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Spherical means and Riesz decomposition for superbiharmonic functions
April, 2006 Spherical means and Riesz decomposition for superbiharmonic functions
Keiji KITAURA, Yoshihiro MIZUTA
The aim in this note is to discuss the behavior at infinity for superbiharmonic functions on
{\mathbf{R}}^{n}
by use of spherical means.
Keiji KITAURA. Yoshihiro MIZUTA. "Spherical means and Riesz decomposition for superbiharmonic functions." J. Math. Soc. Japan 58 (2) 521 - 533, April, 2006. https://doi.org/10.2969/jmsj/1149166786
Keywords: Riesz decomposition , spherical mean , superbiharmonic function
Keiji KITAURA, Yoshihiro MIZUTA "Spherical means and Riesz decomposition for superbiharmonic functions," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 58(2), 521-533, (April, 2006)
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Play Music - Monogatari Documentation
Play music media
'play music <music_id> [with [properties]]'
The play music action let's you, as it name says, play some background music for your game. You can play as many songs as you want simultaneously.
To stop the music, check out the Stop Music documentation.
Action ID: Music
The name of the music you want to play. These assets must be declared beforehand.
The following is a comprehensive list of the properties available for you to modify certain behaviors of the play music action.
The fade property let's you add a fade in effect to the music, it accepts a time in seconds, representing how much time you want it to take until the music reaches it's maximum volume.
The volume property let's you define how high the music will be played.
Make the music loop. This property does not require any value.
To play a song, you must first add the file to your assets/music/ directory and then declare it. To do so, Monogatari has an has a function that will let you declare all kinds of assets for your game.
monogatari.assets ('music', {
'<music_id>': 'musicFileName'
Each browser has it's own format compatibility. MP3 however is the format supported by every browser.
The following will play the song, and once the song ends, it will simply stop.
'play music mainTheme'
'mainTheme': 'mainThemeSong.mp3'
The following will play the song, and once the song ends, it will start over on an infinite loop until it is stopped using the Stop Music Action.
'play music mainTheme with loop'
The following will play the song, and will use a fade in effect.
'play music mainTheme with fade 3'
The following will set the volume of this song to 73%.
'play music mainTheme with volume 73'
Please note however, that the user's preferences regarding volumes are always respected, which means that this percentage is taken from the current player preferences, meaning that if the player has set the volume to 50%, the actual volume value for the song will be the result of:
50 * 0.73 = 36.5%
Of course, you can combine all of this properties, and remember the order doesn't really matter, you can write the properties on the order that feels more natural to you.
'play music mainTheme with volume 100 loop fade 20'
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Generating Higgs Events on the grid - Atlas Wiki
Generating Higgs Events on the grid
1 A specific example
2 The necessary (6) files
3 The main script (Higgs_ShipOff_Everything.py)
4 What input files are produced for each job
5 Checking what is happening on the grid (gridmgr)
6 What output files are produced for each job
7 Changing things: Other Generator/physics process, fast/full simulation, AOD/CBNT .... etc. etc. =
We'll describe here an example where we'll generate
{\displaystyle H\rightarrow ZZ\rightarrow XXYY}
where you can pick your favorite Higgs mass and Z decay channel. This exercise also allows you to test the script on your local ATLAS setup. First make sure this runs before submitting 100's of jobs onto the grid.
The necessary (6) files
For each ATLAS job op the grid we'll need the following files:
1) A joboptions file for our Athena job joboptions_Higgs_BASIC.py
(Here you specify the physics process and details of the output. In our case: Run pythia, atlfast and produce a CBNT output file)
2) A shell script that will run on the remote grid machine ShellScript_Higgs_BASIC.sh
(The ATLAS settings will be set and the athena job will be started by this script)
3) A JDL file containing the names of all required input and output file jdl_Higgs_BASIC.jdl
4) A tar-ball with ATlas software AtlasStuff.tgz
To facilitate the handling of a large number of jobs we have added two more scripts
5) A script that produces all input files: Higgs_ShipOff_Everything.py
6) A general tool from Wouter to manage your jobs on the grid: gridmgr
On your local machine please download these files into a single directory.
The main script (Higgs_ShipOff_Everything.py)
This main task from this script is easily illustrated by the routines that are called for each job:
Create_Joboptions_File() # create joboptions file
Create_JDL_File() # create jdl file
Create_Shell_Script() # craete shell script
Submit_Job_To_Grid() # submit job onto the grid
Cleanup_InputFiles() # save input files
For each job a unique joboptions file, a unique JDL-file and a unique shell script are produced. Then the job is submitted (locally or on the grid) and finally the input files are stored into a separate directory. As input you give the number of jobs and number of events per job followed by a RunType which specifies if you want to run locally or on the grid. If you want to submit onto the grid you need to be on a User Interface machine (At NIKHEF this is for example ui03.nikhef.nl).
How to run. An example:
Submitting a single job with 50 events locally: Higgs_ShipOff_Everything.py 1 50 0
Submitting 20 jobs with 5000 events on the grid: Higgs_ShipOff_Everything.py 20 5000 1
What input files are produced for each job
If you uncomment the 'submit' line in the script (#Submit_Job_To_Grid()) you can check what is produced without running an actual job. In the directory InputFiles_Job1' you'll find the three input files: ShellScript_Higgs_Job1.sh, joboptions_Higgs_Job1.py and jdl_Higgs_Job1.jdl.
Note: If you plan to produce many files you are adviced to store your files on the grid instead of a local disk on the UI machines. Read about the changes you need to make on the web-page by Gustavo: FullChain_on_the_grid
Checking what is happening on the grid (gridmgr)
Using Wouter's tool it is now much easier to follow what is happening to your jobs on the grid.
Check status of all your jobs: ./gridmgr status -a
What output files are produced for each job
Once the job has finished retrieve the output from the job as follows:
Retrieve output for job 1: ./gridmgr retrieve --dir . <Job1>
In a cryptic directory we now find (apart from the standard input and error files):
Higgs.CBNT.Job1.root
Logfile_Higgs_Job1.log
Changing things: Other Generator/physics process, fast/full simulation, AOD/CBNT .... etc. etc. =
This control from the job is in the standard joboptions file. To change the algorithms you'll need to change this file. The settings for this example (AtlFast+CBNT) were taken from RunningAtlfast, but you can also find there settings for AOD output.
Producing full simulation events you might have to submit more ATLAS software. The TestRelease package is not sufficient and you'll need to tar all the code you'd like to use in a new tar-ball (called AtlasStuff.tgz). Note that all code is build on the remote machine.
Retrieved from "https://wiki.nikhef.nl/atlas/index.php?title=Generating_Higgs_Events_on_the_grid&oldid=4719"
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MCatNLO howto - Atlas Wiki
MCatNLO howto
Running MCatNLO
This page describes how to generate
{\displaystyle t{\bar {t}}}
events in using the MCatNLO generator. The description describes what we did for the Rome production and is set up so that you can start from scratch. Since most people will produce various sets of events, some python scripts have been included. They are certainly not brilliant, but provide you with a working starting point.
The three steps to success when using MCatNLO
Producing events using MCatNLO can be split up in three separate steps:
MCatNLO's Integration step
Top decay + Herwig and producing an CBNT and POOL file (Athena)
Retrieved from "https://wiki.nikhef.nl/atlas/index.php?title=MCatNLO_howto&oldid=4667"
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විකිපීඩියා:නිදහස් නොවන අන්තර්ගතය - විකිපීඩියා
විකිපීඩියා:නිදහස් නොවන අන්තර්ගතය
(විකිපීඩියා:Non-free content වෙතින් යළි-යොමු කරන ලදි)
Guideline page about non-free contentසැකිල්ල:SHORTDESC:Guideline page about non-free content
"WP:FU" මෙතැනට යළියොමුවෙයි. Wikipedia:WikiProject Fair use සඳහා, WP:WPFU බලන්න.
"WP:FUI" මෙතැනට යළියොමුවෙයි. Wikipedia:Motto of the day's frequently used idea area සඳහා, WP:MOTD/FUI බලන්න.
"WP:NOTFREE" මෙතැනට යළියොමුවෙයි. එය WP:NOTFREESPEECH සමඟ වරදවා වටහා නොගන්න.
මේවාත් බලන්න: Wikipedia:Non-free content criteria
මෙම පිටුවේ සිංහල විකිපීඩියාවේ ප්රතිපත්ති (policies) කොටස් එකක් හෝ කිහිපයක් අන්තර්ගත වේ. එම කොටස් Policy section template මගින් සලකුණු කර ඇත. එසේ සලකුණු කොට නොමැති මෙම පිටුවේ ඇති කොටස් ප්රතිපත්ති (policies) ලෙස නොසැලකේ.
මෙම ලිපියේ අඩංගු කරුණු සැකෙවින්: Non-free content can be used in articles only if:
මෙම කොටස සිංහල විකිපීඩියාවේ ප්රතිපත්තියක් (policy) එනම් සියළු සංස්කාරකවරුන් සාමාන්යයෙන් අනුගමනය කළයුතු පොදුවේ පිළිගත් සම්මතයක් ලේඛනගත කරයි. Changes made to it should reflect consensus.
For the full non-free content use guideline (including this policy and its criteria) see Wikipedia:Non-free content.
Minimal usage. Multiple items of non-free content are not used if one item can convey equivalent significant information.
Identification of the source of the material, supplemented, where possible, with information about the artist, publisher and copyright holder; this is to help determine the material's potential market value. See: Wikipedia:Citing sources#Multimedia.
The name of each article (a link to each article is also recommended) in which fair use is claimed for the item, and a separate, specific non-free use rationale for each use of the item, as explained at Wikipedia:Non-free use rationale guideline. The rationale is presented in clear, plain language and is relevant to each use.
Enforcement[සංස්කරණය]
Meeting the no free equivalent criterion[සංස්කරණය]
Multiple restrictions[සංස්කරණය]
Meeting the previous publication criterion[සංස්කරණය]
Meeting the contextual significance criterion[සංස්කරණය]
Sourcing[සංස්කරණය]
Meeting the minimal usage criterion[සංස්කරණය]
Number of items[සංස්කරණය]
Image resolution[සංස්කරණය]
{\displaystyle {\text{new width}}={\sqrt {\tfrac {{\text{target pixel count}}\times {\text{original width}}}{\text{original height}}}}}
An original, high resolution image (that can be reasonably scaled down to maintain overall artistic and critical details) may lose some text detail. In such cases, that text should be duplicated on the image description page. Care should be given to the recreation of copyrighted text: for example, while it is appropriate for credits from a movie poster as factual data, such duplication would not be appropriate for an original poem embedded within an image.
Both non-free audio and video files have more explicit metrics for low resolution, which can be found at Creation and usage of media files.
Guideline examples[සංස්කරණය]
Acceptable use[සංස්කරණය]
Text[සංස්කරණය]
Audio clips[සංස්කරණය]
Some non-free images may be used on Wikipedia, providing they meet both the legal criteria for fair use, and Wikipedia's own guidelines for non-free content. Non-free images that reasonably could be replaced by free content images are not suitable for Wikipedia. All non-free images must meet each non-free content criterion; failure to meet those overrides any acceptable allowance here. The following list is not exhaustive but contains the most common cases where non-free images may be used and is subject to the restrictions listed below at unacceptable use of images, notably §7 which forbids the use of press agency or photo agency (e.g., AP, Corbis or Getty Images) images when the image itself is not the subject of commentary.
Iconic and historical images which are not subject of commentary themselves but significantly aid in illustrating historical events may be used if they meet all aspects of the non-free content criteria, particularly no free alternatives, respect for commercial opportunity, and contextual significance. Note that if the image is from a press or photo agency (e.g., AP, Corbis or Getty Images) and is not itself the subject of critical commentary, it is assumed automatically to fail the "respect for commercial opportunity" test.
Pictures of deceased persons, in articles about that person, provided that ever obtaining a free close substitute is not reasonably likely. Note that if the image is from a press or photo agency (e.g., AP, Corbis or Getty Images) and is not itself the subject of critical commentary it is assumed automatically to fail "respect for commercial opportunity".
Unacceptable use[සංස්කරණය]
"WP:TOP100" මෙතැනට යළියොමුවෙයි. categories for members of published lists සඳහා, Wikipedia:Overcategorization § Published_list බලන්න.
Multimedia[සංස්කරණය]
A photo from a press agency or photo agency (e.g., AP, Corbis or Getty Images), unless the photo itself is the subject of sourced commentary in the article.
Non-free image use in list articles[සංස්කරණය]
Non-free image use in galleries or tables[සංස්කරණය]
Pages in userspace consisting solely or almost exclusively of non-free galleries are eligible for speedy deletion per CSD U3.
Exemptions[සංස්කරණය]
Explanation of policy and guidelines[සංස්කරණය]
Background[සංස්කරණය]
Legal position[සංස්කරණය]
In general[සංස්කරණය]
Public Domain periods in the United States. හා සබැඳි මාධ්ය විකිමාධ්ය කොමන්ස් හි ඇත.
Anything published 1927 or later in other countries and still copyrighted there, is typically also copyrighted in the United States. See Wikipedia:Non-U.S. copyrights.[7][පැහැදීම ඇවැසිය]
Applied to Wikipedia[සංස්කරණය]
Handling inappropriate use of non-free content[සංස්කරණය]
Other Wikimedia projects[සංස්කරණය]
↑ The NFCI#2 allowance for logos only applies to the use of the logo on the infobox or lede for the stand-alone article about the entity, and should reflect its most current logo. The use of historical logos for an entity is not allowed, unless the historical logo itself is described in the context of critical commentary about that historical logo.
↑ "A 1961 Copyright Office study found that fewer than 15% of all registered copyrights were renewed. For books, the figure was even lower: 7%. Barbara Ringer, "Study No. 31: Renewal of Copyright" (1960) "Study No. 31: Renewal of Copyright" (1960), reprinted in Library of Congress Copyright Office. Copyright law revision: Studies prepared for the Subcommittee on Patents, Trademarks, and Copyrights of the Committee on the Judiciary, United States Senate, Eighty-sixth Congress, first [-second] session. (Washington: U. S. Govt. Print. Off, 1961), p. 220. A good guide to investigating the copyright and renewal status of published work is Samuel Demas and Jennie L. Brogdon, "Determining Copyright Status for Preservation and Access: Defining Reasonable Effort," Library Resources and Technical Services 41:4 (October, 1997): 323-334." , Hirtle, Peter (2007) Copyright Term and the Public Domain in the United States footnote 7. Of the total US material first published between 1927 and 1963, the percentage of renewed copyrights is far lower, because most published material was never registered at all.
"https://si.wikipedia.org/w/index.php?title=විකිපීඩියා:නිදහස්_නොවන_අන්තර්ගතය&oldid=456903" වෙතින් සම්ප්රවේශනය කෙරිණි
විකිපීඩියාවේ content guideline
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Closure (algebraic structure) - zxc.wiki
Closure (algebraic structure)
In mathematics , in particular algebra , the isolation of a set with respect to a link is understood to mean that the link between any elements of this set results in an element of the set. For example, the set of integers is closed in terms of addition , subtraction and multiplication , but not in terms of division . In the case of algebraic structures with several links, one considers the closeness of all these links accordingly.
Let be a -place inner connection on a set , that is, be a function . A nonempty subset is now called closed with respect to if
{\ displaystyle f}
{\ displaystyle n}
{\ displaystyle A}
{\ displaystyle f}
{\ displaystyle A ^ {n} \ to A}
{\ displaystyle M \ subseteq A}
{\ displaystyle f}
{\ displaystyle f (a_ {1}, \ dotsc, a_ {n}) \ in M}
applies to all . That means, restricted to the definition area , a -digit inner link must also be on .
{\ displaystyle a_ {1}, \ dotsc, a_ {n} \ in M}
{\ displaystyle f}
{\ displaystyle M ^ {n}}
{\ displaystyle n}
{\ displaystyle M}
A subgroup is a non-empty subset of a group that is closed in terms of linkage and inverse formation.
{\ displaystyle (G, +)}
{\ displaystyle +}
A sub-vector space is a non-empty subset of a vector space that is closed with regard to vector addition and scalar multiplication .
{\ displaystyle V}
In general, an algebraic substructure is a (non-empty) subset of an algebraic structure that is closed with regard to all the links in this structure.
The importance of closeness to a link is best understood by looking at examples where it is violated.
As a sub-structure, the group is not closed, i.e. not a sub-group. This subset is closed with regard to the addition, but not with regard to the formation of the inverse: with does not belong .
{\ displaystyle (\ mathbb {N}, +)}
{\ displaystyle (\ mathbb {Z}, +, 0, -)}
{\ displaystyle a \ in \ mathbb {N}}
{\ displaystyle -a}
{\ displaystyle \ mathbb {N}}
The intersection of two sub-vector spaces of a vector space is always itself a sub-vector space, but the union of two sub-vector spaces is not necessarily a sub-vector space. The union is complete with regard to the scalar multiplication, but not necessarily with regard to the vector addition.
Similarly, is a subset also completed over a -digit inner join on if their image is.
{\ displaystyle M}
{\ displaystyle \ infty}
{\ displaystyle f}
{\ displaystyle A}
{\ displaystyle M}
If the power set of an infinite set and the set of all closed sets with respect to a T 1 topology is on , i.e. contains all (infinitely many) one-element subsets of , then a closed set with respect to the set- theoretic average is on .
{\ displaystyle {\ mathcal {P}} (X)}
{\ displaystyle X}
{\ displaystyle {\ mathcal {C}}}
{\ displaystyle X}
{\ displaystyle {\ mathcal {C}} \ subseteq {\ mathcal {P}} (X)}
{\ displaystyle X}
{\ displaystyle {\ mathcal {C}}}
{\ displaystyle \ bigcap}
{\ displaystyle {\ mathcal {P}} (X)}
The property that a link on a set always delivers uniquely certain values in is also referred to as the well-defined nature of this link.
{\ displaystyle f}
{\ displaystyle A}
{\ displaystyle A}
Localization (algebra)
Transitive envelope (relation)
Todd Rowland, Eric W. Weisstein: Set Closure . In: MathWorld (English).
Chi Woo, Michael Slone: Closure of a subset under relations . In: PlanetMath . (English)
This page is based on the copyrighted Wikipedia article "Abgeschlossenheit_%28algebraische_Struktur%29" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Understanding bow speed IBO specification
Archery calculator principles
How fast does an arrow travel?
This arrow speed calculator allows you to accurately determine the speed of an arrow. While it is based on the bow speed IBO specification, you can input the individual settings of your bow. It will give you a rough idea of how fast an arrow travels after changing the bow from the recommended specifications. You can use the results obtained from this archery calculator in the projectile motion calculator to analyze the arrow's path.
Most professional bows use the IBO (International Bowhunting Organization) specification. This specification determines the arrow speed, provided that you keep the following parameters:
Draw length equal to 30 inches;
Draw weight equal to 70 pounds; and
arrow weighing 350 grains.
If you deviate from these parameters, the arrow speed will not equal the one given by the specification. Nevertheless, most archers do not use these exact parameters. This is where this bow speed calculator comes in handy; it allows you to examine how the arrow will behave under a different bow setting.
You should adjust the arrow speed given by the IBO specification according to the following rules:
For every inch of draw length under 30″, subtract 10 ft/s from the IBO value.
For every inch of draw length above 30″, add 10 ft/s to the IBO value.
For every 3 grains of total arrow weight above draw weight multiplied by 5, subtract 1 ft/s from the IBO value.
For every 3 grains of additional weight on the bowstring, subtract 1 ft/s from the IBO value.
All of these rules can be put in one common equation:
\footnotesize \begin{align*} v =\ &\text{IBO} + (L - 30) \times 10 - W\!/3\ +\\ &\min(0, -(A - 5D\!/3) \end{align*}
v
– Actual arrow speed in ft/s;
\text{IBO}
– Arrow speed according to the IBO specification in ft/s;
L
– Draw length in inches;
W
– Additional weight on the bowstring in grains;
A
– Arrow weight in grains; and
D
– Draw weight in pounds.
You can also use the arrow speed calculator to find the momentum and the kinetic energy of the arrow. These are calculated as follows:
\footnotesize \begin{align*} \text{momentum} &= A \cdot v\\\\ \text{kinetic energy} &= \frac{A \cdot v^2}{2} \end{align*}
Our arrow speed calculator converts the units automatically. If you try to do all of these calculations by hand, keep in mind what units you actually use!
Let's consider the following example: you are analyzing the bow of IBO 300. You want to know the arrow's speed when you increase both the draw length and the arrow weight.
Choose the draw length. Let's say it is equal to 32".
Decide on the draw weight and the arrow weight. Let's say you keep the regular peak draw weight of 70 lbs but use arrows weighing 400 grains.
If there is any additional weight on the bowstring, write it down. Let's assume this weight is equal to 5 grains.
Input all of these values into the formula for arrow speed:
\footnotesize \quad \begin{align*} v &=\ \text{IBO} + (L - 30) \times 10 - W\!/3\ +\\ &\qquad\min(0, -(A - 5D/3)\\\\ &=\ 300 + (32 - 30) \times 10 - 5/3\ +\\ &\qquad\min(0, -(400 - 5\times70)/3)\\\\ &=\ 300 + 2 \times 10 - 1.66\ +\\ &\qquad\min(0, -(400 - 350)/3)\\\\ &=\ 300 + 20 - 1.66\ +\\ &\qquad\min(0, -50/3)\\\\ &=\ 318.33 - 16.67\\\\ &=\ 301.67\ \text{ft/s} \end{align*}
Bow IBO rating
Additional weight on string
Use the helical coil calculator to estimate the inductance and volume of coil springs.
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Grade Recognition System - TetrisWiki
Grade Recognition System (GRS) is the system used for determining the awarded grade for a performance in the TGM series.
1.1 GM requirements
2.1 Internal points
2.4 Credit roll
2.4.1 The Absolute Plus
2.4.2 The Absolute
In TGM, grades 9–S9 are entirely determined by score. A grades is awarded to the player when they reach the score threshold of the grade. The final grade, GM, has special requirements that must also be met as well as score.
GM requirements
When the player attains the GM grade, the credits will roll behind the playfield, and the player can continue to play until the end or they top out. Play during the credit roll does not affect the final ranking.
300 12,000 (Grade 1) ≤ 04:15:00
500 40,000 (Grade S4) ≤ 07:30:00
(slightly higher than S9) ≤ 13:30:00
Internal Grade System[a]
Line clear base value:
15 S1 20 2 12 13 30
The Absolute and The Absolute Plus introduces a new system for calculating a player's grade. Unlike TGM, the score is not used to determine the grade.
This new system uses a set of internal grades, which correlate to a displayed grade. Multiple internal grades can correspond to the same displayed grade. For example, internal grades 20 through 22 could be thought of as S4-, S4, and S4+, but TAP does not display these differently. Although it is possible to continue increasing the internal grade beyond 31, the displayed grade will stop at S9. Internal grades use a points system that increase every time the player clears a line. When this counter reaches 100, it is reset, and the player increases one internal grade.
Fighting against the player is a decay rate, which decrements the internal points by one each period. As the player increases internal grades, the decay rate increases, increasing the difficulty of reaching higher grades.
Points will continually drain away, potentially back to zero. The decay rate is dependent on the player's current internal grade and is shown on the table below in frames/1 point. The timer that controls this decay rate will reset every time the internal grade increases, and will count up for every frame of gameplay that the player has more than 0 points, has control of a tetromino, and doesn't have an active combo multiplier (so if a combo starts with a single, the decay will not be stalled until the player makes at least a double).
The number of points awarded is dependent on 4 variables:
Number of lines cleared: A Tetris line clear is worth more than a triple line clear and so on.
Internal grade: A higher internal grade means less points per line clears.
Combo multiplier: The line clear's position in a combo. Just like the CO medal, clearing 2 or more rows will increase the combo, while singles will merely maintain the current position.
Level: The player's level after the line clear.
The following formula determines the points awarded by a particular line clear. The "ceil" indicates that when the Combo Multiplier is applied to the Base Value, the game rounds the multiplication up.
{\displaystyle {\text{Awarded Internal Points}}=\left\lceil {\text{Base Value}}\times {\text{Combo Multiplier}}\right\rceil \times {\text{Level Multiplier}}}
Depending on the number of rows cleared and the current size of the combo, a different combo multiplier is applied in the formula.
Finally, depending on the level after the clear, one of four level multipliers is applied to the awarded grade points. This equals
{\displaystyle 1+\left\lfloor {\text{level}}/250\right\rfloor }
, or equivalently a value from the following lookup table:
E.g., At level 555, Grade 1, clearing 2 doubles in a combo, the first and second doubles respectively will be worth:
{\displaystyle \left\lceil 12\times 1.0\right\rceil \times (1+\left\lfloor 555/250\right\rfloor )=12\times 3=36}
{\displaystyle \left\lceil 12\times 1.4\right\rceil \times (1+\left\lfloor 558/250\right\rfloor )=17\times 3=51}
This system rewards non-singles in steadily increasing speeds.
Immediately following a grade increase, the grade points are at 0. There is therefore nothing to lose from building the stack higher, until you clear a line.
From level 750-999, a Tetris will always increase the internal grade. E.g.,
{\displaystyle \left\lceil 30\times 1.0\right\rceil \times (1+\left\lfloor 750/250\right\rfloor )=30\times 4=120}
The level multiplier is significant. When the player enter section 700 and the music changes, it is a good idea to stack high in order to clear more lines after level 750. Combined with the previous observation, 2 Tetrises will get the player 2 internal grades instead of only 1, doubling the rate of progress.
Combos aside, even though 2 singles are worth much less than a double, and 4 triples are less than 3 Tetrises, 3 doubles are actually worth more than 2 triples.
When a player finishes the game with 999 levels, the end credits roll behind the playfield, similarly to TGM. One of two credit roll types are possible, the Fading Roll, and the M-Roll. The Fading Roll is a semi-invisible roll, where pieces fade out from the stack 4 seconds after being added. Clearing the fading roll will award the player an Orange line ranking, in which an orange line appears under the ranking on the leaderboard screen. These rankings are ranked higher than the regular green line rankings.
The M-Roll is an invisible challenge, as soon as a piece locks it becomes invisible. If the player unlocks the M-Roll and does not clear it, they are awarded the M grade. If the M-Roll is cleared with less than 32 lines cleared, a Green line GM is awarded. Finally, if the M-Roll is cleared with 32 or more lines cleared, an Orange line GM is awarded, the highest possible ranking.[1][2]
If the M-Roll isn't unlocked, Orange line is awarded for simply clearing the credit roll.
The conditions for the M-Roll are the following:
M-Roll Conditions
Tetrises
000-999 ≤ 525 seconds (≤ 08:45:00) S9
000-100 ≤ 65 seconds (≤ 01:05.00) ≥ 2
500-600 ≤ 2 seconds slower than the average of the first 5 section times (rounded down) ≥ 1
600-700 ≤ 2 seconds slower than section 500-600 ≥ 1
900-999 ≤ 2 seconds slower than section 800-900
In the non-Plus version of TGM2, the M grade is awarded as soon as the M-Roll begins, and survival results in the GM grade.
The conditions for the M-Roll are currently believed to include at least the following:
000-500 ≤ 360 seconds (≤ 06:00:00)
000-100 ≤ 90 seconds (≤ 01:30:00) ≥ 1
100-200 ≥ 1
500-600 ≤ average of the first 5 section times (rounded down) ≥ 1
900-999 ≤ 45 seconds (≤ 00:45:00)
Internal Grade: 06079378
Internal Grade Points: 06079379
↑ http://tetrisconcept.net/threads/tap-master.24/page-10#post-1860
↑ http://tetrisconcept.net/threads/the-m-roll-true-conditions.782/#post-24980
Retrieved from "https://tetris.wiki/index.php?title=Grade_Recognition_System&oldid=19964"
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Borrowing - Liquity Docs
Why would I use Liquity for borrowing?
Liquity protocol offers interest-free loans and is more capital efficient than other borrowing systems (i.e. less collateral is needed for the same loan). Instead of selling Ether to have liquid funds, you can use the protocol to lock up your Ether, borrow against the collateral to withdraw LUSD, and then repay your loan at a future date.
For example: Borrowers speculating on future Ether price increases can use the protocol to leverage their Ether positions up to 11 times, increasing their exposure to price changes. This is possible because LUSD can be borrowed against Ether, sold on the open market to purchase more Ether — rinse and repeat.*
*Note: This is not a recommendation for how to use Liquity. Leverage can be risky and should be used only by those with experience.
Collateral is any asset which a borrower must provide to take out a loan, acting as a security for the debt. Currently, Liquity only supports ETH as collateral.
Is Ether (ETH) the only collateral accepted by Liquity?
Yes, ETH is the only collateral type accepted by Liquity.
The protocol charges one-time borrowing and redemption fees that algorithmically adjust based on the last redemption time. For example: If more redemptions are happening (which means LUSD is likely trading at less than 1 USD), the borrowing fee would continue to increase, discouraging borrowing.
Other systems (e.g. MakerDAO) require variable interest rates to make borrowing more or less favorable, but do so implicitly since borrowers would not feel the impact upfront. Given that this also needs to be managed via governance, Liquity instead opts for a fully decentralized and direct feedback mechanism via one-off fees.
How can I borrow with Liquity?
To borrow you must open a Trove and deposit a certain amount of collateral (ETH) to it. Then you can draw LUSD up to a collateral ratio of 110%. A minimum debt of 2,000 LUSD is required.
Troves maintain two balances: one is an asset (ETH) acting as collateral and the other is a debt denominated in LUSD. You can change the amount of each by adding collateral or repaying debt. As you make these balance changes, your Trove’s collateral ratio changes accordingly.
Every time you draw LUSD from your Trove, a one-off borrowing fee is charged on the drawn amount and added to your debt. Please note that the borrowing fee is variable (and determined algorithmically) and has a minimum value of 0.5% under normal operation. The fee is 0% during Recovery Mode. A 200 LUSD Liquidation Reserve charge will be applied as well, but returned to you upon repayment of debt.
For example: The borrowing fee stands at 0.5% and the borrower wants to receive 4,000 LUSD to their wallet. Being charged a borrowing fee of 20.00 LUSD, the borrower will incur a debt of4,220 LUSD after the Liquidation Reserve and issuance fee are added.
This is the ratio between the Dollar value of the collateral in your Trove and its debt in LUSD. The collateral ratio of your Trove will fluctuate over time as the price of Ether changes. You can influence the ratio by adjusting your Trove’s collateral and/or debt — i.e. adding more ETH collateral or paying off some of your debt.
For example: Let’s say the current price of ETH is $3,000 and you decide to deposit 10 ETH. If you borrow 10,000 LUSD, then the collateral ratio for your Trove would be 300%.
If you instead took out 25,000 LUSD that would put your ratio at 120%.
The minimum collateral ratio (or MCR for short) is the lowest ratio of debt to collateral that will not trigger a liquidation under normal operations (aka Normal Mode). This is a protocol parameter that is set to 110%. So if your Trove has a debt 10,000 LUSD, you would need at least $11,000 worth of Ether posted as collateral to avoid being liquidated.
When you open a Trove and draw a loan, 200 LUSD is set aside as a way to compensate gas costs for the transaction sender in the event your Trove being liquidated. The Liquidation Reserve is fully refundable if your Trove is not liquidated, and is given back to you when you close your Trove by repaying your debt. The Liquidation Reserve counts as debt and is taken into account for the calculation of a Trove's collateral ratio, slightly increasing the actual collateral requirements.
What happens if my Trove is redeemed against?
When LUSD is redeemed, the ETH provided to the redeemer is allocated from the Trove(s) with the lowest collateral ratio (even if it is above 110%). If at the time of redemption you have the Trove with the lowest ratio, you will give up some of your collateral, but your debt will be reduced accordingly.
The USD value by which your ETH collateral is reduced corresponds to the nominal LUSD amount by which your Trove’s debt is decreased. You can think of redemptions as if somebody else is repaying your debt and retrieving an equivalent amount of your collateral. As a positive side effect, redemptions improve the collateral ratio of the affected Troves, making them less risky.
Redemptions that do not reduce your debt to 0 are called partial redemptions, while redemptions that fully pay off a Trove’s debt are called full redemptions. In such a case, your Trove is closed, and you can claim your collateral surplus and the Liquidation Reserve at any time.
Let’s say you own a Trove with 2 ETH collateralized and a debt of 3,200 LUSD. The current price of ETH is $2,000. This puts your collateral ratio (CR) at 125% (= 100% * (2 * 2,000) / 3,200). Let’s imagine this is the lowest CR in the Liquity system and look at two examples of a partial redemption and a full redemption:
Somebody redeems 1,200 LUSD for 0.6 ETH and thus repays 1,200 LUSD of your debt, reducing it from 3,200 LUSD to 2,000 LUSD. In return, 0.6 ETH, worth $1,200, is transferred from your Trove to the redeemer. Your collateral goes down from 2 to 1.4 ETH, while your collateral ratio goes up from 125% to 140% (= 100% * (1.4 * 2,000) / 2,000).
Somebody redeems 6,000 LUSD for 3 ETH. Given that the redeemed amount is larger than your debt minus 200 LUSD (set aside as a Liquidation Reserve), your debt of 3,200 LUSD is entirely cleared and your collateral gets reduced by $3,000 of ETH, leaving you with a collateral of0.5 ETH (= 2 - 3,000 / 2,000).
By making liquidation instantaneous and more efficient, Liquity needs less collateral to provide the same security level as similar protocols that rely on lengthy auction mechanisms to sell off collateral in liquidations.
How can I take advantage of leverage?
You can sell the borrowed LUSD on the market for ETH and use the latter to top up the collateral of your Trove. That allows you to draw and sell more LUSD, and by repeating the process you can reach the desired leverage ratio.
Assuming perfect price stability (1 LUSD = $1), the maximum achievable leverage ratio is 11x. It is given by the formula:
maximum leverage ratio =
\frac{MCR}{(MCR - 100\%)}
where MCR is the Minimum Collateral Ratio.
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Generating Higgs To 4 Muons at NIKHEF - Atlas Wiki
Generating Higgs To 4 Muons at NIKHEF
An exercise to simulated Higgs production events at the LHC, where the Higgs boson decays into 2 Z bosons that each decay into 2 muons.
{\displaystyle H\rightarrow ZZ^{*}\rightarrow \mu ^{+}\mu ^{-}\mu ^{+}\mu ^{-}}
The exercise is ment as a starting point for the 'monkey-see monkey-do' technique. It will be easy to plug in your own favorite process. In this example we will use AtlFast for the detector simulation and reconstruction. We will produce an AOD that contains the MC truth and reconstructed AtlFast objects. Since the AOD is in pool format we will also transform the AOD into an Ntuple that allows a simple analysis program to be constructed in Root.
Note: We assume you have the CMT and Athena set-up at NIKHEF in ordnung Starting with CMT and Athena at NIKHEF
1) Setting up the ATLAS environment at NIKHEF
Some packages are required to get the ATLAS software environment ok. As a first time user you should follow steps a) and b). Every time you log on you only have to process c).
a) Setting up the general ATLAS environment at NIKHEF (first time only)
For a fast start follow the following steps:
Login to a SLC3 machine and: source /project/atlas/nikhef/setup/nikhef_setup_10.0.2.csh
Note: If your directory on the project disk is different from your login name you should tell the setup script. Somebody who's login name is 'Tommie', but wants to do all his ATLAS work under /project/atlas/users/pino should use: source /project/atlas/nikhef/setup/nikhef_setup_10.0.2.csh opt slc3 pino.
Get the TestRelease (with some modifications: check the detailed description)
Go to your project directory: cd /project/atlas/users/<your_login_name>
Check out the TestRelease package from the NIKHEF/ATLAS CVS repository: cvs -d /project/atlas/cvs co TestRelease
Go to the cmt directory: cd TestRelease/TestRelease-00-00-18/cmt
Execute cmt config
Execute source setup.csh
For a detailed description please follow the instructions on: ATLAS setup at NIKHEF.
b) Setting up the Package required to produce Ntuples from the AOD (first time only)
To produce Ntuples from an AOD you'll need to add an additional package created at NIKHEF.
Check out the TTBarAnalysis package from the NIKHEF/ATLAS CVS repository: cvs -d /project/atlas/cvs co TTBarAnalysis
Go to the cmt directory: cd TTBarAnalysis/cmt
Build the library: gmake (Note: you might have to do gmake twice)
You can also get a more detailed set of instructions from Installing the AOD->Ntuple (TTBarabalysis) package.
Once this is set-up you can produce TopNtuples from an AOD if you wish to do so.
c) Setting up all required packages ( every time, but not if you have just done a) and b) )
On every login you should now make sure the shell knows where to get the various programs, which means both the ATLAS general and the Ntuple Make program. You can do this by simply sourcing a script similar to init1002.csh. Simply source it in every window where you want to do the generation: source init1002.csh
Note: Again, ... for those of you whose directory on the project disk is different from your login name you should tell the setup script. Edit the init1002.csh file and add the 3 additional parameters to the line in which the general ATLAS setup script in 'sourced'. Look for example in init1002_special.csh.
2) Generating Higgs events decaying into 4 muons
a) Download the scripts
Go again to your project area and check out the Higgs4MuonAnalysis package from the NIKHEF/ATLAS CVS repository:
cd /project/atlas/users/<your_login_name>
cvs -d /project/atlas/cvs co Higgs4MuonAnalysis
cd Higgs4MuonAnalysis
Let's have a look at what files are in the package.
Athena requires steering files telling it what to do. These files are called joboptions files and since this exercise is made up of 2 steps we have 2 (basic) joboptions files. For there rest we have the script and some extra strange file required by Athena:
jobOptions_Pythia_To_Atlfast_To_AOD_BASIC.py joboptions for: Pythia -> AOD:
jobOptions_AOD_to_Ntuple_BASIC.py joboptions for: AOD -> TopNtuple
ShipOff_Pythia.py The script that generates events
PDGTABLE.MeV A steering file required for MC production in Athena (not to be edited)
b) Options in the script
<Nevents> = The number of events per job
<Njobs> = the number of jobs
<f_interactive> = a flag to signal that you want everything on screen (1) instead of logfile (0, default)
The script is called using: ./ShipOff_Pythia.py <Nevents> <Njobs> <f_interactive>
What does the script do. For each job a subdirectory is made called Jobs<JobNr>. In that directory the joboption files specific to this job are created and Athena is run for both steps. The output files (AOD and TopNtuple) are all stored in that directory.
b) Produce 9 events in 1 job in interactive mode
./ShipOff_Pythia.py 9 1 1
Once the run is finished you can find all input and output files in the sub-directory Job1.
./Job1/jobOptions_Pythia_To_Atlfast_To_AOD_Job1.py
./Job1/jobOptions_AOD_to_Ntuple_Job1.py
./Job1/AOD.Job1.pool.root
./Job1/TopNtupleV6.Job1.root
c) Produce 1,000 events in 2 jobs of 500 events using LogFiles
./ShipOff_Pythia.py 500 2
Note: You will again put everything in the subDirectory Job1, so if it still exists you will have to rename it or remove it first.
Once the run is finished you can find in the output files in Job1 and Job2 where not only the AOD and TopNtuple are located, but also the LogFiles for the Athena run for both steps.
Finished! You have now produced 1,000 events with
{\displaystyle H\rightarrow ZZ^{*}\rightarrow \mu ^{+}\mu ^{-}\mu ^{+}\mu ^{-}}
d) Extra: Choosing a different Physics Process:
The Pythia settings that define the process that is generated is given in the file jobOptions_Pythia_To_Atlfast_To_AOD_BASIC.py. If you want to study a different process: simply edit this file and insert your set of pythia parameters.
3) Analysing the content of the Ntuple
To analyse the content of the Ntuple you can either do a MakeClass() yourself or use the Skeleton that was developed at NIKHEF to easily get a handle on the mainobjects and to perform an analysis. It is used in the ATLAS top group and can be found at TopNtuple Analysis Skeleton
Retrieved from "https://wiki.nikhef.nl/atlas/index.php?title=Generating_Higgs_To_4_Muons_at_NIKHEF&oldid=4670"
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Constant Dollar Definition
What is a Constant Dollar?
Constant dollar calculation:
\begin{aligned} &\text{Second Year Constant Dollar Value} = \text{FYDV} \times \frac { \text{CPI}_2 }{ \text{CPI}_1 } \\ &\textbf{where:} \\ &\text{FYDV} = \text{First year dollar value} \\ &\text{CPI}_2 = \text{Consumer price index for second year} \\ &\text{CPI}_1 = \text{Consumer price index for first year} \\ \end{aligned}
Second Year Constant Dollar Value=FYDV×CPI1CPI2where:FYDV=First year dollar valueCPI2=Consumer price index for second yearCPI1=Consumer price index for first year
Basics of Constant Dollars
The constant dollar is often used by companies to compare their recent performance to past performance. Governments also use the constant dollar to track changes in economic indicators, such as wages or GDP. Any kind of financial data represented in dollar terms can be converted into constant dollars by using the consumer price index (CPI) from the relevant years.
Individuals can also use constant dollars to measure the true appreciation of their investments. For example, When calculated in the same currency, the only instance when a constant dollar value is higher in the past than the present is when a country has experienced deflation over that period.
Constant dollar is an adjusted value of currencies to compare dollar values from one period to another.
Constant dollar can be used for multiple calculations. For example, it can be used to calculate growth in economic indicators, such as GDP. It is also used in company financial statements to compare recent performance to past performance.
Example of Constant Dollars
Constant dollars can be used to calculate what $20,000 earned in 1995 would be equal to in 2005. The CPIs for the two years are 152.4 and 195.3, respectively. The value of $20,000 in 1995 would be equal to $25,629.92 in 2005. This is calculated as $20,000 x (195.3/152.4). The calculation can also be done backwards by reversing the numerator and denominator. Doing so reveals that $20,000 in 2005 was equivalent to only $15,606.76 in 1995.
Suppose Eric bought a house in 1992 for $200,000 and sold it in 2012 for $230,000. After paying his real estate agent a 6% commission, he's left with $216,200. Looking at the nominal dollar figures, it appears that Eric has made $16,200. But what happens when we adjust the $200,000 purchase price to 2012 dollars? By using a CPI inflation calculator, we learn that the purchase price of $200,000 in 1992 is the equivalent of $327,290 in 2012. By comparing the constant dollar figures, we discover that Eric has essentially lost $111,090 on the sale of his home.
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A ball rolls off top of a staircase with a horizontal velocity u . If the steps are h metre high and b mere wide, the ball will just hit the edge of nth step if n equals to
\frac{h{u}^{2}}{g{b}^{2}}
\frac{{u}^{2}8}{g{b}^{2}}
\frac{2h{u}^{2}}{g{b}^{2}}
\frac{2{u}^{2}g}{h{b}^{2}}
\sqrt{2gh+{u}^{2}} :u
\sqrt{2gh+{u}^{2}} :\sqrt{2gh}
A particle is projected at an angle
\theta
with horizontal with an initital speed u. When it makes an angle
\alpha
with horizontal, its speed v is-
u\mathrm{cos}\theta
u\mathrm{cos}\theta u\mathrm{cos}\alpha
\frac{u\mathrm{sin}\theta }{\mathrm{sin}\alpha }
\frac{u\mathrm{cos}\theta }{\mathrm{cos}\alpha }
A particle is moving along the path y =
{\mathrm{x}}^{2}
from x = 0 m to x = 2 m. Then the distance traveled by the particle is:
\sqrt{20} \mathrm{m}
>\sqrt{20} \mathrm{m}
<\sqrt{20} \mathrm{m}
\frac{2\mathrm{a}}{\mathrm{v}} \mathrm{sec}
\frac{\mathrm{a}}{\mathrm{v}} \mathrm{sec}
\frac{2\mathrm{a}}{3\mathrm{v}} \mathrm{sec}
\frac{3\mathrm{a}}{\mathrm{v}} \mathrm{sec}
A body is projected with velocity
20\sqrt{3}
m/s with an angle of projection 60
°
with horizontal. Calculate velocity on that point where body makes an angle 30
°
with the horizontal.
\frac{20}{\sqrt{3}} m/s
10\sqrt{3} m/s
A particle is moving with veocity
\stackrel{\to }{\mathrm{v}}=\mathrm{k}\left(\mathrm{y}\stackrel{^}{\mathrm{i}}+\mathrm{x}\stackrel{^}{\mathrm{j}}\right)
; where k is constant. The general equation for the path is:
\mathrm{y}={\mathrm{x}}^{2}+\mathrm{constant}
{\mathrm{y}}^{2}={\mathrm{x}}^{2}+\mathrm{constant}
\mathrm{y}=\mathrm{x}+\mathrm{constant}
A particle is projected with a velocity u making an angle
\mathrm{\theta }
with the horizontal. At any instant, its velocity v is at right angles to its initial velocity u; then v is:
1. ucos
\mathrm{\theta }
2. utan
\mathrm{\theta }
3. ucot
\mathrm{\theta }
4. usec
\mathrm{\theta }
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EUDML | A congruential identity and the 2-adic order of lacunary sums of binomial coefficients. EuDML | A congruential identity and the 2-adic order of lacunary sums of binomial coefficients.
A congruential identity and the 2-adic order of lacunary sums of binomial coefficients.
Tollisen, Gregory; Lengyel, Tamás
Tollisen, Gregory, and Lengyel, Tamás. "A congruential identity and the 2-adic order of lacunary sums of binomial coefficients.." Integers 4 (2004): Paper A04, 8 p., electronic only-Paper A04, 8 p., electronic only. <http://eudml.org/doc/126069>.
@article{Tollisen2004,
author = {Tollisen, Gregory, Lengyel, Tamás},
title = {A congruential identity and the 2-adic order of lacunary sums of binomial coefficients.},
AU - Tollisen, Gregory
TI - A congruential identity and the 2-adic order of lacunary sums of binomial coefficients.
q
Articles by Tollisen
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Earnings growth - Wikipedia
Earnings growth is the annual compound annual growth rate (CAGR) of earnings from investments. For more general discussion see: Sustainable growth rate#From a financial perspective; Stock valuation#Growth rate; Valuation using discounted cash flows#Determine the continuing value; Growth stock; PEG ratio.
2 Other related measures
3 Historical growth rates
4 P/E ratio and growth rate
5 Sustainability of high growth rates
6 Relationship with GDP growth
When the dividend payout ratio is the same, the dividend growth rate is equal to the earnings growth rate.
Earnings growth rate is a key value that is needed when the Discounted cash flow model, or the Gordon's model is used for stock valuation.
The present value is given by:
{\displaystyle P=D\cdot \sum _{i=1}^{\infty }\left({\frac {1+g_{i}}{1+k}}\right)^{i}}
where P = the present value, k = discount rate, D = current dividend and
{\displaystyle g_{i}}
is the revenue growth rate for period i.
If the growth rate is constant for
{\displaystyle i=n+1}
{\displaystyle \infty }
{\displaystyle P=D\cdot {\frac {1+g_{1}}{1+k}}+D\cdot ({\frac {1+g_{2}}{1+k}})^{2}+...+D\cdot ({\frac {1+g_{n}}{1+k}})^{n}+D\cdot \sum _{i=n+1}^{\infty }\left({\frac {1+g_{\infty }}{1+k}}\right)^{i}}
The last term corresponds to the terminal case. When the growth rate is always the same for perpetuity, Gordon's model results:
{\displaystyle P=D\times {\frac {1+g}{k-g}}}
As Gordon's model suggests, the valuation is very sensitive to the value of g used.[1]
Part of the earnings is paid out as dividends and part of it is retained to fund growth, as given by the payout ratio and the plowback ratio. Thus the growth rate is given by
{\displaystyle g={Plowback\ ratio}\times {return\ on\ equity}}
For the S&P 500 Index, the return on equity has ranged between 10 and 15% during the 20th century, the plowback ratio has ranged from 10 to 67% (see payout ratio).
Other related measures[edit]
It is sometimes recommended that revenue growth should be checked to ensure that earnings growth is not coming from special situations like sale of assets.
When the earnings acceleration (rate of change of earnings growth) is positive, it ensures that earnings growth is likely to continue.
Historical growth rates[edit]
According to economist Robert J. Shiller, earnings per share grew at a 3.5% annualized rate over 150 years (inflation-adjusted growth rate was 1.7%).[2] Since 1980, the most bullish period in U.S. stock market history, real earnings growth according to Shiller, has been 2.6%.
12/31/2007 1468.36 17.58 1.4
12/31/2006 1418.30 17.40 14.7
12/31/2002 879.82 31.89 18.5
12/31/2001 1148.08 46.50 -30.8 2001 contraction resulting in P/E Peak
12/31/2000 1320.28 26.41 8.6 Dot-com bubble burst: March 10, 2000
12/31/1997 970.43 24.43 8.3
12/31/1991 417.09 26.12 -14.8
12/31/1990 330.22 15.47 -6.9 July 1990-March 1991 contraction.
12/31/1989 353.40 15.45 .
12/31/1988 277.72 11.69 . Bottom (Black Monday was October 19, 1987)
The Federal Reserve responded to decline in earnings growth by cutting the target Federal funds rate (from 6.00 to 1.75% in 2001) and raising them when the growth rates are high (from 3.25 to 5.50 in 1994, 2.50 to 4.25 in 2005).[3]
P/E ratio and growth rate[edit]
Further information: PEG ratio and PVGO
Growth stocks generally command a higher P/E ratio because their future earnings are expected to be greater. In Stocks for the Long Run, Jeremy Siegel examines the P/E ratios of growth and technology stocks. He examined Nifty Fifty stocks for the duration December 1972 to Nov 2001. He found that
Warranted P/E
Nifty Fifty average 11.62% 41.9 38.7 10.14%
S&P 500 12.14% 18.9 18.9 6.98%
This suggests that the significantly high P/E ratio for the Nifty Fifty as a group in 1972 was actually justified by the returns during the next three decades. However, he found that some individual stocks within the Nifty Fifty were overvalued while others were undervalued.
Sustainability of high growth rates[edit]
High growth rates cannot be sustained indefinitely. Ben McClure[4] suggests that period for which such rates can be sustained can be estimated using the following:
Not very competitive 1 year
Solid company with recognizable brand name 5 years
Company with very high barriers to entry 10 years
Relationship with GDP growth[edit]
It has been suggested that the earnings growth depends on the nominal GDP, since the earnings form a part of the GDP.[5][6] It has been argued that the earnings growth must grow slower than GDP by approximately 2%.[7] See Sustainable growth rate#From a financial perspective.
^ "Discounted Cash Flow (DCF)". Investopedia.
^ "Siegel vs. Shiller: Is the Stock Market Overvalued?". Wharton School of the University of Pennsylvania. September 18, 2018.
^ "Policy Tools". Federal Reserve.
^ Folger, Jean. "DCF Analysis: The Forecast Period & Forecasting Revenue Growth". Investopedia.
^ Fair, Ray C. (April 2000). "Fed Policy and the Effects of a Stock Market Crash on the Economy - Federal Reserve Board unable to offset effects of market crash" (PDF). Business Economics.
^ http://bigpicture.typepad.com/comments/2007/04/earnings_decele.html Earnings Deceleration and Equity Prices, April 08, 2007
^ BERNSTEIN, WILLIAM J.; ARNOTT, ROBERT D. (September–October 2003). "Earnings Growth: The Two Percent Dilution". 59 (5). CFA Institute: 47–55. SSRN 489602. {{cite journal}}: Cite journal requires |journal= (help)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Earnings_growth&oldid=1008241946"
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Cast coefficients of digital filter to double precision - MATLAB double - MathWorks Italia
Lowpass FIR Filter in Single and Double Precision
Cast coefficients of digital filter to double precision
f2 = double(f1)
f2 = double(f1) casts coefficients in a digital filter, f1, to double precision and returns a new digital filter, f2, that contains these coefficients.
Use designfilt to design a 5th-order FIR lowpass filter. Specify a normalized passband frequency of
0.2\pi
rad/sample and a normalized stopband frequency of
0.55\pi
Cast the filter to single precision and cast it back to double precision. Display the first coefficient of each filter.
d = designfilt('lowpassfir','FilterOrder',5, ...
'PassbandFrequency',0.2,'StopbandFrequency', 0.55);
e = single(d);
f = double(e);
coed = d.Coefficients(1)
coed =
coee = e.Coefficients(1)
coee = single
coef = f.Coefficients(1)
Use double to analyze, in double precision, the effects of single-precision quantization of filter coefficients.
f1 — Single-precision digital filter
Single-precision digital filter, specified as a digitalFilter object. Use designfilt to generate a digital filter based on frequency-response specifications and single to cast it to single precision.
Example: f1= single(designfilt('lowpassfir','FilterOrder',3,'HalfPowerFrequency',0.5)) specifies a third-order Butterworth filter with normalized 3-dB frequency 0.5π rad/sample cast in single precision.
f2 — Double-precision digital filter
Double-precision digital filter, returned as a digitalFilter object.
designfilt | digitalFilter | isdouble | issingle | single
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Rayleigh flow - Wikipedia
Rayleigh flow refers to frictionless, non-adiabatic flow through a constant area duct where the effect of heat addition or rejection is considered. Compressibility effects often come into consideration, although the Rayleigh flow model certainly also applies to incompressible flow. For this model, the duct area remains constant and no mass is added within the duct. Therefore, unlike Fanno flow, the stagnation temperature is a variable. The heat addition causes a decrease in stagnation pressure, which is known as the Rayleigh effect and is critical in the design of combustion systems. Heat addition will cause both supersonic and subsonic Mach numbers to approach Mach 1, resulting in choked flow. Conversely, heat rejection decreases a subsonic Mach number and increases a supersonic Mach number along the duct. It can be shown that for calorically perfect flows the maximum entropy occurs at M = 1. Rayleigh flow is named after John Strutt, 3rd Baron Rayleigh.
2 Additional Rayleigh Flow Relations
Figure 1 A Rayleigh Line is plotted on the dimensionless H-ΔS axis.
{\displaystyle \ {\frac {dM^{2}}{M^{2}}}={\frac {1+\gamma M^{2}}{1-M^{2}}}\left(1+{\frac {\gamma -1}{2}}M^{2}\right){\frac {dT_{0}}{T_{0}}}}
{\displaystyle \ {\frac {T_{0}}{T_{0}^{*}}}={\frac {2\left(\gamma +1\right)M^{2}}{\left(1+\gamma M^{2}\right)^{2}}}\left(1+{\frac {\gamma -1}{2}}M^{2}\right)}
{\displaystyle \ \Delta S={\frac {\Delta s}{c_{p}}}=\ln \left[M^{2}\left({\frac {\gamma +1}{1+\gamma M^{2}}}\right)^{\frac {\gamma +1}{\gamma }}\right]}
{\displaystyle {\begin{aligned}H&={\frac {h}{h^{*}}}={\frac {c_{p}T}{c_{p}T^{*}}}={\frac {T}{T^{*}}}\\{\frac {T}{T^{*}}}&={\frac {\left(\gamma +1\right)^{2}M^{2}}{\left(1+\gamma M^{2}\right)^{2}}}\\\end{aligned}}}
The above equation can be manipulated to solve for M as a function of H. However, due to the form of the T/T* equation, a complicated multi-root relation is formed for M = M(T/T*). Instead, M can be chosen as an independent variable where ΔS and H can be matched up in a chart as shown in Figure 1. Figure 1 shows that heating will increase an upstream, subsonic Mach number until M = 1.0 and the flow chokes. Conversely, adding heat to a duct with an upstream, supersonic Mach number will cause the Mach number to decrease until the flow chokes. Cooling produces the opposite result for each of those two cases. The Rayleigh flow model reaches maximum entropy at M = 1.0 For subsonic flow, the maximum value of H occurs at M = 0.845. This indicates that cooling, instead of heating, causes the Mach number to move from 0.845 to 1.0 This is not necessarily correct as the stagnation temperature always increases to move the flow from a subsonic Mach number to M = 1, but from M = 0.845 to M = 1.0 the flow accelerates faster than heat is added to it. Therefore, this is a situation where heat is added but T/T* decreases in that region.
Additional Rayleigh Flow Relations[edit]
The area and mass flow rate are held constant for Rayleigh flow. Unlike Fanno flow, the Fanning friction factor, f, remains constant. These relations are shown below with the * symbol representing the throat location where choking can occur.
{\displaystyle {\begin{aligned}A&=A^{*}={\mbox{constant}}\\{\dot {m}}&={\dot {m}}^{*}={\mbox{constant}}\\\end{aligned}}}
{\displaystyle {\begin{aligned}{\frac {p}{p^{*}}}&={\frac {\gamma +1}{1+\gamma M^{2}}}\\{\frac {\rho }{\rho ^{*}}}&={\frac {1+\gamma M^{2}}{\left(\gamma +1\right)M^{2}}}\\{\frac {T}{T^{*}}}&={\frac {\left(\gamma +1\right)^{2}M^{2}}{\left(1+\gamma M^{2}\right)^{2}}}\\{\frac {v}{v^{*}}}&={\frac {\left(\gamma +1\right)M^{2}}{1+\gamma M^{2}}}\\{\frac {p_{0}}{p_{0}^{*}}}&={\frac {\gamma +1}{1+\gamma M^{2}}}\left[\left({\frac {2}{\gamma +1}}\right)\left(1+{\frac {\gamma -1}{2}}M^{2}\right)\right]^{\frac {\gamma }{\gamma -1}}\end{aligned}}}
Figure 3 Fanno and Rayleigh Line Intersection Chart.
{\displaystyle {\begin{aligned}\Delta S_{F}&={\frac {s-s_{i}}{c_{p}}}=ln\left[\left({\frac {M}{M_{i}}}\right)^{\frac {\gamma -1}{\gamma }}\left({\frac {1+{\frac {\gamma -1}{2}}M_{i}^{2}}{1+{\frac {\gamma -1}{2}}M^{2}}}\right)^{\frac {\gamma +1}{2\gamma }}\right]\\\Delta S_{R}&={\frac {s-s_{i}}{c_{p}}}=ln\left[\left({\frac {M}{M_{i}}}\right)^{2}\left({\frac {1+\gamma M_{i}^{2}}{1+\gamma M^{2}}}\right)^{\frac {\gamma +1}{\gamma }}\right]\end{aligned}}}
Figure 3 shows the Rayleigh and Fanno lines intersecting with each other for initial conditions of si = 0 and Mi = 3.0 The intersection points are calculated by equating the new dimensionless entropy equations with each other, resulting in the relation below.
{\displaystyle \ \left(1+{\frac {\gamma -1}{2}}M_{i}^{2}\right)\left[{\frac {M_{i}^{2}}{\left(1+\gamma M_{i}^{2}\right)^{2}}}\right]=\left(1+{\frac {\gamma -1}{2}}M^{2}\right)\left[{\frac {M^{2}}{\left(1+\gamma M^{2}\right)^{2}}}\right]}
The intersection points occur at the given initial Mach number and its post-normal shock value. For Figure 3, these values are M = 3.0 and 0.4752, which can be found the normal shock tables listed in most compressible flow textbooks. A given flow with a constant duct area can switch between the Rayleigh and Fanno models at these points.
Strutt, John William (Lord Rayleigh) (1910). "Aerial plane waves of finite amplitudes". Proc. R. Soc. Lond. A. 84 (570): 247–284. doi:10.1098/rspa.1910.0075. , also in:
Dover, ed. (1964). Scientific papers of Lord Rayleigh (John William Strutt). Vol. 5. pp. 573–610.
Zucker, Robert D.; Biblarz O. (2002). "Chapter 10. Rayleigh flow". Fundamentals of Gas Dynamics. John Wiley & Sons. pp. 277–313. ISBN 0-471-05967-6.
Emanuel, G. (1986). "Chapter 8.2 Rayleigh flow". Gasdynamics: Theory and Applications. AIAA. pp. 121–133. ISBN 0-930403-12-6.
Wikimedia Commons has media related to Rayleigh flow.
Purdue University Rayleigh flow calculator
University of Kentucky Rayleigh flow Webcalculator
Retrieved from "https://en.wikipedia.org/w/index.php?title=Rayleigh_flow&oldid=1039751706"
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Holland's schema theorem - Wikipedia
Holland's schema theorem, also called the fundamental theorem of genetic algorithms,[1] is an inequality that results from coarse-graining an equation for evolutionary dynamics. The Schema Theorem says that short, low-order schemata with above-average fitness increase exponentially in frequency in successive generations. The theorem was proposed by John Holland in the 1970s. It was initially widely taken to be the foundation for explanations of the power of genetic algorithms. However, this interpretation of its implications has been criticized in several publications reviewed in,[2] where the Schema Theorem is shown to be a special case of the Price equation with the schema indicator function as the macroscopic measurement.
A schema is a template that identifies a subset of strings with similarities at certain string positions. Schemata are a special case of cylinder sets, and hence form a topological space.
Consider binary strings of length 6. The schema 1*10*1 describes the set of all strings of length 6 with 1's at positions 1, 3 and 6 and a 0 at position 4. The * is a wildcard symbol, which means that positions 2 and 5 can have a value of either 1 or 0. The order of a schema
{\displaystyle o(H)}
is defined as the number of fixed positions in the template, while the defining length
{\displaystyle \delta (H)}
is the distance between the first and last specific positions. The order of 1*10*1 is 4 and its defining length is 5. The fitness of a schema is the average fitness of all strings matching the schema. The fitness of a string is a measure of the value of the encoded problem solution, as computed by a problem-specific evaluation function. Using the established methods and genetic operators of genetic algorithms, the schema theorem states that short, low-order schemata with above-average fitness increase exponentially in successive generations. Expressed as an equation:
{\displaystyle \operatorname {E} (m(H,t+1))\geq {m(H,t)f(H) \over a_{t}}[1-p].}
{\displaystyle m(H,t)}
is the number of strings belonging to schema
{\displaystyle H}
at generation
{\displaystyle t}
{\displaystyle f(H)}
is the observed average fitness of schema
{\displaystyle H}
{\displaystyle a_{t}}
is the observed average fitness at generation
{\displaystyle t}
. The probability of disruption
{\displaystyle p}
is the probability that crossover or mutation will destroy the schema
{\displaystyle H}
. It can be expressed as:
{\displaystyle p={\delta (H) \over l-1}p_{c}+o(H)p_{m}}
{\displaystyle o(H)}
is the order of the schema,
{\displaystyle l}
is the length of the code,
{\displaystyle p_{m}}
is the probability of mutation and
{\displaystyle p_{c}}
is the probability of crossover. So a schema with a shorter defining length
{\displaystyle \delta (H)}
is less likely to be disrupted.
An often misunderstood point is why the Schema Theorem is an inequality rather than an equality. The answer is in fact simple: the Theorem neglects the small, yet non-zero, probability that a string belonging to the schema
{\displaystyle H}
will be created "from scratch" by mutation of a single string (or recombination of two strings) that did not belong to
{\displaystyle H}
in the previous generation. Moreover, the expression for
{\displaystyle p}
is clearly pessimistic: depending on the mating partner, recombination may not disrupt the scheme even when a cross point is selected between the first and the last fixed position of
{\displaystyle H}
Plot of a multimodal function in two variables.
The schema theorem holds under the assumption of a genetic algorithm that maintains an infinitely large population, but does not always carry over to (finite) practice: due to sampling error in the initial population, genetic algorithms may converge on schemata that have no selective advantage. This happens in particular in multimodal optimization, where a function can have multiple peaks: the population may drift to prefer one of the peaks, ignoring the others.[3]
The reason that the Schema Theorem cannot explain the power of genetic algorithms is that it holds for all problem instances, and cannot distinguish between problems in which genetic algorithms perform poorly, and problems for which genetic algorithms perform well.
^ Bridges, Clayton L.; Goldberg, David E. (1987). An analysis of reproduction and crossover in a binary-coded genetic algorithm. 2nd Int'l Conf. on Genetic Algorithms and their applications. ISBN 9781134989737.
^ Altenberg, L. (1995). The Schema Theorem and Price’s Theorem. Foundations of genetic algorithms, 3, 23-49.
^ David E., Goldberg; Richardson, Jon (1987). Genetic algorithms with sharing for multimodal function optimization. 2nd Int'l Conf. on Genetic Algorithms and their applications. ISBN 9781134989737.
J. Holland, Adaptation in Natural and Artificial Systems, The MIT Press; Reprint edition 1992 (originally published in 1975).
J. Holland, Hidden Order: How Adaptation Builds Complexity, Helix Books; 1996.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Holland%27s_schema_theorem&oldid=1052474356"
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TIREM propagation model - MATLAB - MathWorks España
Model Coverage Using TIREM™
TIREM propagation model
Model the behavior of electromagnetic radiation from a point of transmission as it travels over irregular terrain, including buildings, by using the Terrain Integrated Rough Earth Model™ (TIREM™) model. Represent the TIREM model by using a TIREM object.
The TIREM model:
Is valid from 1 MHz to 1000 GHz.
TIREM objects require access to an external TIREM library. For more information, see Access TIREM Software.
Create a TIREM object by using the propagationModel function.
Polarization of transmitter and receiver antennas, specified as "horizontal" or "vertical". The object assumes both antennas have the same polarization. The model uses this value to calculate path loss due to ground reflection.
0.005 (default) | numeric scalar in the range [0.0005, 100]
Conductivity of the ground, specified as a numeric scalar in siemens per meter (S/m) in the range [0.0005, 100]. The model uses this value to calculate path loss due to ground reflection. The default value corresponds to average ground.
15 (default) | numeric scalar in the range [1, 100]
Relative permittivity of the ground, specified as a numeric scalar in the range [1, 100]. Relative permittivity is expressed as a ratio of absolute material permittivity to the permittivity of vacuum. The model uses this value to calculate the path loss due to ground reflection. The default value corresponds to average ground.
301 (default) | numeric scalar in the range [250, 400]
Atmospheric refractivity near the ground, specified as a numeric scalar in N-units in the range [250, 400]. The model uses this value to calculate the path loss due to refraction through the atmosphere and tropospheric scatter. The default value corresponds to average atmospheric conditions.
Humidity — Absolute air humidity near ground
9 (default) | numeric scalar in the range [0, 110]
Absolute air humidity near the ground, specified as a numeric scalar in grams per cubic meter (g/m3) in the range [0, 110]. You can use this value to calculate path loss due to atmospheric absorption. The default value corresponds to the absolute humidity of air at 15 degrees Celsius and 70 percent relative humidity.
Display the coverage area for a transmitter using the TIREM model.
pm = propagationModel("tirem");
N=\left(n-1\right)×{10}^{6}
FreeSpace | Rain | Gas | Fog | CloseIn | LongleyRice | RayTracing
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Exponential integral - Wikipedia
Special function defined by an integral
Not to be confused with other integrals of exponential functions.
{\displaystyle E_{1}}
function (top) and
{\displaystyle \operatorname {Ei} }
function (bottom).
In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument.
2.1 Convergent series
2.2 Asymptotic (divergent) series
2.3 Exponential and logarithmic behavior: bracketing
2.4 Definition by Ein
2.5 Relation with other functions
2.8 Exponential integral of imaginary argument
For real non-zero values of x, the exponential integral Ei(x) is defined as
{\displaystyle \operatorname {Ei} (x)=-\int _{-x}^{\infty }{\frac {e^{-t}}{t}}\,dt=\int _{-\infty }^{x}{\frac {e^{t}}{t}}\,dt.}
The Risch algorithm shows that Ei is not an elementary function. The definition above can be used for positive values of x, but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero.
For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and
{\displaystyle \infty }
.[1] Instead of Ei, the following notation is used,[2]
{\displaystyle E_{1}(z)=\int _{z}^{\infty }{\frac {e^{-t}}{t}}\,dt,\qquad |{\rm {Arg}}(z)|<\pi }
For positive values of x, we have
{\displaystyle -E_{1}(x)=\operatorname {Ei} (-x)}
In general, a branch cut is taken on the negative real axis and E1 can be defined by analytic continuation elsewhere on the complex plane.
For positive values of the real part of
{\displaystyle z}
, this can be written[3]
{\displaystyle E_{1}(z)=\int _{1}^{\infty }{\frac {e^{-tz}}{t}}\,dt=\int _{0}^{1}{\frac {e^{-z/u}}{u}}\,du,\qquad \Re (z)\geq 0.}
The behaviour of E1 near the branch cut can be seen by the following relation:[4]
{\displaystyle \lim _{\delta \to 0+}E_{1}(-x\pm i\delta )=-\operatorname {Ei} (x)\mp i\pi ,\qquad x>0.}
Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above.
Convergent series[edit]
For real or complex arguments off the negative real axis,
{\displaystyle E_{1}(z)}
can be expressed as[5]
{\displaystyle E_{1}(z)=-\gamma -\ln z-\sum _{k=1}^{\infty }{\frac {(-z)^{k}}{k\;k!}}\qquad (\left|\operatorname {Arg} (z)\right|<\pi )}
{\displaystyle \gamma }
is the Euler–Mascheroni constant. The sum converges for all complex
{\displaystyle z}
, and we take the usual value of the complex logarithm having a branch cut along the negative real axis.
This formula can be used to compute
{\displaystyle E_{1}(x)}
with floating point operations for real
{\displaystyle x}
between 0 and 2.5. For
{\displaystyle x>2.5}
, the result is inaccurate due to cancellation.
A faster converging series was found by Ramanujan:
{\displaystyle {\rm {Ei}}(x)=\gamma +\ln x+\exp {(x/2)}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}x^{n}}{n!\,2^{n-1}}}\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{\frac {1}{2k+1}}}
These alternating series can also be used to give good asymptotic bounds for small x, e.g.[citation needed]:
{\displaystyle 1-{\frac {3x}{4}}\leq {\rm {Ei}}(x)-\gamma -\ln x\leq 1-{\frac {3x}{4}}+{\frac {11x^{2}}{36}}}
{\displaystyle x\geq 0}
Asymptotic (divergent) series[edit]
Relative error of the asymptotic approximation for different number
{\displaystyle ~N~}
of terms in the truncated sum
Unfortunately, the convergence of the series above is slow for arguments of larger modulus. For example, more than 40 terms are required to get an answer correct to three significant figures for
{\displaystyle E_{1}(10)}
.[6] However, for positive values of x, there is a divergent series approximation that can be obtained by integrating
{\displaystyle xe^{x}E_{1}(x)}
by parts:[7]
{\displaystyle E_{1}(x)={\frac {\exp(-x)}{x}}\left(\sum _{n=0}^{N-1}{\frac {n!}{(-x)^{n}}}+O(N!x^{-N})\right)}
The relative error of the approximation above is plotted on the figure to the right for various values of
{\displaystyle N}
, the number of terms in the truncated sum (
{\displaystyle N=1}
in red,
{\displaystyle N=5}
in pink).
Exponential and logarithmic behavior: bracketing[edit]
Bracketing of
{\displaystyle E_{1}}
by elementary functions
From the two series suggested in previous subsections, it follows that
{\displaystyle E_{1}}
behaves like a negative exponential for large values of the argument and like a logarithm for small values. For positive real values of the argument,
{\displaystyle E_{1}}
can be bracketed by elementary functions as follows:[8]
{\displaystyle {\frac {1}{2}}e^{-x}\,\ln \!\left(1+{\frac {2}{x}}\right)<E_{1}(x)<e^{-x}\,\ln \!\left(1+{\frac {1}{x}}\right)\qquad x>0}
The left-hand side of this inequality is shown in the graph to the left in blue; the central part
{\displaystyle E_{1}(x)}
is shown in black and the right-hand side is shown in red.
Definition by Ein[edit]
{\displaystyle \operatorname {Ei} }
{\displaystyle E_{1}}
can be written more simply using the entire function
{\displaystyle \operatorname {Ein} }
[9] defined as
{\displaystyle \operatorname {Ein} (z)=\int _{0}^{z}(1-e^{-t}){\frac {dt}{t}}=\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}z^{k}}{k\;k!}}}
(note that this is just the alternating series in the above definition of
{\displaystyle \mathrm {E} _{1}}
{\displaystyle E_{1}(z)\,=\,-\gamma -\ln z+{\rm {Ein}}(z)\qquad \left|\operatorname {Arg} (z)\right|<\pi }
{\displaystyle \operatorname {Ei} (x)\,=\,\gamma +\ln {\left|x\right|}-\operatorname {Ein} (-x)\qquad x\neq 0}
Relation with other functions[edit]
Kummer's equation
{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0}
is usually solved by the confluent hypergeometric functions
{\displaystyle M(a,b,z)}
{\displaystyle U(a,b,z).}
But when
{\displaystyle a=0}
{\displaystyle b=1,}
{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(1-z){\frac {dw}{dz}}=0}
{\displaystyle M(0,1,z)=U(0,1,z)=1}
for all z. A second solution is then given by E1(−z). In fact,
{\displaystyle E_{1}(-z)=-\gamma -i\pi +{\frac {\partial [U(a,1,z)-M(a,1,z)]}{\partial a}},\qquad 0<{\rm {Arg}}(z)<2\pi }
with the derivative evaluated at
{\displaystyle a=0.}
Another connexion with the confluent hypergeometric functions is that E1 is an exponential times the function U(1,1,z):
{\displaystyle E_{1}(z)=e^{-z}U(1,1,z)}
The exponential integral is closely related to the logarithmic integral function li(x) by the formula
{\displaystyle \operatorname {li} (e^{x})=\operatorname {Ei} (x)}
for non-zero real values of
{\displaystyle x}
The exponential integral may also be generalized to
{\displaystyle E_{n}(x)=\int _{1}^{\infty }{\frac {e^{-xt}}{t^{n}}}\,dt,}
which can be written as a special case of the incomplete gamma function:[10]
{\displaystyle E_{n}(x)=x^{n-1}\Gamma (1-n,x).}
The generalized form is sometimes called the Misra function[11]
{\displaystyle \varphi _{m}(x)}
{\displaystyle \varphi _{m}(x)=E_{-m}(x).}
Many properties of this generalized form can be found in the NIST Digital Library of Mathematical Functions.
Including a logarithm defines the generalized integro-exponential function[12]
{\displaystyle E_{s}^{j}(z)={\frac {1}{\Gamma (j+1)}}\int _{1}^{\infty }\left(\log t\right)^{j}{\frac {e^{-zt}}{t^{s}}}\,dt.}
The indefinite integral:
{\displaystyle \operatorname {Ei} (a\cdot b)=\iint e^{ab}\,da\,db}
is similar in form to the ordinary generating function for
{\displaystyle d(n)}
, the number of divisors of
{\displaystyle n}
{\displaystyle \sum \limits _{n=1}^{\infty }d(n)x^{n}=\sum \limits _{a=1}^{\infty }\sum \limits _{b=1}^{\infty }x^{ab}}
The derivatives of the generalised functions
{\displaystyle E_{n}}
can be calculated by means of the formula [13]
{\displaystyle E_{n}'(z)=-E_{n-1}(z)\qquad (n=1,2,3,\ldots )}
{\displaystyle E_{0}}
is easy to evaluate (making this recursion useful), since it is just
{\displaystyle e^{-z}/z}
Exponential integral of imaginary argument[edit]
{\displaystyle E_{1}(ix)}
{\displaystyle x}
; real part black, imaginary part red.
{\displaystyle z}
is imaginary, it has a nonnegative real part, so we can use the formula
{\displaystyle E_{1}(z)=\int _{1}^{\infty }{\frac {e^{-tz}}{t}}\,dt}
to get a relation with the trigonometric integrals
{\displaystyle \operatorname {Si} }
{\displaystyle \operatorname {Ci} }
{\displaystyle E_{1}(ix)=i\left[-{\tfrac {1}{2}}\pi +\operatorname {Si} (x)\right]-\operatorname {Ci} (x)\qquad (x>0)}
The real and imaginary parts of
{\displaystyle \mathrm {E} _{1}(ix)}
are plotted in the figure to the right with black and red curves.
There have been a number of approximations for the exponential integral function. These include:
The Swamee and Ohija approximation[15]
{\displaystyle E_{1}(x)=\left(A^{-7.7}+B\right)^{-0.13},}
{\displaystyle {\begin{aligned}A&=\ln \left[\left({\frac {0.56146}{x}}+0.65\right)(1+x)\right]\\B&=x^{4}e^{7.7x}(2+x)^{3.7}\end{aligned}}}
The Allen and Hastings approximation [15][16]
{\displaystyle E_{1}(x)={\begin{cases}-\ln x+{\textbf {a}}^{T}{\textbf {x}}_{5},&x\leq 1\\{\frac {e^{-x}}{x}}{\frac {{\textbf {b}}^{T}{\textbf {x}}_{3}}{{\textbf {c}}^{T}{\textbf {x}}_{3}}},&x\geq 1\end{cases}}}
{\displaystyle {\begin{aligned}{\textbf {a}}&\triangleq [-0.57722,0.99999,-0.24991,0.05519,-0.00976,0.00108]^{T}\\{\textbf {b}}&\triangleq [0.26777,8.63476,18.05902,8.57333]^{T}\\{\textbf {c}}&\triangleq [3.95850,21.09965,25.63296,9.57332]^{T}\\{\textbf {x}}_{k}&\triangleq [x^{0},x^{1},\dots ,x^{k}]^{T}\end{aligned}}}
The continued fraction expansion [16]
{\displaystyle E_{1}(x)={\cfrac {e^{-x}}{x+{\cfrac {1}{1+{\cfrac {1}{x+{\cfrac {2}{1+{\cfrac {2}{x+{\cfrac {3}{\ddots }}}}}}}}}}}}.}
The approximation of Barry et al. [17]
{\displaystyle E_{1}(x)={\frac {e^{-x}}{G+(1-G)e^{-{\frac {x}{1-G}}}}}\ln \left[1+{\frac {G}{x}}-{\frac {1-G}{(h+bx)^{2}}}\right],}
{\displaystyle {\begin{aligned}h&={\frac {1}{1+x{\sqrt {x}}}}+{\frac {h_{\infty }q}{1+q}}\\q&={\frac {20}{47}}x^{\sqrt {\frac {31}{26}}}\\h_{\infty }&={\frac {(1-G)(G^{2}-6G+12)}{3G(2-G)^{2}b}}\\b&={\sqrt {\frac {2(1-G)}{G(2-G)}}}\\G&=e^{-\gamma }\end{aligned}}}
{\displaystyle \gamma }
being the Euler–Mascheroni constant.
Time-dependent heat transfer
Nonequilibrium groundwater flow in the Theis solution (called a well function)
Radial diffusivity equation for transient or unsteady state flow with line sources and sinks
Solutions to the neutron transport equation in simplified 1-D geometries[18]
Bickley–Naylor functions
^ Abramowitz and Stegun, p. 228
^ Abramowitz and Stegun, p. 228, 5.1.1
^ Abramowitz and Stegun, p. 228, 5.1.4 with n = 1
^ Abramowitz and Stegun, p. 229, 5.1.11
^ Bleistein and Handelsman, p. 2
^ Abramowitz and Stegun, p. 228, see footnote 3.
^ After Misra (1940), p. 178
^ Milgram (1985)
^ a b Giao, Pham Huy (2003-05-01). "Revisit of Well Function Approximation and An Easy Graphical Curve Matching Technique for Theis' Solution". Ground Water. 41 (3): 387–390. doi:10.1111/j.1745-6584.2003.tb02608.x. ISSN 1745-6584.
^ a b Tseng, Peng-Hsiang; Lee, Tien-Chang (1998-02-26). "Numerical evaluation of exponential integral: Theis well function approximation". Journal of Hydrology. 205 (1–2): 38–51. Bibcode:1998JHyd..205...38T. doi:10.1016/S0022-1694(97)00134-0.
^ Barry, D. A; Parlange, J. -Y; Li, L (2000-01-31). "Approximation for the exponential integral (Theis well function)". Journal of Hydrology. 227 (1–4): 287–291. Bibcode:2000JHyd..227..287B. doi:10.1016/S0022-1694(99)00184-5.
^ George I. Bell; Samuel Glasstone (1970). Nuclear Reactor Theory. Van Nostrand Reinhold Company.
Abramowitz, Milton; Irene Stegun (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Abramowitz and Stegun. New York: Dover. ISBN 978-0-486-61272-0. , Chapter 5.
Bender, Carl M.; Steven A. Orszag (1978). Advanced mathematical methods for scientists and engineers. McGraw–Hill. ISBN 978-0-07-004452-4.
Bleistein, Norman; Richard A. Handelsman (1986). Asymptotic Expansions of Integrals. Dover. ISBN 978-0-486-65082-1.
Busbridge, Ida W. (1950). "On the integro-exponential function and the evaluation of some integrals involving it". Quart. J. Math. (Oxford). 1 (1): 176–184. Bibcode:1950QJMat...1..176B. doi:10.1093/qmath/1.1.176.
Stankiewicz, A. (1968). "Tables of the integro-exponential functions". Acta Astronomica. 18: 289. Bibcode:1968AcA....18..289S.
Sharma, R. R.; Zohuri, Bahman (1977). "A general method for an accurate evaluation of exponential integrals E1(x), x>0". J. Comput. Phys. 25 (2): 199–204. Bibcode:1977JCoPh..25..199S. doi:10.1016/0021-9991(77)90022-5.
Kölbig, K. S. (1983). "On the integral exp(−μt)tν−1logmt dt". Math. Comput. 41 (163): 171–182. doi:10.1090/S0025-5718-1983-0701632-1.
Milgram, M. S. (1985). "The generalized integro-exponential function". Mathematics of Computation. 44 (170): 443–458. doi:10.1090/S0025-5718-1985-0777276-4. JSTOR 2007964. MR 0777276.
Misra, Rama Dhar; Born, M. (1940). "On the Stability of Crystal Lattices. II". Mathematical Proceedings of the Cambridge Philosophical Society. 36 (2): 173. Bibcode:1940PCPS...36..173M. doi:10.1017/S030500410001714X.
Chiccoli, C.; Lorenzutta, S.; Maino, G. (1988). "On the evaluation of generalized exponential integrals Eν(x)". J. Comput. Phys. 78 (2): 278–287. Bibcode:1988JCoPh..78..278C. doi:10.1016/0021-9991(88)90050-2.
Chiccoli, C.; Lorenzutta, S.; Maino, G. (1990). "Recent results for generalized exponential integrals". Computer Math. Applic. 19 (5): 21–29. doi:10.1016/0898-1221(90)90098-5.
MacLeod, Allan J. (2002). "The efficient computation of some generalised exponential integrals". J. Comput. Appl. Math. 148 (2): 363–374. Bibcode:2002JCoAm.138..363M. doi:10.1016/S0377-0427(02)00556-3.
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.3. Exponential Integrals", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
Temme, N. M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
"Integral exponential function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
NIST documentation on the Generalized Exponential Integral
Weisstein, Eric W. "Exponential Integral". MathWorld.
Weisstein, Eric W. "En-Function". MathWorld.
"Exponential integral Ei". Wolfram Functions Site.
Exponential, Logarithmic, Sine, and Cosine Integrals in DLMF.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Exponential_integral&oldid=1070807809"
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Solvable points on genus one curves
15 April 2008 Solvable points on genus one curves
Andrew Wiles, Mirela Çiperiani
Andrew Wiles,1 Mirela Çiperiani2
A genus one curve defined over
\mathbb{Q}
which has points over
{\mathbb{Q}}_{p}
p
may not have a rational point. It is natural to study the classes of
\mathbb{Q}
-extensions over which all such curves obtain a global point. In this article, we show that every such genus one curve with semistable Jacobian has a point defined over a solvable extension of
\mathbb{Q}
Andrew Wiles. Mirela Çiperiani. "Solvable points on genus one curves." Duke Math. J. 142 (3) 381 - 464, 15 April 2008. https://doi.org/10.1215/00127094-2008-010
Secondary: 11R23 , 11R34 , 14H45 , 14H52
Andrew Wiles, Mirela Çiperiani "Solvable points on genus one curves," Duke Mathematical Journal, Duke Math. J. 142(3), 381-464, (15 April 2008)
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Basic principles and examples of Bayes and naive Bayes - 编程知识
2021-07-20 18:58:37 by pilgrim
Frequency school and Bayes school
Frequentists think that , The distribution parameter to which the sample belongs θ Although unknown , But it's fixed , It's possible to sample
θ
Make an estimate to get
\theta{\hat{}}
. Bayesian school thinks that parameters
θ
It's a random variable , It's not a fixed value , Before the sample is produced , Will be based on experience or other methods to θ Preset a distribution
\pi(\theta)
, It's called a priori distribution . After that, we'll make a comparison of
θ
Adjustment , correct , Write it down as
\pi(\theta|x1,x2,x3,……)
, It's called posterior distribution .
The derivation of Bayes formula
Why naive Bayes
Suppose that the attributes of the training data are determined by n Dimension random vector x Express , The results of classification are random variables y Express , that x and y We can use the joint probability distribution P ( X , Y ) P(X,Y)P(X,Y) describe , Each specific sample ( x i , y i ) (x_i,y_i)(x i ​ ,y i ​ ) Both can pass P ( X , Y ) P(X,Y)P(X,Y) Independent identically distributed generation The starting point of Bayesian classifier is joint probability distribution , According to the properties of conditional probability, we can get P ( X , Y ) = P ( Y ) ∗ P ( X ∣ Y ) = P ( X ) ∗ P ( Y ∣ X ) P(X,Y)=P(Y)*P(X|Y)=P(X)*P(Y|X)P(X,Y)=P(Y)∗P(X∣Y)=P(X)∗P(Y∣X) among P ( Y ) P(Y)P(Y)： Probability of occurrence of each category , This is a priori probability . P ( X ∣ Y ) P(X|Y)P(X∣Y)： The probability of different attributes in a given category , Likelihood probability A priori probability is easy to calculate , Just count the number of samples of different categories , The likelihood probability is affected by the number of attributes , It is difficult to estimate . for example , Each sample contains 100 Attributes , The value of each attribute may have 100 Kind of , Every result of that classification , The conditional probability to calculate is 1002=10000, The amount is huge . therefore , At this time, naive Bayes was introduced .
Naive Bayes , Added a simple , It means simpler Bayes . Naive Bayes assumes that different attributes of samples satisfy the conditional independence hypothesis , On this basis, Bayes theorem is applied to perform the task of classification . For a given item to be classified x, Analyze the posterior probability of the sample in each category , Take the class with the greatest posterior probability as x Category To solve the problem that likelihood probability is difficult to estimate , We need to introduce conditions Independence hypothesis The conditional independence assumption guarantees that all attributes are independent of each other , They don't influence each other , Each attribute has an independent effect on the classification results . In this way, the conditional probability becomes the product of attribute conditional probability
P(X = x|Y = c) = P(X(1)=x(1),X(2)=x(2),……,X(n)=x(n)|Y=c)=\ i=0âˆn​P(Xj=xj∣Y=c)
This is the naive Bayes method , With the training set , We can easily work out a priori probability P ( Y ) P(Y)P(Y) And likelihood probability P ( Y ∣ X ) P(Y|X)P(Y∣X), So we can get the posterior probability P ( X ∣ Y ) P(X|Y)P(X∣Y)
Example – Watermelon book 151 page
First of all, we have a dataset of watermelon 3.0.
We have a problem . The following test set is good or bad ？
We can first calculate a priori probability
And then calculate the conditional probability
Then calculate the probability of good and bad melons
0.063 Significantly larger , So it's probably a good melon .
本文为[pilgrim]所创，转载请带上原文链接，感谢
https://cdmana.com/2021/07/20210718125051383D.html
Tags basic principles examples bayes naive
Copyright © 2020 Basic principles and examples of Bayes and naive Bayes All Rights Reserved.
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OPT Design - Wikibooks, open books for an open world
Output Transformer Design Basics
1 Tube Output Transformer Design
2 Equation Derivation
2.1 Standard OPT Usage
2.2 OPT Small Signal Model
2.2.1 Transformer Basics
2.2.2 Small Signal Model Evaluation
2.2.3 Standard Filter Characteristics
2.3 Field Magnetics
2.3.1.1 A List of the Used Quanteties
2.3.1.2 Integral Form
2.3.1.3 Boundary Limits
2.3.2 Faraday's Law
2.4 Magnetic Flux Density in an Iron Core
2.5 Magnetic Field Intensity in an Iron Core
2.6 Effective Permeability due to Air-Gap
2.7 Toroidal Core Inductance
2.8 C-Core Inductance
2.9 Optimizing Tube Amp Load
Tube Output Transformer DesignEdit
The goal of this book is to show the world how an output transformer or OPT may be designed and what to think of.
The fundamental consideration is that the core must not go into saturation at any voltage or frequency. That means that the core must widthstand:
{\displaystyle B={\frac {U_{rms}}{4,44NAf}}[T]}
Due to DC allmost always being present (especially true in single-end designs) we must also consider the magnetic intensity H:
{\displaystyle H={\frac {NI}{lm}}[A/m]}
where lm is the mean magnetic length around the core.
It can furthermore be shown that the primary inductance need emerges from the fact that
{\displaystyle w_{l}L>r_{p}//R_{L}}
where RL is the reflected load resistance and rp is the plate resistance of the tube(s).
This yields the equation
{\displaystyle L>(r_{p}//n^{2}Z_{L})/w_{l}}
where ZL is the loudspeaker nominal impedance and n the turn-ratio of the transformer.
It can furthermore be shown that maximum output power occurs when each tube is loaded by
{\displaystyle R_{L}=2r_{p}}
Knowing this the minimum inductance L may be calculated.
The small signal model do however also put restrictions on HF parameters. If the OPT is carefully wound (in sectors and not bifilary as well as at least one layer of transformer tape inbetween each layer) the main HF problem will be a so called leakage inductance. The equation for calculating this is:
{\displaystyle w_{h}L_{leak}<(r_{p}+R_{L})}
{\displaystyle L_{leak}<(r_{p}+R_{L})/w_{h}}
This parameter is however very hard to control. But experience has showed that one important thing is that the secondary winding must cover the whole (or both) primaries. Another idea is that the winding ends must not be folded. This last part might mainly eliminate HF resonances.
The inductance of a toroidal transformer can be expressed as:
{\displaystyle L={\frac {\mu _{eff}N^{2}h}{2\pi }}\ln {\frac {b}{a}}[H]}
{\displaystyle \mu _{eff}=\mu _{0}{\frac {\mu _{r}}{1+{\frac {lg}{lm}}\mu _{r}}}}
where lg is the length of the air-gap and lm is the mean magnetic length.
If lg is zero, this simplifies to
{\displaystyle \mu _{eff}=\mu _{0}\mu _{r}}
{\displaystyle \mu _{r}}
is the relative permeability of the iron and
{\displaystyle \mu _{0}}
is the permeability of vacuum.
It can be shown that a single-pole low-pass filter roll-off yields an only -0,5dB impact on the frequency three times lower than fh. This means that if we want -0,5dB at highest audible frequency (20kHz) we would need a fh of 60kHz.
{\displaystyle w_{h}=2\pi f_{h}}
this means a leakage inductance of less than
{\displaystyle L_{leak}<33mH}
if we want to use KT66 tubes in push-pull (PP) where rp=2rp(KT66)=2500 Ohm. Observe that RL in PP-designs is 4 times the load on each tube. Optimum plat-to-plate load is thus 10k Ohms in our case.
The same thing is valid for single-pole high-pass filters, thus fl for -0.5dB@10Hz is 3,3Hz. The inductance therefor needs to be greater than
{\displaystyle L>95H}
If we want to use the suggested core dimensions we get
{\displaystyle A=6,25\cdot 10^{-4}[m^{2}]}
{\displaystyle l_{m}=0,196[m]}
Because P1+P2 needs to widthstand 230V@15Hz and a common maximum flux density for transformer irons is around
{\displaystyle B=1,6T}
the number of turns can now be calculated using the first equation. This gives
{\displaystyle N=3453}
Putting this into the equation for the primary inductance we get
{\displaystyle L=0,041\mu _{r}[H]}
Because we want to use the OPT in Class A push-pull configuration we need not take too much consideration of the DC that will flow due to tube aging and loudspeaker impedance variation with frequency. But it is recommended that the OPT should widthstand at least 10mADC thru both primaries. In SE-configuration a so called air-gap would be needed but this cannot (easily) be realised in toroidal transformers which makes us dependent on the width of the BH-loop of the used iron.
The magnetic intensity for a DC current of 10mA is for our transformer:
{\displaystyle H=176[A/m]}
and here we want to have at least L/3=30H left.
Equation DerivationEdit
In this paragraph the derivation of the above used equations will be explained.
Standard OPT UsageEdit
Standard OPT Usage
This picture shows how an OPT is used in push-pull (PP) configuration. The below theory is however also valid for single-end (SE) configurations.
OPT Small Signal ModelEdit
OPT Small Signal Model
This picture shows the small signal model of the OPT.
Fig.1 shows the trivial OPT connection i.e driven by the generator G thru two plate resistances rp (because of PP).
Fig.2 shows what happens at low frequencies where the OPT works as an ordinary transformer. Reflected impedance is therefor
{\displaystyle R_{L}=n^{2}Z_{L}}
where n is the turn ratio of the OPT and ZL is the loudspeaker impedence.
Fig.3 shows what happens at high frequencies where the leakage inductance LL is dominant over the interlayer capacitance (due to special winding techniques described earlier). The above expression still holds though.
Transformer BasicsEdit
Consider an ideal transformer without iron or copper losses. The output power will then be equal to the input power.
If you transform a high voltage to a low voltage you will be then able to extract a higher current at the secondary than you are putting in on the primary.
{\displaystyle Pin=U1\cdot I1}
{\displaystyle Pout=U2\cdot I2}
{\displaystyle U2=U1/n}
{\displaystyle I2=n\cdot I1}
{\displaystyle U2/I2=1/n^{2}\cdot U1/I1}
{\displaystyle U1/I1=n^{2}\cdot U2/I2}
Small Signal Model EvaluationEdit
While using Norton and Thevenin circuit theory in Fig.2 we get:
{\displaystyle wL_{p}>(2rp//n^{2}Z_{L})}
In Fig.3 we may however just realize the fact that
{\displaystyle wL_{L}<(2rp+n^{2}Z_{L})}
because this is where the reactance of LL becomes dominant.
Standard Filter CharacteristicsEdit
Imagine a single-pole High-Pass filter. Then you might have a capacitor in series with a resistor to ground. The Laplace transfer function then yields
{\displaystyle Uo/Uin=R/(R+1/sC)}
{\displaystyle Uo/Uin=1/(1+1/(sRC))}
{\displaystyle w_{0}=1/RC}
{\displaystyle Uo/Uin=1/(1+w_{0}/s)}
Putting s=jw we get
{\displaystyle Uo/Uin=1/(1+w_{0}/jw)}
and the amplitude of the transfer function gets
{\displaystyle Uo/Uin=1/{\sqrt {1+(w_{0}/w)^{2}}}}
{\displaystyle Uo/Uin=1/{\sqrt {1+(f_{0}/f)^{2}}}}
where f is the frequency.
{\displaystyle f=3f_{0}}
{\displaystyle Uo/Uin=-0,46dB}
Field MagneticsEdit
In this paragraph we will show the world the electromagnetic fundamentals.
The normal form of Maxwell's Equations is
{\displaystyle \nabla \cdot \mathbf {D} =\rho _{f}}
{\displaystyle \nabla \cdot \mathbf {B} =0}
{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}
{\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}}
The first equation, Gauss's Law, describes how electrical fields are caused by electrical charges.
The second equation states that there are no "magnetic charges", or so called magnetic monopoles.
The third equation, Faraday's Law, describes how electrical fields are created due to magnetic field variations.
The fourth equation, Ampere's Law (with Maxwell's correction), describes how magnetic fields are created from electrical field variations.
A List of the Used QuantetiesEdit
E : Electric Field Intensity [V/m]
D : Electric Flux Density [As/m^2]
H : Magnetic Field Intensity [A/m]
B : Magnetic Flux Density [Vs/m^2]
Jf : Free Current Density [A/m^2]
Integral FormEdit
{\displaystyle \oint _{S}\mathbf {D} \cdot d\mathbf {s} =\int _{V}\rho _{f}dv}
{\displaystyle \oint _{S}\mathbf {B} \cdot d\mathbf {s} =0}
{\displaystyle \oint _{C}\mathbf {E} \cdot dl=-\int _{S}{\partial \mathbf {B} \over \partial t}\cdot d\mathbf {s} }
{\displaystyle \oint _{C}\mathbf {H} \cdot dl=\int _{S}\mathbf {J} _{f}\cdot d\mathbf {s} +\int _{S}{\partial \mathbf {D} \over \partial t}\cdot d\mathbf {s} }
Boundary LimitsEdit
Going from medium 1 to medium 2 Maxwell's Equations gives
{\displaystyle \mathbf {\hat {n}} \cdot (\mathbf {D} _{2}-\mathbf {D} _{1})=\sigma _{f}}
{\displaystyle \mathbf {\hat {n}} \times (\mathbf {E} _{2}-\mathbf {E} _{1})=0{\mbox{ or }}\mathbf {E} _{2t}=\mathbf {E} _{1t}}
{\displaystyle \mathbf {\hat {n}} \cdot (\mathbf {B} _{2}-\mathbf {B} _{1})=0}
{\displaystyle \mathbf {\hat {n}} \times (\mathbf {H} _{2}-\mathbf {H} _{1})=\mathbf {K} _{f}{\mbox{ or }}\mathbf {H} _{2t}-\mathbf {H} _{1t}=\mathbf {K} _{f}\times \mathbf {\hat {n}} }
{\displaystyle \sigma _{f}}
is the surface charge density och Kf the free surface current intensity between the mediums.
Faraday's LawEdit
Consider Faraday's Law where we have from Maxwell's Equations:
{\displaystyle \oint _{C}Edl=-{\frac {d}{dt}}\int _{S}BdS}
{\displaystyle V=\oint _{C}Edl}
=emf induced in the curvature C [Volt]
{\displaystyle \Phi =\int _{S}BdS}
=magnetic flux thru the surface S [Vs or Weber]
{\displaystyle V=-{\frac {d\Phi }{dt}}}
If we use several turns N of wire we get
{\displaystyle V=-N{\frac {d\Phi }{dt}}}
And if the magnetic flux flows thru an iron where
{\displaystyle \mu _{r}>>1}
the magnetic flux will stay in the iron only, yielding a secondary voltage proportional to the turn ratio n.
Magnetic Flux Density in an Iron CoreEdit
From Faraday's Law we have
{\displaystyle \Phi =\int _{S}BdS}
And due to no variations in the surface S, we have
{\displaystyle \Phi =BS}
{\displaystyle V=-N{\frac {d\Phi }{dt}}}
{\displaystyle V=-NS{\frac {dB}{dt}}}
A sinusoidal magnetic flux density yields
{\displaystyle B=B_{max}\cdot sin(wt)}
{\displaystyle V=-NSB_{max}\cdot w\cdot cos(wt)}
which maximum occurs when
{\displaystyle V=NSB_{max}w}
{\displaystyle B_{max}={\frac {V}{NSw}}}
{\displaystyle B_{max}={\frac {V}{2\pi NAf}}}
where A has been substituted for S
And if the voltage is sinusoidal
{\displaystyle V={\sqrt {2}}V_{rms}}
{\displaystyle B_{max}={\frac {V_{rms}}{4,44NAf}}}
Magnetic Field Intensity in an Iron CoreEdit
From Maxwell's Equations we get
{\displaystyle \oint _{C}\mathbf {H} \cdot dl=NJ_{f}S=NI}
because we are considering DC only and a homogenous surface.
So if we are using a toroid, then
{\displaystyle H2\pi r=NI}
{\displaystyle l_{m}=2\pi r}
{\displaystyle H={\frac {NI}{l_{m}}}}
Effective Permeability due to Air-GapEdit
Applying Ampere's law we once again get
{\displaystyle \oint _{C}\mathbf {H} \cdot dl=NI}
{\displaystyle B_{f}=B_{g}}
But in the core we will have
{\displaystyle H_{f}=B_{f}/\mu }
and in the air-gap
{\displaystyle H_{g}=B_{f}/\mu _{0}}
{\displaystyle {\frac {B_{f}}{\mu }}(2\pi r-lg)+{\frac {B_{f}}{\mu _{0}}}l_{g}=NI}
{\displaystyle {\frac {B_{f}}{\mu _{0}}}({\frac {lm}{\mu _{r}}}+l_{g})=NI}
{\displaystyle {\frac {B_{f}}{\mu _{eff}}}l_{m}=NI}
{\displaystyle l_{m}=2\pi r-lg}
{\displaystyle \mu _{eff}=\mu _{0}{\frac {l_{m}}{{\frac {l_{m}}{\mu _{r}}}+l_{g}}}}
{\displaystyle \mu _{eff}=\mu _{0}{\frac {\mu _{r}}{1+{\frac {l_{g}}{l_{m}}}\mu _{r}}}}
Toroidal Core InductanceEdit
Consider cylindrical coordinates. Then we get
{\displaystyle B=a_{\phi }B_{\phi }}
{\displaystyle dl=a_{\phi }rd{\phi }}
{\displaystyle \oint _{C}B\cdot dl=\int _{0}^{2\pi }B_{\phi }rd\phi =2\pi rB_{\phi }}
Since the path encircles a total current NI, we have
{\displaystyle 2\pi rB_{\phi }=\mu _{eff}NI}
Knowing the relationship
{\displaystyle B=\mu H}
it is easy to relate to the earlier equations, thus
{\displaystyle B_{\phi }={\frac {\mu _{eff}NI}{2\pi r}}}
{\displaystyle \Phi =\int _{S}Bds=\int _{S}a_{\phi }{\frac {\mu _{eff}NI}{2\pi r}}\cdot a_{\phi }hdr}
{\displaystyle \Phi ={\frac {\mu _{eff}NIh}{2\pi }}\int _{a}^{b}{\frac {dr}{r}}={\frac {\mu _{eff}NIh}{2\pi }}\ln {\frac {b}{a}}}
Using that the flux linkage is
{\displaystyle N\Phi }
and that the small signal inductance is independent of the current, we get
{\displaystyle L={\frac {\mu _{eff}N^{2}h}{2\pi }}ln{\frac {b}{a}}}
C-Core InductanceEdit
This is not so easy to calculate but we can do some approximations if the mean magnetic length could be defined by
{\displaystyle lm=2\cdot c+2\cdot d}
where c is the shortest leg (at the center of the iron) and d the longest dito leg.
Approximating this to a circular toroidal shape, we get
{\displaystyle r_{mean}={\frac {lm}{2\pi }}}
Adding half of the thickness of the iron to this we get b, substracting half of the thickness we get a. Then we might reuse
{\displaystyle L={\frac {\mu _{eff}N^{2}h}{2\pi }}ln{\frac {b}{a}}}
This should be quite valid due to the magnetic flux staying in the iron because of
{\displaystyle \mu _{r_{e}ff}>>1}
Optimizing Tube Amp LoadEdit
Maximum Available Power for Triodes
{\displaystyle i(t)=i_{a}*sin(wt)}
{\displaystyle u(t)=u_{a}*sin(wt)}
{\displaystyle i_{a}=I_{a}}
{\displaystyle u_{a}=U_{a}-2R_{i}I_{a}}
output power may be written
{\displaystyle P=u_{a}*i_{a}/2=U_{a}I_{a}/2-R_{i}I_{a}^{2}}
and derivated by Ia this gives
{\displaystyle {\frac {dP}{dI_{a}}}=U_{a}/2-2R_{i}I_{a}=0}
with maxima for
{\displaystyle I_{a}={\frac {U_{a}}{4R_{i}}}}
{\displaystyle P_{max}={\frac {U_{a}^{2}}{8R_{i}}}-{\frac {U_{a}^{2}}{16R_{i}}}={\frac {U_{a}^{2}}{16Ri}}}
From the image we can see that
{\displaystyle U_{a}=2R_{i}I_{a}+R_{a}I_{a}=I_{a}(2R_{i}+R_{a})={\frac {U_{a}}{4R_{i}}}*(2R_{i}+R_{a})}
where Ua disappears so that
{\displaystyle 1=1/2+{\frac {R_{a}}{4R_{i}}}}
{\displaystyle 2=1+{\frac {R_{a}}{2R_{i}}}}
{\displaystyle 2R_{i}=R_{a}}
A more simple way to prove this is by inspection of image:
{\displaystyle u_{a}=U_{a}-2R_{i}I_{a}=U_{a}-R_{a}I_{a}}
This finally proves that optimum load for a triode is double it's internal resistance. It should however be pointed out that plate voltage should be the limiting factor, for higher voltages where plate dissipation comes into the picture, Ra must be higher.
It is also interesting to note that the efficiency is only 25% in this case, we can prove this by putting:
{\displaystyle P_{a}=U_{a}*I_{a}=U_{a}^{2}/4R_{i}}
{\displaystyle P_{out}=i_{a}*u_{a}/2=I_{a}*(U_{a}-2R_{i}I_{a})/2={\frac {U_{a}}{4R_{i}}}(U_{a}-2R_{i}{\frac {U_{a}}{4R_{i}}})/2={\frac {U_{a}^{2}}{4R_{i}}}(1-1/2)/2}
{\displaystyle {\frac {P_{out}}{P_{a}}}=1/4}
Retrieved from "https://en.wikibooks.org/w/index.php?title=OPT_Design&oldid=3547456"
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Hazard Rate Definition
What Is the Hazard Rate?
The hazard rate refers to the rate of death for an item of a given age (x). It is part of a larger equation called the hazard function, which analyzes the likelihood that an item will survive to a certain point in time based on its survival to an earlier time (t). In other words, it is the likelihood that if something survives to one moment, it will also survive to the next.
The hazard rate only applies to items that cannot be repaired and is sometimes referred to as the failure rate. It is fundamental to the design of safe systems in applications and is often relied on in commerce, engineering, finance, insurance, and regulatory industries.
The hazard rate refers to the rate of death for an item of a given age (x).
It is part of a larger equation called the hazard function, which analyzes the likelihood that an item will survive to a certain point in time based on its survival to an earlier time (t).
The hazard rate cannot be negative, and it is necessary to have a set "lifetime" on which to model the equation.
Understanding the Hazard Rate
The hazard rate measures the propensity of an item to fail or die depending on the age it has reached. It is part of a wider branch of statistics called survival analysis, a set of methods for predicting the amount of time until a certain event occurs, such as the death or failure of an engineering system or component.
The concept is applied to other branches of research under slightly different names, including reliability analysis (engineering), duration analysis (economics), and event history analysis (sociology).
The Hazard Rate Method
The hazard rate for any time can be determined using the following equation:
h(t) = f(t) / R(t)
h(t)=f(t)/R(t)
F(t) is the probability density function (PDF), or the probability that the value (failure or death) will fall in a specified interval, for example, a specific year. R(t), on the other hand, is the survival function, or the probability that something will survive past a certain time (t).
Example of the Hazard Rate
The probability density calculates the probability of failure at any given time. For instance, a person has a certainty of dying eventually. As you get older, you have a greater chance of dying at a specific age, since the average failure rate is calculated as a fraction of the number of units that exist in a specific interval, divided by the number of total units at the beginning of the interval.
If we were to calculate a person's chances of dying at a certain age, we would divide one year by the number of years that person potentially has left to live. This number would grow larger each year. A person aged 60 would have a higher probability of dying at age 65 than a person aged 30 because the person aged 30 still has many more units of time (years) left in his or her life, and the probability that the person will die during one specific unit of time is lower.
In many instances, the hazard rate can resemble the shape of a bathtub. The curve slopes downwards at the beginning, indicating a decreasing hazard rate, then levels out to be constant, before moving upwards as the item in question ages.
Think of it this way: when an auto manufacturer puts together a car, its components are not expected to fail in its first few years of service. However, as the car ages, the probability of malfunction increases. By the time the curve slopes upwards, the useful life period of the product has expired and the chance of non-random issues suddenly occurring becomes much more likely.
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Send signal through CDL channel model - MATLAB - MathWorks 日本
{\mathrm{μ}}_{\mathrm{Φ},\text{desired}}
N-by-M-by-4 numeric array — Use this option to explicitly define the initial phases. N is the number of clusters, equal to the number of path delays, specified by the PathDelays property. M is the number of rays per cluster, equal to 20. The four N-by-M planes, in the third dimension, correspond to the four polarization combinations: θ/θ, θ/Ï•, Ï•/θ, Ï•/Ï•.
For example, this figure shows how the object maps the input signal signalIn to an antenna array of size [2 3 2 2 2]. The antenna array consists of 2-by-2 antenna panels of 2-by-3 elements with 2 polarizations. The object maps the first M = 2 columns of the input signal (s1 and s2) to the first column of antenna elements with the first polarization angle of the first panel. The next M = 2 columns of the input signal (s3 and s4) are mapped to the next column of antenna elements, and so on. Following this pattern, the object maps the first M × N = 6 columns of the input signal (s1 to s6) to the antenna elements with the first polarization angle of the complete first panel. Similarly, the next 6 columns of the input signal (s7 to s12) are mapped to the antenna elements with the second polarization angle of the first panel. Subsequent sets of M × N × P = 12 columns of the input signal (s13 to s24, s25 to s36, s37 to s48) are mapped to consecutive panels, taking panel rows first, then panel columns.
Element spacing, in wavelengths, specified as a row vector of the form [λv λh dgv dgh]. The vector elements represent the vertical and horizontal element spacing and the vertical and horizontal panel spacing, respectively.
Polarization angles in degrees, specified as a row vector of the form [θ Ï].
Mechanical orientation of the array, in degrees, specified as a column vector of the form [α; β; γ]. The vector elements specify the bearing, downtilt, and slant, respectively. The default value indicates that the broadside direction of the array points to the positive x-axis.
Mechanical orientation of the transmit antenna array, specified as a three-element numeric column vector of the form [α; β; γ]. The vector elements specify the bearing, downtilt, and slant rotation angles in degrees, respectively, as specified in TR 38.901 Section 7.1.3. The object applies these rotation angles relative to the default array orientation in the local coordinate system. The default array orientation, corresponding to the value [0; 0; 0], depends on the TransmitAntennaArray property.
For example, this figure shows how the object maps an antenna array of size [2 3 2 2 2] to the output signal signalOut. The antenna array consists of 2-by-2 antenna panels of 2-by-3 elements with 2 polarizations. The first column of antenna elements with the first polarization angle of the first panel are mapped to the first M = 2 columns of the output signal (s1 and s2). The next column of antenna elements are mapped to the next M = 2 columns of the output signal (s3 and s4), and so on. Following this pattern, the object maps the antenna elements with the first polarization angle of the complete first panel to the first M × N = 6 columns of the output signal (s1 to s6). Similarly, the antenna elements with the second polarization angle of the first panel are mapped to the next 6 columns of the output signal (s7 to s12). Consecutive panels are mapped to subsequent sets of M × N × P = 12 columns of the output signal (s13 to s24, s25 to s36, s37 to s48), taking panel rows first, then panel columns.
Mechanical orientation of the receive antenna array, specified as a three-element numeric column vector of the form [α; β; γ]. The vector elements specify the bearing, downtilt, and slant rotation angles in degrees, respectively, as specified in TR 38.901 Section 7.1.3. The object applies these rotation angles relative to the default array orientation in the local coordinate system. The default array orientation, corresponding to the value [0; 0; 0], depends on the ReceiveAntennaArray property.
Fcg = MaximumDopplerShift × 2 × SampleDensity.
{\mathit{F}}_{\mathrm{CS}}=2\mathit{S}Ã\mathit{MaximumDopplerShift}
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Truth against truth: American and Arab history school textbooks portrayal of the Arab–Israeli conflict | QScience.com
oa Truth against truth: American and Arab history school textbooks portrayal of the Arab–Israeli conflict
Authors: Michael H. Romanowski1, Hadeel Alkhateeb1
Affiliations: 1 College of Education, Qatar University
© 2011 Romanowski & Alkhateeb, licensee Bloomsbury Qatar Foundation Journals.
Adwan S, Bar-On D, Musallam A and Naveh E. (2002). Learning each other’s historical narrative: Palestinians and Israelis.
Al Abedlaat M, Badawi F, Badran J, Ismail S, Ahmad H and Azabi I. (2009). The history of the Arabs and the modern world. Amman, Jordan: The Ministry of Education.
Al Tarwneh A, Al Kareem Ahmad A, Al Abedlaat M, Ismail S and Al Tarawneh M. (2009a). The modern and contemporary history of Arabs Part 1 Grade 10. Amman, Jordan: The Ministry of Education.
Al Tarwneh A, Al Abedlaat M, Ismail S and Al Tarawneh M. (2009b). The modern and contemporary history of Arabs Part 2 Grade 10. Amman, Jordan: The Ministry of Education.
Al Wisam in History . Islamic Civilization and Modern Arab History. Dar Gareeb Publishers, Cairo, Eqypt. 2009.
American Textbook Council . (2003). Islam and the textbooks: A report of the american textbook council. Middle East Quarterly Summer. 69–78. http://www.meforum.org/559/islam-and-the-textbooks
Anyon J. (1979). Ideology and United States history textbooks. Harvard Educational Review. 49::3, 361–386.
Apple MW and Christian-Smith LK. The politics of the textbook. Apple MW and Christian-Smith LK (Eds), The Politics of the Textbook. 1991; Routledge, New York. 1–21.
Arida H. (2006). Teaching the Middle East: The perspectives method. Teaching History: A Journal of Methods. 31::2, 74.
Bard M. Rewriting History in Textbooks. 1993; http://www.jewishvirtuallibrary.org/pub/texts.html
. Barlow E (Ed.), Evaluation of Secondary-Level Textbooks for Coverage of the Middle East and North Africa. Third Edition, 1994; Middle East Studies Association/Middle East Outreach Council, Ann Arbor, MI/Tuscon, AZ. .
Bar-Tal D. (2001). The Arab Image in Hebrew School Textbooks: How the Arabs were represented in Hebrew textbooks in Jewish and Israeli schools over one hundred years. Palestine-Israel Journal. 8::2, http://www.pij.org/details.php?id=884
Baroud R. (2010). Teaching the oppressed how to fight oppression, October
1{9}^{th}
Retrieved from http://arabnews.com/opinion/columns/article164908.ece?comments=all.
Boorstin D and Kelley B. A History of the United States. Pearson/Prentice Hall, Boston, MA. 2007.
Boyer P and Stuckey S. The American Nation. Holt, Rinehart, Austin, TX. 2005.
Cayton A, Perry E, Reed L and Winkler A. America: Pathways to the Present. Pearson/Prentice Hall, Boston, MA. 2007.
Cherryholmes CH, Heilman EE and Segal A. Social Studies–the Next Generation: Re-searching in the Postmodern Counterpoints: Studies in the Postmodern Theory of Education. Peter Lang Publishing, New York. 2005.
De Castell Luke, A and Luke C. Editorial introduction: Language authority and criticism. De Castell S, Luke A and Luke C (Eds), Language Authority and Criticism: Readings on the School Textbook. 1989; The Falmer Press, Philadelphia. vii–xi.
Eisner EW. The Educational Imagination: On Design and Evaluation of School Programs. Third Edition, Macmillan, New York. 1994.
Findley P. Deliberate Deceptions. American Educational Trust, Washington, DC. 1995.
FitzGerald F. America Revised. Random House, New York. 1979.
Gannon S. (2003). Who’s afraid of resolution 194? Israel Insider, August. http://web.israelinsider.com/Views/2654.htm.
Giroux HA. Teachers as Intellectuals: Toward a Critical Pedagogy of Learning. Bergin and Garvey Publishers Inc., Massachusetts. 1988.
Hastings D. (2003). U.S. uses UN veto power more than others. Information Clearinghouse. http://www.informationclearinghouse.info/article2016.htm.
Hein L and Selden M. The lessons of war, global power, and social change. Hein L and Selden M (Eds), Censoring History: Citizenship and Memory in Japan, Germany, and the United States. 2000; M.E. Sharpe, Armonk, NY. 3–4.
Ladah MS and Suleiman IA. (2002). Mr. Bush, what about Israel’s defiance of UN resolutions? An open letter to George W. Bush. http://www.mediamonitors.net/michaelsladah&suleimaniajlouni1.html.
Lapsansky-Werner E, Levy P, Roberts R and Taylor A. A United States History. Pearson/Prentice Hall, Boston, MA. 2008.
Lee P. (1998). Making sense of historical accounts. Canadian Social Studies. 32::3, 52–54.
Loewen JW. Lies My Teacher Told Me: Everything Your American History Textbook Got Wrong. The New Press, New York. 1995.
McLaren Peter. Life in Schools: An Introduction To Critical Pedagogy in The Foundations of Education. Third Edition, White Plains, Longman, an imprint off Addison Wesley Longman, NY. 1998.
Mearsheimer JJ and Walt SM. (2007). The Israel lobby and U.S. foreign policy [Kindle Version]. Retrieved from Amazon.com.
Mothfer M and Mohamad KI. The Modern and Contemporary History of Arab Countries. Ed. 15, The Ministry of Education, Baghdad, Iraq. 2009.
Moughrabi F. (2001). The politics of Palestinian textbooks. The Journal of Palestine Studies. 31::1, 5–19.
Nash G. American Odyssey. McGraw-Hill/Glencoe, New York. 2004.
Neff D. (1998). U.S. Vetoes of U.N. Resolutions on Behalf of Israel. http://www.ifamericansknew.org/us_ints/p-neff-veto.html.
Palestine Facts. (2010). http://www.palestinefacts.org/pf_1991to_now_israel_us_support.php.
Romanowski MH. (2009). Excluding Ethical Issues From U.S. History Textbooks: 911 and the War on Terror. American Secondary Education. 37::2,
Snorre L and Wilhelmson L. (2008). Revoking Israel’s UN Membership. Palestine Chronicle. http://www.palestinechronicle.com/view_article_details.php?id=14445.
The Israeli Committee Against Home Demolitions. (2009). Statistics on house demolitions (1967–2009).
United Nations Relief and Works Agency. http://www.unrwa.org/etemplate.php?id=86.
United Nations Resolution 194. (1948). http://domino.un.org/unispal.nsf/0/c758572b78d1cd0085256bcf0077e51a?OpenDocument.
Washington Report on Middle East Affairs. http://wrmea.org/component/content/article/245-2008-november/3845-congress-watch-a-conservative-estimate-of-total-direct-us-aid-to-israel-almost-114-billion.html.
Weber RP. Basic Content Analysis. 2nd ed., Sage, Newbury Park, CA. 1990.
Wiggins G and McTighe J. Understanding by Design. Association for Supervision and Curriculum Development, Alexandria, VA. 2005.
Wingfield M and Karaman B. (1995). Arab Stereotypes and American Educators. http://www.adc.org/education/arab-stereotypes-and-american-educators/.
Keyword(s): content analysis, qualitative research and textbooks
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Using KaTeX With Gatsby and MDX - Trevor Blades
In my spiral and circle posts, I relied heavily on mathematical notation to explain formulas and equations that I used to build those features. Before then, I had never had to use mathematical notation in any web or computer context.
After some searching on the web, I learned that KaTeX was the right tool for the job. It takes a string of text written in TeX syntax and outputs nicely formatted mathematical notation.
TeX's creator Donald Knuth promotes a pronunciation of /ˈtɛx/ (tekh), similar to the last sound of the German word "Bach". So you would say "kay-tekh" rather than "kay-tex".
In Markdown, it's common to see
\TeX
written inside math blocks beginning and ending with $$.
The above Markdown becomes the following when processed by
\KaTeX
c = \sqrt{a^2 + b^2}
Processing the Markdown math blocks with
\KaTeX
is not part of the basic offering of most Markdown renderers, and I needed to configure that part myself.
My website is built with Gatsby and MDX using gatsby-plugin-mdx, which accepts remark and rehype plugins as configuration options. Luckily there's a handful of plugins built for this purpose, such as remark-math, rehype-katex, and gatsby-remark-katex. Sadly, none of these libraries play nicely with the current stable version of MDX (1.6.x at the time of writing), only plain ol' Markdown.
I stumbled upon a GitHub issue related to this topic and experimented with the different "solutions" posted by others. The one that worked for me looks like this:
npm i remark-math@3 remark-html-katex@3 katex
It's important to install older versions of the two remark plugins since the newest versions are ESM only and Gatsby doesn't support ES modules in their gatsby-*.js files.
remarkPlugins: [require('remark-math'), require('remark-html-katex')]
Lastly, add the remark plugins to your gatsby-plugin-mdx configuration and import the stylesheet from the katex package into your client-side bundle.
import 'katex/dist/katex.css';
That's it! I hope you found this short guide to be helpful in setting up
\KaTeX
in your Gatsby site. If not, it better have at least been interesting. 😋
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FullChain on the grid - Atlas Wiki
FullChain on the grid
We'll describe here an example where we'll process the full chain GENERATION-SIMULATION-DIGITIZATION-RECONSTRUCTION-ESDtoAOD for
{\displaystyle H\rightarrow ZZ\rightarrow XXYY}
In the process we POOL output for each step of the chain will be save ON THE GRID with the following names:
gen_Higgs_ZZ_XXYY_XATHENAVERSIONX_JobXJOBNUMBERX.pool.root
hits_Higgs_ZZ_XXYY_XATHENAVERSIONX_JobXJOBNUMBERX.pool.root
digits_Higgs_ZZ_XXYY_XATHENAVERSIONX_JobXJOBNUMBERX.pool.root
esd_Higgs_ZZ_XXYY_XATHENAVERSIONX_JobXJOBNUMBERX.pool.root
aod_Higgs_ZZ_XXYY_XATHENAVERSIONX_JobXJOBNUMBERX.pool.root
Where XATHENAVERSIONX is the athena version and XJOBNUMBERX is the job number
To Change the name of the output Higgs_ZZ_XXYY go to the toShipOff_Everything and change
ChannelName = "Higgs_ZZ_XXYY"
with the name you want
In the ShellScript_BASIC.sh you can add -d SE (Storage Element) to change the default value i.e
lcg-cr --vo atlas -d tbn15.nikhef.nl -l lfn:filenameONTHEGRID file://${PWD}/filenameLOCALLY
1) JobOptions Files:
1.1) A joboptions file for our Generation job Generation_jobOptions_BASIC.py
(Here you specify the physics process and details of the output. In our case: Run pythia, and produce a POOL output file)
1.2) A joboptions file for our Simulation job Simulation_jobOptions_BASIC.py
(Here you specify some features about the simulation)
1.3) A joboptions file for our Digitization job Digitization_jobOptions_BASIC.py
(Here you specify some features about the digitization)
1.4) A joboptions file for our Reconstruction job Recontruction_jobOptions_BASIC.py
(Here you specify some features about the recontruction, the output is a ESD file)
1.5) A joboptions file for our ESDtoAOD job ESDtoAOD_jobOptions_BASIC.py
(Here you specify some features about the ESDtoAOD, the output is a EOD file)
2) A shell script that will run on the remote grid machine ShellScript_BASIC.sh
3) A JDL file containing the names of all required input and output file jdl_BASIC.jdl
This has to be chosen acordingly with the Athena version you want to run. Note as well that there are to files RomeGeo2G4.py and RecExCommon_topOptions.py; which you might have to change for others, or modify, when using other versions of athena rather than 10.0.2
fieldmap.dat contains information about the magnetic field (necesary for simulation)
5) A script that produces all input files: toShipOff_Everything.py
Create_Generation_File() # create joboptions file
Create_Simulation_File() # create joboptions file
Create_Digitization_File() # create joboptions file
Create_Reconstruction_File() # create joboptions file
Create_ESDtoAOD_File() # create joboptions file
Submitting a single job skipping 0 jobs with 50 events locally with athena 10.0.2: Higgs_ShipOff_Everything.py 1 0 50 0 10.0.2
Submitting 20 jobs skiping the first 10 with 5000 events on the grid with athena 10.0.2: Higgs_ShipOff_Everything.py 20 10 50 1 10.0.2
Note: If you choose a different verion of athena you'll have to be sure you're sending along
the piece of code that correspond to that distribution in AtlasStuff.tgz.
"Generation_joboptions_XATHENAVERSIONX_JobXXJobNrXX.py",
"Generation_XATHENAVERSIONX_JobXXJobNrXX.log" ,
"Simulation_joboptions_XATHENAVERSIONX_JobXXJobNrXX.py",
"Simulation_XATHENAVERSIONX_JobXXJobNrXX.log",
"Digitization_joboptions_XATHENAVERSIONX_JobXXJobNrXX.py",
"Digitization_XATHENAVERSIONX_JobXXJobNrXX.log",
"Recosntruction_joboptions_XATHENAVERSIONX_JobXXJobNrXX.py",
"Reconstruction_XATHENAVERSIONX_JobXXJobNrXX.log",
"ESDtoAOD_joboptions_XATHENAVERSIONX_JobXXJobNrXX.py",
"ESDtoAOD_XATHENAVERSIONX_JobXXJobNrXX.log"
With XATHENAVERIONX and XXJobNrXX corresponding to your previous choices
Unless you've change to a different SE (Storage Element) the files will be in the default SE on the grid. Now you can retrieve them to a local machine usig
lcg-cp --vo atlas lfn:<filename> file://<fullpath>/filename
Retrieved from "https://wiki.nikhef.nl/atlas/index.php?title=FullChain_on_the_grid&oldid=4746"
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Alleviating sample imbalance in multi classification - 编程知识
编程知识 cdmana.com
Alleviating sample imbalance in multi classification
2021-01-28 04:20:07 by Walker AI
This article was first published in ： Walker AI
Using deep learning to do multi classification is a common task in industry or research environment . In a research environment , Whether it's NLPã€CV or TTS Series of tasks , The data is rich and clean . And in a real industrial environment , Data problems often become a big problem for practitioners ; Common data problems include ：
The data sample size is small
Lack of data tagging
The data is not clean , There are a lot of disturbances
The distribution of sample number among data classes is not balanced and so on .
besides , There are other problems , This article will not list them one by one . For the above 4 A question ,2020 year 7 month google publish one’s thesis 《 Long-Tail Learning via Logit Adjustment 》 adopt BER ( Balanced Error Rate ) The related reasoning of cross entropy function , On the basis of the original cross entropy , So that the average classification accuracy is higher . This paper will briefly interpret the core inference of this paper , And use keras Deep learning framework to achieve , Finally, through a simple Mnist Experimental results of handwritten numeral classification . This article will be interpreted from the following four aspects ：
Core inference
In the multi classification problem based on deep learning , In order to get better classification results, we often need to analyze the data 〠The structural parameters of neural networks 〠The loss function and training parameters are adjusted ; Especially in the face of data with imbalanced categories , Make more adjustments . In the paper 《 Long-Tail Learning via Logit Adjustment 》 in , In order to alleviate the problem of low classification accuracy caused by imbalanced categories , By adding the prior knowledge of the label to the loss function, we get SOTA effect . therefore , This paper aims at its core inference , First of all, four basic concepts are briefly described ：（1） Long tail distribution ã€ï¼ˆ2）softmaxã€ï¼ˆ3） Cross entropy ã€ï¼ˆ4）BER
1.1 Long tail distribution
If the training data of all categories are sorted from high to low according to the sample size of each category , And show the sorting result on the graph , Then the class imbalance training data will show “ Head †and “ The tail †The distribution form of , As shown in the figure below ：
Category formation with large sample size “ Head †, The category with low sample size is formed “ The tail †, The problem of class imbalance is significant .
softmax Because of its normalization function and easy derivation , It is often used as the activation function of the last layer of neural network in two or more classification problems , It is used to express the prediction output of neural network . This paper deals with softmax Don't go over it , Only the generalized formula is given ：
q\left(c_{j}\right)=\frac{e^{z_{j}}}{\sum_{i=1}^{n} e^{z_{i}}}
In the neural network ,
z_{j}
It's the output of the upper layer ;
q\left(c_{j}\right)
It is the distribution form of the output of this layer ;
\sum_{i=1}^{n} e^{z_{i}}
It's a batch Inside
e^{z_{i}}
In this paper, we don't make too many inferences about the cross entropy function , For details, please refer to the relevant literature of information theory . In the problem of two or more classifications , The cross entropy function and its variants are usually used as the loss function for optimization , Give the basic formula ：
H(p, q)=-\sum_{i} p\left(c_{i}\right) \log q\left(c_{i}\right)
p\left(c_{i}\right)
Is the expected sample distribution , Usually one-hot Coded tags ;
q\left(c_{i}\right)
Is the output of neural network , It can be regarded as the prediction result of neural network to samples .
BER In the second classification, it is the mean value of prediction error rate in positive samples and negative samples ; In the multi classification problem, it is the weighted sum of the error rates of all kinds of samples , It can be expressed in the following form （ Refer to the paper ）：
\operatorname{BER}(f) \doteq \frac{1}{L} \sum_{y \in[L]} \mathbb{P}_{x \mid y}\left(y \notin \operatorname{argmax}_{y^{\prime} \in y} f_{y^{\prime}}(x)\right)
among ,
f
It's the whole neural network ;
f_{y^{\prime}}(x)
Indicates that the input is
x
, Output is
y^{\prime}
The neural network of ;
y \notin \operatorname{argmax}_{y^{\prime} \in y} f_{y^{\prime}}(x)
Represents the label that is wrongly recognized by the neural network
y
\mathbb{P}_{x \mid y}
It is the calculation form of error rate ;
\frac{1}{L}
For all kinds of weights .
2. Core inference
According to the idea of the paper , First, a neural network model is determined ：
f^{*} \in \operatorname{argmin}_{f:} x \rightarrow \mathbb{R}^{L} \text { BER }(f)
f^{*}
To satisfy BER A neural network model of conditions . Then optimize this neural network model
\operatorname{argmax}_{y \in[L]} f_{y}^{*}(x)
, This process is equivalent to
\operatorname{argmax}_{y \in[L]} \mathbb{P}^{\mathrm{bal}}(y \mid x)
, That is, given the training data
x
Get the prediction tag
y
, And the prediction label
y
equalization （ Multiply by their respective weights ） Optimization process of . Shorthand for ：
\operatorname{argmax}_{y \in[L]} f_{y}^{*}(x)=\operatorname{argmax}_{y \in[L]} \mathbb{P}^{\mathrm{bal}}(y \mid x)=\operatorname{argmax}_{y \in[L]} \mathbb{P}(x \mid y)
\mathbb{P}^{\text {bal }}(y \mid x)
\mathbb{P}^{\text {bal }}(y \mid x) \propto \mathbb{P}(y \mid x) / \mathbb{P}(y)
, among
\mathbb{P}(y)
It's a label priori ;
\mathbb{P}(y \mid x)
It's given training data
x
And then the conditional probability of the prediction label . Combined with the essence of training in multi classification neural network ：
According to the above process , Let's say the network outputs logits Write it down as s*：
s^{*}: x \rightarrow \mathbb{R}^{L}
s^{*}
Need to pass through softmax Activation layer , namely
q\left(c_{i}\right)=\frac{e^{s^{*}}}{\sum_{i=1}^{n} e^{s^{*}}}
; So it's not hard to come up with ：
\mathbb{P}(y \mid x) \propto \exp \left(s_{y}^{*}(x)\right)
. combining
\mathbb{P}^{\text {bal }}(y \mid x) \propto \mathbb{P}(y \mid x) / \mathbb{P}(y)
, Can be
\mathbb{P}^{\text {bal }}(y \mid x)
Expressed as ：
\operatorname{argmax}_{y \in[L]} \mathbb{P}^{\text {bal }}(y \mid x)=\operatorname{argmax}_{y \in[L]} \exp \left(s_{y}^{*}(x)\right) / \mathbb{P}(y)=\operatorname{argmax}_{y \in[L]} s_{y}^{*}(x)-\ln \mathbb{P}(y)
Refer to the above formula , The paper gives the optimization
\mathbb{P}^{\text {bal }}(y \mid x)
Two ways of implementing ：
（1） adopt
\operatorname{argmax}_{y \in[L]} \exp \left(s_{y}^{*}(x)\right) / \mathbb{P}(y)
, In the input
x
Through all the neural network layers to get predictions predict after , Divide by a priori
\mathbb{P}(y)
. This method has been used before , And achieved certain results .
\operatorname{argmax}_{y \in[L]} s_{y}^{*}(x)-\ln \mathbb{P}(y)
x
Get a code through the neural network layer logits Then subtract one
\ln \mathbb{P}(y)
. This paper adopts this idea .
Follow the second line of thought , In this paper, we give a general formula directly , be called logit adjustment loss：
\ell(y, f(x))=-\log \frac{e^{f_{y}(x)+\tau \cdot \log \pi_{y}}}{\sum_{y^{\prime} \in[L]} e^{f_{y^{\prime}}(x)+\tau \cdot \log \pi_{y^{\prime}}}}=\log \left[1+\sum_{y^{\prime} \neq y}\left(\frac{\pi_{y^{\prime}}}{\pi_{y}}\right)^{\tau} \cdot e^{\left(f_{y^{\prime}}(x)-f_{y}(x)\right)}\right]
Compared with the regular softmax Cross entropy ：
\ell(y, f(x))=\log \left[\sum_{y^{\prime} \in[L]} e^{f_{y^{\prime}}(x)}\right]-f_{y}(x)=\log \left[1+\sum_{y^{\prime} \neq y} e^{f_{y^{\prime}}(x)-f_{y}(x)}\right]
Essentially, an offset associated with the label prior is applied to each logarithmic output （ That is, through softmax The result before activation ）.
The idea of realization is ： The output of the neural network logits Plus a priori based offset
\log \left(\frac{\pi_{y^{\prime}}}{\pi_{y}}\right)^{\tau}
. In practice, , In order to make it as simple and effective as possible , Take the regulatory factor
\tau
\pi_{y^{\prime}}
=1. be logit adjustment loss Simplified as ：
\ell(y, f(x))=-\log \frac{e^{f_{y}(x)+\tau \cdot \log \pi_{y}}}{\sum_{y^{\prime} \in[L]} e^{f_{y^{\prime}}(x)+\tau \cdot \log \pi_{y^{\prime}}}}=\log \left[1+\sum_{y^{\prime} \neq y} e^{\left(f_{y^{\prime}}(x)-f_{y}(x)-\log \pi_{y}\right)}\right]
stay keras The implementation is as follows ：
def CE_with_prior(one_hot_label, logits, prior, tau=1.0):
param: one_hot_label
param: logits
param: prior: real data distribution obtained by statistics
param: tau: regulator, default is 1
return: loss
log_prior = K.constant(np.log(prior + 1e-8))
# align dim
for _ in range(K.ndim(logits) - 1):
log_prior = K.expand_dims(log_prior, 0)
logits = logits + tau * log_prior
loss = K.categorical_crossentropy(one_hot_label, logits, from_logits=True)
The paper 《 Long-Tail Learning via Logit Adjustment 》 In this paper, we compare several methods to improve the classification accuracy of long tail distribution , And tested with different data sets , Test performance is better than existing methods , Detailed experimental results refer to the paper itself . In order to quickly verify the correctness of the implementation , And the effectiveness of this method , Use mnist A simple experiment of handwritten numeral classification is carried out . The background of the experiment is as follows ：
The training sample 0 ~ 4：5000 Zhang / class ;5 ~ 9 :500 Zhang / class
Test samples 0 ~ 9：500/ class
Running environment Local CPU
Network structure Convolution + Maximum pooling + Full connection
In the above background, comparative experiments are carried out , Compared with the standard multi classification cross entropy and the cross entropy with a priori, they are used as loss Function , The performance of classified networks . Take the same epoch=60, The experimental results are as follows ：
Standard multi class cross entropy
Cross entropy with a priori
Accuracy rate 0.9578 0.9720
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Tags alleviating sample imbalance multi classification
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Train RL Agent for Lane Keeping Assist with Constraint Enforcement - MATLAB & Simulink - MathWorks Switzerland
Train RL Agent with Constraint Enforcement
This example shows how to train an reinforcement learning (RL) agent for lane keeping assist (LKA) with constraints enforced using the Constraint Enforcement block.
In this example, the goal is to keep an ego car traveling along the centerline of a lane by adjusting the front steering angle. This example uses the same vehicle model and parameters as the Train DQN Agent for Lane Keeping Assist Using Parallel Computing (Reinforcement Learning Toolbox) example.
% Paramters
m = 1575; % Total vehicle mass (kg)
Iz = 2875; % Yaw moment of inertia (mNs^2)
lf = 1.2; % Longitudinal distance from center of gravity to front tires (m)
lr = 1.6; % Longitudinal distance from center of gravity to rear tires (m)
Cf = 19000; % Cornering stiffness of front tires (N/rad)
Cr = 33000; % Cornering stiffness of rear tires (N/rad)
Vx = 15; % Longitudinal velocity (m/s)
Ts = 0.1; % Sample time (s)
T = 15; % Duration (s)
rho = 0.001; % Road curvature (1/m)
e1_initial = 0.2; % Initial lateral deviation from center line (m)
e2_initial = -0.1; % Initial yaw angle error (rad)
steerLimit = 0.2618;% Maximum steering angle for driver comfort (rad)
In this example, the constraint function enforced by the Constraint Enforcement block is unknown. To learn the function, you must first collect training data from the environment.
To do so, first create an RL environment using the rlLearnConstraintLKA model. This model applies random external actions through an RL Agent block to the environment.
mdl = 'rlLearnConstraintLKA';
{\mathit{e}}_{1}
{\mathit{e}}_{2}
, their derivatives, and their integrals. Create a continuous observation space for these six signals.
The action from the RL Agent block is the front steering angle, which can take one of 31 possible values from -15 to 15 degrees. Create a discrete action space for this signal.
actInfo = rlFiniteSetSpec((-15:15)*pi/180);
Specify a reset function, which randomly initializes the lateral deviation and relative yaw angle at the start of each training episode or simulation.
Next, create a DQN reinforcement learning agent, which supports discrete actions and continuous observations, using the createDQNAgentLKA helper function. This function creates a critic representation based on the action and observation specifications and uses the representation to create a DQN agent.
agent = createDQNAgentLKA(Ts,obsInfo,actInfo);
In the rlLearnConstraintLKA model, the RL Agent block does not generate actions. Instead, it is configured to pass a random external action to the environment. The purpose for using a data-collection model with an inactive RL Agent block is to ensure that the environment model, action and observation signal configurations, and model reset function used during data collection match those used during subsequent agent training.
In this example, the safety signal is
{\mathit{e}}_{1}
. The constraint for this signal is
{-1\le \text{\hspace{0.17em}}\mathit{e}}_{1}\le 1
; that is, the distance from the centerline of the lane must be less than 1. The constraint depends on the states in
\mathit{x}
: the lateral deviation and its derivative, and the yaw angle error and its derivative. The action
\mathit{u}
is the front steering angle. The relatinship between the states and the lateral deviation is described by the following equation.
{\mathit{e}}_{1}\left(\mathit{k}+1\right)=\mathit{f}\left({\mathit{x}}_{\mathit{k}}\right)+\mathit{g}\left({\mathit{x}}_{\mathit{k}}\right){\mathit{u}}_{\mathit{k}}
To allow for some slack, set the maximum lateral distance to be 0.9.
{\mathit{f}}_{\mathit{x}}+{\mathit{g}}_{\mathit{x}}\mathit{u}\le \mathit{c}
. For the previous equation and constraints, the coefficients of the constraint function are:
{\mathit{f}}_{\mathit{x}}=\left[\begin{array}{c}\mathit{f}\left({\mathit{x}}_{\mathit{k}}\right)\\ -\mathit{f}\left({\mathit{x}}_{\mathit{k}}\right)\end{array}\right],{\text{\hspace{0.17em}}\mathit{g}}_{\mathit{x}}=\left[\begin{array}{c}\mathit{g}\left({\mathit{x}}_{\mathit{k}}\right)\\ -\mathit{g}\left({\mathit{x}}_{\mathit{k}}\right)\end{array}\right],\text{\hspace{0.17em}}\mathit{c}=\left[\begin{array}{c}0.9\\ 0.9\end{array}\right]
{\mathit{f}}_{\mathit{x}}
{\mathit{g}}_{\mathit{x}}
, the RL agent passes a random external action to the environment that is uniformly distributed in the range [–0.2618, 0.2618].
To collect data, use the collectDataLKA helper function. This function simulates the environment and agent and collects the resulting input and output data. The resulting training data has eight columns, the first six of which are the observations for the RL agent.
Integral of lateral deviation
Integral of yaw angle error
Yaw angle error
Derivative of lateral deviation
Derivative of yaw angle error
Lateral deviation in the next time step
data = collectDataLKA(env,agent,count);
load trainingDataLKA data
For this example, the dynamics of the ego car are linear. Therefore, you can find a least-squares solution for the lateral-deviation constraints.
You can apply linear approximations to learn the unknown functions
{\mathit{f}}_{\mathit{x}}
{\mathit{g}}_{\mathit{x}}
inputData = data(1:1000,[2,5,4,6,7]);
% Extract data for the lateral deviation in the next time step.
outputData = data(1:1000,8);
% Compute the relation from the state and input to the lateral deviation.
relation = inputData\outputData;
% Extract the components of the constraint function coefficients.
Rf = relation(1:4)';
Rg = relation(5);
Validate the learned constraints using the validateConstraintLKA helper function. This function processes the input training data using the learned constraints. It then compares the network output with the training output and computes the root mean squared error (RMSE).
validateConstraintLKA(data,Rf,Rg);
The small RMSE value indicates successful constraint learning.
To train the agent with constraint enforcement, use the rlLKAWithConstraint model. This model constrains the actions from the agent before applying them to the environment.
mdl = 'rlLKAwithConstraint';
{\mathit{f}}_{\mathit{i}}
{\mathit{g}}_{\mathit{i}}
Create an RL environment using this model. The observation and action specifications are the same as for the constraint-learning environment.
The Environment subsystem creates an isDone signal that is true when the lateral deviation exceeds a specified constraint. The RL Agent block uses this signal to terminate training episodes early.
Specify options for training the agent. Train the agent for at most 5000 episodes. Stop training if the episode reward exceeds –1.
'StopTrainingValue',-1);
load rlAgentConstraintLKA agent
Since Total Number of Steps equals the product of Episode Number and Episode Steps, each training episode runs to the end without early termination. Therefore, the Constraint Enforcement block ensures that the lateral deviation never violates its constraints.
bdclose('rlLearnConstraintLKA')
bdclose('rlLKAwithConstraint')
% Set initial lateral deviation to random value.
% Set initial relative yaw angle to random value.
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LMIs in Control/Matrix and LMI Properties and Tools/Matrix Inequalities and LMIs - Wikibooks, open books for an open world
LMIs in Control/Matrix and LMI Properties and Tools/Matrix Inequalities and LMIs
2 Linear Matrix Inequality
3 Bilinear Matrix Inequality
A Matrix Inequality,
{\displaystyle G:\mathbb {R} ^{m}\to \mathbb {S} ^{n}}
, in the variable
{\displaystyle x\in \mathbb {R} ^{m}}
{\displaystyle {\begin{aligned}G(x)=G_{0}+\sum _{i=1}^{p}f_{i}(x)G_{i}\leq 0\end{aligned}}}
{\displaystyle x^{T}=[x_{1}\cdots x_{m}],G_{0}\in \mathbb {S} ^{n}}
{\displaystyle G_{i}\in \mathbb {R} ^{n\times n}}
{\displaystyle i=1,\ldots ,p.}
Linear Matrix InequalityEdit
A Linear Matrix Inequality,
{\displaystyle F:\mathbb {R} ^{m}\to \mathbb {S} ^{n}}
{\displaystyle x\in \mathbb {R} ^{m}}
{\displaystyle {\begin{aligned}F(x)=F_{0}+\sum _{i=1}^{m}x_{i}F_{i}\leq 0\end{aligned}}}
{\displaystyle x^{T}=[x_{1}\ldots x_{m}]}
{\displaystyle F_{i}\in \mathbb {S} ^{n}}
{\displaystyle i=0\ldots ,m.}
Bilinear Matrix InequalityEdit
A Bilinear Matrix Inequality (BMI),
{\displaystyle H:\mathbb {R} ^{m}\to \mathbb {S} ^{n}}
{\displaystyle x\in \mathbb {R} ^{m}}
{\displaystyle {\begin{aligned}H(x)=H_{0}+\sum _{i=1}^{m}x_{i}H_{i}+\sum _{i=1}^{m}\sum _{j=1}^{m}x_{i}x_{j}H_{i,j}\leq 0,\end{aligned}}}
{\displaystyle x^{T}=[x_{1}\cdots x_{m}]}
{\displaystyle H_{i}}
{\displaystyle H_{i,j}\in \mathbb {S} ^{n},}
{\displaystyle i=0,\ldots ,m}
{\displaystyle j=0\ldots ,m.}
{\displaystyle A\in \mathbb {R} ^{n\times n}}
{\displaystyle Q\in \mathbb {S} ^{n}}
{\displaystyle Q>0}
. It is desired to find a symmetric matrix
{\displaystyle P\in \mathbb {S} ^{n}}
{\displaystyle {\begin{aligned}PA+A^{T}P+Q<0,\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (1)\end{aligned}}}
{\displaystyle P>0}
{\displaystyle P}
are the design variables in this problem, and although equation
{\displaystyle (1)}
is indeed an LMI in the matrix
{\displaystyle P}
, it does not look like the LMI in definition 3. For simplicity, let us consider the case of
{\displaystyle n=2}
so that each matrix is of dimension
{\displaystyle 2\times 2}
{\displaystyle x=[p_{1}\quad p_{2}\quad p_{3}]^{T}.}
Writing the matrix
{\displaystyle P}
in terms of a basis
{\displaystyle E_{i}\in \mathbb {S} ^{2},}
{\displaystyle i=1,2,3}
{\displaystyle {\begin{aligned}P={\begin{bmatrix}p_{1}&p_{2}\\p_{2}&p_{3}\end{bmatrix}}=p_{1}\underbrace {\begin{bmatrix}1&0\\0&0\end{bmatrix}} _{E_{1}}+p_{2}\underbrace {\begin{bmatrix}0&1\\1&0\end{bmatrix}} _{E_{2}}+p_{3}\underbrace {\begin{bmatrix}0&0\\0&1\end{bmatrix}} _{E_{3}}\end{aligned}}}
Note that the matrices
{\displaystyle E_{i}}
are linearly independent and symmetric, thus forming a basis for the symmetric matrix
{\displaystyle P}
. The matrix inequality in equation
{\displaystyle (1)}
{\displaystyle {\begin{aligned}p_{1}(E_{1}A+A^{T}E_{1})+p_{2}(E_{2}A+A^{T}E_{2})+p_{3}(E_{3}A+A^{T}E_{3}).\end{aligned}}}
{\displaystyle F_{0}=Q}
{\displaystyle F_{i}=E_{i}A+A^{T}E_{i},}
{\displaystyle i=1,2,3,}
{\displaystyle {\begin{aligned}F_{0}+\sum {i=1}^{3}p_{i}F_{i}<0,\end{aligned}}}
which now resembles the definition of LMI given in definition 2. Through out this wiki book, LMIs are typically written in the matrix form of equation
{\displaystyle (1)}
rather than the scalar form of definition 2.
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/Matrix_and_LMI_Properties_and_Tools/Matrix_Inequalities_and_LMIs&oldid=4010118"
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Equivariant K-theory of affine flag manifolds and affine Grothendieck polynomials
15 June 2009 Equivariant
K
-theory of affine flag manifolds and affine Grothendieck polynomials
Masaki Kashiwara, Mark Shimozono
Masaki Kashiwara,1 Mark Shimozono2
2Department of Mathematics, Virginia Polytechnic Institute and State University
We study the equivariant
K
-group of the affine flag manifold with respect to the Borel group action. We prove that the structure sheaf of the (infinite-dimensional) Schubert variety in the K-group is represented by a unique polynomial, which we call the affine Grothendieck polynomial
Masaki Kashiwara. Mark Shimozono. "Equivariant
K
-theory of affine flag manifolds and affine Grothendieck polynomials." Duke Math. J. 148 (3) 501 - 538, 15 June 2009. https://doi.org/10.1215/00127094-2009-032
Secondary: 14M17 , 17B67 , 22E65
Masaki Kashiwara, Mark Shimozono "Equivariant
K
-theory of affine flag manifolds and affine Grothendieck polynomials," Duke Mathematical Journal, Duke Math. J. 148(3), 501-538, (15 June 2009)
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Clique (graph theory) - Wikipedia
For other uses, see Clique (disambiguation).
In the mathematical area of graph theory, a clique (/ˈkliːk/ or /ˈklɪk/) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph
{\displaystyle G}
is an induced subgraph of
{\displaystyle G}
that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied.
23 × 1-vertex cliques (the vertices),
42 × 2-vertex cliques (the edges),
19 × 3-vertex cliques (light and dark blue triangles), and
2 × 4-vertex cliques (dark blue areas).
The 11 light blue triangles form maximal cliques. The two dark blue 4-cliques are both maximum and maximal, and the clique number of the graph is 4.
Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by Erdős & Szekeres (1935),[1] the term clique comes from Luce & Perry (1949), who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinformatics.
A clique,
{\displaystyle C}
, in an undirected graph
{\displaystyle G=(V,E)}
is a subset of the vertices,
{\displaystyle C\subseteq V}
, such that every two distinct vertices are adjacent. This is equivalent to the condition that the induced subgraph of
{\displaystyle G}
{\displaystyle C}
is a complete graph. In some cases, the term clique may also refer to the subgraph directly.
A maximal clique is a clique that cannot be extended by including one more adjacent vertex, that is, a clique which does not exist exclusively within the vertex set of a larger clique. Some authors define cliques in a way that requires them to be maximal, and use other terminology for complete subgraphs that are not maximal.
A maximum clique of a graph,
{\displaystyle G}
, is a clique, such that there is no clique with more vertices. Moreover, the clique number
{\displaystyle \omega (G)}
{\displaystyle G}
is the number of vertices in a maximum clique in
{\displaystyle G}
The intersection number of
{\displaystyle G}
is the smallest number of cliques that together cover all edges of
{\displaystyle G}
The clique cover number of a graph
{\displaystyle G}
is the smallest number of cliques of
{\displaystyle G}
whose union covers the set of vertices
{\displaystyle V}
A maximum clique transversal of a graph is a subset of vertices with the property that each maximum clique of the graph contains at least one vertex in the subset.[2]
The opposite of a clique is an independent set, in the sense that every clique corresponds to an independent set in the complement graph. The clique cover problem concerns finding as few cliques as possible that include every vertex in the graph.
A related concept is a biclique, a complete bipartite subgraph. The bipartite dimension of a graph is the minimum number of bicliques needed to cover all the edges of the graph.
Mathematical results concerning cliques include the following.
Turán's theorem gives a lower bound on the size of a clique in dense graphs.[3] If a graph has sufficiently many edges, it must contain a large clique. For instance, every graph with
{\displaystyle n}
vertices and more than
{\displaystyle \scriptstyle \lfloor {\frac {n}{2}}\rfloor \cdot \lceil {\frac {n}{2}}\rceil }
edges must contain a three-vertex clique.
Ramsey's theorem states that every graph or its complement graph contains a clique with at least a logarithmic number of vertices.[4]
According to a result of Moon & Moser (1965), a graph with 3n vertices can have at most 3n maximal cliques. The graphs meeting this bound are the Moon–Moser graphs K3,3,..., a special case of the Turán graphs arising as the extremal cases in Turán's theorem.
Hadwiger's conjecture, still unproven, relates the size of the largest clique minor in a graph (its Hadwiger number) to its chromatic number.
The Erdős–Faber–Lovász conjecture is another unproven statement relating graph coloring to cliques.
Several important classes of graphs may be defined or characterized by their cliques:
A cluster graph is a graph whose connected components are cliques.
A block graph is a graph whose biconnected components are cliques.
A chordal graph is a graph whose vertices can be ordered into a perfect elimination ordering, an ordering such that the neighbors of each vertex v that come later than v in the ordering form a clique.
An interval graph is a graph whose maximal cliques can be ordered in such a way that, for each vertex v, the cliques containing v are consecutive in the ordering.
A line graph is a graph whose edges can be covered by edge-disjoint cliques in such a way that each vertex belongs to exactly two of the cliques in the cover.
A perfect graph is a graph in which the clique number equals the chromatic number in every induced subgraph.
A split graph is a graph in which some clique contains at least one endpoint of every edge.
A triangle-free graph is a graph that has no cliques other than its vertices and edges.
Additionally, many other mathematical constructions involve cliques in graphs. Among them,
The clique complex of a graph G is an abstract simplicial complex X(G) with a simplex for every clique in G
A simplex graph is an undirected graph κ(G) with a vertex for every clique in a graph G and an edge connecting two cliques that differ by a single vertex. It is an example of median graph, and is associated with a median algebra on the cliques of a graph: the median m(A,B,C) of three cliques A, B, and C is the clique whose vertices belong to at least two of the cliques A, B, and C.[5]
The clique-sum is a method for combining two graphs by merging them along a shared clique.
Clique-width is a notion of the complexity of a graph in terms of the minimum number of distinct vertex labels needed to build up the graph from disjoint unions, relabeling operations, and operations that connect all pairs of vertices with given labels. The graphs with clique-width one are exactly the disjoint unions of cliques.
The intersection number of a graph is the minimum number of cliques needed to cover all the graph's edges.
The clique graph of a graph is the intersection graph of its maximal cliques.
Closely related concepts to complete subgraphs are subdivisions of complete graphs and complete graph minors. In particular, Kuratowski's theorem and Wagner's theorem characterize planar graphs by forbidden complete and complete bipartite subdivisions and minors, respectively.
Main article: Clique problem
In computer science, the clique problem is the computational problem of finding a maximum clique, or all cliques, in a given graph. It is NP-complete, one of Karp's 21 NP-complete problems.[6] It is also fixed-parameter intractable, and hard to approximate. Nevertheless, many algorithms for computing cliques have been developed, either running in exponential time (such as the Bron–Kerbosch algorithm) or specialized to graph families such as planar graphs or perfect graphs for which the problem can be solved in polynomial time.
The word "clique", in its graph-theoretic usage, arose from the work of Luce & Perry (1949), who used complete subgraphs to model cliques (groups of people who all know each other) in social networks. The same definition was used by Festinger (1949) in an article using less technical terms. Both works deal with uncovering cliques in a social network using matrices. For continued efforts to model social cliques graph-theoretically, see e.g. Alba (1973), Peay (1974), and Doreian & Woodard (1994).
Many different problems from bioinformatics have been modeled using cliques. For instance, Ben-Dor, Shamir & Yakhini (1999) model the problem of clustering gene expression data as one of finding the minimum number of changes needed to transform a graph describing the data into a graph formed as the disjoint union of cliques; Tanay, Sharan & Shamir (2002) discuss a similar biclustering problem for expression data in which the clusters are required to be cliques. Sugihara (1984) uses cliques to model ecological niches in food webs. Day & Sankoff (1986) describe the problem of inferring evolutionary trees as one of finding maximum cliques in a graph that has as its vertices characteristics of the species, where two vertices share an edge if there exists a perfect phylogeny combining those two characters. Samudrala & Moult (1998) model protein structure prediction as a problem of finding cliques in a graph whose vertices represent positions of subunits of the protein. And by searching for cliques in a protein-protein interaction network, Spirin & Mirny (2003) found clusters of proteins that interact closely with each other and have few interactions with proteins outside the cluster. Power graph analysis is a method for simplifying complex biological networks by finding cliques and related structures in these networks.
In electrical engineering, Prihar (1956) uses cliques to analyze communications networks, and Paull & Unger (1959) use them to design efficient circuits for computing partially specified Boolean functions. Cliques have also been used in automatic test pattern generation: a large clique in an incompatibility graph of possible faults provides a lower bound on the size of a test set.[7] Cong & Smith (1993) describe an application of cliques in finding a hierarchical partition of an electronic circuit into smaller subunits.
In chemistry, Rhodes et al. (2003) use cliques to describe chemicals in a chemical database that have a high degree of similarity with a target structure. Kuhl, Crippen & Friesen (1983) use cliques to model the positions in which two chemicals will bind to each other.
Clique game
^ The earlier work by Kuratowski (1930) characterizing planar graphs by forbidden complete and complete bipartite subgraphs was originally phrased in topological rather than graph-theoretic terms.
^ Chang, Kloks & Lee (2001).
^ Turán (1941).
^ Graham, Rothschild & Spencer (1990).
^ Barthélemy, Leclerc & Monjardet (1986), page 200.
^ Karp (1972).
^ Hamzaoglu & Patel (1998).
Alba, Richard D. (1973), "A graph-theoretic definition of a sociometric clique" (PDF), Journal of Mathematical Sociology, 3 (1): 113–126, doi:10.1080/0022250X.1973.9989826 .
Barthélemy, J.-P.; Leclerc, B.; Monjardet, B. (1986), "On the use of ordered sets in problems of comparison and consensus of classifications", Journal of Classification, 3 (2): 187–224, doi:10.1007/BF01894188, S2CID 6092438 .
Ben-Dor, Amir; Shamir, Ron; Yakhini, Zohar (1999), "Clustering gene expression patterns.", Journal of Computational Biology, 6 (3–4): 281–297, CiteSeerX 10.1.1.34.5341, doi:10.1089/106652799318274, PMID 10582567 .
Chang, Maw-Shang; Kloks, Ton; Lee, Chuan-Min (2001), "Maximum clique transversals", Graph-theoretic concepts in computer science (Boltenhagen, 2001), Lecture Notes in Comput. Sci., vol. 2204, Springer, Berlin, pp. 32–43, doi:10.1007/3-540-45477-2_5, ISBN 978-3-540-42707-0, MR 1905299 .
Cong, J.; Smith, M. (1993), "A parallel bottom-up clustering algorithm with applications to circuit partitioning in VLSI design", Proc. 30th International Design Automation Conference, pp. 755–760, CiteSeerX 10.1.1.32.735, doi:10.1145/157485.165119, ISBN 978-0897915779, S2CID 525253 .
Day, William H. E.; Sankoff, David (1986), "Computational complexity of inferring phylogenies by compatibility", Systematic Zoology, 35 (2): 224–229, doi:10.2307/2413432, JSTOR 2413432 .
Doreian, Patrick; Woodard, Katherine L. (1994), "Defining and locating cores and boundaries of social networks", Social Networks, 16 (4): 267–293, doi:10.1016/0378-8733(94)90013-2 .
Erdős, Paul; Szekeres, George (1935), "A combinatorial problem in geometry" (PDF), Compositio Mathematica, 2: 463–470 .
Festinger, Leon (1949), "The analysis of sociograms using matrix algebra", Human Relations, 2 (2): 153–158, doi:10.1177/001872674900200205, S2CID 143609308 .
Graham, R.; Rothschild, B.; Spencer, J. H. (1990), Ramsey Theory, New York: John Wiley and Sons, ISBN 978-0-471-50046-9 .
Hamzaoglu, I.; Patel, J. H. (1998), "Test set compaction algorithms for combinational circuits", Proc. 1998 IEEE/ACM International Conference on Computer-Aided Design, pp. 283–289, doi:10.1145/288548.288615, ISBN 978-1581130089, S2CID 12258606 .
Karp, Richard M. (1972), "Reducibility among combinatorial problems", in Miller, R. E.; Thatcher, J. W. (eds.), Complexity of Computer Computations (PDF), New York: Plenum, pp. 85–103, archived from the original (PDF) on 2011-06-29, retrieved 2009-12-13 .
Kuhl, F. S.; Crippen, G. M.; Friesen, D. K. (1983), "A combinatorial algorithm for calculating ligand binding", Journal of Computational Chemistry, 5 (1): 24–34, doi:10.1002/jcc.540050105, S2CID 122923018 .
Kuratowski, Kazimierz (1930), "Sur le problème des courbes gauches en Topologie" (PDF), Fundamenta Mathematicae (in French), 15: 271–283, doi:10.4064/fm-15-1-271-283 .
Luce, R. Duncan; Perry, Albert D. (1949), "A method of matrix analysis of group structure", Psychometrika, 14 (2): 95–116, doi:10.1007/BF02289146, hdl:10.1007/BF02289146, PMID 18152948, S2CID 16186758 .
Moon, J. W.; Moser, L. (1965), "On cliques in graphs", Israel Journal of Mathematics, 3: 23–28, doi:10.1007/BF02760024, MR 0182577 .
Paull, M. C.; Unger, S. H. (1959), "Minimizing the number of states in incompletely specified sequential switching functions", IRE Transactions on Electronic Computers, EC-8 (3): 356–367, doi:10.1109/TEC.1959.5222697 .
Peay, Edmund R. (1974), "Hierarchical clique structures", Sociometry, 37 (1): 54–65, doi:10.2307/2786466, JSTOR 2786466 .
Prihar, Z. (1956), "Topological properties of telecommunications networks", Proceedings of the IRE, 44 (7): 927–933, doi:10.1109/JRPROC.1956.275149, S2CID 51654879 .
Rhodes, Nicholas; Willett, Peter; Calvet, Alain; Dunbar, James B.; Humblet, Christine (2003), "CLIP: similarity searching of 3D databases using clique detection", Journal of Chemical Information and Computer Sciences, 43 (2): 443–448, doi:10.1021/ci025605o, PMID 12653507 .
Samudrala, Ram; Moult, John (1998), "A graph-theoretic algorithm for comparative modeling of protein structure", Journal of Molecular Biology, 279 (1): 287–302, CiteSeerX 10.1.1.64.8918, doi:10.1006/jmbi.1998.1689, PMID 9636717 .
Spirin, Victor; Mirny, Leonid A. (2003), "Protein complexes and functional modules in molecular networks", Proceedings of the National Academy of Sciences, 100 (21): 12123–12128, doi:10.1073/pnas.2032324100, PMC 218723, PMID 14517352 .
Sugihara, George (1984), "Graph theory, homology and food webs", in Levin, Simon A. (ed.), Population Biology, Proc. Symp. Appl. Math., vol. 30, pp. 83–101 .
Tanay, Amos; Sharan, Roded; Shamir, Ron (2002), "Discovering statistically significant biclusters in gene expression data", Bioinformatics, 18 (Suppl. 1): S136–S144, doi:10.1093/bioinformatics/18.suppl_1.S136, PMID 12169541 .
Turán, Paul (1941), "On an extremal problem in graph theory", Matematikai és Fizikai Lapok (in Hungarian), 48: 436–452
Weisstein, Eric W., "Clique", MathWorld
Weisstein, Eric W., "Clique Number", MathWorld
Retrieved from "https://en.wikipedia.org/w/index.php?title=Clique_(graph_theory)&oldid=1057727099"
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The BuyBack Bonus ( BBB ) - Documentation
VERY Important information. Read it!!
“BuyBacks” mean different things to different people and ours is a little special of its own.
This is a big part of the answer to “why would anyone put money on a non-profit decentralization mission?” Other than agreeing with our dream, of course.
The BBB is a collective fund that automatically & continuously accrues and cold-stores a fixed amount (25%) of all platform generated (net) revenue throughout the self-decentralization process.
This is an extra revenue stream for token holders, accrued on top of all other claims, and meant to compensate for the value of their long term agreement with our mission of self-decentralization.
The value of this fund is what allows the mechanism taking tokens out of circulation, thus increasing scarcity and the relative worth of remaining token supply.
Who claims it
Only ARENA Security Token Holders have a claim on the funds accrued in the BuyBack Bonus. Funds are allocated pro rata among existing holders.
Deploying a competitive exchange is a massive task and requires a lot of money.
The BBB is key in rendering our proposal feasible by making it financially attractive to investors, while guaranteeing the delivery of our long term non-profit mission.
This will execute a market order, by which the Holder sells his/her ARENA Security Tokens to CryptoArena Foundation at the current market price. When this transaction is confirmed, automatically, you also "unlock" your share
Token Holders may exit their investment in the above-described methods at any time, at their leisure. Once self-decentralization is complete, when no Holders remain, the BBB ceases to exist and its Token weight (that fixed 25% I mentioned earlier) is allocated to general distributions, thus reaching 100%.
In practical terms, this means shares are worth 150% of their weight, thus, if you own 4% of total supply, your share will actually earn 6% of net profits, but you won’t be able to access the BBB portion (that extra 2%) unless / until you Exit.
ARENA Security Tokens are designed to reward endurance - those who hodl the longest will benefit most.
Company has 100 shares makes 100€ in net revenue in previous period and distributes it. There are 50 Holders with equal shares. Bob the ARENA Holder is one of them
Only ARENA Security Token Holders claim the buybacks, thus, in this example, there are 50 people with have an equal claim on the 25 € of the BuyBack.
€ 0.50 = 25 * (1/50)
€ 0.55 = 25 * (1/45)
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Assigning Variants to Genes (V2G) - Open Targets Genetics Documentation
Overview of Variant-to-Gene (V2G) pipeline aggregation and scoring.
All variants in the variant index are annotated using our Variant-to-Gene (V2G) pipeline. The pipeline integrates V2G evidence that fall into four main data types:
Molecular phenotype quantitative trait loci experiments (QTLs)
Chromatin interaction experiments, e.g. Promoter Capture Hi-C (PCHi-C)
In silico functional predictions, e.g. Variant Effect Predictor (VEP) from Ensembl
Distance between the variant and each gene's canonical transcription start site (TSS)
Within each data type there are multiple sources of information produced by different experimental methods. Some of these sources can further be broken down into separate tissues or cell types (features). A full list of data sources used in the V2G pipeline can be seen on the Data Sources page.
Raw datasets are processed to conform to a standardised format and filtered so that they:
Only contain associations with strong evidence post-multiple testing correction
Only contain cis-regulatory associations
A full list of filters applied to each dataset, and workflows to reproduce the V2G files, can be found on GitHub.
Different data sources use different metrics to measure the association between variants (or genomic intervals) and a gene. For example QTLs provide a p-value from standard linear regression, whereas PCHi-C provides a CHiCAGO score. To harmonise scores across sources, a relevant study-specific metric is extracted followed by quantile transformation using a uniform distribution. If multiple features (tissues/cell types) are available, then the transformation is applied at the feature level. Transformed scores are rounded to the nearest decile, so a score of 1.0 is in the top decile, a score of 0.9 is in the 9th decile, and so on.
Variant-Gene annotation
Next, each variant-gene
(V, G)
pair is annotated with all available functional evidence. QTL and functional prediction data types contain
(V, G)
-centric scores and so are simple to combine. Interaction data types link functional genomic regions (interval A) to gene positions (interval B). Variants that lie within interval A are assigned evidence scores that link it to genes located in interval B. The resulting V2G merge table consists of approximately 1.7 billion evidence strings.
Given the scale of the data, a scoring system was developed so that for a given variant
(V)
we can get a list of genes
(G_1... G_N)
ranked by either (i) the overall V2G score, (ii) a per-source V2G score.
Step 1, Aggregate across features (tissues or cell types). Some data sources (i.e. GTEx and PCHi-C) provide associations measured in multiple tissues or cell lines (features). Where multiple features exist, we aggregate by taking the maximum score across all features for each
(V, G)
pair. This aggregation gives a per-source V2G score for each
(V, G)
Step 2, Aggregate across sources. The next stage is to combine information across the sources to produce an overall V2G score. Given the heterogenous nature of the data, we may have more confidence in evidence from some sources over others. We therefore down-weight some sources before aggregation. Using a prior knowledge we rank evidence from sources in this order [ Transcript functional prediction > QTLs > Interaction based data sets ] and apply the following weights:
Sun et al. (Nature, 2018)
Javierre et al. (Cell, 2016)
Enhancer-TSS correlation
Andersson et al. (Nature, 2014)
DHS-promoter correlation
Thurman et al. (Nature, 2012)
Canonical TSS
After weighting, sources are aggregated across sources by taking the mean weighted-quantile to give an overall V2G score for each
(V, G)
Excluded gene biotypes
The following gene biotypes are excluded from all V2G analysis:
IG_C_pseudogene, IG_J_pseudogene, IG_pseudogene, IG_V_pseudogene, polymorphic_pseudogene, processed_pseudogene, pseudogene, rRNA, rRNA_pseudogene, snoRNA, snRNA, transcribed_processed_pseudogene, transcribed_unitary_pseudogene, transcribed_unprocessed_pseudogene, TR_J_pseudogene, TR_V_pseudogene, unitary_pseudogene, unprocessed_pseudogene
Scoring pseudocode
## Preprocessing (applied across whole dataset)
- Create a score column
* QTL datasets: -log10(p-value)
* Interval datasets: interval score (differs between sources)
* VEP map to v2g_score column in https://github.com/opentargets/v2g_data/blob/master/configs/vep_consequences.tsv
* TSS Distance: 1/distance
- Group by (source, tissue), transform into quantiles
## Aggregate across tissues (per source)
- Group by (variant, source, gene)
- Calculate max score across tissues
- This gives a score per gene for each source
## Aggregate across sources (per variant)
- Group by (variant, gene)
- Caluclate weighted mean over sources using source weights (see below)
- This gives a score per gene for a given variant
source_weights = {
'vep': 1,
'javierre2016': 0.33,
'andersson2014': 0.33,
'thurman2012': 0.33,
'canonical_tss': 0.33,
'gtex_v7': 0.66,
'sun2018': 0.66
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LMIs in Control/Matrix and LMI Properties and Tools/Non Strict Projection Lemma - Wikibooks, open books for an open world
LMIs in Control/Matrix and LMI Properties and Tools/Non Strict Projection Lemma
2 Non Strict Projection Lemma
Non Strict Projection LemmaEdit
{\displaystyle \Psi \in \mathbb {S} ^{n}}
{\displaystyle G\in \mathbb {R} ^{n\times m}}
{\displaystyle \Lambda \in \mathbb {R} ^{m\times p}}
{\displaystyle H\in \mathbb {R} ^{n\times p}.}
{\displaystyle {\mathcal {R}}(G)}
{\displaystyle {\mathcal {R}}(H)}
is linear;y independent. There exists
{\displaystyle \Lambda }
{\displaystyle {\begin{aligned}\ \Psi +G\Lambda H^{T}+H\Lambda ^{T}G^{T}leq0,\end{aligned}}}
{\displaystyle {\begin{aligned}\ N_{G}^{T}\Psi N_{G}\leq 0\end{aligned}}}
{\displaystyle {\begin{aligned}\ N_{H}^{T}\Psi N_{H}\leq 0\end{aligned}}}
{\displaystyle {\mathcal {R}}(N_{G})={\mathcal {N}}(G^{T})}
{\displaystyle {\mathcal {R}}(N_{H})={\mathcal {N}}(H^{T})}
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/Matrix_and_LMI_Properties_and_Tools/Non_Strict_Projection_Lemma&oldid=4009619"
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Invariant Inhomogeneous Bianchi Type-I Cosmological Models with Electromagnetic Fields Using Lie Group Analysis in Lyra Geometry
2014 Invariant Inhomogeneous Bianchi Type-I Cosmological Models with Electromagnetic Fields Using Lie Group Analysis in Lyra Geometry
Ahmad T. Ali
We find a new class of invariant inhomogeneous Bianchi type-I cosmological models in electromagnetic field with variable magnetic permeability. For this, Lie group analysis method is used to identify the generators that leave the given system of nonlinear partial differential equations (NLPDEs) (Einstein field equations) invariant. With the help of canonical variables associated with these generators, the assigned system of PDEs is reduced to ordinary differential equations (ODEs) whose simple solutions provide nontrivial solutions of the original system. A new class of exact (invariant-similarity) solutions have been obtained by considering the potentials of metric and displacement field as functions of coordinates
x
t
. We have assumed that
{F}_{12}
is only nonvanishing component of electromagnetic field tensor
{F}_{ij}
. The Maxwell equations show that
{F}_{12}
x
alone whereas the magnetic permeability
\overline{\mu }
x
t
both. The physical behavior of the obtained model is discussed.
Ahmad T. Ali. "Invariant Inhomogeneous Bianchi Type-I Cosmological Models with Electromagnetic Fields Using Lie Group Analysis in Lyra Geometry." Abstr. Appl. Anal. 2014 (SI55) 1 - 8, 2014. https://doi.org/10.1155/2014/918927
Ahmad T. Ali "Invariant Inhomogeneous Bianchi Type-I Cosmological Models with Electromagnetic Fields Using Lie Group Analysis in Lyra Geometry," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI55), 1-8, (2014)
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Global Dynamics of an HTLV-1 Model with Cell-to-Cell Infection and Mitosis
2014 Global Dynamics of an HTLV-1 Model with Cell-to-Cell Infection and Mitosis
Sumei Li, Yicang Zhou
A mathematical model of human T-cell lymphotropic virus type 1 in vivo with cell-to-cell infection and mitosis is formulated and studied. The basic reproductive number
{R}_{0}
is derived. It is proved that the dynamics of the model can be determined completely by the magnitude of
{R}_{0}
. The infection-free equilibrium is globally asymptotically stable (unstable) if
{R}_{0}<1
\left({R}_{0}>1\right)
. There exists a chronic infection equilibrium and it is globally asymptotically stable if
{R}_{0}>1
Sumei Li. Yicang Zhou. "Global Dynamics of an HTLV-1 Model with Cell-to-Cell Infection and Mitosis." Abstr. Appl. Anal. 2014 (SI66) 1 - 12, 2014. https://doi.org/10.1155/2014/132781
Sumei Li, Yicang Zhou "Global Dynamics of an HTLV-1 Model with Cell-to-Cell Infection and Mitosis," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI66), 1-12, (2014)
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Discussion: “Design Improvements to a Biomass Stirling Engine Using Mathematical Analysis and 3D CFD Modeling” (Mahkamov, K., 2006, ASME J. Energy Resour. Technol., 128, pp. 203–215) | J. Energy Resour. Technol. | ASME Digital Collection
Discussion: “Design Improvements to a Biomass Stirling Engine Using Mathematical Analysis and 3D CFD Modeling” (Mahkamov, K., 2006, ASME J. Energy Resour. Technol., 128, pp. 203–215)
, Whiteknights, Reading RG6 4AY, United Kingdom
J. Energy Resour. Technol. Sep 2007, 129(3): 280 (1 pages)
This is a companion to: Design Improvements to a Biomass Stirling Engine Using Mathematical Analysis and 3D CFD Modeling
Burton, J. D. (September 1, 2007). "Discussion: “Design Improvements to a Biomass Stirling Engine Using Mathematical Analysis and 3D CFD Modeling” (Mahkamov, K., 2006, ASME J. Energy Resour. Technol., 128, pp. 203–215)." ASME. J. Energy Resour. Technol. September 2007; 129(3): 280. https://doi.org/10.1115/1.2751510
Stirling engines, computational fluid dynamics, brakes, mathematical analysis, pistons, bioenergy conversion
Biomass, Computational fluid dynamics, Design, Mathematical analysis, Modeling, Stirling engines, Pistons, Brakes
Some very impressive modeling work lies behind this paper, but leaves the reader with many questions which Dr. Mahkamov may care to answer.
We are told that the Biomass Stirling Engine “has been experimentally tested” and that “experimental data, which is available (where?) of the mechanical brake power output and speed, indicates that”—these “are reasonably close to those predicted by using the 3D CFD model.” Could the author provide the figures so that we can judge for ourselves how close is “reasonably close?”
One observes that, with fluid flow losses included, the
γ
engine achieves an inferior predicted performance to the modified
α
, whether using the second-order model, or the 3D CFD model.
Mode 2nd order 3D CFD
γ
250W
737W
4Hz
3.33Hz
α
2.970W
3.870W
7Hz
5Hz
γ
250W
737W
4Hz
3.33Hz
α
2.970W
3.870W
7Hz
5Hz
α
mode power output would seem to be 6.8 times better than the
γ
using the results of the second-order model, or 3.5 times better using the 3D CFD model. However, this is misleading; we are not comparing like with like.
γ
engine had apparently a simple geometrical error which caused much loss of power owing to the crown of the power piston restricting flow as it approached TDC. Furthermore the
γ
engine model is run with a 31% porosity regenerator, while the
α
engine is run with an improved 40% regenerator. Did the author run his two programs for the
γ
engine with the crown restriction removed and with 40% porosity? Might he be able to provide the figures so that
α
γ
modes can be compared on a level playing field?
On the basis that the pressures,
P7
P5
P1
in Fig. 13 correspond, respectively, to pressures
P2
P2V2
P3
P3V3
P1
(P1V1)
of Fig. 14, then one can make an estimate for how much the gas entrapment is costing the
γ
engine in power output. From Fig. 13 it can be seen that for some 25° either side of the 180° crank angle (where the power piston crown is near TDC) there is a significant difference in pressure between
P7
P5
and the gas is restricted in its passage from compression space 5 to compression space 6 (see Fig. 1).
Transferring this pressure difference so as to modify the anticlockwise compression loop
P2V2
one is able to deduce that the
γ
engine output will be raised at least from
0.737to1.77kW
. Does Dr. Mahkamov feel this is a reasonable estimate?
As a measure of goodness (proportional to the Beale number) one can compare the outputs in the modified Fig. 14 (
γ
mode) and Fig. 17 (
α
mode) on the basis of
kW∕(Chargepressure×rpm×powersweptvolume)
. In changing from
γ
mode to
α
mode the power swept volume changes. In
γ
mode this volume is provided by the power piston (4 in Fig. 1) together with a small contribution from the unbalanced stanchion of the displacer piston. In
α
mode it is the larger expansion piston that becomes effectively the power piston. Using the outputs from the 3D CFD model and a nominal charge pressure of
15bar
then one obtains
kW rpm Power sweptvolume (liter) kW
Bar×liters×rpm
γ
1.77 200 2.46
0.24×10−3
α
0.25×10−3
Bar×liters×rpm
γ
0.24×10−3
α
0.25×10−3
It would seem that using the “measure of goodness factor” in the right-hand column, not very much has been gained in moving from
γ
α
mode. Indeed, much has been lost in terms of the mechanical soundness of the engine crank shaft gas loads, etc.
0.25×10−3kWbar−1liter−1rpm−1
seems to be low. Might the 3D CFD model be significantly underestimating the performance?
Discussion: “Design Improvements to a Biomass Stirling Engine Using Mathematical Analysis and 3D CFD Modeling” [ Mahkamov, K., 2006, ASME J. Energy Resour. Technol., 128, pp 203–215 ]
Closure to “Discussion: ‘Design Improvements to a Biomass Stirling Engine Using Mathematical Analysis and 3D CFD Modeling’ ” ( 2007, ASME J. Energy Resour. Technol., 129, pp. 278, 279, 280 )
Numerical Characterization of Swirl Brakes for High Pressure Centrifugal Compressors
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Semi-Deviation Definition
What Is Semi-Deviation?
Semi-deviation is a method of measuring the below-mean fluctuations in the returns on investment.
Semi-deviation will reveal the worst-case performance to be expected from a risky investment.
Semi-deviation is an alternative measurement to standard deviation or variance. However, unlike those measures, semi-deviation looks only at negative price fluctuations. Thus, semi-deviation is most often used to evaluate the downside risk of an investment.
Understanding Semi-Deviation
In investing, semi-deviation is used to measure the dispersion of an asset's price from an observed mean or target value. In this sense, dispersion means the extent of variation from the mean price.
Semi-deviation is an alternative to the standard deviation for measuring an asset's degree of risk.
Semi-deviation measures only the below-mean, or negative, fluctuations in an asset's price.
This measurement tool is most often used to evaluate risky investments.
The point of the exercise is to determine the severity of the downside risk of an investment. The asset's semi-deviation number can then be compared to a benchmark number, such as an index, to see if it is more or less risky than other potential investments.
The formula for semi-deviation is:
\begin{aligned}&\text{Semi-deviation}\ =\ \sqrt{\frac{1}{n}\ \times\ \sum^n_{r_t\ <\ \text{Average}}(\text{Average}\ -\ r_t)^2}\\&\textbf{where:}\\&n\ =\ \text{the total number of observations below the mean}\\&r_t\ =\ \text{the observed value}\\&\text{average}\ =\text{the mean or target value of a data set}\end{aligned}
Semi-deviation = n1 × rt < Average∑n(Average − rt)2where:n = the total number of observations below the meanrt = the observed value
An investor's entire portfolio could be evaluated according to the semi-deviation in the performance of its assets. Put bluntly, this will show the worst-case performance that can be expected from a portfolio, compared to the losses in an index or whatever comparable is selected.
History of Semi-Deviation in Portfolio Theory
Semi-deviation was introduced in the 1950s specifically to help investors manage risky portfolios. Its development is credited to two leaders in modern portfolio theory.
Harry Markowitz demonstrated how to exploit the averages, variances, and covariances of the return distributions of assets of a portfolio in order to compute an efficient frontier on which every portfolio achieves the expected return for a given variance or minimizes the variance for a given expected return. In Markowitz' explanation, a utility function, defining the investor’s sensitivity to changing wealth and risk, is used to pick an appropriate portfolio on the statistical border.
A.D. Roy, meanwhile, used semi-deviation to determine the optimum trade-off of risk to return. He didn't believe it was feasible to model the sensitivity to risk of a human being with a utility function. Instead, he assumed that investors would want the investment with the smallest likelihood of coming in below a disaster level. Understanding the wisdom of this claim, Markowitz realized two very important principles: Downside risk is relevant for any investor, and return distributions might be skewed, or not symmetrically distributed, in practice. As such, Markowitz recommended using a variability measure, which he called a semivariance, as it only takes into account a subset of the return distribution.
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Fit conditional variance model to data - MATLAB estimate - MathWorks Switzerland
{y}_{t}={\epsilon }_{t},
{\epsilon }_{t}={\sigma }_{t}{z}_{t}
{\sigma }_{t}^{2}=0.0001+0.5{\sigma }_{t-1}^{2}+0.2{\epsilon }_{t-1}^{2}.
{z}_{t}
{y}_{t}={\epsilon }_{t},
{\epsilon }_{t}={\sigma }_{t}{z}_{t},
\mathrm{log}{\sigma }_{t}^{2}=0.001+0.7\mathrm{log}{\sigma }_{t-1}^{2}+0.5\left[\frac{|{\epsilon }_{t-1}|}{{\sigma }_{t-1}}-\sqrt{\frac{2}{\pi }}\right]-0.3\left(\frac{{\epsilon }_{t-1}}{{\sigma }_{t-1}}\right)
{z}_{t}
{y}_{t}={\epsilon }_{t},
{\epsilon }_{t}={\sigma }_{t}{z}_{t}
{\sigma }_{t}^{2}=0.001+0.5{\sigma }_{t-1}^{2}+0.2{\epsilon }_{t-1}^{2}+0.2I\left[{\epsilon }_{t-1}<0\right]{\epsilon }_{t-1}^{2}.
{z}_{t}
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EUDML | Solution of a finite-dimensional problem with -mappings and diagonal multivalued operators. EuDML | Solution of a finite-dimensional problem with -mappings and diagonal multivalued operators.
Solution of a finite-dimensional problem with
M
-mappings and diagonal multivalued operators.
Laitinen, E., and Lapin, A.. "Solution of a finite-dimensional problem with -mappings and diagonal multivalued operators.." Computational Methods in Applied Mathematics 1.3 (2001): 242-264. <http://eudml.org/doc/233408>.
author = {Laitinen, E., Lapin, A.},
keywords = {maximal monotone operator; -mapping; iterative solution; finite difference scheme; variational inequality; -mapping},
title = {Solution of a finite-dimensional problem with -mappings and diagonal multivalued operators.},
TI - Solution of a finite-dimensional problem with -mappings and diagonal multivalued operators.
KW - maximal monotone operator; -mapping; iterative solution; finite difference scheme; variational inequality; -mapping
maximal monotone operator,
M
-mapping, iterative solution, finite difference scheme, variational inequality,
M
-mapping
|
Net Debt Formula and Calculation
What Net Debt Indicates
Comprehensive Debt Analysis
Net Debt vs. Debt-to-Equity
Limitations of Using Net Debt
Net debt is a liquidity metric used to determine how well a company can pay all of its debts if they were due immediately. Net debt shows how much debt a company has on its balance sheet compared to its liquid assets.
\begin{aligned} &\text{Net Debt} = \text{STD} + \text{LTD} - \text{CCE}\\ &\textbf{where:}\\ &\begin{aligned} \text{STD} = &\text{ Debt that is due in 12 months or less}\\ &\text{ and can include short-term bank}\\ &\text{ loans, accounts payable, and lease}\\ &\text{ payments}\end{aligned}\\ &\begin{aligned} \text{LTD} = &\text{ Long-term debt is debt that with a}\\ &\text{ maturity date longer than one year}\\ &\text{ and include bonds, lease payments,}\\ &\text{ term loans, small and notes payable}\end{aligned}\\ &\begin{aligned} \text{CCE} = &\text{ Cash and liquid instruments that can be}\\ &\text{ easily converted to cash.}\end{aligned}\\ &\text{Cash equivalents are liquid investments with a}\\ &\text{maturity of 90 days or less and include}\\ &\text{certificates of deposit, Treasury bills, and}\\ &\text{commercial paper} \end{aligned}
Net Debt=STD+LTD−CCEwhere:STD= Debt that is due in 12 months or less and can include short-term bank loans, accounts payable, and leaseLTD= Long-term debt is debt that with a maturity date longer than one year and include bonds, lease payments,CCE= Cash and liquid instruments that can beCash equivalents are liquid investments with amaturity of 90 days or less and includecertificates of deposit, Treasury bills, and
Total up all short-debt amounts listed on the balance sheet.
Total all long-term debt listed and add the figure to the total short-term debt.
Total all cash and cash equivalents and subtract the result from the total of short-term and long-term debt.
The net debt figure is used as an indication of a business's ability to pay off all of its debts if they became due simultaneously on the date of calculation, using only its available cash and highly liquid assets called cash equivalents.
Net debt helps to determine whether a company is overleveraged or has too much debt given its liquid assets. A negative net debt implies that the company possesses more cash and cash equivalents than its financial obligations and is hence more financially stable.
A negative net debt means a company has little debt and more cash, while a company with a positive net debt means it has more debt on its balance sheet than liquid assets. However, since it's common for companies to have more debt than cash, investors must compare the net debt of a company with other companies in the same industry.
Net debt is in part, calculated by determining the company's total debt. Total debt includes long-term liabilities, such as mortgages and other loans that do not mature for several years, as well as short-term obligations, including loan payments, credit cards, and accounts payable balances.
Net Debt and Total Cash
The net debt calculation also requires figuring out a company's total cash. Unlike the debt figure, the total cash includes cash and highly liquid assets. Cash and cash equivalents would include items such as checking and savings account balances, stocks, and some marketable securities. However, it's important to note that many companies may not include marketable securities as cash equivalents since it depends on the investment vehicle and whether it's liquid enough to be converted within 90 days.
While the net debt figure is a great place to start, a prudent investor must also investigate the company's debt level in more detail. Important factors to consider are the actual debt figures—both short-term and long-term—and what percentage of the total debt needs to be paid off within the coming year.
Debt management is important for companies because if managed properly they should have access to additional funding if needed. For many companies, taking on new debt financing is vital to their long-growth strategy since the proceeds might be used to fund an expansion project, or to repay or refinance older or more expensive debt.
A company might be in financial distress if it has too much debt, but also the maturity of the debt is important to monitor. If the majority of the company's debts are short term, meaning the obligations must be repaid within 12 months, the company must generate enough revenue and have enough liquid assets to cover the upcoming debt maturities. Investors should consider whether the business could afford to cover its short-term debts if the company's sales decreased significantly.
On the other hand, if the company's current revenue stream is only keeping up with paying its short-term debts and isn't able to adequately pay down long-term debt, it's only a matter of time before the company will face hardship or will need an injection of cash or financing. Since companies use debt differently and in many forms, it's best to compare a company's net debt to other companies within the same industry and of comparable size.
Company A has the following financial information listed on its balance sheet. Companies will typically break down whether the debt is short-term or long-term.
Credit Line: $50,000
Term Loan: $200,000
To calculate net debt, we must first total all debt and total all cash and cash equivalents. Next, we subtract the total cash or liquid assets from the total debt amount.
Total debt would be calculated by adding the debt amounts or $100,000 + $50,000 + $200,000 = $350,000.
Cash and cash equivalents are totaled or $30,000 + $20,000 and equal $50,000 for the period.
Net debt is calculated by $350,000 - $50,000 equaling $300,000 in net debt.
The debt-to-equity (D/E) ratio is a leverage ratio, which shows how much of a company's financing or capital structure is made up of debt versus issuing shares of equity. The debt-to-equity ratio is calculated by dividing a company’s total liabilities by its shareholders' equity and is used to determine if a company is using too much or too little debt or equity to finance its growth.
Net debt takes it to another level by measuring how much total debt is on the balance sheet after factoring in cash and cash equivalents. Net debt is a liquidity metric while debt-to-equity is a leverage ratio.
Although it's typically perceived that companies with negative net debt are better able to withstand economic downtrends and deteriorating macroeconomic conditions, too little debt might be a warning sign. If a company is not investing in its long-term growth as a result of the lack of debt, it might struggle against competitors that are investing in its long-term growth.
For example, oil and gas companies are capital intensive meaning they must invest in large fixed assets, which include property, plant, and equipment. As a result, companies in the industry typically have significant portions of long-term debt to finance their oil rigs and drilling equipment.
An oil company should have a positive net debt figure, but investors must compare the company's net debt with other oil companies in the same industry. It doesn't make sense to compare the net debt of an oil and gas company with the net debt of a consulting company with few if any fixed assets. As a result, net debt is not a good financial metric when comparing companies of different industries since the companies might have vastly different borrowing needs and capital structures.
Which Is More Important: Net Debt or Gross Debt?
Gross debt is the nominal value of all of the debts and similar obligations a company has on its balance sheet. If the difference between net debt and gross debt is large, it indicates a large cash balance along with significant debt, which could be a red flag. Net debt removes cash and cash equivalents from the amount of debt, which is useful when calculating enterprise value (EV) or when a company seeks to make an acquisition. This is because a company is not interested in spending cash to acquire cash. Rather, the net debt will give a better estimate of the takeover value.
How Do You Calculate Net Debt in Excel?
To calculate net debt using Microsoft Excel, find the following information on the company's balance sheet: total short-term liabilities; total long-term liabilities; and total current assets. Enter these three items into cells A1 through A3, respectively. In cell A4, enter the formula "=A1+A2−A3" to compute net debt.
What Is Net Debt Per Capita?
Net debt per capita is a country-level metric that looks at a nation's total sovereign debt and divides it by the population size. It is used to understand how much debt a country has in proportion to its population allowing for between-country comparisons in understanding a country's relative solvency.
The cash ratio—total cash and cash equivalents divided by current liabilities—measures a company's ability to repay its short-term debt.
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Continuum Modeling and Simulation of Robotic Appendage Interaction With Granular Material | J. Appl. Mech. | ASME Digital Collection
Continuum Modeling and Simulation of Robotic Appendage Interaction With Granular Material
Guanjin Wang,
Email: gjwang@umd.edu
Amir Riaz,
Email: balab@umd.edu
Balakumar Balachandran Fellow ASME
Wang, G., Riaz, A., and Balachandran, B. (December 4, 2020). "Continuum Modeling and Simulation of Robotic Appendage Interaction With Granular Material." ASME. J. Appl. Mech. February 2021; 88(2): 021013. https://doi.org/10.1115/1.4049069
Legged locomotion has advantages when one is navigating a flowable ground or a terrain with obstacles that are common in nature. With traditional terra-mechanics, one can capture large wheel–terrain interactions. However, legged motion on a granular substrate is difficult to investigate by using classical terra-mechanics due to sharp edge contact. Recent studies have shown that a continuum simulation can serve as an accurate tool for simulating dynamic interactions with granular material at laboratory and field scales. Spurred by this, a computational framework based on the smoothed particle hydrodynamics (SPH) method has been developed for the investigation of single robot appendage interaction with a granular system. This framework has been validated by using experimental results and extended to study robot appendages with different shapes and stride frequencies. The mechanics’ results are expected to help robot navigation and exploration in unknown and complex terrains.
robot locomotion, dynamic interactions, complex terrain, continuum modeling, granular material, constitutive relationship, smoothed particle hydrodynamics, computational mechanics, constitutive modeling of materials, dynamics
Density, Granular materials, Hydrodynamics, Particulate matter, Robots, Simulation, Modeling
Vision-Based Object Detection and Tracking for Autonomous Navigation of Underwater Robots
, “In-Situ Soil Sensing for Planetary Micro-Rovers With Hybrid Wheel-Leg Systems,” Ph.D. thesis,
Experimental Study and Analysis on Driving Wheels’ Performance for Planetary Exploration Rovers Moving in Deformable Soil
. 10.1016/j.jterra.2010.08.001
, “Introduction to Terrain-Vehicle Systems,” Part i: The terrain. Part ii: The Vehicle, Michigan University, Ann Arbor, MI, Technical Report.
Terramechanics-Based High-Fidelity Dynamics Simulation for Wheeled Mobile Robot on Deformable Rough Terrain
A Resistive Force Model for Legged Locomotion on Granular Media,” Adaptive Mobile Robotics
Modeling of the Interaction of Rigid Wheels With Dry Granular Media
Smooth Particle Hydrodynamics Studies of Wet Granular Column Collapses
The Role of Constitutive Models in Mpm Simulations of Granular Column Collapses
Numerical Simulations for Large Deformation of Granular Materials Using Smoothed Particle Hydrodynamics Method
Continuum Modelling and Simulation of Granular Flows Through Their Many Phases
A Constitutive Law for Dense Granular Flows
Continuum Modeling of Rate-Dependent Granular Flows in Sph
Euro. Phys. J. E
. 10.1140/epje/i2003-10153-0
Smoothed Particle Hydrodynamics Modeling of Granular Column Collapse
Improving Convergence in Smoothed Particle Hydrodynamics Simulations Without Pairing Instability
Le Touz ’e
δ
-sph Model for Simulating Violent Impact Flows
A Comprehensive Study on the Parameters Setting in Smoothed Particle Hydrodynamics (SPH) Method Applied to Hydrodynamics Problems
A New Sph-Based Approach to Simulation of Granular Flows Using Viscous Damping and Stress Regularisation
Simulation of Heat Transfer in Moving Granular Material by the Discrete Element Method With Special Emphasis on Inner Particle Heat Transfer
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EUDML | Companion forms and weight one forms. EuDML | Companion forms and weight one forms.
Companion forms and weight one forms.
Buzzard, Kevin; Taylor, Richard
Buzzard, Kevin, and Taylor, Richard. "Companion forms and weight one forms.." Annals of Mathematics. Second Series 149.3 (1999): 905-919. <http://eudml.org/doc/121093>.
@article{Buzzard1999,
author = {Buzzard, Kevin, Taylor, Richard},
keywords = {Artin's conjecture; Galois group; -adic representation; holomorphic weight one newform; overconvergent form of weight one; rigid analytic geometry of modular curves; rigid GAGA; -adic representation},
title = {Companion forms and weight one forms.},
AU - Buzzard, Kevin
TI - Companion forms and weight one forms.
KW - Artin's conjecture; Galois group; -adic representation; holomorphic weight one newform; overconvergent form of weight one; rigid analytic geometry of modular curves; rigid GAGA; -adic representation
Arnaud Jehanne, Michael Müller, Modularity of an odd icosahedral representation
Vincent Pilloni, Overconvergent modular forms
Chris Skinner, Modularity of Galois representations
Eknath Ghate, Vinayak Vatsal, On the local behaviour of ordinary
\Lambda
Mladen Dimitrov, Eknath Ghate, On classical weight one forms in Hida families
Jean-Pierre Wintenberger, La conjecture de modularité de Serre : le cas de conducteur
1
Artin's conjecture, Galois group,
\ell
-adic representation, holomorphic weight one newform, overconvergent form of weight one, rigid analytic geometry of modular curves, rigid GAGA,
\ell
-adic representation
p
Articles by Buzzard
|
What Is an Asset Base?
An asset base refers to the underlying assets that give value to a company, investment, or loan. The asset base is not fixed; it will appreciate or depreciate according to market forces, or increase and decrease as a company sells or acquires new assets.
Although it is completely normal for a company to make changes to its asset base periodically by buying and selling assets, large swings in asset base will affect the company's valuation and can be a red flag for analysts. Lenders use physical assets as a guarantee that at least a portion of money lent can be recouped through the sale of the backed asset in the case that the loan itself cannot be repaid.
An asset base is the underlying value of assets that constitute the basis for the valuation of a firm, loan, or derivative security.
For a firm, the asset base is its book value. For a loan, it is the collateral backing the loan. For a derivative, it is the underlying asset.
Often, the market value of something backed by assets will exceed the implied value of the asset base.
Understanding Asset Base
A company's asset base is included in its valuation and includes tangible, hard assets such as property, plant, equipment, and inventory. It also includes financial assets such as cash, cash equivalents, and securities. Typically, a company's market value will exceed its asset base since market value also includes intangibles as well as expected future growth from cash flows and profits.
With an investment in a futures contract, as an example, the price of the underlying asset used as the asset base of such a derivative contract can increase or decrease rapidly, changing the price that investors are willing to pay for it.
With a loan, the value of a home might increase or decrease over time, affecting the underlying collateral in a mortgage. Margin loans are particularly sensitive to the underlying value of the collateral, as pledged securities whose value fluctuates with the market are often used for this purpose.
A company's asset base is often construed as its book value. The book value of a company literally means the value of a business according to its books (accounts) that is reflected through its financial statements. Theoretically, book value represents the total amount a company is worth if all its assets are sold and all the liabilities are paid back. This is the amount that the company’s creditors and investors can expect to receive if the company is liquidated.
Mathematically, book value is calculated as the difference between a company's total assets and total liabilities.
\text{Book value of a company} = \text{Total assets} - \text{Total liabilities}
Book value of a company=Total assets−Total liabilities
Total assets include all kinds of assets, such as cash and short term investments, total accounts receivable, inventories, net property, plant and equipment (PP&E), investments and advances, intangible assets like goodwill, and tangible assets.
Total liabilities include items like short and long term debt obligations, accounts payable, and deferred taxes.
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Motion in 2D - Test with Audio Solutions 2 - NEET & AIIMS 2019
Motion in 2D - Test with Audio Solutions 2 - NEET & AIIMS 2019Contact Number: 9667591930 / 8527521718
The height y and the distance x along the horizontal plane of a projectile on a certain planet (with no surrounding atmosphere) are given by
y=8t-5{t}^{2} m
and x = 6t m, where t is in seconds. The velocity with which the particle is projected is.
A particle of mass m is projectile with a velocity v making an angle of
{30}^{o}
with the horizontal. The magnitude of angular momentum of the projectile about the point of the projection when particle is at it's maximum height h is
1. \sqrt{\frac{3}{2}}\frac{m{v}^{2}}{g}\phantom{\rule{0ex}{0ex}}2. zero\phantom{\rule{0ex}{0ex}}3. \frac{m{v}^{3}}{\sqrt{2}g}\phantom{\rule{0ex}{0ex}}4. \frac{\sqrt[]{}3}{16}\frac{m{v}^{3}}{g}
if a person can through a stone to maximum height of h meter vertically, then maximum distance through which it can be thrown horizontally by the same person is
1. h/2
At the top of the trajectory of a projectile, direction of its velocity and acceleration are
1. perpendicular to each other
2. parallel to each other
3. inclined to each other at angle of
{45}^{0}
4. antiparallel to each other
These particles A, B and C are projected from the same pair with same initial speed making angles
{30}^{o, }{45}^{o} & {60}^{o}
respectively with the horizontal. Which of the following statements is correct .
1. A, B and C have unequal ranges
2. Range of A and C are less than that of B
3. Range of A and C are equal & greater than that of B
4. A, B and C have equal ranges
At the height 80 m an aeroplane is moving with 150m/s. A bomb is dropped from it, so as to hit a target. At what distance from the target should bomb be dropped ?
Three balls are dropped from the top of a building with equal speed at different angles. When the balls strike ground, their velocities are
{v}_{1}, {v}_{2} & {v}_{3}
respectively , than
1. {v}_{1}>{v}_{2}>{v}_{3}\phantom{\rule{0ex}{0ex}}2. {v}_{3} >{v}_{2}> {v}_{1}\phantom{\rule{0ex}{0ex}}3. {v}_{1 }= {v}_{2}= {v}_{3}\phantom{\rule{0ex}{0ex}}4. {v}_{1} <{v}_{2} <{v}_{3}
An object of mass 2m is projected with a speed of 100 m/s at an angle
\theta = {\mathrm{sin}}^{-1}\left(\frac{3}{5}\right)
to the horizontal. At the highest point, the object breaks into pieces of same mass m & the first one comes to rest . The distance between point of landing of the bigger piece is (given g = 10 m/s2 )
A cart is moving horizontally along a straight line with contant speed of 30 m/s. A projectile is to be fired from the moving cart in such a way that it will return to the cart after the cart has moved 80 m. At what speed (relative to the cart) must the projectile be fired ? (given g = 10 m/s2)
\frac{40}{3} m/s
10\sqrt{8} m/s
A body is projected with velocity u at an angle
\theta
with horizontal from a moving cart then path of the body appears to be
1. Straight line for observer on ground, parabola for observer in cart.
2. Straight line for observer in cart as well as observer on ground.
3. Parabola for observer on cart and parabola of lesse angle for observer on ground.
A cart is moving with speed of 20m/s on a horizontal track. A body is projected with speed 40m/s (relative to cart ) from the cart in such a way that path of body appear to be straight line for an observer on ground. Time for which body remains in air is:
1. 2\sqrt{3} sec \phantom{\rule{0ex}{0ex}}2. 4 sec\phantom{\rule{0ex}{0ex}}3. 4\sqrt{3 }sec\phantom{\rule{0ex}{0ex}}4. 8 sec
Two particles A and B are projected with velocity
{u}_{1}and {u}_{2}
simultaneously. Time after which they start moving perpendicularly to each other is ?
1. \frac{\sqrt{{u}_{1}{u}_{2}}}{g}\phantom{\rule{0ex}{0ex}}2. \frac{2\sqrt{{u}_{1}{u}_{2}}}{g}\phantom{\rule{0ex}{0ex}}3. \frac{\sqrt{{u}_{1}{u}_{2}}}{2g}\phantom{\rule{0ex}{0ex}}4. \frac{\sqrt{2{u}_{1}{u}_{2}}}{g}\phantom{\rule{0ex}{0ex}}
|
Modified duration measures the change in the value of a bond in response to a change in 100-basis-point (1%) change in interest rates.
Modified duration is an extension of the Macaulay duration, and in order to calculate modified duration, the Macaulay duration must first be calculated.
Macaulay duration calculates the weighted average time before a bondholder receives the bond's cash flows.
As a bond's maturity increases, duration increases, and as a bond's coupon and interest rate increases, its duration decreases.
Formula and Calculation of Modified Duration
\begin{aligned}&\text{Modified Duration}=\frac{\text{Macaulay Duration}}{1+\overset{\text{YTM}}{n}}\\&\textbf{where:}\\&\text{Macaulay Duration}=\text{Weighted average term to}\\&\qquad\text{maturity of the cash flows from a bond}\\&\text{YTM}=\text{Yield to maturity}\\&n=\text{Number of coupon periods per year}\end{aligned}
Modified Duration=1+nYTMMacaulay Durationwhere:Macaulay Duration=Weighted average term tomaturity of the cash flows from a bondYTM=Yield to maturityn=Number of coupon periods per year
Modified duration is an extension of the Macaulay duration, which allows investors to measure the sensitivity of a bond to changes in interest rates. Macaulay duration calculates the weighted average time before a bondholder receives the bond's cash flows. In order to calculate modified duration, the Macaulay duration must first be calculated. The formula for the Macaulay duration is:
\begin{aligned}&\text{Macaulay Duration}=\frac{\sum^n_{t=1}(\text{PV}\times \text{CF})\times \text{t}}{\text{Market Price of Bond}}\\&\textbf{where:}\\&\text{PV}\times \text{CF}=\text{Present value of coupon at period }t\\&\text{t}=\text{Time to each cash flow in years}\\&n=\text{Number of coupon periods per year}\end{aligned}
Macaulay Duration=Market Price of Bond∑t=1n(PV×CF)×twhere:PV×CF=Present value of coupon at period tt=Time to each cash flow in yearsn=Number of coupon periods per year
Here, (PV) * (CF) is the present value of a coupon at period t, and T is equal to the time to each cash flow in years. This calculation is performed and summed for the number of periods to maturity.
What Modified Duration Can Tell You
Modified duration measures the average cash-weighted term to maturity of a bond. It is a very important number for portfolio managers, financial advisors, and clients to consider when selecting investments because—all other risk factors equal—bonds with higher durations have greater price volatility than bonds with lower durations. There are many types of duration, and all components of a bond, such as its price, coupon, maturity date, and interest rates, are used to calculate duration.
Here are some principles of duration to keep in mind. First, as maturity increases, duration increases and the bond becomes more volatile. Second, as a bond's coupon increases, its duration decreases and the bond becomes less volatile. Third, as interest rates increase, duration decreases, and the bond's sensitivity to further interest rate increases goes down.
Example of How to Use Modified Duration
Assume a $1,000 bond has a three-year maturity, pays a 10% coupon, and that interest rates are 5%. This bond, following the basic bond pricing formula would have a market price of:
\begin{aligned} &\text{Market Price} = \frac{ \$100 }{ 1.05 } + \frac{ \$100 }{ 1.05 ^ 2 } + \frac{ \$1,100 }{ 1.05 ^ 3 } \\ &\phantom{\text{Market Price} } = \$95.24 + \$90.70 + \$950.22\\ &\phantom{\text{Market Price} } = \$1,136.16 \\ \end{aligned}
Market Price=1.05$100+1.052$100+1.053$1,100Market Price=$95.24+$90.70+$950.22Market Price=$1,136.16
Next, using the Macaulay duration formula, the duration is calculated as:
\begin{aligned}\text{Macaulay Duration}&=\bigg(\$95.24\times\frac{1}{\$1,136.16}\bigg)\\&\quad+\bigg(\$90.70\times\frac{2}{\$1,136.16}\bigg)\\&\quad+\bigg(\$950.22\times\frac{3}{\$1,136.16}\bigg)\\&=2.753\end{aligned}
Macaulay Duration=($95.24×$1,136.161)+($90.70×$1,136.162)+($950.22×$1,136.163)=2.753
This result shows that it takes 2.753 years to recoup the true cost of the bond. With this number, it is now possible to calculate the modified duration.
To find the modified duration, all an investor needs to do is take the Macaulay duration and divide it by 1 + (yield-to-maturity / number of coupon periods per year). In this example that calculation would be 2.753 / (1.05 / 1), or 2.62%. This means that for every 1% movement in interest rates, the bond in this example would inversely move in price by 2.62%.
The Macaulay duration is the weighted average term to maturity of the cash flows from a bond.
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Bending moment - zxc.wiki
circular bending of a rod as a result of a constant bending moment over its length
A moment that can bend a slim ( rod , beam , shaft , etc.) or thin component ( plate , etc.) is referred to as a bending moment .
{\ displaystyle M}
1 bending moment in beam theory
2 examples of bending moment curves on the beam
2.1 Cantilever beam, single force at the free end
2.2 Beams supported at the ends, individual force in between
3 bending moment and bending line
4 bending moment and bending stress
Bending moment in beam theory
Cantilever : tensile and compressive stress in a cross-section near the clamping point
(cut out for illustration)
when subjected to a bending moment (generated by force F at the free end)
The behavior of a slim component or a beam under load is the subject of beam theory . In particular, its behavior under a bending moment that loads it is examined with the aid of strength and elasticity . Instead of the beam theory, we therefore often speak of the bending theory of the beam , or in a narrower sense .
With the help of the theoretical individual disciplines of strength theory and elasticity theory , the bending stresses resulting from the loading bending moment inside the beam and the external elastic deformation (e.g. deflection ) of the beam are calculated and compared with the respective permissible values. The bending stresses should be smaller than the material values permissible for elastic deformation ( proof of strength against plastic deformation or breakage). In some applications there is an additional limitation in the form of a permissible (elastic) deflection. This should not be exceeded by the calculated value.
The total bending stress in a cross-sectional area of the beam is proportional to the bending moment at this point. In the cross-section, it runs from maximum pressure on the inner edge (concave bend) via zero in the neutral zone to maximum tensile stress on the outer edge (convex bend). The proof of strength is i. d. Usually carried out with the maximum tensile stress (the compressive stress that can be tolerated by a beam material is usually the greater).
The bending of the beam is represented by its curvature , which is also proportional to the bending moment acting there at each cross-sectional point. To make a statement about z. B. a permissible deflection is used by the bending line determined from the curvature that is variable over the length of the bar .
Examples of bending moment curves on the beam
Clamped beam ( cantilever beam ) with a force P at the free end
Cantilever beam, single force at the free end
A cantilever beam clamped on one side is loaded by a force at the free end at a distance (see adjacent figure). The bending moment curve is
{\ displaystyle L}
{\ displaystyle P}
{\ displaystyle M (x) = P \ cdot (Lx)}
At the initiation point ( ) of the force, it is zero. It rises linearly to its maximum value up to the clamping point ( ) .
{\ displaystyle x = L}
{\ displaystyle x = 0}
{\ displaystyle M = P \ cdot L}
Beams supported at the ends, individual force in between
Bending moment curve M (x) over bars on two bearings, single force F: max. Bending. at the place of F (e.g. at l / 2)
To calculate the internal moments, the component is mentally cut through at the point of interest , and those moments are considered which act on a section at its intersection . The bending moment at a point is therefore the sum of all torques that are caused by forces on one side of the interface .
{\ displaystyle x}
{\ displaystyle x}
{\ displaystyle x}
In the beam with a single load supported at its ends (see adjacent figure), the left-hand section is subject to a clockwise torque (briefly called a moment in technical mechanics ), which can be described with the help of the contact force F L on the left-hand bearing . The torque increases linearly from zero on the support to the maximum value at the point of load F. On the right of this, a counterclockwise torque that increases linearly from zero to the same maximum value on the right support comes from the load F, so that the sum of moments from Maximum value at the load point decreases linearly to zero at the right end.
{\ displaystyle M (x) = {\ begin {cases} {\ frac {F} {2}} \ cdot x & {\ text {(left of center)}} x <{\ frac {l} {2}} \\ {\ frac {F} {2}} \ cdot (lx) & {\ text {(right of center)}} x> {\ frac {l} {2}} \ end {cases}}}
Special case of central load: The at maximum bending moment has the value
{\ displaystyle x = l / 2}
{\ displaystyle M _ {\ mathrm {max}} = {\ frac {F \ cdot l} {4}}}
Bending moment and bending line
Course of a bending moment on a beam with a central force F, shown here as a point load P, with the maximum bending moment M at l / 2 including the transverse force course Q and the bending line w
Main article : Bend line
The elastic deformation caused by the bending moment load is described by the bending line . For a bar of constant cross-section, the following approximation equation applies to its curvature :
{\ displaystyle w (x)}
{\ displaystyle w '' (x)}
{\ displaystyle w '' (x) = - {\ frac {M_ {y} (x)} {E \ cdot I_ {y}}}}
the curvature (variable x in the direction of the bar)
{\ displaystyle w '' (x)}
the modulus of elasticity (a material property )
{\ displaystyle E}
the axial geometrical moment of inertia (a geometric quantity of the constant cross-section of the beam; index y : bending around the y-axis perpendicular to the x-axis)
{\ displaystyle I_ {y} = constant}
The curvature is proportional to the bending moment , which z. B. can be seen in the bending line shown on the left : bending moment u, curvature in the middle of the beam maximum and zero at the ends (radius of curvature minimal or infinitely large = straight beam end)
{\ displaystyle w ''}
{\ displaystyle M_ {y}}
{\ displaystyle w (x)}
The deflection of the bending line is determined by integrating the curve twice .
{\ displaystyle w (x)}
{\ displaystyle w '' (x)}
Bending moment and bending stress
Main article : Bending stress
The bending stresses to be determined for the strength verification in a beam cross-section are proportional to the bending moment acting there , as indicated in the following approximation equation for a beam with a constant cross-section:
{\ displaystyle \ sigma _ {x} (x, z)}
{\ displaystyle M_ {y} (x)}
{\ displaystyle \ sigma (x, z) = {\ frac {M_ {y} (x)} {I_ {y}}} \ cdot z}
(Variable in the direction of the bar , variable in the direction of the bar height).
{\ displaystyle x}
{\ displaystyle z}
The proportionality with the distance from the neutral beam layer indicates that the bending stress is greatest in the edge layers. The bending stress prevailing there is:
{\ displaystyle z}
{\ displaystyle \ sigma _ {\ mathrm {max}} (x) \, = {\ frac {M_ {y} (x)} {W_ {y}}}}
with ( resistance moment in the beam cross-section to bending about the y-axis).
{\ displaystyle W_ {y} = {\ frac {I_ {y}} {z _ {\ text {Rand}}}}}
↑ So-called "pure bend" (see here ), which rarely occurs. Most of the time, there is a “transverse force bending”: a force multiplied by a part of the length of the beam acts across the beam as a lever arm.
↑ The sign is ignored. Compressive and tensile stress are both the result of a bending moment.
↑ Alfred Böge (Ed.): Manual mechanical engineering: Basics and applications of mechanical engineering . 20th edition. Springer DE, 2011 ( limited preview in the Google book search).
↑ The observation leading from right to left leads to the same result with the help of the right reaction force F R via a left-turning moment.
This page is based on the copyrighted Wikipedia article "Biegemoment" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Coulomb Stress Interactions during the Mw 5.8 Pawnee Sequence | Seismological Research Letters | GeoScienceWorld
Coulomb Stress Interactions during the Mw 5.8 Pawnee Sequence
Colin Pennington;
Sarkeys Energy Center, The University of Oklahoma, 100 East Boyd Street, RM 710, Norman, Oklahoma 73019 U.S.A., colin.n.pennington@ou.edu
Colin Pennington, Xiaowei Chen; Coulomb Stress Interactions during the
Mw
5.8 Pawnee Sequence. Seismological Research Letters 2017;; 88 (4): 1024–1031. doi: https://doi.org/10.1785/0220170011
We investigate the stress interaction between the Watchorn, Labette, and Sooner Lake fault systems and the effect of precursory activities on the 3 September 2016
Mw
5.8 Pawnee earthquake. We obtain fault‐plane solutions for earthquakes with sufficient azimuthal coverage using the HASH algorithm, and then perform coulomb stress analysis on both seismogenic faults and individual nodal planes. Our results found that the three
Mw≥3.0
foreshocks exerted a cumulative coulomb stress change increase of 0.68–1.98 bars at the mainshock hypocenter and also promoted failure for most aftershocks within 2 km of the mainshock. The coulomb stress change of 5 bars exerted by the mainshock also promoted failure for most aftershocks within the conjugate fault system. The results suggest that earthquake interaction should be fully considered in hazard assessment for induced seismicity.
Watchorn Fault
Labette Fault
Sooner Lake Fault
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EUDML | Weak solutions for a viscous -Laplacian equation. EuDML | Weak solutions for a viscous -Laplacian equation.
Weak solutions for a viscous
p
Liu, Changchun. "Weak solutions for a viscous -Laplacian equation.." Electronic Journal of Differential Equations (EJDE) [electronic only] 2003 (2003): Paper No. 63, 11 p., electronic only-Paper No. 63, 11 p., electronic only. <http://eudml.org/doc/123485>.
author = {Liu, Changchun},
keywords = {Pseudoparabolic equations; existence; uniqueness; time-discrete method},
title = {Weak solutions for a viscous -Laplacian equation.},
TI - Weak solutions for a viscous -Laplacian equation.
KW - Pseudoparabolic equations; existence; uniqueness; time-discrete method
Pseudoparabolic equations, existence, uniqueness, time-discrete method
|
EUDML | Hochschild cohomology and moduli spaces of strongly homotopy associative algebras. EuDML | Hochschild cohomology and moduli spaces of strongly homotopy associative algebras.
Hochschild cohomology and moduli spaces of strongly homotopy associative algebras.
Lazarev, A.. "Hochschild cohomology and moduli spaces of strongly homotopy associative algebras.." Homology, Homotopy and Applications 5.1 (2003): 73-100. <http://eudml.org/doc/50598>.
@article{Lazarev2003,
author = {Lazarev, A.},
keywords = {Hochschild cohomology; -algebras; strongly homotopy associative algebras; -algebras},
title = {Hochschild cohomology and moduli spaces of strongly homotopy associative algebras.},
TI - Hochschild cohomology and moduli spaces of strongly homotopy associative algebras.
KW - Hochschild cohomology; -algebras; strongly homotopy associative algebras; -algebras
Hochschild cohomology,
{A}_{\infty }
-algebras, strongly homotopy associative algebras,
{A}_{\infty }
Articles by Lazarev
|
EUDML | Heegner points and -series of automorphic cusp forms of Drinfeld type. EuDML | Heegner points and -series of automorphic cusp forms of Drinfeld type.
Heegner points and
L
-series of automorphic cusp forms of Drinfeld type.
Tipp, Ulrich; Rück, Hans-Georg
Tipp, Ulrich, and Rück, Hans-Georg. "Heegner points and -series of automorphic cusp forms of Drinfeld type.." Documenta Mathematica 5 (2000): 365-444. <http://eudml.org/doc/122641>.
@article{Tipp2000,
author = {Tipp, Ulrich, Rück, Hans-Georg},
keywords = {Heegner points; Drinfeld modular curves; Gross-Zagier formula; derivatives of -series; function field analogue; harmonic functions; Bruhat-Tits tree; conjecture of Birch and Swinnerton-Dyer; derivatives of -series},
title = {Heegner points and -series of automorphic cusp forms of Drinfeld type.},
AU - Tipp, Ulrich
AU - Rück, Hans-Georg
TI - Heegner points and -series of automorphic cusp forms of Drinfeld type.
KW - Heegner points; Drinfeld modular curves; Gross-Zagier formula; derivatives of -series; function field analogue; harmonic functions; Bruhat-Tits tree; conjecture of Birch and Swinnerton-Dyer; derivatives of -series
Heegner points, Drinfeld modular curves, Gross-Zagier formula, derivatives of
L
-series, function field analogue, harmonic functions, Bruhat-Tits tree, conjecture of Birch and Swinnerton-Dyer, derivatives of
L
Modular forms associated to Drinfeld modules
L
Drinfeld modules; higher-dimensional motives, etc.
L
Articles by Tipp
Articles by Rück
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2020 A Variational Method for Multivalued Boundary Value Problems
Droh Arsène Béhi, Assohoun Adjé
In this paper, we investigate the existence of solution for differential systems involving a
\varphi -
Laplacian operator which incorporates as a special case the well-known
p-
Laplacian operator. In this purpose, we use a variational method which relies on Szulkin’s critical point theory. We obtain the existence of solution when the corresponding Euler–Lagrange functional is coercive.
Droh Arsène Béhi. Assohoun Adjé. "A Variational Method for Multivalued Boundary Value Problems." Abstr. Appl. Anal. 2020 1 - 8, 2020. https://doi.org/10.1155/2020/8463263
Received: 18 September 2019; Revised: 18 December 2019; Accepted: 27 December 2019; Published: 2020
Droh Arsène Béhi, Assohoun Adjé "A Variational Method for Multivalued Boundary Value Problems," Abstract and Applied Analysis, Abstr. Appl. Anal. 2020(none), 1-8, (2020)
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Systems | Free Full-Text | DSRP Theory: A Primer
Investigating Market Diffusion of Electric Vehicles with Experimental Design of Agent-Based Modeling Simulation
Optimal Asynchronous Dynamic Policies in Energy-Efficient Data Centers
DSRP Theory: A Primer
Jeb E. Brooks School of Public Policy, Cornell Institute for Public Affairs, SC Johnson College of Business, Cornell University, Ithaca, NY 14850, USA
Cabrera Research Lab, Ithaca, NY 14850, USA
(This article belongs to the Section Complex Systems)
DSRP Theory is now over 25 years old with more empirical evidence supporting it than any other systems thinking framework. Yet, it is often misunderstood and described in ways that are inaccurate. DSRP Theory describes four patterns and their underlying elements—identity (i) and other (o) for Distinctions (D), part (p) and whole (w) for Systems (S), action (a) and reaction (r) for Relationships (R), and point (
\rho
) and view (v) for Perspectives (P)—that are universal in both cognitive complexity (mind) and material complexity (nature). DSRP Theory provides a basis for systems thinking or cognitive complexity as well as material complexity (systems science). This paper, as a relatively short primer on the theory, provides clarity to those wanting to understand DSRP and its implications. View Full-Text
Keywords: DSRP theory; universals; cognitive complexity; systems thinking; complexity; systems science; structural predictions; organization of information DSRP theory; universals; cognitive complexity; systems thinking; complexity; systems science; structural predictions; organization of information
Cabrera, D.; Cabrera, L. DSRP Theory: A Primer. Systems 2022, 10, 26. https://doi.org/10.3390/systems10020026
Cabrera D, Cabrera L. DSRP Theory: A Primer. Systems. 2022; 10(2):26. https://doi.org/10.3390/systems10020026
Cabrera, Derek, and Laura Cabrera. 2022. "DSRP Theory: A Primer" Systems 10, no. 2: 26. https://doi.org/10.3390/systems10020026
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Correlation coefficients - MATLAB corrcoef - MathWorks Switzerland
\rho \left(A,B\right)=\frac{1}{N-1}\sum _{i=1}^{N}\left(\frac{{A}_{i}-{\mu }_{A}}{{\sigma }_{A}}\right)\left(\frac{{B}_{i}-{\mu }_{B}}{{\sigma }_{B}}\right),
{\mu }_{A}
{\sigma }_{A}
{\mu }_{B}
{\sigma }_{B}
\rho \left(A,B\right)=\frac{\mathrm{cov}\left(A,B\right)}{{\sigma }_{A}{\sigma }_{B}}.
R=\left(\begin{array}{cc}\rho \left(A,A\right)& \rho \left(A,B\right)\\ \rho \left(B,A\right)& \rho \left(B,B\right)\end{array}\right).
R=\left(\begin{array}{cc}1& \rho \left(A,B\right)\\ \rho \left(B,A\right)& 1\end{array}\right).
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Bessel function of first kind - MATLAB besselj - MathWorks India
J = besselj(nu,Z)
J = besselj(nu,Z,scale)
J = besselj(nu,Z) computes the Bessel function of the first kind Jν(z) for each element in array Z.
J = besselj(nu,Z,scale) specifies whether to exponentially scale the Bessel function of the first kind to avoid overflow or loss of accuracy. If scale is 1, then the output of besselj is scaled by the factor exp(-abs(imag(Z))).
Calculate the first five Bessel functions of the first kind. Each row of J contains the values of one order of the function evaluated at the points in z.
J = zeros(5,201);
J(i+1,:) = besselj(i,z);
plot(z,J)
legend('J_0','J_1','J_2','J_3','J_4','Location','Best')
title('Bessel Functions of the First Kind for $\nu \in [0, 4]$','interpreter','latex')
ylabel('$J_\nu(z)$','interpreter','latex')
Calculate the unscaled (J) and scaled (Js) Bessel function of the first kind
{\mathit{J}}_{2}\left(\mathit{z}\right)
\mathit{z}
J = besselj(2,z);
Js = besselj(2,z,scale);
Compare the plots of the imaginary part of the scaled and unscaled functions. For large values of abs(imag(z)), the unscaled function quickly overflows the limits of double precision and stops being computable. The scaled function removes this dominant exponential behavior from the calculation, and thus has a larger range of computability compared to the unscaled function.
surf(x,y,imag(J))
title('Bessel Function of the First Kind','interpreter','latex')
surf(x,y,imag(Js))
title('Scaled Bessel Function of the First Kind','interpreter','latex')
Equation order, specified as a scalar, vector, matrix, or multidimensional array. nu is a real number that specifies the order of the Bessel function of the first kind. nu and Z must be the same size, or one of them can be scalar.
Example: besselj(3,0:5)
Functional domain, specified as a scalar, vector, matrix, or multidimensional array. besselj is real-valued where Z is positive. nu and Z must be the same size, or one of them can be scalar.
Example: besselj(1,[1-1i 1+0i 1+1i])
1 — Scale the output of besselj by exp(-abs(imag(Z)))
On the complex plane, the magnitude of besselj grows rapidly as the value of abs(imag(Z)) increases, so exponentially scaling the output is useful for large values of abs(imag(Z)) where the results otherwise quickly lose accuracy or overflow the limits of double precision.
Example: besselj(3,0:5,1)
{z}^{2}\frac{{d}^{2}y}{d{z}^{2}}+z\frac{dy}{dz}+\left({z}^{2}-{\nu }^{2}\right)y=0.
{J}_{\nu }\left(z\right)={\left(\frac{z}{2}\right)}^{\nu }\sum _{\left(k=0\right)}^{\infty }\frac{{\left(\frac{-{z}^{2}}{4}\right)}^{k}}{k!\Gamma \left(\nu +k+1\right)}\text{\hspace{0.17em}}.
{Y}_{\nu }\left(z\right)=\frac{{J}_{\nu }\left(z\right)\mathrm{cos}\left(\nu \pi \right)-{J}_{-\nu }\left(z\right)}{\mathrm{sin}\left(\nu \pi \right)}\text{\hspace{0.17em}}.
You can calculate Bessel functions of the second kind using bessely.
\begin{array}{l}{H}_{\nu }^{\left(1\right)}\left(z\right)={J}_{\nu }\left(z\right)+i\text{\hspace{0.17em}}{Y}_{\nu }\left(z\right)\\ {H}_{\nu }^{\left(2\right)}\left(z\right)={J}_{\nu }\left(z\right)-i\text{\hspace{0.17em}}{Y}_{\nu }\left(z\right).\end{array}
{H}_{\nu }^{\left(K\right)}\left(z\right)
The argument Z must contain real values.
besselh | besseli | besselk | bessely
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We're doing very innovative stuff, only normal to have questions - Can't find yours answered? ask us on Telegram! Thank you to all participants to our Focus Groups - your input is precious to us.
Will you really redistribute 100% of platform revenue to Users?
Yes. Once Self-Decentralization is completed. It starts at 25% at Launch, and grows over time.
CryptoArena's Distribution Blockchain corroborates probabilistic finality ( = node consensus) with deterministic proof to achieve proven finality. ( = verified, irreversible entries on a distributed ledger ).
Mainly through the time component. In brief, our Security Tokens offer a trade-off:
Boosted rewards in the "short term" (defined as from Launch to the completion of Self-Decentralization) in exchange for everyone's consensus to Exit in the distant future at predetermined conditions, thus leaving behind a fully self-sustained, ownerless entity that'll keep operating as if nothing's changed, while distributing 100% of proceedings on an ongoing basis. Thus, fulfilling our mission & promise.
Will it remain sustainable in the long term?
Oh yeah, in fact it will likely grow faster than it ever could while carrying the "burden" of shareholders. The more volume gets distributed the more participation becomes desirable, and greater magnitudes of volume are generated.
Because we use the exchange's own cash flow to power an unprecedented, massive activity incentives program. Simply put, the more money's up for grabs, the more people will want a slice of the pie and be willing to put in more activity than they would otherwise - thus contributing to growing the amount distributed the next period.
Ask yourself: how much more profitable would trading be without predatory intermediaries?
Where will CryptoArena's services be available?
Everywhere, eventually. But we'll likely leave the U.S. for last due to the burden of compliance. More details to come in due time.
When will CryptoArena officially Launch?
It's a little early for announcements but check out the roadmap. CryptoArena is a regulated financial platform featuring high degrees of automation, we will need to be successfully audited, after building our solutions, before we can officially launch the platform. Stay tuned for updates (and for a bunch of smaller but still interesting projects we'll deliver along the way)!
Does CryptoArena require KYC?
Yes. For everything other than basic crypto-to-crypto pairs (not including security tokens). You need a KYC approved account to accrue Glory Points - which you need for revenue distributions.
Why would a Decentralized Exchange require KYC?
You'd be surprised how much more difficult it is to distribute money rather than just take it. Also, we're based in the Netherlands and pride ourselves on the transparency of our promises, rather than operating from a tax haven, and the services we offer require AFM oversight. CryptoArena only shares data as required by law, for compliance with taxation legislation and anti-money laundering purposes, and will never sell or share user data to other third parties.
About Patrons & Champions
Do Patrons also get Glory Points?
Yes. All Users paying any on-platform fee contribute to distributions & earn points, in different weights (trading fees > rest). In fact, both Patrons & Champions earn more points than "normal" Users, due to earning multipliers from cooperative trading.
Yes. Generally speaking, the market doesn't care who you are if you can deliver results. That being said, there is also a "pro" Champion category reserved to certified industry professionals.
Are Champions anonymous?
Pseudonymous. Not Anonymous. Means you use fake identity while acting publicly on the platform, but that the company knows who you are. We don't sell or share this information with anyone other than what is required by law.
Toggle real ID on/off
Select individuals, such as certified pros, influencers or other public figures who may wish to use their real identity can do so by manually toggling a dedicated option in their profile settings.
We generally advise against using your real id regardless of circumstances as a security precaution.
What prevents Champions from working against my interests?
Great question. There's many different ways a representative could various aspects to consider:
All patronages are non-custodial interactions This means they can't "run away" with your money, because it doesn't leave your wallet during a Patronage, it's just locked. It'll only move in case of an unsuccessful patronage resulting in a loss.
Success Fee Based The basic fees of Champions are strictly success fees. Simply put, if they don't make you money they don't get paid. Contracts can be customized into more complex versions which may or may not include additional fee structures, mutual agreement remains required.
Reputation & Metrics To become a Champion, Users must agree to let us collect and publicly display certain key information relating to their trading performance. Their "persona" may change, but not their numbers. If Champions "works to fail" their metrics will plummet, rendering it very difficult for them to acquire enough patrons to profitably continue operating as a Champion.
KYC Compliance This is why their persona may change, but not their numbers. Real ID is required, one per account.
How should I evaluate/pick Champions?
You can set/negotiate your own targets & conditions for each Patronage, so its mostly up to you and what you would consider appropriate. Keep in mind, both parties need to agree on a set of terms & conditions, then confirm them, for a patronage to be officialized.
There are 3 key metrics that Champions work to maximize, and by our systems rank Champions for the purpose of Leaderboards, by which Patrons may evaluate them:
Glory Points - are a weighted indicator of individual network contributions. As a rule of thumb, more points = better. Champions with more points are either more active, more successful or deal in larger volumes - or a combination thereof. It is a valuable indicator, but not very specific.
PPR - stands for Positive Patronage Rate. As is (hopefully) intuitive, this is the ratio, expressed in percentages, at which a Champion has succeeded in deliver the minimum target of the Patronages he has accepted.
PPR = (Successful /Accepted)*100
RoR - stands for Rate of Return (on investment). This is the gain (or loss) compared to the cost of an initial investment, typically expressed in the form of a percentage. When the ROR is positive, it is considered a gain and when the ROR is negative, it reflects a loss on the investment.
What is X? What does Y mean?
Check out Foundational Concepts for info.
Ask us more questions !
.We're (almost) always looking for topics of interests for our community for us to write about.
Next - Self-Decentralization
|
Mach number - zxc.wiki
Physical key figure
Surname Mach number
{\ displaystyle {\ mathit {Ma}}}
dimension dimensionless
{\ displaystyle {\ mathit {Ma}} = {\ frac {v} {c}}}
{\ displaystyle v}
{\ displaystyle c}
Named after Seriously do
scope of application compressible flows
The Mach number (also Mach number, Mach number or Mach number, symbol :) is a dimensionless number of fluid dynamics for speeds . It indicates the ratio of the speed (e.g. of a body or a fluid ) to the speed of sound of the surrounding fluid. It is named after the Austrian physicist and philosopher Ernst Mach . The designation Mach number was introduced in 1929 by the Swiss aerodynamicist Jakob Ackeret . Mach describes the ratio of the speed of a body to the speed of sound. If an aircraft is as fast as sound, it is traveling at Mach 1.
{\ displaystyle {\ mathit {Ma}}}
{\ displaystyle v}
{\ displaystyle c}
{\ displaystyle {\ mathit {Ma}} = {\ frac {v} {c}}}
Whereby one can use the speed of sound in gases in general, which leads to the following expression:
{\ displaystyle c}
{\ displaystyle {\ mathit {Ma}} = {\ frac {v} {\ sqrt {\ kappa \, R _ {\ mathrm {S}} \, T}}}}
{\ displaystyle \ kappa}
the isentropic exponent of the fluid under the given boundary conditions,
{\ displaystyle R _ {\ mathrm {S}} = {\ frac {R} {M}}}
the specific gas constant related to the molar mass and
{\ displaystyle M}
{\ displaystyle R}
{\ displaystyle T}
the temperature of the gas under consideration.
In general, the isentropic exponent also varies for a specific fluid as a function of pressure and temperature . For sufficiently small pressure and temperature changes, it can be approximated as constant.
{\ displaystyle \ kappa}
{\ displaystyle p}
{\ displaystyle T}
Mach number = 1, also known colloquially as “Mach 1”, is understood to be the speed of sound (which, to a good approximation, only depends on the temperature for a certain medium). Correspondingly, “Mach 2” (twice the speed of sound), Mach 3 etc. cannot be converted into “exact” speeds without knowing the reference sound speed.
Using the Mach number, however, flows can be divided into different areas, for example:
{\ displaystyle {\ mathit {Ma}} <0 {,} 8}
subsonic flow,
{\ displaystyle 0 {,} 8 <{\ mathit {Ma}} <1 {,} 2}
transonic flow,
{\ displaystyle {\ mathit {Ma}}> 1 {,} 2}
supersonic flow.
From now on one speaks of hypersonic flow.
{\ displaystyle {\ mathit {Ma}}> 5}
These areas require different approaches, since different physical phenomena occur for each area. For example, compressible effects occur in the flows (compressible flow), while such effects usually do not play a role for them (incompressible flow).
{\ displaystyle {\ mathit {Ma}}> 0 {,} 3}
{\ displaystyle {\ mathit {Ma}} <0 {,} 3}
A McDonnell Douglas F / A-18 Hornet in supersonic flight ; the collision front of the Mach cone is visible as a cloud disk
Standard atmosphere, the values of the speed of sound in the penultimate column are given in knots (unit) . Conversion: 1 knot = 1.852 km / h ≈ 0.514444 m / s
In aviation , the Mach number is used for the dimensionless indication of the airspeed of fast-flying aircraft . It represents the ratio of the airspeed to the speed of sound in the ambient air . Since the speed of sound is primarily dependent on the air temperature , and this in turn depends on the altitude , the display of the Mach number is the only one that is comparable at any cruising altitude and at any ambient temperature Statement. This is particularly important in commercial aircraft to maintain the maximum speed (M MO , Mach Maximum Operating Number ) specified by the aircraft manufacturer with regard to the actual airspeed (TAS, true airspeed) relative to the ambient air . Exceeding the M MO leads to the critical Mach number being reached and thus to boundary layer detachment as the cause of flow breaks and an associated risk of falling, as well as sudden extreme mechanical loads on the aircraft structure. The Mach number is displayed by a special flight instrument, the Machmeter .
Speed of sound in air as a function of temperature
−50 ° C 1080 km / h ≈ 300 m / s
00 ° C 1193 km / h ≈ 331 m / s
At a temperature of −50 ° C and an air pressure of 26 kPa (according to the standard atmosphere, usually at an altitude of approx. 10,000 m), the speed of sound is around 300 m / s = 1080 km / h. A passenger aircraft that flies at a cruising speed of Mach 0.8 under these conditions has a speed of 240 m / s = 864 km / h.
Laval number
Cauchy number (similar to the Laval number in solids)
Ernst Götsch: Aircraft technology . Introduction, basics, aircraft science. Motorbuch-Verlag, Stuttgart 2003, ISBN 3-613-02006-8 .
Michael Grossrubatscher: Pilots reference guide . 7th, revised edition. Self-published by the author, Munich 2008, ISBN 978-3-00-025252-5 (English).
N. Rott: Jakob Ackert and the History of the Mach Number. Annual Review of Fluid Mechanics 17 (1985), pp. 1-9.
N. Rott: J. Ackeret and the history of the Mach number. Swiss engineer and architect 21 (1983), pp. 591–594.
Mach number calculator - Java (Engl.)
Calculator for the speed of sound and Mach number
The speed of sound and the important temperature
Mach Number NASA Glenn Research Center (English)
↑ Jakob Ackeret: The air resistance at very high speeds. Schweizerische Bauzeitung 94 (October 1929), pp. 179–183. See also: N. Rott: Jakob Ackert and the History of the Mach Number. Annual Review of Fluid Mechanics 17 (1985), pp. 1-9; N. Rott: J. Ackeret and the history of the Mach number. Swiss engineer and architect 21 (1983), pp. 591–594.
This page is based on the copyrighted Wikipedia article "Mach-Zahl" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Minkowski addition - Wikipedia
Sums vector sets A and B by adding each vector in A to each vector in B
The red figure is the Minkowski sum of blue and green figures.
In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B, i.e., the set
{\displaystyle A+B=\{\mathbf {a} +\mathbf {b} \,|\,\mathbf {a} \in A,\ \mathbf {b} \in B\}.}
Analogously, the Minkowski difference (or geometric difference)[1] is defined using the complement operation as
{\displaystyle A-B=\left(A^{c}+(-B)\right)^{c}}
{\displaystyle A-B\neq A+(-B)}
. For instance, in a one-dimensional case
{\displaystyle A=[-2,2]}
{\displaystyle B=[-1,1]}
the Minkowski difference
{\displaystyle A-B=[-1,1]}
{\displaystyle A+(-B)=A+B=[-3,3].}
In a two-dimensional case, Minkowski difference is closely related to erosion (morphology) in image processing.
The concept is named for Hermann Minkowski.
2 Convex hulls of Minkowski sums
3.2 Numerical control (NC) machining
3.3 3D solid modeling
3.4 Aggregation theory
4 Algorithms for computing Minkowski sums
4.1 Planar case
4.1.1 Two convex polygons in the plane
5 Essential Minkowski sum
6 Lp Minkowski sum
Minkowski addition of sets. The sum of the squares
{\displaystyle Q_{1}=[0,1]^{2}}
{\displaystyle Q_{2}=[1,2]^{2}}
is the square
{\displaystyle Q_{1}+Q_{2}=[1,3]^{2}.}
Minkowski sum A + B
For example, if we have two sets A and B, each consisting of three position vectors (informally, three points), representing the vertices of two triangles in
{\displaystyle \mathbb {R} ^{2}}
, with coordinates
{\displaystyle A=\{(1,0),(0,1),(0,-1)\}}
{\displaystyle B=\{(0,0),(1,1),(1,-1)\}}
then their Minkowski sum is
{\displaystyle A+B=\{(1,0),(2,1),(2,-1),(0,1),(1,2),(1,0),(0,-1),(1,0),(1,-2)\}}
which comprises the vertices of a hexagon.
For Minkowski addition, the zero set,
{\displaystyle \{0\},}
containing only the zero vector, 0, is an identity element: for every subset S of a vector space,
{\displaystyle S+\{0\}=S.}
The empty set is important in Minkowski addition, because the empty set annihilates every other subset: for every subset S of a vector space, its sum with the empty set is empty:
{\displaystyle S+\emptyset =\emptyset .}
For another example, consider the Minkowski sums of open or closed balls in the field
{\displaystyle \mathbb {K} ,}
which is either the real numbers
{\displaystyle \mathbb {R} }
or complex numbers
{\displaystyle \mathbb {C} .}
{\displaystyle B_{r}:=\{s\in \mathbb {K} :|s|\leq r\}}
{\displaystyle r\in [0,\infty ]}
{\displaystyle 0}
{\displaystyle \mathbb {K} }
{\displaystyle r,s\in [0,\infty ],}
{\displaystyle B_{r}+B_{s}=B_{r+s}}
{\displaystyle cB_{r}=B_{|c|r}}
will hold for any scalar
{\displaystyle c\in \mathbb {K} }
{\displaystyle |c|r}
is defined (which happens when
{\displaystyle c\neq 0}
{\displaystyle r\neq \infty }
{\displaystyle r,s,}
{\displaystyle c}
are all non-zero then the same equalities would still hold had
{\displaystyle B_{r}}
been defined to be the open ball, rather than the closed ball, centered at
{\displaystyle 0}
(the non-zero assumption is needed because the open ball of radius
{\displaystyle 0}
is the empty set). The Minkowski sum of a closed ball and an open ball is an open ball. More generally, the Minkowski sum of an open subset with any other set will be an open subset.
{\displaystyle G=\left\{\left(x,1/x\right):0\neq x\in \mathbb {R} \right\}}
{\displaystyle f(x)={\frac {1}{x}}}
and if and
{\displaystyle Y=\{0\}\times \mathbb {R} }
{\displaystyle y}
{\displaystyle X=\mathbb {R} ^{2}}
then the Minkowski sum of these two closed subsets of the plane is the open set
{\displaystyle G+Y=\{(x,y)\in \mathbb {R} ^{2}:x\neq 0\}=\mathbb {R} ^{2}\setminus Y}
consisting of everything other than the
{\displaystyle y}
-axis. This shows that the Minkowski sum of two closed sets is not necessarily a closed set. However, the Minkowski sum of two closed subsets will be a closed subset if at least one of these sets is also a compact subset.
Convex hulls of Minkowski sums[edit]
Minkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition:
For all non-empty subsets
{\displaystyle S_{1}}
{\displaystyle S_{2}}
of a real vector space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls:
{\displaystyle \operatorname {Conv} (S_{1}+S_{2})=\operatorname {Conv} (S_{1})+\operatorname {Conv} (S_{2}).}
This result holds more generally for any finite collection of non-empty sets:
{\displaystyle \operatorname {Conv} \left(\sum {S_{n}}\right)=\sum \operatorname {Conv} (S_{n}).}
In mathematical terminology, the operations of Minkowski summation and of forming convex hulls are commuting operations.[2][3]
{\displaystyle S}
is a convex set then
{\displaystyle \mu S+\lambda S}
is also a convex set; furthermore
{\displaystyle \mu S+\lambda S=(\mu +\lambda )S}
{\displaystyle \mu ,\lambda \geq 0}
. Conversely, if this "distributive property" holds for all non-negative real numbers,
{\displaystyle \mu ,\lambda }
, then the set is convex.[4]
An example of a non-convex set such that
{\displaystyle A+A\neq 2A.}
The figure to the right shows an example of a non-convex set for which
{\displaystyle A+A\supsetneq 2A.}
An example in
{\displaystyle 1}
dimension is:
{\displaystyle B=[1,2]\cup [4,5].}
It can be easily calculated that
{\displaystyle 2B=[2,4]\cup [8,10]}
{\displaystyle B+B=[2,4]\cup [5,7]\cup [8,10],}
hence again
{\displaystyle B+B\supsetneq 2B.}
Minkowski sums act linearly on the perimeter of two-dimensional convex bodies: the perimeter of the sum equals the sum of perimeters. Additionally, if
{\displaystyle K}
is (the interior of) a curve of constant width, then the Minkowski sum of
{\displaystyle K}
{\displaystyle 180^{\circ }}
rotation is a disk. These two facts can be combined to give a short proof of Barbier's theorem on the perimeter of curves of constant width.[5]
Minkowski addition plays a central role in mathematical morphology. It arises in the brush-and-stroke paradigm of 2D computer graphics (with various uses, notably by Donald E. Knuth in Metafont), and as the solid sweep operation of 3D computer graphics. It has also been shown to be closely connected to the Earth mover's distance, and by extension, optimal transport.[6]
Motion planning[edit]
Minkowski sums are used in motion planning of an object among obstacles. They are used for the computation of the configuration space, which is the set of all admissible positions of the object. In the simple model of translational motion of an object in the plane, where the position of an object may be uniquely specified by the position of a fixed point of this object, the configuration space are the Minkowski sum of the set of obstacles and the movable object placed at the origin and rotated 180 degrees.
Numerical control (NC) machining[edit]
In numerical control machining, the programming of the NC tool exploits the fact that the Minkowski sum of the cutting piece with its trajectory gives the shape of the cut in the material.
3D solid modeling[edit]
In OpenSCAD Minkowski sums are used to outline a shape with another shape creating a composite of both shapes.
Aggregation theory[edit]
Minkowski sums are also frequently used in aggregation theory when individual objects to be aggregated are characterized via sets.[7][8]
Collision detection[edit]
Minkowski sums, specifically Minkowski differences, are often used alongside GJK algorithms to compute collision detection for convex hulls in physics engines.
Algorithms for computing Minkowski sums[edit]
Minkowski addition and convex hulls. The sixteen dark-red points (on the right) form the Minkowski sum of the four non-convex sets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus-signs (+): The right plus-sign is the sum of the left plus-signs.
Planar case[edit]
Two convex polygons in the plane[edit]
For two convex polygons P and Q in the plane with m and n vertices, their Minkowski sum is a convex polygon with at most m + n vertices and may be computed in time O(m + n) by a very simple procedure, which may be informally described as follows. Assume that the edges of a polygon are given and the direction, say, counterclockwise, along the polygon boundary. Then it is easily seen that these edges of the convex polygon are ordered by polar angle. Let us merge the ordered sequences of the directed edges from P and Q into a single ordered sequence S. Imagine that these edges are solid arrows which can be moved freely while keeping them parallel to their original direction. Assemble these arrows in the order of the sequence S by attaching the tail of the next arrow to the head of the previous arrow. It turns out that the resulting polygonal chain will in fact be a convex polygon which is the Minkowski sum of P and Q.
If one polygon is convex and another one is not, the complexity of their Minkowski sum is O(nm). If both of them are nonconvex, their Minkowski sum complexity is O((mn)2).
Essential Minkowski sum[edit]
There is also a notion of the essential Minkowski sum +e of two subsets of Euclidean space. The usual Minkowski sum can be written as
{\displaystyle A+B=\left\{z\in \mathbb {R} ^{n}\,|\,A\cap (z-B)\neq \emptyset \right\}.}
Thus, the essential Minkowski sum is defined by
{\displaystyle A+_{\mathrm {e} }B=\left\{z\in \mathbb {R} ^{n}\,|\,\mu \left[A\cap (z-B)\right]>0\right\},}
where μ denotes the n-dimensional Lebesgue measure. The reason for the term "essential" is the following property of indicator functions: while
{\displaystyle 1_{A\,+\,B}(z)=\sup _{x\,\in \,\mathbb {R} ^{n}}1_{A}(x)1_{B}(z-x),}
{\displaystyle 1_{A\,+_{\mathrm {e} }\,B}(z)=\mathop {\mathrm {ess\,sup} } _{x\,\in \,\mathbb {R} ^{n}}1_{A}(x)1_{B}(z-x),}
where "ess sup" denotes the essential supremum.
Lp Minkowski sum[edit]
For K and L compact convex subsets in
{\displaystyle \mathbb {R} ^{n}}
, the Minkowski sum can be described by the support function of the convex sets:
{\displaystyle h_{K+L}=h_{K}+h_{L}.}
For p ≥ 1, Firey[9] defined the Lp Minkowski sum K +p L of compact convex sets K and L in
{\displaystyle \mathbb {R} ^{n}}
containing the origin as
{\displaystyle h_{K+_{p}L}^{p}=h_{K}^{p}+h_{L}^{p}.}
By the Minkowski inequality, the function hK+pL is again positive homogeneous and convex and hence the support function of a compact convex set. This definition is fundamental in the Lp Brunn-Minkowski theory.
Blaschke sum
Brunn–Minkowski theorem, an inequality on the volumes of Minkowksi sums
Mixed volume (a.k.a. Quermassintegral or intrinsic volume)
Topological vector space#Properties
^ Hadwiger, Hugo (1950), "Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt", Math. Z., 53 (3): 210–218, doi:10.1007/BF01175656
^ Theorem 3 (pages 562–563): Krein, M.; Šmulian, V. (1940). "On regularly convex sets in the space conjugate to a Banach space". Annals of Mathematics. Second Series. Vol. 41. pp. 556–583. doi:10.2307/1968735. JSTOR 1968735. MR 0002009.
^ For the commutativity of Minkowski addition and convexification, see Theorem 1.1.2 (pages 2–3) in Schneider; this reference discusses much of the literature on the convex hulls of Minkowski sumsets in its "Chapter 3 Minkowski addition" (pages 126–196): Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of mathematics and its applications. Vol. 44. Cambridge: Cambridge University Press. pp. xiv+490. ISBN 978-0-521-35220-8. MR 1216521.
^ Chapter 1: Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of mathematics and its applications. Vol. 44. Cambridge: Cambridge University Press. pp. xiv+490. ISBN 978-0-521-35220-8. MR 1216521.
^ The Theorem of Barbier (Java) at cut-the-knot.
^ Zelenyuk, V (2015). "Aggregation of scale efficiency". European Journal of Operational Research. 240 (1): 269–277. doi:10.1016/j.ejor.2014.06.038.
^ Mayer, A.; Zelenyuk, V. (2014). "Aggregation of Malmquist productivity indexes allowing for reallocation of resources". European Journal of Operational Research. 238 (3): 774–785. doi:10.1016/j.ejor.2014.04.003.
^ Firey, William J. (1962), "p-means of convex bodies", Math. Scand., 10: 17–24, doi:10.7146/math.scand.a-10510
Arrow, Kenneth J.; Hahn, Frank H. (1980). General competitive analysis. Advanced textbooks in economics. Vol. 12 (reprint of (1971) San Francisco, CA: Holden-Day, Inc. Mathematical economics texts. 6 ed.). Amsterdam: North-Holland. ISBN 978-0-444-85497-1. MR 0439057.
Gardner, Richard J. (2002), "The Brunn-Minkowski inequality", Bull. Amer. Math. Soc. (N.S.), 39 (3): 355–405 (electronic), doi:10.1090/S0273-0979-02-00941-2
Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In Arrow, Kenneth Joseph; Intriligator, Michael D (eds.). Handbook of mathematical economics, Volume I. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 978-0-444-86126-9. MR 0634800.
Henry Mann (1976), Addition Theorems: The Addition Theorems of Group Theory and Number Theory (Corrected reprint of 1965 Wiley ed.), Huntington, New York: Robert E. Krieger Publishing Company, ISBN 978-0-88275-418-5 – via http://www.krieger-publishing.com/subcats/MathematicsandStatistics/mathematicsandstatistics.html {{citation}}: External link in |via= (help)
Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton landmarks in mathematics (Reprint of the 1979 Princeton mathematical series 28 ed.). Princeton, NJ: Princeton University Press. pp. xviii+451. ISBN 978-0-691-01586-6. MR 1451876.
Nathanson, Melvyn B. (1996), Additive Number Theory: Inverse Problems and Geometry of Sumsets, GTM, vol. 165, Springer, Zbl 0859.11003 .
Oks, Eduard; Sharir, Micha (2006), "Minkowski Sums of Monotone and General Simple Polygons", Discrete and Computational Geometry, 35 (2): 223–240, doi:10.1007/s00454-005-1206-y .
Schneider, Rolf (1993), Convex bodies: the Brunn-Minkowski theory, Cambridge: Cambridge University Press .
Tao, Terence & Vu, Van (2006), Additive Combinatorics, Cambridge University Press .
Mayer, A.; Zelenyuk, V. (2014). "Aggregation of Malmquist productivity indexes allowing for reallocation of resources". European Journal of Operational Research. 238 (3): 774–785. doi:10.1016/j.ejor.2014.04.003.
Zelenyuk, V (2015). "Aggregation of scale efficiency". European Journal of Operational Research. 240 (1): 269–277. doi:10.1016/j.ejor.2014.06.038.
"Minkowski addition", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Howe, Roger (1979), On the tendency toward convexity of the vector sum of sets, Cowles Foundation discussion papers, vol. 538, Cowles Foundation for Research in Economics, Yale University
Minkowski Sums, in Computational Geometry Algorithms Library
The Minkowski Sum of Two Triangles and The Minkowski Sum of a Disk and a Polygon by George Beck, The Wolfram Demonstrations Project.
Minkowski's addition of convex shapes by Alexander Bogomolny: an applet
Wikibooks:OpenSCAD User Manual/Transformations#minkowski by Marius Kintel: Application
Application of Minkowski Addition to robotics by Joan Gerard
Retrieved from "https://en.wikipedia.org/w/index.php?title=Minkowski_addition&oldid=1053681553"
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(Redirected from Socialist millionaires)
Cryptographic problem
In cryptography, the socialist millionaire problem[1] is one in which two millionaires want to determine if their wealth is equal without disclosing any information about their riches to each other. It is a variant of the Millionaire's Problem[2][3] whereby two millionaires wish to compare their riches to determine who has the most wealth without disclosing any information about their riches to each other.
It is often used as a cryptographic protocol that allows two parties to verify the identity of the remote party through the use of a shared secret, avoiding a man-in-the-middle attack without the inconvenience of manually comparing public key fingerprints through an outside channel. In effect, a relatively weak password/passphrase in natural language can be used.
2 Off-the-Record Messaging protocol
Alice and Bob have secret values
{\displaystyle x}
{\displaystyle y}
, respectively. Alice and Bob wish to learn if
{\displaystyle x=y}
without allowing either party to learn anything else about the other's secret value.
A passive attacker simply spying on the messages Alice and Bob exchange learns nothing about
{\displaystyle x}
{\displaystyle y}
, not even whether
{\displaystyle x=y}
Even if one of the parties is dishonest and deviates from the protocol, that person cannot learn anything more than if
{\displaystyle x=y}
An active attacker capable of arbitrarily interfering with Alice and Bob's communication (a man-in-the-middle) cannot learn more than a passive attacker and cannot affect the outcome of the protocol other than to make it fail.
Therefore, the protocol can be used to authenticate whether two parties have the same secret information. Popular instant message cryptography package Off-the-Record Messaging uses the Socialist Millionaire protocol for authentication, in which the secrets
{\displaystyle x}
{\displaystyle y}
contain information about both parties' long-term authentication public keys as well as information entered by the users themselves.
Off-the-Record Messaging protocol[edit]
Main article: Off-the-Record Messaging
State machine of a socialist millionaire protocol (SMP) implementation.
The protocol is based on group theory.
A group of prime order
{\displaystyle p}
{\displaystyle h}
are agreed upon a priori, and in practice are generally fixed in a given implementation. For example, in the Off-the-Record Messaging protocol,
{\displaystyle p}
is a specific fixed 1,536-bit prime.
{\displaystyle h}
is then a generator of a prime-order subgroup of
{\displaystyle (\mathbb {Z} /p\mathbb {Z} )^{*}}
, and all operations are performed modulo
{\displaystyle p}
, or in other words, in a subgroup of the multiplicative group,
{\displaystyle (\mathbb {Z} /p\mathbb {Z} )^{*}}
{\displaystyle \langle h|a,\,b\rangle }
, denote the secure multiparty computation, Diffie–Hellman–Merkle key exchange, which, for the integers,
{\displaystyle a}
{\displaystyle b}
{\displaystyle h^{ab}}
to each party:
Alice calculates
{\displaystyle h^{a}}
and sends it to Bob, who then calculates
{\displaystyle \left(h^{a}\right)^{b}\equiv h^{ab}}
{\displaystyle h^{b}}
and sends it to Alice, who then calculates
{\displaystyle \left(h^{b}\right)^{a}\equiv h^{ba}}
{\displaystyle h^{ab}\equiv h^{ba}}
as multiplication in
{\displaystyle (\mathbb {Z} /p\mathbb {Z} )^{*}}
is associative. Note that this procedure is insecure against man-in-the-middle attacks.
The socialist millionaire protocol[4] only has a few steps that are not part of the above procedure, and the security of each relies on the difficulty of the discrete logarithm problem, just as the above does. All sent values also include zero-knowledge proofs that they were generated according to protocol.
Part of the security also relies on random secrets. However, as written below, the protocol is vulnerable to poisoning if Alice or Bob chooses any of
{\displaystyle a}
{\displaystyle b}
{\displaystyle \alpha }
{\displaystyle \beta }
to be zero. To solve this problem, each party must check during the Diffie-Hellman exchanges that none of the
{\displaystyle h^{a}}
{\displaystyle h^{b}}
{\displaystyle h^{\alpha }}
{\displaystyle h^{\beta }}
that they receive is equal to 1. It is also necessary to check that
{\displaystyle P_{a}\neq P_{b}}
{\displaystyle Q_{a}\neq Q_{b}}
{\displaystyle x}
{\displaystyle a,\alpha ,r}
{\displaystyle p,h}
{\displaystyle y}
{\displaystyle b,\beta ,s}
2 Secure
{\displaystyle g=\langle h|a,b\rangle }
{\displaystyle \gamma =\langle h|\alpha ,\beta \rangle }
{\displaystyle h^{b}\neq 1}
{\displaystyle h^{\beta }\neq 1}
{\displaystyle h^{a}\neq 1}
{\displaystyle h^{\alpha }\neq 1}
{\displaystyle {\begin{aligned}P_{a}&=\gamma ^{r}\\Q_{a}&=h^{r}g^{x}\end{aligned}}}
{\displaystyle {\begin{aligned}P_{b}&=\gamma ^{s}\\Q_{b}&=h^{s}g^{y}\end{aligned}}}
6 Insecure exchange
{\displaystyle P_{a},Q_{a},P_{b},Q_{b}}
{\displaystyle c=\left\langle \left.Q_{a}Q_{b}^{-1}\right|\alpha ,\beta \right\rangle }
{\displaystyle P_{a}\neq P_{b}}
{\displaystyle Q_{a}\neq Q_{b}}
{\displaystyle P_{a}\neq P_{b}}
{\displaystyle Q_{a}\neq Q_{b}}
{\displaystyle c=P_{a}{P_{b}}^{-1}}
{\displaystyle c=P_{a}{P_{b}}^{-1}}
{\displaystyle {\begin{aligned}P_{a}{P_{b}}^{-1}&=\gamma ^{r}\gamma ^{-s}=\gamma ^{r-s}\\&=h^{\alpha \beta (r-s)}\end{aligned}}}
{\displaystyle {\begin{aligned}c&=\left(Q_{a}Q_{b}^{-1}\right)^{\alpha \beta }\\&=\left(h^{r}g^{x}h^{-s}g^{-y}\right)^{\alpha \beta }=\left(h^{r-s}g^{x-y}\right)^{\alpha \beta }\\&=\left(h^{r-s}h^{ab(x-y)}\right)^{\alpha \beta }=h^{\alpha \beta (r-s)}h^{\alpha \beta ab(x-y)}\\&=\left(P_{a}{P_{b}}^{-1}\right)h^{\alpha \beta ab(x-y)}\end{aligned}}}
Because of the random values stored in secret by the other party, neither party can force
{\displaystyle c}
{\displaystyle P_{a}{P_{b}}^{-1}}
to be equal unless
{\displaystyle x}
{\displaystyle y}
{\displaystyle h^{\alpha \beta ab(x-y)}=h^{0}=1}
. This proves correctness.
^ Markus Jakobsson, Moti Yung (1996). "Proving without knowing: On oblivious, agnostic and blindfolded provers.". Advances in Cryptology - CRYPTO '96, volume 1109 of Lecture Notes in Computer Science. Berlin. pp. 186–200. doi:10.1007/3-540-68697-5_15.
^ Andrew Yao (1982). "Protocols for secure communications" (PDF). Proc. 23rd IEEE Symposium on Foundations of Computer Science (FOCS '82). pp. 160–164. doi:10.1109/SFCS.1982.88.
^ Andrew Yao (1986). "How to generate and exchange secrets" (PDF). Proc. 27th IEEE Symposium on Foundations of Computer Science (FOCS '86). pp. 162–167. doi:10.1109/SFCS.1986.25.
^ Fabrice Boudot, Berry Schoenmakers, Jacques Traoré (2001). "A Fair and Efficient Solution to the Socialist Millionaires' Problem" (PDF). Discrete Applied Mathematics. 111 (1): 23–36. doi:10.1016/S0166-218X(00)00342-5. {{cite journal}}: CS1 maint: multiple names: authors list (link)
Description of the OTR-Messaging Protocol version 2
The Socialist Millionaire Problem - Explain it like I'm Five
Goldbug Messenger, which uses an implementation the Socialist Millionaire Protocol
Retrieved from "https://en.wikipedia.org/w/index.php?title=Socialist_millionaire_problem&oldid=1000084733"
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Mathematics | Free Full-Text | Surface Diffusion by Means of Stochastic Wave Functions. The Ballistic Regime
A Longitudinal Study of the Bladder Cancer Applying a State-Space Model with Non-Exponential Staying Time in States
Microscopically Reversible Pathways with Memory
Torres-Miyares, E. E.
Rojas-Lorenzo, G.
Rubayo-Soneira, J.
E. E. Torres-Miyares
G. Rojas-Lorenzo
J. Rubayo-Soneira
S. Miret-Artés
Facultad de Física, Universidad de La Habana, San Lázaro y L, Vedado, La Habana 10400, Cuba
Instituto Superior de Tecnologías y Ciencias Aplicadas (InSTEC), Universidad de La Habana, Avenida Salvador Allende No. 1110, Entre Boyeros e Infanta, Plaza, La Habana 10400, Cuba
Instituto de Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain
Academic Editors: Pedro Fernández de Córdoba, Juan Carlos Castro and Miguel Ángel García March
(This article belongs to the Special Issue On Interdisciplinary Modelling and Numerical Simulation in the Realm of Physics & Engineering)
Stochastic wave function formalism is briefly introduced and applied to study the dynamics of open quantum systems; in particular, the diffusion of Xe atoms adsorbed on a Pt(111) surface. By starting from a Lindblad functional and within the microscopic Caldeira–Leggett model for linear dissipation, a stochastic differential equation (It
\stackrel{^}{o}
-type differential equation) is straightforwardly obtained. The so-called intermediate scattering function within the ballistic regime is obtained, which is observable in Helium spin echo experiments. An ideal two-dimensional gas has been observed in this regime, leading to this function behaving as a Gaussian function. The influence of surface–adsorbate interaction is also analyzed by using the potential of two interactions describing flat and corrugated surfaces. Very low surface coverages are considered and, therefore, the adsorbate–adsorbate interaction is safely neglected. Good agreement is observed when our numerical results are compared with the corresponding experimental results and previous standard Langevin simulations. View Full-Text
Keywords: LINDBLAD approach; Caldeira–Leggett master equation; stochastic differential equation; stocastic wave functions; intermediate scattering function; ballistic regime LINDBLAD approach; Caldeira–Leggett master equation; stochastic differential equation; stocastic wave functions; intermediate scattering function; ballistic regime
Torres-Miyares, E.E.; Rojas-Lorenzo, G.; Rubayo-Soneira, J.; Miret-Artés, S. Surface Diffusion by Means of Stochastic Wave Functions. The Ballistic Regime. Mathematics 2021, 9, 362. https://doi.org/10.3390/math9040362
Torres-Miyares EE, Rojas-Lorenzo G, Rubayo-Soneira J, Miret-Artés S. Surface Diffusion by Means of Stochastic Wave Functions. The Ballistic Regime. Mathematics. 2021; 9(4):362. https://doi.org/10.3390/math9040362
Torres-Miyares, E. E., G. Rojas-Lorenzo, J. Rubayo-Soneira, and S. Miret-Artés. 2021. "Surface Diffusion by Means of Stochastic Wave Functions. The Ballistic Regime" Mathematics 9, no. 4: 362. https://doi.org/10.3390/math9040362
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FAQs - Open Targets Genetics Documentation
What genome build are data in the Genetics Portal based on?
All data are based on GRCh38 from the Genome Reference Consortium.
How is the Genetics Portal related to the Open Targets Platform?
The Genetics Portal is a variant-centric resource that complements the Open Targets Platform. Users can navigate to the Open Targets Platform from Open Targets Genetics and obtain for example drug information in clinical trials (or already marketed) for any target of interest.
The Open Targets Genetics Portal is also one of data sources that provide evidence for target-disease associations available in the Open Targets Platform.
How do I cite discoveries made using Open Targets Genetics?
Please cite our latest paper Open Targets Platform: new developments and updates two years on.
How can I stay informed about new features and developments?
For updates, please subscribe to our newsletter. Questions and feedback can be directed to us via email.
Why are there two variants mapped to the same rsID?
Many multi-allelic sites can be assigned a single rsID, and some rsIDs can point to different positions in the genome. This means that rsIDs are not unique to a single variant. We have mapped all rsIDs from GWAS Catalog to unique variants. A small minority of rsIDs will map to multiple variant IDs (approximately 0.6% of lead variants). When this occurs, variants will be duplicated in the portal.
What is the difference between lead variant and tag variant?
Lead variant is the variant at a given associated locus with the most significant (smallest) p-value whereas Tag variant is the variant that is correlated with the lead variant (r2>0.7) or present in the credible set at a GWAS-associated signal.
What is the effect allele?
The effect allele is the allele whose effects in relation to disease are being studied. In Open Targets Genetics, this is always the alternative allele.
The direction of the effect of the alternative allele can be obtained from the PheWAS plot. If the association has a positive beta coefficient, this means the alternative (effect allele) allele increases the risk. If this value is negative, the alternative allele (effect allele) decreases the risk.
Why are betas and odds ratios displayed inconsistently in the portal?
Effect sizes are derived from summary statistics, where available, otherwise they are taken from GWAS Catalog curated data. All effects have been harmonised to be with respect to the alternative allele.
An effect size may not be shown in the portal if: (1) the effect was not curated for that association by GWAS Catalog; (2) the variant is palindromic as it is not possible to accurately infer the strand, and so direction; (3) The reported risk allele is not-concordant with the alleles in our variant index; (4) the rsID to variant ID mapping was ambiguous (not one-to-one).
Sometimes GWAS Catalog data has been curated from multiple tables in a publication, some with betas, others with odds ratios. In these cases a mixture of betas and odds ratios may be displayed for a single study.
What is the difference between beta coefficient and study beta coefficient?
For every single variant that is independently and significantly associated with one study, we will display individual beta coefficient values with respect to the alternative allele of each of these variants, such as variants 19_44886339_G_A, 19_44908822_C_T, 1_109274968_G_T associated with LDL cholesterol.
On the other hand, we display the study beta coefficient in the colocalisation table of the study locus page e.g. LDL cholesterol (GCST002222) with locus around 19_44886339_G_A (rs7254892). This beta is with respect to the alternative allele of a single variant, the lead variant at the top of the study locus page (i.e. rs7254892 for the LDL cholesterol study).
The reason we have decided to display the study beta it to facilitate the comparison of the direction of effect across different colocalising tissues.
Summary statistics is the aggregate p-values and association data for every variant analysed in a genome-wide association study.
Why is there no LD information for my associated-locus of interest?
Linkage disequilibrium is calculated using the 1000 Genome Phase 3 reference panel. If your variant is not in this panel post-QC (MAF > 1% and max missing rate < 0.05) then we will not provide any LD information for it.
Why is there no credible set information for my associated-locus of interest?
Fine-mapping can only be conducted for studies that we have full summary statistics for. Currently this only consists of UK Biobank summary statistics from the Neale lab. We are currently working with the GWAS Catalog to create a summary statistics repository, which will then be included in the Open Targets Genetics. We encourage the scientific community to submit their full summary statistics to the GWAS Catalog.
Why isn't my variant in Open Targets Genetics?
Our variant index is built from the gnomAD (v2.1) site list, filtered to keep only variants with minor allele frequency > 0.1% in any population (code). If a variant is not in our index, it will not exist in the portal.
Why doesn't my variant report the GTEx QTL?
We apply a multiple testing correction that is different from the GTEx method. We use a method that is applicable across datasets, and not all datasets conduct a permutation analysis. We use a Bonferroni correction based on the number of variants tested per gene, i.e. p < 0.05 / (number of tests per gene). For example, GTEx assigns rs4734621 (8_102432699_T_C) to UBR5 whereas our V2G pipeline assigns it to both ODF1 and NCALD. More details on filtering can be found in the pre-processing help page
How do I download the credible set of variants for an association of interest?
Credible set information is available for all studies that have gone through our fine-mapping pipeline. The full set of variants in the 95% credible set can be downloaded using the Tag Variant table on the Variant page for the lead-variant at your locus of interest.
What Variant-to-Gene (V2G) score threshold should I use as a "significance" cut-off?
The V2G scores are intended as a way to rank genes based on all available functional data. We do not provide an arbitrary cut-off for V2G scores.
The data used to calculate V2G scores are already pre-filtered to remove associations with low evidence after multiple testing procedures are applied. Therefore, any
(V,G)
pair with a non-zero score has at least one good string of evidence in the data. The higher the V2G score, the more evidence there is for a functional association.
Why are case counts missing for some case-control studies?
Sample case counts are stored as part of a text string in GWAS Catalog. This makes the information difficult to parse reliably. We have decided not to show case numbers for these studies.
What is the alternative allele? Why not use the minor allele?
The reference (ref) and alternative (alt) alleles can be determined by looking at the variant ID, which takes the form: chromosome_position_reference_alternative
The ref allele refers to the base that is found in the reference genome, currently GRCh38 in the portal. The alt allele refers to any base, other than the reference, that is found at the locus. The alt allele is not necessarily the minor allele.
For example, if we look at rs2476601 (1_114377568_A_G). A is the ref, and G is the alt. The allele frequencies and effect are with respect to the alt. So the G has frequency of 0.88 in Non-Finnish European, making it the major allele and A the minor allele.
There can be more than 1 alt allele per position in the genome, in which case they will appear as two separate variant IDs in the portal.
Using ref/alt, as opposed to major/minor, keeps things consistent across studies/populations.
Why is the number of independently associated loci different in the portal compared to the study's publication?
We report any association that is curated by the GWAS Catalog (see inclusion criteria), except for a subset of studies (N=162) for which we apply an additional step of distance based clumping (±500kb).
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Chemical_shift Knowpia
{\displaystyle \omega _{0}=\gamma B_{0}\,,}
{\displaystyle \gamma ={\frac {\mu \,\mu _{\mathrm {N} }}{hI}}\,.}
{\displaystyle \omega _{0}=\gamma B_{0}={\frac {2.79\times 5.05\times 10^{-27}\,{\rm {J/T}}}{6.62\times 10^{-34}\,{\rm {Js}}\times {\tfrac {1}{2}}}}\times 1\,{\rm {T}}=42.5\,{\rm {MHz}}\,.}
{\displaystyle \delta ={\frac {\nu _{\mathrm {sample} }-\nu _{\mathrm {ref} }}{\nu _{\mathrm {ref} }}}\,,}
{\displaystyle {\frac {300\,{\rm {Hz}}}{300\times 10^{6}\,{\rm {Hz}}}}=1\times 10^{-6}=1\,{\rm {ppm\,.}}}
"External referencing, involving sample and reference contained separately in coaxial cylindrical tubes."[5] With this procedure, the reference signal is still visible in the spectrum of interest, although the reference and the sample are physically separated by a glass wall. Magnetic susceptibility differences between the sample and the reference phase need to be corrected theoretically,[5] which lowers the practicality of this procedure.
{\displaystyle \Xi [\%]=100(\upsilon _{X}^{obs}/\upsilon _{TMS}^{obs})}
The Knight shift (first reported in 1949) and Shoolery's rule are observed with pure metals and methylene groups, respectively. The NMR chemical shift in its present-day meaning first appeared in journals in 1950. Chemical shifts with a different meaning appear in X-ray photoelectron spectroscopy as the shift in atomic core-level energy due to a specific chemical environment. The term is also used in Mössbauer spectroscopy, where similarly to NMR it refers to a shift in peak position due to the local chemical bonding environment. As is the case for NMR the chemical shift reflects the electron density at the atomic nucleus.[14]
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Introduction to special series — lesson. Mathematics State Board, Class 10.
Carl Friedrich Gauss \((1777-1855)\) is widely acknowledged as one of the finest mathematicians.
There is a most famous story that involved Gauss when he was in primary school. Gauss's teacher once instructed his students to add all the numbers from \(1\) to \(100\), expecting that they would be engaged for a long time. However, he was surprised when young Gauss scribbled down the number \(5050\) after a few seconds of thinking. The teacher was perplexed as to how his student had mentally computed the total so fast, but Gauss pointed out that the issue was rather easy.
As a result, he made the following observation:
\(1 + 2 + 3 + …. + 98 + 99 + 100\).
He added the numbers in pairs. That is:
The first number \((1)\) with last number \((100)\),
The second number \((2)\) with second last number \((99)\),
The third number \((3)\) with the third last number \((98)\), and so on.
\(2 + 99 = 101\)
\(49 + 52 = 101\)
Gauss found that the sum of the numbers within these pairs will always be \(101\).
Total number of pairs \(=\) \(50\)
Sum of these pairs \(= 50 \times 101\) \(=5050\)
So, the sum of all the numbers from \(1\) to \(100\) is \(5050\).
Gauss's approach creates a general formula for the sum of the first '\(n\)' natural numbers, name that:
1+2+3+....+n=\frac{1}{2}n\left(n+1\right)
Thanks to Gauss! This will lead us to explore some more special way of adding numbers.
There are some series whose sum can be expressed by explicit formulae. Such series are called special series.
We will look at some common special series.
(i) Sum of first '\(n\)' natural numbers.
(ii) Sum of first '\(n\)' odd natural number.
(iii) Sum of squares of first '\(n\)' natural numbers.
(iv) Sum of cubes of first '\(n\)' natural numbers.
We can obtain the formula for the sum of any powers of first '\(n\)' natural numbers using the expression \((x + 1)^{k + 1} - x^{k + 1}\).
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Basic terms in Nuclear physics — lesson. Science State Board, Class 10.
Atomic mass unit (u) is defined as \(1/12\)\(^{th}\) of the mass of a neutral carbon atom \(_{6}C^{12}\) (carbon – 12 atom). This unit is used to measure the mass of an atom.
Thus, the mass of a carbon atom is \(12\ u\).
Estimation of mass of helium nucleus:
Consider a helium atom \(_{2}He^{4}\), which has \(2\) electrons, \(2\) protons and \(2\) neutrons.
Mass of a proton \(=\ 1.0078\ u\)
Mass of a neutron \(=\ 1.0087\ u\)
But experimentally, the actual mass of a helium nucleus is \(4.0026\ u\). So, what happened to the remaining mass of \(0.0304\ u\)?
To know that, we need to look at the concept of the mass defect.
The mass of the daughter nucleus formed during a nuclear reaction (fission and fusion) is lesser than the sum of the masses of the two-parent nuclei. This difference in mass is called the mass defect.
In the above example, the mass defect is found to be \(0.0304\ u\).
Unit of energy:
In nuclear physics, the electron volt (\(eV\)) is the unit used to measure the energy of tiny particles. It is the energy of an electron when it is accelerated by an electric potential of one volt.
\(1\ eV\) \(=\) \(1.602 \times{10^{-19}}\) \(joule\)
\(1\ million\ electron\ volt\) \(=\) \(1\ MeV\) \(=\) \(10^6\ eV\) (\(mega\ electron\ volt\))
Generally, the energy of about \(200\ MeV\) is released in a nuclear fission process.
Einstein's mass-energy equivalence:
According to the mass-energy equivalence, the mass is converted into energy and vice versa. Albert Einstein proposed the concept of mass-energy equivalence in \(1905\).
The relation between mass and energy is
Where \(E\) is the energy, \(m\) is the mass, and \(c\) is the velocity of light in a vacuum, which is equal to \(3 \times {10^8}\ ms^{-1}\).
According to Einstein, mass and energy are not independent but are mutually convertible. A body that changes its energy '\(E\)' undergoes a change in its mass, '\(m\)'. For example, if the energy of a particle is increased, its mass also increases.
Binding energy:
Binding energy holds the nucleons (protons and neutrons) of a nucleus together, even from the loss in the total mass of the nucleons (mass defect). It is measured in \(MeV\).
Where \(Δm\) is the mass defect of the nucleus in atomic mass units ‘\(u\)’.
The higher the binding energy per nucleon, the greater is the stability of the atom. Also,
\mathit{Binding}\phantom{\rule{0.147em}{0ex}}\mathit{energy}\phantom{\rule{0.147em}{0ex}}\mathit{per}\phantom{\rule{0.147em}{0ex}}\mathit{nucleon}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{\mathit{Binding}\phantom{\rule{0.147em}{0ex}}\mathit{energy}}{A}
Where \(A\) is the mass number of an element.
https://live.staticflickr.com/5005/5351051490_483f3695b0.jpg
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2015 Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk
Yanyue Shi, Na Zhou
We consider the reducing subspaces of
{M}_{{z}^{N}}
{A}_{\alpha }^{2}\left({\mathbb{D}}^{k}\right)
k\ge 3
{z}^{N}={z}_{1}^{{N}_{1}}\cdots {z}_{k}^{{N}_{k}}
{N}_{i}\ne {N}_{j}
i\ne j
. We prove that each reducing subspace of
{M}_{{z}^{N}}
is a direct sum of some minimal reducing subspaces. We also characterize the minimal reducing subspaces in the cases that
\alpha =0
\alpha \in \left(-1,+\mathrm{\infty }\right)\setminus \mathbb{Q}
, respectively. Finally, we give a complete description of minimal reducing subspaces of
{M}_{{z}^{N}}
{A}_{\alpha }^{2}\left({\mathbb{D}}^{3}\right)
\alpha >-1
Yanyue Shi. Na Zhou. "Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk." Abstr. Appl. Anal. 2015 1 - 12, 2015. https://doi.org/10.1155/2015/209307
Yanyue Shi, Na Zhou "Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk," Abstract and Applied Analysis, Abstr. Appl. Anal. 2015(none), 1-12, (2015)
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Number of wavelet scattering coefficients - MATLAB numCoefficients - MathWorks Benelux
Oversample 1-D Wavelet Scattering Transform
Number of wavelet scattering coefficients
ncf = numCoefficients(sf)
ncf = numCoefficients(sf) returns the number of scattering coefficients for each scattering path in the wavelet time scattering network sf. The number of scattering coefficients depends on the values of the SignalLength, InvarianceScale, and OversamplingFactor properties of sf.
This example shows how to oversample a 1-D wavelet scattering transform.
Load an ECG signal sampled at 180 Hz, and create a wavelet time scattering network to process the signal. To perform a critically downsampled wavelet scattering transform, do not change the value of the OversamplingFactor property in sf. Return the number of scattering coefficients for the scattering network.
sf = waveletScattering('SignalLength',numel(wecg),'SamplingFrequency',Fs);
ncf = 8
Return the 1-D wavelet scattering transform of wecg, and plot the zeroth-order scattering coefficients. Confirm the number of zeroth-order scattering coefficients is equal to ncf.
display(['Number of zeroth-order scattering coefficients: ',...
num2str(numel(s{1}.signals{1}))])
Number of zeroth-order scattering coefficients: 8
plot(s{1}.signals{1},'x-')
title('Zeroth-Order Scattering Coefficients')
To oversample the scattering coefficients by a factor of 2, set the OversamplingFactor property of sf equal to 1 (because
{\mathrm{log}}_{2}2=1
). Return the number of scattering coefficients for the edited network. Confirm the number of scattering coefficients has doubled.
sf.OversamplingFactor = 1;
ncf = 16
Return the wavelet scattering transform of wecg using the edited network, and plot the zeroth-order scattering coefficients. Since the number of coefficients in the critically sampled transform is equal to 8, confirm that the number of zeroth-order coefficients in the oversampled transform is equal to 16.
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EUDML | Orbifold adjunction formula and symplectic cobordisms between lens spaces. EuDML | Orbifold adjunction formula and symplectic cobordisms between lens spaces.
Orbifold adjunction formula and symplectic cobordisms between lens spaces.
Chen, Weimin. "Orbifold adjunction formula and symplectic cobordisms between lens spaces.." Geometry & Topology 8 (2004): 701-734. <http://eudml.org/doc/124502>.
author = {Chen, Weimin},
keywords = {cobordism of lens spaces; orbifold adjunction formula; symplectic 4-orbifolds; pseudoholomorphic curves},
title = {Orbifold adjunction formula and symplectic cobordisms between lens spaces.},
TI - Orbifold adjunction formula and symplectic cobordisms between lens spaces.
KW - cobordism of lens spaces; orbifold adjunction formula; symplectic 4-orbifolds; pseudoholomorphic curves
cobordism of lens spaces, orbifold adjunction formula, symplectic 4-orbifolds, pseudoholomorphic curves
h
s
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Motion in 1D & 2D - Revision Session - NEET & AIIMS 2019, 2020
Motion in 1D & 2D - Revision Session - NEET & AIIMS 2019, 2020Contact Number: 9667591930 / 8527521718
Drops of water fall from the roof of a building 20 m high at regular intervals
of time. The first drop reaching the ground at the same instant fifth drop
starts its fall. What are the distance of second and third drops from roof? (g =
m/{s}^{2}
1. 5.0 m and 1.25 m
3. 11.25 m and 5.0 m
4. 11.25 and 1.25 m
A particle is dropped vertically from rest from a height. The time taken by it to
fall through successive distances of 1 m will the be
(1) All equal, being equal to
\sqrt{2/g}
(2) In the ratio of the square roots of integers 1,2,3....
(3) In the ratio of the difference in the square roots of the integers i.e.
\sqrt{1}
\left(\sqrt{2}-\sqrt{1}\right)
\left(\sqrt{3}-\sqrt{2}\right)
\left(\sqrt{4}-\sqrt{3}\right)
(4) In the ratio of the reciprocal of the square roots of the integers, i.e.,
\frac{1}{\sqrt{1}}
\frac{1}{\sqrt{2}}
\frac{1}{\sqrt{3}}
\frac{1}{\sqrt{4}}
A car drives along straight level frictionless road by an engine delivering constant power. Then velocity is directly proportional to
\frac{1}{\sqrt{t}}
\sqrt{t}
A ball rolls off the top of stairway with a horizontal velocity of magnitude 1.8 m/s. The steps are 0.20 m high and 0.20 m wide. Which step will the ball hit first ?
A car accelerates from rest at a constant rate
\alpha
for some time after which it decelerates at a constant rate
\beta
to come to rest. If the total time elapsed is t, the distance travelled by the car is:
\frac{1}{2}\left(\frac{\alpha \beta }{\alpha +\beta }\right){t}^{2}
\frac{1}{2}\left(\frac{\alpha +\beta }{\alpha \beta }\right){t}^{2}
\frac{1}{2}\left(\frac{{\alpha }^{2}+{\beta }^{2}}{\alpha \beta }\right){t}^{2}
\frac{1}{2}\left(\frac{{\alpha }^{2}-{\beta }^{2}}{\alpha \beta }\right){t}^{2}
A train accelerates from rest at a constant rate
\mathrm{\alpha }
for distance
{\mathrm{x}}_{1}
and time ti. After that, It retards to rest at a constant rate
\mathrm{\beta }
{\mathrm{x}}_{2}
{\mathrm{t}}_{2}
. Then it is found that
\frac{{\mathrm{x}}_{1}}{{\mathrm{x}}_{2}} = \frac{\mathrm{\alpha }}{\mathrm{\beta }} = \frac{{\mathrm{t}}_{1}}{{\mathrm{t}}_{2}}
\frac{{\mathrm{x}}_{1}}{{\mathrm{x}}_{2}} = \frac{\mathrm{\beta }}{\mathrm{\alpha }} = \frac{{\mathrm{t}}_{1}}{{\mathrm{t}}_{2}}
\frac{{\mathrm{x}}_{1}}{{\mathrm{x}}_{2}} = \frac{\mathrm{\beta }}{\mathrm{\alpha }} = \frac{{\mathrm{t}}_{2}}{{\mathrm{t}}_{1}}
\frac{{\mathrm{x}}_{1}}{{\mathrm{x}}_{2}} = \frac{\mathrm{\alpha }}{\mathrm{\beta }} = \frac{{\mathrm{t}}_{2}}{{\mathrm{t}}_{1}}
A smooth square platform ABCD is moving towards right with a uniform speed u. At what angle
\theta
must a particle be projected from A with speed
v
so that it strikes the point D ?
{\mathrm{sin}}^{-1}
\left(\frac{u}{v}\right)
{\mathrm{cos}}^{-1}
\left(\frac{v}{u}\right)
{\mathrm{sin}}^{-1}
\left(\frac{v}{u}\right)
{\mathrm{cos}}^{-1}
\left(\frac{u}{v}\right)
Time taken by the projectile to reach from A to B is t, then the distance AB is equal to:
ut
\sqrt{3} ut
\frac{\sqrt{3}}{2}ut
\frac{ut}{\sqrt{3}}
A man in a lift ascending with an acceleration throws a ball vertically upwards with a velocity u and catches it after time
{\mathrm{t}}_{1}
. Afterwards, when the lift is descending with the same acceleration, the man again throws the ball vertically upwards and catches it after time
{\mathrm{t}}_{2}
. The velocity of the projection of the ball:
1. \frac{{\mathrm{gt}}_{1}{\mathrm{t}}_{2}}{\left({\mathrm{t}}_{1}-{\mathrm{t}}_{2}\right)}\phantom{\rule{0ex}{0ex}}2. \frac{{\mathrm{gt}}_{1}{\mathrm{t}}_{2}}{\left({\mathrm{t}}_{1}+{\mathrm{t}}_{2}\right)}\phantom{\rule{0ex}{0ex}}3. \frac{{\mathrm{gt}}_{1}{\mathrm{t}}_{2}}{2\left({\mathrm{t}}_{1}-{\mathrm{t}}_{2}\right)}\phantom{\rule{0ex}{0ex}}4. \frac{{\mathrm{gt}}_{1}{\mathrm{t}}_{2}}{2\left({\mathrm{t}}_{1}+{\mathrm{t}}_{2}\right)}
Two particles are separated by a horizontal distance R on the ground. Those are projected simultaneously with velocities u and u/
\sqrt{3}
at angles of projection
{60}^{0}
and 150° with the horizontal direction so that they approach each other in the same plane. The time after which they meet at point on the horizontal plane is:
1. u/2 R
2. 2 R/u
4. R/u
A vertical circular disc has different grooves along various chords as shown in fig. The particles are released from upper end O. The times taken by the particles to reach the ends of the grooves OA, OB and OC respectively are in the ratio
2. 1:2:
\sqrt{3}
3. 1:
\sqrt{3}
\sqrt{3}
\sqrt{2}
A boy throws a ball upwards with velocity
{v}_{0}
. The wind imparts a horizontal acceleration of 4
m/{s}^{2}
to the left. The angle
\theta
at which the ball must be thrown so that the ball returns to the boy’s hand is (g = 10
m/{s}^{2}
{\mathrm{tan}}^{-1}
{\mathrm{tan}}^{-1}
(O.2)
{\mathrm{tan}}^{-1}
{\mathrm{tan}}^{-1}
Two inclined planes OA and OB intersect is a horizontal plane having their inclinations
\alpha
and ß with the horizontal as shown in fig. A particle is projected from point P with velocity u along a direction perpendicular to plane OA. The particle strikes plane OB perpendicularly at Q
\alpha
{30}^{0}
\beta
{30}^{0}
, the time of flight from P to Q is
1. u/g
\sqrt{3}
u/g
\sqrt{2}
4. 2 u/g
A ball is dropped vertically from a height above the ground. It hits the ground and bounced up vertically to a height d/2. neglecting subsequent motion and the air resistance, its velocity v with height h above the ground can be represented as
A point P moves in counter-clockwise direction on a circular path as shown in fig. The movement of ‘P’ in such that it sweeps out a length s =
{t}^{3}
+5, where s is in metres and t is in seconds. The radius of the path is 20 m. The acceleration of
‘P’ when t = 2 s is nearly
m/{s}^{2}
m/{s}^{2}
m/{s}^{2}
m/{s}^{2}
A particle is moving with velocity
\stackrel{\to }{v}
K\left(y\stackrel{^}{i}+x\stackrel{^}{j} \right)
, where K is a constant. The
general equation for its path is
1. y =
{x}^{2}
+ constant
2. y2=x + constant
3. xy= constant
4. y2=
{x}^{2}
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Determine spread of credit default swap - MATLAB cdsspread - MathWorks í•œêµ
RPV01=\underset{j=1}{\overset{N}{â}}Z\left(tj\right)\mathrm{Î}\left(tjâ1,tj,B\right)Q\left(tj\right)
RPV01â\frac{1}{2}\underset{j=1}{\overset{N}{â}}Z\left(tj\right)\mathrm{Î}\left(tjâ1,tj,B\right)\left(Q\left(tjâ1\right)+Q\left(tj\right)\right)
when accrued premiums are paid upon default. Here, t0 = 0 is the valuation date, and t1,...,tn = T are the premium payment dates over the life of the contract,T is the maturity of the contract, Z(t) is the discount factor for a payment received at time t, and Δ(tj-1, tj, B) is a day count between dates tj-1 and tj corresponding to a basis B.
ProtectionLeg={â«}_{0}^{T}Z\left(\mathrm{Ï}\right)\left(1âR\right)dPD\left(\mathrm{Ï}\right)
â\left(1âR\right)\underset{i=1}{\overset{M}{â}}Z\left(\mathrm{Ï}i\right)\left(PD\left(\mathrm{Ï}i\right)âPD\left(\mathrm{Ï}iâ1\right)\right)
=\left(1âR\right)\underset{i=1}{\overset{M}{â}}Z\left(\mathrm{Ï}i\right)\left(Q\left(\mathrm{Ï}iâ1\right)âQ\left(\mathrm{Ï}i\right)\right)
where the integral is approximated with a finite sum over the discretization Ï„0 = 0,Ï„1,...,Ï„M = T.
S0=\frac{ProtectionLeg}{RPV01}
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Moving Charges And Magnetism, Popular Questions: ICSE Class 12-science PHYSICS, Physics Part I - Meritnation
please teach me how to see direction in question 27 in each
A uniform magnetic field of 5000 gauss is established along the positive z-direction.A rectangular loop of side 20cm and 5cm carries a current of 10A is suspended in this magnetic field.What is the torque on the loop in the different cases shown in the following figures? What is the force in each case? Which case corresponds to stable equilibrium?
Nasnin asked a question
Show that average energy density of electric field is equal to average energy density of magnetic field
A monochromatic light source of power 5mW emits 8X1015 photons per second. This light ejects photoelectrons from a metal surface. The stopping potential for this set up is 2V. Calculate the work function of the metal.
One mole of an ideal monoatomic gas undergoes a process ABC as shown in the indicator diagram. Heat supplied to the gas during the process is
ans is 7pv..how solve ?
In young experiment if two straight narrow parallel slits 3 mm apart are illuminated with monochromatic light of wavelength 5900 x 10 (raise to power minus 8)cm. Fringes are observed at a distance of 3m from slits. Find width of fringes??
an alpha particle is moving along a circle of radius 0.45m in a magnetic field of 12T.Find (i)the speed of the paticle ,(ii)the time period of the particle.(mass of alpha particle is 6.7*10^-27kg).
ans is (i)2.58*10^8m per secs,(ii)1.09*10^-8s
six wires of current I1 = 1A , I2 = 2A, I3 = 3A , I4 = 1A , I5 = 5A and I6 = 4A cut the page perpendicularly the points 1,2,3,4,5 and 6 respectively as shown in the figure. Find the value of the integral B?dl around the closed path.
Is the mass of deuteron twice the mass of proton and that of alpha particle 4times the proton???
if one moves on the axis of current carrying circular coil starting from the centre of coil then magnetic field induction will be
Force on negative charge is opposite to that on a positive charge. Pls explain.
an electron moving at a speed of 10 kilometre per second in a straight line enters a uniform electric field it experiences a constant acceleration of 2 kilometre per second in a direction opposite to its motion it will come back to initial point where it entered the field after a time
Find the magnetic field at point P. The curved portion is a semi circle and the straight wires are long. Answer this with explaination pls
Let BP and BQ be the magnetic field produced by the wire P and Q which are placed symmetrically in a rectangular loop ABCD as shown in figure. Current in wire P is I directed inward and in Q is 21 directed outwards. If
\underset{A}{\overset{B}{\int }}{\stackrel{\to }{B}}_{Q}·\stackrel{\to }{d\mathcal{l}}=2{µ}_{0}
tesla meter,
\underset{D}{\overset{A}{\int }}{\stackrel{\to }{B}}_{P}·\stackrel{\to }{d\mathcal{l}}=-2{µ}_{0}
tesla meter &
\underset{A}{\overset{B}{\int }}{\stackrel{\to }{B}}_{P}·\stackrel{\to }{d\mathcal{l}}=-{µ}_{0}
tesla meter the value of I will be:
(A) 8 A (B) 4 A (C) 5 A (D) 6 A
Krishanu Saikia asked a question
A charge of 1 C is placed at one end of a non-conducting rod of radius 0.4 m. The rod is rotated in a vertical plane about a horizontal axis passing through the other end of the rod with an angular frequency 2𝜋 × 10^4 rad/sec. The magnetic field at a point on the axis of rotation at a distance 1 m from the center of the path is
A) 5.75 × 10−5 T B) 6.88 × 10−5 T C) 7.25 × 10−5 T D) 8.08 × 10−5 T
a current of 10 A is flowing east to west in a long wire kept horizontally in east west direction. find magnetic field and its direction in the horizontal plane at a distance of
(i)10 cm north
(ii) 20 cm south from the wire.
and in the verical plane at a distance of
(i)40 cm downward
(ii)50 cm upward.
Ina Vashishtha asked a question
Two identical point charges are placed at a separation of I. P is a point on the line joining the charges, at a distance x from any one charge. The field at P is E. E is plotted against x for values of x from close to zero to slightly less than l. Which of the following best represents the resulting curve?
Q26. Work done by thermodynamic system during process AB as shown in the figure, is
(1) P0V0 (2) 2P0V0 (3) 3P0V0 (4) Zero
A proton is moving along the negative direction of x-axis in a magnetic field directed along the positive direction of y-axis. The proton will be deflected along the negative direction of?
Answer the question number 19 by an expart which i easely understand
Check the answers pls and don't ask to send one by one pls.... It just creates spam and is harder for both .....
Pls and fast
Jessica Britto asked a question
Pls solve 23q iii)
Q23. A proton moving towards east in a horizontal plane enters a horizontal magnetic field of 0.34 T directed towards north with speed of 2.0 × 107 ms–1. Calculate (i) the magnitude and direction of the force on the proton, (ii) radius of proton's path and (iii) the lateral displacement of the proton while moving 0.20 m towards east.
Ans. (i) 1.09 × 10–12 N, vertically upwards,
(ii) 0.625 m, (iii) 0.032 m (approx).
Dev Pathak asked a question
Researcher X asked a question
Please answer this...:
3. A uniform magnetic field
\mathrm{B}=\left(3\stackrel{^}{\mathrm{i}}+4\stackrel{^}{\mathrm{j}}+\stackrel{^}{\mathrm{k}}\right)
exists in region of space. A semicircular wire of radius 1 m carrying current 1 A having its centre at (2. 2, 0) placed in x-y plane as shown in fig. The force on semicircular wire will be.
\left(\mathrm{A}\right) \sqrt{2}\left(\stackrel{^}{\mathrm{i}}+\stackrel{^}{\mathrm{j}}+\stackrel{^}{\mathrm{k}}\right) \left(\mathrm{B}\right) \sqrt{2}\left(\stackrel{^}{\mathrm{i}}-\stackrel{^}{\mathrm{j}}+\stackrel{^}{\mathrm{k}}\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{C}\right) \sqrt{2}\left(\stackrel{^}{\mathrm{i}}+\stackrel{^}{\mathrm{j}}-\mathrm{k}\right) \left(\mathrm{D}\right) \sqrt{2}\left(\stackrel{^}{-\mathrm{i}}+\stackrel{^}{\mathrm{j}}+\stackrel{^}{\mathrm{k}}\right)
Arunima asked a question
a) Explain the principle of a moving coil galvanometer with the help of diagram.
b) What is the significance of using (i) radial magnetic field (ii) a cylindrical soft iron core inside the coil of galvanometer.
c) "Increasing the current sensitivity of a galvanometer may not necessarily increase its voltage sensitivity". Justify this statement.
Ankit Tripathi asked a question
State the factors on which the force acting on a charge moving in a magneticfield depends. Write the expression for this force. When is this force maximum andminimum?
A charged particle, having a charge q is moving with a speed v along the x-axis. It enters a region of space where an electric field E(=Ej) and a magnetic field B are both present. The particle on emerging from this region, is observed to be moving along the x-axis only. Obtain an expression for the magnitude of B in terms of v and E. Give the direction of B.
Bharathi C asked a question
define end rule
Derive the expression for the magnetic field at a point on the axis of a current carrying circular loop . Hence find the expression for magnetic moment and writes its S I Unit .
two insulated wires perpendicular to each other in the same plane carry equal currents as shown in the figure. Is there a region where the magnetic field is zero ? If so where is the region? if not explain by the field is not zero.
height insect can crawl?
Prithvi Jaishankar asked a question
If we unwound and rewound a coil will number of turns remain same??
Jeebak Adhikary asked a question
State and explain Maxwell's modification of Ampere's circuital law. Please answer it ASAP.
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Map sequence reads to reference genome using BWA - MATLAB bwamem - MathWorks Italia
Align Reads to Reference Sequence Using BWA
AlternativeHitsThreshold
AppendReadCommentsToSAM
BasesPerBatch
ClipPenalty
DropChainFraction
DropChainLength
FastaHeaderToXR
GapExtensionPenalty
GapOpenPenalty
HeaderInsert
InsertSizeStatistics
MarkShortSplitsSecond
MarkSmallestCoordinatePrimary
MaxMemOccurrence
MaxRoundsMateRescue
MinSeedLength
MismatchPenalty
OutputAllAlignments
OutputScoreThreshold
ReadGroupLine
ReduceSupplementaryMAPQ
SeedSplitRatio
SkipMateRescue
SkipPairing
SoftClipSupplementary
TreatAltAsPrimary
UnpairedReadPenalty
ZDropOff
Map sequence reads to reference genome using BWA
bwamem(indexBaseName,reads1,reads2,outputFileName)
bwamem(___,options)
bwamem(___,Name,Value)
bwamem(indexBaseName,reads1,reads2,outputFileName) maps the sequencing reads from reads1 and reads2 against the reference sequence and writes the results to the output file outputFileName. The input indexBaseName represents the base name (prefix) of the reference index files [1][2].
bwamem requires the BWA Support Package for Bioinformatics Toolbox™. If the support package is not installed, then the function provides a download link. For details, see Bioinformatics Toolbox Software Support Packages.
bwamem(___,options) uses the additional options specified by options. Specify these options after all other input arguments.
bwamem(___,Name,Value) uses additional options specified by one or more name-value pair arguments. For example, 'BandWidth',90 sets the maximum allowable gap length to 90.
BWA Support Package for Bioinformatics ToolboxBWA Support Package for Bioinformatics Toolbox
This example requires the BWA Support Package for Bioinformatics Toolbox™. If the support package is not installed, the software provides a download link. For details, see Bioinformatics Toolbox Software Support Packages.
Build a set of index files for the Drosophila genome. This example uses the reference sequence Dmel_chr4.fa, provided with the toolbox. The 'Prefix' argument lets you define the prefix of the output index files. You can also include the file path information. For this example, define the prefix as Dmel_chr4 and save the index files in the current directory.
bwaindex('Dmel_chr4.fa','Prefix','./Dmel_chr4');
As an alternative to specifying name-value pair arguments, you can use the BWAIndexOptions object to specify the indexing options.
indexOpt = BWAIndexOptions;
indexOpt.Prefix = './Dmel_chr4';
indexOpt.Algorithm = 'bwtsw';
bwaindex('Dmel_chr4.fa',indexOpt);
Once the index files are ready, map the read sequences to the reference using bwamem. Two pair-end read input files are already provided with the toolbox. Using name-value pair arguments, you can specify different alignment options, such as the number of parallel threads to use.
bwamem('Dmel_chr4','SRR6008575_10k_1.fq','SRR6008575_10k_2.fq','SRR6008575_10k_chr4.sam','NumThreads',4);
Alternatively, you can use BWAMEMoptions to specify the alignment options.
alignOpt = BWAMEMOptions;
alignOpt.NumThreads = 4;
bwamem('Dmel_chr4','SRR6008575_10k_1.fq','SRR6008575_10k_2.fq','SRR6008575_10k_chr4.sam',alignOpt)
Base name (prefix) of the reference index files, specified as a character vector or string. For example, the base name of an index file 'Dmel_chr4.bwt' is 'Dmel_chr4'.
The index files are in the AMB, ANN, BWT, PAC, and SA file formats.
reads1 — Name of file with first mate reads or single-end reads
Name of the file with the first mate reads or single-end reads, specified as a character vector or string.
For paired-end data, the sequences in reads1 must correspond read-for-read to sequences in reads2.
Example: 'SRR6008575_10k_1.fq'
reads2 — Name of file with second mate reads
character vector | string | []
Name of the file with the second mate reads, specified as a character vector or string.
Specify reads2 as empty ([], '', or "") if the data consists of single-end reads only.
outputFileName — Output file name
Output file name, specified as a character vector or string. This file contains the mapping results.
Example: 'SRR6008575_10k_chr4.sam'
options — Additional options for mapping
BWAMEMOptions object | character vector | string
Additional options for mapping, specified as a BWAMEMOptions object, character vector, or string. The character vector or string must be in the bwa mem native syntax (prefixed by a dash). If you specify a BWAMEMOptions object, the software uses only those properties that are set or modified.
Example: bwamem(indexbasename,reads1,reads2,outputfile,'BandWidth',90) sets 90 as the maximum allowable gap.
AlternativeHitsThreshold — Threshold for determining which hits receive XA tag in output SAM file
[5 200] (default) | nonnegative integer | two-element numeric vector
Threshold for determining which hits receive an XA tag in the output SAM file, specified as a nonnegative integer n or two-element numeric vector [n m], where n and m must be nonnegative integers.
If a read has less than n hits with a score greater than 80% of the best score for that read, all hits receive an XA tag in the output SAM file.
When you also specify m, the software returns up to m hits if the hit list contains a hit to an ALT contig.
AppendReadCommentsToSAM — Flag to append FASTA or FASTQ comments to output SAM file
Flag to append FASTA or FASTQ comments to the output SAM file, specified as true or false. The comments appear as text after a space in the file header.
BandWidth — Maximum allowable gap length
Maximum allowable gap length, specified as a nonnegative integer.
BasesPerBatch — Number of bases per batch
Number of bases per batch, specified as a positive integer.
If you do not specify BasesPerBatch, the software uses 1e7 * NumThreads by default. NumThreads is the number of parallel threads available when you run bwamem.
If you specify BasesPerBatch, the software uses that exact number and does not multiply the number by NumThreads. This rule applies regardless of whether you explicitly set NumThreads or not.
However, if you specify NumThreads but not BasesPerBatch, the software uses 1e7 * NumThreads.
The batch size is proportional to the number of parallel threads in use. Using different numbers of threads might produce different outputs. Specifying this option helps with the reproducibility of results.
ClipPenalty — Penalty for clipped alignments
[5 5] (default) | nonnegative integer | two-element numeric vector
Penalty for clipped alignments, specified as a nonnegative integer or two-element numeric vector. Each read has the best score for an alignment that spans the length of the read. The software does not clip alignments that do not span the length of the read and do not score higher than the sum of ClipPenalty and the best score of the full-length read.
Specify a nonnegative integer to set the same penalty for both 5' and 3' clipping.
Specify a two-element numeric vector to set different penalties for 5' and 3' clipping.
DropChainFraction — Threshold for dropping chains relative to longest overlapping chain
Threshold for dropping chains relative to the longest overlapping chain, specified as a scalar between 0 and 1.
The software drops chains that are shorter than DropChainFraction * (longest overlapping chain length).
DropChainLength — Minimum number of bases
Minimum number of bases in seeds forming a chain, specified as a nonnegative integer. The software drops chains shorter than DropChainLength.
Example: 'ExtraCommand','-y'
FastaHeaderToXR — Flag to include FASTA header in XR tag
Flag to include the FASTA header in the XR tag, specified as true or false.
GapExtensionPenalty — Gap extension penalty
Gap extension penalty, specified as a nonnegative integer or two-element numeric vector [n m]. n is the penalty for extending a deletion. m is the penalty for extending an insertion.
If you specify a nonnegative integer, the software uses it as the penalty for extending a deletion or an insertion.
GapOpenPenalty — Gap opening penalty
Gap opening penalty, specified as a nonnegative integer or two-element numeric vector [n m]. n is the penalty for opening a deletion. m is the penalty for opening an insertion.
If you specify a nonnegative integer, the software uses it as the penalty for opening a deletion or an insertion.
HeaderInsert — Text to insert into header of output SAM file
[0x0 string] (default) | character vector | string
Text to insert into the header of the output SAM file, specified as a character vector or string.
Character vector or string that starts with @ to insert the exact text to the SAM header
Character vector or string that is a file name, where each line in the file must start with @
Flag to include all available options with the corresponding default values when converting to the original syntax, specified as true or false.
InsertSizeStatistics — Insert size distribution parameters
[1x0 double] (default) | four-element numeric array
Insert size distribution parameters, specified as a four-element numeric array [mean std max min].
mean is the mean insert size.
std is the standard deviation.
max is the maximum insert size.
min is the minimum insert size.
If you specify n elements array, where n is less than four, the elements specify the first n distribution parameters. By default, the software infers unspecified parameters from data.
MarkShortSplitsSecond — Flag to mark shorter split hits as secondary
Flag to mark the shorter split hits as secondary in the SAM flag, specified as true or false.
MarkSmallestCoordinatePrimary — Flag to mark segment with smallest coordinates as primary
Flag to mark the segment with the smallest coordinates as primary when the alignment is split, specified as true or false.
MatchScore — Score for sequence match
Score for a sequence match, specified as a nonegative integer.
MaxMemOccurrence — Maximum number of MEM occurrences
Maximum number of MEM (maximal exact match) occurrences for each read before it is discarded, specified as a positive integer.
MaxRoundsMateRescue — Maximum number of rounds of mate rescue
Maximum number of rounds of mate rescue for each read, specified as a nonnegative integer. The software uses the Smith-Waterman (SW) algorithm for the mate rescue.
MinSeedLength — Minimum seed length
Minimum seed length, specified as a positive integer. The software discards any matches shorter than the minimum seed length.
MismatchPenalty — Penalty for alignment mismatch
Penalty for an alignment mismatch, specified as a nonnegative integer.
NumThreads — Number of parallel threads
OutputAllAlignments — Flag to return all found alignments
Flag to return all found alignments including unpaired and paired-end reads, specified as true or false. If the value is true, the software returns all found alignments and marks them as secondary alignments.
OutputScoreThreshold — Score threshold for returning alignments
Score threshold for returning alignments, specified as a positive integer. Specify the minimum score that alignments must have to be in the output file.
ReadGroupLine — Text to insert into read group header
Text to insert into the read group (RG) header line in the output file, specified as a character vector or string.
ReadType — Type of reads to align
[0x0 string] (default) | 'pacbio | 'ont2d | 'intractg'
Type of reads to align, specified as a character vector or string. Each read type has different default parameter values to use during alignment. You can overwrite any parameters. Valid options are:
'pacbio' — PacBio reads
'ont2d' — Oxford nanopore 2D reads
'intractg' — Intra-species contigs
The parameter values are as follows.
'pacbio'
MinSeedLength = 17
DropChainLength = 40
SeedSplitRatio = 10
MatchScore = 1
MismatchPenalty = 1
GapOpenPenalty = 1
GapExtensionPenalty = 1
ClipPenalty = 0
The equivalent native syntax is '-k17 -W40 -r10 -A1 -B1 -O1 -E1 -L0'.
'ont2d'
'intractg'
GapOpenPenalty = 16
The equivalent native syntax is '-B9 -O16 -L5'.
ReduceSupplementaryMAPQ — Flag to reduce mapping quality (MAPQ) score of supplementary alignments
Flag to reduce the mapping quality (MAPQ) score of supplementary alignments, specified as true or false.
SeedSplitRatio — Threshold for reseeding
1.50 (default) | nonnegative integer
Threshold for reseeding, specified as a nonnegative integer. Specify the seed length at which reseeding happens relative to the minimum seed length MinSeedLength. Specifically, if a MEM (maximal exact match) is longer than MinSeedLength * SeedSplitRatio, reseeding occurs.
SkipMateRescue — Flag to skip mate rescue
Flag to skip mate rescue, specified as true or false. Mate rescue uses the Smith-Waterman (SW) algorithm to align unmapped reads with mates that are properly aligned.
SkipPairing — Flag to skip read pairing
Flag to skip read pairing, specified as true or false. If true, for paired-end reads, the software uses the Smith-Waterman (SW) algorithm to rescue missing hits only and does not try to find hits that fit a proper pair.
SmartPairing — Flag to perform smart pairing
Flag to perform smart pairing, specified as true or false. If the value is true, the software pairs adjacent reads that are in the same file and have the same name. Such FASTQ files are also known as interleaved files.
SoftClipSupplementary — Flag to soft clip supplemental alignments
Flag to soft clip supplemental alignments, specified as true or false. If the value is true, the software soft clips both supplemental alignments and a primary alignment.
The default value is false, which means that the software soft clips the primary alignment and hard clips the supplemental alignments.
TreatAltAsPrimary — Flag to treat ALT contigs as part of primary assembly
Flag to treat ALT contigs as part of the primary assembly, specified as true or false.
UnpairedReadPenalty — Penalty for mapping read pairs as unpaired
Penalty for mapping read pairs as unpaired, specified as a nonnegative integer.
The alignment score for a paired read pair is read1 score + read2 score - insert penalty. The alignment score for an unpaired read pair is read1 score + read2 score - UnpairedReadPenalty. The software compares these two scores to force read pairing. A larger UnpairedReadPenalty value leads to a more aggressive read pairing.
Verbosity — Verbosity level of information printed
Verbosity level of information printed to the MATLAB command line while the software is running, specified as a nonnegative integer. Valid options are:
0 — For disabling all outputs to the command line.
1 — For printing error messages.
2 — For printing warning and error messages.
3 — For printing all messages.
4 — For debugging purposes only.
ZDropOff — Cutoff for Smith-Waterman extension
Cutoff for the Smith-Waterman (SW) extension, specified as a nonnegative integer. The software uses the following expression:
|i-j|\ast MatchScore+ZDropOff
, where i and j are the current positions of the query and reference, respectively. When the difference between the best score and current extension score is larger than this expression value, the software terminates the SW extension.
BWAMEMOptions | bwaindex
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(Redirected from Sierpiński arrowhead curve)
This article is missing information about other Sierpiński curves, see Sierpiński Curve on Wolfram MathWorld. Please expand the article to include this information. Further details may exist on the talk page. (January 2019)
"Sierpinski square snowflake" redirects here. For other uses, see Sierpinski carpet.
Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit
{\displaystyle n\to \infty }
completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling curve.
Because the Sierpiński curve is space-filling, its Hausdorff dimension (in the limit
{\displaystyle n\to \infty }
{\displaystyle 2}
The Euclidean length of the
{\displaystyle n}
th iteration curve
{\displaystyle S_{n}}
{\displaystyle l_{n}={2 \over 3}(1+{\sqrt {2}})2^{n}-{1 \over 3}(2-{\sqrt {2}}){1 \over 2^{n}},}
i.e., it grows exponentially with
{\displaystyle n}
beyond any limit, whereas the limit for
{\displaystyle n\to \infty }
of the area enclosed by
{\displaystyle S_{n}}
{\displaystyle 5/12\,}
that of the square (in Euclidean metric).
Sierpiński curve ("Sierpinski's square snowflake"[1]) of first order
Sierpiński curves of orders 1 and 2
Sierpiński curves of orders 1 to 3
Sierpinski "square curve"[2] of orders 2-4
1 Uses of the curve
3 Arrowhead curve
3.2 Representation as Lindenmayer system
Uses of the curve[edit]
The Sierpiński curve is useful in several practical applications because it is more symmetrical than other commonly studied space-filling curves. For example, it has been used as a basis for the rapid construction of an approximate solution to the Travelling Salesman Problem (which asks for the shortest sequence of a given set of points): The heuristic is simply to visit the points in the same sequence as they appear on the Sierpiński curve.[3] To do this requires two steps: First compute an inverse image of each point to be visited; then sort the values. This idea has been used to build routing systems for commercial vehicles based only on Rolodex card files.[4]
A space-filling curve is a continuous map of the unit interval onto a unit square and so a (pseudo) inverse maps the unit square to the unit interval. One way of constructing a pseudo-inverse is as follows. Let the lower-left corner (0, 0) of the unit square correspond to 0.0 (and 1.0). Then the upper-left corner (0, 1) must correspond to 0.25, the upper-right corner (1, 1) to 0.50, and the lower-right corner (1, 0) to 0.75. The inverse map of interior points are computed by taking advantage of the recursive structure of the curve. Here is a function coded in Java that will compute the relative position of any point on the Sierpiński curve (that is, a pseudo-inverse value). It takes as input the coordinates of the point (x,y) to be inverted, and the corners of an enclosing right isosceles triangle (ax, ay), (bx, by), and (cx, cy). (Note that the unit square is the union of two such triangles.) The remaining parameters specify the level of accuracy to which the inverse should be computed.
static long sierp_pt2code( double ax, double ay, double bx, double by, double cx, double cy,
int currentLevel, int maxLevel, long code, double x, double y )
if (currentLevel <= maxLevel) {
if ((sqr(x-ax) + sqr(y-ay)) < (sqr(x-cx) + sqr(y-cy))) {
code = sierp_pt2code( ax, ay, (ax+cx)/2.0, (ay+cy)/2.0, bx, by,
currentLevel, maxLevel, 2 * code + 0, x, y );
code = sierp_pt2code( bx, by, (ax+cx)/2.0, (ay+cy)/2.0, cx, cy,
The Sierpiński curve can be expressed by a rewrite system (L-system).
Alphabet: F, G, X
Constants: F, G, +, −
Axiom: F−−XF−−F−−XF
X → XF+G+XF−−F−−XF+G+X
Here, both F and G mean “draw forward”, + means “turn left 45°”, and − means “turn right 45°” (see turtle graphics). The curve is usually drawn with different lengths for F and G.
The Sierpiński square curve can be similarly expressed:
Alphabet: F, X
Constants: F, +, −
Axiom: F+XF+F+XF
X → XF−F+F−XF+F+XF−F+F−X
Arrowhead curve[edit]
The Sierpiński arrowhead curve is a fractal curve similar in appearance and identical in limit to the Sierpiński triangle.
Evolution of Sierpiński arrowhead curve
The Sierpiński arrowhead curve draws an equilateral triangle with triangular holes at equal intervals. It can be described with two substituting production rules: (A → B-A-B) and (B → A+B+A). A and B recur and at the bottom do the same thing — draw a line. Plus and minus (+ and -) mean turn 60 degrees either left or right. The terminating point of the Sierpiński arrowhead curve is always the same provided you recur an even number of times and you halve the length of the line at each recursion. If you recur to an odd depth (order is odd) then you end up turned 60 degrees, at a different point in the triangle.
An alternate constriction is given in the article on the de Rham curve: one uses the same technique as the de Rham curves, but instead of using a binary (base-2) expansion, one uses a ternary (base-3) expansion.
Given the drawing functions void draw_line(double distance); and void turn(int angle_in_degrees);, the code to draw an (approximate) Sierpiński arrowhead curve looks like this:
void sierpinski_arrowhead_curve(unsigned order, double length)
// If order is even we can just draw the curve.
if ( 0 == (order & 1) ) {
curve(order, length, +60);
else /* order is odd */ {
turn( +60);
curve(order, length, -60);
void curve(unsigned order, double length, int angle)
if ( 0 == order ) {
draw_line(length);
curve(order - 1, length / 2, -angle);
curve(order - 1, length / 2, angle);
Like many two-dimensional fractal curves, the Sierpiński arrowhead curve can be extended to three dimensions
The Sierpiński arrowhead curve can be expressed by a rewrite system (L-system).
Alphabet: X, Y
Axiom: XF
X → YF + XF + Y
Y → XF − YF − X
Here, F means “draw forward”, + means “turn left 60°”, and − means “turn right 60°” (see turtle graphics).
Wikimedia Commons has media related to Sierpiński curve.
Murray polygon
^ Weisstein, Eric W. "Sierpiński Curve". MathWorld. Retrieved 21 January 2019.
^ Dickau, Robert M. (1996/7)"Two-dimensional L-systems", Robert's Math Figures. MathForum.org. Retrieved 21 January 2019.
^ Platzman, Loren K.; Bartholdi, John J., III (1989). "Spacefilling curves and the planar traveling salesman problem". Journal of the Association for Computing Machinery. 36 (4): 719–737. doi:10.1145/76359.76361.
^ Bartholdi, John J., III. "Some combinatorial applications of spacefilling curves". Georgia Institute of Technology. Archived from the original on 2012-08-03.
Peitgen, H.-O.; Jürgens, H.; Saupe, D. (2013) [1992]. Chaos and Fractals: New Frontiers of Science. Springer. ISBN 978-1-4757-4740-9.
Stevens, Roger T. (1989). Fractal Programming in C. M&T Books. ISBN 9781558510371.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Sierpiński_curve&oldid=1057846828#Arrowhead_curve"
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PropsSI - Maple Help
Home : Support : Online Help : Science and Engineering : ThermophysicalData : CoolProp : PropsSI
PropsSI(output, input1, value1, input2, value2, fluid, opts)
real numbers for the input quantities, optionally with units
The PropsSI function interrogates the CoolProp library for thermophysical data.
The output parameter can, in principle, be any of the numerical thermophysical properties in the Quantity and Maple-specific aliases columns of the following table, whenever that property makes sense for the given fluid.
Only quantities with Yes in the Input? column can be used for input1 and input2, and only some combinations of these inputs will work.
The quantities for input1, input2, and output should be entered as strings or symbols. If a variable with the same name is already in use, it is best to use a string or to use unevaluation quotes to prevent evaluation of the variable name. In almost all circumstances, you can use either one of the names used by the CoolProp library, or an alias defined by the Maple package.
In some situations, the output parameter can be used to compute the partial derivative of one quantity with respect to another, while keeping a third quantity constant. This is done by specifying output in the form "d(OF)/d(WRT)|CONSTANT", where OF, WRT, and CONSTANT are valid CoolProp-recognized quantity names. In this case, OF represents the quantity CoolProp takes the derivative of, WRT is the quantity with respect to which CoolProp takes the derivative, and CONSTANT is the quantity kept constant.
For example, the constant pressure specific heat is the partial derivative of the mass specific enthalpy (Hmass) with respect to the temperature (T) at constant pressure (P); consequently, it can be represented as "d(Hmass)/d(T)|P". (There is also a dedicated representation for this quantity: C.) Specifying a partial derivative is the only situation where the Maple-defined aliases are not recognized.
You should use real constants for value1 and value2. Optionally, you can affix a unit to the value you give; the default unit for any quantity is given in the Unit column of the following table. If you supply a unit with any of the quantities you submit, the answer will have the appropriate unit as well. This behavior can be overridden by using the useunits option: if you supply useunits = true (which can be shortened to just useunits), then the result will always have the appropriate unit, and if you supply useunits = false, the result will never have a unit.
\mathrm{with}\left(\mathrm{ThermophysicalData}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{Atmosphere}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Chemicals}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{CoolProp}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{PHTChart}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Property}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{PsychrometricChart}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{TemperatureEntropyChart}}]
\mathrm{with}\left(\mathrm{CoolProp}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{HAPropsSI}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{PhaseSI}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Property}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Props1SI}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{PropsSI}}]
Determine the saturation temperature of water at 1 atmosphere in kelvin.
\mathrm{PropsSI}\left(T,P,101325,Q,0,\mathrm{Water}\right)
\textcolor[rgb]{0,0,1}{373.124295847684380}
\mathrm{PropsSI}\left(T,P,101325,Q,0,\mathrm{Water},\mathrm{useunits}\right)
\textcolor[rgb]{0,0,1}{373.1242958}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{K}⟧
\mathrm{PropsSI}\left(T,P,1.0\mathrm{Unit}\left(\mathrm{atm}\right),Q,0,\mathrm{Water}\right)
\textcolor[rgb]{0,0,1}{373.1242958}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{K}⟧
\mathrm{PropsSI}\left(T,P,1.0\mathrm{Unit}\left(\mathrm{atm}\right),Q,0,\mathrm{Water},\mathrm{useunits}=\mathrm{false}\right)
\textcolor[rgb]{0,0,1}{373.124295847684380}
Determine the constant pressure specific heat of water at 300 kelvin and 1 atmosphere, in two ways. The first way uses the dedicated representation of this quantity, C, for output. The second way uses the the partial derivative of the mass specific enthalpy (Hmass) with respect to the temperature (T) at constant pressure (P), "d(Hmass)/d(T)|P".
\mathrm{PropsSI}\left(C,P,1.0\mathrm{Unit}\left(\mathrm{atm}\right),T,300,\mathrm{Water}\right)
\textcolor[rgb]{0,0,1}{4180.635777}\textcolor[rgb]{0,0,1}{}⟦\frac{\textcolor[rgb]{0,0,1}{J}}{\textcolor[rgb]{0,0,1}{\mathrm{kg}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{K}}⟧
\mathrm{PropsSI}\left("d\left(Hmass\right)/d\left(T\right)|P",P,1.0\mathrm{Unit}\left(\mathrm{atm}\right),T,300,\mathrm{Water}\right)
\textcolor[rgb]{0,0,1}{4180.635777}\textcolor[rgb]{0,0,1}{}⟦\frac{\textcolor[rgb]{0,0,1}{J}}{\textcolor[rgb]{0,0,1}{\mathrm{kg}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{K}}⟧
The ThermophysicalData[CoolProp][PropsSI] command was introduced in Maple 2016.
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EUDML | Deviations of stationary curves in the bundle . EuDML | Deviations of stationary curves in the bundle .
Deviations of stationary curves in the bundle
Os{c}^{\left(2\right)}\left(M\right)
Miron, R.; Balan, V.; Stavrinos, P.C.; Tsagas, Gr.
Miron, R., et al. "Deviations of stationary curves in the bundle .." Balkan Journal of Geometry and its Applications (BJGA) 2.1 (1997): 51-60. <http://eudml.org/doc/222521>.
@article{Miron1997,
author = {Miron, R., Balan, V., Stavrinos, P.C., Tsagas, Gr.},
keywords = {osculator bundle; -linear connections; stationary curves; -linear connections},
title = {Deviations of stationary curves in the bundle .},
AU - Miron, R.
AU - Stavrinos, P.C.
TI - Deviations of stationary curves in the bundle .
KW - osculator bundle; -linear connections; stationary curves; -linear connections
osculator bundle,
N
-linear connections, stationary curves,
N
-linear connections
Articles by Miron
Articles by Stavrinos
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EUDML | Finite groups of bounded rank with an almost regular automorphism of prime order. EuDML | Finite groups of bounded rank with an almost regular automorphism of prime order.
Finite groups of bounded rank with an almost regular automorphism of prime order.
Khukhro, E.I.
Khukhro, E.I.. "Finite groups of bounded rank with an almost regular automorphism of prime order.." Sibirskij Matematicheskij Zhurnal 43.5 (2002): 1182-1191 (2002); translation in Sib. Math. J. 43. <http://eudml.org/doc/50238>.
@article{Khukhro2002,
author = {Khukhro, E.I.},
keywords = {regular automorphisms; rank of finite groups; nilpotent subgroups; nilpotency classes; associated Lie rings},
title = {Finite groups of bounded rank with an almost regular automorphism of prime order.},
AU - Khukhro, E.I.
TI - Finite groups of bounded rank with an almost regular automorphism of prime order.
KW - regular automorphisms; rank of finite groups; nilpotent subgroups; nilpotency classes; associated Lie rings
regular automorphisms, rank of finite groups, nilpotent subgroups, nilpotency classes, associated Lie rings
p
Articles by Khukhro
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EUDML | solutions of systems of finite difference equations. EuDML | solutions of systems of finite difference equations.
{C}^{m}
solutions of systems of finite difference equations.
Liu, Xinhe; Zhao, Xiuli; Ma, Jianmin
Liu, Xinhe, Zhao, Xiuli, and Ma, Jianmin. " solutions of systems of finite difference equations.." International Journal of Mathematics and Mathematical Sciences 2003.36 (2003): 2315-2326. <http://eudml.org/doc/50733>.
author = {Liu, Xinhe, Zhao, Xiuli, Ma, Jianmin},
keywords = {system of difference equations},
title = { solutions of systems of finite difference equations.},
TI - solutions of systems of finite difference equations.
KW - system of difference equations
Articles by Ma
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what is meant by acidic strength of HA - Chemistry - Equilibrium - 7102700 | Meritnation.com
what is meant by acidic strength of HA?
In HA, H is for hydrogen and A is used as hypothetical acid. Acidic strength of HA means the tendency of HA to lose H+ ions. The more readily it loses H+ more will be the acidic strength.
If HA is strong acid then dissociation will be complete.
\to
For example: Hydrochloric acid
HCl (aq)
\to
H+(aq) + Cl-(aq)
If HA is a weak acid then dissociation is not completed an ions remain in equilibrium with acid.
⇔
For example, acetic acid
CH3COOH(aq)
⇔
H+(aq) + CH3COO-(aq)
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Convert ARMA model to AR model - MATLAB arma2ar - MathWorks Australia
{y}_{t}=0.2{y}_{t-1}-0.1{y}_{t-2}+{\epsilon }_{t}+0.5{\epsilon }_{t-1}.
{y}_{t}
\begin{array}{lll}{y}_{t}& =& 0.7{y}_{t-1}-0.45{y}_{t-2}+0.225{y}_{t-3}-0.1125{y}_{t-4}+0.0562{y}_{t-5}+...\\ & & -0.0281{y}_{t-6}+0.0141{y}_{t-7}+{\epsilon }_{t}\end{array}
{y}_{t}={\epsilon }_{t}-0.2{\epsilon }_{t-1}+0.5{\epsilon }_{t-3}.
The MA model is in difference-equation notation because the left side contains only
{y}_{t}
and its coefficient of 1. Create a cell vector containing the MA lag term coefficient in order starting from t - 1. Because the second lag term of the MA model is missing, specify a 0 for its coefficient.
{y}_{t}=-0.2{y}_{t-1}-0.04{y}_{t-2}+0.492{y}_{t-3}+0.1984{y}_{t-4}+0.0597{y}_{t-5}+{\epsilon }_{t}
\begin{array}{l}\left\{\left[\begin{array}{ccc}1& 0.2& -0.1\\ 0.03& 1& -0.15\\ 0.9& -0.25& 1\end{array}\right]-\left[\begin{array}{ccc}-0.5& 0.2& 0.1\\ 0.3& 0.1& -0.1\\ -0.4& 0.2& 0.05\end{array}\right]{L}^{4}-\left[\begin{array}{ccc}-0.05& 0.02& 0.01\\ 0.1& 0.01& 0.001\\ -0.04& 0.02& 0.005\end{array}\right]{L}^{8}\right\}{y}_{t}=\\ \left\{\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+\left[\begin{array}{ccc}-0.02& 0.03& 0.3\\ 0.003& 0.001& 0.01\\ 0.3& 0.01& 0.01\end{array}\right]{L}^{4}\right\}{\epsilon }_{t}\end{array}
{y}_{t}={\left[{y}_{1t}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{y}_{2t}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{y}_{3t}\right]}^{\prime }
{\epsilon }_{t}={\left[{\epsilon }_{1t}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{\epsilon }_{2t}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{\epsilon }_{3t}\right]}^{\prime }
{y}_{t}
and enter the rest in order by lag. Because the equation is in lag operator notation, include the sign in front of each matrix. Construct a vector that indicates the degree of the lag term for the corresponding coefficients.
{\epsilon }_{t}
{y}_{t}=1.5+0.2{y}_{t-1}-0.1{y}_{t-2}+{\epsilon }_{t}+0.5{\epsilon }_{t-1}.
{y}_{t}
\left(1-0.2L+0.1{L}^{2}\right){y}_{t}=1.5+\left(1+0.5L\right){\epsilon }_{t}
\Phi \left(L\right){y}_{t}=1.5+\Theta \left(L\right){\epsilon }_{t}
{\Theta }^{-1}\left(L\right)\Phi \left(L\right){y}_{t}={\Theta }^{-1}\left(L\right)1.5+{\epsilon }_{t}
{y}_{t}=1+0.7{y}_{t-1}-0.45{y}_{t-2}+0.225{y}_{t-3}-0.1125{y}_{t-4}+0.0562{y}_{t-5}+{\epsilon }_{t}.
When you work from a model in difference-equation notation, negate the AR coefficients of the lagged responses to construct the lag-operator polynomial equivalent. For example, consider
{y}_{t}=0.5{y}_{t-1}-0.8{y}_{t-2}+{\epsilon }_{t}-0.6{\epsilon }_{t-1}+0.08{\epsilon }_{t-2}
. The model is in difference-equation form. To convert to an AR model, enter the following into the command window.
\left(1-0.5L+0.8{L}^{2}\right){y}_{t}=\left(1-0.6L+0.08{L}^{2}\right){\epsilon }_{t}.
{\Phi }_{0}{y}_{t}=c+{\Phi }_{1}{y}_{t-1}+...+{\Phi }_{p}{y}_{t-p}+{\Theta }_{0}{\epsilon }_{t}+{\Theta }_{1}{\epsilon }_{t-1}+...+{\Theta }_{q}{\epsilon }_{t-q},
\Phi \left(L\right){y}_{t}=c+\Theta \left(L\right){\epsilon }_{t},
\Phi \left(L\right)={\Phi }_{0}-{\Phi }_{1}L-{\Phi }_{2}{L}^{2}-...-{\Phi }_{p}{L}^{p}
{L}^{j}{y}_{t}={y}_{t-j}
\Theta \left(L\right)={\Theta }_{0}+{\Theta }_{1}L+{\Theta }_{2}{L}^{2}+...+{\Theta }_{q}{L}^{q}
{\Theta }^{-1}\left(L\right)\Phi \left(L\right){y}_{t}={\epsilon }_{t},
\Phi \left(L\right)=\sum _{j=0}^{p}{\Phi }_{j}{L}^{j}
\Theta \left(L\right)=\sum _{k=0}^{q}{\Theta }_{k}{L}^{k}.
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Home > WikiCS > Articles > Optimization Problems
The area of interest of applied computer science is the solving of problems. There are different kinds of problems, but the ones that we are interested in are the optimization problems where the aim is to find the best solution among all possible ones.
Section 1: Definition
Section 2: Algorithms
Section 3: Heuristics
Section 4: Meta-heuristics
Section 5: Multiple Objective Problems
Subsection 5.1: Definition
Subsection 5.2: Exact and Approximate Problems
Subsection 5.3: Solve Multiple Objective Problems
Section 6: Related
An optimization problem consists to find the best solution among all possible ones. For example, in the Bin Packing Problem (BPP) the aim is to find the right number of boxes of a given size to store a set of objects of given sizes; optimization involves, for example, finding the smallest number of boxes.
It is important to make two distinctions. First, between a problem which refers to a general class, e.g. “Bin Packing”, and an instance representing a special type of a problem, e.g. “the Bin Packing problem involving size 5 boxes for 25 objects of different sizes”. Secondly, two categories of problem classes exist: the abstract problem classes and the concrete problem classes. As its name suggests, the second category refers to problems that have a “concrete existing”, i.e. problems for which instances can be created. The BPP corresponds to this category. Together, it is part also of a more abstract class: grouping problems. With abstract problem classes only, it is impossible to define instances. In fact, as shown in Figure 1↓, abstract and concrete problem classes form a hierarchy of optimizing problems.
Figure 1 Some optimization problems.
An optimization problem can be defined as a finite set of variables, where the correct values for the variables specify the optimal solution. If the variables range over real numbers, the problem is called continuous, and if they can only take a finite set of distinct values, the problem is called combinatorial. In the case of the grouping of user’s profiles, we are dealing with combinatorial optimization problems because the number of communities of interests is finite.
A combinatorial optimization problem is defined [1] as the set of all the instances of the problem, with each instance,
I
, being defined by a pair
(\mathcal{F},c)
\mathcal{F}
is called the search space, i.e. the set of all possible solutions for an instance, and
c
is a cost function calculated for each solution of
\mathcal{F}
and used to determine the performances of each solution.
2 Algorithms
To solve problems it is necessary to develop methods, often called algorithms in computer science, that describe the set of actions to be performed under given circumstances. In one of the definitions found in the literature [2], an algorithm is stated as the list of precise rules that specify “what to do” under all possible conditions. This definition includes the one of the Turing Machine [3], which is an abstract representation of a computing device. Another definition is to describe an algorithm as a finite set of instructions (evaluations and assignations), which leads to a solution.
The complexity,
O
, of an algorithm defines a relationship between the size of an instance, such as the number of objects in the Bin Packing Problem, and the necessary resources to solve it, i.e. the amount of memory and number of CPU cycles required. A complexity of
O(n^{2})
for example signifies that the resources required evolute as the square of the size of the instance, i.e. an instance two times larger than another one needs four times more resources.
An important category of problems consists of the NP-hard ones, for which no polynomial time algorithm has been found so far. With these problems, the CPU time increases exponentially with the size of an instance. In other words, when the size of the problem increases, it becomes impossible to compute all the valid solutions. For example, in a Bin Packing Problem involving 5 objects it is possible to compute all the different solutions to determine the best one. Whereas for 500 objects, it is no longer possible. NP-hard problems can only be solved by specific algorithms which try to reach an optimal solution, or at least a solution as close as possible to one optimal in a reasonable time.
3 Heuristics
When dealing with NP-hard problems it is often necessary to use algorithms that do not guarantee an optimal solution. This class of algorithms is known as heuristics.
A heuristic is an “intuitive” way to find a valid and often reasonably good solution for a given problem in a “reasonable” lapse of time, i.e. a heuristic is based on “rules-of-thumb”, ideas that seem to be helpful in some typical instances though, without providing any guarantee of the quality of the solution.
For example, in the Bin Packing Problem, the first-fit descending heuristic which consists in treating objects in descending order of size and putting one into the first group that can take it, is a well-known heuristic giving good results in such cases (depending on the sizes of the bins), though not necessary the optimal one. The biggest problem about heuristics is that it is strongly instance- and problem-dependent, and that the results may be very poor.
When the essential factor is the time of execution, heuristics is the best solution. For example, a first-fit heuristic is used in an operating system to find a memory zone when a program allocates a given amount of data. In this case, the fact that the solution proposed is not the best one is less important than the time needed. But, because of their drawback, heuristics rarely reach the status of being able to provide a solution when the quality of the solution is important, and cannot be considered as a general approach.
4 Meta-heuristics
The two disadvantages of heuristics are that the solutions proposed can often be very low-quality and are strongly instance- and problem-dependent. Computer science has developed several methods to work-around these disadvantages. All these methods use heuristics in some way or another, but enable the entire search space to be search in; this is the reason why the term meta-heuristics is employed. Most meta-heuristics need cost functions which are, most of the time, a mathematical expression that represents the quality of a solution for a given problem. A brief introduction to simulated annealing and tabu search ends this section. Another meta-heuristic known as the genetic algorithms will be presented in the next sections.
5 Multiple Objective Problems
Many real-world problems to be optimized depend on multiple objectives to reach. Increasing the performance of a solution for one objective usually leads to a decrease in the performances for the others. In such situations, it is not possible to construct a suitable mathematical model to represent the problem, i.e. it is not possible to find a cost function representing a single measure of quality for a solution. But, a value for each criterion to be optimized can be computed and the difficulty is to choose a solution which “is good for each criterion”. In fact, a solution which optimizes one criterion may be the worst possible one for the others. Moreover, each criterion may have a particular importance which is expressed through a weight assigned to it. These types of problems are known as multi-criteria decision problems.
An example of a multi-criteria problem is the choice of a car. Different criteria are defined, the price (to be minimized), the consumption (to be minimized), the comfort (to be maximized) and the power (to be maximized). Table 1↓ shows the values for different cars; the best value for each criterion is in bold: no solution seems to be better than any other. Moreover, each person may have his or her preferences, for example the power may be very important while the comfort is not.
Price (k€) Cons. (
1/100
km) Comfort Power (kW
\mathrm{Car}\,1
\mathbf{8.75}
\mathbf{6.2}
30
\mathrm{Car}\,2
13.75
7.5
50
\mathrm{Car}\,3
25
8
80
\mathrm{Car}\,4
62.5
20
\mathbf{120}
Table 1 Example of a multi-criteria decision problem.
5.2 Exact and Approximate Problems
I define an exact multi-criteria problem as a problem where the aim is characterized by a set of well-defined criteria; each criterion can be accurately expressed by a function and each function must either be maximized or minimized.
But, it is not always possible to express the quality by a set of well-defined measures. As already pointed out by Kleinberg [4], in the field of information retrieval problems, there is a lack in objective functions that are both concretely defined and correspond to human notions of quality. So, I define a,
So, in the case of the problem dealt with in this thesis, we are facing with an approximate multi-criteria problem as a problem where the aim cannot be accurately characterized in any way. A set of criteria is defined to approximate the characteristics of the target without guaranteeing an exact match between the criteria and the characteristics. Each criterion defines a function that must either be maximized or minimized.
5.3 Solve Multiple Objective Problems
A simple approach to solving multiple objective problems is to transform a multi-criteria decision in a single-criterion problem by aggregating the different objectives to form a single one. One of the techniques used is to compute a linear combination of the criteria values. Once this cost function has been constructed, a method is used to solve the problem by optimizing this single function. This type of solution works when the objectives do not compete, i.e. an improvement of one criterion does not lead to the negative influence of any other criteria. Moreover, it is impossible to apply this approach when the criteria do not have the same “physical existence”, i.e. when the price criterion is expressed in k€ and the power criterion in kW.
Another approach widely used in research into multiple objective optimization is due to Vilfredo Pareto [5]: given a set of solutions, a multi-objective decision is used to select a solution in this reduced space search. The concept of the Pareto front is illustrated in Figure 2↓, where five solutions
A
B
C
D
O
are represented for a problem with two criteria,
f_{1}
f_{2}
. The solution,
O
, is not dominated by any other solution, i.e. it is not possible to improve one criterion without downgrading another. This solution is called a Pareto optimal. All the Pareto optimum form the so-called Pareto-optimal front.
Figure 2 The Pareto optimal front (a) and dominance relations in objective space (b).
The PROMETHEE method [6] is another approach to solve multi-criteria problems.
6 Related
Hierarchical Problems
[1] Christos H. Papadimitriou & Kenneth Steiglitz, Combinatorial Optimization, Algorithms and Complexity, Prentice-Hall, 1982.
[2] Michael Garey & David Johnson, Computers and Intractability — A Guide to the Theory of NP-completeness, W.H. Freeman Co., 1979.
[3] Alan Turing, “On Computable Numbers, With an Application to the Entscheidungsproblem”, In Proceedings of the London Mathematical Society, pp. 230—265, 1936.
[4] Jon Kleinberg, “Authoritative Sources in a Hyperlinked Environment”, Journal of the ACM, 46(5), pp. 604—632, 1998.
[5] Vilfredo Pareto, Cours d’économie politique, F. Rouge, 1988.
[6] Jean-Pierre Brans & Bertrand Mareschal, ”The PROMCALC & GAIA Decision Support System for Multicriteria Decision Aid”, Decision Support Systems, 12(4‑5), pp. 297—310, 1994.
Copyright © 2021 Paul Otlet Institute •• designed by Rose Avril
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Neher–McGrath method - Wikipedia
Neher–McGrath method
(Redirected from Neher–McGrath)
In electrical engineering, Neher–McGrath is a method of estimating the steady-state temperature of electrical power cables for some commonly encountered configurations. By estimating the temperature of the cables, the safe long-term current-carrying capacity of the cables can be calculated.
J. H. Neher and M. H. McGrath were two electrical engineers who wrote a paper in 1957 about how to calculate the capacity of current (ampacity) of cables.[1] The paper described two-dimensional highly symmetric simplified calculations which have formed the basis for many cable application guidelines and regulations. Complex geometries, or configurations that require three-dimensional analysis of heat flow, require more complex tools such as finite element analysis. Their article became used as reference for the ampacity in most of the standard tables.
The Neher–McGrath paper summarized years of research into analytical treatment of the practical problem of heat transfer from power cables. The methods described included all the heat generation mechanisms from a power cable (conductor loss, dielectric loss and shield loss).[2]
From the basic principles that electric current leads to thermal heating and thermal power transfer to the ambient environment requires some temperature difference, it follows that the current leads to a temperature rise in the conductors. The ampacity, or maximum allowable current, of an electric power cable depends on the allowable temperatures of the cable and any adjacent materials such as insulation or termination equipment. For insulated cables, the insulation maximum temperature is normally the limiting material property that constrains ampacity. For uninsulated cables (typically used in outdoor overhead installations), the tensile strength of the cable (as affected by temperature) is normally the limiting material property. The Neher–McGrath method is the electrical industry standard for calculating cable ampacity, most often employed via lookup in tables of precomputed results for common configurations.
US National Electrical Code use[edit]
The equation in section 310-15(C) of the National Electrical Code, called the Neher–McGrath equation (NM) (given below), may be used to estimate the effective ampacity of a cable.[3]
{\displaystyle I={\sqrt {Tc-(Ta+{\Delta }Td) \over Rdc(1+Yc)Rca}}}
{\displaystyle Tc}
is normally the limiting conductor temperature derived from the insulation or tensile strength limitations.
{\displaystyle {\Delta }TD}
is a term added to the ambient temperature
{\displaystyle Ta}
to compensate for heat generated in the jacket and insulation for higher voltages.
{\displaystyle {\Delta }TD}
is called the dielectric loss temperature rise and is generally regarded as insignificant for voltages below 2000 V. Term
{\displaystyle (1+YC)}
is a multiplier used to convert direct current resistance (
{\displaystyle Rdc}
) to the effective alternating current resistance (which typically includes conductor skin effects and eddy current losses). For wire sizes smaller than AWG No. 2 (33.6 mm2, 0.0521 sq in), this term is generally regarded as insignificant.
{\displaystyle Rca}
is the effective thermal resistance between the conductor and the ambient conditions, which can require significant empirical or theoretical effort to estimate. With respect to the AC-sensitive terms, tabular presentation of the NM equation results in the National Electrical Code was developed assuming the standard North American power frequency of 60 hertz and sinusoidal wave forms for current and voltage.
The challenges posed by the complexity of estimating
{\displaystyle Rca}
and of estimating the local increase in ambient temperature obtained by co-locating many cables (in a duct bank) create a market niche in the electric power industry for software dedicated to ampacity estimation.
^ Neher, J. H.; McGrath, M. H. (October 1957). "The Calculation of the Temperature Rise and Load Capability of Cable Systems". AIEE Transactions. 76 (III): 752–772.
^ Anders, George J. (1997). Rating of Electric Power Cables: Ampacity Computations for Transmission, Distribution, and Industrial Applications. McGraw-Hill Professional. pp. 17–20. ISBN 0-07-001791-3.
^ Lane, Keith. "Heating" (PDF). Pure Power. Consulting-Specifying Engineer (Spring 2008): 15–19.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Neher–McGrath_method&oldid=1061950523"
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ACV synthetase - Wikipedia
ACV synthetase
ACV synthetase (ACVS, L-δ-(α-aminoadipoyl)-L-cysteinyl-D-valine synthetase, N-(5-amino-5-carboxypentanoyl)-L-cysteinyl-D-valine synthase, EC 6.3.2.26) is an enzyme that catalyzes the chemical reaction
3 ATP + L-2-aminohexanedioate + L-cysteine + L-valine + H2O
{\displaystyle \rightleftharpoons }
3 AMP + 3 PPi + N-[L-5-amino-5-carboxypentanoyl]-L-cysteinyl-D-valine
The five substrates of this enzyme are ATP, L-2-aminohexanedioate, L-cysteine, L-valine, and H2O, whereas its three products are AMP, diphosphate, and N-[L-5-amino-5-carboxypentanoyl]-L-cysteinyl-D-valine.
ACVS is an example of a nonribosomal peptide synthetase (NRPS). It participates in penicillin and cephalosporin biosyntheses.
Byford MF, Baldwin JE, Shiau CY, Schofield CJ (1997). "The Mechanism of ACV Synthetase". Chem. Rev. 97 (7): 2631–2650. doi:10.1021/cr960018l. PMID 11851475.
Theilgaard HB, Kristiansen KN, Henriksen CM, Nielsen J (1997). "Purification and characterization of delta-(L-alpha-aminoadipyl)-L-cysteinyl-D-valine synthetase from Penicillium chrysogenum". Biochem. J. 327 (Pt 1): 185–91. doi:10.1042/bj3270185. PMC 1218779. PMID 9355751.
Retrieved from "https://en.wikipedia.org/w/index.php?title=ACV_synthetase&oldid=1079675214"
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Pre Video Test - Hydrocarbons
Pre Video Test - HydrocarbonsContact Number: 9667591930 / 8527521718
has configuration
(4) Can't be predicted
A hydrocarbon has molecular formula
{C}_{8}{H}_{12}
. In this hydrocarbon only one
{H}_{2}
molecule can be added. On ozonolysis it gives symmetrical diketone. The hydrocarbon is
How many halogen derivatives are possible for
{C}_{6}{H}_{6}
The order of melting point for isomeric pentanes is
(1) n-pentane > Isopentane > Neopentane
(2) Neopentane > Isopentane > n-pentane
(3) Neopentane > n-pentane > Isopentane
(4) Isopentane > n-pentane > Neopentane
Which one of the following compounds can be used to distinguish propyne from propene?
(1) Aqueous
KMn{O}_{4}
(2) Dilute
{H}_{2}S{O}_{4}
B{r}_{2}
(4) Ammonical
AgN{O}_{3}
The compound 'A' is
{\mathrm{CH}}_{3}-\underset{{\mathrm{CH}}_{3}}{\overset{{\mathrm{CH}}_{3}}{\underset{|}{\overset{|}{\mathrm{C}}}}}-{\mathrm{CH}}_{2}-\mathrm{OH}\underset{160-170°\mathrm{C}}{\overset{\mathrm{Conc}. {\mathrm{H}}_{2}{\mathrm{SO}}_{4}}{\to }} \mathrm{A}\left(\mathrm{Major}\right)
C{H}_{3}\phantom{\rule{0ex}{0ex}} l\phantom{\rule{0ex}{0ex}}C{H}_{3}-C=CH-C{H}_{3}
C{H}_{3}\phantom{\rule{0ex}{0ex}} l\phantom{\rule{0ex}{0ex}}C{H}_{3}-CH-CH=C{H}_{2}
C{H}_{3}\phantom{\rule{0ex}{0ex}} l\phantom{\rule{0ex}{0ex}}C{H}_{2}=C-C{H}_{2}-C{H}_{3}
C{H}_{3}-CH-C{H}_{2}-C{H}_{3}\stackrel{{\left(CH3\right)}_{3}C-\overline{)O}K}{\to }A \left(Major\right)
C{H}_{2}=CH-C{H}_{2}-C{H}_{3}
C{H}_{3}-CH=CH-C{H}_{3}
C{H}_{3}\phantom{\rule{0ex}{0ex}} l\phantom{\rule{0ex}{0ex}}C{H}_{2}=C-C{H}_{3}
(4) No reaction
Which compound is most reactive towards electrophilic substitution reaction?
In the nitrating mixture
\left({H}_{2}S{O}_{4} and HN{O}_{3}\right), HN{O}_{3}
3. Acid as well as base
4. Neither acid nor base
Compound 'A' and 'B' respectively
(1) Both are Cis-butene-2
(2) Cis-butene-2, Trans-butene-2
(3) Both are Trans-butene-2
(4) Trans-butene-2, Cis-butene-2
{\mathrm{CH}}_{3}-\underset{{\mathrm{CH}}_{3}}{\underset{|}{\mathrm{CH}}}-\mathrm{CH}={\mathrm{CH}}_{2}+\mathrm{HBr}\to \left(\mathrm{Major}\right)
{\mathrm{CH}}_{3}-\underset{{\mathrm{CH}}_{3}}{\underset{|}{\mathrm{CH}}}-{\mathrm{CH}}_{2}-{\mathrm{CH}}_{2}-\mathrm{Br}
{\mathrm{CH}}_{3}-\underset{{\mathrm{CH}}_{3}}{\underset{|}{\mathrm{CH}}}-\underset{ \mathrm{Br}}{\underset{|}{\mathrm{CH}}}-{\mathrm{CH}}_{3}
{\mathrm{CH}}_{3}-\underset{{\mathrm{CH}}_{3}}{\underset{|}{\stackrel{\mathrm{Br}}{\stackrel{|}{\mathrm{C}}}}}-{\mathrm{CH}}_{2}-{\mathrm{CH}}_{3}
\underset{ \mathrm{Br}}{\underset{|}{{\mathrm{CH}}_{2}}}-\underset{{\mathrm{CH}}_{3}}{\underset{|}{\mathrm{CH}}}-{\mathrm{CH}}_{2}-{\mathrm{CH}}_{3}
'A' is
In which of the following homolytic bond fission takes place.
1. Alkaline hydrolysis of ethyl chloride
2. Addition of HBr to double bond
3. Photochlorination of methane
4. Nitration of benzene
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Predict responses for new observations from naive Bayes incremental learning classification model - MATLAB predict - MathWorks India
{c}_{k}=\sum _{j=1}^{K}\stackrel{^}{P}\left(Y=j|{x}_{1},...,{x}_{P}\right)Cos{t}_{jk}.
\stackrel{^}{P}\left(Y=k|{x}_{1},..,{x}_{P}\right)=\frac{P\left({X}_{1},...,{X}_{P}|y=k\right)\pi \left(Y=k\right)}{P\left({X}_{1},...,{X}_{P}\right)},
P\left({X}_{1},...,{X}_{P}|y=k\right)
P\left({X}_{1},..,{X}_{P}\right)
P\left({X}_{1},...,{X}_{P}\right)=\sum _{k=1}^{K}P\left({X}_{1},...,{X}_{P}|y=k\right)\pi \left(Y=k\right).
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2013 Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means
Zai-Yin He, Yu-Ming Chu, Miao-Kun Wang
a,b>\mathrm{0}
a\ne b
, the Schwab-Borchardt mean
\text{S}\text{B}\left(a,b\right)
\text{S}\text{B}\left(a,b\right)=\left\{\sqrt{{b}^{\mathrm{2}}-{a}^{\mathrm{2}}}/{\text{c}\text{o}\text{s}}^{-\mathrm{1}}\left(a/b\text{) if}\mathrm{}a<b,{\sqrt{{a}^{\mathrm{2}}-{b}^{\mathrm{2}}}/\text{c}\text{o}\text{s}\text{h}}^{-\mathrm{1}}\left(a/b\text{) if}a>b
. In this paper, we find the greatest values of
{\alpha }_{\mathrm{1}}
{\alpha }_{\mathrm{2}}
and the least values of
{\beta }_{\mathrm{1}}
{\beta }_{\mathrm{2}}
\left[\mathrm{0,1}/\mathrm{2}\right]
H\left({\alpha }_{\mathrm{1}}a+\left(\mathrm{1}-{\alpha }_{\mathrm{1}}\right)b,{\alpha }_{\mathrm{1}}b+\left(\mathrm{1}-{\alpha }_{\mathrm{1}}\right)a\right)<{S}_{AH}\left(a,b\right)<H\left({\beta }_{\mathrm{1}}a+\left(\mathrm{1}-{\beta }_{\mathrm{1}}\right)b,{\beta }_{\mathrm{1}}b+\left(\mathrm{1}-{\beta }_{\mathrm{1}}\right)a\right)
H\left({\alpha }_{\mathrm{2}}a+\left(\mathrm{1}-{\alpha }_{\mathrm{2}}\right)b,{\alpha }_{\mathrm{2}}b+\left(\mathrm{1}-{\alpha }_{\mathrm{2}}\right)a\right)<{S}_{HA}\left(a,b\right)<H\left({\beta }_{\mathrm{2}}a+\left(\mathrm{1}-{\beta }_{\mathrm{2}}\right)b,{\beta }_{\mathrm{2}}b+\left(\mathrm{1}-{\beta }_{\mathrm{2}}\right)a\right)
. Similarly, we also find the greatest values of
{\alpha }_{\mathrm{3}}
{\alpha }_{\mathrm{4}}
{\beta }_{\mathrm{3}}
{\beta }_{\mathrm{4}}
\left[\mathrm{1}/\mathrm{2,1}\right]
C\left({\alpha }_{\mathrm{3}}a+\left(\mathrm{1}-{\alpha }_{\mathrm{3}}\right)b,{\alpha }_{\mathrm{3}}b+\left(\mathrm{1}-{\alpha }_{\mathrm{3}}\right)a\right)<{S}_{CA}\left(a,b\right)<C\left({\beta }_{\mathrm{3}}a+\left(\mathrm{1}-{\beta }_{\mathrm{3}}\right)b,{\beta }_{\mathrm{3}}b+\left(\mathrm{1}-{\beta }_{\mathrm{3}}\right)a\right)
C\left({\alpha }_{\mathrm{4}}a+\left(\mathrm{1}-{\alpha }_{\mathrm{4}}\right)b,{\alpha }_{\mathrm{4}}b+\left(\mathrm{1}-{\alpha }_{\mathrm{4}}\right)a\right)<{S}_{AC}\left(a,b\right)<C\left({\beta }_{\mathrm{4}}a+\left(\mathrm{1}-{\beta }_{\mathrm{4}}\right)b,{\beta }_{\mathrm{4}}b+\left(\mathrm{1}-{\beta }_{\mathrm{4}}\right)a\right)
H\left(a,b\right)=\mathrm{2}ab/\left(a+b\right)
A\left(a,b\right)=\left(a+b\right)/\mathrm{2}
C\left(a,b\right)=\left({a}^{\mathrm{2}}+{b}^{\mathrm{2}}\right)/\left(a+b\right)
are the harmonic, arithmetic, and contraharmonic means, respectively, and
{S}_{HA}\left(a,b\right)=\text{S}\text{B}\left(H,A\right)
{S}_{AH}\left(a,b\right)=\text{S}\text{B}\left(A,H\right)
{S}_{CA}\left(a,b\right)=\text{S}\text{B}\left(C,A\right)
{S}_{AC}\left(a,b\right)=\text{S}\text{B}\left(A,C\right)
are four Neuman means derived from the Schwab-Borchardt mean.
Zai-Yin He. Yu-Ming Chu. Miao-Kun Wang. "Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means." J. Appl. Math. 2013 1 - 4, 2013. https://doi.org/10.1155/2013/807623
Zai-Yin He, Yu-Ming Chu, Miao-Kun Wang "Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-4, (2013)
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The slice determined by moduli equation x=y¯ in the deformation space of once punctured tori
April, 1999 The slice determined by moduli equation
x=\overline{y}
in the deformation space of once punctured tori
In the deformation space of once punctured tori, we investigate the slice determined by moduli equation such that the first and the second moduli are complex conjugate. We show the figure of the slice to some extent.
Takehiko SASAKI. "The slice determined by moduli equation
x=\overline{y}
in the deformation space of once punctured tori." J. Math. Soc. Japan 51 (2) 371 - 386, April, 1999. https://doi.org/10.2969/jmsj/05120371
Keywords: deformation space , Punctured torus , Quasi-Fuchsian groups
Takehiko SASAKI "The slice determined by moduli equation
x=\overline{y}
in the deformation space of once punctured tori," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 51(2), 371-386, (April, 1999)
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PROCEEDINGS OF THE THIRTEENTH SYMPOSIUM
THE THIRTEENTH SUMMER SYMPOSIUM ON REAL ANALYSIS
Density Continuous Functions
Krzysztof Ciesielski, Lee Larson, Krzysztof Ostaszewski
Concerning Two Properties of Connectivity Functions
Richard G. Gibson
Fractional Hadamard Powers of Positive Definite Matrices
TD Howroyd, CJF Upton, WW Wood
Restriction and Intersection Theorems in Real Analysis
ON A CERTAIN CONVERSE OF HÖLDER’S INEQUALITY FOR LORENTZ SPACES
LEONARD Y. H. YAP
Approximating Hausdorff Measures
Remarks on Laczkovich’s Circle-Squaring Proof
This paper, as advertised, contains none of my own work, but is an attempt to relay some of the flavor of the remarkable solution my friend Miklós Laczkovich has given to a 1925 problem of Alfred Tarski. As most of my knowledge of the history of this problem is directly attributable to Mik, I’ll quote him rather liberally throughout this paper. I’ll begin with the introductory remarks to his paper Equidecomposability and discrepancy; a solution to Tarski’s circle squaring problem which will appear (or perhaps already has appeared) in Crelle’s Journal.
BOLYAI-GERWIEN THEOREM AND HILBERT’S THIRD PROBLEM
Lifting: the connection between functional representations of vector lattices (summary)
SYMMETRIC DERIVATIVES AND SYMMETRIC INTEGRALS
LIMITS UNDER THE INTEGRAL SIGN
Pitt’s dimensionless Cantor set Don Spear
BOREL MEASURABLE SELECTIONS AND APPLICATIONS OF THE BOUNDEDNESS PRINCIPLE
R. Daniel Mauldin, Glen A. Schlee
“A Lower Bound for the Packing Measure which is a Multiple of the Hausdorff Measure”
Sandra Meinershagen
THE ω-LIMIT SETS FOR SELF MAPS OF AN INTERVAL
A. M. Bruckner
RADIAL CLUSTER SET AND INTERPOLATION
Change of variable in the semigroup valued refinement integral
The standard change of variable formula, for
T
measurable from a measure space
\left(S,S,\mathrm{\lambda }\right)
\left(\overline{S},\overline{S}\right)
on which a measurable
\overline{t}
is defined (see e.g.[H]),
{\int }_{S}\overline{t}T\text{\hspace{0.17em}}d\mathrm{\lambda }={\int }_{\overline{S}}\overline{t}\text{\hspace{0.17em}}d\left(\mathrm{\lambda }{T}^{-1}\right)
, is developed for a semigroup valued refinement integral. (The integral on the right can be rewritten in “Jacobian” form,
{\int }_{\overline{S}}\overline{t}\frac{d\mathrm{\lambda }{T}^{-1}}{d\overline{\mathrm{\lambda }}}d\overline{\lambda }
\lambda {T}^{-1}
is differentiable with respect to a measure
\overline{\mathrm{\lambda }}
\overline{S}
; the attempt in [DS] Lemma III.10.8 to develop the result by starting from
\overline{\mathrm{\lambda }}
is incorrect). This yields a result for the order-convergent integral of real-valued integrands against positive finitely additive measures in an Archimedian ordered vector space as well as a (correct) result for the [DS] Banach space valued integral of a vector valued integrand against a real-valued finitely additive measure.
A MULTIDIMENSIONAL VARIATIONAL INTEGRAL AND ITS EXTENSIONS
Washek F. Pfeffer, Wei-Chi Yang
We define a variational integral in the m–dimensional Euclidean space so that the Gauss–Green theorem holds for each vector field which is everywhere differentiable (not necessary continuously). The variational integral is then extended by a transfinite sequence of improper integrals, and the Gauss–Green theorem is proved for vector fields which are differentiable only outside fairly large exceptional sets. The variational integral and its extensions are invariant with respect to a continuously differentiable change of coordinates, and hence suitable for integration on differentiable manifolds.
INTERSECTION CONDITIONS FOR SOME DENSITY AND I-DENSITY LOCAL SYSTEMS
VARIATIONS ON PRODUCTS AND QUOTIENTS OF DARBOUX FUNCTIONS
Tomasz Natkaniec, Waldemar Orwat
THE CHI FUNCTIONS IN GENERALIZED SUMMABILITY
F. C. Leary
THE SEMI-BOREL CLASSIFICATION OF THE EXTREME PATH DERIVATIVES
The goal of this paper is to investigate of the semi-Borel and Baire classification of the multifunction of all path derived numbers of a semi-Borel and Baire function of the class
\alpha
. Consequently the classification of the extreme path derivatives is given. The results hold in the setting of ordinary qualitative and approximate path differentiation, and the proofs are based on a classification of the collection of paths which is considered as a multifunction of the semi-Borel class
\alpha
Differentiability and Density Continuity
Separate and Joint Continuity II
ON THE EQUIVALENCE OF HENSTOCK-KURZWEL AND RESTRICTED DENJOY NTEGRALS IN
{\text{R}}^{n}
Chew Tuan Seng
Some Higher Dimensional Marcinkiewicz Theorems
SPECTRAL RADIUS OF NONSINGULAR TRANSFORMATIONS
M. G. NADKARNI, J. B. ROBERTSON
POROUS SETS AND ADDITIVITY OF LEBESGUE MEASURE
M. Repický
Functions with all singular sets of Hausdorff dimension bigger than one
ON THE DANIÉLL INTEGRAL
Piotr Mikusiński
WEIGHTED SYMMETRIC FUNCTIONS
Tan Cao Tran
CONVERGENCE THEOREMS FOR THE VARIATIONAL INTEGRAL
Li Baoling
A CHARACTERIZATION OF NON-ATOMIC PROBABILITIES ON [0,1] WITH NOWHERE DENSE SUPPORTS
ON DISCONTINUITY POINTS FOR CLOSED GRAPH FUNCTIONS
An Answer to a Question of R.G. Gibson and F. Roush
Jan Jastrzȩbski
THROWING A DART AT FREILING’S ARGUMENT AGAINST THE CONTINUUM HYPOTHESIS
SOME SYMMETRIC COVERING LEMMAS
A REMARK ON ABSOLUTELY CONTINUOUS FUNCTIONS
C. Goffman, F. C. Liu
ANOTHER PROOF OF THE MEASURABILITY OF δ FOR THE GENERALIZED RIEMANN INTEGRAL
THE RADON–NIKODYM DERIVATIVE IN EUCLIDEAN SPACES
A DESCRIPTIVE CHARACTERIZATION OF THE GENERALIZED RIEMANN INTEGRAL
MARTINGALE PROOF OF THE EXISTENCE OF LEBESGUE POINTS
Michał Morayne, Sławomir Solecki
ON EXTREMAL VALUES OF CONTINUOUS MONOTONE FUNCTIONS
NOTE ON POINT SET THEORY
UNPUBLISHED RESULTS OF K. PEKÁR AND H. ZLONICKÁ ON PREPONDERANT DERIVATIVES AND
{\text{M}}_{4}
- SETS
L. Zajíček
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Convert decimal number to character vector representing binary number - MATLAB dec2bin - MathWorks Italia
Convert Integer to Binary Representation
Convert Array of Integers to Binary Representation
Convert decimal number to character vector representing binary number
str = dec2bin(d)
str = dec2bin(d,n)
str = dec2bin(d) returns the binary representation of symbolic number d as a character vector. d must be a nonnegative integer.
If d is a matrix or multidimensional array of symbolic numbers with N elements, dec2bin returns a character array with N rows. Each row of the output str corresponds to an element of d accessed with linear indexing.
str = dec2bin(d,n) returns a binary representation with at least n bits.
{2}^{60}
d = sym(2)^60
1152921504606846976
Convert the decimal number to binary representation.
'1000000000000000000000000000000000000000000000000000000000000'
\left(\begin{array}{cc}64& 123\\ 54& 11\end{array}\right)
Convert the integers to binary representation using dec2bin. dec2bin returns 4 rows of character vectors. Each row contains a 7-digit binary number.
Return a binary representation with at least 8 digits by specifying the number of digits.
str = dec2bin(d,8)
Number of bits, specified as a scalar positive integer.
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Genetic Algorithm Options - MATLAB & Simulink - MathWorks 日本
1/\sqrt{r}
1/\sqrt{2}
The shrink parameter controls how the standard deviation shrinks as generations go by. If you set InitialPopulationRange to be a 2-by-1 vector, the standard deviation at the kth generation, σk, is the same at all coordinates of the parent vector, and is given by the recursive formula
{\mathrm{Ï}}_{k}={\mathrm{Ï}}_{kâ1}\left(1â\text{Shrink}\frac{k}{\text{Generations}}\right).
If you set InitialPopulationRange to be a vector with two rows and nvars columns, the standard deviation at coordinate i of the parent vector at the kth generation, σi,k, is given by the recursive formula
{\mathrm{Ï}}_{i,k}={\mathrm{Ï}}_{i,kâ1}\left(1â\text{Shrink}\frac{k}{\text{Generations}}\right).
“Distance†measures a crowding of each individual in a population. Choose between the following:
\begin{array}{c}{F}_{\mathrm{max}}\left(j\right)=\underset{k}{\mathrm{max}}{F}_{k}\left(j\right)\\ {F}_{\mathrm{min}}\left(j\right)=\underset{k}{\mathrm{min}}{F}_{k}\left(j\right).\end{array}
w\left(k\right)=\underset{j}{â}\frac{{F}_{\mathrm{max}}\left(j\right)â{F}_{k}\left(j\right)}{1+{F}_{\mathrm{max}}\left(j\right)â{F}_{\mathrm{min}}\left(j\right)}.
p\left(j,k\right)=w\left(k\right)\frac{{F}_{\mathrm{max}}\left(j\right)â{F}_{k}\left(j\right)}{1+{F}_{\mathrm{max}}\left(j\right)â{F}_{\mathrm{min}}\left(j\right)}.
For gamultiobj, if the geometric average of the relative change in the spread of the Pareto solutions over MaxStallGenerations is less than FunctionTolerance, and the final spread is smaller than the average spread over the last MaxStallGenerations, then the algorithm stops. The geometric average coefficient is ½. The spread is a measure of the movement of the Pareto front. See gamultiobj Algorithm.
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{\displaystyle E={\frac {p^{2}}{2m_{e}}}+U(z)}
Rectangular barrier modelEdit
{\displaystyle \psi (z)}
{\displaystyle -{\frac {\hbar ^{2}}{2m_{e}}}{\frac {\partial ^{2}\psi (z)}{\partial z^{2}}}+U(z)\,\psi (z)=E\,\psi (z)}
{\displaystyle \psi _{L}(z)=e^{ikz}+r\,e^{-ikz}}
{\displaystyle \psi _{R}(z)=t\,e^{ikz}}
{\displaystyle k={\tfrac {1}{\hbar }}{\sqrt {2m_{e}E}}}
{\displaystyle \psi _{B}(z)=\xi e^{-\kappa z}+\zeta e^{\kappa z}}
{\displaystyle \kappa ={\tfrac {1}{\hbar }}{\sqrt {2m_{e}(U-E)}}}
{\displaystyle j_{i}={\tfrac {\hbar k}{m_{e}}}}
{\displaystyle j_{t}=|t|^{2}\,j_{i}}
{\displaystyle j_{t}=-i{\tfrac {\hbar }{2m_{e}}}\left\{\psi _{R}^{*}{\tfrac {\partial }{\partial z}}\psi _{R}-\psi _{R}{\tfrac {\partial }{\partial z}}\psi _{R}^{*}\right\}}
{\displaystyle j_{t}={\tfrac {\hbar k}{m_{e}}}|t|^{2}}
{\displaystyle |t|^{2}=[1+{\tfrac {1}{4}}{\varepsilon ^{-1}(1-\varepsilon )^{-1}}\sinh ^{2}\kappa w]^{-1}}
{\displaystyle \varepsilon =E/U}
{\displaystyle \kappa }
{\displaystyle |t|^{2}=16\,\varepsilon (1-\varepsilon )\,e^{-2\kappa w}}
{\displaystyle j_{t}=\left[{\tfrac {4k\kappa }{k^{2}+\kappa ^{2}}}\right]^{2}\,{\tfrac {\hbar k}{m_{e}}}\,e^{-2\kappa w}}
{\displaystyle k={\tfrac {1}{\hbar }}{\sqrt {2m_{e}E}}}
{\displaystyle \kappa ={\tfrac {1}{\hbar }}{\sqrt {2m_{e}(U-E)}}}
Tunneling between two conductorsEdit
{\displaystyle I_{i}={\tfrac {1}{2}}e^{2}v\,\rho (E_{F})\,V}
{\displaystyle I_{t}={\tfrac {1}{2}}e^{2}v\,\rho (E_{F})\,V\,T}
Bardeen's formalismEdit
{\displaystyle E_{\mu }^{S}}
{\displaystyle E_{\nu }^{T}}
{\displaystyle \psi _{\mu }^{S}(t)=\psi _{\mu }^{S}\exp(-{\tfrac {i}{\hbar }}E_{\mu }^{S}t)}
{\displaystyle \psi _{\nu }^{T}(t)=\psi _{\nu }^{T}\exp(-{\tfrac {i}{\hbar }}E_{\nu }^{T}t)}
{\displaystyle \psi _{\mu }^{S}(t)}
{\displaystyle \psi _{\nu }^{T}(t)}
{\displaystyle \psi (t)=\psi _{\mu }^{S}(t)\,+\,\sum _{\nu }{c_{\nu }(t)\,\psi _{\nu }^{T}(t)}}
{\displaystyle c_{\nu }(0)=0}
{\displaystyle \psi _{\nu }^{T}}
{\displaystyle {\psi _{\nu }^{T}}^{*}}
{\displaystyle c_{\nu }}
{\displaystyle \psi _{\mu }^{S}}
{\displaystyle \psi _{\nu }^{T}}
{\displaystyle {\tfrac {\textrm {d}}{{\textrm {d}}t}}c_{\nu }(t)=-{\tfrac {i}{\hbar }}\int \psi _{\mu }^{S}\,U_{T}\,{\psi _{\nu }^{T}}^{*}{\textrm {d}}x\,{\textrm {d}}y\,{\textrm {d}}z\,\exp[-{\tfrac {i}{\hbar }}(E_{\mu }^{S}-E_{\nu }^{T})t]}
{\displaystyle M_{\mu \nu }=\int _{z>z_{o}}\psi _{\mu }^{S}\,U_{T}\,{\psi _{\nu }^{T}}^{*}{\textrm {d}}x\,{\textrm {d}}y\,{\textrm {d}}z}
{\displaystyle |c_{\nu }(t)|^{2}=|M_{\mu \nu }|^{2}\,{\frac {4\sin ^{2}[{\tfrac {1}{2\hbar }}(E_{\mu }^{S}-E_{\nu }^{T})t]}{(E_{\mu }^{S}-E_{\nu }^{T})^{2}}}}
{\displaystyle |c_{\nu }(t)|^{2}}
{\displaystyle |c_{\nu }(t+{\textrm {d}}t)|^{2}}
{\displaystyle |c_{\nu }(t+{\textrm {d}}t)|^{2}-|c_{\nu }(t)|^{2}}
{\displaystyle {\textrm {d}}t}
{\displaystyle |c_{\nu }(t)|^{2}}
{\displaystyle \Gamma _{\mu \rightarrow \nu }\;{\overset {\underset {\mathrm {def} }{}}{=}}\;{\frac {{\textrm {d}}\,}{{\textrm {d}}t}}|c_{\nu }(t)|^{2}={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}\,{\frac {\sin[(E_{\mu }^{S}-E_{\nu }^{T}){\tfrac {t}{\hbar }}]}{\pi (E_{\mu }^{S}-E_{\nu }^{T})}}}
{\displaystyle {\tfrac {1}{\hbar }}t}
{\displaystyle (E_{\mu }^{S}-E_{\nu }^{T})}
{\displaystyle E_{\mu }^{S}=E_{\nu }^{T}}
{\displaystyle \Gamma _{\mu \rightarrow \nu }={\tfrac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}\,\delta (E_{\mu }^{S}-E_{\nu }^{T})}
{\displaystyle \delta (E_{\mu }^{S}-E_{\nu }^{T})}
{\displaystyle E_{\mu }^{S}}
{\displaystyle \Gamma _{\mu \rightarrow \nu }={\tfrac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}\,\rho _{T}(E_{\mu }^{S})}
{\displaystyle \varepsilon }
{\displaystyle \varepsilon +\mathrm {d} \varepsilon }
{\displaystyle \rho _{S}(\varepsilon )\mathrm {d} \varepsilon }
{\displaystyle 2e\cdot \rho _{S}(\varepsilon )\mathrm {d} \varepsilon }
{\displaystyle V}
{\displaystyle f}
{\displaystyle \varepsilon }
{\displaystyle f(E_{F}-eV+\varepsilon )-f(E_{F}+\varepsilon )}
{\displaystyle E_{F}-eV}
{\displaystyle E_{F}}
{\displaystyle \varepsilon =0}
{\displaystyle E_{F}}
{\displaystyle E_{F}+eV}
{\displaystyle \varepsilon =eV}
{\displaystyle 2e\cdot \rho _{S}(E_{F}-eV+\varepsilon )\mathrm {d} \varepsilon }
{\displaystyle f(E_{F}-eV+\varepsilon )-f(E_{F}+\varepsilon )}
{\displaystyle \Gamma }
{\displaystyle I_{t}={\frac {4\pi e}{\hbar }}\int _{-\infty }^{+\infty }[f(E_{F}-eV+\varepsilon )-f(E_{F}+\varepsilon )]\,\rho _{S}(E_{F}-eV+\varepsilon )\,\rho _{T}(E_{F}+\varepsilon )\,|M|^{2}\,d\varepsilon }
{\displaystyle I_{t}={\frac {4\pi e}{\hbar }}\int _{0}^{eV}\rho _{S}(E_{F}-eV+\varepsilon )\,\rho _{T}(E_{F}+\varepsilon )\,|M|^{2}\,d\varepsilon }
{\displaystyle I_{t}\propto \int _{0}^{eV}\rho _{S}(E_{F}-eV+\varepsilon )\,\rho _{T}(E_{F}+\varepsilon )\,d\varepsilon }
{\displaystyle M_{\mu \nu }=\int _{z>z_{o}}\psi _{\mu }^{S}\,U_{T}\,{\psi _{\nu }^{T}}^{*}{\textrm {d}}x\,{\textrm {d}}y\,{\textrm {d}}z}
{\displaystyle U_{T}\,{\psi _{\nu }^{T}}^{*}}
{\displaystyle M_{\mu \nu }=\int _{z>z_{o}}\left({\psi _{\nu }^{T}}^{*}E_{\mu }\psi _{\mu }^{S}+\psi _{\mu }^{S}{\tfrac {\hbar ^{2}}{2m}}{\tfrac {\partial ^{2}}{\partial z^{2}}}{\psi _{\nu }^{T}}^{*}\right){\textrm {d}}x\,{\textrm {d}}y\,{\textrm {d}}z}
{\displaystyle E_{\mu }\,{\psi _{\mu }^{S}}}
{\displaystyle \psi _{\mu }^{S}}
{\displaystyle M_{\mu \nu }=-{\tfrac {\hbar ^{2}}{2m}}\int _{z>z_{o}}\left({\psi _{\nu }^{T}}^{*}{\tfrac {\partial ^{2}}{\partial z^{2}}}{\psi _{\mu }^{S}}-{\psi _{\mu }^{S}}{\tfrac {\partial ^{2}}{\partial z^{2}}}{\psi _{\nu }^{T}}^{*}\right){\textrm {d}}x\,{\textrm {d}}y\,{\textrm {d}}z}
{\displaystyle \partial _{z}\left({\psi _{\nu }^{T}}^{*}\,\partial _{z}\psi _{\mu }^{S}-{\psi _{\mu }^{S}}\,\partial _{z}{\psi _{\nu }^{T}}^{*}\right)}
{\displaystyle M_{\mu \nu }={\tfrac {\hbar ^{2}}{2m}}\int _{z=z_{o}}\left({\psi _{\mu }^{S}}{\tfrac {\partial }{\partial z}}{\psi _{\nu }^{T}}^{*}-{\psi _{\nu }^{T}}^{*}{\tfrac {\partial }{\partial z}}{\psi _{\mu }^{S}}\right){\textrm {d}}x\,{\textrm {d}}y}
{\displaystyle \Gamma _{\mu \rightarrow \nu }}
Gallery of STM imagesEdit
Early inventionEdit
Other related techniquesEdit
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{\textstyle {\mathrm{\Δ}}_{}\quad }
{\textstyle {\mathrm{\Δ}}_{c\quad }\=\quad \{\quad \mathrm{\α}\quad \in {\mathrm{\Δ}}_{}\quad \|\quad {\mathrm{\α}}_{\|\mathrm{\𝔞}}\quad \=\quad 0\}\.}
{\textstyle \left\{{a}_{1}\,\quad {a}_{2\quad }\,\quad ..\.\quad \,\quad {a}_{s}\right\}}
{\textstyle \left\{{a}_{1}\,\quad {a}_{2\quad }\,\quad ..\.\quad \,\quad {a}_{s}\,\quad {h}_{s\+1}\,\quad ..\.\quad \,\quad {h}_{m}\right\}}
{\textstyle \left\{{a}_{1}\,\quad {a}_{2\quad }\,\quad ..\.\quad \,\quad {a}_{s}\right\}\quad }
{\textstyle x\quad \in \quad \mathrm{\𝔤}}
{\textstyle {\mathrm{\α}}_{}\,\quad }
{\textstyle \mathrm{ad}\left({a}_{i}\right)\left(x\right)\quad \=\quad 0}
{\textstyle i\quad \=\quad 1\,2\,\quad ..\.\quad \,s\.\quad }
{\mathrm{Δ}}^{+}
{\mathrm{Δ}}^{+}/{\mathrm{Δ}}_{{c}^{}}^{+}
{\textstyle {\mathrm{\Δ}}_{0}\quad }
{\textstyle {\mathrm{\Δ}}_{0\quad c}\quad \=\quad {\mathrm{\Δ}}_{0}\quad \bigcap {\mathrm{\Δ}}_{c}\quad \.}
{\textstyle {\mathrm{\α}}_{}\in {\mathrm{\Δ}}_{0}\quad \/{\mathrm{\Δ}}_{0\quad c}\,\quad }
{\textstyle \mathrm{\α}\'\quad }
{\textstyle \overline{\mathrm{\α}}\quad -\quad \mathrm{\α}\'\in \mathrm{span}\left({\mathrm{\Δ}}_{0}\right)\.}
{\textstyle {\mathrm{\α}}_{}\'\quad }
{\textstyle \mathrm{\α}\quad \quad }
{\textstyle {\mathrm{\alpha }}_{}\in {\mathrm{\Delta }}_{0}\quad \/{\mathrm{\Delta }}_{0\quad c}}
{\textstyle \quad }
{\textstyle \quad }
{\textstyle \mathrm{sl}\left(n\right)\,\quad \mathrm{su}\left(p\,\quad q\right)\,\quad \mathrm{su}\ast \left(n\right)\,\quad \mathrm{so}\left(p\,\quad q\right)\,\quad \quad \mathrm{so}\ast \left(n\right)\,\quad \mathrm{sp}\left(n\,\quad \mathrm{\ℝ}\right)\,\quad \mathrm{sp}\left(p\,\quad q\right)\,\quad \mathrm{sp}\left(n\right)\.}
{\textstyle \mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:}
{\textstyle \mathrm{with}\left(\mathrm{LieAlgebras}\right)\:}
{\textstyle \mathrm{SatakeDiagram}\left(''su(9,\; 4)''\right)}
{\textstyle \mathrm{SatakeDiagram}\left(''su(6,\; 6)''\right)}
{\textstyle \mathrm{SatakeDiagram}\left(''sp(10,\; 6)''\right)}
{\textstyle \mathrm{SatakeDiagram}\left(''so(12,\; 4)''\right)}
{\textstyle \mathrm{su}\left(6\,\quad 2\right)\.\quad }
{\textstyle \mathrm{SatakeDiagram}\left(''su(6,\; 2)''\right)}
{\textstyle \mathrm{su}\left(6\,\quad 2\right)\quad }
{A}_{7}
{\textstyle \mathrm{DynkinDiagram}\left(''A''\,7\right)}
{\textstyle \mathrm{su}\left(6\,\quad 2\right)\.\quad }
{\textstyle \quad \mathrm{su}\left(6\,\quad 2\right)\.}
{\textstyle \mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left(''su(6,\; 2)''\,\mathrm{su62}\,\mathrm{labelformat}=''gl''\,\mathrm{labels}=\left['E'\,'\mathrm{\theta }'\right]\right)\:}
{\textstyle \mathrm{DGsetup}\left(\mathrm{LD}\right)}
\textcolor[rgb]{0,0,1}{\mathrm{Lie algebra: su62}}
{\textstyle \quad \mathrm{su}\left(6\,2\right)}
{\textstyle \mathrm{CSA}≔\left[\mathrm{E11}\,\mathrm{E22}\,\mathrm{Ei11}\,\mathrm{Ei22}\,\mathrm{Ei55}\,\mathrm{Ei66}\,\mathrm{Ei77}\right]}
\textcolor[rgb]{0,0,1}{\mathrm{CSA}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{E11}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{E22}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Ei11}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Ei22}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Ei55}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Ei66}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Ei77}}\right]
{\textstyle \mathrm{K1}≔\mathrm{Killing}\left(\left[\mathrm{Ei11}\,\mathrm{Ei22}\,\mathrm{Ei55}\,\mathrm{Ei66}\,\mathrm{Ei77}\right]\right)}
{\textstyle \textcolor[rgb]{0,0,1}{\mathrm{K1}}\textcolor[rgb]{0,0,1}{:=}\left[\begin{array}{rrrrr}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{64}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{32}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{16}& \textcolor[rgb]{0,0,1}{16}\\ \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{32}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{64}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{16}& \textcolor[rgb]{0,0,1}{16}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{32}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{16}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{16}\\ \textcolor[rgb]{0,0,1}{16}& \textcolor[rgb]{0,0,1}{16}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{16}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{32}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{16}\\ \textcolor[rgb]{0,0,1}{16}& \textcolor[rgb]{0,0,1}{16}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{16}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{16}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{32}\end{array}\right]}
{\textstyle \mathrm{LinearAlgebra}:-\mathrm{IsDefinite}\left(\mathrm{K1}\,\mathrm{query}='\mathrm{negative\_definite}'\right)}
\textcolor[rgb]{0,0,1}{\mathrm{true}}
{\textstyle A≔\left[\mathrm{E11}\,\mathrm{E22}\right]}
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{E11}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{E22}}\right]
{\textstyle \mathrm{K2}≔\mathrm{Killing}\left(A\right)}
\textcolor[rgb]{0,0,1}{\mathrm{K2}}\textcolor[rgb]{0,0,1}{:=}\left[\begin{array}{rr}\textcolor[rgb]{0,0,1}{32}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{32}\end{array}\right]
{\textstyle \mathrm{\𝔞}\mathit{\,}\mathit{\quad }}
\text{# RSD := RootSpaceDecomposition(CSA):}
xmlns="http://www.w3.org/1998/Math/MathML">RSD≔mapevalDG,table0,1,0,I,0,0,I=E28−IEi28,1,1,I,−I,0,0,0=E14+IEi14,1,0,3I,2I,−I,0,0=E15−IEi15,0,−1,0,−I,0,−I,I=E47+IEi47,0,0,2I,2I,−I,−I,I=E57−IEi57,0,0,0,0,I,−I,−I=E68−IEi68,−1,−1,I,−I,0,0,0=E32+IEi32,0,−1,0,I,I,−I,0=E46−IEi46,2,0,0,0,0,0,0=Ei13,0,−1,0,−I,−I,I,0=E46+IEi46,0,0,0,0,0,−I,2I=E78+IEi78,−1,0,−3I,−2I,I,0,0=E35+IEi35,−1,−1,−I,I,0,0,0=E32−IEi32,0,0,0,0,−I,I,I=E68+IEi68,−1,0,−I,0,0,−I,I=E37+IEi37,1,−1,−I,I,0,0,0=E12+IEi12,0,2,0,0,0,0,0=Ei24,−1,1,−I,I,0,0,0=E21−IEi21,0,0,2I,2I,−I,0,−I=E58−IEi58,0,−1,0,I,0,I,−I=E47−IEi47,0,0,0,0,I,−2I,I=E67−IEi67,0,1,0,−I,0,0,−I=E28+IEi28,0,0,0,0,0,I,−2I=E78−<
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Resistances in parallel — lesson. Science State Board, Class 10.
Components in parallel circuits are connected to the source in two or more loops. There are multiple paths for the electric charge to flow in a parallel circuit.
Even if the circuit is broken at any point in the loop, the current can flow through the circuit. Any electric appliances linked to the circuit will also function. Hence, the electrical wiring in our houses is made of parallel circuits.
The sum of the individual currents in each parallel branch in a parallel circuit equals the main current flowing into or out of the parallel branches. The potential difference across separate parallel branches are the same.
Parallel connections of resistors
Let the resistances of three resistors be \(R_1\), \(R_2\) and \(R_3\), connected parallel across point \(A\) and \(B\). Let '\(I\)' be the current flowing through the circuit.
The potential difference across the three resistances are the same and equal to the potential difference between point \(A\) and \(B\). This potential difference is measured using the voltmeter.
The current \(I\) starts from the positive terminal of the battery and reaches point \(A\). The current passing through the resistors \(R_1\), \(R_2\) and \(R_3\) is divided into three branches \(I_1\), \(I_2\) and \(I_3\), respectively.
According to Ohm's law, the current \(I_1\), \(I_2\) and \(I_3\) are given as,
\begin{array}{l}{I}_{1}=\frac{V}{{R}_{1}}\\ \\ {I}_{2}=\frac{V}{{R}_{2}}\\ \\ {I}_{3}=\frac{V}{{R}_{3}}\end{array}
Then, the total current passing through the circuit is
Substituting the values of \(I_1\), \(I_2\) and \(I_3\) in the above equation, we get
---- (eq. 1)
Let \(R_P\) be the effective resistance of the parallel combination of resistors in the circuit. Then, the (eq. 1) becomes
I\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{V}{{R}_{P}}
On combining (eq. 1) and (eq. 2), we get
Hence, the sum of the reciprocals of the individual resistances is equal to the reciprocal of the effective or equivalent resistance when several resistors are connected in parallel.
The effective or equivalent resistance is
\frac{R}{n}
when the '\(n\)' number of resistors having an equal resistances '\(R\)' are connected in parallel.
The above statement can be given in the form of an equation as,
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Dirac sea - Wikipedia
The Dirac sea is a theoretical model of the void as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930[1] to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons (electrons traveling near the speed of light).[2] The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, before its experimental discovery in 1932.[nb 1]
In hole theory, the solutions with negative time evolution factors[clarification needed] are reinterpreted as representing the positron, discovered by Carl Anderson. The interpretation of this result requires a Dirac sea, showing that the Dirac equation is not merely a combination of special relativity and quantum mechanics, but it also implies that the number of particles cannot be conserved.[3]
Dirac sea theory has been displaced by quantum field theory, though they are mathematically compatible.
2 Inelegance of Dirac sea
5 Revival in the theory of causal fermion systems
Similar ideas on holes in crystals had been developed by soviet physicist Yakov Frenkel in 1926, but there is no indication the concept was discussed with Dirac when the two met in a soviet physics Congress in the summer of 1928.
The origins of the Dirac sea lie in the energy spectrum of the Dirac equation, an extension of the Schrödinger equation consistent with special relativity, an equation that Dirac had formulated in 1928. Although this equation was extremely successful in describing electron dynamics, it possesses a rather peculiar feature: for each quantum state possessing a positive energy E, there is a corresponding state with energy -E. This is not a big difficulty when an isolated electron is considered, because its energy is conserved and negative-energy electrons may be left out. However, difficulties arise when effects of the electromagnetic field are considered, because a positive-energy electron would be able to shed energy by continuously emitting photons, a process that could continue without limit as the electron descends into ever lower energy states. However, real electrons clearly do not behave in this way.
Dirac's solution to this was to rely on the Pauli exclusion principle. Electrons are fermions, and obey the exclusion principle, which means that no two electrons can share a single energy state within an atom. Dirac hypothesized that what we think of as the "vacuum" is actually the state in which all the negative-energy states are filled, and none of the positive-energy states. Therefore, if we want to introduce a single electron we would have to put it in a positive-energy state, as all the negative-energy states are occupied. Furthermore, even if the electron loses energy by emitting photons it would be forbidden from dropping below zero energy.
Dirac further pointed out that a situation might exist in which all the negative-energy states are occupied except one. This "hole" in the sea of negative-energy electrons would respond to electric fields as though it were a positively charged particle. Initially, Dirac identified this hole as a proton. However, Robert Oppenheimer pointed out that an electron and its hole would be able to annihilate each other, releasing energy on the order of the electron's rest energy in the form of energetic photons; if holes were protons, stable atoms would not exist.[4] Hermann Weyl also noted that a hole should act as though it has the same mass as an electron, whereas the proton is about two thousand times heavier. The issue was finally resolved in 1932, when the positron was discovered by Carl Anderson, with all the physical properties predicted for the Dirac hole.
Inelegance of Dirac seaEdit
Despite its success, the idea of the Dirac sea tends not to strike people as very elegant. The existence of the sea implies an infinite negative electric charge filling all of space. In order to make any sense out of this, one must assume that the "bare vacuum" must have an infinite positive charge density which is exactly cancelled by the Dirac sea. Since the absolute energy density is unobservable—the cosmological constant aside—the infinite energy density of the vacuum does not represent a problem. Only changes in the energy density are observable. Geoffrey Landis (author of "Ripples in the Dirac Sea", a hard science fiction short story) also notes[citation needed] that Pauli exclusion does not definitively mean that a filled Dirac sea cannot accept more electrons, since, as Hilbert elucidated, a sea of infinite extent can accept new particles even if it is filled. This happens when we have a chiral anomaly and a gauge instanton.
The development of quantum field theory (QFT) in the 1930s made it possible to reformulate the Dirac equation in a way that treats the positron as a "real" particle rather than the absence of a particle, and makes the vacuum the state in which no particles exist instead of an infinite sea of particles. This picture is much more convincing, especially since it recaptures all the valid predictions of the Dirac sea, such as electron-positron annihilation. On the other hand, the field formulation does not eliminate all the difficulties raised by the Dirac sea; in particular the problem of the vacuum possessing infinite energy.
Mathematical expressionEdit
Upon solving the free Dirac equation,
{\displaystyle i\hbar {\frac {\partial \Psi }{\partial t}}=(c{\hat {\boldsymbol {\alpha }}}\cdot {\hat {\boldsymbol {p}}}+mc^{2}{\hat {\beta }})\Psi ,}
one finds[5]
{\displaystyle \Psi _{\mathbf {p} \lambda }=N\left({\begin{matrix}U\\{\frac {(c{\hat {\boldsymbol {\sigma }}}\cdot {\boldsymbol {p}})}{mc^{2}+\lambda E_{p}}}U\end{matrix}}\right){\frac {\exp[i(\mathbf {p} \cdot \mathbf {x} -\varepsilon t)/\hbar ]}{{\sqrt {2\pi \hbar }}^{3}}},}
{\displaystyle \varepsilon =\pm E_{p},\quad E_{p}=+c{\sqrt {\mathbf {p} ^{2}+m^{2}c^{2}}},\quad \lambda =\operatorname {sgn} \varepsilon }
for plane wave solutions with 3-momentum p. This is a direct consequence of the relativistic energy-momentum relation
{\displaystyle E^{2}=p^{2}c^{2}+m^{2}c^{4}}
upon which the Dirac equation is built. The quantity U is a constant 2 × 1 column vector and N is a normalization constant. The quantity ε is called the time evolution factor, and its interpretation in similar roles in, for example, the plane wave solutions of the Schrödinger equation, is the energy of the wave (particle). This interpretation is not immediately available here since it may acquire negative values. A similar situation prevails for the Klein–Gordon equation. In that case, the absolute value of ε can be interpreted as the energy of the wave since in the canonical formalism, waves with negative ε actually have positive energy Ep.[6] But this is not the case with the Dirac equation. The energy in the canonical formalism associated with negative ε is –Ep.[7]
The Dirac sea interpretation and the modern QFT interpretation are related by what may be thought of as a very simple Bogoliubov transformation, an identification between the creation and annihilation operators of two different free field theories.[citation needed] In the modern interpretation, the field operator for a Dirac spinor is a sum of creation operators and annihilation operators, in a schematic notation:
{\displaystyle \psi (x)=\sum a^{\dagger }(k)e^{ikx}+a(k)e^{-ikx}}
An operator with negative frequency lowers the energy of any state by an amount proportional to the frequency, while operators with positive frequency raise the energy of any state.
In the modern interpretation, the positive frequency operators add a positive energy particle, adding to the energy, while the negative frequency operators annihilate a positive energy particle, and lower the energy. For a fermionic field, the creation operator
{\displaystyle a^{\dagger }(k)}
gives zero when the state with momentum k is already filled, while the annihilation operator
{\displaystyle a(k)}
gives zero when the state with momentum k is empty.
But then it is possible to reinterpret the annihilation operator as a creation operator for a negative energy particle. It still lowers the energy of the vacuum, but in this point of view it does so by creating a negative energy object. This reinterpretation only affects the philosophy. To reproduce the rules for when annihilation in the vacuum gives zero, the notion of "empty" and "filled" must be reversed for the negative energy states. Instead of being states with no antiparticle, these are states that are already filled with a negative energy particle.
The price is that there is a nonuniformity in certain expressions, because replacing annihilation with creation adds a constant to the negative energy particle number. The number operator for a Fermi field[8] is:
{\displaystyle N=a^{\dagger }a=1-aa^{\dagger }}
which means that if one replaces N by 1−N for negative energy states, there is a constant shift in quantities like the energy and the charge density, quantities that count the total number of particles. The infinite constant gives the Dirac sea an infinite energy and charge density. The vacuum charge density should be zero, since the vacuum is Lorentz invariant, but this is artificial to arrange in Dirac's picture. The way it is done is by passing to the modern interpretation.
Dirac's idea is more directly applicable to solid state physics, where the valence band in a solid can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductors, though they are never referred to as "positrons". Unlike in particle physics, there is an underlying positive charge—the charge of the ionic lattice—that cancels out the electric charge of the sea.
Revival in the theory of causal fermion systemsEdit
Dirac's original concept of a sea of particles was revived in the theory of causal fermion systems, a recent proposal for a unified physical theory. In this approach, the problems of the infinite vacuum energy and infinite charge density of the Dirac sea disappear because these divergences drop out of the physical equations formulated via the causal action principle.[9] These equations do not require a preexisting space-time, making it possible to realize the concept that space-time and all structures therein arise as a result of the collective interaction of the sea states with each other and with the additional particles and "holes" in the sea.
^ This was not the original intent of Dirac though, as the title of his 1930 paper (A Theory of Electrons and Protons) indicates. But it soon afterwards became clear that the mass of holes must be that of the electron.
^ Greiner 2000
^ Alvarez-Gaume & Vazquez-Mozo 2005
^ Greiner 2000, pp. 107–109
^ Greiner 2000, p. 15
^ Greiner 2000, p. 117
^ Sattler 2010
^ Finster 2011
Alvarez-Gaume, Luis; Vazquez-Mozo, Miguel A. (2005). "Introductory Lectures on Quantum Field Theory". CERN Yellow Report CERN. 1 (96): 2010–001. arXiv:hep-th/0510040. Bibcode:2005hep.th...10040A.
Dirac, P. A. M. (1930). "A Theory of Electrons and Protons". Proc. R. Soc. Lond. A. 126 (801): 360–365. Bibcode:1930RSPSA.126..360D. doi:10.1098/rspa.1930.0013. JSTOR 95359.
Dirac, P. A. M. (1931). "Quantized Singularities In The Electromagnetic Fields". Proc. Roy. Soc. A. 133 (821): 60–72. Bibcode:1931RSPSA.133...60D. doi:10.1098/rspa.1931.0130. JSTOR 95639.
Finster, F. (2011). "A formulation of quantum field theory realizing a sea of interacting Dirac particles". Lett. Math. Phys. 97 (2): 165–183. arXiv:0911.2102. Bibcode:2011LMaPh..97..165F. doi:10.1007/s11005-011-0473-1. ISSN 0377-9017. S2CID 39764396.
Greiner, W. (2000). Relativistic Quantum Mechanics. Wave Equations (3rd ed.). Springer Verlag. ISBN 978-3-5406-74573. (Chapter 12 is dedicate to hole theory.)
Sattler, K. D. (2010). Handbook of Nanophysics: Principles and Methods. CRC Press. pp. 10–4. ISBN 978-1-4200-7540-3. Retrieved 2011-10-24.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Dirac_sea&oldid=1083913589"
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EUDML | Augmented group systems and n-knots. EuDML | Augmented group systems and n-knots.
Silver, Daniel S.. "Augmented group systems and n-knots.." Mathematische Annalen 296.4 (1993): 585-594. <http://eudml.org/doc/165102>.
author = {Silver, Daniel S.},
keywords = {-knot; satellite construction; satellite -knots; incompressible Seifert surface; augmented group system; finitely presented group},
title = {Augmented group systems and n-knots.},
AU - Silver, Daniel S.
TI - Augmented group systems and n-knots.
KW - -knot; satellite construction; satellite -knots; incompressible Seifert surface; augmented group system; finitely presented group
n
-knot, satellite construction, satellite
n
-knots, incompressible Seifert surface, augmented group system, finitely presented group
Articles by Daniel S. Silver
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\eta
Galina Valentinovna Voskresenskaya1
1 Work position: Samara State University, the chair of algebra and geometry. Work address: 443011, Russia, Samara, acad.Pavlova street, house 1, room 406. Tel. (846-2) 34-54-38
\eta -
p
p\ne 2,
\eta
SL\left(5,ℂ\right).
\eta
p
p\ne 2,
\eta
SL\left(5,ℂ\right).
Galina Valentinovna Voskresenskaya 1
author = {Galina Valentinovna Voskresenskaya},
%T Multiplicative Dedekind $\eta $-function and representations of finite groups
Galina Valentinovna Voskresenskaya. Multiplicative Dedekind $\eta $-function and representations of finite groups. Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 359-380. doi : 10.5802/jtnb.495. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.495/
[1] A.J.F. Biagioli, The construction of modular forms as products of transforms of the Dedekind eta-function. Acta Arith. LIV (1990), 274–300. | MR: 1058891 | Zbl: 0718.11017
[2] H.S.M. Coxeter, W.O.J. Moser, Generators and relations for discrete groups. Springer-Verlag (1965), 161 pp. | MR: 174618 | Zbl: 0133.28002
\eta
-functions. Contemp. Math. 45 (1985), 89–98. | Zbl: 0578.10028
\eta
-products. Lecture Notes in Math. 1395 (1989), 173–200. (Springer-Verlag) | Zbl: 0688.10023
[5] M. Hall,jr, The theory of groups. The Macmillan Company. New York (1959). | MR: 103215 | Zbl: 0084.02202
[6] K. Harada, Another look at the Frame shapes of finite groups. J. Fac. Sci. Univ. Tokyo. Sect. IA. Math. 34 (1987), 491–512. | MR: 927599 | Zbl: 0653.10024
[7] T. Hiramatsu Theory of automorphic forms of weight 1. Advanced Studies in Pure Math. 13 (1988), 503–584. | MR: 971528 | Zbl: 0658.10031
[8] N. Ishii, Cusp forms of weight one, quartic reciprocity and elliptic curves. Nagoya Math. J. 98 (1985), 117–137. | MR: 792776 | Zbl: 0556.10019
[9] M. Koike, On McKay’s conjecture. Nagoya Math. J. 95 (1984), 85–89. | MR: 759465 | Zbl: 0548.10018
\eta
-products. Sci. Pap. Coll. Arts and Sci. Univ. Tokyo 35 (1986), 133–149 . | Zbl: 0597.10025
[11] Y. Martin, K. Ono, Eta-quotients and elliptic curves. Proc. Amer. Math. Soc. 125 (1997), 3169–3176. | MR: 1401749 | Zbl: 0894.11020
[12] G. Mason, Finite groups and Hecke operators. Math. Ann. 282 (1989), 381–409. | MR: 985239 | Zbl: 0636.10021
{M}_{24}
and certain automorphic forms. Contemp. Math. 45 (1985), 223–244. | MR: 822240 | Zbl: 0578.10029
[14] K. Ono, Shimura sums related to imaginary quadratic fields. Proc. Japan Acad. 70 (A) (1994), 146–151. | MR: 1291170 | Zbl: 0813.11031
[15] G.V. Voskresenskaya, Modular forms and representations of the dihedral group. Math. Notes. 63 (1998), 130–133. | MR: 1631789 | Zbl: 0923.11070
SL\left(5,ℂ\right).
Functional Anal. Appl. 29 (1995), 71–73. | MR: 1340307 | Zbl: 0847.11022
[17] G.V. Voskresenskaya, Modular forms and regular representations of groups of order 24. Math. Notes 60 (1996), 292–294. | MR: 1429128 | Zbl: 0923.11069
[18] G.V. Voskresenskaya, One special class of modular forms and group representations. J. Th. Nombres Bordeaux 11(1999), 247–262. | Numdam | MR: 1730443 | Zbl: 0954.11014
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An Odyssey to Discover Round and Squared Pizzas! – Math is in the Air
An Odyssey to Discover Round and Squared Pizzas!
April 16, 2016 Leave a Comment Written by Francesco Bonesi
The Geometry of a pizza
Hi, everyone. Today we’re going to speak about pizza!
Yes, I’m not crazy! I perfectly know it’s a blog on maths!
Indeed, we want to talk of geometry of pizza and why there are some people who prepare it round and some who prepare it squared…who is earning money from this?
But the real answer I want you to propone today is: do you prefer a round or a squared pizza?
I imagine that the average reaction is another question: “What’s the difference? It’s pizza!”
The difference is huge and I’ll show what it is. Have fun!
Dido and oxhide
There are plenty of mathematical problems called maximum/minimum problems. In general they can be solved in many ways but I have no intentions of explaining the general theory. Some of you probably knew about it from school or at University. To people who reads it for the first time: “don’t be scared!”
In a few words, we try to understand how to minimize or maximize quantities under certain circumstances. An example from history: once upon a time there was Dido, Tyre’s queen, arrived in North Africa at the court of King Iarbas as a refugee. Iarbas decided to give her as much and as could be encompassed by an oxhide. Apart from the strange offering, Dido made lots of strips from the oxhide and made a very tiny rope. Finally, she used it to cover an huge land, the land in which Cartago was founded. Now, the question is: what’s the best geometrical form, having a rope, to obtain the maximum area?
In this example, the question is equivalent to: having fixed the perimeter, what’s the figure with maximum area? The answer is: the circle!
Another example: in an opposite way, fixing the area, what is the figure with smaller perimeter? The answer is: the circle again!
Do you know how to show this fact? To simplify the problem let us consider regular polygons, that are polygons with equal edges. It is possible to show that, fixed area A, the perimeter of a regular n-gon is: <<formula>>
So, this is a very well known formula, don’t you know it? 😛
We see that the perimeter decreases as n increases, so we could say that the minimum perimeter is when there are an infinite number of edges…so? The circle!
If you compute the limit as n goes to infinity, you get <<formula>> and it’s the perimeter of a circle of area A!
So, we answered the question!
Come back to Pizza! We discovered that, fixed area, a round pizza has less perimeter than any other shape. But…perimeter is a thing, border is another. The border is an area, not a line!!
So, now the real question is? Who has the minimal border? Who maximal?
Let us start writing some formulas. A is the fixed area, n is the number of the edges of the regular polygon. Let b be the measure of the border.
The n-gon can be decomposed in n isosceles triangles. The angle in front of the base is
\frac{2\pi}{n}
and the area of this triangle is
A^{,} = \frac{a^2}{2} \sin\left(\frac{2\pi}{n}\right)
a
is the apothem. Knowing that
A=n A^{,}
A =\frac{n a^2}{2} \sin\left(\frac{2\pi}{n}\right)
a = \sqrt{ \frac{2A}{n\sin\left(\frac{2\pi}{n}\right)} }
If the pizza was a circle of area
A
A=\pi r^2
r
is the radius. Hence
r=\sqrt{\frac{A}{\pi}}
. Comparing
r
a
a = \sqrt{ \frac{2\pi}{n\sin\left(\frac{2\pi}{n}\right)} } r
. That’s the relation between a and r.
Now, either to the n-gon and to the circle, we cut a border thick
b
. We avoid to make all computations, the final formula for the internal areas of the little triangles are
A_{int}^{,} = \tan\left( \frac{\pi}{n} \right) \left( \sqrt{ \frac{\pi}{n \tan\left( \frac{\pi}{n} \right) } } - b \right)^2
. So we get:
A_{int}^P = n A_{int}^{,} = n\tan\left( \frac{\pi}{n} \right) \left( \sqrt{ \frac{\pi}{n \tan\left( \frac{\pi}{n} \right) } } - b \right)^2
For the circle, the situation is similar:
A_{int}^C = \pi(r-b)^2
.We compare now the two areas computing:
A_{int}^C ? A_{int}^P
. Using the relation between r and a, we get that the difference is positive if
b
has a value between 0 and
\frac{2\left( \sqrt{\pi n \tan\left( \frac{\pi}{n} \right)} \right)}{ n \tan\left( \frac{\pi}{n} \right) - \pi} r
\frac{2\left( \sqrt{\pi n \tan\left( \frac{\pi}{n} \right)} \right)}{ n \tan\left( \frac{\pi}{n} \right) - \pi}
n\geq 3
0.8749
and, as n goes to infinity, it goes to
1
This means that if b is less than the 87% of the radius, then the internal area of the circle is greater that the n-gon’s one, for all
n\geq 3
. The converse holds for borders!
So, the question “do you prefer a round or a squared pizza? ” has now an answer. Since the square is a 4-gon the answer is: if you like borders, choose a squared pizza, otherwise a round one!
Conclusion (of the average man)
With all those letters, we lost the gist. Let us fix some values.
An average pizza has an area of
A=1000 cm^2
. It turns out that, if the border is between 0 and 17 centimeters, the internal area of the square is smaller than the circle’s one. Since the border is 1 or 2 cm thick, than our result holds!
If you’re not crazy/deviated/maniac/sociopathic/my-cousin/curious, it’s enough for today. See you next time! 😛
Conclusion (of an applied mathematician, of a economist, of a financial man and similar)
There are other questions: if a pizza company makes round pizzas, how much money does it lose?
Indeed, having a greater area to cover with ingredients, it should spend more.
I made some computations: fixing
A=1000cm^2
b=1cm
, the internal area of a squared pizza is
877,5 cm^2
and of a round pizza is
891 cm^2
. So a squared pizza has 1.5% less ingredients than a round one.
Simplifying, the pizza company would save the 1.5% of the price of ingredients making squared pizzas instead of round pizzas. This means that, if making 100 pizzas costs 400 euros (100 euros for bases and 300 euros for ingredients), supposing that he sells pizzas at a price of 6 euros, he would earn 600-400=200 euros.
Leaving the price to 6 euros, but making squared pizzas, he would save the 1.5% of the costs of ingredients. Hence he would save 4.50 euros. It’s not too much, but it’s only 100 pizzas! Do you want to save more? Make bigger pizzas!
Conclusion (of a pure mathematician)
What? Have I used numbers?
Well, this is the end! Sayonara!
Analysis, Maximum-minimum problems
areas, borders, circle, Dido, ingredients, maximum minimum, oxhide, perimeters, pizza, polygons
Euclidean distance and others: metric spaces and topology [Part 2]
This sentence is false: if yuo can raed tihs, yuo msut be raelly smrat - logic and paradoxes in Probability
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Check coprime relation - MATLAB iscoprime - MathWorks France
Coprime Array Elements
Check coprime relation
iscp = iscoprime(x)
[iscp,ispcp,pidx,pgcd] = iscoprime(x)
iscp = iscoprime(x) returns true if all elements of x are coprime and false if two or more elements of x have a greatest common divisor (gcd) greater than 1.
[iscp,ispcp,pidx,pgcd] = iscoprime(x) checks if pairs of elements of x have a greatest common divisor greater than 1. This syntax also returns the indices of all pairs of elements of x and the greatest common divisor of each pair.
Create an array x whose elements are
9=3×3
15=3×5
25=5×5
. Verify that all elements of x are coprime.
x = [9 15 25];
iscp = logical
Verify that at least one pair of elements of x has a greatest common divisor greater than 1. Output the pairs and their greatest common divisors.
[~,ispcp,pidx,pgcd] = iscoprime(x)
ispcp = logical
pidx = 2×3
pgcd = 1×3
Input array, specified as a row vector of positive integers.
iscp — True if all elements are coprime
True if all elements are coprime, returned as a logical scalar.
ispcp — True if elements are pairwise coprime
True if all elements are pairwise coprime, returned as a logical scalar. ispcp is true if x has no two elements whose greatest common divisor is greater than 1. ispcp is false if any two elements of x have as greatest common divisor a number greater than 1.
pidx — Array pair indices
Array pair indices, returned as a two-row matrix. pidx has
\left(\begin{array}{c}n\\ 2\end{array}\right)=\frac{1}{2}n\left(n-1\right)
columns. Each column of pidx specifies the indices of a pair of elements in x.
pgcd — Pair greatest common divisors
Pair greatest common divisors, returned as a row vector with a number of elements equal to the number of columns of pidx. Each element of pgcd is the greatest common divisor of the two elements of x identified by the indices in the corresponding column of pidx.
coincidence | crt
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Compact multiclass model for support vector machines (SVMs) and other classifiers - MATLAB - MathWorks India
\begin{array}{cccc}& \text{Learner 1}& \text{Learner 2}& \text{Learner 3}\\ \text{Class 1}& 1& 1& 0\\ \text{Class 2}& -1& 0& 1\\ \text{Class 3}& 0& -1& -1\end{array}
\stackrel{^}{k}
\stackrel{^}{k}=\underset{k}{\text{argmin}}\frac{\sum _{l=1}^{B}|{m}_{kl}|g\left({m}_{kl},{s}_{l}\right)}{\sum _{l=1}^{B}|{m}_{kl}|}.
{L}_{d}\approx ⌈10{\mathrm{log}}_{2}K⌉
{L}_{s}\approx ⌈15{\mathrm{log}}_{2}K⌉
\Delta \left({k}_{1},{k}_{2}\right)=0.5\sum _{l=1}^{L}|{m}_{{k}_{1}l}||{m}_{{k}_{2}l}||{m}_{{k}_{1}l}-{m}_{{k}_{2}l}|,
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Remote Sensing | Free Full-Text | Consistency between Satellite Ocean Colour Products under High Coloured Dissolved Organic Matter Absorption in the Baltic Sea
Arctic Sea Ice Classification Based on CFOSAT SWIM Data at Multiple Small Incidence Angles
Algorithm of Additional Correction of Level 2 Remote Sensing Reflectance Data Using Modelling of the Optical Properties of the Black Sea Waters
Motion Blur Removal for Uav-Based Wind Turbine Blade Images Using Synthetic Datasets
Evolution of Ocean Color Atmospheric Correction: 1970–2005
Selmes, N.
Kwiatkowska, E.
Consistency between Satellite Ocean Colour Products under High Coloured Dissolved Organic Matter Absorption in the Baltic Sea
Stefan G. H. Simis
Nick Selmes
European Organisation for the Exploitation of Meteorological Satellites, Eumetsat Allee 1, 64295 Darmstadt, Germany
Academic Editors: Cédric Jamet and Jae-Hyun Ahn
(This article belongs to the Special Issue Atmospheric Correction for Remotely Sensed Ocean Color Data)
Ocean colour (OC) remote sensing is an important tool for monitoring phytoplankton in the global ocean. In optically complex waters such as the Baltic Sea, relatively efficient light absorption by substances other than phytoplankton increases product uncertainty. Sentinel-3 OLCI-A, Suomi-NPP VIIRS and MODIS-Aqua OC radiometric products were assessed using Baltic Sea in situ remote sensing reflectance (
{R}_{rs}
) from ferry tracks ([email protected]) and at two Aerosol Robotic Network for Ocean Colour (AERONET-OC) sites from April 2016 to September 2018. A range of atmospheric correction (AC) processors for OLCI-A were evaluated. POLYMER performed best with <23 relative % difference at 443, 490 and 560 nm compared to in situ
{R}_{rs}
and 28% at 665 nm, suggesting that using this AC for deriving Chl a will be the most accurate. Suomi-VIIRS and MODIS-Aqua underestimated
{R}_{rs}
by 35, 29, 22 and 39% and 34, 22, 17 and 33% at 442, 486, 560 and 671 nm, respectively. The consistency between different AC processors for OLCI-A and MODIS-Aqua and VIIRS products was relatively poor. Applying the POLYMER AC to OLCI-A, MODIS-Aqua and VIIRS may produce the most accurate
{R}_{rs}
and Chl a products and OC time series for the Baltic Sea. View Full-Text
Keywords: Sentinel-3; OLCI; validation; remote sensing reflectance; atmospheric correction; Baltic Sea; MODIS-Aqua; Suomi-VIIRS Sentinel-3; OLCI; validation; remote sensing reflectance; atmospheric correction; Baltic Sea; MODIS-Aqua; Suomi-VIIRS
Tilstone, G.H.; Pardo, S.; Simis, S.G.H.; Qin, P.; Selmes, N.; Dessailly, D.; Kwiatkowska, E. Consistency between Satellite Ocean Colour Products under High Coloured Dissolved Organic Matter Absorption in the Baltic Sea. Remote Sens. 2022, 14, 89. https://doi.org/10.3390/rs14010089
Tilstone GH, Pardo S, Simis SGH, Qin P, Selmes N, Dessailly D, Kwiatkowska E. Consistency between Satellite Ocean Colour Products under High Coloured Dissolved Organic Matter Absorption in the Baltic Sea. Remote Sensing. 2022; 14(1):89. https://doi.org/10.3390/rs14010089
Tilstone, Gavin H., Silvia Pardo, Stefan G.H. Simis, Ping Qin, Nick Selmes, David Dessailly, and Ewa Kwiatkowska. 2022. "Consistency between Satellite Ocean Colour Products under High Coloured Dissolved Organic Matter Absorption in the Baltic Sea" Remote Sensing 14, no. 1: 89. https://doi.org/10.3390/rs14010089
Dr. Gavin Tilstone is a NERC Merit Scientist with over 25 years’ experience in the fields of Inherent and Apparent Optical properties, phytoplankton, photosynthesis, primary production and ocean colour remote sensing. He leads research on satellite ocean colour validation and algorithm development, on water quality including detecting harmful algal blooms and the use of satellite ocean colour in quantifying the marine carbon cycle. He works Earth Observation Science and Applications group. Gavin’s role is to expand the use of Earth Observation data for studying the environment and quantifying the marine carbon cycle by bringing in research funding and leading projects. Dr. Tilstone manages a research team who work on optics and ocean colour remote sensing within International, European and UKRI research projects. The results from these projects contribute to impact in policy using ocean colour space observations to address key questions on climate change, eutrophication and harmful algal blooms. Dr. Tilstone has worked on >30 International and EU research projects, has considerable management experience as Principal Investigator and Co-Investigator on >20 ESA, EU, NERC and Commercial projects, has employed a wide range post-doctoral and visiting fellowships and has supervised nine PhD projects. Dr. Tilstone has published over 90 peer-reviewed papers, with an H-index of 36 and i-10 index of 72.
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Tropical_monsoon_climate Knowpia
An area of tropical monsoon climate (occasionally known as a tropical wet climate or a tropical monsoon and trade-wind littoral climate) is a type of climate that corresponds to the Köppen climate classification category subtype "Am". Tropical monsoon climates have monthly mean temperatures above 18 °C (64 °F) in every month of the year and a dry season.[1]: 200–1 Tropical monsoon climates is the intermediate climate between the wet Af (or tropical rainforest climate) and the drier Aw (or tropical savanna climate) in terms of dryness.
{\textstyle 100-\left({\frac {Total\ Annual\ Precipitation\ (mm)}{25}}\right)}
.[1] This is in direct contrast to the drier tropical savanna climate, whose driest month has less than 60 mm of precipitation and also less than
{\textstyle 100-\left({\frac {Total\ Annual\ Precipitation\ (mm)}{25}}\right)}
of average monthly precipitation, as well as the wetter tropical rainforest climate with the driest month's rainfall above 60mm. In essence, a tropical monsoon climate tends to either have more rainfall than a tropical savanna climate or have less pronounced dry seasons, but still has a dry season unlike a tropical rainforest climate. A tropical monsoon climate tends to vary less in temperature during a year than does a tropical savanna climate because they are found closer to the equator. This climate has a driest month which nearly always occurs at or soon after the winter solstice, with monsoons normally giving precipitation in the summer instead .[1]
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