.1, the claimed results hold as:
$E\left({s}_{T}-{s}_{0}|\Omega \right)=0$ and $E\left[{\left({s}_{T}-{s}_{0}\right)}^{2}|\Omega \right]=T{\sigma }^{2}$ . (2)
Proof of Theorem 2.3: Note the expected return for forward is the value of following:
$\frac{{\Theta }^{\prime }\left({F}_{T}-{S}_{0}-C\right)}{{\Theta }^{\prime }{S}_{0}}\cong {f}_{T}-{s}_{0}-c$(3)
assuming ${\Theta }^{\prime }\left({F}_{T}-{S}_{0}-C\right)$ is small. Its variance is obviously zero since the return is not random. $\square$
Proof of Theorem 2.4: (a) Note the inflow of selling weighted put option at time T is given as ${\Theta }^{\prime }\left[\mathrm{max}\left({S}_{T},K\right)-P\right]$ . Thus its return is given as
$\begin{array}{l}\frac{{\Theta }^{\prime }\left[\mathrm{max}\left({S}_{T},K\right)-P-{S}_{0}\right]}{{\Theta }^{\prime }{S}_{0}}\\ =\mathrm{max}\left(\frac{{\Theta }^{\prime }\left[{S}_{T}-{S}_{0}\
right]}{{\Theta }^{\prime }{S}_{0}},\frac{{\Theta }^{\prime }\left[K-{S}_{0}\right]}{{\Theta }^{\prime }{S}_{0}}\right)-\frac{{\Theta }^{\prime }P}{{\Theta }^{\prime }{S}_{0}}\\ \cong \mathrm{max}\
left({s}_{T}-{s}_{0},k-{s}_{0}\right)-p\equiv \mathrm{max}\left({x}_{T},{x}_{0}\right)-p\end{array}$(4)
assuming ${S}_{T}-{S}_{0}$ and $K-{S}_{0}$ are small.
Now the expected return conditional on $\Omega$ in (4) is computed as:
$E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)|\Omega \right]-p={x}_{0}\Phi \left({z}_{0}\right)+\sigma \sqrt{T}\varphi \left({z}_{0}\right)-p$
from (4) where ${x}_{T}\equiv {s}_{T}-{s}_{0}$ , since^18
$\begin{array}{c}E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)|\Omega \right]=E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)|{x}_{T}<{x}_{0},\Omega \right]\mathrm{Pr}\left({x}_{T}<{x}_{0}\right)\\
\text{\hspace{0.17em}}\text{\hspace{0.17em}}+E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)|{x}_{T}\ge {x}_{0},\Omega \right]\mathrm{Pr}\left({x}_{T}\ge {x}_{0}\right)\\ ={x}_{0}\mathrm{Pr}\left({x}
_{T}<{x}_{0}\right)+E\left({x}_{T}|{x}_{T}\ge {x}_{0},\Omega \right)\mathrm{Pr}\left({x}_{T}\ge {x}_{0}\right)\\ ={x}_{0}\Phi \left({z}_{0}\right)+\sigma \sqrt{T}\frac{\varphi \left({z}_{0}\right)}
{1-\Phi \left({z}_{0}\right)}\left[1-\Phi \left({z}_{0}\right)\right]\\ ={x}_{0}\Phi \left({z}_{0}\right)+\sigma \sqrt{T}\varphi \left({z}_{0}\right)\end{array}$(5)
from the definition of conditional expectation, where ${x}_{T}~N\left(0,T{\sigma }^{2}\right)$ from Assumption 2.1 and
$E\left({x}_{T}|{x}_{T}\ge {x}_{0},\Omega \right)=\sigma \sqrt{T}\frac{\varphi \left({z}_{0}\right)}{1-\Phi \left({z}_{0}\right)}$(6)
for the third equality (5) from Greene ( [6] : p. 759), and
$\mathrm{Pr}\left({x}_{T}\ge {x}_{0}\right)=\mathrm{Pr}\left(\frac{{x}_{T}}{\sigma \sqrt{T}}\ge \frac{{x}_{0}}{\sigma \sqrt{T}}\right)=\mathrm{Pr}\left({z}_{T}\ge {z}_{0}\right)\equiv 1-\Phi \left
where ${z}_{T}={x}_{T}/\sigma \sqrt{T}$ .
(b) The return’s variance of (4) is defined as:
$\begin{array}{l}E{\left(\mathrm{max}\left({x}_{T},{x}_{0}\right)-E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)|\Omega \right]|\Omega \right)}^{2}\\ =E\left({\left[\mathrm{max}\left({x}_{T},{x}_{0}
\right)\right]}^{2}|\Omega \right)-{\left(E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)|\Omega \right]\right)}^{2}\end{array}$(8)
Note the second term of right hand side in (8) is derived from (5) directly. Then the first term of right hand side in (8) is arranged as:
$\begin{array}{c}E\left({\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)\right]}^{2}|\Omega \right)=E\left({\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)\right]}^{2}|{x}_{T}<{x}_{0},\Omega \right)\
mathrm{Pr}\left({x}_{T}<{x}_{0}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E\left({\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)\right]}^{2}|{x}_{T}\ge {x}_{0},\Omega \right)\mathrm{Pr}\
left({x}_{T}\ge {x}_{0}\right)\\ ={x}_{0}^{2}\mathrm{Pr}\left({x}_{T}<{x}_{0}\right)+E\left({x}_{T}^{2}|{x}_{T}\ge {x}_{0},\Omega \right)\mathrm{Pr}\left({x}_{T}\ge {x}_{0}\right)\\ ={x}_{0}^{2}\Phi
\left({z}_{0}\right)+E\left({x}_{T}^{2}|{x}_{T}\ge {x}_{0},\Omega \right)\left[1-\Phi \left({z}_{0}\right)\right]\\ ={x}_{0}^{2}\Phi \left({z}_{0}\right)+T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}\ge
{z}_{0}\right)\left[1-\Phi \left({z}_{0}\right)\right]\end{array}$ (9)
$\begin{array}{c}E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)=\frac{1-{F}_{3,0}\left({z}_{0}^{2}\right)}{1-{F}_{1,0}\left({z}_{0}^{2}\right)}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text
{\hspace{0.17em}}{z}_{0}\ge 0\\ =\frac{{F}_{3,0}\left({z}_{0}^{2}\right)}{{F}_{1,0}\left({z}_{0}^{2}\right)}\left[1-2\Phi \left({z}_{0}\right)\right]+\frac{1-{F}_{3,0}\left({z}_{0}^{2}\right)}{1-{F}_
{1,0}\left({z}_{0}^{2}\right)}\left[1-\Phi \left(-{z}_{0}\right)\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{z}_{0}<0\end{array}$
because, for the second term in last equation in (9), we may show that
$E\left({x}_{T}^{2}|{x}_{T}\ge {x}_{0}\right)=T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)$(10)
from $\begin{array}{c}E\left({x}_{T}^{2}|{x}_{T}\ge {x}_{0}\right)={\int }_{{x}_{T}\ge {x}_{0}}{x}_{T}^{2}\frac{g\left({x}_{T}\right)}{G\left({x}_{T}\ge {x}_{0}\right)}\text{d}{x}_{T}\\ =T{\sigma }^
{2}{\int }_{{z}_{T}\ge {z}_{0}}{z}_{T}^{2}\frac{g\left(\sigma \sqrt{T}{z}_{T}\right)}{G\left({z}_{T}\ge {z}_{0}\right)}\sigma \sqrt{T}\text{d}{z}_{T}\\ =T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}\ge
since $\frac{g\left(\sigma \sqrt{T}{z}_{T}\right)}{G\left({z}_{T}\ge {z}_{0}\right)}\sigma \sqrt{T}$ is the truncated density function of variable ${z}_{T}$ where
$1={\int }_{{x}_{T}\ge {x}_{0}}\frac{g\left({x}_{T}\right)}{G\left({x}_{T}\ge {x}_{0}\right)}\text{d}{x}_{T}={\int }_{{z}_{T}\ge {z}_{0}}\frac{g\left(\sigma \sqrt{T}{z}_{T}\right)}{G\left({z}_{T}\ge
{z}_{0}\right)}\sigma \sqrt{T}\text{d}{z}_{T}$
from the change of variable formula where g and G denote the density and distribution functions of ${x}_{T}$ respectively, and $\sigma \sqrt{T}\text{d}{z}_{T}=\text{d}{x}_{T}$ since ${z}_{T}={x}_{T}/
\sigma \sqrt{T}$ by definition. Note ${z}_{T}$ has a standard normal, ${z}_{T}^{2}$ has a central ${\chi }_{\left(1\right)}^{2}$ distribution respectively.
Case 1: if ${z}_{0}\ge 0$ , then
$E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)=E\left({z}_{T}^{2}|{z}_{T}^{2}\ge {z}_{0}^{2}\right)=\frac{1-E\left({z}_{T}^{2}|{z}_{T}^{2}<{z}_{0}^{2}\right)\text{}{F}_{\text{1,0}}\left({z}_{0}^{2}\
$\begin{array}{c}E\left({z}_{T}^{2}|{z}_{T}^{2}\ge {z}_{0}^{2}\right)=E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}{z}_{T}<-{z}_{0}\right)\\ =E\left({z}_
{T}^{2}|{z}_{T}\ge {z}_{0}\right)\frac{\mathrm{Pr}\left[{z}_{T}>{z}_{0}\right]}{\mathrm{Pr}\left[{z}_{T}^{2}\ge {z}_{0}^{2}\right]}+E\left({z}_{T}^{2}|{z}_{T}<-{z}_{0}\right)\frac{\mathrm{Pr}\left
[{z}_{T}<-{z}_{0}\right]}{\mathrm{Pr}\left[{z}_{T}^{2}\ge {z}_{0}^{2}\right]}\\ =E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)\end{array}$(12)
from $\frac{\mathrm{Pr}\left[{z}_{T}\ge {z}_{0}\right]}{\mathrm{Pr}\left[{z}_{T}^{2}\ge {z}_{0}^{2}\right]}=\frac{\mathrm{Pr}\left[{z}_{T}<-{z}_{0}\right]}{\mathrm{Pr}\left[{z}_{T}^{2}\ge {z}_{0}^{2}
\right]}=\frac{1}{2}$ and
$E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)={\int }_{{z}_{0}}^{\infty }{z}_{T}^{2}\varphi \left({z}_{T}\right)\text{d}{z}_{T}={\int }_{-\infty }^{-{z}_{0}}{z}_{T}^{2}\varphi \left({z}_{T}\right)\
using the symmetry of normal distribution; and
$E\left({z}_{T}^{2}|{z}_{T}^{2}\ge {z}_{0}^{2}\right)=\frac{1-E\left({z}_{T}^{2}|{z}_{T}^{2}<{z}_{0}^{2}\right)\text{}{F}_{\text{1,0}}\left({z}_{0}^{2}\right)}{\text{1}-{F}_{\text{1,0}}\left({z}_{0}^
solving following equation for $E\left({z}_{T}^{2}|{z}_{T}^{2}\ge {z}_{0}^{2}\right)$
$1=E\left({z}_{T}^{2}\right)=E\left({z}_{T}^{2}|{z}_{T}^{2}<{z}_{0}^{2}\right)\text{}{F}_{\text{1,0}}\left({z}_{0}^{2}\right)+E\left({z}_{T}^{2}|{z}_{T}^{2}\ge {z}_{0}^{2}\right)\left[\text{1}-{F}_{\
for the final equality of (11). Further note
$E\left({z}_{T}^{2}|{z}_{T}^{2}<{z}_{0}^{2}\right)=2\frac{\Gamma \left(\frac{3}{2}\right)}{\Gamma \left(\frac{1}{2}\right)}\frac{{F}_{3,0}\left({z}_{0}^{2}\right)-{F}_{3,0}\left(0\right)}{{F}_{1,0}\
from Marchand ( [10] : p. 26 and Remark 4), where $\Gamma \left(\frac{1}{2}\right)=\sqrt{\text{π}}$ and $\Gamma \left(\frac{3}{2}\right)=\frac{\sqrt{\text{π}}}{2}$ , where $h\left(0,1,0\right)={F}_
{1,0}\left({z}_{0}^{2}\right)-{F}_{1,0}\left(0\right)$ and $h\left(0,3,0\right)={F}_{3,0}\left({z}_{0}^{2}\right)-{F}_{3,0}\left(0\right)$ in Marchand ( [10] : p. 26 and Remark 4) where ${F}_{1,0}\
left(0\right)={F}_{3,0}\left(0\right)=0$ with p = 1, $\alpha =1$ and $\lambda =0$ that is a non-centrality parameter.
Plugging (15) into (11) results in
$E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)=\frac{1-\frac{{F}_{3,0}\left({z}_{0}^{2}\right)}{{F}_{1,0}\left({z}_{0}^{2}\right)}{F}_{\text{1,0}}\left({z}_{0}^{2}\right)}{\text{1}-{F}_{\text{1,0}}\
left({z}_{0}^{2}\right)}=\frac{1-{F}_{3,0}\left({z}_{0}^{2}\right)}{\text{1}-{F}_{\text{1,0}}\left({z}_{0}^{2}\right)}$ . (16)
Case 2: ${z}_{0}<0$
$\begin{array}{l}E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)\\ =E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0},{z}_{0}\le {z}_{T}<-{z}_{0}\right)\mathrm{Pr}\left[{z}_{0}\le {z}_{T}<-{z}_{0}\right]\\ \text{\
hspace{0.17em}}\text{\hspace{0.17em}}+E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0},-{z}_{0}<{z}_{T}\right)\mathrm{Pr}\left[-{z}_{0}<{z}_{T}\right]\\ =E\left({z}_{T}^{2}|{z}_{T}^{2}<{z}_{0}^{2}\right)\left
[1-2\Phi \left({z}_{0}\right)\right]+E\left({z}_{T}^{2}|-{z}_{0}<{z}_{T}\right)\left[1-\Phi \left(-{z}_{0}\right)\right]\end{array}$
$\begin{array}{l}=E\left({z}_{T}^{2}|{z}_{T}^{2}<{z}_{0}^{2}\right)\left[1-2\Phi \left({z}_{0}\right)\right]+E\left({z}_{T}^{2}|{z}_{0}^{2}<{z}_{T}^{2}\right)\left[1-\Phi \left(-{z}_{0}\right)\right]
\\ =E\left({z}_{T}^{2}|{z}_{T}^{2}<{z}_{0}^{2}\right)\left[1-2\Phi \left({z}_{0}\right)\right]+\frac{1-E\left({z}_{T}^{2}|{z}_{T}^{2}<{z}_{0}^{2}\right){F}_{1,0}\left({z}_{0}^{2}\right)}{1-{F}_{1,0}\
left({z}_{0}^{2}\right)}\left[1-\Phi \left(-{z}_{0}\right)\right]\\ =\frac{{F}_{3,0}\left({z}_{0}^{2}\right)}{{F}_{1,0}\left({z}_{0}^{2}\right)}\left[1-2\Phi \left({z}_{0}\right)\right]+\frac{1-{F}_
{3,0}\left({z}_{0}^{2}\right)}{1-{F}_{1,0}\left({z}_{0}^{2}\right)}\left[1-\Phi \left(-{z}_{0}\right)\right]\end{array}$(17)
from (12) for the third equality, from (14) for the fourth equality and from (16) for the final equality.
Consequently we get,
$\begin{array}{c}{V}_{p}^{2}=E\left({\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)\right]}^{2}|\Omega \right)-{\left(E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)|\Omega \right]\right)}^{2}\\ ={x}
_{0}^{2}\Phi \left({z}_{0}\right)+T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)\left[1-\Phi \left({z}_{0}\right)\right]-{\left({x}_{0}\Phi \left({z}_{0}\right)+\sigma \sqrt{T}\varphi \
from (5) and (9). $\square$
Proof of Theorem 2.5: Note the covariance between non-hedging and put option conditional on $\Omega$ is defined as:
$\begin{array}{l}E\left[\left(\mathrm{max}\left({x}_{T},{x}_{0}\right)-p-E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)-p|\Omega \right]\right){x}_{T}|\Omega \right]\\ =E\left[\left(\mathrm{max}\
left({x}_{T},{x}_{0}\right)-E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)|\Omega \right]\right){x}_{T}|\Omega \right]\\ =E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right){x}_{T}|\Omega \right]-E\left
[\mathrm{max}\left({x}_{T},{x}_{0}\right)|\Omega \right]E\left({x}_{T}|\Omega \right)\\ =E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right){x}_{T}|\Omega \right]\end{array}$
since $E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right)|\Omega \right]$ is constant conditional on $\Omega$ for the second equality and the fourth equality holds from $E\left({x}_{T}|\Omega \right)=0$
Now the claimed result is derived since
$\begin{array}{l}E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right){x}_{T}|\Omega \right]\\ =E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right){x}_{T}|{x}_{T}<{x}_{0},\Omega \right]\mathrm{Pr}\left({x}_{T}
<{x}_{0}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+E\left[\mathrm{max}\left({x}_{T},{x}_{0}\right){x}_{T}|{x}_{T}\ge {x}_{0},\Omega \right]\mathrm{Pr}\left({x}_{T}\ge {x}_
{0}\right)\\ ={x}_{0}E\left({x}_{T}|{x}_{T}<{x}_{0},\Omega \right)\mathrm{Pr}\left({x}_{T}<{x}_{0}\right)+E\left({x}_{T}^{2}|{x}_{T}\ge {x}_{0},\Omega \right)\mathrm{Pr}\left({x}_{T}\ge {x}_{0}\
right)\\ ={x}_{0}E\left({x}_{T}|{x}_{T}<{x}_{0},\Omega \right)\Phi \left({z}_{0}\right)+E\left({x}_{T}^{2}|{x}_{T}\ge {x}_{0},\Omega \right)\left[1-\Phi \left({z}_{0}\right)\right]\\ =-{x}_{0}\sigma
\sqrt{T}\varphi \left({z}_{0}\right)+T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}\ge {z}_{0}\right)\left[1-\Phi \left({z}_{0}\right)\right].\end{array}$
from (10) and
$E\left({x}_{T}|{x}_{T}<{x}_{0},\Omega \right)=-\sigma \sqrt{T}\frac{\varphi \left({z}_{0}\right)}{\Phi \left({z}_{0}\right)}$(18)
from Greene ( [6] : p. 759) for the last two equations. $\square$
Proof of Theorem 2.6: Note the expected return for forward is the value of following:
$\frac{{\Theta }^{\prime }\left({S}_{0}-{F}_{T}-C\right)}{{\Theta }^{\prime }{S}_{0}}\cong {s}_{0}-{f}_{T}-c$(19)
assuming ${F}_{T}-{S}_{0}$ is small. Its variance is obviously zero since the return is not random. $\square$
Proof of Theorem 2.7: (a) Note the outflow of buying call option at time T is given as $\mathrm{min}\left({S}_{T},K\right)+P$ . Thus its return normalized by ${S}_{0}$ is given as the negative value
of following:
$\begin{array}{l}-\frac{{\Theta }^{\prime }\left[\mathrm{min}\left({S}_{T},K\right)+P-{S}_{0}\right]}{{\Theta }^{\prime }{S}_{0}}\\ =-\mathrm{min}\left(\frac{{\Theta }^{\prime }\left[{S}_{T}-{S}_{0}\
right]}{{\Theta }^{\prime }{S}_{0}},\frac{{\Theta }^{\prime }\left[K-{S}_{0}\right]}{{\Theta }^{\prime }{S}_{0}}\right)-\frac{{\Theta }^{\prime }P}{{\Theta }^{\prime }{S}_{0}}\\ \cong -\mathrm{min}\
left({s}_{T}-{s}_{0},k-{s}_{0}\right)-p\equiv -\mathrm{min}\left({x}_{T},{x}_{0}\right)-p\end{array}$(20)
assuming ${S}_{T}-{S}_{0}$ and $K-{S}_{0}$ are small.
Now the expected return conditional on $\Omega$ is value of following:
$-E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)|\Omega \right]-p=\sigma \sqrt{T}\varphi \left({z}_{0}\right)-{x}_{0}\left[1-\Phi \left({z}_{0}\right)\right]-p$(21)
from (17) where ${x}_{T}\equiv {s}_{T}-{s}_{0}$ , since
$\begin{array}{c}E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)|\Omega \right]=E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)|{x}_{T}<{x}_{0},\Omega \right]\mathrm{Pr}\left({x}_{T}<{x}_{0}\right)\\
\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)|{x}_{T}\ge {x}_{0},\Omega \right]\mathrm{Pr}\left({x}_{T}\ge {x}_{0}\right)\\ =E\left({x}_{T}|{x}_
{T}<{x}_{0},\Omega \right)\mathrm{Pr}\left({x}_{T}<{x}_{0}\right)+{x}_{0}\mathrm{Pr}\left({x}_{T}\ge {x}_{0}\right)\\ =-\sigma \sqrt{T}\frac{\varphi \left({z}_{0}\right)}{\Phi \left({z}_{0}\right)}\
Phi \left({z}_{0}\right)+{x}_{0}\left[1-\Phi \left({z}_{0}\right)\right]\\ =-\sigma \sqrt{T}\varphi \left({z}_{0}\right)+{x}_{0}\left[1-\Phi \left({z}_{0}\right)\right]\end{array}$(22)
from the definition of conditional expectation, where ${x}_{T}~N\left(0,T{\sigma }^{2}\right)$ from Assumption 2.1 and (18)
$\mathrm{Pr}\left({x}_{T}<{x}_{0}\right)=\mathrm{Pr}\left(\frac{{x}_{T}}{\sigma \sqrt{T}}<\frac{{x}_{0}}{\sigma \sqrt{T}}\right)=\mathrm{Pr}\left({z}_{T}<{z}_{0}\right)\equiv \Phi \left({z}_{0}\
where ${z}_{0}={x}_{0}/\sigma \sqrt{T}$ and ${z}_{T}={x}_{T}/\sigma \sqrt{T}$ .
(b) The return’s variance of call option conditional on $\Omega$ is given as:
$\begin{array}{l}E{\left(\mathrm{min}\left({x}_{T},{x}_{0}\right)-E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)|\Omega \right]|\Omega \right)}^{2}\\ =E\left({\left[\mathrm{min}\left({x}_{T},{x}_{0}
\right)\right]}^{2}|\Omega \right)-{\left(E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)|\Omega \right]\right)}^{2}\end{array}$(24)
Note the second term of right hand side in (24) is derived from (21) directly. Then the first term of right hand side in (24) is arranged as:
$\begin{array}{c}E\left({\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)\right]}^{2}|\Omega \right)=E\left({\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)\right]}^{2}|{x}_{T}<{x}_{0},\Omega \right)\
mathrm{Pr}\left({x}_{T}<{x}_{0}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E\left({\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)\right]}^{2}|{x}_{T}\ge {x}_{0},\Omega \right)\mathrm{Pr}\
left({x}_{T}\ge {x}_{0}\right)\\ =E\left({x}_{T}^{2}|{x}_{T}<{x}_{0},\Omega \right)\mathrm{Pr}\left({x}_{T}<{x}_{0}\right)+{x}_{0}^{2}\mathrm{Pr}\left({x}_{T}\ge {x}_{0}\right)\\ =E\left({x}_{T}^{2}|
{x}_{T}<{x}_{0},\Omega \right)\Phi \left({z}_{0}\right)+{x}_{0}^{2}\left[1-\Phi \left({z}_{0}\right)\right]\\ =T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}<{z}_{0}\right)\Phi \left({z}_{0}\right)+{x}_{0}
^{2}\left[1-\Phi \left({z}_{0}\right)\right]\end{array}$ (25)
text{if}\text{\hspace{0.17em}}{z}_{0}<0\\ =\frac{{F}_{3,0}\left({z}_{0}^{2}\right)}{{F}_{1,0}\left({z}_{0}^{2}\right)}\left[1-2\Phi \left(-{z}_{0}\right)\right]+\frac{1-{F}_{3,0}\left({z}_{0}^{2}\
right)}{1-{F}_{1,0}\left({z}_{0}^{2}\right)}\Phi \left(-{z}_{0}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{z}_{0}\ge 0\end{array}$
from $E\left({x}_{T}^{2}|{x}_{T}<{x}_{0}\right)=T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}<{z}_{0}\right)$ as similarly in (10) and
Case 1: ${z}_{0}<0$
$E\left({z}_{T}^{2}|{z}_{T}<{z}_{0}\right)=E\left({\left(-{z}_{T}\right)}^{2}|-{z}_{T}\ge -{z}_{0}\right)=E\left({z}_{T}^{2}|{z}_{T}^{2}\ge {z}_{0}^{2}\right)=\frac{1-{F}_{3,0}\left({z}_{0}^{2}\
right)}{1-{F}_{\text{1,0}}\left({z}_{0}^{2}\right)}$ (26)
From symmetry and ${z}_{T}$ has a standard normal distribution, from (12) for the second equality, from (11) and (16) for the final equality.
Case 2: ${z}_{0}\ge 0$
$\begin{array}{c}E\left({z}_{T}^{2}|{z}_{T}<{z}_{0}\right)=E\left({z}_{T}^{2}|{z}_{T}<{z}_{0},-{z}_{0}<{z}_{T}<{z}_{0}\right)\mathrm{Pr}\left[-{z}_{0}<{z}_{T}<{z}_{0}\right]\\ \text{\hspace{0.17em}}\
text{\hspace{0.17em}}+E\left({z}_{T}^{2}|{z}_{T}<{z}_{0},{z}_{T}<-{z}_{0}\right)\mathrm{Pr}\left[{z}_{T}<-{z}_{0}\right]\\ =E\left({z}_{T}^{2}|{z}_{T}^{2}<{z}_{0}^{2}\right)\left[1-2\Phi \left(-{z}_
{0}\right)\right]+E\left({z}_{T}^{2}|{z}_{T}<-{z}_{0}\right)\Phi \left(-{z}_{0}\right)\\ =\frac{{F}_{3,0}\left({z}_{0}^{2}\right)}{{F}_{1,0}\left({z}_{0}^{2}\right)}\left[1-2\Phi \left(-{z}_{0}\
right)\right]+\frac{1-{F}_{3,0}\left({z}_{0}^{2}\right)}{1-{F}_{1,0}\left({z}_{0}^{2}\right)}\Phi \left(-{z}_{0}\right)\end{array}$(27)
from (15) and (16) for the final equality. Consequently, we get,
$\begin{array}{c}{V}_{c}^{2}=E\left({\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)\right]}^{2}|\Omega \right)-{\left(E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)|\Omega \right]\right)}^{2}\\ =T{\
sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}<{z}_{0}\right)\Phi \left({z}_{0}\right)+{x}_{0}^{2}\left[1-\Phi \left({z}_{0}\right)\right]-{\left(-\sigma \sqrt{T}\varphi \left({z}_{0}\right)+{x}_{0}\left[1-\
Phi \left({z}_{0}\right)\right]\right)}^{2}\end{array}$
from (25) and (22). $\square$
Proof of Theorem 2.8: Note the covariance between non-hedging and call option conditional on $\Omega$ is defined as:
$\begin{array}{l}E\left[\left(-\mathrm{min}\left({x}_{T},{x}_{0}\right)-p-E\left[-\mathrm{min}\left({x}_{T},{x}_{0}\right)-p|\Omega \right]\right)\left(-{x}_{T}\right)|\Omega \right]\\ =E\left[\left
(\mathrm{min}\left({x}_{T},{x}_{0}\right)-E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)|\Omega \right]\right){x}_{T}|\Omega \right]\\ =E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right){x}_{T}|\Omega
\right]-E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right)|\Omega \right]E\left({x}_{T}|\Omega \right)\\ =E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right){x}_{T}|\Omega \right]\end{array}$
since the fourth equality holds from $E\left({x}_{T}|\Omega \right)=0$ .
Now the claimed result is derived since
$\begin{array}{l}E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right){x}_{T}|\Omega \right]\\ =E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right){x}_{T}|{x}_{T}<{x}_{0},\Omega \right]\mathrm{Pr}\left({x}_{T}
<{x}_{0}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+E\left[\mathrm{min}\left({x}_{T},{x}_{0}\right){x}_{T}|{x}_{T}\ge {x}_{0},\Omega \right]\mathrm{Pr}\left({x}_{T}\ge {x}_{0}\
right)\\ =E\left({x}_{T}^{2}|{x}_{T}<{x}_{0},\Omega \right)\mathrm{Pr}\left({x}_{T}<{x}_{0}\right)+{x}_{0}E\left({x}_{T}|{x}_{T}\ge {x}_{0},\Omega \right)\mathrm{Pr}\left({x}_{T}\ge {x}_{0}\right)\
$\begin{array}{l}=E\left({x}_{T}^{2}|{x}_{T}<{x}_{0},\Omega \right)\Phi \left({z}_{0}\right)+{x}_{0}E\left({x}_{T}|{x}_{T}\ge {x}_{0},\Omega \right)\left[1-\Phi \left({z}_{0}\right)\right]\\ =T{\
sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}<{z}_{0}\right)\Phi \left({z}_{0}\right)+{x}_{0}\sigma \sqrt{T}\frac{\varphi \left({z}_{0}\right)}{1-\Phi \left({z}_{0}\right)}\left[1-\Phi \left({z}_{0}\right)\
right]\\ =T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}<{z}_{0}\right)\Phi \left({z}_{0}\right)+{x}_{0}\sigma \sqrt{T}\varphi \left( z 0 \right)\end{array}$
from (6) and from $E\left({x}_{T}^{2}|{x}_{T}<{x}_{0}\right)=T{\sigma }^{2}E\left({z}_{T}^{2}|{z}_{T}<{z}_{0}\right)$ as similarly in (10) for the last two equations. $\square$
Proof of Theorem 4.1: To get such a solution point, we solve following three equations:
$V=\frac{R-\alpha }{\beta }$(30)
from the UML $R=\alpha +\beta \text{}V$ . Note
$w\equiv \frac{R-{R}_{p}}{-{R}_{p}}$ and $1-w\equiv \frac{R}{{R}_{p}}$(31)
from (28). Therefore
${\left(\frac{R-\alpha }{\beta }\right)}^{2}={\left(\frac{R-{R}_{p}}{-{R}_{p}}\right)}^{2}{V}_{n}^{2}+{\left(\frac{R}{{R}_{p}}\right)}^{2}{V}_{p}^{2}+2Co{v}_{pn}\left(\frac{R-{R}_{p}}{-{R}_{p}}\
by plugging (30) and (31) into (29). Then we solve (32) as
$\frac{{R}_{p}{}^{2}}{{\beta }^{2}}{\left(R-\alpha \right)}^{2}={\left(R-{R}_{p}\right)}^{2}{V}_{n}^{2}+{R}^{2}{V}_{p}^{2}-2Co{v}_{pn}\left(R-{R}_{p}\right)R$ .
$\begin{array}{l}\left[\frac{{R}_{p}^{2}}{{\beta }^{2}}-{V}_{n}^{2}-{V}_{p}^{2}+2Co{v}_{pn}\right]{R}^{2}+2\left[-\alpha \frac{{R}_{p}^{2}}{{\beta }^{2}}+{R}_{p}{V}_{n}^{2}-Co{v}_{pn}{R}_{p}\right]R\
\ \text{ }+\frac{{\alpha }^{2}{R}_{p}^{2}}{{\beta }^{2}}-{R}_{p}^{2}{V}_{n}^{2}=0\end{array}$(33)
which is the second order polynomial equation of unknown R.
Now solving (33) results in two roots
${\overline{R}}_{1}=\frac{-b+\sqrt{{b}^{2}-ac}}{a}$ and ${\overline{R}}_{2}=\frac{-b-\sqrt{{b}^{2}-ac}}{a}$
assuming ${b}^{2}-ac\ge 0$ where
$a=\frac{{R}_{p}^{2}}{{\beta }^{2}}-{V}_{n}^{2}-{V}_{p}^{2}+2Co{v}_{pn}$ , $b=-\alpha \frac{{R}_{p}^{2}}{{\beta }^{2}}+{R}_{p}{V}_{n}^{2}-Co{v}_{pn}{R}_{p}$ , $c=\frac{{\alpha }^{2}{R}_{p}^{2}}{{\
beta }^{2}}-{R}_{p}^{2}{V}_{n}^{2}$ .
Then we select ${\overline{V}}_{i}=\frac{{\overline{R}}_{i}-\alpha }{\beta }$ for $i=1,2$ . $\square$
^1The number of countries with floating and free floating arrangements are 36 and 29 by 2014, respectively, according to IMF (https://www.imf.org/external/pubs/nft/2014/areaers/ar2014.pdf).
^2U.S. Department of Commerce, Trade Finance Guide, Ch. 12, “Foreign Exchange Risk Management,” http://trade.gov/publications/pdfs/tfg2008ch12.pdf.
^3Kim (2013) [9] considered a single currency case and just for selling case. So its practical applications are very limited.
^4American currency option and currency future are not considered in this paper because of the speculative nature. Thus, the focus is solely on the hedging of the FX risk.
^5It is a non-hedging and to buy the foreign currency at time T.
^6The value of the put option was derived by Garman and Kohlhagen (1983) [5] .
^7See Diebold and Nason (1990) [3] for this issue.
^8See Theorem 2.4 for the definitions.
^9Buying the foreign exchange means outflow of domestic currency. So, a negative of the forward amount is taken.
^10In case of call option, ${R}_{p}$ , ${V}_{p}^{2}$ and $Co{v}_{pn}$ are replaced by ${R}_{c}$ , ${V}_{c}^{2}$ and $Co{v}_{cn}$ respectively.
^11It is called as the capital allocation line in the portfolio theory.
^12It is called as the capital market line in the portfolio theory.
^13It is also equivalently written as $\rho =1-\frac{\stackrel{˜}{R}-{R}_{f}}{R\left({w}^{*}\right)-\stackrel{˜}{R}}$ from the property of the proportional triangular.
^14It is when the marginal cost of V is small and thus a riskless forward contract is not chosen.
^15Remind that any forward is not used in this case.
^16EXCEL code for optimal hedging ratio computation is available at https://blog.naver.com/yunyeongkim
^17It is negative for buying of foreign currency (also for the forward contract) because it means the outflow of domestic currency.
^18Note $E\left(x\right)={\int }_{A}^{}x\frac{f\left(x\right)}{\text{Pr}\left[A\right]}\text{d}x\text{Pr}\left[A\right]+{\int }_{B}^{}x\frac{f\left(x\right)}{\text{Pr}\left[B\right]}\text{d}x\text
{Pr}\left[B\right]=E\left(x|A\right)+E\left(x|B\right)$ where $x\in A\cup B$ . | {"url":"https://file.scirp.org/Html/4-1501426_87878.htm","timestamp":"2024-11-06T01:04:45Z","content_type":"application/xhtml+xml","content_length":"393923","record_id":"<urn:uuid:1785ca3a-6b49-45c2-a144-80f82abdc0dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00752.warc.gz"} |
How to Calculate the Odds of Winning a Lottery Prize - nightofthedayofthedawn
How to Calculate the Odds of Winning a Lottery Prize
A Toto Sidney is a form of gambling that uses the process of drawing numbers to determine which prize will be awarded. It is a popular and often legal way to raise money for various causes. The first
recorded lottery was in 15th-century France, where towns used the game to raise money to fortify their defenses or provide aid to the poor. The lottery became increasingly common in England and the
United States, where it was used to fund college education and public projects.
The Gambling Law of 1894 made lottery a crime in the United States, but it has since been revived and is now regulated by several state governments. In the US, most lottery games are state-sponsored
and require a small amount of money to purchase a ticket for the chance to win.
Lottery Math
The mathematics of lotteries can be challenging to understand, especially if you are unfamiliar with probability theory or factorials. This is because lottery prizes can be large and may have
multiple winners, each of whom receives a different number. In order to calculate the odds of winning a prize, you must multiply each number against every other number in the pool. This requires a
significant amount of time and effort to complete.
Many people enjoy playing the lottery because they think it is fun and they think it will give them a chance to become rich. However, if you are considering buying a lottery ticket, you should
consider the cost and monetary value of the prize before making your decision.
There are a few ways to calculate the odds of winning a prize, and they can all be found in the laws of probability. In addition, the probability of winning a prize is determined by how many people
are participating in the lottery.
If the probability of winning a prize is low enough, then the lottery will not be a wise financial decision. On the other hand, if the total utility of a lottery prize exceeds the monetary loss
caused by purchasing a lottery ticket, then the purchase of a lottery ticket can be considered to be a rational decision.
It is possible to account for the purchase of a lottery ticket by a model that models the expected utility maximization curve. A decision model that assumes a curvature of this curve can account for
the purchase of a lottery ticket because it can capture risk-seeking behavior and incorporate other non-monetary gains in determining the utility of a prize.
The lottery was very popular in the United States and England during the colonial era. It played an important role in financing roads, libraries, churches, colleges, canals, and bridges. It was also
used by local militias to pay soldiers and other workers.
The lottery is a type of gambling that uses the process of drawing numbers at random to determine which prize will be awarded. It can be used to raise money for various causes and is commonly
regulated by several state governments. The majority of state lotteries are organized by charitable organizations, but private companies often sponsor them as well. The lottery has also been used in
the sports world as a way to select players for specific teams. | {"url":"https://nightofthedayofthedawn.org/how-to-calculate-the-odds-of-winning-a-lottery-prize/","timestamp":"2024-11-13T21:55:28Z","content_type":"text/html","content_length":"31057","record_id":"<urn:uuid:67e44b0d-6fe3-4f8b-b917-6a372b345e63>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00870.warc.gz"} |
Resolution of the Maxwell equations in a domain with reentrant corners
In the case when the computational domain is a polygon with reentrant corners, we give a decomposition of the solution of Maxwell's equations into the sum of a regular part and a singular part. It is
proved that the space to which the singular part belongs is spanned by the solutions of a steady state problem. The precise regularity of the solution is given depending on the angle of the reentrant
corners. The mathematical decomposition is then used to introduce an algorithm for the numerical resolution of Maxwell's equations in presence of reentrant corners. This paper is a continuation of
the work exposed in [3]. The same methodology can be applied to the Helmholtz equation or to the Lamé system as well.
Dive into the research topics of 'Resolution of the Maxwell equations in a domain with reentrant corners'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/resolution-of-the-maxwell-equations-in-a-domain-with-reentrant-co-3","timestamp":"2024-11-02T15:29:12Z","content_type":"text/html","content_length":"52880","record_id":"<urn:uuid:a2c214ad-d926-4f16-8c30-8d332775d12a>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00328.warc.gz"} |