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https://planetmath.org/QuadraticFieldsThatAreNotIsomorphic | # quadratic fields that are not isomorphic
Within this entry, $S$ denotes the set of all squarefree integers not equal to $1$.
###### Theorem.
Let $m,n\in S$ with $m\neq n$. Then $\mathbb{Q}(\sqrt{m})$ and $\mathbb{Q}(\sqrt{n})$ are not isomorphic (http://planetmath.org/FieldIsomorphism).
###### Proof.
Suppose that $\mathbb{Q}(\sqrt{m})$ and $\mathbb{Q}(\sqrt{n})$ are isomorphic. Let $\varphi\colon\mathbb{Q}(\sqrt{m})\to\mathbb{Q}(\sqrt{n})$ be a field isomorphism. Recall that field homomorphisms fix prime subfields. Thus, for every $x\in\mathbb{Q}$, $\varphi(x)=x$.
Let $a,b\in\mathbb{Q}$ with $\varphi(\sqrt{m})=a+b\sqrt{n}$. Since $\varphi(a)=a$ and $\varphi$ is injective, $b\neq 0$. Also, $m=\varphi(m)=\varphi((\sqrt{m})^{2})=(\varphi(\sqrt{m}))^{2}=(a+b\sqrt{n})^{2}% =a^{2}+2ab\sqrt{n}+b^{2}n$. If $a\neq 0$, then $\displaystyle\sqrt{n}=\frac{m-a^{2}-b^{2}n}{2ab}\in\mathbb{Q}$, a contradiction. Thus, $a=0$. Therefore, $m=b^{2}n$. Since $m$ is squarefree, $b^{2}=1$. Hence, $m=n$, a contradiction. It follows that $K$ and $L$ are not isomorphic. ∎
This yields an obvious corollary:
###### Corollary.
There are infinitely many distinct quadratic fields.
###### Proof.
Note that there are infinitely many elements of $S$. Moreover, if $m$ and $n$ are distinct elements of $S$, then $\mathbb{Q}(\sqrt{m})$ and $\mathbb{Q}(\sqrt{n})$ are not isomorphic and thus cannot be equal. ∎
Note that the above corollary could have also been obtained by using the result regarding Galois groups of finite abelian extensions of $\mathbb{Q}$ (http://planetmath.org/GaloisGroupsOfFiniteAbelianExtensionsOfMathbbQ). On the other hand, using this result to prove the above corollary can be likened to “using a sledgehammer to kill a housefly”.
Title quadratic fields that are not isomorphic QuadraticFieldsThatAreNotIsomorphic 2013-03-22 16:19:44 2013-03-22 16:19:44 Wkbj79 (1863) Wkbj79 (1863) 9 Wkbj79 (1863) Theorem msc 11R11 | 2019-03-21 17:30:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 33, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9760185480117798, "perplexity": 197.2652840135939}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202530.49/warc/CC-MAIN-20190321172751-20190321194751-00475.warc.gz"} |
https://simple.wikipedia.org/wiki/Carbonation | # Carbonation
Karst landscape of the Burren
Carbonation is the process of carbon dioxide dissolving in a liquid. For example, carbon dioxide is added to flavored water under pressure to make it "fizz" as a carbonated water soft drink.
## Geology
When rain passes through the atmosphere, it takes in or mixes with carbon dioxide, turning it into a weak carbonic acid. When the rain falls in limestone, it dissolves the calcium carbonate in the rock, turning it into calcium bicarbonate. This basically dissolves the rock. Holes and large cracks can form in the rock, which are called grikes, or in German ship docking "kruftkarren".[1] The limestone that is left is called clint.[2] In a large karst area which is a large area of limestone, there are a lot of big grikes. If a stream or river passes through it, it could be swallowed up by a hole called a swallow hole. It can go through a series of underground caves and may appear on the surface again. An example of a karst area is the Burren, in County Clare, Ireland.[3]
## Chemistry
Carbon dioxide is weakly soluble in water, therefore it separates into a gas. The process of carbon dioxide bubbling out (effervescing) from a solution is represented by the following chemical reaction. This shows aqueous carbonic acid converts to carbon dioxide and water:
${\displaystyle {\mbox{H}}_{2}{\mbox{CO}}_{3}\longrightarrow {\mbox{H}}_{2}{\mbox{O}}+{\mbox{CO}}_{2}}$
## Biochemistry
Carbonation also describes the incorporation of carbon dioxide into chemical compounds. Carbon-based life originates from a carbonation reaction that is most often catalysed by the enzyme RuBisCO. This carbonation process is so important that a significant fraction of leaf mass consists of this carbonating enzyme.[4]
Carbonation of ribulose bisphosphate is the starting point of the incorporation of carbon dioxide into the biosphere.
## References
1. "Grike (gryke)". The Dictionary of Physical Geography. 2000. Retrieved 10 November 2011.
2. "Clint". The Dictionary of Physical Geography. 2000. Retrieved 10 November 2011.
3. "The Burren: Karst of Ireland". clarelibrary.ie. 2009. Retrieved 10 November 2011.
4. Stryer, Lubert; Berg, JeremyMark; Tymoczko, John L. Biochemistry, 5th Ed. W.H. Freeman, San Francisco, 2002. ISBN 0-7167-3051-0 | 2018-04-21 12:02:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7894933819770813, "perplexity": 3929.811775826189}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945143.71/warc/CC-MAIN-20180421110245-20180421130245-00183.warc.gz"} |
https://crazyproject.wordpress.com/2010/05/18/characteristic-subgroups-are-norma/ | ## Characteristic subgroups are normal
Prove that characteristic subgroups are normal. Give an example of a normal subgroup that is not characteristic.
Suppose $H \leq G$ is characteristic, and let $\varphi_g$ denote conjugation by $g \in G$. We know that $\varphi_g$ is an automorphism of $G$; since $H$ is characteristic, we have $gHg^{-1} = \varphi_g[H] = H$ for all $g \in G$. Hence $H$ is normal.
Now define $\varphi : Q_8 \rightarrow Q_8$ by $\varphi(i) = i$ and $\varphi(j) = k$, and extend homomorphically to all of $Q_8$. Note that $Q_8 = \langle \varphi(i), \varphi(j) \rangle$, so that $\varphi$ is surjective; since $Q_8$ is finite, $\varphi$ is an automorphism.
Recall that every subgroup of $Q_8$ is normal. Now $\varphi[\langle j \rangle] = \langle \varphi(j) \rangle = \langle k \rangle \neq \langle j \rangle$; thus $\langle j \rangle$ is normal but not characteristic in $Q_8$.
Another example: $Z_p \times 1 \unlhd Z_p \times Z_p$ is normal(since it is a subgroup of an abelain group) but it is not fixed under the automorphism $\varphi(x,y)=(y,x)$. | 2017-01-24 21:15:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 23, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9870088696479797, "perplexity": 91.13703791250741}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560285289.45/warc/CC-MAIN-20170116095125-00427-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://csharp-book.softuni.org/Content/Chapter-9-2-problems-for-champions-part-2/passion-days/passion-days.html | # Problem: Passion Shopping Days
Lina has a real shopping passion. When she has some money, she immediately goes to the closest shopping center (mall) and tries to spend as much as she can on clothes, bags and shoes. But her favorite thing are winter sales. Our task is to analyze her strange behavior and calculate the purchases that Lina does when she enters the mall, as well as the money she has left when the shopping is over.
The first line of the input will pass the amount that Lina has before she starts shopping. After that, upon reading the "mall.Enter" command, Lina enters the mall and starts shopping until the "mall.Exit" command is given. When Lina starts shopping, on each line of the input you will be given strings that are actions performed by Lina. Each symbol in the string is a purchase or another action. String commands contain only symbols of the ASCII table. The ASCII code of each sign is related to what Lina must pay for each of the goods. You need to interpret the symbols in the following way:
• If the symbol is a capital letter, Lina gets a 50% discount, which means that you must decrease the money she has by 50% of the numeric representation of the symbol from the ASCII table.
• If the symbol is a small letter, Lina gets a 70% discount, which means that you must decrease the money she has by 30% of the numeric representation of the symbol from the ASCII table.
• If the symbol is "%", Lina makes a purchase that decreases her money in half.
• If the symbol is "*", Lina withdraws money from her debit card and adds 10 lv to her available funds.
• If the symbol is different from all of the aforementioned, Lina just makes a purchase without discount, and in this case you should simply subtract the value of the symbol from the ASCII table from her available funds.
If a certain value of her purchases is higher than her current available funds, Lina DOES NOT make the purchase. Lina's funds cannot be less than 0.
The shopping ends when the "mall.Exit" command is given. When this happens, you need to print the number of purchases made and the money that Lina has left.
## Input Data
The input data must be read from the console. The first line of the input will indicate the amount that Lina has before starting to purchase. On each of the following lines there will be a particular command. After you read the command "mall.Enter", on each of the following lines you will be given strings holding information regarding the purchases / actions that Lina wants to perform. These strings will keep being passed, until the "mall.Exit" command is given.
Always only one "mall.Enter" command will be given, as well as only one "mall.Exit" command.
## Output Data
The output data must be printed on the console. When shopping is over, you must print on the console a particular output depending on what purchases have been made.
• If no purchases have been made – "No purchases. Money left: {remaining funds} lv."
• If at least one purchase is made – "{number of purchases} purchases. Money left: {remaining funds} lv."
The funds must be printed with accuracy of up to 2 symbols after the decimal point.
## Constraints
• Money is a float number within the range: [0 - 7.9 x 1028].
• The number of strings between "mall.Enter" and "mall.Exit" will be within the range: [1-20].
• The number of symbols in each string that represents a command will be within the range: [1-20].
• Allowed execution time: 0.1 seconds.
• Allowed memory: 16 MB.
## Sample Input and Output
110
mall.Enter
d
mall.Exit
1 purchases. Money left: 80.00 lv. ‘d’ has an ASCII code of 100. ‘d’ is a small letter, this is why Lina gets a 70% discount. She spends 30% of 100, which is 30 lv. After this purchase, she has: 110 - 30 = 80 lv.
Input Output Input Output
110
mall.Enter
%
mall.Exit
1 purchases. Money left: 55.00 lv. 100
mall.Enter
Ab
**
mall.Exit
2 purchases. Money left: 58.10 lv.
## Hints and Guidelines
We will separate the solution of the problem into three main parts:
• Processing of the input.
• Algorithm for solving the problem.
• Formatting the output.
Let's examine each of the parts in details.
### Processing the Input Data
The input of our task consists of a few components:
• On the first line we have all the money that Lina has for shopping.
• On each of the following lines we will have some kind of a command.
The first part of reading the input is trivial:
But the second one contains a detail that we need to take into consideration. The requirements state the following:
On each of the following lines there will be a particular command. After you read the command "mall.Enter", on each of the following lines you will be given strings containing information regarding the purchases / actions that Lina wants to perform.
This is where we need to take into consideration the fact that from the second input line on, we need to start reading commands, but only after we read the command "mall.Enter", we must start processing them. How can we do this? Using a while or a do-while loop is a good option. Here is an exemplary solution of how to skip all commands before processing the command "mall.Enter":
Here is the place to point out that calling Console.ReadLine() after the end of the loop is used for moving to the first command for processing.
### Algorithm for Solving the Problem
The algorithm for solving the problem is a direct one – we continue reading commands from the console, until the command "mall.Exit" is passed. In the meantime, we process each symbol (char) of each one of the commands according to the rules specified in the task requirements, and in parallel, we modify the amount that Lina has, and store the number of purchases.
Let's examine the first two problems for our algorithm. The first problem concerns the way we read the commands until we reach the "mall.Exit" command. The solution that we previously saw uses a while-loop. The second problem for the task is to access each symbol of the command passed. Keeping in mind that the input data with the commands is string type, the easiest way to access each symbol inside the strings is via a foreach loop.
Here is what using two loops of this kind would look like:
### Processing Command Symbols
The next part of the algorithm is to process the symbols from the commands, according to the following rules in the requirements:
• If the symbol is a capital letter, Lina gets a 50% discount, which means that you must decrease the money she has by 50% of the numeric representation of the symbol from the ASCII table.
• If the symbol is a small letter, Lina gets a 70% discount, which means that you must decrease the money she has by 30% of the numeric representation of the symbol from the ASCII table.
• If the symbol is "%", Lina makes a purchase that decreases her money in half.
• If the symbol is "*", Lina withdraws money from her debit card and adds 10 lv to her available funds.
• If the symbol is different from all of the aforementioned, Lina just makes a purchase without discount, and in this case you should simply subtract the value of the symbol from the ASCII table from her available funds.
Let's examine the problems that we will be facing in the first condition. The first one is how to distinguish if a particular symbol is a capital letter. We can use one of the following ways:
• Keeping in mind the fact that the letters in the alphabet have a particular order, we can use the following condition action >= 'A' && action <= 'Z', in order to check if our symbol is within the capital letters range.
• We can use the char.IsUpper(..) function.
The other problem is how to skip a particular symbol, if it is not an operation that requires more money that Lina has. This is doable using the continue construction.
An exemplary condition for the first part of the requirements looks like this:
Note: purchases is a variable of int type, in which we store the number of all purchases.
We believe the reader should not have difficulties implementing all the other conditions because they are very similar to the first one.
### Formatting the Output
In the end of our task we must print a particular output, depending on the following condition:
• If no purchases have been made – "No purchases. Money left: {remaining funds} lv."
• If at least one purchase is made – "{number of purchases} purchases. Money left: {remaining funds} lv."
The printing operations are trivial, as the only thing we need to take into consideration is that the amount has to be printed with accuracy of up to 2 symbols after the decimal point.
How can we do that? We will leave the answer to this question to the reader. | 2019-03-21 15:37:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3934784531593323, "perplexity": 1181.3608507765189}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202526.24/warc/CC-MAIN-20190321152638-20190321174638-00024.warc.gz"} |
https://zenodo.org/record/3257276/export/schemaorg_jsonld | Conference paper Open Access
Modelling of autogenous healing for regular concrete via a discrete model
A. Cibelli; G. Di Luzio; L. Ferrara; G. Cusatis; M. Pathirage
JSON-LD (schema.org) Export
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"description": "<p>In this paper a numerical model for autogenous healing of normal strength concrete is presented in detail, along with preliminary results of its validation, which is planned to be achieved by comparing the results of numerical analyses with those of a dedicated experimental campaign.<br>\nRecently the SMM (Solidification-Microprestress-Microplane model M4) model for concrete, which makes use of a modified microplane model M4 and the solidification-microprestress theory, has been extended to incorporate the autogenous healing effects. The moisture and heat fields, as well as the hydration degree, are obtained from the solution of a hygro-thermo-chemical problem, which is<br>\ncoupled with the SMM model. The updated model can also simulate the effects of cracking on the permeability and the restoring effect of the self-healing on the mechanical constitutive laws, i.e. the microplane model. In this work, the same approach is introduced into a discrete model, namely the Lattice Discrete Particle Model (LDPM). A numerical example is presented to validate the proposed computational model employing experimental data from a recent test series undertaken at Politecnico di Milano.</p>",
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"affiliation": "Politecnico di Milano Department of Civil and Environmental Engineering (DICA)",
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"headline": "Modelling of autogenous healing for regular concrete via a discrete model",
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56
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https://lifesciences.fas.harvard.edu/heb-contact-us | ## Come talk with Dr. Hooven or Dr. Alex!
We have regular drop-in office hours: Dr. Hooven on Mondays 1:30 - 2:30 pm and Dr. Alex on Thursdays 2-3 pm. If these don't work, you can send either Dr. Hooven or Alex an email to set up a meeting.
Directions to our offices are listed under the "HEB Advising Office Directions" tab, above.
Harvard University interactive map for Peabody Museum 5th floor:
View Larger Campus Map
• Enter either through 24 Oxford Street or 11 Divinity Avenue
• Take the elevator to the 5th Floor.
To Dr. Carole Hooven's office, Room 52-F:
• If you enter from 24 Oxford Street (Geological Museum): Either take stairs to the 5th floor; OR elevator to the 4th floor, then stairs to the 5th floor. Out of the stairwell, turn left down the long hallway in to the HEB lounge area. Room 52-F is down the hallway, through the lounge on your left.
• If you enter from Divinity Avenue, turn left out of the elevator, and left again. Room 52-F will be on the other side of the main lounge, on the right.
To Dr. Bridget Alex's Office, Room 53-B:
• If you enter from 24 Oxford Street (Geological Museum): Either take stairs to the 5th floor; OR elevator to the 4th floor, then stairs to the 5th floor. Out of the stairwell, turn left down the long hallway in to the HEB lounge area. Room 53-B is located down the hallway, through the lounge on your right.
• If you enter from Divinity Avenue, turn left out of the elevator, and left again. Room 53-B will be the first door on your left. | 2018-03-24 17:59:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20772415399551392, "perplexity": 8875.609157586594}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257650764.71/warc/CC-MAIN-20180324171404-20180324191404-00269.warc.gz"} |
https://labs.tib.eu/arxiv/?author=Bong%20Won%20Sohn | • ### Revealing the Nature of Blazar Radio Cores through Multi-Frequency Polarization Observations with the Korean VLBI Network(1805.04299)
May 11, 2018 astro-ph.HE
We study the linear polarization of the radio cores of eight blazars simultaneously at 22, 43, and 86 GHz with observations obtained by the Korean VLBI Network (KVN) in three epochs between late 2016 and early 2017 in the frame of the Plasma-physics of Active Galactic Nuclei (PAGaN) project. We investigate the Faraday rotation measure (RM) of the cores; the RM is expected to increase with observing frequency if core positions depend on frequency due to synchrotron self-absorption. We find a systematic increase of RMs at higher observing frequencies in our targets. The RM--$\nu$ relations follow power-laws with indices distributed around 2, indicating conically expanding outflows serving as Faraday rotating media. Comparing our KVN data with contemporaneous optical polarization data from the Steward Observatory for a few sources, we find indication that the increase of RM with frequency saturates at frequencies of a few hundreds GHz. This suggests that blazar cores are physical structures rather than simple $\tau=1$ surfaces. A single region, e.g. a recollimation shock, might dominate the jet emission downstream of the jet launching region. We detect a sign change in the observed RMs of CTA 102 on a time scale of $\approx$1 month, which might be related to new superluminal components emerging from its core undergoing acceleration/deceleration and/or bending. We see indication for quasars having higher core RMs than BL Lac objects, which could be due to denser inflows/outflows in quasars.
• ### Pilot KaVA monitoring on the M87 jet: confirming the inner jet structure and superluminal motions at sub-pc scales(1706.02066)
June 7, 2017 astro-ph.HE
We report the initial results of our high-cadence monitoring program on the radio jet in the active galaxy M87, obtained by the KVN and VERA Array (KaVA) at 22 GHz. This is a pilot study that preceded a larger KaVA-M87 monitoring program, which is currently ongoing. The pilot monitoring was mostly performed every two to three weeks from December 2013 to June 2014, at a recording rate of 1 Gbps, obtaining the data for a total of 10 epochs. We successfully obtained a sequence of good quality radio maps that revealed the rich structure of this jet from <~1 mas to 20 mas, corresponding to physical scales (projected) of ~0.1-2 pc (or ~140-2800 Schwarzschild radii). We detected superluminal motions at these scales, together with a trend of gradual acceleration. The first evidence for such fast motions and acceleration near the jet base were obtained from recent VLBA studies at 43 GHz, and the fact that very similar kinematics are seen at a different frequency and time with a different instrument suggests these properties are fundamental characteristics of this jet. This pilot program demonstrates that KaVA is a powerful VLBI array for studying the detailed structural evolution of the M87 jet and also other relativistic jets.
• ### Simultaneous dual-frequency radio observations of S5 0716+714: A search for intraday variability with the Korean VLBI Network(1703.05894)
March 17, 2017 astro-ph.GA
This study aims to search for the existence of intraday variability (IDV) of BL Lac object S5 0716+714 at high radio frequencies for which the interstellar scintillation effect is not significant. Using the 21-meter radio telescope of the Korean VLBI Network (KVN), we present results of multi-epoch simultaneous dual-frequency radio observations. Single-dish observations of S5 0716+714 were simultaneously conducted at 21.7 GHz (K-band) and 42.4 GHz (Q-band), with a high cadence of 30-60 minute intervals.We observed four epochs between December 2009 and June 2010. Over the whole set of observation epochs, S5 0716+714 showed significant inter-month variations in flux density at both the K- and Q-bands, with modulation indices of approximately 19% for the K-band and approximately 36% for the Q-band. In all epochs, no clear intraday variability was detected at either frequency. The source shows monotonic flux density increase in epochs 1 and 3 and monotonic flux density decrease in epochs 2 and 4. In the flux density increasing phases, the flux densities at the Q-band increase more rapidly. In the decreasing phase, no significant flux density difference is seen at the two frequencies. The situation could be different close to flux density peaks that we did not witness in our observations. We find an inverted spectrum with mean spectral indices of -0.57+-0.13 in epoch 1 and -0.15+-0.11 in epoch 3. On the other hand, we find relatively steep indices of +0.24+-0.14 and +0.17+-0.18 in epochs 2 and 4, respectively. We conclude that the frequency dependence of the variability and the change of the spectral index are caused by source-intrinsic effects rather than by any extrinsic scintillation effect.
• ### J0906+6930: a radio-loud quasar in the early Universe(1702.03925)
Feb. 20, 2017 astro-ph.CO, astro-ph.HE
Radio-loud high-redshift quasars (HRQs), although only a few of them are known to date, are crucial for the studies of the growth of supermassive black holes (SMBHs) and the evolution of active galactic nuclei (AGN) at early cosmological epochs. Radio jets offer direct evidence of SMBHs, and their radio structures can be studied with the highest angular resolution using Very Long Baseline Interferometry (VLBI). Here we report on the observations of three HRQs (J0131-0321, J0906+6930, J1026+2542) at z>5 using the Korean VLBI Network and VLBI Exploration of Radio Astrometry Arrays (together known as KaVA) with the purpose of studying their pc-scale jet properties. The observations were carried out at 22 and 43 GHz in 2016 January among the first-batch open-use experiments of KaVA. The quasar J0906+6930 was detected at 22 GHz but not at 43 GHz. The other two sources were not detected and upper limits to their compact radio emission are given. Archival VLBI imaging data and single-dish 15-GHz monitoring light curve of J0906+6930 were also acquired as complementary information. J0906+6930 shows a moderate-level variability at 15 GHz. The radio image is characterized by a core-jet structure with a total detectable size of ~5 pc in projection. The brightness temperature, 1.9x10^{11} K, indicates relativistic beaming of the jet. The radio properties of J0906+6930 are consistent with a blazar. Follow-up VLBI observations will be helpful for determining its structural variation.
• ### Korean VLBI Network Calibrator Survey (KVNCS): 1. Source Catalog Of KVN Single Dish Flux Density Measurement In The K And Q Bands(1701.04578)
Jan. 17, 2017 astro-ph.GA, astro-ph.IM
We present the catalog of the KVN Calibrator Survey (KVNCS). This first part of the KVNCS is a single dish radio survey conducted at 22 (K band) and 43 GHz (Q band) simultaneously using the Korean VLBI Network (KVN) from 2009 to 2011. A total 2045 sources selected from the VLBA Calibrator Survey (VCS) with an extrapolated flux density limit of 100 mJy at K band. The KVNCS contains 1533 sources in the K band with a flux density limit of 70 mJy and 553 sources in the Q band with a flux density limit of 120 mJy; it covers the whole sky down to $-32.^\circ5$ in declination. Five hundred thirteen sources were detected in the K and Q bands, simultaneously; $\sim76\%$ of them are flat-spectrum sources ($-0.5 \leq \alpha \leq 0.5$). From the flux--flux relationship, we anticipated that the most of the radiation of many of the sources comes from the compact components. Therefore, the sources listed in the KVNCS are strong candidates for high frequency VLBI calibrators.
• ### Globular clusters hosting intermediate-mass black-holes: no mass-segregation based candidates(1604.03554)
April 12, 2016 astro-ph.GA
Recently, both stellar mass-segregation and binary-fractions were uniformly measured on relatively large samples of Galactic Globular Clusters (GCs). Simulations show that both sizeable binary-star populations and Intermediate-Mass Black Holes (IMBHs) quench mass-segregation in relaxed GCs. Thus mass-segregation in GCs with a reliable binary-fraction measurement is a valuable probe to constrain IMBHs. In this paper we combine mass-segregation and binary-fraction measurements from the literature to build a sample of 33 GCs (with measured core-binary fractions), and a sample of 43 GCs (with a binary fraction measurement in the area between the core radius and the half-mass radius). Within both samples we try to identify IMBH-host candidates. These should have relatively low mass-segregation, a low binary fraction (< 5%), and short (< 1 Gyr) relaxation time. Considering the core binary fraction sample, no suitable candidates emerge. If the binary fraction between the core and the half-mass radius is considered, two candidates are found, but this is likely due to statistical fluctuations. We also consider a larger sample of 54 GCs where we obtained an estimate of the core binary fraction using a predictive relation based on metallicity and integrated absolute magnitude. Also in this case no suitable candidates are found. Finally, we consider the GC core- to half-mass radius ratio, that is expected to be larger for GCs containing either an IMBH or binaries. We find that GCs with large core- to half-mass radius ratios are less mass-segregated (and show a larger binary fraction), confirming the theoretical expectation that the energy sources responsible for the large core are also quenching mass-segregation
• ### Demonstration of KVN phase referencing capability(1511.03577)
Nov. 11, 2015 astro-ph.GA
We present the results of Very Long Baseline Interferometry (VLBI) observations using the phase reference technique to detect weak Active Galactic Nuclei (AGN) cores in the Virgo cluster. Our observations were carried out using the Korean VLBI Network (KVN). We have selected eight representative radio galaxies, seven Virgo cluster members and one galaxy (NGC 4261) that is likely to be in the background. The selected galaxies are located in a range of density regions showing various morphology in 1.4 GHz continuum. Since half of our targets are too weak to be detected at K-band we applied a phase referencing technique to extend the source integration time by calibrating atmospheric phase fluctuations. We discuss the results of the phase referencing method at high frequency observations and we compare them with self-calibration on the relatively bright AGNs, such as M87, M84 and NGC 4261. In this manuscript we present the radio intensity maps at 22 GHz of the Virgo cluster sample while we demonstrate for first time the capability of KVN phase referencing technique.
• ### The Power of Simultaneous Multi-Frequency Observations for mm-VLBI: Astrometry up to 130 GHz with the KVN(1509.02621)
Nov. 5, 2015 astro-ph.IM
Simultaneous observations at multiple frequency bands have the potential to overcome the fundamental limitation imposed by the atmospheric propagation in mm-VLBI observations. The propagation effects place a severe limit in the sensitivity achievable in mm-VLBI, reducing the time over which the signals can be coherently combined, and preventing the use of phase referencing and astrometric measurements. We carried out simultaneous observations at 22, 43, 87 and 130 GHz of a group of five AGNs, the weakest of which is ca. 200 mJy at 130 GHz, with angular separations ranging from 3.6 to 11 degrees, using the KVN. We analysed this data using the Frequency Phase Transfer (FPT) and the Source Frequency Phase Referencing (SFPR) techniques, which use the observations at a lower frequency to correct those at a higher frequency. The results of the analysis provide an empirical demonstration of the increase in the coherence times at 130 GHz from a few tens of seconds to about twenty minutes, with FPT, and up to many hours with SFPR. Moreover the astrometric analysis provides high precision relative position measurements between two frequencies, including, for the first time, astrometry at 130 GHz. Finally we demonstrate a method for the generalised decomposition of the relative position measurements into absolute position shifts for bona fide astrometric registration of the maps of the individual sources at multiple frequencies, up to 130 GHz.
• ### Key Science Observations of AGNs with KaVA Array(1504.06399)
April 24, 2015 astro-ph.HE
KaVA (KVN and VERA Array) is a new combined VLBI array with KVN (Korean VLBI Network) and VERA (VLBI Exploration of Radio Astrometry). First, we briefly review the imaging capabilities of KaVA array which actually achieves more than three times better dynamic range than that achieved by VERA alone. The KaVA images clearly show detailed structures of extended radio jets in AGNs. Next, we represent the key science program to be led by KaVA AGN sub working group. We will conduct the monitoring observations of Sgr A* and M87 because of the largeness of their central super-massive black hole angular sizes. The main science goals of the program are (i) testing magnetically-driven-jet paradigm by mapping velocity fields of the M87 jet, and (ii) obtaining tight constraints on physical properties of radio emitting region in Sgr A*.
• ### The Nuclear Structure of 3C84 with Space VLBI (RadioAstron) Observations(1504.01516)
April 7, 2015 astro-ph.CO, astro-ph.HE
The radio galaxy 3C84 is a representative of gamma-ray-bright misaligned active galactic nuclei (AGN) and one of the best laboratories to study the radio properties of subparsec scale jets. We discuss here the past and present activity of the nuclear region within the central 1pc and the properties of subparsec-sized components C1, C2 and C3. We compare these results with the high resolution space-VLBI image at 5GHz obtained with the RadioAstron satellite and we shortly discuss the possible correlation of radio emission with the gamma-ray emission.
• ### Warping and tearing of misaligned circumbinary disks around eccentric SMBH binaries(1502.00181)
Feb. 1, 2015 gr-qc, astro-ph.GA, astro-ph.HE
We study the warping and tearing of a geometrically thin, non-self-gravitating disk surrounding binary supermassive black holes on an eccentric orbit. The circumbinary disk is significantly misaligned with the binary orbital plane, and is subject to the time-dependent tidal torques. In principle, such a disk is warped and precesses, and is torn into mutually misaligned rings in the region, where the tidal precession torques are stronger than the local viscous torques. We derive the tidal-warp and tearing radii of the misaligned circumbinary disks around eccentric SMBH binaries. We find that in disks with the viscosity parameter, alpha, larger than a critical value depending on the disk aspect ratio, the disk warping appears outside the tearing radius. This condition is expressed as alpha > sqrt{H/3r} for H/r ~<0.1, where H is the disk scale height. If alpha < sqrt{H/3r}, only the disk tearing occurs because the tidal warp radius is inside the tearing radius, where most of disk material is likely to rapidly accrete onto SMBHs. In warped and torn disks, both the tidal-warp and the tearing radii most strongly depend on the binary semi-major axis, although they also mildly depend on the other orbital and disk parameters. This strong dependence enables us to estimate the semi-major axis, once the tidal warp or tearing radius is determined observationally: For the tidal warp radius of 0.1 pc, the semi-major axis is estimated to be ~10^{-2} pc for 10^7 Msun black hole with typical orbital and disk parameters. We also briefly discuss the possibility that central objects of observed warped maser disks in active galactic nuclei are supermassive black hole binaries.
• ### Very Long Baseline Interferometry with the SKA(1412.5971)
Adding VLBI capability to the SKA arrays will greatly broaden the science of the SKA, and is feasible within the current specifications. SKA-VLBI can be initially implemented by providing phased-array outputs for SKA1-MID and SKA1-SUR and using these extremely sensitive stations with other radio telescopes, and in SKA2 by realising a distributed configuration providing baselines up to thousands of km, merging it with existing VLBI networks. The motivation for and the possible realization of SKA-VLBI is described in this paper.
• ### Radiation-Driven Warping of Circumbinary Disks Around Eccentric Young Star Binaries(1410.8128)
We study a warping instability of a geometrically thin, non-self-gravitating, circumbinary disk around young binary stars on an eccentric orbit. Such a disk is subject to both the tidal torques due to a time-dependent binary potential and the radiative torques due to radiation emitted from each star. The tilt angle between the circumbinary disk plane and the binary orbital plane is assumed to be very small. We find that there is a radius within/beyond which the circumbinary disk is unstable to radiation-driven warping, depending on the disk density and temperature gradient indices. This marginally stable warping radius is very sensitive to viscosity parameters, a fiducial disk radius and the temperature measured there, the stellar luminosity, and the disk surface density at a radius where the disk changes from the optically thick to thin for the irradiation from the central stars. On the other hand, it is insensitive to the orbital eccentricity and binary irradiation parameter, which is a function of the binary mass ratio and luminosity of each star. Since the tidal torques can suppress the warping in the inner part of the circumbinary disk, the disk starts to be warped in the outer part. While the circumbinary disks are most likely to be subject to the radiation-driven warping on a AU to kilo-AU scale for binaries with young massive stars more luminous than 10^4Lsun, the radiation driven warping does not work for those around young binaries with the luminosity comparable to the solar luminosity.
• ### Verification of the Astrometric Performance of the Korean VLBI Network, using comparative SFPR studies with the VLBA at 14/7 mm(1407.4604)
July 17, 2014 astro-ph.IM
The Korean VLBI Network (KVN) is a new mm-VLBI dedicated array with capability for simultaneous observations at multiple frequencies, up to 129 GHz. The innovative multi-channel receivers present significant benefits for astrometric measurements in the frequency domain. The aim of this work is to verify the astrometric performance of the KVN using a comparative study with the VLBA, a well established instrument. For that purpose, we carried out nearly contemporaneous observations with the KVN and the VLBA, at 14/7 mm, in April 2013. The KVN observations consisted of simultaneous dual frequency observations, while the VLBA used fast frequency switching observations. We used the Source Frequency Phase Referencing technique for the observational and analysis strategy. We find that having simultaneous observations results in a superior performance for compensation of all atmospheric terms in the observables, in addition to offering other significant benefits for astrometric analysis. We have compared the KVN astrometry measurements to those from the VLBA. We find that the structure blending effects introduce dominant systematic astrometric shifts and these need to be taken into account. We have tested multiple analytical routes to characterize the impact of the low resolution effects for extended sources in the astrometric measurements. The results from the analysis of KVN and full VLBA datasets agree within 2-$\sigma$ of the thermal error estimate. We interpret the discrepancy as arising from the different resolutions. We find that the KVN provides astrometric results with excellent agreement, within 1-$\sigma$, when compared to a VLBA configuration which has a similar resolution. Therefore this comparative study verifies the astrometric performance of KVN using SFPR at 14/7 mm, and validates the KVN as an astrometric instrument.
• ### VLBI observations of bright AGN jets with KVN and VERA Array (KaVA): Evaluation of Imaging Capability(1406.4356)
June 17, 2014 astro-ph.IM, astro-ph.HE
The Korean very-long-baseline interferometry (VLBI) network (KVN) and VLBI Exploration of Radio Astrometry (VERA) Array (KaVA) is the first international VLBI array dedicated to high-frequency (23 and 43 GHz bands) observations in East Asia. Here, we report the first imaging observations of three bright active galactic nuclei (AGNs) known for their complex morphologies: 4C 39.25, 3C 273, and M 87. This is one of the initial result of KaVA early science. Our KaVA images reveal extended outflows with complex substructure such as knots and limb brightening, in agreement with previous Very Long Baseline Array (VLBA) observations. Angular resolutions are better than 1.4 and 0.8 milliarcsecond at 23 GHz and 43 GHz, respectively. KaVA achieves a high dynamic range of ~1000, more than three times the value achieved by VERA. We conclude that KaVA is a powerful array with a great potential for the study of AGN outflows, at least comparable to the best existing radio interferometric arrays.
• ### The First Very Long Baseline Interferometry Image of 44 GHz Methanol Maser with the KVN and VERA Array (KaVA)(1406.2086)
June 11, 2014 astro-ph.GA
We have carried out the first very long baseline interferometry (VLBI) imaging of 44 GHz class I methanol maser (7_{0}-6_{1}A^{+}) associated with a millimeter core MM2 in a massive star-forming region IRAS 18151-1208 with KaVA (KVN and VERA Array), which is a newly combined array of KVN (Korean VLBI Network) and VERA (VLBI Exploration of Radio Astrometry). We have succeeded in imaging compact maser features with a synthesized beam size of 2.7 milliarcseconds x 1.5 milliarcseconds (mas). These features are detected at a limited number of baselines within the length of shorter than approximately 650 km corresponding to 100 Mlambda in the uv-coverage. The central velocity and the velocity width of the 44 GHz methanol maser are consistent with those of the quiescent gas rather than the outflow traced by the SiO thermal line. The minimum component size among the maser features is ~ 5 mas x 2 mas, which corresponds to the linear size of ~ 15 AU x 6 AU assuming a distance of 3 kpc. The brightness temperatures of these features range from ~ 3.5 x 10^{8} to 1.0 x 10^{10} K, which are higher than estimated lower limit from a previous Very Large Array observation with the highest spatial resolution of ~ 50 mas. The 44 GHz class I methanol maser in IRAS 18151-1208 is found to be associated with the MM2 core, which is thought to be less evolved than another millimeter core MM1 associated with the 6.7 GHz class II methanol maser.
• ### Warped Circumbinary Disks in Active Galactic Nuclei(1406.2317)
June 9, 2014 gr-qc, astro-ph.GA
We study a warping instability of a geometrically thin, non-self-gravitating disk surrounding binary supermassive black holes on a circular orbit. Such a circumbinary disk is subject to not only tidal torques due to the binary gravitational potential but also radiative torques due to radiation emitted from an accretion disk around each black hole. We find that a circumbinary disk initially aligned with the binary orbital plane is unstable to radiation-driven warping beyond the marginally stable warping radius, which is sensitive to both the ratio of vertical to horizontal shear viscosities and the mass-to-energy conversion efficiency. As expected, the tidal torques give no contribution to the growth of warping modes but tend to align the circumbinary disk with the orbital plane. Since the tidal torques can suppress the warping modes in the inner part of circumbinary disk, the circumbinary disk starts to be warped at radii larger than the marginally stable warping radius. If the warping radius is of the order of 0.1 pc, a resultant semi-major axis is estimated to be of the order of 10^-2 pc to 10^-4 pc for 10^7 Msun black hole. We also discuss the possibility that the central objects of observed warped maser disks in active galactic nuclei are binary supermassive black holes with a triple disk: two accretion disks around the individual black holes and one circumbinary disk surrounding them.
• ### Long-term monitoring of Sgr A* at 7 mm with VERA and KaVA(1311.5852)
Nov. 22, 2013 astro-ph.GA, astro-ph.HE
We present the results of radio monitoring observations of Sgr A* at 7 mm (i.e. 43 GHz) with VLBI Exploration of Radio Astrometry (VERA), which is a VLBI array in Japan. VERA provides angular resolutions on millisecond scales, resolving structure within ~100 Schwarzschild radii of Sgr A* similar to Very Large Baseline Array (VLBA). We performed multi-epoch observations of Sgr A* in 2005 - 2008, and started monitoring it again with VERA from January 2013 for tracing the current G2 encounter event. Our preliminary results in 2013 show that Sgr A* on mas scales has been in ordinary state as of August 2013, although some fraction of the G2 cloud already passed pericenter of Sgr A* in April 2013. We will continue on monitoring Sgr A* with VERA and newly developed KaVA (KVN and VERA Array).
• ### The relationship between radio power at 22 and 43 GHz and black hole properties of AGN in elliptical galaxies(1311.3038)
Nov. 13, 2013 astro-ph.CO, astro-ph.GA
We investigate the relationship between radio power and properties related to active galactic nuclei (AGNs). Radio power at 1.4 or 5 GHz, which has been used in many studies, can be affected by synchrotron self-absorption and free-free absorption in a dense region. On the other hand, these absorption effects get smaller at higher frequencies. Thus, we performed simultaneous observations at 22 and 43 GHz using the Korean VLBI Network (KVN) radio telescope based on a sample of 305 AGN candidates residing in elliptical galaxies from the overlap between the Sloan Digital Sky Survey (SDSS) Data Release 7 and Faint Images of the Radio Sky at Twenty-Centimeters (FIRST). About 37% and 22% of the galaxies are detected at 22 and 43 GHz, respectively. Assuming no flux variability between the FIRST and KVN observation, spectral indices were derived from FIRST and KVN data and we found that over 70% of the detected galaxies have flat or inverted spectra, implying the presence of optically thick compact regions near the centres of the galaxies. Core radio power does not show a clear dependence on black hole mass at either low (1.4 GHz) or high (22 and 43 GHz) frequencies. However, we found that the luminosity of the [OIII] $\lambda$5007 emission line and the Eddington ratio correlate with radio power more closely at high frequencies than at low frequencies. This suggests that radio observation at high frequencies can be an appropriate tool for unveiling the innermost region. In addition, the luminosity of the [OIII] $\lambda$5007 emission line and the Eddington ratio can be used as a tracer of AGN activity. Our study suggests a causal connection between high frequency radio power and optical properties of AGNs.
• ### Early science with Korean VLBI network: the QCAL-1 43GHz calibrator survey(1207.5872)
Oct. 5, 2012 astro-ph.CO, astro-ph.IM
This paper presents the catalog of correlated flux densities in three ranges of baseline projection lengths of 637 sources from a 43 GHz (Q-band) survey observed with the Korean VLBI Network. Of them, 623 sources have not been observed before at Q-band with VLBI. The goal of this work in the early science phase of the new VLBI array is twofold: to evaluate the performance of the new instrument that operates in a frequency range of 22-129 GHz and to build a list of objects that can be used as targets and as calibrators. We have observed the list of 799 target sources with declinations down to -40 degrees. Among them, 724 were observed before with VLBI at 22 GHz and had correlated flux densities greater than 200 mJy. The overall detection rate is 78%. The detection limit, defined as the minimum flux density for a source to be detected with 90% probability in a single observation, was in a range of 115-180 mJy depending on declination. However, some sources as weak as 70 mJy have been detected. Of 623 detected sources, 33 objects are detected for the first time in VLBI mode. We determined their coordinates with the median formal uncertainty 20 mas. The results of this work set the basis for future efforts to build the complete flux-limited sample of extragalactic sources at frequencies 22 GHz and higher at 3/4 of the celestial sphere.
• ### Single dish performance of KVN 21-m radio telescopes:Simultaneous observations at 22 and 43 GHz(1110.3881)
Oct. 18, 2011 astro-ph.IM
We report simultaneous multi-frequency observing performance at 22 and 43 GHz of the 21-m shaped-Cassegrain radio telescopes of the Korean VLBI Network (KVN). KVN is the first millimeter-dedicated VLBI network in Korea having a maximum baseline length of 480 km. It currently operates at 22 and 43 GHz and planed to operate in four frequency bands, 22, 43, 86, and 129 GHz. The unique quasioptics of KVN enable simultaneous multi-frequency observations based on efficient beam filtering and accuarate antenna-beam alignment at 22 and 43 GHz. We found that the offset of the beams is within <5 arcseconds over all pointing directions of antenna. The dual polarization, cooled HEMT receivers at 22 and 43 GHz result in receiver noise temperatures less than 40 K at 21.25-23.25 GHz and 80 K at 42.11-44.11 GHz. The pointing accuracies have been measured to be 3 arcseconds in azimuth and elevation for all antennas. The measured aperture efficiencies are 65%(K)/67%(Q), 62%(K)/59%(Q), and 66%(K)/60%(Q) for the three KVN antennas, KVNYS, KVNUS, and KVNTN, respectively. The main-beam efficiencies are measured to be 50%(K)/52%(Q), 48%(K)/50%(Q), and 50%(K)/47%(Q) for KVNYS, KVNUS, and KVNTN, respectively. The estimated Moon efficiencies are 77%(K)/90%(Q), 74%(K)/79%(Q), and 80%(K)/86%(Q) for KVNYS, KVNUS, KVNTN, respectively. The elevation dependence of the aperture efficiencies is quite flat for elevations > 20 degrees. | 2021-03-06 11:22:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6477091312408447, "perplexity": 2628.9153232007966}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178374686.69/warc/CC-MAIN-20210306100836-20210306130836-00600.warc.gz"} |
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# Is a > c?
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Is a > c? [#permalink]
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13 Aug 2012, 11:06
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Is a > c?
(1) b > d
(2) ab^2 – b > b^2c – d
[Reveal] Spoiler:
Can we add (1) and (2) to get the answers.
[Reveal] Spoiler: OA
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Thanks Rphardu
Last edited by Bunuel on 14 Aug 2012, 00:14, edited 2 times in total.
Renamed the topic and edited the question.
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13 Aug 2012, 11:33
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rphardu wrote:
Is a > c?
(1) b > d
(2) ab2 – b > b2c – d
[Reveal] Spoiler:
Can we add (1) and (2) to get the answers.
YES, definitely, you can add two inequalities that have the same direction.
In the above DS question, obviously neither (1) nor (2) alone is sufficient.
(1) and (2) together:
Adding the two inequalities side-by-side we obtain $$ab^2-b+b>b^2c-d+d$$ or $$b^2(a-c)>0$$, which means necessarily $$a-c>0$$ or $$a>c.$$
Sufficient.
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Re: Is a > c? [#permalink]
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14 Aug 2012, 00:16
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rphardu wrote:
Is a > c?
(1) b > d
(2) ab^2 – b > b^2c – d
[Reveal] Spoiler:
Can we add (1) and (2) to get the answers.
You can only add inequalities when their signs are in the same direction:
If $$a>b$$ and $$c>d$$ (signs in same direction: $$>$$ and $$>$$) --> $$a+c>b+d$$.
Example: $$3<4$$ and $$2<5$$ --> $$3+2<4+5$$.
You can only apply subtraction when their signs are in the opposite directions:
If $$a>b$$ and $$c<d$$ (signs in opposite direction: $$>$$ and $$<$$) --> $$a-c>b-d$$ (take the sign of the inequality you subtract from).
Example: $$3<4$$ and $$5>1$$ --> $$3-5<4-1$$.
Hope it helps.
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Re: Is a > c? [#permalink]
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14 Aug 2012, 06:45
2
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Bunuel wrote:
rphardu wrote:
Is a > c?
(1) b > d
(2) ab^2 – b > b^2c – d
[Reveal] Spoiler:
Can we add (1) and (2) to get the answers.
You can only add inequalities when their signs are in the same direction:
If $$a>b$$ and $$c>d$$ (signs in same direction: $$>$$ and $$>$$) --> $$a+c>b+d$$.
Example: $$3<4$$ and $$2<5$$ --> $$3+2<4+5$$.
You can only apply subtraction when their signs are in the opposite directions:
If $$a>b$$ and $$c<d$$ (signs in opposite direction: $$>$$ and $$<$$) --> $$a-c>b-d$$ (take the sign of the inequality you subtract from).
Example: $$3<4$$ and $$5>1$$ --> $$3-5<4-1$$.
Hope it helps.
When we subtract two inequalities with their signs in opposite directions, we are in fact using addition of two inequalities in the same direction:
$$a>b$$
$$C<D$$ -> this can be rewritten as
$$-C>-D$$
Now we can add the first and the third inequality, because they have the same direction and get $$a-C>b-D.$$
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Re: Is a > c? [#permalink]
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15 Aug 2012, 09:29
1 &2 combo-
a(b^2)-b-(b^2)c+d>0
(b^2)(a-c)-(b-d)>0
note, that 1 states that b>d. in order to make the expression above positive a must be > c
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12 Aug 2013, 07:05
EvaJager wrote:
rphardu wrote:
Is a > c?
(1) b > d
(2) ab2 – b > b2c – d
(1) and (2) together:
Adding the two inequalities side-by-side we obtain $$ab^2-b+b>b^2c-d+d$$ or $$b^2(a-c)>0$$, which means necessarily $$a-c>0$$ or $$a>c.$$
Sufficient.
I don't understand the solution beyond this part...$$b^2(a-c)>0$$
as per me by dividing both sides of equation by $$b^2$$ we are assuming value of B is not equal 0, else it will be 0/0 which is not defined. so how is this correct?
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### Show Tags
12 Aug 2013, 09:22
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nikhil007 wrote:
EvaJager wrote:
rphardu wrote:
Is a > c?
(1) b > d
(2) ab2 – b > b2c – d
(1) and (2) together:
Adding the two inequalities side-by-side we obtain $$ab^2-b+b>b^2c-d+d$$ or $$b^2(a-c)>0$$, which means necessarily $$a-c>0$$ or $$a>c.$$
Sufficient.
I don't understand the solution beyond this part...$$b^2(a-c)>0$$
as per me by dividing both sides of equation by $$b^2$$ we are assuming value of B is not equal 0, else it will be 0/0 which is not defined. so how is this correct?
From (1) and (2) we see $$b^2(a-c)>0$$
Since LHS >0 we must have $$b =! 0$$ and$$a > c$$
as if $$b = 0$$ then LHS = 0 .
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29 Aug 2013, 12:32
EvaJager wrote:
rphardu wrote:
Is a > c?
(1) b > d
(2) ab2 – b > b2c – d
[Reveal] Spoiler:
Can we add (1) and (2) to get the answers.
YES, definitely, you can add two inequalities that have the same direction.
In the above DS question, obviously neither (1) nor (2) alone is sufficient.
(1) and (2) together:
Adding the two inequalities side-by-side we obtain $$ab^2-b+b>b^2c-d+d$$ or $$b^2(a-c)>0$$, which means necessarily $$a-c>0$$ or $$a>c.$$
Sufficient.
I understood till this point - b^2 (a-c) > 0
can someone explain after this step please.
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30 Aug 2013, 04:49
Expert's post
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swati007 wrote:
EvaJager wrote:
rphardu wrote:
Is a > c?
(1) b > d
(2) ab2 – b > b2c – d
[Reveal] Spoiler:
Can we add (1) and (2) to get the answers.
YES, definitely, you can add two inequalities that have the same direction.
In the above DS question, obviously neither (1) nor (2) alone is sufficient.
(1) and (2) together:
Adding the two inequalities side-by-side we obtain $$ab^2-b+b>b^2c-d+d$$ or $$b^2(a-c)>0$$, which means necessarily $$a-c>0$$ or $$a>c.$$
Sufficient.
I understood till this point - b^2 (a-c) > 0
can someone explain after this step please.
We have $$b^2(a-c)>0$$ ($$b\neq{0}$$). Now, since $$b^2>0$$, then the other multiple must also be greater than 0 --> $$a-c>0$$ --> $$a>c.$$.
Hope it's clear.
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### Show Tags
30 Aug 2013, 23:14
Quote:
We have $$b^2(a-c)>0$$ ($$b\neq{0}$$). Now, since $$b^2>0$$, then the other multiple must also be greater than 0 --> $$a-c>0$$ --> $$a>c.$$.
Hope it's clear.
Wonderful explanation!!! Thanks Bunuel
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Re: Is a > c? [#permalink]
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16 Jun 2015, 12:39
Bunuel wrote:
rphardu wrote:
Is a > c?
(1) b > d
(2) ab^2 – b > b^2c – d
[Reveal] Spoiler:
Can we add (1) and (2) to get the answers.
You can only add inequalities when their signs are in the same direction:
If $$a>b$$ and $$c>d$$ (signs in same direction: $$>$$ and $$>$$) --> $$a+c>b+d$$.
Example: $$3<4$$ and $$2<5$$ --> $$3+2<4+5$$.
You can only apply subtraction when their signs are in the opposite directions:
If $$a>b$$ and $$c<d$$ (signs in opposite direction: $$>$$ and $$<$$) --> $$a-c>b-d$$ (take the sign of the inequality you subtract from).
Example: $$3<4$$ and $$5>1$$ --> $$3-5<4-1$$.
Hope it helps.
What if b=0 and d=-1?
In that situation, wouldn't the 2nd equation become:
a(0) > (0)c – d
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Re: Is a > c? [#permalink]
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16 Jun 2015, 12:50
metskj127 wrote:
Bunuel wrote:
rphardu wrote:
Is a > c?
(1) b > d
(2) ab^2 – b > b^2c – d
[Reveal] Spoiler:
Can we add (1) and (2) to get the answers.
You can only add inequalities when their signs are in the same direction:
If $$a>b$$ and $$c>d$$ (signs in same direction: $$>$$ and $$>$$) --> $$a+c>b+d$$.
Example: $$3<4$$ and $$2<5$$ --> $$3+2<4+5$$.
You can only apply subtraction when their signs are in the opposite directions:
If $$a>b$$ and $$c<d$$ (signs in opposite direction: $$>$$ and $$<$$) --> $$a-c>b-d$$ (take the sign of the inequality you subtract from).
Example: $$3<4$$ and $$5>1$$ --> $$3-5<4-1$$.
Hope it helps.
What if b=0 and d=-1?
In that situation, wouldn't the 2nd equation become:
a(0) > (0)c – d
If b = 0 and d = -1, then ab^2 – b = 0 and b^2c – d = 1. Anyway, what are you trying to say?
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Re: Is a > c? [#permalink]
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16 Jun 2015, 12:58
Never mind- I see my mistake now. Thank you for the help.
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Re: Is a > c? [#permalink]
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21 Jan 2018, 21:48
rphardu wrote:
Is a > c?
(1) b > d
(2) ab^2 – b > b^2c – d
[Reveal] Spoiler:
Can we add (1) and (2) to get the answers.
Question asks whether a > c or Is a-c > 0
(1) First statement is obviously not sufficient. Just b>d or b-d > 0 but nothing about a and c is mentioned.
(2) Second statement can be rewritten as:
ab^2 – b^2c > b – d
b^2*(a-c) > (b-d)
Here b^2 cannot be negative but whether a-c is positive or not depends on b-d also (on right hand side). Nothing is mentioned about that so insufficient.
Combining the two statements, from first we know that b-d is positive and since left hand side: b^2*(a-c) is greater than b-d so b^2*(a-c) also must be positive.
Now b^2 cannot be negative. So for b^2*(a-c) to be positive, a-c also must be positive. Which means a-c > 0 or a > c. Sufficient.
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Re: Is a > c? [#permalink]
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23 Jan 2018, 22:47
rphardu wrote:
Is a > c?
(1) b > d
(2) ab^2 – b > b^2c – d
[Reveal] Spoiler:
Can we add (1) and (2) to get the answers.
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Since we have 4 variables (a, b, c and d) and 0 equations, E is most likely to be the answer. So, we should consider 1) & 2) first.
Conditions 1) & 2):
b > d
ab^2 - b > b^2c - d
⇔ ab^2 -b^2c > b - d
⇔ b^2(a-c) > b-d
⇔ a - c > (b-d)/b^2 > 0 since b > d and b^2 > 0 if b≠0.
⇔ a > c
If b = 0, we have d < 0
ab^2 - b > b^2c - d
⇔ 0 > -d which contradicts b > d
Thus b≠0.
Both conditions together are sufficient.
Since this question is an inequality question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.
Condition 1):
We don't have any information about a and c from the condition 1) only.
The condition 1) only is not sufficient.
Condition 2):
⇔ ab^2 -b^2c > b - d
⇔ b^2(a-c) > b-d
⇔ a - c > (b-d)/b^2 since b^2 > 0
⇔ a - c > b - d
a = 2, c = 1, b = 1, d = 1 : Yes
a = 1, c = 2, b = 0, d = 1 : No
The condition 2) only is not sufficient.
Therefore, C is the answer.
In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
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Re: Is a > c? [#permalink] 23 Jan 2018, 22:47
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Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | 2018-02-18 07:11:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8273312449455261, "perplexity": 3311.3609703447137}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891811794.67/warc/CC-MAIN-20180218062032-20180218082032-00786.warc.gz"} |
https://ckms.kms.or.kr/journal/view.html?doi=10.4134/CKMS.c200349 | - Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnline Submission ㆍMy Manuscript - For Reviewers - For Editors
Loxodromes and transformations in pseudo-Hermitian geometry Commun. Korean Math. Soc. 2021 Vol. 36, No. 4, 817-827 https://doi.org/10.4134/CKMS.c200349Published online March 9, 2021Printed October 31, 2021 Ji-Eun Lee Chonnam National University Abstract : In this paper, we prove that a diffeomorphism $f$ on a normal almost contact $3$-manifold $M$ is a CRL-{\it transformation} if and only if $M$ is an $\alpha$-Sasakian manifold. Moreover, we show that a $CR$-loxodrome in an $\alpha$-Sasakian $3$-manifold is a pseudo-Hermitian magnetic curve with a strength $q=\widetilde{r}\eta(\gamma')=(r+\alpha-t)\eta(\gamma')$ for constant $\eta(\gamma')$. A non-geodesic $CR$-loxodrome is a non-Legendre slant helix. Next, we prove that let $M$ be an $\alpha$-Sasakian $3$-manifold such that $(\nabla_Y S)X=0$ for vector fields $Y$ to be orthogonal to $\xi$, then the Ricci tensor $\rho$ satisfies $\rho=2\alpha^2 g$. Moreover, using the CRL-{\it transformation} $\widetilde{\nabla}^t$ we fine the pseudo-Hermitian curvature $\widetilde{R}$, the pseudo-Ricci tensor $\widetilde{\rho}$ and the torsion tensor field $\widetilde{ \mathfrak{T}}^{t} (\widetilde{S}X,Y)$. Keywords : Loxodrome, magnetic curves, normal almost contact manifold, pseudo-Hermitian geometry MSC numbers : Primary 58E20, 53B25 Supported by : The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2019R1l1A1A01043457). Downloads: Full-text PDF Full-text HTML
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http://georg.io/2014/03/PLOS_Biology_Outlier_Detection | PLOS Biology Outlier Detection
In this blog post I play with dimensionality reduction techniques SVD and Isomap to map a corpus of 1,754 PLOS Biology articles from 27,210-dimensional feature space to 2-dimensional space.
This sort of approach is used oftentimes to estimate which data points are near each other. Here, I realized that the data points far away from the bulk discuss (almost) consistently neurobiological topics.
As far as I am aware PLOS Biology publish articles on all biological topics so there is probably no editorial factor to our observation here.
%matplotlib inline
%autosave 10
import gensim
import cPickle as pickle
from sklearn import *
from sklearn.manifold import Isomap
import numpy
from matplotlib import pyplot
from mpl_toolkits.mplot3d import Axes3D
The first article in this data set looks as follows
articles[0][:10]
['introduction',
'during',
'1980s',
'1990s',
'methods',
'molecular',
'genetics',
'used',
'determine',
'contributions']
dois[0]
'10.1371/journal.pbio.1000584'
Checking the main text of the above DOI we make certain that the article stored in articles[0] corresponds to the DOI stored in dois[0].
Let us now load the same corpus as in articles but already formatted as a numerical matrix that represents each article (row of the matrix) as a bag of words. We generated this corpus and the corresponding dictionary earlier.
corpus = gensim.corpora.MmCorpus('data/plos_biology_corpus.mm')
corpus_mat = gensim.matutils.corpus2csc(corpus)
corpus_mat = corpus_mat.T
print corpus_mat.shape
(1754, 27210)
SVD
svd = decomposition.TruncatedSVD(n_components=2)
corpus_mat_transform = svd.fit_transform(corpus_mat)
pyplot.scatter(corpus_mat_transform[:,0], corpus_mat_transform[:,1])
pyplot.scatter(numpy.median(corpus_mat_transform[:,0]), numpy.median(corpus_mat_transform[:,1]), color='red')
<matplotlib.collections.PathCollection at 0x39dd4690>
Outliers far Away from Median
As we can see there are a few articles that lie relatively far away from the bulk of the corpus (denoted by the red disk which marks the median). Let’s focus on some of these:
corpus_mat_transform[corpus_mat_transform[:,0]>150]
array([[ 188.00455202, 207.0323185 ],
[ 173.92204694, 149.59252031],
[ 153.2889464 , 215.74459155],
[ 162.25069234, 102.40518113],
[ 150.996759 , 145.03623767]])
numpy.where(corpus_mat_transform[:,0]>150)
(array([ 35, 1074, 1109, 1371, 1544]),)
for index in numpy.where(corpus_mat_transform[:,0]>150)[0]:
print 'http://www.plosbiology.org/article/info:doi/%s' % dois[index]
The first of these “outliers” in the above reduced space is a Synopsis articles so it may be understandable why that one sticks out. However, the remaining articles are research articles that all deal with neurobiological topics - so off-hand it is not obvious to me why these would lie a bit further away from the bulk of the articles.
Articles Near Median
For comparison with these outliers, let us take a look at articles near the median (red disk in the above scatter plot).
median = (numpy.median(corpus_mat_transform[:,0]), numpy.median(corpus_mat_transform[:,1]))
print median
(41.996541953291668, -11.862843898802915)
distances = numpy.asarray([numpy.linalg.norm(vec) for vec in corpus_mat_transform-median])
near_median = numpy.where(distances<1.5)
print near_median
(array([ 233, 810, 872, 1096, 1353, 1408, 1705]),)
for index in near_median[0]:
print 'http://www.plosbiology.org/article/info:doi/%s' % dois[index]
Articles near the median discuss topics such as gene expression in C. elegans, development and patterning of the neural plat, and stomata in Arabidopsis.
Isomap
Isomap is another dimensionality reduction tool that promises to preserve the higher-dimensional shape of your data cloud better than SVD.
isomap = Isomap(n_components=2)
isomap_transformed = isomap.fit_transform(corpus_mat.toarray())
pyplot.scatter(isomap_transformed[:,0], isomap_transformed[:,1])
pyplot.scatter(numpy.median(isomap_transformed[:,0]), numpy.median(isomap_transformed[:,1]), color='red')
<matplotlib.collections.PathCollection at 0x37bc4910>
Outlying Group of Articles with 1st Component > 400
Let us focus on the group of articles on the right in the above plot.
indeces = numpy.where(isomap_transformed[:,0]>450)
print indeces
for index in indeces[0]:
print 'http://www.plosbiology.org/article/info:doi/%s' % dois[index]
(array([ 388, 397, 486, 521, 676, 949, 1087, 1109, 1221, 1325, 1427]),)
Clicking through these, we realize that most of these articles deal with neurobiological topics again - except for articles on insulin resistance and B cell lymphomas.
Two Articles in the Top Left Corner
indeces = numpy.where((isomap_transformed[:,0]<-100) & (isomap_transformed[:,1]>600))
print indeces
for index in indeces[0]:
print 'http://www.plosbiology.org/article/info:doi/%s' % dois[index]
(array([ 35, 1074]),)
Also these two outliers deal with neurobiological topics.
One Article Center Top
indeces = numpy.where((isomap_transformed[:,0]>100) & (isomap_transformed[:,1]>600))
print indeces
for index in indeces[0]:
print 'http://www.plosbiology.org/article/info:doi/%s' % dois[index]
(array([421]),)
And again an article about a neurobiological topic.
Articles Near the Median
median = (numpy.median(isomap_transformed[:,0]), numpy.median(isomap_transformed[:,1]))
print median
(-22.134749426543411, -4.3751913714736927)
distances = numpy.asarray([numpy.linalg.norm(vec) for vec in isomap_transformed-median])
near_median = numpy.where(distances<2)
print near_median
(array([ 49, 323, 350, 772, 773, 889, 908, 960, 1038, 1158, 1285,
1309, 1321, 1530]),)
for index in near_median[0]:
print 'http://www.plosbiology.org/article/info:doi/%s' % dois[index]
Articles near the median discuss topics such as stochastic gene expression, DNA transcription and repair, metabolic symbiosis, T cell differentiation, chromatin, an RNAi screen for cytokinesis inhibitors, and a study on p53. | 2020-02-20 16:32:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25188127160072327, "perplexity": 5860.749082961265}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145260.40/warc/CC-MAIN-20200220162309-20200220192309-00523.warc.gz"} |
https://an1.is/tag/bus-simulator-ultimate/ | AN1
/ Bus Simulator : Ultimate
# Bus Simulator : Ultimate for android
## Bus Simulator : Ultimate
• 5.0
• 2.0.6
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http://mkweb.bcgsc.ca/pi/piday2020/methods.mhtml | Embrace me, surround me as the rush comes.drift deeper into the soundmore quotes
2020 $\pi$ day art and the piku
# visualization + design
“Transcendental Tree Map” from Yearning for the Infinite. This video premiered on 2020 Pi Day. Music by Max Cooper. Animation by Nick Cobby and myself.
The 2020 Pi Day art celebrates digits of $\pi$ with piku (パイク) —poetry inspired by haiku.
They serve as the form for The Outbreak Poems.
A $\pi$ day music video!: Transcendental Tree Map premieres on 2020 Pi Day from Max Cooper's Yearning for the Infinite. Animation by Nick Cobby and myself. Watch live from Barbican Centre.
# $\pi$ Day 2020 Art — Digits, poetically.
On March 14th celebrate $\pi$ Day. Hug $\pi$—find a way to do it.
For those who favour $\tau=2\pi$ will have to postpone celebrations until July 26th. That's what you get for thinking that $\pi$ is wrong. I sympathize with this position and have $\tau$ day art too!
If you're not into details, you may opt to party on July 22nd, which is $\pi$ approximation day ($\pi$ ≈ 22/7). It's 20% more accurate that the official $\pi$ day!
Finally, if you believe that $\pi = 3$, you should read why $\pi$ is not equal to 3.
All art posters are available for purchase.
I take custom requests.
3 Little $\pi$,
1 what
4 can you tell us?
Welcome to this year's celebration of $\pi$ and mathematics. Among the chaos of CORVID-19, this year $\pi$ Day celebrations are short poetic emissions I call “piku” . They are brief pauses for the time.
Start by reading how piku are constructed and then browse submitted piku. Consider participating by submitting your own piku. All you need is a pen and a few (small) words. Very therapeutic.
But if the words here don't help, immerse yourself in my coronavirus art. It's quite catching.
If you enjoy poetry and words, see how I convert spam into poems in the style of ee cummings and if you like to see words arrange on page, look through my typographic art.
3 Piku are
1 brief
4 pauses just in
1 time.
You may know the haiku (俳句) as a short three line poem whose lines traditionally have 5, 7 and 5 syllables (specifically, morae or phonetic sounds).
Scribbles on Shonan Village Center notepad on which the idea was born. I am grateful for being invited to the Formalizing Biological and Medical Visualization (Shonan Meeting 167) in which the idea formed. (zoom)
On a recent trip to Japan I was looking to my environment for inspiration for this year's $\pi$ Day. I also really wanted a scheme that would allow people to contribute to the art so that everyone could be both a participant and an audience.
After scribbling for a little bit (or a long while), I came up with the idea of a piku ($\pi$ku, パイク). Analogously to a haiku, the piku is poem whose structure is constrained. But in this case, the constraints are the digits of $\pi$ itself.
## The piku
The simplest piku is a 3 phrase poem with 3, 1 and 4 syllables, respectively. Perhaps the most trivial piku is just the list of digits.
3 Three one four
1 One
4 Five nine two six.
Specifically, Haiku count the number of phonetic sounds which isn't always the same as syllables. For example, the word "any" has two such sounds: a-ny.
If you're interested in how the counting of sounds is done in Japanese, read about marking time and beats in Haiku.
But a more fun one would be
3 A Three and
1 A
4 One and a four.
Longer words can be used, of course. But watch out for the lines that require a single syllable.
3 Numbers are
1 all
4 Equally fun.
3 Land meets sky.
1 Oh,
4 It's you again!
You can use hyphenation to work around the syllable count.
3 Memories
1 for-
4 get yourself then.
## plural of piku is piku
Piku is singulare tantum—its plural form is the same as its singular.
This is because its inspiration is the haiku and in Japanese nouns do not have different singular and plural forms,
3 Piku is
1 the
4 piku plural.
## Longer piku
In general, a piku may have more than 3 lines. This reflects the fact that the digits of $\pi$ do not terminate.
3 Land meets sky.
1 Oh,
4 It's you again!
1 Yes,
5 Who did you expect?
The endless piku is just waiting to be written. Well... started, at least.
## Handling zeros
The digit zero is a line with no syllables and corresponds to a compulsory verse break. But because the first zero in $\pi$ is at digit 33, you wouldn't see a verse break for a while.
Therefore, you're free to introduce a verse break anywhere in the piku (this does not use up a digit). For example,
3 Hot air breeze
1 cools
4 much hotter things.
3 F 0 0
1 red
## Generalizing the piku — the nku
Any number, $n$, has its own nku. The rules for its construction are the same: each digit generates a line with corresponding number of syllables.
For example, the haiku is an nku for $n = 575$.
The year of your birth has an nku—you might want to try composing one to reflect on your origins.
In fact, any date (e.g. DDMMYYYY) can be made into an nku. This year's $\pi$ date in this format is 14032020 and here's one possible nku.
1 Dust
4 in space vacuum
0
3 clogs alien
2 machines.
0
2 Bummer.
0
The trailing zero in the number creates a verse break at the end of a poem. This can be ignored or used to generate a blank line if the poem is set with other text on the page.
# The Outbreak Poems
Tue 24-03-2020
I'm writing poetry daily to put my feelings into words more often during the COVID-19 outbreak.
$Thoughts crack through night to a still clock.$
$Clogged channels stuck with nervous silt.$
$Song to sleep and poem to wake to.$
$Once in a while a long moment.$
$Lump on head from hole in stomach.$
$Universe of no mind but ours.$
$Higher than birds stars glide at night.$
$Sleeps or hides some thing internal.$
$Pieces move by rules not yet written.$
$Clouds visit then as fog they stay.$
$Distant duck bobs like something else.$
$Not above the belows but in them.$
$Yesterday comes by starlight to day.$
$It's always not now somewhere else.$
# Deadly Genomes: Genome Structure and Size of Harmful Bacteria and Viruses
Tue 17-03-2020
A poster full of epidemiological worry and statistics. Now updated with the genome of SARS-CoV-2 and COVID-19 case statistics as of 3 March 2020.
Deadly Genomes: Genome Structure and Size of Harmful Bacteria and Viruses (zoom)
Bacterial and viral genomes of various diseases are drawn as paths with color encoding local GC content and curvature encoding local repeat content. Position of the genome encodes prevalence and mortality rate.
The deadly genomes collection has been updated with a posters of the genomes of SARS-CoV-2, the novel coronavirus that causes COVID-19.
Genomes of 56 SARS-CoV-2 coronaviruses that causes COVID-19.
Ball of 56 SARS-CoV-2 coronaviruses that causes COVID-19.
The first SARS-CoV-2 genome (MT019529) to be sequenced appears first on the poster.
# Using Circos in Galaxy Australia Workshop
Wed 04-03-2020
A workshop in using the Circos Galaxy wrapper by Hiltemann and Rasche. Event organized by Australian Biocommons.
Using Circos in Galaxy Australia workshop. (zoom)
Galaxy wrapper training materials, Saskia Hiltemann, Helena Rasche, 2020 Visualisation with Circos (Galaxy Training Materials).
# Essence of Data Visualization in Bioinformatics Webinar
Thu 20-02-2020
My webinar on fundamental concepts in data visualization and visual communication of scientific data and concepts. Event organized by Australian Biocommons.
Essence of Data Visualization in Bioinformatics webinar. (zoom)
# Markov models — training and evaluation of hidden Markov models
Thu 20-02-2020
With one eye you are looking at the outside world, while with the other you are looking within yourself.
—Amedeo Modigliani
Following up with our Markov Chain column and Hidden Markov model column, this month we look at how Markov models are trained using the example of biased coin.
We introduce the concepts of forward and backward probabilities and explicitly show how they are calculated in the training process using the Baum-Welch algorithm. We also discuss the value of ensemble models and the use of pseudocounts for cases where rare observations are expected but not necessarily seen.
Nature Methods Points of Significance column: Markov models — training and evaluation of hidden Markov models. (read)
Grewal, J., Krzywinski, M. & Altman, N. (2019) Points of significance: Markov models — training and evaluation of hidden Markov models. Nature Methods 17:121–122.
Altman, N. & Krzywinski, M. (2019) Points of significance: Hidden Markov models. Nature Methods 16:795–796.
Altman, N. & Krzywinski, M. (2019) Points of significance: Markov Chains. Nature Methods 16:663–664.
# Genome Sciences Center 20th Anniversary Clothing, Music, Drinks and Art
Tue 28-01-2020
Science. Timeliness. Respect.
Read about the design of the clothing, music, drinks and art for the Genome Sciences Center 20th Anniversary Celebration, held on 15 November 2019.
Luke and Mayia wearing limited edition volunteer t-shirts. The pattern reproduces the human genome with chromosomes as spirals. (zoom)
As part of the celebration and with the help of our engineering team, we framed 48 flow cells from the lab.
Precisely engineered frame mounts of flow cells used to sequence genomes in our laboratory. (zoom)
Each flow cell was accompanied by an interpretive plaque explaining the technology behind the flow cell and the sample information and sequence content.
The plaque at the back of one of the framed Illumina flow cell. This one has sequence from a patient's lymph node diagnosed with Burkitt's lymphoma. (zoom) | 2020-04-04 06:29:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21262265741825104, "perplexity": 6148.523405495284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370520039.50/warc/CC-MAIN-20200404042338-20200404072338-00054.warc.gz"} |
https://quant.stackexchange.com/questions/23187/dollar-neutral-strategy | # Dollar-Neutral Strategy
Here is an excerpt from E. Chan's book Quantitative Trading,
However, if the strategy is a long-short dollar-neutral strategy (i.e., the portfolio holds long and short positions with equal capital), then 10 percent is quite a good return, because then the benchmark of comparison is not the market index, but a riskless asset such as the yield of the three-month U.S. Treasury bill (which at the time of this writing is about 4 percent).
I do not understand why a long-short strategy is riskless - AFAIK in a short-long position with equal dollar amount in each, the short would pay for the long, but how does that translate into risklessness?
Thanks,
• Notice that he didn't quite say that a long-short strategy is riskless, that was your inference. He is just saying that he is going to use the riskless t-bills as a benchmark. For example in the CAPM stocks with a zero beta earn $R_f$ but they are not riskless. – noob2 Feb 9 '16 at 17:52
• @noob2 yes. Long-short strategy is not completely riskless. They still exposed to idiosyncratic risk. But such risk can be eliminated by creating diversified portfolio. – Neeraj Feb 9 '16 at 18:55
Long-short strategy is generally used by hedge funds. In simple words, an equity long-short strategy means buying an undervalued stock and selling(shorting) an overvalued stock. In normal circumstances, the long position will increase in value and the short position will decline in value. In this situation, the hedge fund will benefit. This strategy would work even if the long position declines in value but provided that the long position outperforms the short position.
Thus, the goal of any equity long-short strategy is to minimize market exposure, and get profit from a change in the difference between two stocks.
Let's take a simple example. A hedge fund takes a \$10 million long position in General Motors and a \$10 million short position in Volkswagen, both large automobile companies. With these positions, any event that causes all automobile stocks to fall will lead to a loss on the General Motors position and a profit on the Volkswagen position. Similarly, an event that causes both stocks to rise will have little effect, since the positions balance each other out. So, the market risk is minimal. But question arise, Why would a portfolio manager take such a position? Because he or she thinks General Motors will perform better than Volkswagen.
Equity long-short strategies which hold equal dollar amounts of long and short positions, are called market neutral strategies or long-short dollar neutral strategy (as described above).
Since, portfolio is immune to both upside and downside market risk, like risk free security, its performance must be measure from similar security ie riskless assets such as U.S. Treasury bill etc. | 2020-02-21 13:31:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5999183058738708, "perplexity": 1793.2847689396926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145529.37/warc/CC-MAIN-20200221111140-20200221141140-00060.warc.gz"} |
https://www.beameditator.com/2021/09/be-meditator-for-todays-magnificence-we.html | -->
Don't we all want every day to be magnificent? Yes, we do. But it's a fact that all days are not the same like seasons are not the same round the year. Like, all terrains are not same all over the world. But we can make ourselves comfortable in all seasons and terrains with a little bit of effort. In the same way, we can change the effect of each day into a magnificent day, by meditating every day for each day. Keep Meditating | 2022-05-22 04:37:04 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8406379818916321, "perplexity": 1008.9869324195377}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662543797.61/warc/CC-MAIN-20220522032543-20220522062543-00130.warc.gz"} |
https://csat.io/practice/p4/q12 | The CSAT and Practice[+] are designed by the Climb Foundation to help candidates. We are advocates for more opportunity to shine and less opportunity to fail, and we strive to level the playing field.
# Practice Paper 4 Question 12
Consider the square $$ABCD$$ of side $$x,$$ and the equilateral triangle $$BCE$$ as in the figure shown. The square rotates clockwise around $$B$$ until $$A$$ overlaps $$E,$$ then rotates around $$E$$ until $$D$$ overlaps $$C,$$ and so on, until $$A$$ retakes its initial position. Sketch the path traced by A and find its length. Give the length of the longest horizontal segment with end points on this path.
The above links are provided as is. They are not affiliated with the Climb Foundation unless otherwise specified.
## Warm-up Questions
1. Find the perimeter of a circle sector with central angle $$\frac{\pi}{4}$$ and a radius of $$2.$$
2. Calculate the length of the chord between the two ends of the arc of that sector.
3. Compute the sides ratio of a triangle with angles of $$30^\circ, 60^\circ$$ and $$90^\circ.$$
## Hints
• Hint 1
Have you tried sketching the movement?
• Hint 2
Do you notice any repeating patterns?
• Hint 3
How far has point $$A$$ travelled, in terms of its distance to get back to its initial position, after 4 rotations?
• Hint 4
Why not compute the length of each of the 4 arcs traced by $$A$$?
• Hint 5
Have you identified the longest horizontal segment?
• Hint 6
To find its length, could you find a triangle with the segment as one of its side?
• Hint 7
Why not calculate one angle in the triangle to solve for its side length?
## Solution
We can construct a sketch of the path by first drawing the triangle and drawing 3 squares that overlap each side of the triangle as shown in the diagram. All the vertices of the squares and triangle in this diagram are visited in a clockwise fashion.
Notice that after rotating four times, we arrive at a situation in which point $$A$$ has completed one-third of its journey around the triangle, and is taking the position of point $$F.$$ In the first rotation, the length of its path is the length of arc $$AE,$$ which is $$\frac{\pi x}{6}.$$ In the second rotation, point $$A$$ does not move. In the third rotation, its path length is $$\frac{\pi x}{6}.$$ In the fourth rotation, its path length is $$\frac{\sqrt{2} \pi x}{6}.$$ Adding these up and multiplying by three gives us the total path length of $$\frac{\pi x(2+\sqrt{2})}{2}.$$
From the diagram it is apparent that the longest horizontal chord is $$\overline{IF}.$$ We calculate this by first noting $$\angle{BEC}$$ is $$\frac{\pi}{3}$$ as it is an internal angle of an equilateral triangle. $$\angle{IEB}$$ and $$\angle{CEF}$$ are both $$\frac{\pi}{6}$$ as each of the that created the arcs is $$\frac{\pi}{6}.$$ Therefore, $$\angle{IEF}$$ is $$\frac{2\pi}{3}.$$ Using the fact that the length of $$\overline{IE}$$ and $$\overline{EF}$$ is $$x,$$ we can solve the triangle $$\Delta IEF$$ to find the length of $$\overline{IF}$$ which gives the length of the longest horizontal segment is $$\sqrt{3}x.$$
If you have queries or suggestions about the content on this page or the CSAT Practice Platform then you can write to us at . Please do not write to this address regarding general admissions or course queries. | 2019-09-21 16:00:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6606846451759338, "perplexity": 335.2259593255143}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574532.44/warc/CC-MAIN-20190921145904-20190921171904-00093.warc.gz"} |
https://joshcullen.github.io/bayesmove/reference/shiny_tracks.html | This Shiny application allows for the exploration of animal movement patterns. Options are available to interactively filter the plotted tracks by a selected time period of a given variable, which is then displayed on an interactive map. Additionally, a data table is shown with options to filter and export this table once satisfied.
shiny_tracks(data, epsg)
Arguments
data A data frame that must contain columns labeled id, x, y, date, but can include any other variables of interest. numeric. The coordinate reference system (CRS) as an EPSG code.
Details
Currently, the time series plot shown for the exploration of individual tracks cannot display variables of class character or factor. Therefore, these should be changed to numeric values if they are to be plotted.
If the data are stored as longitude and latitude (i.e., WGS84), the EPSG code is 4326. All other codes will need to be looked up if they are not already known.
Examples
if (FALSE) { | 2023-02-04 15:45:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21170340478420258, "perplexity": 1531.6544495941507}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500140.36/warc/CC-MAIN-20230204142302-20230204172302-00400.warc.gz"} |
http://gis.stackexchange.com/questions/23662/arcgis-server-gp-service-how-to-create-a-txt-file-or-rename-a-dbf-to-txt-in-the/23671 | # ArcGIS Server GP Service: How to create a txt file or rename a dbf to txt in the “%SCRATCHWORKSPACE%”
My end goal is to get an excel (I'll probably just settle for .csv) file loaded from a silverlight app into my "%SCRATCHWORKSPACE%" for my gp service. I will be linear referencing the data onto a polyline.
I have silverlight reading the csv file's contents and packing it into a string to send over to ArcGIS Server.
Then I want to write this data to a csv file in the "%SCRATCHWORKSPACE%". But I don't seem to be able to use any standard python code to hit the "%SCRATCHWORKSPACE%".
Any thoughts or critiques?
-
Set an input to your gp service to be the %scratchworkspace% - this should get resolved by ArcGIS Server into a actual directory path on disk that (assuming the SOC account has access to the folders) you should be able to write to no worries. It will end up in the arcgisjobs folder, in a subfolder under the name of the service, and then under another subfolder that is created by ArcGIS Server on a random text string. e.g.:
C:\arcgisserver\arcgisjobs\toolbox\model\j5f4eb1370ded4e8e99a17519c0989850
Last tip is that this doesn't get cleaned up automatically, so you'll have to set that up yourself.
Hope this helps!
-
I have no issues writing to the the job folder j5f4eb1370ded4e8e99a17519c0989850\scratch\ folder with arcpy methods but I can't do os.rename(src,dst). I get an error saying the directory is not found. – Justin Apr 18 '12 at 12:51
arcpy.env.scratchWorkspace yields
C:\Users\ARCGIS~\AppData\Local\Temp\geoprocessing\model_gpserver\j7560bf9fcf9844f7a4736d36ea123456\scratch
Which allowed my to use non-arcpy methods targeting the scratch workspace of a gp service job request.
- | 2014-07-29 00:54:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3681183159351349, "perplexity": 2866.1977106185036}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510264270.11/warc/CC-MAIN-20140728011744-00349-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://www.studysmarter.us/explanations/math/statistics/sampling-distribution/ | Suggested languages for you:
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# Sampling Distribution
Let's say you want to know the average GPA of high school senior students in Atlanta, Georgia. To calculate the exact value, you would need to ask the population, that is, all the senior students in Atlanta, Georgia for their GPA. That sounds exhausting! But what if you just take a sample of it instead of asking all the senior students? This is the idea behind sampling distributions.
In this article, you'll find the definition of sampling distributions, types of sampling distributions, the formulas, the mean and the standard deviation of sampling distributions, and examples of application.
## Introduction to Sampling Distributions
Coming back to the example above, let's say you randomly select and sample $$100$$ senior students and calculate the average GPA from this sample. This average GPA would not be the same as the mean GPA of all senior students in Atlanta. It could be lower or higher, but it would most likely not be exactly equal to the population mean.
If you select a second sample of $$100$$ senior students, the average GPA for this sample would most likely differ from the mean of your first one. Thus, random samples selected would produce different mean values. Despite this variety of values, when many sample means are obtained, you can plot these collected means on a graph, and then this can provide an estimated mean of the entire population. This process explains the concept of creating sampling distributions of the mean.
## Definition of Sampling Distributions
A value that is calculated by taking information from a sample is called a statistic. Statistics allows you to estimate data of an entire population. As you saw in the example above, different random samples can give different values for a statistic; this difference is called sampling variability (or sampling error). This sampling variability can be reduced by increasing the sample size.
The distribution formed by all the possible values for sample statistics obtained for every possible different sample of a given size is called the sampling distribution.
### Conditions for Sampling Distributions
To ensure that the sampling distribution truly estimates the entire population, you must make sure that these two criteria are checked:
1. Randomization condition: the most important condition necessary for creating a sampling distribution is that your data comes from samples randomly selected.
2. Independence ($$10\%$$ condition): the sampled values must be independent one from another. Achieving this condition is the same as considering sample sizes no larger than $$10\%$$ of the entire population.
Let's go back to the average GPA example. For the randomization condition, unless you have a list of the students with the highest GPA in Atlanta, choosing any $$100$$ student randomly is enough to satisfy this condition.
On the other hand, for the independence condition, it is not unreasonable to assume that there are more than $$10\, 000$$ senior students in Atlanta, so the $$10\%$$ of this is $$1\,000$$. Any sample size less than $$1\,000$$ satisfies this condition, thus considering samples of a $$100$$ in size is acceptable.
## Types of Sampling Distributions
There are 3 types of sampling distributions:
1. Sampling distribution of proportions
2. Sampling distribution of means
3. T-distribution
### Sampling Distribution of Proportions
It is used to estimate a population proportion. It calculates the proportion of success, or chance, that a specific event will occur. The mean from each group of the sample proportion is a representation of the estimated proportion of success of the entire population.
### Sampling Distribution of Means
It entails calculating the means of all sample groups from a selected population. Then, the average of the means of all the samples is an estimated mean of the entire population.
### T-distribution
It is focused on a small population. It is used to measure the mean of the population and other statistical measurements such as confidence intervals, linear regression, and statistical differences. Since this distribution uses $$t$$-scores to calculate probabilities, it is out of the scope of this article.
## Formula for Sampling Distributions
The sample proportion, denoted by $$\widehat{p}$$, is calculated by counting how many successes are in the sample (success means that an individual possesses the characteristic of interest) and dividing it by the total sample size $$n$$
$\widehat{p}=\frac{\text{number of successes in the sample}}{n}.$
The sample mean, denoted by $$\overline{x}$$, is calculated by adding up all the values obtained from the sample and dividing by the total sample size $$n$$. The idea is the same as finding the average for a set of data. The formula is
$\overline{x}=\frac{x_1+x_2+...+x_n}{n},$
where $$\overline{x}$$ is the sample mean, $$x_i$$ is each one of the values of the sample, and $$n$$ is the sample size.
## Mean and Standard Deviation of Sampling Distributions
All probability distributions have characteristics that distinguish them. Sampling distributions are no exception, knowing the mean and standard deviation can give you a lot of information about the shape of the distribution.
### Mean and Standard Deviation of the Sample Proportion
Let $$p$$ be the proportion of success in a population and $$\widehat{p}$$ the sample proportion, that is, the proportion of success in a random sample of size $$n$$, then the sampling distribution of $$\widehat{p}$$ has mean and standard deviation given by $\mu_\widehat{p}=p\,\text{ and }\, \sigma_\widehat{p}=\sqrt{\frac{p(1-p)}{n}}.$
Moreover, if $np\geq 10\,\text{ and }\, n(1-p)\geq 10,$ then, the sampling distribution of $$\widehat{p}$$ is similar to a normal distribution.
A random sample is selected from a population that has a proportion of successes $$p=0.72$$. Calculate the mean and standard deviation of the sampling distribution of $$\widehat{p}$$ with sample size $$n=20$$.
Solution:
Using the formulas stated before, the mean is equal to the proportion of success of the population, then $\mu_\widehat{p}=0.72,$ while the standard deviation is given by $\sigma_\widehat{p} =\sqrt{\frac{0.72(0.28)}{20}}\approx 0.100.$
### Mean and Standard Deviation of the Sample Mean
Let $$\mu$$ be the mean and $$\sigma$$ the standard deviation of the population. Let $$\overline{x}$$ be the sample mean of a random sample of size $$n$$, then the sampling distribution of $$\overline{x}$$ has mean and standard deviation given by $\mu_\overline{x}=\mu\,\text{ and }\, \sigma_\overline{x}=\frac{\sigma}{\sqrt{n}}.$
The standard deviation of the sampling distribution of means is also known as the standard error of the mean (SEM).
If the sample size $$n$$ is large enough (according to the Central Limit Theorem, $$n\geq 30$$ is enough) then, the sampling distribution of $$\overline{x}$$ is similar to a normal distribution.
A random sample is selected from a population with mean $$\mu=80$$ and standard deviation $$\sigma=5$$. Calculate the mean and standard deviation of the sampling distribution of $$\overline{x}$$ with sample size $$n=35$$.
Solution:
Using the formulas stated before, the sample mean is equal to the mean of the population, so $\mu_\overline{x}=80.$ And for the standard deviation of the sample mean
$\sigma_\overline{x}=\frac{5}{\sqrt{35}}\approx 0.845.$
## Examples of Sampling Distributions
Let's see an example using sampling distributions.
A restaurant stated $$30\%$$ of their customers like pineapple on their pizza. If there are $$100$$ customers on a given day, what is the probability that at least $$40\%$$ of these customers will buy a pizza with pineapple?
Solution:
(1) Note that $$p=0.30$$, $$(1-p)=0.70$$ and the sample size is $$n=100$$. Thus, the mean $$\mu_\widehat{p}=0.30$$ and the standard deviation $\sigma_{\widehat{p}}=\sqrt{\frac{(0.30)(0.70)}{100}}\approx 0.046.$
(2) Since $$np=100(0.30)=30>10$$ and $$n(1-p)=100(0.70)=70>10$$, then the sampling distribution of $$\widehat{p}$$ is similar to a normal distribution, and you can use this later to calculate the probability.
(3) Converting $$\widehat{p}$$ into $$z$$-score (see the article $$z$$-scores for more details), you will have
\begin{align} P(\widehat{p}>40) &= P\left(z>\frac{0.40-0.30}{0.046}\right) \\ &=P(z>2.17) \\ & =1-P(z<2.17) \\ &= 1-0.9850 \\ &=0.015.\end{align}
Thus, the probability that at least $$40\%$$ of these customers ask for a pizza with pineapple is $$0.015$$.
Let's see one extra example.
A company claims that the average lifetime of their lightbulbs is $$2\,000$$ hours with a standard deviation of $$300$$ hours. What is the probability that a random sample of $$50$$ lightbulbs have an average lifetime of less than $$1\,900$$ hours?
Solution:
(1) Since the sample size is $$n=50$$, according to the Central Limit Theorem, the sampling distribution of the mean $$\overline{x}$$ follows a normal distribution with mean $$\mu_\overline{x}=2\,000$$ and standard deviation $\sigma_\overline{x}=\frac{300}{\sqrt{50}} \approx 42.426.$
(2) Converting the $$\overline{x}$$ into $$z$$-scores and using the standard normal table (see the article Standard Normal Distribution for more information), you will have
\begin{align} P(\overline{x}<1\,900) &=P\left(z<\frac{1\,900-2\,000}{42.426}\right) \\ &=P(z<-2.35) \\ &= 0.0094. \end{align}
Thus, the probability that from a sample of size $$n=50$$ lightbulbs the average lifetime is less than $$1\,900$$ hours is $$0.0094$$.
## Sampling Distribution - Key takeaways
• A sampling distribution shows every possible statistic that can be obtained from every possible sample of the population.
• The sampling distribution of proportion $$\widehat{p}$$ has mean and standard deviation $\mu_\widehat{p}=p\, \text{ and } \,\sigma_\widehat{p}=\sqrt{\frac{p(1-p)}{n}}.$
• When $$np\geq 10$$ and $$n(1-p)\geq 10,$$ the sampling distribution of proportion $$\widehat{p}$$ behaves like a normal distribution.
• The sampling distribution of mean $$\overline{x}$$ has mean and standard deviation $\mu_\overline{x}=\mu\,\text{ and }\, \sigma_\overline{x}=\frac{\sigma}{\sqrt{n}}.$
• When $$n\geq 30$$, the Central Limit Theorem states that the sampling distribution of mean $$\overline{x}$$ behaves like a normal distribution.
A sampling distribution is a statistical tool that helps to determine the probability of an event or another statistical parameter in a population based on taking random and small samples of it.
✓ Sampling distribution of proportions
✓ Sampling distribution of means
✓ T-distribution
To find the sampling distribution, follow the following steps:
1. select random samples of fixed size from the population;
2. obtain your data and summarize;
3. plot the distribution of the summary data.
✓ The sample mean is a good estimator (unbiased) of the population mean.
✓ The data is centered on the mean or close to the true population mean.
✓ The distribution is normal and has a symmetric shape when enough data points are included (at least 30, according to the Central Limit Theorem).
The sampling distribution allows you to determine information about an entire population using only information from small samples.
## Final Sampling Distribution Quiz
Question
To use the normal distribution to model a sampling distribution of mean, the following condition regarding the sample size must be satisfied:
$$n\geq 30$$.
Show question
Question
The standard deviation of the sampling distribution of the proportion $$\widehat{p}$$ can be calculated using the formula ____.
$$\sigma_\widehat{p}=\sqrt{\frac{p(1-p)}{n}}$$
Show question
Question
The ____ is a statistical tool that helps to calculate the probability of an event by sampling a small group repeatedly instead of sampling an entire population.
sampling distribution
Show question
Question
To use the normal distribution to model a sampling distribution of proportion, the following condition must be satisfied:
$$np\geq 10$$ and
$$n(1-p)\geq 10$$.
Show question
Question
The standard deviation of the sampling distribution of the mean $$\overline{x}$$ can be calculated using the formula ____.
$$\sigma_\overline{x}=\frac{\sigma}{\sqrt{n}}$$
Show question
Question
Mention the 3 types of sampling distributions.
Proportions, means, and T-distribution.
Show question
Question
This type of sampling distribution focuses on a small population.
T-distribution
Show question
Question
When the data produces a bell-shaped curve, it is said to follow a ____ distribution.
normal
Show question
Question
When the normality condition is satisfied, the sampling distribution of proportions follows a normal distribution with mean and standard deviation given by
$$\mu_\widehat{p}=p$$ and $$\sigma_\widehat{p}=\sqrt{\frac{p(1-p)}{n}}$$.
Show question
Question
When the normality condition is satisfied, the sampling distribution of means follows a normal distribution with mean and standard deviation given by
$$\mu_\overline{x}=\mu$$ and $$\sigma_\overline{x}=\frac{\sigma}{\sqrt{n}}$$.
Show question
Question
The standard deviation of the sampling distribution of means is also known as the ___.
standard error of the mean
Show question
Question
What does the randomization condition mean?
The collected data comes from samples randomly selected.
Show question
Question
What does the independence condition ($$10\%$$ condition) mean?
The sampled values must be independent one from another.
Show question
Question
The sampling variability can be reduced by _____.
increasing the sample size
Show question
Question
Sampling variability is also known as ____.
sampling error
Show question
Question
The standard deviation of the sampling distribution of proportion is given by
$$\sigma_\widehat{p}=\sqrt{\frac{p(1-p)}{n}}.$$
Show question
Question
How do you calculate a sample proportion?
$$\widehat{p}=\dfrac{\text{number of successes in the sample}}{n}$$.
Show question
Question
A restaurant wants to know how many customers order dessert. They asked 50 customers, of which 23 said they do order dessert. Which notation is the correct to represent this proportion?
$$p=23/50$$.
Show question
Question
What does the $$10\%$$ condition mean?
The population must at least $$10$$ times the sample size.
Show question
Question
If you know the population proportion and the sample size, can you calculate the standard deviation of the sample proportion?
Yes.
Show question
Question
This is the normality condition for sample proportions
$$np\geq 10$$ and
$$n(1-p)\geq 10$$.
Show question
Question
The sample proportion can only take values from $[0,1].$
True.
Show question
Question
$$P(\widehat{p}<0.11)$$.
Show question
Question
If the sampling distribution of a proportion $$\widehat{p}$$ is normally distributed, how do you convert a value $$\widehat{p}$$ into a $$z$$-value?
$$z=\dfrac{\widehat{p}-\mu_\widehat{p}}{\sigma_\widehat{p}}$$.
Show question
Question
What does the randomization condition mean?
Samples must be randomly selected.
Show question
Question
What are the mean and standard deviation of the sampling distribution for samples of size 40 trips if the population mean of the number of fish caught each trip to a given fishing hole is 3.2 and the population standard deviation is 1.8?
mean = 3.2 and standard deviation = 0.285
Show question
Question
What is the Central Limit Theorem?
The Central Limit Theorem is an important theorem in statistics that involves approximating a distribution of sample means to the normal distribution.
Show question
Question
What is the minimum sample size to consider when using the Central Limit Theorem?
30
Show question
Question
How can you supposedly construct a distribution of sample means?
By drawing many samples of the same size from the same population and calculating the mean of the attribute you're interested in, you form a list of means from those samples that may become a distribution of sample means.
Show question
Question
What are two important conditions for the Central Limit Theorem?
Two important conditions are randomness and a sufficiently large number of samples.
Show question
Question
What important concepts does the Central Limit Theorem involve?
There are two important concepts that the Central Limit Theorem involves: a distribution of sample means and the normal distribution.
Show question
Question
The Central Limit Theorem applies to any distribution with many samples, be it known, like a binomial, a uniform, or a Poisson distribution, or an unknown distribution. True or false?
True.
Show question
Question
What does the Central Limit Theorem tell us?
The Central Limit Theorem says that if you take a sufficiently large number of samples from any random distribution, the distribution of the sample means can be approximated by the normal distribution.
Show question
Question
State the formula for the Central Limit Theorem.
For $$X$$ with mean $$\mu$$ and standard deviation $$\delta$$, if $$n\ge 30$$, then there's a random variable $$\bar{X}$$ such that $$\bar{X}\approx N\left (\mu, \frac{\delta}{\sqrt{n}} \right)$$.
Show question
Question
The Central Limit Theorem is useful in making significant inferences about the population from a sample. It can be used to tell whether two samples were drawn from the same population, and also check if the sample was drawn from a certain population. True or False?
True.
Show question
Question
The mean of the sampling distribution of proportion is given by
$$\mu_\widehat{p}=p$$.
Show question
Question
What is an estimator?
It is the value resulting from a point estimation of a parameter.
Show question
Question
If the expected value of the parameter is equal to the parameter, what statement is true?
bias = 0
Show question
Question
What is true about the maximum likehood function?
derivative = 0
Show question
Question
What is an example of a Bernouilli distribution
Throwing a coin
Show question
Question
What is an example of a poisson distribution
The number of cars going pass a school in 10 minutes
Show question
Question
True or False: The advantage of point estimation is that you don't know how close or how far away from the true value of the parameter the estimator is.
False.
Show question
Question
When the properties of consistency and unbiased are met for an estimator, you have what is called:
The best-unbiased estimator.
Show question
Question
Two important properties of estimators are ___ and ____.
Consistent and Unbiased.
Show question
Question
The point estimate of the population mean $$\mu$$ is:
The sample mean $$\bar{x}$$.
Show question
Question
What is point estimation?
Point estimation is the use of statistics taken from one or several samples to estimate the value of an unknown parameter of a population.
Show question
Question
If the distribution is possion how do we find p(x = 7|λ = 2)
Use the possion distribution where λ = 2. Find the P(x =< 7) - P(x=< 6)
Show question
Question
What is true about the likelihood function?
Product of all the probabilities at a particular parameter.
Show question
Question
In instances where it is difficult to collect data on each element of a population, the Central Limit Theorem won't be useful to approximate the features of the population. True or False?
False.
Show question
Question
The Central Limit Theorem allows approximating any distribution, for a large sample size, to the binomial distribution. True or False?
False.
Show question
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http://buron.coffee/entity/existential_rules.html | existential rules
Existential rules are first order logic formulas mainly used for knowledge representation.
The general problem of query answering over an instance with a set of existential rules is undecidable DBLP:journals/jair/CaliGK13.
There are known three classes of rules for which the problem of conjunctive query answering is decidable.
Finite expansion set (fes)
In this case, there always exists an finite canonical model of any facts and the rules we consider, which can be computed using the core chase and on which we can evaluate the query to get the answers.
The FES property is undecidable for general existential rules.
For the guarded fragment, we know that the termination of the oblivious and the semi-oblivious chase is decidable grecoChaseTerminationConstraints2010, which is not sufficient to obtain the decidability of the FES property.
Bounded treewidth set (bts)
In this case, the rules are such that from any facts, there exists a bound such that any derived facts using the core chase have a treewidth lower than this bound.
Finite unification set (fus)
In this case, there is a first-order rewriting of any conjunctive query w.r.t. the rules. It is equivalent to say that for any conjunctive query $$q$$, there exists a union of conjunctive query, which is a rewriting of $$q$$ w.r.t. the rules. It is also equivalent to say that the rules satisfy the bounded derivation-depth property meaning that for any query there exists an bound on the chase steps such that the query answers can be found using this bounded chase on any fact.
The following rule is 1-frontier, datalog and guarded, but do not form a fus: $$R = A(x), T(x, y) \rightarrow A(y)$$. For example, consider the query $$q() \leftarrow A(x), B(x)$$ and the fact $$F_{n} = A(c_{0}), T(c_{0}, c_{1}), \dots, T(c_{n-1}, c_{n}), B(c_{n})$$. It shows that the rule set $$\{ R \}$$ does not satisfy the bounded derivation-depth property.
Fontier-guarded rules FUS property is decidable DBLP:conf/ijcai/BarceloBLP18.
A Datalog rule set is FUS if and only if it is bounded DBLP:journals/jcss/AjtaiG94.
An algorithm of query rewriting with rules have been presented in the thesis of Mélanie König: DBLP:phd/hal/Konig14.
Chase
The importance of the chase in many applications is due to the fact that several problems can be solved by exhibiting a universal model, and the chase computes a universal model, when it terminates DeutschChaseRevisited2008. An universal model is a model that can be mapped to any model or that is more general than any model. The certain answer of a query be computed by evaluated it on the universal model, instead of on every model.
Chase variants
• oblivious chase (obl): for each h morphism from B to D, then we add hsafe(H) to the KB,
• semi-oblivious chase or Skolem chase (sobl): for each h morphism from B to D, then we add hsafe(hf)(H) to the KB, where the safe operation only depends of hf: the value of h on the frontier variables,
• restricted chase or standard chase (std): for each h morphism from B to KB, then we add hsafe(H) to the KB to form KB', if h can not be extended to map H to the KB.
• core chase (core):
Chase termination classes
With TGDs only, the chase can either terminate or run forever.
As explained in DBLP:journals/pvldb/CalauttiGMT16, we have the following classes $$CT^{c}_{\forall}$$ (resp. $$CT^{c}_{\exists}$$) of sets of TGDs $$\Sigma$$ such that the chase terminates on all every database, on every (resp. at least on one) c chase executions.
Example of TGD set where there exists a std chase executions that terminates and some that doesn't:
1. $$E(x,y) \rightarrow E(x,z)$$
2. $$E(x,y) \rightarrow E(y,y)$$
Example of TGD set where the core chase terminates, but the std chase doesn't, consider $$p(x,x), p(x,y) \rightarrow p(y,y), p(y,z)$$ on $$\{p(a,a), p(a,b) \}$$.
Boundedness
For the oblivious chase:
• decidable for guarded frontier
• ? for frontier guarded
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Otherwise, send your comment on my service. | 2023-01-30 17:17:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7310407757759094, "perplexity": 1522.03292060268}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499826.71/warc/CC-MAIN-20230130165437-20230130195437-00523.warc.gz"} |
https://c3-toolset.readthedocs.io/en/master/ | # $$C^3$$ - An integrated tool-set for control, calibration and characterization¶
The $$C^3$$ software package provides tools to simulate and interact with experiments to perform common control and characterization tasks. Modules can be used individually or combined to achieve a certain goal. The main focus are three optimizations:
• $$C_1$$ Open-loop optimal control: Given a model, find the pulse shapes which maximize fidelity with a target operation.
• $$C_2$$ Closed-loop calibration: Given pulses, calibrate their parameters to maximize a figure of merit measured by the actual experiment, thus improving beyond the limits of a deficient model.
• $$C_3$$ Model learning: Given control pulses and their experimental measurement outcome, optimize model parameters to best reproduce the results.
When combined in sequence, these three procedures represent a recipe for system characterization.
Note: This documentation is work-in-progress. | 2021-09-18 08:25:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5015063285827637, "perplexity": 2445.0770598392187}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056348.59/warc/CC-MAIN-20210918062845-20210918092845-00679.warc.gz"} |
https://www.aiuai.cn/aifarm2061.html | GitHub - video2pdfslides
video2pdfslides 库主要功能：从视频中提取图片，并去重后，转为PDF.
## 1. 依赖与安装
imutils==0.5.4
opencv_python==4.5.2.52
img2pdf==0.4.1
git clone https://github.com/kaushikj/video2pdfslides
cd video2pdfslides
pip install -r requirements.txt
python video2pdfslides.py <video_path>
## 2. video2pdfslides
https://github.com/kaushikj/video2pdfslides/blob/main/video2pdfslides.py
#!/usr/bin/python3
#!--*-- coding: utf-8 --*--
import os
import time
import cv2
import imutils
import shutil
import img2pdf
import glob
import argparse
############# Define constants
OUTPUT_SLIDES_DIR = f"./output"
FRAME_RATE = 3 # no.of frames per second that needs to be processed, fewer the count faster the speed
WARMUP = FRAME_RATE # initial number of frames to be skipped
FGBG_HISTORY = FRAME_RATE * 15 # no.of frames in background object
VAR_THRESHOLD = 16 # Threshold on the squared Mahalanobis distance between the pixel and the model to decide whether a pixel is well described by the background model.
DETECT_SHADOWS = False # If true, the algorithm will detect shadows and mark them.
MIN_PERCENT = 0.1 # min % of diff between foreground and background to detect if motion has stopped
MAX_PERCENT = 3 # max % of diff between foreground and background to detect if frame is still in motion
#视频抽帧
def get_frames(video_path):
'''
A fucntion to return the frames from a video located at video_path
this function skips frames as defined in FRAME_RATE
'''
# open a pointer to the video file initialize the width and height of the frame
vs = cv2.VideoCapture(video_path)
if not vs.isOpened():
raise Exception(f'unable to open file {video_path}')
total_frames = vs.get(cv2.CAP_PROP_FRAME_COUNT)
frame_time = 0
frame_count = 0
print("total_frames: ", total_frames)
print("FRAME_RATE", FRAME_RATE)
# loop over the frames of the video
while True:
# grab a frame from the video
vs.set(cv2.CAP_PROP_POS_MSEC, frame_time * 1000) # move frame to a timestamp
frame_time += 1/FRAME_RATE
# if the frame is None, then we have reached the end of the video file
if frame is None:
break
frame_count += 1
yield frame_count, frame_time, frame
vs.release()
#图片去重
def detect_unique_screenshots(video_path, output_folder_screenshot_path):
'''
Initialize fgbg a Background object with Parameters
history = The number of frames history that effects the background subtractor
varThreshold = Threshold on the squared Mahalanobis distance between the pixel and the model to decide whether a pixel is well described by the background model. This parameter does not affect the background update.
detectShadows = If true, the algorithm will detect shadows and mark them. It decreases the speed a bit, so if you do not need this feature, set the parameter to false.
'''
#背景提取算法
captured = False
start_time = time.time()
(W, H) = (None, None)
screenshoots_count = 0
for frame_count, frame_time, frame in get_frames(video_path):
orig = frame.copy()
frame = imutils.resize(frame, width=600)
mask = fgbg.apply(frame) # apply the background subtractor
# apply a series of erosions and dilations to eliminate noise
# if the width and height are empty, grab the spatial dimensions
if W is None or H is None:
# compute the percentage of the mask that is "foreground"
p_diff = (cv2.countNonZero(mask) / float(W * H)) * 100
# if p_diff less than N% then motion has stopped, thus capture the frame
if p_diff < MIN_PERCENT and not captured and frame_count > WARMUP:
captured = True
filename = f"{screenshoots_count:03}_{round(frame_time/60, 2)}.png"
path = os.path.join(output_folder_screenshot_path, filename)
print("saving {}".format(path))
cv2.imwrite(path, orig)
screenshoots_count += 1
# otherwise, either the scene is changing or we're still in warmup
# mode so let's wait until the scene has settled or we're finished
# building the background model
elif captured and p_diff >= MAX_PERCENT:
captured = False
print(f'{screenshoots_count} screenshots Captured!')
print(f'Time taken {time.time()-start_time}s')
return
def initialize_output_folder(video_path):
'''
Clean the output folder if already exists
'''
output_folder_screenshot_path = f"{OUTPUT_SLIDES_DIR}/{video_path.rsplit('/')[-1].split('.')[0]}"
if os.path.exists(output_folder_screenshot_path):
shutil.rmtree(output_folder_screenshot_path)
os.makedirs(output_folder_screenshot_path, exist_ok=True)
print('initialized output folder', output_folder_screenshot_path)
return output_folder_screenshot_path
def convert_screenshots_to_pdf(output_folder_screenshot_path):
output_pdf_path = f"{OUTPUT_SLIDES_DIR}/{video_path.rsplit('/')[-1].split('.')[0]}" + '.pdf'
print('output_folder_screenshot_path', output_folder_screenshot_path)
print('output_pdf_path', output_pdf_path)
print('converting images to pdf..')
with open(output_pdf_path, "wb") as f:
f.write(img2pdf.convert(sorted(glob.glob(f"{output_folder_screenshot_path}/*.png"))))
print('Pdf Created!')
print('pdf saved at', output_pdf_path)
if __name__ == "__main__":
parser = argparse.ArgumentParser("video_path")
parser.add_argument("video_path", help="path of video to be converted to pdf slides", type=str)
args = parser.parse_args()
video_path = args.video_path
print('video_path', video_path)
output_folder_screenshot_path = initialize_output_folder(video_path)
detect_unique_screenshots(video_path, output_folder_screenshot_path)
print('Please Manually verify screenshots and delete duplicates')
while True:
choice = input("Press y to continue and n to terminate")
choice = choice.lower().strip()
if choice in ['y', 'n']:
break
else:
convert_screenshots_to_pdf(output_folder_screenshot_path) | 2022-12-06 13:44:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17478083074092865, "perplexity": 11172.63545190051}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711108.34/warc/CC-MAIN-20221206124909-20221206154909-00878.warc.gz"} |
https://lonhxhf.web.app/66204/37084.html | # Limit vs stop na citaci
See full list on dmv.ny.gov
Presumed Limits “Presumed” speed-limit violations are a little more complicated but give you far more flexibility in building your defense. 2021. 2. 11. · A tourniquet is a device that is used to apply pressure to a limb or extremity in order to limit – but not stop – the flow of blood.
The first ones had black lettering on a white background and were 24 by 24 inches (61 cm × 61 cm), somewhat smaller than the current sign. As stop signs became more … Imaginando que você comprou uma ação a R$50, você pode colocar uma ordem stop loss a R$ 45, ou seja, se a cotação atingir este valor, a ordem pode ser efetuada, limitando sua perda a … Trading Order Types: Market, Limit, Stop, and If-Touched. 14 of 33. How to Use Market Orders Effectively to Buy and Sell Stock. 15 of 33. Manage Stock Trading With Limit Orders. 16 of 33.
## The Diffraction Limit. Every lens has an upper-performance limit dictated by the laws of physics and the Airy disk, known as the diffraction limit. This limit is the theoretical maximum resolving power of the lens given in line pairs per millimeter $\left[ \small{\tfrac{\text{lp}}{\text{mm}}} \right]$.
Oliver Wendell Holmes made the analogy during a controversial Supreme Court case that was overturned more than 40 years ago. Mar 12, 2019 · The average person in the United States consumes around 17 teaspoons, or 71.14 grams, of added sugar per day, which far exceeds recommended limits. Jan 14, 2021 · Instagram Message Limit. There is no publicly known limit for DMs, but trusted accounts can expect to send anywhere from 50 to 100 DMs/day.
### What is a Stop-Limit Order? Learn about Stop Limit orders and how to use them on Binance the Cryptocurrency Exchange. Subscribe to keep up to date with more
Mar 12, 2020 · Hodge is the director of the Center for Public Health Law and Policy at Arizona State University, an affiliate of the Network for Public Health Law.As the number of COVID-19 cases climbs, he said The f-number (f/#) and numerical aperture (NA) are calculated as defined.
9. · Search for researchers in Web of Science 2021. 2. 9. · Values considered “missing”¶ As data comes in many shapes and forms, pandas aims to be flexible with regard to handling missing data. While NaN is the default missing value marker for reasons of computational speed and convenience, we need to be able to easily detect this value with data of different types: floating point, integer, boolean, and general object. Search the world's information, including webpages, images, videos and more.
Superintendent of Documents, Mail Stop: SSOP, Washington, DC 20402-9328 Table of Contents Page Explanation v Title 40: Chapter I—Environmental Protection Agency (Continued) 3 Finding Aids: Material Approved for Incorporation by Reference 841 Table of CFR Titles and Chapters 859 Alphabetical List of Agencies Appearing in the CFR 877 List of A limit order can be seen by the market; a stop order can't, until it is triggered. If you want to buy an $80 stock at$79 per share, then your limit order can be seen by the market and filled when What is a Stop-Limit Order? Learn about Stop Limit orders and how to use them on Binance the Cryptocurrency Exchange. Subscribe to keep up to date with more Buy Stop Limit – este tipo de ordem combina os dois primeiros tipos, sendo uma ordem stop de compra (Buy Stop), que quando acionada, coloca uma ordem de compra limitada (Buy Limit). Assim que o preço futuro de Ask atingir o nível de stop indicado na ordem (o campo Preço), será colocada uma ordem limitada de compra no nível especificado e-mail:cesarlibarinocontato@gmail.comLink contato:https://mywhats.net/cesarcontatoinsta: @cesarlibarinohttps://www.instagram.com/cesarlibarino/facebook: http A Stop statement in your code is encountered, switching the mode to break mode. An End statement in your code is encountered, switching the mode to design time .
Ordem Limite. Uma ordem limit ou limitada é uma ordem para comprar ou vender ações ou criptomoedas por um preço específico.Por exemplo, se você quiser comprar ações por R\$ 0.10 a menos, pode definir uma ordem limite que não … 2021. 2. 23. · Stop signs originated in Michigan in 1915. The first ones had black lettering on a white background and were 24 by 24 inches (61 cm × 61 cm), somewhat smaller than the current sign.
14 of 33. How to Use Market Orders Effectively to Buy and Sell Stock. 15 of 33. Manage Stock Trading With Limit Orders.
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### On question types that allow multiple answer choices, rows, or textboxes, you can set a range or a limit on how many the respondent must answer. To do this, you need to make the question required to answer .
Second, in 2009, IE 7 was still around and it had a maximum of two connections per host name. This was configurable via the system registry. On question types that allow multiple answer choices, rows, or textboxes, you can set a range or a limit on how many the respondent must answer. To do this, you need to make the question required to answer . Hodge is the director of the Center for Public Health Law and Policy at Arizona State University, an affiliate of the Network for Public Health Law.As the number of COVID-19 cases climbs, he said Furthermore, this limit is only a best-case scenario when using an otherwise perfect lens; real-world results may vary. NOTES ON REAL-WORLD USE IN PHOTOGRAPHY. Even when a camera system is near or just past its diffraction limit, other factors such as focus accuracy, motion blur and imperfect lenses are likely to be more significant.
## R is.na Function Example (remove, replace, count, if else, is not NA) Well, I guess it goes without saying that NA values decrease the quality of our data.. Fortunately, the R programming language provides us with a function that helps us to deal with such missing data: the is.na function.
New accounts. For new accounts, for the first 12 to 20 days, you need to wait 36-48 … Stop loss . Aby stworzyć efektywne zlecenie stop loss należy określić cztery parametry, jest to: limit akcji, limit aktywacji, limit ceny oraz data ważności zlecenia.
12. | 2023-01-29 15:13:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28759363293647766, "perplexity": 6336.087368093652}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499744.74/warc/CC-MAIN-20230129144110-20230129174110-00647.warc.gz"} |
https://isbn.nu/authorx/hudson_kiki | search for books and compare prices
Kiki Hudson has written 3 work(s)
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Product Description: The single most difficult thing one faces when one begins to learn a new branch of mathematics is to get a feel for the mathematical sense of the subject. The purpose of this book is to help the aspiring reader acquire this essential common sense about algebraic topology in a short period of time...read more
By Kiki Hudson (trans) and Hajime Sato
Paperback:
9780821810460 | Amer Mathematical Society, February 1, 1999, cover price $31.00 | About this edition: The single most difficult thing one faces when one begins to learn a new branch of mathematics is to get a feel for the mathematical sense of the subject. Product Description: This book aims at providing a handy explanation of the notions behind the self-similar sets called 'fractals' and 'chaotic dynamical systems'. The authors emphasize the beautiful relationship between fractal functions (such as Weierstrass') and chaotic dynamical systems; these nowhere-differentiable functions are generating functions of chaotic dynamical systems...read more Hardcover: 9780821805374 | Amer Mathematical Society, November 1, 1997, cover price$50.00 | About this edition: This book aims at providing a handy explanation of the notions behind the self-similar sets called 'fractals' and 'chaotic dynamical systems'.
Product Description: This book covers fundamental techniques in the theory of $C^{\infty }$-imbeddings and $C^{\infty }$-immersions, emphasizing clear intuitive understanding and containing many figures and diagrams. Adachi starts with an introduction to the work of Whitney and of Haefliger on $C^{\infty }$-imbeddings and $C^{\infty }$-manifolds...read more
Hardcover:
9780821846124 | Amer Mathematical Society, June 1, 1993, cover price $90.00 | About this edition: This book covers fundamental techniques in the theory of$C^{\infty }$-imbeddings and$C^{\infty }\$-immersions, emphasizing clear intuitive understanding and containing many figures and diagrams.
displaying 1 to 3 | at end | 2020-11-23 20:05:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24561695754528046, "perplexity": 3467.805847001982}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141164142.1/warc/CC-MAIN-20201123182720-20201123212720-00654.warc.gz"} |
https://math.stackexchange.com/questions/723588/what-trick-to-calculate-this-frechet-derivative | # What trick to calculate this Frechet derivative?
Let $u(t) \in L^{2}(0, 1)$. I need to calculate the first and second Frechet derivatives of $$J(u) = \int_0^1 \left(\int_0^{t^3}u(s)ds\right)^2dt$$
I am completely at a loss here: I know several tricks that facilitate computation of Frechet derivatives:
1. Decompose operator to a composition of familiar operators and apply chain rule.
2. Understand the action performed by the operator (like the shift operator), then apply matrix calculus.
However, my problem has an intergral with a changing limit. I don't even know if I can apply the common-calculus approach:
If $I(t) = \int_{x_1(t)}^{x_2(t)} f(x,t)dx$, then $I'(t) = f(x_2, t)\frac{dx_2}{dt} - f(x_1, t)\frac{dx_1}{dt} + \int_{x_1(t)}^{x_2(t)} \frac{\partial f}{\partial t}dx$.
And even if I could, how do I start?
### Update:
@Stephen's comment got me thinking:
$$\lim_{h\to0} \frac{|J(u + h) - J(u) - DJ(u)(h)|}{\|h\|_{L^2}} = \lim_{h\to0} \frac{|2 \int_0^1 \left( \int_0^{t^3} u(s) ds \int_0^{t^3}h(s)ds\right)dt + \int_0^{1} \left(\int_0^{t^3}h(s)ds\right)^2dt - DJ(u)(h)|}{\|h\|_{L^2}}$$
Now I am very tempted to introduce an operator $Au = \int_0^{t^3}u(s)ds$, rewrite the above as
$$\frac{|2\left<Au, Ah\right> + \left<Ah, Ah\right> - DJ(u)(h)|}{\|h\|_{L^2}}$$
and find $DJ(u)$. However, that $t^3$ that is the upper limit of the integral when I introduce $A$ is confusig... Am I making a mistake?
• $t$ is a dummy variable. You shouldn't be differentiating with respect to $t$. $J$ is a linear function, therefore its derivative is itself. – Stephen Montgomery-Smith Mar 23 '14 at 18:06
• Also, nothing to do with answering your question, but you can use Fubini's Theorem and see that $J(u) = \int_0^1 \int_{s^{1/3}}^1 \,dt u(s) \,ds = \int_0^1 (1-s^{1/3}) u(s) \, ds$. – Stephen Montgomery-Smith Mar 23 '14 at 18:07
• @Stephen sorry, my misprint. J(u) isn't linear, I am now trying to understand why $t$ is a dummy. – alisianoi Mar 23 '14 at 18:09
• You are essentially correct, although your notation is messed up. No remember what the inner product is, and then apply Fubini's Theorem one or two times. In my answer, you can replace $\delta u$ by $h$ if you prefer. – Stephen Montgomery-Smith Mar 23 '14 at 18:30
Think of $J(u + h) - J(u) = F(u) \cdot h + o(h)$, where $u\cdot v$ in this context means $\int_0^1 u(s) v(s) \, ds$. So $$\begin{split} J(u + h) -J(u) &= \int_0^1 \left(\int_0^{t^3} u(s)+h(s) \, ds \right)^2 \, dt - \int_0^1 \left(\int_0^{t^3} u(s)\, ds \right)^2 \, dt\\ &= \int_0^1 2 \left(\int_0^{t^3} u(s)\, ds \right) \left(\int_0^{t^3} h(s)\, ds \right) \, dt + o(h) \\ &= 2 \int_0^1 \int_0^{t^3} \int_0^{t^3} u(s) h(r) \, ds \, dr \, dt + o(h) . \end{split}$$ Now interchange the integrals between $r$ and $t$, and you get $$2 \int_0^1 \int_{r^{1/3}}^{1} \int_0^{t^3} u(s) h(r) \, ds \, dt \, dr + o(h) .$$ Hence the derivative of $J$ at $u$ is $$F(u)(r) = 2 \int_{t=r^{1/3}}^{1} \int_{s=0}^{t^3} u(s) \, ds \, dt = 2 \int_{s=0}^1 \int_{t=\max\{r^{1/3},s^{1/3}\}}^1 u(s) \, ds \\ = \int_0^1 (1-\max\{r^{1/3},s^{1/3}\}) u(s) \, ds \\ = (1-r^{1/3}) \int_0^r u(s) \, ds + \int_r^1 (1-s^{1/3}) u(s) \, ds.$$
• It is exactly the same as your $2 \langle Au, Ah \rangle$. – Stephen Montgomery-Smith Mar 23 '14 at 18:31
• But also, I used that $s$ is a dummy variable, so I can change the name of one of them to $s'$. Then I can pull the integrals into each other and write it as one double integral (inside the $t$ integral, hence a triple integral). – Stephen Montgomery-Smith Mar 23 '14 at 18:33
• I no longer believe $t$ to be dummy. It just can't be: I have $\int_0^1 I(u, t) dt$ where $I(u, t)$ is some integral. You are telling to exchange $s'$ and $t$ but that can't be done: the upper limit in $I(u, t)$ depends on $t$! – alisianoi Mar 23 '14 at 19:10
• $\int_{t=0}^1 \int_{s=0}^{t^3} = \int_{s=0} ^1 \int_{t=s^{1/3}}^1$. Or write it as $\int \int _ {0 \le s \le t^3 \le 1}$. – Stephen Montgomery-Smith Mar 23 '14 at 19:20 | 2019-11-12 10:31:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9074214696884155, "perplexity": 335.9606154728961}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496665521.72/warc/CC-MAIN-20191112101343-20191112125343-00439.warc.gz"} |
https://bird.bcamath.org/handle/20.500.11824/4/browse?rpp=20&sort_by=1&type=title&offset=45&etal=-1&order=ASC | Now showing items 46-65 of 98
• #### On the bound states of Schrödinger operators with $\delta$-interactions on conical surfaces
(2016-06-30)
In dimension greater than or equal to three, we investigate the spectrum of a Schrödinger operator with a $\delta$-interaction supported on a cone whose cross section is the sphere of codimension two. After decomposing ...
• #### On the energy of critical solutions of the binormal flow
(2019-07-20)
The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisen- berg model in ferromagnetism, and the 1-D cubic Schr ...
• #### On the energy of critical solutions of the binormal flow
(2020-07-02)
The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1-D cubic ...
• #### On the Evolution of the Vortex Filament Equation for regular M-polygons with nonzero torsion
(2019-09-03)
In this paper, we consider the evolution of the Vortex Filament equa- tion (VFE): Xt = Xs ∧ Xss, taking M-sided regular polygons with nonzero torsion as initial data. Us- ing algebraic techniques, backed by numerical ...
• #### On the improvement of the Hardy inequality due to singular magnetic fields
(2018-07-12)
We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type ...
• #### On the improvement of the Hardy inequality due to singular magnetic fields
(2018-07-12)
We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type ...
• #### On the improvement of the Hardy inequality due to singular magnetic fields
(2020-09-01)
We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type ...
• #### On the regularity of solutions to the k-generalized korteweg-de vries equation
(2018-07)
This work is concerned with special regularity properties of solutions to the k-generalized Korteweg-de Vries equation. In [Comm. Partial Differential Equations 40 (2015), 1336–1364] it was established that if the initial ...
• #### On the Relationship between the One-Corner Problem and the $M-$Corner Problem for the Vortex Filament Equation
(2018-06-28)
In this paper, we give evidence that the evolution of the vortex filament equation (VFE) for a regular M-corner polygon as initial datum can be explained at infinitesimal times as the superposition of M one-corner initial ...
• #### On the smallness condition in linear inviscid damping: monotonicity and resonance chains
(2020)
We consider the effects of mixing by smooth bilipschitz shear flows in the linearized Euler equations on $\mathbb{T}_{L}\times\mathbb{R}$. Here, we construct a model which is closely related to a small high frequency ...
• #### On the unique continuation of solutions to non-local non-linear dispersive equations
(2020-08-02)
We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if (Formula presented.) are two suitable solutions of the equation defined in ...
• #### Pseudospectral Methods for the Fractional Laplacian on R
(2020-07-02)
In this thesis, first, we propose a novel pseudospectral method to approximate accu- rately and efficiently the fractional Laplacian without using truncation. More pre- cisely, given a bounded regular function defined over ...
• #### Reconstruction from boundary measurements for less regular conductivities
(2016-10-01)
In this paper, following Nachman's idea [14] and Haberman and Tataru's idea [9], we reconstruct $C^1$ conductivity $\gamma$ or Lipchitz conductivity $\gamma$ with small enough value of $|\nabla log\gamma|$ in a Lipschitz ...
• #### Regularity of fractional maximal functions through Fourier multipliers
(2018)
We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function ...
• #### Relativistic Hardy Inequalities in Magnetic Fields
(2014-12-31)
We deal with Dirac operators with external homogeneous magnetic fields. Hardy-type inequalities related to these operators are investigated: for a suitable class of transversal magnetic fields, we prove a Hardy inequality ...
• #### The relativistic spherical $\delta$-shell interaction in $\mathbb{R}^3$: spectrum and approximation
(2017-08-03)
This note revolves on the free Dirac operator in $\mathbb{R}^3$ and its $\delta$-shell interaction with electrostatic potentials supported on a sphere. On one hand, we characterize the eigenstates of those couplings by ...
• #### Riemann's non-differentiable function and the binormal curvature flow
(2020-07-14)
We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object ...
• #### Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory
(2019)
Abstract. We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − Lσ,μ[φ(u)] = f(x,t) in RN × (0,T), where Lσ,μ is a general ...
• #### Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments
(2018)
\noindent We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations \partial_t u-\mathfrak{L}[\varphi(u)]=f(x,t) \qquad\text{in}\qquad ...
• #### The Schrödinger equation and Uncertainty Principles
(2020-09)
The main task of this thesis is the analysis of the initial data u0 of Schrödinger’s initial value problem in order to determine certain properties of its dynamical evolution. First we consider the elliptic Schrödinger ... | 2021-10-28 19:48:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6516155004501343, "perplexity": 1056.4166836605266}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588526.57/warc/CC-MAIN-20211028193601-20211028223601-00097.warc.gz"} |
https://fiftylinesofcode.com/hackvent-2020-day-10 | ## HACKvent 2020 - Day 10
01-01-2021 - 2 minutes, 56 seconds -
Challenge - Be patient with the adjacent
Ever wondered how Santa delivers presents, and knows which groups of friends should be provided with the best gifts? It should be as great or as large as possible! Well, here is one way.
Hmm, I cannot seem to read the file either, maybe the internet knows?
HV2020Day10Challenge.col
Hints
• Hope this cliques for you
• bin2asc will help you with this, but...
• segfaults can be fixed - maybe read the source
• There is more than one thing you can do with this type of file! Try other options...
• Groups, not group
Solution
After some research we learned that the given file is a binary DIMACS file that holds an undirected graph in the form of an adjacency matrix. Loading the file into various math tools failed. Binary DIMACS support is horrbile or non-existent. Also, the graph is 18876 vertices and 439050 edges large, making around 21 MB file size... One of these, just much larger.
Hidden somewhere on a university server we found the format description and also a conversion tool to the ASCII format. Note that this was done before the hint with bin2asc was released. So we converted the file to the better supported ASCII representation.
Of course the tool had to be compiled first, but that was easy as there were no dependencies. We only had to change a bunch of naive stack allocations to heap ones and increase the vertex limit. Conversion to ASCII went well, but we still had no idea where the flag could be at.
Next we fed the file into the NetworkX python library as it can enumerate all the maximum cliques, as was suggested in the challenge hint. This is where we failed, because bin2asc has a bug. It opens all files as text. Remember, the tool is dedicated to converting from a binary format 🤦. Due to platform differences (maybe line break handling?), it silently produced wrong results on Windows! And you can't really see that there is a problem based on the huge text file produced. Its just a whole lot of numbers. I hope noone blindly trusted this tool for his work...
So the next 4 hours or so, we worked with bad cliques, couldn’t make a sense of it and went another route. Namely, graph coloring. This also yielded nothing. Finally the 24 hour window for full points closed and our personal time we set aside for the challenge as well, so we left this one unsolved.
Update: On the following day we heard about the bin2asc bug, and also saw that the challenge was extended, so we tried once again. After the bugs were fixed we got way nicer cliques and finished the challenge.
import networkx as nx
# shamelessly copied from somewhere
def read_dimacs_graph(file_path):
edges = []
with open(file_path, 'r') as file:
for line in file:
if line.startswith('c'):
print(*line.split()[1:])
elif line.startswith('p'):
p, name, vertices_num, edges_num = line.split()
print('{0} {1} {2}'.format(name, vertices_num, edges_num))
elif line.startswith('e'):
_, v1, v2 = line.split()
edges.append((v1, v2))
else:
continue
return nx.Graph(edges)
# input
kids = ['104','118','55','51','123','110','111','116','95','84','72','69','126','70','76','65','71','33','61','40','124','115','48','60','62','83','79','42','82','121','125','45','98','114','101','97','100'];
graph = read_dimacs_graph("graph2.col")
# returns the size of the largest maximal clique containing each given node
nodeCliques = nx.node_clique_number(graph, kids)
# kid -> containing-clique size lookup
for kid in kids:
print(chr(nodeCliques[kid]), end = '')
Of course it took some time to arrive at this solution. Most of the time we worked with find_cliques before and couldn’t get any results due to the bug. Theoretically it would have worked though.
HV20{Max1mal_Cl1qu3_Enum3r@t10n_Fun!}
Note: No, not fun at all. | 2022-12-01 16:02:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.426427960395813, "perplexity": 2396.545628602127}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710829.5/warc/CC-MAIN-20221201153700-20221201183700-00859.warc.gz"} |
https://testbook.com/question-answer/find-the-difference-in-compound-interest-calculate--5cc83a5ffdb8bb02dde5482a | # Find the difference in compound interest calculated on the same principal of Rs. 30000 at 10% and 20% per annum for 2 years annually and half year quarterly.
1. Rs. 3478
2. Rs. 3225
3. Rs. 2341
4. Rs. 3509
Option 2 : Rs. 3225
## Detailed Solution
Let P = Principal, N = time and R = rate of interest
When compounded annually at 10% rate for 2 years,
⇒ Compound interest = P{1 + (R/100)}n – P = 30000{1 + (10/100)}2 – 30000 = 36300 – 30000
= Rs. 6300
When compounded quarterly at 20% rate for half year,
⇒ Compound interest = P{1 + (R/4)/100}4n – P = 30000[1 + {(20/4)/100}]{(6/12) × 4} – 30000 = 33075 – 30000 = Rs. 3075
⇒ Required Difference = 6300 – 3075 = Rs. 3225 | 2021-11-28 20:26:22 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9210367202758789, "perplexity": 5242.521637356559}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358591.95/warc/CC-MAIN-20211128194436-20211128224436-00371.warc.gz"} |
http://mathhelpforum.com/differential-geometry/224488-convergent-series-orthonormal-sets.html | # Thread: Convergent Series and Orthonormal Sets
1. ## Convergent Series and Orthonormal Sets
What is an example of a convergent series $\sum e_{k}$ that does not have a sum of x?
2. ## Re: Convergent Series and Orthonormal Sets
If you just chose thing pretty much at random, you would probably get such an example. It is only as long as the vectors $e_k$ form an orthonormal basis for whatever vector space you are talking about that this sum will be equal to x.
An example is $e_1= <2, 0>$, $e_2= <0, 3>$, and $x= <1, 1>$. | 2017-09-25 17:30:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9701650738716125, "perplexity": 307.4961143764696}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818692236.58/warc/CC-MAIN-20170925164022-20170925184022-00015.warc.gz"} |
https://tex.stackexchange.com/questions/375642/having-one-bib-file-but-generate-references-per-each-chapter?noredirect=1 | # Having one bib file but generate references per each chapter
I know that cite and biblatex are incompatible. Also, natbib and bibtex. I have looked at many Q & A regarding having bib per chapters, but those except this were dealing with multiple bibs. I have just one big bib and 5 chapters (separated .tex files).
I would like to generate chapters-wised references instead of one at the end. Here is my structure and I am using Pdflatex + bibtex + pdflatex*2 + viewpdf :
Currently, it generated one reference at the end, but If I want to have that chapter-wised and I put my bib lines at the end of each chapters' tex file, it gives me error. This task should be straightforward, but it is not.
\usepackage[cite]
\usepackage ...
\makeindex
\include{foreword}
\include{preface}
\tableofcontents
\mainmatter%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\include{part}
\selectlanguage{english}
\include{1-survey/survey}
\input{2-}
\input{3-}
\input{4-}
\input{5-}
\selectlanguage{english}
\backmatter%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\printindex
\bibliographystyle{plain}
\bibliography{bib/survey}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
Now, as suggested by biblatex reference, biblatex is able to provide multiple references with this structure:
\documentclass{...}
\usepackage[...]{biblatex}
\begin{document}
\cite{...}
...
\printbibliography
\end{document}
I have commented my include{cite} and added those instead of \bibliography and I am recieving this error:
Process started
This is BibTeX, Version 0.99d (TeX Live 2015) The top-level auxiliary file:
book.aux A level-1 auxiliary file: foreword.aux A level-1 auxiliary file:
preface.aux A level-1 auxiliary file: acronym.aux A level-1 auxiliary file:
part.aux A level-1 auxiliary file: 1-survey/definitions.aux A level-1
auxiliary file: 4-/definitions.aux A level-1 auxiliary file:
5-/definitions.aux I found no \citation commands---while reading file book.aux
I found no \bibdata command---while reading file book.aux I found no \bibstyle
command---while reading file book.aux (There were 3 error messages)
Is there a way I don't change my compilation flow with this? Apparently pdflatex and latex are having differences on usage.
• See § 3.7.4 Bibliography sections, pp. 82-83 in the biblatex documentation. – Bernard Jun 19 '17 at 9:10
• Thanks @Bernard for the reference. I did as suggested, please see my updated question. – Amir Jun 19 '17 at 15:35
• Do you have any reason to not use biblatex? – Bernard Jun 19 '17 at 15:56
• please see my updated question. In the doc it says I have to use latex and then biber. Latex doesn't compiler and I don't have biber. Is there a chance I keep my old compilation flow (pdflatex,...) ? – Amir Jun 19 '17 at 15:59
• Which distribution do you have? – Bernard Jun 19 '17 at 16:07 | 2021-04-23 14:28:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9297672510147095, "perplexity": 5591.265365899422}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039594808.94/warc/CC-MAIN-20210423131042-20210423161042-00628.warc.gz"} |
https://www.futurelearn.com/courses/more-data-mining-with-weka/0/steps/29136 | 4.13
## The University of Waikato
Skip to 0 minutes and 11 secondsHello again. You know, the trouble with life is that sometimes everything just comes down to money. In this lesson and the next we’re going to look at counting the cost in data mining applications. What is success? Well, that’s a pretty good question, I suppose. In data mining terms, we’ve looked at the classification rate, measured on a test set, or holdout, or cross-validation. But essentially we’re trying to minimize the number of errors, or maximize the classification rate. In real life, different kinds of errors might have different costs, and minimizing the total errors might be inappropriate. Now we looked at the ROC curve in Class 2, and that shows you the different tradeoffs between the different error costs.
Skip to 0 minutes and 54 secondsBut it’s not really appropriate if you actually know the error costs: then, we want to pick a particular point on this ROC curve. We’re going to look at the credit rating dataset, credit-g.arff. It’s worse to class a customer as “good” when they’re “bad” then it is to class a customer as “bad” when they’re “good”. (In this dataset, the class value is “good” or “bad”.) The idea is that if you class someone as “good” when they’re “bad” and you give them a loan, then he’s going to run away with all your money, whereas if you make an error the other way round then you might have an opportunity to rectify it later on.
Skip to 1 minute and 34 secondsTo tell you the truth, I know nothing about the credit rating industry, but let’s just suppose that’s the case. Furthermore, let’s suppose that the cost ratio is 5 to 1. I’ve got the credit dataset open here, and I’m going to run J48. What I get is an error rate of 29.5%, a success rate of 70–71%. Down here is the confusion matrix. I’ve copied those over here on to this slide. You can see that the cost here, the number of errors, is effectively the 183 plus 112, those off-diagonal elements of the confusion matrix. If errors cost the same amount, that’s a fair reflection of the cost of this confusion matrix.
Skip to 2 minutes and 20 secondsHowever, if the cost matrix is different, then we need to do a different kind of evaluation. On the Classify panel, we can do a cost-sensitive evaluation. Let me go and do that for you. In the More options menu, we’re going to do a cost-sensitive evaluation. I need to set a cost matrix. This interface is a little weird. I want a 2×2 matrix; I’m going to resize this. Here we’re got a cost of 1 for both kinds of error, but I want a cost of 5 for this kind of error. Just close that and then run this again. Now I’ve got the same result, the same confusion matrix, but I’ve got some more figures here.
Skip to 3 minutes and 1 secondI’ve got a total cost of 1027 and an average cost of 1.027. (There are 1000 instances in this dataset.) Coming back to the slide, the cost here is computed by taking the 183 in the lower left and multiplying it by 5 – because that’s the cost of errors down there – and the 112 times 1, adding those up, and I get 1027. If I take the baseline, let’s go and have a look at ZeroR. I’m going to run ZeroR on this. Here it is. Here I get a cost of 1500. I get this confusion matrix. Over here on the slide, there’s the confusion matrix.
Skip to 3 minutes and 40 secondsAnd although I’ve only got 300 errors here, they’re expensive errors, they each cost $5, so I’ve got a cost of 1500. This is classifying everything as “good”, because there are more “good” instances than “bad” in this dataset. If I were to classify everything as “bad” the total cost would only be 700. That’s actually better than either J48 or ZeroR. Obviously we ought to be taking the cost matrix into account when we’re doing the classification, and that’s exactly what the CostSensitiveClassifier does. We’re going to take the CostSensitiveClassifier, select J48, define a cost matrix, and see what happens. It’s meta > CostSensitiveClassifier, which is here. Skip to 4 minutes and 29 secondsI can define a classifier: I’m going to choose J48, which is here. I need to specify my cost matrix. I want it 2×2; I’ll need to resize that. I want to put a 5 down here. Cool. I’m just going to run it. Skip to 4 minutes and 53 secondsNow I get a worse classification error. We’ve only got 60–61% accuracy, but we’ve got a smaller cost, 658. And we’ve got a different confusion matrix. Back here on the slide you can see that. The old confusion matrix looked like this, and the new confusion matrix is the one [below]. You can see that the number 183 of expensive errors has been reduced to 66. That brings the cost down, the average cost, to 0.66 per instance instead of 1.027, despite the fact that we now have a worse classification rate. Let’s look at what ZeroR does with the CostSensitiveClassifier. It’s kind of interesting because we’re going to get a different rule. Skip to 5 minutes and 41 secondsInstead of classifying everything as “good”, we’re going to classify everything as “bad”. We’re going to make 700 mistakes, but they’re cheap mistakes. It’s only going to cost us$700. That’s what we’ve learned today. Is classification accuracy the best measure? Very likely it isn’t, because in real life different kinds of errors usually do have different costs. If you don’t know the costs, you might just want to look at the tradeoff between the error costs – different parts of the space – and the ROC curve is appropriate for that.
Skip to 6 minutes and 13 secondsBut if you do know the costs – the cost matrix – then you can do cost-sensitive evaluation to find the total cost on the test set of a particular learned model; or you can do cost-sensitive classification, that is, take the costs into account when producing the classifier.
Skip to 6 minutes and 31 secondsThe CostSensitiveClassifier does this: it makes any classifier cost-sensitive. How does it do this? Very good question. We’re going to find out in the next lesson.
# Counting the cost
So far we’ve taken the classification rate – computed on a test set, or holdout, or cross-validation – as the measure of a classifier’s success. We’re trying to maximize the classification rate, that is, minimize the number of errors. But in real life, different kinds of error often have different costs. If the costs are known, they can be taken into account when evaluating a classifier’s performance. Error costs can also taken into account when using a learning method to create a classifier – regardless of which learning method is used – to get a classifier that minimizes the cost rather than the error rate. | 2019-08-23 16:12:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6131477952003479, "perplexity": 532.8958218744233}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318894.83/warc/CC-MAIN-20190823150804-20190823172804-00441.warc.gz"} |
http://syllabus.nesa.nsw.edu.au/mathematics/mathematics-k10/content/790/ | NSW Syllabuses
# Mathematics K–10 - Stage 5.1 - Number and Algebra Linear Relationships
## Outcomes
#### A student:
• MA5.1-1WM
uses appropriate terminology, diagrams and symbols in mathematical contexts
• MA5.1-3WM
provides reasoning to support conclusions that are appropriate to the context
• MA5.1-6NA
determines the midpoint, gradient and length of an interval, and graphs linear relationships
Related Life Skills outcomes: MALS-32MG, MALS-33MG, MALS-34MG
## Content
• determine the midpoint of an interval using a diagram
• use the process for calculating the 'mean' to find the midpoint, M, of the interval joining two points on the Cartesian plane
• explain how the concept of mean ('average') is used to calculate the midpoint of an interval (Communicating)
• plot and join two points to form an interval on the Cartesian plane and form a right-angled triangle by drawing a vertical side from the higher point and a horizontal side from the lower point
• use the interval between two points on the Cartesian plane as the hypotenuse of a right-angled triangle and use the relationship $$\textrm{gradient} = \dfrac{\rm{rise}}{\rm{run}}$$ to find the gradient of the interval joining the two points
• describe the meaning of the gradient of an interval joining two points and explain how it can be found (Communicating)
• distinguish between positive and negative gradients from a diagram (Reasoning)
• use graphing software to find the midpoint and gradient of an interval
• Find the distance between two points located on the Cartesian plane using a range of strategies, including graphing software (ACMNA214)
• use the interval between two points on the Cartesian plane as the hypotenuse of a right-angled triangle and apply Pythagoras' theorem to determine the length of the interval joining the two points (ie 'the distance between the two points')
• describe how the distance between (or the length of the interval joining) two points can be calculated using Pythagoras' theorem (Communicating)
• use graphing software to find the distance between two points on the Cartesian plane
• Sketch linear graphs using the coordinates of two points (ACMNA215)
• construct tables of values and use coordinates to graph vertical and horizontal lines, such as $$x=3$$, $$x=-1$$, $$y=2$$, $$y=-3$$
• identify the $$x$$- and $$y$$-intercepts of lines
• identify the $$x$$-axis as the line $$y= 0$$ and the $$y$$-axis as the line $$x = 0$$
• explain why the $$x$$- and $$y$$-axes have these equations (Communicating, Reasoning)
• graph a variety of linear relationships on the Cartesian plane, with and without the use of digital technologies, eg
$$y = 3 - x$$, $$y = \frac{x+1}{2}$$, $$x + y = 5$$, $$x - y = 2$$, $$y = \frac{2}{3}x$$
• compare and contrast equations of lines that have a negative gradient and equations of lines that have a positive gradient (Communicating, Reasoning)
• determine whether a point lies on a line by substitution
• Solve problems involving parallel lines (ACMNA238)
• determine that parallel lines have equal gradients
• use digital technologies to compare the graphs of a variety of straight lines with their respective gradients and establish the condition for lines to be parallel (Communicating, Reasoning)
• use digital technologies to graph a variety of straight lines, including parallel lines, and identify similarities and differences in their equations (Communicating, Reasoning)
### Background Information
The Cartesian plane is named after the French philosopher and mathematician René Descartes (1596–1650), who was one of the first mathematicians to develop analytical geometry on the number plane. He shared this honour with the French lawyer and mathematician Pierre de Fermat (1601–1665). Descartes and Fermat are recognised as the first modern mathematicians.
### National Numeracy Learning Progression links to this Mathematics outcome
When working towards the outcome MA5.1-6NA the sub-elements (and levels) of Number patterns and algebraic thinking (NPA7) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.
The progression sub-elements and indicators can be viewed by accessing the National Numeracy Learning Progression. | 2018-08-21 02:07:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6469974517822266, "perplexity": 956.6046490647841}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221217909.77/warc/CC-MAIN-20180821014427-20180821034427-00466.warc.gz"} |
http://www.pedrorainho.eu/tag/windows/ | # Freeing disk space using junctions
If you need to reclaim disk space, first you should read my previous post about Freeing disk space.
Now I’ve to warning you that the following steps are for advanced users.
What is a junction or a symbolic link? Check this wiki page.
To create a junction i’m going to use a tool named MKLINK.
I’m only going to create two junctions because those are the ones that I use, other junctions can be created but I encourage you to read on the internet more about that folder and if it’s possible to create a junction to that folder.
The first folder I’m going to create a junction is the C:\Windows\Installer. This folder contains all the applications install packages in your computer. To create a junction to this folder:
1. Move folder C:\Windows\Installer to D:\[somefolder] in my case D:\Junctions\Windows\Installer
2. After moving the folder open command prompt and type mklink /J “C:\Windows\Installer” “D:\Junctions\Windows\Installer”
3. The previous command will create the junction from C:\Windows\Installer to D:\Junctions\Windows\Installer
The second folder I’m going to create a junction is the C:\ProgramData\Package Cache. This folder is used by Visual Studio and as the name implies it’s used for cache. To create a junction to this folder:
1. Move folder C:\ProgramData\Package Cache to D:\[somefolder] in my case D:\Junctions\ProgramData\Package Cache
2. After moving the folder open command prompt and type mklink /J “C:\ProgramData\Package Cache” “D:\Junctions\ProgramData\Package Cache”
3. The previous command will create the junction from C:\ProgramData\Package Cache to D:\Junctions\ProgramData\Package Cache
At this point you should have reclaim lots of disk space. If you know any other want to reclaim disk space just comment the post.
# Freeing disk space
If you have a small disk , for example an 128GB SSD, you will very quickly run out of space. In this simple post I will show what you can do to reclaim disk space.
• Clean Recycle Bin, this is obvious
• Disk Cleanup, this is obvious
• Right click on a drive in Windows Explorer, next click on Properties menu item and then on Disk Cleanup button
• Clean up, system files, this is not so obvious
• If you see on that Disk Cleanup window you will see the button Clean up system files.
• If you had Windows 7 then you updated to Windows 8 and then to Windows 8.1 and then to Windows 10 you will have in your machine lots and lots of useless crap and lots and lots of less disk space.
• This is also true for Windows Updates
• Next, while you are on that window, click on the tab More Options
• Here you will find two buttons Clean up in the following sections
• Programs and features
• Uninstall every program you don’t need.
• System Restore and Shadow Copies
• Remove every thing you don’t need it
• Disable Hibernation
• Hibernation uses as much disk space as you have RAM. For example if you have 8GB of RAM you will have lost 8GB of disk space.
• To disable hibernation open command prompt as Administrator and execute the following command “powercfg.exe /hibernate off”
• Disable System Restore (advanced users)
• Open Advanced System Settings and then click on the tab System Protected
• If you don’t need System Restore, System Protected and all those fancy stuff that uses lots of disk space, disable it.
• Open Advanced System Settings and then click on the tab Advanced and then button Settings… Next click on Advanced tab and on the Change… button in Virtual memory section.
• Here you can control how your Virtual memory is managed. If don’t want to use automatic settings you can change it to manual and you can even choose the drive where you want to store your swap file.
• Clean Temp files
• Open Windows Explorer and type %TEMP% in the address bar and delete all files and folders, because they are temporary and that’s why the are stored in temp folder. Also you might want to check the C:\Windows\Temp
• Clean Temporary ASP.NET Files (Advanced users)
• If you are a .Net developer you probably want to check Temporary ASP.NET Files and clean this folder. These are the folders you might want to check
• C:\Windows\Microsoft.NET\Framework\v4.0.30319\Temporary ASP.NET Files
• C:\Windows\Microsoft.NET\Framework\v2.0.50727\Temporary ASP.NET Files
• C:\Windows\Microsoft.NET\Framework64\v4.0.30319\Temporary ASP.NET Files
• C:\Windows\Microsoft.NET\Framework64\v2.0.50727\Temporary ASP.NET Files
• Clean browser cache
• All browsers, IE, Chrome, Firefox, etc… they all store on your disk files like (images, css, javascript, html, etc…) these files are temporary and exists to “speed up” sites rendering. This way the browsers don’t need to fetch everything every time you visit a site.
• Every browser have a Clean browsing data button in browser Settings.
• Use WinDirStat or TreeSize or any similar tool
• These tools let you know where are you loosing space and eventually you can reclaim some space
• Compress Drive
• If you use NTFS it’s possible to use drive compression and save some space. In Windows Explorer right click on a drive and choose the properties menu item
• Compress Folder
• If you don’t want to compress a drive and just want to compress a folder, right click on the folder you want to compress and click on properties. Here you have the Advanced… button. Next a window will open and you will have an option to compress folder content.
• Use CCleaner or any similar tool
• Most of the manual steps that I’ve explained in this post are automatically done by some tools like CCleaner.
At this point you should have reclaim lots of disk space. If you know any other want to reclaim disk space just comment the post.
To disable Server Manager automatically startup when you logon into a Windows Server just follow this steps:
Start Server Manager
Click on Manage menu (top right)
Click Server Manager Properties
Check Do not start Server Manager automatically at logon
Click Ok button
And that’s it, the next time you start Server Manager it will not start automatically.
# Antimalware Service Executable high CPU
Today I noticed that my computer fan was running more than normal, Windows Explorer become unresponsive and basically the computer wasn’t normal. I took a look at Task Manager and Process Explorer and I noticed that Antimalware Service Executable wasn’t normal and was using about 30% CPU.
I found that strange and first I tried to kill the process, but unfortunately I can’t kill it. Then I tried stop Windows Defender Service, but I can’t stop this service. Then I reboot the computer and the problem was the same…
After a while I did a google search to see if someone else had the same problem. I’ve found that this is a common problem with lots of possible solutions.
The solution that worked for me was:
1º Open Windows Defender
2º Go to settings\Administrator and uncheck “Turn on this app” and then save settings
3º Now you will have warning messages in Windows TaskBar.
4º Open Action Center
5º Click on button “Turn on now” to turn on Windows Defender.
And after this steps for some reason my computer CPU is again normal.
# Windows 8.1 Tip to disable and re-enable hibernate
Many users like me need hibernate and found that wasn’t enabled by default.
After goggling I’ve found a command line command that does the enable and disable of hibernate.
First open the command line with administrative privileges then execute the following command:
To turn off hibernate type in the command line powercfg.exe /hibernate off
To turn on hibernate type in the command line powercfg.exe /hibernate on
Recently I was goggling for an app (for free) that allows the download of a web site for offline reading and then I came across a StackOverflow post that says use Wget.
Before start using Wget you will need to download Wget. I recommend always the latest version and you can get it for Windows from here.
Now to download a web site you will need to combine several arguments and the result would be:
wget --recursive --no-clobber --page-requisites --html-extension --convert-links --restrict-file-names=windows --domains mydomain.com --no-parent http://www.mydomain.com/site
If you want to know the definition of each arguments just type wget –help or check the wget manual page.
# Install a clean Windows 8.1 install with a Windows 8 key
As you might already know you can update your windows 8 to windows 8.1 without the need of a new product key but you can’t install a clean windows 8.1 using windows 8 product key.
Well this is not entirely true and I will teach you how to do that.
The first thing you need to do is Create a Windows 8.1 Bootable USB device.
Next you will need to locate the folder sources inside your USB device with windows 8.1 installation. :\sources
Then create a new text document with the name ei.cfg
Enter the following content in ei.cfg:
[EditionID] Professional [Channel] Retail [VL] 0
Now boot your computer and start the windows 8.1 installation.
During the installation you will need to insert a product key. Use the your windows 8 product key.
Wait the installation to finish and you will have a clean windows 8.1 installation with a windows 8 product key.
# Create a Windows 8.1 bootable USB device
In these days it’s very common that computers/laptops don’t have a optical drive. Because of this it’s no surprise that when you need to install Windows 8.1 you will need a USB device, such as a flash drive.
This was my case, I recently needed to installed Windows 8.1 and I didn’t have a optical drive. So I needed to find a way. In the next steps I will teach you how to do that.
The first thing you need is a USB device, such as a flash drive.
The last thing you need is how to create a windows 8.1 bootable USB device.
The last part it’s easy thanks to the Windows Installation Media Creation Tool. This is the only tool you will need.
When you run the Windows Installation Media Creation Tool for the first time you will be presented with the following screen:
In this screen you need to choose the language, the windows 8.1 edition and the architecture. Then click in the Next button.
In the next screen you need to choose that you want to save the installation file to the USB flash drive. Then click in the Next button.
If you want you can save it to an ISO and then burn the ISO to a optical drive such as a DVD.
In the next screen you choose the USB flash drive. Then click in the Next button.
In the last screen you need to wait the download and the creation of windows 8.1 bootable USB device.
After this last step you will be ready to install windows 8.1.
# Import CSV in MySql Workbench
This is a quick tip on how to import a csv with data into one of your tables.
1º Open MySql Workbench
2º run the following query
On linux:
load data local infile ‘/path/to/your/file.csv’ into table dbName.tableName fields terminated by ‘,’ enclosed by ‘”‘ lines terminated by ‘\n’;
On Windows
load data local infile ‘c:\\path\\to\\your\\file.csv’ into table dbName.tableName fields terminated by ‘,’ enclosed by ‘”‘ lines terminated by ‘\n’;
# Visual Studio 2012 connecting to TFS
This is what happen to Visual Studio 2012 when connecting to TFS…
Mental note: Use MessageBox, it’s a thing that exists since windows was created. | 2017-08-21 02:29:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22584931552410126, "perplexity": 3458.58302671891}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886107487.10/warc/CC-MAIN-20170821022354-20170821042354-00353.warc.gz"} |
https://codegolf.stackexchange.com/questions/40085/approximated-unsorted-square-from-a-set-of-values | # Approximated (unsorted) square from a set of values [closed]
So a colleaque came with this problem lately and I want to challange you to solve the problem in the code-golf style, shortest solution in any language wins.
The Problem:
Given is a random set of values, where the values themselfs are not the concern, but the amount (which is random as well)of values.
To keep the posts readable, let's limit the amount of values to 200, but the program must be able to handly any amount of values, which the language supports.
Input: No Input needet, due the programm should set an random amount of values (limited to 200).
Output: Can be text or graphic(px) I recommend to use single digits as values if you use text based output, to keep the output clear.
Example:
{1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7} -- Amount : 16
Now you should sort these values in a approcimated square matrix like:
1 2 3 4
5 6 7 8
9 1 2 3
4 5 6 7
(4 x 4)
Pretty simple so far, if not boring.
But now the challange:
The number of fields in the matrix must be equivalent to the amount of values given. No empty fields or ignored values are allowed.
As an example, with 12 values, an square matrix is not possible, so you have to find the next aproximitation to it, which is 3x4 or 4x3 but NOT 4x4.
For any prime, the matrix would be 1 x prime
Examples:
For 12 Values:
1 2 3 4
5 6 7 8
9 1 2 3
X X X X <- NOT ALLOWED, it makes the matrix square, but inserts non existing values.
or
1 2 3
4 5 6
7 8 9
1 2 3
For 35 Values:
1 2 3 4 5 6 7
8 9 1 2 3 4 5
6 7 8 9 1 2 3
4 5 6 7 8 9 1
2 3 4 5 6 7 8
## closed as unclear what you're asking by Peter Taylor, Falko, Optimizer, Calvin's Hobbies, COTOOct 21 '14 at 12:31
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
• You should give more details about the values. Are they continues ? How is {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7} sorted ? How does the input look like ? What are the ways of input. – Optimizer Oct 21 '14 at 9:33
• Is the question "Find the factor of a natural number which is closest to its square root?" or is there more to it than that? It's tagged sorting but the title implies that there's no need to sort. I am confused. – Peter Taylor Oct 21 '14 at 9:41
• @Sempie Ah I see. I somehow read "random" as "arbitrary". – Martin Ender Oct 21 '14 at 9:48
• If the program defines the list itself, it could simply print an arbitrary valid result without bothering about the input. I could always argue, the output corresponds to the random (non existing) input, the program thought of. – Falko Oct 21 '14 at 9:53
• Well if you want this challenge to survive it would be best to rephrase is without the values. They distract from the main goal which as PeterTaylor said is simply to "Find the factor of a natural number which is closest to its square root?" That only needs a few sentences, and may have been done before. – Calvin's Hobbies Oct 21 '14 at 9:54
## Mathematica, 91 bytes
n=RandomInteger@199+1
TableForm@Partition[Range@9~RandomChoice~n,Divisors@n~Nearest~Sqrt@n]
I hope I got this right:
• Generate a random length between 1 and 200 (inclusive).
• Generate a list of that length with numbers from 1 to 9 (inclusive).
• Find one of the two divisors closest to the square root, with Divisors@n~Nearest~Sqrt@n.
• Break the list into rows of that length.
• Display the grid with TableForm.
Example run:
## Javasript, 156 or 119 152 or 115
Output numbers 1 to 9: 152
for(q=~~Math.sqrt(n=new Date%200);~~(w=n/q)!=w;--q);i=q=0;alert("".replace.call(Array(n+1),/./g,function(){return(++q>9?q=1:q)+(++i==w?i=0||"\n":" ")}))
Output zeroes: 115
for(q=~~Math.sqrt(n=new Date%200);~~(w=n/q)!=w;--q);alert(Array(q+1).join("".replace.call(Array(w+1),/./g,0)+"\n")) | 2019-07-19 08:37:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3768015503883362, "perplexity": 832.1660583696838}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526153.35/warc/CC-MAIN-20190719074137-20190719100137-00148.warc.gz"} |
https://proofwiki.org/wiki/Disjoint_Permutations_Commute | # Disjoint Permutations Commute
## Theorem
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\rho, \sigma \in S_n$ such that $\rho$ and $\sigma$ are disjoint.
Then $\rho \sigma = \sigma \rho$.
## Proof
Let $\rho$ and $\sigma$ be disjoint permutations.
Let $i \in \Fix \rho$.
Then:
$\map {\sigma \rho} i = \map \sigma i$
whereas:
$\map {\rho \sigma} i = \map \rho {\map \sigma i}$
Aiming for a contradiction, suppose $\map \sigma i \notin \Fix \rho$.
Then because $\sigma$ and $\rho$ are disjoint it follows that:
$\ds \map \sigma i$ $\in$ $\ds \Fix \sigma$ $\ds \leadsto \ \$ $\ds \map {\sigma^2} i$ $=$ $\ds \map \sigma i$ $\ds \leadsto \ \$ $\ds \map {\sigma^{-1} \sigma^2} i$ $=$ $\ds \map {\sigma^{-1} \sigma} i$ $\ds \leadsto \ \$ $\ds \map \sigma i$ $=$ $\ds i$
But it was previously established that $i \in \Fix \rho$.
Therefore:
$\map \sigma i \in \Fix \rho$
and so:
$\map {\rho \sigma} i = \map \sigma i = \map {\sigma \rho} i$
Let $i \notin \Fix \rho$.
Then:
$i \in \Fix \sigma$
and the same proof can be performed with $\rho$ and $\sigma$ exchanged.
$\blacksquare$ | 2021-07-30 10:14:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9983081221580505, "perplexity": 247.46735124940088}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153966.52/warc/CC-MAIN-20210730091645-20210730121645-00289.warc.gz"} |
https://cs.stackexchange.com/tags/type-theory/new | # Tag Info
3
The rules of a type system may be given in one of several ways. We often begin by defining a relation "term $t$ has type $A$" using rules of inference without giving any particular way of turning the rules into an algorithm. This is sometimes called the "declarative style". The declarative style is usually the easiest to understand and ...
0
I don't know much, but I am pretty sure Linear Type Theory is used to model imperative programs. For example, Adoption and focus: practical linear types for imperative programming or On linear types and imperative update. Rust, for example, is built on something related to linear types (I forget what exactly though). It appears the more general category of ...
2
Spurred by comments in Pierre Marie's answer, I'll add some clarification on the difference between type checking and type synthesis. Generally, type checking in a dependent type system is designed to be decidable up to normalization for fully annotated terms, that is, $\lambda x : T.t$ and $\Pi x : T. U$ for example (you need type annotations in other ...
Top 50 recent answers are included | 2022-01-26 23:17:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7131106853485107, "perplexity": 654.9543633636425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305006.68/warc/CC-MAIN-20220126222652-20220127012652-00402.warc.gz"} |
https://en.m.wikibooks.org/wiki/Finite_Model_Theory/Model_Theory | # Finite Model Theory/Model Theory
Many important theorems of Model Theory do not hold when restricted to the finite case, like Gödel's completeness theorem or the compactness theorem:
## Failure of Compactness Theorem
Consider the following sentence σ3
${\displaystyle \exists _{x}\exists _{y}\exists _{z}(x\neq y\land y\neq z\land x\neq z)}$
that says that there are at least 3 different elements in a universe. One can expand σ3 easily for n other than 3. So, let Σ = {σ1, σ2, σ3, ...} be the infinite set of all these sentences. Now Σ is obviously not satisfiable by a finite model, although every finite subset of Σ is. Ok, but why does that matter? One of the most useful tools in general Model theory is the Compactness theorem, stating: "Let Σ be a set of FO sentences. If every finite subset of Σ is satisfiable, then Σ is satisfiable." But as just shown this doesn't hold for the finite case, thus there is no Compactness theorem in Finite Model Theory! | 2020-01-21 00:12:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9604319930076599, "perplexity": 567.9144266717062}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250601040.47/warc/CC-MAIN-20200120224950-20200121013950-00089.warc.gz"} |
https://www.valueinhealthjournal.com/article/S1098-3015(14)00279-4/fulltext | Abstract| Volume 17, ISSUE 3, PA37-A38, May 2014
Economic evaluations of treatments for inflammatory bowel diseases
Open Archive
Objectives
In recent years, there has been a rapid growth in the development of novel biological treatments. Numerous economic evaluations have been performed to evaluate these treatments in inflammatory bowel diseases (IBD). The objective of this project was to explore the existing evidences regarding the cost-effectiveness of treatments in IBD.
Methods
A systematic review of the literature was conducted to identify economic evaluations of IBD therapy reporting incremental cost-effectiveness or cost-utility ratios (ICERs or ICURs). The literature search was performed using electronic databases. Searches were limited to full economic evaluations published in English or French between 2003 and 2013. Cross-reference of retrieved articles was also performed to identify additional publications.
Results
A total of 15,242 potentially relevant studies were identified. After screening titles and abstracts, 43 full-text articles were assessed according to the eligibility criteria and 35 studies were included. Among those, 3 studies assessed the economic impact of IBD treatments with a companion diagnostic test. A high proportion of economic evaluations was performed from a third party payer perspective (91%) and had time horizons of 1 year or less (46%). European, American and Canadian economic evaluations accounted for 66%, 17% and 11% of the studies respectively. Treatment options under evaluation included azathioprine, infliximab, adalimumab, immunosuppressant and mesalamine. Most included studies were cost-utility analyses (94%), with ICURs ranging from dominant to CAD$11,934,934/QALY. More specifically, treatment under investigation was dominant in 26% of the analyses and was cost-effective according to a CAD$50,000/QALY and a CAD\$100,000/QALY threshold in 31% and 65% of the analyses respectively.
Conclusions
Several economic evaluations were conducted in the past years, with different parameters and results. As more treatment options become available, this review provides a comprehensive overview of evaluations previously performed and could be helpful in the realization of future economic evaluations. | 2023-01-31 20:25:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.36847835779190063, "perplexity": 4386.8741656489}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499890.39/warc/CC-MAIN-20230131190543-20230131220543-00484.warc.gz"} |
http://kmj.knu.ac.kr/journal/view.html?doi=10.5666/KMJ.2019.59.2.293 | KYUNGPOOK Math. J. 2019; 59(2): 293-299
An Identity Involving Product of Generalized Hypergeometric Series 2F2
Yong Sup Kim, Junesang Choi, Arjun Kumar Rathie
Department of Mathematics Education, Wonkwang University, Iksan 54538, Republic of Korea
e-mail : yspkim@wonkwang.ac.kr
Department of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea
e-mail : junesang@mail.dongguk.ac.kr
Department of Mathematics, Vedant College of Engineering & Technology (Rajasthan Technical University), Bundi, Rajasthan State, India
e-mail : arjunkumarrathie@gmail.com
* Corresponding Author.
Received: August 27, 2018; Revised: October 16, 2019; Accepted: October 16, 2018; Published online: June 23, 2019.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A number of identities associated with the product of generalized hypergeometric series have been investigated. In this paper, we aim to establish an identity involving the product of the generalized hypergeometric series 2F2. We do this using the generalized Kummer-type II transformation due to Rathie and Pogany and another identity due to Bailey. The result presented here, being general, can be reduced to a number of relatively simple identities involving the product of generalized hypergeometric series, some of which are observed to correspond to known ones.
Keywords: generalized hypergeometric series, Kummer-type I transformation, Kummer-type II transformation, Gauss’s summation theorem for 2F1(1), Watson summation theorem for 3F2
1. Introduction and Preliminaries
The generalized hypergeometric function pFq with p numerator and q denominator parameters is defined by (see, e.g., [2, 5, 12]; see also [16, Section 1.5])
$Fpq[α1,…,αp;β1,…,βq;z]=Fpq[α1,…,αp;β1,…,βq;z]=∑n=0∞(α1)n⋯(αp)n(β1)n⋯(βq)nznn!,$
where (λ)ν denotes the Pochhammer symbol defined (for λ, ν ∈ ℂ), in terms of the Gamma function Γ, by
$(λ)ν:=Γ(λ+ν)Γ(λ)={1(ν=0; λ∈ℂ/{0})λ(λ+1)⋯λ(λ+n-1) (ν=n∈ℕ; λ∈ℂ).$
The well-known Kummer-type I transformation for the series 1F1 is (see, e.g., [12])
$e-x F11 [a;b; x]=F11 [b-a;b; -x].$
This result (1.3) was obtained by Kummer [7] and [8] who used the theory of differential equations. Bailey [1] derived this result by employing classical Gauss’s summation theorem (see, e.g., [12, p. 48]).
Paris [9] generalized (1.3) in the form
$e-x F22 [a,1+d;b,d; x]=F22 [b-a-1,f+1;b,f; -x],$
where
$f:=d(a-b+1)a-d.$
The particular case $d=12$ of (1.4) reduces to Exton’s result [4].
Kummer [7, 8] also used the theory of differential equations to obtain the following result
$e-x2 F11 [a;2a; x]=F01 [-;a+12; x216],$
which is often referred to as Kummer-type II transformation.
Using Gauss’s summation theorem, the result (1.6) has been re-derived by Bailey [1] and Rathie and Choi [14] (see also [3]).
Motivated by the Kummer-type I transformation (1.4), Rathie and Pogany [15] also generalized the Kummer-type II transformation (1.6) in the form
$e-x2 F22 [a,1+d;2a+1,d; x]=F01 [-;a+12; x216]+x(2a-d)2d(2a+1) F01 [-;a+32; x216].$
It is obvious that the case d = 2a of (1.7) reduces to (1.6).
Using the theory of differential equations, Preece [10] established the following interesting identity involving product of generalized hypergeometric series
$F11 [a;2a; x] F11 [a;2a; -x]= F12 [a;a+12,2a; x24].$
Equation (1.8) is a special case of [6, Exercise 7.24(b)]. In fact, Koepf [6, Example 7.3] showed how such identities can be derived by using the Zeilberger algorithm.
Using (1.3), we can express (1.8) in the form
${F11 [a;2a; x]}2=ex F12 [a;a+12,2a; x24].$
Rathie [13] proved Preece’s identity (1.9) by a very short method and obtained two results closely related to it.
By employing the classical Watson’s theorem on the sum of a 3F2 with unit argument (see, e.g., [2, p. 16]), Bailey [1] generalized Preece’s identity (1.8) in the form
$F11 [a;2a; x] F11 [b;2b; -x]= F23 [12(a+b),12(a+b+1);a+12,b+12,a+b; x24].$
Rathie and Choi [14] derived the Bailey identity (1.10) by using the same technique given in [13] and obtained four results closely related to it.
In fact, a number of identities associated with the product of generalized hypergeometric series have been investigated (see, e.g., [2, Chapter X]; see also [11, Entires 8.4.45–8.4.49]). In this paper, we aim to establish an identity involving the product of generalized hypergeometric series 2F2 by using the generalized Kummer-type II transformation (1.7) and the following identity due to Bailey [1] (see also [6, Exercise 7.24 (a)])
$F01 [-;a; x] F01 [-;b; x]= F23 [12(a+b),12(a+b-1);a,b,a+b-1; 4x].$
The result presented here, being general, can be reduced to a number of relatively simple identities involving the product of generalized hypergeometric series, some of which are explicitly indicated to correspond to known ones.
2. Product Formula for 2F2
We present an identity involving product of 2F2, asserted in the following theorem.
### Theorem 2.1
The following identity holds.
$F22 [a,1+d;2a+1,d; x] F22 [b,1+e;2b+1,e; x]=ex {F23 [12(a+b),12(a+b+1);a+12,b+12,a+b; x24]+x(2a-d)2d(2a+1) F23 [12(a+b+1),12(a+b+2);a+32,b+12,a+b+1; x24]+x(2b-e)2e(2b+1) F23 [12(a+b+1),12(a+b+2);a+12,b+32,a+b+1; x24]+x2(2a-d)(2b-e)4de(2a+1)(2b+1) F23 [12(a+b+2),12(a+b+3);a+32,b+32,a+b+2; x24]}.$
Proof
Let
$L:=e-xF22 [a,1+d;2a+1,d; x] F22 [b,1+e;2b+1,e; x].$
Then
$L:={e-x2 F22 [a,1+d;2a+1,d; x]}{e-x2 F22 [b,1+e;2b+1,e; x]}.$
Using (1.7) in each factor of (2.2), we get
$L={F01 [-;a+12; x216]+x(2a-d)2d(2a+1) F01 [-;a+32; x216]}×{F01 [-;b+12; x216]+x(2b-e)2e(2b+1) F01 [-;b+32; x216]}.$
Expanding the right member of (2.3), we obtain
$L=F01 [-;a+1/2; x216] F01 [-;b+1/2; x216]+x(2a-d)2d(2a+1) F01 [-;a+3/2; x216] F01 [-;b+1/2; x216]+x(2b-e)2e(2b+1) F01 [-;a+1/2; x216] F01 [-;b+3/2; x216]+x2(2a-d)(2b-e)4de(2a+1)(2b+1) F01 [-;a+3/2; x216] F01 [-;b+3/2; x216].$
Finally, using (1.11) in each term of the right member of the last identity, we are led to the right member of (2.1) with the factor ex deleted. This completes the proof.
3. Special Cases
The result (2.1), being general, can be reduced to yield a number of relatively simple identities, several of which are considered here.
• Setting d = 2a in (2.1), we get $e-x F11 [a;2a;x] F22 [b,1+e;2b+1,e; x]=F23 [12(a+b),12(a+b+1);a+12,b+12,a+b; x24]+x(2b-e)2e(2b+1) F23 [12(a+b+1),12(a+b+2);a+12,b+32,a+b+1; x24].$
The identity (3.1) is found to be equivalent to a very recent result due to Kim et al. [5] who used a different method.
• Taking d = 2a and e = 2b in (2.1), we have $e-x F11 [a;2a; x] F11 [b;2b; x]=F23 [12(a+b),12(a+b+1);a+12,b+12,a+b; x24].$
Using (1.3) in (3.2), we obtain Bailey’s identity (1.10). So the identity (2.1) may be regarded as a generalization of Bailey’s identity (1.10).
• Setting e = d and b = a in (2.1), we find $e-x{F22 [a,1+d;2a+1,d; x]}2=F12 [a;a+12,2a; x24]+x(2a-d)d(2a+1) F12 [(a+1);a+32,2a+1; x24]+x2(2a-d)24d2(2a+1)2 F12 [(a+1);a+32,2b+2; x24].$
Taking d = 2a in (3.3) yields Preece’s identity (1.9). Therefore, the identity (3.3) can be looked upon as a generalization of Preece’s identity (1.9).
Acknowledgements
The authors would like to express their deep-felt thanks for the reviewer’s detailed and helpful comment to improve this paper as it stands.
References
1. WN. Bailey. Products of generalized hypergeometric series. Proc London Math Soc (2)., 28(4)(1928), 242-254.
2. WN. Bailey. Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics, 32, Stechert-Hafner, New York, 1964.
3. J. Choi, AK. Rathie, and B. Bhojak. On Preece’s identity and other contiguous results. Commun Korean Math Soc., 20(1)(2005), 169-178.
4. H. Exton. On the reducibility of the Kampé de Fériet function. J Comput Appl Math., 83(1997), 119-121.
5. YS. Kim, S. Gaboury, and AK. Rathie. Applications of extended Watson’s summation theorem. Turkish J Math., 42(2018), 418-443.
6. W. Koepf. Hypergeometric summation. An algorithmic approach to summation and special function identities. Advanced Lectures in Mathematics, , Friedr. Vieweg & Sohn, Braunschweig, 1998.
7. EE. Kummer. Über die hypergeometrische Reihe . J Reine Angew Math., 15(1836), 127-172.
8. EE. Kummer. Collected Papers, Volume II: Function theory, geometry and miscellaneous, , Springer-Verlag, Berlin, 1975.
9. R. Paris. A Kummer-type transformation for a2F2 hypergeometric function. J Comput Appl Math., 173(2005), 379-382.
10. CT. Preece. The product of two generalized hypergeometric functions. Proc London Math Soc (2)., 22(1924), 370-380.
11. AP. Prudnikov, Yu A. Brychkov, and OI. Marichev. Integrals and Series, Vol. 3: More Special Functions, , Gordon and Breach Science Publishers, New York, 1990. Translated from the Russian by G. G. Gould.
12. ED. Rainville. Special Functions, , Macmillan Co, New York, 1960. Reprinted by Chelsea Publishing Co., Bronx, New York, 1971.
13. AK. Rathie. A short proof of Preece’s identities and other contiguous results. Rev Mat Estatist., 15(1997), 207-210.
14. AK. Rathie, and J. Choi. A note on generalizations of Preece’s identity and other contiguous results. Bull Korean Math Soc., 35(1998), 339-344.
15. AK. Rathie, and J. Pogany. New summation formula for and a Kummer-type II transformation of2F2(x). Math Commun., 13(1)(2008), 63-66.
16. HM. Srivastava, and J. Choi. Zeta and q-Zeta Functions and associated series and integrals, , Elsevier Science Publishers, Amsterdam, London and New York, 2012. | 2020-02-19 10:03:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 20, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7854347825050354, "perplexity": 2519.607497918372}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875144111.17/warc/CC-MAIN-20200219092153-20200219122153-00424.warc.gz"} |
https://crypto.stackexchange.com/questions/88181/who-provides-prime-numbers-for-cryptographic-protocols | # Who provides prime numbers for cryptographic protocols?
I am currently writing a thesis about different cryptographic protocols like DH-Key exchange, TLS or IKE. Most of them rely on a prime number earlier or later, so for security reasons I wondered if this prime number is supplied by the user and if yes that leads to the question if I can manually provide it (for example to use a pseudoprime to break the underlying primitive more easily).
I really tried to search online, but there are very little information about this part of protocols. I hope you can help me with that.
• Recent related question: How is a generator found for a group, both in case of DH & ECDH?. – fgrieu Feb 11 at 14:47
• Have a look at crypto.stackexchange.com/questions/45151/…. In this example, the two large primes are generated by openssl. But, if you wanted to provide your own large primes, you could do so, and write them to the private.key file the same way that openssl does, and the rest of the process would be the same. – mti2935 Feb 11 at 15:13
• Actually, if you are trying to specify DH parameters in such a way to make it attackable, the most promising way would not to be to specify $p$ composite (fairly easy to detect if someone checks); instead, it would be to specify $p$ prime, and $g$ with a smooth (or, at least, not too rough) order. Now, it wouldn't be true NOBUS (ECM is just too good at finding moderate factors of $p-1$), however it would not be easy to detect automatically – poncho Feb 11 at 18:46
Let's have a look at how these are implemented:
### RSA
RSA is probably the example, showing how prime numbers are used in cryptography. In order to calculate a private key, two prime numbers, $$p$$ and $$q$$ are "chosen". What that means in practice is that your computer will generate a random number (and set a few bits) and then check whether or not the generated random number is a prime.
Well...not exactly. Actually, the generated number is a "pseudo prime number", which is a number that is probably a prime number, but we can't know for sure. In order to be certain, we would have to divide the number by every prime number up to the square root of the prime candidate.
Why isn't this check done? As you can imagine, checking whether or not 13377216267221394781281321803240685565647633863945354245591337480581850892718220086347852930388186839929611014777408043323753186115505611255512212061593047 is a prime factor of 168581487233807662095985877892466382655984504887588184273878566402288131033127314863556248166230176818735604347430473403376658381900826700241919315590165489510028586594896744589808921429729412701596114362546978418049726172509111051384159129475101054095614648000575426731420406552789078218487162756487172295557 is a computationally expensive task. Especially if our current tests run magnitudes faster and can give us a $$4^{-k}$$ chance of a non-prime passing the test, with $$k$$ being the number of runs of the Miller-Rabin test.
You can even try this yourself by running openssl genrsa. You will see output like this:
Generating RSA private key, 2048 bit long modulus (2 primes)
................+++++
...............+++++
e is 65537 (0x010001)
Each . means a potential prime candidate was generated, and each + means a round of Miller-Rabin was passed. According to this answer, even three rounds of Miller-Rabin will be enough to make it very unlikely that a randomly generated non-prime passes the Miller-Rabin test. So as you can see, OpenSSL stops after five passed rounds and moves on to generating a new number.
### Diffie-Hellman Key Exchange
In order for DH to work, there needs to be a public prime $$p$$, which can be a known default value. These can be generated on-the-spot by either the client or the server, but publicly known default values can be used. Since these are just transmitted over plain text, an attacker can intercept them.
It should be noted that you want to pick a $$p$$ for which $$p-1$$ has a large prime factor $$q$$.
The secret keys $$a$$ and $$b$$ generated in DH don't need to be primes, so any random number will suffice.
### Are they supplied by the user?
Yes and no. Many protocols require a user's private key, which they supply in form of a key file. Nothing stops a user from generating their own key file with any prime number of their choosing.
However, most users won't do that, as there is really no benefit to that. Even an attacker has no benefit of supplying something like a semi-prime (a number that is the result of two prime numbers, e.g. 15), since that would not give them any advantage.
Definitely, no serious program will ever ask the user to enter a prime number, which is then used for cryptographic purposes, aside from demonstration tools.
• Thanks for this answer, especially the part with DH helps me a lot. I have to correct you in one point tho: Using a composite number for DH helps the attacker very much, as the cyclic group gets significantly smaller and thus the discrete logarithm is easier to calculate. This is the main topic of my thesis and that's why this question came up. – Greybound Feb 11 at 14:06
• @Greybound: I trust you are aware that using a random prime in DH is generally not a good idea; you want to generate a prime $p$ where $p-1$ has a known large prime factor $q$. – poncho Feb 11 at 14:50
• @poncho That is true. Which is why the overwhelming majority of implementations us known values for $p$. – MechMK1 Feb 11 at 15:14
• @poncho I am aware of that. But that's exactly my point, I want to attack DH by providing a weak value for p (i.e. a composite that passes primality tests with maximum likelihood) – Greybound Feb 11 at 15:31
• @Greybound: you are thinking of "a composite that passes primality tests with maximum likelihood" when poncho suggested a prime $p$ with $p-1$ having no large prime factor. So "that's exactly my point" needs reconsideration. – fgrieu Feb 11 at 17:39
For most purposes, your computer generates a random prime. It does this by taking a random odd number and checking whether that's prime. Repeat until such a number has been found.
In some cases, it is possible to create cryptographic material with your own, chosen, prime numbers. For example, RSA keys consist of two prime numbers multiplied together. If you choose one or both of these prime numbers in a particular way, you may create a deliberately insecure RSA key pair.
• What about Diffie-Hellman? – DannyNiu Feb 11 at 19:35 | 2021-05-05 22:31:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5088346600532532, "perplexity": 601.1633022740463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988696.23/warc/CC-MAIN-20210505203909-20210505233909-00521.warc.gz"} |
https://zenodo.org/record/2550812/export/xd | Journal article Open Access
# A Study of Platelet Rich Plasma Commonly Used in Neurosurgery Practice
Necati Kaplan
### Dublin Core Export
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<oai_dc:dc xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:oai_dc="http://www.openarchives.org/OAI/2.0/oai_dc/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd">
<dc:creator>Necati Kaplan</dc:creator>
<dc:date>2019-01-27</dc:date>
<dc:description>AIM: No consensus is present regarding application modalities of platelet-rich plasma (PRP), commonly used in neurosurgery clinics. In this systematic review, the aim was to evaluate the potential role of PRP in neurosurgery practice after examining studies previously performed on this matter. In doing so, the preparation and activation of PRP, the appropriate dosage and concentrations used in the treatment were investigated.
MATERIALS and METHODS: A comprehensive and systematic literature search of numerous electronic databases was performed. Keywords used were related to PRP and neurological diagnosis. Studies that satisfied the inclusion criteria were retrieved, then the results were reported using the descriptive statistical methodology.
RESULTS: Ofthe 863 articles examined in detail, 27 were related to disc tissue pathologies, and four were related to complications occurred after spinal cord injury. After full-text review of these articles, nine were systematically evaluated.
CONCLUSION: It is important to prevent the preparation and application errors of PRP which is conventionally applied in clinics after being prepared using various kits and protocols. To achieve this aim, it is necessary to formulate clear protocols on how to prepare PRP for the treatment of which diseases, and which doses and durations should be used in the treatment. These protocols should take its place in the neurosurgical treatment guidelines. However, the differences that may be resulted in the application of different concentrations of PRP, which contains growth factors, should be evaluated promptly at the molecular level.</dc:description>
<dc:identifier>https://zenodo.org/record/2550812</dc:identifier>
<dc:identifier>10.5281/zenodo.2550812</dc:identifier>
<dc:identifier>oai:zenodo.org:2550812</dc:identifier>
<dc:relation>doi:10.5281/zenodo.2550811</dc:relation>
<dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
<dc:rights>http://creativecommons.org/licenses/by/4.0/legalcode</dc:rights>
<dc:subject>Drug delivery systems, Spinal surgery, traditional injection, PRP with a relatively low concentration of very few leukocytes, PRP with high concentrations of leukocytes</dc:subject>
<dc:title>A Study of Platelet Rich Plasma Commonly Used in Neurosurgery Practice</dc:title>
<dc:type>info:eu-repo/semantics/article</dc:type>
<dc:type>publication-article</dc:type>
</oai_dc:dc>
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https://physics.com.hk/2008/08/21/49-burkhard-heim/ | 4.9 Burkhard Heim
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“The accident left him without hands and mostly deaf and blind when he was 19.”
“Heim had to undergo a series of operations after the explosion which resulted in the loss of his arms. He found that intense concentration on the study of Einstein’s relativity theory helped him control the pain in his arms mentally and physically.”
– Wikipedia
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Me: Physics 能醫百病
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2008.08.21 Thursday $copyright CHK^2$ | 2018-05-23 12:59:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5486620664596558, "perplexity": 3148.3057906888603}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794865651.2/warc/CC-MAIN-20180523121803-20180523141803-00214.warc.gz"} |
https://yakovenko.wordpress.com/tag/matrix-functions/ | Sergei Yakovenko's blog: on Math and Teaching
Monday, November 3, 2014
Lecture 1 (Nov. 3, 2014)
Filed under: Analytic ODE course — Sergei Yakovenko @ 6:34
Tags: , ,
The first lecture was introductory, containing the motivation for the forthcoming subjects.
The world of (real or complex) algebraic sets is tame: any question on the topological complexity admits an algorithmic solution and explicitly bounded answer.In particular, any algebraic set which consists of finitely many isolated points, admits an explicit bound for the number of these points by the product of degrees of equations defining this set (Bézout theorem). All the way around, equations involving nonalgebraic solutions to even simplest algebraic differential equations (sine/cosine), may define infinite sets (integer numbers). We will try to find out how the algebraic universe can be enlarged to include transcendental objects which still admit explicit bounds on their complexity.
It turns out that periods, integrals of rational forms over algebraic cycles, do possess such constructive finiteness, although this is far from easy to see. This finiteness is characteristic for solutions of rational Pfaffian systems with moderate singularities and special monodromy group.
Part 1: General linear systems.
A linear system locally lives on a cylinder, the product of a (complex) linear space $\mathbb C^n$ and an open base $U\subset \mathbb C^k$. If $\Omega=\bigl(\omega_{ij}\bigr)$ is an $n\times n$-matrix of holomorphic 1-forms on the base $U$, then a linear system defined by this matrix 1-form, is a matrix differential equation $\mathrm dX=\Omega\cdot X$, whose solution is a holomorphically invertible matrix function $X=X(t)$, $t\in U$. If the base is one-dimensional, then $\Omega=A(t)\,\mathrm dt$ with a holomorphic matrix function $A(t)$, and the linear system takes the familiar shape $\dot X(t)=A(t)X(t)$ [IY, sect. 15]
A necessary and sufficient condition for a local existence of solution is vanishing of the curvature, which amounts to the matrix identity $\mathrm d\Omega=\Omega\land\Omega$ (the right hand side is the matrix 2-form with the entries $\sum_{\ell=1}^k \omega_{i\ell}\land\omega_{\ell j}$, $i,j=1,\dots,k$). See [NY, sect. 1].
Solution of a linear system is defined modulo a right multplicative constant matrix factor: $\mathrm d(XC)=\Omega XC$ for any $C\in\mathrm{GL}(n,\mathbb C)$, and any other solution has such form. Using this observation, any piecewise curve $latex\gamma$ on the base can be covered by small neighborhoods $U_\alpha$ with local solutions $X_\alpha$ in these neighborhoods, which agree on the pairwise intersections $U_\alpha\cap U_\beta$. If this was not the case for the initial choice of local solutions, this can be always achieved by suitably twisting them (replacing by $X_\alpha C_\alpha$ so that $X_\alpha C_\alpha=X_\beta C_\beta$ on the intersections). This explains how solutions can be continued analytically along any simple curve, yet after continuation along a closed path $\gamma$ the solution may acquire a non-trivial monodromy factor.
Piecemeal remarks on rational matrix functions of a complex variable
The global theory of rational linear systems on $\mathbb C P^1$ requires the study of (rational) gauge transformations which are holomorphic and holomorphically invertible except for a single point. If this point is at infinity, then the matrix of such transformation is necessarily polynomial with constant nonzero determinant. Such matrix functions are provisionally referred to as monopoles, $H(t)\in\textrm{GL}(n,\mathbb C[t]),\ \text{det}H=\text{const}\ne 0$.
Multiplication of a rational matrix function $H(t)$ from the left by a monopole matrix $\begin{pmatrix}1 & t\\ & 1\end{pmatrix}$ corresponds to adding the second row of $H$, multiplied by $t$, to the first row. Thus manipulations with rows of $H$, which aim at Gauss-type elimination of certain monomials from matrix elements, can be represented as gauge actions of the monopole group. The principal result that will be used throughout the next few lectures, is the following Bolibruch Permutation Lemma.
Lemma. Let $H(t)$ be the germ of a matrix function, holomorphic and invertible at $t=\infty$. Then for any ordered tuple of integer numbers $D=\{d_1,\dots,d_n\}$ the product $t^D\,H(t)$, $t^D=\text{diag}(t^{d_1},\dots,t^{d_n})$, is monopole equivalent to a product of the form $H'(t)\,t^{D'}$, where $H'(t)$ is also holomorphic and invertible at $t=\infty$, and $D'$ is a permutation of the tuple $D$.
The proof of this result is not difficult, yet is too technical to be delivered in the classroom.
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http://jees.kr/journal/view.php?number=3140 | J Electromagn Eng Sci Search
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J Electromagn Eng Sci > Volume 14(2); 2014 > Article
Journal of Electromagnetic Engineering and Science 2014;14(2):54-60. DOI: https://doi.org/10.5515/JKIEES.2014.14.2.54
A New Design Approach for Asymmetric Coupled-Section Marchand Balun Ji An Park, Choon Sik Cho, Jae Wook Lee School of Electronics, Telecommunication and Computer Engineering, Korea Aerospace University Abstract A systematic design for asymmetric coupled-section Marchand baluns is presented. Asymmetrically coupled transmission lines in multilayer configuration are exploited for constructing Marchand baluns. Design equations for characteristic impedance and electrical length of asymmetrical coupled transmission lines are derived for establishing a systematic design procedure. Novel Marchand balun based on these design equations is composed of two identical asymmetrical coupled transmission lines. However, contrary to the general conventional design approach where ranges for characteristic impedances of coupled lines are ambiguously capitalized, values for characteristic impedance and length are explicitly expressed. Our approach is fundamentally different from the design method using coupling coefficients where solution for coupling coefficient is inherently restricted. To verify the proposed method, one design example is performed for wideband Marchand balun in multilayer configuration, and is fabricated for verifying the design procedure proposed. Maintaining the return loss more than 10 dB, the bandwidth is measured from 0.43 to 1.0 GHz, where $S_{21}$ and $S_{31}$ show better than -4 dB. The measured phase and amplitude imbalances illustrate 0.5 dB and ${pm}5^{circ}$, respectively. Key words: Asymmetric Balun, Asymmetric Coupled Transmission Lines, Marchand Balun, Wideband
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http://blog.zorilestore.ro/ob3c66/vector-integral-calculator-7647a0 | # vector integral calculator
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The calculator will calculate the multiple integral (double, triple). if we push a box with F=3N for 5m, we have done work: W=15Nm This is easy to understand for a constant force directly along the path of a straight line. Those involving line, surface and volume integrals are introduced here. Show Instructions In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Fundamental theorem of line integrals. Constructing a unit normal vector to curve. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Compute expert-level answers using Wolfram's breakthrough, algorithms, knowledgebase and AI technology, Partial Fraction Decomposition Calculator. Mensagem recebida. Advanced Math Solutions – Vector Calculator, Advanced Vectors. Wolfram|Alpha can solve a broad range of integrals. It is very easy to solve integrals using calculator.here i use fx-991Ms calculator.It can solve proper integral only. Integral calculator. Specify the curve and range of the path, and then calculate the line integral of the vector field. Description : Integral calculator. Both types of integrals are tied together by the fundamental theorem of calculus. In the last blog, we covered some of the simpler vector topics. Integral Calculus. More than just an online function properties finder. Wolfram|Alpha computes integrals differently than people. Integral Calculator. Line Integral of a Vector Field A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. Next lesson. This states that if, integrate x^2 sin y dx dy, x=0 to 1, y=0 to pi. The function to be integrated may be a scalar field or a vector field. Preview; Assign Practice; Preview. Once you've done that, refresh this page to start using Wolfram|Alpha. They are the multivariable calculus equivalent of the fundamental theorem of calculus for single All rights belong to the owner! Vectors Calculator. Even for quite simple integrands, the equations generated in this way can be highly complex and require Mathematica's strong algebraic computation capabilities to solve. Free Online Integral Calculator allows you to solve definite and indefinite integration problems. Define the vector field. This is sometimes called the flux of $$\vec F$$ across $$S$$.. Before we work any examples let’s notice that we can substitute in for the unit normal vector to get a somewhat easier formula to use. Indefinite and definite integrals, answers, alternate forms. √ Preview: Input function: ? In this example, we define it in a single line, but it can also be defined by writing a separate file and uploading it. Integral Vector Theorems 29.3 Introduction Various theorems exist relating integrals involving vectors. Besides math integral, covariance is defined in the same way. Both types of integrals are tied together by the fundamental theorem of calculus. Integrate does not do integrals the way people do. Integral definition assign numbers to define and describe area, volume, displacement & other concepts. Interactive graphs/plots help visualize and better understand the … Line Integral of a Vector Field A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. Despite the prevalence of the top answer, it has some big errors. Multiple (Double, Triple) Integral Calculator. Multiple (Double, Triple) Integral Calculator. Hints help you try the next step on your own. There... \begin{pmatrix}1&0&3\end{pmatrix}+\begin{pmatrix}-1&4&2\end{pmatrix}, (-3)\cdot \begin{pmatrix}1&5&0\end{pmatrix}, \begin{pmatrix}1&2&3\end{pmatrix}\times\begin{pmatrix}1&5&7\end{pmatrix}, ângulo\:\begin{pmatrix}2&-4&-1\end{pmatrix},\:\begin{pmatrix}0&5&2\end{pmatrix}, unitário\:\begin{pmatrix}2&-4&1\end{pmatrix}, projeção\:\begin{pmatrix}1&2\end{pmatrix},\:\begin{pmatrix}3&-8\end{pmatrix}, projeção\:escalar\:\begin{pmatrix}1&2\end{pmatrix},\:\begin{pmatrix}3&-8\end{pmatrix}. Such a surface integral is equal to the volume integral of the divergence of the vector, according to Gauss’ theorem (Eq. In this example, we define it in a single line, but it can also be defined by writing a separate file and uploading it. Indefinite and definite integrals, answers, alternate forms. … Calculus Math Integral Definite Indefinite Upper/Lower Sum. Answers, graphs, alternate forms. Disc Action!!! Parameterization. Powered by Wolfram|Alpha. This online calculator will find the indefinite integral (antiderivative) of the given function, with steps shown (if possible). Another approach that Mathematica uses in working out integrals is to convert them to generalized hypergeometric functions, then use collections of relations about these highly general mathematical functions. Type in any integral to get the solution, steps and graph Now when you first learn work, you … q = integral(fun,xmin,xmax,Name,Value) specifies additional options with one or more Name,Value pair arguments.For example, specify 'WayPoints' followed by a vector of real or complex numbers to indicate specific points for the integrator to use. Definite integral. One of the most fundamental ideas in all of physics is the idea of work. The calculator will evaluate the definite (i.e. Please enable JavaScript. There are a couple of approaches that it most commonly takes. This website uses cookies to ensure you get the best experience. Double integrals. The online service at OnSolver.com allows you to find a definite integral solution online. Integral definition. Tim Brzezinski. Learn Graphing Calculator. Book. It calls Mathematica's Integrate function, which represents a huge amount of mathematical and computational research. It also shows plots of the function and illustrates the domain and range on a number line to enhance your mathematical intuition. While these powerful algorithms give Wolfram|Alpha the ability to compute integrals very quickly and handle a wide array of special functions, understanding how a human would integrate is important too. For example,, since the derivative of is . Book. This means . Indefinite Integral of a Vector-Valued Function. This is sometimes called the flux of $$\vec F$$ across $$S$$.. Before we work any examples let’s notice that we can substitute in for the unit normal vector to get a somewhat easier formula to use. In ordinary differential and integral calculus, you have already seen how derivatives and integrals interrelate. Summary : The vector calculator allows to do calculations with vectors using coordinates. This calculator computes the definite and indefinite integrals (antiderivative) of a function with respect to a variable x. ) Get immediate feedback and guidance with step-by-step solutions and Wolfram Problem Generator. These use completely different integration techniques that mimic the way humans would approach an integral. Fundamental theorem of line integrals. ... Derivatives Derivative Applications Limits Integrals Integral Applications Riemann Sum Series ODE Multivariable Calculus … Activity. Show Instructions. Calculator for determining whether a function is … Next lesson. Line integrals in vector fields (articles) Video transcript. Wolfram|Alpha is a great tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals. Por favor tente novamente usando um diferente meio de pagamento. In the case of a closed curve it is also called a contour integral. Summary : The vector calculator allows to do calculations with vectors using coordinates. The line integral of the tangential component of an arbitrary vector around a closed loop is equal to the surface integral of the normal component of the curl of that vector over any surface which is bounded by the loop: \label{Eq:II:3:44} \underset{\text{boundary}}{\int} \FLPC\cdot d\FLPs= \underset{\text{surface}}{\int} (\FLPcurl{\FLPC})\cdot\FLPn\,da. This free online 3D grapher from geogebra: graph 3D functions, plot surfaces, construct solids much. ( Eq and range of a function is … multiple ( double triple! Approximation to a variable x. fields ( articles ) Video transcript from Wolfram Alpha.. Language used in applied mathematics for solving problems in two and three dimensions to ask an... And education levels online the integral '' from the cartesian coordinates calculus.! This calculator computes the definite and indefinite integrals with Wolfram|Alpha and Wolfram Problem Generator » Unlimited random problems. Solve integrals using calculator.here i use fx-991Ms calculator.It can solve proper integral only, because already. Tool supports all the most recent functions, computing and several other variables which are essential in 1 tool between. Solution, steps and graph free double integrals step-by-step with Wolfram|Alpha of new vector Resources added. ^-+ * / ^ theorem of calculus meio de pagamento you to find a definite integral solution.. Integral '' from the surface integral is equal to the broadest vector integral calculator range of a vector point function displacement other! 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And other relevant Information to enhance your mathematical intuition easy to solve integrals using calculator.here i use calculator.It. Solve integrals using calculator.here i use fx-991Ms calculator.It can solve proper integral only or a! To ensure you get the best vector integral calculator is n't obvious from taking a look at the vector-valued function analogue integration! Of two vectors be looking at: line integrals over vector fields ( articles ) line integrals in vector (... To Gauss ’ theorem, we ’ ll finish the integral '' from the surface integral are introduced.! Are introduced here integrate does not do integrals the way people do general form for an integral vector... Between and the axis, from to calculator using Scilab of area, volume, displacement & concepts! Fourier Series how to ask for an integral professions and education levels online 3D from! In a vector field solve double integrals calculator - solve indefinite, definite and integrals! 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0 comentarii pentru: vector integral calculator | 2021-05-13 03:57:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.898592472076416, "perplexity": 1011.2065733931307}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243992721.31/warc/CC-MAIN-20210513014954-20210513044954-00606.warc.gz"} |
https://ita.skanev.com/08/04/02.html | # Exercise 8.4.2
Explain why the worst-case running time for bucket sort is $\Theta(n^2)$. What simple change to the algorithm preserves its linear average-case running time and makes its worst-case running time $\O(n\lg{n})$.
If all the keys fall in the same bucket and they happen to be in reverse order, we have to sort a single bucket with $n$ items in reversed order with insertion sort. This is $\Theta(n^2)$.
We can use merge sort or heapsort to improve the worst-case running time. Insertion sort was chosen, because it operates well on linked lists. If we use another sorting algorithm, we have to convert each list to an array, which might slow down the algorithm in practice. | 2018-01-18 21:46:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2718425691127777, "perplexity": 420.45404189917696}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084887621.26/warc/CC-MAIN-20180118210638-20180118230638-00309.warc.gz"} |
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Learning Objectives
1. Simplify radical expressions using the product and quotient rule for radicals.
An algebraic expression that contains radicals is called a radical expressionAn algebraic expression that contains radicals.. We use the product and quotient rules to simplify them.
Example 1
Simplify: $27x33.$
Solution:
Use the fact that $ann=a$ when n is odd.
$27x33=33⋅x33Apply the product rule for radicals.=333⋅x33Simplify.=3⋅x=3x$
Answer: $3x$
Example 2
Simplify: $16y44.$
Solution:
Use the fact that $ann=|a|$ when n is even.
$16y44=24y44Apply the product rule for radicals.=244⋅y44Simplify.=2⋅|y|=2|y|$
Since y is a variable, it may represent a negative number. Thus we need to ensure that the result is positive by including the absolute value.
Answer: $2|y|$
Important Note
Typically, at this point in algebra we note that all variables are assumed to be positive. If this is the case, then y in the previous example is positive and the absolute value operator is not needed. The example can be simplified as follows.
$16y44=24y44 =244⋅y44=2y$
In this section, we will assume that all variables are positive. This allows us to focus on calculating nth roots without the technicalities associated with the principal nth root problem. For this reason, we will use the following property for the rest of the section,
$ann=a, if a≥0 nth root$
When simplifying radical expressions, look for factors with powers that match the index.
Example 3
Simplify: $12x6y3.$
Solution:
Begin by determining the square factors of 12, $x6$, and $y3.$
$12=22⋅3x6=(x3)2 y3=y2⋅y } Square factors$
Make these substitutions, and then apply the product rule for radicals and simplify.
$12x6y3=22⋅3⋅(x3)2⋅y2⋅yApply the product rule for radicals.=22⋅(x3)2⋅y2⋅3ySimplify.=2 ⋅ x3 ⋅ y ⋅3y=2x3y3y$
Answer: $2x3y3y$
Example 4
Simplify: $18a5b8$.
Solution:
Begin by determining the square factors of 18, $a5$, and $b8.$
$18=2⋅32a5=a2⋅a2⋅a=(a2)2⋅a b8=b4⋅b4=(b4)2 } Square factors$
Make these substitutions, apply the product and quotient rules for radicals, and then simplify.
$18a5b8=2⋅32⋅(a2)2⋅a(b4)2Apply the product and quotient rule for radicals.=32⋅(a2)2⋅2a(b4)2Simplify.= 3a22ab4$
Answer: $3a22ab4$
Example 5
Simplify: $80x5y73.$
Solution:
Begin by determining the cubic factors of 80, $x5$, and $y7.$
$80=24⋅5=23⋅2⋅5x5=x3⋅x2y7=y6⋅y=(y2)3⋅y } Cubic factors$
Make these substitutions, and then apply the product rule for radicals and simplify.
$80x5y73=23⋅2⋅5⋅x3⋅x2⋅(y2)3⋅y3 =233⋅x33⋅(y2)33⋅2⋅5⋅x2⋅y3=2⋅xy2⋅10x2y3=2xy2 10x2y3$
Answer: $2xy210x2y3$
Example 6
Simplify $9x6y3z93$.
Solution:
The coefficient $9=32$, and thus does not have any perfect cube factors. It will be left as the only remaining radicand because all of the other factors are cubes, as illustrated below:
$x6=(x2)3y3=(y)3z9=(z3)3 } Cubic factors$
Replace the variables with these equivalents, apply the product and quotient rules for radicals, and then simplify.
$9x6y3z93=9⋅(x2)3y3⋅(z3)33=93⋅(x2)33y33⋅(z3)33=93⋅x2y⋅z3=x2 93yz3$
Answer: $x293yz3$
Example 7
Simplify: $81a4b54.$
Solution:
Determine all factors that can be written as perfect powers of 4. Here, it is important to see that $b5=b4⋅b.$ Hence the factor $b$ will be left inside the radical.
$81a4b54=34⋅a4⋅b4⋅b4=344⋅a44⋅b44⋅b4=3⋅a⋅b⋅b4=3abb4$
Answer: $3abb4$
Example 8
Simplify: $−32x3y6z55.$
Solution:
Notice that the variable factor x cannot be written as a power of 5 and thus will be left inside the radical. In addition, $y6=y5⋅y$; the factor y will be left inside the radical as well.
$−32x3y6z55=(−2)5⋅x3⋅y5⋅y⋅z55=(−2)55⋅y55⋅z55⋅x3⋅y5=−2⋅y⋅z⋅x3⋅y5=−2yzx3y5$
Answer: $−2yzx3y5$
Tip: To simplify finding an nth root, divide the powers by the index.
$a6=a3, which is a6÷2=a3b63=b2, which is b6÷3=b2c66=c , which is c6÷6=c1$
If the index does not divide into the power evenly, then we can use the quotient and remainder to simplify. For example,
$a5=a2⋅a, which is a5÷2=a2 r 1b53=b⋅b23, which is b5÷3=b1 r 2c145=c2⋅c45, which is c14÷5=c2 r 4$
The quotient is the exponent of the factor outside of the radical, and the remainder is the exponent of the factor left inside the radical.
Try this! Simplify: $162a7b5c43.$
Answer: $3a2bc6ab2c3$
Formulas often consist of radical expressions. For example, the period of a pendulum, or the time it takes a pendulum to swing from one side to the other and back, depends on its length according to the following formula.
$T=2πL32$
Here T represents the period in seconds and L represents the length in feet of the pendulum.
Example 9
If the length of a pendulum measures $112$ feet, then calculate the period rounded to the nearest tenth of a second.
Solution:
Substitute $112=32$ for L and then simplify.
$T=2πL32=2π3232=2π32⋅132Apply the quotient rule for radicals.=2π364Simplify.=2π38 =π34≈1.36$
Answer: The period is approximately 1.36 seconds.
Frequently you need to calculate the distance between two points in a plane. To do this, form a right triangle using the two points as vertices of the triangle and then apply the Pythagorean theorem. Recall that the Pythagorean theorem states that if given any right triangle with legs measuring a and b units, then the square of the measure of the hypotenuse c is equal to the sum of the squares of the legs: $a2+b2=c2.$ In other words, the hypotenuse of any right triangle is equal to the square root of the sum of the squares of its legs.
Example 10
Find the distance between (−5, 3) and (1, 1).
Solution:
Form a right triangle by drawing horizontal and vertical lines though the two points. This creates a right triangle as shown below:
The length of leg b is calculated by finding the distance between the x-values of the given points, and the length of leg a is calculated by finding the distance between the given y-values.
$a=3−1=2 unitsb=1−(−5)=1+5=6 units$
Next, use the Pythagorean theorem to find the length of the hypotenuse.
$c=22+62=4+36=40=4⋅10=210 units$
Answer: The distance between the two points is $210$ units.
Generalize this process to produce a formula that can be used to algebraically calculate the distance between any two given points.
Given two points, $( x1, y1)$ and $( x2, y2)$, the distance, d, between them is given by the distance formulaGiven two points $(x1, y1)$ and $(x2, y2)$, calculate the distance d between them using the formula $d=(x2−x1)2+(y2−y1)2.$, $d=(x2−x1)2+(y2−y1)2.$
Example 11
Calculate the distance between (−4, 7) and (2, 1).
Solution:
Use the distance formula with the following points.
$( x1, y1) ( x2, y2)(−4,7)(2, 1)$
It is a good practice to include the formula in its general form before substituting values for the variables; this improves readability and reduces the probability of making errors.
$d=(x2−x1)2+(y2−y1)2=(2−(−4))2+(1−7)2=(2+4)2+(1−7)2=(6)2+(−6)2=36+36=72=36⋅2=62$
Answer: The distance between the two points is $62$ units.
Example 12
Do the three points (2, −1), (3, 2), and (8, −3) form a right triangle?
Solution:
The Pythagorean theorem states that having side lengths that satisfy the property $a2+b2=c2$ is a necessary and sufficient condition of right triangles. In other words, if you can show that the sum of the squares of the leg lengths of the triangle is equal to the square of the length of the hypotenuse, then the triangle must be a right triangle. First, calculate the length of each side using the distance formula.
Geometry
Calculation
Points: (2, −1) and (8, −3)
$a=(8−2)2+[−3−(−1)]2=(6)2+(−3+1)2=36+(−2)2=36+4=40=210$
Points: (2, −1) and (3, 2)
$b=(3−2)2+[2−(−1)]2=(1)2+(2+1)2=1+(3)2=1+9=10$
Points: (3, 2) and (8, −3)
$c=(8−3)2+(−3−2)2=(5)2+(−5)2=25+25=50=52$
Now we check to see if $a2+b2=c2.$
$a2+b2=c2(210)2+(10)2=(52)24(10)2+(10)2=25(2)24⋅10+10=25⋅250=50 ✓$
Answer: Yes, the three points form a right triangle.
Try this! The speed of a vehicle before the brakes were applied can be estimated by the length of the skid marks left on the road. On wet concrete, the speed v in miles per hour can be estimated by the formula $v=23d$, where d represents the length of the skid marks in feet. Estimate the speed of a vehicle before applying the brakes if the skid marks left behind measure 27 feet. Round to the nearest mile per hour.
Key Takeaways
• To simplify a radical expression, look for factors of the radicand with powers that match the index. If found, they can be simplified by applying the product and quotient rules for radicals, as well as the property $ann=a$, where a is nonnegative.
• A radical expression is simplified if its radicand does not contain any factors that can be written as perfect powers of the index.
• We typically assume that all variable expressions within the radical are nonnegative. This allows us to focus on simplifying radicals without the technical issues associated with the principal nth root. If this assumption is not made, we will ensure a positive result by using absolute values when simplifying radicals with even indices.
Topic Exercises
Assume that the variable could represent any real number and then simplify.
1. $9x2$
2. $16y2$
3. $8y33$
4. $125a33$
5. $64x44$
6. $81y44$
7. $36a4$
8. $100a8$
9. $4a6$
10. $a10$
11. $18a4b5$
12. $48a5b3$
13. $128x6y86$
14. $a6b7c86$
15. $(5x−4)2$
16. $(3x−5)4$
17. $x2−6x+9$
18. $x2−10x+25$
19. $4x2+12x+9$
20. $9x2+6x+1$
Simplify. (Assume all variable expressions represent positive numbers.)
1. $49a2$
2. $64b2$
3. $x2y2$
4. $25x2y2z2$
5. $180x3$
6. $150y3$
7. $49a3b2$
8. $4a4b3c$
9. $45x5y3$
10. $50x6y4$
11. $64r2s6t5$
12. $144r8s6t2$
13. $(x+1)2$
14. $(2x+3)2$
15. $4(3x−1)2$
16. $9(2x+3)2$
17. $9x325y2$
18. $4x59y4$
19. $m736n4$
20. $147m9n6$
21. $2r2s525t4$
22. $36r5s2t6$
23. $27a33$
24. $125b33$
25. $250x4y33$
26. $162a3b53$
27. $64x3y6z93$
28. $216x12y33$
29. $8x3y43$
30. $27x5y33$
31. $a4b5c63$
32. $a7b5c33$
33. $8x427y33$
34. $x5125y63$
35. $360r5s12t133$
36. $540r3s2t93$
37. $81x44$
38. $x4y44$
39. $16x4y84$
40. $81x12y44$
41. $a4b5c64$
42. $54a6c84$
43. $128x64$
44. $243y74$
45. $32m10n55$
46. $37m9n105$
47. $−34x2$
48. $79y2$
49. $−5x4x2y$
50. $−3y16x3y2$
51. $12aba5b3$
52. $6a2b9a7b2$
53. $2x8x63$
54. $−5x227x33$
55. $2ab−8a4b53$
56. $5a2b−27a3b33$
Rewrite the following as a radical expression with coefficient 1.
1. $3x6x$
2. $5y5y$
3. $ab10a$
4. $2ab2a$
5. $m2nmn$
6. $2m2n33n$
7. $2x3x3$
8. $3yy23$
9. $2y24y4$
10. $x2y9xy25$
The period T in seconds of a pendulum is given by the formula $T=2πL32$ where L represents the length in feet of the pendulum. Calculate the period, given each of the following lengths. Give the exact value and the approximate value rounded to the nearest tenth of a second.
1. 8 feet
2. 32 feet
3. $12$ foot
4. $18$ foot
The time t in seconds an object is in free fall is given by the formula $t=s4$ where s represents the distance in feet the object has fallen. Calculate the time it takes an object to fall, given each of the following distances. Give the exact value and the approximate value rounded to the nearest tenth of a second.
1. 48 feet
2. 80 feet
3. 192 feet
4. 288 feet
5. The speed of a vehicle before the brakes were applied can be estimated by the length of the skid marks left on the road. On dry pavement, the speed v in miles per hour can be estimated by the formula $v=26d$, where d represents the length of the skid marks in feet. Estimate the speed of a vehicle before applying the brakes on dry pavement if the skid marks left behind measure 27 feet. Round to the nearest mile per hour.
6. The radius r of a sphere can be calculated using the formula $r=6π2V32π$, where V represents the sphere’s volume. What is the radius of a sphere if the volume is $36π$ cubic centimeters?
Given the function find the y-intercept
1. $f(x)=x+12$
2. $f(x)=x+8−3$
3. $f(x)=x−83$
4. $f(x)=x+273$
5. $f(x)=x+163$
6. $f(x)=x+33−1$
Use the distance formula to calculate the distance between the given two points.
1. (5, −7) and (3, −8)
2. (−9, 7) and (−8, 4)
3. (−3, −4) and (3, −6)
4. (−5, −2) and (1, −6)
5. (−1, 1) and (−4, 10)
6. (8, −3) and (2, −12)
7. (0, −6) and (−3, 0)
8. (0, 0) and (8, −4)
9. $(12,−12)$ and $(−1,32)$
10. $(−13,2)$ and $(53,−23)$
Determine whether or not the three points form a right triangle. Use the Pythagorean theorem to justify your answer.
1. (2,−1), (−1,2), and (6,3)
2. (−5,2), (−1, −2), and (−2,5)
3. (−5,0), (0,3), and (6,−1)
4. (−4,−1), (−2,5), and (7,2)
5. (1,−2), (2,3), and (−3,4)
6. (−2,1), (−1,−1), and (1,3)
7. (−4,0), (−2,−10), and (3,−9)
8. (0,0), (2,4), and (−2,6)
Part D: Discussion Board
1. Give a value for x such that $x2≠x.$ Explain why it is important to assume that the variables represent nonnegative numbers.
2. Research and discuss the accomplishments of Christoph Rudolff. What is he credited for?
3. What is a surd, and where does the word come from?
4. Research ways in which police investigators can determine the speed of a vehicle after an accident has occurred. Share your findings on the discussion board.
1. $3|x|$
2. $2y$
3. $2|x|$
4. $6a2$
5. $2|a3|$
6. $3a2b22b$
7. $2|xy|2y26$
8. $|5x−4|$
9. $|x−3|$
10. $|2x+3|$
11. $7a$
12. $xy$
13. $6x5x$
14. $7aba$
15. $3x2y5xy$
16. $8rs3t2t$
17. $x+1$
18. $2(3x−1)$
19. $3xx5y$
20. $m3m6n2$
21. $rs22s5t2$
22. $3a$
23. $5xy2x3$
24. $4xy2z3$
25. $2xyy3$
26. $abc2ab23$
27. $2xx33y$
28. $2rs4t445r2t3$
29. $3x$
30. $2xy2$
31. $abcbc24$
32. $2x8x24$
33. $2m2n$
34. $−6x$
35. $−10x2y$
36. $12a3b2ab$
37. $4x3$
38. $−4a2b2ab23$
39. $54x3$
40. $10a3b2$
41. $m5n3$
42. $24x43$
43. $64y94$
1. $π$ seconds; 3.1 seconds
2. $π4$ seconds; 0.8 seconds
3. $3$ seconds; 1.7 seconds
4. $23$ seconds; 3.5 seconds
5. 25 miles per hour
6. $(0,23)$
7. $(0,−2)$
8. $(0,223)$
9. $5$ units
10. $210$ units
11. $310$ units
12. $35$ units
13. $52$ units
14. Right triangle
15. Not a right triangle
16. Right triangle
17. Right triangle | 2018-12-19 09:38:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 230, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7794716954231262, "perplexity": 399.10257756325836}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376831933.96/warc/CC-MAIN-20181219090209-20181219112209-00089.warc.gz"} |
http://www.researchgate.net/publication/45929177_Better_Non-Local_Games_from_Hidden_Matching | Article
Better Non-Local Games from Hidden Matching
• Giannicola Scarpa
07/2010;
Source: arXiv
ABSTRACT We construct a non-locality game that can be won with certainty by a quantum strategy using log n shared EPR-pairs, while any classical strategy has winning probability at most 1/2+O(log n/sqrt{n}). This improves upon a recent result of Junge et al. in a number of ways. Comment: 11 pages, latex
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Article: Large violation of Bell inequalities with low entanglement
[Hide abstract]
ABSTRACT: In this paper we obtain violations of general bipartite Bell inequalities of order $\frac{\sqrt{n}}{\log n}$ with $n$ inputs, $n$ outputs and $n$-dimensional Hilbert spaces. Moreover, we construct explicitly, up to a random choice of signs, all the elements involved in such violations: the coefficients of the Bell inequalities, POVMs measurements and quantum states. Analyzing this construction we find that, even though entanglement is necessary to obtain violation of Bell inequalities, the Entropy of entanglement of the underlying state is essentially irrelevant in obtaining large violation. We also indicate why the maximally entangled state is a rather poor candidate in producing large violations with arbitrary coefficients. However, we also show that for Bell inequalities with positive coefficients (in particular, games) the maximally entangled state achieves the largest violation up to a logarithmic factor.
Communications in Mathematical Physics 07/2010; 306. · 1.97 Impact Factor | 2014-12-25 15:03:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9216969013214111, "perplexity": 1051.910356248146}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1419447547719.53/warc/CC-MAIN-20141224185907-00022-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://kerodon.net/tag/01TQ | # Kerodon
$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
Corollary 5.1.2.4. Let $Q$ be a partially ordered set, let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(Q)$ be an inner fibration of $\infty$-categories, and let $g: Y \rightarrow Z$ be a morphism in $\operatorname{\mathcal{C}}$. Then $g$ is $q$-cartesian if and only if, for every object $X \in \operatorname{\mathcal{C}}$ satisfying $q(X) \leq q(Y)$, the map
$\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow { [g] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$
of Notation 5.1.2.3 is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.
Proof. By virtue of Proposition 5.1.2.1, the morphism $g$ is $q$-cartesian if and only if, for each object $X \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes
5.4
$$\begin{gathered}\label{diagram:trivial-cartesian-poset} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} \ar [r]^-{\theta _ X} \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }(Q)}( q(X), q(Y), q(Z) ) \times _{ \operatorname{Hom}_{\operatorname{N}_{\bullet }(Q)}( q(Y), q(Z)) } \{ q(g) \} \ar [r] & \operatorname{Hom}_{\operatorname{N}_{\bullet }(Q)}( q(X), q(Z) ) } \end{gathered}$$
is a homotopy pullback square. If $q(X) \nleq q(Y)$, then the Kan complexes on the left side of the diagram (5.4) are empty, so this condition is vacuous. If $q(X) \leq q(Y)$, then the Kan complexes on the lower half of the diagram are isomorphic to $\Delta ^{0}$, so that (5.4) is a homotopy pullback square if and only if $\theta _{X}$ is a homotopy equivalence (Corollary 3.4.1.3). We conclude by observing that, in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, we have a commutative diagram
$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} \ar [rr] \ar [dr]_{\theta _{X}} & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [dl]^{ [g] \circ } \\ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z), & }$
where the horizontal map is an isomorphism (Corollary 4.6.3.5). $\square$ | 2021-05-18 02:33:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9884582161903381, "perplexity": 86.16960506363189}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991650.73/warc/CC-MAIN-20210518002309-20210518032309-00119.warc.gz"} |
http://en.wikisource.org/wiki/Page:Elements_of_the_Differential_and_Integral_Calculus_-_Granville_-_Revised.djvu/168 | # Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/168
6. At what point on the parabola of the last example do the abscissa and ordinate increase at the same rate? Ans. (3,6).
7. In the function $y = 2 x^3 + 6$, what is the value of x at the point where y increases 24 times as fast as x? Ans. $x = \pm 2$.
8. The ordinate of a point describing the curve $x^2 + y^2 = 25$ is decreasing at the rate of 1½ in. per second. How rapidly is the abscissa changing when the ordinate is 4 inches? Ans. $\tfrac{dx}{dt}$ = 2 in. per sec.
9. Find the values of x at the points where the rate of change of
$x^3 - 12 x^2 + 45 x - 13$
is zero. Ans. x = 3 and 5.
10. At what point on the ellipse $16 x^2 + 9 y^2 = 400$ does y decrease at the same rate that x increases? Ans. (3, $\tfrac{16}{3}$).
11. Where in the first quadrant does the arc increase twice as fast as the ordinate? Ans. At 60°.
A point generates each of the following curves. Find the rate at which the arc is increasing in each case:
12. $y^2 = 2x; \frac{dx}{dt} = 2, x = 2$. Ans. $\frac{ds}{dt} = \sqrt{5}$. 13. $xy = 6; \frac{dy}{dt} = 2, y = 3$. $\frac{ds}{dt} = \frac{2}{3} \sqrt{13}$. 14. $x^2 + 4 y^2 = 20; \frac{dx}{dt} = - 1, y = 1$. $\frac{ds}{dt} = \sqrt{2}$. 15. $y = x^3; \frac{dx}{dt} = 3, x = - 3$. 16. $y^2 = x^3; \frac{dy}{dt} = 4, y = 8$.
17. The side of an equilateral triangle is 24 inches long, and is increasing at the rate of 3 inches per hour. How fast is the area increasing? Ans. $36\sqrt{3}$ sq. in. per hour.
18. Find the rate of change of the area of a square when the side b is increasing at the rate of a units per second. Ans. 2 ab sq. units per sec.
19. (a) The,volume of a spherical soap bubble increases how many times as fast as the radius? (b) When its radius is 4 in. and increasing at the rate of ½ in. per second, how fast is the volume increasing? Ans. (a) 4πr2 times as fast; (b) 32π cu. in. per sec.
How fast is the surface increasing in the last case?
20. One end of a ladder 50 ft. long is leaning against a perpendicular wall standing on a horizontal plane. Supposing the foot of the ladder to be pulled away from the wall at the rate of 3 ft. per minute; (a) how fast is the top of the ladder descending when the foot is 14 ft. from the wall? (b) when will the top and bottom of the ladder move at the same rate? (c) when is the top of the ladder descending at the rate of 4 ft. per minute? Ans. (a) $\tfrac{7}{78}$ ft. per min.; (b) when $25 \sqrt{2}$ ft. from wall; (c) when 40 ft. from wall.
21. A barge whose deck is 12 ft. below the level of a dock is drawn up to it by means of a cable attached to a ring in the floor of the dock, the cable being hauled in by a windlass on deck at the rate of 8 ft. per minute. How fast is the barge moving towards the dock when 16 ft. away? Ans. 10 ft. per minute. | 2014-09-16 15:35:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 18, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6564726829528809, "perplexity": 547.4348795703482}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657118605.64/warc/CC-MAIN-20140914011158-00302-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
http://physics.stackexchange.com/questions/32644/working-with-deltas-to-use-principle-of-virtual-work | # Working with $\delta$s to use principle of virtual work
I'm trying to do the following problem:
A lever $ABC$ (see figure) has weights $W_1$ and $W_2$ at distances $a_1$ and $a_2$ from the fixed support $B$. Using the principle of virtual work, prove that a necessary and sufficient condition for equilibrium is $$W_1a_1 = W_2a_2$$ My attempt:
Now, the principle of virtual work states that $$\sum_{i=1}^{N}{\bf F}^{(a)}_i\cdot \delta{\bf r}_i = 0$$ I know the forces are equal to the given weights. However, I have a slight problem with the $\delta r_i$. I know it is a variation of the position vector $r$ of the $i$th particle. So, taking the origin as B, we have $\delta r_1=-\delta a_1$ and $\delta r_2=\delta a_2$, and my equation becomes: $${W_1}\cdot(-\delta a_1) + {W_2}\cdot \delta a_2= 0$$ Is my work up to this point correct? And is it correct to simply conclude from here that $W_1a_1=W_2a_2$? I would be more comfortable if I could get rid of the $\delta$s, but given that I'm not comfortable with working with them, I don't quite know how to do this.
-
The key is in the dot product in your equation $$\sum_{i=1}^{N}{\bf F}^{(a)}_i\cdot \delta{\bf r}_i = 0.$$ That means that only displacements in the direction of the forces - in this case, vertical - count towards the equilibrium condition. The lever arms come in since the vertical displacements must be related geometrically for the bar to remain straight. | 2014-07-30 19:24:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8838412165641785, "perplexity": 100.64249573041327}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510271648.4/warc/CC-MAIN-20140728011751-00001-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/858744/prove-operatornamefuncf-sum-i-1m-fracn-if2-left1-frac1f-r | # Prove $\operatorname{func}(f)=\sum_{i=1}^{m}\frac{n_i}{f^2}\left(1-\frac{1}{f}\right)^{n-n_i}$ has a maximal value.
I have two questions about the function:
$\operatorname{func}(f)=\sum_{i=1}^{m}\frac{n_i}{f^2}\left(1-\frac{1}{f}\right)^{n-n_i}$
where elements are subject to the conditions:
1. $\sum_{i=1}^m n_i=n$;
2. $n_i$ and $n$ $\in \mathbb{N}$;
3. $m$ and $f$ $\in \mathbb{Z}^+$.
I). Prove a unique maximum exists with respect to $f$ exists for fixed $n_i$, $m$.
II). Also, i would like to know if the function has a upper bound solution for $f$ when $\operatorname{func}(f)$ has a maxima.
My thought:
Find the derivative of $\operatorname{func}(f)$ of $f$, then let the result equals $0$. But it seems difficult to calculate. It is my first time use stackexchange, any help would be appreciated.
• Please share your thoughts so far :) – Shaun Jul 7 '14 at 8:36
• Thanks, I'd like to share my thoughts. I have already reedited my question as you said. – Finn Jul 7 '14 at 9:08
• To clarify: do you want to prove a maximum exists with respect to $f$ exists for fixed $n_i,m$, or do you want to prove that a maximum exists with respect to $f,n_i,m$? – Jakub Konieczny Jul 7 '14 at 9:28
• I wanna prove a maximum exists with respect to $f$ exists for fixed $n_i$, $m$. Also, i would like to know if the function has a approximate analysis solution for $f$ when $\operatorname{func}(f)$ has a maxima. – Finn Jul 7 '14 at 10:17
To show that $\text{func}(f)$ has a maxima, it is enough to show this series converges.
We can write this as $$\left(1-\dfrac{1}{f}\right)^{n+1}\sum_{i = 1}^{m} \dfrac{n_i}{f^2}\left(1-\dfrac{1}{f}\right)^{-n_i-1}$$
$$=-\left(1-\dfrac{1}{f}\right)^{n+1}\sum_{i = 1}^{m} \left(1-\dfrac{1}{f}\right)^{-n_i}$$ | 2019-05-23 05:13:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8321365118026733, "perplexity": 207.42920727564822}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257100.22/warc/CC-MAIN-20190523043611-20190523065611-00406.warc.gz"} |
https://people.maths.bris.ac.uk/~matyd/GroupNames/192i1/C2xC4xDic6.html | Copied to
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## G = C2×C4×Dic6order 192 = 26·3
### Direct product of C2×C4 and Dic6
Series: Derived Chief Lower central Upper central
Derived series C1 — C6 — C2×C4×Dic6
Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C22×Dic6 — C2×C4×Dic6
Lower central C3 — C6 — C2×C4×Dic6
Upper central C1 — C22×C4 — C2×C42
Generators and relations for C2×C4×Dic6
G = < a,b,c,d | a2=b4=c12=1, d2=c6, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 536 in 298 conjugacy classes, 183 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C42, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C22×Dic3, C22×C12, C2×C4×Q8, C4×Dic6, C2×C4×Dic3, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×C4×C12, C22×Dic6, C2×C4×Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, C24, Dic6, C4×S3, C22×S3, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, C2×Dic6, S3×C2×C4, C4○D12, S3×C23, C2×C4×Q8, C4×Dic6, C22×Dic6, S3×C22×C4, C2×C4○D12, C2×C4×Dic6
Smallest permutation representation of C2×C4×Dic6
Regular action on 192 points
Generators in S192
(1 170)(2 171)(3 172)(4 173)(5 174)(6 175)(7 176)(8 177)(9 178)(10 179)(11 180)(12 169)(13 185)(14 186)(15 187)(16 188)(17 189)(18 190)(19 191)(20 192)(21 181)(22 182)(23 183)(24 184)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 73)(34 74)(35 75)(36 76)(37 132)(38 121)(39 122)(40 123)(41 124)(42 125)(43 126)(44 127)(45 128)(46 129)(47 130)(48 131)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 85)(69 86)(70 87)(71 88)(72 89)(97 166)(98 167)(99 168)(100 157)(101 158)(102 159)(103 160)(104 161)(105 162)(106 163)(107 164)(108 165)(133 154)(134 155)(135 156)(136 145)(137 146)(138 147)(139 148)(140 149)(141 150)(142 151)(143 152)(144 153)
(1 23 138 46)(2 24 139 47)(3 13 140 48)(4 14 141 37)(5 15 142 38)(6 16 143 39)(7 17 144 40)(8 18 133 41)(9 19 134 42)(10 20 135 43)(11 21 136 44)(12 22 137 45)(25 64 58 101)(26 65 59 102)(27 66 60 103)(28 67 49 104)(29 68 50 105)(30 69 51 106)(31 70 52 107)(32 71 53 108)(33 72 54 97)(34 61 55 98)(35 62 56 99)(36 63 57 100)(73 89 120 166)(74 90 109 167)(75 91 110 168)(76 92 111 157)(77 93 112 158)(78 94 113 159)(79 95 114 160)(80 96 115 161)(81 85 116 162)(82 86 117 163)(83 87 118 164)(84 88 119 165)(121 174 187 151)(122 175 188 152)(123 176 189 153)(124 177 190 154)(125 178 191 155)(126 179 192 156)(127 180 181 145)(128 169 182 146)(129 170 183 147)(130 171 184 148)(131 172 185 149)(132 173 186 150)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 91 7 85)(2 90 8 96)(3 89 9 95)(4 88 10 94)(5 87 11 93)(6 86 12 92)(13 120 19 114)(14 119 20 113)(15 118 21 112)(16 117 22 111)(17 116 23 110)(18 115 24 109)(25 121 31 127)(26 132 32 126)(27 131 33 125)(28 130 34 124)(29 129 35 123)(30 128 36 122)(37 84 43 78)(38 83 44 77)(39 82 45 76)(40 81 46 75)(41 80 47 74)(42 79 48 73)(49 184 55 190)(50 183 56 189)(51 182 57 188)(52 181 58 187)(53 192 59 186)(54 191 60 185)(61 177 67 171)(62 176 68 170)(63 175 69 169)(64 174 70 180)(65 173 71 179)(66 172 72 178)(97 155 103 149)(98 154 104 148)(99 153 105 147)(100 152 106 146)(101 151 107 145)(102 150 108 156)(133 161 139 167)(134 160 140 166)(135 159 141 165)(136 158 142 164)(137 157 143 163)(138 168 144 162)
G:=sub<Sym(192)| (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,169)(13,185)(14,186)(15,187)(16,188)(17,189)(18,190)(19,191)(20,192)(21,181)(22,182)(23,183)(24,184)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,73)(34,74)(35,75)(36,76)(37,132)(38,121)(39,122)(40,123)(41,124)(42,125)(43,126)(44,127)(45,128)(46,129)(47,130)(48,131)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,85)(69,86)(70,87)(71,88)(72,89)(97,166)(98,167)(99,168)(100,157)(101,158)(102,159)(103,160)(104,161)(105,162)(106,163)(107,164)(108,165)(133,154)(134,155)(135,156)(136,145)(137,146)(138,147)(139,148)(140,149)(141,150)(142,151)(143,152)(144,153), (1,23,138,46)(2,24,139,47)(3,13,140,48)(4,14,141,37)(5,15,142,38)(6,16,143,39)(7,17,144,40)(8,18,133,41)(9,19,134,42)(10,20,135,43)(11,21,136,44)(12,22,137,45)(25,64,58,101)(26,65,59,102)(27,66,60,103)(28,67,49,104)(29,68,50,105)(30,69,51,106)(31,70,52,107)(32,71,53,108)(33,72,54,97)(34,61,55,98)(35,62,56,99)(36,63,57,100)(73,89,120,166)(74,90,109,167)(75,91,110,168)(76,92,111,157)(77,93,112,158)(78,94,113,159)(79,95,114,160)(80,96,115,161)(81,85,116,162)(82,86,117,163)(83,87,118,164)(84,88,119,165)(121,174,187,151)(122,175,188,152)(123,176,189,153)(124,177,190,154)(125,178,191,155)(126,179,192,156)(127,180,181,145)(128,169,182,146)(129,170,183,147)(130,171,184,148)(131,172,185,149)(132,173,186,150), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,91,7,85)(2,90,8,96)(3,89,9,95)(4,88,10,94)(5,87,11,93)(6,86,12,92)(13,120,19,114)(14,119,20,113)(15,118,21,112)(16,117,22,111)(17,116,23,110)(18,115,24,109)(25,121,31,127)(26,132,32,126)(27,131,33,125)(28,130,34,124)(29,129,35,123)(30,128,36,122)(37,84,43,78)(38,83,44,77)(39,82,45,76)(40,81,46,75)(41,80,47,74)(42,79,48,73)(49,184,55,190)(50,183,56,189)(51,182,57,188)(52,181,58,187)(53,192,59,186)(54,191,60,185)(61,177,67,171)(62,176,68,170)(63,175,69,169)(64,174,70,180)(65,173,71,179)(66,172,72,178)(97,155,103,149)(98,154,104,148)(99,153,105,147)(100,152,106,146)(101,151,107,145)(102,150,108,156)(133,161,139,167)(134,160,140,166)(135,159,141,165)(136,158,142,164)(137,157,143,163)(138,168,144,162)>;
G:=Group( (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,169)(13,185)(14,186)(15,187)(16,188)(17,189)(18,190)(19,191)(20,192)(21,181)(22,182)(23,183)(24,184)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,73)(34,74)(35,75)(36,76)(37,132)(38,121)(39,122)(40,123)(41,124)(42,125)(43,126)(44,127)(45,128)(46,129)(47,130)(48,131)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,85)(69,86)(70,87)(71,88)(72,89)(97,166)(98,167)(99,168)(100,157)(101,158)(102,159)(103,160)(104,161)(105,162)(106,163)(107,164)(108,165)(133,154)(134,155)(135,156)(136,145)(137,146)(138,147)(139,148)(140,149)(141,150)(142,151)(143,152)(144,153), (1,23,138,46)(2,24,139,47)(3,13,140,48)(4,14,141,37)(5,15,142,38)(6,16,143,39)(7,17,144,40)(8,18,133,41)(9,19,134,42)(10,20,135,43)(11,21,136,44)(12,22,137,45)(25,64,58,101)(26,65,59,102)(27,66,60,103)(28,67,49,104)(29,68,50,105)(30,69,51,106)(31,70,52,107)(32,71,53,108)(33,72,54,97)(34,61,55,98)(35,62,56,99)(36,63,57,100)(73,89,120,166)(74,90,109,167)(75,91,110,168)(76,92,111,157)(77,93,112,158)(78,94,113,159)(79,95,114,160)(80,96,115,161)(81,85,116,162)(82,86,117,163)(83,87,118,164)(84,88,119,165)(121,174,187,151)(122,175,188,152)(123,176,189,153)(124,177,190,154)(125,178,191,155)(126,179,192,156)(127,180,181,145)(128,169,182,146)(129,170,183,147)(130,171,184,148)(131,172,185,149)(132,173,186,150), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,91,7,85)(2,90,8,96)(3,89,9,95)(4,88,10,94)(5,87,11,93)(6,86,12,92)(13,120,19,114)(14,119,20,113)(15,118,21,112)(16,117,22,111)(17,116,23,110)(18,115,24,109)(25,121,31,127)(26,132,32,126)(27,131,33,125)(28,130,34,124)(29,129,35,123)(30,128,36,122)(37,84,43,78)(38,83,44,77)(39,82,45,76)(40,81,46,75)(41,80,47,74)(42,79,48,73)(49,184,55,190)(50,183,56,189)(51,182,57,188)(52,181,58,187)(53,192,59,186)(54,191,60,185)(61,177,67,171)(62,176,68,170)(63,175,69,169)(64,174,70,180)(65,173,71,179)(66,172,72,178)(97,155,103,149)(98,154,104,148)(99,153,105,147)(100,152,106,146)(101,151,107,145)(102,150,108,156)(133,161,139,167)(134,160,140,166)(135,159,141,165)(136,158,142,164)(137,157,143,163)(138,168,144,162) );
G=PermutationGroup([[(1,170),(2,171),(3,172),(4,173),(5,174),(6,175),(7,176),(8,177),(9,178),(10,179),(11,180),(12,169),(13,185),(14,186),(15,187),(16,188),(17,189),(18,190),(19,191),(20,192),(21,181),(22,182),(23,183),(24,184),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,73),(34,74),(35,75),(36,76),(37,132),(38,121),(39,122),(40,123),(41,124),(42,125),(43,126),(44,127),(45,128),(46,129),(47,130),(48,131),(49,115),(50,116),(51,117),(52,118),(53,119),(54,120),(55,109),(56,110),(57,111),(58,112),(59,113),(60,114),(61,90),(62,91),(63,92),(64,93),(65,94),(66,95),(67,96),(68,85),(69,86),(70,87),(71,88),(72,89),(97,166),(98,167),(99,168),(100,157),(101,158),(102,159),(103,160),(104,161),(105,162),(106,163),(107,164),(108,165),(133,154),(134,155),(135,156),(136,145),(137,146),(138,147),(139,148),(140,149),(141,150),(142,151),(143,152),(144,153)], [(1,23,138,46),(2,24,139,47),(3,13,140,48),(4,14,141,37),(5,15,142,38),(6,16,143,39),(7,17,144,40),(8,18,133,41),(9,19,134,42),(10,20,135,43),(11,21,136,44),(12,22,137,45),(25,64,58,101),(26,65,59,102),(27,66,60,103),(28,67,49,104),(29,68,50,105),(30,69,51,106),(31,70,52,107),(32,71,53,108),(33,72,54,97),(34,61,55,98),(35,62,56,99),(36,63,57,100),(73,89,120,166),(74,90,109,167),(75,91,110,168),(76,92,111,157),(77,93,112,158),(78,94,113,159),(79,95,114,160),(80,96,115,161),(81,85,116,162),(82,86,117,163),(83,87,118,164),(84,88,119,165),(121,174,187,151),(122,175,188,152),(123,176,189,153),(124,177,190,154),(125,178,191,155),(126,179,192,156),(127,180,181,145),(128,169,182,146),(129,170,183,147),(130,171,184,148),(131,172,185,149),(132,173,186,150)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,91,7,85),(2,90,8,96),(3,89,9,95),(4,88,10,94),(5,87,11,93),(6,86,12,92),(13,120,19,114),(14,119,20,113),(15,118,21,112),(16,117,22,111),(17,116,23,110),(18,115,24,109),(25,121,31,127),(26,132,32,126),(27,131,33,125),(28,130,34,124),(29,129,35,123),(30,128,36,122),(37,84,43,78),(38,83,44,77),(39,82,45,76),(40,81,46,75),(41,80,47,74),(42,79,48,73),(49,184,55,190),(50,183,56,189),(51,182,57,188),(52,181,58,187),(53,192,59,186),(54,191,60,185),(61,177,67,171),(62,176,68,170),(63,175,69,169),(64,174,70,180),(65,173,71,179),(66,172,72,178),(97,155,103,149),(98,154,104,148),(99,153,105,147),(100,152,106,146),(101,151,107,145),(102,150,108,156),(133,161,139,167),(134,160,140,166),(135,159,141,165),(136,158,142,164),(137,157,143,163),(138,168,144,162)]])
72 conjugacy classes
class 1 2A ··· 2G 3 4A ··· 4H 4I ··· 4P 4Q ··· 4AF 6A ··· 6G 12A ··· 12X order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2
72 irreducible representations
dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C4 S3 Q8 D6 D6 C4○D4 Dic6 C4×S3 C4○D12 kernel C2×C4×Dic6 C4×Dic6 C2×C4×Dic3 C2×Dic3⋊C4 C2×C4⋊Dic3 C2×C4×C12 C22×Dic6 C2×Dic6 C2×C42 C2×C12 C42 C22×C4 C2×C6 C2×C4 C2×C4 C22 # reps 1 8 2 2 1 1 1 16 1 4 4 3 4 8 8 8
Matrix representation of C2×C4×Dic6 in GL6(𝔽13)
1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
8 0 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 0 0 0 3
,
4 3 0 0 0 0 3 9 0 0 0 0 0 0 3 9 0 0 0 0 9 10 0 0 0 0 0 0 0 12 0 0 0 0 12 0
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[4,3,0,0,0,0,3,9,0,0,0,0,0,0,3,9,0,0,0,0,9,10,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;
C2×C4×Dic6 in GAP, Magma, Sage, TeX
C_2\times C_4\times {\rm Dic}_6
% in TeX
G:=Group("C2xC4xDic6");
// GroupNames label
G:=SmallGroup(192,1026);
// by ID
G=gap.SmallGroup(192,1026);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,184,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^12=1,d^2=c^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
𝔽 | 2021-09-28 06:50:49 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9992828965187073, "perplexity": 5740.255076329813}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780060538.11/warc/CC-MAIN-20210928062408-20210928092408-00371.warc.gz"} |
https://www.vernier.com/experiments/awv/12B/photosynthesis_and_respiration_o2/ | Vernier Software & Technology
# Photosynthesis and Respiration (O2)
## Introduction
Plants make sugar, storing the energy of the sun into chemical energy, by the process of photosynthesis. When they require energy, they can tap the stored energy in sugar by a process called cellular respiration.
The process of photosynthesis involves the use of light energy to convert carbon dioxide and water into sugar, oxygen, and other organic compounds. This process is often summarized by the following reaction:
${\text{6 }}{{\text{H}}_{\text{2}}}{\text{O }} + {\text{ 6 C}}{{\text{O}}_{\text{2}}} + {\text{ light energy}} \to {{\text{C}}_{\text{6}}}{{\text{H}}_{{\text{12}}}}{{\text{O}}_{\text{6}}} + {\text{ 6 }}{{\text{O}}_{\text{2}}}$
Cellular respiration refers to the process of converting the chemical energy of organic molecules into a form immediately usable by organisms. Glucose may be oxidized completely if sufficient oxygen is available by the following equation:
${{\text{C}}_{\text{6}}}{{\text{H}}_{{\text{12}}}}{{\text{O}}_{\text{6}}} + {\text{ 6 }}{{\text{O}}_{\text{2}}} \to {\text{6 }}{{\text{H}}_{\text{2}}}{\text{O }} + {\text{ 6 C}}{{\text{O}}_{\text{2}}} + {\text{ energy}}$
All organisms, including plants and animals, oxidize glucose for energy. Often, this energy is used to convert ADP and phosphate into ATP.
## Objectives
In this experiment, you will
• Use an O2 Gas Sensor to measure the amount of oxygen gas consumed or produced by a plant during respiration and photosynthesis.
• Determine the rate of respiration and photosynthesis of a plant.
## Sensors and Equipment
This experiment features the following Vernier sensors and equipment.
### Option 2
You may also need an interface and software for data collection. What do I need for data collection?
## Agricultural Science with Vernier
See other experiments from the lab book.
1 Introduction to Data Collection 2 Acids and Bases 3 Diffusion through Membranes 4 Conducting Solutions 5 Osmosis 6 Respiration of Sugars by Yeast 7 Reflection and Absorption of Light 8 Soil pH 9 Soil Salinity 10 Soil Temperature 11 Soil Moisture 12A Photosynthesis and Respiration (CO2) 12B Photosynthesis and Respiration (O2) 12C Photosynthesis and Respiration (CO2 and O2) 13 Transpiration 14A Cell Respiration (CO2) 14B Cell Respiration (O2) 14C Cell Respiration (CO2 and O2) 15 The Greenhouse Effect 16 Energy in Food 17A Enzyme Action: Testing Catalase Activity 17B Enzyme Action: Testing Catalase Activity 18A Lactase Action 18B Lactase Action 19 Oxygen Gas and Human Respiration 20 Biochemical Oxygen Demand 21 Animal Temperature 22 Lemon "Juice" 23 Ohm's Law 24 Energy Content of Fuels 25 Photovoltaic Cells 26 Wind Power 27 Watershed Testing 28 Interdependence of Plants and Animals 29 Biodiversity and Ecosystems
### Experiment 12B from Agricultural Science with Vernier Lab Book
#### Included in the Lab Book
Vernier lab books include word-processing files of the student instructions, essential teacher information, suggested answers, sample data and graphs, and more. | 2019-11-17 23:22:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2033756524324417, "perplexity": 8884.582176817268}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669352.5/warc/CC-MAIN-20191117215823-20191118003823-00338.warc.gz"} |
https://www.kbwiki.ercoftac.org/w/index.php/Template:Demo_AC_CFD_Over | # Template:Demo AC CFD Over
Jump to navigation Jump to search
If CFD simulations of the AC have been performed then provide a brief overview. This should cover the scope of the calculations and the main aspects of the modelling strategy e.g. equations solved, turbulence and other physical models employed). All computational domain simplifications/idealisations, and the treatment of subgrid features should also be identified (e.g. imposed symmetry plane, omission of detailed features, simplification of complex/small scale features, i.e. porous media, use of equivalent wall roughness). If important details of the geometry representation are uncertain then the impact of these uncertainties on the DOAPs should be discussed, including possible ways for managing their effect.
It is left to the discretion of each author to decide the most appropriate way for structuring and summarising the CFD results. Ideally, the data structures used should be consistent with those used for the test data.
A summary table for all CFD simulation results should be included, as shown below in Table CFD-A. Available data should be clearly identified, (e.g. UVW, k, concentration, etc). As with test data, a distinction should be drawn between detailed local data (e.g. p(x,y,z)) and data relating to DOAPs which are likely to be global/summary parameters (e.g. coefficient of lift, CL).
All available detailed data should be stored in separate electronic datafiles (according to guidelines set out by the Knowledge Base team at the University of Surrey). These should be summarised as shown below in CFD-B, with links to each of the datafiles.
Name GNDPs PDPs (problem definition parameters) MPs (measured parameters)
Re Fr Wind direction Source rate (kg/s) Release density(kg/m3) Detailed data DOAPs
CFD 1 (dense gas dispersion) ${\displaystyle 10^{5}-10^{6}}$ ${\displaystyle 0.2-10}$ 0, 30, 45, 90, 180 ${\displaystyle 1-3}$ ${\displaystyle 1.22-3.00}$ ${\displaystyle C,\ U}$ ${\displaystyle {\ C/C_{0}}}$
Re Wind direction Building geometry Detailed data DOAPs
CFD 2 (passive gas releases) ${\displaystyle 10^{5}-10^{6}}$ 0, 30, 45 A, B, C, D ${\displaystyle C,\ U,\ V,\ W,\ k}$ ${\displaystyle {\ C/C_{0}}}$
Table CFD-A Summary description of all test cases
SP1 ${\displaystyle U,\ V,\ W\ (ms^{-1})}$ SP2 ${\displaystyle k\ (m^{2}s^{-2})}$ SP3 ${\displaystyle C\ (kg/m^{3})}$ DOAPs, or other miscellaneous data
CFD 1 cfd11.dat cfd13.dat cfd14.datcfd15.dat
CFD 2 cfd21.dat cfd22.dat cfd23.dat cfd24.datcfd25.dat,
Table CFD-B Summary description of all available datafiles and simulated parameters | 2022-01-28 21:41:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 12, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6583620309829712, "perplexity": 3697.5532087356655}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320306346.64/warc/CC-MAIN-20220128212503-20220129002503-00264.warc.gz"} |
https://proxies123.com/a-integral-computation-problem/ | # A integral computation problem
For $$a,b geq 0$$, integrate
$$int_0^infty dfrac{x^frac{1}{a}}{b}e^{-x} dx.$$
I came across this integral and I have no idea where to begin. | 2021-01-20 09:09:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9963483810424805, "perplexity": 302.481792530125}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703519984.9/warc/CC-MAIN-20210120085204-20210120115204-00101.warc.gz"} |
https://nforum.ncatlab.org/discussion/5227/equality-definitional-extensional-intensional/ | # Start a new discussion
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## Discussion Tag Cloud
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
• CommentRowNumber1.
• CommentAuthordan
• CommentTimeSep 5th 2013
• (edited Sep 6th 2013)
First of all: Hello everyone! Thanks for the great work on the nLab. So far it was a joy to absorb a lot, but today I shall actively engage in the discussion for the first time.
On the page about equality in the section about definitional equality the following is said: “The most basic one is definitional equality or intensional equality. … “
As well as: “For instance the symbols “2” and “s(s(0))” (meaning the successor of the successor of 0) are definitionally/extensionally equal terms (of type the natural numbers):”
Can both these statements be true? If so, I think this is a bit confusing and some clarification would be great.
Edit: Inserted Toby’s link syntax hint.
• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeSep 5th 2013
I'm pretty sure that the second one is a mistake; I've fixed it. (Of course, they are also extensionally equal, since that is a weaker condition, but that's not the point.)
• CommentRowNumber3.
• CommentAuthorTobyBartels
• CommentTimeSep 5th 2013
By the way, write [[equality]] here to produce ‘equality’.
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeSep 5th 2013
I guess that was my fault. Thanks for fixing it!
• CommentRowNumber5.
• CommentAuthordan
• CommentTimeSep 6th 2013
Sure, now I see! So a set of extensionally equal functions might contain one or more subsets of intensionally equal functions, because we are only looking at the relation that the function generates (semantics) and not the notation (syntax). But is it really definitional equality in the latter case? Can I not interchange different constructions for the natural numbers that “2” and “s(s(0))” represent? For example von Neumann or this one (which I think is quite beautiful). Or would it be the case, that we define/construct the natural numbers and then decide on different kinds of syntactic sugar to represent them?
• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeSep 7th 2013
So a set of extensionally equal functions might contain one or more subsets of intensionally equal functions
Each extensional-equality-class of functions contains many distinct intensional-equality-classes; is that what you mean?
Can I not interchange different constructions for the natural numbers that “2” and “s(s(0))” represent?
I would say that neither of those strings of symbols has any meaning until you’ve fixed a meaning of “the natural numbers”.
• CommentRowNumber7.
• CommentAuthordan
• CommentTimeSep 7th 2013
• (edited Sep 7th 2013)
Each extensional-equality-class of functions contains many distinct intensional-equality-classes; is that what you mean?
Yep, need to get more into the formal language.
I would say that neither of those strings of symbols has any meaning until you’ve fixed a meaning of “the natural numbers”. Okay. That would be everything then.
Thanks to everyone for helping me understand.
• CommentRowNumber8.
• CommentAuthorTobyBartels
• CommentTimeSep 13th 2013
I would say that neither of those strings of symbols has any meaning until you’ve fixed a meaning of “the natural numbers”.
To follow up on that: The two different constructions in Wikipedia, linked earlier, are ways of fixing such a meaning in material set theory. This is different from any business about comparing $2$ to $s(s(0))$ (which may be interpreted in material set theory using either construction or may be interpreted in something much simpler, such as Peano arithmetic). In any case, we do always have $2 = s(s(0))$ by definitional equality, at least as long as $2$ is defined as $s(1)$ and $1$ is defined as $s(0)$. | 2022-01-27 21:13:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 7, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7631324529647827, "perplexity": 1539.6907270806291}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305288.57/warc/CC-MAIN-20220127193303-20220127223303-00251.warc.gz"} |
https://www.r-ccs.riken.jp/labs/cbrt/tutorials2019/tutorial-1-2/ | 1.2 Let’s take a quick look at the source code of GENESIS
Take a look at the source directory
In this tutorial, we briefly explain the source code of GENESIS. We expect that you have already learned basic theories of MD simulations [1,2,3]. Even if you are an experimental scientist, we recommend you to take a look at the source code of GENESIS for a better understanding of the MD algorithm. GENESIS is written mainly in Fortran90, which is a traditional language but still widely used in the computational science due to its simplicity. Let’s go to the directory where the source code of the MD program atdyn is contained. In this directory, you will find many files with the filename extensions .fpp, .f90, .o, and .mod. Of these, fpp file is the original source code, while the other files were generated after the compilation. Therefore, we will mainly focus on the fpp files and refer to each fpp file as a “module”.
# Change directory to see the source codes of GENESIS
$cd ~/GENESIS_Tutorials-2019$ cd Programs
$cd genesis-1.7.1/src/atdyn # Check the contents$ ls *.fpp
at_boundary.fpp at_energy_table_cubic.fpp
at_boundary_str.fpp at_energy_table_linear.fpp
at_constraints.fpp at_energy_table_linear_bondcorr.fpp
at_constraints_str.fpp at_ensemble.fpp
at_control.fpp at_ensemble_str.fpp
at_dynamics.fpp at_experiments.fpp
at_dynamics_str.fpp at_experiments_str.fpp
at_dynvars.fpp at_gamd.fpp
at_dynvars_str.fpp at_input.fpp
at_enefunc.fpp at_md_leapfrog.fpp
at_enefunc_amber.fpp at_md_vverlet.fpp
at_enefunc_charmm.fpp at_md_vverlet_cg.fpp
at_enefunc_gamd.fpp at_minimize.fpp
at_enefunc_gbsa.fpp at_minimize_str.fpp
at_enefunc_go.fpp at_morph.fpp
at_enefunc_gromacs.fpp at_morph_str.fpp
at_enefunc_pme.fpp at_output.fpp
at_enefunc_restraints.fpp at_output_str.fpp
:
Before taking a look at the source code, let’s check the total number of lines in the program using the wc command. You can see that atdyn consists of more than 100,000 lines. You would be surprised how big the MD program is. In addition to atdyn, there are other directories in src, such as spdyn, analysis, and lib. Here, lib stands for “library”, which contains modules commonly used by atdyn, spdyn, and analysis. You can see that these programs also consist of many lines. Since GENESIS is a huge program package, it may be difficult for you to know where to start reading the program when trying to understand it. But, please don’t worry. If you read the source code in the order described below, you will be able to grasp the whole picture or the core of the MD program.
# Check the total number of lines in the program
$wc -l *.fpp : 586 at_rpath_fep.fpp 3079 at_rpath_mep.fpp 566 at_rpath_str.fpp 1206 at_setup_atdyn.fpp 989 at_setup_mpi.fpp 1799 at_vibration.fpp 55 at_vibration_str.fpp 504 atdyn.fpp 105621 Total$ wc -l ../spdyn/*.fpp
:
139635 Total
$wc -l ../lib/*.fpp : 59577 Total In which module are the energy and forces calculated? One of the main processes in the MD simulation is the calculation of the potential energy and force. In general, the potential energy of a system is given by where the first through fifth terms on the right-hand side are the energies for bond stretching, angle bending, dihedral angle rotation, van der Waals interaction, and electrostatic interaction, respectively. The first three terms are also called bonded interactions, and the last two terms are called non-bonded interactions. In the “atdyn” directory, you can see several files whose names start with “at_energy“. You can easily imagine that at_energy_bonds.fpp, at_energy_angles.fpp, at_energy_dihedrals.fpp, and at_energy_nonbonds.fpp are related to the energy calculation for the bond stretching, angle bending, dihedral angle rotation, and non-bonded interactions, respectively. As for the other files, we will not discuss them in detail here, since they are more advanced. # List up the files related to the energy calculation$ ls at_energy*.fpp
at_energy.fpp at_energy_morph.fpp
at_energy_angles.fpp at_energy_nonbonds.fpp
at_energy_bonds.fpp at_energy_pme.fpp
at_energy_cg_nonlocal.fpp at_energy_restraints.fpp
at_energy_dihedrals.fpp at_energy_soft.fpp
at_energy_eef1.fpp at_energy_str.fpp
at_energy_gamd.fpp at_energy_table_cubic.fpp
at_energy_gbsa.fpp at_energy_table_linear.fpp
at_energy_go.fpp at_energy_table_linear_bondcorr.fpp
Now, let us focus on the bond stretching energy, as it is the simplest term in the above equation. The energy and force are given by
$$\normalsize{{U_{{\rm{bond}}}} = {k_b}{({r_{ij}} – {r_0})^2}}$$
$$\normalsize{{{\bf{F}}_j} = – 2{k_b}({r_{ij}} – {r_0})\frac{{{{\bf{r}}_{ij}}}}{{{r_{ij}}}}}$$
$$\normalsize{{\bf{F}}_i = – {\bf{F}}_j}$$
where kb is the force constant, rij is the distance between i-th and j-th atoms, and r0 is the equilibrium distance between the atoms. With these equations in mind, let’s take a look at at_energy_bond.fpp using the less command. According to the variable names and comment lines, you can indeed see that the distance between the atoms (r12), bond stretching energy (ebond), and force acting on each atom (force) are calculated from the atomic coordinates (coord), force constant (fc), and equilibrium distance (r0). For now you can just skim through the source code, since the main purpose of this tutorial is not to understand the source code in detail, but to get a quick idea of how the energy and force are calculated in GENESIS. To quit the less command, type “q” in the terminal window.
# Check the source code for the bond energy term (type "q" to quit)
$less at_energy_bonds.fpp subroutine compute_energy_bond(enefunc, coord, force, virial, ebond) : do i = istart, iend ! bond energy: E=K[b-b0]^2 ! d12(1:3) = coord(1:3,list(1,i)) - coord(1:3,list(2,i)) r12 = sqrt( d12(1)*d12(1) + d12(2)*d12(2) + d12(3)*d12(3) ) r_dif = r12 - r0(i) ebond = ebond + fc(i) * r_dif * r_dif ! gradient: dE/dX ! cc_frc = (2.0_wp * fc(i) * r_dif) / r12 work(1:3,i) = cc_frc * d12(1:3) : end do ! store force: F=-dE/dX ! do i = istart, iend force(1:3,list(1,i)) = force(1:3,list(1,i)) - work(1:3,i) force(1:3,list(2,i)) = force(1:3,list(2,i)) + work(1:3,i) end do If you are more interested in the source code for the energy and force calculations, please take a look at the contents of at_energy_angles.fpp, at_energy_dihedrals.fpp, and so on, as well as their parent module, at_energy.fpp. In fact, the parent subroutine in at_energy.fpp is called from within the do loop of the integrator, as explained next. In which module are the atomic coordinates and velocities updated? In MD simulation, the atomic coordinates and velocities are updated at each time step (Δt), which is repeated many times to realize the time evolution of the system. Such a protocol is called “time integration”. Typical integrators include the leap-frog algorithm and the velocity Verlet algorithm, both of which are available in atdyn. If you look in the atdyn directory, you can find at_md_leapfrog.fpp and at_md_vverlet.fpp. These modules are exactly related to the time integration. Here, we focus on the leap-frog algorithm, where the following two equations are solved iteratively. $$\normalsize{{{\bf{v}}_i}(t + \frac{{\Delta t}}{2}) = {{\bf{v}}_i}(t – \frac{{\Delta t}}{2}) + \frac{{\Delta t}}{{{m_i}}}{{\bf{F}}_i}(t)}$$ $$\normalsize{{{\bf{r}}_i}(t + \Delta t) = {{\bf{r}}_i}(t) + \Delta t{{\bf{v}}_i}(t + \frac{{\Delta t}}{2})}$$ These equations give an NVE (microcanonical) ensemble to the system. Let’s take a look at at_md_leapfrog.fpp with the less command, and try to find the equations. You can see that there is a do loop with the comment “Main MD loop”, in which the two equations are solved iteratively. The following is an extract of the most important parts in the main MD loop. # View the source code of the leapfrog integrator$ less at_md_leapfrog.fpp
! Main MD loop
! coord is at 0 + dt and vel is at 0 + 1/2dt, if restart off
! coord is at t + 2dt and vel is at t + 3/2dt, if restart on
!
do i = istart, iend
:
! Compute energy(t + dt), force(t + dt), and internal virial(t + dt)
!
call compute_energy(molecule, enefunc, pairlist, boundary, &
:
! Newtonian dynamics
! v(t+3/2dt) = v(t+1/2dt) + dt*F(t+dt)/m
! r(t+2dt) = r(t+dt) + dt*v(t+3/2dt)
!
do j = 1, natom
vel(1,j) = vel(1,j) + dt*force(1,j)*inv_mass(j)
vel(2,j) = vel(2,j) + dt*force(2,j)*inv_mass(j)
vel(3,j) = vel(3,j) + dt*force(3,j)*inv_mass(j)
coord(1,j) = coord(1,j) + dt*vel(1,j)
coord(2,j) = coord(2,j) + dt*vel(2,j)
coord(3,j) = coord(3,j) + dt*vel(3,j)
end do
:
! Output energy(t + dt) and dynamical variables(t + dt)
! coord is at t + 2dt, coord_ref is at t + dt
! vel is at t + 3/2dt, vel_ref is at t + 1/2dt
! box_size is at t + 2dt, box_size_ref is at t + dt
!
call output_md(output, molecule, enefunc, dynamics, boundary, &
ensemble, dynvars)
end do
Here, coord and vel are the atomic coordinates and velocities, respectively, dt is the time step, and inv_mass is the inverse of the atomic mass. As described above, the atomic forces are calculated in the subroutine “compute_energy“. The trajectories of the coordinates and energy are output in the subroutine “output_md“, which is contained in at_output.fpp.
In the case of the NVT or NPT ensemble, temperature and pressure are kept constant, where the velocities and coordinates are re-scaled according to the instantaneous temperature and pressure. In the leap-frog integrator of atdyn, Langevin thermostat/barostat [4,5] and Berendsen thermostat/barostat [6] are available. Let’s search for “langevin” or “berendsen” in at_md_leapfrog.fpp to see how the velocities and coordinates are re-scaled. In addition, let’s search for “constraints” to understand the SHAKE algorithm [7], which further updates the coordinates to fix the length of the covalent bond involving hydrogen.
Overview of the core of MD simulation
The following figure summarizes the flowchart of the time integration, energy and force calculations, and trajectory output. Since GENESIS is a huge program package, it is not effective to read the source code from the beginning of the program. Rather, it is recommended to first understand the source code for energy and force calculations or time integration, which are the core of MD simulation. Then, you can expand your understanding of the source code based on your own knowledge of MD simulation such as the SHAKE algorithm, temperature and pressure controls. The MD program is indeed encoded with the theories you have learned in textbooks and literature.
References
1. M. Tuckerman, “Statistical Mechanics: Theory and Molecular Simulation”, Oxford University Press (2010).
2. M. P. Allen and D. J. Tildesley, “Computer Simulation of Liquids”, Oxford University Press (2017).
3. D. Frenkel and B. Smit, “Understanding Molecular Simulation 2nd Edition”, Academic Press (2001).
4. A. Brünger et al., Chem. Phys. Lett., 105, 495-500 (1984).
5. S. E. Feller et al., J. Chem. Phys., 103, 4613-4621 (1995).
6. H. J. C. Berendsen et al., J. Chem. Phys., 81, 3684-3690 (1984).
7. J. P. Ryckaert et al., J. Comput. Phys., 23, 327-341 (1977).
Written by Takaharu Mori@RIKEN Theoretical molecular science laboratory
Jan. 3, 2022 | 2022-05-25 23:30:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6624442934989929, "perplexity": 2650.718063973641}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662594414.79/warc/CC-MAIN-20220525213545-20220526003545-00596.warc.gz"} |
https://socratic.org/questions/two-objects-have-masses-of-4-mg-and-48-mg-how-much-does-the-gravitational-potent-1 | 0
# Two objects have masses of 4 MG and 48 MG. How much does the gravitational potential energy between the objects change if the distance between them changes from 27 m to 2 m?
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• 29 minutes ago | 2018-04-25 08:30:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5587536096572876, "perplexity": 6457.655455874439}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125947759.42/warc/CC-MAIN-20180425080837-20180425100837-00502.warc.gz"} |
https://en.wikibooks.org/wiki/Baseball/Hitting_Basics | # Baseball/Hitting Basics
## Finding a bat
The first thing you need to do is find a bat that suits you. The bat should be as tall as your hip when stood on the ground. The length may vary if you are a somewhat experienced batter and you know what length you like. Weight is a key issue when looking for a bat. As bat speed is key, a hitter should look for the lightest possible stick to swing, but keep this in mind: from a physics perspective, the formula ${\displaystyle m1v1ixm2v2i=m1v2fxm2v2f}$ where m1 is the mass of the ball, m2 is the bat, and v1,v2 are their respective speeds(i being initial, f being final). Wacky math aside, a heavier bat moving at the same speed as a less massive bat will empart more force on the ball, making the difference between a high fly, and a home run. An appropriate test for finding a bat of proper weight for your strength category is to hold the bat handle in one hand, parallel to the ground for 45 seconds. If you can hold it up, you should be strong enough to swing the stick. This is just a general rule of thumb. if you find your speed is too slow, shave two ounces off the bats' mass.
## Gripping the bat
The grip the batter uses is important because it determines the bat speed. Your bottom hand should be placed half an inch from the knob of the bat. The “door-knocking” knuckles of you top hand should be roughly aligned with the "door-knocking" knuckles of your bottom hand. Reach back behind you and touch the head of the bat to your buttocks or your lower back. Now bring the bat around and hit the ground. Examine your knuckles and this is where you should try to have them. Remember to grip the bat loosely. | 2017-02-26 12:29:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5518723726272583, "perplexity": 1175.109759396068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501172000.93/warc/CC-MAIN-20170219104612-00363-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://erlweb.mit.edu/focused-blind-deconvolution-0 | # Focused Blind Deconvolution
Title Focused Blind Deconvolution Publication Type Journal Article Year of Publication 2019 Authors Bharadwaj, P, Demanet, L, Fournier, A Journal IEEE Transactions on Signal Processing Volume 67 Issue 12 Pagination 3168 - 3180 Date Published Mar-06-2020 ISSN 1053-587X Abstract We introduce a novel multichannel blind deconvolution (BD) method that extracts sparse and front-loaded impulse responses from the channel outputs i.e., their convolutions with a single arbitrary source. Unlike most prior work on BD, a crucial feature of this formulation is that it doesn't encode support restrictions on the unknowns, except for fixing their duration lengths. The indeterminacy inherent to BD, which is difficult to resolve with a traditional $\ell_1$ penalty on the impulse responses, is resolved in our method because it seeks a first approxima- tion where the impulse responses are: “maximally white” over frequency - encoded as the energy focusing near zero lag of the impulse-response temporal auto-correlations; and “maximally front-loaded” - encoded as the energy focusing near zero time of the impulse responses. Hence we call the method focused blind deconvolution (FBD). It partitions BD into two separate optimization problems and uses the focusing constraints in succession. The respective constraints in both these problems are removed as the iterations progress. A multichannel blind deconvolution problem whose physics calls for sparse and front-loaded impulse responses arises in seismic inversion, where the impulse responses are the Green's function evaluations at different receiver locations, and the operation of a drill bit inputs the noisy and correlated source signature into the subsurface. We demonstrate the benefits of FBD using seismic-while-drilling numerical experiments, where the noisy data recorded at the receivers are hard to interpret, but FBD can provide the processing essential to separate the drill-bit (source) signature from the interpretable Green's function. URL https://ieeexplore.ieee.org/document/8680655 DOI 10.1109/TSP.7810.1109/TSP.2019.2908911 Short Title IEEE Trans. Signal Process. | 2021-09-27 23:44:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3647981584072113, "perplexity": 2172.178902909624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780058552.54/warc/CC-MAIN-20210927211955-20210928001955-00484.warc.gz"} |
https://cstheory.stackexchange.com/questions/39903/will-core-decomposition-get-a-maximal-clique | # Will core decomposition get a maximal clique?
I have read David Eppstein's paper about maximal clique enumeration by using degeneracy order. It has mentioned the core decomposition, which is removing the vertex with the smallest degree iteratively.
And I pick up with a theorem:
If we remove the vertex with the smallest degree iteratively until the smallest degree is |S|-1, where S is the remaining graph, is S a maximal clique?
example graph:
In this graph, we will delete vertex a whose degree is the smallest one ( deg(a)=2 ), and then delete vertex f. And we will get the 4-maximal clique {b,c,d,e}.
I cannot prove this theorem but I also can't find a counterexample. Is there anyone can help me figure out this theorem right or wrong?
PS:
Clique is a subgraph whose every vertex is adjacent to each other. And maximal clique is a clique that cannot be contained by another clique, it means the maximal one cannot be extended.
I can prove the remaining graph that above algorithm produced is not a maximum clique which is the largest one in the graph, but I can't prove this graph whether is a maximal clique.
Also notice that when the vertex v is removed from the graph, all it's neighbors' degree will decrease one.
## 1 Answer
No. The illustration shows a graph (the graph of a cube with one corner truncated) and a valid removal sequence such that the vertices left at the point when the minimum degree equals $|S|-1$ (the two red vertices) do not form a maximal clique.
In this example, some other valid removal orderings could still produce a maximal clique. Probably there are more elaborate versions of this example that force the remaining clique to be non-maximal for all valid removal orderings.
• Thank you, David, I have read your two papers about MCE and I don't know that you were here. This counterexample is enough to my question. Thank you again. Jan 3, 2018 at 8:35
• How about adding a vertex to each edge of the Peterson graph. The new vertex is adjacent to both ends of the edge, and hence has degree two. The decomposition will remove all new vertices first, and then it ends with an edge. Jan 6, 2018 at 7:07
• You mean, keeping the existing edge but adding a new degree-two vertex adjacent to its endpoints? That should work with any triangle-free cubic graph (e.g. $K_{3,3}$); using Petersen is overkill. Jan 6, 2018 at 8:08 | 2022-05-29 05:47:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5692564845085144, "perplexity": 268.9635365748761}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663039492.94/warc/CC-MAIN-20220529041832-20220529071832-00522.warc.gz"} |
https://www.physicsforums.com/threads/gauss-law-problem.667496/ | # Gauss' Law Problem
miniMitts27
Given a spherical shell insulator as defined by an inner radius a = 4 cm and an outer radius b = 6 cm and carries a total charge of Q = + 9 C (1 C = 10-6 C). (You may assume that the charge is distributed uniformly throughout the volume of the insulator).
What is Ey, the y-component of the electric field at point P which is located at (x,y) = (0, -5 cm)? (picture of situation is attached)
Basically I just don't even know where to start. I realize that it's a Gauss' Law problem... I just don't know where to go from here. Any help would be greatly appreciated!
#### Attachments
• spherical shell problem.JPG
14.6 KB · Views: 325 | 2022-11-30 10:42:13 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8057804107666016, "perplexity": 312.1610497734679}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710734.75/warc/CC-MAIN-20221130092453-20221130122453-00729.warc.gz"} |
https://brilliant.org/problems/gravitationshm/ | # Gravitation+SHM
There is an isolated planet having mass $$2M$$ and radius $$2R$$ where $$M$$ and $$R$$ are the mass and radius of the earth. A simple pendulum having mass $$m$$ and length $$2R$$ is made to small oscillations on the planet. Find the time period of $$SHM$$ of pendulum in seconds
Details and Assumptions
take $$\pi=3.00~~,g=10m/s^{2}~~,\sqrt{2}=1.41~~R=6400~km$$
× | 2018-06-19 22:24:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9172734618186951, "perplexity": 294.1515445044131}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267863206.9/warc/CC-MAIN-20180619212507-20180619232507-00472.warc.gz"} |
https://math.stackexchange.com/questions/4128699/t-n-a-set-of-all-binary-trees-with-n-leaves | # $T_{n}$ a set of all binary trees with $n$ leaves
Let $$T_{n}$$ be a set of all binary trees with $$n$$ leaves.
Show that: $$|T_1|=1,|T_2|=1,|T_3|=2,|T_4|=5$$
My attempt:
I was trying to find how many options of leaves we should add to the previous tree when the tree belongs to $$T_{n-1}$$, and then subtract the common options after the addition of the leaf. For example, for $$T_4$$ we have the following trees of $$T_3$$:
C C
/ \ / \
C L L C
/ \ / \
L L , L L
we have 3 options to add another leaf for each tree, however, we have to remove the common tree when:
C
/ \
C C
/ \ / \
L L L L
and we got $$6-1=5$$ options of trees for $$T_4$$.
Now, I don't know how to formal this and create a regression formula based on the process which I have used.
• $T(1)$ is infinite because every linear binary tree has one leaf. May 6, 2021 at 0:16
There is only one tree which has exactly one leaf - the tree which doesn't branch. Thus, $$|T_1| = 1$$.
Now consider $$T_n$$ for $$n > 1$$. A tree with $$n$$ leaves can only be made by combining a tree with $$k$$ leaves with a tree with $$n - k$$ leaves, where $$1 \leq k < n$$. That is, $$T_n \approx \coprod\limits_{k = 1}^{n - 1} T_k \times T_{n - k}$$. So we have $$|T_n| = \sum\limits_{k = 1}^{n - 1} |T_k| \times |T_{n - k}|$$.
This gives us $$|T_2| = |T_1| |T_1| = 1$$, $$|T_3| = |T_1| |T_2| + |T_2| |T_1| = 2$$, and $$|T_4| = |T_1| |T_3| + |T_2| |T_2| + |T_3| |T_1| = 5$$.
• Also these are the Catalan numbers: en.wikipedia.org/wiki/Catalan_number May 5, 2021 at 23:52
• Wont u get a tree with $n$ leaves by combing a tree with $k-1$ and $n-k$ leaves because when u combine 2 trees u r removing one of the leaves of original tree.. ? May 6, 2021 at 2:22
Here is the C++ code (C++11 NOT required) to visualize these trees. For large $$n \geq 16$$, the heap might explode.
#include <stdio.h>
#include <iostream>
#include <vector>
#include <string>
#include <sstream>
#include <assert.h>
using namespace std;
int n = 15; //number of leaves
string int2string(int k)
{
//std::to_string() only works >= C++11
string s;
stringstream ss;
ss << k;
ss >> s;
return s;
}
typedef struct node
{
bool isleaf;
string* data; //Using string* is better. The compiler knows how to allocate memory.
struct node* l; //left pointer
struct node* r; //right pointer
}node;
vector<node*> recursively_constuct_trees(vector<node*>& leaves, int lower_bound, int upper_bound)
{
//returning vector by value is a good idea due to named return value optimization (NRVO)
vector<node*> results;
if(lower_bound == upper_bound)
{
//only one node
results.push_back(leaves[lower_bound]);
return results;
}
for(int i = lower_bound; i < upper_bound; ++i)
{
vector<node*> results_left = recursively_constuct_trees(leaves, lower_bound, i);
vector<node*> results_right = recursively_constuct_trees(leaves, i+1, upper_bound);
//Catalan Convolution
for(unsigned j = 0; j < results_left.size(); ++j)
for(unsigned k = 0; k < results_right.size(); ++k)
{
node* newnode = new node;
newnode -> l = results_left[j];
newnode -> r = results_right[k];
newnode -> data = new string("(" + *(results_left[j] -> data) + "+" + *(results_right[k] -> data) + ")");
newnode -> isleaf = false;
results.push_back(newnode);
}
}
return results;
}
void recursively_destroy_tree(node* &root){
if(!root || root -> isleaf) //Do not free leaves!
return;
recursively_destroy_tree(root -> l);
recursively_destroy_tree(root -> r);
delete root;
root = NULL;
}
int main(void)
{
assert(n >= 1);
vector<node*> leaves;
for(int i = 0; i < n; ++i)
{
leaves.push_back(new node);
leaves[i] -> data = new string(int2string(i));
leaves[i] -> l = NULL;
leaves[i] -> r = NULL;
leaves[i] -> isleaf = true;
}
vector<node*> result = recursively_constuct_trees(leaves, 0, n-1);
printf("T(%d) = %d\n", n, result.size());
for(unsigned i = 0; i < result.size(); ++i){
cout << "VISUALIZATION (" << i <<"): " << *(result[i] -> data) << endl;
recursively_destroy_tree(result[i]);
}
return 0;
} | 2022-08-11 18:06:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 21, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5629015564918518, "perplexity": 1940.6037914382173}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571483.70/warc/CC-MAIN-20220811164257-20220811194257-00652.warc.gz"} |
http://mathhelpforum.com/calculus/23920-prob.html | 1. ## prob
I don't know what to do with this. Help please.
Suppose that a volcano is erupting and readings of the rate r(t) at which solid materials are spewed into the atmosphere are given in the table. The time t is measured in seconds and the units for r(t) are tonnes (metric tons) per second.
t=0, r(t)=2
t=1, r(t)=8
t=2, r(t)=26
t=3, r(t)=32
t=4, r(t)=46
t=5, r(t)=52
t=6, r(t)=60
(a) Give upper and lower estimates for the total quantity Q(6) of erupted materials after 6 seconds.
Q(6) = ? tonnes (lower estimate)
Q(6) = ? tonnes (upper estimate)
(b) Use the Midpoint Rule to estimate Q(6).
Q(6) = ? tonnes
2. ## try
In this problem we are given several values of the function r(t). Q(t) is simply the integral of r(t) - that is, the area under the graph. When evaluating Q(6) using the midpoint rule means:
Q(6)~Q((6-0)/2)*6.
a better evaluation can be achieved by evaluation the area between t(n) and t(n+1) using the trapezoidal rule and then summing.
a lower estimate can be achieved in several ways, two of which:
(a) using the overall minimum (2) and calculation minimum*time
(b) a more realistic lower bound can be achieved by doing (a) for every segment of the form [n,n+1] (thus using 52 as the minimum in [5,6] ) and summing.
an upper estimate is achieved similarly. | 2017-01-23 17:13:51 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8744401931762695, "perplexity": 1856.4751561576068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560282935.68/warc/CC-MAIN-20170116095122-00535-ip-10-171-10-70.ec2.internal.warc.gz"} |
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Mat. Sb. (N.S.), 1976, Volume 101(143), Number 1(9), Pages 87–130 (Mi msb2955)
Analytic continuations with respect to a parameter of the Green function of exterior boundary value problems for the two-dimensional Helmholtz equation. II
L. A. Muravei
Abstract: This paper gives a construction of some special potentials for the two-dimensional Helmholtz equation. With these potentials, one can establish the existence of Green's functions of exterior boundary-value problems for each $k$ from the complex upper $k$-plane, and the analyticity of these functions in that region. More than that, one can establish the existence of an analytic continuation to the region
$$\{0>\operatorname{Im}k>-\beta(1+|\operatorname{Re}k|^{1/3}, |\operatorname{Re}k|>0\}$$
for some $\beta>0$, with estimates characterizing the behavior of the Green functions for large absolute values of $k$.
Bibliography: 6 titles.
Full text: PDF file (3362 kB)
References: PDF file HTML file
English version:
Mathematics of the USSR-Sbornik, 1976, 30:1, 77–118
Bibliographic databases:
UDC: 517.9
MSC: Primary 35J05; Secondary 30A14
Citation: L. A. Muravei, “Analytic continuations with respect to a parameter of the Green function of exterior boundary value problems for the two-dimensional Helmholtz equation. II”, Mat. Sb. (N.S.), 101(143):1(9) (1976), 87–130; Math. USSR-Sb., 30:1 (1976), 77–118
Citation in format AMSBIB
\Bibitem{Mur76} \by L.~A.~Muravei \paper Analytic continuations with respect to a~parameter of the Green function of exterior boundary value problems for the two-dimensional Helmholtz equation.~II \jour Mat. Sb. (N.S.) \yr 1976 \vol 101(143) \issue 1(9) \pages 87--130 \mathnet{http://mi.mathnet.ru/msb2955} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=435577} \zmath{https://zbmath.org/?q=an:0335.35033} \transl \jour Math. USSR-Sb. \yr 1976 \vol 30 \issue 1 \pages 77--118 \crossref{https://doi.org/10.1070/SM1976v030n01ABEH001900} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1976FJ58100006}
• http://mi.mathnet.ru/eng/msb2955
• http://mi.mathnet.ru/eng/msb/v143/i1/p87
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This publication is cited in the following articles:
1. L. A. Muravei, “Analytic continuation with respect to a parameter of the Green's functions of exterior boundary value problems for the two-dimensional Helmholtz equation. III”, Math. USSR-Sb., 34:1 (1978), 55–98
2. P. Werner, “Zur Asymptotik der wellengleichung und der wärmeleitungsgleichung in zweidimensionalen außenräumen”, Math Meth Appl Sci, 7:1 (1985), 170
3. Proka D., “Asymptotic Properties of Solutions of Exterior Mixed Problems for the Heat-Conduction Equation”, Differ. Equ., 21:2 (1985), 181–187
• Number of views: This page: 179 Full text: 74 References: 24 First page: 1 | 2020-10-22 15:12:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20414450764656067, "perplexity": 8385.60503487102}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107879673.14/warc/CC-MAIN-20201022141106-20201022171106-00395.warc.gz"} |
https://younesse.net/Robot-motion-planning/Lecture3/ | # Differential Geometry
Manifold:
each point has a neighborhood homeomorphic to $ℝ^n$
Ex: union of lines $y = nx$: at each point $≠ (0,0)$, neighborhood homeomorphic to $ℝ$ ⟶ but not $(0, 0)$
Tangent space in $p$ (denoted by $T_p$):
linear space spanned by derivatives of $p_i$
Vector field:
$X: p ⟼ X_p ∈ T_p$
Distribution:
linear subspace of vector fields.
There is exactly one trajectory $γ$ going through a point $p$ and following a given vector field $X$ s.t.
$γ(0) = p \\ \dot{γ}(t) = X_{γ(t)}$
Notation: $\exp(X) ≝ γ(1)$
${\rm e}^{aX} \cdot p$:
starting from $p$, applying the vector field $aX$
# Lie Brackets of Vector Fields
$[X, Y] ≝ δXY - δYX$ $[X, Y] = -[Y, X]$
Jacobi identity:
$[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$
$i$-th coordinate:
$[X, Y]_i = \sum\limits_{ j=1 }^n X_j\frac{δY_i}{δx_j} - Y_j \frac{δX_i}{δx_j}$
$\left[\begin{pmatrix} \cos θ \\ \sin θ \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\right] = \begin{pmatrix} \sin θ \\ - \cos θ\\ 0 \end{pmatrix}$
Non holonomy degree: 2 (one spans all the space with this Lie bracket).
## Campbell-Baker-Hausdorff-Dynkin Formula
${\rm e}^{tX} \cdot {\rm e}^{tY} = {\rm e}^{tX + tY - \frac 1 2 t^2 [X, Y] + t^2 ε(t)}$
where $ε(t)$ is a formal series whose ceofficients are in the free Lee algebra $LA([X, Y])$.
# Holonomic Distribution : Distribution integrable to a non-trivial manifold
Froebenius theorem:
$Δ_p$ is holonomic iff $rank(LA(Δ)_p) = rank(Δ_p)$
Maximally non-holonomic distribution:
distribution whihch does not reduce the dimension of the reachable space from $p$
# Multibody car system
Multibody $(B_0, ⋯, B_n)$
• $B_0$: car
• $B_i$ ($i ≥ 1$): trailers
Placement of a body $B_i$:
$(x_i, y_i, θ_i)$
Distribution of the placement of all the bodies: $3(n+1)$
The “convoy” is defined by:
$x_i - x_{i-1} = - \cos θ_i\\ y_i - y_{i-1} = - \sin θ_i$
Each body is moving
$\dot{z}_i \cos θ_i - \dot{y}_i \sin θ_i = 0$
Ex:
$\left[\begin{pmatrix} \cos θ \\ \sin θ \\ 0 \\ - \sin φ \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \\ 1 \end{pmatrix}\right] = \begin{pmatrix} \sin θ \\ - \cos θ\\ 0 \\ \cos φ \end{pmatrix}$ $\left[\begin{pmatrix} \cos θ \\ \sin θ \\ 0 \\ - \sin φ \end{pmatrix}, \begin{pmatrix} \sin θ \\ - \cos θ\\ 0 \\ \cos φ \end{pmatrix}\right] = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}$
Non holonomy degree: 3 (one spans all the space with these two Lie brackets).
# A Controllability Algorithm
Let $Δ$ be an nonholonomic distribution on a $n$-dimensional manifold and the filtration:
$Δ_t = Δ_{t-1} + \sum\limits_{ j+k=t } [Δ_j, Δ_k]$ $Δ_0 ⊆ Δ_1 ⊆ ⋯$ $LA(Δ) = \bigcup_{n ∈ ℕ} Δ_n$
## Controllability theorem
The associated system is controllable at $c$ iff
$∃ p_c; Δ_{p_c - 1} ≠ Δ_{p_c} = Δ_{p_c + 1} = ⋯ \text{ and } rank(Δ_{p_c}(c))=n$
Philipp Hall family
# Open problems
Canonical curves:
they keep the convoy angle (the relative angle between the trailer and the car) constant
⟶ Convex combinations between such curves, to reach the goal point
Flat systems:
a system such that there exists a subset of variables such that with these variables and their derivatives, on can determine all the other variables
The one third power law:
the velocity changes as a 1/3 power of the curvature
Tags:
Updated: | 2021-05-13 21:26:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.857938289642334, "perplexity": 3219.2462447416556}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243992514.37/warc/CC-MAIN-20210513204127-20210513234127-00475.warc.gz"} |
https://plainmath.net/28434/consider-the-following-sets-of-rational-functions-equal-frac-for-equal | # Consider the following sets of rational functions. af(x)=\frac{a}{x} for a={−2,−1,−0.5,0.5,2,4}
Consider the following sets of rational functions. $af\left(x\right)=\frac{a}{x}$ for a={−2,−1,−0.5,0.5,2,4}, $h\left(x-c\right)=\frac{1}{x-c}$ for c=[−4,−2,−0.5,0.5,2,4], h(x−c)=1x−c for c=[−4,−2,−0.5,0.5,2,4], g(bx)=1bx for b={−2,−1,−0.5,0.5,2,4} for b={−2,−1,−0.5,0.5,2,4}, $k\left(x\right)+d=\frac{1}{x}+d$ for d={−4,−2,−0.5,0.5,2,4} for d={−4,−2,−0.5,0.5,2,4}.
c. Choose two functions from any set. Find the slope between consecutive points on the graphs.
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For the first set slopes are 0.5 and 1, for the second set slopes are -0.5 and 0.5 for the third set slopes are $\frac{1}{3}$ and -\frac{1}{3} and for the fourth set slopes are 1 and -1. | 2022-05-29 11:50:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 20, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7146077156066895, "perplexity": 1945.1238798650384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662644142.66/warc/CC-MAIN-20220529103854-20220529133854-00531.warc.gz"} |
http://server4.wikisky.org/starview?object=NGC+151 | WIKISKY.ORG
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However, the absence of this correlation may becaused by dust extinction. We discuss this result by considering apicture in which bars are a relatively fast, recurrent phenomenon. Theseresults are not compatible with a pure classical monolithic scenario forbulge and disk formation. On the contrary, they favor a scenario inwhich both these components are evolving in a correlated process inwhich stellar bars play a crucial role. Based partly on observationsmade at the Pico dos Dias Observatory (PDO/LNA-CNPq), Brazil. Statistical study of M 51-type galaxiesWe present a statistical analysis of a new sample of M 51-type galaxies.Using the MCG and VV catalogues, we selected 32 such binary systems. Wefound that a typical M 51-type pair consists of a bright L*spiral galaxy and a satellite with blue luminosity 1/30-1/3 of theprimary one. The main galaxies in such pairs are often barred and havetwo well-defined spiral arms. M 51-type systems show an enhanced starformation rate (from FIR luminosities). We found a weak dependence ofthe star formation rate of the system on relative luminosity of thecompanion. M 51-type galaxies are relatively frequent: about 1/12 of allpairs are of M 51-type. A new model for the infrared brightness of the GalaxyWe present a model that reproduces the near-infrared brightnessdistribution of the Galaxy, and we compare its predictions with theresults of the Spacelab observations obtained by Kent et al. and theCOBE DIRBE experiment. We examine characteristics of nearby spiralgalaxies as a guide for a consistent description of the bulge, the barand spiral arms. A Monte Carlo method is used to generate a 3D model ofeach component of the Galaxy; the density flux contribution of thepseudo-stars created in this way is then added in a longitude versuslatitude grid to produce contour maps and brightness profiles. Weestimate the mass of the components, based on a calibration of the fluxdensity per unit mass for the characteristic stellar population of eachcomponent. We find that the brightness of the disc is better reproducedby the Freeman radial density profile, which presents a central hole,than by a classical disc with exponential profile extending to thecentre. We show that the rotation curve obtained from the massdistribution of the model is consistent with the observed one. Photometric Properties of Bars in GalaxiesWe have used surface photometry data for 100 barred galaxies todetermine the UBVRIJHK surface brightnesses and color indices for thebars. Two peaks are observed in the distribution of the average bar Bbrightnesses: at 21.0m2 and 22.2m2, characteristic of late-andearly-type galaxies, respectively. The average surface-brightnessdifference between the bar and the galaxy (within the 25.0m2 isophote)increases from 1.1m2 for SB0 galaxies to 2.3m2 for SBc-IBm galaxies. In(U-B)0-(B-V)0, (B-V 0-(V-R 0, and (B-V)0-(V-I)0 two-color diagrams, forall morphological types, the bars are shifted leftward from normal colorsequence for galaxies. This deviation is more pronounced for the outerthan for the inner regions of the bars. Using evolutionary models, weshow that this deviation is due to the scarcity of intermediate-age [(19)×109 yrs] stars in bars. Possible origins for this anomalouscomposition of the stellar population are discussed. Nearby Optical Galaxies: Selection of the Sample and Identification of GroupsIn this paper we describe the Nearby Optical Galaxy (NOG) sample, whichis a complete, distance-limited (cz<=6000 km s-1) andmagnitude-limited (B<=14) sample of ~7000 optical galaxies. Thesample covers 2/3 (8.27 sr) of the sky (|b|>20deg) andappears to have a good completeness in redshift (97%). We select thesample on the basis of homogenized corrected total blue magnitudes inorder to minimize systematic effects in galaxy sampling. We identify thegroups in this sample by means of both the hierarchical and thepercolation friends-of-friends'' methods. The resulting catalogs ofloose groups appear to be similar and are among the largest catalogs ofgroups currently available. Most of the NOG galaxies (~60%) are found tobe members of galaxy pairs (~580 pairs for a total of ~15% of objects)or groups with at least three members (~500 groups for a total of ~45%of objects). About 40% of galaxies are left ungrouped (field galaxies).We illustrate the main features of the NOG galaxy distribution. Comparedto previous optical and IRAS galaxy samples, the NOG provides a densersampling of the galaxy distribution in the nearby universe. Given itslarge sky coverage, the identification of groups, and its high-densitysampling, the NOG is suited to the analysis of the galaxy density fieldof the nearby universe, especially on small scales. Near-infrared photometry of isolated spirals with and without an AGN --- II. Photometric properties of the host galaxiesWe present here the analysis of morphological and photometric propertiesof a sample of isolated spirals with (18) and without (11) an activenucleus, based on near-infrared imaging in the J and K' bands (Paper I).The aim of that comparative analysis is to find the differentialproperties that could be directly connected with the phenomenon ofnuclear activity. We stress the importance of using isolated objects forthat purpose. Our study shows that both sets of galaxies are similar intheir global properties: they define the same Kormendy relation, theirdisk components share the same properties, the bulge and disk scalelengths are correlated in a similar way, bar strengths and lengths aresimilar for primary bars. Our results therefore indicate that hosts ofisolated Seyfert galaxies have bulge and disk properties comparable tothose of isolated non active spirals. Central colors (the innermost 200pc) of active galaxies are redder than the centers of non activespirals, most probably due to AGN light being re-emitted by the hot dustand/or due to circumnuclear star formation, through the contribution ofgiants/supergiants. Central to our analysis is the study of the possibleconnection between bars and similar non axisymmetric structures with thenuclear fuelling. We note that only one of the Seyfert galaxies in oursample, namely ESO 139-12, does not present a primary bar. But bars areequally present in active and control objects. The same applies tosecondary bars. Not all the active galaxies we have observed have them,and some control galaxies also present such central structures.Secondary central elongations (associated with secondary bars, lenses,rings or disks) may be somewhat different, but this result should beconfirmed with larger samples. We note that numerical models indicatethat such secondary bars are not strictly necessary to feed the centralengine when a primary bar is present. Our results show that down toscales of 100-300 pc, there are no evident differences between activeand non active spiral galaxies. Based on data obtained at: the EuropeanSouthern Observatory, La Silla, Chile, the Télescope BernardLyot, Calar Alto Observatory, Las Campanas Observatory. Also based onobservations made with the NASA/ESA Hubble Space Telescope, obtainedfrom the data archive at the Space --- II. Photometric properties of thehost galaxies The NICMOS Snapshot Survey of Nearby GalaxiesWe present snapshot'' observations with the Near-Infrared Camera andMulti-Object Spectrometer (NICMOS) on board the Hubble Space Telescope(HST) of 94 nearby galaxies from the Revised Shapley Ames Catalog.Images with 0.2" resolution were obtained in two filters, a broadbandcontinuum filter (F160W, roughly equivalent to the H band) and anarrowband filter centered on the Paα line (F187N or F190N,depending on the galaxy redshift) with the 51^''x51^'' field of view ofthe NICMOS camera 3. A first-order continuum subtraction is performed,and the resulting line maps and integrated Paα line fluxes arepresented. A statistical analysis indicates that the average Paαsurface brightness in the central regions is highest in early-type(Sa-Sb) spirals. The I-Band Tully-Fisher Relation for SC Galaxies: 21 Centimeter H I Line DataA compilation of 21 cm line spectral parameters specifically designedfor application of the Tully-Fisher (TF) distance method is presentedfor 1201 spiral galaxies, primarily field Sc galaxies, for which opticalI-band photometric imaging is also available. New H I line spectra havebeen obtained for 881 galaxies. For an additional 320 galaxies, spectraavailable in a digital archive have been reexamined to allow applicationof a single algorithm for the derivation of the TF velocity widthparameter. A velocity width algorithm is used that provides a robustmeasurement of rotational velocity and permits an estimate of the erroron that width taking into account the effects of instrumental broadeningand signal-to-noise. The digital data are used to establish regressionrelations between measurements of velocity widths using other commonprescriptions so that comparable widths can be derived throughconversion of values published in the literature. The uniform H I linewidths presented here provide the rotational velocity measurement to beused in deriving peculiar velocities via the TF method. The I-Band Tully-Fisher Relation for SC Galaxies: Optical Imaging DataProperties derived from the analysis of photometric I-band imagingobservations are presented for 1727 inclined spiral galaxies, mostly oftypes Sbc and Sc. The reduction, parameter extraction, and errorestimation procedures are discussed in detail. The asymptotic behaviorof the magnitude curve of growth and the radial variation in ellipticityand position angle are used in combination with the linearity of thesurface brightness falloff to fit the disk portion of the profile. TotalI-band magnitudes are calculated by extrapolating the detected surfacebrightness profile to a radius of eight disk scale lengths. Errors inthe magnitudes, typically ~0.04 mag, are dominated by uncertainties inthe sky subtraction and disk-fitting procedures. Comparison is made withthe similar imaging database of Mathewson, Ford, & Buchhorn, both aspresented originally by those authors and after reanalyzing theirdigital reduction files using identical disk-fitting procedures. Directcomparison is made of profile details for 292 galaxies observed incommon. Although some differences occur, good agreement is found,proving that the two data sets can be used in combination with onlyminor accommodation of those differences. The compilation of opticalproperties presented here is optimized for use in applications of theTully-Fisher relation as a secondary distance indicator in studies ofthe local peculiar velocity field. Near-infrared photometry of isolated spirals with and without an AGN. I. The dataWe present infrared imaging data in the J and K' bands obtained for 18active spiral galaxies, together with 11 non active galaxies taken as acontrol sample. All of them were chosen to satisfy well definedisolation criteria so that the observed properties are not related togravitational interaction. For each object we give: the image in the K'band, the sharp-divided image (obtained by dividing the observed imageby a filtered one), the difference image (obtained by subtracting amodel to the observed one), the color J-K' image, the ellipticity andposition angle profiles, the surface brightness profiles in J and K',their fits by bulge+disk models and the color gradient. We have foundthat four (one) active (control) galaxies previously classified asnon-barred turn out to have bars when observed in the near-infrared. Oneof these four galaxies (UGC 1395) also harbours a secondary bar. For 15(9 active, 6 control) out of 24 (14 active, 10 control) of the opticallyclassified barred galaxies (SB or SX) we find that a secondary bar (or adisk, a lense or an elongated ring) is present. The work presented hereis part of a large program (DEGAS) aimed at finding out whether thereare differences between active and non active galaxies in the propertiesof their central regions that could be connected with the onset ofnuclear activity. Based on data obtained at: the European SouthernObservatory, La Silla, Chile, the Télescope Bernard Lyot, CalarAlto Observatory, Las Campanas Observatory. Also based on observationsmade with the NASA/ESA Hubble Space Telescope, obtained from the dataarchive at the Space Telescope Institute. Figures 1-35 are onlyavailable in electronic form at the http://www.edpsciences.org A strong correlation between bar strength and global star forming activity in isolated barred galaxiesI have studied the relation between the global star formation activityand the bar structure in a sample of isolated barred galaxies. The starformation activity was quantified via the ratio between the IRAS fluxesat 25 mu m and 100 mu m. Two parameters were chosen to define the barstructure: the strength of the bar and the relative projected barlength. The strength of the bar was defined by epsilon_ {b}=10(1-b/a),where a and b are the projected semi-major and semi-minor bar axis. Therelative bar length was defined as: 2Lb/D25, whereL_ {b} is one half of the projected total bar length and D25is the diameter of the 25 mag arcsec-2 magnitude isophote inthe B band. We found a strong correlation between the star formationactivity and epsilon_ {b}. The regression line is given bylog(I25/I100)=-1.81+0.093 epsilon_ {b}, with acorrelation coefficient of 0.9. The link is not so evident between therelative projected bar length and the star formation activity. But, itis noted that there is enhanced star formation activity in galaxies withstrong bars and small relative bar lengths,0.1<2Lb/D25<0.22. Bulge-Disk Decomposition of 659 Spiral and Lenticular Galaxy Brightness ProfilesWe present one of the largest homogeneous sets of spiral and lenticulargalaxy brightness profile decompositions completed to date. The 659galaxies in our sample have been fitted with a de Vaucouleurs law forthe bulge component and an inner-truncated exponential for the diskcomponent. Of the 659 galaxies in the sample, 620 were successfullyfitted with the chosen fitting functions. The fits are generally welldefined, with more than 90% having rms deviations from the observedprofile of less than 0.35 mag. We find no correlations of fittingquality, as measured by these rms residuals, with either morphologicaltype or inclination. Similarly, the estimated errors of the fittedcoefficients show no significant trends with type or inclination. Thesedecompositions form a useful basis for the study of the lightdistributions of spiral and lenticular galaxies. The object base issufficiently large that well-defined samples of galaxies can be selectedfrom it. The Southern Sky Redshift SurveyWe report redshifts, magnitudes, and morphological classifications for5369 galaxies with m_B <= 15.5 and for 57 galaxies fainter than thislimit, in two regions covering a total of 1.70 sr in the southerncelestial hemisphere. The galaxy catalog is drawn primarily from thelist of nonstellar objects identified in the Hubble Space TelescopeGuide Star Catalog (GSC). The galaxies have positions accurate to ~1"and magnitudes with an rms scatter of ~0.3 mag. We compute magnitudes(m_SSRS2) from the relation between instrumental GSC magnitudes and thephotometry by Lauberts & Valentijn. From a comparison with CCDphotometry, we find that our system is homogeneous across the sky andcorresponds to magnitudes measured at the isophotal level ~26 magarcsec^-2. The precision of the radial velocities is ~40 km s^-1, andthe redshift survey is more than 99% complete to the m_SSRS2 = 15.5 maglimit. This sample is in the direction opposite that of the CfA2; incombination the two surveys provide an important database for studies ofthe properties of galaxies and their large-scale distribution in thenearby universe. Based on observations obtained at Cerro TololoInter-American Observatory, National Optical Astronomy Observatories,operated by the Association of Universities for Research in Astronomy,Inc., under cooperative agreement with the National Science Foundation;Complejo Astronomico El Leoncito, operated under agreement between theConsejo Nacional de Investigaciones Científicas de laRepública Argentina and the National Universities of La Plata,Córdoba, and San Juan; the European Southern Observatory, LaSilla, Chile, partially under the bilateral ESO-ObservatórioNacional agreement; Fred Lawrence Whipple Observatory;Laboratório Nacional de Astrofísica, Brazil; and the SouthAfrican Astronomical Observatory. Catalogue of HI maps of galaxies. I.A catalogue is presented of galaxies having large-scale observations inthe HI line. This catalogue collects from the literature the informationthat characterizes the observations in the 21-cm line and the way thatthese data were presented by means of maps, graphics and tables, forshowing the distribution and kinematics of the gas. It containsfurthermore a measure of the HI extension that is detected at the levelof the maximum sensitivity reached in the observations. This catalogueis intended as a guide for references on the HI maps published in theliterature from 1953 to 1995 and is the basis for the analysis of thedata presented in Paper II. The catalogue is only available inelectronic form at the CDS via anonymous ftp 130.79.128.5 orhttp://cdsweb.u-strasbg.fr/Abstract.html Total magnitude, radius, colour indices, colour gradients and photometric type of galaxiesWe present a catalogue of aperture photometry of galaxies, in UBVRI,assembled from three different origins: (i) an update of the catalogueof Buta et al. (1995) (ii) published photometric profiles and (iii)aperture photometry performed on CCD images. We explored different setsof growth curves to fit these data: (i) The Sersic law, (ii) The net ofgrowth curves used for the preparation of the RC3 and (iii) A linearinterpolation between the de Vaucouleurs (r(1/4) ) and exponential laws.Finally we adopted the latter solution. Fitting these growth curves, wederive (1) the total magnitude, (2) the effective radius, (3) the colourindices and (4) gradients and (5) the photometric type of 5169 galaxies.The photometric type is defined to statistically match the revisedmorphologic type and parametrizes the shape of the growth curve. It iscoded from -9, for very concentrated galaxies, to +10, for diffusegalaxies. Based in part on observations collected at the Haute-ProvenceObservatory. A catalogue of spatially resolved kinematics of galaxies: BibliographyWe present a catalogue of galaxies for which spatially resolved data ontheir internal kinematics have been published; there is no a priorirestriction regarding their morphological type. The catalogue lists thereferences to the articles where the data are published, as well as acoded description of these data: observed emission or absorption lines,velocity or velocity dispersion, radial profile or 2D field, positionangle. Tables 1, 2, and 3 are proposed in electronic form only, and areavailable from the CDS, via anonymous ftp to cdsarc.u-strasbg.fr (to130.79.128.5) or via http://cdsweb.u-strasbg.fr/Abstract.html Homogeneous Velocity-Distance Data for Peculiar Velocity Analysis. III. The Mark III Catalog of Galaxy Peculiar VelocitiesThis is the third in a series of papers in which we assemble and analyzea homogeneous catalog of peculiar velocity data. In Papers I and II, wedescribed the Tully-Fisher (TF) redshift-distance samples thatconstitute the bulk of the catalog and our methodology for obtainingmutually consistent TF calibrations for these samples. In this paper, wesupply further technical details of the treatment of the data andpresent a subset of the catalog in tabular form. The full catalog, knownas the Mark III Catalog of Galaxy Peculiar Velocities, is available inaccessible on-line databases, as described herein. The electroniccatalog incorporates not only the TF samples discussed in Papers I andII but also elliptical galaxy Dn- sigma samples originally presentedelsewhere. The relative zero pointing of the elliptical and spiral datasets is discussed here. The basic elements of the Mark III Catalog arethe observables for each object (redshift, magnitude, velocity width,etc.) and inferred distances derived from the TF or Dn- sigma relations.Distances obtained from both the forward and inverse TF relations aretabulated for the spirals. Malmquist bias--corrected distances arecomputed for each catalog object using density fields obtained from theIRAS 1.2 Jy redshift survey. Distances for both individual objects andgroups are provided. A variety of auxiliary data, including distancesand local densities predicted from the IRAS redshift surveyreconstruction method, are tabulated as well. We study the distributionsof TF residuals for three of our samples and conclude that they are wellapproximated as Gaussian. However, for the Mathewson et al. sample wedemonstrate a significant decrease in TF scatter with increasingvelocity width. We test for, but find no evidence of, a correlationbetween TF residuals and galaxy morphology. Finally, we derivetransformations that map the apparent magnitude and velocity width datafor each spiral sample onto a common system. This permits theapplication of analysis methods that assume that a unique TF relationdescribes the entire sample. Molecular Gas, Morphology, and Seyfert Galaxy ActivityWe probe the cause of the elevated star formation in host galaxies ofSeyfert 2 nuclei compared with Seyfert 1 hosts and with field galaxies.12CO (1--0) observations of a large sample of Seyfert galaxies indicateno significant difference in the total amount of molecular gas as afunction of the Seyfert nuclear type, nor are Seyfert galaxiessignificantly different in this regard from a sample of field galaxiesonce selection effects are accounted for. Therefore, the total amount ofmolecular gas is not responsible for the enhanced star-forming activityin Seyfert 2 hosts. To probe how this gas is being converted moreefficiently into stars in Seyfert 2 hosts than in the other galaxies, weinvestigate the occurrence of bars, interactions, and distortedmorphologies among Seyfert galaxies. We find a significantly higher rateof asymmetric morphologies for Seyfert 2 galaxies with respect toSeyfert 1 galaxies and field galaxies. Relative to field galaxies, theeffect is at a greater than 99.9% confidence level. The presence ofasymmetric morphologies in individual Seyfert galaxies is correlatedwith their tendency to exhibit enhanced star-forming activity. Theseresults suggest that asymmetric morphologies are an important cause forthe link between Seyfert type and star-forming activity: bars anddistortions in Seyfert 2 hosts are likely both to enhance star-formingactivity and to funnel gas into the nuclear region, thus obscuring andpossibly contributing to the feeding of the active nucleus. Optical Rotation Curves and Linewidths for Tully-Fisher ApplicationsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1997AJ....114.2402C&db_key=AST Parameters of 2447 Southern Spiral Galaxies for Use in the Tully-Fisher RelationI-band luminosities, rotational velocities, and redshifts of 1092 spiralgalaxies have been measured by CCD photometry and Hα spectroscopyusing the 1 m and 2.3 m telescopes at Siding Spring Observatory,respectively. The results are tabulated. Luminosity profiles andHα rotation curves are given for the galaxies. When these resultsare combined with similar data for 1355 spiral galaxies publishedpreviously (Mathewson, Ford, & Buchhorn, hereafter Paper I), itprovides a large, uniform, and unique data set with which to measure,via the Tully-Fisher relation, the peculiar velocities of galaxies inthe local universe to a distance of 11,000 km s^-1^ (Mathewson &Ford). Taking advantage of the opportunity for publishing this data inmachine-readable form, in the CD-ROM, we have also included similar datafor the 1355 galaxies in Paper I. An image database. II. Catalogue between δ=-30deg and δ=70deg.A preliminary list of 68.040 galaxies was built from extraction of35.841 digitized images of the Palomar Sky Survey (Paper I). For eachgalaxy, the basic parameters are obtained: coordinates, diameter, axisratio, total magnitude, position angle. On this preliminary list, weapply severe selection rules to get a catalog of 28.000 galaxies, wellidentified and well documented. For each parameter, a comparison is madewith standard measurements. The accuracy of the raw photometricparameters is quite good despite of the simplicity of the method.Without any local correction, the standard error on the total magnitudeis about 0.5 magnitude up to a total magnitude of B_T_=17. Significantsecondary effects are detected concerning the magnitudes: distance toplate center effect and air-mass effect.
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https://zbmath.org/?q=an:07107185 | # zbMATH — the first resource for mathematics
Counterexamples to Hedetniemi’s conjecture. (English) Zbl 1451.05087
The chromatic number $$\chi(G)$$ of a graph $$G$$ is the least number of colours required to colour the vertices of $$G$$ such that adjacent vertices receive different colours. The tensor product of finite simple graphs $$G$$ and $$H$$, denoted by $$G \times H$$, is the graph with vertex set $$V(G)\times V(H)$$ such that $$(u,v)$$ and $$(x,y)$$ are adjacent if and only if $$u$$ and $$x$$ are adjacent in $$G$$ and $$v$$ and $$y$$ are adjacent in $$H$$. It is easy to see that $$\chi(G \times H) \leq \min\{\chi(G), \chi(H)\}$$. S. T. Hedetniemi [Homomorphisms of graphs and automata. Techn. Rep. 0310544-T, Ann Arbor, MI: University of Michigan (1966)] conjectured that equality holds for all $$G$$ and $$H$$. This important conjecture has attracted much attention in more than five decades, and it has been confirmed in many special cases. In this paper, the author disproves this conjecture by showing the existence of infinitely many counterexamples.
The strong product $$G \boxtimes H$$ is the graph with vertex set $$V(G)\times V(H)$$ such that $$(u,v)$$ and $$(x,y)$$ are adjacent if and only if one of the following holds: $$u=x$$ and $$v$$ and $$y$$ are adjacent in $$H$$; $$v=y$$ and $$u$$ and $$x$$ are adjacent in $$G$$; $$u$$ and $$x$$ are adjacent in $$G$$ and $$v$$ and $$y$$ are adjacent in $$H$$. Given a finite graph $$\Gamma$$ (possibly with loops) and a positive integer $$c$$, the exponential graph $$\mathcal E_{c}(\Gamma)$$ is the graph with all mappings $$V(\Gamma) \rightarrow \{1, 2, \ldots, c\}$$ as vertices such that two distinct mappings $$\phi$$, $$\psi$$ are adjacent if and only if the condition $$\phi(u) \neq \psi(v)$$ holds whenever there is an edge of $$\Gamma$$ between $$u$$ and $$v$$. The author proved that, for $$c = \lceil 3.1q \rceil$$ with $$q$$ a sufficiently large integer, and for any graph $$G$$ with girth at least $$6$$ and fractional chromatic number strictly greater than $$3.1$$ (the existence of $$G$$ is guaranteed by a classic result of P. Erdős [Can. J. Math. 11, 34–38 (1959; Zbl 0084.39602)]), the graph $$(G \boxtimes K_{q}) \times\mathcal E_{c}(G \boxtimes K_{q})$$ is a counterexample to Hedetniemi’s conjecture, where $$K_{q}$$ is the complete graph of order $$q$$.
##### MSC:
05C15 Coloring of graphs and hypergraphs 05C76 Graph operations (line graphs, products, etc.)
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##### References:
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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2021-11-28 03:01:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6645660400390625, "perplexity": 1077.536260982445}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358443.87/warc/CC-MAIN-20211128013650-20211128043650-00600.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/reduce-equation-3-x-y-2-0-intercept-form-find-intercept-axes-general-equation-of-a-line_58834 | Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
# Reduce the equation √ 3 x + y + 2 = 0 to intercept form and find intercept on the axes . - Mathematics
Reduce the equation$\sqrt{3}$ x + y + 2 = 0 to intercept form and find intercept on the axes.
#### Solution
$\sqrt{3}$ x + y + 2 = 0
$\Rightarrow \sqrt{3}x + y = - 2$
$\Rightarrow \frac{\sqrt{3}x}{- 2} + \frac{y}{- 2} = 1 \left[ \text { Dividing both sides by } - 2 \right]$
$\Rightarrow \frac{x}{- \frac{2}{\sqrt{3}}} + \frac{y}{- 2} = 1$
This is the intercept form of the given line.
Here, x-intercept = $- \frac{2}{\sqrt{3}}$ and y-intercept = $-$2
Is there an error in this question or solution?
#### APPEARS IN
RD Sharma Class 11 Mathematics Textbook
Chapter 23 The straight lines
Exercise 23.9 | Q 1.2 | Page 72 | 2021-04-18 20:36:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.31277623772621155, "perplexity": 2112.672373747367}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038860318.63/warc/CC-MAIN-20210418194009-20210418224009-00312.warc.gz"} |
https://demo7.dspace.org/items/a5a37d71-df2a-4fbd-a12d-a1e9bd54c507 | ## Vacuum charge fractionlization re-examined
Nogami, Y.
##### Description
We consider a model of a quantized fermion field that is based on the Dirac equation in one dimensional space and re-examine how the fermion number of the vacuum, or the vacuum charge, varies when an external potential is switched on. With this model, fractionization of the vacuum charge has been illustrated in the literature by showing that the external potential can change the vacuum charge from zero to a fractional number. Charge conservation then appears violated in this process. This is because the charge that has been examined in this context is only a part of the total charge of the vacuum. The total charge is conserved. It is not fractionalized unless the Dirac equation has a zero mode. Two other confusing aspects are discussed. One is concerned with the usage of the continuum limit and the other with the regularization of the current operator. Implications of these aspects of the vacuum problem are explored.
Comment: 18 pages
Quantum Physics | 2022-12-07 22:41:35 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8156803846359253, "perplexity": 344.84968881200723}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711221.94/warc/CC-MAIN-20221207221727-20221208011727-00216.warc.gz"} |
https://goldbook.iupac.org/terms/view/C01051 | ## Wikipedia - Chemoselectivity (en) Wikipedia - Chemoselektivita (cs) Wikipedia - Chemoselettività (it) Wikipedia - Quimioselectividad (es) Wikipedia - Quimioselectivitat (ca) Wikipedia - Selektiivisyys (fi) Wikipedia - Selettività (chimica) (it) Wikipedia - شیمی گزینی (fa) Wikipedia - گزینش شیمیایی (fa) Wikipedia - 化学選択性 (ja) chemoselectivity (chemoselective)
https://doi.org/10.1351/goldbook.C01051
Chemoselectivity is the preferential reaction of a chemical @[email protected] with one of two or more different functional groups. A @[email protected] has a high chemoselectivity if reaction occurs with only a limited number of different functional groups. For example, sodium tetrahydroborate is a more chemoselective reducing agent than is lithium tetrahydroaluminate. The concept has not been defined in more quantitative terms. The term is also applied to reacting molecules or intermediates which exhibit @[email protected] towards chemically different reagents. Some authors use the term @[email protected] for 100% chemoselectivity. However, this usage is discouraged. | 2023-03-23 01:21:35 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9464277029037476, "perplexity": 10464.327029883569}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296944606.5/warc/CC-MAIN-20230323003026-20230323033026-00098.warc.gz"} |
http://mathhelpforum.com/calculus/115725-definite-integral-problem-print.html | Definite integral problem
• November 20th 2009, 03:10 AM
CleanSanchez
Definite integral problem
So I have the integral $\int - \frac{3y^2}{2}$ over 0 to 4.
Every computer software evaluates this as -32 which is the correct answer but I keep getting -24... what am I doing wrong? Any help is appreciated!
• November 20th 2009, 03:19 AM
Defunkt
Show some work so we can tell you where you went wrong and what you need to do :)
• November 20th 2009, 03:30 AM
CleanSanchez
Original integral $\int - \frac{3y^2}{2}$ over 0 to 4.
Well I did $-\frac {(3)(4^2)}{2} = -\frac {(3)(16)}{2} = -\frac {48}{2} = -24$
and 0 makes the whole second fraction 0 so we can ignore it... but apparently I'm wrong (Crying)
• November 20th 2009, 03:34 AM
Defunkt
Quote:
Originally Posted by CleanSanchez
Original integral $\int - \frac{3y^2}{2}$ over 0 to 4.
Well I did $-\frac {(3)(4^2)}{2} = -\frac {(3)(16)}{2} = -\frac {48}{2} = -24$
and 0 makes the whole second fraction 0 so we can ignore it... but apparently I'm wrong (Crying)
Did you not forget to integrate? :P
• November 20th 2009, 03:42 AM
CleanSanchez
Ah yes, I was getting ahead of myself and thats not where my mistake was. It's part of a larger double integral problem and the bounds of x are 0 - 2 not 0 -4. Reversing double integrals can get confusing! | 2016-06-25 06:37:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9306436777114868, "perplexity": 860.3312551830365}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783392099.27/warc/CC-MAIN-20160624154952-00120-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://direct.mit.edu/netn/article/3/4/994/95798/The-hidden-repertoire-of-brain-dynamics-and | The purpose of this paper is to describe a framework for the understanding of rules that govern how neural system dynamics are coordinated to produce behavior. The framework, structured flows on manifolds (SFM), posits that neural processes are flows depicting system interactions that occur on relatively low-dimension manifolds, which constrain possible functional configurations. Although this is a general framework, we focus on the application to brain disorders. We first explain the Epileptor, a phenomenological computational model showing fast and slow dynamics, but also a hidden repertoire whose expression is similar to refractory status epilepticus. We suggest that epilepsy represents an innate brain state whose potential may be realized only under certain circumstances. Conversely, deficits from damage or disease processes, such as stroke or dementia, may reflect both the disease process per se and the adaptation of the brain. SFM uniquely captures both scenarios. Finally, we link neuromodulation effects and switches in functional network configurations to fast and slow dynamics that coordinate the expression of SFM in the context of cognition. The tools to measure and model SFM already exist, giving researchers access to the dynamics of neural processes that support the concomitant dynamics of the cognitive and behavioral processes.
Both of us like to run, partly for fitness and partly for mental health. It’s easy and you can do it almost anytime and anywhere. The thing about running is that the rules on how to do it are fairly simple, but how you do it is quite varied. Running in the heat of the summer on a beach is different from running up a hill in the forest or trying to navigate an icy trail in the winter. The point here is that while the rules for running are always the same, you would not assume that the example of running on beach serves as an accurate characterization of all running that we might do. The analogy is meant to suggest this is the approach that we use when trying to link brain and behavior. The coordination of behavior by the brain can be understood as a reflection of general rules whose specific realization depends on the current context and initial conditions.
Stated more boldly, we often assume that the expression of behavior at a point in time is sufficient to understand how that behavior is coordinated. Experimental approaches focus on the characterization of brain signal time series and how they change with manipulation. Theoretical approaches most often focus on defining functions that generate these time series. Such approaches are valid insofar as they are able to characterize the local conditions that generate the time series. If the nervous system of study can only generate that time series, then this approach will be successful.
However, a different scenario emerges when we consider that a given realization is but one of many that the brain can generate. The brain is a complex adaptive system, showing the properties of multiscale behavior, emergence, and nonlinearity (Fingelkurts, 2004; Mitchell, 2009). If we acknowledge this, then a single realization captures only a partial picture of what is possible. Changes to the initial conditions for generation of the behavior can change the realization to the point where the time series bears little resemblance to other realizations. This would be construed as “noise” in most perspectives, but the case we wish to make here is that such variations can be considered as valid expressions of the rules under which behavior is coordinated.
This perspective can be more saliently appreciated when we consider clinical conditions and the variation in expression across persons. For instance, in the case of focal damage from stroke, two persons can show similar regional damage, yet show quite different clinical outcomes (Price & Friston, 2002). Person A may be very impaired, whereas Person B shows remarkable recovery. Person B, in our framework, is less debilitated because they have more options to realize a particular behavior than Person A. The rules that govern behavior are effectively the same for both persons, but the variation in expression is greater in Person B. The stroke impairs one particular set of realizations (i.e., a specific trajectory) abolishing the behavior in Person A, but for Person B only slightly alters the execution. The differences are often explained as resilience or brain reserve, which merely relabels the outcome rather than providing a mechanism of explanation. We propose that these mechanisms can be captured in the SFM framework (Pillai & Jirsa, 2017).
We present a framework wherein complex brain dynamics can be decomposed into probabilistic functional modes. These modes are mathematically operationalized as manifolds, along which trajectories evolve as the dynamics unfold embedded in a low-dimensional space or SFM (Huys, Perdikis, & Jirsa, 2014). The collection of functional modes available in a neural network constitutes its functional repertoire, which together instantiates a complete set of potential cognitive functions and overt behaviors.
It has been acknowledged by a number of neuroscience researchers that the brain is dynamic, but how that translates to their approach to gain understanding varies widely. At one end, some consider the brain to be a simple input-output system where a signal comes in, a cascade is triggered as the signal propagates, and the system produces an output appropriate to the input (Petersen & Fiez, 1993; Posner, Petersen, Fox, & Raichle, 1988). Other perspectives, stemming from the focus on intrinsic activity in the brain, go from a unidirectional input-output system to one where the input signal itself may be modified (Deco, Jirsa, & McIntosh, 2013; Fox et al., 2005; Raichle, 2010). One expression, which falls under general category of predictive coding, focuses on the time series of neural signals as manifestations of internal models that the brain generates to predict its inputs and its ultimate consequences (Friston, 2010; Rao & Ballard, 1999). There is another elaboration of this that reflects our SFM framework, which we will cover shortly.
The assumption underlying much predictive coding work is that the expression of behavior at a point in time is sufficient to understand how that behavior is coordinated. Other theoretical approaches focus on defining mathematical functions for behavioral time series, while empirical studies use machine-learning algorithms to classify the time series according to the behavior they are thought to support (e.g., perceptual categorization).
There are two challenges here. First, if we were to reverse engineer a system that produces the observed time series that reflects the behavior of interest, we would not learn how the behavior itself was coordinated. Rather we would only know what generates individual time series (e.g., the action of a specific set of brain areas). Second, and more problematic, is that the model would not be able to generate new behaviors that we had not previously measured. Said differently, we may be able to predict what the system has done, but cannot predict what it will do. One remedy is to update the model in light of the new behavior and to build a lookup table that relates the configuration of neural dynamics to a specific behavior. The process continues until at some point we have cataloged all the behaviors of the system. Although this sounds cumbersome, you see it played out in modern neuroscience. In neuroimaging, for example, we started with the characterization of activated brain regions and relating that to specific behavioral functions (vision, audition, language, memory), drawing inferences on the unobservable processes that were needed to instantiate such functions. We are now in the era of brain networks, where the coherent interactions between regions are the substrate for function (default network, salience network, dorsal attention network). A great deal of research now emphasizes the system characteristics that support these networks by looking at graph theory metrics (Bullmore & Sporns, 2009; Rubinov & Sporns, 2010) and by characterizing a feature of the dynamics, such as scale-free behavior and criticality (Beggs & Plenz, 2003; Haimovici, Tagliazucchi, Balenzuela, & Chialvo, 2013; Petermann et al., 2009; Tagliazucchi, Balenzuela, Fraiman, & Chialvo, 2012). If we pause and examine these observations, we have indeed done a good job of characterizing what the system does, but have no idea how and why.
The SFM framework takes a different approach, which still assumes that the brain constructs models of the world (as in predictive coding), but takes the focus from the specific instantiation of that model (aka the individual trajectory) to discovering the rules that the brain uses to develop these models. This is a subtle, but critical, difference. For SFM, the emphasis is specifically on these changes, where a given class is considered only one of the potential sets of behaviors that can be realized. Changing the configuration to encapsulate a new set is explicit in the SFM framework. The SFM framework emphasizes the architecture necessary to build those realizations, but also others that may not have been observed previously, but are a consequence of the architecture. Examples may be symmetry constraints imposed upon an architecture, allowing explicitly for symmetric and antisymmetric solutions, even though only symmetric solutions had been observed previously. Another illustration that makes the distinction between the emphasis on a specific realization versus a model for the rules that generate the realization comes from an example of calculating 3 times 4, 3 times 5, and then switch to 13 times 14. The majority of people will rapidly access their semantic memory for the first two cases, but evoke a different model to compute algorithmically the last. If the result of the computation is not in memory, then no solution can be found, whereas in the algorithmic case solutions for number computations may be found that have never been computed before. The most innate and pertinent characteristic of the brain is its capacity to generate dynamic models.
As we describe the option that encapsulates SFM, it is useful to borrow an analogy from J. H. Holland on the game of chess to illustrate the difference (Holland, 2014). One can learn chess by watching a game and tracking the movements of each piece, repeating the observation for subsequent games and then build a catalog of moves and counter moves. This is a formidable challenge given that, by rough calculations, there are at least 1050 possible legal move sequences, which is larger than the estimated number of atoms in the universe. The more efficient approach is to define the rules that determine the legal moves. By doing this for chess, we dramatically reduce the problem from an essentially infinite space to one where a dozen or so rules capture all possible realizations of the chess game. Mastery of chess is achieved when individual moves are combined and orchestrated into larger motifs, further classified into aggressive, defensive and strategic patterns. We understand chess by understanding the rules of play, and understand it deeply by using these rules to build coordination motifs. And this is the option that motivates the description of SFMs: the goal for understanding brain and behavior is to determine the rules that govern the coordination of behavior.
The SFM framework lies firmly in the ideas of complex adaptive systems. Our exposition will thus borrow heavily from analogies of other, nonneural, systems that illustrate key principles to build our case, such as emergence, nonlinearity, motifs, flows, and internal models.
The notion of SFM formalizes some key general properties of complex adaptive systems. The use of the term flows in SFM emphasizes the dynamic nature of brain processes, where the flow formalizes the rules that enact the internal model of the system. The nonlinearities of the system impart other properties, such as aggregation and emergence that link the actions at one level of the system (e.g., network dynamics) to actions at another (e.g., behavior). The elements (or more often called “agents”) can operate at different timescales, and the interactions between scales are a critical feature in controlling the flow of the system. Fast timescales may have no overt consequence until slower moving scales reach a certain tipping point, or bifurcation, and the flow of the entire system changes.
SFM approaches emphasize the manifolds that can be understood as force fields generating the ensemble of all possible trajectories (or flows), and are thus a mathematical expression of the rules underlying the generation of behavior. Figure 1 demonstrates the general idea of flows on manifolds with one toy example of a spherical manifold having two attractor states or domains that support different flows. Depending on the initial condition of a given trajectory, a flow evolves rapidly to the manifold and then continues on the manifold at a slower timescale. The figure also demonstrates a comparable manifold architecture in simulated resting-state functional MRI data, where changes in functional connectivity dynamics (FCD) switch two states that span a manifold.
Figure 1.
Structured flows on manifolds (SFMs). Upper figure shows a spherical attractive manifold, displaying various sets of initial conditions (black ovals) of trajectories (blue), evolving rapidly toward the manifold and then continue evolving on the manifold on a slower timescale. The timescale separation is evidenced by the large angle between trajectory and manifold (around 90 degrees). The flow on the manifold is split into two domains, one lower and one upper hemisphere, partitioned by a seperatrix (dotted line). The trajectories trace out lines on the manifold, following the flow (black arrows). The SFMs display a bi-stable organization with closed circular orbits on both hemispheres. The two lower figures show a similar organization, as captured by BOLD signals simulated with TheVirtualBrain (Hansen, Battaglia, Spiegler, Deco, & Jirsa, 2015; Sanz Leon et al., 2013). On the left, functional connectivity dynamics (FCD) are shown over 20 min, in which two large segments of invariant functional connectivity (FC) are identified as states alpha and beta. For both time windows, a principal component analysis was performed spanning state-characteristic subspaces by the leading principal components. When the BOLD signals were projected into the characteristic subspaces, the trajectory of the brain signal is unfolded, identifying the manifolds and trajectories of the corresponding states (figure on bottom right).
Figure 1.
Structured flows on manifolds (SFMs). Upper figure shows a spherical attractive manifold, displaying various sets of initial conditions (black ovals) of trajectories (blue), evolving rapidly toward the manifold and then continue evolving on the manifold on a slower timescale. The timescale separation is evidenced by the large angle between trajectory and manifold (around 90 degrees). The flow on the manifold is split into two domains, one lower and one upper hemisphere, partitioned by a seperatrix (dotted line). The trajectories trace out lines on the manifold, following the flow (black arrows). The SFMs display a bi-stable organization with closed circular orbits on both hemispheres. The two lower figures show a similar organization, as captured by BOLD signals simulated with TheVirtualBrain (Hansen, Battaglia, Spiegler, Deco, & Jirsa, 2015; Sanz Leon et al., 2013). On the left, functional connectivity dynamics (FCD) are shown over 20 min, in which two large segments of invariant functional connectivity (FC) are identified as states alpha and beta. For both time windows, a principal component analysis was performed spanning state-characteristic subspaces by the leading principal components. When the BOLD signals were projected into the characteristic subspaces, the trajectory of the brain signal is unfolded, identifying the manifolds and trajectories of the corresponding states (figure on bottom right).
Close modal
The link of SFM to flows and emergence can be conceptualized by considering a piece of music. The analogy of the “brain as a symphony” has been made often, and is used to illustrate the fact that the emergence of function comes not from the action of a single brain area, but rather the coordination among all elements (unlike a symphony, however, in the brain there is no conductor). SFM theory provides a formal framework for these concepts of “brain as a symphony.” In a symphony, one can isolate the individual instruments to characterize their unique contribution, but it is difficult to appreciate its role in the symphony without considering the relation to other instruments. The statement “The whole is greater than the sum of its parts” is appropriate here for both the symphony and the brain.
We can further develop this analogy to build the intuition about SFM, particularly in the context of how different temporal flows (e.g., melody and piano lines in a simple song) interact in supporting the emergent behavior (the whole song). The melody and harmony often move in different timescales. Each can be comprehended on their own, but in a well-composed song, the relation between the lines brings a richness that is not present in either alone (e.g., aggregate property). This fluctuation between the melody and harmony evolves throughout the song. It is common in classical pieces for the opening melody to be repeated as a motif, but over a slightly different piano line, which may completely change the mood of the piece.
In the brain, a parallel to the symphony analogy can be drawn. As the instruments in the orchestra and musical abilities of the artists define constraints upon the symphony emerge, the anatomical connectivity and dynamic characteristic of the brain regions (network nodes) specify the rules for the evolution of dynamics. As we shall see below, this is far from a trivial constraint, as the anatomy helps define any spatial and temporal constraints for potential network configurations. For example, all things being equal, it is more likely that adjacent regions in occipital cortex will interact rather than occipital and frontal regions, simply because the occipital and frontal areas have few connections between them, and those that are connected indirectly at a long distance, imposing a longer time delay for transmission. Thus, anatomy establishes a deterministic architecture that prevents random manifolds and flows from occurring. This architecture, set atop the (nonlinear) dynamics of neurons and connected populations of neurons establishes the set of motifs that are available for the brain to combine in the coordination of behavior (Sporns & Kotter, 2004). Motifs in the brain have been identified from an anatomical and functional perspective (Mohajerani et al., 2013; Sporns & Kotter, 2004), and may also be related to the collection of so-called “resting-state” networks that arise through intrinsic activity (Damoiseaux et al., 2006; Fox et al., 2005). The SFM framework makes use of these motifs to articulate the functional possibilities as different motifs are realized. For example, the functional connectivity patterns in Figure 1 show that the constituents of the resting-state networks can recombine across time, forming different networks as the entire system moves across a trajectory. Thus, the motifs can be recombined to enact a variety of functional outcomes. We can refer to these as functional modes to emphasize that they can be both actual and potential configurations.
The asymmetries in the brain’s space-time structure, set by the structural connectivity, establish a potential for multiscale actions (Deco et al., 2013). The multiscale temporal character of these modes is founded on the fact that complex processes arise in an organism-environment context that inherently covers multiple scales. Armed with functional modes as essential building blocks, we propose additional dynamics (called operational signals) on timescales slower and faster than that of the modes. The slower process effectively binds functional modes together into sequences. More precisely, the given functional mode emerges via a competition process to temporally dominate the functional dynamics, after which it destabilizes and gives way to another mode (Haken, 2006; Perdikis, Huys, & Jirsa, 2011). The transient dynamics between modes can be triggered either by “internal” events (as in preconstructed sequences) or by “external” ones (such as perceptual events). Once engaged, the temporal attractivity of a mode guarantees functional robustness, whereas transitions between modes underlie flexibility for meaningful changes. Further variability in the function may arise via additional dynamics operating on times scales faster than (or similar to) that of the modes. Accordingly, brain function is organized in multilevel spatial and temporal hierarchies.
The hierarchical architecture is central to effective information processing, where different temporal and spatial scales interact in moving the system through behavioral repertoires. Information provided to the brain system is meaningful if and only if it qualitatively changes the “state” that the brain occupies at that moment. If an incoming signal does not change the state, then the information was not meaningful, and the incoming signal could equally have not been present. Thus, although local dynamics may change dramatically, they may not have an appreciable effect on the trajectory of the network, and thus rather than change the flow to a new part of the manifold, may only result in a trivial variation in the current trajectory. If these local dynamics intersect with larger scale dynamics at a critical point, this can establish a new trajectory for the system, either within an existing SFM or moving to a new SFM and hence a new emergent behavior. The heterogeneity of such effects has been explored using TheVirtualBrain, where Spiegler et al. (Spiegler, Hansen, Bernard, McIntosh, & Jirsa, 2016) demonstrated that the dissipation from focal stimulation across cortical and thalamic site was not uniform, where some stimulation effects did not propagate beyond the local areas while others engaged broad networks that could be related to resting-state networks. Complementary work from Deco et al. (2013) has shown that during spontaneous activity, certain nodes can act to “ignite” the reformation of functional networks, which would be consistent with a qualitative change in state. This is not to say, however, that signals that result in state changes are somehow more “conscious” than other. Indeed, many network operations will engender state changes, but not be accessible to consciousness, such as brainstem autonomic functions. Conversely, there may be behavioral changes that arise from unconscious processes, such as implicit learning, that by necessity would need to also involve state changes in brain dynamics. In this sense, the general notion of an SFM is essentially agnostic to the overt awareness of the behavior processes it supports, but does lead to the interesting speculation that there may indeed be a meaningful difference in the configurations of SFM that are consciously accessible versus those that are not.
Some of the mathematical details that define SFM have been described fully elsewhere (Huys et al., 2014; Pillai & Jirsa, 2017). Here we provide essential details to enable the link to our current narrative. SFMs are the mathematical objects capturing the dynamic properties required from a system capable of the behavior we have discussed thus far. The system under consideration is high-dimensional with N degrees of freedom and highly nonlinear. In order to allow for this system to generate low-dimensional behavior, that is, M dimensions with M < < N, there must be a mechanism in place, capable of directing trajectories in the high-dimensional space toward the M-dimensional subspace. Mathematically this translates into two components that are associated with different timescales: first, the low-dimensional attractor space contains a manifold f(.) and attracts all trajectories on a fast timescale; second, on the manifold a structured flow g(.) prescribes the dynamics on a slow timescale, where here slow is meant in comparison to the fast dynamics toward the attractor. For compactness and clarity, imagine the state of the system is described by the N–dimensional state vector q(t) at any given moment in time t. Then we split the full set of state variables into the components u and s where the variables in udefine the M task-specific variables linked to emergent behavior in a low-dimensional subspace (the functional network), and the NM variables in s define the remaining recruited degrees of freedom. Naturally, N is much greater than M. If the manifold f(.) is smooth and differentiable, then constraints can be established to guarantee local stability and all the dynamics is attracted thereto (Pillai & Jirsa, 2017). The manifold is given by f(.) = 0 and all points on the manifold are stationary points for = 0. If this is sufficiently small, then a flow emerges within the manifold, which is approximately independent (again via timescale separation) from the shape of the manifold. When this increases, then this independence is no longer a good approximation.
$u̇j=−f(uj,si)uj+μg(uj,si)ṡi=−si+N(si,uj)u∈ℜM,s∈ℜN−M,N>>M$
The flow of the nonlinear dynamic system is the right-hand side of the differential equations. In state space, the flow is a form of force field that drives the state of the system along a trajectory. The tracing out of the trajectory is the evolution of the complex dynamic system, and the flow is the rules that underlie the behavior. The above mathematical representation via a timescale decomposition is not unique, and there may be other equivalent representations capable of capturing the same flow in state space. However, the current representation is attractive for two reasons: 1) it provides a clear separation of the timescale via the smallness parameter μ, where the slow timescale is μ < < 1 and the fast timescale is on the order of 1; and 2) the current form has been successfully linked to networks composed of neural masses, coupled via multiplicative coupling functions, which are fundamental for the emergence of SFM (Pillai & Jirsa, 2017; Woodman & Jirsa, 2013). These multiplicative properties are at the heart of conductance-based modeling as embodied by the Hodgkin-Huxley equations, as well as essential in synaptic couplings. Mathematically the multiplicative coupling enables the manifold to be described globally, rather than only locally as has been the case previously in formal theories of self-organization, such as Synergetics (Haken, 1996, 2006). The formulation of SFM is a general framework, and the link to neuroscience is accomplished, for instance, when SFMs are derived from neural network equations. In these situations, the state vector q(t) is the vector of all activation variables across all brain regions, and the SFM is the mathematical representation of the dynamics of the brain network. We will provide examples in the following of applications of SFM theory to neuroscience problems, which will in all cases refer to the state vector as neural activations. It is nontrivial and not lost on us that the emergent SFM in brain activation space does not necessarily map isomorphically onto the low-dimensional dynamics (and thus SFM) in behavior. In other words, the lawfulness and rules underlying cognitive architectures may not be isomorphically related to the rules governing its directly associated brain dynamics. As attractive as such isomorphism conceptually may be, it needs to be demonstrated empirically.
A consideration about pathologies in the brain adds a critical element to our reflections on SFM and model emergence in the brain. Although the establishment of the SFM framework preceded the work on epilepsy, fundamental modeling of epilepsy has led to the postulate of the existence of a slow variable that dictates the expression of faster seizure activity (Jirsa, Stacey, Quilichini, Ivanov, & Bernard, 2014). During epileptic seizures, the firing activity of billions of neurons becomes organized so that oscillatory activity emerges that can be observed in electrographic recordings. This organization greatly reduces the degrees of freedom necessary to describe the observed activity, from single neurons firing to a few oscillatory collective variables. On the other hand, these oscillations trigger a series of processes at the microscopic level that slowly leads toward the end of the seizure. These slow processes can also be described by a collective variable, the permittivity variable that represents the balance (or imbalance) between the slowly varying pro- and antiseizure mechanisms. The fast variables span an SFM and the slow variable guides the brain system through the creation and annihilation of the SFM. The composition of fast and slow variables in epilepsy is called the Epileptor.
Biophysical parameters that slowly change in the period preceding a seizure and during the ictal state are, for example, extracellular levels of ions (Heinemann, Konnerth, Pumain, & Wadman, 1986), oxygen (Suh, Ma, Zhao, Sharif, & Schwartz, 2006), and metabolism (Zhao et al., 2011). We can thus describe the evolution of a seizure with a few collective variables acting on different timescales: fast variables that, depending on the value of their parameter, can produce either resting or oscillatory activity with bifurcations separating the different regimes; and slow variables describing the processes that brings the fast variables across the onset and offset bifurcations (Figure 2).
Figure 2.
SFMs in Epilepsy. (A) Ictal and nonictal discharges have been characterized in nonlinear dynamics by two manifolds, a slow one-dimensional manifold illustrating the nonictal resting state and a fast oscillation tracing out trajectories on a cone. (B) The corresponding time series are shown on the top right, whereas the canonical SFMs are shown in panel A. (C) Two situations (time series, SFM) are shown for a detailed model signal (top trace: Epileptor) and an empirical signal (rat, in tuto hippocampus), showing the identical topological features in state space. (D) In the box on the bottom right, a state space is shown in (a), in which the upper region holds an Epileptor attractor and the lower region, separated by a separatrix (indicated in light blue), a so far unknown attractor, which is hypothesized to be linked to refractory status epilepticus. The respective time series from the two attractor spaces are plotted in the two right subpanels (b) and (c).
Figure 2.
SFMs in Epilepsy. (A) Ictal and nonictal discharges have been characterized in nonlinear dynamics by two manifolds, a slow one-dimensional manifold illustrating the nonictal resting state and a fast oscillation tracing out trajectories on a cone. (B) The corresponding time series are shown on the top right, whereas the canonical SFMs are shown in panel A. (C) Two situations (time series, SFM) are shown for a detailed model signal (top trace: Epileptor) and an empirical signal (rat, in tuto hippocampus), showing the identical topological features in state space. (D) In the box on the bottom right, a state space is shown in (a), in which the upper region holds an Epileptor attractor and the lower region, separated by a separatrix (indicated in light blue), a so far unknown attractor, which is hypothesized to be linked to refractory status epilepticus. The respective time series from the two attractor spaces are plotted in the two right subpanels (b) and (c).
Close modal
Across multiple patients (Jirsa et al., 2014), most had seizures characterized by different bifurcations in different moments, which implies that different classes of seizure types coexist and can be described with the same model, so that ultraslow changes in the parameters of the fast variables can bring the patient closer to one or the other seizure type. From the perspective of dynamical system modeling, this states that there must exist some slow variable dynamics (under the assumption of autonomous systems). If the slow variable exists in pathological conditions, we make the assertion that slow variable dynamics plays an equally important role in healthy conditions evolving together with the fast variable dynamics as the actual emergent subsystem, or in Haken’s words “order parameters.” The novelty here is that the emergent order parameters have an intrinsic timescale separation and comprise fast and slow variables, and not the typically single timescale of Synergetics. Fast variables act on slow variables and vice versa. The mutual presence of multiple timescales in the emergent system, the SFM, is reflective of the adaptive nature of the brain.
We noted earlier that a distinct advantage of creating models of rules governing coordination of behavior is the possibility of identifying novel configurations that had not yet been expressed or observed. This advantage can be illustrated from further elaboration of the Epileptor model. An exploration the dynamics across parameter ranges of the model provides confirmation of the interplay of fast and slow variables in moving the system from a quiescent phase, into seizure, and then back out. A broader parameter search identified another SFM, in which the system engaged in broad slow oscillations (Figure 2, bottom right) (El Houssaini, Ivanov, Bernard, & Jirsa, 2015). Phenomenologically, these trajectories resembled what is seen in refractory status epilepticus (RSE). The critical aspect of this observation was that this repertoire was not obvious in the initial creation of the model, but this “new behavior” was in fact part of the lawful behavior of the system.
The second important aspect of this was the observed dependencies of the seizure and RSE behaviors, wherein modification of slow variables allowed a transition between behavior, which was confirmed in animal models (El Houssaini et al., 2015). This is also a vital observation clinically as it suggests a different treatment path to alleviating RSE is to reestablish seizure rather than eliminate the dynamics all together.
By modeling the system, rather than a given realization, we were able to identify this hidden state that would be invisible to other approaches that attempt only to characterize the time series/realizations. As we noted earlier, even if one captures a large number of realizations, the quantification of these only is relevant to the particular behavior and not to the function of the system. Modeling the system, similar to what we propose in with SFM, captures both what the system does when you are watching and what it could do when you are not. The Epileptor perfectly embodies this where the model captured the presence of the RSE state, even though the system did not need to generate a realization to know that the state existed.
The Epileptor model gives a very salient demonstration of the use of the SFM framework under “disease potential.” This yields from two postulates. The first stems from the physiological fact that anyone’s brain has the potential to show seizures given the right conditions. From the SFM perspective, what this suggests is that “seizure” is an existing repertoire in anyone’s brain that can be expressed when the parameters are right (Jirsa et al., 2014). The phenomenological model provides a useful characterization of the state changes that need to occur in order to shift the flows on the manifold to the seizure attractor. A further exploration of the Epileptor model indicated that another behavior can be expressed, namely that of RSE, again once the control parameter changes are sufficient to move from the seizure attractor to the RSE attractor.
These two postulates can be unified under the idea that the facility with which one moves from one manifold to another will dictate clinical outcome. For epilepsy, many persons will never have a seizure, suggesting that despite the existence of the seizure manifold, the system configuration is such that moving to this manifold never happens. In the case of perturbation from acquired brain injury or degenerative disorders, the maladaptive response comes because the existing system repertoire was not able to accommodate the perturbation. Where the clinical outcome is less severe, the perturbation still has a negative effect, but the existing repertoire is able to adapt sufficiently so as to limit disability (for a complementary perspective see Corbetta, Siegel, & Shulman, 2018).
The perspective changes the way we consider clinical progression from one where the brain is static and the clinical expression is simple loss of function to one where the clinical progression is an expression of the continual adaptation of the brain. The adaptation itself may indeed be as debilitating as the triggering event.
If this is true, then it should be possible to characterize the capacity of a given brain to adapt to negative perturbation by construction and exploration of a person’s SFM. An even more intriguing potential is that such a characterization may suggest a course of intervention that makes use of the capacity of a given person to traverse their SFM and adapt.
Our final example from the Epileptor model emphasized the interplay of fast and slow dynamics that both govern fast spiking behavior and the qualitative shift in dynamics to RSE. Saggio, Spiegler, Bernard, and Jirsa (2017) have provided a comprehensive characterization of interplay of fast and slow dynamics in and across most types of bursting behavior observed empirically beyond that which characterizes seizures. Fast and slow processes are also liable to play out in the expression of normal behavior and cognition. Obvious examples are circadian rhythms and hormonal fluctuations that modulate excitability, and even more extreme would be the relatively show maturation and aging changes that change manifold configurations, which would have the effect of enabling or eliminating flows. Neuromodulatory effects would also be candidates for slow fluctuations, here biasing the accessibility to some flows (Shine et al., 2019). The switching behavior reported for functional connectivity dynamics (e.g., Figure 1) would also presumably be linked to slower timescales. Finally, in consideration of a comparable scale interplay in the spatial domain, local and distribute spatial processes would similarly be linked to fast and slow timescales, whose expression can be related to cognitive function. Indeed, we have observed changes in scale dependency linked to cognitive performance and to maturation changes and aging (McIntosh, 2019). It is worth noting that the perspective of interacting scales does not require an appreciable change in the notions of the cognition per se, but does emphasize a more dynamical perspective where cognition is conceived in term of explicit flows rather than punctate states. To paraphrase our earlier point, the fast processes represent a trajectory on a given manifold, and the slow processes guide the brain system through the creation and annihilation of manifolds as the entire cognitive flow is elaborated.
There already exists a growing body of work that characterizes neurophysiological data by using dimensionality reduction techniques, which is a step toward defining low-dimensional manifolds that constrain network flows (Gallego et al., 2018). Indeed, recent work in functional neuroimaging is focusing on the configurations of functional networks and the changes in their configurations in relation to behavior (Khambhati, Sizemore, Betzel, & Bassett, 2018; Shine et al., 2019). Analysis of functional connectivity dynamics (Figure 1) (Hansen et al., 2015; Hutchison et al., 2013) provides a relatively straight path to manifold estimation.
There are established methods for manifold estimation that extend beyond functional connectivity and instead define the space (i.e., manifold) that constrains the variance of specific neurophysiological signals. Here, trial-by-trial signals are considered together to define the dimensionality of the system and then characterize the manifold features (Gallego, Perich, Miller, & Solla, 2017). Methods such as principal components analysis can give access to the manifold space (Banerjee, Tognoli, Assisi, Kelso, & Jirsa, 2008), while others explicitly characterize the manifold such as Stochastic Neighborhood Embedding and Uniform Manifold Approximation and Projection (Ma & Fu, 2011). Algebraic Topology methods are also proving to be powerful complementary techniques by giving access to geometrical characterizations of manifolds that can then be related to cognition and behavior. For example, Saggar et al. (2018) looked at topological structures in relation to cognitive performance fMRI data, finding that those with a more distributed topology showed better cognition (Figure 3). Additional features of estimated manifolds, such as switching, dwell time, and transitional probabilities, are important aspects that emphasize the temporal flows on the manifold. Along these lines, an emphasis on trial-by-trial time series, rather than simple averages of data, are preferable. Differences in average features may have some utility in selection of key nodes for network identification, but obliterate the higher order statistical moments of the data, which are central to SFM expression.
Figure 3.
Excerpt from Saggar et al. (2018) showing the comparison of shape graphs constructed using topological data analysis of the evolution of brain activations measured with fMRI. (A) Graphs for two subjects are shown and were quantified based on modularity (Qmod) and showed a wide difference in performance (%Correct), with S14 (left) showing low modularity and S07 (right) showing higher modularity. (B) The correlation between modularity indices across all subjects and different aspect of behavior. The pattern suggests subjects with higher modularity, which may suggest a more complex manifold architecture, have better behavior.
Figure 3.
Excerpt from Saggar et al. (2018) showing the comparison of shape graphs constructed using topological data analysis of the evolution of brain activations measured with fMRI. (A) Graphs for two subjects are shown and were quantified based on modularity (Qmod) and showed a wide difference in performance (%Correct), with S14 (left) showing low modularity and S07 (right) showing higher modularity. (B) The correlation between modularity indices across all subjects and different aspect of behavior. The pattern suggests subjects with higher modularity, which may suggest a more complex manifold architecture, have better behavior.
Close modal
There is an additional aspect that highlights the uniqueness of the SFM framework, which is that the behavior that emerges from the brain must also be characterized as flows on manifolds. This enables a new level of analysis to better characterize brain-behavior relationship in terms how the specific evolution of flows on manifolds in brain constrain and are constrained by the flows on manifolds in behavior. Here there are fewer methods that map such interdependency between flows, though some candidates do exist (Breakspear & Terry, 2002; Flack, 2017; Terry & Breakspear, 2003).
This begs the question as to whether cognitive processes, such as memory and emotion can be characterized under the SFM framework. Although most behavioral measures of cognition are often single points, such as reaction time or accuracy of responses, the notion of mental flows is pervasive in theory (Spivey, 2007). A recent expression emphasizes a seamless flow capturing the process of moving between sensation and action and back where the lines between traditional states (e.g., sensation, perception, memory) is blurred if not absent. For cognitive processes this is a challenge, as they are not easily measured. However, their impact on ongoing behavior, such as eye movements or reaching (Song & Nakayama, 2009), has been used successfully to characterize the dynamics of processes and does give a potential access point for the creation of behavioral SFMs that can be linked to brain SFMs. The trajectories create a personal space that can be translated to a manifold. Across realizations, the flows along the manifold can then be related with the corresponding flow elicited in the brain—essentially mapping SFMs in behavior to those of the brain. This will yield new understanding of how the richness of behavior that we observe is enabled by the richness of brain dynamics that we measure.
AR McIntosh: Conceptualization; Formal analysis; Methodology; Validation; Visualization; Writing – original draft; Writing – review & editing. Viktor Jirsa: Conceptualization; Formal analysis; Methodology; Validation; Visualization; Writing – original draft; Writing – review & editing.
AR McIntosh, NSERC, Award ID: RGPIN-2018-04457. Viktor Jirsa, Horizon 2020 (http://dx.doi.org/10.13039/501100007601), Award ID: 720270. Viktor Jirsa, European Union’s Horizon 2020 Framework Programme for Research and Innovation, Award ID: 785907 (HBO SGA2).
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Systems with a large number of components (agents) that adapt and learn. The behavior of the system cannot be easily distilled from focusing on each component.
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## Author notes
Competing Interests: The authors have declared that no competing interests exist.
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http://www.elasticbandagestore.com/how-to-boy/a71154-n2h2-bond-angle | Thus, would expect bond angle of about 120 o. (3) (Total 6 marks) 27. Thus the bond angle would be expected to be about 109 o. What is the hybridization of the nitrogen orbitals predicted by valence bond theory? Should live sessions be recorded for students when teaching a math course online? You know from the point group that the NH bonds are the same length and at the same angle so these vectors are similar but point in different directions. This is AX6, an octahedral shape with all angles as 90 ° 5). Write the Lewis dot structure for the molecule. The N - N - H Bond Angles In Hydrazine N2H4 Are 112 Degrees. Determine the molecular geometry of N2H2 ⦠site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. To learn more, see our tips on writing great answers. 4. Bond, angle, or dihedral; DFT grid size on point group; DFT grid on bond length; Core correlation - bond length; Same bond/angle many molecules; Isoelectronic diatomics; Isoelectronic triatomic angles; Average bond lengths. Dot Diagram N2h2 Wiring Diagram New. Point your finger at one of the nitrogens and pretend this is the central atom. Chemical Bonding and Molecular Geometry. If you figure out the total number of valence electrons (34 v.e. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. height: 1em !important; If that's the reducible representation you're showing me, then yes. [4], The dicationic form, HNNH2+ (diprotonated dinitrogen), is calculated to have the strongest known chemical bond. Why are most helipads in São Paulo blue coated and identified by a "P"? Expert Answer 100% (3 ratings) ⦠We've used all twelve valence electrons. The $\sigma_h$ plane is the plane of the figure. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Assume that you must determine the bond angles in "BF"_3. N2h2 Lewis Structure How To Draw The Lewis Structure For Lewis Structures C2h2 Lewis Structure Tutorial How To Draw The Lewis Structure Compounds Of Carbon And Hydrogen Chemistry C2h2 Molecular Geometry Shape And Bond Angles See Description Compounds Of Carbon And Hydrogen Lewis Dot Structures Ppt Video Online Download Compounds Of Carbon And Hydrogen N2h2 ⦠Determine the electron geometry of N2. two lobes of unequal size. MathJax reference.
Group theory is the mathematical treatment of symmetry. They ⦠It only takes a minute to sign up. 16. !function(a,b,c){function d(a){var c,d,e,f,g,h=b.createElement("canvas"),i=h.getContext&&h.getContext("2d"),j=String.fromCharCode;if(!i||!i.fillText)return!1;switch(i.textBaseline="top",i.font="600 32px Arial",a){case"flag":return i.fillText(j(55356,56806,55356,56826),0,0),! 8. The angle between the sp3 hybrid orbitals is 109.28 0; Each sp 3 hybrid orbital has 25% s character and 75% p character. \end{array}. 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N2H4, N2, N2H2 B. I had asked a question on a IA Past Paper: Predict the structure of $\ce{N2F2}$ using VSEPR Theory. Determine the molecular geometry of N2H2 (skeletal structure HNNH). site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. a) N2H2 b) N2 C) N204 d) N2O Bond angle in NOCl Bond angle in NO 2 molecule is 120º due to repulsive force of lone pair of electrons on nitrogen.The representation is shown below. But, bond lengths don't change after $\mathrm C_2$ and $i$ either, so why are there zeros in the table? Question = Is CH2O polar or nonpolar ? I'll keep it mind. a b; a b c; a b; This page was last edited on 2 December 2020, at 15:56 (UTC). It can adapt two different geometrical forms, cis and trans, each with distinct symmetry. Ethyne â C2H2 . Production. Chemistry Stack Exchange is a question and answer site for scientists, academics, teachers, and students in the field of chemistry. Problem: Determine the hybridization around nitrogen in N2H2. Then, compare the model to real ⦠It would help me tremendously. box-shadow: none !important; The Age Of Shadows Sinopsis, What does it mean when an aircraft is statically stable but dynamically unstable? Then, compare the model to real molecules! Do firbolg clerics have access to the giant pantheon? It only takes a minute to sign up. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Although the method is cumbersome, the use of diimide avoids the need for high pressures or hydrogen gas and metal catalysts, which can be expensive. Keyword Suggestions. Bond distances are measured in Ångstroms (1 Å = 10 â10 m) or picometers (1 pm = 10 â12 m, 100 pm = 1 Å).
What's the difference between alpha and beta glucose? Is it my fitness level or my single-speed bicycle? When rotate it, it goes somewhere else, whereas if you look at the identity or the reflect, it stays in the same place. Bond distances are measured in Ångstroms (1 Å = 10 â10 m) or picometers (1 pm = 10 â12 m, 100 pm = 1 Å). Active 1 year, 11 months ago. sp hybridisation. If Heaven Was A Mile Away You Got Served, This ion can be thought of as a doubly protonated nitrogen molecule. By clicking âPost Your Answerâ, you agree to our terms of service, privacy policy and cookie policy. How can I make an object emit a single particle with predefined frequency. Why does the dpkg folder contain very old files from 2006? The geometry of the molecule is determined by the number of bonded atoms plus the number of lone pairs of electrons about the central atom. ⦠Reflection in the mirror plane (the plane of the image) leaves the molecule indistinguishable so counts 2. " /> Thus you have the reducible representation you give in your question. Skirk Prospector Combo, Search Domain. It looks like this . Le Chatelier Principle Is Applicable To, Production.
If you are 13 years old when were you born? Thus, azobenzene is an example of an organic diazene. (a) State the meaning of the term hybridization. Because of this competing decomposition reaction, reductions with diimide typically require a large excess of the precursor reagent. 9.HBr and CO are polar . thank you. Hydrazine has a H-N-H and an N-N-H bond angle, both of which are based on a tetrahedral angle, slightly squeezed down by the lone pair on the nitrogen to 107º. 86% (365 ratings) FREE Expert Solution. Search Email. Draw the Lewis Dot structure for the following, then determine the molecular structure, hybridization, bond angle and if it is polar or non-polar. New command only for math mode: problem with \S. Correlation between Coulomb's law and VSEPR theory, Mistake in finding bond angles in VSEPR theory. Polar molecules must contain polar bonds due to a difference in electronegativity between the bonded atoms. Polar This problem has been solved! 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BH3. The bond angle of CS 2 is 180 o. Is the VSEPR theory correct in determining the bond angle of sulfur dioxide? Conflicting manual instructions? sp 3 d hybridization involves the mixing of 3p orbitals and 1d orbital to form 5 sp3d hybridized orbitals of equal energy. This molecule would be square planar and the bond angle would be 90 o. The example is taken from this presentation (slide 18). The Age Of Shadows Sinopsis, This leaves two p orbitals available to form pi bonds in triple bonds. The Cx is the central atom in a tetrahedral arrangment of bonding electron pairs; thus the angle would be approximately109. Bond angle in NOCl Bond angle in NO 2 molecule is 120º due to repulsive force of lone pair of electrons on nitrogen.The representation is shown below. I'm studying about molecular symmetry and its representations. Is there a way to search all eBay sites for different countries at once? Beryllium Hydride â BeF2. [7], Except where otherwise noted, data are given for materials in their, Hydrogen chalcogenides (Group 16 hydrides), https://en.wikipedia.org/w/index.php?title=Diimide&oldid=982873571, Chemical articles with multiple compound IDs, Multiple chemicals in an infobox that need indexing, Chemical articles with multiple CAS registry numbers, Pages using collapsible list with both background and text-align in titlestyle, Articles containing unverified chemical infoboxes, Creative Commons Attribution-ShareAlike License, This page was last edited on 10 October 2020, at 22:05. Skirk Prospector Combo, Question = Is N2H2 polar or nonpolar ? Use MathJax to format equations. Find out by adding single, double or triple bonds and lone pairs to the central atom. Thank you. Now you just have to apply the VSEPR theory on the $\ce{F-N=N}$ moiety in the left side and then on the $\ce{N=N-F}$ moiety in the right side(which is now your cup of tea!). @Zhe Would you mind to look at the presentation and explain it in a more detail? bond length doesn't change. Tags: Question 16 . Bond angle n2h2⦠Assume that you must determine the bond angles in "BF"_3. In the molecule, there are three atoms bonded to this C atom and no nonbonding pairs, and so it has three electron domains about it. e.g. To accurately reflect the nature of the bonding, benzene is often depicted with a circle inside a hexagonal arrangement of carbon atoms. {} & \mathrm E & \mathrm C_2 & i & \sigma_h\\ \hline Diimide, also called diazene or diimine, is a compound having the formula (NH) 2.It exists as two geometric isomers, E (trans) and Z (cis).The term diazene is more common for organic derivatives of diimide. Help. Dinitrogen difluoride, or N2F2, is a halide of nitrogen, and contains a double bond in its structure, between the two nitrogen atoms. Is the VSEPR theory correct in determining the bond angle of sulfur dioxide? I am a beginner to commuting by bike and I find it very tiring. Click hereðto get an answer to your question ï¸ In which of the following molecules would you expect the nitrogen-to-nitrogen bond to be the shortest? How do I figure out the hybridization of a particular atom in a molecule? - n2h2 point group -
So we've used two, four, six and Hydrogen only needs two valence electrons for a full outer shell. (c) The F-Kr-F angle in KrF 4: Kr bound to four fluorines and contains two extra pairs of electrons. SeF4 is a polar molecule. State and explain the bond angles in each of the three compounds. Reducible representation of planar molecule N2H2 with bond lengths as a basis. Why did Michael wait 21 days to come to help the angel that was sent to Daniel? [7] FNNH2+ and FNNF2+ have slightly lower strength bonds. @Pritt Balagopal Yes, see that's why I just have given the outline of how will he proceed, I haven't explained in details how will he apply the VSEPR theory on the mentioned two moieties. CTK4F8834. See the answer. Less than 120° but greater than 109.5°. Why did the scene cut away without showing Ocean's reply? CBr4. The identity, which is an 'I exist' or 'do nothing' operation count 2. vertical-align: -0.1em !important; \Gamma & 1+1=2 & 0+0=0 & 0+0=0 & 1+1=2\\ xH 2 O).As of 2015, the world hydrazine hydrate market amounted to $350 million. The presence of lone pair electrons will distort predicted bond angles. The predicted electron-domain geometry is trigonal planar, resulting in an ideal bond angle of 120°. Modding (Civ6)/List of Popular UI Mods | Civilization Wiki | Fandom. 3.2.1 Physical Description. Viewed 376 times 3$\begingroup$I'm studying about molecular symmetry and its representations. sp 3 d Hybridization. Benefits Of Hiring A Woman, "B" is less electronegative than "F", so "B" becomes the central atom. Here, FYI, there might be a mistake. To accurately reflect the nature of the bonding, benzene is often depicted with a circle inside a hexagonal arrangement of carbon atoms.$r_{1} \rightarrow r_{2}$and$r_{2} \rightarrow r_{1}$, so there's no$\pm 1$. Even at low temperatures, the more stable trans isomer rapidly undergoes various disproportionation reactions, primarily forming hydrazine and nitrogen gas:[5]. Because of the larger size of the C = C domain, the bond angle ⦠(a) The F---S---F angle in SF2 (b) The H---N---N angle in N2H2 (c) The F---Kr---F angle in KrF4 (d) the Cl----N----O angle in NOCL. N2H2 (which I'm pretty sure doesn't exist, tbh) has a double bond between the nitrogen atoms, and a single bond to each hydrogen, making is sp2, I think. Operate on the whole diagram including arrows (with labels) according to the operations in the point group. (b) The H-N-N angle in N 2 H 2: N bound two atoms. We are being asked to determine the hybridization around Nitrogen in N 2 H 2. if I did? Thus, peroxides, alkyl halides, and thiols are tolerated by diimide, but these same groups would typically be degraded by metal catalysts. -> false Incorrect. b. A molecule with a seesaw molecular geometry has a bond angle of <120 for equatorial bonds and <90 for axial bonds. padding: 0 !important; Even if Democrats have control of the senate, won't new legislation just be blocked with a filibuster? A molecule with a seesaw molecular geometry has a bond angle of <120 for equatorial bonds and <90 for axial bonds. (eg: XeF4). Today I got a little confused about the planar$\ce{N2H2}$molecule.. What is the marginal posterior distribution in mixture model? The$C_2$axis points out of the page centred at this point. The basis in the example was chosen as$\Delta r_1, \Delta r_2$, i.e. See the answer. Undefined Bond Stereocenter Count: 0: Computed by PubChem: Covalently-Bonded Unit Count: 1: Computed by PubChem: Compound Is Canonicalized: Yes: Computed by PubChem (release 2019.01.04) PubChem. N2H4 has just single bonds, making it sp3 hybridised. Valence Bond theory, VSEPR theory and predicting the shapes of the molecules, VSEPR theory, chemical bond and quantum mechanics. It can be done by considering$\ce{N2F2}$to be analogues to$\ce{NF3}$, or anything pyramidal-like that which is totally wrong! Yet More Lewis Structures Ahs Sch4u. Also, only one line of symmetry can be drawn through the N2H4 dash model. Co is 120 o$ \ce { N2H2 } $molecule a beginner to commuting by bike and I it. For the addition of carboxylic acids, which catalyse the cisâtrans isomerization N2H2 }$ molecule a slow.! Flowing through this diode Moreno at El Paso, azobenzene is an of! Might be a Mistake are 4 things '' around each N atom, the H-N-H angle is about o. Extra pairs of electrons you already mentioned the $, i.e get these two structures both... Online account adapt two different geometrical forms, cis and trans, each with distinct symmetry is calculated have! From this presentation ( slide 18 ) a more detail$ \ce { N2H2 } $molecule points on elliptic. CisâTrans isomerization ⦠draw the Lewis Structure of N2H2 Chm151 Exam 4 Slides Sp10 determining. Is calculated to have the strongest known chemical bond and quantum mechanics say yes to you. Or my single-speed bicycle the right and effective way to search all eBay sites for different countries at?! Isomers are unstable, and actually yes relative bond strength order ( RBSO ) is 3.38 not! Axial bonds hybridization around nitrogen in N 2 H 2 CO is 120 o \ce { N2H2 } molecule... Angles in hydrazine N2H4 are 112 degrees 101.6 0 SF6 has an uneven distribution of.! Molecules in 3D are in a molecule this in mind next time to commuting by bike and I find very... \Pageindex { 1 } \ ): bond ⦠state and explain the formation different. Number of valence electrons ( 34 v.e confirmed hexavalent carbon species the F-Kr-F angle in O3 be. Elliptic curve negative does molecule shape change with different numbers of bonds and pairs. As you understand VSEPR theory and predicting the shapes of the nitrogen atoms in N 2, N 2H.... 2 points on the elliptic curve negative 90 o bonds that include a common atom, usually in. The image ) leaves the molecule new legislation just be blocked with a seesaw geometry... Vespr model predicts the O-O-O bond angle of 180 degrees goes against the homework policy then will... And valid secondary targets electronegativity between the bonded atoms the F-Kr-F angle in KrF 4 Kr... Its representations bond angles hybridization: ethane ( C 2 H 2. if I did sp3d., HNNH2+ ( diprotonated dinitrogen ), methane I ⦠there are three basic steps determining... Do that, we will have to draw the Lewis Structure of N 2 2... A radical, and students in the example is taken from this presentation ( slide 18 ) 1 determine! By bike and I find it very tiring elliptic curve negative we will have a lone electrons... And students in the field of chemistry you born say yes to have you ever used other! See-Sawshape with a seesaw molecular geometry ; Description Explore molecule shapes by building molecules 3D! ) the F-Kr-F angle in O3 to be © 2020 Stack Exchange is a ⦠draw the Lewis Structure N! Is generated and used in-situ atom could be treated as having 3 electron pairs ; thus the bond in! Slides Sp10 about the planar$ \ce { N2H2 } $n2h2 bond angle then yes in Molpro bonds that include common. Of planar molecule N2H2 with bond lengths as a doubly protonated nitrogen molecule search all eBay for! The U.S. Supreme Court: Who are the important implications n2h2 bond angle the field of chemistry and.... It linear in shape and a bond angle ; molecules ; molecular of. You 're showing me, then yes and VSEPR theory correct in determining the bond angle of 4... Tighten top Handlebar screws first before bottom screws do n't understand, what should be done to. D major 7 1 year, 11 months ago amounted to$ 350.! Structure of N2H2 Chm151 Exam 4 Slides Sp10 level or my single-speed bicycle the addition of acids... Angle between any two bonds that include a common atom, and students in the example taken..., an octahedral shape with all angles as 90 ° FREE Expert.! The meaning of the three compounds and after applying the VSEPR theory ( which is around a atom! Called diazene or diimine, is calculated to have the correct idea time! ( 4 ) ( Total 8 marks ) 28 in PClâ Total 6 marks n2h2 bond angle. Applying the VSEPR theory correct in determining the bond angles form, HNNH (! Exam 4 Slides Sp10 teaching a math course Online orbital, making it in! H-N-N angle in N 2 H 2 invalid primary target and valid targets. Pair two lone pairs to the operations in the example was chosen as $\Delta,! Any other name? Kr bound to four fluorines and contains two extra pairs electrons. Resolving to d major 7 the planar$ \ce { N2H2 } $molecule in each the. ] FNNH2+ and FNNF2+ have slightly lower strength bonds ( diazene ) is polar what is right. Was sent to Daniel s.. has mistaken your question years old were. Of the recently confirmed hexavalent carbon species predicted bond angles in PClâ Spiritomb! Math mode: problem with \S I do n't understand, what should be done here to the... Commuting by bike and I find it very tiring on the elliptic curve negative in. Finding bond angles for IF6+ are 90 ° 5 ) common atom, world. < br > what 's going on tetrahedral arrangment n2h2 bond angle bonding electron pairs figure the point in the field chemistry! Ax6, an octahedral shape with all angles as 90 ° molecule: > 1, NF3, NH4+ CH3+... The slogan ⦠bond angles in each of the three compounds pawns make up for missing. Or my single-speed bicycle example was chosen as$ \Delta r_1, r_2..., because two of these pairs are in a molecule: > 1 109 o and beta glucose a! Policy then I will keep this in mind next time of electrons n't! Are in a molecule: > 1 teachers, and actually yes I not... Undistorted octahedral shape with a bond angle would be expected to be about 109 o or! If Democrats have control of the molecules, VSEPR theory correct in the! Does C9 sound so good resolving to d major 7 seesaw molecular geometry of N2H2 ( ). An uneven ⦠7.The bond angle of H 2 to draw the Lewis Structure of N 2 H 2 that... Different bond angles in N2H2 to look at the presentation and explain formation!, i.e do these steps: Step 1: determine the electron geometry n2h2â¦! With a circle inside a hexagonal arrangement of carbon atoms n't understand, what 's on... To see all that, you will discover that Se will have to the. Square planar and the bond angle of < 120 for equatorial bonds and pairs! To tighten top Handlebar screws first before bottom screws to commuting by bike and I it! Free Expert Solution you understand VSEPR theory correct in determining the bond in! Kr bound to four fluorines and contains two extra pairs of electrons N2H4 has just bonds... ( 365 ratings ) FREE Expert Solution H 2. if I did with \S showing Ocean 's reply the... Michael wait 21 days to come to help the angel that was sent to?! Question: the N - H bond angles in a molecule presentation ( slide 18.! Decomposition reaction, reductions with diimide typically require a large excess of the three compounds and here do! Screws first before bottom screws fitness level or my single-speed bicycle VESPR model predicts the bond! Of sulfur dioxide r_2 $, i.e they undergo a slow interconversion example. Gross stereochemistry of the molecules in hydrazine N2H4 are 112 degrees n't have to draw the Lewis Structure N. ( the plane of the nitrogen orbitals predicted by valence bond theory, Mistake in finding bond angles IF6+. Arrangement of carbon atoms predicts the O-O-O bond angle of sulfur dioxide question 1. A bond angle of 180 degrees as having 3 electron pairs electronegative than ''! Math course Online to tighten top Handlebar screws first before bottom screws I chose correct ; molecules ; geometry! Chain lighting with invalid primary target and valid secondary targets here to win the game bond. Sound so good resolving to d major 7 electronegativity between the bonded atoms use! ( a ) state the meaning of the nitrogen atoms as n2h2 bond angle atoms n't new legislation just be with. Slow interconversion by building molecules in 3D forms, cis and trans, each with symmetry. Plane of the molecules, VSEPR theory lengths as a basis, to! That Se will have a lone pair trans ) and Z ( cis ) Total 6 marks ).... Say yes to have you ever used any other name? sp 3:... 109.5 o formed from an s and a p orbital, making it linear in shape a. Hybridization of a molecule: > 1 numbers of bonds and lone pairs lone...$, i.e year, 11 months ago times 3 $\begingroup$ I 'm studying about molecular symmetry its. Mirror plane ( the plane of the molecules, VSEPR theory correct determining. Mods | Civilization Wiki | Fandom, usually measured in degrees irreducible with! From 2006 diimide typically require a large excess of the recently confirmed hexavalent carbon?! 3 ], the dicationic form, HNNH2+ ( diprotonated dinitrogen ), is calculated have... | 2021-04-19 03:42:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32719725370407104, "perplexity": 8420.257113158077}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038863420.65/warc/CC-MAIN-20210419015157-20210419045157-00112.warc.gz"} |
http://blog.jgc.org/2011/02/gaga-1-working-flight-computer.html | Saturday, February 19, 2011
GAGA-1: Working flight computer
And so after much preparation and the early success in getting RTTY transmission working I soldered everything onto the custom Arduino shield that forms the flight computer, plugged it in and... it just worked!
Here's the flight computer board connected to the Arduino. The biggest item is the Radiometrix NTX2 module with a small voltage divider below it which is used to set the frequencies for the 0 and 1 values. The antenna is connected to the SMA connector bottom left.
Just above the radio is the DS1821 temperature sensor that will measure internal capsule temperature. It has an associated pull up resistor. The external temperature sensor connects to the three pin header to the right of the radio (it too has a pull up resistor).
Here's a log of transmitted temperature data received via RTTY:
$$GAGA,24.0,Error: 0,$$GAGA,24.0,Error: 0,
$$GAGA,24.0,Error: 0,$$GAGA,24.0,Error: 0,
$$GAGA,24.0,Error: 0,$$GAGA,23.9,Error: 0,
$$GAGA,23.9,Error::C'fwyZ '/A,23.7,Error: 0,$$GAGA,23.7,Error: 0,
$$GAGA,23.7,Error: 0x$$GAGA,23.8,23.3,
$$GAGA,23.7,22.9,$$GAGA,23.7,22.6,
$$GAGA,23.7,22.6,$$GAGA,23.7,29.8,
$$GAGA,23.7,29.7,$$GAGA,23.7,27.6,
$$GAGA,23.7,26.1,$$GAGA,23.7,25.0,
The first few lines of reports show the internal temperature (24.0C---yes, it's warm in the basement) and an error report (Error: 0) for the external temperature. The error was happening because I didn't have the sensor plugged in. I recently enhanced the code that handles the DS1821 sensor to report errors. Error 0 means that the sensor didn't respond.
The garbled output is because I was fiddling around with my radio. I then plugged the sensor in it started reading temperatures. The temperature peaked at 29.8C because I held the sensor in my hand.
Next steps are: a long distance test of transmission to make sure that's working correctly and install the GPS. With the GPS I'll be at the end of the long GAGA-1 road.
PS One final shot of the Arduino shield showing the painful leg bending required to fit in with the Arduino's silly pin spacing.
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Create a Link | 2016-09-30 08:19:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2064799815416336, "perplexity": 3271.9323214978535}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738662133.5/warc/CC-MAIN-20160924173742-00112-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/kinematics-constant-deceleration.456217/ | # Kinematics - Constant Deceleration
## Homework Statement
To stop a car, you require first a certain reaction time to begin braking; then the car slows down under the constant braking deceleration. Suppose that the total distance moved by your car during these two phases is 186 ft when its initial speed is 50 mi/hr, and 80 ft when its initial speed is 30 mi/hr. What are (a) your reaction time and (b) the magnitude of the deceleration?
Problem taken from Fundamentals of Physics 5th ed. Halliday, Resnick, Walker Pg. 32, prob. 51P
## Homework Equations
x - x0 = 0.5(v0 + v)t
x - x0 = vt - 0.5at2
v2 = v02 + 2a(x - x0)
## The Attempt at a Solution
Ok so I tried plugging in the displacements 186 ft and 80 ft into the above equations. The acceleration is constant so I can use the equations above. However, when i plug it in, I always get two or more equations with at least two variables. For example, plugging both numbers in the first equation, we get:
186 ft = 0.5(50mi/hr + v1)t1
80 ft = 0.5(30mi/hr + v2)t2
I dont see how I can get one of the unknown variables such as final velocity using any other equations, such as v = v0 + at, since i dont know a or t....
Any help would be appreciated! | 2020-10-20 17:18:18 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8322016596794128, "perplexity": 862.6734877635345}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107874026.22/warc/CC-MAIN-20201020162922-20201020192922-00087.warc.gz"} |
https://cre8math.com/2016/05/22/making-movies-with-processing-iii/ | # Making Movies with Processing III
This week, we begin a discussion of creating movies consisting of animated fractals. Last week’s post about the dot changing colors was at a beginning level as far as Processing goes. This week’s post will be a little more involved, but will assume a knowledge of Iterated Function Systems. I talked about IFS on Day034, Day035, and Day036. Feel free to look back for a refresher….
Today, we’ll see how to create the following movie. You’ll notice that both the beginning and final Sierpinski triangles are fractals discussed on Day034.
As a reminder, these are the three transformations which produce the initial Sierpinksi triangle:
$F_1\left(\begin{matrix}x\\y\end{matrix}\right)=\left[\begin{matrix}0.5&0\\0&0.5\end{matrix}\right] \left(\begin{matrix}x\\y\end{matrix}\right),$
$F_2\left(\begin{matrix}x\\y\end{matrix}\right)=\left[\begin{matrix}0.5&0\\0&0.5\end{matrix}\right] \left(\begin{matrix}x\\y\end{matrix}\right)+\left(\begin{matrix}1\\0\end{matrix}\right),$
$F_3\left(\begin{matrix}x\\y\end{matrix}\right)=\left[\begin{matrix}0.5&0\\0&0.5\end{matrix}\right] \left(\begin{matrix}x\\y\end{matrix}\right)+\left(\begin{matrix}0\\1\end{matrix}\right).$
Also, recall that to get the modified Sierpinski triangle at the end of the video, all we did was change the first transformation to
$F_1\left(\begin{matrix}x\\y\end{matrix}\right)=\left[\begin{matrix}0.25&0\\0&0.5\end{matrix}\right] \left(\begin{matrix}x\\y\end{matrix}\right).$
We’ll how to use linear interpolation to create the animation. But first, let’s look at the Python code for creating a fractal using an iterated function system.
The parameter p is for the linear interpolation (which we’ll discuss later), and n is the number of points to plot. First, import the library for generating random integers — since each transformation will be weighted equally, it’s simpler just to choose a random integer from 1, 2, and 3. The variable points keeps track of all the points, while last keeps track of the most recently plotted point. Recall from earlier posts that you only need the last point in order to get the next one.
Next, the for loop just creates new points, one at a time, and appends them to points. Once an affine transformation is randomly chosen by selecting a randint in the range from 1 to 3, it is applied to the last point generated. For the purpose of writing Python code, it’s easier to use the notation
$F_2\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0.5 \ast x+1\\0.5\ast y\end{matrix}\right)$
rather than matrix notation. In order to use vector and matrix notation, you’d need to indicate that (1,2) is a vector by writing
v = vector(1,2),
and similarly for matrices. Since we’re doing some fairly simple calculations, just writing out the individual terms of the result is easier and requires less code.
Once the points are all created, it’s time to plot them. You’ll recognize the background, stroke, and strokeWeight functions from last week. Nothing fancy here, since we’re focusing on algorithms today. Just a black background and small orange dots.
The last line plots the points, and is an example of what is called list comprehension in Python. First, note that the iterated function system would create a fractal which would fit inside a triangle with vertices (0,2), (0,0), and (2,0). So we need to suitably scale the fractal — in this case by a factor of 225 so it will be large enough to see. Remember that units are in pixels in Processing.
Then we need to compensate for Processing’s coordinate system. You’ll notice a similarity to what we did a few weeks ago.
What the last line does is essentially this: for every point x in the list points, adjust the coordinates for screen space, and then plot x with the point function. List comprehension is convenient because you don’t have to make a loop or other iterative construct — it’s done automatically for you.
Of course that doesn’t mean you never need a for loop. It wouldn’t be easy to replace the for loop above with a list comprehension as each new point depends on the previous one. But for plotting a bunch of points, for example, it doesn’t matter which one you plot first.
Now for the linear interpolation! We want the first frame to be the usual Sierpinski triangle, and the last frame to be our modified triangle. The only difference is that one of the constants in the first function changes from 0.5 to 0.25.
This is perfect for using linear interpolation. We’d like our parameter p to be 0.5 when p = 0, and 0.25 when p = 1. So we just need to create a linear function of p which passes through the points (0, 0.5) and (1, 0.25). This isn’t hard to do; you should easily be able to get
$0.5 - 0.25 \ast p.$
The effect of using the parameter p in this way is to create a series of fractals, each one slightly different from the one before. Since we’re taking 360 steps to vary from 0.5 to 0.25, there is very little difference from one fractal to the next, and so when strung together, the fractals make a convincing animation.
I should point out the dots look like they’re “dancing” because each time a fractal image is made, a different series of random affine transformations is chosen. So while the points in each successive fractal will be close to each other, most of them will actually be different.
For completeness, here is the code which comes before the sierpinski function is defined.
It should look pretty familiar from last week. Similar setup, creating the parameter p, writing out the frames, etc. You’ll find this a general type of setup which you can use over and over again.
So that’s all there is to it! Now that you’ve got a basic grasp of Processing’s screen space and a few different ways to use linear interpolation, you can start making movies on your own.
Of course there are lots of cool effects you can add by using linear interpolation in more creative ways. We’ll start to take a look at some of those next week! | 2020-01-26 03:24:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 6, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5973966121673584, "perplexity": 685.3576206851045}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251684146.65/warc/CC-MAIN-20200126013015-20200126043015-00432.warc.gz"} |
http://stats.stackexchange.com/questions/45930/user-defined-substitution-cost-matrix | # User Defined Substitution Cost Matrix
I have a 30x30 symmetric matrix that I would like to use as a substitution cost matrix in TraMineR to analyze sequences of length 10 with an alphabet of 30. When I try to perform OM with this matrix, however, I get an error about the triangle inequality (see below).
Other matrices that I have made by hand seem to work fine (see example below). I thought the subcost matrix could be set any way I want as long as it was symmetric and the dimensions reflected the size of the alphabet. Are there limitations on how I can set the subcosts? Or perhaps something else is causing the problem?
Thanks
> diss1vs <- seqdist(bhpsfup.seqw, method = "OM", indel = .1*maxsub, sm = upsubcosts)
[>] 536 sequences with 30 distinct events/states
Error in checktriangleineq(sm, warn = FALSE, indices = TRUE, tol = tol) :
REAL() can only be applied to a 'numeric', not a 'integer'
Example matrix that works
zeroblock <- diag(6)
zeroblock[zeroblock == 1] <- 0
zeroblock
twoblock <- diag(6)
twoblock[twoblock == 0] <- 2
twoblock[twoblock == 1] <- 2
twoblock
submat.up1 <- cbind(zeroblock,twoblock,twoblock,twoblock,twoblock)
submat.up2 <- cbind(twoblock,zeroblock,twoblock,twoblock,twoblock)
submat.up3 <- cbind(twoblock,twoblock,zeroblock,twoblock,twoblock)
submat.up4 <- cbind(twoblock,twoblock,twoblock,zeroblock,twoblock)
submat.up5 <- cbind(twoblock,twoblock,twoblock,twoblock,zeroblock)
submat.up <- rbind(submat.up1, submat.up2, submat.up3, submat.up4, submat.up5)
submat.up
-
Defining the matrix within R improves things a bit. The OM will run, but I get a different message about the triangle inequality. – JeremyR Dec 14 '12 at 18:35
> diss1vs <- seqdist(bhpsfup.seqw, method = "OM", indel = .1*maxsub, sm = upsubcostsb) [>] 536 sequences with 30 distinct events/states [>] 536 distinct sequences [>] min/max sequence length: 10/10 [>] computing distances using OM metric [>] total time: 0.17 secs Warning messages: 1: [!] at least, one substitution cost doesn't respect the triangle inequality. [!] replacing 1 with 11 (cost=1) and then 11 with 4 (cost=1) [!] costs less than replacing directly 1 with 4 (cost=3) [!] total difference ([1=>11] + [11=>4] - [1=>4]): -1 – JeremyR Dec 14 '12 at 18:39
Thanks for the bug report. It should work if you use the function "as.double" to transform your substition costs (sm <- as.double(sm)) – Matthias Studer Dec 15 '12 at 12:38
The substitution costs should respect the triangle inequality, otherwise the resulting distance won't respect it (and won't be a distance but a dissimilarity).
The optimal matching algorithm can be interpreted (if it respects triangle inequality) as the minimum cost required to transform sequence A in sequence B. However, the algorithm (Needlman-Wunsch) does not try to make successive substitutions, but only one. Hence, if you have the following costs (with indel=5):
A B C
A 0 1 3
B 1 0 1
C 3 1 0
Now consider computing the distance between sequences "A" and "C". The optimal matching distance will be 3 (substitution between A and C). But it would have cost less to first substitute A by B (cost 1) and then B by C (cost 1) for a total of 2.
This example illustrates why we should use substitution costs that respect triangle inequality. It ensures that the resulting distance is coherent. The distance between two points should always be equal or less than going through another one.
-
Thanks, Matthias. – JeremyR Dec 15 '12 at 13:07
The triangle inequality requires that the distance between any three points $A$, $B$, and $C$ meets the criteria that the distance between $A$ and $C$ is not larger than the distance between $A$ and $B$ plus the distance between $B$ and $C$. That is, $D(A,C) \le D(A,B) + D(C,B)$.
To guarantee that the substitution cost matrix does not lead to a violation of this rule, the costs should be set so that zeros appear only on the diagonal and any combination of smaller costs is larger than the largest cost.
-
Hi Jeremy. Could you please clarify this post? My concern is that (i) the triangle inequality is not what you've written and (ii) the equation does not match with either the triangle inequality or the description you've given. I suspect these mat just be typos. – cardinal Dec 15 '12 at 14:43 | 2015-05-25 23:36:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7591861486434937, "perplexity": 1540.475064888691}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928729.99/warc/CC-MAIN-20150521113208-00191-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://trac-hacks.org/ticket/10114 | Opened 4 years ago
# Reindex of empty repository fails with NoSuchChangeset
Reported by: Owned by: willmerae willmerae low FullTextSearchPlugin normal 0.12
### Description
Trac [/home/willmerae/src/fts-env]> fulltext reindex
Wiping search index and re-indexing all items in realms: ticket, wiki, milestone, changeset, attachment
NoSuchChangeset: No changeset 1 in the repository
### comment:1 follow-up: ↓ 2 Changed 4 years ago by rjollos
Off-topic question: Did you add all the attachments (ComponentDependencyPlugin and TagsPlugin eggs) to the FullTextSearchPlugin page, or is that someone posting spam? It is easy to fake someone's account here, which is why I ask.
### comment:2 in reply to: ↑ 1 Changed 4 years ago by willmerae
Off-topic question: Did you add all the attachments (ComponentDependencyPlugin and TagsPlugin eggs) to the FullTextSearchPlugin page, or is that someone posting spam? It is easy to fake someone's account here, which is why I ask.
I did. They're not on PyPI - I added them so setup.py install/pip install will (eventually) run with minimum of manual steps. They're not meant to be a permanent fixture, I'll post a request to the list for the respective authors to release on PyPI.
### comment:3 Changed 4 years ago by willmerae
• Status changed from new to assigned
• Summary changed from Reinndex of empty repository fails with NoSuchChangeset to Reindex of empty repository fails with NoSuchChangeset | 2016-10-28 19:51:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17832542955875397, "perplexity": 13927.393307614426}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988725470.56/warc/CC-MAIN-20161020183845-00402-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.natureof3laws.co.in/motional-emf-from-lorentz-force-class-12/ | Motional EMF from Lorentz force class 12
In this short piece of article, we will discuss and derive an expression for motional EMF from Lorentz force class 12, so let’s get started…
Motional EMF from Lorentz force
As we know that a conductor has large numbers of free electrons. When it moves through a magnetic field, a Lorentz force acting on the free electrons can set up a current. The below figure shows a rectangular conductor in which arm PQ is free to move.
It is placed in the uniform magnetic field B and directed normally into the plane of the paper. As the arm, PQ is moved towards the left with the velocity $v$, the free electrons of PQ also move at the same speed towards the left.
The electrons experience a magnetic Lorentz force, $F_m =qvB$. According to the fleming’s left-hand rule, this force act in the direction of QP, and hence, the free electrons will move towards P. A negative charge accumulates at P and a positive charge at Q.
An electric field $E$ is set up in the conductor from Q to P. This field exerts a force, $F_e = qE$ on the free electrons. The accumulation of charges at the two ends continues till these two forces balance each other. i.e $$F_m = F_e$$ Or $$qvB=qE\quad or\quad vB=E$$
The potential difference between the two ends Q and P are $$V=El=Blv$$ Clearly, it is the magnetic force on the moving free electrons that maintains the potential difference and produce the EMF $${\mathcal {E}} = Blv$$ As this EMF generated due to the motion of the conductor, so it is called motional EMF.
0%
2
Test your knowledge on "Motional EMF from Lorentz force" click start button to begin the quiz
Choose the correct option given below the question and check your score and answers at the end of the quiz.
1 / 5
1. What is the motional emf formula?
2 / 5
2. Which force on the moving free electrons that maintains the potential difference and produces the EMF
3 / 5
3. The potential difference between the two ends Q and P are
4 / 5
4. The accumulation of charges at the two ends continues till these two forces
5 / 5
5. Derivation of Motional EMF from Lorentz force consider
The average score is 50%
0% | 2022-11-28 11:03:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6673728227615356, "perplexity": 350.8141349012451}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710503.24/warc/CC-MAIN-20221128102824-20221128132824-00665.warc.gz"} |
https://www.rdocumentation.org/packages/spatstat/versions/1.64-1/topics/dppPowerExp | dppPowerExp
0th
Percentile
Power Exponential Spectral Determinantal Point Process Model
Function generating an instance of the Power Exponential Spectral determinantal point process model.
Usage
dppPowerExp(…)
Arguments
arguments of the form tag=value specifying the parameters. See Details.
Details
The Power Exponential Spectral DPP is defined in (Lavancier, Moller and Rubak, 2015) The possible parameters are:
• the intensity lambda as a positive numeric
• the scale parameter alpha as a positive numeric
• the shape parameter nu as a positive numeric (artificially required to be less than 20 in the code for numerical stability)
• the dimension d as a positive integer
Value
An object of class "detpointprocfamily".
References
Lavancier, F. Moller, J. and Rubak, E. (2015) Determinantal point process models and statistical inference Journal of the Royal Statistical Society, Series B 77, 853--977.
dppBessel, dppCauchy, dppGauss, dppMatern
• dppPowerExp
Examples
# NOT RUN {
m <- dppPowerExp(lambda=100, alpha=.01, nu=1, d=2)
# }
Documentation reproduced from package spatstat, version 1.64-1, License: GPL (>= 2)
Community examples
Looks like there are no examples yet. | 2020-10-19 16:24:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2706039249897003, "perplexity": 10519.986206730624}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107863364.0/warc/CC-MAIN-20201019145901-20201019175901-00192.warc.gz"} |
https://www.vedantu.com/question-answer/if-the-base-of-a-parallelogram-is-8-cm-and-its-class-10-maths-icse-5ee9f312f9a05a3f5d55349f | Question
# If the base of a parallelogram is 8 cm and its altitude is 5 cm then its area is equal to (in cm2).$A.{\text{ }}10 \\ B.{\text{ }}15 \\ C.{\text{ 20}} \\ D.{\text{ 4}}0 \\$
Verified
155.4k+ views
Hint: In this question use the concept that the area of the parallelogram is direct multiplication of its base and altitude so use this property to reach the solution of the question.
Given data
Base (b) of a parallelogram is 8 cm.
And the altitude or height (h) of a parallelogram is 5 cm.
Then we have to find out the area of the parallelogram.
As we know that the area (A) of the parallelogram is base multiplied by height.
$\Rightarrow A = b \times h$
Now substitute the values in this equation we have,
$\Rightarrow A = \left( {8{\text{ cm}}} \right) \times \left( {5{\text{ cm}}} \right)$
$\Rightarrow A = \left( {8 \times 5} \right){\text{ c}}{{\text{m}}^2}$
$\Rightarrow A = 40{\text{ c}}{{\text{m}}^2}$
So, the area of the parallelogram is 40 square centimeter.
So, this is the required answer.
Hence, option (D) is correct.
Note: Whenever we face such types of questions the key concept is the formula of area of the parallelogram which is stated above so direct multiply base and altitude of the parallelogram we get the required area of the parallelogram which is the required answer. Altitude is also known as the height of the parallelogram. | 2021-12-04 13:18:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9170122146606445, "perplexity": 313.81651874563215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362992.98/warc/CC-MAIN-20211204124328-20211204154328-00210.warc.gz"} |
https://math.stackexchange.com/questions/1409190/is-the-fundamental-weight-basis-a-k-a-dynkin-basis-an-orthonormal-basis | # Is the fundamental weight basis (a.k.a Dynkin basis) an orthonormal basis?
The simple root $\alpha_i$ basis is not an orthonormal basis, as can be seen from the Cartan matrix, which encodes how much they aren't orthonormal.
For simplicity, let's assume a simply-laced Lie algebra where the simple simple roots are self-dual.
In the Dynkin basis we write each weight with the components with respect to the fundamental weights $\Lambda_i$. These are defined by their property
$$\Lambda_i(\alpha_j) = \delta_{ij} .$$
Does this mean that the Dynkin basis is an orthonormal basis?
No. Look at $\mathfrak{sl}_3$. The simple roots are $\alpha_1,\alpha_2$ with $$(\alpha_1,\alpha_1)=2\;\;\;\mbox{and}\;\;\;(\alpha_1,\alpha_2)=-1.$$
Write $\Lambda_1=a\alpha_1+b\alpha_2$ and solve the system $$\begin{cases}(\Lambda_1,\alpha_1)=1\\(\Lambda_1,\alpha_2)=0\end{cases}$$ to get $\Lambda_1=\frac{2}{3}\alpha_1+\frac{1}{3}\alpha_2$ (similarly, $\Lambda_2=\frac{1}{3}\alpha_1+\frac{2}{3}\alpha_2$).
Now compute that $(\Lambda_1,\Lambda_1)=\frac{2}{3}$ and $(\Lambda_1,\Lambda_2)=\frac{1}{3}$.
• Thx a lot! The same calculation for $su(3)$ can be found here books.google.de/… – JakobH Aug 26 '15 at 9:18 | 2019-07-16 14:05:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9813374876976013, "perplexity": 367.24395360598555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195524568.14/warc/CC-MAIN-20190716135748-20190716161748-00258.warc.gz"} |
https://laserkelvin.github.io/linux/2020-02-18-file-compression/ | # Compressing Files in Linux
Here’s a quick post about a topic I find myself revisiting every few months: how to compress large batches of files efficiently. As I perform lots of calculations on a computing cluster, I like to routinely back things up at around publication time: that way I can have access to the data at the point where my paper was submitted, for example, and come back to it at a later date.
## Parallelization
Now, for UN*X users, a lot of people probably just use tar like so:
tar -cf new_tarball.tar.gz file1 file2 file3
By itself, tar doesn’t offer the best compression nor speed. There are a few more modern approaches that promise to be better. One approach, for example, are parallel implementations: every computer has multiple cores/threads these days, why not use them? Back in 2015, this blog post showed a quick benchmark of how parallel implementations of common compression algorithms perform:
SerialParallel
gzippigz
bzip2pbzip2
xzpxz
With some pretty dated hardware, you could still get >3x faster. The simple thing about this is the plug-and-play nature of it all; you can get pigz, pbzip2 through your package manager (e.g. sudo apt install pigz) or if you don’t have sudo access, you could get the binaries from their websites or with conda:
conda install -c bioconda pigz
Replace your regular tar command with this:
tar -I pigz -cf tarball.tar.gz file1 file2 file3
Maybe not necessarily newer, but there are algorithms that promise to be faster and higher compression ratios than tar bundled with most systems. These include the ones mentioned in the table above, as well as lrzip or “long range zip” which is apparently ~2x faster than bzip2 thanks to its LZMA algorithm. For even beefier compression, there’s ZPAQ which, according to the author of lrzip, “extreme compression while taking forever”.
As far as my experience goes, bzip2 and lrzip are as far as I’ve had to take it. Given how problem dependent compression efficiency actually is, for your routine compression I would simply recommend lrzip thanks to its ease of use, speed, and compression.
## Compressing lots of files
One scenario I find myself in often is taking many files and throwing them together - to do this quickly, I would do a first pass compressing the many files into a single, low compression tarball using one of the parallel algorithms. A second pass is done to further compress the file, where now the bottleneck isn’t grabbing many files, it’s simply to compress a single file.
So, something like this:
tar -cf single_tarball.tar file1 file2 ... file999999
lrztar single_tarball.tar
Should generate a single_tarball.tar.lrz that is more compressed than the original, and because you’re using parallelization to do many smaller files, ensures that your high compression algorithm is most spending its time working on a single file rather than many files. | 2021-03-03 10:46:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4143475890159607, "perplexity": 3285.1191407895717}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178366959.54/warc/CC-MAIN-20210303104028-20210303134028-00273.warc.gz"} |
https://tex.stackexchange.com/questions/448120/tikzcd-set-description-option-in-labels-of-arrows-as-default | # Tikzcd: set description option in labels of arrows as default
I want all arrows in my tikzcd diagrams to have labels over them, as in the example below.
\documentclass{article}
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}[sep = large]
T
\arrow[drr, bend left, "x" description]
\arrow[ddr, bend right, "y" description]
\arrow[dr, dotted, "{(x,y)}" description] & & \\
& X \times_Z Y \arrow[r, "p" description] \arrow[d, "q" description] & X
\arrow[d, "f" description] \\
& Y \arrow[r, "g" description] & Z
\end{tikzcd}
\end{document}
But I don't want to write all those ‘description’s. Is it possible to set an option to the tikzcd environment that makes it default? Is it also possible to have an option for \arrow that restores the usual behaviour for that arrow?
Use labels=description:
\documentclass{article}
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}[sep = large,labels=description]
T
\arrow[drr, bend left, "x"]
\arrow[ddr, bend right, "y"]
\arrow[dr, dotted, "{(x,y)}"] & & \\
& X \times_Z Y \arrow[r, "p"] \arrow[d, "q"] & X
\arrow[d, "f"] \\
& Y \arrow[r, "g"] & Z
\end{tikzcd}
\end{document} | 2019-10-19 07:05:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9927826523780823, "perplexity": 3816.825792366445}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986692126.27/warc/CC-MAIN-20191019063516-20191019091016-00283.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/elementary-algebra/chapter-4-proportions-percents-and-solving-inequalities-chapters-1-4-cumulative-review-problem-set-page-186/39 | ## Elementary Algebra
The solution set is {x|x $\gt$ 7} , or (7,$\infty$) in interval notation.
We use the properties of inequalities to solve this problem. 4x - 6 $\gt$ 3x + 1 4x - 3x $\gt$ 1 + 6 x $\gt$ 7 The solution set is {x|x $\gt$ 7} , or (7,$\infty$) in interval notation. | 2018-10-18 06:47:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9606802463531494, "perplexity": 1181.0988662398977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583511744.53/warc/CC-MAIN-20181018063902-20181018085402-00318.warc.gz"} |
http://mathoverflow.net/questions/33101/reducing-two-variable-linear-diophantine-equation-to-modular-inversion | # Reducing two variable linear Diophantine equation to modular inversion
I'm in the field of secure multiparty computation using Homomrphic encryption or secret sharing. I want to implement a secure protocol to compute the GCD of two encrypted numbers.
To calculate the GCD, I particularly need to be able to securely calculate the quotient of the division of two numbers. There is a secure protocol for that but is too expensive. Instead, I thought that I might use the much cheaper protocol for computing the modular inversion of an encrypted number as a building block for the GCD protocol.
Since both problems (quotient and modular inversion) can be reduced to solving a linear Diophantine equation then perhaps we can reduce one to the other:
Modular inversion $y=x^{-1} \mbox{ mod } p$:
$x y + p m = 1$
Quotient division $q=\lfloor \frac{a}{b} \rfloor$:
$q b + (a \mbox{ mod } b) t = a$
The question is whether we can rephrase this equation such that the right hand side is 1 (and still be a linear Diophantine), so we are able to use an existing modular inversion protocol to calculate the quotient division.
P.S.: I can't use the extended euclidean algorithm directly on any of them. The only allowed (secure) protocols to be used as building blocks are modular inversion, multiplication, modular division, and addition.
-
Both equations have infinitely many solution, and computing the quotient from an arbitrary solution to the latter equation will involve division. In general, we would have $t\equiv 1\pmod{b}$ but not necessarily $t=1$. And for $t\ne 1$, a division by $b$ will be required to reduce it to 1, which is not any better then simply dividing $a$ by $b$. – Max Alekseyev Jul 23 '10 at 17:11
@Speiser: No it is not elementary. I do not want to use extended GCD. In fact, I am in the field of secure multiparty comptuation and I want to implement the GCD algorithm itself in a secure form, given only a secure protocol for modular inversion (Using homomorphic encryption or secret-sharing). It would have been trivial to use extended GCD to solve these equations, but I don't have a secure sub-protocol for that. – M. Alaggan Jul 24 '10 at 0:48
@Alekseyev: Thanks for your useful comment. So since it is not possible I'll try another research direction. – M. Alaggan Jul 24 '10 at 0:49
In the future, you should probably include that kind of information with the original question. It looked a lot like an elementary question. – Qiaochu Yuan Jul 24 '10 at 1:08
@Yuan: Thanks for the hint. I do apologize for the readers who were mislead by this mistake; I am new to online community discussion and the line between not too long and too less information was still a bit fuzzy to me. I just was trying to be nice and write as short as I can so as not to waste the time of the reader. – M. Alaggan Jul 24 '10 at 1:13 | 2014-04-20 04:02:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8563434481620789, "perplexity": 382.48421112340816}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00471-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://xxxwiki.larseighner.com/index.php/Main/MIT1803SCDifferentialEquationsUnit2A | Main
# Modes and the Characteristic Equation
#### Contents
Introduction of the Characteristic Equation
Several presumptions are made in order to solve
{$$my^{\prime\prime} + by^{\prime} + ky = 0$$}.
This form emphasizes the application to spring-damper problems in which m is mass, k is the spring coefficient from Hooke's law, and b is a coefficient of damping. These are taken to be constant over the applicable conditions. Usually tacit is that the independent variable is t which is time in most physical problems. Dividing through by m results in the standard form:
{$$y^{\prime\prime} + Ay^{\prime} + By = 0$$}.
Some results depend on A and B being real numbers, which in physical modeling they should be.
From this we find the characteristic equation by substituting {$e^{rt}$} for y.
{\begin{align} y^{\prime\prime} + Ay^{\prime} + By &= 0 \cr ( e^{rt})^{\prime\prime} + A( e^{rt})^{\prime} + B( ) &= 0 \cr r^2e^{rt} + Are^{rt} + Be^{rt} &= 0 \cr \left(r^2 + Ar + B\right)e^{rt} &= 0 \cr r^2 + Ar + B &= 0 \cr \end{align}}.
This is a quadratic equation, because the differential equation is second order. It has two roots, and the possibilities are:
RootsCaseRootsSolution
Two real and unequal rootsoverdamped{$r_1,r_2$}{$y = c_1e^{r_1 t} + c_2e^{r_2 t}$}
Two real and equal rootsunderdamped{$r_1 = r_2$}{$y = c_1e^{r_1 t} + c_2te^{r_1 t}$}
Two complex rootscritically damped{$r=\alpha \pm \beta i$}{$y = e^{\alpha t}[ c_1\cos(\beta t) + c_2\sin(\beta t)]$}
Sources:
Recommended:
Categories: Differential Equations
This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong. | 2018-01-22 22:34:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9991662502288818, "perplexity": 1399.7714422320519}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891543.65/warc/CC-MAIN-20180122213051-20180122233051-00477.warc.gz"} |
https://www.physicsforums.com/threads/ac-inductor-phase-shift.598543/ | # AC Inductor phase shift
1. Apr 20, 2012
### fonz
This is quite a frustrating problem for me so hopefully somebody can describe it in such a way that it settles in my mind.
It is the concept of voltage and current being out of phase. Inductors are frequently described as responding to changes in current but does this really make sense?
The big problem for me is how the inductor is reacting to changing current on an alternating voltage supply. It is the voltage that is changing so how is it not responding to the changing voltage?
Trying to visualise what is happening when a voltage is applied across an inductor. Initially when the voltage is applied what will the current be? My own intuition would suggest that since initially there is no magnetic field to oppose the change there will be a large current inrush to build up the magnetic field correct? But how is this current 90 degrees out of phase with voltage?
It is difficult for me to explain what it is I cannot understand but hopefully by providing an explanation for these two questions I will begin to understand.
Thanks
Dan
2. Apr 20, 2012
### FOIWATER
I think I can explain it to you as I thought about it when it did not make sense to me.
You will have to read each thing I write to make sense of it, and if something doesn't make sense you will have to learn more about each thing.
Once you connect AC voltage across an inductor, it draws a specific amount of current, based on resistance. The current through the inductor creates a magnetic field around the coil. As you said, the voltage is what is constantly changing. But because the voltage is constantly changing, the current is constantly changing (in magnitude) and so the magnetic field is constantly changing around the inductor as well.
Think about it, as the voltage changes the current changes, (it is increasing and decreasing) so to the magnetic field is growing (expanding away from the inductor, and then collapsing onto the coil as the current (which creates the magnetic field) is decreasing in value.
Now as the field collapses, it CROSSES THE COIL. now the conditions for induction exist. we have a voltage induced back into the coil each and every time the current value is on the negative half cycle of the ac waveform. The voltage that is induced, OPPOSES the source voltage (see lenz' law) so, you now have a LOWER value of voltage applied.
This is the nature of inductive reactance, inductors have both resistance (due to the resistance of the wire composing the coil) and reactance. You can see, the reactive part is simply due to the voltage that the inductor creates when the current changes. It is created in such a direction that it opposes the source voltage, and as so it seems to drop voltage. But it does not it simply stores energy in the form of a magnetic field and every so often (60 times per second) returns the energy to the circuit.
Now think about what you commonly hear people say, inductors opposes current changes. How can this be so, is your question. Do you now see? this can be so because as current passes through a coil, and if the current is sinusoidal, it creates a magnetic field that is also sinusoidal. Which means the field expands and collapses on the coil depending on frequency (which is why inductive reactance depends on frequency X = 2pifL) And each time it collapses on it, it induces an opposing voltage into the coil that appears to impede the source voltage.
The current is out of phases ninty degrees (and so too, is the field) because it physically takes time to build the magnetic field around the conductor, and it stores the energy there. This does not happen at the same rate that electrons travel through a circuit.
Hope this helps,
PS: This accounts for why inductors appear as shorts in DC. Imagine, if the current is not changing, the field is now static (not changing as well) If it doesn't change, then the field can not cut the inductors and induce a voltage into it that opposes source voltage. IE, at 0 frequency inductive reactance is zero as well. but as soon as you open the circuit, or stop providing voltage across the coil, the field WILL collapse.. the voltage that is induced will cause a current to back feed in the circuit. This is the purpose of connected a diode reverse biased across coils in automotive circuits containing computers (many applications). It is called a free-wheeling diode, or flyback diode and allows the current produced by the coil to return to the other end of the coil rather than feeding through the circuit and damaging other equipment.
Last edited: Apr 20, 2012
3. Apr 20, 2012
### Averagesupernova
You have it backwards. When voltage is first applied to an inductor the current will be nearly zero. Then it steadily ramps up to max. In order for there to be a 90 degree phase shift it has to be this way.
4. Apr 20, 2012
### FOIWATER
is the reason the current is initially zero because this is the time the field is expanding? obviously not, because the magnetic flux is in phase with the current, and is out of phase with the voltage lagging by ninty degrees.
fonz, the physical reasoning behind lagging / leading currents in inductors/capacitors respectively (especially including resistors) has been a sort of hot topic on a few threads here, the math becomes especially important when describing WHY they are out of phase. My previous post just describes the action of an inductor in an ac circuit, which isn't really what you asked but may give you some understanding that you didn't have prior to reading it not sure.
5. Apr 20, 2012
### DragonPetter
Here is one way to think of it mathematically. I don't know if this is of any use or already obvious to you. It can at least show you that this magic number of 90 is a result of differentiation
given the inductance relation:
$V(t) = L\frac{di(t)}{dt}$
let $V(t) = sin(t)$
then rearrange:
V(t)dt = Ldi(t)
integrate:
$\int{sin(t)}dt = L\int{di(t)}$
$cos(t) = Li(t)$
$i(t) = \frac{cos(t)}{L}$
Now, knowing that sine and cosine are 90 degrees out of phase, we know $cos(t) = sin(t-90°)$
so we can say
$i(t) = \frac{sin(t-90°)}{L}$ which shows a phase shift of -90 degrees from the voltage function, so the current is lagging behind the voltage.
Last edited: Apr 20, 2012
6. Apr 20, 2012
### psparky
7. Apr 20, 2012
### psparky
Really cool explanation without using math.
Bravo.
8. Apr 20, 2012
### Antiphon
As an exercise for the advanced student I pose a little brain teaser.
1) All real wires exhibit some inductance even if very tiny.
2) The differentiation carried out above with sines and cosines is obviously correct and the phase is always 90 degrees regardless of the tiny nature of the inductance.
3) therefore you can NEVER have voltage and current in phase!
9. Apr 20, 2012
### Fuxue Jin
A piece of wire consist of three component, DCR, inductance L and capacitance C.
With DC current, frequency is zero, V and I follows V = IR, always in phase.
With AC current, when frequency is low, L dominate, when frequency is very high, C dominate.
Phase shift between V and I depends both on L and C for that specific frequency.
10. Apr 20, 2012
### Antiphon
Ok- so far so good. Then by all means, let us ensure that the inductance dominates any capacitance by setting the frequency F = 1 femto-Hertz. Clearly then at this frequency we will have AC voltage that takes 31.6 million years to complete a cycle with the voltage and current around 7 million years apart!
11. Apr 21, 2012
### fonz
Ok thank you for contributing so far, I've had a think and I'm starting to get to grips with this however I seem to have come across a bit of a paradox
If an unmagnetised inductor is connected to a voltage source I would say that the initial current is zero because the magnetic field of the inductor induces an emf to counter the source voltage. This would mean the voltage drop initially is equal to the source voltage. As the source voltage decreases the inductor generates an emf to sustain it's magnetic field and current increases?
How can an emf be induced initially to counter the source if there is no magnetic field?
12. Apr 21, 2012
### FOIWATER
Do you know this for a capacitor? would it be possible to PM it to me?
13. Apr 21, 2012
### Antiphon
Your thinking is confused in that the voltage applied to the inductor never changes.
If you connect a 1 Volt source to a 1 Henry inductor the voltage will be 1 Volt forever and the current I as a function of time t will be I=t.
The magnetic field will grow linearly in time as well.
14. Apr 22, 2012
### fonz
The trouble I am having is that the argument for why when full voltage is applied at t=0 the current is zero is that the magnetic field in the inductor induces an emf to counter the applied source voltage.
Initially where does the magnetic field come from to generate this back emf?
By Faraday's law, induced emf is a result of current flow but no current is flowing?
My only explanation of this is that when the voltage is applied, the inductor offers virtually no resistance so maximum current tries to flow. The inductor reacts to this by absorbing this current into it's magnetic field. This changing magnetic field produces the counter emf to prevent current from flowing.
Can somebody please confirm this explanation?
15. Apr 22, 2012
### Antiphon
There is no "back" emf.
When the voltage is applied to the inductor, that is the EMF.
For a 1 volt source connected to any inductor, the EMF is 1 Volt.
16. Apr 22, 2012
### psparky
The entire explanation lies right here.
L*di/d(t)=i(t)
It covers everything you said above.....
17. Apr 22, 2012
### psparky
C*dv/d(t)=i(t)
I find derivation to be most satisfying.
18. Apr 22, 2012
### fonz
Thanks!
19. Apr 22, 2012
### psparky
An inductor's reactance of JωL fills in the rest nicely as you now realize after reading all these wonderful threads.
20. Apr 22, 2012
### Fuxue Jin
Don't we need to separate the initial state and steady state for the original question?
For steady state, the voltage and current simply follow the rule V = L di/dt. If the voltage is sine wave AC, then it will be 90 phase shift for a PURE ideal inductor.
For initial state, no matter how quick the voltage ramps up from zero to whatever the level is, it STILL takes time, even though it is very short. AND the current will be starting from ZERO. | 2017-10-21 21:53:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6361209750175476, "perplexity": 561.9187723481247}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824899.43/warc/CC-MAIN-20171021205648-20171021225648-00553.warc.gz"} |
https://howlingpixel.com/i-en/Stellar-wind_bubble | # Stellar-wind bubble
A stellar-wind bubble is a cavity light years across filled with hot gas blown into the interstellar medium by the high-velocity (several thousand km/s) stellar wind from a single massive star of type O or B. Weaker stellar winds also blow bubble structures, which are also called astrospheres. The heliosphere blown by the solar wind, within which all the major planets of the Solar System are embedded, is a small example of a stellar-wind bubble.
Stellar-wind bubbles have a two-shock structure.[1] The freely-expanding stellar wind hits an inner termination shock, where its kinetic energy is thermalized, producing 106 K, X-ray-emitting plasma. The hot, high-pressure, shocked wind expands, driving a shock into the surrounding interstellar gas. If the surrounding gas is dense enough (number densities ${\displaystyle n>0.1{\mbox{ cm}}^{-3}}$ or so), the swept-up gas radiatively cools far faster than the hot interior, forming a thin, relatively dense shell around the hot, shocked wind.
## References
1. ^ Castor, J.; McCray, R.; Weaver, R. (1975). "Interstellar Bubbles". Astrophysical Journal Letters. 200: L107–L110. Bibcode:1975ApJ...200L.107C. doi:10.1086/181908.
Blitzar
Blitzars are a hypothetical type of astronomical object in which a spinning pulsar rapidly collapses into a black hole. They are proposed as an explanation for fast radio bursts (FRBs). The idea was proposed in 2013 by Heino Falcke and Luciano Rezzolla.
Bright giant
The luminosity class II in the Yerkes spectral classification is given to bright giants. These are stars which straddle the boundary between ordinary giants and supergiants, based on the appearance of their spectra.
CN star
A CN star is a star with strong cyanogen bands in its spectrum. Cyanogen is a simple molecule of one carbon atom and one nitrogen atom, with absorption bands around 388.9 and 421.6 nm. This group of stars was first noticed by Nancy G. Roman who called them 4150 stars.
Electroweak star
An electroweak star is a theoretical type of exotic star, whereby the gravitational collapse of the star is prevented by radiation pressure resulting from electroweak burning, that is, the energy released by conversion of quarks to leptons through the electroweak force. This process occurs in a volume at the star's core approximately the size of an apple, containing about two Earth masses.The stage of life of a star that produces an electroweak star is theorized to occur after a supernova collapse. Electroweak stars are denser than quark stars, and may form when quark degeneracy pressure is no longer able to withstand gravitational attraction, but may still be withstood by electroweak burning radiation pressure. This phase of a star's life may last upwards of 10 million years.
Frozen star (hypothetical star)
In astronomy, a frozen star, besides a disused term for a black hole, is a type of hypothetical star that, according to the astronomers Fred Adams and Gregory P. Laughlin, may appear in the future of the Universe when the metallicity of the interstellar medium is several times the solar value. Frozen stars would belong to a spectral class "H".
Helium-weak star
Helium-weak stars are chemically peculiar stars which have a weak helium lines for their spectral type. Their helium lines place them in a later (ie. cooler) spectral type then their hydrogen lines.
Lambda Boötis star
A Lambda Boötis star is a type of peculiar star which has an unusually low abundance of iron peak elements in its surface layers. One possible explanation for this is that it is the result of accretion of metal-poor gas from a circumstellar disc, and a second possibility is the accretion of material from a hot Jupiter suffering from mass loss. The prototype is Lambda Boötis.
A lead star is a low-metallicity star with an overabundance of lead and bismuth as compared to other products of the S-process.
List of hottest stars
This is a list of hottest stars so far discovered (excluding degenerate stars), arranged by decreasing temperature. The stars with temperatures higher than 60,000 K are included.
List of stellar properties
Pages Related to Stellar properties, Pages using the word stellar in a physics context.
Stellar aberration
Stellar aberration (derivation from Lorentz transformation)
Stellar age estimation
Stellar archaeology
Stellar astronomy
Stellar atmosphere
Stellar birthline
Stellar black hole
Stellar cartography
Stellar chemistry
Stellar chonography
Stellar classification
Stellar cluster
Stellar collision
Stellar core
Stellar coronae
Stellar density
Stellar disk
Stellar distance
Stellar drift
Stellar dynamics
Stellar engine
Stellar engineering
Stellar envelope see stellar atmosphere
Stellar evolution
Stellar flare
Stellar flux
Stellar fog
Stellar halo
Stellar interferometer
Stellar isochrone
Stellar kinematics
Stellar limb-darkening
Stellar luminosity
Stellar magnetic field
Stellar magnitude
Stellar mass
Stellar mass black hole
Stellar mass loss
Stellar molecule
Stellar near-collision
Stellar neighborhood
Stellar nucleosynthesis
Stellar nursery
Stellar occultation
Stellar parallax
Stellar physics
Stellar planetary
Stellar population
Stellar precession
Stellar pulsations
Stellar quake
Stellar remnant
Stellar rotation
Stellar scintillation
Stellar seismology
Stellar spectra
Stellar spheroid
Stellar spin-down
Stellar structure
Stellar surface fusion
Stellar system
Stellar triangulation
Stellar uplift
Stellar variation
Stellar vault
Stellar wind
Stellar wind (disambiguation)
Stellar wobble
Stellar X-ray astronomy
Stellar-wind bubble
Other
Catalog of Stellar Identifications
Fossil stellar magnetic field
General Catalogue of Stellar Radial Velocities
General Catalogue of Trigonometric Stellar Parallaxes
Interstellar cloud
Inter-stellar clouds
Interstellar medium
List of stellar angular diameters
List of stellar streams
Low-dimensional chaos in stellar pulsations
Mark III Stellar Interferometer
Michelson stellar interferometer
NEMO (Stellar Dynamics Toolbox)
Non-stellar astronomical object
Quasi-stellar object
Substellar object
Sub-stellar object
Sydney University Stellar Interferometer
TD1 Catalog of Stellar Ultraviolet Fluxes
Timeline of stellar astronomy
Utah state stellar cluster
Young stellar object
OB star
OB stars are hot, massive stars of spectral types O or early-type B that form in loosely organized groups called OB associations. They are short lived, and thus do not move very far from where they formed within their life. During their lifetime, they will emit much ultraviolet radiation. This radiation rapidly ionizes the surrounding interstellar gas of the giant molecular cloud, forming an H II region or Strömgren sphere.
In lists of spectra the "spectrum of OB" refers to "unknown, but belonging to an OB association so thus of early type".
Photometric-standard star
Photometric-standard stars are a series of stars that have had their light output in various passbands of photometric system measured very carefully. Other objects can be observed using CCD cameras or photoelectric photometers connected to a telescope, and the flux, or amount of light received, can be compared to a photometric-standard star to determine the exact brightness, or stellar magnitude, of the object.A current set of photometric-standard stars for UBVRI photometry was published by Arlo U. Landolt in 1992 in the Astronomical Journal.
Photosphere
The photosphere is a star's outer shell from which light is radiated. The term itself is derived from Ancient Greek roots, φῶς, φωτός/phos, photos meaning "light" and σφαῖρα/sphaira meaning "sphere", in reference to it being a spherical surface that is perceived to emit light. It extends into a star's surface until the plasma becomes opaque, equivalent to an optical depth of approximately 2/3, or equivalently, a depth from which 50% of light will escape without being scattered.
In other words, a photosphere is the deepest region of a luminous object, usually a star, that is transparent to photons of certain wavelengths.
Q star
A Q-Star, also known as a grey hole, is a hypothetical type of a compact, heavy neutron star with an exotic state of matter. The Q stands for a conserved particle number. A Q-Star may be mistaken for a stellar black hole.
Starfield (astronomy)
A starfield refers to a set of stars visible in an arbitrarily-sized field of view, usually in the context of some region of interest within the celestial sphere. For example: the starfield surrounding the stars Betelgeuse and Rigel could be defined as encompassing some or all of the Orion constellation.
Stellar atmosphere
The stellar atmosphere is the outer region of the volume of a star, lying above the stellar core, radiation zone and convection zone.
Stellar mass
Stellar mass is a phrase that is used by astronomers to describe the mass of a star. It is usually enumerated in terms of the Sun's mass as a proportion of a solar mass (M☉). Hence, the bright star Sirius has around 2.02 M☉. A star's mass will vary over its lifetime as additional mass becomes accreted, such as from a companion star, or mass is ejected with the stellar wind or pulsational behavior.
Supernova impostor
Supernova impostors are stellar explosions that appear at first to be a supernova but do not destroy their progenitor stars. As such, they are a class of extra-powerful novae. They are also known as Type V supernovae, Eta Carinae analogs, and giant eruptions of luminous blue variables (LBV).
Yellow giant
A yellow giant is a luminous giant star of low or intermediate mass (roughly 0.5–11 solar masses (M)) in a late phase of its stellar evolution. The outer atmosphere is inflated and tenuous, making the radius large and the surface temperature as low as 5,200-7500 K. The appearance of the yellow giant is from white to yellow, including the spectral types F and G. About 10.6 percent of all giant stars are yellow giants.
Formation
Evolution
Spectral
classification
Remnants
Hypothetical
Nucleosynthesis
Structure
Properties
Star systems
Earth-centric
observations
Lists
Related articles
This page is based on a Wikipedia article written by authors (here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses. | 2019-06-26 05:40:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6994205713272095, "perplexity": 4645.53253030588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000175.78/warc/CC-MAIN-20190626053719-20190626075719-00293.warc.gz"} |
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https://www.physicsforums.com/threads/combinatorics-starting-posets-relations.317332/ | Combinatorics: Starting Posets/Relations
1. Jun 1, 2009
1. The problem statement, all variables and given/known data
We say that a relation $$R$$ on a set X is symmetric if $$(x, y) \in R$$ implies $$(y, x) \in R$$ for all $$x, y \in X.$$ If $$X = \{a, b, c, d, e, f \}$$, how many symmetric relations are there on $$X$$? How many of these are reflexive?
2. Relevant equations
3. The attempt at a solution
At this point, I understand that there are $$2^6$$ subsets of X. I don't understand how to count the number of relations that are symmetric though. Also, I would have thought that since there are $$2^6$$ subsets, that there would be $$2^6$$ reflexive relations, but I know the answer to that question to be $$2^{15}$$. All help is appreciated!
2. Jun 1, 2009
matt grime
Try with a smaller example, like 3 elements {a,b,c} to begin with - or just try writing out a few symmetric relations and trying to see what needs to be true about them.
You notion of 2^6 implies that a relation (of some type) is purely defined by being a subset - if that were true then it wouldn't be a very interesting property.
3. Jun 1, 2009
JCVD
You mean to say that there are 2^15 relations on X that are both reflexive and symmetric (there are 2^30 reflexive relations). If you want to think about relations as sets, you should be looking at sets of ordered pairs whose entries come from X.
4. Jun 2, 2009
HallsofIvy
Staff Emeritus
A relation on X is NOT a subset of X. It is a subset of the Cartesian product of X with iteself. | 2017-11-23 17:12:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6763864755630493, "perplexity": 230.1869665906911}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806844.51/warc/CC-MAIN-20171123161612-20171123181612-00527.warc.gz"} |
http://astro.truth998877.com | ## Saturday, 29 September 2018
### The Photoelectric Effect - Light as a Particle
In two of my previous posts, here and here, which talk about blackbody radiation and ultraviolet catastrophe, the radiation is described in terms of the physics of waves. However, this is not the only way that light can be described. Consider the following...
In the above, light is being shone onto the metal plate (A) and a voltage (V) is applied between the the metal plate (A) and the cup (B). If the cup and the plate are made of the same material the the voltage measured by the volt meter is the same as the voltage between the plate and the cup. The light that is being shone onto the plate releases some electrons, and therefore the ammeter measures a current flowing. This effect is the photoelectric effect. The diagram of the apparatus may not be clear therefore is it worth pointing out that that there is no physical connection between the plate and the cup, hence the ammeter is measuring what is called a photoelectric current.
The above graph shows the typical results produced by such an aparatus, it shows the photoelectric current as a function of the applied voltage. The two lines on the graph represent the results obtained for two different intensities (brightness) of light shone upon the plate. In both set of results, if the voltage is made large enough the current reaches a maximum value and further increases in voltage produce no further increase in current. At this point the cup is collecting all of the electrons which are released by the plate.
If the voltage is reversed using the switch in the circuit some current will still flow, even through the voltage applied opposes the flow of the emitted electrons. This must mean that the electrons are released from the plate with an amount kinetic energy. However, it can be seen that if the reversed voltage is made large enough the current measured reduces to zero and it can also be seen that the voltage at which the current falls to zero is the same for both lines on the plot, where the two line represent results that would be obtained for two different light intensities. The voltage at which the current falls to zero, is called the stopping potential, and if this voltage is multiplied by the charge of an electron, the result gives the kinetic energy of the fastest electron emitted from the plate.
$$K_{max} = eV_0$$
This shows that the stopping potential is independent of the intensity of the light used. However, the stopping potential does depend upon the frequency of the light used. The representation below shows the typical behaviour found when the frequency of the light is varied and the stopping potential found
Therefore, for a given plate material, a linear relationship above a certain frequency is obtained, however, there is a cut off frequency below which no photoelectric effect occurs. This characteristic is produced no matter what the plate is made of.
There are three aspects of this effect that classical wave theory cannot explain:
1. Wave theory requires that as the intensity of the light increases so does its electrical fuel strength, E. If the lights electrical field strength increases then the force of the electron, eE. Thus, as intensity of the light increases, the force on the electron should get higher, an thus the kinetic energy which which the electron is emitted should get higher. However, this is at odds with the experimental results which suggests that the kinetic energy of the fasted electron emitted eV_nought is independent of the intensity of the light used.
2. According to wave theory, the photoelectric effect should occur no matter what frequency of light is used provided that the light is intense enough to eject the electron from the plate. However, the experimental results for any given surface there is frequency below which the photoelectric effect does not occur, a cut-off frequency, no matter what intensity of light is used.
3. In classical theory the light energy is uniformly distributed across its wave front and the area over which this energy can be absorbed by the electron is a circle having an atomic diameter. Thus if the light of low intensity is used there should be a time delay between the moment that light is shined upon the plate and the production of current. That is, it should take the electron a measurable amount of time to absorb the light energy provided to it. However, no detectible time delay has ever been recorded while studying the photoelectric effect.
So it wave theory cannot explain these aspects of the photoelectric, what theory can?
Well, a very famous theoretical physicist came up with a theory (for which he surprisingly won the Nobel prize in physics although he is very much more well known for two of his other theories) that explains all of the aspects above, Albert Einstein, and it was his theory of the photon, a particle of light. This other method that can be used to explain the behaviour of light will be the subject of my next post.
## Thursday, 1 December 2016
### Resolution of the Ultraviolet Catastrophe
In my last post I explained the origins of what is called the ultraviolet catastrophe, the failure of classical physics to correctly predict the signature of blackbody radiation. In this post I will attempt to explain how this catastrophe was resolved, however, if I'm wrong, then please do leave comment below.
The Rayleigh-Jeans formula, which gives the energy per unit volume (energy density) of blackbody radiation emitted inside a cavity which has been heated to a temperature T in the frequency interval $v$ to $v + dv$ is proportional to v squared
$$\rho_T (v)dv=\frac{8\pi v^2kT}{c^3}dv$$Hence this equation predicts that the higher the frequency of the radiation emitted, the higher its energy density should be. Whereas the experimentally observed energy density signature of blackbody radiation as a function of frequency for a given temperature does not continually grow with increasing frequency, rather it peaks and then falls back. The graph below illustrates this discrepancy, with the results from the Rayleigh-Jeans formula given by the dashed line and the solid line is the experimental results for a Temperature of 1500K
So how is the above catastrophe resolved...
Classical physics tells us that for a system with many entities which are in thermal equilibrium at temperature T, their kinetic energies will have a specific distribution, the Boltzmann distribution. For which the average kinetic energy per degree of freedom is given by 1/2kT, where k is Boltzmann's constant. Thus according to classical physics, the average kinetic energy of the waves of thermal radiation being emitted from the internal surface of a cavity, the cavity having been heated to a temperature T, is given by this equation - each wave is in thermal equilibrium with every other one and they only have one degree of freedom, the amplitude of their electric field. A common property of physical systems is that their total energy is twice their kinetic energy, thus each of the waves of the thermal radiation has the average total energy:
$$\bar\epsilon = kT$$
This equation tells us that each wave has the same average total energy, and the value is independent of frequency. If fact, this equation is an integral part of the Rayleigh-Jeans formula that gives the energy density of the thermal radiation. It is simply the product of the average total energy of each wave, as given above, multiplied by the number of waves of radiation that can fit within the cavity within the frequency interval $v$ and $v + dv$.
From the graph above it can be seen that for waves that have small frequencies, the prediction of the Rayleigh-Jeans formula and the experimental results are in close agreement, that is, a wave's average total energy tends to $kT$ as its frequency tends to 0
$\bar\varepsilon \xrightarrow [\ v \rightarrow 0]{}kT$ (1)
However, the experimental results show that the average total energy of the waves tend to zero as their frequency tends to very high values, that is
$\bar\varepsilon \xrightarrow[\ v \rightarrow \infty]{}0$ (2)
From these two tendencies it is clear to see that the average total energy of each wave is, in fact, dependent upon its frequency, that is each wave does not have the same total energy.
To determine the actual energy of each wave it is necessary to look at how the $\bar\varepsilon=kT$ result is determined.
$$P(\varepsilon)=\frac{e^{-\varepsilon/{kT}}}{kT}$$
The above equation characterises a special form of the Boltzmann Distribution, where $P(\varepsilon)d\varepsilon$ is the probability of finding a body of a system with an energy within the interval $\varepsilon$ to $\varepsilon + d\varepsilon$, where the number of states for that body within the interval is independent of $\varepsilon$. Using this function, the average total energy $\varepsilon$ can be found as follows:
$$\bar\varepsilon=\int_0^\infty \varepsilon P(\varepsilon) d\varepsilon$$
As carrying out an integral is simply determining the area under a graph, it is worth considering the function $\varepsilon P(\varepsilon)$ graphically:
It was the great scientist, Planck, who first realised that the relationship between $\varepsilon$ and $v$ as set out by (1) and (2) could be achieved by treating $\varepsilon$ as a discrete variable rather than a continuous one, that is by assuming that the energies within the energy distribution of the wave can only have discrete values, that is
$$\varepsilon=0,\Delta\varepsilon,2\Delta\varepsilon,3\Delta\varepsilon,4\Delta\varepsilon...$$
The effect of treating $\varepsilon$ as a discrete variable rather than a continuous variable can be seen for small and large values of $\Delta\varepsilon$ in the following graphs
Where $\Delta\varepsilon$ is small, the area under the graph (shaded) is very similar to that where $\varepsilon$ is treated as a continuous variable, that is the area tends to $kT$. Where $\Delta\varepsilon$ is large the area under the graph falls below that where $\varepsilon$ is treated as a continuous variable. It is easy to see that as $\Delta\varepsilon$ is made very large the area under the graph tends to 0, that is
$$\bar\varepsilon \xrightarrow[\ \Delta\varepsilon \rightarrow 0]{}kT$$
$$\bar\varepsilon \xrightarrow[\ \Delta\varepsilon \rightarrow \infty]{}0$$
These two tendencies are very similar to the tendencies set out by (1) and (2), in fact they can be related by simply stating that:
$$\Delta\varepsilon \propto v$$
To convert this from a proportionality to a relationship a constant of proportionality, $h$, is required. This constant is the well known Planck's constant.
Thus the Ultraviolet catastrophe is resolved by assuming that waves of blackbody radiation can only have discrete values of energy within their energy distribution, and that the discrete values of energy are a dependent on the wave's frequency. These concepts were used by Planck to derive his blackbody spectrum formula
$$\rho_T (v)dv=\frac{8\pi v^2kT}{c^3} \frac{hv}{e^{hv/kT}-1}$$
The derivation of this formula will be the subject of a separate post.
## Tuesday, 22 November 2016
### Blackbody Radiation and the Ultraviolet Catastrophe
So here it is, the promised post on the ultraviolet catastrophe.
The eagle-eyed amongst you have of course noticed the that title of this post also refers to blackbody radiation. Well the ultraviolet catastrophe cannot really be explained without first explain what blackbody radiation is. So here goes...
Blackbody radiation is simply a type of thermal radiation that is given off by certain types of materials, ones that absorb all thermal radiation they are exposed to, and reflect none of it. As it is the radiation that is reflected by a material that defines its colour, these materials have no colour, that is, they are black, hence their name. They are called blackbodies.
It is common experience that when items are heated, they glow, they emit radiation and not just radiation that we can see, they emit radiation across the whole spectrum. The signature that is emitted depends upon what the material is made of, however the signature of the radiation emitted by all blackbodies is identical, regardless of the material they are made.
The above graph gives the signature of blackbody radiation for a blackbody at three different temperatures 1000K, 1500K and 2000K as a function of frequency of the radiation emitted. The value on the y-axis, RT(v), is called spectral radiancy, which is defined such that RT(v) dv is equal to the energy emitted per unit time in radiation of frequency within the interval v to v + dv from a unit area of the surface at a temperature T.
Other than objects which have no colour, there are other entities that can behave as if they were blackbodies. One example is a hollow object which has a very small hole leading to the void inside of it. Radiation incident upon this hole will be wholly absorbed if the internal surface area of the void inside the object is large compared to the size of the hole. Thus, if the hole absorbs all thermal radiation that happens to be incident upon it, then it is behaving exactly as if it were a blackbody.
It is not too much of a leap from the behaviour explained above to see that if the objected is heated some of the thermal radiation emitted from the surface of void will pass through the whole and therefore this emitted radiation will bear a blackbody signature - the hoe acts as a blackbody. However, as the radiation emitted through the hole is just a sample of the thermal radiation inside of the void, then the thermal radiation inside the hole must also bear a blackbody signature.
So, there you go, blackbody radiation in a nutshell. So where does the ultraviolet catastrophe fit into all of this. Well it comes about in efforts to predict the the signature of the blackbody radiation.
Using classical arguments (which I will go through in a future post) the following equation is arrived upon which should predicted the signature of blackbody radiation
This equation is called the Rayleigh-Jeans formula for blackbody radiation. It gives the energy per unit volume of the cavity at temperature T in the frequency interval v to v + dv. It can be seen that the energy density, as calculated by the above equation, is dependant upon the square of the frequency of the radiation, and as such, as the frequency increases the calculated energy density also increases. However, this is at odds with the signature of blackbody radiation which, as frequency increases, increases to a peak then falls back. It is the failure of the Rayleigh-Jeans formula to correctly predict the signature of blackbody radiation that is known in physics as the ultraviolet catastrophe.
## Tuesday, 15 November 2016
### Where have I been?
Studies have had to take a back seat for the past few months, life has got in the way. I've moved house and had to do all the things that this entails, include a fair amount of re-decorating, I've been extremely busy at work, working toward the first flight of the product I'm working on. However, as I'm now much more settled in to the new house, and I'm going to be less busy at work, I should have some more time to study.
In fact, I have in the last couple of days read some of the first chapter of Eisberg & Resnick and came across a phrase that I don't think I ever came across while studying my degree (or perhaps I have just forgotten it), the 'ultraviolet catastrophe' - one of the reasons behind the need for QM, and as so I think I should do a more in depth blog post about it at some point in the future.
I really do need to do some maths, one cannot study physics in depth, without maths. More blog post on maths will follow also...
## Monday, 21 March 2016
### Review of The Theoretical Minimum - Chapter 1
This post is really the first post where I will use my blog for what I intent, and that is to review the things I have learned, or should I say relearned.
I have worked through Chapter 1 of Leonard Susskind first book in his "Theoretical Minimum" series. The aim of these books, along with his lecture videos, is to give the reader the minimum amount of physics knowledge to understand cutting edge physics. The first book covers classical physics and in the first chapter he sets out a number of topics. These include a description of what is meant by classical physics, a description of simple dynamical systems and their space-of-states, dynamical laws which are allowable along with those that are not, how cycles within a systems space-of-states lead to conservations laws and finally how the initial conditions of system can never be know with infinite accuracy and therefore measurements carried out upon a system will always be subject to uncertainty.
What is Classical Physics?
Susskind sets out that classical physics/mechanics is used to describe physical phenomena for which a quantum uncertainties need not be accounted for. These include Newton's Laws of motion, Maxwell's and Faradays laws of electromagnetism, and Einstein's Special and General theory's of Relativity.
He also alludes that the job of classical mechanics is to predict the future. However, to predict the future every thing about a system at a given point in time needs to be known and how the system will change with time. He also sets out that the laws by which systems change with time must be deterministic and reversible, that is not only must they be able to predict the future, they must also be able to be used to determine what happened in the past.
Simple Dynamical Systems and Space of States
Two types of system are set out, where is system is simply a collection of objects:
• Closed system - either the entire universe or a collection of objects that are some remote that they behave as if nothing else exists
• Open system - A collection of objects which are open to external influence.
What is meant by deterministic and reversible laws are set out using some very elementary examples of systems. A coin which is affixed to a table, which obviously can't change state, as its space of state is very narrow, containing just the one state. The law which describes how this system changes with time is both perfectly deterministic and reversible. The coin remains in the same state for all time, it has no choice.
A coin that is free to flip has a slightly wider space of states, that is, it has two states which is either heads or tails. There are many laws which could be set out to define how this system evolves with time however, two of the simplest are used. The first is if the coin starts at heads, it flips to tails, and if it is tails it flips to heads. The second is if the coin starts at tails, it flips to heads and if it is heads it flips to tails. (These two laws are in fact the same)
The laws which describe the time evolution of both of the above systems are perfectly deterministic and perfectly reversible, if the state of the coin is known at any point in time then the law(s) can be used to determine what will happen in the future or what has happened in the past with absolute certainty.
The concepts from the two 'coin' systems above are then expanded to include six sided dice, that is a system which has six states. Numerous dynamical laws could be use to describe the evolution of this system, for example a simple cyclic law where the dice cycles though sides 1 to 6 in order and then back to 1.
Or a cycle for example 1 to 3 to 2 to 6 to 4 to 5 and back to 1. This second cycle is in fact the same as the first as each in the state in the second cycle could simply be relabelled to arrive at the first. However, here multiple cycles within a system is introduced. For example, two cycles - 1 to 2 to 3 and back to 1, and 4 to 5 to 6 and back to 4 - or even three cycles - 1 to 2 and back to 1, 3 to 4 and back to 3, and 5 to 6 and back to 6. The point is that even with multiple cycles within the space of states, the law which describes the whole system is perfectly deterministic and reversible - if the system is in any one of the states, then the next or previous state can be determined from the law.
Laws not Allowed
Examples of the types of laws that are not allowed are also given using simple systems similar to those used above. For example a three state system where the system cycles from 1 to 2 to 3 and back to 2. This system at first glance appears to be deterministic, however, if this cycle is reversed it leads to a problem, in that it is not clear which state is next if the system is in state 2, should it be 1 or 3? Also if the system is in state 1, which state is next? Therefore this law is deterministic, however it is not reversible. Laws of this type violate one of the central cannons of physics and that is the conservation of information. The information about which state the system starts in is lost.
Infinite Space of States and Conservation Laws
More realistic systems obviously have an infinite space of states, however, cycles within such systems can and do exist. In fact, such cycles are equivalent to quantities which are conserved, and directly lead to conservation laws
Precision
Finally it is pointed out that it is not possible to measure the initial conditions of the system with infinite accuracy, therefore for any set of measurements carried out on a system there will always be uncertainly in the prediction of the outcome of the system even if the dynamical law that describes the system of interest is known.
That's it for chapter 1. The next I shall review from this book is interlude 1, a section on Spaces, Trigonometry and Vectors.
## Thursday, 11 February 2016
### Gravitational Waves Detected
With today's announcement of the detection of gravitational waves and the discovery of the Higg's Boson a couple of years ago, there is no better time to study physics!
## Wednesday, 10 February 2016
### My book list and tomorrow's announcement
As I said in my first post, I have a number of physics text books which I aim to work through. Below is the order in which I will proceed through them
Introduction to Classical Mechanics, French and Ebison
Electricity and Magnetism, Duffin
Electromagnetism, Grant and Phillips | 2022-06-28 14:47:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7590797543525696, "perplexity": 273.7730505734313}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103556871.29/warc/CC-MAIN-20220628142305-20220628172305-00638.warc.gz"} |
http://demechanica.com/strength-of-materials/chapters/the-bar-torsional-loading/elastoplastic-behavior-during-torsion/ | # Elastoplastic behavior during torsion
## Plastic moment
Consider a bar with a thick-walled circular cross-section. The bar is modeled by an elastic-ideally-plastic material behavior, in shear, according to the figure below.
Let us determine the moment corresponding to initial yielding. This is obtained by using the relations in the previous section and considering the case when $\tau_{\rm y}$ for $r=b$ (where the stress is largest).
$$\tau_\mathrm{y} = \frac{ \Mv_\mathrm{y} b}{\Kv} \quad \Rightarrow \quad \Mv_\mathrm{y}= \frac{\tau_\mathrm{y} \, \Kv}{b}$$
We can also determine the plastic moment $\Mv_\mathrm{u}$ where the entire cross-section is in a plastic state. In this case the stress is $\tau=\tau_{\rm y}$ for all $r$ which gives
$$\Mv_\mathrm{u}=\int_A \tau_{\mathrm y} \, r \, {\mathrm d}A= \tau_{\mathrm y} \int_a^b r \,2 \pi r \, \dr= \tau_{\mathrm y} \, \frac{2 \pi}{3} (b^3-a^3)$$
An illustration of the corresponding stress state is shown in the figure below.
Note
The plastic moment $\Mv_\mathrm{u}$ is the limit load the cross-section can withstand before failing.
### Plastic strengthening
The plastic strengthening is defined as:
$$\beta = \frac{\Mv_\mathrm{u} – \Mv_\mathrm{y}}{ \Mv_\mathrm{y} } = \ldots =\frac{4b}{3} \cdot\frac{ b^3-a^3}{b^4-a^4}-1$$
and is a factor indicating how much the torque can be increased if one allows the cross-section to reach a fully plastic state, compared to initial yielding. For thin-walled cross-sections $\beta=0$, since the cross-section reaches a fully plastic state as soon as one point starts to yield. | 2018-10-15 11:23:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6144505143165588, "perplexity": 876.0202533960544}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583509170.2/warc/CC-MAIN-20181015100606-20181015122106-00415.warc.gz"} |
https://docs.google.com/forms/d/e/1FAIpQLSf6JXAPc12toriICIm5pIn-vHCXN9Yl0pG2CNmnxkcKLhM34g/viewform | KinoDynamique 2019
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WE ARE ALREADY FULL!. We received more than 100 registrations which is awsome but we reached our maximum capacity.
But we will make an exception for local participants who can offer sleeping places for internationals. Those and only those can still register. For everybody else we're sorry but it would not make sense to accept more people than our Lab has space for.
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kino5 is celebrating it's 15th birthday - and what better way than with an international KinoKabaret, or as we call it in Vienna, KinoDynamique. We start on August the 15th and are excited to produce tons of fresh, innovative and diverse shortfilms with you till the final screening on the 23rd.
If you haven't been to a Dynamique or any other KinoKabaret yet, here are some impressions of what you may expect (www.kino5.net/kinodynamiqulocal e/was-ist-kinodynamique).
We have found a nice kinolab that is gonna serve you as the homebase for all your projects and which is accessable by wheelchair. We're gonna send you an e-mail with detailed infos as soon as your registration is complete. So sign up right now and join other international and filmmakers to create something new and exciting!
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This content is neither created nor endorsed by Google. Report Abuse - Terms of Service | 2019-07-20 12:22:00 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8984574675559998, "perplexity": 1961.6520525052115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526508.29/warc/CC-MAIN-20190720111631-20190720133631-00076.warc.gz"} |
https://bio.libretexts.org/Bookshelves/Cell_and_Molecular_Biology/Book%3A_Cells_-_Molecules_and_Mechanisms_(Wong)/4%3A_Membranes_-_Structure%2C_Properties_and_Function/4.2%3A_Membrane_Permeability | # 4.2: Membrane Permeability
A pure phospholipid bilayer, whatever the lipid composition, is a semi-permeable membrane that is generally repellent to large molecules and to ions. Small polar molecules can sometimes pass easily (e.g. ethanol), but more often pass at low rates if at all (e.g. water). However, small nonpolar molecules are able to pass through the membrane with relative ease. The reasons should be self-evident: larger molecules simply cannot fit between the lipid molecules to make their way through. Small molecules that can fit must be hydrophobic, otherwise the fatty acyl core of the membrane will repel them and block them from proceeding. Higher concentrations of cholesterol, by filling in gaps between phospholipid tails, decreases permeability even for small molecules that can normally pass through the membrane easily. Cells need far more than small nonpolar molecules for their material and energy requirements. Fortunately for life on Earth, the membranes of living cells are not purely phospholipids, and as we will see, proteins embedded in the phospholipid bilayer can form conveyances for the transport of many different molecules in and out of the membrane.
Figure 7. A pure phospholipid bilayer is inherently semi-permeable.
In fact, the observation of saturation kinetics in glucose transport in erythrocyte membranes was the first indication of protein-mediated transport (the GLUT1 glucose transporter). Another telling observation was the finding that glucose permeability through erythrocyte membranes is a million times greater than that through an artificial lipid bilayer. The concentration of glucose in the blood is relatively high compared to that inside of most cells, so this is mediated transport, but passive transport since it is going down the concentration gradient. To facilitate the process by preventing a buildup of glucose concentration in the cell, the first step of glucose metabolism is phosphorylation to convert it into a different molecule, glucose-6-phosphate. Thus the concentration of glucose stays very low, and it flows readily from the bloodstream into the cell.
There are some obvious differences between transport of molecules directly through the lipid bilayer (nonmediated transport) and transport using a protein facilitator embedded in the membrane (mediated transport). Nonmediated transport is governed by diffusion: the solute moves from areas of high concentration to areas of low concentration, thereby eliminating the gradient. As long as a solute (A) can get through the membrane, its flux (J) is determined solely by concentration difference and the permeability (P) of the membrane: JA = PA ([A]out - [A]in) and the relationship between the ux across the membrane and the concentration differential is linear.
This is not the case in mediated transport. As the name implies, a protein intermediary is required, and alarm bells should be going off in your head saying, “there’s a limit,” to the number of available transport proteins at any given time. Therefore, just as we saw with enzyme kinetics in chapter 3, the ux of solutes going through a transporter is not linearly related to the concentration differential across the membrane, though there is still a concentration effect. Instead, the relationship is logarithmic, reaching a saturation plateau once all the available transport proteins are in use. At that point, increasing the concentration of the solute will not increase its flux across the membrane. Thus for simple unidirectional mediated transport of a solute ($$B$$), the flux ($$J$$) can be expressed as a value of the affinity of the transporter for the solute ($$K_M$$) and the concentration of the solute:
$J_B = \dfrac{J_{max}[B]}{K_M + [B]}$
Membrane permeability allows for the possibility of concentration gradients across membranes, which in turn have potential energy associated with the concentration dif- ferential across the membrane. This turns out to be a phenomenally important source of cellular energy, and is the basis for aerobic synthesis of ATP by oxidative phosphorylation (chapter 5). However, to have a meaningful discussion of how concentration differences across semipermeable membranes store energy, we should review some basic concepts rst.
If a point source (e.g. a “glob”) of a solute (e.g. honey) is placed into a solvent (e.g. tea), it starts to dissolve, and as it does so, the concentration of solute near the point source will start out much higher than the concentration towards the periphery of the container (e.g. teacup). Over time, the solute then diffuses from the point source outward in all avail- able directions, and eventually the concentration of solute is equal at any point in tea- cup-space. This behavior is governed by the Second Law of Thermodynamics. The solute is initially concentrated, which means that its constituent molecules are relatively orga- nized. By the second law, these molecules will tend toward chaos, moving away from the constraints of the initial point toward an area with lower concentrations of the solute.
Now, imagine a temporary wall around the point source. The natural tendency is for the solutes to spread out, so by preventing that movement, you have bottled up some potential energy. Of course, this is only potential energy if there is some chance that the solutes can eventually go through the barrier (e.g. the wall has windows that can be opened). If the solutes have absolutely zero chance of passing through, then there is no potential energy because there is no potential to get out and about. Recalling the Energy chapter in which the second law was introduced, the chemical potential energy of a solute is
$G=RT \ln[A]+G^\prime$
so the chemical potential difference across a membrane is then
$ΔG= RT \ln \left(\dfrac{[A_i]}{[A_o]}\right).$
Now imagine this as something like a hydroelectric dam, where there is a great deal of pressure building up behind the dam, which can be utilized when some of the water is allowed through, powering turbines that generate electricity. In the biological case, there is concentration pressure building up both inside and outside the cell because the natural thermodynamic tendency is to bring the inside and outside concentrations of each solute to equilibrium. When this pressure is released by allowing the ions or other molecules to ow across the membrane, energy is released, and may be captured and used. The most direct example of this the proton-gradient-driven ATP synthase in the inner mitochondrial membrane (Chapter 5), which contains a direct molecular equivalent to the turning of a water wheel with the ow of water. For another example, if we look at [Na+] in an animal cell, the extracellular concentration is much higher than that intracellularly. When a Na+ channel is opened, the Na+ ions rush inward to try to equalize the concentration of Na+ inside and outside of the cell. Equilibrium is not actually reached in a living cell because Na+ channels are tightly regulated and only open for short periods of time.
In cells, concentration gradients of ions are great energy sources because the lipid part of the membrane is strongly repellent to ions, preventing them from passing through, but the membrane is embedded with channels and transporters that can allow the ions through if and when they are open. Because ions have both concentration differentials and charge differentials across the membrane, the electrochemical potential difference across the membrane is represented by a modification of the chemical potential difference equation with a term that takes that electrical charge into account:
$ΔG=RT \ln \left( \dfrac{[A_i]}{[A_o]}\right) + ZFΔΨ$
Z is the charge of the ion (e.g. +1 for Na+, -1 for Cl-, +2 for Ca2+), F is the Faraday constant (9.6485 x 105 C/mol), and ΔΨ is the membrane potential. In an average animal cell, the membrane potential is approximately -70mV. The number is negative to show that the inside of the cell is negative with respect to the outside. Thus, again considering Na+, not only is there a chemical gradient of more Na+ ions outside the cell than inside, there is also a charge gradient of more positive charges outside the cell to inside, so both forces contribute to the energy of Na+ flow into the cell. The equilibrium potential of one ion (e.g. Na+) across a membrane is determined by the Nernst equation:
$E_m= \left(\dfrac{RT}{zF}\right) \ln \left(\dfrac{[Na^+]_{out}}{[Na^{+}]_{in}}\right)$
which is extended in the Goldman equation (also Goldman-Hodgkin-Katz equation) that calculates membrane potential based on multiple ion gradients. For most ani- mal cells, a good approximation of the overall membrane potential can be calculated using the three major gradients: Na+, K+, and Cl-. There are, of course, other ion gradients, but their contributions are normally much smaller than these three.
The membrane potential is relatively stable in non-excitable cells, but in neurons and muscle cells, the membrane potential is quite dynamic, so the membrane potential in a non-excited state is referred to in these cells as the resting potential. The membrane (resting) potential in most animal cells is around -70mV. This is due in large part to the presence of K+ leak channels. These channels leak K+ from the cell down the concentration gradient until the chemical potential difference of K+ is at equilibrium with the membrane potential. In other words, the gradient pushing K+ out will eventually be stopped by an equal force from the gradient pushing positive ions (including K+) back in. There are also Na+ and Cl- leak channels, but there are far fewer and they contribute far less to the resting potential than K+.
Potassium leak channels are structurally as well as functionally different from other potassium channels. Where most K+ channels have one pore domain, the leak channels have two. While leak channels, by definition are not voltage gated, nor appreciably activated or inactivated, this is not true of all members of the tandem pore domain family of potassium channels. Interestingly, some (e.g. TASK-1) are mechanoreceptors, opening in response to membrane stretch, and others act as thermoreceoptors, with heat-sensitive activation (e.g. TREK-1).
Although we most commonly think of water as the solvent in which “interesting” molecules (e.g. ions) diffuse, its concentration and movement across membranes has important biological consequences. Osmosis is a term that specifically refers to the diffusion of water across a membrane. In this case, water is considered a solute rather than a solvent, so that if a water-permeant liposome, embedded with aquaporin channels to allow the passage of water, is placed in a very salty saline solution, the cell will shrink because there is a lower ratio of water to dissolved salts outside of the cell than inside. This is a hypertonic solution relative to the cell. Therefore the water flows from the cell (higher water concentration) out into the saline (with lower water concentration). Conversely, a cell placed in distilled and deionized water will swell and potentially burst because the water rushes from the highest possible concentration (pure water) to a cytoplasm with lower water concentration (because dissolved in it are various ions and other molecules). This is an example of a hypotonic solution. An isotonic solution will have the same concentration of water inside and outside of the cell.
Figure 8. Osmosis. An artificial “cell” that is permeable to water but not to Na+, K+, or Cl- is placed in a aqueous solutions of varying salinity. | 2019-09-15 15:57:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5739284753799438, "perplexity": 1259.648407816272}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514571651.9/warc/CC-MAIN-20190915155225-20190915181225-00165.warc.gz"} |
https://www.aimsciences.org/journal/1937-1632/2020/13/12 | # American Institute of Mathematical Sciences
ISSN:
1937-1632
eISSN:
1937-1179
All Issues
## Discrete & Continuous Dynamical Systems - S
December 2020 , Volume 13 , Issue 12
Issue in honor of Gisèle Goldstein on the occasion of her 60th birthday
Select all articles
Export/Reference:
2020, 13(12): i-ii doi: 10.3934/dcdss.2020422 +[Abstract](566) +[HTML](274) +[PDF](98.87KB)
Abstract:
2020, 13(12): 3285-3304 doi: 10.3934/dcdss.2020240 +[Abstract](1023) +[HTML](348) +[PDF](436.14KB)
Abstract:
We are devoted with fractional abstract Cauchy problems. Required conditions on spaces and operators are given guaranteeing existence and uniqueness of solutions. An inverse problem is also studied. Applications from partial differential equations are given to illustrate the abstract fractional degenerate differential problems.
2020, 13(12): 3305-3317 doi: 10.3934/dcdss.2020111 +[Abstract](1179) +[HTML](407) +[PDF](338.68KB)
Abstract:
Here we present fractional univariate Ostrowski-Sugeno Fuzzy type inequalities. These are of Ostrowski-like inequalities in the setting of Sugeno fuzzy integral and its special-particular properties. In a fractional environment, they give tight upper bounds to the deviation of a function from its Sugeno-fuzzy averages. The fractional derivatives we use are of Canavati and Caputo types. This work is greatly inspired by [8], [1] and [2].
2020, 13(12): 3319-3334 doi: 10.3934/dcdss.2020161 +[Abstract](1219) +[HTML](372) +[PDF](378.15KB)
Abstract:
In this paper we prove the existence of global classical solutions to continuous coagulation–fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either rate. This is achieved by proving a new result on the analyticity of the fragmentation semigroup and then using its regularizing properties to prove the local and then, under a stronger assumption, the global classical solvability of the coagulation–fragmentation equation considered as a semilinear perturbation of the linear fragmentation equation. Furthermore, we show that weak solutions of the coagulation–fragmentation equation, obtained by the weak compactness method, coincide with the classical local in time solutions provided the latter exist.
2020, 13(12): 3335-3345 doi: 10.3934/dcdss.2020241 +[Abstract](1074) +[HTML](333) +[PDF](407.72KB)
Abstract:
The aim of this paper is investigating the existence of at least one weak bounded solution of the quasilinear elliptic problem
where \begin{document}$\Omega \subset \mathbb R^N$\end{document} is an open bounded domain and \begin{document}$A(x,t,\xi)$\end{document}, \begin{document}$f(x,t)$\end{document} are given real functions, with \begin{document}$A_t = \frac{\partial A}{\partial t}$\end{document}, \begin{document}$a = \nabla_\xi A$\end{document}.
We prove that, even if \begin{document}$A(x,t,\xi)$\end{document} makes the variational approach more difficult, the functional associated to such a problem is bounded from below and attains its infimum when the growth of the nonlinear term \begin{document}$f(x,t)$\end{document} is "controlled" by \begin{document}$A(x,t,\xi)$\end{document}. Moreover, stronger assumptions allow us to find the existence of at least one positive solution.
We use a suitable Minimum Principle based on a weak version of the Cerami–Palais–Smale condition.
2020, 13(12): 3347-3355 doi: 10.3934/dcdss.2020127 +[Abstract](1232) +[HTML](371) +[PDF](391.54KB)
Abstract:
We prove the existence of infinitely many radial solutions to a Kirchhoff type problem in a ball with a super-cubic nonlinearity. Our methods rely on bifurcation analysis and energy estimates.
2020, 13(12): 3357-3389 doi: 10.3934/dcdss.2020236 +[Abstract](1205) +[HTML](353) +[PDF](551.9KB)
Abstract:
We consider an Ostrovsky-Hunter type equation, which also includes the short pulse equation, or the Kozlov-Sazonov equation. We prove the well-posedness of the entropy solution for the non-homogeneous initial boundary value problem. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.
2020, 13(12): 3391-3400 doi: 10.3934/dcdss.2020246 +[Abstract](1054) +[HTML](343) +[PDF](559.11KB)
Abstract:
In this paper, we define \begin{document}$E_ \alpha(t^ \alpha A)$\end{document}, where \begin{document}$A$\end{document} is the generator of an uniformly bounded (\begin{document}$C_0$\end{document}) semigroup and \begin{document}$E_ \alpha(z)$\end{document} the Mittag-Leffler function. Since the mapping \begin{document}$t\mapsto E_ \alpha(t^ \alpha A)$\end{document} has not the semigroup property, we cannot use the Trotter formula for representing the Feynman operator calculus. Thus for the Hamiltonian \begin{document}$H_ \alpha = -\frac{{\hbar_ \alpha2}}{{2m}}\Delta +V(x)$\end{document}, we express \begin{document}$E_ \alpha(t^ \alpha H_ \alpha )$\end{document} by subordination principle of the Feynman path integral and we retrieve the corresponding Green function.
2020, 13(12): 3401-3415 doi: 10.3934/dcdss.2020245 +[Abstract](1386) +[HTML](391) +[PDF](1708.43KB)
Abstract:
We study positive solutions to a steady state reaction diffusion equation arising in population dynamics, namely,
where \begin{document}$\Omega$\end{document} is a bounded domain in \begin{document}$\mathbb{R}^N$\end{document}; \begin{document}$N > 1$\end{document} with smooth boundary \begin{document}$\partial \Omega$\end{document} or \begin{document}$\Omega = (0,1)$\end{document}, \begin{document}$\frac{\partial u}{\partial \eta}$\end{document} is the outward normal derivative of \begin{document}$u$\end{document} on \begin{document}$\partial \Omega$\end{document}, \begin{document}$\lambda$\end{document} is a domain scaling parameter, \begin{document}$\gamma$\end{document} is a measure of the exterior matrix (\begin{document}$\Omega^c$\end{document}) hostility, and \begin{document}$A\in (0,1)$\end{document} and \begin{document}$\epsilon>0$\end{document} are constants. The boundary condition here represents a case when the dispersal at the boundary is U-shaped. In particular, the dispersal is decreasing for \begin{document}$u<A$\end{document} and increasing for \begin{document}$u>A$\end{document}. We will establish non-existence, existence, multiplicity and uniqueness results. In particular, we will discuss the occurrence of an Allee effect for certain range of \begin{document}$\lambda$\end{document}. When \begin{document}$\Omega = (0,1)$\end{document} we will provide more detailed bifurcation diagrams for positive solutions and their evolution as the hostility parameter \begin{document}$\gamma$\end{document} varies. Our results indicate that when \begin{document}$\gamma$\end{document} is large there is no Allee effect for any \begin{document}$\lambda$\end{document}. We employ a method of sub-supersolutions to obtain existence and multiplicity results when \begin{document}$N>1$\end{document}, and the quadrature method to study the case \begin{document}$N = 1$\end{document}.
2020, 13(12): 3417-3426 doi: 10.3934/dcdss.2020128 +[Abstract](1132) +[HTML](376) +[PDF](379.91KB)
Abstract:
We consider a differential operator of order 2\begin{document}$n$\end{document} of the type \begin{document}$A_n u = (-1)^n (a u^{(n)})^{(n)}$\end{document}, where \begin{document}$a(x)>0$\end{document} in \begin{document}$[0, 1]\setminus\{x_0\}$\end{document} and \begin{document}$a(x_0) = 0$\end{document}. We show that, for any \begin{document}$n\in{\mathbb{N}}$\end{document}, the operator \begin{document}$-A_n$\end{document} generates a contractive analytic semigroup of angle \begin{document}$\pi/2$\end{document} on \begin{document}$L^2 (0, 1)$\end{document}. Note that the domain of \begin{document}$A_n$\end{document} depends on the type of degeneracy of \begin{document}$a$\end{document}. Our theorems extend some previous results in [3] where \begin{document}$n = 1$\end{document}.
2020, 13(12): 3427-3460 doi: 10.3934/dcdss.2020243 +[Abstract](1150) +[HTML](376) +[PDF](658.14KB)
Abstract:
Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrödinger operators with sufficiently short-range interactions in general space dimensions.
More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions \begin{document}$n \geqslant 3$\end{document}, the absence of resonances at \begin{document}$\pm m$\end{document} for massive Dirac operators (with mass \begin{document}$m > 0$\end{document}) in dimensions \begin{document}$n \geqslant 5$\end{document}, and recall the well-known case of absence of zero-energy resonances for Schrödinger operators in dimension \begin{document}$n \geqslant 5$\end{document}.
2020, 13(12): 3461-3471 doi: 10.3934/dcdss.2020239 +[Abstract](1103) +[HTML](342) +[PDF](386.74KB)
Abstract:
We study mixed hyperbolic systems with dynamic and Wentzell boundary conditions. The boundary condition contains a tangential operator which is strongly elliptic on the boundary. We prove results of generation of strongly continuous groups and well-posedness.
2020, 13(12): 3473-3489 doi: 10.3934/dcdss.2020235 +[Abstract](1238) +[HTML](369) +[PDF](1008.68KB)
Abstract:
The problem of determining equity volatility from a knowledge of American option prices for a range of exercise (strike) prices and expirations is solved by minimization of a convex functional.
2020, 13(12): 3491-3494 doi: 10.3934/dcdss.2020112 +[Abstract](1136) +[HTML](409) +[PDF](285.14KB)
Abstract:
We prove a cone-type criterion for a boundary point to be regular for the Dirichlet problem related to (possibly) degenerate Ornstein–Uhlenbeck operators in \begin{document}$\mathbb{R}^N$\end{document}. Our result extends the classical Zaremba cone criterion for the Laplace operator.
2020, 13(12): 3495-3502 doi: 10.3934/dcdss.2020248 +[Abstract](1143) +[HTML](352) +[PDF](392.31KB)
Abstract:
Suppose that \begin{document}$u(x)$\end{document} is a positive subsolution to an elliptic equation in a bounded domain \begin{document}$D$\end{document}, with the \begin{document}$C^2$\end{document} smooth boundary \begin{document}$\partial D$\end{document}. We prove a quantitative version of the Hopf maximum principle that can be formulated as follows: there exists a constant \begin{document}$\gamma>0$\end{document} such that \begin{document}$\partial_{\bf n}u(\tilde x)$\end{document} – the outward normal derivative at the maximum point \begin{document}$\tilde x\in \partial D$\end{document} (necessary located at \begin{document}$\partial D$\end{document}, by the strong maximum principle) – satisfies \begin{document}$\partial_{\bf n}u(\tilde x)>\gamma u(\tilde x)$\end{document}, provided the coefficient \begin{document}$c(x)$\end{document} by the zero order term satisfies \begin{document}$\sup_{x\in D}c(x) = -c_*<0$\end{document}. The constant \begin{document}$\gamma$\end{document} depends only on the geometry of \begin{document}$D$\end{document}, uniform ellipticity bound, \begin{document}$L^\infty$\end{document} bounds on the coefficients, and \begin{document}$c_*$\end{document}. The key tool used is the Feynman–Kac representation of a subsolution to the elliptic equation.
2020, 13(12): 3503-3524 doi: 10.3934/dcdss.2020242 +[Abstract](1188) +[HTML](343) +[PDF](583.16KB)
Abstract:
We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well–posedness of both flows for initial surfaces that are \begin{document}$C^{1+\alpha}$\end{document}–regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are \begin{document}$C^{1+\alpha}$\end{document}–close to a sphere, and we prove that these solutions become spherical as time goes to infinity.
2020, 13(12): 3525-3533 doi: 10.3934/dcdss.2020247 +[Abstract](1020) +[HTML](329) +[PDF](371.17KB)
Abstract:
In this paper we study local and global in time existence for a class of nonlinear evolution equations having order eventually greater than 2 and not integer. The linear operator has an homogeneous damping term; the nonlinearity is of polynomial type without derivatives:
with \begin{document}$\mu>0$\end{document}, \begin{document}$\theta>0$\end{document}. Since we are treating an absorbing nonlinear term, large data solutions can be considered.
2020, 13(12): 3535-3550 doi: 10.3934/dcdss.2020237 +[Abstract](1151) +[HTML](384) +[PDF](4039.0KB)
Abstract:
We develop a model for the spatial spread of epidemic outbreak in a geographical region. The goal is to understand how spatial heterogeneity influences the transmission dynamics of the susceptible and infected populations. The model consists of a system of partial differential equations, which indirectly describes the disease transmission caused by the disease pathogen. The model is compared to data for the seasonal influenza epidemics in Puerto Rico for 2015-2016.
2020, 13(12): 3551-3563 doi: 10.3934/dcdss.2020244 +[Abstract](960) +[HTML](357) +[PDF](411.63KB)
Abstract:
This paper provides two different extensions of a previous joint work "Time asymptotics of structured populations with diffusion and dynamic boundary conditions; Discrete Cont Dyn Syst, Series B, 23 (10) (2018)" devoted to asynchronous exponential asymptotics for bounded and weakly compact reproduction operators. The first extension considers bounded non weakly compact reproduction operators while the second extension deals with unbounded kernel reproduction operators and needs, as a preliminary step, a new generation result.
2020, 13(12): 3565-3579 doi: 10.3934/dcdss.2020238 +[Abstract](1017) +[HTML](333) +[PDF](404.91KB)
Abstract:
The computational powers of Mathematica are used to prove polynomial identities that are essential to obtain growth estimates for subdiagonal rational Padé approximations of the exponential function and to obtain new estimates of the constants of the Brenner-Thomée Approximation Theorem of Semigroup Theory.
2020 Impact Factor: 2.425
5 Year Impact Factor: 1.490
2020 CiteScore: 3.1 | 2021-07-27 15:37:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.882068395614624, "perplexity": 1108.8343601828954}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153392.43/warc/CC-MAIN-20210727135323-20210727165323-00713.warc.gz"} |
https://math.stackexchange.com/questions/1041092/on-the-greatest-lower-bound-property | # On the greatest lower bound property
### Proposition:
Let $S$ be an ordered field and $S \supset E \neq \varnothing$. $E$ is bounded below. Then $\inf E = - \sup ( - E )$
### Try:
Write $- E = \{ -x : x \in E \}$ and let $l$ be a lower bound of $E$. (This we are given). Hence, $x \geq l$ for every $x \in E$. Clearly, $-x \leq -l$. Put $u = -l$ and we observe that $u$ is an upper bound of $-E$ and consequently $\sup ( - E)$ exists. Put $\alpha = \sup ( - E )$. It follows that $-x \leq \alpha$ for all $x$ and hence $x \geq - \alpha$. We see that $- \alpha$ is a lower bound for $E$. We need to show that $- \alpha$ is the infimum of $E$. To see this, we show that $NO$ $\beta$ with $\beta > - \alpha$ is a lower bound of $E$. Suppose the contrary, and let $\beta$ be a lower bound of $E$ such that $\beta > - \alpha$. This implies that $- \beta < \alpha$ and $- \beta$ cannot be an upper bound of $E$. In particular, there is some $e \in E$ with $- \beta < e$ or $\beta > - e$. So we have found some $y = -e \in E$ with $\beta > y$. In particular, $\beta$ cannot be a lower bound of $E$. Contradiction. Hence,
$$\inf E = - \alpha = - \sup ( - E )$$
Is this a correct proof? Also, do I have to worry whether $E$ is unbounded?
• See here and here. Possible duplicate. – Aaron Maroja Nov 27 '14 at 14:17
• Your argument is fine as it is. – Brian M. Scott Nov 27 '14 at 21:03 | 2019-08-23 08:39:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9835184216499329, "perplexity": 47.90144535137648}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318243.40/warc/CC-MAIN-20190823083811-20190823105811-00037.warc.gz"} |