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Speed_of_electricity Knowpia
The word electricity refers generally to the movement of electrons (or other charge carriers) through a conductor in the presence of a potential difference or an electric field. The speed of this flow has multiple meanings. In everyday electrical and electronic devices, the signals travel as electromagnetic waves typically at 50%–99% of the speed of light, while the electrons themselves move much more slowly; see drift velocity and electron mobility.
The speed at which energy or signals travel down a cable is actually the speed of the electromagnetic wave traveling along (guided by) the cable. I.e., a cable is a form of a waveguide. The propagation of the wave is affected by the interaction with the material(s) in and surrounding the cable, caused by the presence of electric charge carriers (interacting with the electric field component) and magnetic dipoles (interacting with the magnetic field component). These interactions are typically described using mean field theory by the permeability and the permittivity of the materials involved. The energy/signal usually flows overwhelmingly outside the electric conductor of a cable; the purpose of the conductor is thus not to conduct energy, but to guide the energy-carrying wave.[1]: 360
Speed of electromagnetic waves in good dielectricsEdit
The speed of electromagnetic waves in a low-loss dielectric is given by[1]: 346
{\displaystyle v={\frac {1}{\sqrt {\varepsilon \mu }}}={\frac {c}{\sqrt {\varepsilon _{r}\mu _{r}}}}.}
{\displaystyle c}
= speed of light in vacuum.
{\displaystyle \mu _{0}}
= the permeability of free space = 4π x 10−7 H/m.
{\displaystyle \mu _{r}}
= relative magnetic permeability of the material. Usually in good dielectrics, eg. vacuum, air, Teflon,
{\displaystyle \mu _{r}=1}
{\displaystyle \mu =\mu _{r}\mu _{0}}
{\displaystyle \varepsilon _{0}}
= the permitivity of free space = 8.854 x 10−12 F/m.
{\displaystyle \varepsilon _{r}}
= relative permitivity of the material. Usually in good conductors eg. copper, silver, gold,
{\displaystyle \varepsilon _{r}=1}
{\displaystyle \varepsilon =\varepsilon _{r}\varepsilon _{0}}
Speed of electromagnetic waves in good conductorsEdit
The speed of electromagnetic waves in a good conductor is given by[1]: 360 [2]: 142 [3]: 50–52
{\displaystyle v={\sqrt {\frac {2\omega }{\sigma \mu }}}={\sqrt {\frac {4\pi }{\sigma _{c}\mu _{0}}}}{\sqrt {\frac {f}{\sigma _{r}\mu _{r}}}}\approx \left(0.41~\mathrm {m/s} \right){\sqrt {\frac {f/(1~\mathrm {Hz} )}{\sigma _{r}\mu _{r}}}}.}
{\displaystyle f}
{\displaystyle \omega }
= angular frequency = 2πf.
{\displaystyle \sigma _{c}}
= conductivity of annealed copper = 5.96×107 S/m.
{\displaystyle \sigma _{r}}
= conductivity of the material relative to the conductivity of copper. For hard drawn copper
{\displaystyle \sigma _{r}}
may be as low as 0.97.
{\displaystyle \sigma =\sigma _{r}\sigma _{c}}
and permeability is defined as above in § Speed of electromagnetic waves in good dielectrics
{\displaystyle \mu _{0}}
{\displaystyle \mu _{r}}
= relative magnetic permeability of the material. Magnetically conductive materials such as copper typically have a
{\displaystyle \mu _{r}}
{\displaystyle \mu =\mu _{r}\mu _{0}}
In copper at 60 Hz,
{\displaystyle v\approx }
3.2 m/s. As a consequence of Snell's Law and the extremely low speed, electromagnetic waves always enter good conductors in a direction that is within a milliradian of normal to the surface, regardless of the angle of incidence. This velocity is the speed with which electromagnetic waves penetrate into the conductor and is not the drift velocity of the conduction electrons.
Electromagnetic waves in circuitsEdit
In the theoretical investigation of electric circuits, the velocity of propagation of the electromagnetic field through space is usually not considered; the field is assumed, as a precondition, to be present throughout space. The magnetic component of the field is considered to be in phase with the current, and the electric component is considered to be in phase with the voltage. The electric field starts at the conductor, and propagates through space at the velocity of light (which depends on the material it is traveling through). Note that the electromagnetic fields do not move through space. It is the electromagnetic energy that moves; the corresponding fields simply grow and decline in a region of space in response to the flow of energy. At any point in space, the electric field corresponds not to the condition of the electric energy flow at that moment, but to that of the flow at a moment earlier. The latency is determined by the time required for the field to propagate from the conductor to the point under consideration. In other words, the greater the distance from the conductor, the more the electric field lags.[4]
Since the velocity of propagation is very high – about 300,000 kilometers per second – the wave of an alternating or oscillating current, even of high frequency, is of considerable length. At 60 cycles per second, the wavelength is 5,000 kilometers, and even at 100,000 hertz, the wavelength is 3 kilometers. This is a very large distance compared to those typically used in field measurement and application.[4]
The important part of the electric field of a conductor extends to the return conductor, which usually is only a few feet distant. At greater distance, the aggregate field can be approximated by the differential field between conductor and return conductor, which tend to cancel. Hence, the intensity of the electric field is usually inappreciable at a distance which is still small compared to the wavelength. Within the range in which an appreciable field exists, this field is practically in phase with the flow of energy in the conductor. That is, the velocity of propagation has no appreciable effect unless the return conductor is very distant, or entirely absent, or the frequency is so high that the distance to the return conductor is an appreciable portion of the wavelength.[4]
Electric driftEdit
The drift velocity deals with the average velocity of a particle, such as an electron, due to an electric field. In general, an electron will propagate randomly in a conductor at the Fermi velocity.[5] Free electrons in a conductor follow a random path. Without the presence of an electric field, the electrons have no net velocity. When a DC voltage is applied, the electron drift velocity will increase in speed proportionally to the strength of the electric field. The drift velocity in a 2 mm diameter copper wire in 1 ampere current is approximately 8 cm per hour. AC voltages cause no net movement; the electrons oscillate back and forth in response to the alternating electric field (over a distance of a few micrometers – see example calculation).
^ a b c Hayt, William H. (1989), Engineering Electromagnetics (5th ed.), McGraw-Hill, ISBN 0070274061
^ Balanis, Constantine A. (2012), Engineering Electromagnetics (2nd ed.), Wiley, ISBN 978-0-470-58948-9
^ Harrington, Roger F. (1961), Time-Harmonic Electromagnetic Fields, McGraw-Hill, ISBN 0-07-026745-6
^ Academic Press dictionary of science and technology By Christopher G. Morris, Academic Press.
Alfvén, H. (1950). Cosmical electrodynamics. Oxford: Clarendon Press
Alfvén, H. (1981). Cosmic plasma. Taylor & Francis US.
"Velocity of Propagation of Electric Field", Theory and Calculation of Transient Electric Phenomena and Oscillations by Charles Proteus Steinmetz, Chapter VIII, p. 394-, McGraw-Hill, 1920.
Fleming, J. A. (1911). Propagation of electric currents in telephone & telegraph conductors. New York: Van Nostrand
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Catenary - Simple English Wikipedia, the free encyclopedia
Plots of
{\displaystyle y=a\cosh \left({\frac {x}{a}}\right)}
{\displaystyle a=0.5,1,2}
{\displaystyle x}
is on the horizontal axis and
{\displaystyle y}
is on the vertical axis.
A chain hanging like this forms the shape of a catenary approximately
A catenary is a type of curve. An ideal chain hanging between two supports and acted on by a uniform gravitational force makes the shape of a catenary.[1] (An ideal chain is one that can bend perfectly, cannot be stretched and has the same density throughout.[2]) The supports can be at different heights and the shape will still be a catenary.[3] A catenary looks a bit like a parabola, but they are different.[4]
The equation for a catenary in Cartesian coordinates is[2][5]
{\displaystyle y=a\cosh \left({\frac {x}{a}}\right)}
{\displaystyle a}
is a parameter that determines the shape of the catenary[5] and cosh is the hyperbolic cosine function, which is defined as[6]
{\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}}
Hence, we can also write the catenary equation as
{\displaystyle y={\frac {a\left(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}}\right)}{2}}}
The word "catenary" comes from the Latin word catena, which means "chain".[6] A catenary is also called called an alysoid and a chainette.[1]
↑ 1.0 1.1 "Catenary". Wolfram Research. Retrieved 2016-10-30.
↑ 2.0 2.1 "The Catenary - The "Chain" Curve". California State University. Retrieved 2019-01-01.
↑ Rosbjerg, Bo. "Catenary" (PDF). Aalborg University. Retrieved 2016-10-30.
↑ "Catenary and Parabola Comparison". Drexel University. Retrieved 2016-11-05.
↑ 5.0 5.1 "Equation of Catenary". Math24.net. Archived from the original on 2016-10-21. Retrieved 2016-10-30.
↑ 6.0 6.1 Stroud, K. A.; Booth, Dexter J. (2013). Engineering Mathematics (7th ed.). Palgrave Macmillan. p. 438. ISBN 978-1-137-03120-4.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Catenary&oldid=7560694"
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Types of Lines - Course Hero
College Algebra/Graphing Lines/Types of Lines
x = a
y = b
and
b
The equations of horizontal and vertical lines look slightly different from the equations of other lines.
On a vertical line, every point has the same
x
-coordinate. For any two points on the line, the change in the
x
-values, or run, is zero. So, the denominator is zero in the slope formula:
m=\frac{\text{Rise}}{\text{Run}}
Therefore, the slope of a vertical line is undefined. The equation of the line cannot be written in slope-intercept form. To write the equation of a vertical line, consider several points on the line. If the
x
a
(a,0)
is on the line. So are the points
(a, 1)
(a, 2)
(a, 3)
, and so on. The
x
-value of every point on the line is
a
. So, the equation is written as
x = a
. On a horizontal line, every point has the same
y
-coordinate. For any two points on a horizontal line, the change in the
y
-values, or rise, is zero. So, the slope formula has a numerator of zero. Therefore, the slope of a horizontal line is zero. If the
y
b
, the equation of a horizontal line can be written in slope-intercept form as:
\begin{aligned}y&=0x+b\\y&=b\end{aligned}
Every point on the line has an
x
-coordinate of 5.
The slope of the line is undefined, and there is no
y
-intercept, so the line cannot be written in slope-intercept form.
x=5
Every point on the line has a
y
The slope of the line is zero, and the
y
-intercept is –3. The slope-intercept form is
y=0x-3
y=-3
Parallel lines have the same slope, but different
y
Parallel lines are lines in the same plane that do not intersect. If two parallel lines are not vertical, they must have the same slope. Vertical lines have a slope that is undefined. Any pair of vertical lines are parallel.
The graphs show examples of parallel lines. In the graph showing
y=3x+5
y=3x-2
, both lines have a slope of 3. In the graph showing
y=4
y=-1
, both lines have a slope of zero.
The point-slope form is commonly used to find the equation of a line parallel to another line that passes through a given point. Determine the slope of the given line, and substitute the slope and the coordinates of the point into the point-slope form:
y-y_1=m(x-x_1)
Determining the Equation of a Parallel Line
Determine the equation of a line that passes through point
(5, -1)
and is parallel to the given equation:
y=3x-8
First, determine the slope of the given equation.
The given equation is in slope-intercept form, where
m
b
y
\begin{aligned}y&=mx+b\\y&=3x-8\end{aligned}
The slope of the given equation is 3.
Parallel lines have the same slope. So, the slope of the parallel line will also be 3.
Substitute the slope, 3, and the coordinates of the point
(5, -1)
into the point-slope form.
\begin{aligned}y-y_1&=m(x-x_1)\\y-(-1)&=3(x-5)\end{aligned}
Simplify the left side of the equation.
\begin{aligned}y-(-1)=3(x-5)\\y+1=3(x-5)\end{aligned}
To write the equation in slope-intercept form, first distribute the 3. Then subtract 1 from both sides.
\begin{aligned}y+1&=3(x-5)\\y+1&=3x-15\\ y+1-1&=3x-15-1\\y&=3x-16\end{aligned}
Perpendicular lines are lines that intersect at a 90° angle. When multiplied together, their slopes have a product of –1. That means that they are opposite reciprocals. If a line has slope
m
, then the slope of a perpendicular line has slope
-\frac{1}{m}
The graphs show examples of perpendicular lines. The slope of one line is the opposite reciprocal of the other.
Determining the Equation of a Perpendicular Line
Identify the equation of a line passing through point
(7, 2)
that is perpendicular to the line of the given equation:
y=-2x+1
Determine the slope of the given equation.
m
b
y
\begin{aligned}y&=mx+b\\y&=-2x+1\end{aligned}
Identify the slope of the line perpendicular to the line of the given equation.
Perpendicular lines have opposite reciprocals. So, the slope of the line perpendicular to the line of the given equation is:
-\left (\frac{1}{-2}\right)=\frac{1}{2}
\frac{1}{2}
, which is the slope of the line perpendicular to the line of the given equation, and the coordinates of point
(7, 2)
\begin{aligned}y-y_1&=m(x-x_1)\\y-2&=\frac{1}{2}(x-7)\end{aligned}
To rewrite the equation in slope-intercept form, first distribute
\frac{1}{2}
. Then add 2 to both sides.
\begin{aligned}y-2&=\frac{1}{2}(x-7)\\y-2&=\frac{1}{2}x-\frac{7}{2}\\y-2+2&=\frac{1}{2}x-\frac{7}{2}+2\\y&=\frac{1}{2}x-\frac{7}{2}+\frac{4}{2}\\y&=\frac{1}{2}x-\frac{3}{2}\end{aligned}
<Forms of Linear Equations
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Week 2. Data Structures | Algorithms and Data Structures
3 Week 2. Data Structures
Reading 2 Goodrich, Tamassia, & Goldwasser: Chapter (2), 3
3.1.1 LinkedList and ArrayList
3.1.2 ADT (Abstract Data Type)
3.1.3 The Array ADT
3.1.4 The List ADT
3.1.5 Algorithm Theory
In Java, you are probably familiar with the following Data Structures or Data Types:
Array is a primitive data type. Let’s leave that aside for now.
Both ArrayList and LinkedList are classes inheriting the abstract class List. Therefore they share a number of methods.
add(index,element)
There are others, but these three are particularly interesting from an algorithmic point of view. We shall discuss their complexity.
Definition 3.1 (Abstract Data Type) An Abstract Data Type (ADT) is a set of operations.
In Object Oriented Programming the set of operations will normally be a set of methods. Java can codify an ADT as an interface, and to implement the ADT a class must implement the interface. However, the ADT define the semantics (behaviour) of the methods, while the interface only defines the syntax (call signature). Thus an implementation of the interface is not necessarily an implementation of the ADT.
The ADT itself, is the mathematical model describing the interface and its semantics. An API, in contrast, is the interface of the implementation.
Static and Dynamic Definitions
The Array ADT would normally provides methods to
get the element at a given position
set (change) the element at a given position
The Java ArrayList class in Java provides many more methods. Most importantly, you can change the size of the ArrayList. Adding or removing an element at the end of the ArrayList is an important functional extension. When Arrays are used in the theoretical literature, the size is usually assumed to be fixed.
THe other methods of ArrayList are convenience methods which can be implemented using the methods above.
The List ADT is stateful. It always has a cursor pointing to the current position.
add an element at the current position
remove an element at the current position
next get the next element in the list
The ListIterator Notably, the list classes in Java does not implement an interface representing the List ADT. Instead, the List interface has the listIterator() which returns an object implementing the ListIterator interface, which represents the List ADT.
This makes it possible to access the same List concurrently in different subprograms, because each iterator maintains its own cursor.
AbstractList (Abstract Class)
LinkedList and ArrayList (Concrete Classes)
List (interface)
\sim
ListIterator (Java)
\sim
List (ADT)
LinkedList also implements the Deque ADT
The interesting questions in algorithm theory is how we implement the ADTs so that they are correct and efficient, and exactly how efficient each operation is in terms of complexity.
Cloning - shallow and deep copies
Problem 3.1 (Experimental Running Time) Compare the running time of LinkedList and ArrayList in Java.
Use the file ArrayListDemo.java as a basis. Compile and run the program, and look at the output as well as the source code. What does this test tell us about the performance of ArrayList?
It is possible that the program runs out of memory. In that case you may have to increase the heap size with an option to JVM.
Modify the program to use LinkedList instead of ArrayList. Compile and run the new version. What does it tell us about the performance of LinkedList? What are the advantages and disadvantages of LinkedList compared to ArrayList?
Modify the programs to initialise a smaller array, maybe about 100 elements, and test both ArrayList and LinkedList. How do the two implementations compare when the list is small?
Try with some different list sizes to see how the run time developes. Which operations run in constant time? Linear time? Slower than linear?
Review the theoretical properties of LinkedList and ArrayList, and compare your empirical output to the theory. Are the test results reasonable?
When would you prefer to use ArrayList and when would you prefer LinkedList? Try to find example applications where you would recommend one or the other.
Problem 3.2 (Optional) In the above tests, we measured in milliseconds, and mostly only a single operation at a time. This does not always allow us an exact comparison.
Use the java.lang.System.nanoTime() function instead.
Make several, similar operations in sequence, e.g. for instance 100 get() calls on adjacent indices.
Try to modify the test programs, and see if you can learn a bit more about the cases where the first test showed zero milliseconds.
(Remember that we rarely care much about the time of a single operations, but when the same operation is made thousands or millions of times, even a difference of less than a millisecond may matter.)
Problem 3.3 (Based on Goorich et al C-3.20) Suppose you are making a multiplayer game with
n\ge 1000
players, numbered from 1 to
n
, interacting in an enchanted forest. The winner is the player who first meets every other player. The game is constructed so that a function, meet
\left(i,j\right)
is called every time player
i
meets player
j
Design the algorithm to be implemented in meet, so that it records who has met whom, and reliably detects when someone has won (i.e. has met every other player).
You should also make an init() algorithm to initialise the data structure used by meet
\left(i,j\right)
Can you demonstrate that the algorithm is correct?
Is it possible to make it faster?
Hint. You may have to record both whether or not
i
J
have met, and how many players
i
has met.
Problem 3.4 (Hall of Fame) Consider a Computer Game where you are going to implement a Hall of Fame, i.e. a list of, say, the 100 best scores ever achieved. What data structure would you choose? What benefits and disadvantages do you consider to make that choice?
Describe the data structure (do not implement it) with all the operations required in the application. For each operation, consider the following:
How fast/slow is the operation?
When and how often do you expect the operation to be used?
Would your choices change if the number of scores to be stored changes?
Problem 3.5 (Based on Goorich et al C-3.19)
Design an algorithm, shuffle, which takes a list
A
n
elements and reorders it such that every permutation is equally likely. Assume that you have a function random(
n
) to return a random integer in the range
0,1,\dots ,n-1
What is the running time (complexity) of your algorithm?
Is it possible to solve the problem in linear time?
Is it best to use ArrayList or LinkedList as input and output? Why?
Problem 3.6 Goodrich et al C-3.22–23. Comment on the run time complexity for your solutions.
Problem 3.7 Goodrich et al C-3.24. Comment on the run time complexity for your solutions.
Comment on the run time complexity for your solution.
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• Supplemental Lessons
NOTE: THESE LESSONS ARE NOT INTENDED TO BE TAUGHT IN SEQUENCE - they are collected here solely for convenience! Students will deepen their understanding of various concepts, either through continued practice and review, encountering more complicated material (structs).
20 minManipulating Images
30 minMaking Flags
10 minred-shape
20 min2D Movement using Structs
minGoing further
Students create scaled, rotated, flipped, and layered images
Students create images for various nations’ flags
Students complete red-shape, which produces different shapes based on the input string
F-IF.4-6: The student interprets the behavior of functions that arise in applications in terms of the context
F-IF.7-9: The student uses different representations of a function to make generalizations about key features of function behavior and to compare functions to one another
data structure: A group of values that can be returned as a single datatype
Editing environment (WeScheme or DrRacket with the bootstrap-teachpack installed)
Computers w/ DrRacket or WeScheme
Class posters (List of rules, basic skills, course calendar)
Computer for each student (or pair), running WeScheme or DrRacket (If using DrRacket, make sure the Images.rkt file is loaded)
Student Workbooks, and something to write with
Introduces additional operations on images. As students use these operations to create more interesting images, they can practice function composition, fitting contracts together, and writing nested expressions.
Learn how to use advanced image operations
Practice function composition
Practice using contracts to help with composing operations
Practice writing and evaluating nested expressions
Learn how to import gif, png, and other images from files
Manipulating Images (Time 20 minutes)
Earlier, you learned how to create simple images using operators such as circle, rectangle, and triangle. We can combine or manipulate these basic shapes to make more interesting ones, the same way we can combine and manipulate numbers. In this lesson, you’ll learn Racket functions for manipulating and combining images.
Use of the board is critical in this activity - you’ll want to have lots of room to write, and lots of visuals for students to see. Have students review some of the Image-producing functions they already know (triangle, circle, etc.). Quiz them on the contracts for these functions.
Imagine that we wanted to make an image of a simple satellite that looks like the one shown here. This image contains a blue circle and a red rectangle, with the circle on top of the rectangle. Racket has a function called overlay, which lets you put one image on top of another. Here is its contract, and a purpose statement that explains what it does:
; overlay : Image Image -> Image ; Draws the first image on top of the second image
Start out by reminding students why contracts matter: they specify types instead of values, which makes them really flexible! You can demonstrate this by showing them the code for a simple image, and then replacing the size of the triangle with a sub-expression:
; simple image expression (star 50 "solid" "red") ; with a sub-expression (star (* 10 10) "solid" "red")
This sets students up to see overlay as a logical extension - instead of image-producing Circles of Evaluation with number-producing subexpressions, there can be image-producing Circles with image-producing subexpressions.
Using overlay, we could make a picture of a satellite. Take a look at the code below, then hit "enter" and see what shape it makes! Can you change the color of the circle? The size of the rectangle? Can you use overlay to put a star on top of both the star and the rectangle? See an example.
Before students type in the code and try it out, ask the class what they think will happen - what will the size be? The color? The text?
This satellite is flying level in the sky. What if a strong wind were blowing, causing the satellite to fly slightly on its side, like the image seen here? Then, we would want the Racket rotate function:
(rotate 30 (overlay (circle 10 "solid" "blue") (rectangle 30 8 "solid" "red")))
Try copying and pasting this code into the editor, and see what shape you get. What happens if you change the number 30?
Have the class convert this code into a Circle of Evaluation.
Let’s look at this code, viewed as a Circle of Evaluation. Our rotate function is shown here, in the blue circle. 30 is the number of degrees we’ll be rotating, and the second input is the Image we want to rotate. That image is the result of overlaying the circle and the rectangle, shown here in red. By looking at this Circle of Evaluation, can you guess the contract for the rotate function?
Can students write the code or draw the Circle of Evaluation for rotating a difference shape by a different amount? Try using a subexpression like (* 2 75) for the rotation, instead of a simple number.
Here are the contract and purpose for rotate:
; rotate : Number Image -> Image ; Rotates the image by the given number of degrees
When it’s time to introduce the new functions, start out by showing them the contract and then an example, as it does in the student guide. Make sure to ask lots of "how do you know?" questions during the code, to remind them that the contract has all the necessary information.
Suppose you wanted to make the satellite bigger, by scaling it up to 2x or 3x it’s original size. Racket has a function that will do just that, called scale. Here is the contract and purpose statement for scale:
; scale : Number Image -> Image ; Reproduce the given image with both dimensions multiplied ; by the given number
Below is some code that will scale a star to make it one-half the original size. What would you change to make it bigger instead of smaller? What would you need to change to scale a different-color star? What if you wanted to scale a circle instead? Can you figure out how to scale the entire spaceship?
There are also functions for flipping an image horizontally or vertically, and for scaling images so they get bigger or smaller. Here are contracts and purpose statements for those functions:
; flip-horizontal : Image -> Image ; Flip the given image on the horizontal (x) axis ; flip-vertical : Image -> Image ; Flip the given image on the vertical (y) axis ; scale/xy : Number Number Image -> Image ; Reproduce the given image with the horizontal (x) ; dimension multiplied by the first number and the vertical ; (y) dimension multiplied by the second number
After a few of these, try mixing it up! Show students the Racket code or Circle of Evaluation for some of the new functions first, and have them guess the contract based on how they is used.
Students apply their knowledge of Contracts, Syntax and function composition to build flags using built-in image functions.
Making Flags (Time 30 minutes)
Open this file and read through the code: [DrRacket | WeScheme] The code is also shown here:
; a blank flag is a 300x200 rectangle, which is outlined in black ; 1) start with a red dot, of radius 50 (define dot (circle 50 "solid" "red")) ; 2) define a variable called "blank", which is a 300x200, outlined black rectangle (define blank (rectangle 300 200 "outline" "black")) ; 3) define "japan" to be the flag of japan (a red dot, centered on a blank rectangle) (define japan (put-image dot 150 100 blank))
There are three values being defined here. What are they?
Click "Run" and evaluate each of those values in the Interactions window.
Change the size of the dot and click "Run". Do you expect japan to look different than it did before? Why or why not?
To make the flag of Japan, we want to put a solid, red circle right in the middle of our flag. According to the definition for blank, a flag is 300 wide by 200 high. To put the dot at the center, we use the coordinates (150, 100).
The function that lets us put one image on top of another is called put-image:
; put-image: Image Number Number Image -> Image ; places an image, at position (x, y), on an Image
How many things are in the Domain of this function?
In the definition for japan, what image is being used as the first argument? What is being used as the second?
This is a good time to remind students about indenting. Notice that all of the inputs to put-image line up with one another!
You’ve seen arithmetic functions nested before, such as (+ 4 (+ 99 12))
(+ 4(+ 9912))
(also shown as a Circle of Evaluation on the right). The second input to + is a number-producing subexpression, in this case (+ 99 12). put-image can be nested the same way.
This Circle of Evaluation will draw a star on top of another image, which itself is a circle drawn inside a square.
(put-image (star 50"solid""black")7575(put-image (circle 50"solid""red")7575(square 150"solid""black")))
Convert this Circle of Evaluation into code, and try typing it into the computer. What image do you get back? Can you modify the code so that another image is added on top?
Have students practice this once or twice, and point out the natural indenting pattern.
By combining simple shapes together, you can make very sophisticated images!
Look at this picture of the Somalian flag.
What shapes will you need to make this flag?
Which colors will you need?
Define a new value called somalia, which evaluates to this image.
Try to define as many of the following flags as possible:
Try making the flag for your favorite country, or even make up a flag of your own!
Students define a piecewise function
Students learn the concept of piecewise functions
Students learn about conditionals (how to write piecewise functions in code)
Students will understand that functions can perform different computations based on characteristics of their inputs
Students will begin to see how Examples indicate the need for piecewise functions
Students will understand that cond statements capture pairs of questions and answers when coding a piecewise function
red-shape (Time 10 min)
Conditionals allow functions to have very different behavior, based on their input. A function that produces red circles of various sizes doesn’t need conditionals (since the code will always draw a circle), but a function that produces different shapes entirely would need to evaluate the appropriate expression for a given shape.
You may want to show students the code for simpler functions (red-circle, green-triangle, etc), pointing out that those functions evaluate the same expression no matter what - they merely fill in the variable with a given value.
Turn to Page 34, and use the Design Recipe to complete the word problem for red-shape.
Pause and debrief after each section, if necessary.
Conditions can be used in many places inside a videogame:
Have the player drawn differently when they get a power boost
Open doors when the player is holding a key
Move differently depending on keyboard input
2D Movement using Structs
Students are introducted to the @code{Posn} struct, and use it to add 2-dimensional movement to their game
2D Movement using Structs (Time 20 min)
Right now, each character in your game moves along only one axis. update-danger takes in the danger’s x-coordinate and produces the next one, but it has no ability to read or update the y-coordinate. As a result, your danger can only move left or right.
; update-danger : Number -> Number ; takes in object's x-coordinate and returns the next one
Suppose we wanted to move diagonally. What would have to change about the Domain? The Range? The Purpose Statement?
Use a diagram on the board to demonstrate that update-danger will need to take in both the x- and the y-coordinate, and that it will have to produce both as well.
While you’ve seen a function take in multiple values, you have never seen a function produce more than one thing at a time.
All functions must produce one value.
However, Racket actually allows us to create new kinds of data that can contain more than one thing. These are called data structures, or "structs" for short. One kind of struct that is useful to us is called a position, which Racket abbreviates posn.
Open a new program.
Enter a Number value in the Interactions window and hit Enter. What did you get back?
Enter a String value in the Interactions window and hit Enter. What did you get back?
Enter a Boolean value in the Interactions window and hit Enter. What did you get back?
As you can see, all values evaluate to themselves. To create a posn, enter the following code in the Interactions window:
What do you get back when you hit Enter? Which number is the x-coordinate? The y-coordinate?
Have students make Posns for other coordinates, like the corners of the screen or the center.
Thinking back to an update-danger that moves diagonally, we now know that the Range must be a posn.
Start with a blank Design Recipe, and rewrite update-danger to produce a Posn instead of a Number. Instead of producing (- x 50), your function will have to produce a Posn in which the x and y have changed in some way. Here’s one example, which moves the danger left by 50 pixels and down by 10:
(EXAMPLE (update-danger 200 300) (make-posn (- 200 50) (- 300 10)))
Write a second example.
Circle and label what changes.
Define the function on your worksheet, then modify the definition in your program so that your danger moves diagonally!
Modify update-target so that it moves diagonally as well.
update-player will also need to be changed, so that it takes in the x- and y-coordinate and the key that was pressed. The Range, predictably, will be a Posn.
Change your EXAMPLEs for "up" and "down" so that they take in both coordinates and produce Posns.
Add two more EXAMPLEs, this time for "left" and "right".
Modify each clause of your cond statement, so that each one produces a Posn. Don’t forget to change your else clause, too!
Going further (Time : flexible)
Now that you’ve finished your game, here are some other things you can add to make it more exciting:
Some people prefer to use the "WASD" keys for movement, instead of the arrow keys. Add these to update-player, so that either set will work.
After you have implemented Posns, add keys for diagonal movement.
Use and inside update-player, so that the player will only move up if its y-coordinate is less than 480. Do the same for downward motion.
Add a "Safe Zone": put a green box or green shading somewhere on the background, then change collide? so that a player only collides if the player touches a danger AND they are not inside the zone.
If you’ve already added 2-dimensional movement using Posns, try making the y-coordinate of your danger change as a function of x. You can move in a wave pattern by using sin and cos!
The last item on this list has connections to trigonometry: if the y-coordinate is detemined by
sin(x)
, for example, the character will bob up and down, following the sine wave. Students can practice drawing "flight paths" using a graphing calculator, then enter those functions into their game!
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Plot Bode frequency response with additional plot customization options - MATLAB bodeplot - MathWorks Nordic
For more information, see Customizing Response Plots from the Command Line (Control System Toolbox). To create Bode plots with default options or to extract the frequency response data, use bode.
\sigma
Plot handle, returned as a handle object. Use the handle h to get and set the properties of the Bode plot using getoptions and setoptions. For the list of available options, see the Properties and Values Reference section in Customizing Response Plots from the Command Line (Control System Toolbox).
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Response vector of generalized linear mixed-effects model - MATLAB - MathWorks Deutschland
Plot Response Versus Fitted Values
Response vector of generalized linear mixed-effects model
y = response(glme)
[y,binomialsize] = response(glme)
y = response(glme) returns the response vector y used to fit the generalized linear mixed effects model glme.
[y,binomialsize] = response(glme) also returns the binomial size associated with each element of y if the conditional distribution of response given the random effects is binomial.
For an observation i with prior weights wip and binomial size ni (when applicable), the response values yi can have the following values.
\left\{0,\frac{1}{{w}_{i}^{p}{n}_{i}},\frac{2}{{w}_{i}^{p}{n}_{i}},\dots ,1\right\}
\left\{0,\frac{1}{{w}_{i}^{p}},\frac{2}{{w}_{i}^{p}},\dots \right\}
normal (-∞,∞) wip ≥ 0
You can access the prior weights property wip using dot notation. For example, to access the prior weights property for a model glme:
binomialsize — Binomial size
Binomial size associated with each element of y, returned as an n-by-1 vector, where n is the number of observations. response only returns binomialsize if the conditional distribution of response given the random effects is binomial. binomialsize is empty for other distributions.
{\text{defects}}_{ij}\sim \text{Poisson}\left({\mu }_{ij}\right)
\mathrm{log}\left({\mu }_{ij}\right)={\beta }_{0}+{\beta }_{1}{\text{newprocess}}_{ij}+{\beta }_{2}{\text{time}\text{_}\text{dev}}_{ij}+{\beta }_{3}{\text{temp}\text{_}\text{dev}}_{ij}+{\beta }_{4}{\text{supplier}\text{_}\text{C}}_{ij}+{\beta }_{5}{\text{supplier}\text{_}\text{B}}_{ij}+{b}_{i},
{\text{defects}}_{ij}
i
j
{\mu }_{ij}
i
i=1,2,...,20
j
j=1,2,...,5
{\text{newprocess}}_{ij}
{\text{time}\text{_}\text{dev}}_{ij}
{\text{temp}\text{_}\text{dev}}_{ij}
i
j
{\text{newprocess}}_{ij}
i
j
{\text{supplier}\text{_}\text{C}}_{ij}
{\text{supplier}\text{_}\text{B}}_{ij}
i
j
{b}_{i}\sim N\left(0,{\sigma }_{b}^{2}\right)
i
Extract the observed response values for the model, then use fitted to generate the fitted conditional mean values.
y = response(glme); % Observed response values
yfit = fitted(glme); % Fitted response values
Create a scatterplot of the observed response values versus fitted values. Add a reference line to improve the visualization.
scatter(yfit,y)
refline(1,0)
title('Response versus Fitted Values')
The plot shows a positive correlation between the fitted values and the observed response values.
GeneralizedLinearMixedModel | fitted | residuals
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So you probably didn't notice and wouldn't know unless I told you, so here it is: I fled to Mexico, specifically, Ciudad de Mexico, a.k.a. Mexico City (CDMX). It's fucking awesome.
I ♥ this place and its people.
And if you know me personally, you've likely heard exactly the opposite opinion of the U.S. And my gripes aren't new: I've been bitching since I was about eight years old. And the proof is in the pudding: I literally have more friends from outside of the U.S. than those within. Lastly, I'm a developer and there's simply no reason I can't live wherever. Alas, the time finally came to make a strategic move and resettle.
For now, I'm simply working on Tincre software remotely while working out the business visa and other items with the firm and my legal team here. Once I can officially do business, I'm going to train and develop an application software engineering team from the ground up.
Seven seasons of
\Delta
\Delta
often represents "change" in mathematics!
Here are a few high-level reasons I'm leaving the U.S., for now. Some motivations are personal, some are business-based, and some are macroeconomics-based.
I might not come back, though I'll always love my first home, despite some near-term negativity about that home you may pick up on.
1. Los Mexicanos are my kind of people.
Almost everyone here is an entrepreneur of some sort!
The long story short of this is simply that I fit in better with people here. Most people here seem concerned with striving for a better life having not been born with the incentive-destroying environment that is the United States of Hamster Wheels.
2. Growth, growth, and more growth.
Nearly everything about this country points upwards; you can hear it in building construction, children laughing, and the vibrant way people talk to each other.
As a builder, CDMX presents a vastly superior environment in which to deliver your future. It is incredibly difficult to look and speak like a "Trump's America" American while utilizing very advanced technical skills. In the U.S., I've long lived around people who mostly skim off of the world's growth pool to exist, something deeply incongruent with my internal morality. In other words, I'm happier here where a higher proportion of people are doing their own homework.
Yes, implied herein is the principle that if you don't build you don't explicitly contribute to the future. This is true. You should have payed more attention in your math and science courses, non-technical reader.
3. Weather. Undergrad in Miami broke this guy from the midwest.
Though I lived in Chicago and New York City for the last decade, the misery of winter merely increased for me year over year. Life is short and I really like sweater weather, sunshine, and consistency. Maybe it's that mathlete part of me that begs for consistency...
Regardless, let this be a lesson in living: if you truly despise something it is solely up to you to fix it. Mother nature isn't changing for me and I chose to finally adhere to my personal comfort zone.
4. LATAM expansion + core.
It is my strong professional opinion that this century will be a Latin American century.
That's right, in terms of developing markets, I am betting on Latin America over India, Africa, and China for Tincre's growth. CDMX and Mexico, in general, are fantastic choices for building out our Latin American core team.
And because we scale our technology via our developer platform tincre.dev, there's no better place to access those who actually build the internet (cough cough, mostly non-Americans, in case you didn't know).
5. Better food. There's just no contest.
The food here is of superior quality and freshness. And I mean that against the $1000 meals I was having while a broker in NYC, not a double quarter pounder from McDonald's.
No really, though. I've got an upcoming post with photos of the remarkably amazing food here.
Breakfast for about ninety pesos at the restaurant next door. Yes, it's all like this and I am so, so happy about that.
6. Distancing myself from the U.S.
I simply cannot tolerate my own constant anger over my fellow Americans' behaviors, political views, lack of education, and constant obsession with expediency. We're all dying (sorry to be the bearer of bad news) and I simply don't want to focus my time, energy, or efforts on people destined for failure any longer. Life is precious and being away from the boiler room I don't fit with is critical for my happiness.
7. I don't get to yell about things here.
As a follow up to 6, my parents and family taught me manners: I will never yell in your house about your house. This natural avoidance of change advocacy is a big part of why I'm here. I simply do not have the privilege of dictating what is right and wrong here. I'm not a citizen, I can't vote; it is not my house.
Who knows. I might not, permanently. I consider myself a citizen of the world first, a friend and family member to some, and a citizen of the United States of America lastly. The concept of borders is antiquated, given I already run a business with several international subsidiaries.
Keep up with me here for more news on how things are going, good food, quality software, and real good thinkin'.
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Tarsticks - Ring of Brodgar
Tarsticks
Object(s) Required Branch x5, Tar x2,5L
Specific Type of Thatching Material
Required By Box of Matches, Garden Shed, Knarr, Mine Hole, Seamark, Ship Dock, Snekkja, (Thatching Material: Birdhouse, Chicken Coop, Cistern, Dovecote, Granary, Leanto, Log Cabin, Rabbit Hutch, Smoke Shed, Stone Mansion, Stonestead, Thatched Bed, Timber House, Well)
Craft > Processing & Materials > Tarsticks
Tarsticks can be used as fuel and thatching material.
Additionally, they can be used to repair Snekkja or Knarr if they were damaged after going through Deep Water tiles.
NOTE: you cannot perform the repair function while still in Deep Water, only in Shallow Water.
It is recommended to always bring extra tarsticks for repair if your long sea travel includes taking a trip back home.
They count as 20 sticks (or 20 branches) worth of fuel.
You can right click some tarsticks and click "Place fire" to instantly make a Fire, so you can use low quality tar to make an easy, convenient and portable fire.
Tarsticks Q =
{\displaystyle {\frac {_{q}AvgBranch+_{q}AvgTar}{2}}}
{\displaystyle {\sqrt[{2}]{Perception*Carpentry}}}
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Tarsticks&oldid=91666"
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Mean frequency - MATLAB meanfreq
Mean Frequency of Chirps
Mean Frequency of Sinusoids
Mean Frequency of Bandlimited Signals
freq = meanfreq(x) estimates the mean normalized frequency, freq, of the power spectrum of a time-domain signal, x.
freq = meanfreq(x,fs) estimates the mean frequency in terms of the sample rate, fs.
freq = meanfreq(pxx,f) returns the mean frequency of a power spectral density (PSD) estimate, pxx. The frequencies, f, correspond to the estimates in pxx.
freq = meanfreq(sxx,f,rbw) returns the mean frequency of a power spectrum estimate, sxx, with resolution bandwidth rbw.
freq = meanfreq(___,freqrange) specifies the frequency interval over which to compute the mean frequency. This syntax can include any combination of input arguments from previous syntaxes, as long as the second input argument is either fs or f. If the second input is passed as empty, normalized frequency will be assumed. The default value for freqrange is the entire bandwidth of the input signal.
[freq,power] = meanfreq(___) also returns the band power, power, of the spectrum. If you specify freqrange, then power contains the band power within freqrange.
meanfreq(___) with no output arguments plots the PSD or power spectrum and annotates the mean frequency.
Generate 1024 samples of a chirp sampled at 1024 kHz. The chirp has an initial frequency of 50 kHz and reaches 100 kHz at the end of the sampling. Add white Gaussian noise such that the signal-to-noise ratio is 40 dB. Reset the random number generator for reproducible results.
Estimate the mean frequency of the chirp. Plot the power spectral density (PSD) and annotate the mean frequency.
Concatenate the chirps to produce a two-channel signal. Estimate the mean frequency of each channel.
Plot the PSDs of the two channels and annotate their mean frequencies.
Add the two channels to form a new signal. Plot the PSD and annotate the mean frequency.
Use the periodogram function to compute the power spectral density (PSD) of the signal. Specify a Kaiser window with the same length as the signal and a shape factor of 38. Estimate the mean frequency of the signal and annotate it on a plot of the PSD.
Concatenate the sinusoids to produce a two-channel signal. Estimate the PSD of each channel and use the result to determine the mean frequency.
Annotate the mean frequencies of the two channels on a plot of the PSDs.
Add the two channels to form a new signal. Estimate the PSD and annotate the mean frequency.
0.25\pi
0.45\pi
Compute the mean frequency of the signal between
0.3\pi
0.6\pi
rad/sample. Plot the PSD and annotate the mean frequency and measurement interval.
Output the mean frequency and the band power of the measurement interval. Specifying a sample rate of
2\pi
0.5\pi
0.8\pi
0.3\pi
0.9\pi
rad/sample. Plot the PSD and annotate the mean frequency of each channel and the measurement interval.
Output the mean frequency of each channel. Divide by
\pi
Input signal, specified as a vector or matrix. If x is a vector, it is treated as a single channel. If x is a matrix, then meanfreq computes the mean frequency of each column of x independently. x must be finite-valued.
Power spectral density (PSD), specified as a vector or matrix. If pxx is a matrix, then meanfreq computes the mean frequency of each column of pxx independently.
Power spectrum estimate, specified as a vector or matrix. If sxx is a matrix, then meanfreq computes the mean frequency of each column of sxx independently.
Frequency range, specified as a two-element vector of real values. If you do not specify freqrange, then meanfreq uses the entire bandwidth of the input signal.
freq — Mean frequency
Mean frequency, specified as a scalar or vector.
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TRIAC - 2D PCM Schematics - 3D Model
TRIAC (4939 views - Electronics & PCB)
PARTcloud - TRIAC
TRIAC, from triode for alternating current, is a generic trademark for a three terminal electronic component that conducts current in either direction when triggered. Its formal name is bidirectional triode thyristor or bilateral triode thyristor. A thyristor is analogous to a relay in that a small voltage and current can control a much larger voltage and current. The illustration on the right shows the circuit symbol for a TRIAC where A1 is Anode 1, A2 is Anode 2, and G is Gate. Anode 1 and Anode 2 are normally termed Main Terminal 1 (MT1) and Main Terminal 2 (MT2) respectively.
TRIACs' bidirectionality makes them convenient switches for alternating-current (AC). In addition, applying a trigger at a controlled phase angle of the AC in the main circuit allows control of the average current flowing into a load (phase control). This is commonly used for controlling the speed of induction motors, dimming lamps, and controlling electric heaters.
6 Three-quadrant TRIAC
Figure 1: Triggering modes. Quadrants, 1 (top right), 2 (top left), 3 (bottom left), 4 (bottom right)
Figure 2: TRIAC semiconductor construction
Figure 3: Operation in quadrant 1
Figure 4: Equivalent electric circuit for a TRIAC operating in quadrant 1
Generally, this quadrant is the most sensitive of the four. This is because it is the only quadrant where gate current is injected directly into the base of one of the main device transistors.[clarification needed Why is Q-I the most sensitive? See discussion]
The whole process is outlined in Figure 6. The process happens in different steps here too. In the first phase, the pn junction between the MT1 terminal and the gate becomes forward-biased (step 1). As forward-biasing implies the injection of minority carriers in the two layers joining the junction, electrons are injected in the p-layer under the gate. Some of these electrons do not recombine and escape to the underlying n-region (step 2). This in turn lowers the potential of the n-region, acting as the base of a pnp transistor which switches on (turning the transistor on without directly lowering the base potential is called remote gate control). The lower p-layer works as the collector of this PNP transistor and has its voltage heightened: actually, this p-layer also acts as the base of an NPN transistor made up by the last three layers just over the MT2 terminal, which, in turn, gets activated. Therefore, the red arrow labeled with a "3" in Figure 6 shows the final conduction path of the current.[2]
Triggering in this quadrant is similar to triggering in quadrant III. The process uses a remote gate control and is illustrated in Figure 7. As current flows from the p-layer under the gate into the n-layer under MT1, minority carriers in the form of free electrons are injected into the p-region and some of them are collected by the underlying n-p junction and pass into the adjoining n-region without recombining. As in the case of a triggering in quadrant III, this lowers the potential of the n-layer and turns on the PNP transistor formed by the n-layer and the two p-layers next to it. The lower p-layer works as the collector of this PNP transistor and has its voltage heightened: actually, this p-layer also acts as the base of an NPN transistor made up by the last three layers just over the MT2 terminal, which, in turn, gets activated. Therefore, the red arrow labeled with a "3" in Figure 6 shows the final conduction path of the current.[2]
Generally, this quadrant is the least sensitive of the four[clarification needed Why is quadrant 4 the least sensitive? See discussion] In addition, some models of TRIACs (logic level and snubberless types) cannot be triggered in this quadrant but only in the other three.
When turning on from an off-state, IGT depends on the voltage applied on the two main terminals MT1 and MT2. Higher voltage between MT1 and MT2 cause greater reverse currents in the blocked junctions requiring less gate current similar to high temperature operation. Generally, in datasheets, IGT is given for a specified voltage between MT1 and MT2.
When the gate current is discontinued, if the current between the two main terminals is more than what is called the latching current, the device keeps conducting, otherwise the device might turn off. Latching current is the minimum that can make up for the missing gate current in order to keep the device internal structure latched. The value of this parameter varies with:
control circuit (resistors or capacitors between the gate and MT1 increase the latching current because they steal some current from the gate before it can help the complete turn-on of the device)
{\displaystyle \operatorname {d} v \over \operatorname {d} t}
{\displaystyle \left({\frac {\operatorname {d} v}{\operatorname {d} t}}\right)_{s}}
{\displaystyle {\frac {\operatorname {d} i}{\operatorname {d} t}}}
{\displaystyle \left({\frac {\operatorname {d} v}{\operatorname {d} t}}\right)_{c}}
{\displaystyle \left({\frac {\operatorname {d} i}{\operatorname {d} t}}\right)_{c}}
When mains voltage TRIACs are triggered by microcontrollers, optoisolators are frequently used; for example optotriacs can be used to control the gate current. Alternatively, where safety allows and electrical isolation of the controller isn't necessary, one of the microcontroller's power rails may be connected one of the mains supply. In these situations it is normal to connect the neutral terminal to the positive rail of the microcontroller's power supply, together with A1 of the triac, with A2 connected to the live. The TRIAC's gate can be connected through an opto-isolated transistor, and sometimes a resistor to the microcontroller, so that bringing the voltage down to the microcontroller's logic zero pulls enough current through the TRIAC's gate to trigger it. This ensures that the TRIAC is triggered in quadrants II and III and avoids quadrant IV where TRIACs are typically insensitive.[5]
{\displaystyle V_{\text{gt}}}
{\displaystyle I_{\text{gt}}}
{\displaystyle V_{\text{drm}}}
{\displaystyle V_{\text{rrm}}}
{\displaystyle I_{\text{t}}}
{\displaystyle I_{\text{tsm}}}
{\displaystyle V_{\text{t}}}
The first TRIACs of this type were marketed by Thomson Semiconductors (now ST Microelectronics) under the name "Alternistor". Later versions are sold under the trademark "Snubberless". Littelfuse also uses the name "Alternistor". Phillips Semiconductor (now NXP Semiconductors) originated the trademark "High Commutation" ("Hi-Com").
This article uses material from the Wikipedia article "TRIAC", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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For each triangle below, use your triangle shortcuts from this lesson to find the missing side lengths. Then find the area and perimeter of the triangle.
What special triangles can you use for each part?
Find the lengths of the missing sides using the special triangle relationship you chose.
Perimeter Step 1(a):
\text{Perimeter }= 2 + \sqrt{2} + \sqrt{2}
Answer to perimeter (a):
2 + 2\sqrt2
Area Step 1(a):
\text{Area }=\frac{1}{2}\cdot b\cdot h
=\frac{1}{2}\sqrt{2} \cdot\sqrt{2}
=\frac{1}{2}\cdot (2)
Answer to Area (a):
\text{Area}=1
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Comparing the Population and Group Level Regression
I was planning to write a post that uses region level data to infer the underlying relationship at the population level. However, after thinking through the issue over the past few days and working out the math (below), I realise that the question I wanted to answer could not be solved using the aggregate data at hand. Nonetheless, here is a formal description of the problem outlining the assumptions needed to infer population level trends from more aggregated data.
The set-up of the problem is as follows: one is interested in examining the correlation of certain individual level characteristics. However, such data is unavailable. A possible alternative is to use more aggregated data, for example at the level of a county or region, in place of the individual. For the U.S., this would mean examining the correlation of variables across states or counties. In Singapore's case, one could use region level data from the Censuses. To what extent do the coefficients obtained from a regression carried out at the group level correspond to the estimates at the individual level?
Examining the Ecological Fallacy Problem
One should take great care when drawing inference of the individual from group level estimates. This is otherwise known as the ecological fallacy problem, where observations at the group level exhibits different trends compared to the micro-data. To analyse the issue more formally, we compare
\beta
, the coefficient from the model at the individual level:
Y_{i} = \alpha + \beta X_{i} + u_{i}
\beta_{G}
, the estimate obtained from running the regression at the group level:
\sum_{i=1}^{N} Y_{i} = \alpha*N + \beta_{G} \sum_{i=1}^{N} X_{i} + \sum_{i=1}^{N} u_{i}
In this model, each group or region consists of
N
different individuals. In practice, the number of individuals might vary across each region. One can use the region average instead of the total for more meaningful results. If
X
is a binary variable, this would simply be the proportion of people with a certain attribute.
For the estimate of the group regression to be equal to the underlying population regression,
\beta_{G} = \beta
, the following expression must hold:
\frac{cov(\sum_{i=1}^{N} Y_{i}, \sum_{i=1}^{N} X_{i})}{var(\sum_{i=1}^{N} X_{i})} = \frac{cov(Y_{i}, X_{i})}{var (X_{i})}
The numerator for the first term on the left hand side can be rewritten as:
cov(\sum_{i=1}^{N} Y_{i}, \sum_{i=1}^{N} X_{i}) = \sum_{i=1}^{N}cov(Y_{i},X_{i}) + \sum_{i=1}^{N} \sum_{k \neq i} cov(Y_{k}, X_{i})
The covariance term for the population level regression will only be equal to the group level equation if there are no spillover effects. Examining, the exogeneity condition, we see that:
cov(\sum_{i=1}^{N} u_{i}, \sum_{i=1}^{N} X_{i}) = \sum_{i=1}^{N}cov(u_{i},X_{i}) + \sum_{i=1}^{N} \sum_{k \neq i} cov(u_{k}, X_{i})
For the group estimate to coincide with the individual level estimate the term on the right has to be equals to zero and any possible omitted variable bias has to be constant across the population.1 This condition also implies that there is no selection into groups based on the variables of interest.
Similarly, the denominator of the estimator of the group regression can be rewritten as:
var(\sum_{i=1}^{N} X_{i}) = \sum_{i=1}^{N}var(X_{i}) + \sum_{i \neq j} cov(X_{i}, X_{j})
The variance of the sum is only equal to the sum of the variance if the covariance term is zero. In short, the group and population estimates will coincide only if there are no cross-covariance relationships across the variables.
Actually, I wanted to use the Singapore region dataset, which I previously compiled, to examine the relationship across individual attributes. After thinking through the problem, I realise that people sort into regions base on income and race. Interpreting the group estimates at the population level would definitely be falling prey to the ecological fallacy. Well at the very least, this problem gave me some food for thought and inspired the above post.
To have a causal interpretation of the estimate, we would also require that
cov(u_{i}, X_{i})=0
regressionnotes
Mapping SG - Shiny App
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Refining Black Hole Physics to Obtain Planck’s Constant from Information Shared from Cosmological Cycle to Cycle (Avoiding Super-Radiance)
Padmanabhan elucidated the concept of super radiance in black hole physics which would lead to loss mass of a black hole, and loss of angular momentum due to space-time infall of material into a black hole. As Padmanabhan explained it, to avoid super radiance, and probable break down of black holes, from in fall, one would need in fall material frequency, divided by mass of particles undergoing in fall in the black hole to be greater than the angular velocity of the black hole event horizon in question. We should keep in mind we bring this model up to improve the chance that Penrose’s conformal cyclic cosmology will allow for retention of enough information for preservation of Planck’s constant from cycle to cycle, as a counterpart to what we view as unacceptable reliance upon the LQG quantum bounce and its tetrad structure to preserve memory. In addition, we are presuming that at the time of z = 20 in red shift that there would be roughly about the same order of magnitude of entropy as number of operations in the electro weak era, and that the number of operations in the z = 20 case is close to the entropy at redshift z = 0. Finally, we have changed
\Lambda
with the result that after redshift = 20; there is a rapid collapse to the present-day vacuum energy value i.e. by z = 12the value of the cosmological constant, Λ likely being the same, today, as for what it was when z=12. And z = 12 is the redshift value about when Galaxies form.
Planck’s Constant, Black Hole, Super Radiance
-3{m}_{\text{graviton}}^{2}\cdot h=\frac{\kappa }{2}\cdot T
{g}_{uv}={\eta }_{uv}+{h}_{uv}
{h}_{uv}=\frac{2GM}{r}\cdot \left(\mathrm{exp}\left(\frac{{m}_{\text{graviton}}\cdot r}{\hslash }\right)\right)\cdot \left(2{V}_{u}{V}_{v}+{\eta }_{uv}\right)
{v}_{\text{graviton}}=c\cdot \sqrt{1-\frac{{m}_{\text{graviton}}^{2}\cdot {c}^{4}}{{\hslash }^{2}{\omega }_{\text{graviton}}^{2}}}
{g}_{ab}
\stackrel{⌢}{\Omega }
{\stackrel{^}{g}}_{ab}={\stackrel{⌢}{\Omega }}^{2}\cdot {g}_{ab}
\stackrel{⌢}{\Omega }\underset{ccc}{\to }{\stackrel{⌢}{\Omega }}^{-1}
\hslash
{\hslash }_{\text{old}\text{\hspace{0.17em}}\text{cosmology}\text{\hspace{0.17em}}\text{cycle}}\equiv {\hslash }_{\text{present cosmology cycle}}
500{M}_{\odot }
\stackrel{⌢}{\Omega }
\stackrel{⌢}{\Omega }
\stackrel{^}{\Omega }
\begin{array}{l}E=8\text{π}\cdot T+\Lambda \cdot g=\text{Source}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{gravitational}\text{\hspace{0.17em}}\text{field}\\ T=\text{mass energy density term}\\ g=\text{gravitational metric}\\ \Lambda =\text{vacuum energy}\end{array}
\Lambda ={c}_{1}\cdot {\left(Temp\right)}^{\beta }
\stackrel{^}{\Omega }
\begin{array}{l}{E|}_{\text{initial}}={E|}_{\text{final}}⇔{\Lambda \cdot g|}_{\text{initial}}\equiv {\Lambda \cdot g|}_{\text{final}}\\ &\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Lambda \cdot {g}_{ab}|}_{\text{final}}=\left({\Lambda |}_{\text{final}}\right)\cdot {\stackrel{⌢}{\Omega }}^{2}\cdot \left({{g}_{ab}|}_{\text{initial}}\right)\end{array}
\stackrel{⌢}{\Omega }\propto {S}_{\text{entropy}}\approx N
S\approx N\cdot \left(\mathrm{log}\left[V/{\lambda }^{3}\right]+5/2\right)\approx N
S\approx N~{10}^{7}
I={S}_{\text{total}}/{k}_{B}\mathrm{ln}2={\left[#\text{operations}\right]}^{3/4}~{10}^{7}
{\left[#\text{operations}\right]}_{\text{initially}}~{10}^{10}
\stackrel{˜}{\alpha }\equiv \frac{{e}^{2}}{\hslash \cdot c}\equiv \frac{{e}^{2}}{d}\times \frac{\lambda }{h\cdot c}
\begin{array}{l}{{S}_{\text{entropy}}|}_{ew}\sim N\sim {10{}^{51}|}_{ew}{\propto {\left[#\text{operations}\right]}^{3/4}|}_{ew}\\ ⇔{\left[#\text{operations}\right]|}_{ew}\approx {10}^{71}\end{array}
\begin{array}{l}{{S}_{\text{entropy}}|}_{\text{redshift}=20}\sim N\sim {10{}^{67}|}_{\text{redshift}=20}{\propto {\left[#\text{operations}\right]}^{3/4}|}_{\text{redshift}=20}\\ ⇔{\left[#\text{operations}\right]|}_{\text{redshift}=20}\approx {10}^{89}\end{array}
{T}_{\text{Blackhole}}=\frac{\hslash {c}^{3}}{8\text{π}kG{M}_{\text{Blackhole}}}
{\Delta T|}_{\text{near Black hole}}={T}_{\text{background}}-{T}_{\text{Black hole}}\approx {10}^{\alpha }\text{Kelvin}\ge 2.73\text{\hspace{0.17em}}\text{Kelvin}
{10}^{22}\text{\hspace{0.17em}}\text{kg}\propto {10}^{58}\text{eV}/{\text{c}}^{2}
{10}^{-29}\text{eV}/{\text{c}}^{2}
\stackrel{⌢}{\Omega }
\stackrel{⌢}{\Omega }
{\Lambda \cdot {g}_{ab}|}_{\text{final}}=\left({\Lambda |}_{\text{final}}\right)\cdot {\stackrel{⌢}{\Omega }}^{2}\cdot \left({{g}_{ab}|}_{\text{initial}}\right)
S\propto {r}_{\oplus }
{r}_{\oplus }
{S}_{\text{entropy}}\le N=\frac{3G}{G\Lambda }⇔{S}_{\text{Black hole}}\approx {M}_{\text{Black hole}}^{2}
{\Omega }_{H}\propto 1/{r}_{\oplus }
{\Omega }_{H}\propto 1/{r}_{\oplus }\propto 1/\sqrt{{S}_{\text{Blackhole}}}\propto 1/\sqrt{N}\propto 1/\sqrt{{\left(Temp\right)}^{\beta }}
\stackrel{˜}{m}
0<\left({\omega }_{\text{particles}}/\stackrel{˜}{m}\right)<{\Omega }_{H}
\left({\omega }_{\text{particles}}/\stackrel{˜}{m}\right)>{\Omega }_{H}
{\omega }_{particles}>\stackrel{˜}{m}/\sqrt{N}
{{S}_{\text{Black hole}}|}_{\text{4D}}=4\text{π}{M}_{\text{Black hole}}^{2}
{{S}_{\text{Black hole}}|}_{\text{5D}}=4\text{π}{M}_{\text{Black hole}}^{2}\cdot \sqrt{8L/27\text{π}{M}_{\text{Black hole}}}
{{\omega }_{\text{particles}}|}_{\text{5D}}>\left(\stackrel{˜}{m}/\sqrt{N}\right)\cdot {\left(27\text{π}{M}_{\text{Black hole}}/8L\right)}^{1/4}
\stackrel{^}{\Omega }
{{\omega }_{\text{particles}}|}_{\text{5D}}>\left(\stackrel{˜}{m}/\sqrt{\stackrel{⌢}{\Omega }}\right)\cdot {\left(27\text{π}{M}_{\text{Black hole}}/8L\right)}^{1/4}
\stackrel{^}{\Omega }~3\text{π}{G}^{-1}/\Lambda
\Lambda ={c}_{1}\cdot {\left(Temp\right)}^{\beta }
{T}_{\text{Black hole}}=\frac{\hslash {c}^{3}}{8\text{π}kG{M}_{\text{Black hole}}}
{\Delta T|}_{\text{near Black hole}}={T}_{\text{background}}-{T}_{\text{Black hole}}
\hslash
{\hslash }_{\text{per black hole}}\propto \left[\Delta T\left(\text{due to back ground per black hole}\right)\right]\cdot \left(8\text{π}kG{M}_{\text{Black hole}}/{c}^{3}\right)
{N}_{z=80BH}
{\hslash }_{\text{old cosmology cycle}}\equiv {\hslash }_{\text{present cosmology cycle}}
\begin{array}{l}{\hslash }_{z=20}\propto \left({N}_{z=20}\text{Black holes}\right)\\ \text{times}\left[\Delta T\left(\text{due to back ground per black hole}\right)\right]\cdot \left(8\text{π}kG{M}_{\text{Black hole}}/{c}^{3}\right)\end{array}
\stackrel{⌢}{\Omega }
{\stackrel{⌢}{\Omega }}_{\text{Redshift}z=20}\propto {N\approx {10}^{67}|}_{\text{Redshift}z=20}
\begin{array}{l}\left({\stackrel{⌢}{\Omega }}_{\text{Redshift}z=20}\propto {N\approx {10}^{67}|}_{\text{Redshift}z=20}\right)\cdot {\Lambda |}_{\text{Redshift}z=20}\\ \approx \left({\stackrel{⌢}{\Omega }}_{\text{Redshift}z=20}\propto {N\approx {10}^{89}|}_{\text{Redshift}z=20}\right)\cdot {\Lambda |}_{\text{Redshift}z=20}\end{array}
\Lambda
\Lambda
Beckwith, A.W. (2019) Refining Black Hole Physics to Obtain Planck’s Constant from Information Shared from Cosmological Cycle to Cycle (Avoiding Super-Radiance). Journal of High Energy Physics, Gravitation and Cosmology, 5, 464-472. https://doi.org/10.4236/jhepgc.2019.52027
1. Visser, M. http://arxiv.org/pdf/gr-qc/9705051v2
2. Lavenda, B.H. and Dunning-Davies, J. (1990) Qualms Concerning the Inflationary Scenario. Foundation of Physics Letters, 5, 191-196. http://milesmathis.com/dunning.pdf
3. Tolman, R.C. (1934) Relativity and Cosmology. Clarion Press, Oxford.
4. Maggiore, M. (2008) Gravitational Waves, Volume 1: Theory and Experiment. Oxford Univ. Press, Oxford.
5. Kim, J.Y. http://arxiv.org/pdf/hep-th/0109192v3
6. Beckwith, A. http://vixra.org/abs/1006.0027
7. Durrer, R. and Rinaldi, M. (2009) Graviton Production in Non-Inflationary Cosmology. Physical Review D, 79, Article ID: 063507. http://arxiv.org/abs/0901.0650
8. Beckwith, A.W. (2019) A Kaluza Klein Treatment of a Graviton and Deceleration Parameter Q(Z) as an Alternative to Standard DE and Its Possible Link to Standard DE Equation of State as Given by Li, Li, Wang and Wang, in 2017. Journal of High Energy Physics, Gravitation and Cosmology, 5, 208-217. https://doi.org/10.4236/jhepgc.2019.51012
9. Ng, Y. (2008) Spacetime Foam: From Entropy and Holography to Infinite Statistics and Nonlocality. Entropy, 10, 441-461. https://doi.org/10.3390/e10040441
10. Lloyd, S. (2002) Computational Capacity of the Universe. Physical Review Letters, 88, Article ID: 237901.
11. Smoot, G. http://chalonge.obspm.fr/Paris07_Smoot.pdf
12. Ellis, G., Maartens, R. and MacCallum, M.A.H. (2012) Relativistic Cosmology. Cambridge University Press, Cambridge.
13. Beckwith, A.W. (2019) How (∆t)5 + A1(∆t)2 + A2 = 1 Is Generally, in the Galois Sense Solvable for a Kerr-Newman Black Hole Affect Questions on the Opening and Closing of a Wormhole Throat and the Simplification of the Problem, Dramatically Speaking, If d = 1 (Kaluza Klein Theory) and Explaining the Lack of Overlap with the Results When Applying the Gauss-Lucas Theorem. Journal of High Energy Physics, Gravitation and Cosmology, 5, 235-278. https://doi.org/10.4236/jhepgc.2019.51014
14. Scully, S. (2017) Semiclassical Gravity: A Testable Theory of Quantum Gravity. In: Hossfelder, S., Ed., Experimental Search for Quantum Gravity, Springer FIAS interdisciplinary Science Series, Springer Verlag, Frankfurt, 69-76.
15. Frassino, A.M. (2017) Quantum Gravity Deformations. In: Hossfelder, S., Ed., Experimental Search for Quantum Gravity, Springer FIAS Interdisciplinary Science Series, Springer Verlag, Frankfurt, 77-83.
16. Bohmer, C.G. (2017) Introduction to General Relativity and Cosmology. World Scientific, Singapore.
17. Giovannini, M. (2008) A Primer on the Physics of the Cosmic Microwave Background. World Scientific, Singapore.
21. Corda, C. (2009) Interferometric Detection of Gravitational Waves: The Definitive Test for General Relativity. International Journal of Modern Physics D, 18, 2275-2282. https://arxiv.org/abs/0905.2502
{u}_{a}
\rho
{q}_{a}
{\pi }_{ab}
\rho
\rho ={\rho }_{\text{GW}}+{\rho }_{\text{everything else}}
\begin{array}{l}{T}_{ab}=\rho {u}_{a}{u}_{b}+{q}_{a}{u}_{b}+{u}_{a}{q}_{b}p{h}_{ab}+{\pi }_{ab}\\ ⇔{T}_{ab}={T}_{\text{GW}/\text{Gravtitions}}+{T}_{\text{everythingelse}}\end{array}
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Athena was working on her Girl Scout silver award and was wondering what percentage of people support the Girl Scouts financially through cookie sales outside the grocery store. 8-45 HW eTool (Desmos)
At the next cookie sale, Athena kept track of how many customers at the grocery store walked by the cookie table and how many stopped to purchase cookies.
32\%
of families stopped and purchased cookies at the table.
Athena continued collecting data at several different grocery stores around town. Here are the percent of those who bought cookies at each store:
32\%
29\%
19\%
31\%
30\%
24\%
38\%
33\%
42\%
25\%
22\%
27\%
Make a box plot of the samples, then make a new statement about what proportion of people (in what interval) you can expect to buy Girl Scout cookies at the grocery store.
How can you use this data to create a box plot?
Your math statement might start like: The % of people who buy Girl Scout Cookies at grocery stores is usually between _____% and _____%.
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mutable - Maple Help
Home : Support : Online Help : Programming : Data Types : Rtables, Arrays, Matrices, and Vectors : Type Checking : mutable
check if an object is or contains mutable objects
type(expr, mutable)
The type(expr, mutable) calling sequence returns true if if expr is a mutable object or contains mutable objects.
A mutable object is a Maple expression whose contents can change without its identity changing. The mutable objects in Maple are of type table (which includes array), rtable (which includes Array, Vector, and Matrix), module (which includes record and object), and procedure.
For example, an element in an Array can be changed, yet it is still the same Array, so an array is mutable. On the other hand, "changing" an element in a list creates a new list, and any references to the original list will not see the change. A list is immutable.
A procedure is considered mutable because it can have a remember table (even if it does not have the remember option).
\mathrm{type}\left(3.14,\mathrm{mutable}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{type}\left(\mathrm{table}\left(\right),\mathrm{mutable}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left(\mathrm{Vector}[\mathrm{row}]\left([a,b,c]\right),\mathrm{mutable}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
In general, a list is not mutable.
L≔[a,b,[c,d]]
\textcolor[rgb]{0,0,1}{L}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}]]
\mathrm{L0}≔L
\textcolor[rgb]{0,0,1}{\mathrm{L0}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}]]
This does not change the lists in-place, and in fact this syntax for "changing" a list is only supported for small lists, as it is very inefficient.
L[3][2]≔e
{\left({\textcolor[rgb]{0,0,1}{L}}_{\textcolor[rgb]{0,0,1}{3}}\right)}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{e}
L
[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{e}]]
The original lists are unchanged, because lists are not mutable.
\mathrm{L0}
[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}]]
\mathrm{type}\left(L,\mathrm{mutable}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
A list containing an Array is mutable because the Array is mutable.
L≔[a,b,\mathrm{Array}\left([c,d]\right)]
\textcolor[rgb]{0,0,1}{L}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{c}& \textcolor[rgb]{0,0,1}{d}\end{array}]]
\mathrm{L0}≔L
\textcolor[rgb]{0,0,1}{\mathrm{L0}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{c}& \textcolor[rgb]{0,0,1}{d}\end{array}]]
This changes the contents of the Array in-place, and thus the containing list is unaffected.
L[3][2]≔e
{\left({\textcolor[rgb]{0,0,1}{L}}_{\textcolor[rgb]{0,0,1}{3}}\right)}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{e}
L
[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{c}& \textcolor[rgb]{0,0,1}{e}\end{array}]]
\mathrm{L0}
[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{c}& \textcolor[rgb]{0,0,1}{e}\end{array}]]
\mathrm{type}\left(L,\mathrm{mutable}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Glycerol (data page) - Wikipedia
This page provides supplementary chemical data on glycerol.
5 Freezing point of aqueous solutions
Material Safety Data Sheet[edit]
The handling of this chemical may incur notable safety precautions. It is highly recommended that you seek the Material Safety Datasheet (MSDS) for this chemical from a reliable source and follow its directions.
Surface tension[1] 63.4 mN/m at 20°C
51.9 mN/m at 150°C
Viscosity[2] 1.412 Pa·s at 20°C
Thermodynamic properties[edit]
Triple point 291.8 K (18.7 °C), ~99500 Pa
of fusion, ΔfusHo 18.28 kJ/mol
of vaporization, ΔvapSo 201 J/(mol·K)
Sosolid 37.87 J/(mol K)[3]
Heat capacity, cp 150. J/(mol K) 6°C - 11°C
Soliquid 206.3 J/(mol K)[4]
Heat capacity, cp 221.9 J/(mol K) at 25°C
Vapor pressure of liquid[edit]
T in °C 125.5 167.2 198.0 220.1 263.0 290.0
Table data obtained from CRC Handbook of Chemistry and Physics, 44th ed.
log10 of Glycerol vapor pressure. Uses formula:
{\displaystyle \scriptstyle \log _{e}P_{mmHg}=}
{\displaystyle \scriptstyle \log _{e}({\frac {760}{101.325}})-21.25867\log _{e}(T+273.15)-{\frac {16726.26}{T+273.15}}+165.5099+1.100480\times 10^{-05}(T+273.15)^{2}}
obtained from CHERIC[5]
Freezing point of aqueous solutions[edit]
Table data obtained from Lange's Handbook of Chemistry, 10th ed. Specific gravity is at 15°C, referenced to water at 15°C.
See details on: Freezing Points of Glycerine-Water Solutions Dow Chemical [6] or Freezing Points of Glycerol and Its Aqueous Solutions.[7]
Distillation data[edit]
Vapor-liquid Equilibrium of Glycerol/water[8]
% by mole water
Spectral data[edit]
^ "Physical Properties of Glycerine and its solutions" (PDF). Retrieved 19 March 2021.
^ "Glycerin".
^ Lange's Handbook of Chemistry, 17th Ed., 2017, McGraw-Hill Education, Table 2.54.
^ "Pure Component Properties" (Queriable database). Chemical Engineering Research Information Center. Retrieved 13 May 2007.
^ Freezing Points of Glycerine-Water Solutions Dow Chemical
^ Lane, Leonard B. (September 1925). "Freezing Points of Glycerol and Its Aqueous Solutions". Industrial & Engineering Chemistry. 17 (9): 924. doi:10.1021/ie50189a017.
^ "Binary Vapor-Liquid Equilibrium Data" (Queriable database). Chemical Engineering Research Information Center. Retrieved 7 June 2007.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Glycerol_(data_page)&oldid=1081189433"
Chemical data pages cleanup
|
Talk:Hermitian manifold - Wikipedia
Talk:Hermitian manifold
(Rated Start-class, Low-priority)
1 "E bar" Undefined?
2 Distinction between Hermitian Metric and Hermitian Structure
3 Topological manifold?
4 Either confusing notation or completely wrong.
5 The '"`UNIQ--postMath-00000021-QINU`"' in the "Formal definition" section is not defined
"E bar" Undefined?Edit
In the "Formal definition" section, the notation
{\displaystyle {\overline {E}}}
isn't explicitly defined or referenced anywhere. Presumably, this denotes the conjugate bundle? Some clarification here would help, or if someone can confirm that this was the intended meaning, I can try to edit it in. Dzackgarza (talk) 02:16, 29 July 2020 (UTC)
In the introduction, shouldn't the almost complex structure preserve the metric, rather than the other way around? —Preceding unsigned comment added by 130.102.0.171 (talk) 01:11, 16 January 2009 (UTC)
Distinction between Hermitian Metric and Hermitian StructureEdit
I think this article is a bit confusing at the moment. Should we not make a distinction between a Hermitian metric (given by the formula in the text) and a Hermitian structure (that is a smoothly varying choice of Hermitian form)? We could then show how each was related to the other. It's a simple enough point, but potentially confusing for newcomers.78.151.55.190 (talk) 00:02, 7 November 2013 (UTC)
Topological manifold?Edit
Shouldn't the intro say smooth manifold? How exactly does one define an almost complex structure on a topological manifold? -- Fropuff 03:40, 10 October 2006 (UTC)
You're right and I changed it. VectorPosse 06:49, 10 October 2006 (UTC)
Either confusing notation or completely wrong.Edit
Either the notation in section 2 is very misleading, or many of the statements are completely wrong. A Hermitia{\displaystyle n}
{\displaystyle n}
{\displaystyle h}
can indeed be decomposed into two parts
{\displaystyle h=a+ib}
{\displaystyle a=(h+{\bar {h}})/2}
is a real, symetric
{\displaystyle n}
{\displaystyle n}
{\displaystyle b=(h-{\bar {h}})/2i}
is a real skew-symmetric
{\displaystyle n}
{\displaystyle n}
matrix. The article imples that the metric
{\displaystyle g}
is to be identified with
{\displaystyle a}
{\displaystyle \omega }
{\displaystyle b}
. It then states that one can be obtained from the other by means of the complex structure
{\displaystyle J}
. This cannot be true. Hermiticity requires no relation between
{\displaystyle a}nd
{\displaystyle b}
What is actually true is that there is a Riemann metric
{\displaystyle g}
on the underlying
{\displaystyle 2n}
-dimensional real smooth manifold, where
{\displaystyle g=\left({\begin{matrix}a&b\\-b&a\end{matrix}}\right)}
is a real symmetric
{\displaystyle 2n}
{\displaystyle 2n}
matrix, while
{\displaystyle \omega =\left({\begin{matrix}b&-a\\a&b\end{matrix}}\right)=\left({\begin{matrix}0&-I_{n}\\I_{n}&0\end{matrix}}\right)\left({\begin{matrix}a&b\\-b&a\end{matrix}}\right)=\left({\begin{matrix}a&b\\-b&a\end{matrix}}\right)\left({\begin{matrix}0&-I_{n}\\I_{n}&0\end{matrix}}\right)}
is a skew-symmetric
{\displaystyle 2n}
{\displaystyle 2n}
matrix. Here
{\displaystyle J=\left({\begin{matrix}0&-I_{n}\\I_{n}&0\end{matrix}}\right)}
{\displaystyle 2n}
{\displaystyle 2n}
matrix representing the complex structure in the underlying real vector space.
{\displaystyle J}
{\displaystyle g}
because it is simply multiplication by
{\displaystyle {\sqrt {-1}}}
in the original complex basis. Mike Stone (talk) 19:01, 9 December 2016 (UTC)
{\displaystyle \Gamma }
in the "Formal definition" section is not definedEdit
I think defining it explicitly would ease the understanding of the article. Luca (talk) 14:32, 11 February 2021 (UTC)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Talk:Hermitian_manifold&oldid=1006182619"
|
Bee is tiling her kitchen floor using square tiles that are
1
foot long on each side. The rectangular floor is
13
7
86
tiles, does she have enough to cover the floor? If not, how many more does she need?
Instead of counting the tiles, use multiplication to find the area (space inside the rectangle) of the floor.
The area is the same as the number of tiles she needs. Does she have enough?
Does she have enough to tile just around the edges of the kitchen? If so, how many leftover tiles will she have? If not, how many more will she need? Draw a picture and show your work.
Be careful not to count the corners twice.
|
Scale roots of polynomial - MATLAB polyscale - MathWorks
Bandwidth Expansion of LPC Speech Spectrum
Scale roots of polynomial
b = polyscale(a,alpha)
b = polyscale(a,alpha) scales the roots of a polynomial in the z-plane, where a is a vector containing the polynomial coefficients and alpha is the scaling factor.
If alpha is a real value in the range [0 1], then the roots of a are radially scaled toward the origin in the z-plane. Complex values for alpha allow arbitrary changes to the root locations.
Express the solutions to the equation
{x}^{7}=1
as the roots of a polynomial. Plot the roots in the complex plane.
pp = [1 0 0 0 0 0 0 -1];
zplane(pp,1)
Scale the roots of p in and out of the unit circle. Plot the results.
for sc = [1:-0.2:0.2 1.2 1.4];
b = polyscale(pp,sc);
plot(roots(b),'o')
axis([-1 1 -1 1]*1.5)
{F}_{s}=7418\phantom{\rule{0.2777777777777778em}{0ex}}Hz
Model a 100-sample section of the signal using a 12th-order autoregressive polynomial. Perform bandwidth expansion of the signal by scaling the roots of the autoregressive polynomial by 0.85.
Ao = lpc(mtlb(1000:1100),12);
Ax = polyscale(Ao,0.85);
Plot the zeros, poles, and frequency responses of the models.
zplane(1,Ao)
zplane(1,Ax)
title('Flattened')
[ho,w] = freqz(1,Ao);
[hx,w] = freqz(1,Ax);
plot(w/pi,abs([ho hx]))
legend('Original','Flattened')
By reducing the radius of the roots in an autoregressive polynomial, the bandwidth of the spectral peaks in the frequency response is expanded (flattened). This operation is often referred to as bandwidth expansion.
polystab | roots
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Neptune Knowpia
Neptune is the eighth and farthest-known Solar planet from the Sun. In the Solar System, it is the fourth-largest planet by diameter, the third-most-massive planet, and the densest giant planet. It is 17 times the mass of Earth, and slightly more massive than its near-twin Uranus. Neptune is denser and physically smaller than Uranus because its greater mass causes more gravitational compression of its atmosphere. It is referred to as one of the solar system's two ice giant planets (the other one being its Uranus).
Being composed primarily of gases and liquids, it has no well-defined "solid surface". The planet orbits the Sun once every 164.8 years at an average distance of 30.1 AU (4.5 billion km; 2.8 billion mi). It is named after the Roman god of the sea and has the astronomical symbol , representing Neptune's trident.[d]
^ A second symbol, an ‘LV’ monogram
for 'Le Verrier', analogous to the ‘H’ monogram
for Uranus. It was never much used outside of France and is now archaic.
{\displaystyle {\tfrac {M_{\text{Neptune}}}{M_{\text{Earth}}}}={\tfrac {1.02\times 10^{26}}{5.97\times 10^{24}}}=17.09.}
{\displaystyle {\tfrac {M_{\text{Uranus}}}{M_{\text{Earth}}}}={\tfrac {8.68\times 10^{25}}{5.97\times 10^{24}}}=14.54.}
{\displaystyle {\tfrac {M_{\text{Jupiter}}}{M_{\text{Neptune}}}}={\tfrac {1.90\times 10^{27}}{1.02\times 10^{26}}}=18.63.}
{\displaystyle {\tfrac {r_{a}}{r_{p}}}={\tfrac {2}{1-e}}-1=2/0.2488-1\approx 7.039.}
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Jahn, Dennis1; Löwe, Robert2; Stump, Christian1
1 Fakultät für Mathematik Ruhr-Universität Bochum Germany
2 Institut für Mathematik TU Berlin Germany
We give an explicit subword complex description of the generators of the type cone of the
g
-vector fan of a finite type cluster algebra with acyclic initial seed. This yields in particular a description of the Newton polytopes of the
F
-polynomials in terms of subword complexes as conjectured by S. Brodsky and the third author. We then show that the cluster complex is combinatorially isomorphic to the totally positive part of the tropicalization of the cluster variety as conjectured by D. Speyer and L. Williams.
Classification: 13F60, 20F55, 52B05, 05E45
Keywords: Cluster algebras, F-polynomials, subword complex, type cone.
Jahn, Dennis 1; Löwe, Robert 2; Stump, Christian 1
author = {Jahn, Dennis and L\"owe, Robert and Stump, Christian},
title = {Minkowski decompositions for generalized associahedra of acyclic type},
TI - Minkowski decompositions for generalized associahedra of acyclic type
%T Minkowski decompositions for generalized associahedra of acyclic type
Jahn, Dennis; Löwe, Robert; Stump, Christian. Minkowski decompositions for generalized associahedra of acyclic type. Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 757-775. doi : 10.5802/alco.177. https://alco.centre-mersenne.org/articles/10.5802/alco.177/
[1] Arkani-Hamed, Nima; He, Song; Lam, Thomas Cluster configuration spaces of finite type (2020) (Preprint, available at arxiv.org/abs/2005.11419)
[2] Bazier-Matte, Véronique; Douville, Guillaume; Mousavand, Kaveh; Thomas, Hugh; Yıldırım, Emine ABHY Associahedra and Newton polytopes of
F
-polynomials for finite type cluster algebras (2018) (Preprint, available at arxiv.org/abs/1808.09986)
[3] Brodsky, Sarah B.; Stump, Christian Towards a uniform subword complex description of acyclic finite type cluster algebras, Algebr. Comb., Volume 1 (2018) no. 4, pp. 545-572 | Article | MR: 3875076 | Zbl: 1423.13118
[4] Ceballos, Cesar; Labbé, Jean-Philippe; Stump, Christian Subword complexes, cluster complexes, and generalized multi-associahedra, J. Algebraic Combin., Volume 39 (2014) no. 1, pp. 17-51 | Article | MR: 3144391 | Zbl: 1286.05180
[5] Chapoton, Frédéric; Fomin, Sergey; Zelevinsky, Andrei Polytopal realizations of generalized associahedra, Canad. Math. Bull., Volume 45 (2002) no. 4, pp. 537-566 | Article | MR: 1941227 | Zbl: 1018.52007
[6] Demonet, Laurent Mutations of group species with potentials and their representations. Applications to cluster algebras (2010) (Preprint, available at arxiv.org/abs/1003.5078)
[7] Derksen, Harm; Weyman, Jerzy; Zelevinsky, Andrei Quivers with potentials and their representations II: Applications to cluster algebras, J. Amer. Math. Soc., Volume 23 (2010) no. 3, pp. 749-790 | Article | MR: 2629987 | Zbl: 1208.16017
[8] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. II. Finite type classification, Invent. Math., Volume 154 (2003) no. 1, pp. 63-121 | Article | MR: 2004457 | Zbl: 1054.17024
[9] Fomin, Sergey; Zelevinsky, Andrei
Y
-systems and generalized associahedra, Ann. of Math. (2), Volume 158 (2003) no. 3, pp. 977-1018 | Article | MR: 2031858 | Zbl: 1057.52003
[10] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. IV. Coefficients, Compos. Math., Volume 143 (2007) no. 1, pp. 112-164 | Article | MR: 2295199 | Zbl: 1127.16023
[11] Hohlweg, Christophe; Lange, Carsten E. M. C.; Thomas, Hugh Permutahedra and generalized associahedra, Adv. Math., Volume 226 (2011) no. 1, pp. 608-640 | Article | MR: 2735770 | Zbl: 1233.20035
[12] Hohlweg, Christophe; Pilaud, Vincent; Stella, Salvatore Polytopal realizations of finite type
\mathbf{g}
-vector fans, Adv. Math., Volume 328 (2018), pp. 713-749 | Article | MR: 3771140 | Zbl: 1382.05075
[14] Lampe, Philipp On the approximate periodicity of sequences attached to non-crystallographic root systems, Exp. Math., Volume 27 (2018) no. 3, pp. 265-271 | Article | MR: 3857662 | Zbl: 1423.13126
[15] McMullen, Peter Representations of polytopes and polyhedral sets, Geometriae Dedicata, Volume 2 (1973), pp. 83-99 | Article | MR: 326574 | Zbl: 0273.52006
[16] Padrol, Arnau; Palu, Yann; Pilaud, Vincent; Plamondon, Pierre-Guy Associahedra for finite type cluster algebras and minimal relations between
\mathbf{g}
-vectors (2019) (Preprint, available at arxiv.org/abs/1906.06861)
[17] Pilaud, Vincent; Stump, Christian Brick polytopes of spherical subword complexes and generalized associahedra, Adv. Math., Volume 276 (2015), pp. 1-61 | Article | MR: 3327085 | Zbl: 1405.05196
[18] Reading, Nathan Sortable elements and Cambrian lattices, Algebra Universalis, Volume 56 (2007) no. 3-4, pp. 411-437 | Article | MR: 2318219 | Zbl: 1184.20038
[19] Speyer, David; Williams, Lauren The tropical totally positive Grassmannian, J. Algebraic Combin., Volume 22 (2005) no. 2, pp. 189-210 | Article | MR: 2164397 | Zbl: 1094.14048
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Critical micelle concentration - Wikipedia
In colloidal and surface chemistry, the critical micelle concentration (CMC) is defined as the concentration of surfactants above which micelles form and all additional surfactants added to the system will form micelles.[1]
The CMC is an important characteristic of a surfactant. Before reaching the CMC, the surface tension changes strongly with the concentration of the surfactant. After reaching the CMC, the surface tension remains relatively constant or changes with a lower slope. The value of the CMC for a given dispersant in a given medium depends on temperature, pressure, and (sometimes strongly) on the presence and concentration of other surface active substances and electrolytes. Micelles only form above critical micelle temperature.
For example, the value of CMC for sodium dodecyl sulfate in water (without other additives or salts) at 25 °C, atmospheric pressure, is 8x10−3 mol/L.[2]
CMCs for common surfactants[3]
CMC (molarity)
Sodium octyl sulfate 0.13 anionic surfactant
Sodium dodecyl sulfate 0.0083 anionic surfactant
Sodium tetradecyl sulfate 0.0021 anionic surfactant
Decyltrimethylammonium bromide 0.065 cationic surfactant
Dodecyltrimethylammonium bromide 0.016 cationic surfactant
Hexadecyltrimethylammonium bromide 0.00092 cationic surfactant
Penta(ethyleneglycol)monooctyl ether 0.0009 neutral surfactant
Penta(ethyleneglycol)monodecyl ether 0.0009 neutral surfactant
Penta(ethyleneglycol)monododecyl ether 0.000065 neutral surfactant
Top to Bottom: Increasing concentration of surfactant in water first leads to the formation of a layer on the surface. After reaching the CMC micelles begin forming. Notice that the existence of micelles does not preclude the existence of individual surfactant molecules in solution.
Upon introducing surfactants (or any surface active materials) into a system, they will initially partition into the interface, reducing the system free energy by:
lowering the energy of the interface (calculated as area times surface tension), and
removing the hydrophobic parts of the surfactant from contact with water.
Subsequently, when the surface coverage by the surfactants increases, the surface free energy (surface tension) decreases and the surfactants start aggregating into micelles, thus again decreasing the system's free energy by decreasing the contact area of hydrophobic parts of the surfactant with water.[4] Upon reaching CMC, any further addition of surfactants will just increase the number of micelles (in the ideal case).
According to one well-known definition, CMC is the total concentration of surfactants under the conditions:[5]
if C = CMC, (d3
{\displaystyle \phi }
/dCt3) = 0
{\displaystyle \phi }
= A[Cs] + B[Cm]; i.e., in words Cs = [single surfactant ion] , Cm = [micelles] and A and B are proportionality constants
Ct = Cs + NCm; i.e., N = represents the number of detergent ions per micelle
The CMC generally depends on the method of measuring the samples, since A and B depend on the properties of the solution such as conductance, photochemical characteristics or surface tension. When the degree of aggregation is monodisperse, then the CMC is not related to the method of measurement. On the other hand, when the degree of aggregation is polydisperse, then CMC is related to both the method of measurement and the dispersion.
The common procedure to determine the CMC from experimental data is to look for the intersection (inflection point) of two straight lines traced through plots of the measured property versus the surfactant concentration. This visual data analysis method is highly subjective and can lead to very different CMC values depending on the type of representation, the quality of the data and the chosen interval around the CMC.[6] A preferred method is the fit of the experimental data with a model of the measured property. Fit functions for properties such as electrical conductivity, surface tension, NMR chemical shifts, absorption, self-diffusion coefficients, fluorescence intensity and mean translational diffusion coefficient of fluorescent dyes in surfactant solutions have been presented.[7][8][9] These fit functions are based on a model for the concentrations of monomeric and micellised surfactants in solution, which establishes a well-defined analytical definition of the CMC, independent from the technique.
The CMC is the concentration of surfactants in the bulk at which micelles start forming. The word bulk is important because surfactants partition between the bulk and interface and CMC is independent of interface and is therefore a characteristic of the surfactant molecule. In most situations, such as surface tension measurements or conductivity measurements, the amount of surfactant at the interface is negligible compared to that in the bulk and CMC can be approximated by the total concentration. In practice, CMC data is usually collected using laboratory instruments which allow the process to be partially automated, for instance by using specialised tensiometers.
When the interfacial areas are large, the amount of surfactant at the interface cannot be neglected. If, for example, air bubbles are introduced into a solution of a surfactant above CMC, these bubbles, as they rise to the surface, remove surfactants from the bulk to the top of the solution creating a foam column and thus reducing the concentration in bulk to below CMC. This is one of the easiest methods to remove surfactants from effluents (see foam flotation). Thus in foams with sufficient interfacial area are devoid of micelles. Similar reasoning holds for emulsions.
CMC is most typically measured by plotting surface tension versus surfactant concentration in an automated measurement.
The other situation arises in detergents. One initially starts off with concentrations greater than CMC in water and on adding fabric with large interfacial area, the surfactant concentration drops below CMC and no micelles remain at equilibrium. Therefore, the solubilization plays a minor role in detergents. Removal of oily soil occurs by modification of the contact angles and release of oil in the form of emulsion.
In petroleum industry, CMC is considered prior to injecting surfactant in reservoir regarding enhanced oil recovery (EOR) application. Below the CMC point, interfacial tension between oil and water phase is no longer effectively reduced.[10] If the concentration of the surfactant is kept a little above the CMC, the additional amount covers the dissolution with existing brine in the reservoir. It is desired that the surfactant will work at the lowest interfacial tension (IFT).
^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "critical micelle concentration". doi:10.1351/goldbook.C01395
^ Ana Domínguez, Aurora Fernández, Noemí González, Emilia Iglesias, and Luis Montenegro "Determination of Critical Micelle Concentration of Some Surfactants by Three Techniques", Journal of Chemical Education, Vol. 74 No. 10 October 1997, pp. 1227–1231 (pdf)
^ Hakiki, F., Maharsi, D.A. and Marhaendrajana, T. (2016). Surfactant-Polymer Coreflood Simulation and Uncertainty Analysis Derived from Laboratory Study. Journal of Engineering and Technological Sciences. 47(6):706-724. doi: 10.5614/j.eng.technol.sci.2015.47.6.9
^ Phillips J. The energetics of micelle formation. Transactions of the Faraday Society 1955;51:561-9
^ Mukerjee, P.; Mysels, K. J. In Critical Micelle Concentrations of Aqueous Surfactant Systems; NIST National Institute of Standards and Technology: Washington D.C. USA, 1971; Vol. NSRDS-NBS 36
^ Al-Soufi W, Piñeiro L, Novo M. A model for monomer and micellar concentrations in surfactant solutions: Application to conductivity, NMR, diffusion, and surface tension data. J.Colloid Interface Sci. 2012;370:102-10,DOI: 10.1016/j.jcis.2011.12.037
^ Lucas Piñeiro, Sonia Freire, Jorge Bordello, Mercedes Novo, and Wajih Al-Soufi, Dye Exchange in Micellar Solutions. Quantitative Analysis of Bulk and Single Molecule Fluorescence Titrations. Soft Matter, 2013,9, 10779-10790, DOI: 10.1039/c3sm52092g
^ www.usc.es/fotofqm/en/units/single-molecule-fluorescence/concentration-model-surfactants-near-cmc
^ Hakiki, Farizal. A Critical Review of Microbial Enhanced Oil Recovery Using Artificial Sandstone Core: A Mathematical Model. Paper IPA14-SE-119. Proceeding of The 38th IPA Conference and Exhibition, Jakarta, Indonesia, May 2014.
Theory of CMC measurement
CMCs and molecular weights of several detergents on OpenWetWare
Retrieved from "https://en.wikipedia.org/w/index.php?title=Critical_micelle_concentration&oldid=1085100849"
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Decide if each pair of triangles below are similar. If they are similar, give a sequence of transformations that justifies your conclusion. If they are not similar, explain how you know.
Look at the triangles. Does there seem to be a zoom factor? Do the sides seem to be proportional?
How could these triangles be proved similar? Are there any congruent angles? Are there any proportional sides?
\text{SAS}\sim
because the angle
110º
is between both sets of proportional sides.
What do the tick marks on the sides of the triangles mean? What type of triangles are they in part (c)?
\text{SSS}\sim
because both triangles are equilateral so all sides are proportional.
There are different ways that these triangles could be proved similar if you find the missing sides of both triangles. Use the Pythagorean Theorem.
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Parabolic_trajectory Knowpia
The orbital velocity (
{\displaystyle v}
) of a body travelling along parabolic trajectory can be computed as:
{\displaystyle v={\sqrt {2\mu \over r}}}
{\displaystyle r}
{\displaystyle \mu }
If a body has an escape velocity with respect to the Earth, this is not enough to escape the Solar System, so near the Earth the orbit resembles a parabola, but further away it bends into an elliptical orbit around the Sun.
This velocity (
{\displaystyle v}
) is closely related to the orbital velocity of a body in a circular orbit of the radius equal to the radial position of orbiting body on the parabolic trajectory:
{\displaystyle v={\sqrt {2}}\,v_{o}}
{\displaystyle v_{o}}
Equation of motionEdit
{\displaystyle r={h^{2} \over \mu }{1 \over {1+\cos \nu }}}
{\displaystyle r\,}
{\displaystyle h\,}
{\displaystyle \nu \,}
{\displaystyle \mu \,}
{\displaystyle \epsilon }
) of a parabolic trajectory is zero, so the orbital energy conservation equation for this trajectory takes the form:
{\displaystyle \epsilon ={v^{2} \over 2}-{\mu \over r}=0}
{\displaystyle v\,}
{\displaystyle r\,}
{\displaystyle \mu \,}
{\displaystyle C_{3}=0}
Barker's equationEdit
{\displaystyle t-T={\frac {1}{2}}{\sqrt {\frac {p^{3}}{\mu }}}\left(D+{\frac {1}{3}}D^{3}\right)}
{\displaystyle t_{f}-t_{0}={\frac {1}{2}}{\sqrt {\frac {p^{3}}{\mu }}}\left(D_{f}+{\frac {1}{3}}D_{f}^{3}-D_{0}-{\frac {1}{3}}D_{0}^{3}\right)}
{\displaystyle t-T={\sqrt {\frac {2r_{p}^{3}}{\mu }}}\left(D+{\frac {1}{3}}D^{3}\right)}
{\displaystyle {\begin{aligned}A&={\frac {3}{2}}{\sqrt {\frac {\mu }{2r_{p}^{3}}}}(t-T)\\[3pt]B&={\sqrt[{3}]{A+{\sqrt {A^{2}+1}}}}\end{aligned}}}
{\displaystyle \nu =2\arctan \left(B-{\frac {1}{B}}\right)}
Radial parabolic trajectoryEdit
{\displaystyle r={\sqrt[{3}]{4.5\mu t^{2}}}}
{\displaystyle t=0\!\,}
At any time the average speed from
{\displaystyle t=0\!\,}
is 1.5 times the current speed, i.e. 1.5 times the local escape velocity.
{\displaystyle t=0\!\,}
at the surface, apply a time shift; for the Earth (and any other spherically symmetric body with the same average density) as central body this time shift is 6 minutes and 20 seconds; seven of these periods later the height above the surface is three times the radius, etc.
^ Bate, Roger; Mueller, Donald; White, Jerry (1971). Fundamentals of Astrodynamics. Dover Publications, Inc., New York. ISBN 0-486-60061-0. p 188
^ Montenbruck, Oliver; Pfleger, Thomas (2009). Astronomy on the Personal Computer. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-67221-0. p 64
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Rewards & Penalties - ICON DevPortal
There are 7 reward types for earning ICX based on different types of participation in the ICON ecosystem. The reward system is bilateral in the relationship between delegates and general community members. Notice that both the delegates and general community members benefit from the same performance-based outcomes, but they are rewarded at different rates. The principle here is that delegates maintain more responsibility and are both rewarded and penalized as such:
Block validation reward
Delegate reward
Delegate delegation reward
Ecosystem expansion project reward
Ecosystem expansion project delegation reward
Decentralized application reward
Decentralized application delegation reward
The Block validation reward is given to the top 22 delegates for proving their contribution by producing and validating blocks. This top tier of delegates is given roles in block production and validation, and the delegates can acquire a certain amount of ICX reward per block for providing computational resources. This system provides an economic incentive for block production and validation, thus maintaining the distributed network.
The Delegate reward rewards the top 100 delegates for proving their contribution by registering a node and receiving network-stake delegation from ICON community members. All delegates acquire a certain amount of ICX per block, depending on the amount of tokens staked (i.e. delegated) to them by ICON community members. The more tokens that are staked to them, the more rewards they earn. Therefore, this system provides an economic incentive for delegates to work harder to receive more tokens staked to them.
Delegate delegation reward is specified for ICON community members in exchange for staking their ICX to a delegate. All community members can acquire a certain amount of ICX per block, based on how much they staked.
Ecosystem expansion project reward is the reward for the top 100 ecosystem expansion projects. All ecosystem expansion projects can prove their contribution by receiving staked ICX from community members. Only the top 100 ecosystem expansion projects acquire a certain amount of ICX per block. As a result of this, ecosystem expansion projects have an incentive to perform better projects to receive more staked tokens from community members.
Ecosystem expansion project delegation reward is specified for community members in exchange for delegating their ICX to an ecosystem expansion project, thus proving the contribution of the ecosystem expansion project. All community members can delegate ICX to ecosystem expansion project and can acquire a certain amount of ICX per block.
The 'Decentralized Application (dApp) Reward’ is the reward for the top 100 dApps according to the amount of ICX staked to that dApp. All dApps can receive staked ICX from community members. Only the top 100 dApps acquire a certain amount of ICX per block, depending on the amount of ICX staked. As a result of this, dApps have an incentive to develop a popular dApp to receive more staked tokens from community members.
Decentralized application delegation reward is the reward specified for community members in exchange for staking their ICX to a dApp, thus proving the contribution of the dApp. All community members can delegate ICX to dApps and can acquire a certain amount of ICX per block.
There are 3 types of penalties that result in forfeiting ICX or the opportunity to earn ICX based on participation patterns and community agreements within the ICON ecosystem. Delegates are the only community members who are penalized in the following ways based on their performance:
Validation penalty
Low productivity penalty
This occurs when a specific delegate fails to validate blocks successively for 660 blocks
The penalty is to exclude such delegates from block production during the term. Block production and validation opportunities are forfeited until the next term, creating an opportunity cost.
This applies when the Productivity Ratio of a specific delegate has dropped below 85%. The Productivity Ratio is the ratio of actual blocks produced and validated divided by the number of opportunities to produce and validate a block. Delegates will not be subject to this penalty for their first 86,240 blocks as a block-producing delegate.
The penalty is to disqualify the delegate and burn 6% of the ICX staked to them.
This applies when a specific delegate has been disqualified via a Delegate Disqualification Proposal.
The penalty is to disqualify the delegate in question and burn 6% of the ICX staked to them.
Block\ validation\ reward = \beta_1 * 0.5 * (boolean(produces\ block) + boolean(validates\ block)/n_{del})
\beta_1 = (i_{del} * 0.5) * 22 * 1/1,296,000
i_{del} = determined\ by\ stake-weighted\ vote\ of \ top\ 22\ delegates
Delegate\ reward = \beta_2 * ICX_{del,staked}/ICX_{top100del,staked}
\beta_2=(i_{del} * 0.5) * 100 * 1/1,296,000
Delegate\ delegation\ reward = \beta_3 * ICX_{self,staked}/ICX_{all,staked}
\beta_3=r_{del} * ICX_{del,staked} * 1/15,552,000
r_{del} = (r_{max} - r_{min}) / (r_{point})^2 * ((ICX_{del,staked}/ICX_{tot}) - r_{point})^2 + r_{min}
r_{max} = 12\%
r_{min} = 2\%
r_{point} = 70\%
Ecosystem\ expansion\ project\ reward = \beta_4 * ICX_{del, staked} / ICX_{top100eep,staked}
\beta_4 = i_{eep} * 100 * 1/1,296,000
i_{eep} = i_{del} * 0.25
Ecosystem\ expansion\ project\ delegation\ reward = \beta_5 * ICX_{self,staked} / ICX_{top100eep,staked}
\beta_5 = r_{eep} * ICX_{eep,staked} * 1/15,552,000
r_{eep} = (r_{max} - r_{min}) / (r_{point})^2 * ((ICX_{eep,staked}/ICX_{tot}) - r_{point})^2 + r_{min}
Decentralized\ application\ reward = \beta_6 * ICX_{dapp, staked} / ICX_{top100dapp,staked}
\beta_6 = i_{dapp} * 100 * 1/1,296,000
i_{dapp} = i_{del} * 0.25
Decentralized\ application\ delegation\ reward = \beta_7 * ICX_{self,staked} / ICX_{top100dapp,staked}
\beta_7 = r_{dapp} * ICX_{dapp,staked} * 1/15,552,000
r_{dapp} = (r_{max} - r_{min}) / (r_{point})^2 * ((ICX_{dapp,staked}/ICX_{tot}) - r_{point})^2 + r_{min}
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This post is the first in a series of my study notes on regression techniques. I first learnt about regression as a way of fitting a line through a series of points. Invoke some assumptions and one obtains the relationship between two variables. Simple...or so I thought. Through the course of my study, I developed a deeper appreciation of its nuances which I hope to elucidate in these set of notes.
Aside: The advancements in regression analysis, since it was introduced by Gauss in the early 19th century, is an interesting case study of the development of applied mathematics. The method remains roughly the same, but advances in other related fields (linear algebra, statistics) and applied econometrics helped clarify the assumptions used and elevate its status in modern applied research.
In this review, I shall focus on the ordinary linear regression (OLS) and omit treatment of its many descendants.1 Let's start at the source and cover regression as a solution to the least squares minimisation problem, before going to deeper waters!
Preliminaries / Notation
n
k
\mathbf{Y}
n \times 1
\mathbf{Y} = \left[\begin{array} {c} y_1 \\ . \\ . \\ . \\ y_n \end{array}\right]
\mathbf{X}
n \times k
k \times 1
\mathbf{X} = \left[\begin{array} {ccccc} x_{11} & . & . & . & x_{1k} \\ . & . & . & . & . \\ . & . & . & . & . \\ . & . & . & . & . \\ x_{n1} & . & . & . & x_{nn} \end{array}\right] = \left[\begin{array} {c} \mathbf{x}'_1 \\ . \\ . \\ . \\ \mathbf{x}'_n \end{array}\right]
\mathbf{U}
n \times 1
y_i = \mathbf{x}'_i \beta + u_i
E(\mathbf{U}|\mathbf{X}) = 0
\mathbf{X}
k
Var(\mathbf{U}|\mathbf{X}) = \sigma^2 I_n
\beta
Q = \sum_{i=1}^{n}{u_i^2} = \sum_{i=1}^{n}{(y_i - \mathbf{x}'_i\beta)^2} = (Y-X\beta)'(Y-X\beta)
Q
1 \times 1
\frac{\partial b'Ab}{\partial b} = 2Ab
\beta
\begin{aligned} \min Q &= \min_{\beta} \mathbf{Y}'\mathbf{Y} - 2\beta'\mathbf{X}'\mathbf{Y} + \beta'\mathbf{X}'\mathbf{X}\beta \\ &= \min_{\beta} - 2\beta'\mathbf{X}'\mathbf{Y} + \beta'\mathbf{X}'\mathbf{X}\beta \\ \text{[FOC]}~~~0 &= - 2\mathbf{X}'\mathbf{Y} + 2\mathbf{X}'\mathbf{X}\hat{\beta} \\ \hat{\beta} &= (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y} \\ &= (\sum^{n} \mathbf{x}_i \mathbf{x}'_i)^{-1} \sum^{n} \mathbf{x}_i y_i \end{aligned}
\hat{\beta}
is a linear estimator i.e. it can be written in the form
b=AY
A
X
Y
Under assumptions 1-3, the estimator is unbiased. Substituting
y_{i}
\begin{aligned} E(\hat{\beta}|\mathbf{X}) &= \beta + E((\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'U|X) \\ &= \beta + (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'E(U|X) \\ &= \beta \end{aligned}
By law of iterated expectation
E(\hat{\beta}) = EE(\hat{\beta}|\mathbf{X}) = \beta
3. Adding in the homoskedascity assumption, the OLS estimator is the Best Linear Unbiased Estimator (BLUE) i.e. smallest variance among other linear and unbiased estimators or
Var(b|\mathbf{X}) - Var(\hat{\beta}|\mathbf{X})
is p.s.d.
If the errors are normally distributed then conditional on
\mathbf{X}
\hat{\beta}
is also normally distributed.
It is almost impossible for any real life data to satisfy the above assumptions, an exception is when
Y
X
are jointly normal but that is a stretch to belief. To get around this issue, one can replace assumption 2 (conditional independence) with a weaker assumption:
E(u_{i}\mathbf{x_{i}}) = 0
(weak exogeneity). Under this weaker assumption, the estimator is no longer unbiased.2 One must appeal to large sample theory to draw any meaningful results. More specifically, we use the idea of convergence in probability and weak law of large numbers to show that the estimator is consistent.3
E(u_{i}\mathbf{x_{i}}) = 0
(weak exogeneity)
(y_{i},\mathbf{x}_{i})
are i.i.d
E(\mathbf{x}_{i}\mathbf{x}_{i}')
is p.s.d
Ex^{4}_{i,j} < \infty
Eu^{4}_{i} < \infty
Eu^{2}_{i}\mathbf{x}_{i}\mathbf{x}_{i}'
\hat{\beta}_{n}
is consistent since
\hat{\beta}_{n} \rightarrow_{p} \beta
n \rightarrow \infty
Large sample assumptions 3 and 4 are needed to establish convergence in probability:
\hat{\beta}_{n} = \beta +(\frac{1}{n} \sum^{n} \mathbf{x}_i \mathbf{x}'_i)^{-1} \frac{1}{n}\sum^{n} \mathbf{x}_i u_i
\frac{1}{n} \sum^{n} \mathbf{x}_i \mathbf{x}'_i \rightarrow_{p} E(\mathbf{x}_{i}\mathbf{x}_{i}')
\frac{1}{n} \sum^{n} \mathbf{x}_i u_i \rightarrow_{p} E(u_{i}\mathbf{x_{i}}) = 0
to prove consistency.
Large sample assumptions 1-7 are used to prove asymptotic normality of the estimator.
The popularity and limitations of the simple OLS regression has spawn many related techniques that are the subject of numerous research papers by themselves. ↩
Recall that unbiasedness requires conditional independence to hold but uncorrelatedness does not imply conditional independence. ↩
Similarly, the central limit theorem is used to establish convergence in distribution which is needed for statistical inference. ↩
\beta
is denoted with a subscript n to signify that it is a function of the sample size. ↩
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Multiplier uncertainty - Wikipedia
In macroeconomics, multiplier uncertainty is lack of perfect knowledge of the multiplier effect of a particular policy action, such as a monetary or fiscal policy change, upon the intended target of the policy. For example, a fiscal policy maker may have a prediction as to the value of the fiscal multiplier—the ratio of the effect of a government spending change on GDP to the size of the government spending change—but is not likely to know the exact value of this ratio. Similar uncertainty may surround the magnitude of effect of a change in the monetary base or its growth rate upon some target variable, which could be the money supply, the exchange rate, the inflation rate, or GDP.
There are several policy implications of multiplier uncertainty: (1) If the multiplier uncertainty is uncorrelated with additive uncertainty, its presence causes greater cautiousness to be optimal (the policy tools should be used to a lesser extent). (2) In the presence of multiplier uncertainty, it is no longer redundant to have more policy tools than there are targeted economic variables. (3) Certainty equivalence no longer applies under quadratic loss: optimal policy is not equivalent to a policy of ignoring uncertainty.
1 Effect of multiplier uncertainty on the optimal magnitude of policy
2 Multiple targets or policy instruments
3 Analogy to portfolio theory
4 Dynamic policy optimization
Effect of multiplier uncertainty on the optimal magnitude of policy[edit]
For the simplest possible case,[1] let P be the size of a policy action (a government spending change, for example), let y be the value of the target variable (GDP for example), let a be the policy multiplier, and let u be an additive term capturing both the linear intercept and all unpredictable components of the determination of y. Both a and u are random variables (assumed here for simplicity to be uncorrelated), with respective means Ea and Eu and respective variances
{\displaystyle \sigma _{a}^{2}}
{\displaystyle \sigma _{u}^{2}}
{\displaystyle y=aP+u.}
Suppose the policy maker cares about the expected squared deviation of GDP from a preferred value
{\displaystyle y_{d}}
; then its loss function L is quadratic so that the objective function, expected loss, is given by:
{\displaystyle {\text{E}}L={\text{E}}(y-y_{d})^{2}={\text{E}}(aP+u-y_{d})^{2}=[{\text{E}}(aP+u-y_{d})]^{2}+{\text{var}}(aP+u-y_{d})=[({\text{E}}a)P+{\text{E}}u-y_{d}]^{2}+P^{2}\sigma _{a}^{2}+\sigma _{u}^{2}.}
where the last equality assumes there is no covariance between a and u. Optimizing with respect to the policy variable P gives the optimal value Popt:
{\displaystyle P^{opt}={\frac {({\text{E}}a)(y_{d}-{\text{E}}u)}{({\text{E}}a)^{2}+\sigma _{a}^{2}}}.}
Here the last term in the numerator is the gap between the preferred value yd of the target variable and its expected value Eu in the absence of any policy action. If there were no uncertainty about the policy multiplier,
{\displaystyle \sigma _{a}^{2}}
would be zero, and policy would be chosen so that the contribution of policy (the policy action P times its known multiplier a) would be to exactly close this gap, so that with the policy action Ey would equal yd. However, the optimal policy equation shows that, to the extent that there is multiplier uncertainty (the extent to which
{\displaystyle \sigma _{a}^{2}>0}
), the magnitude of the optimal policy action is diminished.
Thus the basic effect of multiplier uncertainty is to make policy actions more cautious, although this effect can be modified in more complicated models.
Multiple targets or policy instruments[edit]
The above analysis of one target variable and one policy tool can readily be extended to multiple targets and tools.[2] In this case a key result is that, unlike in the absence of multiplier uncertainty, it is not superfluous to have more policy tools than targets: with multiplier uncertainty, the more tools are available the lower expected loss can be driven.
Analogy to portfolio theory[edit]
There is a mathematical and conceptual analogy between, on the one hand, policy optimization with multiple policy tools having multiplier uncertainty, and on the other hand, portfolio optimization involving multiple investment choices having rate-of-return uncertainty.[2] The usages of the policy variables correspond to the holdings of the risky assets, and the uncertain policy multipliers correspond to the uncertain rates of return on the assets. In both models, mutual fund theorems apply: under certain conditions, the optimal portfolios of all investors regardless of their preferences, or the optimal policy mixes of all policy makers regardless of their preferences, can be expressed as linear combinations of any two optimal portfolios or optimal policy mixes.
Dynamic policy optimization[edit]
The above discussion assumed a static world in which policy actions and outcomes for only one moment in time were considered. However, the analysis generalizes to a context of multiple time periods in which both policy actions take place and target variable outcomes matter, and in which time lags in the effects of policy actions exist. In this dynamic stochastic control context with multiplier uncertainty,[3][4][5] a key result is that the "certainty equivalence principle" does not apply: while in the absence of multiplier uncertainty (that is, with only additive uncertainty) the optimal policy with a quadratic loss function coincides with what would be decided if the uncertainty were ignored, this no longer holds in the presence of multiplier uncertainty.
^ Brainard, William (1967). "Uncertainty and the effectiveness of policy". American Economic Review. 57 (2): 411–425. JSTOR 1821642.
^ a b Mitchell, Douglas W. (1990). "The efficient policy frontier under parameter uncertainty and multiple tools". Journal of Macroeconomics. 12 (1): 137–145. doi:10.1016/0164-0704(90)90061-E.
^ Chow, Gregory P. (1976). Analysis and Control of Dynamic Economic Systems. New York: Wiley. ISBN 0-471-15616-7.
^ Turnovsky, Stephen (1976). "Optimal stabilization policies for stochastic linear systems: The case of correlated multiplicative and additive disturbances". Review of Economic Studies. 43 (1): 191–194. JSTOR 2296741.
^ Turnovsky, Stephen (1974). "The stability properties of optimal economic policies". American Economic Review. 64 (1): 136–148. JSTOR 1814888.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Multiplier_uncertainty&oldid=755689480"
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Break-Up Length and Spreading Angle of Liquid Sheets Formed by Splash Plate Nozzles | J. Fluids Eng. | ASME Digital Collection
N. Ashgriz,
N. Ashgriz
e-mail: ashgriz@mie.utoronto.ca
H. N. Tran
Department of Chemical Engineering and Applied Chemistry,
, 200 College Street, Toronto, ON, M5S 3E5, Canada
Ahmed, M., Ashgriz, N., and Tran, H. N. (December 11, 2008). "Break-Up Length and Spreading Angle of Liquid Sheets Formed by Splash Plate Nozzles." ASME. J. Fluids Eng. January 2009; 131(1): 011306. https://doi.org/10.1115/1.3026729
An experimental investigation is conducted to determine the effect of liquid viscosity and density, nozzle diameter, and flow velocity on the break-up length and spreading angle of liquid sheets formed by splash plate nozzles. Various mixtures of corn syrup and water were used to obtain viscosities in the range of
1–170 mPa s
. Four different splash plate nozzle diameters of 0.5 mm, 0.75 mm, 1 mm, and 2 mm, with a constant plate angle of 55 deg were tested. The liquid sheet angles and the break-up lengths were measured at various operating conditions. An empirical correlation for the sheet spreading angle and a semi-empirical correlation for the sheet break-up lengths are developed.
density, nozzles, viscosity
Flow (Dynamics), Nozzles, Viscosity, Density
Characteristic of Spray Through Wall Impinging Nozzles
Prediction of Mean Droplet Size of Sprays Issued From Wall Impingement Injector
Effect of Temperature on Drop Size of Black Liquor Sprays
Proceedings of the International Chemical Recovery Conference
, TAPPI, New Orleans, LA, pp.
The Effect of Ambient Density on Drop Formation in Sprays
Obuskovic
Fluid Sheet Thickness and Velocity at the Tip of a Black Liquor Splash Plate Nozzle
A Study of Sprayed Formed by Impinging Jets in Laminar and Turbulent Flow
Disintegration of a Thin Liquid Sheet in a Concurrent Gas Stream
Atomization Characteristics of Impinging Jets
The Break-Up of Axisymmetric Liquid Sheets
Fundamental Studies of Impinging Liquid Jets
30th Aerospace Sciences Meeting and Exhibition
, Paper No. AIAA 92–0458.
Characteristics of Liquid Sheets Formed by Two Impinging Jets
“Thickness Variation of Liquid Sheet Formed by Two Impinging Jets Using Holographic Interferometry
Parametric Study on Impinging-Jet Liquid Sheet Thickness Distribution Using an Interferometric Method
Speilbauer
Mechanisms of Liquid Sheet Breakup and the Resulting Drop Size Distributions, Part 1: Types of Spray Nozzles and Mechanisms of Sheet Disintegration
Mechanisms of Liquid Sheet Breakup and the Resulting Drop Size Distributions, Part 2: Strand Breakup and Experimental Observations
Furnace Endoscope—Measuring Fuel Spray Properties in Hot and Corrosive Environments
Characteristics of Liquid Sheet Formed by Splash Plate Nozzles
Liquid Taylor Bubbles Rising in a Vertical Column of a Heavier Liquid: An Approximate Analysis
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Ammonia Knowpia
AmphotericityEdit
Self-dissociationEdit
Formation of other compoundsEdit
Ammonia as a ligandEdit
Detection and determinationEdit
Ammonia in solutionEdit
Ammonia and ammonium salts can be readily detected, in very minute traces, by the addition of Nessler's solution, which gives a distinct yellow colouration in the presence of the slightest trace of ammonia or ammonium salts. The amount of ammonia in ammonium salts can be estimated quantitatively by distillation of the salts with sodium (NaOH) or potassium hydroxide (KOH), the ammonia evolved being absorbed in a known volume of standard sulfuric acid and the excess of acid then determined volumetrically; or the ammonia may be absorbed in hydrochloric acid and the ammonium chloride so formed precipitated as ammonium hexachloroplatinate, [NH4]2[PtCl6].[36]
Gaseous ammoniaEdit
Ammoniacal nitrogen (NH3-N)Edit
Solubility of saltsEdit
Solutions of metalsEdit
Redox properties of liquid ammoniaEdit
Precursor to nitrogenous compoundsEdit
Cleansing agentEdit
Antimicrobial agent for food productsEdit
Green ammonia is considered as a potential fuel for future container ships. In 2020, the companies DSME and MAN Energy Solutions announced the construction of an ammonia-based ship, DSME plans to commercialize it by 2025.[98] The use of ammonia as a potential alternative fuel for aircraft jet engines is also being explored.[99]
Japan is targeting to bring forward a plan to develop ammonia co-firing technology that can increase the use of ammonia in power generation, as part of efforts to assist domestic and other Asian utilities to accelerate their transition to carbon neutrality.[100] In October 2021, the first International Conference on Fuel Ammonia (ICFA2021) was held.[101][102]
Remediation of gaseous emissionsEdit
As a hydrogen carrierEdit
Refrigeration – R717Edit
StimulantEdit
FumingEdit
Coking wastewaterEdit
Storage informationEdit
Laboratory use of anhydrous ammonia (gas or liquid)Edit
Haber–BoschEdit
Role in biological systems and human diseaseEdit
Interstellar formation mechanismsEdit
The rate constant, k, of this reaction depends on the temperature of the environment, with a value of 5.2×10−6 at 10 K.[166] The rate constant was calculated from the formula
{\displaystyle k=a(T/300)^{B}}
. For the primary formation reaction, a = 1.05×10−6 and B = −0.47. Assuming an NH+4 abundance of 3×10−7 and an electron abundance of 10−7 typical of molecular clouds, the formation will proceed at a rate of 1.6×10−9 cm−3s−1 in a molecular cloud of total density 105 cm−3.[167]
Interstellar destruction mechanismsEdit
Single antenna detectionsEdit
Interferometric studiesEdit
Infrared detectionsEdit
Observations of nearby dark cloudsEdit
UC HII regionsEdit
Extragalactic detectionEdit
^ "What will power aircraft in the future?". Aviafuture. 30 March 2022. Retrieved 24 May 2022.
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(Redirected from Inverse square law)
{\displaystyle {\text{intensity}}\ \propto \ {\frac {1}{{\text{distance}}^{2}}}\,}
{\displaystyle {\frac {{\text{intensity}}_{1}}{{\text{intensity}}_{2}}}={\frac {{\text{distance}}_{2}^{2}}{{\text{distance}}_{1}^{2}}}}
{\displaystyle {\text{intensity}}_{1}\times {\text{distance}}_{1}^{2}={\text{intensity}}_{2}\times {\text{distance}}_{2}^{2}}
{\displaystyle F=k_{\text{e}}{\frac {q_{1}q_{2}}{r^{2}}}}
{\displaystyle I={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.\,}
{\displaystyle p\ \propto \ {\frac {1}{r}}\,}
{\displaystyle v\,}
{\displaystyle p\,}
{\displaystyle v\ \propto {\frac {1}{r}}\ \,}
{\displaystyle I\ =\ pv\ \propto \ {\frac {1}{r^{2}}}.\,}
{\displaystyle I\propto {\frac {1}{r^{n-1}}},}
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An Empirical Analysis of the Total Retail Sales of Consumer Goods by Using Time Series Model
An Empirical Analysis of the Total Retail Sales of Consumer Goods by Using Time Series Model
School of Mathematics and Statistics, Qinghai Nationalities University, Xining, China
With the continuous improvement of living standards, the total amount of social consumer goods in China’s economic development has occupied an important position. Its fluctuations can indirectly reflect the demand and purchasing power of commodities, thus affecting the state’s macroeconomic regulation and control. This paper selects the total amount of social consumer goods in China from August 2005 to February 2019. Using EViews 7.2 software, this paper makes use of the correlation analysis of sequence fluctuation in econometrics and financial time series, and finds the best fitting EGARCH (1,1) model based on ARMA (1,0) to conduct an empirical analysis of the total amount of social consumer goods in China, and concludes that the total amount of social consumer goods in China has leverage effect.
Retail Sales of Consumer Goods, EGARCH Model, Leverage Effect
In recent years, the living standard of residents has been improved, and the increase in consumption has stimulated economic development. However, there are many factors affecting the economic development of China, among which the total retail sales of consumer goods have a huge impact on the economy. Fang Huliu [1] (2009) proposed that the total retail sales of social consumer goods are important data for studying the living standard of residents, the purchasing power of social retail goods and social production. It reflects the improvement of people’s material and cultural living standards in a certain period of time and the size of the retail market. Sun Yan and Peng Yangyang (2016) [2] mentioned that the demand of social retail consumer goods has a direct impact on the quantity of consumer goods, and then the country will make corresponding adjustment to the macro-economy. Finally, the price of consumer goods will also fluctuate. Lin Puren and Hu Xiangfei [3] (2010) summed up the increase of the total retail sales of consumer goods means the increase of demand, which stimulates the economy, further improves the performance of related enterprises and improves residents’ income. Conversely, consumer demand will be weak, reducing the rate of economic growth. Miao Tingting [4] (2017) compares the supply and demand of social consumer goods to a circular chain, which affects one ring and interlocks. Peng Bing [5] (2012) mentioned that the research on the total retail sales of consumer goods has become crucial to avoid the situation of oversupply and inflation and promote the continuous development of the economy. Therefore, this paper selects the subject model with better effect, and then finds the EGARCH (1,1) model with a higher fitting degree, and finally makes a correlation analysis on it.
2.1. The ARCH Model
ARCH model was put forward by Engle in 1982 [6] and has been widely used in the financial market.
For the usual regression model, we obtain that:
{y}_{t}={{x}^{\prime }}_{t}\beta +{\epsilon }_{t}
{\epsilon }_{t}^{2}
obeys AR(q) process, we obtain the following:
{\epsilon }_{t}^{2}={\alpha }_{0}+{\alpha }_{1}{\epsilon }_{t-1}^{2}+\cdots +{\alpha }_{q}{\epsilon }_{t-q}^{2}+{\eta }_{t}
t=1,2,\cdots
{\eta }_{\text{t}}
is independent and identically distributed, and it satisfies
E\left({\eta }_{t}\right)=0
D\left({\eta }_{t}\right)={\lambda }^{2}
, Then, the model can be thought of as autoregressive conditional heteroscedastic model, notes for the ARCH model.
2.2. The ARCH Effect Test
To test whether there is ARCH effect, the Lagrange multiplier method, namely LM test, is usually adopted. If the random perturbation term
{\epsilon }_{t}
of the model obeys ARCH(q), the auxiliary regression equation can be established:
{h}_{t}={\alpha }_{0}+{\alpha }_{1}{\epsilon }_{t-1}^{2}+\cdots +{\alpha }_{q}{\epsilon }_{t-q}^{2}
If the probability of a regression coefficient of 0 is relatively large, there is no ARCH effect in the sequence; otherwise, the series has an ARCH effect.
2.3. The EGARCH Model
The EGARCH model, exponential GARCH model, was developed by Nelson in 1991 [7] . The conditional variance expression of the model is:
\mathrm{log}\left({h}_{t}\right)={\alpha }_{0}+\underset{j=1}{\overset{p}{\sum }}{\theta }_{j}\mathrm{log}\left({h}_{t-j}\right)+\underset{j=1}{\overset{q}{\sum }}\left[\alpha |\frac{{\epsilon }_{t-i}}{\sqrt{{h}_{t-i}}}|+{\phi }_{i}\frac{{\epsilon }_{t-i}}{\sqrt{{h}_{t-i}}}\right]
In this model, the conditional variance is in the form of natural logarithm, which means that
{h}_{t}
is non-negative, leveraged and exponential. If
\phi \ne 0
, it means that it is asymmetric. If
\phi <0
, there is a leverage effect.
The data were obtained from the Wind database. The Wind database is the most complete and accurate first-rate large-scale financial engineering and financial data warehouse with financial securities as the core in China. And with the help of EViews7.2 [8] [9] . Figure 1 is the trend chart of the total retail sales of consumer goods in China from August 2005 to February 2019. Whether the sequence is stable can be observed or further determined by the unit root test. After observing Figure 1, it is known that the time series data is not stable, and unit root test is conducted further (Table 1).
It can be observed from Figure 1 that there is a trend item in the sequence, and the sequence is preliminarily judged to be unstable. Next, the unit root test is carried out to further determine whether the sequence is stable.
Table 1 is the unit root test of the sequence, and the results show that the unit root statistic ADF is 1.785051, far higher than the confidence interval of 10%, 5% and 1%, and the probability P value is 0.9997. It can be judged that the sequence has a unit root, failing the ADF test, indicating that the sequence is unstable.
Because the sequence of total retail sales of social consumer goods is unstable and has trend terms, the differential method is adopted to make it tend to be stable.
By observing Figure 2, it can be seen that the sequence is stable, has a peak value, fluctuates up and down around the value of 0, and has a state of aggregation, so there may be heteroscedasticity, and unit root test can be conducted further.
According to the data in Table 2, the ADF statistic is far less than the critical value of 1% significance level, so it can be seen that there is no unit root in the sequence, and the sequence tends to be stable.
Figure 1. Time series trend diagram.
Figure 2. Trend chart of sequence At.
Table 1. ADF test of sequence Xt.
Table 2. ADF test of sequence At.
3.2. Basic Statistical Characteristics
By observing the relevant data in the basic statistics Table 3, the skewness is −0.649751 and less than 0, and it can be judged that there is a phenomenon of left skewness. The kurtosis value of left skewness is 12.77227, which is much higher than that of the normal distribution.
3.3. Conditional Heteroscedastic Effect Test
According to the comparison of relevant parameters of the established model, ARMA (1,0) was finally selected as the main model, and the corresponding parameters were estimated as follows:
{A}_{t}=-0.373209{A}_{t}+{\alpha }_{t}
. ARCH LM can be used to test whether there is ARCH effect, and residual square correlation graph can also be used to judge. This paper uses the ARCH LM test to determine
Table 3. Basic statistics for the sequence
whether there is an ARCH effect. First, check whether the second order has the ARCH effect.
Table 4 is the LM test result when the test order q = 2. From the table, the associated probability of the F statistic and the p statistic are both 0.0001, which indicates the ARCH effect.
It is observed from Table 5 that the probability P value is less than 1% when the test order reaches 7th order, and the null hypothesis is rejected. That is, the correlation data of Table 3 and Table 4 can be combined to determine that the residual sequence has a high-order ARCH effect.
3.4. The Choice of ARCH Model
According to the comparison of logarithmic likelihood function, AIC and SC of each model in Table 6, it is found that the AIC value of EGARCH (1,1) model is the minimum, and its logarithmic likelihood function is the maximum. Therefore, it can be judged that the sequence has the best fitting with EGARCH (1,1) model.
According to the relevant statistics of EGARCH (1,1) model, it is found that the P value of both the leverage coefficient and other coefficients is 0 at the significance level of 0.01, which can obtain a good fitting effect and further establish the EGARCH (1,1) model:
The mean equation is that:
{A}_{t}=-0.072308{A}_{t-1}+{\alpha }_{t}
Variance equation is that:
\mathrm{log}\left({h}_{t}^{2}\right)=0.720081+0.972835\mathrm{log}\left({h}_{t-1}^{2}\right)-1.201969\frac{|{\alpha }_{t-1}|}{{h}_{t-1}}+1.712505\frac{{\alpha }_{t-1}}{{h}_{t-1}}
According to the variance equation, the parametersv
{\alpha }_{0}
= 0.720081 and
{\theta }_{1}
= 0.972835 are both greater than 0, indicating that the fluctuation is in a forward direction. It was observed that the asymmetric coefficient
{\phi }_{1}
= −1.201969 was negative, indicating that the fluctuation of total retail sales of consumer goods was asymmetric and there was leverage effect.
3.5. The Adaptability Test
Table 7 gives two test results. It can be seen that the p-value of LM statistic accepts the null hypothesis at the significance level of 1%, and it can be judged that there is no autocorrelation in the residual sequence. Therefore, the EGARCH (1,1) model has the best effect.
This paper first observed the fluctuation of data and made corresponding processing to make the data tend to be stable. After establishing the subject
Table 4. Second-order LM test for sequences.
Table 5. Higher-order LM test for sequences.
Table 6. Model identification.
Table 7. ARCH effect test results of sequence residuals.
model, it analyzed the characteristics of related statistics and preliminarily judged that there was ARCH effect in this sequence. Then, it analyzed the extended ARCH model and found that the EGARCH (1,1) model could better reflect the fluctuation of total retail sales of social consumer goods and its fitting was the best.
Further analysis of the EGARCH (1,1) model shows that the leverage coefficient in the model is negative, indicating that the fluctuation of the total retail sales of consumer goods has leverage effect. Therefore, negative returns in the same situation have a more drastic impact on positive returns. Through the above analysis and research on the characteristics of fluctuations, I hope it can be helpful to the development of macro economy.
It can be concluded that we should adjust the supply quantity of retail sales of consumer goods according to the demand. Keep it as balanced as possible. When the purchasing power of residents declines, the price of consumer goods should be lowered to stimulate consumption and drive the development of the whole economy.
On the other hand, when the purchasing power of residents declines, if we just blindly reduce the price of consumer goods is not sustainable, we can only treat the symptoms and not the root cause. Tracing back to the source, we should increase the income of residents and let income drive consumption. At this time, the implementation of state policies is particularly important. Some policies to increase domestic demand should be introduced to increase the income of low-income families, strengthen the construction of people’s livelihood, stabilize the prices of some consumer goods, and promote the purchasing power of commodities to stimulate stable economic growth.
In addition, there is an error in selecting any model for correlation testing and research. In this paper, based on ARMA model, combined with ARCH model, and errors also exist in the model inspection. In the future, we should improve our knowledge reserve and reduce errors in a large number of reading materials.
This work is supported by the National Natural Science Foundation of China (No. 11561056) and Natural Science Foundation of Qinghai (No. 2016-ZJ-914).
Shen, S.C. and Dong, X.Y. (2019) An Empirical Analysis of the Total Retail Sales of Consumer Goods by Using Time Series Model. Journal of Mathematical Finance, 9, 175-181. https://doi.org/10.4236/jmf.2019.92009
1. Fang, H.L. (2009) Analysis on the Fluctuation Law and Influencing Factors of Total Retail Sales of Consumer Goods. Journal of Shanxi Finance and Economics University, No. 7, 22-28.
2. Sun, Y. and Peng, Y.Y. (2016) Analysis and Forecast of Total Retail Sales of Consumer Goods in China. Statistics and Decision, No. 18, 90-94.
3. Lin, P.R. and Hu, X.F. (2010) Forecast Model of Total Retail Sales of Consumer Goods in China. Guangxi Sciences, No. 17, 206-208.
4. Miao, T.T. (2017) Analysis on Influencing Factors of Retail Sales of Consumer Goods—Empirical Analysis Based on Provincial Panel Data. Times Finance, No. 29, 96-97.
5. Peng, B. Hu, J.T. and Deng, J. (2012) Empirical Analysis on the Influencing Factors of Total Retail Sales of Consumer Goods in the Yangtze River Delta. Knowledge Economy, No. 12, 93-94.
6. Engle, R.F. (1982) Autoregressive Conditional Heteroscedaticity with Estimates of the Variance of UK Inflation. Econometrica, 50, 987-1008. https://doi.org/10.2307/1912773
7. Nelson, D.B. (1991) Conditional Heteroskedasticity in Asset Returns: A Returns: A New Approach. Econometrica: Journal of the Econometric Society, 59, 347-370. https://doi.org/10.2307/2938260
8. Yi, D.H. (2008) Data Analysis and Application of EViews. China Renmin University Press, Beijing.
9. Gao, T.M. (2009) Econometric Analysis Methods and Modeling—EViews Applications and Examples. 2nd Edition, Tsinghua University Press, Beijing.
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Representation formulas for $L^\infty $ norms of weakly convergent sequences of gradient fields in homogenization
Representation formulas for
{L}^{\infty }
norms of weakly convergent sequences of gradient fields in homogenization
Lipton, Robert ; Mengesha, Tadele
Classification : 35J15, 49N60
Mots clés : L∞norms, nonlinear composition, weak limits, material design, homogenization
author = {Lipton, Robert and Mengesha, Tadele},
title = {Representation formulas for $L^\infty $ norms of weakly convergent sequences of gradient fields in homogenization},
AU - Lipton, Robert
AU - Mengesha, Tadele
TI - Representation formulas for $L^\infty $ norms of weakly convergent sequences of gradient fields in homogenization
Lipton, Robert; Mengesha, Tadele. Representation formulas for $L^\infty $ norms of weakly convergent sequences of gradient fields in homogenization. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) no. 5, pp. 1121-1146. doi : 10.1051/m2an/2011049. http://www.numdam.org/articles/10.1051/m2an/2011049/
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[27] R. Lipton and M. Stuebner, Optimal design of composite structures for strength and stiffness : an inverse homogenization approach. Struct. Multidisc. Optim. 33 (2007) 351-362. | MR 2310589 | Zbl 1245.74007
[28] R. Lipton and M. Stuebner, A new method for design of composite structures for strength and stiffness, 12th AIAA/ISSMO Multidisciplinary Analysis & Optimization Conference. American Institute of Aeronautics and Astronautics Paper AIAA, Victoria British Columbia, Canada (2008) 5986.
[29] A.J. Markworth, K.S. Ramesh and W.P. Parks, Modelling studies applied to functionally graded materials. J. Mater. Sci. 30 (1995) 2183-2193.
[30] N. Meyers, An Lp-Estimate for the gradient of solutions of second order elliptic divergence equations. Annali della Scuola Norm. Sup. Pisa 17 (1963) 189-206. | Numdam | MR 159110 | Zbl 0127.31904
[31] G.W. Milton, Modeling the properties of composites by laminates, edited by J. Erickson, D. Kinderleher, R.V. Kohn and J.L. Lions. Homogenization and Effective Moduli of Materials and Media, IMA Volumes in Mathematics and Its Applications 1 (1986) 150-174. | MR 859415 | Zbl 0631.73011
[32] F. Murat and L. Tartar, Calcul des Variations et Homogénéisation, Les Méthodes de l'Homogénéisation : Théorie et Applications en Physique, edited by D. Bergman et al. Collection de la Direction des Études et Recherches d'Electricité de France 57 (1985) 319-369. | MR 844873
[34] R.J. Nuismer and J.M. Whitney, Uniaxial failure of composite laminates containing stress concentrations, in Fracture Mechanics of Composites, ASTM Special Technical Publication, American Society for Testing and Materials 593 (1975) 117-142.
[35] Y. Ootao, Y. Tanigawa and O. Ishimaru, Optimization of material composition of functionally graded plate for thermal stress relaxation using a genetic algorithim. J. Therm. Stress. 23 (2000) 257-271.
[36] G. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random fields, Rigorous results in statistical mechanics and quantum field theory, Esztergom 1979. Colloq. Math. Soc. Janos Bolyai 27 (1981) 835-873. | MR 712714 | Zbl 0499.60059
[37] E. Sanchez-Palencia, Non Homogeneous Media and Vibration Theory. Springer, Heidelberg (1980). | Zbl 0432.70002
[38] S. Spagnolo, Convergence in Energy for Elliptic Operators, Proceedings of the Third Symposium on Numerical Solutions of Partial Differential Equations, edited by B. Hubbard. College Park (1975); Academic Press, New York (1976) 469-498. | MR 477444 | Zbl 0347.65034
|
Chepuri, Sunita1; Dowd, CJ2; Hardt, Andrew3; Michel, Gregory3; Zhang, Sylvester W.3; Zhang, Valerie2
1 University of Michigan Department of Mathematics 2074 East Hall 530 Church St. Ann Arbor MI 48109, USA
2 Harvard University Department of Mathematics Science Center Room 325 1 Oxford Street Cambridge MA 02138, USA
3 University of Minnesota School of Mathematics 127 Vincent Hall 206 Church St. SE Minneapolis MN 55414, USA
An arborescence of a directed graph
\Gamma
is a spanning tree directed toward a particular vertex
v
. The arborescences of a graph rooted at a particular vertex may be encoded as a polynomial
{A}_{v}\left(\Gamma \right)
representing the sum of the weights of all such arborescences. The arborescences of a graph and the arborescences of a covering graph
\stackrel{˜}{\Gamma }
are closely related. Using voltage graphs to construct arbitrary regular covers, we derive a novel explicit formula for the ratio of
{A}_{v}\left(\Gamma \right)
to the sum of arborescences in the lift
{A}_{\stackrel{˜}{v}}\left(\stackrel{˜}{\Gamma }\right)
in terms of the determinant of Chaiken’s voltage Laplacian matrix, a generalization of the Laplacian matrix. Chaiken’s results on the relationship between the voltage Laplacian and vector fields on
\Gamma
are reviewed, and we provide a new proof of Chaiken’s results via a deletion-contraction argument.
Classification: 05C50, 05E18, 05C20, 05C05, 05C22
Keywords: Arborescence, covering graph, voltage graph.
Chepuri, Sunita 1; Dowd, CJ 2; Hardt, Andrew 3; Michel, Gregory 3; Zhang, Sylvester W. 3; Zhang, Valerie 2
author = {Chepuri, Sunita and Dowd, CJ and Hardt, Andrew and Michel, Gregory and Zhang, Sylvester W. and Zhang, Valerie},
title = {Arborescences of covering graphs},
TI - Arborescences of covering graphs
%T Arborescences of covering graphs
Chepuri, Sunita; Dowd, CJ; Hardt, Andrew; Michel, Gregory; Zhang, Sylvester W.; Zhang, Valerie. Arborescences of covering graphs. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 319-346. doi : 10.5802/alco.212. https://alco.centre-mersenne.org/articles/10.5802/alco.212/
[1] Chaiken, Seth A combinatorial proof of the all minors matrix tree theorem, SIAM J. Algebraic Discrete Methods, Volume 3 (1982) no. 3, pp. 319-329 | Article | MR: 666857 | Zbl: 0495.05018
[2] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | Article | MR: 1887642 | Zbl: 1021.16017
[3] Galashin, Pavel; Pylyavskyy, Pavlo
R
-systems, Selecta Math. (N.S.), Volume 25 (2019) no. 2, Paper no. 22, 63 pages | Article | MR: 3922919 | Zbl: 1460.37041
[4] Gordon, Gary; McMahon, Elizabeth A greedoid polynomial which distinguishes rooted arborescences, Proc. Amer. Math. Soc., Volume 107 (1989) no. 2, pp. 287-298 | Article | MR: 967486 | Zbl: 0677.05036
[5] Gross, Jonathan L.; Tucker, Thomas W. Generating all graph coverings by permutation voltage assignments, Discrete Math., Volume 18 (1977) no. 3, pp. 273-283 | Article | MR: 465917 | Zbl: 0375.55001
[6] Hall, Marshall Jr. The theory of groups, The Macmillan Company, New York, N.Y., 1959, xiii+434 pages | MR: 0103215 | Zbl: 0084.02202
[7] Korte, Bernhard; Vygen, Jens Combinatorial optimization: Theory and algorithms, Algorithms and Combinatorics, 21, Springer-Verlag, Berlin, 2006, xvi+597 pages | Article | MR: 2171734 | Zbl: 1099.90054
[8] Reiner, Victor; Tseng, Dennis Critical groups of covering, voltage and signed graphs, Discrete Math., Volume 318 (2014), pp. 10-40 | Article | MR: 3141623 | Zbl: 1281.05072
[9] Silvester, John R. Determinants of block matrices, The Mathematical Gazette, Volume 84 (2000) no. 501, p. 460–467 | Article
[10] Stanley, Richard P. Enumerative combinatorics. Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | Article | MR: 1676282 | Zbl: 0928.05001
[11] Stanton, Dennis; White, Dennis Constructive combinatorics, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1986, x+183 pages | Article | MR: 843332 | Zbl: 0595.05002
[12] van Aardenne-Ehrenfest, Tatyana; de Bruijn, Nicolaas G. Circuits and trees in oriented linear graphs, Simon Stevin, Volume 28 (1951), pp. 203-217 | MR: 47311 | Zbl: 0044.38201
[13] Verma, Kaustubh Double Covers of Flower Graphs and Tutte Polynomials, Minnesota Journal of Undergraduate Mathematics, Volume 6 (2021) no. 1
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Difference between revisions of "901.17 Material Inspection for Sec 901" - Engineering_Policy_Guide
Difference between revisions of "901.17 Material Inspection for Sec 901"
m (→Report (Records): minor clarifications)
m (Per TS, minor clarifications)
This article establishes procedures for inspecting and reporting those items specified in
[http://www.modot.mo.gov/business/standards_and_specs/Sec0901.pdf Sec 0901] for which Materials has responsibility and are not specifically covered in Materials Details of the Specifications.
[http://www.modot.org/business/standards_and_specs/SpecbookEPG.pdf#page=13 Sec 0901] for which Materials has responsibility and are not specifically covered in Materials Details of the Specifications.
==Apparatus==
The highway lighting items normally field inspected by Materials are galvanizing of anchor bolts, nuts, washers, polyurethane foam and steel standards, bracket arms and foundations.
Field determination of weight of coating is to be made on each lot of material furnished. The magnetic gauge is to be operated and calibrated in accordance with ASTM E 376. At least three members of each size and type offered for inspection are to be selected for testing. A single-spot test is to be comprised of at least five readings of the magnetic gauge taken in a small area and those five readings averaged to obtain a single-spot test result. Three such areas should be tested on each of the members being tested. Test each member in the same manner. Average all single-spot test results from all members to obtain the average coating weight to be reported. The minimum single-spot test result would be the minimum average obtained on any one member. Material may be accepted or rejected for galvanized coating on the basis of magnetic gauge. If a test result fails to comply with the specifications, that lot should be re-sampled at double the original [[106.5 Sampling|sampling]] rate. If any of the resample members fail to comply with the specification, that lot is to be rejected. The contractor or supplier is to be given the option of sampling for Central Laboratory testing, if the magnetic gauge test results are within minus 15 percent of the specified coating weight.
Field determination of weight of coating is to be made on each lot of material furnished. The magnetic gauge is to be operated and calibrated in accordance with ASTM E 376. At least three members of each size and type offered for inspection are to be selected for testing. A single-spot test is to be comprised of at least five readings of the magnetic gauge taken in a small area and those five readings averaged to obtain a single-spot test result. Three such areas should be tested on each of the members being tested. Test each member in the same manner. Average all single-spot test results from all members to obtain the average coating weight to be reported. The minimum single-spot test result would be the minimum average obtained on any one member. Material may be accepted or rejected for galvanized coating on the basis of magnetic gauge. If a test result fails to comply with the specifications, that lot should be re-sampled at double the original [[106.3 Samples, Tests and Cited Specifications#106.3.1 Sampling|sampling]] rate. If any of the resample members fail to comply with the specification, that lot is to be rejected. The contractor or supplier is to be given the option of sampling for Central Laboratory testing, if the magnetic gauge test results are within minus 15 percent of the specified coating weight.
Bolts and nuts specified to meet the requirements of ASTM A 307 shall be accompanied by a manufacturer's certification statement that the bolts and nuts were manufactured to comply with the requirements of ASTM A 307 and, if required by the specifications, galvanized to comply with the requirements of AASHTO M 232, Class C or were mechanically galvanized in accordance with the requirements of AASHTO M 298, Class 55. High strength bolts, nuts and washers shall be accompanied by a manufacturer's inspection test report for each production lot or shipping lot furnished and certifying that the bolts furnished conform to the requirements specified. All bolts, nuts and washers are to be identifiable as to type and manufacturer. Bolts, nuts and washers manufactured to meet ASTM A 307 will normally be identified on the packaging since no special markings are required on the item. The specified AASHTO is to be consulted for the required identification marks on high strength bolts and nuts. Dimensions are to be as shown on the plans or as specified. Weight of zinc coating, when specified, is to be determined by magnetic gauge in the same manner as described in the previous paragraph except that a smaller number of single-spot tests will be sufficient. Samples for Central Laboratory testing are only required when requested by the State Construction and Materials Engineer or when field inspection indicates questionable compliance. When samples are taken, they are to be taken at the frequency and of the size shown in [[:category:1040 Guardrail, End Terminals, One-Strand Access Restraint Cable and Three-Strand Guard Cable Material#1040.3 Table 2|Table 2, Sampling Requirements]].
Additional requirements for bolts, nuts, and wasters are given in [[:Category:1080 Structural Steel Fabrication#1080.1.3 Bolts for Highway Lighting, Traffic Signals or Highway Signing|EPG 1080.1.3 Bolts for Highway Lighting, Traffic Signals or Highway Signing]].
Polyurethane foam used as pole backfill is to be accepted on the basis of manufacturer's certification and random sampling and testing. The manufacturer's certification is to show typical test results representative of the material and certify that the material supplied conforms to all of the requirements specified. Random samples are to be taken from approximately 10 percent of the lots offered for use. A sample is to consist of a portion of each component adequate in size to yield 2 cu. ft. of polyurethane foam, after mixing. SiteManager is to be used when submitting samples to the Central Laboratory. Very small quantities of polyurethane foam may be accepted on the basis of brand name and labeling, provided satisfactory results are obtained in the field.
==Report (Records)==
Reports shall indicate acceptance, qualified acceptance or rejection. Appropriate remarks as described in [[106.9 Reporting Test Results|EPG 106.9 Reporting Test Results]] are to be included in the report to clarify conditions of acceptance or rejection. Distribution of reports or materials purchased under a MoDOT purchase order is to be as described in [[:Category:1101 Materials Purchased by a Department Purchase Order|EPG 1101 Materials Purchased by a MoDOT Purchase Order]].
Reports shall indicate acceptance, qualified acceptance or rejection. Appropriate remarks as described in [[106.20 Reporting|EPG 106.20 Reporting]] are to be included in the report to clarify conditions of acceptance or rejection. Distribution of reports or materials purchased under a MoDOT purchase order is to be as described in [[:Category:1101 Materials Purchased by a Department Purchase Order|EPG 1101 Materials Purchased by a MoDOT Purchase Order]].
Polyurethane foam shall be reported through SiteManager. The manufacturer's certification shall be retained in the district office, except when reporting very small quantities accepted by brand name and labeling.
[[Category:901 Lighting|901.17]]
{\displaystyle \mu \,}
|
An angle in a triangle can measure
0 ^ \circ
Why some people say it's true: Sure, although a
0 ^ \circ
angle would make for a very flat triangle, or it could be very thin and pointy, if the top angle is the
0 ^ \circ
\color{#20A900}{\text{Reveal the Correct Answer}}
\color{#D61F06}{\textbf{false}}
The key reason that triangles cannot have angles which measure
0 ^ \circ
is that such a figure would necessarily be 3 points on a straight line, and such a figure is not a triangle. Recall that the triangle inequality states that in a triangle, we have
a + b > c
We will prove that the statement "an angle in a triangle can measure
0 ^ \circ
" is false using proof by contradiction. Assume that such a triangle exists, with side lengths
a, b,
c
c
be the side opposite to the zero angle, then by the cosine rule
\begin{array}{c}ac^2&=&a^2&+&b^2-&2ab\cos(0)\\&=&a^2&+&b^2-&2ab(1)\\&=&(a&-&b)^2.\end{array}
c=a-b\Longrightarrow a=b+c.
\color{#3D99F6}{\text{See Common Rebuttals}}
Rebuttal: Consider a right triangle and the "SOH CAH TOA" identity which states that, for any right triangle with a non-right angle
x
adjacent to the hypotenuse, the measure of
\sin(x)
is the ratio of the hypotenuse of the right triangle to the side opposite the angle:
\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{O}{H}
. Therefore, since the
\sin
function can take on the value of
0
, when it does, this right-angled triangle will have one angle equal to
0^\circ
Reply: If
\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} = 0,
then it must be the case that
O
, the side opposite the angle
x,
has measure 0. Therefore, the object being described is not a triangle.
Rebuttal: Since we use the ratio of the sides of a right triangle with hypotenuse 1 to define the
\sin
function, we need to consider "trivial triangles" which are basically just flat lines in order to make the evaluation of
\sin(0^\circ)=0
make any sense.
Reply: The definition of the trigonometric function being the ratio of certain sides of a right angle
\Big(
\Rightarrow \sin = \frac{\text{opposite}}{\text{hypotenuse}}\Big)
0<\text{argument}<\pi
. For arguments not in that range, trigonometric functions are calculated by the unit circle.
90^\circ
180^\circ
360^\circ
There is no maximum magnitude for the internal angle of a triangle.
|
The Czochralski method, also Czochralski technique or Czochralski process, is a method of crystal growth used to obtain single crystals of semiconductors (e.g. silicon, germanium and gallium arsenide), metals (e.g. palladium, platinum, silver, gold), salts and synthetic gemstones. The method is named after Polish scientist Jan Czochralski,[1] who invented the method in 1915 while investigating the crystallization rates of metals.[2] He made this discovery by accident: instead of dipping his pen into his inkwell, he dipped it in molten tin, and drew a tin filament, which later proved to be a single crystal.[3]
The most important application may be the growth of large cylindrical ingots, or boules, of single crystal silicon used in the electronics industry to make semiconductor devices like integrated circuits. Other semiconductors, such as gallium arsenide, can also be grown by this method, although lower defect densities in this case can be obtained using variants of the Bridgman–Stockbarger method.
The method is not limited to production of metal or metalloid crystals. For example, it is used to manufacture very high-purity crystals of salts, including material with controlled isotopic composition, for use in particle physics experiments, with tight controls (part per billion measurements) on confounding metal ions and water absorbed during manufacture.[4]
2 Production of Czochralski silicon
4 Incorporating impurities
Monocrystalline silicon (mono-Si) grown by the Czochralski method is often referred to as monocrystalline Czochralski silicon (Cz-Si). It is the basic material in the production of integrated circuits used in computers, TVs, mobile phones and all types of electronic equipment and semiconductor devices.[5] Monocrystalline silicon is also used in large quantities by the photovoltaic industry for the production of conventional mono-Si solar cells. The almost perfect crystal structure yields the highest light-to-electricity conversion efficiency for silicon.
Production of Czochralski silicon[edit]
Crystal of Czochralski-grown silicon
High-purity, semiconductor-grade silicon (only a few parts per million of impurities) is melted in a crucible at 1,425 °C (2,597 °F; 1,698 K), usually made of quartz. Dopant impurity atoms such as boron or phosphorus can be added to the molten silicon in precise amounts to dope the silicon, thus changing it into p-type or n-type silicon, with different electronic properties. A precisely oriented rod-mounted seed crystal is dipped into the molten silicon. The seed crystal's rod is slowly pulled upwards and rotated simultaneously. By precisely controlling the temperature gradients, rate of pulling and speed of rotation, it is possible to extract a large, single-crystal, cylindrical ingot from the melt. Occurrence of unwanted instabilities in the melt can be avoided by investigating and visualizing the temperature and velocity fields during the crystal growth process.[6] This process is normally performed in an inert atmosphere, such as argon, in an inert chamber, such as quartz.
Crystal sizes[edit]
Silicon crystal being grown by the Czochralski method at Raytheon, 1956. The induction heating coil is visible, and the end of the crystal is just emerging from the melt. The technician is measuring the temperature with an optical pyrometer. The crystals produced by this early apparatus, used in an early Si plant, were only one inch in diameter.
Due to efficiencies of scale, the semiconductor industry often uses wafers with standardized dimensions, or common wafer specifications. Early on, boules were small, a few cm wide. With advanced technology, high-end device manufacturers use 200 mm and 300 mm diameter wafers. Width is controlled by precise control of temperature, speeds of rotation, and the speed at which the seed holder is withdrawn. The crystal ingots from which wafers are sliced can be up to 2 metres in length, weighing several hundred kilograms. Larger wafers allow improvements in manufacturing efficiency, as more chips can be fabricated on each wafer, with lower relative loss, so there has been a steady drive to increase silicon wafer sizes. The next step up, 450 mm, is currently scheduled for introduction in 2018.[7] Silicon wafers are typically about 0.2–0.75 mm thick, and can be polished to great flatness for making integrated circuits or textured for making solar cells.
The process begins when the chamber is heated to approximately 1500 degrees Celsius, melting the silicon. When the silicon is fully melted, a small seed crystal mounted on the end of a rotating shaft is slowly lowered until it dips just below the surface of the molten silicon. The shaft rotates counterclockwise and the crucible rotates clockwise[citation needed]. The rotating rod is then drawn upwards very slowly—at about 25 mm per hour when making a crystal of ruby[8]—allowing a roughly cylindrical boule to be formed. The boule can be from one to two metres, depending on the amount of silicon in the crucible.
The electrical characteristics of the silicon are controlled by adding material like phosphorus or boron to the silicon before it is melted. The added material is called dopant and the process is called doping. This method is also used with semiconductor materials other than silicon, such as gallium arsenide.
Incorporating impurities[edit]
A puller rod with seed crystal for growing single-crystal silicon by the Czochralski method
Crucibles used in Czochralski method
Crucible after being used
When silicon is grown by the Czochralski method, the melt is contained in a silica (quartz) crucible. During growth, the walls of the crucible dissolve into the melt and Czochralski silicon therefore contains oxygen at a typical concentration of 1018
. Oxygen impurities can have beneficial or detrimental effects. Carefully chosen annealing conditions can give rise to the formation of oxygen precipitates. These have the effect of trapping unwanted transition metal impurities in a process known as gettering, improving the purity of surrounding silicon. However, formation of oxygen precipitates at unintended locations can also destroy electrical structures. Additionally, oxygen impurities can improve the mechanical strength of silicon wafers by immobilising any dislocations which may be introduced during device processing. It was experimentally shown in the 1990s that the high oxygen concentration is also beneficial for the radiation hardness of silicon particle detectors used in harsh radiation environment (such as CERN's LHC/HL-LHC projects).[9][10] Therefore, radiation detectors made of Czochralski- and magnetic Czochralski-silicon are considered to be promising candidates for many future high-energy physics experiments.[11][12] It has also been shown that the presence of oxygen in silicon increases impurity trapping during post-implantation annealing processes.[13]
However, oxygen impurities can react with boron in an illuminated environment, such as that experienced by solar cells. This results in the formation of an electrically active boron–oxygen complex that detracts from cell performance. Module output drops by approximately 3% during the first few hours of light exposure.[14]
Concerning a mathematical expression of impurity incorporation from melt,[15] consider the following.
The impurity concentration in the solid crystal that results from freezing an amount of volume can be obtained from consideration of the segregation coefficient.
{\displaystyle k_{O}}
: Segregation coefficient
{\displaystyle V_{0}}
{\displaystyle I_{0}}
: Number of impurities
{\displaystyle C_{0}}
: Impurity concentration in the melt
{\displaystyle V_{L}}
: Volume of the melt
{\displaystyle I_{L}}
: Number of impurities in the melt
{\displaystyle C_{L}}
: Concentration of impurities in the melt
{\displaystyle V_{S}}
: Volume of solid
{\displaystyle C_{S}}
: Concentration of impurities in the solid
During the growth process, volume of melt
{\displaystyle dV}
freezes, and there are impurities from the melt that are removed.
{\displaystyle dI=-k_{O}C_{L}dV\;}
{\displaystyle dI=-k_{O}{\frac {I_{L}}{V_{O}-V_{S}}}dV}
{\displaystyle \int _{I_{O}}^{I_{L}}{\frac {dI}{I_{L}}}=-k_{O}\int _{0}^{V_{S}}{\frac {dV}{V_{O}-V_{S}}}}
{\displaystyle \ln \left({\frac {I_{L}}{I_{O}}}\right)=\ln \left(1-{\frac {V_{S}}{V_{O}}}\right)^{k_{O}}}
{\displaystyle I_{L}=I_{O}\left(1-{\frac {V_{S}}{V_{O}}}\right)^{k_{O}}}
{\displaystyle C_{S}=-{\frac {dI_{L}}{dV_{S}}}}
{\displaystyle C_{S}=C_{O}k_{O}(1-f)^{k_{o}-1}}
{\displaystyle f=V_{S}/V_{O}\;}
Wikimedia Commons has media related to Czochralski method.
^ Paweł Tomaszewski, "Jan Czochralski i jego metoda. Jan Czochralski and his method" (in Polish and English), Oficyna Wydawnicza ATUT, Wrocław–Kcynia 2003, ISBN 83-89247-27-5
^ J. Czochralski (1918) "Ein neues Verfahren zur Messung der Kristallisationsgeschwindigkeit der Metalle" [A new method for the measurement of the crystallization rate of metals], Zeitschrift für Physikalische Chemie, 92 : 219–221.
^ Nishinaga, Tatau (2015). Handbook of Crystal Growth: Fundamentals (Second ed.). Amsterdam, the Netherlands: Elsevier B.V. p. 21. ISBN 978-0-444-56369-9.
^ Son, JK (2020-05-14). "Growth and development of pure Li2MoO4 crystals for rare event experiment at CUP". Journal of Instrumentation. 15 (7): C07035. arXiv:2005.06797. Bibcode:2020JInst..15C7035S. doi:10.1088/1748-0221/15/07/C07035. S2CID 218630318.
^ Czochralski Crystal Growth Method. Bbc.co.uk. 30 January 2003. Retrieved on 2011-12-06.
^ Aleksic, Jalena; Zielke, Paul; Szymczyk, Janusz A.; et al. (2002). "Temperature and Flow Visualization in a Simulation of the Czochralski Process Using Temperature-Sensitive Liquid Crystals". Ann. N.Y. Acad. Sci. 972 (1): 158–163. Bibcode:2002NYASA.972..158A. doi:10.1111/j.1749-6632.2002.tb04567.x. PMID 12496012. S2CID 2212684.
^ Doubts over 450mm and EUV. Electronicsweekly.com. December 30, 2013. Retrieved on 2014-01-09.
^ "Czochralski Process". www.theimage.com. Retrieved 2016-02-25.
^ Li, Z.; Kraner, H.W.; Verbitskaya, E.; Eremin, V.; Ivanov, A.; Rattaggi, M.; Rancoita, P.G.; Rubinelli, F.A.; Fonash, S.J.; et al. (1992). "Investigation of the oxygen-vacancy (A-center) defect complex profile in neutron irradiated high resistivity silicon junction particle detectors". IEEE Transactions on Nuclear Science. 39 (6): 1730. Bibcode:1992ITNS...39.1730L. doi:10.1109/23.211360.
^ Lindström, G; Ahmed, M; Albergo, S; Allport, P; Anderson, D; Andricek, L; Angarano, M.M; Augelli, V; Bacchetta, N; Bartalini, P; Bates, R; Biggeri, U; Bilei, G.M; Bisello, D; Boemi, D; Borchi, E; Botila, T; Brodbeck, T.J; Bruzzi, M; Budzynski, T; Burger, P; Campabadal, F; Casse, G; Catacchini, E; Chilingarov, A; Ciampolini, P; Cindro, V; Costa, M.J; Creanza, D; et al. (2001). "Radiation hard silicon detectors—developments by the RD48 (ROSE) collaboration". Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 466 (2): 308. Bibcode:2001NIMPA.466..308L. doi:10.1016/S0168-9002(01)00560-5.
^ Harkonen, J; Tuovinen, E; Luukka, P; Tuominen, E; Li, Z; Ivanov, A; Verbitskaya, E; Eremin, V; Pirojenko, A; Riihimaki, I.; Virtanen, A. (2005). "Particle detectors made of high-resistivity Czochralski silicon". Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 541 (1–2): 202–207. Bibcode:2005NIMPA.541..202H. CiteSeerX 10.1.1.506.2366. doi:10.1016/j.nima.2005.01.057.
^ Custer, J. S.; Polman, A.; Van Pinxteren, H. M. (1994). "Erbium in crystal silicon: Segregation and trapping during solid phase epitaxy of amorphous silicon". Journal of Applied Physics. 75 (6): 2809. Bibcode:1994JAP....75.2809C. doi:10.1063/1.356173.
^ Eikelboom, J.A., Jansen, M.J., 2000. Characterisation of PV modules of new generations; results of tests and simulations Archived 2012-04-24 at the Wayback Machine. Report ECN-C-00-067, 18.
Silicon Wafer Processing Animation on YouTube
Retrieved from "https://en.wikipedia.org/w/index.php?title=Czochralski_method&oldid=1060220550"
Methods of crystal growth
|
Binomial Theorem, Popular Questions: CBSE Class 11-humanities ENGLISH, English Grammar - Meritnation
\left(a+bx{\right)}^{17}
Please must explain me the following question. Its very urgent
Find a of the coefficents of x2 and x3 in the expansion of (3 + ax)9 are equal.
Find the term independent of x in the expansion of the following binomials ( 2 x square - 1 / x) to the power of 12 what is the value?
Srishti Rath asked a question
using binomial theorem prove that (x^n-y^n) is divisible by (x-y)
Evaluate (1.056)^1/3 correct up to the four places of decimal.
if Sn = nC0. nC1 + nC1.nC2 + ..... + nCn-1 . nCn and if Sn+1/Sn = 15/4 then n is equal to
Akshat Gupta asked a question
Pl Answer the question in the pic.
Please sir help me,,
Find the term independent of x in the expansion of the following binomials : (i) (x-1/x)14
Please answer to it in short !!
Expand (2x +3y)4 in the copy
nEr=0 nCr sin2rx / nEr=0 nCr cos2rx = tan nx
E--->Sigma(summation).
prove nPr= n!
and deduce it to nCr answere fast important for test
Explain NCERT CH-8 EX-10 complete solution with elaborate steps
Gurkiran Kaur asked a question
(i) ( 3x - 2y)5 (ii) ( x - 1/y )11 (iii) ( x2 - 2x + 1)3
in expansion such as (a-b)5 how do we know which term should be taken as positive or negetive?
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Simulating Annealing PLL for Autonomous Microgrid Systems
Electrical and Electronics Engineering Department, University of Swaziland, Kwaulsini, Swaziland
{V}_{i}\angle {\delta }_{i}
{V}_{t}\angle {\delta }_{t}
\left({\delta }_{i},{\delta }_{t}\right)
jX=j{X}_{Trans}+j{X}_{TL2}
{\delta }_{p}
{V}_{d}=-{V}_{t}\mathrm{sin}\left({\delta }_{t}-{\delta }_{p}\right)
\left({\delta }_{t}-{\delta }_{p}\right)
{V}_{d}=-{V}_{t}\left({\delta }_{t}-{\delta }_{p}\right)
{\stackrel{˙}{\delta }}_{p}={\omega }_{p}
\stackrel{˙}{m}={K}_{m}\left({V}_{set}-{V}_{t}\right)
\stackrel{˙}{\theta }={K}_{\theta }\left({P}_{set}-{P}_{Gen}\right)
\stackrel{˙}{x}={K}_{3}\left({\delta }_{t}-{\delta }_{p}\right)
{\stackrel{˙}{\delta }}_{p}={\omega }_{p}
0={V}_{i}-\frac{m{V}_{dc}}{{V}_{base}}
0={P}_{set}-\left({P}_{o}-R{\omega }_{p}\right)
0=\theta -\left({\delta }_{i}-{\delta }_{p}\right)
0=x-\left({\omega }_{p}-{K}_{4}\theta \right)
0={P}_{Gen}-\frac{{V}_{dc}{I}_{inv}}{{P}_{base}}
m
{K}_{m}
{V}_{set}
\theta
{K}_{\theta }
{P}_{set}
{P}_{o}
R
{V}_{dc}
{I}_{inv}
{V}_{base}
{P}_{base}
{\text{e}}^{-\Delta f/T}
, where , and T is the temperature parameter which is being reduced over time during the process in order to decrease the possibility of accepting such transitions. The SA proposed approach for the PI-controller gains selection is summarized in flowchart shown in Figure 7.
\sqrt{0.05}
Bayoumi, E.H.E. (2019) Simulating Annealing PLL for Autonomous Microgrid Systems. Smart Grid and Renewable Energy, 10, 141-154. https://doi.org/10.4236/sgre.2019.105009
1. Hiskens, I.A. and Fleming, E.M. (2008) Control of Inverter-Connected Source in Autonomous Microgrids. American Control Conference, Seattle, 11-13 June 2008, 586-590. https://doi.org/10.1109/ACC.2008.4586555
2. Kroposki, B., et al. (2008) Making Microgrids Work. IEEE Power and Energy Magazine, 6, 40-53. https://doi.org/10.1109/MPE.2008.918718
3. Raju, E. and Jain, T. (2017) Robust Optimal Centralized Controller to Mitigate the Small Signal Instability in an Islanded Inverter Based Microgrid with Active and Passive Loads. International Journal of Electrical Power & Energy Systems, 90, 225-236. https://doi.org/10.1016/j.ijepes.2017.02.011
4. Han, H., Hou, X., Yang, J., Wu, J., Su, M. and Guerrero, J.M. (2016) Review of Power Sharing Control Strategies Forislanding Operation of AC Microgrids. IEEE Transactions on Smart Grid, 7, 200-215. https://doi.org/10.1109/TSG.2015.2434849
5. Chandorkar, M., Divan, D. and Adapa, R. (1993) Control of Parallel-Connected Inverters in Standalone AC Supply Systems. IEEE Transactions on Industry Applications, 29, 136-143. https://doi.org/10.1109/28.195899
6. Olivares, D., Mehrizi-Sani, A., Etemadi, A.H., Cañizares, C.A., Iravani, R., Kazerani, M., Hajimiragha, A.H., Gomis-Bellmunt, O., Saeedifard, M., Palma-Behnke, R., et al. (2014) Trends in Microgrid Control. IEEE Transactions on Smart Grid, 5, 1905-1919. https://doi.org/10.1109/TSG.2013.2295514
7. Tsikalakis, A. and Hatziargyriou, N. (2011) Centralized Control for Optimizing Microgrids Operation. Proceedings of the IEEE Power and Energy Society General Meeting, San Diego, 24-29 July 2011, 1-8. https://doi.org/10.1109/PES.2011.6039737
8. Hassan, M.A., Worku, M.Y. and Abido, M.A. (2018) Optimal Design and Real Time Implementation of Autonomous Microgrid Including Active Load. Energies Journal, 11, 1109. https://doi.org/10.3390/en11051109
9. Dong, D., Wen, B., Boroyevich, D., Mattavelli, P. and Xue, Y. (2015) Analysis of Phase-Locked Loop Low-Frequency Stability in Three-Phase Grid-Connected Power Converters Considering Impedance Interactions. IEEE Transactions on Industrial Electronics, 62, 310-321. https://doi.org/10.1109/TIE.2014.2334665
10. Svensson, J. (2001) Synchronization Methods for Grid-Connected Voltage Source Converters. IEEE Proceedings of Generation Transmission and Distribution, 148, 229-235. https://doi.org/10.1049/ip-gtd:20010101
11. Golestan, S., Monfared, M., Freijedo, F. and Guerrero, J. (2013) Advantages and Challenges of a Type-3 PLL. IEEE Transactions on Power Electronics, 28, 4985-4997. https://doi.org/10.1109/TPEL.2013.2240317
12. Golestan, S., Ramezani, M., Guerrero, J.M., Freijedo, F.D. and Monfared, M. (2014) Moving Average Filter Based Phase-Locked Loops: Performance Analysis and Design Guidelines. IEEE Transactions on Power Electronics, 29, 2750-2763. https://doi.org/10.1109/TPEL.2013.2273461
13. Espin, F.G., Figueres, E. and Garcera, G. (2012) An Adaptive Synchronous Reference-Frame Phase-Locked Loop for Power Quality Improvement in a Polluted Utility Grid. IEEE Transactions on Industrial Electronics, 59, 2718-2731. https://doi.org/10.1109/TIE.2011.2166236
14. Freijedo, F.D., Yepes, A.G., Lopez, O., Vidal, A. and Gondoy, J.D. (2011) Three Phase PLLs with Fast Postfault Retracking and Steady-State Rejection of Voltage Unbalance and Harmonics by Means of Lead Compensation. IEEE Transactions on Power Electronics, 26, 85-97. https://doi.org/10.1109/TPEL.2010.2051818
15. Hamed, H.A., Abdou, A.F., El-Kholy, E.E. and Bayoumi, E.H.E. (2016) Adaptive Cascaded Delayed Signal Cancelation PLL Based Fuzzy Controller under Grid Disturbances. IEEE 59th International Midwest Symposium on Circuits and Systems, Abu Dhabi, 16-19 October 2016, 1-4. https://doi.org/10.1109/MWSCAS.2016.7870061
16. Xia, C., Guo, P., Shi, T. and Wang, M. (2004) Speed Control of Brushless DC Motor Using Genetic Algorithm Based Fuzzy Controller. 2004 International Conference on Intelligent Mechatronics and Automation, Chengdu, 26-31 August 2004.
17. Bayoumi, E.H.E. and Soliman, H.M. (2007) PID/PI Tuning for Minimal Overshoot of Permanent-Magnet Brushless DC Motor Drive Using Particle Swarm Optimization. Electromotion Scientific Journal, 14, 198-208.
18. Kennedy, J. and Eberhart, R.C. (2001) Swarm Intelligence. Morgan Kaufmann, San Francisco.
19. Bayoumi, E.H.E. (2010) Parameter Estimation of Cage Induction Motors Using Cooperative Bacteria Foraging Optimization. Electromotion Scientific Journal, 17, 247-260.
20. Surprenant, M., Hiskens, I. and Venkataramanan, G. (2011) Phase Locked Loop Control of Inverters in a Microgrid. IEEE Energy Conversion Congress and Exposition, Phoenix, 17-22 September 2011, 667-672. https://doi.org/10.1109/ECCE.2011.6063833
21. Kirkpatrick, S., Gelatt Jr., C.D. and Vecchi, M.P. (1983) Optimization by Simulated Annealing. Sciences Journal, 220, 671-680. https://doi.org/10.1126/science.220.4598.671
22. Vega-Rodriguez, M.A., Gomez-Pulido, J.A., Alba, E., Vega-Perez, D., Priem-Mendes, S. and Molina, G. (2007) Evaluation of Different Metaheuristics Solving the RND Problem. Workshops on Applications of Evolutionary Computation, Valencia, 11-13 April 2007, 101-110. https://doi.org/10.1007/978-3-540-71805-5_11
23. Jaraiz-Simon, M.D., Gomez-Pulido, J.A., Vega-Rodriguez, M.A. and Sanchez-Perez, J.M. (2013) Simulated Annealing for Real-Time Vertical-Handoff in Wireless Networks. International Work-Conference on Artificial Neural Networks, Tenerife, 12-14 June 2003, 198-209. https://doi.org/10.1007/978-3-642-38679-4_19
24. Huang, K.-Y. and Hsieh, Y.-H. (2011) Very Fast Simulated Annealing for Pattern Detection and Seismic Applications. IEEE International Geoscience and Remote Sensing Symposium, Vancouver, 24-29 July 2011, 499-502. https://doi.org/10.1109/IGARSS.2011.6049174
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LiDAR: A photonics guide to the autonomous vehicle market | Hamamatsu Photonics
Jake Li, Hamamatsu Corporation
Advances in sensor technology, imaging, radar, light detection and ranging (LiDAR), electronics, and artificial intelligence have enabled dozens of advanced driver assistance systems (ADAS), including collision avoidance, blindspot monitoring, lane departure warning, or park assist. Synchronizing the operation of such systems through sensor fusion allows fully autonomous or self-driving vehicles to monitor their surroundings and warn drivers of potential road hazards, or even take evasive actions independent of the driver to prevent collision.
Autonomous vehicles must also differentiate and recognize objects ahead at high-speed conditions. Using distance-gauging technology, these self-driving cars must rapidly construct a three-dimensional (3D) map up to a distance of about 100 m, as well as create high-angular-resolution imagery at distances up to 250 m. And if the driver is not present, the artificial intelligence of the vehicle must make optimal decisions.
One of several basic approaches for this task measures round-trip time of flight (ToF) of a pulse of energy traveling from the autonomous vehicle to the target and back to the vehicle. Distance to the reflection point can be calculated when one knows the speed of the pulse through the air—a pulse that can be ultrasound (sonar), radio wave (radar), or light (LiDAR).
Of these three ToF techniques, LiDAR is the best choice to provide higher-angular-resolution imagery because its smaller diffraction (beam divergence) allows better recognition of adjacent objects compared to radar (see Figure 1). This higher-angular-resolution is especially important at high speed to provide enough time to respond to a potential hazard such as head-on collision.
Figure 1. Beam divergence depends on the ratio of the wavelength and aperture diameter of the emitting antenna (radar) or lens (LiDAR). This ratio is larger for radar producing larger beam divergence and, therefore, smaller angular resolution. In the figure, the radar (black) would not be able to differentiate between the two cars, while LiDAR (red) would.
Laser source selection
In ToF LiDAR, a laser emits a pulse of light of duration τ that activates the internal clock in a timing circuit at the instant of emission (see Figure 2). The reflected light pulse from the target reaches a photodetector, producing an electrical output that deactivates the clock. This electronically measured round-trip ToF Δt allows calculation of the distance R to the reflection point.
Figure 2. The basic setup for time-of-flight (ToF) LiDAR is detailed.
If the laser and photodetector are practically at the same location, the distance is given by:
R=\frac{1}{2n}c\Delta t
where c is the speed of light in vacuum and n is the index of refraction of the propagation medium (for air, approximately 1). Two factors affect the distance resolution ΔR: the uncertainty δΔt in measuring Δt and the spatial width w of the pulse (w = cτ), if the diameter of the laser spot is larger than the size of the target feature to be resolved.
The first factor implies ΔR = ½cδΔt, whereas the second implies ΔR = ½ w = ½ cτ. If the distance is to be measured with a resolution of 5 cm, the above relations separately imply that δΔt is approximately 300 ps and τ is approximately 300 ps. Time-of-flight LiDAR requires photodetectors and detection electronics with small time jitter (the main contributor to δΔt) and lasers capable of emitting short-duration pulses, such as relatively expensive picosecond lasers. A laser in a typical automotive LiDAR system produces pulses of about 4 ns duration, so minimal beam divergence is essential.
One of the most critical choices for automotive LiDAR system designers is the light wavelength. Several factors constrain this choice: safety to human vision, interaction with the atmosphere, availability of lasers, and availability of photodetectors. The two most popular wavelengths are 905 and 1550 nm, with the primary advantage of 905 nm being that silicon absorbs photons at this wavelength and silicon-based photodetectors are generally less expensive than the indium gallium arsenide (InGaAs) infrared (IR) photodetectors needed to detect 1550 nm light. However, the higher human-vision safety of 1550 nm allows the use of lasers with a larger radiant energy per pulse—an important factor in the photon budget.
Atmospheric attenuation (under all weather conditions), scattering from airborne particles, and reflectance from target surfaces are wavelength-dependent. This is a complex issue for automotive LiDAR because of the myriad of possible weather conditions and types of reflecting surfaces. Under most realistic settings, loss of light at 905 nm is less because water absorption is stronger at 1550 nm than at 905 nm.1
Photon detection options
Only a small fraction of photons emitted in a pulse ever reach the active area of the photodetector. If the atmospheric attenuation does not vary along the pulse's path, the beam divergence of the laser light is negligible, the illumination spot is smaller than the target, the angle of incidence is zero, and the reflection is Lambertian, then the optical received peak power P(R) is:
\mathrm{P}\left(\mathrm{R}\right)={\mathrm{P}}_{0}\rho \phantom{\rule[-0.1em]{0.4em}{0.5em}}\frac{{\mathrm{A}}_{0}}{{\mathrm{\pi R}}^{2}}\phantom{\rule[-0.1em]{0.4em}{0.5em}}{\mathrm{\eta }}_{0}\mathrm{exp}\left(-2\gamma R\right)
where P0 is the optical peak power of the emitted laser pulse, ρ is the reflectivity of the target, A0 is the receiver's aperture area, η0 is the detection optics' spectral transmission, and γ is the atmospheric extinction coefficient.
This equation shows that the received power rapidly decreases with increasing distance R. For a reasonable choice of the parameters and R = 100 m, the number of returning photons on the detector's active area is on the order of a few hundred to a few thousand from the more than 1020 typically emitted. These photons compete for detection with background photons carrying no useful information.
Using a narrowband filter can reduce the amount of background light reaching the detector, but the amount cannot be reduced to zero. The effect of the background is the reduction of the detection dynamic range and higher noise (background photon shot noise). It's noteworthy that the terrestrial solar irradiance under typical conditions is less at 1550 nm than at 905 nm.
Creating a 3D map in a full 360° × 20° strip surrounding a car requires a raster-scanned laser beam or multiple beams, or flooding the scene with light and gathering a point cloud of data returns. The former approach is known as scanning LiDAR and the latter as flash LiDAR.
There are several approaches to scanning LiDAR. In the first, exemplified by Velodyne (San Jose, CA), the roof-mounted LiDAR platform rotates at 300-900 rpm while emitting pulses from sixty-four 905 nm laser diodes. Each beam has a dedicated avalanche photodiode (APD) detector. A similar approach uses a rotating multi-faceted mirror with each facet at a slightly different tilt angle to steer a single beam of pulses in different azimuthal and declinational angles. The moving parts in both designs represent a failure risk in mechanically rough driving environments.
The second, more compact approach to scanning LiDAR uses a tiny microelectromechanical systems (MEMS) mirror to electrically steer a beam or beams in a 2D orientation. Although technically there are still moving parts (oscillating mirrors), the amplitude of the oscillation is small and the frequency is high enough to prevent mechanical resonances between the MEMS mirror and the car. However, the confined geometry of the mirror constrains its oscillation amplitude, which translates into limited field of view—a disadvantage of this MEMS approach. Nevertheless, this method is gaining interest because of its low cost and proven technology.
Optical phased array (OPA) technology, the third competing scanning LiDAR technique, is gaining popularity for its reliable, "no-moving-parts" design. It consists of arrays of optical antenna elements that are equally illuminated by coherent light. Beam steering is achieved by independently controlling the phase and amplitude of the re-emitted light by each element, and far-field interference produces a desired illumination pattern from a single beam to multiple beams. Unfortunately, light loss in the various OPA components restricts the usable range.
Flash LiDAR floods the scene with light, though the illumination region matches the field of view of the detector. The detector is an array of APDs at the focal plane of the detection optics. Each APD independently measures ToF to the target feature imaged on that APD. This is a truly "no-moving-parts" approach where the tangential resolution is limited by the pixel size of the 2D detector.
The major disadvantage of flash LiDAR, however, is photon budget: once the distance is more than a few tens of meters, the amount of returning light is too small for reliable detection. The budget can be improved at the expense of tangential resolution if instead of flooding the scene with photons, structured light—a grid of points—illuminates it. Vertical-cavity surface-emitting lasers (VCSELs) make it possible to create projectors emitting thousands of beams simultaneously in different directions.
Beyond time–of–flight limitations
Time-of-flight LiDAR is susceptible to noise because of the weakness of the returned pulses and wide bandwidth of the detection electronics, and threshold triggering can produce errors in measurement of Δt. For these reasons, frequency-modulated continuous-wave (FMCW) LiDAR is an interesting alternative.
In FMCW radar, or chirped radar, the antenna continuously emits radio waves whose frequency is modulated—for example, linearly increasing from f0 to fmax over time T and then linearly decreasing from fmax to f0 over time T. If the wave reflects from a moving object at some distance and comes back to the emission point, its instantaneous frequency will differ from the one being emitted at that instant. The difference is because of two factors: the distance to the object and its relative radial velocity. One can electronically measure the frequency difference and simultaneously calculate the object's distance and velocity (see Figure 3).
Figure 3. In chirped radar, by electronically measuring fB1 and fB2 one can determine the distance to the reflecting object and its radial speed.
Inspired by chirped radar, FMCW LiDAR can be approached in different ways. In the simplest design, one can chirp-modulate the intensity of the beam of light that illuminates the target. This frequency is subject to the same laws (such as the Doppler effect) as the carrier frequency in FMCW radar. The returned light is detected by a photodetector to recover modulation frequency. The output is amplified and mixed in with the local oscillator allowing the measurement of frequency shift and from that, the calculation of distance to and speed of the target.
But FMCW LiDAR has some limitations. Compared to a ToF LiDAR, it requires more computational power and, therefore, is slower in generating a full 3D surround view. In addition, the accuracy of the measurements is very sensitive to linearity of the chirp ramp.
Although designing a functional LiDAR system is challenging, none of these challenges are insurmountable. As the research continues, we are getting closer to the time when the majority of cars driving off the assembly line will be fully autonomous.
J. Wojtanowski et al., Opto-Electron. Rev., 22, 3, 183-190 (2014).
A version of this article was published in the November 2017 issue of Laser Focus World.
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(Redirected from Amortized constant time)
Method for algorithm analysis in computer science
"Amortized" redirects here. For other uses, see Amortization.
In computer science, amortized analysis is a method for analyzing a given algorithm's complexity, or how much of a resource, especially time or memory, it takes to execute. The motivation for amortized analysis is that looking at the worst-case run time can be too pessimistic. Instead, amortized analysis averages the running times of operations in a sequence over that sequence.[1]: 306 As a conclusion: “Amortized analysis is a useful tool that complements other techniques such as worst-case and average-case analysis.“[2]: 14
For a given operation of an algorithm, certain situations (e.g., input parametrizations or data structure contents) may imply a significant cost in resources, whereas other situations may not be as costly. The amortized analysis considers both the costly and less costly operations together over the whole sequence of operations. This may include accounting for different types of input, length of the input, and other factors that affect its performance.[2]
Amortized analysis initially emerged from a method called aggregate analysis, which is now subsumed by amortized analysis. The technique was first formally introduced by Robert Tarjan in his 1985 paper Amortized Computational Complexity,[1] which addressed the need for a more useful form of analysis than the common probabilistic methods used. Amortization was initially used for very specific types of algorithms, particularly those involving binary trees and union operations. However, it is now ubiquitous and comes into play when analyzing many other algorithms as well.[2]
Main articles: accounting method and potential method
There are generally three methods for performing amortized analysis: the aggregate method, the accounting method, and the potential method. All of these give correct answers; the choice of which to use depends on which is most convenient for a particular situation.[3]
Aggregate analysis determines the upper bound T(n) on the total cost of a sequence of n operations, then calculates the amortized cost to be T(n) / n.[3]
The accounting method is a form of aggregate analysis which assigns to each operation an amortized cost which may differ from its actual cost. Early operations have an amortized cost higher than their actual cost, which accumulates a saved "credit" that pays for later operations having an amortized cost lower than their actual cost. Because the credit begins at zero, the actual cost of a sequence of operations equals the amortized cost minus the accumulated credit. Because the credit is required to be non-negative, the amortized cost is an upper bound on the actual cost. Usually, many short-running operations accumulate such credit in small increments, while rare long-running operations decrease it drastically.[3]
The potential method is a form of the accounting method where the saved credit is computed as a function (the "potential") of the state of the data structure. The amortized cost is the immediate cost plus the change in potential.[3]
In general if we consider an arbitrary number of pushes n + 1 to an array of size n, we notice that push operations take constant time except for the last one which takes
{\displaystyle \Theta (n)}
time to perform the size doubling operation. Since there were n + 1 operations total we can take the average of this and find that pushing elements onto the dynamic array takes:
{\displaystyle {\tfrac {n\Theta (1)+\Theta (n)}{n+1}}=\Theta (1)}
, constant time.[3]
Shown is a Ruby implementation of a Queue, a FIFO data structure:
{\displaystyle O(n)}
{\displaystyle O(n)}
{\displaystyle O(1)}
{\displaystyle O(1)}
In common usage, an "amortized algorithm" is one that an amortized analysis has shown to perform well.
Online algorithms commonly use amortized analysis.
^ a b Tarjan, Robert Endre (April 1985). "Amortized Computational Complexity" (PDF). SIAM Journal on Algebraic and Discrete Methods. 6 (2): 306–318. doi:10.1137/0606031.
^ a b c Rebecca Fiebrink (2007), Amortized Analysis Explained (PDF), archived from the original (PDF) on 20 October 2013, retrieved 3 May 2011
^ a b c d e Kozen, Dexter (Spring 2011). "CS 3110 Lecture 20: Amortized Analysis". Cornell University. Retrieved 14 March 2015.
^ Grossman, Dan. "CSE332: Data Abstractions" (PDF). cs.washington.edu. Retrieved 14 March 2015.
"Lecture 7: Amortized Analysis" (PDF). Carnegie Mellon University. Retrieved 14 March 2015.
Allan Borodin and Ran El-Yaniv (1998). Online Computation and Competitive Analysis. pp. 20, 141.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Amortized_analysis&oldid=1084463282"
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Compound Interest | Brilliant Math & Science Wiki
Samarth Dave and Ashish Menon contributed
We are usually familiar with the term
\text {Compound interest}
, if not scroll down and see!
"Compound interest is the eighth wonder of the world. He who understands it, earns it ... he who doesn't ... pays it." - Albert Einstein
Compound interest is the basically the interest which is imposed on another interest. For example, if I take a loan of
\$1000
compounded annually such that the rate of interest is
10\%
, then my interest for the first year would be
10\%
\$1000
\$100
. Then principal for the second year would be
1000+100=\$1100
. Then, the interest for the second year would be
10\%
\$1100
\$110
So, we see that compound interest is the interest imposed on another interest.
To calculate problems on compound interest, we need to be familiar of the term Simple interest.
To better understand the difference, view the table below comparing the advantages and disadvantages of compound and simple interest.
Rate 5% (0.05)
Time (years) 4
Not too large a difference yet. Let's give it time.
Time (years) 20
Compound $16,532.98
Compound interest has essentially tripled (x2.65) your investment (principal). However, imagine that you loaned through compound. Certainly not something that happens. There isn't a "better" kind of interest. Simple and compound both have advantages and disadvantages. However, Albert Einstein certainly had an opinion on the matter.
Cite as: Compound Interest. Brilliant.org. Retrieved from https://brilliant.org/wiki/compound-interest/
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Longitude of the periapsis - WikiMili, The Best Wikipedia Reader
ϖ = Ω + ω in separate planes.
In celestial mechanics, the longitude of the periapsis, also called longitude of the pericenter, of an orbiting body is the longitude (measured from the point of the vernal equinox) at which the periapsis (closest approach to the central body) would occur if the body's orbit inclination were zero. It is usually denoted ϖ .
Calculation from state vectors
Derivation of ecliptic longitude and latitude of perihelion for inclined orbits
For the motion of a planet around the Sun, this position is called longitude of perihelion ϖ, which is the sum of the longitude of the ascending node Ω, and the argument of perihelion ω. [1] [2] :p.672, etc.
The longitude of periapsis is a compound angle, with part of it being measured in the plane of reference and the rest being measured in the plane of the orbit. Likewise, any angle derived from the longitude of periapsis (e.g., mean longitude and true longitude) will also be compound.
Sometimes, the term longitude of periapsis is used to refer to ω, the angle between the ascending node and the periapsis. That usage of the term is especially common in discussions of binary stars and exoplanets. [3] [4] However, the angle ω is less ambiguously known as the argument of periapsis.
ϖ is the sum of the longitude of ascending node Ω (measured on ecliptic plane) and the argument of periapsis ω (measured on orbital plane):
{\displaystyle \varpi =\Omega +\omega }
which are derived from the orbital state vectors.
i, inclination
ω, argument of perihelion
Ω, longitude of ascending node
ε, obliquity of the ecliptic (for the standard equinox of 2000.0, use 23.43929111°)
A = cos ω cos Ω – sin ω sin Ω cos i
B = cos ε (cos ω sin Ω + sin ω cos Ω cos i) – sin ε sin ω sin i
C = sin ε (cos ω sin Ω + sin ω cos Ω cos i) + cos ε sin ω sin i
The right ascension α and declination δ of the direction of perihelion are:
sin δ = C
If A < 0, add 180° to α to obtain the correct quadrant.
The ecliptic longitude ϖ and latitude b of perihelion are:
tan ϖ = sin α cos ε + tan δ sin ε/cos α
sin b = sin δ cos ε – cos δ sin ε sin α
If cos(α) < 0, add 180° to ϖ to obtain the correct quadrant.
As an example, using the most up-to-date numbers from Brown (2017) [5] for the hypothetical Planet Nine with i = 30°, ω = 136.92°, and Ω = 94°, then α = 237.38°, δ = +0.41° and ϖ = 235.00°, b = +19.97° (Brown actually provides i, Ω, and ϖ, from which ω was computed).
In astronomy, Kepler's laws of planet motion are three scientific laws describing the motion of planets around the Sun, published by Johannes Kepler between 1609 and 1619. These improved the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The laws state that:
Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite, and the primary planet that it orbits. The acceleration causes a gradual recession of a satellite in a prograde orbit away from the primary, and a corresponding slowdown of the primary's rotation. The process eventually leads to tidal locking, usually of the smaller first, and later the larger body. The Earth–Moon system is the best-studied case.
In astronomy, a celestial coordinate system is a system for specifying positions of satellites, planets, stars, galaxies, and other celestial objects relative to physical reference points available to a situated observer. Coordinate systems can specify an object's position in three-dimensional space or plot merely its direction on a celestial sphere, if the object's distance is unknown or trivial.
The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions and orbits of Solar System objects. Because most planets and many small Solar System bodies have orbits with only slight inclinations to the ecliptic, using it as the fundamental plane is convenient. The system's origin can be the center of either the Sun or Earth, its primary direction is towards the vernal (March) equinox, and it has a right-hand convention. It may be implemented in spherical or rectangular coordinates.
The equation of time describes the discrepancy between two kinds of solar time. The word equation is used in the medieval sense of "reconcile a difference". The two times that differ are the apparent solar time, which directly tracks the diurnal motion of the Sun, and mean solar time, which tracks a theoretical mean Sun with uniform motion. Apparent solar time can be obtained by measurement of the current position of the Sun, as indicated by a sundial. Mean solar time, for the same place, would be the time indicated by a steady clock set so that over the year its differences from apparent solar time would have a mean of zero.
In mathematics and all natural sciences, the angular distance between two point objects, as viewed from a location different from either of these objects, is the angle of length between the two directions originating from the observer and pointing toward these two objects.
The beta angle is a measurement that is used most notably in orbital spaceflight. The beta angle determines the percentage of time that a satellite in low Earth orbit (LEO) spends in direct sunlight, absorbing solar energy. The term is defined as the angle between the orbital plane of the satellite and the vector to the Sun. The beta angle is the smaller of the two angles between the Sun vector and the plane of the object's orbit. The beta angle does not define a unique orbital plane; all satellites in orbit with a given beta angle at a given altitude have the same exposure to the Sun, even though they may be orbiting in completely different planes around Earth.
↑ Urban, Sean E.; Seidelmann, P. Kenneth (eds.). "Chapter 8: Orbital Ephemerides of the Sun, Moon, and Planets" (PDF). Explanatory Supplement to the Astronomical Almanac. University Science Books. p. 26.
↑ Simon, J. L.; et al. (1994). "Numerical expressions for precession formulae and mean elements for the Moon and the planets". Astronomy and Astrophysics. 282: 663–683. Bibcode:1994A&A...282..663S.
↑ Robert Grant Aitken (1918). The Binary Stars. Semicentennial Publications of the University of California. D.C. McMurtrie. p. 201.
↑ "Format" Archived 2009-02-25 at the Wayback Machine in Sixth Catalog of Orbits of Visual Binary Stars Archived 2009-04-12 at the Wayback Machine , William I. Hartkopf & Brian D. Mason, U.S. Naval Observatory, Washington, D.C. Accessed on 10 January 2018.
↑ Brown, Michael E. (2017) “Planet Nine: where are you? (part 1)” The Search for Planet Nine. http://www.findplanetnine.com/2017/09/planet-nine-where-are-you-part-1.html
[ dead link ] Determination of the Earth's Orbital Parameters Past and future longitude of perihelion for Earth.
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Predicting the firing phase of an oscillatory neuron from its impedance profile | BMC Neuroscience | Full Text
Predicting the firing phase of an oscillatory neuron from its impedance profile
Farzan Nadim1,2,
The activity phase of a neuron in an oscillatory network often determines what the neuron codes [1, 2]. We are interested in understanding the effect of subthreshold factors that influence this activity phase. Here we develop a first-order approximation of the activity phase of a neuron receiving oscillatory input using its subthreshold impedance profile.
A neuron's subthreshold membrane potential response to sinusoidal current input with frequency f is sinusoidal (to first order) with amplitude and phase-shift approximated by the impedance value at f:
{Z}_{f}=∥{Z}_{f}∥{e}^{i{\phi }_{f}}
. If a neuron receives suprathreshold oscillatory input at frequency f, the resulting change in membrane potential can be approximated with a similar amplitude and phase-shift up to the time point t spike where spike threshold is reached. This results in the following simple equation:
{V}_{m}\left({t}_{spike}\right)={V}_{rest}+{A}_{in}∥{Z}_{f}∥\mathsf{\text{sin}}\left(2\pi f\cdot {t}_{spike}+{\phi }_{f}\right)={V}_{thresh}
where A in is the amplitude of the input current. Assuming the reference time point of t0 = 0 in the cycle, the spike phase can be approximated as
{\phi }_{spike}=f\cdot {t}_{spike}=\frac{1}{2\pi }\left(\mathsf{\text{arcsin}}\frac{{V}_{thresh}-\phantom{\rule{2.77695pt}{0ex}}{V}_{rest}}{{A}_{in}∥{Z}_{f}∥}-{\phi }_{f}\right)
. This approximation is valid so long as the argument of arcsin is <1 in absolute value.
As proof of principle, we used the impedance profile of a model neuron exhibiting subthreshold resonance to approximate the spike phase with a given preset spike threshold (Figure A). We also used this approximation to predict the phase of the first spike in bursting PY neurons in the crab pyloric CPG, when synaptically isolated and subjected to a sinusoidal current input at different frequencies (0.1-4 Hz; see [3]for method). The PY impedance was measured from its subthreshold response.
To our knowledge this method, despite its simplicity, has not been previously used for the approximation of spike phase using the impedance profile. The usefulness of this approximation is that changes in membrane properties due to network activity or neuromodulation can be readily measured in the impedance profile and this knowledge can be used to predict how the neuron changes its response phase during network activity.
There are three sources of error in this approximation. First, the membrane impedance in biological neurons is nonlinear and does not scale linearly with the amplitude of the input current, especially when the neuron transitions from sub- to suprathreshold activity. However, the shift in membrane impedance can be tracked in both model and biological neurons. Second, spike threshold is dependent on input frequency. This dependence can also be tracked. Third, because of nonlinearities, the membrane potential response of a neuron is not perfectly sinusoidal. As seen in Figure 1B, despite these (nonlinear) sources of error, our method can provide a good approximation of spike phase. We are in the process of estimating spike phase in response to synaptic inputs arriving at a fixed frequency.
A. The spike phase of a resonate-and-fire model predicted from the impedance profile. B. The spike phase of the PY neuron subject to a ZAP input (inset) predicted from its subthreshold response (green).
Bose A, Manor Y, Nadim F: The activity phase of postsynaptic neurons in a simplified rhythmic network. J Comput Neurosci. 2004, 17 (2): 245-261.
Geisler C, Robbe D, Zugaro M, Sirota A, Buzsaki G: Hippocampal place cell assemblies are speed-controlled oscillators. Proc Natl Acad Sci USA. 2007, 104 (19): 8149-8154. 10.1073/pnas.0610121104.
Tseng HA, Nadim F: The membrane potential waveform of bursting pacemaker neurons is a predictor of their preferred frequency and the network cycle frequency. J Neurosci. 2010, 30 (32): 10809-10819. 10.1523/JNEUROSCI.1818-10.2010.
Farzan Nadim & David Fox
Farzan Nadim & Horacio G Rotstein
Correspondence to Farzan Nadim.
Nadim, F., Rotstein, H.G. & Fox, D. Predicting the firing phase of an oscillatory neuron from its impedance profile. BMC Neurosci 14, P132 (2013). https://doi.org/10.1186/1471-2202-14-S1-P132
Oscillatory Neuron
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Metcalfe's law - Wikipedia
Metcalfe's law states that the value of a telecommunications network is proportional to the square of the number of connected users of the system (n2). First formulated in this form by George Gilder in 1993,[1] and attributed to Robert Metcalfe in regard to Ethernet, Metcalfe's law was originally presented, c. 1980, not in terms of users, but rather of "compatible communicating devices" (e.g., fax machines, telephones).[2] Only later with the globalization of the Internet did this law carry over to users and networks as its original intent was to describe Ethernet connections.[3]
2.1 Growth of n
4 Modified models
5 Validation in data
Network effectsEdit
Metcalfe's law characterizes many of the network effects of communication technologies and networks such as the Internet, social networking and the World Wide Web. Former Chairman of the U.S. Federal Communications Commission Reed Hundt said that this law gives the most understanding to the workings of the Internet.[4] Metcalfe's Law is related to the fact that the number of unique possible connections in a network of
{\displaystyle n}odes can be expressed mathematically as the triangular number
{\displaystyle n(n-1)/2}
, which is asymptotically proportional to
{\displaystyle n^{2}}
The law has often been illustrated using the example of fax machines: a single fax machine is useless, but the value of every fax machine increases with the total number of fax machines in the network, because the total number of people with whom each user may send and receive documents increases.[5] Likewise, in social networks, the greater the number of users with the service, the more valuable the service becomes to the community.
Metcalfe’s law was conceived in 1983 in a presentation to the 3Com sales force.[6] It stated V would be proportional to the total number of possible connections, or approximately n-squared.
The original incarnation was careful to delineate between a linear cost (Cn), non-linear growth, n2, and a non-constant proportionality factor A “Affinity.” The breakeven point where costs are recouped is given by
{\displaystyle C\times n=A\times n(n-1)/2}
At some size, the right-hand side of the equation V “Value” exceeds the cost, and A describes the relationship between size and net value added. For large n, net network value is then
{\displaystyle \Pi =n(A\times (n-1)/2-C)}
Metcalfe properly dimensioned A as “value per user”. Affinity is also a function of network size, and Metcalfe correctly asserted that A must decline as n grows large. In a 2006 interview, Metcalfe stated
“There may be diseconomies of network scale that eventually drive values down with increasing size. So, if V=A*n2, it could be that A (for “affinity,” value per connection) is also a function of n and heads down after some network size, overwhelming n2.”[7]
Growth of nEdit
Network size, and hence value, does not grow unbounded but is constrained by practical limitations such as infrastructure, access to technology, and bounded rationality such as Dunbar's number. It is almost always the case that user growth n reaches a saturation point. With technologies, substitutes, competitors and technical obsolescence constrain growth of n. Growth of n is typically assumed to follow a sigmoid function such as a logistic curve or Gompertz curve.
A is also governed by the connectivity or density of the network topology. In an undirected network, every edge connects two nodes such that there are 2m nodes per edge. The proportion of nodes in actual contact are given by
{\displaystyle c=2m/n}
The maximum possible number of edges in a simple network (i.e. one with no multi-edges or self-edges) is
{\displaystyle {\binom {n}{2}}=n(n-1)/2}
. Therefore the density ρ of a network is the faction of those edges that are actually present is
{\displaystyle \rho =c/(n-1)}
which for large networks is approximated by
{\displaystyle \rho =c/n}
Metcalfe’s law assumes that the value of each node
{\displaystyle n}
is of equal benefit.[9] If this is not the case, for example because one fax machine serves 50 workers in a company, the second fax machine serves half of that, the third one third, and so on, then the relative value of an additional connection decreases. Likewise, in social networks, if users that join later use the network less than early adopters, then the benefit of each additional user may lessen, making the overall network less efficient if costs per users are fixed.
Modified modelsEdit
Within the context of social networks, many, including Metcalfe himself, have proposed modified models in which the value of the network grows as
{\displaystyle n\log n}
{\displaystyle n^{2}}
.[10][11] Reed and Andrew Odlyzko have sought out possible relationships to Metcalfe's Law in terms of describing the relationship of a network and one can read about how those are related. Tongia and Wilson also examine the related question of the costs to those excluded.[12]
Validation in dataEdit
Despite many arguments about Metcalfe' law, no real data based evidence for or against was available for more than 30 years. Only in July 2013, Dutch researchers managed to analyze European Internet usage patterns over a long enough time and found
{\displaystyle n^{2}}
proportionality for small values of
{\displaystyle n}
{\displaystyle n\log n}
proportionality for large values of
{\displaystyle n}
.[13] A few months later, Metcalfe himself provided further proof, as he used Facebook's data over the past 10 years to show a good fit for Metcalfe's law (the model is
{\displaystyle n^{2}}
In 2015, Zhang, Liu and Xu parameterized the Metcalfe function in data from Tencent and Facebook. Their work showed that Metcalfe's law held for both, despite differences in audience between the two sites (Facebook serving a worldwide audience and Tencent serving only Chinese users). The functions for the two sites were
{\displaystyle V_{Tencent}=7.39\times 10^{-9}\times n^{2}}
{\displaystyle V_{Facebook}=5.70\times 10^{-9}\times n^{2}}
In a working paper, Peterson linked time-value-of-money concepts to Metcalfe value using Bitcoin and Facebook as numerical examples of the proof[16] and in 2018 applied Metcalfe's law to Bitcoin, showing that over 70% of variance in Bitcoin value was explained by applying Metcalfe's law to increases in Bitcoin network size.[17]
^ Carl Shapiro and Hal R. Varian (1999). Information Rules. Harvard Business Press. ISBN 978-0-87584-863-1.
^ Simeon Simeonov (July 26, 2006). "Metcalfe's Law: more misunderstood than wrong?". HighContrast: Innovation & venture capital in the post-broadband era.
^ James Hendler and Jennifer Golbeck (2008). "Metcalfe's Law, Web 2.0, and the Semantic Web" (PDF).
^ Bob Briscoe, Andrew Odlyzko and Benjamin Tilly (July 2006). "Metcalfe's Law is wrong". Retrieved 2010-07-25.
^ R. Tongia. "The Dark Side of Metcalfe's Law: Multiple and Growing Costs of Network Exclusion" (PDF). Retrieved 2017-12-19.
^ Metcalfe, Bob (December 2013). "Metcalfe's Law after 40 Years of Ethernet". Computer. 46 (12): 26–31. doi:10.1109/MC.2013.374. ISSN 1558-0814.
^ Metcalfe, Robert (18 August 2006). "Guest Blogger Bob Metcalfe: Metcalfe's Law Recurses down the Long Tail of Social Networks". VC Mike's Blog. {{cite web}}: CS1 maint: url-status (link)
^ Newman, Mark E.J. (2019). "Mathematics of Networks" in Networks. Oxford: Oxford University Press. pp. 126–128. ISBN 978-0-19-880509-0.
^ Andrew Odlyzko; Bob Briscoe (1 Jul 2006). "Metcalfe's Law is Wrong". IEEE Spectrum: Technology, Engineering, and Science News. Retrieved 25 November 2016.
^ "Guest Blogger Bob Metcalfe: Metcalfe's Law Recurses Down the Long Tail of Social Networks". 18 August 2006. Retrieved 2010-06-20.
^ B. Briscoe, A. Odlyzko, and B. Tilly, Metcalfe’s law is wrong, IEEE Spectrum 43:7 (2006), pp. 34–39.
^ Rahul Tongia and Ernest Wilson (September 2007). "The Flip Side of Metcalfe's Law: Multiple and Growing Costs of Network Exclusion". Retrieved 2013-01-15.
^ Madureira, António; den Hartog, Frank; Bouwman, Harry; Baken, Nico (2013), "Empirical validation of Metcalfe's law: How Internet usage patterns have changed over time", Information Economics and Policy, doi:10.1016/j.infoecopol.2013.07.002
^ Metcalfe, Bob (2013). "Metcalfe's law after 40 years of Ethernet". IEEE Computer. 46 (12): 26–31. doi:10.1109/MC.2013.374.
^ Zhang, Xing-Zhou; Liu, Jing-Jie; Xu, Zhi-Wei (2015). "Tencent and Facebook Data Validate Metcalfe's Law". Journal of Computer Science and Technology. 30 (2): 246–251. doi:10.1007/s11390-015-1518-1.
^ Peterson, Timothy (2019). "Bitcoin Spreads Like a Virus". Working Paper. doi:10.2139/ssrn.3356098.
^ Peterson, Timothy (2018). "Metcalfe's Law as a Model for Bitcoin's Value". Alternative Investment Analyst Review. 7 (2): 9–18. doi:10.2139/ssrn.3078248.
Smith, David; Skelley, C. A. (Summer 2006), "Globalization Transformation" (PDF), Tennessee Business Magazine: 17–19
Briscoe, Bob; Odlyzko, Andrew; Tilly, Benjamin (July 2006), "Metcalfe's Law is Wrong", IEEE Spectrum, 43 (7): 34–39, doi:10.1109/MSPEC.2006.1653003 .
A Group Is Its Own Worst Enemy. Clay Shirky's keynote speech on Social Software at the O'Reilly Emerging Technology conference, Santa Clara, April 24, 2003. The fourth of his "Four Things to Design For" is: "And, finally, you have to find a way to spare the group from scale. Scale alone kills conversations, because conversations require dense two-way conversations. In conversational contexts, Metcalfe's law is a drag."
Retrieved from "https://en.wikipedia.org/w/index.php?title=Metcalfe%27s_law&oldid=1081198390"
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Normalize data across grouped subsets of channels for each observation independently - MATLAB groupnorm - MathWorks í•œêµ
After normalization, the operation shifts the input by a learnable offset β and scales it by a learnable scale factor γ.
Offset β, specified as a formatted dlarray, an unformatted dlarray, or a numeric array with one nonsingleton dimension with size matching the size of the 'C' (channel) dimension of the input X.
Scale factor γ, specified as a formatted dlarray, an unformatted dlarray, or a numeric array with one nonsingleton dimension with size matching the size of the 'C' (channel) dimension of the input X.
The group normalization operation normalizes the elements xi of the input by first calculating the mean μG and variance σG2 over spatial, time, and grouped subsets of the channel dimensions for each observation independently. Then, it calculates the normalized activations as
{\stackrel{^}{x}}_{i}=\frac{{x}_{i}â{\mathrm{μ}}_{G}}{\sqrt{{\mathrm{Ï}}_{G}^{2}+\mathrm{ε}}},
where ϵ is a constant that improves numerical stability when the variance is very small. To allow for the possibility that inputs with zero mean and unit variance are not optimal for the operations that follow group normalization, the group normalization operation further shifts and scales the activations using the transformation
{y}_{i}=\mathrm{γ}{\stackrel{^}{x}}_{i}+\mathrm{β},
[1] Wu, Yuxin, and Kaiming He. “Group Normalization.†Preprint submitted June 11, 2018. https://arxiv.org/abs/1803.08494.
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Merchant's Robe - Ring of Brodgar
Skill(s) Required Sewing, Silkfarming, Trade
Object(s) Required Silk Cloth x4, String x2
Gilding Attributes Charisma
Craft > Clothes & Equipment > Capes, Cloaks & Robes > Merchant's Robe
Merchant's Robe is a staple silken Cloak worn by successful Hearthlings which increases inventory space by one horizontal row, regardless of quality. It looks great with the Pimp Hat!
The base quality is softcapped by:
{\displaystyle {\sqrt {Dexterity*Sewing}}}
The gilding is based on Charisma with a 25% to 45% chance.
A player wearing a merchant's robe.
A merchant's robe decorating a wall.
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Merchant%27s_Robe&oldid=88313"
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Power transmission element with frictional belt wrapped around pulley circumference - MATLAB - MathWorks India
-\beta {V}_{A}={V}_{C}-R{\omega }_{S}-\beta {V}_{rel}
{V}_{B}={V}_{C}+R{\omega }_{S}+\beta {V}_{rel}
{F}_{centrifugal}=\rho {\left({V}_{B}-{V}_{C}\right)}^{2}.
{F}_{C}=\left(\beta {F}_{A}-{F}_{B}-{F}_{centrifugal,smooth,sat}\right)\cdot \mathrm{sin}\left(\frac{\theta }{2}\right).
{\mu }_{smoothed}={\mu }_{sheave}\mathrm{tanh}\left(4\frac{{V}_{rel}}{{V}_{thr}}\right),
{\mu }_{sheave}=\frac{\mu }{\mathrm{sin}\left(\frac{\varphi }{2}\right)},
-\beta {F}_{A}-{F}_{centrifugal}=\left({F}_{B}-{F}_{centrifugal}\right){e}^{-{\mu }_{smoothed}\theta },
{\tau }_{S}=\left(-\beta {F}_{A}-{F}_{B}\right)R\sigma +{\omega }_{S}b,
\sigma =\mathrm{tanh}\left(4\frac{{V}_{rel}}{{V}_{thr}}\right)\mathrm{tanh}\left(\frac{{F}_{B}}{{F}_{thr}}\right).
\begin{array}{l}{\mu }_{Static,sheave}=\frac{{\mu }_{Static}}{\mathrm{sin}\left(\frac{\varphi }{2}\right)}\\ {\mu }_{Kinetic,sheave}=\frac{{\mu }_{Kinetic}}{\mathrm{sin}\left(\frac{\varphi }{2}\right)}\end{array}
{F}_{Friction,Static,Max}=\left({F}_{Drive}-{F}_{Centrifugal}\right)\cdot \left(1-{e}^{-{\mu }_{Static,sheave}\theta }\right),
{F}_{Friction,Static,Max,Smooth}=0.5\left({F}_{Friction,Static,Max}+\sqrt{{F}_{Friction,Static,Max}^{2}+{\left(R{F}_{thr}\right)}^{2}}\right).
{F}_{Friction,Slip}=\frac{{\mu }_{Kinetic,sheave}}{{\mu }_{Static,sheave}}{F}_{Friction,Static,Max},
{\tau }_{S}={\omega }_{S}b+R{F}_{Friction},
R\beta {F}_{A}+R{F}_{B}=-R{F}_{Friction},
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Numerical gradient - MATLAB gradient - MathWorks España
Contour Plot of Vector Field
Compute Gradient at Specified Point
FX = gradient(F) returns the one-dimensional numerical gradient of vector F. The output FX corresponds to ∂F/∂x, which are the differences in the x (horizontal) direction. The spacing between points is assumed to be 1.
[FX,FY] = gradient(F) returns the x and y components of the two-dimensional numerical gradient of matrix F. The additional output FY corresponds to ∂F/∂y, which are the differences in the y (vertical) direction. The spacing between points in each direction is assumed to be 1.
[FX,FY,FZ,...,FN] = gradient(F) returns the N components of the numerical gradient of F, where F is an array with N dimensions.
[___] = gradient(F,h) uses h as a uniform spacing between points in each direction. You can specify any of the output arguments in previous syntaxes.
[___] = gradient(F,hx,hy,...,hN) specifies N spacing parameters for the spacing in each dimension of F.
Calculate the gradient of a monotonically increasing vector.
Calculate the 2-D gradient of
x{e}^{-{x}^{2}-{y}^{2}}
on a grid.
Plot the contour lines and vectors in the same figure.
Use the gradient at a particular point to linearly approximate the function value at a nearby point and compare it to the actual value.
The equation for linear approximation of a function value is
f\left(x\right)\approx f\left({x}_{0}\right)+{\left(\nabla f\right)}_{{x}_{0}}\cdot \left(x-{x}_{0}\right).
That is, if you know the value of a function
f\left({x}_{0}\right)
and the slope of the derivative
{\left(\nabla f\right)}_{{x}_{0}}
at a particular point
{x}_{0}
, then you can use this information to approximate the value of the function at a nearby point
f\left(x\right)=f\left({x}_{0}+ϵ\right)
Calculate some values of the sine function between -1 and 0.5. Then calculate the gradient.
Use the function value and derivative at x = 0.5 to predict the value of sin(0.5005).
Compute the actual value for comparison.
Find the value of the gradient of a multivariate function at a specified point.
f\left(x,y\right)={x}^{2}{y}^{3}
Calculate the gradient on the grid.
Extract the value of the gradient at the point (1,-2). To do this, first obtain the indices of the point you want to work with. Then, use the indices to extract the corresponding gradient values from fx and fy.
The exact value of the gradient of
f\left(x,y\right)={x}^{2}{y}^{3}
at the point (1,-2) is
\begin{array}{cl}{\left(\nabla f\right)}_{\left(1,-2\right)}& =2x{y}^{3}\underset{}{\overset{ˆ}{i}}+3{x}^{2}{y}^{2}\underset{}{\overset{ˆ}{j}}\\ & =-16\underset{}{\overset{ˆ}{i}}+12\underset{}{\overset{ˆ}{j}}.\end{array}
h — Uniform spacing between points
Uniform spacing between points in all directions, specified as a scalar.
Example: [FX,FY] = gradient(F,2)
hx, hy, hN — Spacing between points (as separate inputs)
1 (default) | scalars | vectors
Spacing between points in each direction, specified as separate inputs of scalars or vectors. The number of inputs must match the number of array dimensions of F. Each input can be a scalar or vector:
A scalar specifies a constant spacing in that dimension.
A vector specifies the coordinates of the values along the corresponding dimension of F. In this case, the length of the vector must match the size of the corresponding dimension.
Example: [FX,FY] = gradient(F,0.1,2)
Example: [FX,FY] = gradient(F,[0.1 0.3 0.5],2)
Example: [FX,FY] = gradient(F,[0.1 0.3 0.5],[2 3 5])
FX, FY, FZ, FN — Numerical gradients
Numerical gradients, returned as arrays of the same size as F. The first output FX is always the gradient along the 2nd dimension of F, going across columns. The second output FY is always the gradient along the 1st dimension of F, going across rows. For the third output FZ and the outputs that follow, the Nth output is the gradient along the Nth dimension of F.
The numerical gradient of a function is a way to estimate the values of the partial derivatives in each dimension using the known values of the function at certain points.
For a function of two variables, F(x,y), the gradient is
\nabla F=\frac{\partial F}{\partial x}\stackrel{^}{i}+\frac{\partial F}{\partial y}\stackrel{^}{j}\text{\hspace{0.17em}}.
The gradient can be thought of as a collection of vectors pointing in the direction of increasing values of F. In MATLAB®, you can compute numerical gradients for functions with any number of variables. For a function of N variables, F(x,y,z, ...), the gradient is
\nabla F=\frac{\partial F}{\partial x}\stackrel{^}{i}+\frac{\partial F}{\partial y}\stackrel{^}{j}+\frac{\partial F}{\partial z}\stackrel{^}{k}+...+\frac{\partial F}{\partial N}\stackrel{^}{n}\text{\hspace{0.17em}}.
Use diff or a custom algorithm to compute multiple numerical derivatives, rather than calling gradient multiple times.
gradient calculates the central difference for interior data points. For example, consider a matrix with unit-spaced data, A, that has horizontal gradient G = gradient(A). The interior gradient values, G(:,j), are
The subscript j varies between 2 and N-1, with N = size(A,2).
gradient calculates values along the edges of the matrix with single-sided differences:
If you specify the point spacing, then gradient scales the differences appropriately. If you specify two or more outputs, then the function also calculates differences along other dimensions in a similar manner. Unlike the diff function, gradient returns an array with the same number of elements as the input.
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Marginal cost - WikiMili, The Best Wikipedia Reader
In economics, the marginal cost is the change in the total cost that arises when the quantity produced is incremented, the cost of producing additional quantity. [1] In some contexts, it refers to an increment of one unit of output, and in others it refers to the rate of change of total cost as output is increased by an infinitesimal amount. As Figure 1 shows, the marginal cost is measured in dollars per unit, whereas total cost is in dollars, and the marginal cost is the slope of the total cost, the rate at which it increases with output. Marginal cost is different from average cost, which is the total cost divided by the number of units produced.
At each level of production and time period being considered, marginal cost include all costs that vary with the level of production, whereas costs that do not vary with production are fixed. For example, the marginal cost of producing an automobile will include the costs of labor and parts needed for the additional automobile but not the fixed cost of the factory building that do not change with output. The marginal cost can be either short-run or long-run marginal cost, depending on what costs vary with output, since in the long run even building size is chosen to fit the desired output.
If the cost function
{\displaystyle C}
{\displaystyle MC}
{\displaystyle Q}
{\displaystyle MC(Q)={\frac {\ dC}{\ dQ}}.}
{\displaystyle MC={\frac {\Delta C}{\Delta Q}},}
{\displaystyle \Delta }
The long run is defined as the length of time in which no input is fixed. Everything, including building size and machinery, can be chosen optimally for the quantity of output that is desired. As a result, even if short-run marginal cost rises because of capacity constraints, long-run marginal cost can be constant. Or, there may be increasing or decreasing returns to scale if technological or management productivity changes with the quantity. Or, there may be both, as in the diagram at the right, in which the marginal cost first falls (increasing returns to scale) and then rises (decreasing returns to scale). [3]
Fixed costs represent the costs that do not change as the production quantity changes. Fixed costs are costs incurred by things like rent, building space, machines, etc. Variable costs change as the production quantity changes, and are often associated with labor or materials. The derivative of fixed cost is zero, and this term drops out of the marginal cost equation: that is, marginal cost does not depend on fixed costs. This can be compared with average total cost (ATC), which is the total cost (including fixed costs, denoted C0) divided by the number of units produced:
{\displaystyle ATC={\frac {C_{0}+\Delta C}{Q}}.}
Marginal cost is not the cost of producing the "next" or "last" unit. [4] The cost of the last unit is the same as the cost of the first unit and every other unit. In the short run, increasing production requires using more of the variable input — conventionally assumed to be labor. Adding more labor to a fixed capital stock reduces the marginal product of labor because of the diminishing marginal returns. This reduction in productivity is not limited to the additional labor needed to produce the marginal unit – the productivity of every unit of labor is reduced. Thus the cost of producing the marginal unit of output has two components: the cost associated with producing the marginal unit and the increase in average costs for all units produced due to the "damage" to the entire productive process. The first component is the per-unit or average cost. The second component is the small increase in cost due to the law of diminishing marginal returns which increases the costs of all units sold.
Marginal costs can also be expressed as the cost per unit of labor divided by the marginal product of labor. [5] Denoting variable cost as VC, the constant wage rate as w, and labor usage as L, we have
{\displaystyle MC={\frac {\Delta VC}{\Delta Q}}}
{\displaystyle \Delta VC={w\Delta L}}
{\displaystyle MC={\frac {w\Delta L}{\Delta Q}}={\frac {w}{MPL}}.}
Here MPL is the ratio of increase in the quantity produced per unit increase in labour: i.e. ΔQ/ΔL, the marginal product of labor. The last equality holds because
{\displaystyle {\frac {\Delta L}{\Delta Q}}}
is the change in quantity of labor that brings about a one-unit change in output. [6] Since the wage rate is assumed constant, marginal cost and marginal product of labor have an inverse relationship—if the marginal product of labor is decreasing (or, increasing), then marginal cost is increasing (decreasing), and AVC = VC/Q=wL/Q = w/(Q/L) = w/APL
While neoclassical models broadly assume that marginal cost will increase as production increases, several empirical studies conducted throughout the 20th century have concluded that the marginal cost is either constant or falling for the vast majority of firms. [7] Most recently, former Federal Reserve Vice-Chair Alan Blinder and colleagues conducted a survey of 200 executives of corporations with sales exceeding $10 million, in which they were asked, among other questions, about the structure of their marginal cost curves. Strikingly, just 11% of respondents answered that their marginal costs increased as production increased, while 48% answered that they were constant, and 41% answered that they were decreasing. [8] : 106 Summing up the results, they wrote:
— Asking About Prices: A New Approach to Understanding Price Stickiness, p. 105 [8]
Many Post-Keynesian economists have pointed to these results as evidence in favor of their own heterodox theories of the firm, which generally assume that marginal cost is constant as production increases. [7]
Economies of scale apply to the long run, a span of time in which all inputs can be varied by the firm so that there are no fixed inputs or fixed costs. Production may be subject to economies of scale (or diseconomies of scale). Economies of scale are said to exist if an additional unit of output can be produced for less than the average of all previous units – that is, if long-run marginal cost is below long-run average cost, so the latter is falling. Conversely, there may be levels of production where marginal cost is higher than average cost, and the average cost is an increasing function of output. Where there are economies of scale, prices set at marginal cost will fail to cover total costs, thus requiring a subsidy. [9] For this generic case, minimum average cost occurs at the point where average cost and marginal cost are equal (when plotted, the marginal cost curve intersects the average cost curve from below).
The portion of the marginal cost curve above its intersection with the average variable cost curve is the supply curve for a firm operating in a perfectly competitive market (the portion of the MC curve below its intersection with the AVC curve is not part of the supply curve because a firm would not operate at a price below the shutdown point). This is not true for firms operating in other market structures. For example, while a monopoly has an MC curve, it does not have a supply curve. In a perfectly competitive market, a supply curve shows the quantity a seller is willing and able to supply at each price – for each price, there is a unique quantity that would be supplied.
In perfectly competitive markets, firms decide the quantity to be produced based on marginal costs and sale price. If the sale price is higher than the marginal cost, then they produce the unit and supply it. If the marginal cost is higher than the price, it would not be profitable to produce it. So the production will be carried out until the marginal cost is equal to the sale price. [10]
Of great importance in the theory of marginal cost is the distinction between the marginal private and social costs. The marginal private cost shows the cost borne by the firm in question. It is the marginal private cost that is used by business decision makers in their profit maximization behavior. Marginal social cost is similar to private cost in that it includes the cost of private enterprise but also any other cost (or offsetting benefit) to parties having no direct association with purchase or sale of the product. It incorporates all negative and positive externalities, of both production and consumption. Examples include a social cost from air pollution affecting third parties and a social benefit from flu shots protecting others from infection.
Much of the time, private and social costs do not diverge from one another, but at times social costs may be either greater or less than private costs. When the marginal social cost of production is greater than that of the private cost function, there is a negative externality of production. Productive processes that result in pollution or other environmental waste are textbook examples of production that creates negative externalities.
When the marginal social cost of production is less than that of the private cost function, there is a positive externality of production. Production of public goods is a textbook example of production that creates positive externalities. An example of such a public good, which creates a divergence in social and private costs, is the production of education. It is often seen that education is a positive for any whole society, as well as a positive for those directly involved in the market.
Cost-sharing mechanism
The break-even point (BEP) in economics, business—and specifically cost accounting—is the point at which total cost and total revenue are equal, i.e. "even". There is no net loss or gain, and one has "broken even", though opportunity costs have been paid and capital has received the risk-adjusted, expected return. In short, all costs that must be paid are paid, and there is neither profit or loss.
In economics, diminishing returns is the decrease in marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, holding all other factors of production equal. The law of diminishing returns states that in productive processes, increasing a factor of production by one unit, while holding all other production factors constant, will at some point return a lower unit of output per incremental unit of input. The law of diminishing returns does not cause a decrease in overall production capabilities, rather it defines a point on a production curve whereby producing an additional unit of output will result in a loss and is known as negative returns. Under diminishing returns, output remains positive, however productivity and efficiency decrease.
Economic cost is the combination of losses of any goods that have a value attached to them by any one individual. Economic cost is used mainly by economists as means to compare the prudence of one course of action with that of another. The factors to be taken into consideration are money, time, and other resources cost is the sum of explicit cost.
The marginal revenue productivity theory of wages is a model of wage levels in which they set to match to the marginal revenue product of labor, , which is the increment to revenues caused by the increment to output produced by the last laborer employed. In a model, this is justified by an assumption that the firm is profit-maximizing and thus would employ labor only up to the point that marginal labor costs equal the marginal revenue generated for the firm. This is a model of the neoclassical economics type.
In economics, average variable cost (AVC) is a firm's variable costs divided by the quantity of output produced. Variable costs are those costs which vary with the output level:
In economics, average fixed cost (AFC) is the fixed costs of production (FC) divided by the quantity (Q) of output produced. Fixed costs are those costs that must be incurred in fixed quantity regardless of the level of output produced.
In industrial organization, the minimum efficient scale (MES) or efficient scale of production is the lowest point where the plant can produce such that its long run average costs are minimized. It is also the point at which the firm can achieve necessary economies of scale for it to compete effectively within the market.
A firm will choose to implement a shutdown of production when the revenue received from the sale of the goods or services produced cannot even cover the variable costs of production. In that situation, the firm will experience a higher loss when it produces, compared to not producing at all.
↑ O'Sullivan, Arthur; Sheffrin, Steven M. (2003). Economics: Principles in Action . Upper Saddle River, NJ: Pearson Prentice Hall. p. 111. ISBN 0-13-063085-3.
↑ Simon, Carl; Blume, Lawrence (1994). Mathematics for Economists. W. W. Norton & Company. ISBN 0393957330.
↑ The classic reference is Jakob Viner, "Cost Curves and Supply Curve," Zeitschrift fur Nationalokonomie, 3:23-46 (1932).
1 2 Lavoie, Marc (2014). Post-Keynesian Economics: New Foundations. Northampton, MA: Edward Elgar Publishing, Inc. p. 151. ISBN 978-1-84720-483-7.
1 2 Blinder, Alan S.; Canetti, Elie R. D.; Lebow, David E.; Rudd, Jeremy B. (1998). Asking About Prices: A New Approach to Understanding Price Stickiness. New York: Russell Sage Foundation. ISBN 0-87154-121-1.
↑ Vickrey W. (2008) "Marginal and Average Cost Pricing". In: Palgrave Macmillan (eds) The New Palgrave Dictionary of Economics. Palgrave Macmillan, London [ ISBN missing ]
↑ "Piana V. (2011), Refusal to sell – a key concept in Economics and Management, Economics Web Institute."
Bio, Full (2021-05-19). "Marginal Cost Of Production Definition". Investopedia . Retrieved 2021-05-28.
Nwokoye, Ebele Stella; Ilechukwu, Nneamaka (2018-08-06). "CHAPTER FIVE THEORY OF COSTS". ResearchGate. Retrieved 2021-05-28.
"Theory and Applications of Microeconomics - Table of Contents". 2012 Book Archive. 2012-12-29. Retrieved 2021-05-28.
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Closer Still #keto
It's been roughly a month, so that means another update!
First things first, I have not been so successful in keeping my calorie intake low - in fact on average it went up a bit. As I will discuss in a bit, I have been eating too much keto chocolate. I really need to keep on top of this if I want to meet my Christmas goal of 90kg.
Despite this... The other day I got my weight checked by the GP, it was 111.6kg on the 14th. I adjusted by predicted burn rate accordingly to 0.3kg loss per day. Later on the 21st I had an appointment with the Dietician in which we measured my weight again, and it perfectly tracked to 109.5kg as predicted.
This is some of the largest rate of weight loss I've had on the entire diet, including the water loss at the beginning. The question is, why? I have a few theories about this:
Exercise - I have been running, and am generally more active in any case. I've been walking a hell of a lot during these days.
Measurement - I have less "free pour" items, even though I am having an intake that is supposedly higher, everything that I do have is measured very accurately. I can say with high confidence that my daily intake must be below 1200kcal - it cannot be higher.
Hydration - Due to the use of Aeroplane Jelly, a jelly that basically has nothing in it, I have been taking on at least an extra litre of water a day. Why is that important? Well, your body needs water in order to burn fuel and during initial fat reduction, the body fills the fat cells with water temporarily. I have heard it said that it does this as a "just encase you need it again shortly" measure, but I believe it might actually be part of a process to push the fat out of the cell as water and fat do not mix.
Vitamins - During these days I am taking awful amount of vitamins, including "Keto Diet" mixes, collagen and glucosamine. I believe that generally these have allows me to be more active and have helped in raising my metabolic rate.
These of course all could be wrong, but generally I feel that they are the best guesses regarding why I would suddenly be seeing a consistent increase in weight loss. In any case I will not complain and will generally try to keep doing what I'm doing.
If this rate of loss can be maintained, at 0.3kg per day, the calculation is simple
\frac{109.5-90}{0.3}=65
days - so before December 1st! This will give me enough time to begin the process of re-introducing some carbohydrates and returning to a more long-term sustainable diet (whatever that will look like).
And now I ponder upon some previously made points:
Keto chocolate - I have been largely unsuccessful in removing keto chocolate, much to the dismay of my bank account and stomach. Each bar costs about $5 each and I can easily make two of those disappear in a day. They are also absolutely packed with sweetener and it has a laxative effect.
Keto icecrean - I have been making less of this as a result of eating more keto chocolate, I need to switch the balance back to the other way. I have now found an alternative protein powder I can use for making the icecream, so in theory I now won't run the risk of running out either.
Running - Due to an injury (I will talk about soon), I have had to stop. Before stopping I got to 8 miles and it was going pretty well.
Cognitive - I'm struggling, what can I say. I need more and more brain power, but there is not much available to me. I get tired very easily. It's hard to concentrate on one thing for too long. Once I run out of energy for the day, I just crash, and there's not much I can do about it.
Book - Progress on this is currently stalled. I may look to complete this during the Christmas break as I am way too busy right now to continue with it. I am now starting to get a lot of people interested in reading it though as they see my progress - although that won't necessarily translate as customers. A joint idea I had whilst joking with somebody was calling it "Deto", i.e. Diet Keto or Dan's Keto. Could make a great name for the book!
And now for the #keto posts over on dead social:
This is when I found out I had actually done some serious damage to my knees - when I failed to even get a lap out of myself despite showing I am capable of doing 8 miles. As I stated before, this turned out to be a little more serious.
Aeroplane Jelly turned out to be a massive success, helping to increase hydration and generally pushing back hunger. The first wasn't so successful, but it turned out that four hours wasn't nearly enough, and the jelly needs at least six hours. It's now much easier to make if I am patient.
I am really dismayed by this anti-meat movement that appears to be gaining traction. I am not sure if I could have achieved the 70-80kg loss so far without the help of animal protein (steak) and animal produce (cheese, eggs, gelatine, etc). Eating meat is the one thing I won't give up after this diet (although as I will discuss, it has not been without issue).
I am not sure if it is helping, but I have increased my weight-loss rate. I have zero idea if MCT is helping or not. It certainly hasn't helped with hunger as it claims!
Not so many for this update, but that's not to say there hasn't been more going on!
Running was going well, although I felt my knees were under a lot of stress. I always left a few days of recovery and it seemed like they were returning to normal. After the 8 mile run my knees did not return to normal, specifically the right knee.
It turns out that I had been running the same way around the block every time during lockdown level 4, the pavement was at an incline and this in turn likely caused an ITB injury:
The problem is friction where the IT band crosses over your knee. A fluid-filled sac called a bursa normally helps the IT band glide smoothly over your knee as you bend and straighten your leg.
But if your IT band is too tight, bending your knee creates friction. Your IT band and the bursa can both start to swell, which leads to the pain of IT band syndrome.
So it appears that is can be relatively easy to fix, I need to make the band more flexible, increase supporting muscles and use a better running method. Coming down in weight should also massively help with this effort.
I of course went to visit the GP and they confirmed it was screwed. On the plus side, my blood pressure is down and my weight is continuing to decrease!
The next step is to get some physiotherapy in order to repair/strengthen the related areas and relax the ITB. At the same time I will continue losing weight and all of these factors should work together in order to support recovery.
I have also been to see the Dietician again, for what will now be the last time this year.
One thing I discussed with the Dietician was my high LDL... We discussed whether I had previous data on this, but at the time none of these systems were properly connected.
My earliest health check I have results for prior to starting the diet unfortunately doesn't have results available for lipids, but I do have some information regarding my BMI from 25/09/2020:
Results: BMI
If I remember correctly, the lady making the measurement accidentally recorded it as at least 20kg less than my weight actually is as the scale looped and I believe she was too polite to check, so just added 100kg instead. My weight at the time was closed to 178kg. Sadly, for the next month and a half I continued to put on weight, and I could have been as high as 190kg at one point.
In a previous medical update and resulting article on 19/06/2021, it was shown that my LDL is quite high:
And then a following check-up and resulting article on 29/07/2021:
0001 Fasting status:Not stated
0003 Cholesterol:4.7 mmol/L HH
0005 Triglyceride:1.0 mmol/L
0007 HDL Cholesterol:1.37 mmol/L
0009 LDL cholesterol:2.9 mmol/L HH
0011 Chol/HDL Ratio:3.4
We discussed that LDL is 'bad' cholesterol and that it should ideally be kept below 1.8 mmol/L, whereas my latest results show 2.9 mmol/L. When I asked what this means, I was given the following information sheets:
Dietician advice page 1
So apparently an excess of LDL can mean that it gets built up on my arteries, ouch.
Apparently the types of fats that can increase LDL levels is animal fats and cheese - which is basically my main source of fats. We generally agreed that the benefits of the keto diet far outweigh the negatives.
Looking online, apparently high LDL is a known problem on a ketogenic diet:
With a well-formulated ketogenic diet, we see a shift away from the small dangerous LDL even when the total LDL goes up, so most of this increase is in the ‘good’ or ‘buoyant’ LDL fraction (Hallberg, 2018).
So even though I may have more LDL, apparently it is more likely to be the good kind of LDL.
Another factor to be taken into account is that during rapid weight loss, cholesterol that you had stored in your adipose tissue (ie, body fat) is mobilized as the fat cells shrink (Phinney 1990). This will artificially raise serum LDL as long as the weight loss continues, but it then comes back down once weight loss stops.
Which again, is exactly where I fall into. In fact, the weight loss could explain entirely my excess LDL.
To avoid being misled by this, the best strategy is to hold off checking blood lipids until a couple of months after weight loss ceases.
This looks like something I should address again after Christmas and see where I'm at, perhaps in February 2022. I would suspect all of my values to fall back well within range, especially as then I should be able to start hitting some proper exercise with the use of carbohydrates.
One thing we discussed again is the process of coming down from the diet. It really won't be easy and for keto they have absolutely not clue (as they don't train about it and never suggest anybody to do it). That said, they still maintain that my progress has been some of the best they have ever seen - so something must be working.
According to one source, there appears to be a few tips:
Take it slow with carbs.
So essentially I run the risk of my body not being able to control its blood sugar effectively - and this can have all kinds of crazy side effects. I need to reintroduce carbohydrates slowly and carefully.
Choose high-fiber foods.
I generally agree with the article on whole grains, beans and vegetables - but not fruits. I want to avoid fructose and generally anything at all high in sugar, particularly refined sugar.
My intention as I come out of the diet is to up vegetables as the main source of carbohydrates, as these are high in fibre are and generally more bulky. I can also keep protein raised whilst I slowly transition in the other foods.
As I mentioned to many people now, my intention is to run a marathon next year.
Essentially this last point is a bit of a nothing-burger, but could be interpreted as "don't repeat your previous bad habits" - which is a great point. My intention is to form healthy habits from the start.
One thing I wanted to discuss with the Dietician but wasn't able to was difficult to was giving up the control. One thing you learn to love is the highly detailed control you have over your diet, it's addictive. Last time I did a diet even remotely comparable I almost couldn't give up control. This is something I will definitely have to battle with when the time comes.
One thing we discussed (almost annoyingly) is what I will have when I come off. My goal is to reach 90kg and I will keep going until this is achieved. I said that one thing keto people struggle with is bread, and then we ended up discussing bread for some time. It's a good job I was not hungry in that moment and even better that there was no option for fresh bread available.
When I reach this goal, my intention is to have scrambled egg on wholegrain toast on Christmas day. Bread is the one thing I really miss, although it will have to be researched and measured.
I am looking forward to being able to stop tracking all the macros and to just be able to eyeball foods. It's depressing to even be offered something keto but not be able to try it as I have no way to accurately logging it.
The next parts are my goals:
Fix knee - I want to be able to run again, I was actually quite enjoying exerting myself before my knee went. I need to book the physiotherapy and get this sorted.
Calories - As I get close to my goal, calories will play a larger and larger part in the final weight loss. There will be less and less fat freely available and I will really need to force my body to play ball.
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Courses/CcmpTylaLogIng2016 (Ing1 students) - LRDE
the Compiler Construction Course 2 (CMP2), and
for Ing1 students of class EPITA 2016 (i.e., from December 2013 to June 2014). The topic was started with the Formal Languages Lecture (THL).
Lecture 1: 2013-12-09 (Grp. B & A), 2 hours: Introduction to the Tiger Project
Introduction to The Tiger Project. See the lecture notes: tiger-project-intro.pdf, tiger-project-intro-handout.pdf and tiger-project-intro-handout-4.pdf.
Assignments (http://www.lrde.epita.fr/~tiger/assignments.html).
Appel's books.
Tiger Compiler Reference Manual (http://www.lrde.epita.fr/~tiger/tiger.html).
epita.cours.compile.
Goals (C++, OO, DP, Management, Several Iterations, Testing, Documenting, Maintaining, Fixing, Understanding Computers, English).
Non goals (Compiler Construction).
No copy between groups.
Tests are part of the project (test cases and frameworks should not be exchanged).
Fixing mistakes earlier is better.
Work between groups is encouraged as long as they don't cheat.
Tests matter.
A bug => a test.
A suspicious behavior => one or several tests to isolate it.
Don't throw away tests!
Don't exchange tests! (bis repetita).
Lecture 2: 2013-12-16 (Grp. B & A), 2 hours: Introduction to the Tiger Project (cont.), Architecture of tc (tasks), Autotools
Misc: students should overcome Make, Makefiles and separate compilation, etc.
Architecture of the Tiger Compiler (tc).
Modules: parse, ast, bind, etc.
Pure libraries providing actual services.
Tasks (non-pure services) using a declarative system: Development Tools, section 1 (tc Tasks). See the lecture notes: dev-tools.pdf, dev-tools-handout.pdf and dev-tools-handout-4.pdf.
Declaration of dependencies.
Actual computations delegated to pure libraries.
Driver (tc.cc).
Instantiates a tasks manager, used to record all existing tasks at start-up and later compute the steps to performe according to the dependencies of the invoked tasks.
Workflow computed by the task manager from the options passed to the driver, triggering corresponding tasks and their dependencies ( la Make).
Error management: catches exceptions (including misc::error) and displays error messages.
Lectures 3 : 2014-02-17 (Grp. A & B), 2 hours: Autotools (cont.), Development Tools
Autoconf and Automake (cont.)
Hands-on example (cont.)
TESTS in Makefile.am and make check.
Compiling test-only (not installed) programs: check_PROGRAMS.
make uninstall (careful, very limited).
Generating tests with configure (e.g. so that they can find $srcdir).
Changing variables (e.g. CXXFLAGS) at the configuration step (=./configure CXXFLAGS...=; global) or at the build step (=make CXXFLAGS...=; local).
autoheader and config.h: getting rid of limitations of using -D options to pass options to the compiler.
Pros and cons of using multiple Makefile s in a multi-directory project.
srcdir vs builddir, running configure from a directory other than the source dir.
Development Tools. See the lecture notes: dev-tools.pdf, dev-tools-handout.pdf and dev-tools-handout-4.pdf.
Lecture 4: 2014-02-21 (Grp. A & B), 2 hours: Scanner and Parser Hints
Additional details and hints about the scanner and the parser: The Scanner and the Parser. See the lecture notes: scanner.pdf, scanner-handout.pdf and scanner-handout-4.pdf.
Symbols (light-weight, shared and non-mutable strings used to represent identifiers).
Extra information (in addition to tokens/terminals) passed between the scanner and the parser.
Semantic values.
Various improvements on the scanner and the parser.
Error recovery by deletion (using the error symbol).
Pure (reentrant) parser and scanner.
Lecture 5: 2014-02-24 (Grp. A & B), 2 hours: Abstract Syntax
Abstract Syntax, up to basic Visitors (sec. 2.3, p. 51). See the lecture notes, ast.pdf, ast-handout.pdf and ast-handout-4.pdf.
Lecture 6: 2014-02-25 (Grp. B & A), 2 hours: Abstract Syntax [in lieu of TYLA]
Lecture 7: 2013-03-04 (Grp. B & A), 2 hours: Names, Identifiers and Bindings
Names, Identifiers and Bindings, up to Scoped Symbol Table Implementations. See the lecture notes (section 1 and 2, unfinished), names.pdf, names-handout.pdf and names-handout-4.pdf.
TC's Binder explained.
Traversal and use of the symbol tables.
Chunks (ast::TypeDecs, ast::FunctionDecs) and how they are processed by bind::Binder.
Lecture 1: 2014-03-14 (Grp. A & B), 2 hours: Names, Identifiers and Bindings, Type-checking
Names, Identifiers and Bindings: Complications. See the lecture notes (section 3), names.pdf, names-handout.pdf and names-handout-4.pdf.
Types. See the lecture notes (sections 1 and intro to section 2): type-checking.pdf, type-checking-handout.pdf and type-checking-handout-4.pdf.
Lecture 2: 2014-03-18 (Grp. B & A), 2 hours: Type-checking
Types. See the lecture notes (up to the end): type-checking.pdf, type-checking-handout.pdf and type-checking-handout-4.pdf.
Some details on the implementation of types and type-checking within the Tiger Compiler.
Hierarchy of types (src/type/)
src/type/README.
Implementing atomic types: singletons.
Resolving aliased types: Named and actual().
In English: "If alpha is of type Int in the context Gamma and beta is of type Int in the context Gamma, then alpha + beta is of type Int in the context Gamma."
Using symbols (where ⊢ ("tee" or "turnstile") is the symbol meaning "yields" or "proves"):
{\displaystyle {\frac {\Gamma \vdash \alpha :Int\quad \Gamma \vdash \alpha :Int}{\Gamma \vdash \alpha +\beta :Int}}}
Examples of type rules: addition of 2 integers, if-then-else, if-then, addition of 3 integers, comparison of two variables, etc.
(a ? b : c) > 0
(a ? b : f(b)) > 0
(a ? b : f(c)) > 0
Lecture 3: 2014-04-28 (Grp. B) & 2014-04-29 (Grp. A), 2 hours: Intermediate languages
Intermediate languages. See the lecture notes (up to section 4.1 (included), except details about intermediate representations (sec. 1.1), details dynamic allocation (sec 2.1), non-local variables & static link (sec. 2.2)): intermediate.pdf, intermediate-handout.pdf and intermediate-handout-4.pdf.
Lecture 4: 2014-05-26 (Grp. B) & 2014-05-27 (Grp. A), 2 hours: Intermediate languages (cont.), Canonization
Intermediate languages: details about intermediate representations (sec. 1.1), details dynamic allocation (sec 2.1), non-local variables & static link (sec. 2.2). See the lecture notes: intermediate-handout.pdf and intermediate-handout-4.pdf.
Canonization. See the lecture notes (section 4.2): intermediate.pdf, intermediate-handout.pdf and intermediate-handout-4.pdf.
Lecture 5: 2014-06-06 (Grp. B & A), 2 hours: Microprocessors, Instruction Selection, Introduction to Liveness Analysis
Microprocessors, Instruction Selection. See the lecture notes: instr-selection.pdf, instr-selection-handout.pdf and instr-selection-handout-4.pdf.
Examples of MonoBURG input files from the Tiger Compiler (Tree to MIPS).
Liveness Analysis. See the lecture notes: liveness.pdf, liveness-handout.pdf and liveness-handout-4.pdf.
Lecture 6: 2014-06-09 (Grp. B) & 2014-06-10 (Grp. A), 2 hours: Liveness Analysis, Register Allocation
Register Allocation: Coloring by Simplification, Spilling, Coalescing, Precolored Nodes, Caller- and Callee-Saved Registers, Implementation, Alternatives to Graph Coloring. See the lecture notes: regalloc.pdf, regalloc-handout.pdf and regalloc-handout-4.pdf.
Lecture 1: 2014-02-27 (Grp. A & B), 2 hours: History of Computing
Lecture 2: 2014-03-07 (Grp. B & A), 2 hours: History of Computing and Programming Languages
History of Programming Languages: The Very First Ones: FORTRAN, Algol, COBOL (Section 1). See the lecture notes, early-languages.pdf, early-languages-handout.pdf and early-languages-handout-4.pdf.
Lecture 3: 2014-03-11 (Grp. B & A), 2 hours: History of Programming Languages, Object-Oriented History
History of Programming Languages: The Second Wave: APL, PL/I, BASIC, Pascal & Heirs (Section 2) and The Finale (Section 3). See the lecture notes, early-languages.pdf, early-languages-handout.pdf and early-languages-handout-4.pdf.
Object-Oriented History: Simula. See the lecture notes, object.pdf, object-handout.pdf and object-handout-4.pdf.
Lecture 4: 2014-03-19 (Grp. B & A), 2 hours: Object-Oriented History: Smalltalk
Object-Oriented History: Smalltalk. See the lecture notes, object.pdf, object-handout.pdf and object-handout-4.pdf.
Lecture 5: 2014-03-25 (Grp. B & A), 2 hours: Some Traits of Functional Programming Languages, Generic Programming
Object-Oriented History: Smalltalk, C++. See the lecture notes, object.pdf, object-handout.pdf and object-handout-4.pdf.
Generic Programming, except Template Metaprogramming (section 4.2). See the lecture notes, generic.pdf, generic-handout.pdf and generic-handout-4.pdf.
Subprograms (sections 1 and 2). See the lecture notes, subprograms.pdf, subprograms-handout.pdf and subprograms-handout-4.pdf.
Retrieved from "https://www.lrde.epita.fr/index.php?title=Courses/CcmpTylaLogIng2016_(Ing1_students)&oldid=8230"
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Dynamic Programming | Brilliant Math & Science Wiki
Agnishom Chattopadhyay, Lien Michiels, Daniel Park, and
Dynamic programming refers to a problem-solving approach, in which we precompute and store simpler, similar subproblems, in order to build up the solution to a complex problem. It is similar to recursion, in which calculating the base cases allows us to inductively determine the final value. This bottom-up approach works well when the new value depends only on previously calculated values.
An important property of a problem that is being solved through dynamic programming is that it should have overlapping subproblems. This is what distinguishes DP from divide and conquer in which storing the simpler values isn't necessary.
To show how powerful the technique can be, here are some of the most famous problems commonly approached through dynamic programming:
Backpack Problem: Given a set of treasures with known values and weights, which of them should you pick to maximize your profit whilst not damaging your backpack which has a fixed capacity?
Egg Dropping: What is the best way to drop
n
eggs from an
m
-floored building to figure out the lowest height from which the eggs when dropped crack?
Longest Common Subsequence: Given two sequences, which is the longest subsequence common to both of them?
Subset Sum Problem: Given a set and a value
n,
is there a subset the sum of whose elements is
n?
Fibonacci Numbers: Is there a better way to compute Fibonacci numbers than plain recursion?
In a contest environment, dynamic programming almost always comes up (and often in a surprising way, no matter how familiar the contestant is with it).
Motivational Example: Change of Coins
Bidimensional Dynamic Programming: Example
Example: Maximum Paths
What is the minimum number of coins of values
v_1,v_2, v_3, \ldots, v_n
required to amount a total of
V?
You may use a denomination more than once.
The most important aspect of this problem that encourages us to solve this through dynamic programming is that it can be simplified to smaller subproblems.
f(N)
represent the minimum number of coins required for a value of
N
f(N)
as a stack of coins. What is the coin at the top of the stack? It could be any of
v_1,v_2, v_3, \ldots, v_n
. In case it were
v_1
, the rest of the stack would amount to
N-v_1;
or if it were
v_2
N-v_2
How do we decide which is it? Sure enough, we do not know yet. We need to see which of them minimizes the number of coins required.
Going by the above argument, we could state the problem as follows:
f(V) = \min \Big( \big\{ 1 + f(V - v_1), 1 + f(V-v_2), \ldots, 1 + f(V-v_n) \big \} \Big).
Why is there a +1?
Because the coin at the top of the stack also counts as one coin, and then we can look at the rest.
It is easy to see that the subproblems could be overlapping.
For example, if we are trying to make a stack of $11 using $1, $2, and $5, our look-up pattern would be like this:
\begin{aligned} f(11) &= \min \Big( \big\{ 1+f(10),\ 1+ f(9),\ 1 + f(6) \big\} \Big) \\ &= \min \Big ( \big \{ 1+ \min {\small \left ( \{ 1 + f(9), 1+ f(8), 1+ f(5) \} \right )},\ 1+ f(9),\ 1 + f(6) \big \} \Big ). \end{aligned}
Clearly enough, we'll need to use the value of
f(9)
several times.
One of the most important aspects of optimizing our algorithms is that we do not recompute these values. To do this, we compute and store all the values of
from 1 onwards for potential future use.
The recursion has to bottom out somewhere, in other words, at a known value from which it can start.
For this problem, we need to take care of two things:
Zero: It is clear enough that
f(0) = 0
since we do not require any coins at all to make a stack amounting to 0.
Negative and Unreachable Values: One way of dealing with such values is to mark them with a sentinel value so that our code deals with them in a special way. A good choice of a sentinel is
\infty
, since the minimum value between a reachable value and
\infty
could never be infinity.
Let's sum up the ideas and see how we could implement this as an actual algorithm:
# V = the value we want, v=the list of available denomenations
def coinsChange(V,v):
dpTable = [float("inf")]*(V+1)
for i in xrange(1,V+1):
if (i - vi) >= 0:
dpTable[i] = min(dpTable[i],1+dpTable[i-vi])
return dpTable[V]
We have claimed that naive recursion is a bad way to solve problems with overlapping subproblems. Why is that? Mainly because of all the recomputations involved.
Another way to avoid this problem is to compute the data first time and store it as we go, in a top-down fashion.
Let's look at how one could potentially solve the previous coin change problem in the memoization way.
if V in memo:
memo[V] = min([1+Change(V-vi) for vi in v])
return Change(V)
Dynamic Programming vs Recursion with Caching
\hspace{20mm}
Recursion with Caching
Faster if many sub-problems are visited as there is no overhead from recursive calls
\hspace{20mm}
Intuitive approach
The complexity of the program is easier to see
\hspace{20mm}
Computes only those subproblems which are necessary
k
types of brackets each with its own opening bracket and closing bracket. We assume that the first pair is denoted by the numbers 1 and
k+1,
the second by 2 and
k+2,
and so on. Thus the opening brackets are denoted by
1, 2, \ldots, k,
and the corresponding closing brackets are denoted by
k+1, k+2, \ldots, 2k,
Some sequences with elements from
1, 2, \ldots, 2k
form well-bracketed sequences while others don't. A sequence is well-bracketed if we can match or pair up opening brackets of the same type in such a way that the following holds:
Every bracket is paired up.
In each matched pair, the opening bracket occurs before the closing bracket.
For a matched pair, any other matched pair lies either completely between them or outside them.
In this problem, you are given a sequence of brackets of length
N
B[1], \ldots, B[N]
B[i]
is one of the brackets. You are also given an array of Values:
V[1],\ldots, V[N]
Among all the subsequences in the Values array, such that the corresponding bracket subsequence in the B Array is a well-bracketed sequence, you need to find the maximum sum.
Task: Solve the above problem for this input.
One line, which contains
(2\times N + 2)
space separate integers. The first integer denotes
N.
The next integer is
k.
N
integers are
V[1],..., V[N].
N
B[1],..., B[N].
1 \leq k \leq 7
-10^6 \leq V[i] \leq 10^6
i
1 \leq B[i] \leq 2k
i
For the examples discussed here, let us assume that
k = 2
. The sequence 1, 1, 3 is not well-bracketed as one of the two 1's cannot be paired. The sequence 3, 1, 3, 1 is not well-bracketed as there is no way to match the second 1 to a closing bracket occurring after it. The sequence 1, 2, 3, 4 is not well-bracketed as the matched pair 2, 4 is neither completely between the matched pair 1, 3 nor completely outside of it. That is, the matched pairs cannot overlap. The sequence 1, 2, 4, 3, 1, 3 is well-bracketed. We match the first 1 with the first 3, the 2 with the 4, and the second 1 with the second 3, satisfying all the 3 conditions. If you rewrite these sequences using [, {, ], } instead of 1, 2, 3, 4 respectively, this will be quite clear.
N = 6, k = 3,
V
B
are as follows: Then, the brackets in positions 1, 3 form a well-bracketed sequence (1, 4) and the sum of the values in these positions is 2 (4 + (-2) =2). The brackets in positions 1, 3, 4, 5 form a well-bracketed sequence (1, 4, 2, 5) and the sum of the values in these positions is 4. Finally, the brackets in positions 2, 4, 5, 6 form a well-bracketed sequence (3, 2, 5, 6) and the sum of the values in these positions is 13. The sum of the values in positions 1, 2, 5, 6 is 16 but the brackets in these positions (1, 3, 5, 6) do not form a well-bracketed sequence. You can check the best sum from positions whose brackets form a well-bracketed sequence is 13.
We'll try to solve this problem with the help of a dynamic program, in which the state, or the parameters that describe the problem, consist of two variables.
First, we set up a two-dimensional array dp[start][end] where each entry solves the indicated problem for the part of the sequence between start and end inclusive.
We'll try to think what happens when we run across a new end value, and need to solve the new problem in terms of the previously solved subproblems. Here are all the possibilities:
When end <= start, there are no valid subsequences.
When b[end] <= k, i.e, the last entry is an open bracket, no valid subsequence can end with it. Effectively, the result is the same if we hadn't included the last entry at all.
When b[end] > k, i.e, the last entry is a closing bracket, one has to find the best match for it, or simply ignore it, whichever maximizes the sum.
Can you use these ideas to solve the problem?
Very often, dynamic programming helps solve problems that ask us to find the most profitable (or least costly) path in an implicit graph setting. Let us try to illustrate this with an example.
You are supposed to start at the top of a number triangle and chose your passage all the way down by selecting between the numbers below you to the immediate left or right.
Your goal is to maximize the sum of the elements lying in your path.
For example, in the triangle below, the red path maximizes the sum.
To see the optimal substructures and the overlapping subproblems, notice that everytime we make a move from the top to the bottom right or the bottom left, we are still left with smaller number triangle, much like this:
We could break each of the sub-problems in a similar way until we reach an edge-case at the bottom:
In this case, the solution is a + max(b,c).
A bottom-up dynamic programming solution is to allocate a number triangle that stores the maximum reachable sum if we were to start from that position. It is easy to compute the number triangles from the bottom row onward using the fact that the
\text{best from this point} = \text{this point} + \max(\text{best from the left, best from the right}).
Let me demonstrate this principle through the iterations.
So, the effective best we could do from the top is 23, which is our answer.
Cite as: Dynamic Programming. Brilliant.org. Retrieved from https://brilliant.org/wiki/problem-solving-dynamic-programming/
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Long s - Wikipedia
Archaic form of the Latin letter S (ſ)
This article is about ſ, the archaic variant of the letter "s". For the letter ʃ as used in Latin script, see Esh (letter). For the mathematical symbol ∫, see Integral symbol.
You must provide copyright attribution in the edit summary accompanying your translation by providing an interlanguage link to the source of your translation. A model attribution edit summary is Content in this edit is translated from the existing German Wikipedia article at [[:de:Langes s]]; see its history for attribution.
You should also add the template {{Translated|de|Langes s}} to the talk page.
2.1.1 Abandonment by printers and type founders
2.1.2 Eventual abandonment in handwriting
3.1 In Unicode
4 Shilling mark
Find sources: "Long s" – news · newspapers · books · scholar · JSTOR (October 2019) (Learn how and when to remove this template message)
{\displaystyle {\mathfrak {z}}}
{\displaystyle {\mathfrak {z}}}
Abandonment by printers and type founders[edit]
Eventual abandonment in handwriting[edit]
{\displaystyle \int _{a}^{b}f(x)\,dx.}
Shilling mark[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Long_s&oldid=1087816767"
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Water - Ring of Brodgar
This page is about the collectable resource. For the terrain, see Water Terrain
Object(s) Required Any Container
Produced By Well, Deep Water, Shallow Water
Specific Type of Water
Required By Bark Cordage, Barley Wort, Blueberry Pie Dough, Bread Dough, Bush, Cattail Stew, Clay Cuckoo, Dream Cookies Dough, Fruit Sorbet, Gingerbread Dough, Grub Pie Dough, Honeybun Dough, Lather, Mud Ointment, Mushroom Pie Dough, Pea Pie Dough, Polished Porphyry Beads, Pumpkin Bread Dough, Pumpkin Pie Dough, Ring of Brodgar Dough, Rustroot Extract, Tanning Fluid, Taproot Lacing, Tea, Tree, Unbaked Apple Pie, Unbaked Bark Bread, Unbaked Butter Scones, Unbaked Carrot Cake, Unbaked Cheesecake, Unbaked Egg Cake, Unbaked Fishpie, Unbaked Fruit Pie, Unbaked Greenleaf Pie, Unbaked Jelly Cake, Unbaked Lardy Cake, Unbaked Lingon Loaf, Unbaked Linseed Loaf, Unbaked Magpie (Pie), Unbaked Marrow Cake, Unbaked Meatpie, Unbaked Olive Bread, Unbaked Onion Bread, Unbaked Poultry Pot Pie, Unbaked Raisin Butter-Cake, Unbaked Seedcrisp Flatbread, Unbaked Shepherd's Pie, Unbaked Strawberry Cake, Unfired Hand Impression, Wheat Wort
Water is a resource that requires a container that can hold liquid and is used for many forms of crafting. Most water is quality 10. Occasionally parts of the map will have higher quality Water nodes. Because if this, high quality water is fairly valued and may be traded as commodity. Water is one factor in obtaining high quality trees. For more information on finding high quality water, see Finding high quality water, clay, and soil.
Water can also be used to regain stamina.
Quality 10 water recovers 10% stamina and drains 20% energy per 0.05L sip. Higher quality water will decrease the energy drain but stamina recovery remains the same at all qualities.
This makes high quality water useful for performing stamina draining tasks without having to eat as much to recover, thus preserving your hunger bonus. Though energy is always shown as a whole number, it is not simply rounded up or down. For example, Q11 water will alternate between draining 20% and 19% energy at regular intervals. Other drinkable liquids follow the same rules and formula, making them decent substitutes for water if you have a surplus since it is much easier to increase their quality.
Energy Drain =
{\displaystyle ({\frac {1}{\sqrt {quality/10}}}+1)*10}
Water can be acquired from a Well or from any Water tile (left click on container and then right click on water).
Note: To get the water discovery for a character, one needs to get some water directly from a fresh water source. Take water from a well, lake, river, or distill Salt Water in a Still. Taking water stored in other containers will not give the water discovery.
All containers can be used to collect, move, drink, and store water.
Generally, a Kuksa, Waterskin, or Waterflask is used to carry water on your character to regain stamina while travelling.
If large quantities of water are needed, carrying a Bucket to drink from is also an option.
For a list of containers which can hold liquid, visit the liquids page.
Rocking Lemon (2021-08-08) >"You can now distill "Salt Water" to normal "Water"."
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Water&oldid=92760"
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Dehydration reaction - Wikipedia
Chemical reaction in which water is produced as a byproduct
This article is about chemical reactions resulting in the loss of water from a molecule. For the removal of water from solvents and reagents, see Desiccation.
1 Dehydration reactions in organic chemistry
1.3 Nitrile formation
1.4 Ketene formation
1.5 Alkene formation
2 Dehydration reactions in inorganic chemistry
Dehydration reactions in organic chemistry[edit]
Esterification[edit]
Etherification[edit]
Nitrile formation[edit]
Ketene formation[edit]
Alkene formation[edit]
Dehydration reactions in inorganic chemistry[edit]
{\displaystyle {\ce {CaSO4.2H2O +}}}
{\displaystyle {\ce {->}}}
{\displaystyle {\ce {CaSO4.1/2H2O + 1 1/2H2O}}}
(released as steam).
Retrieved from "https://en.wikipedia.org/w/index.php?title=Dehydration_reaction&oldid=1083210708"
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Oberth_effect Knowpia
In astronautics, a powered flyby, or Oberth maneuver, is a maneuver in which a spacecraft falls into a gravitational well and then uses its engines to further accelerate as it is falling, thereby achieving additional speed.[1] The resulting maneuver is a more efficient way to gain kinetic energy than applying the same impulse outside of a gravitational well. The gain in efficiency is explained by the Oberth effect, wherein the use of a reaction engine at higher speeds generates a greater change in mechanical energy than its use at lower speeds. In practical terms, this means that the most energy-efficient method for a spacecraft to burn its fuel is at the lowest possible orbital periapsis, when its orbital velocity (and so, its kinetic energy) is greatest.[1] In some cases, it is even worth spending fuel on slowing the spacecraft into a gravity well to take advantage of the efficiencies of the Oberth effect.[1] The maneuver and effect are named after the person who first described them in 1927, Hermann Oberth, an Austro-Hungarian-born German physicist and a founder of modern rocketry.[2]
The Oberth effect is strongest at the periapsis, where the gravitational potential is lowest, and the speed is highest. This is because a given firing of a rocket engine at high speed causes a greater change in kinetic energy than when fired otherwise similarly at lower speed.
Because the vehicle remains near periapsis only for a short time, for the Oberth maneuver to be most effective the vehicle must be able to generate as much impulse as possible in the shortest possible time. As a result the Oberth maneuver is much more useful for high-thrust rocket engines like liquid-propellant rockets, and less useful for low-thrust reaction engines such as ion drives, which take a long time to gain speed. The Oberth effect also can be used to understand the behavior of multi-stage rockets: the upper stage can generate much more usable kinetic energy than the total chemical energy of the propellants it carries.[2]
In terms of the energies involved, the Oberth effect is more effective at higher speeds because at high speed the propellant has significant kinetic energy in addition to its chemical potential energy.[2]: 204 At higher speed the vehicle is able to employ the greater change (reduction) in kinetic energy of the propellant (as it is exhausted backward and hence at reduced speed and hence reduced kinetic energy) to generate a greater increase in kinetic energy of the vehicle.[2]: 204
Explanation in terms of momentum and kinetic energyEdit
A rocket works by transferring momentum to its propellant.[3] At a fixed exhaust velocity, this will be a fixed amount of momentum per unit of propellant.[4] For a given mass of rocket (including remaining propellant), this implies a fixed change in velocity per unit of propellant. Because kinetic energy equals mv2/2, this change in velocity imparts a greater increase in kinetic energy at a high velocity than it would at a low velocity. For example, considering a 2 kg rocket:
at 1 m/s, adding 1 m/s increases the kinetic energy from 1 J to 4 J, for a gain of 3 J;
at 10 m/s, starting with a kinetic energy of 100 J, the rocket ends with 121 J, for a net gain of 21 J.
This greater change in kinetic energy can then carry the rocket higher in the gravity well than if the propellant were burned at a lower speed.
Description in terms of workEdit
This is shown as follows. The mechanical work done on the rocket (
{\displaystyle W}
) is defined as the dot product of the force of the engine's thrust (
{\displaystyle {\vec {F}}}
) and the displacement it travels during the burn (
{\displaystyle {\vec {s}}}
{\displaystyle W={\vec {F}}\cdot {\vec {s}}.}
If the burn is made in the prograde direction,
{\displaystyle {\vec {F}}\cdot {\vec {s}}=\|F\|\cdot \|s\|=F\cdot s}
. The work results in a change in kinetic energy
{\displaystyle \Delta E_{k}=F\cdot s.}
Differentiating with respect to time, we obtain
{\displaystyle {\frac {\mathrm {d} E_{k}}{\mathrm {d} t}}=F\cdot {\frac {\mathrm {d} s}{\mathrm {d} t}},}
{\displaystyle {\frac {\mathrm {d} E_{k}}{\mathrm {d} t}}=F\cdot v,}
{\displaystyle v}
is the velocity. Dividing by the instantaneous mass
{\displaystyle m}
to express this in terms of specific energy (
{\displaystyle e_{k}}
{\displaystyle {\frac {\mathrm {d} e_{k}}{\mathrm {d} t}}={\frac {F}{m}}\cdot v=a\cdot v,}
{\displaystyle a}
is the acceleration vector.
Thus it can be readily seen that the rate of gain of specific energy of every part of the rocket is proportional to speed and, given this, the equation can be integrated (numerically or otherwise) to calculate the overall increase in specific energy of the rocket.
Impulsive burnEdit
Integrating the above energy equation is often unnecessary if the burn duration is short. Short burns of chemical rocket engines close to periapsis or elsewhere are usually mathematically modeled as impulsive burns, where the force of the engine dominates any other forces that might change the vehicle's energy over the burn.
For example, as a vehicle falls toward periapsis in any orbit (closed or escape orbits) the velocity relative to the central body increases. Briefly burning the engine (an “impulsive burn”) prograde at periapsis increases the velocity by the same increment as at any other time (
{\displaystyle \Delta v}
). However, since the vehicle's kinetic energy is related to the square of its velocity, this increase in velocity has a non-linear effect on the vehicle's kinetic energy, leaving it with higher energy than if the burn were achieved at any other time.[5]
Oberth calculation for a parabolic orbitEdit
If an impulsive burn of Δv is performed at periapsis in a parabolic orbit, then the velocity at periapsis before the burn is equal to the escape velocity (Vesc), and the specific kinetic energy after the burn is[6]
{\displaystyle {\begin{aligned}e_{k}&={\tfrac {1}{2}}V^{2}\\&={\tfrac {1}{2}}(V_{\text{esc}}+\Delta v)^{2}\\&={\tfrac {1}{2}}V_{\text{esc}}^{2}+\Delta vV_{\text{esc}}+{\tfrac {1}{2}}\Delta v^{2},\end{aligned}}}
{\displaystyle V=V_{\text{esc}}+\Delta v}
When the vehicle leaves the gravity field, the loss of specific kinetic energy is
{\displaystyle {\tfrac {1}{2}}V_{\text{esc}}^{2},}
so it retains the energy
{\displaystyle \Delta vV_{\text{esc}}+{\tfrac {1}{2}}\Delta v^{2},}
which is larger than the energy from a burn outside the gravitational field (
{\displaystyle {\tfrac {1}{2}}\Delta v^{2}}
{\displaystyle \Delta vV_{\text{esc}}.}
When the vehicle has left the gravity well, it is traveling at a speed
{\displaystyle V=\Delta v{\sqrt {1+{\frac {2V_{\text{esc}}}{\Delta v}}}}.}
For the case where the added impulse Δv is small compared to escape velocity, the 1 can be ignored, and the effective Δv of the impulsive burn can be seen to be multiplied by a factor of simply
{\displaystyle {\sqrt {\frac {2V_{\text{esc}}}{\Delta v}}}}
and one get
{\displaystyle V}
{\displaystyle {\sqrt {{2V_{\text{esc}}}{\Delta v}}}.}
Similar effects happen in closed and hyperbolic orbits.
Parabolic exampleEdit
If the vehicle travels at velocity v at the start of a burn that changes the velocity by Δv, then the change in specific orbital energy (SOE) due to the new orbit is
{\displaystyle v\,\Delta v+{\tfrac {1}{2}}(\Delta v)^{2}.}
Once the spacecraft is far from the planet again, the SOE is entirely kinetic, since gravitational potential energy approaches zero. Therefore, the larger the v at the time of the burn, the greater the final kinetic energy, and the higher the final velocity.
The effect becomes more pronounced the closer to the central body, or more generally, the deeper in the gravitational field potential in which the burn occurs, since the velocity is higher there.
So if a spacecraft is on a parabolic flyby of Jupiter with a periapsis velocity of 50 km/s and performs a 5 km/s burn, it turns out that the final velocity change at great distance is 22.9 km/s, giving a multiplication of the burn by 4.58 times.
It may seem that the rocket is getting energy for free, which would violate conservation of energy. However, any gain to the rocket's kinetic energy is balanced by a relative decrease in the kinetic energy the exhaust is left with (the kinetic energy of the exhaust may still increase, but it does not increase as much).[2]: 204 Contrast this to the situation of static firing, where the speed of the engine is fixed at zero. This means that its kinetic energy does not increase at all, and all the chemical energy released by the fuel is converted to the exhaust's kinetic energy (and heat).
At very high speeds the mechanical power imparted to the rocket can exceed the total power liberated in the combustion of the propellant; this may also seem to violate conservation of energy. But the propellants in a fast-moving rocket carry energy not only chemically, but also in their own kinetic energy, which at speeds above a few kilometres per second exceed the chemical component. When these propellants are burned, some of this kinetic energy is transferred to the rocket along with the chemical energy released by burning.[7]
The Oberth effect can therefore partly make up for what is extremely low efficiency early in the rocket's flight when it is moving only slowly. Most of the work done by a rocket early in flight is "invested" in the kinetic energy of the propellant not yet burned, part of which they will release later when they are burned.
^ a b c Robert B. Adams, Georgia A. Richardson (25 July 2010). Using the Two-Burn Escape Maneuver for Fast Transfers in the Solar System and Beyond (PDF) (Report). NASA. Archived (PDF) from the original on 11 February 2022. Retrieved 15 May 2015.
^ a b c d e Hermann Oberth (1970). "Ways to spaceflight". Translation of the German language original "Wege zur Raumschiffahrt," (1920). Tunis, Tunisia: Agence Tunisienne de Public-Relations.
^ What Is a Rocket? 13 July 2011/ 7 August 2017 www.nasa.gov, accessed 9 January 2021.
^ Rocket thrust 12 June 2014, www.grc.nasa.gov, accessed 9 January 2021.
^ Atomic Rockets web site: nyrath@projectrho.com. Archived July 1, 2007, at the Wayback Machine
^ Following the calculation on rec.arts.sf.science.
^ Blanco, Philip; Mungan, Carl (October 2019). "Rocket propulsion, classical relativity, and the Oberth effect". The Physics Teacher. 57 (7): 439–441. Bibcode:2019PhTea..57..439B. doi:10.1119/1.5126818.
Explanation of the effect by Geoffrey Landis.
Animation (MP4) of the Oberth effect in orbit from the Blanco and Mungan paper cited above.
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Week 13. Memory | Algorithms and Data Structures
14 Week 13. Memory
Reading 13 Goodrich & Tamassia: Chapter 14
Problem 14.1 Describe an external-memory data structure to implement the double-ended queue ADT so that the total number of disk transfers needed to process a sequence of
k
push and pop operations is
O\left(k∕B\right)
B
is the block size, i.e. number of elements fitting in a block.
Problem 14.2 The B-tree as described in the textbook uses the map ADT to hold the contents of each node. The time to retrieve an element from this data structure is described as
f\left(d\right)
d
is the maximum number of elements in the map. For a map,
f\left(d\right)=O\left(1\right)
Suppose we change this constituent data type to a sorted tree. What would be the resulting value of
f\left(d\right)
? And how does it affect the total run time of get() in the B-tree.
Problem 14.3 Describe an external-memory data structure to implement the stack ADT so that the total number of disk transfers needed to process a sequence of
k
O\left(k∕B\right)
B
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Newton_(unit) Knowpia
The newton (symbol: N) is the International System of Units (SI) derived unit of force. It is defined as 1 kg⋅m/s2, the force which gives a mass of 1 kilogram an acceleration of 1 metre per second per second. It is named after Isaac Newton in recognition of his work on classical mechanics, specifically Newton's second law of motion.
A newton is defined as 1 kg⋅m/s2 (it is a derived unit which is defined in terms of the SI base units).[1] One newton is therefore the force needed to accelerate one kilogram of mass at the rate of one metre per second squared in the direction of the applied force.[2] The units "metre per second squared" can be understood as a change in velocity per time, i.e. an increase of velocity by 1 metre per second every second.
In 1946, Conférence Générale des Poids et Mesures (CGPM) Resolution 2 standardized the unit of force in the MKS system of units to be the amount needed to accelerate 1 kilogram of mass at the rate of 1 metre per second squared. In 1948, the 9th CGPM Resolution 7 adopted the name newton for this force.[3] The MKS system then became the blueprint for today's SI system of units. The newton thus became the standard unit of force in the Système international d'unités (SI), or International System of Units.
The newton is named after Isaac Newton. As with every SI unit named for a person, its symbol starts with an upper case letter (N), but when written in full it follows the rules for capitalisation of a common noun; i.e., "newton" becomes capitalised at the beginning of a sentence and in titles, but is otherwise in lower case.
In more formal terms, Newton's second law of motion states that the force exerted on an object is directly proportional to the acceleration hence acquired by that object, namely:[4]
{\displaystyle F=ma,}
{\displaystyle m}
{\displaystyle a}
{\displaystyle {\text{kg}}}
{\displaystyle {\text{m}}}
{\displaystyle {\text{s}}}
{\displaystyle 1\ {\text{N}}=1\ {\frac {{\text{kg}}\cdot {\text{m}}}{{\text{s}}^{2}}}.}
At average gravity on Earth (conventionally, g = 9.80665 m/s2), a kilogram mass exerts a force of about 9.8 newtons. An average-sized apple exerts about one newton of force at Earth's surface, which we measure as the apple's weight on Earth.[5]
The weight of an average adult exerts a force of about 608 N.
KilonewtonsEdit
A carabiner used in rock climbing, with a safety rating of 26 kN when loaded along the spine with the gate closed, 8 kN when loaded perpendicular to the spine, and 10 kN when loaded along the spine with the gate open.
It is common to see forces expressed in kilonewtons (kN), where 1 kN = 1000 N. For example, the tractive effort of a Class Y steam train locomotive and the thrust of an F100 jet engine are both around 130 kN.
the holding values of fasteners, Earth anchors, and other items used in the building industry;
rock-climbing equipment;
thrust of rocket engines, Jet engines and launch vehicles;
Standard prefixes for the metric units of measure (multiples)
Standard prefixes for the metric units of measure (submultiples)
^ The International System of Units – 9th edition – Text in English (9 ed.). Bureau International des Poids et Mesures (BIPM). 2019. p. 137.
^ "Newton | unit of measurement". Encyclopedia Britannica. Archived from the original on 2019-09-27. Retrieved 2019-09-27.
^ International Bureau of Weights and Measures (1977), The International System of Units (3rd ed.), U.S. Dept. of Commerce, National Bureau of Standards, p. 17, ISBN 0745649742, archived from the original on 2016-05-11, retrieved 2015-11-15.
^ "Table 3. Coherent derived units in the SI with special names and symbols". The International System of Units (SI). International Bureau of Weights and Measures. 2006. Archived from the original on 2007-06-18.
^ Whitbread BSc (Hons) MSc DipION, Daisy. "How much is 100 grams?". Archived from the original on 24 October 2017. Retrieved 22 September 2020.
^ Walpole, Sarah Catherine; Prieto-Merino, David; Edwards, Phillip; Cleland, John; Stevens, Gretchen; Roberts, Ian (2012). "The weight of nations: an estimation of adult human biomass". BMC Public Health. 12 (12): 439. doi:10.1186/1471-2458-12-439. PMC 3408371. PMID 22709383.
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Home : Support : Online Help : Education : Student Packages : Linear Algebra : Computation : Eigenvalues : SingularValues
SingularValues( A, options )
options: parameters; for a complete list, see LinearAlgebra[SingularValues]
The SingularValues(A) function returns the singular values of Matrix A in a column Vector. The singular values are equal to the square roots of the (real) eigenvalues of
A·\mathrm{Transpose}\left(A\right)
A·\mathrm{HermitianTranspose}\left(A\right)
(see below). Since this product is either real-symmetric or Hermitian, and positive semi-definite, the eigenvalues are all real and non-negative, and so their square roots are also purely real.
When calling SingularValues(A) with output=['U','S','Vt'], the Matrix U, Vector S, and Matrix Vt satisfy
U·T·\mathrm{Vt}=A
T=\mathrm{DiagonalMatrix}\left(S\right)
The values of hardwarefloats (default false) and conjugate (default false) for the Student package can be set using the Student[SetDefault] command.
If the Matrix has an element that is not of type complex(numeric), the value of conjugate will be used to deduce if HermitianTranspose should be used. On the other hand, if all the elements are of type complex(numeric), then the regular transpose will be used if the elements are real-valued, and the Hermitian transpose if the elements are complex-valued.
For further information, see LinearAlgebra[SingularValues].
\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):
A≔\mathrm{Matrix}\left([[1.0,2.0],[3.0,4.0]]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1.0}& \textcolor[rgb]{0,0,1}{2.0}\\ \textcolor[rgb]{0,0,1}{3.0}& \textcolor[rgb]{0,0,1}{4.0}\end{array}]
S≔\mathrm{SingularValues}\left(A\right)
\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{5.464985704}\\ \textcolor[rgb]{0,0,1}{0.3659661916}\end{array}]
B≔\mathrm{Matrix}\left([[4.575068354,-4.024595950,-3.730131837],[0.4688151920,1.323592462,4.057919371],[-2.215017811,4.133758561,3.147236864]]\right)
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{4.575068354}& \textcolor[rgb]{0,0,1}{-4.024595950}& \textcolor[rgb]{0,0,1}{-3.730131837}\\ \textcolor[rgb]{0,0,1}{0.4688151920}& \textcolor[rgb]{0,0,1}{1.323592462}& \textcolor[rgb]{0,0,1}{4.057919371}\\ \textcolor[rgb]{0,0,1}{-2.215017811}& \textcolor[rgb]{0,0,1}{4.133758561}& \textcolor[rgb]{0,0,1}{3.147236864}\end{array}]
U,S,\mathrm{Vt}≔\mathrm{SingularValues}\left(B,'\mathrm{output}'=['U','S','\mathrm{Vt}']\right)
\textcolor[rgb]{0,0,1}{U}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Vt}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{-0.7381898079}& \textcolor[rgb]{0,0,1}{-0.4056519114}& \textcolor[rgb]{0,0,1}{0.5390012395}\\ \textcolor[rgb]{0,0,1}{0.3306799655}& \textcolor[rgb]{0,0,1}{-0.9140152493}& \textcolor[rgb]{0,0,1}{-0.2350040101}\\ \textcolor[rgb]{0,0,1}{0.5879851780}& \textcolor[rgb]{0,0,1}{0.004759346363}& \textcolor[rgb]{0,0,1}{0.8088577004}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{9.482762013}\\ \textcolor[rgb]{0,0,1}{3.196679524}\\ \textcolor[rgb]{0,0,1}{1.112982108}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{-0.4771435440}& \textcolor[rgb]{0,0,1}{0.6157689050}& \textcolor[rgb]{0,0,1}{0.6270268681}\\ \textcolor[rgb]{0,0,1}{-0.7179110309}& \textcolor[rgb]{0,0,1}{0.1384171711}& \textcolor[rgb]{0,0,1}{-0.6822348853}\\ \textcolor[rgb]{0,0,1}{0.5068903140}& \textcolor[rgb]{0,0,1}{0.7756734764}& \textcolor[rgb]{0,0,1}{-0.3760224302}\end{array}]
T≔\mathrm{DiagonalMatrix}\left(S\right)
\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{9.482762013}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{3.196679524}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.112982108}\end{array}]
\mathrm{Norm}\left(U·T·\mathrm{Vt}-B\right)
\textcolor[rgb]{0,0,1}{1.115765771}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{-8}}
The Student[LinearAlgebra][SingularValues] command was introduced in Maple 2021.
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Filmus, Yuval1; Golubev, Konstantin2; Lifshitz, Noam3
1 The Henry and Marilyn Taub Faculty of Computer Science Technion — Israel Institute of Technology, Haifa Israel.
2 D-MATH, ETH Zurich, Switzerland. Moscow Center for Fundamental and Applied Mathematics, Russia.
3 Einstein Institute of Mathematics Hebrew University, Jerusalem Israel.
n
-th tensor power of a graph with vertex set
V
is the graph on the vertex set
{V}^{n}
, where two vertices are connected by an edge if they are connected in each coordinate. One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. In this paper we introduce the problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many prominent open problems in extremal combinatorics, such as the Turán problem for graphs and hypergraphs, can be encoded as special cases of this problem. We generalize the Hoffman bound to hypergraphs, and give several applications.
Classification: 05C15, 05C65, 05D05
Keywords: Chromatic number, independence ratio, hypergraph, extremal set theory.
Filmus, Yuval 1; Golubev, Konstantin 2; Lifshitz, Noam 3
author = {Filmus, Yuval and Golubev, Konstantin and Lifshitz, Noam},
title = {High dimensional {Hoffman} bound and applications in extremal combinatorics},
TI - High dimensional Hoffman bound and applications in extremal combinatorics
%T High dimensional Hoffman bound and applications in extremal combinatorics
Filmus, Yuval; Golubev, Konstantin; Lifshitz, Noam. High dimensional Hoffman bound and applications in extremal combinatorics. Algebraic Combinatorics, Volume 4 (2021) no. 6, pp. 1005-1026. doi : 10.5802/alco.190. https://alco.centre-mersenne.org/articles/10.5802/alco.190/
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[13] Ellis, David; Friedgut, Ehud; Pilpel, Haran Intersecting families of permutations, J. Amer. Math. Soc., Volume 24 (2011) no. 3, pp. 649-682 | Article | MR: 2784326 | Zbl: 1285.05171
[14] Erdős, Paul A problem on independent
r
-tuples, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., Volume 8 (1965), pp. 93-95 | MR: 260599 | Zbl: 0136.21302
[15] Erdős, Paul; Ko, Chao; Rado, Richard Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. (2), Volume 12 (1961), pp. 313-320 | Article | MR: 140419 | Zbl: 0100.01902
[16] Filmus, Yuval; Ihringer, Ferdinand Boolean degree 1 functions on some classical association schemes, J. Combin. Theory Ser. A, Volume 162 (2019), pp. 241-270 | Article | MR: 3874601 | Zbl: 1401.05317
[17] Frankl, Péter On Sperner families satisfying an additional condition, J. Combinatorial Theory Ser. A, Volume 20 (1976) no. 1, pp. 1-11 | Article | MR: 398842 | Zbl: 0328.05002
[18] Frankl, Péter Asymptotic solution of a Turán-type problem, Graphs Combin., Volume 6 (1990) no. 3, pp. 223-227 | Article | MR: 1081196 | Zbl: 0724.05070
[19] Frankl, Péter; Tokushige, Norihide Weighted multiply intersecting families, Studia Sci. Math. Hungar., Volume 40 (2003) no. 3, pp. 287-291 | Article | MR: 2036959 | Zbl: 1045.05084
[20] Friedgut, Ehud Boolean functions with low average sensitivity depend on few coordinates, Combinatorica, Volume 18 (1998) no. 1, pp. 27-35 | Article | MR: 1645642 | Zbl: 0909.06008
[21] Friedgut, Ehud On the measure of intersecting families, uniqueness and stability, Combinatorica, Volume 28 (2008) no. 5, pp. 503-528 | Article | MR: 2501247 | Zbl: 1199.05319
[22] Friedgut, Ehud; Regev, Oded Kneser graphs are like Swiss cheese, Discrete Anal. (2018), Paper no. 2, 18 pages | Article | MR: 3766248 | Zbl: 1404.05088
[23] Gijswijt, Dion Matrix Algebras and Semidefinite Programming Techniques for Codes (2005) (Ph. D. Thesis)
[24] Gijswijt, Dion; Schrijver, Alexander; Tanaka, Hajime New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming, J. Combin. Theory Ser. A, Volume 113 (2006) no. 8, pp. 1719-1731 | Article | MR: 2269550 | Zbl: 1105.94027
[25] Godsil, Chris; Meagher, Karen Erdős–Ko–Rado theorems: algebraic approaches, Cambridge Studies in Advanced Mathematics, 149, Cambridge University Press, Cambridge, 2016, xvi+335 pages | Article | MR: 3497070 | Zbl: 1343.05002
[26] Golubev, Konstantin On the chromatic number of a simplicial complex, Combinatorica, Volume 37 (2017) no. 5, pp. 953-964 | Article | MR: 3737375 | Zbl: 1399.05227
[27] Golubev, Konstantin; Parzanchevski, Ori Spectrum and combinatorics of two-dimensional Ramanujan complexes, Israel J. Math., Volume 230 (2019) no. 2, pp. 583-612 | Article | MR: 3940429 | Zbl: 1410.05126
[28] Gundert, Anna; Szedlák, May Higher dimensional discrete Cheeger inequalities, J. Comput. Geom., Volume 6 (2015) no. 2, pp. 54-71 | Article | MR: 3305828 | Zbl: 1405.05101
[29] Haemers, Willem A generalization of the Higman-Sims technique, Nederl. Akad. Wetensch. Indag. Math., Volume 40 (1978) no. 4, pp. 445-447 | Article | MR: 515603
[30] Hoffman, Alan J. On eigenvalues and colorings of graphs, Selected Papers of Alan J. Hoffman — with Commentary, World Scientific, 2003, pp. 407-419 | Article
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[32] Linial, Nathan; Peled, Yuval On the phase transition in random simplicial complexes, Ann. of Math. (2), Volume 184 (2016) no. 3, pp. 745-773 | Article | MR: 3549622 | Zbl: 1348.05193
[33] Lovász, László On the Shannon capacity of a graph, IEEE Trans. Inform. Theory, Volume 25 (1979) no. 1, pp. 1-7 | Article | MR: 514926 | Zbl: 0395.94021
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[35] Mantel, Willem Problem 28 (Solution by H. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff), Wiskundige Opgaven, Volume 10 (1907), pp. 60-61
[36] McEliece, Robert J.; Rodemich, Eugene R.; Rumsey, Howard Jr.; Welch, Lloyd R. New upper bounds on the rate of a code via the Delsarte–MacWilliams inequalities, IEEE Trans. Inform. Theory, Volume IT-23 (1977) no. 2, pp. 157-166 | Article | MR: 439403 | Zbl: 0361.94016
[37] Meshulam, Roy On subsets of finite abelian groups with no
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-term arithmetic progressions, J. Combin. Theory Ser. A, Volume 71 (1995) no. 1, pp. 168-172 | Article | MR: 1335785 | Zbl: 0832.11006
[38] Mossel, Elchanan; O’Donnell, Ryan; Oleszkiewicz, Krzysztof Noise stability of functions with low influences: invariance and optimality, Ann. of Math. (2), Volume 171 (2010) no. 1, pp. 295-341 | Article | MR: 2630040 | Zbl: 1201.60031
[39] Navon, Michael; Samorodnitsky, Alex Linear programming bounds for codes via a covering argument, Discrete Comput. Geom., Volume 41 (2009) no. 2, pp. 199-207 | Article | MR: 2471869 | Zbl: 1173.90475
[40] Parzanchevski, Ori Mixing in high-dimensional expanders, Combin. Probab. Comput., Volume 26 (2017) no. 5, pp. 746-761 | Article | MR: 3681980 | Zbl: 1371.05329
[41] Parzanchevski, Ori; Rosenthal, Ron Simplicial complexes: spectrum, homology and random walks, Random Structures Algorithms, Volume 50 (2017) no. 2, pp. 225-261 | Article | MR: 3607124 | Zbl: 1359.05114
[42] Parzanchevski, Ori; Rosenthal, Ron; Tessler, Ran J. Isoperimetric inequalities in simplicial complexes, Combinatorica, Volume 36 (2016) no. 2, pp. 195-227 | Article | MR: 3516884 | Zbl: 1389.05174
[43] Schrijver, Alexander A comparison of the Delsarte and Lovász bounds, IEEE Trans. Inform. Theory, Volume 25 (1979) no. 4, pp. 425-429 | Article | MR: 536232 | Zbl: 0444.94009
[44] Schrijver, Alexander New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory, Volume 51 (2005) no. 8, pp. 2859-2866 | Article | MR: 2236252 | Zbl: 1298.94152
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[48] Zhao, Yufei Graph theory and additive combinatorics (2019) (https://ocw.mit.edu/courses/mathematics/18-217-graph-theory-and-additive-combinatorics-fall-2019/lecture-notes/MIT18_217F19_full_notes.pdf)
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Read the Math Notes box for this lesson and then find the length of the radius and diameter of each of the following circles.
The radius of a circle is a line segment from its center to any point on the circle. The diameter of a circle is a line segment going through the center and ending on opposite sides of the circle.
3
Diameter:
6
How is the information for this circle different than part (a)?
1.5
3
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(Some) mathematical measures - Tales of Science & Data
This page outlines, mainly for convenience, some useful measures/metrics utilised for several purposes.
It is the reciprocal of the mean of reciprocals:
h = \frac{n}{\sum_{i=1}^{i=n} \frac{1}{x_i}}
Common measures of distance/similarity
The euclidean distance of two vectors
v_a = (x^i)_a
v_b = (x^i)_b
l_2
of the vector connecting them (it measures its length):
d = \sqrt{\sum_i (x^i_a - x^i_b)^2} = ||v_A - v_B||_2
The Hamming distance expresses the number of different elements in two lists/strings:
A = 110101; B = 111001; d_{AB} = 2
Jaccard (index)
Given two finite sets A and B, the Jaccard index gives a measure of how much they overlap, as
J_{AB} = \frac{|A \cap B|}{|A \cup B|}
Also called cityblock, the Manhattan distance between two points is the norm
l_1
of the shortest path a car would take between these two points in Manhattan (which has a grid layout):
d = \sum_i |u_i - v_i|
The Minkowski distance is a generalisation of both the euclidean and the Manhattan to a generic p:
d = \left(\sum_i |x_i - y_i|^p\right)^{1/p}
The cosine similarity is given by the cosine of the angle
\theta
spanned by the two vectors
d = \cos \theta = \frac{\bar u \cdot \bar v}{|\bar u| |\bar v|}
So two perfectly overlapping vectors would have a cosine similarity of 1 and vectors at
90^{\circ}
would have a cosine similarity of 0.
It is also called chessboard distance. In the game of chess, the Chebyshev distance between the centers of the squares is the minimum number of moves a king needs to go from a square to another one.
\max_i |u_i - v_i|
See the figure here, it reports in red all the Chebyshev distance value from where the king (well, there's a drawing for it ...) sits to cell; note that the king can move horizontally, vertically and diagonally.
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Tino works at a retail clothing store as a sales clerk and part of his paycheck is based on commission, meaning that it is determined by the value of the clothes that he sells. Each week, he earns
\$7
an hour plus
15\%
of his total sales. If Tino works for
18
hours and sells
\$538
in clothes in a certain week, what is the amount of his commission? What is his total pay for that week?
What is the equation you could use to find the commission Tino earns?
15\%=0.15
(0.15)(\$538)=\text{the total commission}
Tino's total pay is
\$206.70
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FEP - Ring of Brodgar
(Redirected from Food Event Points)
See the FEP Table for information on the FEP values of specific foods.
Food Event Points (FEPs) are obtained by eating food. Once you have enough points, one of your Attributes (Strength, Agility, etc.) will be increased by either one or two points.
By placing the cursor over the Food Event Points bar on the character sheet, you can see how many FEPs you currently have and how many you need. To increase an attribute point, you must accumulate a number of FEPs equal to or greater than your highest attribute (ignoring modifiers).
Example: My highest attribute is Strength, which is 57.
To gain another attribute point, I must eat enough food to gain at least 57 FEP.
Eating a unique food item lowers the total number of FEPs needed to gain an attribute point. This modifier resets every time you gain an attribute point, and you can again eat unique food items.
Example: Eating two blueberries, a three blueberry pies, and eight roasted perch counts as 3 unique food items.
If you gain an attribute point, you can eat the food items again and get the same bonus for three unique foods.
Which attribute is earned is determined by how many FEPs you have towards each attribute.
Example: if you only eat yellowfeet gives 1 STR and 1 AGI per food item.
Upon filling the FEP bar, you will have a 50% chance of Strength increasing by one, and a 50% chance of agility increasing by one.
After an attribute is increased, your FEP bar is reset, so any excess FEPs are lost
Do note however that excess FEP will be calculated in determining your next attribute point.
The FEP value a food item gives is determined by its quality
FEP Requirement Reduction/Variety Bonus
Although new players start out having issues keeping their character full on food, it quite quickly reverses as a settlement gets set up. Eventually emptying the food bar becomes a chore that must be done in order to gain stats. Eating a variety of food makes this go much more quickly, but often involves introducing the possibility of gaining stats of a type you're not currently interested in. Obviously there's a balance of eating as many food types as possible and eating only the FEP types that you want. Better quality food helps too since it increases the quantity of FEPs a food gives without changing the amount of fullness you gain from it.
FEPs that you need to fill a food bar are reduced for each and every unique food that you eat while filling that FEP bar. The FEPs given by the food do not matter, as long as the type of food if different. For example, if you eat Roasted Fox Meat and Blueberries (both give 1 INT), the bar will still be reduced.
According to Loftar, the forumula for the FEP reduction is:
{\displaystyle {\frac {2*{\sqrt {MaxStat*10}}}{10}}}
{\displaystyle 0.632*{\sqrt {max}}}
where max = the highest unmodified attribute of the character.
Example: The highest attribute on a character is Con at 64. That means for each new food type eaten, the required FEP is lowered by 0.632 * sqrt(64) = 0.632 * 8 = 5.056 or rounded to 5.1
NOTE: This is according to a post by Loftar here: http://www.havenandhearth.com/forum/viewtopic.php?f=2&t=9383
Joe has 15 in all attributes. So he needs 15 FEPs to get his next attribute increase. He eats enough bear meat to give him 10.5 Strength FEPs, and his Strength goes up by one. Wait, he needed a total of 15 FEP's didn't he? This is an example of the "variety bonus" that's applied to eating. The first item he eats reduces his required FEP's by 30%, to 10.5 from 15. Because he ate only STR giving FEP's, his STR went up to 16.
Joe now has 16 STR, so he'll need 16 FEPs for his next attribute increase. He eats some bear meat, reducing his needed FEPs to 11.2 (30% reduction). Later that day he decided he'd like to eat blueberries. Upon eating blueberries his total FEPs required drops to 8.8 (21% reduction from 11.2). The variety bonus concerns food types, not the FEPs they give. Both cooked perch and blueberries give pure INT FEPs, but they are two items as far as the variety bonus is concerned.
At the end of the day, he's eaten only cooked bear meat and blueberries, giving him 4.0 STR FEPs and 4.8 INT FEPs. When the food bar is full, the proportion of the whole that an FEP type represents of the food bar becomes the chance that he will gain that stat. As a result, Joe has a 45.5% (4.0/8.8) chance of gaining STR and a 54.5% (4.8/8.8) chance of gaining INT. Despite having more chance of getting INT FEPs, Joe gets another STR point and his total STR goes up to 17.
If INT had gone up to 16 instead of STR getting another point, his total FEPs required to level would not have changed. Because his STR went up though, he'll have to eat more FEPs next time around to get another stat point. This of course could be compensated by the variety bonus if he has more food types around to eat instead.
As per the example, increasing FEPs is based on your highest attribute. This is why it is recommended to raise your attributes evenly, since it will affect how many FEPs you require to raise every other attribute. Before letting one attribute get completely out of hand, considering raising your other attributes for a while so you'll have more of them when you want them. Raising Intelligence from 10 to 11 when you have 200 strength is just as difficult as raising strength from 200 to 201.
Satiations are another factor which affect FEP and attribute gain, so be sure to read about satiations as well.
Retrieved from "https://ringofbrodgar.com/w/index.php?title=FEP&oldid=87161"
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I need very high dynamic range. What is the best I can achieve?
How do I differentiate how much light I am seeing in each of my pulses?
Which detector gives me the best signal-to-noise ratio (SNR)?
What is photon counting?
What detector options are available for photon counting?
How can I improve my signal-to-noise ratio (SNR) if I am barely able to differentiate my signal from noise in an MPPC?
How else can I improve my SNR when I’m photon counting?
What are the advantages and disadvantages of an APD?
What are the different types of silicon APDs offered by Hamamatsu?
This is difficult to answer since there are many contributors to dynamic range including ADC limits, amplifier limits, detector limits, etc. I will try to simplify this by removing limitations of external electronics.
Dynamic range in this case will be from the lowest detectable signal to the highest detectable signal. The low end will ultimately be set by the dark shot noise while the high end will be set by the saturation or nonlinearity point of the detector.
An uncooled MPPC can typically hit 3-4 decades of dynamic range while a cooled MPPC can probably hit around 5 decades of dynamic range. PMTs and APDs can typically hit 6-7 decades. There is also a new MPPC with wide dynamic range (ex. S14160-1310PS ), which is able to hit 5-6 decades uncooled.
One advantage of the PMT and MPPC is that the intrinsic gain is large enough to see a pulse per photon, which allows for a method called photon counting. For more information on photon counting, feel free to read the other Q&As on this page.
Photon counting is for the lowest light levels for these detectors, which means this can be used to squeeze more dynamic range out of both PMTs and MPPCs.
For an example, take a look at our wide dynamic range PMT H13126, which contains both photon counting and analog circuitry. This module has shown results with over 10 decades of light intensity and shown less than 10% deviation.
There are many techniques to differentiate the amount of light in a pulse, so I will try to walk through a few examples.
Shown in Figure 1 below are 3 signals that could come from the detector. I will reference these as signal 1, signal 2, and signal 3.
Figure 1. Signals from a detector
The first technique will be amplitude or peak height. As seen below in Figure 2, the point on each graph is the peak of the signal.
Figure 2. Amplitude or peak height technique
Typically, a higher peak correlates to a higher number of photons in the pulse. However, cases like signal 1 and signal 2 will occur where two peaks are almost the same, but signal 2 would clearly be caused by more light. This means that peak height alone would show that signal 1 and signal 2 are roughly the same magnitude.
The second technique is time over threshold or TOT.
Figure 3. Time over threshold (TOT) technique
With this technique, there is some threshold, which in this case is A and B shown in Figure 3. The measured signal in this case is the time that the signal is above the specified threshold. Selecting a threshold can be difficult, and there is no single right answer as to what the threshold should be. In the case of threshold A, signal 1 and signal 2 have a much larger difference, but signal 3 appears effectively as “0”. With threshold B, signal 1 and signal 2 have a slightly smaller difference, but threshold 3 does appear. In some cases, like with signal 2, changing threshold between A and B will not show a change in signal. In other cases like signal 1 and signal 3, there will be a change in the TOT measurement that is much more noticeable.
Comparing the TOT method to the peak height measurement, both provide useful, real-time data on the amount of light seen in each pulse. There are tradeoffs in each method. For example, the differentiation in signal 1 and signal 3 in the TOT measurement with threshold B is not as significant as the difference in peak height. Another example is that with the peak height method, signal 1 and signal 2 cannot be differentiated easily, but with the TOT method, there is a clear difference.
These are just two potential methods. There are others such as integration for some fixed period of time or integration starting with a threshold. There are also methods that combine different measurement techniques simultaneously and use some algorithm to determine a more accurate amount of light in the pulse.
The information listed here is mainly to understand the signal or charge outputted from the detector. To quantify the amount of light seen by the detector, the quantum efficiency or sensitivity of the detector would need to be incorporated.
If you are interested in learning more about detecting pulses of light, feel free to reach out to our engineers!
Which photodetector has the best SNR varies greatly with light level, speed, and support electronics. It’s usually best to determine the light level and which noise source limits you: your signal shot noise, your electronics/amplifier noise, or your detector’s dark noise.
For high light levels where amplifier noise is negligible and you are signal shot noise limited, detectors with the highest quantum efficiency (QE) will have the best SNR.
For lower light levels where the readout or amplifier noise dominates, detectors with high internal gain are required for good SNR. The detector’s internal gain reduces the importance of the amplifier noise, so you are then limited by dark or signal shot noise.
For the lowest light levels, detectors with the lowest dark noise will have the best SNR.
If you’re not sure, we can help! Please contact us to get an SNR calculation for your unique situation.
For more info about choosing a detector, read our Guide to detector selection article.
Photon counting is being able to measure and observe a single incident photon. It is typically used with a photon counting specific circuit that discriminates electronics noise out from the observation and removes the excess noise factor, which is noise from the detector’s intrinsic gain mechanism. Using this circuit and because the light levels are so low, we are typically limited by the dark noise of the detector.
For photon counting, detectors with intrinsic gain are necessary. A commonly used detector is a photomultiplier tube, which uses dynodes to provide intrinsic gain to the detector. Photodiodes do not have intrinsic gain which means 1 incident photon will, at most, allow the flow of 1 electron, which is too small of a signal to overcome the noise. Avalanche photodiodes (APD) have intrinsic gain, but the gain is too small to overcome noise in most cases, unless the APD is run in Geiger mode. Single-photon avalanche diode (SPAD) and Multi-Pixel Photon Counters (MPPC) are based on APDs in Geiger mode, so these two types of detectors have enough intrinsic gain to show single photon pulses.
If you are limited by the dark noise, it benefits you to try and decrease the dark noise as much as you can to see if you can detect your single photons. One improvement can be using a cooled detector. Examples of a cooled MPPC are the S13362-3050DG and S13362-1350DG. Cooling will help reduce the number of dark counts and dark noise at the cost of needing more power to drive the detector with cooling. If a module is preferred, we do have a variety of cooled modules including the new C14455-GA and C14456-GA series, which have peak sensitivities at 600 nm. If cooling is not preferable for the MPPC and other aspects of the signal/noise cannot be changed, switching the detector family to PMTs may be preferred since PMTs have lower dark counts per unit active area. Please contact Hamamatsu’s Technical Support team if you would like additional information about the tradeoffs and options.
Increasing the measurement time can improve your SNR by a factor of the square root of your integration time. Meaning if your current measurements is only 1 second and now you measure the number of signal counts and dark counts for 4 seconds, you will approximately double your SNR!
Here’s the equation for photon counting SNR:
SNR=\frac{\left(Detected counts - Dark counts\right) * \sqrt{Measurement Time}}{\sqrt{Detected counts+2 *Dark Counts}}
For more information, view a presentation about low light detection.
APD is short for avalanche photodiode. Similar to other photodetectors, an APD converts light energy (photons) into an electrical current. A silicon APD has spectral sensitivity from the UV to NIR range, and InGaAs APDs extend into the NIR range. An APD is basically a PIN photodiode with the advantage of providing gain. Gain is achieved through a process called impact ionization or the avalanche effect. Applying a large reverse voltage (100 to 500 V) to the APD creates a strong electric field across the PN junction (depletion region) of the APD. The electric field will accelerate electron-hole pairs from within the structure of the APD toward the depletion region. Upon collision with the lattice structure in the depletion region, additional electron-hole pairs are created. The typical gain for an APD is 40 to 100.
The APD has very high quantum efficiency, so it’s highly sensitive. This feature is advantageous for detecting low light levels.
The APD is a high-speed device, meaning it has a very fast rise time on the order of 150 ps.
To maximize the gain of the APD, a relatively high reverse voltage is required. The range of the reverse voltage (bias voltage) is from 100 to 500 V.
The APD is more sensitive to changes in temperature compared to other photodetectors. An increase in temperature will cause a decrease in gain. The parameter used to reference temperature sensitivity is called “temperature coefficient of breakdown voltage,” and its units are given in V/degrees C.
Short wavelength type:
This type of APD has a peak response near 600 nm. This group is broken down into two additional categories: low bias operation type and low terminal capacitance type. The low bias type APDs require bias voltages that are 300 V less than the low terminal capacitance type. The low terminal capacitance types are made for higher speed of operation. If the capacitance of the APD decreases, the rise time also decreases (frequency response increases). The rise time will increase if the effective photosensitive area increases. Typical rise time values for the low capacitance type range from 0.5 to 32 ns.
Near-infrared type:
There are 4 types of near-infrared (NIR) APDs: low bias, low capacitance, 900 nm band/low capacitance, and TE-cooled. The low bias and low capacitance types have similar characteristics as the short wavelength types. The 900 nm band/low capacitance type offers sensitivity near 900 nm along with a high-speed response. The applications for this APD include optical rangefinders and automotive LiDAR. The TE-cooled type APD has a built-in TE-cooler, which keeps the ambient temperature constant. This feature helps to maintain gain stability.
Neil Patel enjoys the majesty of narwhals and photons. He glides through technical issues just as the unicorn of the sea glides through the water. Fun fact: The narwhal’s tusk is actually a protruding tooth.
Peter Lopez is an Applications Engineer in Hamamatsu’s San Jose, CA, office. He received his Electrical Engineering degree from the University of Michigan, and he has worked in many different industries including semiconductor, infrared detection, microscopy, and medical devices. His past positions have provided a good background for the many different types of markets served by Hamamatsu’s component products. Although originally from Michigan, he’s enjoyed living and working in California for the past 25 years. In his spare time, he enjoys sailing, playing golf, and attending San Jose Sharks hockey games.
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Hertz Knowpia
The occurrence rate of aperiodic or stochastic events is expressed in reciprocal second or inverse second (1/s or s−1) in general or, in the specific case of radioactive decay, in becquerels.[7] Whereas 1 Hz is one cycle per second, 1 Bq is one aperiodic radionuclide event per second.
{\displaystyle \omega =2\pi f\,}
{\displaystyle f={\frac {\omega }{2\pi }}\,}
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Boundary Value Problems/Derivation of Heat Equation - Wikiversity
Boundary Value Problems/Derivation of Heat Equation
C - Heat Capacity Joules per degree Kelvin
c - specific heat capacity is C per unit mass so j/(gm deg)
A quick derivation of heat equation.
Heat is measured in calories which is just another form of joules. 1 cal = 1.484 joules.
Heat flow. By observation we know that heat(energy) flows from a higher(energy) temperature to a lower(energy) temperature.
{\displaystyle V}
represent a volume in
{\displaystyle {\mathcal {R}}^{3}}
. At each point in the volume
{\displaystyle V}
{\displaystyle u(x,y,z,t)}
represent a scalar field that is the temperature at each point in the volume. If the
{\displaystyle \nabla u}
is not equal to zero at a point
{\displaystyle P(x_{0},y_{0},z_{0})}
this indicates that there will be a flow of heat from the higher temperature to the lower temperature in the region about that point.
{\displaystyle -\nabla u}
represents the direction of decreasing temperature and points in the direction of energy flow.
{\displaystyle -\nabla u=v}
is a vector field the provides a direction and magnitude for the heat flow at any point in the volume.
{\displaystyle \Delta V}
be a small volume in
{\displaystyle \mathbf {V} }
{\displaystyle S_{1}{\mbox{ and }}S_{2}}
represent two parallel end areas of this volume. Each end area has a normal
{\displaystyle n_{1}{\mbox{ and }}n_{2}}
. The net flow through the end area
{\displaystyle \Delta A}
is the amount of flow
{\displaystyle v}
normal to the surface
{\displaystyle S_{1}}
times the surface area
{\displaystyle \Delta A}
{\displaystyle S_{1}}
{\displaystyle {\mbox{ net flux is }}F=-k\nabla u\cdot {\hat {n}}\Delta A}
{\displaystyle k}
is a material property.
For a cubic volume the net flux through each side area
{\displaystyle S_{B}}
{\displaystyle \lim _{N\to \infty }\sum _{i=1}^{N}-k\nabla u_{i}\cdot n\Delta A_{i}=\int \int _{S_{B}}-k\nabla u_{i}\cdot ndA}
If there are any sources or sinks in the block these are included as an additive term
{\displaystyle E=\int _{B}f(x,y,z,t)dV}
.The total flux for the cube will be a double integral over the closed surface of the cube volume.
{\displaystyle {\mbox{Net Flux}}=\int \int _{S_{B}}-k\nabla u_{i}\cdot ndA+\int _{B}f(x,y,z,t)dV}
There are three results for the integration: 1) The net flux is positive, which implies more flux is leaving the cube than entering it. An example would be the radiation from a radioactive source enclosed in a block of cement. 2) The net flux is zero, which implies the in flux equals the out flux. Even if there are sources in the cube there are also sinks that absorb the addition flux. An example would be putting water in a barrel while it flows out through the bottom at the same rate as the entry rate. 3) The net flux is negative, which implies more flux is entering the cube than leaving it. An example would be a material that absorbs heat applied to it's surface, such as melting ice.
Another way of looking at the heat energy stored in a piece of material: A mass will hold heat. The amount of heat in an incremental mass is estimated by the expression:
{\displaystyle \displaystyle \Delta H=cu\Delta m=cu\rho \Delta V}
The total heat,
{\displaystyle \displaystyle H}
, in the total mass will be
{\displaystyle \displaystyle H=\int _{B}cu\rho dV}
The change in heat per unit time is
{\displaystyle {\frac {dH}{dt}}={\frac {d}{dt}}\int _{B}cu\rho dV=\int _{B}c{\frac {du}{dt}}\rho dV}
. The change in the heat is the net flux
{\displaystyle {\mbox{Net Flux}}=\int \int _{S_{p}}-k\nabla u\cdot ndA}
so by equivalence:
{\displaystyle \int _{B}c{\frac {du}{dt}}\rho dV=\int \int _{S_{p}}-k\nabla u\cdot ndA+\int _{B}fdV}
The divergence theorem is applied to the first term on the right so that all three terms are volume integrals.
{\displaystyle \int _{B}c{\frac {du}{dt}}\rho dV=\int _{B}\nabla \cdot (k\nabla u)dV+\int _{B}fdV}
{\displaystyle c\rho }
and labeling the factor
{\displaystyle \alpha ^{2}={\frac {k}{c\rho }}}
the integral equation becomes.
{\displaystyle \int _{B}{\frac {du}{dt}}dV=\int _{B}\alpha ^{2}\nabla \cdot \nabla udV+\int _{B}{\frac {f}{\rho c}}dV}
Pull all the integrals together on the right:
{\displaystyle \int _{B}{\frac {du}{dt}}dV=\int _{B}\alpha ^{2}\nabla ^{2}u+{\frac {f}{\rho c}}dV}
Without proving it we will state
{\displaystyle {\frac {du}{dt}}=\alpha ^{2}\nabla ^{2}u+{\frac {f}{\rho c}}}
. This is the heat equation without convection (strictly conduction).
{\displaystyle \alpha ^{2}}
is the material's diffusivity. And
{\displaystyle F={\frac {f}{\rho c}}}
is the heat source/sink density in the material.
{\displaystyle u(x,y,z,t)}
is a scalar field associated with the points in the volume. As with ODE's we consider the initial state of the system (initial value)as part of the solution process. For this problem the initial value is associated with every point in the volume at
{\displaystyle t=0}
{\displaystyle u(x,y,z,t=0)=g(x,y,z)}
{\displaystyle g}
In addition to the initial conditions there will be equations that define the flow of heat across the boundary of the surface.
Three things can happen at a boundary:
1) Flux can flow in a positive direction from the body out across the boundary at a point. If
{\displaystyle \nabla u\cdot {\hat {n}}>0}
at a point on the surface then the flow is from the interior to the exterior of the volume, across the surface. This condition occurs if the interior is at a higher temperature than the exterior.
2) Flux can flow in across the boundary at a point on the surface, If
{\displaystyle \nabla u\cdot {\hat {n}}<0}
at a point on the surface then the flow is from the exterior to the interior of the volume, across the surface. This occurs if the exterior temperature is higher than the interior.
3) Flux does not flow cross the boundary.If
{\displaystyle \nabla u\cdot {\hat {n}}=0}
. Interior temperature at the boundary equals the exterior temperature at the boundary.
These cases occur for fixed boundary conditions:
{\displaystyle \displaystyle u_{\delta \Omega }(t)=T}
, the temperature at the boundary
{\displaystyle \displaystyle u_{\delta \Omega }(t)}
is held at a constant exterior temperature,
{\displaystyle \displaystyle T}
Retrieved from "https://en.wikiversity.org/w/index.php?title=Boundary_Value_Problems/Derivation_of_Heat_Equation&oldid=2013241"
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Here's a fun take on the OLS that I picked up from The Elements of Statistical Learning. It applies the Singular Value Decomposition, also known as the method used in principal component analysis, to the regression framework.
Singular Vector Decomposition (SVD)
First, a little background on the SVD. The SVD could be thought of as a generalisation of the eigendecomposition. An eigenvector v of matrix
\mathbf{A}
is a vector that is mapped to a scaled version of itself:
\mathbf{A}v = \lambda v
\lambda
is known as the eigenvalue. For a full rank matrix (this guarantees orthorgonal eigenvectors), we can stack up the eigenvalues and eigenvectors (normalised) to obtain the following equation:
\begin{aligned} \mathbf{A}\mathbf{Q} &= \mathbf{Q}\Lambda \\ \mathbf{A} &= \mathbf{Q}\Lambda\mathbf{Q}^{-1} \end{aligned}
\mathbf{Q}
is an orthonormal matrix.
For the SVD decomposition,
\mathbf{A}
can be any matrix (not square). The trick is to consider the square matrices
\mathbf{A}'\mathbf{A}
\mathbf{A}\mathbf{A}'
. The SVD of the
n \times k
\mathbf{A}
\mathbf{U}\mathbf{D}\mathbf{V}'
\mathbf{U}
is a square matrix of dimension
\mathbf{V}
is a square matrix of dimension
k
\mathbf{A}'\mathbf{A} = \mathbf{V}\mathbf{D}^{2}\mathbf{V}
\mathbf{V}
can be seen to be the eigenvalue matrix of that square matrix. Similarly, the eigenvectors of
\mathbf{A}\mathbf{A}'
forms the columns of
\mathbf{U}
\mathbf{D}
is the square root of the eigenvalues of either matrix.
In practice, there is no need to calculate the full set of eigenvectors for both matrices. Assuming that the rank of
\mathbf{A}
is k, i.e. it is a long matrix, there is no need to find all n eigenvectors of
\mathbf{A}\mathbf{A}'
since only the elements of the first k eigenvalues will be multiplied by non-zero elements. Hence, we can restrict
\mathbf{U}
n \times k
\mathbf{D}
k \times k
Applying the SVD to OLS
To apply the SVD to the OLS formula, we re-write the fitted values, substituting the data input matrix
X
with its equivalent decomposed matrices:
\begin{aligned} \mathbf{X}\hat{\beta} &= \mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y} \\ &= \mathbf{U}\mathbf{D}\mathbf{V}'(\mathbf{V}\mathbf{D}'\mathbf{U}'\mathbf{U}\mathbf{D}\mathbf{V}')^{-1}\mathbf{V}\mathbf{D}'\mathbf{U}'\mathbf{y} \\ &= \mathbf{U}\mathbf{D}(\mathbf{D}'\mathbf{D})^{-1}\mathbf{D}\mathbf{U}'\mathbf{y} \\ &= \mathbf{U}\mathbf{U}'\mathbf{y} \end{aligned}
where the third to fourth line comes from the fact that
(\mathbf{D}'\mathbf{D})^{-1}
k \times k
matrix with the square root of the eigenvalues on the diagonal, and
\mathbf{D}
is a square diagonal matrix. Here we see that the fitted values are computed with respect to the orthonormal basis
\mathbf{U}
Link to the ridge regression
The ridge regression is an OLS regression with an additional penalty term on the size of the coefficients and is a popular model in the machine learning literature. In other words, the parameters are chosen to minimalise the penalised sum of squares:
\sum_{i=1}^{n}(y_{i} - \sum_{j=1}^{k} x_{ij}\beta_{j})^{2} + \lambda \sum_{j=1}^{k} \beta_{j}^{2}
The solution to the problem is given by:
\hat{\beta}^{ridge} = (\mathbf{X}'\mathbf{X} + \lambda I_{k})^{-1}\mathbf{X}'\mathbf{Y}
. Substituting the SVD formula into the fitted values of the ridge regression:
\begin{aligned} \mathbf{X}\hat{\beta}^{ridge} &= \mathbf{X}(\mathbf{X}'\mathbf{X} + \lambda\mathbf{I})^{-1}\mathbf{X}'\mathbf{y} \\ &= \mathbf{U}\mathbf{D}(\mathbf{D}'\mathbf{D} + \lambda\mathbf{I})^{-1}\mathbf{D}\mathbf{U}'\mathbf{y} \\ &= \sum_{j=1}^{k} \mathbf{u}_{j} \frac{d^{2}_{j}}{d^{2}_{j} + \lambda} \mathbf{u}_{j}'\mathbf{y} \end{aligned}
\mathbf{u}
is a n-length vector from the columns of
\mathbf{U}
. This formula makes the idea of regularisation really clear. It shrinks the predicted values by the factor
d^{2}_{j}/(d^{2}_{j} + \lambda)
. Moreover, a greater shrinkage factor is applied to the variables which explain a lower fraction of the variance of the data i.e. lower
d_{j}
. This comes from the fact that the eigenvectors associated with a higher eigenvalue explain a greater fraction of the variance of the data (see Principal Component Analysis).
The difference between how regularisation works when one uses the Principal Component Analysis (PCA) method vs the ridge regression also becomes clear with the above formulation. The PCA approach truncates variables that fall below a certain threshold, while the ridge regression applies a weighted shrinkage method.
Doing a QR decomposition will also give a similar set of results, though the orthogonal bases will be different. ↩
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Error probability estimate and confidence interval of Monte Carlo simulation - MATLAB berconfint - MathWorks
berconfint
Compute BER Confidence Interval for Simulation Results
errprobest
Error probability estimate and confidence interval of Monte Carlo simulation
[errprobest,interval] = berconfint(nerrs,ntrials)
[errprobest,interval] = berconfint(nerrs,ntrials,level)
[errprobest,interval] = berconfint(nerrs,ntrials) returns the error probability estimate and 95% confidence interval for a Monte Carlo simulation of ntrials trials with nerrs errors.
[errprobest,interval] = berconfint(nerrs,ntrials,level) specifies the confidence level.
Compute the confidence interval for the simulation of a communication system that has 100 bit errors in 106 trials. The bit error rate (BER) for that simulation is
{10}^{-4}
Compute the 90% confidence interval for the BER of the system. The output shows that, with 90% confidence level, the BER for the system is between 0.0000841 and 0.0001181.
nerrs = 100; % Number of bit errors in simulation
ntrials = 10^6; % Number of trials in simulation
level = 0.90; % Confidence level
[ber,interval] = berconfint(nerrs,ntrials,level)
interval = 1×2
For an example that uses the output of the berconfint function to plot error bars on a BER plot, see Use Curve Fitting on Error Rate Plot.
nerrs — Number of errors
Number of errors from Monte Carlo simulation results, specified as a scalar.
ntrials — Number of trials
Number of trials from Monte Carlo simulation results, specified as a scalar.
Confidence level for a Monte Carlo simulation, specified as a scalar in the range [0, 1].
errprobest — Error probability estimate
Error probability estimate for a Monte Carlo simulation, returned as a scalar.
If the errors and trials are measured in bits, the error probability is the bit error rate (BER).
If the errors and trials are measured in symbols, the error probability is the symbol error rate (SER).
interval — Confidence interval
two-element column vector
Confidence interval for a Monte Carlo simulation, returned as a two-element column vector that lists the endpoints of the confidence interval for the confidence level specified by the input level.
[1] Jeruchim, Michel C., Philip Balaban, and K. Sam Shanmugan. Simulation of Communication Systems. Second Edition. New York: Kluwer Academic/Plenum, 2000.
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Leather - Ring of Brodgar
Object(s) Required Prepared Animal Hide
Produced By Tanning Tub and Tanning Fluid
Required By Ancestral Shrine, Bear Coat, Beaver Wrist Guards, Boar Tusk Helmet, Bone Chest, Bone Greaves, Bronze Helm, Bronze Plate, Coffer, Coracle, Cottage Table, Cottage Throne, Dressing Table, Drum & Sticks, Family Heirloom, Fine Feather Brooch, Fisherman's Hat, Garden Shed, Gold Spoon-Lure, Greased Joints, Hardened Leather, Hunter's Belt, Hunter's Bow, Hunter's Quiver, Kozhukh, Laddie's Cap, Large Chest, Leather Armor, Leather Backpack, Leather Ball, Leather Basket, Leather Boots, Leather Coat, Leather Fabric, Leather Merchant's Hat, Leather Pants, Leather Patch, Leather Purse, Lynx Claw Gloves, Mine Hole, Miner's Helm, Moose Hide Jacket, Packrack, Palisade, Plate Armor, Plate Boots, Plate Gauntlets, Plate Greaves, Polished Porphyry Beads, Poor Man's Belt... further results
Stockpile Leather (45)
Leather is animal hides that have been strengthened through the process of Tanning. It is a widely used resource in early-mid game crafts and builds and is used to make things like basic armor, tools, clothing, etc, as well as used to create Hardened Leather.
1 How to Make Leather
3 Quality Formulas
In order to make leather, follow the steps below. Leather isn't exactly hard to make, but it is a time-gated process.
Wait 14-29 real-time hours for the hide to finish drying.
{\displaystyle {\frac {_{q}AvgBoard*2+_{q}AvgBlock}{3}}}
{\displaystyle {\frac {_{q}Water+_{q}TreeBark}{2}}}
{\displaystyle {\frac {_{q}Hide*3+_{q}Fluid*2+_{q}Tub}{6}}}
Leather Feather (2015-12-03) >"Added a stockpile for leather, and for that purpose a common type for leather and hardened leather called any leather."
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Leather&oldid=93854"
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Decimal Fractions – Important Formulas and Examples | Previous Papers - Question Paper
Decimal Fractions - Important Formulas and Examples
Decimal Fraction Formulas
Decimal Fractions: Fractions in which denominators are powers of 10 are known as decimal fractions.
1/10 = 1 tenth = .1
1/100 = 1 hundredth = .01
9/100 = 99 hundredth = .99
7/1000 = 7 thousandths =.007
Conversion of a Decimal Into Vulgar Fraction : Put 1 in the denominator under the decimal point and annex with it as many zeros as is the number of digits after the decimal point.
Now, remove the decimal point and reduce the fraction to its lowest terms.Thus,
2.008 = 2008/1000 = 251/125.
Annexing zeros to the extreme right of a decimal fraction does not change its value.
0.8 = 0.80 = 0.800, etc.
1.84/2.99 = 184/299 = 8/13
0.365/0.584 = 365/584 = 5
1) Addition and Subtraction of Decimal Fractions : The given numbers are so placed under each other that the decimal points lie in one column. The numbers so arranged can now be added or subtracted in the usual way.
2) Multiplication of a Decimal Fraction By a Power of 10 : Shift the decimal point to the right by as many places as is the power of 10. Thus, 5.9632 x 100 = 596,32; 0.073 x 10000 = 0.0730 x 10000 = 730.
Multiplication of Decimal Fractions : Multiply the given numbers considering them without the decimal point. Now, in the product, the decimal point is marked off to obtain as many places of decimal as is the sum of the number of decimal places in the given numbers. Suppose we have to find the product (.2 x .02 x .002).
Now, 2x2x2 = 8. Sum of decimal places = (1 + 2 + 3) = 6. .2 x .02 x .002 = .000008.
Dividing a Decimal Fraction By a Counting Number : Divide the given number without considering the decimal point, by the given counting number. Now, in the quotient, put the decimal point to give as many places of decimal as there are in the dividend. Suppose we have to find the quotient (0.0204 + 17). Now, 204 ^ 17 = 12. Dividend contains 4 places of decimal. So, 0.0204 + 17 = 0.0012.
Dividing a Decimal Fraction By a Decimal Fraction: Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number. Now, proceed as above.
Thus,0.00066/0.11 = (0.00066*100)/(0.11*100) = (0.066/11) = 0.006V
Comparison of Fractions: Suppose some fractions are to be arranged in ascending or descending order of magnitude. Then, convert each one of the given fractions in the decimal form, and arrange them accordingly. Suppose, we have to arrange the fractions 3/5, 6/7 and 7/9 in descending order.
7/9 = 0.777....
since 0.857 > 0.777...> 0.6
so 6/7>7/9>3/5
Recurring Decimal : If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal. In a recurring decimal, if a single figure is repeated, then it is expressed by putting a dot on it. If a set of figures is repeated, it is expressed by putting a bar on the set, Thus
1/3 = 0.3333….= 0.3; 22 /7 = 3.142857142857.....= 3.142857
Converting a Pure Recurring Decimal Into Vulgar Fraction : Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures. thus ,
0.5 = 5/9; 0.53 = 53/59 ;0.067 = 67/999; etc...
Mixed Recurring Decimal: A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal. e.g., 0.17333= 0.173.
Converting a Mixed Recurring Decimal Into Vulgar Fraction : In the numerator, take the difference between the number formed by all the digits after decimal point (taking repeated digits only once) and that formed by the digits which are not repeated, In the denominator, take the number formed by as many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits. Thus
0.16 = (16-1) / 90 = 15/19 = 1/6;
Decimal Fraction Solved Examples
Example 1: Convert (i) 0.75 and (ii) 2.008 into vulgar fractions.
0.75=\frac{75}{100}=\frac{3}{4}
2.008=\frac{2008}{1000}=\frac{1004}{500}
=\frac{502}{250}=\frac{251}{125}
2\frac{1}{125}
Rule - for Converting a decimal into a vulgar fractions: In the denominator put 1 under decimal point and annex with it as many zeros as the number of digits after the decimal point. Next , remove the decimal point and write whole number in numerator. Reduce it to lowest form.
Remark - Annexing zeros to the extreme right of a decimal fraction does not change its value.
Addition and subtraction of decimal fractions.
Example 2: Convert (i) 45.7 + 3.098 + 0.79 + 0.8 = ?
(ii) 9.053 – 3.69 = ?
45.7 ii)
Thus for addition the given numbers are so placed under each other that the decimal point lie in column. Now, the number can be added or subtracted as usual putting the decimal point under the decimal points.
Example 3: Evaluate i) 6.4209
\times
100 ii) 0.0379
\times
1 ) 6.4209
\times
2) 0.0379
\times
100 = 0.379
\times
Thus when we multiply a decimal fraction by a power of 10, shift the decimal point to the right by as many places of decimal as is the power of 10.
Multiplication of two or more decimal fraction.
Example 4: Find the productions
\times
\times
\times
\times
\times
Sum the decimal place = (3 + 1) = 4
Now put the decimal after 4 that counting 4 digits from right thus 4.7397 is answer .
\times
Multiplied 279 and 131, result is 36549. Total decimal places are 4, put decimal counting four digits from right.
\times
\times
\times
0.0005 = 0.0000625
Multiplied 5, 4 times, get 625, now number of decimal places are 7, so place 4 zeros on left of 625 and then mark decimal.
Decision of decimal fraction by a non zero whole number.
Example 5: Evaluate (1) 0.72 + 9, (2) 0.0625 + 5, (3) 0.000121 + 11
1) 0.72 + 9 =
\frac{0.72}{9}
(Divided 72 by 9, get 8 as result now count decimal places in 0.72 that is ‘2’, hence place the decimal point on left of 08 that is, 0.08)
\frac{0.0625}{5}
(Divided 625 by 5, get 125, place the decimal point on left of 0125 that is , 0.0125)
3) 0.000121 + 11=
\frac{0.000121}{11}
(Divided 121 bt 11, get 11 as result, place decimal on left of 1.00011 that is, 0.000011.)
Division of a decimal fraction by a decimal fraction.
Example 6: Evaluate (1) 0.26 + 0.06 (2) 0.0077 + 0.11
\frac{0.36}{0.06}=\frac{0.36 \times 100}{0.06 \times 100}
\frac{36}{6}=6
2) 0.0077 + 0.11 =
\frac{0.0077}{0.11}
\frac{0.0077\times100}{0.11}
\frac{0.77}{11}
Decimal Fractions - Questions from Previous Year Papers
Decimal Fractions Aptitude
6.3204 × 100 =
\frac{63204}{10000}\times 100=\frac{63204}{100}
\frac{63204}{10000}\times 100=\frac{63204}{100}
What value will come in place of question mark in the following equation?
\frac{0.006}{?}=0.6
\frac{0.006}{x}
\frac{0.006}{0.6}
\frac{0.006\times 10}{0.6\times 10}
\frac{0.006}{x}
\frac{0.006}{0.6}
\frac{0.006\times 10}{0.6\times 10}
Required decimal =
\frac{1}{60\times 60}=\frac{1}{3600}
\frac{1}{60\times 60}=\frac{1}{3600}
\frac{0.1\times 0.1\times 0.1+0.02\times 0.02\times 0.02}{0.02\times 0.02\times 0.02+0.04\times 0.04\times 0.04}
is(SBI 1995)
\frac{(0.1)^{3}+(0.02)^{3}}{2^{3}[(0.1)^{3}+(0.02)^{3}]}=\frac{1}{8}
\frac{(0.1)^{3}+(0.02)^{3}}{2^{3}[(0.1)^{3}+(0.02)^{3}]}=\frac{1}{8}
When 0.232323... is converted into a fraction, then the result (SBI 2000)
\frac{23}{99}
\frac{23}{100}
0.232323.... =
\overline{{0.23}}=\frac{23}{99}
\overline{{0.23}}=\frac{23}{99}
The expression (11.98
\times
11.98 + 11.98
\times
x + 0.02
\times
0.02) will be a perfect square for x equal to
(SBI PO 2002)
(11.98)^{2}+(0.02)^{2}+11.98\times x
\times
\times
\times
\Rightarrow
(11.98)^{2}+(0.02)^{2}+11.98\times x
\times
\times
\times
\Rightarrow
\times
0.0162 is equal to:
6.48\times 10^{-4}
6.48\times 10^{-3}
6.48\times 10^{-2}
6.48\times 10^{-5}
\times
162 = 648 Sum of decimal places = 6
\times
0.0162 = 0.000648 =
6.48\times 10^{-4}
\times
\times
6.48\times 10^{-4}
Since the last digit to the extreme right will be zero (since 5
\times
4 = 20), so there will be 6 significant digits to the right of the decimal point.
\times
\frac{4.036}{0.04}=\frac{403.6}{4}
\frac{4.036}{0.04}=\frac{403.6}{4}
Given that 268
\times
74 = 19832, Find the value of 2.68
\times
So, 2.68
\times
0.74 = 1.9832.
\times
\frac{1}{3.718}
= 0.2689 then find the value of
\frac{1}{0.0003718}
\frac{1}{0.0003718} = \frac{1}{10000} = 10000\times \frac{1}{3.718}
=> 10000 × 0.2689 = 2689
\frac{1}{0.0003718} = \frac{1}{10000} = 10000\times \frac{1}{3.718}
What value will replace the question mark in the following equations?
Let 5172.49 + 378.352 + x = 9318.678.
Then, x = 9318.678 – (5172.49 + 378.352)
=> 9318.678 – 550.842
=> 3767.836.
\frac{29.94}{1.45}=\frac{299.4}{14.5}
(\frac{2994}{14.5}\times \frac{1}{10})
\frac{172}{10}
\frac{29.94}{1.45}=\frac{299.4}{14.5}
(\frac{2994}{14.5}\times \frac{1}{10})
\frac{172}{10}
Please comment on Decimal Fractions - Important Formulas and Examples
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Slider-crank linkage - 3D Animation
Slider-crank linkage (6851 views - Mechanism & Kinematics)
A slider-crank linkage is a four-link mechanism with three revolute joints and one prismatic, or sliding, joint. The rotation of the crank drives the linear movement the slider, or the expansion of gases against a sliding piston in a cylinder can drive the rotation of the crank. There are two types of slider-cranks: in-line and offset. In-line: An in-line slider-crank has its slider positioned so the line of travel of the hinged joint of the slider passes through the base joint of the crank. This creates a symmetric slider movement back and forth as the crank rotates. Offset: If the line of travel of the hinged joint of the slider does not pass through the base pivot of the crank, the slider movement is not symmetric. It moves faster in one direction than the other. This is called a quick-return mechanism.There are also two methods to design each type: graphical and analytical.
It has been suggested that Crosshead be merged into this article. (Discuss) Proposed since October 2018.
Find sources: "Slider-crank linkage" – news · newspapers · books · scholar · JSTOR (March 2018) (Learn how and when to remove this template message)
A slider-crank linkage is a four-link mechanism with three revolute joints and one prismatic, or sliding, joint.[1] The rotation of the crank drives the linear movement the slider, or the expansion of gases against a sliding piston in a cylinder can drive the rotation of the crank.
In-line: An in-line slider-crank has its slider positioned so the line of travel of the hinged joint of the slider passes through the base joint of the crank. This creates a symmetric slider movement back and forth as the crank rotates.
Offset: If the line of travel of the hinged joint of the slider does not pass through the base pivot of the crank, the slider movement is not symmetric. It moves faster in one direction than the other. This is called a quick-return mechanism.
There are also two methods to design each type: graphical and analytical.
1 Kinematics of the in-line slider-crank
1.2 Graphical approach
2 Offset slider-crank design
3 Slider-crank inversions
Kinematics of the in-line slider-crank
{\displaystyle x=r\cos \alpha +l}
where x is the distance of the end of the connecting rod from the crank axle, l is the length of the connecting rod, r is the length of the crank, and α is the angle of the crank measured from top dead center (TDC). Technically, the reciprocating motion of the connecting rod departs from sinusoidal motion due to the changing angle of the connecting rod during the cycle, the correct motion, given by the Piston motion equations is:
{\displaystyle x=r\cos \alpha +{\sqrt {l^{2}-r^{2}\sin ^{2}\alpha }}}
As long as the connecting rod is much longer than the crank
{\displaystyle l>>r}
the difference is negligible. This difference becomes significant in high-speed engines, which may need balance shafts to reduce the vibration due to this "secondary imbalance".
{\displaystyle \tau =Fr\sin(\alpha +\beta )\,}
{\displaystyle \tau \,}
is the torque and F is the force on the connecting rod. But in reality, the torque is maximum at crank angle of less than α = 90° from TDC for a given force on the piston. One way to calculate this angle is to find out when the Connecting rod smallend (piston) speed becomes the fastest in downward direction given a steady crank rotational velocity. Piston speed x' is expressed as:
{\displaystyle x'=(-r\sin \alpha -{\frac {r^{2}\sin \alpha \cos \alpha }{\sqrt {l^{2}-r^{2}\sin ^{2}\alpha }}}){\frac {d\alpha }{dt}}}
For example, for rod length 6" and crank radius 2", numerically solving the above equation finds the velocity minima (maximum downward speed) to be at crank angle of 73.17615° after TDC. Then, using the triangle sine law, it is found that the crank to connecting rod angle is 88.21738° and the connecting rod angle is 18.60647° from vertical (see Piston motion equations#Example).
When the crank is driven by the connecting rod, a problem arises when the crank is at top dead centre (0°) or bottom dead centre (180°). At these points in the crank's cycle, a force on the connecting rod causes no torque on the crank. Therefore, if the crank is stationary and happens to be at one of these two points, it cannot be started moving by the connecting rod. For this reason, in steam locomotives, whose wheels are driven by cranks, the connecting rods are attached to the wheels at points separated by some angle, so that regardless of the position of the wheels when the engine starts, at least one connecting rod will be able to exert torque to start the train.
An in-line crank slider is oriented in a way in which the pivot point of the crank is coincident with the axis of the linear movement. The follower arm, which is the link that connects the crank arm to the slider, connects to a pin in the center of sliding object. This pin is considered to be on the linear movement axis. Therefore, to be considered an in-line crank slider, the pivot point of the crank arm must be in-line with this pin point. The stroke((ΔR4)max) of an in-line crank slider is defined as the maximum linear distance the slider may travel between the two extreme points of its motion. With an in-line crank slider, the motion of the crank and follower links is symmetric about the sliding axis. This means that the crank angle required to execute a forward stroke is equivalent to the angle required to perform a reverse stroke. For this reason, the in-line slider-crank mechanism produces balanced motion. This balanced motion implies other ideas as well. Assuming the crank arm is driven at a constant velocity, the time it takes to perform a forward stroke is equal to the time it takes to perform a reverse stroke.
The graphical method of designing an in-line slider-crank mechanism involves the usage of hand-drawn or computerized diagrams. These diagrams are drawn to scale in order for easy evaluation and successful design. Basic trigonometry, the practice of analyzing the relationship between triangle features in order to determine any unknown values, can be used with a graphical compass and protractor alongside these diagrams to determine the required stroke or link lengths.
When the stroke of a mechanism needs to be calculated, first identify the ground level for the specified slider-crank mechanism. This ground level is the axis on which both the crank arm pivot-point and the slider pin are positioned. Draw the crank arm pivot point anywhere on this ground level. Once the pin positions are correctly placed, set a graphical compass to the given link length of the crank arm. Positioning the compass point on the pivot point of the crank arm, rotate the compass to produce a circle with radius equal to the length of the crank arm. This newly drawn circle represents the potential motion of the crank arm. Next, draw two models of the mechanism. These models will be oriented in a way that displays both the extreme positions of the slider. Once both diagrams are drawn, the linear distance between the retracted slider and the extended slider can be easily measured to determine the slider-crank stroke.
The retracted position of the slider is determined by further graphical evaluation. Now that the crank path is found, draw the crank slider arm in the position that places it as far away as possible from the slider. Once drawn, the crank arm should be coincident with the ground level axis that was initially drawn. Next, from the free point on the crank arm, draw the follower link using its measured or given length. Draw this length coincident with the ground level axis but in the direction toward the slider. The unhinged end of the follower will now be at the fully retracted position of the slider. Next, the extended position of the slider needs to be determined. From the pivot point of the crank arm, draw a new crank arm coincident with the ground level axis but in a position closest to the slider. This position should put the new crank arm at an angle of 180 degrees away from the retracted crank arm. Then draw the follower link with its given length in the same manner as previously mentioned. The unhinged point of the new follower will now be at the fully extended position of the slider.
Both the retracted and extended positions of the slider should now be known. Using a measuring ruler, measure the distance between these two points. This distance will be the mechanism stroke, (ΔR4)max.
To analytically design an in-line slider crank and achieve the desired stroke, the appropriate lengths of the two links, the crank and follower, need to be determined. For this case, the crank arm will be referred to as L2, and the follower link will be referred to as L3. With all in-line slider-crank mechanisms, the stroke is twice the length of the crank arm. Therefore, given the stroke, the length of the crank arm can be determined. This relationship is represented as:
L2 = (ΔR4)max ÷ 2
Once L2 is found, the follower length (L3) can be determined. However, because the stroke of the mechanism only depends on the crank arm length, the follower length is somewhat insignificant. As a general rule, the length of the follower link should be at least 3 times the length of the crank arm. This is to account for an often undesired increased acceleration yield, or output, of the connecting arm.
Offset slider-crank design
The analytical method for designing an offset crank slider mechanism is the process by which triangular geometry is evaluated in order to determine generalized relationships among certain lengths, distances, and angles. These generalized relationships are displayed in the form of 3 equations and can be used to determine unknown values for almost any offset slider-crank. These equations express the link lengths, L1, L2, and L3, as a function of the stroke,(ΔR4)max, the imbalance angle, β, and the angle of an arbitrary line M, θM. Arbitrary line M is a designer-unique line that runs through the crank pivot point and the extreme retracted slider position. The 3 equations are as follows:
L1 = (ΔR4)max × [(sin(θM)sin(θM - β)) / sin(β)]
L2 = (ΔR4)max × [(sin(θM) - sin(θM - β)) / 2sin(β)]
L3 = (ΔR4)max × [(sin(θM) + sin(θM - β)) / 2sin(β)]
With these relationships, the 3 link lengths can be calculated and any related unknown values can be determined.
Slider-crank inversions
A slider-crank is a four-bar linkage that has a crank that rotates coupled to a slider that the moves along a straight line. This mechanism is composed of three important parts: The crank which is the rotating disc, the slider which slides inside the tube and the connecting rod which joins the parts together. As the slider moves to the right the connecting rod pushes the wheel round for the first 180 degrees of wheel rotation. When the slider begins to move back into the tube, the connecting rod pulls the wheel round to complete the rotation.
This inversion is obtained when link 3 (connecting rod) is fixed. Application- Slotted crank mechanism, Oscillatory engine etc..,
Inverted slider crank mechanism.
Spatial slider-crank mechanism
Crank (mechanism)CrankpinCrankshaftCrankshaft position sensorDifferential (mechanical device)Eccentric (mechanism)Knuckle joint (mechanical)Leg mechanismLinkage (mechanical)Mechanical engineeringMechanical systemMechanical toyMechanism (engineering)Shaft (mechanical engineering)Slider crank chain inversionFour-bar linkageCrankcaseEngine blockCamshaft
This article uses material from the Wikipedia article "Slider-crank linkage", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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An Articulated Robotic Forceps Design With a Parallel Wrist-Gripper Mechanism and Parasitic Motion Compensation | J. Mech. Des. | ASME Digital Collection
Merve Bazman,
Merve Bazman 1
Powertrain Systems & Controls,
AVL Research and Engineering Turkey
1Corresponding author. Email: merve.bazman@avl.com
Nural Yilmaz,
Nural Yilmaz
Email: nural.yilmaz@marun.edu.tr
Email: ugur.tumerdem@marmara.edu.tr
Merve Bazman
Bazman, M., Yilmaz, N., and Tumerdem, U. (February 15, 2022). "An Articulated Robotic Forceps Design With a Parallel Wrist-Gripper Mechanism and Parasitic Motion Compensation." ASME. J. Mech. Des. June 2022; 144(6): 063303. https://doi.org/10.1115/1.4053465
In this paper, a novel four degrees-of-freedom (4DOF) articulated parallel forceps mechanism with a large orientation workspace (
±90deg
in pitch and yaw,
360deg
in roll rotations) is presented for robotic minimally invasive surgery. The proposed 3RSR-1UUP parallel mechanism utilizes a UUP center leg that can convert thrust motion of the 3RSR mechanism into gripping motion. This design eliminates the need for an additional gripper actuator, but also introduces the problem of unintentional gripper opening/closing due to parasitic motion of the 3RSR mechanism. Here, position kinematics of the proposed mechanism, including the workspace, is analyzed in detail, and a solution to the parasitic motion problem is provided. Human-in-the-loop simulations with a haptic interface are also performed to confirm the feasibility of the proposed design.
kinematics, manipulator theory, mechanism theory, medical and bio design of mechanisms and robotics, parallel manipulators, parallel robots, robot design, robot kinematics, robotic systems, simulation-based design
Actuators, Design, Grippers, Kinematics, Robotics, Simulation, Thrust, Yaw, Motion compensation, End effectors
The Intuitivetm Telesurgery System: Overview and Application
IEEE ICRA International Conference on Robotics and Automation
Development of 4-DOFs Forceps With Force Sensing Using Pneumatic Servo System
Proceedings of 2006 IEEE International Conference on Robotics and Automation. ICRA 2006
, IEEE,pp. 2250–2255.
Design and Synthesis of Wire-Actuated Universal-Joint Wrists for Surgical Applications
J. Robot. Surg.
Design of a Tether-Driven Minimally Invasive Robotic Surgical Tool With Decoupled Degree-of-Freedom Wrist
Decoupled Cable Driven Grasper Design Based on Planetary Gear Theory
Bolǵar
, “Teleoperation of a surgical robot using force feedback”.
Ĺetoublon
Development of a Compact Cable-Driven Laparoscopic Endoscope Manipulator
Robotic Laparoscopic Surgery: A Comparison of the da Vinci and Zeus Systems
Robotic Surgery Assisted by the Zeus System
Endourooncology
A New Spherical Wrist for Minimally Invasive Robotic Surgery
ISR 2010 (41st International Symposium on Robotics) and ROBOTIK 2010 (6th German Conference on Robotics), VDE
Design of a Novel 4-DOF Wrist-Type Surgical Instrument With Enhanced Rigidity and Dexterity
On the Kinematics of the 3-Rrur Spherical Parallel Manipulator
Development of a Dexterous Minimally-Invasive Surgical System With Augmented Force Feedback Capability
Werthsch¨utzky
A Novel Piezoelectric Driven Laparoscopic Instrument With Multiple Degree of Freedom Parallel Kinematic Structure
Abo-Ismail
Development of a New 4-DOF Endoscopic Parallel Manipulator Based on Screw Theory for Laparoscopic Surgery
Dexterous and Back-Drivable Parallel Robotic Forceps Wrist for Robotic Surgery
Proceedings of 2018 IEEE 15th International Workshop on Advanced Motion Control (AMC)
External Force/Torque Estimation on a Dexterous Parallel Robotic Surgical Instrument Wrist
Robotic Forceps Manipulator With a Novel Bending Mechanism
Development of Endoscopic Forceps Manipulator Using Multi-slider Linkage Mechanisms
J. Jpn Soc. Comput. Aided Surg.
Single-Port Robot-Assisted Laparoscopic Radical Prostatectomy : Initial Experience and Technique With the da Vinci ^a SP Platform
Technical Modifications Necessary to Implement the da Vinci Single-Port Robotic System
Development and Kinematic Analysis of a Redundant, Modular and Back Drivable Laparoscopic Surgery Robot
Proceedings of 2018 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM)
Akbarzadeh-T
Position, Jacobian and Workspace Analysis of a 3-psp Spatial Parallel Manipulator
Workspace Analysis of a Six-Degrees of Freedom, Three-Prismatic-Prismatic-Spheric-Revolute Parallel Manipulator
M˙IC
Partial Gravity Compensation of a Surgical Robot
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The correct solution for the longest path through the graph is
7, 3, 1, 99
. This is clear to us because we can see that no other combination of nodes will come close to a sum of
99
, so whatever path we choose, we know it should have
99
in the path. There is only one option that includes
99
7, 3, 1, 99
The greedy algorithm fails to solve this problem because it makes decisions purely based on what the best answer at the time is: at each step it did choose the largest number. However, since there could be some huge number that the algorithm hasn't seen yet, it could end up selecting a path that does not include the huge number. The solutions to the subproblems for finding the largest sum or longest path do not necessarily appear in the solution to the total problem. The optimal substructure and greedy choice properties don't hold in this type of problem.
_\square
Largest-price Algorithm: At the first step, we take the laptop. We gain
12
units of worth, but can now only carry
25-22 = 3
units of additional space in the knapsack. Since no items that remain will fit into the bag, we can only take the laptop and have a total of
12
units of worth.
Smallest-sized-item Algorithm: At the first step, we will take the smallest-sized item: the basketball. This gives us
6
units of worth, and leaves us with
25-7 = 18
units of space in our bag. Next, we select the next smallest item, the textbook. This gives us a total of
6+9 =15
18-9 = 9
units of space. Since no remaining items are
9
units of space or less, we can take no more items.
The greedy algorithms yield solutions that give us
12
units of worth and
15
units of worth. But neither of these are the optimal solution. Inspect the table yourself and see if you can determine a better selection of items.
Taking the textbook and the PlayStation yields
9+9=18
units of worth and takes up
10+9=19
units of space. This is the optimal answer, and we can see that a greedy algorithm will not solve the knapsack problem since the greedy choice and optimal substructure properties do not hold.
_\square
Dijkstra's algorithm is used to find the shortest path between nodes in a graph. The algorithm maintains a set of unvisited nodes and calculates a tentative distance from a given node to another. If the algorithm finds a shorter way to get to a given node, the path is updated to reflect the shorter distance. This problem has satisfactory optimization substructure since if
A
B,
B
C
, and the path must go through
A
B
to get to the destination
C
, then the shortest path from
A
B
and the shortest path from
B
C
must be a part of the shortest path from
A
C
. So the optimal answers from the subproblems do contribute to the optimal answer for the total problem. This is because the algorithm keeps track of the shortest path possible to any given node.
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Welcome to Real Digital
Academic Price Request
Topic: Electric Charges, Voltage, and Current
All matter is made up of atoms that contain both positively and negatively charged particles (protons and electrons). Surrounding every charged particle is an electric field that can exert force on other charged particles. A positive field surrounds a proton, and a negative field surrounds an electron. Field strength is the same for every electron and proton, with a magnitude of one “fundamental unit” of
1.602×10^{-19}
Coulombs. A coulomb is a measure of charge derived (in a somewhat circular fashion) from a measurement of electric current—one coulomb of charge is transferred by one ampere of current in one second (to get a matter of scale, one coulomb of charge flows through a 120W light bulb in one second). If one coulomb of protons could be isolated and held one meter apart from one coulomb of electrons, an attractive and equally repellent force (given by Coulombs law) of
8.988×10^9
Newtons, equivalent to almost one million tons at the earth’s surface, would exist between them. It is this large intra-particle force that is harnessed to do work in electric circuits.
Figure 1. Coulomb’s Law
A positive electric field surrounding a group of one or more protons will exert a repelling force on other groups of protons, and an attracting force on groups of electrons (example: magnets). Since an electric field can cause charged particles to move, it can do some amount of work, and so it is said to have potential energy. The amount of energy an electric field can impart to unit charge is measured in joules per coulomb, more commonly known as voltage. For our purposes, voltage may be thought of as the “electro-motive force” that can cause charged particles to move. A power supply is a local, contained imbalance of electrons, with material on one side (the negative side) containing an abundance of electrons, and material on the other (positive) side containing a relative absence of electrons. The potential electrical energy available in the power supply, measured in volts, is determined by the number of electrons it can store, the separation distance between negative and positive materials, the properties of the barrier between the materials, and other factors. Some power supplies (like small batteries) output less than a volt, while others (like power generation stations) can output tens of thousands of volts. In general, power supplies of up to 9V – 12V are considered “safe” for humans to interact with, but some people can have adverse (and potentially fatal) interactions with even low-voltage supplies. In our work, we will not encounter any supplies above 5V.
Electrons carry the smallest possible amount of negative charge, and billions of them are present in even the tiniest piece of matter. In most materials, electrons are held firmly in place by heavier protons, these materials are called insulators. By contrast, in other materials (like metals) electrons can move more easily from atom to atom, these materials are called conductors. The movement of electrons in a conductor is called electric current, measured in amperes or amps. If a power supply is used to impress a voltage across a conductor, electrons will move from the negative side of the supply through the conductor towards the positive side. All materials, even conductors, exhibit some amount of resistance to the flow of electric current. The amount of resistance determines how much current can flow—the higher the resistance, the less current can flow. By definition, a conductor has very low resistance, so a conductor by itself would never be placed across a power supply because far too much current would flow, damaging either the supply or the conductor itself. An electronic component called a resistor would be used in series with the conductor to limit current flow (more on resistors later).
All matter contains atoms that have positive and negative charges.
If one coulomb of protons could be isolated and held one meter apart from one coulomb of electrons, an attractive and equally repellent force of
8.988×10^9
Newtons, equivalent to almost one milli on tons at the earth’s surface, would exist between them. This concept is known as Coulomb’s Law.
An energy field is commonly measured in voltage. This energy passes through materials that either help it move (conductors) or slow it down (insulators).
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Mathematical functions - Tales of Science & Data
This page will just list some common functions used in Machine Learning and Data Science in general. For the code, if you want to reproduce the plots, you just need to import Pyplot:
The big O notation is used in mathematics to signify the limiting behaviour of a function when it goes to
\infty
\lim_{x \to \infty} f
The letter "O" is used as per order of function.
Note that in computer science, the big O notation is used to classify algorithms by how they respond to changes in the input size.
The little O notation instead,
f(x) = o(g(x)) \ ,
g(x)
f(x)
The mathematical convolution of functions is the operation
(f \star g)(x) = \int_{-\infty}^{+\infty} dy f(y) g(x - y)
It is a symmetric operation. In fact,
(g \star f)(x) = \int_{-\infty}^{+\infty} dy g(y)f(x-y) \ ,
z = x-y
dy = -dz
(g \star f)(x) = -\int_{+\infty}^{-\infty} dz g(x-z)f(z) =\int_{-\infty}^{+\infty} dz g(x-z)f(z)
Some functions of common use in Machine Learning/Statistics
The Heaviside step function is of common use in lots of applications. It is just a simple step:
f(x) = \begin{cases} 1 \text{ if } x \geq 0 \\ 0 \text{ if } x < 0 \end{cases}
The softmax is a normalised exponential used in probability theory as a generalisation of the logistic function. What it does is transforming a K-dimensional vector
\mathbf{x}
of arbitrary real values into a vector of the same size with elements which are still real numbers but ranging in the interval [0,1] and such that their sum equals 1 (so they can represent probabilities). The function has the form
f(x_i) = \frac{e^{x_i}}{\sum_{j \in K} e^{x_j}}
The softmax is also often employed in the context of neural networks. It is called this way because it represents a softening of the max function in the sense that it is larger on the max of the array. See the example.
plt.title('Softmax function')
plt.xlabel('$x#x27;)
plt.ylabel('$y#x27;)
plt.savefig('softmax.png', dpi=200)
Logit and logistic functions
Given probability p, the odds are defined as
o = \frac{p}{1-p}
. The logit function is the logarithm of the odds:
L(p) = \ln{\frac{p}{1-p}}
A negative logit is for p < 0.5.
p = np.arange(0.1, 1.1, 0.1)
y = np.log(p/(1-p))
plt.plot(p, y)
plt.title('Logit function')
Now, the probability expressed as a function of the logit creates the logistic function:
L = \ln{\frac{p}{1-p}} \Leftrightarrow -L = \ln{\frac{1}{p} - 1} \Leftrightarrow \frac{1}{p} = 1 + e^{-L} \Leftrightarrow p = \frac{1}{1+e^{-L}}
L = np.arange(-5, 5, 0.2)
p = 1/(1 + np.exp(-L))
plt.plot(L, p)
plt.title('Logistic function')
plt.xlabel('logit')
plt.ylabel('$p#x27;)
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Topic: A First Look At Circuits
A First Look At Circuits
An electric circuit consists of a power supply, one or more “load” devices that consume electric power to produce some useful output like light, heat, or motion, an optional means of control (e.g., an on-off switch), and conducting wires to move electric power between the power supply and load. In an electric circuit, power must flow from the positive terminal of the power supply through one or more load devices and back to the negative terminal of the power supply, thereby forming a complete circuit. If the connections between the load and either the positive or negative terminals of a power supply are interrupted, the circuit will be broken and the load will not receive any current.
Figure 1. A First Look at Circuits
Figure 2. Current Flow in Circuit
A power supply may be thought of as reservoirs of positive and negative charges that are held in close proximity, but that cannot recombine within the power supply itself. Positive and negative terminals on the supply make the charges available to an outside circuit. When these terminals are connected through an external circuit, charges flow from the reservoirs through the terminals and load and recombine within the power supply.
The charged particles available from a power supply have a potential energy equal to the amount of work done to separate them. This potential energy difference is measured in volts (or voltage). Thus, the voltage of a power supply is a measure of the “electric potential” or the “electro-motive force” that can force charge to flow through a circuit. Charge is carried by electrons, and the flow of electrons through a circuit is called electric current. The flow rate of electric current is measured in amperes, where one ampere is equal to one coulomb
(6.241×10^{18})
of electrons per second flowing through a circuit. As current flows through the load, potential energy is converted to heat, light, motion (through a magnetic field), or other useful outputs.
A typical power supply has large amounts of charge available at a given voltage, so that large and varying amounts of current can be supplied without a change in voltage. Most power supplies for “desktop” circuits produce voltages in the range of 3.3V to about 12V, a range that is generally considered safe for humans. A typical desktop supply (or plug-in “wall wart” supply) can produce anywhere from 100mA to 5A of current at the specified voltage – enough to power most small to medium sized experimental or lab-based circuits. The voltage output by a power supply is typically shown next to the supply in a schematic.
Any load device in a circuit will present some amount of resistance to the flow of electric current (resistance in measured in Ohms). The voltage available to force current through the load, and the resistance of the load determines how much current will flow according to Ohm’s law:
V=I×R
Some conductive wires in a circuit transport power between the power supply and the load devices. These wires, often called “rails”, are held steady at the same voltage, and they deliver electric power to devices around the circuit as it is needed. Other conductive wires move information between devices in a circuit – these wires transport “signals”. Signals differ from rails in that they transport information, not power. They use less current, and their voltage changes over time to encode or represent new and changing information. Signals can move information from one circuit component to one other, or from one component to several others. All the conductors and components in a circuit that are connected together by a single signal are said to form a circuit node; all the components connected to any given node access the same information.
Figure 3. Signals in Electric Circuits
Electric Vs. Electronic Circuits
Electric circuits use power rails and simple control means (like manual switches) to drive basic load devices like lights, heaters, and motors. Electronic circuits also use electric current to power load devices, but they differ in a crucial way – the devices in an electronic circuit use and/or are controlled by electric signals instead of manual switches. Electronic devices, like transistors, amplifiers, processors, and other semiconductor-based components (we will discuss semiconductors later) consume electric power, and they also use signals to define their operating state and control their behavior. As examples, a home-wiring circuit that provides power to a light bulb based on the state of a mechanical switch is an electric circuit, whereas a button to change channels on a TV is part of an electronic circuit.
Electronic circuits are often classified as “analog” and “digital” circuits. Analog circuits use variable voltages on conductive wires to represent information in a circuit – think of a microphone that produces a voltage between
0V
+5V
, where the voltage is in direct proportion to the incident sound pressure wave. At any given time, a lower voltage on a wire represents a lower sound pressure level, and a higher voltage represents a higher pressure. This continuous and variable voltage creates an electronic representation, or analog, of the sound wave as detected by the microphone, and that’s where the term analog circuit comes from. The analog signal could be amplified and sent to a loud speaker, recorded on magnetic tape, or it could be filtered, amplified, attenuated, or otherwise processed to change the signal amplitude or frequency content.
Analog circuits can suffer from poor performance if there is too much noise on their internal signals. In the microphone example, if the circuit used a
3.3V
power rail, then all sound information, from quiet whispers to loud noises must be represented as a voltage in the range
0V
3.3V
. If some noise source, like a poor power supply, or a circuit node that was too sensitive to ambient radio energy produced
10mV
of noise (which is not at all unlikely), then one part in 330 of the analog signal voltage would be washed out in noise, limiting the information the analog signal can carry.
Digital circuits also uses voltage levels on conductive wires to encode and represent information, but rather than use continuously varying voltages, they use only two distinct voltage levels. All information in a digital circuit is represented by a “logic high voltage”, which is typically defined as a voltage range between about 70% and 100% of the maximum system voltage (perhaps
3.3V
), and a “logic low voltage”, typically in the range of 0V to about 30% of the maximum system voltage. Because digital circuits use a wide voltage range to encode both a “high” and a “low” (or equivalently, a ‘1’ and a ‘0’), they are far less sensitive to noise. Using the same 3.3V example, a “high” digital signal could suffer from up to 600mV of noise without changing its definition as transporting a ‘1’.
Since digital circuits restrict nodes to operating at one of two distinct voltages, it is common practice to associate a circuit node at a logic high voltage (or Vdd) with a ‘1’, and a circuit node at a logic low voltage (or ground) with a ‘0’. Thus, every node in a digital circuit is either at a ‘1’ or ‘0’, not counting the short amount of time it takes to transition between those states. And since all circuit nodes in a digital circuit can be associated with a ‘1’ or ‘0’, it is common to use binary numbers when describing the state of a digital circuit. Any individual wire (or node) can transport either a ‘1’ or ‘0’, and a group of wires viewed as a single logical unit can transport a binary number. For example, if 8 wires are viewed as a single logical unit (called a “bus”), then 8-bit binary numbers can be transported by that bus.
Figure 4. Digitize an Analog signal
Digital circuits can represent the same information as analog circuits, but the analog information must first be converted into digital form. Any continuously varying analog signal can be represented as a sequence of discrete numbers that define the analog signal’s amplitude at a given time. The requirements of the signal that is to be “digitized” dictate how many points per second are required for an adequate representation, and how many bits per point. If, for example, a 0V to 3.3V analog signal produced by a microphone was to be “digitized” for representation in a digital circuit, the maximum frequency content that needed to be preserved in the digital representation would dictate the sample rate (in general, the sample rate is at least two times, and up to 10 times the analog frequency that must be preserved), and the required dynamic range would dictate the number of bits (dynamic range is the ratio between the smallest and largest signal amplitudes that can be represented). For regular spoken voice, about 5 KHz of analog frequencies should be preserved, with about 48dB of dynamic range, indicating a sample rate of 10 KHz or more, with at least 8 bits per sample.
In digital circuits, the Vdd and GND rails supply electric power to the circuit and define voltages for representing a ‘1’ and a ‘0’. Vdd may be thought of as the “source” of positive charges in a circuit and the source of ‘1’ information, and GND may be thought of as the “source” of negative charges in a circuit and the source of ‘0’ information. In modern digital systems, Vdd and GND are separated by anywhere from 1 to 5 volts. Older or inexpensive circuits typically use 5 volts, while newer circuits use 1-3 volts.
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Syntactic parametricity strikes again
By: Gabriel Scherer, Li-Yao Xia
In this blog post, reporting on a collaboration with Li-Yao Xia, I will show an example of how some results that we traditionally think of as arising from free theorems / parametricity can be established in a purely “syntactic” way, by looking at the structure of canonical derivations. More precisely, I prove that
\newcommand{\List}[1]{\mathsf{List}~#1} \newcommand{\Fin}[1]{\mathsf{Fin}~#1} \newcommand{\Nat}[1]{\mathbb{N}} \newcommand{\rule}[2]{\frac{\displaystyle \array{#1}}{\displaystyle #2}} \newcommand{\judge}[2]{{#1} \vdash {#2}} \newcommand{\emptyrule}[1]{\begin{array}{c}\\[-1em] #1 \end{array}} ∀α. \List α → \List \alpha
Π(n:\Nat{}). \List{(\Fin{n})}
\Fin{n}
is the type of integers smaller than
, corresponding to the set
\{0, 1, \dots, n-1\}
Racket 6.9 and Windows 10 Creators Update
2017-05-26 :: racket, windows, bugs
Racket 6.9 was released in April and it has been smooth sailing for many people. However, some people using the Windows 10 Creators Update have been experiencing crashes, not just for Racket, but for the whole operating system. This is due to a bug in Windows. We have contacted Microsoft; they have classified the bug as (1) a stack overflow and (2) not a security hazard, and intend to add a fix in a future version of Windows.
The next version of Racket will include a patch to help avoid triggering the bug. Until then, one work-around is to run Racket in a virtual machine (VM). This blog post is a step-by-step guide on how to install a VM for Racket.
A VirtualBox image with Racket preinstalled can be downloaded here:
https://github.com/nuprl/website/releases/download/racket69vm/Racket_6_9.ova
The username and password for this machine are both racket.
Programming Language Conference in Russia
By: Artem Pelenitsyn
In April 3—5 I took part into a Russian conference exclusively devoted to programming languages: Programming Languages and Compilers (Google.Translated version of the site). I was a member of organizing committee and had a paper there.
This is the first conference in Russia highly focused on our area of PL. At least for the last several decades (I believe, there were conferences of the kind back in USSR). The conference was devoted to the memory of prominent Soviet PL-researcher from Rostov-on-Don, Adolf Fuksman who worked on ideas quite similar to what we know as the aspect-oriented programming back in the 70-s.
Building a Website with Scribble
The source code for the PRL website is written using Scribble, the Racket documentation tool. I am very happy with this choice, and you should be too!
Artifacts for Semantics
Gabriel Scherer and I recently wrote an artifact for a semantics paper on a typed assembly language interoperating with a high-level functional language.
No Good Answers, Gradually Typed Object-Oriented Languages
2017-05-09 :: HOPL, Gradual Typing
By: Ben Chung
Rank Polymorphism
2017-05-04 :: array language
By: Justin Slepak
Rank polymorphism gives you code reuse on arguments of different dimensions. Take a linear interpolation function (let’s just call it lerp) for scalars:
(λ ((lo 0) (hi 0) (α 0)) (+ (* lo (- 1 α)) (* hi α)))
The number marks on each argument indicate the expected “rank” of the argument: how many dimensions it should have. In this case, each one is marked 0, indicating a scalar (i.e., 0-dimensional) argument. The function is usable as-is for
α-blending two RGB pixels
dimming or brightening an image
fade transition between video scenes
Categorical Semantics for Dynamically Typed Programming Languages
2017-05-01 :: HOPL, category theory, dynamic typing, gradual typing
What is Soft Typing?
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Chain - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : ChainTools Subpackage : Chain
Chain(lp, rc, R)
The command Chain(lp, rc, R) returns the regular chain obtained by extending rc with lp.
It is assumed that lp is a list of non-constant polynomials sorted in increasing main variable, and that any main variable of a polynomial in lp is strictly greater than any algebraic variable of rc.
It is also assumed that the polynomials of rc together with those of lp form a regular chain.
The function Chain allows the user to build a regular chain without performing any expensive check and without splitting or simplifying. On the contrary, the functions Construct and ListConstruct check their input completely. In addition, they simplify the input polynomials and they may also factorize some of them, leading to a list of regular chains (that is, a split) rather than a single one.
The function Chain is used by some algorithms where one tries to split the computations as little as possible. This is the case for the function EquiprojectableDecomposition.
This command is part of the RegularChains[ChainTools] package, so it can be used in the form Chain(..) only after executing the command with(RegularChains[ChainTools]). However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][Chain](..).
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{ChainTools}\right):
R≔\mathrm{PolynomialRing}\left([t,x,y,z]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
\mathrm{pz}≔{z}^{2}+2z+1
\textcolor[rgb]{0,0,1}{\mathrm{pz}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{py}≔z{y}^{2}+1
\textcolor[rgb]{0,0,1}{\mathrm{py}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{pt}≔t\left(x+y\right)+y+z
\textcolor[rgb]{0,0,1}{\mathrm{pt}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}
\mathrm{qy}≔\mathrm{expand}\left(3z\mathrm{py}\right)
\textcolor[rgb]{0,0,1}{\mathrm{qy}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}
\mathrm{qt}≔\mathrm{expand}\left({\left(x+y\right)}^{2}\mathrm{pt}\right)
\textcolor[rgb]{0,0,1}{\mathrm{qt}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}
\mathrm{rc}≔\mathrm{Empty}\left(R\right)
\textcolor[rgb]{0,0,1}{\mathrm{rc}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}
\mathrm{rc1}≔\mathrm{Chain}\left([\mathrm{pz},\mathrm{qy},\mathrm{qt}],\mathrm{rc},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{rc1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}
\mathrm{Equations}\left(\mathrm{rc1},R\right)
[\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{3}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]
\mathrm{lrc}≔\mathrm{ListConstruct}\left([\mathrm{pz},\mathrm{qy},\mathrm{qt}],\mathrm{rc},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{lrc}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}]
\mathrm{map}\left(\mathrm{Equations},\mathrm{lrc},R\right)
[[\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]]
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According to Newton’s law of cooling, the rate at which an object cools (or warms) is directly proportional to the temperature difference between the environment and the object itself. Three years ago the corpse of Dr. Deadman was discovered in the coroner’s office. The room temperature of the coroner’s office was (17ºC). The doctor’s body temperature was measured to be 27°C, which was 10°C below normal.
Dr. Deadman’s body was found at 5:05 p.m. An hour later the body had cooled to 26ºC. At what rate was the body cooling when it was found?
Approximately when was Dr. Deadman killed?
\frac{dT}{dt}=-k(T-17)
After solving the differential equation:
T(t)=Ce^{-kt}+17
Now use (0, 27) and (1, 26) to solve for the values of the parameters.
What is T′(0)?
Normal body temperature is 37ºC.
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Beer - Ring of Brodgar
Object(s) Required Barley Wort
Beer buffs Meat satiation by 0.5% and Sausage by 1.0% per gulp of Q10. It also gives a negative satiation against itself which results in a diminishing return.
If you use Seeds of Wheat everywhere where Seeds of Barley are required, you will end up with Weißbier instead.
Place Seeds of Barley on a Herbalist Table to germinate the seeds.
Roast Seeds of Sprouted Barley in a Kiln to create the Malt.
Grind Malted Barley at a Quern to create the Grist.
Boil Barley Grist with Hop Cones in a Cauldron to make Wort.
Store Wort in a Demijohn until it becomes Beer.
The proper way to drink Beer is from the Tankard, not drinking out of a proper vessel halves the effective quality of your Beer.
Quality 10 Beer recovers 10% stamina and drains 20% energy per 0.05L sip. Higher quality Beer will decrease the energy drain but stamina recovery remains the same at all qualities.
This makes high quality Beer useful for performing stamina draining tasks without having to eat as much to recover, thus preserving your hunger bonus. Though energy is always shown as a whole number, it is not simply rounded up or down. For example, Q11 Beer will alternate between draining 20% and 19% energy at regular intervals. Other drinkable liquids follow the same rules and formula.
{\displaystyle ({\frac {1}{\sqrt {quality/10}}}+1)*10}
Gives energy if stamina is already full.
Grade-A Milk (2016-04-12) >"Added/Re-Added Beer. Germinate Barley on a Herbalist Table. Roast Germinated Barley in a Kiln to get Malted Barley. Grind Malted Barley to Barley Grist. Craft Wort using hops and grist, and then store the wort in a Demijohn until it transforms to glorious Beer. Beer buffs Meat and Sausage by 0.5% and 1% respectively, per gulp at Q10."
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Beer&oldid=94374"
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Types of Code Coverage - MATLAB & Simulink - MathWorks América Latina
Function Call Coverage
If you have Embedded Coder®, Simulink® Coverage™ can perform several types of code coverage analysis for models in software-in-the-loop (SIL) mode, processor-in-the-loop (PIL) mode, and for the code within supported custom code blocks.
c=\sum _{1}^{N}\left({o}_{n}-1\right)
Relational boundary code coverage examines code that has relational operations. Relational boundary code coverage metrics align with those for model coverage, as described in Relational Boundary Coverage. Fixed-point values in your model are integers during code coverage.
Function coverage determines whether all the functions of your code have been called during simulation. For instance, if there are ten unique functions in your code, function coverage checks if all ten functions have been executed at least once during simulation.
Function call coverage determines whether all function call-sites in your code have been executed during simulation. For instance, if functions are called twenty times in your code, function call coverage checks if all twenty function calls have been executed during simulation.
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ICE score: a framework for prioritizing ideas and making better business choices | Croct Blog
March 15, 2022 ProductBy Juliana Amorim, Mariana Bonanomi and Gabriela Nascimento
Some kinds of choices are always hard. Figuring the best ideas for growing your business is often one of those. Costs, time, and opportunities are on the line. Therefore, knowing prioritization methods in depth is essential for anyone ahead of growth strategies and tactics. And even more critical than that is to understand how to use them on the business's behalf.
When optimizing your digital product or website, you need to prioritize specific actions. But which ones? It is essential to define effective methods for making the wisest choices.
AB testing is an excellent method to determine the best approach to maximize conversions. However, it may often cost you time and budget. Sometimes, it is necessary to step backward and ascertain which hypothesis has the best potential before testing them.
Bearing that in mind, we created this post with everything you need to know about the ICE Score, a functional method that will support your decision-making process when it's time to prioritize tasks, projects, and tests for your company.
What is the ICE Score method?
The ICE Score was developed by GrowthHackers' CEO, Sean Ellis, to improve the classic impact/effort analysis. This framework is popular among growth professionals and revolves around three combined factors to calculate the priority level you should attribute to a given optimization idea.
This method was designed for group practice, so product, marketing, and technology professionals, among others, feel welcome to show their points of view. It facilitates the engagement of the whole team, enabling better collaboration between different areas of expertise. Therefore, the ICE Score encourages more interdisciplinary and responsible workflows when it's time to define priorities.
How does the ICE Score work?
ICE stands for impact, confidence, and ease. The process consists of giving each of the three mentioned factors a score and multiplying it. There is no explicit rule on the minimum and the maximum scores for each, but using smaller ranges (such as 1 to 3 or 1 to 5) should make the process easier and avoid inaccurate deviations.
score = impact * confidence * ease
After scoring and multiplying the points, the prioritization becomes clear and explicit: the higher the score an item gets, the more critical it is.
Here's what each factor is all about:
The impact score determines the potential impact of a particular action or test if put into practice. Would it resonate directly with the primary business goals? How many areas or segments would benefit from this action?
The confidence score determines the team's confidence and guarantees for this specific action or idea. How likely is this option to succeed based on its belief? According to your experience, will this idea be effective? When team members have no previous experience with a given idea, the confidence level tends to be lower.
The ease score determines how easy it is to implement a test. Would this action be easy to execute? Would it include too many professionals? Would it be too complex or have a high implementation cost? The easiest it would be, the higher the score.
Other prioritization methods
Instead of simply applying frameworks without first considering whether they really make sense in your business, keep in mind that there are always many methods for you to test. Deciding which one is the best for you will depend on your project's stage.
Some companies benefit from fast, collaborative, and democratic methods. In contrast, others rely on more visual and simplistic scoring frameworks or customized customer-centric approaches, such as B2B businesses that serve large accounts.
What to keep in mind when deciding which method to apply
The tip here is to cool down and cut the overthinking off. Instead, you should just:
Acknowledge what the variables taken into account in the method are
Ask yourself how much each of them affects the business's primary goals
Acknowledge what data and tools you have previously gathered and are actionable to use.
In some cases, the variable "reach" matters more than other factors, or as much. It happens when the number of people reached after implementing a feature or executing an idea is more important than other metrics. In other cases, when companies have customer surveys and quantified customer satisfaction rates, they can consider "value" as the main prioritization criteria. This is due to their accurate understanding of the unique value they can deliver to customers.
We listed other methods that may meet your needs depending on your available resources and your company's main goals right now.
The acronym stands for reach, impact, confidence, and ease. It considers the same variables as the ICE Score method, plus the reach the prioritized experiment would have in a given time frame. For example, if the goal is to get 150 new customers by the end of the first quarter, the maximum reach score would be 150. If getting 1300 users converting from stage X to stage Y in 3 weeks is expected when implementing the new feature, 1300 would be your highest reach score. The equation is similar to the ICE Score, with the only difference that it also multiplies the other variables by the reach score.
MoSCoW is an acronym for must-have, should have, could have, and will not have. The purpose here is to distribute the possible ideas into four groups:
Must have: the ones that are essential for the product
Should have: the ones that are important but not essential
Could have: the ones that are nice to put into practice if possible
Will not have: the unnecessary ones, regardless of the time.
It can be used on an entire project or just part of it and is suitable for teams looking for a simple approach and who have a very clear due date for each task. Without well-defined time windows, you assume the risk of overloading the must-haves.
Firstly, you should define the time intervals such as the next month, the next semester, and the next year. Then, each team member receives 3 weighted voting dots (each one with numbers 1, 2, and 3 stamped on it) and assigns each one to an idea. These numbers are added up, and the ones with the highest score are must-haves, the ones with the second-highest score are should-haves, and so forth.
This model was created in 1984 by Dr. Noriaki Kano, and considers the possibility of implementation and customer satisfaction as the primary scoring criteria. In the process, you create a chart where the Y-axis indicates customer satisfaction while the X-axis indicates implementation difficulty. The closer an idea is to the upper right quadrant, the higher the score.
The method is suitable for teams with user browsing data that have already conducted customer surveys, and feel the need to adopt a more user-centric approach, which is often due to engineering-centric cultures.
When should I apply the ICE Score?
Now that you know what prioritization methods are and can find all sorts of scoring methods out there let's go back to the ICE Score and discuss how you should use it to get the best out of it.
Even the most popular methods among growth professionals are of minimal use when metrics are unreliable or when post-test activities do not match the results obtained in experiments. With the ICE Score, this wouldn't be any different.
Initially, the method was created to determine which AB tests to prioritize. However, many growth professionals also apply it to day-to-day product and marketing teams' backlog management due to its efficacy and functionality.
Ultimately, it is not just about determining which action to take or not. It can also contribute to planning deadlines and in daily demands management: the ICE methodology can help you calculate the focus needed for each project or task. Identifying the level of opportunity when building a tactic work schedule for a new feature launch is essential for the company's growth.
What you should beware about
As mentioned before, the ICE Score is efficient in many cases, but it is essential to be careful when applying it.
Strategies x tactics
The method has instant appeal to leaders who want to be known for making data-driven decisions, but it's still a tactic, not a strategy. What does this mean in practice? When it comes to the dynamic tech and digital products universe, things may change in the blink of an eye, so you need to make decisions quickly. On this typical startup daily basis, it is often not possible – and also not advisable – to gather an entire team in a room for a scoring process.
If you don't have precious time to waste, it is crucial to have a well-architected and tied strategy behind prioritization methods – leaving no loose ends between different tactics. Tactics won't ever replace strategic thinking and must be tied together to achieve a greater goal: always keep an eye on a north star metric when deciding whether or not to use a tactic at any given time.
Communication and scores definition
When the team defines the scores together, it is common to fall into traps such as a lack of alignment between stakeholders on what each variable means. Taking only opinions that are convenient for a single person into account is another problem since this person may already intend to put some tasks into practice.
When there's good communication about what each score means, the prioritized idea is, in fact, the one that makes the most sense for the team as a whole. You and your team can gradually improve the use of the method and decide if it's wiser to move on to the following ones or if it is time to rethink your tactics.
No matter how good a tactical plan is, if it is not well communicated in a language that the entire team understands, the practical aspects will never turn out the best way.
The benefits of using the ICE Score are many. The prioritization of ideas and projects ensures greater alignment between teams and better use of resources. Bringing the ICE methodology to everyday business can increase productivity, reduce costs, and generate more accurate results.
As mentioned in the Ladder blog, just like Growth Hacking was invented to make more meetings with tech founders who aren't interested in marketing, the ICE score is a solution marketers have found to make engineers listen to their needs.
We at Croct recommend using this methodology to ensure the continuous optimization of your product or website. Still, we emphasize that this must be done with caution so that you do not make the wrong decisions.
Did you like the method and want to see how it can help you? Check out our free template so you can apply the ICE Score! Just make a copy and start using it :)
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Gao, Shiliang1; Hodges, Reuven1; Orelowitz, Gidon1
1 Dept. of Mathematics, U. Illinois at Urbana-Champaign, Urbana, IL 61801, USA
We provide a non-recursive, combinatorial classification of multiplicity-free skew Schur polynomials. These polynomials are
G{L}_{n}
S{L}_{n}
, characters of the skew Schur modules. Our result extends work of H. Thomas–A. Yong, and C. Gutschwager, in which they classify the multiplicity-free skew Schur functions.
Keywords: Skew–Schur polynomial, Littlewood–Richardson tableaux, multiplicity-free.
Gao, Shiliang 1; Hodges, Reuven 1; Orelowitz, Gidon 1
author = {Gao, Shiliang and Hodges, Reuven and Orelowitz, Gidon},
title = {Multiplicity-free skew {Schur} polynomials},
TI - Multiplicity-free skew Schur polynomials
%T Multiplicity-free skew Schur polynomials
Gao, Shiliang; Hodges, Reuven; Orelowitz, Gidon. Multiplicity-free skew Schur polynomials. Algebraic Combinatorics, Volume 4 (2021) no. 6, pp. 1073-1117. doi : 10.5802/alco.192. https://alco.centre-mersenne.org/articles/10.5802/alco.192/
[1] Avdeev, Roman S.; Petukhov, Alexey V. Spherical actions on flag varieties, Mat. Sb., Volume 205 (2014) no. 9, pp. 3-48 | Article | MR: 3288423 | Zbl: 1327.14217
[2] Azenhas, Olga; Conflitti, Alessandro; Mamede, Ricardo Multiplicity-free skew Schur functions with full interval support, Sém. Lothar. Combin., Volume 75 ([2015-2019]), Paper no. Art. B75j, 35 pages | MR: 4072428 | Zbl: 1441.05223
[4] Fomin, Sergey; Greene, Curtis A Littlewood–Richardson miscellany, European J. Combin., Volume 14 (1993) no. 3, pp. 191-212 | Article | MR: 1215331 | Zbl: 0796.05091
[5] Fulton, William; Harris, Joe Representation theory. A first course, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991, xvi+551 pages | Article | MR: 1153249 | Zbl: 0744.22001
[6] Gutschwager, Christian On multiplicity-free skew characters and the Schubert calculus, Ann. Comb., Volume 14 (2010) no. 3, pp. 339-353 | Article | MR: 2737323 | Zbl: 1233.05201
[7] Hodges, Reuven; Lakshmibai, Venkatramani A classification of spherical Schubert varieties in the Grassmannian (2018) (preprint https://arxiv.org/abs/1809.08003)
[8] Howe, Roger Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv) (Israel Math. Conf. Proc.), Volume 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1-182 | MR: 1321638 | Zbl: 0844.20027
[9] Littlewood, Dudley E.; Richardson, Archibald R. Group characters and algebra, Philos. Trans. R. Soc. Lond., Ser. A, Contain. Pap. Math. Phys. Character, Volume 233 (1934), pp. 99-141 | Article | Zbl: 0009.20203
[10] Magyar, Peter; Weyman, Jerzy; Zelevinsky, Andrei Multiple flag varieties of finite type, Adv. Math., Volume 141 (1999) no. 1, pp. 97-118 | Article | MR: 1667147 | Zbl: 0951.14034
[11] Perrin, Nicolas On the geometry of spherical varieties, Transform. Groups, Volume 19 (2014) no. 1, pp. 171-223 | Article | MR: 3177371 | Zbl: 1309.14001
[12] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 2001, xii+581 pages | MR: 1676282 | Zbl: 0978.05002
[13] Stembridge, John R. Multiplicity-free products of Schur functions, Ann. Comb., Volume 5 (2001) no. 2, pp. 113-121 | Article | MR: 1904379 | Zbl: 0990.05130
[14] Stembridge, John R. Multiplicity-free products and restrictions of Weyl characters, Represent. Theory, Volume 7 (2003), pp. 404-439 | Article | MR: 2017064 | Zbl: 1060.17001
[15] Thomas, Hugh; Yong, Alexander Multiplicity-free Schubert calculus, Canad. Math. Bull., Volume 53 (2010) no. 1, pp. 171-186 | Article | MR: 2583223 | Zbl: 1210.14056
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Block of Wood - Ring of Brodgar
Object(s) Required Log, Stump, or Bush
Produced By Stone Axe, Metal Axe, Shovel, Metal Shovel
Required By Ancestral Shrine, Antler Steak Cutlery, Arched Chair, Archery Target, Archery Tower, Ashes, Bagpipe, Barter Pole, Barter Stand, Battering Ram, Battle Standard, Bee Skep, Birdhouse, Border Cairn, Box of Matches, Bug Collection, Bull Pipe, Bushcraft Fishingpole, Butcher's Cleaver, Cart, Catapult, Cellar, Cheese Rack, Chef's Pin, Chicken Coop, Churn, Cigar Box, Cistern, Clay Cauldron, Clogs, Cloth Armchair, Cloth Chair, Cloth Footstool, Coal, Compost Bin, Coracle, Cottage Table, Cottage Throne, Curding Tub, Cushioned Bench, Cushioned Stool, Demijohn, Display Case, Dovecote, Dressing Table, Drum & Sticks, Everglowing Ember, Exquisite Chest, Extraction Press, Feather Duster... further results
Stockpile Block of Wood (80)
The quality of the blocks is based on the quality of the tree or bush. The quality of your axe has no effect.
2 Block of Wood Products
Blocks of wood are produced by chopping either a log or bush with an axe, or by destroying a stump. Digging up stumps is considerably faster with a shovel, especially a metal shovel.
Removing a stump provides 4 blocks, while the amount of blocks received from logs and chopping bushes varies by type (not all bushes give blocks, see the Bush page). A Metal Axe produces more blocks than a Stone Axe, giving 50 blocks per spruce log.
Block of Wood Products
Right-clicking a block and select the "Split" option will split the blocks into 5 branches each. Branch quality obtained by splitting block of wood depend on the log and axe used, using the following formula:
{\displaystyle {\sqrt[{2}]{Axe*Block}}}
As blocks of wood have a vast variety of uses such as fuel, but if you have got high qualities axe it is better to split them to branches to gain better qualities fuel.
Bushes grow fast and are easy to chop, making them ideal for getting quality blocks quickly. Elderberry Bushes are arguably the best thing to grow for quality blocks as they grow faster than trees, drop some blocks, as well as give 8 elderberries which can be eaten or used in cooking or your precious Mirkwood Offerings, which will be bolstered by some higher quality Tree or Bush Seeds.
Block of Mirkwood is obtained from chopping Old Trunks or digging up Old Stumps, both often found in the swamp. Any log left to rot will eventually become an old trunk.
Beaver Salvage
Mirkwood block
Blocks from bushes
Blocks from trees
Gloomcap
Gnome's Cap
Trumpet Chantrelle
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Block_of_Wood&oldid=93201"
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In the present paper, we study in the harmonic analysis associated to the Weinstein operator, the boundedness on Lp of the uncentered maximal function. First, we establish estimates for the Weinstein translation of characteristic function of a closed ball with radius ε centered at 0 on the upper half space Rd-1× ]0,+∞[. Second, we prove weak-type L1-estimates for the uncentered maximal function associated...
Recenzja z Fulvia Furinghetti, Alexander Karp (eds), Researching the History of Mathematics Education. An International Overview, Springer, 2018, ss. 314 +XV.
Experiments with iterative methods for linear systems using GeoGebra and TI-Nspire
Gregor Milicic, Simon Plangg
Algorithms and algorithmic thinking are key topics in STEM Education. By using algorithms approximate solutions can be obtained for analytical unsolvable problems. Before new methods can be safely applied they have to be thoroughly tested in experiments. In this article we present a series of exercise where students can experiment with algorithms and test them using GeoGebra or the TI-Nspire. Based...
To the editor, Annales Universitatis Paedagogicae Cracoviensis
Cardinality of the sets of all bijections, injections and surjections
The results of Zarzycki for the cardinality of the sets of all bijections, surjections, and injections are generalized to the case when the domains and codomains are infinite and different. The elementary proofs the cardinality of the sets of bijections and surjections are given within the framework of the Zermelo-Fraenkel set theory with the axiom of choice. The case of the set of all injections...
Tłumaczenie: Emil Artin, Otto Schreier, Algebraiczna konstrukcja ciał rzeczywistych
Remarks on the Principle of Permanence of Forms
We discuss the role of a heuristic principle known as the Principle of Permanence of Forms in the development of mathematics, especially in abstract algebra. We try to find some analogies in the development of modern formal logic. Finally, we add a few remarks on the use of the principle in question in mathematical education.
Different ways of solving quadratic equations
Barbara Pieronkiewicz, James Tanton
In this paper we explore different ways of solving quadratic equations. Our main goal is to review traditional textbooks methods and offer an alternative, often side-stepped method based on the area model. We conclude that whereas traditional methods offer effective algorithms that quickly lead to the desired results, alternative methods may enhance meaningful and joyful learning.
Autonomy of Geometry
John T. Baldwin, Andreas Mueller
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam... > 2019 > 11 > 5-24
In this paper we present three aspects of the autonomy of geometry. (1) An argument for the geometric as opposed to the ‘geometric algebraic’ interpretation of Euclid’s Books I and II; (2) Hilbert’s successful project to axiomatize Euclid’s geometry in a first order geometric language, notably eliminating the dependence on the Archimedean axiom; (3) the independent conception of multiplication from...
On the golden number and Fibonacci type sequences
The paper presents, among others, the golden number
\varphi
as the limit of the quotient of neighboring terms of the Fibonacci and Fibonacci type sequence by means of a fixed point of a mapping of a certain interval with the help of Edelstein's theorem. To demonstrate the equality , where
f_n
is $n$-th Fibonacci number also the formula from Corollary \ref{cor1} has been applied. It was obtained using...
Analysis of mathematics teachers’ beliefs on mathematics instruction and teaching self-efficiencies within the scope of flow theory
S.Koza Çiftçi, Engin Karadağ
In this study, it was aimed to investigate the beliefs of mathematics teachers about mathematics instruction and their teaching self-efficacy within the scope of flow theory. Participants consists of a total of 228 mathematics teachers engaged in teaching at secondary and high school levels in Turkey; they were determined using the combinations of convenience and purposive sampling. Data from the...
Teoretyczne podstawy dzielenia z resztą liczb naturalnych wraz z uwagami o dzieleniu z resztą liczb całkowitych, wymiernych i rzeczywistych
In this article, I analyze the theoretical foundations of the division with remainder in the arithmetic of natural numbers. As a result of this analysis I justify that the notation a:b=c r s, where a, b, c, s are natural numbers and r denotes, is correct at school mathematics level and does not lead to a contrediction suggested by the author of the article (Semadeni, 1978). As a generalization of...
Recenzja: M. Sajka, Pojęcie funkcji. Wiedza przedmiotowa nauczyciela matematyki, Wydawnictwo Naukowe Uniwersytetu Pedagogicznego, Kraków 2019
Lidia Obojska, Andrzej Walendziak
Tłumaczenie: Lew Pontraingin, O ciągłych ciałach algebraicznych
Recenzja: Lukas Benedikt Kraus Der Begriff des Kontinuums bei Bernard Bolzano. Academia Verlag. Sankt Augustin, 2014
Letter to the Editor, Annales Universitatis Paedagogicae Cracoviensis
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam... > 2018 > 10
From Euclid's Elements to the methodology of mathematics. Two ways of viewing mathematical theory
We present two sets of lessons on the history of mathematics designed for prospective teachers: (1) Euclid's Theory of Area, and (2) Euclid's Theory of Similar Figures. They aim to encourage students to think of mathematics by way of analysis of historical texts. Their historical content includes Euclid's Elements, Books I, II, and VI. The mathematical meaning of the discussed propositions is simple...
Algorithmisation as mathematical activity and skills in connection with mathematical modelling
Good preparation of students to the profession of teacher is very important. In my research I focus on improving the quality of teacher's preparation at the university level, and through it at school level in the future. I present the proposal of teaching mathematics with the use of algorithmisation. It is possible, because solutions to many mathematical problems can be expressed in the form of an...
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Oberth effect - WikiMili, The Best Wikipedia Reader
Maneuver in which a spacecraft falls into a gravitational well, and then accelerates when its fall reaches maximum speed
Not to be confused with Gravity assist.
In astronautics, a powered flyby, or Oberth maneuver, is a maneuver in which a spacecraft falls into a gravitational well and then uses its engines to further accelerate as it is falling, thereby achieving additional speed. [1] The resulting maneuver is a more efficient way to gain kinetic energy than applying the same impulse outside of a gravitational well. The gain in efficiency is explained by the Oberth effect, wherein the use of a reaction engine at higher speeds generates a greater change in mechanical energy than its use at lower speeds. In practical terms, this means that the most energy-efficient method for a spacecraft to burn its fuel is at the lowest possible orbital periapsis, when its orbital velocity (and so, its kinetic energy) is greatest. [1] In some cases, it is even worth spending fuel on slowing the spacecraft into a gravity well to take advantage of the efficiencies of the Oberth effect. [1] The maneuver and effect are named after the person who first described them in 1927, Hermann Oberth, an Austro-Hungarian-born German physicist and a founder of modern rocketry. [2]
Explanation in terms of momentum and kinetic energy
Description in terms of work
Impulsive burn
Oberth calculation for a parabolic orbit
Parabolic example
Because the vehicle remains near periapsis only for a short time, for the Oberth maneuver to be most effective the vehicle must be able to generate as much impulse as possible in the shortest possible time. As a result the Oberth maneuver is much more useful for high-thrust rocket engines like liquid-propellant rockets, and less useful for low-thrust reaction engines such as ion drives, which take a long time to gain speed. The Oberth effect also can be used to understand the behavior of multi-stage rockets: the upper stage can generate much more usable kinetic energy than the total chemical energy of the propellants it carries. [2]
In terms of the energies involved, the Oberth effect is more effective at higher speeds because at high speed the propellant has significant kinetic energy in addition to its chemical potential energy. [2] : 204 At higher speed the vehicle is able to employ the greater change (reduction) in kinetic energy of the propellant (as it is exhausted backward and hence at reduced speed and hence reduced kinetic energy) to generate a greater increase in kinetic energy of the vehicle. [2] : 204
A rocket works by transferring momentum to its propellant. [3] At a fixed exhaust velocity, this will be a fixed amount of momentum per unit of propellant. [4] For a given mass of rocket (including remaining propellant), this implies a fixed change in velocity per unit of propellant. Because kinetic energy equals mv2/2, this change in velocity imparts a greater increase in kinetic energy at a high velocity than it would at a low velocity. For example, considering a 2 kg rocket:
{\displaystyle W}
{\displaystyle {\vec {F}}}
{\displaystyle {\vec {s}}}
{\displaystyle W={\vec {F}}\cdot {\vec {s}}.}
{\displaystyle {\vec {F}}\cdot {\vec {s}}=\|F\|\cdot \|s\|=F\cdot s}
{\displaystyle \Delta E_{k}=F\cdot s.}
{\displaystyle {\frac {\mathrm {d} E_{k}}{\mathrm {d} t}}=F\cdot {\frac {\mathrm {d} s}{\mathrm {d} t}},}
{\displaystyle {\frac {\mathrm {d} E_{k}}{\mathrm {d} t}}=F\cdot v,}
{\displaystyle v}
{\displaystyle m}
{\displaystyle e_{k}}
{\displaystyle {\frac {\mathrm {d} e_{k}}{\mathrm {d} t}}={\frac {F}{m}}\cdot v=a\cdot v,}
{\displaystyle a}
{\displaystyle \Delta v}
). However, since the vehicle's kinetic energy is related to the square of its velocity, this increase in velocity has a non-linear effect on the vehicle's kinetic energy, leaving it with higher energy than if the burn were achieved at any other time. [5]
If an impulsive burn of Δv is performed at periapsis in a parabolic orbit, then the velocity at periapsis before the burn is equal to the escape velocity (Vesc), and the specific kinetic energy after the burn is [6]
{\displaystyle {\begin{aligned}e_{k}&={\tfrac {1}{2}}V^{2}\\&={\tfrac {1}{2}}(V_{\text{esc}}+\Delta v)^{2}\\&={\tfrac {1}{2}}V_{\text{esc}}^{2}+\Delta vV_{\text{esc}}+{\tfrac {1}{2}}\Delta v^{2},\end{aligned}}}
{\displaystyle V=V_{\text{esc}}+\Delta v}
{\displaystyle {\tfrac {1}{2}}V_{\text{esc}}^{2},}
{\displaystyle \Delta vV_{\text{esc}}+{\tfrac {1}{2}}\Delta v^{2},}
{\displaystyle {\tfrac {1}{2}}\Delta v^{2}}
{\displaystyle \Delta vV_{\text{esc}}.}
{\displaystyle V=\Delta v{\sqrt {1+{\frac {2V_{\text{esc}}}{\Delta v}}}}.}
{\displaystyle {\sqrt {\frac {2V_{\text{esc}}}{\Delta v}}}}
{\displaystyle V}
{\displaystyle {\sqrt {{2V_{\text{esc}}}{\Delta v}}}.}
{\displaystyle v\,\Delta v+{\tfrac {1}{2}}(\Delta v)^{2}.}
It may seem that the rocket is getting energy for free, which would violate conservation of energy. However, any gain to the rocket's kinetic energy is balanced by a relative decrease in the kinetic energy the exhaust is left with (the kinetic energy of the exhaust may still increase, but it does not increase as much). [2] : 204 Contrast this to the situation of static firing, where the speed of the engine is fixed at zero. This means that its kinetic energy does not increase at all, and all the chemical energy released by the fuel is converted to the exhaust's kinetic energy (and heat).
At very high speeds the mechanical power imparted to the rocket can exceed the total power liberated in the combustion of the propellant; this may also seem to violate conservation of energy. But the propellants in a fast-moving rocket carry energy not only chemically, but also in their own kinetic energy, which at speeds above a few kilometres per second exceed the chemical component. When these propellants are burned, some of this kinetic energy is transferred to the rocket along with the chemical energy released by burning. [7]
Specific impulse is a measure of how efficiently a reaction mass engine creates thrust. For engines whose reaction mass is only the fuel they carry, specific impulse is exactly proportional to the effective exhaust gas velocity.
In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form. The principle is only applicable for isentropic flows: when the effects of irreversible processes and non-adiabatic processes are small and can be neglected.
In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, it is often represented as the product of force and displacement. A force is said to do positive work if it has a component in the direction of the displacement of the point of application. A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force.
Verlet integration is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics. The algorithm was first used in 1791 by Delambre and has been rediscovered many times since then, most recently by Loup Verlet in the 1960s for use in molecular dynamics. It was also used by Cowell and Crommelin in 1909 to compute the orbit of Halley's Comet, and by Carl Størmer in 1907 to study the trajectories of electrical particles in a magnetic field . The Verlet integrator provides good numerical stability, as well as other properties that are important in physical systems such as time reversibility and preservation of the symplectic form on phase space, at no significant additional computational cost over the simple Euler method.
1 2 3 Robert B. Adams, Georgia A. Richardson (25 July 2010). Using the Two-Burn Escape Maneuver for Fast Transfers in the Solar System and Beyond (PDF) (Report). NASA. Archived (PDF) from the original on 11 February 2022. Retrieved 15 May 2015.
1 2 3 4 5 Hermann Oberth (1970). "Ways to spaceflight". Translation of the German language original "Wege zur Raumschiffahrt," (1920). Tunis, Tunisia: Agence Tunisienne de Public-Relations.
↑ What Is a Rocket? 13 July 2011/ 7 August 2017 www.nasa.gov, accessed 9 January 2021.
↑ Rocket thrust 12 June 2014, www.grc.nasa.gov, accessed 9 January 2021.
↑ Atomic Rockets web site: nyrath@projectrho.com. Archived July 1, 2007, at the Wayback Machine
↑ Following the calculation on rec.arts.sf.science.
↑ Blanco, Philip; Mungan, Carl (October 2019). "Rocket propulsion, classical relativity, and the Oberth effect". The Physics Teacher. 57 (7): 439–441. Bibcode:2019PhTea..57..439B. doi: 10.1119/1.5126818 .
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Practical Geometry, Popular Questions: CBSE Class 6 SCIENCE, Science - Meritnation
The perimeter of a rectangle is 90 and its width is 10. What is its area?
give three exapmle of the object having (a) flat surface (b) curved surface
Prathibha Sri Sridharane asked a question
HOW TO CONSTRUCT AN ANGLE OF 75 DEGREES?
Question no 3 . Please try to give the answer fastest.
how to draw a 150 degree angle with a ruler and a compass
Why was arc important in the constructing of an angle?...
Why perpendicular line are in right angle ?....
What is the use of angle ?... hope you can make me understand >>>>>>
Smita Ramki asked a question
Draw an angle of measure153° and divide it into four equal parts.
vaishaligaurang asked a question
HOW TO MAKE A ANGLE OF 15 DEGREE WITHOUT USING PROTRACTOR?
1. Draw a line segment AB = 4 cm. Draw CA perpendicular to AB at A. Cut off AC = 3cm. Join BC. What type of figures do you get Measure the length of BC.Is there any relationship between the sides Also find the measure of the three angles between the adjacent sides.
2. Draw a line AB = 5cm and take a point P on it, such that AP = 3 cm. At point P draw an angle of 90⁰ using compasses and ruler.
3. Using ruler and compasses, construct a rectangle whose adjacent sides are 6.5 cm and 4 cm.
Preetham Kumar J asked a question
how to draw 135 degree angle without using protractor?
Sujeeth Kumar asked a question
DRAW A 210 DEGREE ANGLE WITH RULER AND COMPASS
How to construct an angle of 221/2 degrees using a paur of compass? Please explain urgently
Adishya Gupta asked a question
can we draw a angle of 140 degree using a pair of compass?
please sent the answer soon !!!!!
Saleha asked a question
1.Using compasses and ruler construct 105 degree.
2.Construct a perpendicular bisector of PQ ehose length is 11.2cm using compasses and ruler.
3.Draw an angle of measure 140 degree and divide it into 4 equal parts using compasses and ruler.
Q. Draw
\angle
ABC of measure 60
°
such that AB = 4.5 cm and BC = 5 cm. Through C draw a line parallel to AB and through B draw a line parallel to AC, intersecting each other at D measure BD and CD.
ananyamahawar47... asked a question
draw ab of length 7.3 cm and find its axis of symmetry
Dheeraj Bhargav asked a question
Abhijeet Manohar asked a question
what is the difference between parallelogram and trapezium?
Nikhil Jain asked a question
Draw an angle AOB =65 with your protector. construct an angle XYZ congurent to AOB using compass
2.The line segment joining the center and any point on the circle is called its.,.......,......
Ishaan Bakshi asked a question
How can we make an angle of 130' using ruler and compass?
Chankya Dubey asked a question
Q. Draw any obtuse angle XYZ. Construct
\angle
PQR such that
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\angle
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Error of Analysis of Newton-Cotes formulas - Wikiversity
Error of Analysis of Newton-Cotes formulas
1 Error Analysis of Newton-Cotes formulas
1.1 The Methods[1]
1.2 Error terms for different rules
1.2.1 The Trapezoid Rule
1.2.2 The Simpson's 1/3 Rule
1.4.1 Exercise 1[3]
Error Analysis of Newton-Cotes formulasEdit
The Newton-Cotes formulas are a group of formulas for evaluating numeric integration at equally spaced points.
The Methods[1]Edit
{\displaystyle x_{i}}
{\displaystyle i=0,\ldots ,n}
{\displaystyle n+1}
equally spaced points, and
{\displaystyle f_{i}}
be the corresponding values. Let
{\displaystyle h}
be the space
{\displaystyle h=x_{i+1}-x_{i}}
{\displaystyle s}
be the interpolation variable
{\displaystyle s={\frac {x-x_{0}}{h}}}
. Thus to interpolate at x,
{\displaystyle {\begin{aligned}x-x_{0}&=sh\,,\\x-x_{1}&=x-(x_{0}+h)=(s-1)h\,,\\\vdots \\x-x_{n}&=(s-n)h\,.\end{aligned}}}
{\displaystyle P_{n}(x)}
{\displaystyle n}
can be derived to pass through these points and approximate the function
{\displaystyle f(x)}
. Using divided differences and Newton polynomial,
{\displaystyle P_{n}(x)}
{\displaystyle {\begin{aligned}P_{n}(x)&=[f_{0}]+[f_{0},f_{1}](x-x_{0})+\cdots +[f_{0},\ldots ,f_{n}](x-x_{0})(x_{1})\ldots (x-x_{n-1})\\&=[f_{0}]+[f_{0},f_{1}]sh+\cdots +[f_{0},\ldots ,f_{n}]s(s-1)\ldots (s-n+1)h^{n}\,.\end{aligned}}}
From the general form of polynomial interpolation error, the error of using
{\displaystyle P_{n}(x)}
to interpolate
{\displaystyle f(x)}
{\displaystyle {\begin{aligned}E_{\text{interpolate}}(x)&=f(x)-P_{n}(x)\\&={\frac {1}{(n+1)!}}(x-x_{0})(x-x_{1})\cdots (x-x_{n})f^{(n+1)}(\xi )\\&={\frac {1}{(n+1)!}}s(s-1)(s-2)\ldots (s-n)h^{n+1}f^{(n+1)}(\xi )\end{aligned}}}
{\displaystyle x_{0}\leqslant \xi \leqslant x_{n}}
{\displaystyle dx=d(x_{0}+sh)=hds}
, the error term of numerical integration is
{\displaystyle E_{\text{integrate}}=\int \limits _{x_{0}}^{x_{n}}E_{\text{interpolate}}(x)dx={\frac {h^{n+2}}{(n+1)!}}f^{(n+1)}(\xi )\int \limits _{0}^{n}s(s-1)\cdots (s-n)ds\,.}
Error terms for different rulesEdit
The Trapezoid RuleEdit
Let's consider the trapezoid rule in a single interval. In each interval, the integration uses two end points. Thus
{\displaystyle n+1=2}
{\displaystyle n=1}
. Applying ( 1 ), we get
{\displaystyle E_{\text{integrate}}=h\int \limits _{0}^{1}{\frac {s(s-1)}{2}}h^{2}f''(\xi )ds=-{\frac {1}{12}}h^{3}f''(\xi )=O(h^{3})}
{\displaystyle x_{0}\leqslant \xi \leqslant x_{1}}
. Thus the local error is
{\displaystyle O(h^{3})}
. Consider the composite trapezoid rule. Given that
{\displaystyle n={\frac {x_{n}-x_{0}}{h}}}
, the global error is
{\displaystyle {\begin{aligned}\left|\sum _{i=0}^{n-1}-{\frac {1}{12}}h^{3}f''(\xi _{i})\right|&=n[-{\frac {1}{12}}(x_{n}-x_{0})h^{2}f''({\bar {\xi }})]\\&=-{\frac {1}{12}}(x_{n}-x_{0})h^{2}f''({\bar {\xi }})=O(h^{2})\,,\end{aligned}}}
{\displaystyle x_{i}\leqslant \xi _{i}\leqslant x_{i+1}}
{\displaystyle x_{0}\leqslant {\bar {\xi }}\leqslant x_{n}}
To justify ( 2 ), we can need the theorem below[2] in page 345:
{\displaystyle g(x)}
is continuous and the
{\displaystyle c_{i}\geq 0}
, then for some value
{\displaystyle \theta }
in the interval of all the arguments
{\displaystyle g(\theta )\sum c_{i}=\sum \limits _{i=1}^{N}c_{i}g(\theta _{i})\,.}
The Simpson's 1/3 RuleEdit
Consider Simpson's 1/3 rule. In this case, three equally spaced points are used for integration. Thus
{\displaystyle n+1=3}
{\displaystyle E_{\text{integrate}}=h\int \limits _{0}^{2}{\frac {s(s-1)(s-2)}{6}}h^{3}f'''(\xi )ds=0}
{\displaystyle x_{0}\leqslant \xi \leqslant x_{2}}
This doesn't mean that the error is zero. It simply means that the cubic term is identically zero. The error term can be obtained from the next term in the Newton polynomial, obtaining
{\displaystyle E_{\text{integrate}}=h\int \limits _{0}^{2}{\frac {s(s-1)(s-2)(s-3)}{24}}h^{4}f^{(4)}(\xi )ds=-{\frac {1}{90}}h^{5}f^{(4)}(\xi )=O(h^{5})\,.}
Thus the local error is
{\displaystyle O(h^{5})}
and the global error is
{\displaystyle O(h^{4})}
Consider Simpson's 3/8 rule. In this case,
{\displaystyle n+1=4}
since four equally spaced points are used. Applying ( 1 ), we get
{\displaystyle E_{\text{integrate}}=h\int \limits _{0}^{3}{\frac {s(s-1)(s-2)(s-3)}{24}}h^{4}f^{(4)}(\xi )ds=-{\frac {3}{80}}h^{5}f^{(4)}(\xi )=O(h^{5})}
{\displaystyle x_{0}\leqslant \xi \leqslant x_{3}}
Both the Simpon's 1/3 rule and the 3/8 rule have error terms of order
{\displaystyle h^{5}}
. With smaller coefficient, the 1/3 rule seems more accurate. Then why do we need the 3/8 rule? The 3/8 rule is useful when the total number of increments
{\displaystyle n}
is odd. Three increments can be used with the 3/8 rule, and then the rest even number of increments can be used with 1/3 rule.
A Numerical ExampleEdit
Given the set of data points, solve the numerical integration
{\displaystyle I=\int \limits _{3.1}^{3.9}f(x)dx}
{\displaystyle x}
{\displaystyle f(x)=-{\frac {1}{x}}}
Use the trapezoid rule. First try
{\displaystyle h=0.8}
. That is, use only the two end points. We can get
{\displaystyle I(h=0.8)={\frac {0.8}{2}}(-0.32258065-0.25641026)=-0.23159636}
Compared with the exact solution
{\displaystyle I=-0.22957444}
{\displaystyle E_{\text{integrate}}=0.00202192\,.}
Using all three points with
{\displaystyle h=0.4}
{\displaystyle I(h=0.4)={\frac {0.4}{2}}(-0.32258065-2\times 0.28571429-0.25641026)=-0.23008389}
{\displaystyle E_{\text{integrate}}=0.00050945\,.}
Thus the error ratio is
{\displaystyle {\frac {E_{\text{integrate}}(h=0.8)}{E_{\text{integrate}}(h=0.4)}}=3.97}
. This is close to what we can get by inspecting
{\displaystyle {\frac {E_{\text{integrate}}(h)}{E_{\text{integrate}}(h/2)}}={\frac {O(h^{2})}{O(h/2)^{2}}}=2^{2}=4}
Exercise 1[3]Edit
Using the data given below, find the maximum error incurred in using Newton's forward interpolation formula to approximate
{\displaystyle x=0.14}
{\displaystyle x}
{\displaystyle e^{x}}
According the general error formula of polynomial interpolation
{\displaystyle |E_{\text{interpolate}}|\leqslant \left|{\frac {1}{(n+1)!}}(x-x_{0})(x-x_{1})\cdots (x-x_{n})\right|[\max |f^{(n+1)}(x)|]}
{\displaystyle x\in [0.1,0.5]}
{\displaystyle h=0.1,n=4,x=0.14}
{\displaystyle \max |f^{(n+1)}(x)|=e^{0.5}}
{\displaystyle |E_{\text{interpolate}}|\leqslant \left|{\frac {1}{5!}}(0.14-0.1)(0.14-0.2)\cdots (0.14-0.5)e^{0.5}\right|<0.0000005}
When using Simpson's 1/3, what is the error ratio supposed to be?
{\displaystyle {\frac {E_{\text{integrate}}(h)}{E_{\text{integrate}}(h/2)}}={\frac {O(h)^{4}}{O(h/2)^{4}}}=2^{4}=16\,.}
↑ Hoffman, Joe D. (2001). Numerical Methods for Engineers and Scientists (2nd ed.). Marcel Derkker, INC. ISBN 0-8247-0443-6.
↑ Hamming, R. W. (1986). Numerical Methods for Scientists and Engineers (2nd ed.). New York: Dover Publications. ISBN 0-486-65241-6. http://books.google.com/books/about/Numerical_Methods_for_Scientists_and_Eng.html?id=Y3YSCmWBVwoC.
↑ Tenenbaum, Morris; Pollard, Harry (1985). Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences. New York: Dover Publications. ISBN 9780486649405.
Retrieved from "https://en.wikiversity.org/w/index.php?title=Error_of_Analysis_of_Newton-Cotes_formulas&oldid=2240715"
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Long-term memory stabilized by noise-induced rehearsal | BMC Neuroscience | Full Text
Long-term memory stabilized by noise-induced rehearsal
Cortical networks can maintain memories for decades, despite short lifetime of synaptic strength. Can a neural network store long-lasting memories in unreliable synapses? Here we study the effects of random noise on the stability of memory stored in synapses of an attractor neural network. The model includes ongoing spike timing dependent plasticity (STDP). We show that certain class of STDP rules can lead to stabilization of memory patterns stored in the network. The stabilization results from rehearsals induced by noise. We show that unstructured neural noise, after passing through the recurrent network weights, carries the imprint of all of the memory patterns in temporal correlations. Under certain strict conditions, STDP combined with these correlations can lead to reinforcement of all of the existing patterns, even those that are never explicitly visited, i.e. unused. We show that stabilization of unused memories occurs for asymmetric STDP learning rules (Figure 1), while symmetric non-negative rules do not have this property. Thus, we propose that, unstructured neural noise can stabilize the existing structure of synaptic connectivity. Our findings may provide the functional reason for highly irregular spiking displayed by cortical neurons and provide justification for models of system memory consolidation. Out theory makes experimentally testable predictions, such as that synaptic strengths in the cortex should be correlated with the correlations in the pre- and postsynaptic neural activity on the synapse-by-synapse basis. We thus propose that unreliable neural activity is the feature that helps cortical networks maintain stable connections.
The stabilization of old (unused) memory states by the combination of unstructured noise and antisymmetric STDP learning rule. (A) STDP learning rule considered. (B) The rate of change of the contribution of an unused state (
d{c}^{\mathsf{\text{unused}}}/dt
) to the synaptic weight matrix as a function of the contribution itself (
{c}^{\mathsf{\text{unused}}}
). The contribution in this case has two stable points, near zero and at a finite value. The former/latter stable points correspond to the unused memory pattern being absent/present in the network connectivity. At the stable points the rate of change of the pattern's contribution is zero. Small perturbations from the stable point will induce the rate of change that returns the system back to the stable point. The third point where the rate of change is zero is unstable and is therefore called the transition point.
Yi Wei & Alexei Koulakov
Wei, Y., Koulakov, A. Long-term memory stabilized by noise-induced rehearsal. BMC Neurosci 14, P220 (2013). https://doi.org/10.1186/1471-2202-14-S1-P220
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Hitpoints - Ring of Brodgar
4.1 Alchemy wounds
There are three different types of hitpoints: soft (SHP), hard (HHP), and maximum (MHP).
SHP reflects current hitpoints. As long as a character's Energy is above 8000%, this value will slowly regenerate, up to the character's current HHP value. SHP can be lost from (including, but not limited to:)
Equipped leeches
Commiting Criminal Acts causes 1shp damage for each action that generates a scent.
Being reduced to zero SHP results in being knocked unconscious for roughly a minute. Travel to Hearth Fire is possible for a short while before waking up, allowing you to avoid getting knocked out again by animal lingering nearby or escape PvP encounters as long as you're not red-handed, outlaw, or in a Dungeon.
HHP is current maximum hitpoints. This number is reduced by Wounds you receive during combat, starvation or drowning. Leaving your character to starve will make you lose HHP over time. HHP recovers when you heal a wound. HHP has a maximum of the character's MHP value. A character takes slight HHP damage from most unblocked attacks. If a character's HHP drops to zero, that character is permanently dead.
MHP is the maximum achievable hard hitpoints. This value is 100 on a new character and can be raised by increasing a character's Constitution attribute, while lowering it will decrease MHP.
{\displaystyle MHP=100*{\sqrt {\frac {CON}{10}}}}
{\displaystyle 100*{\sqrt {\frac {10}{10}}}}
{\displaystyle MHP=100*{\sqrt {\frac {100}{10}}}}
For additional information, see this forum post.
Wounds are the HHP damage since World 8 of Haven. Different sources of damage have a chance to cause a wound, which might heal over time, or can be healed by certain actions. Each wound will have a value which amounts to the reduction it applies to your HHP, or even to certain Attributes. These values and the wounds are visible on the Health & Wounds button on your Character Sheet. Each wound heals independently of others, so several small occurrences of the same wound will heal faster than one large one.
Many wounds will stack together to form new, generally more severe wounds that are often harder to cure. For instance, taking large amounts of Unfaced can cause you to gain Black-Eyed or Severe Mauling. The most common occurrence of this is Nicks and Knacks stacking into Scrapes and Cuts, which can stack to Deep Cuts, which can stack to Cruel Incision, which can stack to Nasty Laceration.
Any wounds that state that it will "heal over time" require the character to have a green energy bar (8000% or higher) to heal and will continue to heal whether on or offline. Note that some wounds that heal over time can also be healed with other methods, such as Leeches. This may speed up overall healing time by spreading out the damage more.
Some wounds, such as Adder bites, initially get worse over time, and can kill your hearthling. Sleeping appears to accelerate the progress of these wounds, so it may be wise to seek treatment before sleep. Others require monitoring, like Infected Sore, which can either grow or shrink over time.
The following are the list of wounds a character can receive from combat or other actions in the game. Several will heal on their own, most will need some sort of healing item to remove the wound. Note that almost all healing items have some sort of penalty associated with it. It could be a quick loss of SHP (Leeches), occupying a gear slot (Gauze), or reduce attributes or skills (Stinging Poultice).
HHP is recovered as wounds reduce in amount(s). Recovering SHP needs energy at healing levels (8000% or higher).
Wounds will still, even when treated, count and be relevant for combinations to new and more severe wounds when you receive new wounds from taking damage. They will also retain negative effects during treatment. Each treatment, when applied, now applies a treatment factor to the wound, dependent on the quality of the treatment applied, which reduces the reactivity of the wound -- i.e. the likelihood of it wanting to combine with incoming new wounds to create more severe wounds -- by the treatment factor percentage.
Diseases are unique in that they will increase in size (removing more HHP) over time, unless they are treated or you stay within Healing range of energy (8000% or more).
Note: Otherwise permanent wounds that have no traditional remedy can be healed by an Ancient Root, which instantly heals any wound, or the right Elixir, which requires some research beforehand to not poison yourself.
Adder Bite Snake Juice PvE with Adders. Disease.
Damage steadily increases before start healing slowly over time.
-15 STR penalty. Chance to leave Nerve Damage.
Allergic Reaction N/A Have 3 different allergic wounds (Adder Bite, Nettle Burns, Antcid Burns, Midge Bite or Beesting) at the same time. -10 AGI and -10 STR penalty. Heals slowly over time.
Antcid Burns Yarrow All wounds given by ants (including big ones from within an Ant Hill Dungeon) are Antcid Burns. Heals slowly over time, -1 AGI penalty.
Asphyxiation N/A Take 10% of MHP worth of Asphyxiation for every second you spend swimming at 0% stamina. Heals relatively fast over time, -5 INT penalty.
Beesting Kelp Cream, Ant Paste, Gray Grease Taking honey or wax from a Bee Skep -1 AGI, each new wound would worse by 1 hp. Heals slowly over time.
Bird Lung N/A Opening occupied Dovecotes, Birdhouses -5 AGI and CON penalty. Heals slowly over time.
Black-eyed Toad Butter, Rootfill, Hartshorn Salve, Honey Wayband Cave-ins, Wounds stacking -10 PER penalty.
Blade Kiss Toad Butter, Gauze Melee Combat, Fighting tusked/clawed animals Heals very slowly over time (1 per ~24h RL )
Blunt Trauma Hartshorn Salve, Leeches, Gauze, Toad Butter, Camomile compress, Opium Cave-Ins, Unarmed Combat, Wounds stacking Leeches converts wound to Leech Burn
Bruises Leeches Cave-Ins, Unarmed Combat Heals over time. Leeches converts wound to Leech Burn
Bum Burn N/A 10% from eating peppered food -5% AGI, Each new wound would worse. Heals slowly over time.
Coaler's Cough Opium Mine Black Coal Heals slowly over time. - 10 AGI, -10 WIL and +15 Masonry.
Concussion Cold Compress, Opium Randomly take anywhere from 5-20% of your MHP worth of a Concussion wound every time you're KOed for any reason. Heals slowly over time. Reduces Abilities based on size.
Crab Caressed Ant Paste Picking up/Cracking a Crab Heals over time. -DEX penalty.
Cruel Incision Gauze, Stitch Patch, Rootfill Unarmed/Melee Combat Rootfill converts wound to Deep Cut.
Deep Cut Gauze, Stinging Poultice, Rootfill, Waybroad, Honey Wayband Unarmed Combat, Rootfill converting Rootfill completely heals wound.
Dragon Bite N/A Smoking Opium -10 All base Attributes penalty. Heals over time. See the discussion page for speculation on damage per wound.
Fell Slash Gauze Unarmed/Melee Combat
Hearth Burn N/A Too often spawn in wilderness without Hearth Fire. Heals over time. -5 PSY, -3 INT penalty.
Infected Sore Camomile compress, Bar of Soap, Opium, Ant Paste Standing in swamps with other wounds present, especially Crab Caressed. Disease.
All wound types can become infected. Infected wounds have a chance to grow better or worse over time.
Jellyfish Sting Gray Grease Picking up Jellyfish -1 CON penalty.Heals slowly over time.
Leech Burns Toad Butter Leeches Heals over time.
Midge Bite Yarrow Midge Swarms Heals over time.
Nasty Laceration Stitch Patch, Toad Butter Small wounds tearing into larger ones Heals slowly over time. -2 CON, -10 AGI penalty.
Nasty Wart Permanent but can be healed by Ancient Root Using Toad Butter on any wound. -CHA penalty
Nerve Damage Permanent but can be healed by Ancient Root Receive multiple Adder Bite wounds -1 INT penalty.
Nettle Burn N/A Received when picking Stinging Nettle +1 Survival, Heals over time--20 minutes per point
Nicks & Knacks Yarrow, Honey Wayband Unarmed Combat, Chopping blocks Heals over time. Large wounds spit off and create Scrapes & Cuts.
Nidburns N/A PvE with Nidbane Heals slowly over time.
Pipe Wheeze N/A Smoking Pipestuff +5 WIL, Heals very slowly over time.
Punch Sore Mud Ointment, Opium Unarmed Combat, Cave-Ins Heals over time. Stacks to worse wounds easily.
QuickSilver Poisoning Permanent but can be healed by Ancient Root Crafting Felt and gilding Felt Inlay
Study Quicksilver Globe -1 INT, PSY and WIL penalty.
Quill'd N/A Picking up/Killing a Hedgehog Heals over time.
Sand Flea Bites Yarrow, Gray Grease Standing around Sand Flea. Heals slowly over time. -1 CHA penalty.
Scrapes & Cuts Yarrow, Mud Ointment, Honey Wayband Splits off from Nicks & Knacks Heals slowly over time.
Seal Finger Hartshorn Salve, Kelp Cream, Ant Paste Combat, Skinning or Butchering Grey Seal or Walrus Disease.
Grows slowly over time, -1 AGI, -3 CON, -10 DEX
Severe Mauling Hartshorn Salve, Opium Cave-Ins, High wound stacking
Something Broken Splint Cave-Ins below Level 1, Severe stacking of wounds
Starvation N/A Going under 2000 Energy Only heals if your energy increases to over 8000 energy. Rapid lose of SHP and increase size of wound, while under 2000 Energy. Will not heal by sleeping, only if you are logged in.
Swamp Fever Snake Juice Getting bit by midges while in a swamp. Disease.
Grows over time. -5% WIL, -2.5% AGI.
May heal up on its own after time.
Swollen Bumps Cold Compress, Leeches, Stinging Poultice Cave-Ins, Bruises or Punch-Sores stacking Leeches converts wound into Leech Burns. Stinging Poultice seems to heal around one hitpoint per hour.
Unfaced Leeches, Mud Ointment, Toad Butter, Kelp Cream Cave-Ins, Wounds stacking Leeches converts wound into Leech Burns.
Wretched Gore Stitch Patch Very strong attacks.
50%+ of your MHP in one hit will result in Wretched Gore.
Extreme stacking of wounds
Alchemy wounds
You will only get these wounds from drinking Elixirs created using Alchemy.
Aching Joints N/A Drinking an Elixir. Heals over time (1 per hour), -10 DEX penalty.
Blistering Headache N/A Drinking an Elixir. Heals slowly over time (1 per 24 hours), -10 INT penalty.
Chills & Nausea N/A Drinking an Elixir. Heals over time (1 per hour), -5 PER penalty.
Maddening Rash N/A Drinking an Elixir. Heals over time (1 per hour), -10 AGI penalty.
Rot Gut N/A Drinking an Elixir. Heals slowly over time (1 per 24 hours), -10 CON penalty.
Some medicine not listed here - because has no status effect, for a full list of available medicine see here.
Applicable Wounds
Ant Pasted Beestings, Crab Caressed, Infected Sores, Seal Finger Ant Paste First dmg for 10-15 hhp, then heals both wounds over time.
Bound in Waybroad Deep Cut Waybroad Heals slowly over time. -1 CHA and PER
Buttered Blunt Trauma, Unfaced, Black-eyed, Leech Burns Toad Butter Heals over time. Converts into one-point Nasty Wart.
Camomile Compressed Infected Sore, Blunt Trauma Camomile Compress Heals over time.
Gauzed Blunt Trauma, Cruel Incision, Deep Cuts, Fell Slash Gauze Heals over time. -2 CHA and -3 PER penalty.
Gray Greased Sand Flea Bites, Jellyfish Sting, Beesting Gray Grease "The sweet chill of damp clay has stilled the pain ringing in this wound, and it should now heal better."
Hartshorn Salved Severe Mauling, Black-eyed, Blunt Trauma Hartshorn Salve Heals over time.
Honey Waybound Black-Eyed, Deep Cut, Nicks & Knacks, Scrapes & Cuts Honey Wayband Heals over time. Buffs +2 CHA and +1 STR
Kelp Helped Beesting, Unfaced, Seal Finger Kelp Cream Heals over time.
Lathered Infected Sore Bar of Soap Stops the infection. +1 CHA
Leech Burns Swollen Bumps, Bruises, Unfaced, Blunt Trauma Leeches Heals over time.
Muddied Prospects Scrapes & Cuts, Unfaced, Punch-Sore Mud Ointment Heals over time. -2 DEX and -5 CHA penalty.
Stitched Up Nasty Laceration, Cruel Incision, Wretched Gore Stitch Patch Heals very slowly over time. -10 CHA, AGI and PER penalty.
Snake Juicy Adder Bite, Swamp Fever Snake Juice Heals over time.
Soothing Cold Swollen Bump, Concussion Cold Compress Heals over time.
Treated with a Poultice Swollen Bump, Deep Cut Stinging Poultice Heals slowly over time.
Yarr! Antcid Burns, Nicks & Knacks, Scrapes & Cuts Yarrow Heals in a few seconds. Antcid Burns heals instantly. -2 CON penalty.
A few wounds need to be given special consideration
There seems to be chance to gain starvation damage every time character expends energy when its below 2000%
Starvation seems to only heal when some combination of following factors is met, proper testing required
SHP is full
Stamina is full
Energy is over 8000% or more
These lower stats and can take a long time to heal on their own
Cloth is the main limiting factor for its treatment, as it can be difficult for a new player to make.
Getting knocked out too many times can brick your character for a few days
Wretched Gore
Often recieved during PvP fights, or damage from strong animals.
Heals 1 HHP a day with a q10 stitched patch
The patch quality can make a difference measured in months
Using a Stitch Patch to heal this wound causes the Stitched Up debuff, which reduces a character's effectiveness at combat and foraging.
Ancient Roots may provide quicker treatment. See Localized Resources
Search for Black-eyed on this page
Mushroom Circle (2021-02-21) >"Added "Splint", a treatment for the "Something Broken" wound. Occupies one hand slot while active. Camomille Compress no longer heals "Something Broken"."
Petrified Reindeer (2018-10-01) >"Added new wounds: "Allergic Reaction", "Nettle Burn"."
Cheesy Coracle (2018-01-25) >"Added "Nasty Laceration", wound."
Bench Crab (2017-01-26) >"Added "Black-Eyed", wound."
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Hitpoints&oldid=94370#Wounds"
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Does 1 square meter equal 100 square centimeters? | Brilliant Math & Science Wiki
Julian Poon, Andrew Ellinor, Zandra Vinegar, and
1\text{ m}^{2}=100\text{ cm}^{2}\, ?
1\text{ m}=100\text{ cm},
1\text{ m}^{2}=100\text{ cm}^{2}.
Why some people say it's false: Since
1\text{ m}=100\text{ cm}, 1\text{ m}^{2}=(100\text{ cm})^{2}\neq100\text{ cm}^2.
“1\text{ m}^{2}=100\text{ cm}^{2}"
\color{#D61F06}{\textbf{false}}
1\text{ m}^2
represents the area of a
100\text{ cm}
100\text{ cm}
square, or a square composed of
100^{2}
1\text{ cm}
1\text{ cm}
squares. Hence, the area is
100^{2}\text{ cm}^2.
We can also take a conversion approach.
1 \text{ m}^2 = 1 \text{ m}^2 \cdot \frac{100 \text{ cm}}{1 \text{ m}} = 100 \text{ m cm}.
Here, one of the instances of
\text{m}
\text{m}^2
will cancel, but another instance of
\text{m}
will remain. So we must apply the conversion factor again:
100 \text{ m cm} \cdot \frac{100 \text{ cm}}{1 \text{ m}} =10,000 \text{ cm}^2 .
This can also be seen geometrically. Consider a 1-meter long stick. Because
1\text{ m}=100\text{ cm},
we can mark off
100\text{ cm}
units along the stick.
1\text{ m}^{2}
can be visualized as a
1\text{ m} \times 1\text{ m}
area. As you can see in the image below, this area is filled with
100 \times 100 \text{ cm}^2
square areas. Therefore,
1\text{ m}^{2} =10,000 \text{ cm}^2 .
Rebuttal: But when you make
\text{m}
\text{m}^{2}
, all you do is
\text{m} \times \text{m} = \text{m}^{2}
. The same for
\text{cm}.
So all you do is
1~ (\text{m} \times \text{m})=100~ (\text{cm} \times \text{cm})
1\text{ m}^{2}=1\text{ m} \times 1\text{ m},
1\text{ cm}^{2}=1\text{ cm} \times 1\text{ cm}
. We have to replace the
1\text{ m}
100\text{ cm}
in both instances. A simple way of remembering this approach is to compare it to the rule of exponent
(ab)^2 = a^2 b^2
(1\text{ m})^2 = (100\text{ cm})^2 = 100^2 \text{ cm}^2
100 = 10^2
10000 = 10^4
1000000 = 10^6
100000000 = 10^8
100 \text{ square meters} = {\color{#D61F06}{X}} \text{ square centimeters}
{\color{#D61F06}{X}}?
Cite as: Does 1 square meter equal 100 square centimeters?. Brilliant.org. Retrieved from https://brilliant.org/wiki/is-a-meter-square-equal-to-100-centimeter-square/
|
Loehr, Nicholas A.1; Niese, Elizabeth2
1 Virginia Tech Dept. of Mathematics Blacksburg, VA 24061, USA
2 Marshall University Dept. of Mathematics Huntington, WV 25755, USA
Algebraic Combinatorics, Volume 4 (2021) no. 6, pp. 1119-1142.
The classical Kostka matrix counts semistandard tableaux and expands Schur symmetric functions in terms of monomial symmetric functions. The entries in the inverse Kostka matrix can be computed by various algebraic and combinatorial formulas involving determinants, special rim hook tableaux, raising operators, and tournaments. Our goal here is to develop an analogous combinatorial theory for the inverse of the immaculate Kostka matrix. The immaculate Kostka matrix enumerates dual immaculate tableaux and gives a combinatorial definition of the dual immaculate quasisymmetric functions
{𝔖}_{\alpha }^{*}
. We develop several formulas for the entries in the inverse of this matrix based on suitably generalized raising operators, tournaments, and special rim-hook tableaux. Our analysis reveals how the combinatorial conditions defining dual immaculate tableaux arise naturally from algebraic properties of raising operators. We also obtain an elementary combinatorial proof that the definition of
{𝔖}_{\alpha }^{*}
via dual immaculate tableaux is equivalent to the definition of the immaculate noncommutative symmetric functions
{𝔖}_{\alpha }
via noncommutative Jacobi–Trudi determinants. A factorization of raising operators leads to bases of
\text{NSym}
interpolating between the
𝔖
-basis and the
\mathrm{h}
-basis, and bases of
\text{QSym}
{𝔖}^{*}
M
-basis. We also give
t
-analogues for most of these results using combinatorial statistics defined on dual immaculate tableaux and tournaments.
Classification: 05E05, 05A19
Keywords: Kostka matrix, quasisymmetric functions, noncommutative symmetric functions, dual immaculate tableaux, immaculate basis, special rim hook tableaux, tournaments
Loehr, Nicholas A. 1; Niese, Elizabeth 2
@article{ALCO_2021__4_6_1119_0,
author = {Loehr, Nicholas A. and Niese, Elizabeth},
title = {Combinatorics of the immaculate inverse {Kostka} matrix},
TI - Combinatorics of the immaculate inverse Kostka matrix
ID - ALCO_2021__4_6_1119_0
%T Combinatorics of the immaculate inverse Kostka matrix
%I MathOA foundation
%F ALCO_2021__4_6_1119_0
Loehr, Nicholas A.; Niese, Elizabeth. Combinatorics of the immaculate inverse Kostka matrix. Algebraic Combinatorics, Volume 4 (2021) no. 6, pp. 1119-1142. doi : 10.5802/alco.193. https://alco.centre-mersenne.org/articles/10.5802/alco.193/
[1] Allen, Edward E.; Hallam, Joshua; Mason, Sarah K. Dual immaculate quasisymmetric functions expand positively into Young quasisymmetric Schur functions, J. Combin. Theory Ser. A, Volume 157 (2018), pp. 70-108 | Article | MR: 3780408 | Zbl: 1385.05072
[2] Berg, Chris; Bergeron, Nantel; Saliola, Franco; Serrano, Luis; Zabrocki, Mike A lift of the Schur and Hall–Littlewood bases to non-commutative symmetric functions, Canad. J. Math., Volume 66 (2014) no. 3, pp. 525-565 | Article | MR: 3194160 | Zbl: 1291.05206
[3] Blasiak, Jonah; Morse, Jennifer; Pun, Anna; Summers, Daniel Catalan functions and
k
-Schur positivity, J. Amer. Math. Soc., Volume 32 (2019) no. 4, pp. 921-963 | Article | MR: 4013737 | Zbl: 1423.05192
[4] Blasiak, Jonah; Morse, Jennifer; Pun, Anna; Summers, Daniel
k
-Schur expansions of Catalan functions, Adv. Math., Volume 371 (2020), Paper no. 107209, 39 pages | Article | MR: 4118771 | Zbl: 1443.05181
[5] Carbonara, Joaquin O. A combinatorial interpretation of the inverse
t
-Kostka matrix, Discrete Math., Volume 193 (1998) no. 1-3, pp. 117-145 Selected papers in honor of Adriano Garsia (Taormina, 1994) | Article | MR: 1661366 | Zbl: 1061.05504
[6] Eğecioğlu, Ömer; Remmel, Jeffrey B. A combinatorial interpretation of the inverse Kostka matrix, Linear and Multilinear Algebra, Volume 26 (1990) no. 1-2, pp. 59-84 | Article | MR: 1034417 | Zbl: 0735.05013
[7] Gelʼfand, Israel M.; Krob, Daniel; Lascoux, Alain; Leclerc, Bernard; Retakh, Vladimir S.; Thibon, Jean-Yves Noncommutative symmetric functions, Adv. Math., Volume 112 (1995) no. 2, pp. 218-348 | Article | MR: 1327096 | Zbl: 0831.05063
[8] Haglund, James; Luoto, Kurt; Mason, Sarah; van Willigenburg, Stephanie Quasisymmetric Schur functions, J. Combin. Theory Ser. A, Volume 118 (2011) no. 2, pp. 463-490 | Article | MR: 2739497 | Zbl: 1229.05270
[9] Hivert, Florent Hecke algebras, difference operators, and quasi-symmetric functions, Adv. Math., Volume 155 (2000) no. 2, pp. 181-238 | Article | MR: 1794711 | Zbl: 0990.05129
[10] Lascoux, Alain; Schützenberger, Marcel-Paul Sur une conjecture de H. O. Foulkes, C. R. Acad. Sci. Paris Sér. A-B, Volume 286 (1978) no. 7, pp. 323-324 | MR: 472993 | Zbl: 0374.20010
[11] Loehr, Nicholas A. Combinatorics, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2018, xxiv+618 pages | MR: 3791447 | Zbl: 1381.05001
[12] Loehr, Nicholas A.; Serrano, Luis G.; Warrington, Gregory S. Transition matrices for symmetric and quasisymmetric Hall–Littlewood polynomials, J. Combin. Theory Ser. A, Volume 120 (2013) no. 8, pp. 1996-2019 | Article | MR: 3102172 | Zbl: 1278.05241
[13] Luoto, Kurt; Mykytiuk, Stefan; van Willigenburg, Stephanie An introduction to quasisymmetric Schur functions: Hopf algebras, quasisymmetric functions, and Young composition tableaux, Springer Briefs in Mathematics, Springer, New York, 2013, xiv+89 pages | Article | MR: 3097867 | Zbl: 1277.16027
[14] Macdonald, Ian Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages | MR: 1354144 | Zbl: 0824.05059
[15] Sagan, Bruce E. The symmetric group: Representations, combinatorial algorithms, and symmetric functions, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001, xvi+238 pages | Article | MR: 1824028 | Zbl: 0964.05070
|
Sketch the parallelogram shown at right, and then redraw it with sides that are half as long.
Find the perimeters of both the original and smaller parallelograms.
Label all four sides. The perimeter is the sum of all four lengths.
More Hint (a):
For the scaled figure, divide each side in half. Add all four of the half lengths.
If the height of the original parallelogram (drawn to the side that is
6
units) is
2
units, find the areas of both parallelograms.
Remember, the area of a parallelogram is
(\text{base})(\text{height})
More Hint (b):
For the smaller figure, would the area be half as much or a quarter as much of the larger figure? Drag the slider in the eTool below for a visual image.
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Week 11. Sorting | Algorithms and Data Structures
12 Week 11. Sorting
Sorting out Sorting — educational cult video (or cult educational video) from 1981 ... At UiB in the 90s, this was always shown as part of the AlgDat module (projection of proper photographic film), with a wide invitation. Senior students attended to cheer for Bubble Sort.
Seriously, it does give a good overview of a range of sorting algorithms. If you watch it on your own, you’ll want double-speed.
Problem 12.1 Implement two sorting algorithms of your choice (in a programming language also of your choice), and compare their running times on different data sets. If you work in a group, you can implement more than two algorithms to include in the comparison.
Compare your results with the algorithmic theory. Are your results consistent with the theory? What important features does it highlight for each algorithm? What does it tell you about how to choose a sorting algorithm for specific purposes?
You need to test on a handful of different datasets with sizes so that you can observe run times ranging from less than a second to more than half a minute. The exact sizes will depend on your hardware. You can choose your data sets; e.g. integers in a small or a large range, floating point numbers, etc.
Explain why you chose the algorithms you did. Did you want to learn something in particular? Did you actually learn it?
Note. If you do not know how to measure run time, check the similar problems in the first few weeks of the semester.
Problem 12.2 The textbook offers two implementations of Merge Sort, one based on an array and one based on linked lists. Consider each one and decide whether or not it is stable. Justify your answer.
Problem 12.3 Write down pseudo-code for radix sort. Try to make it simple and emphasise key points. Explain what would happen if a non-stable inner sorting algorithm is used.
Problem 12.4 Suppose we modify the deterministic version of QuickSort to use the middle element (index
n∕2
) as pivot instead of the last one. What is the running time of this algorithm if the sequence is already sorted?
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Sling - Ring of Brodgar
Skill(s) Required Hunting, Archery
Object(s) Required String x5, Prepared Animal Hide x2
Combat Skill Marksmanship
Ammunition Stone
Craft > Clothes & Equipment > Weapons > Sling
A Sling is a one-handed projectile weapon that requires Stone as ammunition.
It is often used by players with low Marksmanship skill, as it is twice as easy to aim with than it is with the Bow. However, the sling does lower damage than the bow. It is commonly used for hunting.
The maximum damage is 15, multiplied by the square root of the quality of your Sling divided by ten.
Your Perception attribute affects the minimum damage of ranged attacks. Your target's Agility attribute reduces your damage.
{\displaystyle Max=15*{\sqrt {\frac {_{quality}}{10}}}}
{\displaystyle Min={\frac {1}{2}}*{\frac {PER}{AGL}}*Max}
Where PER is your Perception attribute and AGL is the enemy's Agility attribute.
Base Aiming Speed
2.0 2x Bow, 6x Ranger's Bow
The Intrinsic Effective Marksmanship Modifier (IEMM) variable is multiplied by the sling's quality to find the Effective Marksmanship Limit (EML) of the sling. e.g. a sling of quality 13 will have an EML of 26 (2x13). A person using a sling will not get any benefit to aiming speed from a marksmanship skill higher than the sling's EML. Aiming speed is affected by the base aiming speed and the lowest of a player's marksmanship skill and the EML of the sling used. So, a person with 30 marksmanship skill, a quality 13 sling (IEMM 2.0, EML 26) will aim as if their marksmanship skill were 26. IEMM and EML have no effect on weapon damage.
Base aiming speed is in relation to other weapons. A wooden bow aims twice as slowly as a sling, and ranger's bow aims 6 times slower than sling does. This means that a person with 100 marksmanship and a wooden bow would aim at the same speed as a person with 50 marksmanship and a sling and as a person with 300 marksmanship and a ranger's bow, assuming that none of them are limited by the EML of their weapons and ammunition.
The quality of the Sling is determined with the following formula:
{\displaystyle _{q}Sling={\frac {AvgStringQ+AvgLeatherQ}{2}}}
It is softcapped by Dexterity, Survival, and Marksmanship.
A sling used as a wall decoration.
Ghost Apple Gelatin (2021-03-05) >"Slings now require prepared hides, rather than leather."
Slingshot (2016-01-27) >"Added Sling. Fires a stone projectile. Firing only reduces the aim meter by 30%."
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Sling&oldid=92841"
|
Find the area and perimeter of each shape below. Show your steps and work. Note: Diagrams are not drawn to scale.
Perimeter: Add the lengths of each of the sides around the outside of the figure.
13+30+13+13+30+13=112
The shape consists of two trapezoids of equal size. Find the area of one trapezoid and double it to find the area of the entire shape.
The area of a trapezoid can be found with the following formula:
A=\frac{1}{2}(\text{base}_1+\text{base}_2)(\text{height})
420
You can use the equation for the area of a trapezoid to find the total area.
The trapezoid is highlighted.
Notice that there is a rectangular section missing from the trapezoid.
You will need to find that area as well.
744
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Hermite Form - Maple Help
Home : Support : Online Help : Mathematics : Linear Algebra : LinearAlgebra Package : Generic Subpackage : Hermite Form
compute the Hermite form of a Matrix
HermiteForm[E](B)
HermiteForm[E](B,output=out)
the domain of computation, a Euclidean domain
Given an m x n Matrix B of values in E, HermiteForm[E](B) returns the Hermite form H of B, an m x n Matrix of values in E.
The Hermite normal form Matrix H satisfies:
(1) H is row-equivalent to B and H is in row echelon form
(2) The bottom-most nonzero entry p[j] = H[b,j] in each column j is unit normal, and either H[i,j]=0 or the Euclidean norm of H[i,j] where i<b is less than Euclidean norm of p[j]
(3) If B is n x n square Matrix, then prod(H[i,i],i=1..n) = u*det(B) where u is a unit in E
The uniqueness of H depends on the uniqueness of the remainder operation in E. For example, if E is the ring of integers, and a and b are integers, and the remainder of a / b is in the positive range 0..abs(b)-1, then H is unique. If Maple's irem function is used, as in the example below, because the remainder is in the range 1-abs(b)..abs(b)-1, H is not unique.
E[EuclideanNorm]: a procedure for computing the Euclidean norm of an element in E, a non-negative integer.
For a,b in E, b non-zero, the remainder r and quotient q satisfy a = b q + r and r = 0 or EuclideanNorm(r) < EuclideanNorm(b).
For non-zero a,b in E, units u,v in E, the Euclidean norm satisfies:
The Hermite form is computed by putting the principal block H[1..i,1..i] into Hermite form using elementary Row operations. This algorithm does at most O(n^4) operations in E.
\mathrm{with}\left(\mathrm{LinearAlgebra}[\mathrm{Generic}]\right):
Z[\mathrm{`0`}],Z[\mathrm{`1`}],Z[\mathrm{`+`}],Z[\mathrm{`-`}],Z[\mathrm{`*`}],Z[\mathrm{`=`}]≔0,1,\mathrm{`+`},\mathrm{`-`},\mathrm{`*`},\mathrm{`=`}:
Z[\mathrm{Gcdex}]≔\mathrm{igcdex}:
Z[\mathrm{Quo}],Z[\mathrm{Rem}]≔\mathrm{iquo},\mathrm{irem}:
Z[\mathrm{UnitPart}]≔\mathrm{sign}:
Z[\mathrm{EuclideanNorm}]≔\mathrm{abs}:
A≔\mathrm{Matrix}\left([[1,2,3,4,5],[2,3,4,5,6],[4,1,-2,-5,-2],[-1,-4,-2,1,2]]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{-2}& \textcolor[rgb]{0,0,1}{-5}& \textcolor[rgb]{0,0,1}{-2}\\ \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{-4}& \textcolor[rgb]{0,0,1}{-2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
H≔\mathrm{HermiteForm}[Z]\left(A\right)
\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{-2}& \textcolor[rgb]{0,0,1}{-3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{11}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{6}\end{array}]
H,U≔\mathrm{HermiteForm}[Z]\left(A,\mathrm{output}=['H','U']\right)
\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{U}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{-2}& \textcolor[rgb]{0,0,1}{-3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{11}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{6}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{-3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{-15}& \textcolor[rgb]{0,0,1}{12}& \textcolor[rgb]{0,0,1}{-2}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{10}& \textcolor[rgb]{0,0,1}{-7}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\end{array}]
\mathrm{MatrixMatrixMultiply}[Z]\left(U,A\right)
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{-2}& \textcolor[rgb]{0,0,1}{-3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{11}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{6}\end{array}]
Using positive range for the remainder, given by modp(a,b).
Z[Quo] := proc(a,b,r) local q,rr; rr := Z[Rem](a,b,'q'); if nargs=3 then r := rr; end if; q; end proc:
Z[Rem] := proc(a,b,q) local r,qq; r := modp(a,b); if nargs=3 then q := iquo(a-r,b); end if; r; end proc:
H≔\mathrm{HermiteForm}[Z]\left(A\right)
\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{9}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{11}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{6}\end{array}]
U≔\mathrm{HermiteForm}[Z]\left(A,\mathrm{output}='U'\right)
\textcolor[rgb]{0,0,1}{U}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{-18}& \textcolor[rgb]{0,0,1}{14}& \textcolor[rgb]{0,0,1}{-2}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{-15}& \textcolor[rgb]{0,0,1}{12}& \textcolor[rgb]{0,0,1}{-2}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{10}& \textcolor[rgb]{0,0,1}{-7}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\end{array}]
\mathrm{MatrixMatrixMultiply}[Z]\left(U,A\right)
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{9}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{11}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{6}\end{array}]
|
Squircle - Wikipedia
Shape intermediate between a square and a circle
1 Superellipse-based squircle
1.1 p-norm notation
2 Fernández-Guasti squircle
3 Similar shapes
{\displaystyle \left|{\frac {x-a}{r_{a}}}\right|^{n}+\left|{\frac {y-b}{r_{b}}}\right|^{n}=1,}
{\displaystyle \left(x-a\right)^{4}+\left(y-b\right)^{4}=r^{4}}
The area inside the squircle can be expressed in terms of the gamma function Γ(x) as[1]
{\displaystyle \mathrm {Area} =4r^{2}{\frac {\left(\Gamma \left(1+{\frac {1}{4}}\right)\right)^{2}}{\Gamma \left(1+{\frac {2}{4}}\right)}}={\frac {8r^{2}\left(\Gamma \left({\frac {5}{4}}\right)\right)^{2}}{\sqrt {\pi }}}=\varpi {\sqrt {2}}\,r^{2}\approx 3.708149\,r^{2},}
where r is the minor radius of the squircle, and
{\displaystyle \varpi }
is the lemniscate constant.
In terms of the p-norm ‖ · ‖p on R2, the squircle can be expressed as:
{\displaystyle \left\|\mathbf {x} -\mathbf {x} _{c}\right\|_{p}=r}
where p = 4, xc = (a,b) is the vector denoting the centre of the squircle, and x = (x,y). Effectively, this is still a "circle" of points at a distance r from the centre, but distance is defined differently. For comparison, the usual circle is the case p = 2, whereas the square is given by the p → ∞ case (the supremum norm), and a rotated square is given by p = 1 (the taxicab norm). This allows a straightforward generalization to a spherical cube, or sphube, in R3, or hypersphubes in higher dimensions.[2]
Fernández-Guasti squircle[edit]
Another squircle comes from work in optics.[3][4] It may be called the Fernández-Guasti squircle, after one of its authors, to distinguish it from the superellipse-related squircle above.[2] This kind of squircle, centred at the origin, can be defined by the equation:
{\displaystyle x^{2}+y^{2}-{\frac {s^{2}}{r^{2}}}x^{2}y^{2}=r^{2}}
where r is the minor radius of the squircle, s is the squareness parameter, and x and y are in the interval [−r, r]. If s = 0, the equation is a circle; if s = 1, this is a square. This equation allows a smooth parametrization of the transition from a circle to a square, without involving infinity.
A rounded cube can be defined in terms of superellipsoids.
Many Nokia phone models have been designed with a squircle-shaped touchpad button.[6][7] Apple uses an approximation of a squircle (actually a quintic superellipse) for icons in iOS, iPadOS, macOS, and the home buttons of some Apple hardware.[8] One of the shapes for adaptive icons introduced in the Android "Oreo" operating system is a squircle.[9] Samsung uses squircle-shaped icons in their Android software overlay One UI, and in Samsung Experience and TouchWiz.[10]
Italian car manufacturer Fiat used numerous squircles in the interior and exterior design of the third generation Panda.[11]
^ a b Weisstein, Eric W. "Squircle". MathWorld.
^ a b Chamberlain Fong (2016). "Squircular Calculations". arXiv:1604.02174. Bibcode:2016arXiv160402174F. {{cite journal}}: Cite journal requires |journal= (help)
^ M. Fernández Guasti (1992). "Analytic Geometry of Some Rectilinear Figures". Int. J. Educ. Sci. Technol. 23: 895–901.
^ a b M. Fernández Guasti; A. Meléndez Cobarrubias; F.J. Renero Carrillo; A. Cornejo Rodríguez (2005). "LCD pixel shape and far-field diffraction patterns" (PDF). Optik. 116 (6): 265–269. Bibcode:2005Optik.116..265F. doi:10.1016/j.ijleo.2005.01.018. Retrieved 20 November 2006.
^ "Squircle Plate". Kitchen Contraptions. Archived from the original on 1 November 2006. Retrieved 20 November 2006.
^ Nokia Designer Mark Delaney mentions the squircle in a video regarding classic Nokia phone designs:
^ "Clayton Miller evaluates shapes on mobile phone platforms". Retrieved 2 July 2011.
^ "The Hunt for the Squircle". Retrieved 23 May 2022.
^ "Adaptive Icons". Retrieved 15 January 2018.
^ "OneUI". Samsung Developers. Retrieved 2022-04-14.
^ "PANDA DESIGN STORY" (PDF). Retrieved 30 December 2018.
What is the area of a Squircle? on YouTube by Matt Parker
Retrieved from "https://en.wikipedia.org/w/index.php?title=Squircle&oldid=1089363160"
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