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https://www.gamedev.net/forums/topic/203596-super-mario-jump-on-head-killing/ | Archived
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hy, i''m looking for a technique for solving super mario like collision detection (killing by jumping on an object). i''ve already tried to implement it, but my approach has some flaws. do you know some tutorials about this, or a piece of code to look at? thanks in advance, no_skill
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Assuming -Y is down, +Y is up...
You want the player horizontally above the object...
((PLAYERX + PLAYERWIDTH) > ENEMYX) && (PLAYERX < (ENEMYX + ENEMYWIDTH))
... you want the player coming downwards only...
(PLAYERVEL_Y <= 0)
... and you want the player''s "feet" above a certain "head" margin...
((PLAYERY + PLAYERHEIGHT) > (ENEMYY + HEADTHICKNESS))
Logical AND all those together, and you''ve got yourself a "head-hit" collision test. If that fails, but the player is still colliding with the enemy, handle it like otherwise (usually injure or kill the player).
Mind you, re-write it first. I just jotted some psuedocode so I know exactly what I''m talking about.
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implementing that is no problem, but i''m having two moving objects and if the player is moving down and the enemy up sometimes i miss the collision.
i tested the whole path the player traveled so far against the enemy''s new position, but the problem still occurs (of course not as often as before).
i''ll try testing the two full paths against, but i wonder how it was done on the snes etc. i don''t think it has enough power for such calculations.
(...maybe it had, as it''s only a if with different parameters....)
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Since we know that Mario must always be on top of a monster for a head on collision, why not make the first test vertical. That is:
If Mario.Y > Monster.Y, then check if (Mario.Y - Monster.Y) = 1 (or however maximum distance for an actual kill).
Then, check simply if he is within horizontal bounds:
If (abs(Monster.X - Mario.X) < Mario.Width)) then go run the monster dead routine. abs() means absolute function.
I think this is how I implemented top-down collision technique in my game. Tell me if I''m wrong.
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but what if mario moves through this area in one frame?
that''s the main problem.
how do games in general handle collision detection between two moving objects?
do they compare only the end position of both movements.
or test the whole path of one object against the end position of another.
how?
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I dont know why you have problems...
if they move more then 1000 pixels per second then you may have this problem, but there is no problem doing something like this:
if (player.x+player.w > enemy.x) && (player.x<enemy.x+enemy.w) && (player.h+player.y-enemy.y<=0)enemy.dead=true;
[edited by - ilankt on January 27, 2004 2:15:16 AM]
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thanks, i'll try that (along with a lot of ideas i got in the meantime )
i'm wondering how games in general deal with collision detection with two moving objects
a)
1. test every object against the map -> new position
2. only compare the new positions of each object against each other
b)
1. test the whole movement of every object against the whole movement of each other object + the map
c)
i don't know
how do games solve this nowadays?
[edited by - no_skill on January 27, 2004 6:18:54 AM]
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Dont use a single point x,y for checking, or you could pass up the collision.
Use bounding boxes x1,y1 x2,y2 that way you have a larger area to hit in.
If your running at 30 frames per second, like most games do, then there is no way you should pass up any collisions.
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if you have bounding rectangles, make rays from each corner from where the object started to where it ended for both objects, see if any of the rays intersect the other objects rays. Sounds good to me.
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i''ve already solved the problem by using bounding boxes.
i''ll put the code on jnrdev when it''s complete.
http://jnrdev.weed-crew.net
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• 11 | 2017-11-21 12:19:55 | {"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": 0.18169772624969482, "perplexity": 3074.0591128568753}, "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/1510934806353.62/warc/CC-MAIN-20171121113222-20171121133222-00567.warc.gz"} |
https://zbmath.org/?q=an%3A0724.34029 | # zbMATH — the first resource for mathematics
On a certain boundary value problem. (English) Zbl 0724.34029
An existence result for the BVP $$x''-k^ 2x+f(t,x,x')=0,\quad t\geq 0,$$ $$x(0)=0,\quad x(\infty)=0$$ is proved by using the Leray-Schauder topological principle.
##### MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
##### Keywords:
existence result; Leray-Schauder topological principle | 2021-04-11 20:07:28 | {"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": 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": 0.5292868614196777, "perplexity": 2899.838532939364}, "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/1618038064898.14/warc/CC-MAIN-20210411174053-20210411204053-00069.warc.gz"} |
https://cs.stackexchange.com/questions/98602/is-it-possible-to-add-every-word-in-a-file-to-a-set-in-mathrmon-time | # Is it possible to add every word in a file to a set in $\mathrm{O}(n)$ time?
The Problem:
I am currently analyzing a simple program that takes a file of length $$n$$, splits it into its individual words (seperated by white space) and adds those words to a set:
def file_word_set(name):
with open(name) as f:
res = set()
res.update(words)
return res
My Analysis:
Splitting a file of length $$n$$ into individual words takes $$\mathrm{O}(n)$$ time. The splitting will produce a list of $$k$$ strings where $$0 \leq k \leq n$$. These strings will be of varying length. Inserting a string into a set can be done in $$\mathrm{O}(m)$$ time where $$m$$ is the length of that string. This is because we must iterate through each of the characters of the array in order to determine its location in the underlying hash-table. Since the total length of all $$k$$ strings is no more than $$\mathrm{O}(n)$$, we will need to consider $$\mathrm{O}(n)$$ characters in total when building the set. Therefore the function takes $$\mathrm{O}(n)$$ time.
Is this correct?
• What about the set insert operation? What if it is $\mathcal{O}(\log n)$? – kelalaka Oct 15 '18 at 5:56
• The overall complexity obviously depends on the complexity of the methods that you're calling (especially split and update). Without knowing what they do, your question is unanswerable. – David Richerby Oct 15 '18 at 11:28
Splitting
While splitting you have to touch every element once to see that is a delimiter or not. So it $$\in \mathcal{O}(n)$$
Assuming a Hash-based Map is used;
cost of Inserting a string
It is $$\in \mathcal{O}(1)$$ but if you consider the hash function as not a constant then you have to analyze the hash function itself. When we are talking a hashmap, the string size, usually, is not considered.
Total insert cost
It is $$\in \mathcal{O}(n)$$,
Total cost
It is the sum of all = $$\mathcal{O}(n) + \mathcal{O}(n)= \mathcal{O}(n)$$ | 2019-12-06 04:36:41 | {"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": 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": 15, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5296652913093567, "perplexity": 538.4869263746265}, "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-2019-51/segments/1575540484477.5/warc/CC-MAIN-20191206023204-20191206051204-00051.warc.gz"} |
http://www.w3.org/TR/2007/WD-MathML3-20071005/chapter7.html | # 7 MathML interactions with the Wide World
Because MathML is, typically, embedded in a wider context, it is important to describe the conditions that processors should acknowledge in order to recognize XML fragments as MathML. This chapter describes the fundamental mechanisms to recognize and transfer MathML markup fragments within a larger environment such as an XML document or a desktop file-system, it raises the issues of combining external markup within MathML, then indicates how cascading style sheets can be used within MathML.
This chapter applies to both content and presentation MathML and indicates a particular processing model to the semantics, annotation and annotation-xml elements defined in Section 5.3 Attributions in Strict Content MathML.
## 7.1 Invoking MathML Processors: namespace, extensions, and mime-types
### 7.1.1 Recognizing MathML in an XML Model
Within an XML document supporting namespaces (TODO: cite xmlns and xml specs), the preferred method to recognize MathML markup is by the identification of the math element in the appropriate namespace, i.e. that of URI http://www.w3.org/1998/Math/MathML.
This is the recommended method to embed MathML within [XHTML] documents. Some user-agents' setup may require supplementary information to be available, such as the MicroSoft behaviour specification (TODO: quote) used in the MathType browser-extension (TODO:quote).
Markup-language specifications that wish to embed MathML may provide special conditions independent of this recommendation. The conditions should be equivalent and the elements' local-names should remain the same.
### 7.1.2 Resource Types for MathML Documents
Although rendering MathML expressions often occurs in place in a Web browser, other MathML processing functions take place more naturally in other applications. Particularly common tasks include opening a MathML expression in an equation editor or computer algebra system. It is important therefore to specify the encoding-names that MathML fragments should be called with:
MIME types [RFC2045], [RFC2046] offer a strategy that can be used in current user agents to invoke a MathML processor. This is primarily useful when referencing separate files containing MathML markup from an embed or object element, or within a desktop environment. (TODO: check that this still applies)
[RFC3023] assigns MathML the MIME type application/mathml+xml which is the official mime-type. The W3C Math Working Group recommends the standard file extension .mml within a registry associating file formats to file-extension. In MathML 1.0, text/mathml was given as the suggested MIME type. This has been superceded by RFC3023. In the next section, alternate encoding names are provided for the purposes of desktop transfers.
Issue specify-encoding-names-in-details wiki (member only) Encoding names are specified in the section below, and are described in the chapter 5 as attribute values of the annotation* elements. Moreover, content-types and encoding-names have a fairly similar semantic and are even used exchangeably in some environments (e.g. Java's DataFlavor). It might be worth trying to homogenize our list and maybe specify mime-type equivalence for each encoding names. None recorded
## 7.2 Transferring MathML in Desktop Environments
MathML expressions are often exchanged between applications using the familiar copy-and-paste or drag-and-drop paradigms. This section provides recommended ways to process MathML while applying these paradigms.
Applying them will transfer MathML fragments between the contexts of two applications by making them available in several flavors, often called clipboard formats or data flavors. The copy-and-paste paradigm lets application place content in a central clipboard, one data-stream per clipboard format; consuming applications negotiate by choose to read the data of the format they elect. The drag-and-drop pardigm lets application offer content by declaring the available formats and potential recipients accept or reject a drop based on this list; the drop action then lets the receiving application request the delivery of the format in the indicated format. The list of flavors is generally ordered, going from the most wishable to the least wishable flavor.
Current desktop platforms offer both of these transfer paradigms using similar transfer architectures. In this section we specify what applications should provide as transfer-flavors, how they should be named, and how they should handle the special semantics, annotation, and annotation-xml elements.
To summarize the two negotiation mechanisms, we shall, here, be talking of flavors, each having a name (a character string) and a content (a stream of binary data), which are exported.
### 7.2.1 Basic Flavors' Names and Contents
MathML contains two distinct vocabularies: one for encoding mathematical semantics called Chapter 4 Content Markup and one for encoding visual presentation called Chapter 3 Presentation Markup. Some MathML-aware applications import and export only one of these vocabularies, while other may be capable of producing and consuming both. Consequently, we propose three distinct MathML flavors:
Flavor Name Description
MathML Content Instance contains content MathML markup only
MathML Presentation Instance contains presentation MathML markup only
MathML Any well-formed MathML instance presentation markup, content markup, or a mixture of the two is allowed
Note that Content MathML, Presentation MathML and MathML are the exact strings that should be used to describe the flavors described above. On operating systems that allow such, applications should register such names (e.g. Windows' RegisterClipboardFormat).
When transferring MathML, for example when placing it within a clipboard, an application MUST ensure the content is a well-formed XML instance of a MathML schema. Specifically:
1. The instance MUST begin with a XML processing instruction, e.g. <?xml version="1.0">
2. The instance MUST contain exactly one root math element.
3. Since MathML is frequently embedded within other XML document types, the instance MUST declare the MathML namespace on the root math element. In addition, the instance SHOULD use a schemaLocation attribute on the math element to indicate the location of MathML schema documents against which the instance is valid. Note that the presence of the schemaLocation attribute does not require a consumer of the MathML instance to obtain or use the cited schema documents.
4. The instance MUST use numeric character references (e.g. α) rather than character entity names (e.g. α) for greater interoperability.
5. The character encoding for the instance MUST be either specified in the XML header, UTF-16, or UTF-8. UTF-16-encoded data MUST begin with a byte-order mark (BOM). If no BOM or encoding is given, the character encoding will be assumed to be UTF-8.
### 7.2.2 Recommended Behaviours when Transferring
Applications that transfer MathML SHOULD adhere to the following conventions:
1. Applications that have pure presentation markup and/or pure content markup versions of an expression SHOULD offer as many of these two flavors as are available.
2. When both presentation and content are exported, recipients should consider it equivalent to a single MathML instance in which presentation and content are combined at the top level using MathML's semantics element (see Section 5.5.1 Top-level Parallel Markup). (TODO: issue: in DnD you can't read several, at least in java) The order between flavors determines whether presentation wraps content, or vice-versa. Usually, Presentation MathML should be offered first so that it wraps the Content MathML.
3. When an application has a mixed presentation and content version in addition to pure presentation and/or content versions, it should export the mixed versionafter the pure presentation and/or content markup versions, and mark it as the generic MathML flavor.
4. When an application cannot produce pure presentation and/or content markup versions, or cannot determine whether MathML data is pure presentation or content markup (e.g. data being passed through from a third application,) it should export only one version marked as the generic MathML flavor.
5. An application that only has pure presentation and/or content markup versions of an expression available SHOULD NOT export a second copy of the data marked as the generic MathML flavor.
6. When an application exports a MathML fragment whose root element is a semantics element, it SHOULD offer, after the flavors above, a flavor for each annotation or annotation-xml element: the flavor should be given by the encoding attribute value, and the content should be the child text in UTF-8 (if the annotation element contains only textual data), a valid XML fragment (if the annotation-xml element contains children), or the data resulting of requesting the URL given by the href attribute.
7. As a final fallback applications SHOULD export a version of the data in plain-text flavor (such as CF_UNICODETEXT, UnicodeText, NSStringPboardType, text/plain, ...). When an application has multiple versions of an expression available, it may choose the version to export as text at its discretion. Since some older MathML-aware programs expect MathML instances transferred as text to begin with a math element, the text version should generally omit the XML processing instruction, DOCTYPE declaration and other XML prolog material before the math element. Similarly, the BOM should be omitted for Unicode text encoded as UTF-16. Note, the Unicode text version of the data should always be the last flavor exported, following the principle that exported flavors should be ordered with the most specific flavor first and the least specific flavor last.
### 7.2.3 Discussion
For purposes of determining whether a MathML instance is pure content markup or pure presentation markup, the math element and the semantics, annotation and annotation-xml elements should be regarded as belonging to both the presentation and content markup vocabularies. This is obvious for the root math element which is required for all MathML expressions. However, the semantics element and its child annotation elements comprise an arbitrary annotation mechanism within MathML, and are not tied to either presentation or content markup. Consequently, applications consuming MathML should always process these four elements even if the application only implements one of the two vocabularies.
It is worth noting that the above recommendations allow agents producing MathML to provide binary data for the clipboard, for example as an image or an application-specific format. The sole method to do so is to reference the binary data by the href attribute since XML child-text does not allow arbitrary byte-streams.
While the above recommendations are intended to improve interoperability between MathML-aware applications utilizing the transfer flavors, it should be noted that they do not guarantee interoperablility. For example, references to external resources (e.g. stylesheets, etc.) in MathML data can also cause interoperability problems if the consumer of the data is unable to locate them, just as can happen when cutting and pasting HTML or many other data types. Applications that make use of references to external resources are encouraged to make users aware of potential problems and provide alternate ways for obtaining the referenced resources. In general, consumers of MathML data containing references they cannot resolve or do not understand should ignore them.
### 7.2.4 Examples
#### 7.2.4.1 Example 1
An e-Learning application has a database of quiz questions, some of which contain MathML. The MathML comes from multiple sources, and the e-Learning application merely passes the data on for display, but does not have sophisticated MathML analysis capabilities. Consequently, the application is not aware whether a given MathML instance is pure presentation or pure content markup, nor does it know whether the instance is valid with respect to a particular version of the MathML schema. It therefore places the following data formats on the clipboard:
Flavour Name Flavor Content
MathML
<?xml version="1.0"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">...[/itex]
Unicode Text
<math xmlns="http://www.w3.org/1998/Math/MathML">...[/itex]
#### 7.2.4.2 Example 2
An equation editor is able to generate pure presentation markup, valid with respect to MathML 2.0, 2nd Edition. Consequently, it exports the following flavors:
Flavour Name Flavor Content
Presentation MathML
<?xml version="1.0"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">...[/itex]
Tiff (a rendering sample)
Unicode Text
<math xmlns="http://www.w3.org/1998/Math/MathML">...[/itex]
#### 7.2.4.3 Example 3
A schema-based content management system contains multiple MathML representations of a collection of mathematical expressions, including mixed markup from authors, pure content markup for interfacing to symbolic computation engines, and pure presentation markup for print publication. Due to the system's use of schemas, markup is stored with a namespace prefix. The system therefore can transfer the following data:
Flavour Name Flavor Content
Presentation MathML
<?xml version="1.0"?>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://www.w3.org/Math/XMLSchema/mathml2/mathml2.xsd">
<mml:mrow>
...
<mml:mrow>
</mml:math>
Content MathML
<?xml version="1.0"?>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://www.w3.org/Math/XMLSchema/mathml2/mathml2.xsd">
<mml:apply>
...
<mml:apply>
</mml:math>
MathML
<?xml version="1.0"?>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://www.w3.org/Math/XMLSchema/mathml2/mathml2.xsd">
<mml:mrow>
<mml:apply> ... content markup within presentation markup ... </mml:apply>
...
</mml:mrow>
</mml:math>
TeX
{x \over x-1}
Unicode Text
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://www.w3.org/Math/XMLSchema/mathml2/mathml2.xsd">
<mml:mrow>
...
<mml:mrow>
</mml:math>
#### 7.2.4.4 Example 4
A similar content management system is web-based and delivers MathML representations of mathematiacly expressions. The system is able to produce presentation MathML, content MathML, TeX and pictures in PNG format. In web-pages being browsed, it could produce a MathML fragment such as the following:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:semantics>
<mml:mrow>...</mml:mrow>
<mml:annotation-xml encoding="MathML content">...</mml:annotation-xml>
<mml:annotation encoding="TeX">{1 \over x}</mml:annotation>
<mml:annotation encoding="image/png" href="formula3848.png"/>
</mml:semantics>
</mml:math>
A web-browser that receives such a fragment and tries to export it as part of a drag-and-drop action, can offer the following flavors:
Flavour Name Flavor Content
Presentation MathML
<?xml version="1.0"?>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://www.w3.org/Math/XMLSchema/mathml2/mathml2.xsd">
<mml:mrow>
...
<mml:mrow>
</mml:math>
Content MathML
<?xml version="1.0"?>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://www.w3.org/Math/XMLSchema/mathml2/mathml2.xsd">
<mml:apply>
...
<mml:apply>
</mml:math>
MathML
<?xml version="1.0"?>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://www.w3.org/Math/XMLSchema/mathml2/mathml2.xsd">
<mml:mrow>
<mml:apply> ... content markup within presentation markup ... </mml:apply>
...
</mml:mrow>
</mml:math>
TeX
{x \over x-1}
image/png (the content of the picture file, requested from formula3848.png
Unicode Text
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://www.w3.org/Math/XMLSchema/mathml2/mathml2.xsd">
<mml:mrow>
...
<mml:mrow>
</mml:math>
## 7.3 Combining MathML and Other Formats
Since MathML is most often generated by authoring tools, it is particularly important that opening a MathML expression in an editor should be easy to do and to implement. In many cases, it will be desirable for an authoring tool to record some information about its internal state along with a MathML expression, so that an author can pick up editing where he or she left off. The following markup is proposed:
1. For any extra information that is expected to be semantically equivalent MathML-3 proposes the usage of the semantics element presented in Section 5.3 Attributions in Strict Content MathML.
2. For any extra information that cannot be declared as such, and is, expectedly, private to the application. MathML-3 suggests to use the maction, see Section 3.6.1 Bind Action to Sub-Expression (maction).
### 7.3.1 Mixing MathML and HTML
In order to fully integrate MathML into XHTML, it should be possible not only to embed MathML in XHTML, as described in Section 7.1.1 Recognizing MathML in an XML Model, but also to embed XHTML in MathML. However, the problem of supporting XHTML in MathML presents many difficulties. Therefore, at present, the MathML specification does not permit any XHTML elements within a MathML expression, although this may be subject to change in a future revision of MathML.
In most cases, XHTML elements (headings, paragraphs, lists, etc.) either do not apply in mathematical contexts, or MathML already provides equivalent or better functionality specifically tailored to mathematical content (tables, mathematics style changes, etc.). However, there are two notable exceptions, the XHTML anchor and image elements. For this functionality, MathML relies on the general XML linking and graphics mechanisms being developed by other W3C Activities.
Issue Linking-and-marking-ids wiki (member only) We wish to stop using xlink for links since it seems unimplemented and add the necessary attributes at presentation elements. None recorded
MathML has no element that corresponds to the XHTML anchor element a. In XHTML, anchors are used both to make links, and to provide locations to which a link can be made. MathML, as an XML application, defines links by the use of the mechanism described in the W3C Recommendation "XML Linking Language" [XLink].
A MathML element is designated as a link by the presence of the attribute xlink:href. To use the attribute xlink:href, it is also necessary to declare the appropriate namespace. Thus, a typical MathML link might look like:
<mrow xmlns:xlink="http://www.w3.org/1999/xlink"
...
</mrow>
MathML designates that almost all elements can be used as XML linking elements. The only elements that cannot serve as linking elements are those which exist primarily to disambiguate other MathML constructs and in general do not correspond to any part of a typical visual rendering. The full list of exceptional elements that cannot be used as linking elements is given in the table below.
MathML elements that cannot be linking elements
mprescripts none
malignmark maligngroup
Note that the XML Linking [XLink] and XML Pointer Language [XPointer] specifications also define how to link into a MathML expressions. Be aware, however, that such links may or may not be properly interpreted in current software.
### 7.3.3 Images
The img element has no MathML equivalent. The decision to omit a general mechanism for image inclusion from MathML was based on several factors. However, the main reason for not providing an image facility is that MathML takes great pains to make the notational structure and mathematical content it encodes easily available to processors, whereas information contained in images is only available to a human reader looking at a visual representation. Thus, for example, in the MathML paradigm, it would be preferable to introduce new glyphs via the mglyph element which at a minimum identifies them as glyphs, rather than simply including them as images.
### 7.3.4 MathML and Graphical Markup
Apart from the introduction of new glyphs, many of the situations where one might be inclined to use an image amount to displaying labeled diagrams. For example, knot diagrams, Venn diagrams, Dynkin diagrams, Feynman diagrams and commutative diagrams all fall into this category. As such, their content would be better encoded via some combination of structured graphics and MathML markup. However, at the time of this writing, it is beyond the scope of the W3C Math Activity to define a markup language to encode such a general concept as "labeled diagrams." (See http://www.w3.org/Math for current W3C activity in mathematics and http://www.w3.org/Graphics for the W3C graphics activity.)
One mechanism for embedding additional graphical content is via the semantics element, as in the following example:
<semantics>
<apply>
<intersect/>
<ci>A</ci>
<ci>B</ci>
</apply>
<annotation-xml encoding="SVG1.1">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 290 180">
<clipPath id="a">
<circle cy="90" cx="100" r="60"/>
</clipPath>
<circle fill="#AAAAAA" cy="90" cx="190"
r="60" style="clip-path:url(#a)"/>
<circle stroke="black" fill="none" cy="90" cx="100" r="60"/>
<circle stroke="black" fill="none" cy="90" cx="190" r="60"/>
</svg>
</annotation-xml>
<annotation-xml encoding="application/xhtml+xml">
<img xmlns="http://www.w3.org/1999/xhtml" src="intersect.gif" alt="A intersect B"/>
</annotation-xml>
</semantics>
Here, the annotation-xml elements are used to indicate alternative representations of the Content MathML depiction of the intersection of two sets. The first one is in the "Scalable Vector Graphics" format [SVG1.1] (see [XHTML-MathML-SVG] for the definition of an XHTML profile integrating MathML and SVG), the second one uses the XHTML img element embedded as an XHTML fragment. In this situation, a MathML processor can use any of these representations for display, perhaps producing a graphical format such as the image below.
Note that the semantics representation of this example is given in the Content MathML markup, as the first child of the semantics element. In this regard, it is the representation most analogous to the alt attribute of the img element in XHTML, and would likely be the best choice for non-visual rendering.
## 7.4 Using CSS with MathML
When MathML is rendered in an environment that supports [CSS21], controlling mathematics style properties with a CSS stylesheet is obviously desirable. MathML 2.0 has significantly redesigned the way presentation element style properties are organized to facilitate better interaction between MathML renderers and CSS style mechanisms. It introduces four new mathematics style attributes with logical values. Roughly speaking, these attributes can be viewed as the proper selectors for CSS rules that affect MathML.
Controlling mathematics styling is not as simple as it might first appear because mathematics styling and text styling are quite different in character. In text, meaning is primarily carried by the relative positioning of characters next to one another to form words. Thus, although the font used to render text may impart nuances to the meaning, transforming the typographic properties of the individual characters leaves the meaning of text basically intact. By contrast, in mathematical expressions, individual characters in specific typefaces tend to function as atomic symbols. Thus, in the same equation, a bold italic 'x' and a normal italic 'x' are almost always intended to be two distinct symbols that mean different things. In traditional usage, there are eight basic typographical categories of symbols. These categories are described by mathematics style attributes, primarily the mathvariant attribute.
Text and mathematics layout also obviously differ in that mathematics uses 2-dimensional layout. As a result, many of the style parameters that affect mathematics layout have no textual analogs. Even in cases where there are analogous properties, the sensible values for these properties may not correspond. For example, traditional mathematical typography usually uses italic fonts for single character identifiers, and upright fonts for multicharacter identifier. In text, italicization does not usually depend on the number of letters in a word. Thus although a font-slant property makes sense for both mathematics and text, the natural default values are quite different.
Because of the difference between text and mathematics styling, only the styling aspects that do not affect layout are good candidates for CSS control. MathML 3.0 captures the most important properties with the new mathematics style attributes, and users should try to use them whenever possible over more direct, but less robust, approaches. A sample CSS stylesheet illustrating the use of the mathematical style attributes is available in Appendix E Sample CSS Style Sheet for MathML. Users should not count on MathML implementations to implement any other properties than those in the Font, Colors, and Outlines families of properties described in [CSS2] and implementations should only implement these properties within MathML-elements. Note that these prohibitions do not apply to CSS stylesheets that implement the MathML-CSS profile. (TODO: quote).
TODO: add equivalence statements and conflict resolution and stress that CSS changes should not be considered meaningful.
Generally speaking, the model for CSS interaction with the math style attributes runs as follows. A CSS style sheet might provide a style rule such as:
math *.[mathsize="small"] {
font-size: 80%
}
This rule sets the CSS font-size properties for all children of the math element that have the mathsize attribute set to small. A MathML renderer would then query the style engine for the CSS environment, and use the values returned as input to its own layout algorithms. MathML does not specify the mechanism by which style information is inherited from the environment. However, some suggested rendering rules for the interaction between properties of the ambient style environment and MathML-specific rendering rules are discussed in Section 3.2.2 Mathematics style attributes common to token elements, and more generally throughout Chapter 3 Presentation Markup.
It should be stressed, however, that some caution is required in writing CSS stylesheets for MathML. Because changing typographic properties of mathematics symbols can change the meaning of an equation, stylesheet should be written in a way such that changes to document-wide typographic styles do not affect embedded MathML expressions. By using the MathML 2.0 mathematics style attributes as selectors for CSS rules, this danger is minimized.
Another pitfall to be avoided is using CSS to provide typographic style information necessary to the proper understanding of an expression. Expressions dependent on CSS for meaning will not be portable to non-CSS environments such as computer algebra systems. By using the logical values of the new MathML 3.0 mathematics style attributes as selectors for CSS rules, it can be assured that style information necessary to the sense of an expression is encoded directly in the MathML.
MathML 3.0 does not specify how a user agent should process style information, because there are many non-CSS MathML environments, and because different users agents and renderers have widely varying degrees of access to CSS information. In general, however, developers are urged to provide as much CSS support for MathML as possible.
Overview: Mathematical Markup Language (MathML) Version 3.0
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https://lucatrevisan.wordpress.com/category/math/ | # Talagrand’s Generic Chaining
Welcome to phase two of in theory, in which we again talk about math. I spent last Fall teaching two courses and getting settled, I mostly traveled in January and February, and I have spent the last two months on my sofa catching up on TV series. Hence I will reach back to last Spring, when I learned about Talagrand’s machinery of generic chaining and majorizing measures from Nikhil Bansal, in the context of our work with Ola Svensson on graph and hypergraph sparsification. Here I would like to record what I understood about the machinery, and in a follow-up post I plan to explain the application to hypergraph sparsification.
# Lies, Damns Lies, and Herbert London
I am grading the final projects of my class, I am trying the clear the backlog of publishing all the class notes, I am way behind on my STOC reviews, and in two days I am taking off for a complicated two-week trips involving planes, trains and a rented automobile, as well as an ambitious plan of doing no work whatsoever from December 20 to December 31.
So, today I was browsing Facebook, and when I saw a post containing an incredibly blatant arithmetic mistake (which none of the several comments seemed to notice) I spent the rest of the morning looking up where it came from.
The goal of the post was to make the wrong claim that people have been paying more than enough money into social security (through payroll taxes) to support the current level of benefits. Indeed, since the beginning, social security has been paying individuals more than they put in, and now that population and salaries have stop growing, social security is also paying out retired people more than it gets from working people, so that the “trust fund” (whether one believes it is a real thing or an accounting fiction) will run out in the 2030s unless some change is made.
This is a complicated matter, but the post included a sentence to the extent that $4,500 a year, with an interest of 1% per year “compounded monthly”, would add up to$1,3 million after 40 years. This is not even in the right order of magnitude (it adds up to about $220k) and it should be obvious without making the calculation. Who would write such a thing, and why? My first stop was a July 2012 post on snopes, which commented on a very similar viral email. Snopes points out various mistakes (including the rate of social security payroll taxes), but the calculation in the snopes email, while based on wrong assumptions, has correct arithmetic: it says that$4,500 a year, with a 5% interest, become about $890k after 49 years. So how did the viral email with the wrong assumptions and correct arithmetic morph into the Facebook post with the same wrong assumptions but also the wrong arithmetic? I don’t know, but here is an August 2012 post on, you can’t make this stuff up, Accuracy in Media, which wikipedia describes as a “media watchdog.” The post is attributed to Herbert London, who has PhD from Columbia, is a member of the Council on Foreign Relation and used to be the president of a conservative think-tank. Currently, he has an affiliation with King’s College in New York. London’s post has the sentence I saw in the Facebook post: (…) an employer’s contribution of$375 per month at a modest one percent rate compounded over a 40 year work experience the total would be $1.3 million. The rest of the post is almost identical to the July 2012 message reported by Snopes. Where did Dr. London get his numbers? Maybe he compounded this hypothetical saving as 1% per month? No, because that would give more than$4 million. One does get about $1.3 million if one saves$375 a month for thirty years with a return of 1% per month, though.
Perhaps a more interesting question is why this “fake math” is coming back after five years. In 2012, Paul Ryan put forward a plan to “privatize” Social Security, and such a plan is now being revived. The only way to sell such a plan is to convince people that if they saved in a private account the amount of payroll taxes that “goes into” Social Security, they would get better benefits. This may be factually wrong, but that’s hardly the point.
# Ellenberg’s announcement of a solution to the cap-set problem
Jordan Ellenberg has just announced a resolution of the “cap problem” using techniques of Croot, Lev and Pach, in a self-contained three-page paper. This is a quite unexpected development for a long-standing open problem in the core of additive combinatorics.
Perhaps the right starting point for this story is 1936, when Erdos and Turan conjectured that, for every ${k}$, if ${A}$ is a subset of ${\{1,\ldots, N\}}$ without ${k}$-terms arithmetic progressions, then ${|A|= o_k(n)}$, or, equivalently, that if ${A}$ is a subset of the integers of positive density, then it must have arbitrarily long arithmetic progressions. Their goal in stating this conjecture was that resolving it would be a stepping stone to proving that the prime numbers have arbitrarily long arithmetic progressions. This vision came true several decades later. Szemeredi proved the conjecture in 1975, and Green and Tao proved that the primes contain arbitrarily long arithmetic progressions in 2004, with Szemeredi’s theorem being a key ingredient in their proof.
Rewinding a bit, the first progress on the Erdos-Turan conjecture came from Roth, who proved the ${k=3}$ case In 1955. Roth’s proof establishes that if ${A \subseteq \{ 1,\ldots, N\}}$ does not have length-3 arithmetic progressions, then ${|A|}$ is at most, roughly ${N/\log\log N}$. Erdos also conjectured that the bound should be ${o(N/\log N)}$, and if this were true it would imply that the primes have infinitely many length-3 arithmetic progressions simply because of their density.
Roth’s proof uses Fourier analysis, and Meshulam, in 1995, noted that the proof becomes much cleaner, and it leads to better bounds, if one looks at the analog problem in ${{\mathbb F}_p^n}$, where ${{\mathbb F}_p}$ is a finite field (of characteristic different from 2). In this case, the question is how big can ${A\subseteq {\mathbb F}_p^n}$ be if it does not have three points on a line. An adaptation of Roth’s techniques gives an upper bound of the order of ${p^n/n}$, which, for constant ${p}$, is of the order of ${N/\log N}$ if ${N}$ is the size of the universe of which ${A}$ is a subset.
Bourgain introduced a technique to work on ${{\mathbb Z}/N{\mathbb Z}}$ “as if” it where a vector space over a finite field, and proved upper bounds of the order of ${N/\sqrt {\log N}}$ and then ${N/(\log N)^{3/4}}$ to the size of a subset of ${\{ 1,\ldots , N\}}$ without length-3 arithmetic progressions. The latest result in this line is by Sanders, who proved a bound of ${(N poly\log\log N)/\log N}$, very close to Erdos’s stronger conjecture.
How far can these results be pushed? A construction of Behrend’s shows that there is a set ${A\subseteq \{ 1,\ldots, N\}}$ with no length-3 arithmetic progression and size roughly ${N/2^{\sqrt {\log N}}}$. The construction is simple (it is a discretization of a sphere in ${\sqrt {\log N}}$ dimensions) and it has some unexpected other applications. This means that the right bound in Roth’s theorem is of the form ${N^{1-o(1)}}$ and that the “only” question is what is the ${N^{-o(1)}}$ term.
In the finite vector space case, there is no analog of Behrend’s construction, and so the size of say, the largest subset of ${{\mathbb F}_3^n}$ without three points on a line, was completely open, with an upper bound of the order of ${3^n/n}$ and lower bounds of the order of ${c^n}$ for some constant ${c<3}$. The cap problem was the question of whether the right bound is of the form ${3^{(1-o(1)) }}$ or not.
Two weeks ago, Croot, Lev and Pach proved that if ${A}$ is a subset of ${({\mathbb Z}/4{\mathbb Z})^n}$ without length-3 arithmetic progressions, then ${|A|}$ is at most of the order of ${4^{.926 \cdot n}}$. This was a strong indication that the right bound in the cap problem should be sub-exponential.
This was done a couple of days ago by Ellenberg, who proved an upper bound of the form ${(2.756)^n}$ holds in ${{\mathbb F}_3^n}$. The proof is not specific to ${{\mathbb F}_3}$ and generalizes to all finite fields.
Both proofs use the polynomial method. Roughly speaking, the method is to associate a polynomial to a set of interest (for example, by finding a non-zero low-degree polynomial that is zero for all points in the set), and then to proceed with the use of simple properties of polynomials (such as the fact that the space of polynomials of a certain degree has a bounded dimension, or that the set of zeroes of a univariate non-zero polynomial is at most the degree) applied either to the polynomial that we constructed or to the terms of its factorization.
Let ${P_d}$ be the vector space of ${n}$-variate polynomials over ${{\mathbb F}_3}$ of total degree ${d}$ that are cube-free (that is, such that all variables occur in monomials with degree 0, 1, or 2), and let ${m_d}$ be its dimension.
If ${A\subseteq {\mathbb F}_3^n}$ is a set such that there are no distinct ${a,b,c}$ such that ${a+b+c=0}$ (a different property from being on a line, but the draft claims that the same argument works for the property of not having three points on a line as well), then Ellenberg shows that
$\displaystyle m_d - 3^n + |A| \leq 3 m_{d/2}$
then the bound follows from computing that ${m_{2n/3} \geq 3^n - c^n}$ and ${m_{n/3} \leq c^n}$ for ${c \approx 2.756\cdots}$.
The finite field Kakeya problem is another example of a problem that had resisted attacks from powerful Fourier-analytic proofs, and was solved by Zeev Dvir with a relatively simple application of the polynomial method. One may hope that the method has not yet exhausted its applicability.
Gil Kalai has posted about further consequence of the results of Croot, Lev, Pach and Ellenberg.
# Paul Erdös’s 102-ennial
Paul Erdös would be 102 year old this year, and in celebration of this the Notices of the AMS have published a two-part series of essays on his life and his work: [part 1] and [part 2].
Of particular interest to me is the story of the problem of finding large gaps between primes; recently Maynard, Ford, Green, Konyagin, and Tao solved an Erdös $10,000 question in this direction. It is probably the Erdös open question with the highest associated reward ever solved (I don’t know where to look up this information — for comparison, Szemeredi’s theorem was a$1,000 question), and it is certainly the question whose statement involves the most occurrences of “$\log$“.
# Harald Helfgott on Growth in Groups
The Bulletin of the AMS is going to publish a 57-page survey on growth in groups, which is already online, and which touches several topics of interest to readers of in theory, including the recent work of Bourgain and Gamburd on expander Cayley graphs of $SL_2(p)$ and the work of Helfgott and Seress on the diameter of permutation groups.
# Alexander Grothendieck
Alexander Grothendieck died on Thursday at age 86 in Saint-Girons.
Grothendieck, who started his work in functional analysis, is known for his far-reaching (and still incomplete) program of creating new foundations for algebraic geometry, a work that he led at IHES in the 50s and 60s and that is documented in the thousands of pages of EGA and SGA. If modern algebraic geometry is built on schemes, has applications to number theory, has definitions given in a category-theoretic language, and is completely incomprehensible to the novice, it is all thanks to Grothendieck’s vision.
In his 40s, and the top of his game, Grothendieck left IHES over the issue of military funding, and progressively detached from the mathematical community, while embracing environmental and anti-war causes.
Grothendieck’s life story, from his escaping Nazi Germany, to his revolution in mathematics, to his radical politics and later hermit life, is spellbinding. Some of it is told in a great two-part article in the Notices of the AMS (part 1, part 2) and I would also recommend this more technical essay and this more philosophical one.
Grothendieck has a flair for effective naming, he had a way with words, and he liked elaborate metaphors. Here is his famous “how to break a nut” analogy describing his style of mathematical research
I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible
through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!
# Old and new allegiances
The Italian prime minister is in the United States for the UN general assembly meeting, and he was in the San Francisco bay area on Sunday and Monday. (No, this is not the one who paid money to an underage prostitute and had she released from custody by falsely claiming she was the niece of Mubarak, it’s the new one.)
Those two days were busy, and he met Italian start-up founders in the area, went to a dinner at Stanford hosted by John Hennessy, presided the inauguration of a new Italian-language school, went to Twitter, Google, Facebook and Yahoo, and he met the local Italian research community.
For the last event, a few colleagues and I were asked to give a short presentation. Being not sure what to say to a prime minister, I asked a colleague who is the department chair at an Italian computer science department for some data on state funding of university research in computer science, and if there was a way to turn this data into a recommendation, and his response could be summarized as “we cannot be saved, there is no hope.” This might have made a good theme for a presentation, but instead I talked about the importance of fundamental research, and of working on ideas for which the technology is not ready yet, so that when the technology is ready the ideas are mature. Politicians are good at feigning attention when their mind is elsewhere, and he feigned it well.
Yesterday I was “interviewed” as part of the process to naturalize as an American citizen. Part of the interview is a test of the knowledge that an American is supposed to have. I liked to think that the officer would bring up a map of the world and ask me to point to France, then I would point to Brazil, and he would say, “great, now you are ready to be an American!” (Instead he asked how many US senators there are, when was the constitution written, and things like that.) The vibe was very different from any other interaction I have had with the American immigration system before; now it’s not any more “who are you, why are you stealing our jobs, and how do we know you are not a terrorist,” but it’s all “yay, you are going to be one us.”
# Riemann zeta functions and linear operators
A conjectural approaches to the Riemann hypothesis is to find a correspondence between the zeroes of the zeta function and the eigenvalues of a linear operator. This was first conceived by Hilbert and by Pólya in the 1910s. (See this correspondence with Odlyzko.) In the subsequent century, there have been a number of rigorous results relating the zeroes of zeta functions in other settings to eigenvalues of linear operators. There is also a connection between the distribution of known zeroes of the Riemann zeta function, as derived from explicit calculations, and the distribution of eigenvalues of the Gaussian unitary ensemble.
In a previous post we talked about the definition of the Ihara zeta function of a graph, and Ihara’s explicit formula for it, in terms of a determinant that is closely related to the characteristic polynomial of the adjacency matrix of the graph, so that the zeroes of the zeta function determine the eigenvalue of the graph. And if the zeta function of the graph satisfies a “Riemann hypothesis,” meaning that, after a change of variable, all zeroes and poles of the zeta function are ${1/\sqrt {d-1}}$ in absolute value, then the graph is Ramanujan. Conversely, if the graph is Ramnujan then its Ihara zeta function must satisfy the Riemann hypothesis.
As I learned from notes by Pete Clark, the zeta functions of curves and varieties in finite fields also involve a determinant, and the “Riemann hypothesis” for such curves is a theorem (proved by Weil for curves and by Deligne for varieties), which says that (after an analogous change of variable) all zeroes and poles of the zeta function must be ${\sqrt q}$ in absolute value, where ${q}$ is the size of the finite field. This means that one way to prove that a family of ${(q+1)}$-regular graphs is Ramanujan is to construct for each graph ${G_n}$ in the family a variety ${V_n}$ over ${{\mathbb F}_q}$ such that the determinant that comes up in the zeta function of ${G_n}$ is the “same” (up to terms that don’t affect the roots, and to the fact that if ${x}$ is a root for one characteristic polynomial, then ${1/x}$ has to be a root for the other) as the determinant that comes up in the zeta function of the variety ${V_n}$. Clark shows that one can constructs such families of varieties (in fact, curves are enough) for all the known algebraic constructions of Ramanujan graphs, and one can use families of varieties for which the corresponding determinant is well understood to give new constructions. For example, for every ${d}$, if ${k}$ is the number of distinct prime divisors of ${d-1}$ (which would be one if ${d-1}$ is a prime power) Clark gets a family of graphs with second eigenvalue ${2^k \sqrt{d-1}}$. (The notes call ${k}$ the number of distinct prime divisors of ${d}$, but it must be a typo.)
So spectral techniques underlie fundamental results in number theory and algebraic geometry, which then give us expander graphs. Sounds like something that we (theoretical computer scientists) should know more about.
The purpose of these notes is to explore a bit more the statements of these results, although we will not even begin to discuss their proofs.
One more reason why I find this material very interesting is that all the several applications of polynomials in computer science (to constructing error-correcting codes, secret sharing schemes, ${k}$-wise independent generators, expanders of polylogarithmic degree, giving the codes used in PCP constructions, self-reducibility of the permanent, proving combinatorial bounds via the polynomial method, and so on), always eventually rely on three things:
• A multivariate polynomial restricted to a line can be seen as a univariate polynomial (and restricting to a ${k}$-dimensional subspace gives a ${k}$-variate polynomial); this means that results about multivariate polynomials can often be proved via a “randomized reduction” to the univariate case;
• A univariate polynomial of degree ${d}$ has at most ${d}$ roots, which follows from unique factorization
• Given ${d}$ desired roots ${a_1,\ldots,a_d}$ we can always find an univariate polynomial of degree ${d}$ which has those roots, by defining it as ${\prod_i (x-a_i)}$.
This is enough to explain the remarkable pseudo-randomness of constructions that we get out of polynomials, and it is the set of principles that underlie much of what is below, except that we are going to restrict polynomials to the set of zeroes of another polynomial, instead of a line, and this is where things get much more complicated, and interesting.
Before getting started, however, I would like to work out a toy example (which is closely related to what goes on with the Ihara zeta function) showing how an expression that looks like the one defining zeta functions can be expressed as a characteristic polynomial of a linear operator, and how its roots (which are then the eigenvalues of the operator) help give bound to a counting problem, and how one can get such bounds directly from the trace of the operator.
If we have quantities ${C_n}$ that we want to bound, then the “zeta function” of the sequence is
$\displaystyle z(T) := \exp\left( \sum_n \frac {C_n}{n} T^n \right)$
For example, and this will be discussed in more detail below, if ${P(\cdot,\cdot)}$ if a bivariate polynomial over ${{\mathbb F}_q}$, we may be interested in the number ${C_1}$ of solutions of the equation ${P(x,y) = 0}$ over ${{\mathbb F}_q}$, as well as the number of solutions ${C_n}$ over the extension ${{\mathbb F}_{q^n}}$. In this case, the zeta function of the curve defined by ${P}$ is precisely
$\displaystyle z(T) = \exp\left( \sum_n \frac {C_n}{n} T^n \right) \ \ \ \ \ (1)$
and Weil proved that ${z(\cdot)}$ is a rational function, and that if ${\alpha_1,\ldots,\alpha_{2g}}$ are the zeroes and poles of ${z()}$, that is, the roots of the numerator and the denominator, then they are all at most ${\sqrt q}$ in absolute value, and one has
$\displaystyle | C_n - q^n | \leq \sum_i |\alpha_i|^n \leq 2g q^{n/2} \ \ \ \ \ (2)$
How does one get an approximate formula for ${C_n}$ as in (2) from the zeroes of a function like (1), and what do determinants and traces have got to do with it?
Here is our toy example: suppose that we have a graph ${G=(V,E)}$ and that ${c_\ell}$ is the number of cycles of the graph of length ${\ell}$. Here by “cycle of length ${\ell}$” we mean a sequence ${v_0,\ldots,v_\ell}$ of vertices such that ${v_0=v_\ell}$ and, for ${i=0,\ldots,\ell-1}$, ${\{ v_i,v_{i-1} \} \in E}$. So, for example, in a graph in which the vertices ${a,b,c}$ form a triangle, ${a,b,c,a}$ and ${a,c,b,a}$ are two distinct cycles of length 3, and ${a,b,a}$ is a cycle of length 2.
We could define the function
$\displaystyle z(T) := \exp\left( \sum_\ell \frac {c_\ell}{\ell} T^\ell \right)$
and hope that its zeroes have to do with the computation of ${c_\ell}$, and that ${z}$ can be defined in terms of the characteristic polynomial.
Indeed, we have (remember, ${T}$ is a complex number, not a matrix):
$\displaystyle z(T) = \frac 1 {\det (I - TA) }$
and, if ${\alpha_1,\ldots,\alpha_n}$ are the zeroes and poles of ${z(T)}$, then
$\displaystyle c_\ell \leq \sum_i |\alpha_i|^{-\ell}$
How did we get these bounds? Well, we cheated because we already knew that ${c_\ell = {\rm Tr}(A^\ell) = \sum_i \lambda_i^\ell}$, where ${\lambda_i}$ are the eigenvalues of ${A}$. This means that
$\displaystyle z(T) = \exp \left( \sum_\ell \sum_i \frac {\lambda_i^\ell}{\ell} T^\ell \right)$
$\displaystyle = \prod_i \exp \left( \sum_\ell \frac {\lambda_i^\ell}{\ell} T^\ell \right)$
$\displaystyle = \prod_i \frac 1 {1- T\lambda_i }$
$\displaystyle = \frac 1 { \det(I - TA) }$
Where we used ${\frac 1 {1-x} = \exp\left( \sum_n \frac { x^n}{n} \right)}$. Now we see that the poles of ${z}$ are precisely the inverses of the eigenvalues of ${A}$.
Now we are ready to look more closely at various zeta functions.
I would like to thank Ken Ribet and Jared Weinstein for the time they spent answering questions. Part of this post will follow, almost word for word, this post of Terry Tao. All the errors and misconceptions below, however, are my own original creation. | 2020-08-14 13:39:25 | {"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": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 129, "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": 0.7198135852813721, "perplexity": 315.14174856482225}, "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-2020-34/segments/1596439739328.66/warc/CC-MAIN-20200814130401-20200814160401-00010.warc.gz"} |
https://www.gradesaver.com/textbooks/math/other-math/thinking-mathematically-6th-edition/chapter-5-number-theory-and-the-real-number-system-chapter-summary-review-and-test-review-exercises-page-335/45 | ## Thinking Mathematically (6th Edition)
$0.\overline{5} = \frac{5}{9}$
RECALL: The repeating decimal $0.\overline{a}$, where $a$ is an integer between 0 and 10, is equal to $\frac{a}{9}$ Thus, the given repeating decimal when expressed as a quotient of two integers in lowest terms is: $0.\overline{5} = \frac{5}{9}$ | 2018-09-22 06:52:55 | {"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": 0.8022042512893677, "perplexity": 212.74880358572997}, "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-2018-39/segments/1537267158205.30/warc/CC-MAIN-20180922064457-20180922084857-00473.warc.gz"} |
https://www.physicsforums.com/threads/do-the-efes-physically-describe-something.881933/ | # B Do the EFEs physically describe something
1. Aug 13, 2016
### Megaton
Hi I'm new to this so please don't butcher me. I am just a enthusiastic individual with a huge interest in theoretical physics.
In GR, the Einstein field equations relate energy and momentum (I think) to curvature, then to local flat geometry (I think) my question is do the EFEs physically describe something (ie: a rotating spherical object like earth or a large nebula etc...) or are they a purely mathematical tool used to describe local geometry in relation to energy and momentum. I know the geodesic equation is needed to explain the "force" of gravity which results, i'm just wondering if they also to describe the topology as well.
I have always found this confusing, and can not seem to get a clear answer on it, thanks to anyone would cold help.
2. Aug 14, 2016
### Staff: Mentor
More precisely, they relate stress-energy, which includes energy and momentum (density) but also includes pressure and other stresses, to a portion of curvature, the Einstein tensor.
I'm not sure what you mean by this. The fact that spacetime can be considered flat locally is not due to the EFE (although the EFE is certainly consistent with it).
Solutions to the EFE, i.e., metric tensors, physically describe spacetimes that can contain various kinds of gravitating objects, like planets, stars, black holes, etc.
The geodesic equation is not the same as the EFE.
The topology is something else again; topology is not the same as geometry.
3. Aug 14, 2016
### Ibix
You can think of the Field Equations as like Newton's law of gravity (note: the EFEs aren't quite equivalent to the Newtonian force equation - but it'll do for this explanation). Both tell you how gravity works in a general sense, but you need to feed in specific details in order to describe a specific situation. So, for example, to describe the solar system in Newtonian gravity you would write the gravitational force on a small mass m as
$$\vec F=-Gm\left ( \frac {M_{sun}}{|\vec r- \vec r_{sun}|^3}(\vec r- \vec r_{sun})+ \frac {M_{mercury}}{|\vec r- \vec r_{mercury}|^3}(\vec r- \vec r_{mercury})+ \frac {M_{venus}}{|\vec r- \vec r_{venus}|^3}(\vec r- \vec r_{venus})+\ldots\right)$$
plus terms for all other planets, moons, etc. Which is rather more complicated than $F=GMm/r^2$.
The EFEs are (mathematically) worse than Newtonian gravity for a number of reasons. First, as you and Peter noted, there are more things than just mass that go in to the "source" term. Secondly, the equations are non-linear, so the solution for mass 1 plus the solution for mass 2 is not the same as the solution for mass 1 plus mass 2. The combination of the two (and other factors, like an extra dimension) is why I can write the Newtonian solution for multiple point masses off the top of my head but you need computer support to do the same thing for the full GR solution.
As Peter says, topology and geometry are different things. A good example is the Asteroids computer game. The whole thing takes place in flat space described by Euclidean geometry. If you couldn't move your spaceship off the edge of the screen, that would be a finite bounded Euclidean space - the topology and geometry of a sheet of paper. But the game lets you move off one edge of the screen and on to the opposite edge. The geometry is the same (triangles have 180 degree interior angles, circles have $c=2\pi r$ etc.), but the topology is toroidal - like the surface of a donut.
Last edited: Aug 14, 2016
4. Aug 14, 2016
### pervect
Staff Emeritus
GR as a theory definitely makes experimental predictions. I'd go so far as to call them "physical" predictions, though it's not quite clear what that term means in abstract or to you.
I believe I recall reading that one doesn't need any more assumptions than the EFE to get the complete theory of GR, but perhaps I'm missing some small seemingly innocuous assumptions. For instance, I think I recall it being said that one doesn't need to assume geodesic motion of test particles as a separate assumption, but I also recall that there may be some seemingly innocuous assumptions required to prove that, related to the positivity of the energy density. I'm not sure if such seemingly fine points are of interest to you, though I suspect you're just looking for the "big picture".
The big picture would be that if you have a ball of matter, like a rotating Earth, you can get experimental predictions out of the math, for instance the "frame dragging" results of GPB, and other, less subtle, predictions.
You might find some situations where you would require information about the properties of the matter itself which is separate from the EFE's to make those predictions, and there might be some situations where the information is not known for a certanity. The structure of neutron stars would, I think, fall into this category, there are some reasonable theories about them, but I believe there may be some uncertanities in their structure relating not to the EFE's, but to the "physical" properties of neutron degenerate matter itself.
Sorry if this isn't an exact answer, but the question is general, and a lot depends on whether your trying to get a razor-sharp answer or just a general "feel".
5. Aug 14, 2016 | 2017-12-16 23:28:14 | {"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": 0.6045114994049072, "perplexity": 384.72732183113965}, "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-2017-51/segments/1512948589512.63/warc/CC-MAIN-20171216220904-20171217002904-00010.warc.gz"} |
https://forum.allaboutcircuits.com/threads/battery-tester-how-do-these-things-work-and-why-should-i-buy-a-more-expensive-one.188990/ | # Battery Tester - How do these things work, and why should I buy a more expensive one?
#### Joe Stavitsky
Joined Apr 5, 2020
107
I'd like to see if I can build one of these myself.
This is this the first Google result I got for a circuit
Is it accurate? Meaning, is it the kind of thing I'm likely to find in a commercial model? If not, is a commercial model better in any respect? In what way is the $79 dollar unit "better" than the$11 dollar unit?
Thanks so much
Joe
ETA: My strategic concern is not specifically car batteries (although I may get one eventually) but large batteries in general, like UPS and jump starters.
Last edited:
#### MrChips
Joined Oct 2, 2009
27,144
How does one test a battery?
What is the criteria for differentiating between a good and bad battery?
#### Joe Stavitsky
Joined Apr 5, 2020
107
How does one test a battery?
What is the criteria for differentiating between a good and bad battery?
A good battery holds charge for optimally infinite time and passes current with no internal resistance, and a bad battery... doesn't.
#### MrChips
Joined Oct 2, 2009
27,144
A good battery holds charge for optimally infinite time and passes current with no internal resistance, and a bad battery... doesn't.
Then that is what the instrument should measure.
#### strantor
Joined Oct 3, 2010
6,112
The devices you linked to are load testers. They place a high current draw on the battery and see how much current it can source, measure voltage sag, measure internal resistance of the battery, and some of them test other external factors such as alternator output, starter current, chassis ground resistance, etc. They are relative complex compared to the circuit you posted, which merely measures voltage level of a battery.
The $11 unit is probably the commercial equivalent of the circuit you posted and won't test any of the things I mentioned above. You can DIY this simple circuit with relative ease, but it will probably not cost you less than$11. You're better off buying the $11 cheap China tester if all you care about is testing battery voltage. If what you care about is diagnosing automotive battery charging and starting systems then your best bet is to buy one of the more expensive commercial units. If what you care about is learning electronics with a somewhat practical starting point, building the circuit you posted is a good starting point. If you're a masochist and what you care about is learning how to swim in electronics by diving off a cliff into shark infested rocky waters, rolling your automated load tester is a good place to start. If I have a good understanding of what you're after (a good load tester that provides useful data for a decent price) then I recommend letting go of all notions of DIY because you already found your answer and it costs >$50.
#### crutschow
Joined Mar 14, 2008
30,806
Just measuring its voltage doesn't necessarily tell you whether the battery is good or bad.
#### Joe Stavitsky
Joined Apr 5, 2020
107
The devices you linked to are load testers. They place a high current draw on the battery and see how much current it can source, measure voltage sag, measure internal resistance of the battery, and some of them test other external factors such as alternator output, starter current, chassis ground resistance, etc. They are relative complex compared to the circuit you posted, which merely measures voltage level of a battery.
The $11 unit is probably the commercial equivalent of the circuit you posted and won't test any of the things I mentioned above. You can DIY this simple circuit with relative ease, but it will probably not cost you less than$11. You're better off buying the $11 cheap China tester if all you care about is testing battery voltage. If what you care about is diagnosing automotive battery charging and starting systems then your best bet is to buy one of the more expensive commercial units. If what you care about is learning electronics with a somewhat practical starting point, building the circuit you posted is a good starting point. If you're a masochist and what you care about is learning how to swim in electronics by diving off a cliff into shark infested rocky waters, rolling your automated load tester is a good place to start. If I have a good understanding of what you're after (a good load tester that provides useful data for a decent price) then I recommend letting go of all notions of DIY because you already found your answer and it costs >$50.
See edit. Thanks for the suggestion, and I'll look into what you said. I have a retail location nearby that lends tools, I'll see if they have anything.
#### MrChips
Joined Oct 2, 2009
27,144
Simple.
I take a 12V automotive headlight bulb and socket and mount in on a metal box with two toggle switches for low and high beam. I put a shield over the lamp because the light output is intense.
You can leave it on as long as it takes to get an estimate of battery capacity.
#### Joe Stavitsky
Joined Apr 5, 2020
107
Simple.
I take a 12V automotive headlight bulb and socket and mount in on a metal box with two toggle switches for low and high beam. I put a shield over the lamp because the light output is intense.
You can leave it on as long as it takes to get an estimate of battery capacity.
This is also a solid plan.
#### dl324
Joined Mar 30, 2015
15,117
A good battery holds charge for optimally infinite time and passes current with no internal resistance, and a bad battery... doesn't.
You're describing an ideal battery that doesn't exist in the real world.
#### strantor
Joined Oct 3, 2010
6,112
Simple.
I take a 12V automotive headlight bulb and socket and mount in on a metal box with two toggle switches for low and high beam. I put a shield over the lamp because the light output is intense.
You can leave it on as long as it takes to get an estimate of battery capacity.
The testers he showed are automotive battery testers so I assume the application is an automotive battery. A car head lamp won't be near enough load.
#### debe
Joined Sep 21, 2010
1,320
This is what i use & seems to work for me. I test a new fully charged battery when i buy it & record the results. That is then my reference as the battery ages. ( most new batterys i have ever purchased are not fully charged when bought}
#### LowQCab
Joined Nov 6, 2012
2,506
If the device You are using does not provide at least
the "average" Load the Battery will experience in normal service,
then it leaves-out a lot of very valuable information.
A tester that digitally displays the "Cold-Cranking-Amps" of a Battery
using ~20-gauge Test-Leads, is an absolute joke.
It might give You some various useful bits of information, but don't believe the display.
.
.
.
Last edited:
#### debe
Joined Sep 21, 2010
1,320
On those displays as the years go by the cold cranking amp display actualy drops, as does internal resistance. In the Ford dealership i used to work in you had to use one of these testers with a print out. This was used for Ford warranty claims, if the tester did not come up with battery fail, then no warranty claim. | 2022-12-05 07:52:06 | {"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": 0.24212773144245148, "perplexity": 2617.225631807159}, "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/1669446711013.11/warc/CC-MAIN-20221205064509-20221205094509-00000.warc.gz"} |
https://quantumfrontiers.com/2015/02/28/always-look-on-the-bright-side-of-cptp-maps/ | # Always look on the bright side…of CPTP maps.
Once upon a time, I worked with a postdoc who shaped my views of mathematical physics, research, and life. Each week, I’d email him a PDF of the calculations and insights I’d accrued. He’d respond along the lines of, “Thanks so much for your notes. They look great! I think they’re mostly correct; there are just a few details that might need fixing.” My postdoc would point out the “details” over espresso, at a café table by a window. “Are you familiar with…?” he’d begin, and pull out of his back pocket some bit of math I’d never heard of. My calculations appeared to crumble like biscotti.
Some of the math involved CPTP maps. “CPTP” stands for a phrase little more enlightening than the acronym: “completely positive trace-preserving”. CPTP maps represent processes undergone by quantum systems. Imagine preparing some system—an electron, a photon, a superconductor, etc.—in a state I’ll call “$\rho$“. Imagine turning on a magnetic field, or coupling one electron to another, or letting the superconductor sit untouched. A CPTP map, labeled as $\mathcal{E}$, represents every such evolution.
“Trace-preserving” means the following: Imagine that, instead of switching on the magnetic field, you measured some property of $\rho$. If your measurement device (your photodetector, spectrometer, etc.) worked perfectly, you’d read out one of several possible numbers. Let $p_i$ denote the probability that you read out the $i^{\rm{th}}$ possible number. Because your device outputs some number, the probabilities sum to one: $\sum_i p_i = 1$. We say that $\rho$ “has trace one.” But you don’t measure $\rho$; you switch on the magnetic field. $\rho$ undergoes the process $\mathcal{E}$, becoming a quantum state $\mathcal{E(\rho)}$. Imagine that, after the process ended, you measured a property of $\mathcal{E(\rho)}$. If your measurement device worked perfectly, you’d read out one of several possible numbers. Let $q_a$ denote the probability that you read out the $a^{\rm{th}}$ possible number. The probabilities sum to one: $\sum_a q_a =1$. $\mathcal{E(\rho)}$ “has trace one”, so the map $\mathcal{E}$ is “trace preserving”.
Now that we understand trace preservation, we can understand positivity. The probabilities $p_i$ are positive (actually, nonnegative) because they lie between zero and one. Since the $p_i$ characterize a crucial aspect of $\rho$, we call $\rho$ “positive” (though we should call $\rho$ “nonnegative”). $\mathcal{E}$ turns the positive $\rho$ into the positive $\mathcal{E(\rho)}$. Since $\mathcal{E}$ maps positive objects to positive objects, we call $\mathcal{E}$ “positive”. $\mathcal{E}$ also satisfies a stronger condition, so we call such maps “completely positive.”**
So I called my postdoc. “It’s almost right,” he’d repeat, nudging aside his espresso and pulling out a pencil. We’d patch the holes in my calculations. We might rewrite my conclusions, strengthen my assumptions, or prove another lemma. Always, we salvaged cargo. Always, I learned.
I no longer email weekly updates to a postdoc. But I apply what I learned at that café table, about entanglement and monotones and complete positivity. “It’s almost right,” I tell myself when a hole yawns in my calculations and a week’s work appears to fly out the window. “I have to fix a few details.”
Am I certain? No. But I remain positive.
*Experts: “Trace-preserving” means $\rm{Tr}(\rho) =1 \Rightarrow \rm{Tr}(\mathcal{E}(\rho)) = 1$.
**Experts: Suppose that ρ is defined on a Hilbert space and that is defined on . “ is positive” means
To understand what “completely positive” means, imagine that our quantum system interacts with an environment. For example, suppose the system consists of photons in a box. If the box leaks, the photons interact with the electromagnetic field outside the box. Suppose the system-and-environment composite begins in a state defined on a Hilbert space . acts on the system’s part of state. Let denote the identity operation that maps every possible environment state to itself. Suppose that changes the system’s state while preserves the environment’s state. The system-and-environment composite ends up in the state . This state is positive, so we call “completely positive”:
This entry was posted in Reflections, Theoretical highlights by Nicole Yunger Halpern. Bookmark the permalink.
I'm a theoretical physicist and an ITAMP Postdoctoral Fellow at the Harvard-Smithsonian Institute for Theoretical Atomic, Molecular, and Optical Physics (ITAMP). Catch me at ITAMP, Harvard physics, or MIT. Before moving here, I completed a physics PhD at Caltech's Institute for Quantum Information and Matter, under John Preskill's auspices. I write one article per month for Quantum Frontiers. My research consists of what I call "quantum steampunk" (https://quantumfrontiers.com/2018/07/29/so-long-and-thanks-for-all-the-fourier-transforms/): I re-envision 19th-century thermodynamics with 21st-century quantum information theory, and I use the combination as a new lens through which to view various fields of science.
## 7 thoughts on “Always look on the bright side…of CPTP maps.”
1. First comment! Also, a personal note. I’ve seen the frustration in students when they first take QM and they realize there are so many things to explore about its principles. One sacrifices generality for a coherent introduction. It is just reasonable to go for incremental understanding and it might be too soon at that point to talk about mixed states and quantum transformations. But it is fun to tease them a bit with this kind of topics. It not only sparks their curiosity but makes them realize QM is a field that continuously challenges the student and the professor. The only choice is then, as you say, to remain positive. Thanks for the article!
• Thanks for your thoughts, Sergio! At least your students have an advantage, having an instructor who notices what challenges them and who tries to motivate them. If your students need more “teasing . . . with this kind of topics,” feel free to send them to Quantum Frontiers!
2. thanks Nicole food for thought here!
3. we have to remember that there is no boundary between the classical and quantum read Bell’s papers especially ‘Introduction to the hidden variable question.’
4. Read Feynman on spinning plate and wobble it is fundamental and so deep. What is spin in particles? It’s not classical spin is it so what exactly is it? Any thoughts let me know help me please!
jonti
5. Carlton Caves’ on-line notes “Completely positive maps, positive maps, and the Lindblad form” (2002) provide a student-friendly survey of the differential limit of CPTP maps (to my knowledge this material is not found elsewhere in the literature).
And thank you, Nicole, for your outstanding work and sustained commitment to quantum community-building.
@unpublished{cite-key, Author = {C.
M. Caves}, Note = {On-line memorandum
to C. A. Fuchs and J. Renes:
\url{http://info.phys.unm.edu/~caves/
</code | 2020-11-27 23:28:02 | {"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": 29, "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": 0.6170908212661743, "perplexity": 1857.300357271081}, "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/1606141194634.29/warc/CC-MAIN-20201127221446-20201128011446-00181.warc.gz"} |
https://proxies-free.com/tag/langmi/ | formal languages - \$ L = {a ^ i ;: ; ( exists x in Lang (M_i)) ;[ xx notin Lang(M_i) ] } \$ not recursively listable
Paper question of the past year:
To let $$M_i$$ identify the Turing machine with code $$i$$ with the alphabet $$Sigma = {a, b }$$,
Show that the following language can not be recursively enumerated:
$$L = {a ^ i ;: ; ( exists x in Lang (M_i)) ;[ xx notin Lang(M_i) ] }$$
I tried to show that we can reduce $$L_e$$, the set of Turing machines that accept no conditions to $$L$$, For that, I have to find a transformation $$f$$ That's how it is $$Long (M)$$ is empty, then the machine is represented by $$f (M)$$ has a string $$x$$ but not $$xx$$, and if $$Long (M)$$ is not empty, then the machine is represented by $$f (M)$$ contains $$xx$$ if it contains $$x$$,
But I am not sure how to find such a transformation. | 2019-07-17 11:06:01 | {"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": 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": 15, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6457475423812866, "perplexity": 385.960620995215}, "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-2019-30/segments/1563195525136.58/warc/CC-MAIN-20190717101524-20190717123524-00481.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-10th-edition/chapter-10-analytic-geometry-10-6-polar-equations-of-conics-10-6-assess-your-understanding-page-685/45 | ## Precalculus (10th Edition)
Use the fact that $d(F,P)=e\cdot d(D,P)$ and that $d(D,P)=p+r\sin\theta$ to derive and have $r=\dfrac{ep}{1-e\sin\theta}$ Refer to the step-bystep part for the actual derivation.
We know that $d(F,P)=e\cdot d(D,P)$ and that $d(D,P)=p+r\sin\theta$. Hence, $r=e\cdot(p+r\sin\theta)\\r=ep+er\sin\theta\\r-er\sin\theta=ep\\r(1-e\sin\theta)=ep\\r=\frac{ep}{1-e\sin\theta}$ | 2021-10-21 15:48:41 | {"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": 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": 0.8853316903114319, "perplexity": 497.0319586610298}, "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/1634323585424.97/warc/CC-MAIN-20211021133500-20211021163500-00600.warc.gz"} |
https://brilliant.org/problems/the-medium-equality/ | # The Medium Equality
Algebra Level 1
If $x$ is a real number satisfying $x^3 - \dfrac1{x^3} = 14$, compute $x - \dfrac1x$.
× | 2021-02-28 22:18:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "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": 0.8059644103050232, "perplexity": 2340.9445326290697}, "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-2021-10/segments/1614178361776.13/warc/CC-MAIN-20210228205741-20210228235741-00194.warc.gz"} |
https://khufkens.com/2017/05/24/level-theta-s-images/ | # Level Theta S images
My Virtual Forest project is still running strong and generates tons of spherical images (currently ~50GB). However, the post on which the camera sits is not perfectly level. The Theta S camera normally compensates for this using an internal gyroscope which detects pitch and roll of the camera. Yet, when downloading images directly from the camera no adjustments are made and the pitch and roll data is merely recorded in the EXIF data of the image.
As such I wrote a small bash script which rectifies (levels the horizon) in Theta S spherical images using this internal EXIF data. This is an alternative implementation to the THETA EXIF Library by Regen. I use his cute Lama test images for reference. All credit for the funky images go to Regen. Below is the quick install guide to using my script. I hope it helps speed up people’s Theta S workflow.
## Install
Download, fork or copy paste the script from my github repository to your machine and make it executable.
$chmod +x theta_rectify.sh ## Use $ theta_rectify.sh image.jpg
The above command will rectify the image.jpg file and output a new file called image_rectified.jpg.
Visual comparison between my results and those of Regen’s python script show good correspondence.
## Requirements
The script depends on a running copy of exiftools, imagemagick and POVRay. These tools are commonly available in most Linux distros, and can be installed on OSX using tools such as homebrew. I lack a MS Windows system, but the script should be easily adjusted to cover similar functionality. | 2023-02-06 05:39:20 | {"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": 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": 0.19609622657299042, "perplexity": 3651.315001399074}, "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/1674764500304.90/warc/CC-MAIN-20230206051215-20230206081215-00518.warc.gz"} |
https://www.omscs-notes.com/operating-systems/distributed-file-systems/ | # Operating Systems
## Distributed File Systems
Notice a tyop typo? Please submit an issue or open a PR.
# Distributed File Systems
## Distributed File Systems
Modern operating systems export a high-level filesystem interface to abstract all of the (potentially) different types of storage devices present on a machine and unify them under a common API.
The OS can hide the fact that there isn't even local physical storage on a machine; rather, the files are maintained on a remote filesystem that is being accessed over the network.
Environments that involve multiple machines for the delivery of the filesystem service are called distributed filesystems (DFS).
## DFS Models
A simple DFS model involves clients being served files from a server running on a different machine.
Often, the server is not running on a single machine, but rather is distributed across multiple machines. Files may be replicated across every machine or partitioned amongst machines.
In replicated systems, all the files are replicated and available on every server machine. If one machine fails, other machines can continue to service requests. Replicated systems are fault tolerant. In addition, requests entering a replicated system can be serviced by any of the replicas. Replicated systems are highly available.
In partitioned systems, each server holds only some subset of the total files. If you need to support more files, you can just partition across more machines. In the replicated model, you will need to upgrade your hardware. This makes partitioned systems more scalable.
It's common to see a DFS that utilizes both partitioning and replication; for example, partitioning all the files, and replicating each partition.
Finally, files can be stored on and served from all machines. This blurs the line between servers and clients, because all the nodes in the system are peers. Every node is responsible for maintaining the files and providing the filesystem service. Each peer will take some portion of the load by servicing some of the requests, often those that for files local to that peer.
## Remote File Service: Extremes
At one extreme, we have the upload/download model. When a client wants to access a file, it downloads the entire file, performs the modifications and then uploads the entire file back to the server.
The benefit of this model is that all of the modifications can be done locally, which means they can be done quickly, without incurring any network cost.
One downside of this model is that the client has to download the entire file, even for small modifications. A second downside of this model is that it takes away file access control from the server. Once the server gives the file to the client, it has no idea what the client is doing with the file or when it will give it back.
At the other extreme, we have the true remote file access. In this model, the file remains on the server and every single operation has to pass through the server. The client makes no attempt to leverage any kind of local caching or buffering.
The benefit of this extreme is that the server now has full control and knowledge of how the clients are accessing and modifying a file. This makes it easier to ensure that the state of the filesystem is consistent.
The downside of this model is that every file operations pays a network cost. In addition, this model is limited in its scalability. Since every file operation goes to the server, the server will get overloaded more quickly, and will not be able to service as many clients.
## Remote File Service: A Compromise
We should allow clients to benefit from using their local memory/disk and to store at least some part of the file they are accessing. This may include downloading blocks of the file they are accessing, as well as prefetching blocks that they may soon be accessing.
When files can be served from a local cache, lower latencies on file operations can be achieved since no additional network cost in incurred. In addition, server scalability is increased as local service decreases server load.
Once we allow clients to cache portions of the file, however, it becomes necessary for the clients to start interacting with the server more frequently. On the one hand, clients need to notify the server of any modifications they make. In addition, they need to query the server at some reasonable frequency to detect if the files the are accessing from their cache have been changed by someone else.
This approach is beneficial because the server continues to have insight into what the clients are doing and retains some control over which accesses can be permitted. This makes it easier to maintain consistency. At the same time, the server can be somewhat out of the way, allowing some proportion of operations to occur on the client. The helps to reduce load.
The downside with this compromise is that the server becomes more complex. The server needs to perform additional tasks and maintain additional state to make sure it can provide consistency guarantees. This also means that the client has to understand file sharing semantics that are different from what they are used to in a normal filesystem.
## Stateless vs Stateful File Server
A stateless server keeps no state.
It has no notion of:
• which files/blocks are being accessed
• which operations are being performed
• how many clients are accessing how many files
As a result, every request has to be completely self-contained. This type of server is suitable for the extreme models, but it cannot be used for any model that relies on caching. Without state, we cannot achieve consistency management. In addition, since every request has to be self-contained, more bits need to be transferred on the wire for each request.
One positive of this approach is that since there is no state on the server, there is no CPU/memory utilization required to manage that state. Another positive is that this design is very resilient. Since requests cannot rely on any internal state, the server can be restarted if it fails at no detriment to the client.
A stateful server maintains information about:
• the clients in the system
• which files are being accessed
• which types of accesses are being performed
• which clients have a file cached
• which clients have read/written the file
Because of the state, the server can allow data to be cached and can guarantee consistency. In addition, state management allows for other functionality like locking. Since accesses are known, clients can request relative blocks ("next kB of data") instead of having to specify absolute offsets.
On failure, however, all that state needs to be recovered so that the filesystem remains consistent. This requires that the state must be incrementally check-pointed to prevent too much loss. In addition, there are runtime overheads incurred by the server to maintain state and enforce consistency protocols and by the client to perform caching.
## Caching State in a DFS
Caching state in a DFS involves allowing clients to locally maintain a portion of the state - a file block, for example - and also allows them to perform operations on this cached state: open/read/write,etc.
Keeping the cached portions of the file consistent with the server's representation of the file requires cache coherence mechanisms.
For example, two clients may cache the same portion of file. If one client modifies their local portion, and sends the update to the file server, how does the other client become aware of those changes?
For client/server systems, these coherence mechanisms may be trigged in different ways. For example, the mechanism may be triggered on demand any time a client tries to access a file, or it may be triggered periodically, or it may triggered when a client tries to open a file. In addition, the coherence mechanisms may be driven by the client or the server.
## File Sharing Semantics in DFS
### Single Node, UNIX
Whenever a file is modified by any process, that change is immediately visible to any other process in the system. This will be the case even if the change isn't pushed out to disk because both processes have access to the same buffer cache.
### DFS
Even if a write gets pushed to the file server immediately, it will take some time before that update is actually delivered to the file server. It is possible that another client will not see that update for a while, and every time it performs a read operation it will continue seeing "stale" data. Given that message latencies may vary, we have no way of determining how long to delay any possible read operation in order to make sure that any write from anywhere in the system arrives at the file servers so that we can guarantee no staleness.
In order to maintain acceptable performance, a DFS will typically sacrifice some of the consistency, and will accept more relaxed file sharing semantics.
### Session Semantics
In session semantics, the client writes back whatever data was modified on close. Whenever a client needs to open a file, the cache is skipped, and the client checks to see if a more recent version is present on the file server.
With session semantics, it is very possible for a client to be reading a stale version of a file. However, we know that when we close a file or open a file that we are consistent with the rest of the filesystem at that moment.
Session semantics are easy to reason about, but they are not great for situations where clients want to concurrently share a file. For example, for two clients to be able to write to a file and see each other's updates, they will constantly need to close and re-open the file. Also, files that stay open for longer periods of time may become severely inconsistent.
In order to avoid long periods of inconsistency, the client may write back changes periodically to the server. In addition, the server can send invalidation notifications periodically, which can enforce time bounds on inconsistency. Once the server notification is received, a client has to sync with the most recent version of the file.
The filesystem can provide explicit operations to let the client flush its updates to the server, or sync its state with the remote server.
### Other strategies
With immutable files, you never modify an old file, but rather create a new file instead.
With transactions, the filesystem exports some API to allow clients to group file updates into a single batch to be applied atomically.
## File Vs Directory Service
Filesystems have two different types of files: regular files and directories. These two types of files often have very different access patterns. As a result, it is not uncommon to adopt one type of semantics for files, and another for directories. For example, we have may session semantics for files, and UNIX semantics for directories.
## Replication and Partitioning
Fileserver distribution can be accomplished via replication and/or partitioning.
With replication, the filesystem can be replicated onto multiple machines, such that each machine holds all the files.
The benefit of this is that client requests can be load balanced across all machines, which can lead to better performance. In addition, the system overall can be more available. Finally, this system is more fault tolerant. When one replica fails, the other replicas can serve clients with no degradation of service.
The downside of this approach is that writes become more complex. One solution is to force clients to write synchronously to all replicas. Another solution involves writing to one replica, and then having a background job asynchronously propagate the write to the other replicas.
If replicas get out of sync, they must be reconciled. One simple reconciliation technique is voting, where the majority wins.
With partitioning, every machine has a subset of all of the files in the system. Partitioning may be based on filename, file type, directory, or some other grouping.
The main benefit of this approach is that it allows for better scalability as the size of the filesystem grows. With the replicated approach, adding more machines does not mean that you can support a larger filesystem. With the partitioned approach, more machines does mean more files.
As well, writes are simple. Since a single file only lives on a single machine, writes to that file are localized to that machine.
A main downside of this approach is that when a partition goes down, the data stored on that partition can no longer be accessed.
In addition, balancing the system is more difficult, because we have to take into consideration how the particular files are accessed. If there is a particular file that is more frequently accessed by most clients in the system, we can run into the issue of hotspots.
Finally, we can have a solution where the filesystem is partitioned across some axis, and then each partition is replicated some number of times.
## Networking File System (NFS) Design
In a Networking File System (NFS), clients access files across the network, hence the name.
Clients request and access files via the VFS, using the same types of file descriptors and operations that they use to access files in their local storage. The VFS layer determines if the file belongs to the local filesystem or whether the request needs to be pushed to the NFS client so that it can be passed to the remote filesystem.
The NFS client interacts with the NFS server via RPC. The NFS server receives the request, forms it into a proper filesystem operation that is delivered to the local VFS. From there, the request is passed to the local file system interface. On the server machine, requests coming from the NFS server look identical to filesystem requests coming from any other application running on the server machine.
When an open request comes to an NFS server, it will create a file handle. This file handle will contain information about the server machine as well as information about the file. This handle will be returned back to the client machine. Whenever the client wants to access files via NFS, it can pass this handle. If the file is deleted or the server machine dies, this handle becomes stale.
## NFS Versions
NFSv3 is stateless, whereas NFSv4 is stateful. Since NFSv4 is stateful, it can support operations like file caching and file locking. Even though NFSv3 is stateless, the implementation typically includes modules that provide certain stateful functionality.
For files that are not accessed concurrently, NFS behaves with session semantics. On close, all of the changes made to a file are flushed to the server, and on open a check is performed and, if necessary the cached portions of the files are updated.
NFS also support periodic updates. Using these updates will break the session semantics when there are multiple clients that are concurrently updating a file.
The periods, by default, are 3 seconds for files and 30 seconds for directories. The rationale is that directories are modified less frequently, and when modified, the changes are more easily resolved.
NFSv4 incorporates a delegation mechanism where the server delegates to the client all rights to manage a file for a given period of time. This avoids the periodic update checks mentioned above.
With server-side state, NFS can support locking. NFS uses a lease-based mechanism to support locking. When a client acquires a lock, the server assigns a certain time period to the client during which the lock is valid. It is then the clients responsibility to either release the lock within the specified time frame or explicitly extend the lock. If the client goes down, the server isn't locked forever.
NFSv4 also supports a reader/writer lock called "share reservation". It also support mechanisms for upgrading from being a reader to a writer, and vice versa.
## Sprite DFS Access Pattern Analysis
In the paper on caching in the sprite system, the authors performed a study to see how files are accessed in the production filesystem used by their department.
They found that 33% of all file accesses are writes. Caching can be an important mechanism to improve performance: 66% of all operations can be improved. In order to leverage the cache for writes as well, a write-through policy is not sufficient.
Session semantics may have been a good strategy, but the authors also noticed that 75% of files were open less than 0.5 seconds and 90% of files were open less than 10 seconds. This means that the overheads associated with session semantics were still too high.
They also observed that 20-30% of new data was deleted within 30 seconds, with 50% of new data being deleted within 5 minutes. The also observed that file sharing in their system is rare. As a result, the authors decided that write-back on close was not really necessary.
Of course, the authors needed to support concurrent access even though they didn't need to optimize for it.
## Sprint DFS from Analysis to Design
The authors decided that Sprite should support caching, and use a write-back policy to send changes to the server.
Every 30 seconds, a client will write back all of the blocks that have not been modified within the last 30 seconds. Note that a file may be opened and closed multiple times by the client before any data is sent to the server.
The intuition behind this is that blocks that have been more recently modified will continue to be modified. It doesn't make sense to force write-backs on blocks that are likely to be worked on more. This strategy avoids sending the same blocks to the server over and over.
Note that this 30 second threshold is directly related to authors' observation that 20-30% of new data was deleted within 30 seconds.
When a client comes along and wants to open a file that is being written by another client, the server will contact the writer and collect all of the outstanding dirty blocks.
Every open operation has to go to the server. This means that directories cannot be cached on the client.
Finally, caching is completely disabled on concurrent writes. In this case, all of the writes will be serialized on the server-side. Because concurrent writes do not happen frequently, this cost will not be significant.
## File Access Operations in Sprite
Assume that we have multiple clients that are accessing a file for reading, and one writer client.
All open operations will go through the server.
All of the clients will be allowed to cache blocks. The writer will also have to keep timestamps on each modified block in order to enforce the 30 second write-back policy.
When the writer closes the file, the contents will be stored in the writers' cache. Of course, the next open will still have to go through the server. This means that the server and the client need to keep some sort of version number in order to stay in sync.
On a per file basis, the client keeps track of:
• cache (overall yes/no)
• cached blocks
• timer for each dirty block
• version number
On a per file basis, the server keeps track of:
• writer
• version
Let's assume that a new writer comes along after the first writer has closed the file. This is referred to as a sequential writer.
In this case, the server contacts the old writer to gather all of the dirty blocks, which is convenient given that the client keeps track of all dirty blocks.
If the old writer has closed the file, the server will update the value of the writer it holds to point to the new writer. At this point, the new writer can proceed and cache the file.
Let's assume that a third writer comes along and wants to write to the file concurrently with the current writer. This is concurrent sharing.
When the write request comes, the server contacts the current writer to gather the dirty blocks. When the server realizes that this writer hasn't closed the file, it will disable the caching of the file for both writers.
To enable/disable caching, it makes sense for the server to maintain a cacheable flag on a per file basis.
Both writers will continue to have access to the file, but all operations will have to go through the server.
When a writer closes a file, the server will see it - as every operation goes through the server - and will make the file cacheable again.
OMSCS Notes is made with in NYC by Matt Schlenker. | 2021-07-26 13:39:32 | {"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": 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": 0.2864585220813751, "perplexity": 1403.3613015932165}, "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/1627046152129.33/warc/CC-MAIN-20210726120442-20210726150442-00481.warc.gz"} |
https://www.minichemistry.com/molar-volume-molar-concentration.html | Molar Volume & Molar Concentration
Show/Hide Sub-topics (Stoichiometry | O Level)
Molar Volume is the volume occupied by one mole of gas.
• Molar volume changes with temperature and pressure. (Higher temperature = higher volume, higher pressure = smaller volume)
• Molar volume at a fixed temperature and pressure is the same for ALL gases.
• At room temperature ($25^{\circ}\text{C}$) and pressure (1 atmosphere, atm), also known as r.t.p., the molar volume of ANY gas is 24 $\text{dm}^{3}$.
• At standard temperature ($0^{\circ}\text{C}$) and pressure (1 atm), also known as s.t.p., the molar volume of ANY gas is 22.4 $\text{dm}^{3}$.
• $\text{dm}^{3}$ is known as cubic decimetre. 1 $\text{dm}^{3}$ = 1 litre = 1000 $\text{cm}^{3}$ = $\frac{1}{1000} \, \text{m}^{3}$
Note 1: It is important to know how to convert among $\text{cm}^{3}$, $\text{dm}^{3}$ and $\text{m}^{3}$ A short tutorial on the conversion process will be given at the end of this post. (For those who do not want to memorise the conversion numbers above)
Note 2: It is important to take note of the temperature and pressure in the given question to determine if r.t.p or s.t.p should be used.
At r.t.p,
$$\text{Vol. of gas} = \text{No. of moles} \times 24 \, \text{dm}^{3}$$
At s.t.p,
$$\text{Vol. of gas} = \text{No. of moles} \times 22.4 \, \text{dm}^{3}$$
Avogadro’s law states that, at the SAME temperature and pressure, equal volumes of all gases contain the same number of gas particles.
• $24 \, \text{dm}^{3}$ of helium gas at r.t.p. contains 1 mole of He atoms ($6 \times 10^{23}$ atoms)
• $24 \, \text{dm}^{3}$ of carbon dioxide gas at r.t.p. contains 1 mole of $\text{CO}_{2}$ atoms ($6 \times 10^{23}$ atoms)
Example
Calculate the number of moles of gas particles for $200 \, \text{cm}^{3}$ of carbon dioxide gas at r.t.p.
• Convert to $\text{dm}^{3}$: \begin{aligned} 200 \, \text{cm}^{3} &= \frac{200}{1000} \, \text{dm}^{3} \\ &= 0.20 \, \text{dm}^{3} \end{aligned}
• Number of moles: $\frac{0.20}{24} = 0.0083 \, \text{mol}$
Calculate the number of gas particles for $10 \, \text{dm}^{3}$ of hydrogen gas at s.t.p.
• No need to convert since it is already in $\text{dm}^{3}$
• Number of moles: $\frac{10}{22.4} = 0.446 \, \text{mol}$
• Number of gas particles: $0.446 \times \left( 6 \times 10^{23} \right) = 2.67 \times 10^{23}$
Molar Concentration
Molar concentration is the amount of solute (Typically in grams or moles) present in a given volume of solution. (Typically, $1 \, \text{dm}^{3}$ in Chemistry)
Hence,
$$\text{Conc.} \,(\text{g dm}^{-3}) = \frac{\text{Mass of solute in grams}}{\text{Vol. of solution in dm}^{3}}$$
$$\text{Conc.} \,(\text{mol dm}^{-3}) = \frac{\text{No. of moles of solute}}{\text{Vol. of solution in dm}^{3}}$$
Note: Concentration in $\text{mol dm}^{-3}$ is called molarity.
A molar solution refers to a solution that contains 1 mole of solute in 1 $\text{dm}^{3}$ of solution.
Important: If a solution with X $\text{mol dm}^{-3}$ is diluted with more solution, the number of moles of solutes does not change! The concentration will drop but the number of moles of solutes does not change. You can find the number of moles of the solutes and divide by the final volume of the solution to find the final concentration.
Example
10.4 g of potassium hydroxide was dissolved to formed $250 \, \text{cm}^{3}$ of solution. Calculate the concentration of solution in $\text{g dm}^{-3}$ and $\text{mol dm}^{-3}$
• Convert to $\text{dm}^{3}$: $250 \, \text{cm}^{3} = \frac{250}{1000} = 0.250 \, \text{dm}^{3}$
• Conc. in $\text{g dm}^{-3}$: $\frac{10.4}{0.250} = 41.6 \, \text{g dm}^{-3}$
• Molar mass of potassium hydroxide: $39 + 16 + 1 = 56 \, \text{g mol}^{-1}$
• No. of moles of potassium hydroxide present: $\frac{10.4}{56} = 0.186 \, \text{mol}$
• Molarity: $\frac{0.186}{0.250} = 0.743 \, \text{mol dm}^{-3}$
[olevelnextprev]
How To Convert Among $\text{dm}^{3}$, $\text{cm}^{3}$ and $\text{m}^{3}$
In order to know how to convert among those units, you will need to learn what is prefixes. (Visit that link to learn more)
From prefixes, we know that: $1 \, \text{m} = 10 \, \text{dm} = 100 \, \text{cm}$
Recall that volume is $\text{length} \times \text{length} \times \text{length}$.
$1 \, \text{m} \times 1 \, \text{m} \times 1 \, \text{m} = 1 \, \text{m}^{3}$
$10 \, \text{dm} \times 10 \, \text{dm} \times 10 \, \text{dm} = 1000 \, \text{dm}^{3}$
$100 \, \text{cm} \times 100 \, \text{cm} \times 100 \, \text{cm} = 1000000 \, \text{cm}^{3}$
Hence, $1 \, \text{m}^{3} = 1000 \, \text{dm}^{3} = 1000000 \, \text{cm}^{3}$ | 2021-09-24 12:17:12 | {"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": 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": 0.9373854994773865, "perplexity": 999.7067702165004}, "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-2021-39/segments/1631780057524.58/warc/CC-MAIN-20210924110455-20210924140455-00025.warc.gz"} |
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# S95-06
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16 Sep 2014, 01:49
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If the average (arithmetic mean) of six different numbers is 25, how many of the numbers are greater than 25?
(1) None of the six numbers is greater than 50.
(2) Three of the six numbers are 7, 8, and 9, respectively.
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16 Sep 2014, 01:49
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Official Solution:
We must determine how many numbers in a set of six distinct numbers are greater than 25. Since the average of the six numbers is 25, these six numbers must sum to $$6 \times 25 = 150$$.
Statement 1 tells us that none of the six numbers is greater than 50. Many combinations of numbers satisfy this statement. For example, the six numbers could be 19, 20, 21, 22, 23, and 45. In this case, only 1 number is greater than 25. On the other hand, if the six numbers are 1, 2, 3, 47, 48, and 49, then 3 of the numbers are greater than 25. Since we cannot find a unique value, statement 1 alone is NOT sufficient to answer the question. Eliminate answer choices A and D. The correct answer choice is B, C, or E.
Statement 2 tells us that three of the six numbers are 7, 8, and 9. This means that the other three numbers must sum to $$150 - (7 + 8 + 9) = 150 - 24 = 126$$. However, since there is no limit to what any of the numbers can be, we can have different combinations. For example, the three numbers could be 1, 2, and 123. In this case, only 1 number is greater than 25. But if the three numbers are 41, 42, and 43, then all 3 are greater than 25. Statement 2 alone is also NOT sufficient. Eliminate answer choice B. The correct answer choice is either C or E.
Both statements together tell us that three of the numbers are 7, 8, and 9, that the other three numbers must sum to 126, and that none of the numbers is greater than 50. If all three remaining numbers are 25 or less (remember, the numbers must be different), then these remaining numbers can sum to, at maximum, $$25 + 24 + 23 = 72$$. If two numbers are 25 or less, and one is 50 (the upper bound for numbers), the sum is at most $$25 + 24 + 50 = 99$$. If one number is 25 or less, and the other two numbers are 50 and 49, then the maximum sum is $$25 + 50 + 49 = 124$$. In order to reach 126, all three numbers MUST be greater than 25. Both statements together are sufficient.
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13 Apr 2017, 19:56
Very hard question. I understand the official answer, but I don't know how it is possible to do this within two minutes unless you have incredible number sense
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14 Apr 2017, 00:36
joondez wrote:
Very hard question. I understand the official answer, but I don't know how it is possible to do this within two minutes unless you have incredible number sense
You can check other solutions here: https://gmatclub.com/forum/if-the-avera ... 88461.html
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# S95-06
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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®. | 2018-09-21 17:48:41 | {"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": 0.5807172656059265, "perplexity": 690.1333077410296}, "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-2018-39/segments/1537267157351.3/warc/CC-MAIN-20180921170920-20180921191320-00498.warc.gz"} |
https://www.zbmath.org/?q=se%3A00002447+cc%3A92 | # zbMATH — the first resource for mathematics
Rapid prediction of refractive index and density of aqueous lithium iodate solutions. (English) Zbl 0898.92036
Summary: Quadratic equations to describe density and refractive index of aqueous Lithium Iodate solutions, which were originally proposed by J. Szewczyk and K. Sangwal [J. Chem. Eng. Data 33, 418-423 (1988)] do not adequately describe the effect of temperature as well as solute concentration. For accurate predictions of refractive indices and densities of aqueous Lithium Iodate solutions over the temperature range $$10^\circ$$C to $$60^\circ$$C and solute concentration range 0.01 to 3.0, a simple polynomial $$Y= K_1+ K_2(X-X_1)+ K_3(X-X_1)^p$$ has been used.
##### MSC:
92F05 Other natural sciences (mathematical treatment) | 2021-02-25 11:57:38 | {"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": 0.2706097662448883, "perplexity": 3994.9120530701794}, "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-2021-10/segments/1614178350942.3/warc/CC-MAIN-20210225095141-20210225125141-00094.warc.gz"} |
https://web2.0calc.com/questions/parabola_93 | +0
# Parabola
0
42
1
Find the equation whose graph is a parabola with vertex (2,4), vertical axis of symmetry, and contains the point (-1,-1). Express your answer in the form "ax^2 + bx + c".
Nov 19, 2021
#1
+12710
+1
Find the equation whose graph is a parabola with vertex (2,4), vertical axis of symmetry, and contains the point (-1,-1).
Hello Guest!
$$f(x)=a (x-2)^2+4\\ a ((-1)-2)^2+4=-1\\ 9a=-5\\ \color{blue}a=-\frac{5}{9}\\ f(x)=-\frac{5}{9}(x^2-4x+4)+4$$
$$f(x)= -\dfrac{5}{9}x^2+\dfrac{20}{9}x+\dfrac{16}{9}$$
!
Nov 19, 2021
edited by asinus Nov 19, 2021 | 2021-12-02 21:42:54 | {"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": 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": 0.8235592246055603, "perplexity": 1429.010106619478}, "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/1637964362297.22/warc/CC-MAIN-20211202205828-20211202235828-00478.warc.gz"} |
http://tug.org/pipermail/texworks/2010q4/003201.html | Stefan Löffler st.loeffler at gmail.com
Mon Oct 18 08:12:59 CEST 2010
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Hi,
Am 2010-10-14 20:54, schrieb Arno Trautmann:
> this just entered my mind when opening a document that was \input into
> another one: Is it possible or would it be possible to implement a
> feature that opens a document that is \input or \included into another
> document by clicking on it? (Assuming LaTeX syntax, i.e. \input{filename})
> That would be a nice and useful feature!
Sure, scripting is your friend. I've attached a preliminary version of
such a script. It should support both \include{} and \input{}. I haven't
tested it extensively, though. So problems may arise on Windows
(different path syntax?), or with \include'd files in subdirectories,
parent directories, etc. But it's a start :).
HTH
Stefan | 2018-09-22 13:28:50 | {"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": 0.8715319037437439, "perplexity": 11054.722955032415}, "config": {"markdown_headings": true, "markdown_code": false, "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-2018-39/segments/1537267158429.55/warc/CC-MAIN-20180922123228-20180922143628-00454.warc.gz"} |
https://studyqas.com/is-cos-theta-15-17-what-is-equal-to-it/ | # Is cos theta = 15/17 what is equal to it
Is cos theta = 15/17 what is equal to it
## This Post Has 10 Comments
1. maggie123433 says:
Since cos theta is 15/17, let a right triangle have a hypotenuse of 17 and a leg of 15. This means that the other is 8.
Sin 2 theta is a double angle identity.
Sin 2 theta = 2 sin theta cos theta = 2(8/17)(15/17) = 240/289
Cos 2theta = cos^2 – sin^2 = (15/17)^2 – (8/17)^2 = 161/289
2. mettababeeeee says:
its A & B
Step-by-step explanation:
3. coliver15 says:
Hello there!
The only thing we have to do here is remember our trigonometric functions!
sin=y/r
cos=x/r
tan=y/x
sec=r/x
csc=r/y
cot=x/y
Knowing that tanᎾ=15/8, we know that y=15 and x=8
Now we need to look at the answer choices and see if they are correct.
A. secᎾ=17/8
Since secᎾ=r/x and r=17 and x=8, this is correct.
B. cosᎾ=15/17
Since cosᎾ=x/r and x=8 and r=17, this is wrong.
C. cotᎾ=8/15
Since cotᎾ=x/y and x=8 and y=15, this is also correct!
D. cscᎾ=17/15
Since cscᎾ=r/y and r=17 and y=15, this is right too.
I really hope this helps!
Best wishes 🙂
4. eeeeee7891 says:
$\sin( \theta) = - \frac{8}{17}$
EXPLANATION
We draw the diagram as shown in the attachment.
We can see that the angle is in the 4th quadrant.
We use the Pythagoras Theorem to determine the hypotenuse of the right angle triangle as follows:
${h}^{2} = {8}^{2} + {15}^{2}$
${h}^{2} = 64 + 225$
${h}^{2} = 289$
$h = \sqrt{289}$
$= 17units$
We know that the sine ratio is the ratio of the length of the opposite side to the length of the hypotenuse. Since the angle is in the 4th quadrant the sine ratio is negative.
$\sin( \theta) = - \frac{8}{17}$
Note that the negative sign of the 8 and the positive sign of the 15 helped us to determine the quadrant in which the angle is, and that the negative sign has nothing to do with the length. I know this confuses most students.
$The terminal side of an angle in standard position passes through p(15,-8) . what is the value of si$
5. alondrachon says:
The correct option is $\boxed{{\mathbf{Option B}}}$ .
Further explanation:
The sine ratio is the ratio of the length of the side that is opposite to the given angle to the length of the hypotenuse.
It can be expressed as,
$\sin \theta =\frac{p}{h}$
Pythagoras theorem is always used in a right angle tri
angle to obtain the one of the side of the triangle.
Hypotenuse is the longest side in the right angle triangle.
Pythagoras theorem can be expressed as,
${H^2}={P^2}+{B^2}$
Here, $H$ is the hypotenuse, $B$ is the base and $P$ is the perpendicular.
Step by step explanation:
Step 1:
The terminal side of an angle passes through the point $p\left({15,-8}\right)$ as shown in the attached figure.
It can be observed that the given point lies in the fourth quadrant as $x$ coordinate is positive and $y$ coordinate is negative.
As we can see in the attached figure the perpendicular is the side of the triangle that is opposite the angle $\theta$ .
Therefore, the perpendicular is 8 and the base is $15$ .
Now find the hypotenuse by the use of Pythagoras theorem.
$\begin{gathered}{H^2}={8^2}+{15^2}\hfill\\{H^2}=64+225\hfill\\H=\sqrt{289}\hfill\\H=17\hfill\\\end{gathered}$
Therefore, the length of the hypotenuse is ${\text{17 units}}$ .
We know that the sine function is negative in fourth quadrant.
Therefore, $\sin \theta$ cab be expressed as,
$\sin \theta=-\frac{8}{{17}}$
Thus, option B $\sin \theta=-\frac{8}{{17}}$ is correct.
Subject: Mathematics
Chapter: Triangles
Keywords: side, lengths, distance, Pythagoras theorem, triangle, hypotenuse, base, perpendicular, right angle triangle, longest side, sine function, fourth quadrant, positive sign, negative sign, ratio.
$The terminal side of an angle in standard position passes through p(15,-8) . what is the value of si$
6. graciearany says:
$a.tan(\theta)=\frac{15}{8}$
$b.csc(\theta)=\frac{17}{15}$
$d.sec(\theta)=\frac{17}{8}$
Step-by-step explanation:
The trigonometry functions on a right triangle are given by:
$sin(\theta)=\frac{opposite}{hypotenuse}$
$csc(\theta)=\frac{hypotenuse}{opposite}$
$cos(\theta)=\frac{adjacent}{hypotenuse}$
$sec(\theta)=\frac{hypotenuse}{adjacent}$
$tan(\theta)=\frac{opposite}{adjacent}$
$cot(\theta)=\frac{adjacent}{opposite}$
Let's calculate the adjacent side using pythagorean theorem:
$a^2+b^2=c^2$
Where:
a=opposite
c=hypotenuse
The problem provides us a and c because:
$sin(\theta)=\frac{opposite}{hypotenuse}=\frac{15}{17}$
So:
$15^2+b^2=17^2$
Solving for b:
$b^2=17^2-15^2\\b^2=289-225\\b^2=64\\b=\sqrt{64} \\b=8$
Therefore:
a=opposite=15
c=hypotenuse=17
Finally, let's see if the given options are correct:
$a.tan(\theta)=\frac{15}{8}=\frac{opposite}{adjacent}=\frac{15}{8}$ , This is correct.
$b.csc(\theta)=\frac{17}{15}=\frac{adjacent}{hypotenuse}=\frac{17}{15}$ , This is correct.
$c.cot(\theta)=\frac{17}{8}=\frac{adjacent}{opposite}=\frac{8}{15}$ , This is incorrect.
$d.sec(\theta)=\frac{17}{8}=\frac{hypotenuse}{adjacent}=\frac{17}{8}$ , This is correct.
$Check all that apply: if sin theta = 15/17 then: a. tan theta = 15/8 b. csc theta = 17/15 c. coc t$
7. julielebo8 says:
Hope this helps.......
$Given cos theta = 15/17. Find Sine Theta and tangent theta. Use the Pythagorean Identity for Trig to$
8. bbb4951 says:
it is A, C and D
Step-by-step explanation:
I just did it, have a good one 🙂
9. siyah87 says:
If Cos theta = 15/17, then
Sec theta = 17/15 because cosine and secant are reciprocal functions.
Letter A
10. loraine8401 says:
secᎾ=17/8
cotᎾ=8/15
cscᎾ=17/15
Step-by-step explanation: | 2022-11-29 11:46:31 | {"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": 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": 0.9193106293678284, "perplexity": 1358.9738011714792}, "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/1669446710691.77/warc/CC-MAIN-20221129100233-20221129130233-00777.warc.gz"} |
http://developmentalsystems.org/teacher_algorithms_for_drl_learners | TL;DR
Teaching algorithms are becoming a key ingredient to scaffold Deep RL agents into their learning odyssey, leading to policies that can generalize to multiple environments. This blog presents this emerging Deep RL sub-field and showcases recent work done in the Flowers Lab on designing a Learning Progress based teacher algorithm.
Building Deep RL agents that generalize. Leveraging advances in both training procedures and scalability, the Deep Reinforcement Learning community is now able to tackle ever more challenging control problems. This dynamic is shifting the sets of environments that are considered by researchers for benchmarking, leading to a drift from simple deterministic environments towards complex, more stochastic alternatives. This shift is key if we want to bring Deep RL outside of the lab, where robustness and generalization to previously unseen changes of the environment is essential for successful operation.
Training in procedurally generated environment. One emerging method as a step towards this objective is to train Deep RL agents on proceduraly generated environments and assess whether the considered approach is able to generalize to the space of possible environments. In Open AI’s paper Quantifying Generalization in Reinforcement Learning, authors present CoinRun, a mario-like platform game whose levels are proceduraly generated. They show that increasing the training set of proceduraly generated levels directly correlates with an increase in test-time performances on previously unseen levels, showing that training diversity is key for generalization.
Links with domain randomization. Similar procedures (termed Domain Randomization) have also been used as a way to enable Sim2Real transfer, which recently led to the training of a Deep RL agent able to robustly solve a Rubik’s cube with a real-world robotic hand, leveraging intensive training in simulation. To facilitate the development and comparative analysis of Deep RL algorithms that generalize, propositions of benchmark environments emerges from the community, such as Open AI’s ProcGen, a suite of 16 procedural 2D games, or Unity’s Obstacle Tower Challenge, a complex 3D procedural escape game. However, a limitation of these benchmarks is that they do not allow to study whether the ability to control the procedural generation can be beneficial for training Deep RL agents.
Teacher algorithms that dynamically adapt the distribution of procedurally generated environments. Given such a possibility, how can a teacher algorithm control the sequence of training environments enabling a Deep RL agent to acquire fastly skills that generalize to many environments? A simple approach is to propose randomly generated environments. If the amount of unfeasible environments from the set is not too high for the given learner, this might work reasonably well (as in CoinRun). But for more challenging scenarios, one would need smarter training strategies that avoid proposing too easy or too difficult environments, while ensuring forgetting does not happen. This brings us in the domain of Curriculum Learning (CL).
Automated Curriculum Learning. Given enough time and expert knowledge on the given benchmark, one could design a learning curriculum tailored for the considered learner. This is likely to work well, however, for real-world applications of Deep RL, it might be interesting to try to relax the time (i.e money) and expert knowledge (i.e money) required to design the curriculum. And there might also be cases where acquiring expert knowledge is not possible (as when choosing tasks from a task space embedding learned online). One more versatile approach which alleviates this need for domain knowledge is to rely on Automatic Curriculum Learning.
In our recent CoRL 2019 paper, we propose such an approach based on the use of Absolute Learning Progress as a signal to organise the curriculum. We frame this generalization challenge as a Strategic Student problem, formalizing a setting where one has to sequentially select distributions of environments to train on, and where the objective is to maximize average competence over the whole space of environments after a given number of interactions. Importantly, we explicitly do not assume to know the learnable subset of environments that can be generated, and we consider the learner as a black-box whose abilites are unknown. Thus, a teacher algorithm needs to incrementally learn a model of how the Deep RL agent performs in various parameterized environments, and to use this model to organize the learning curriculum.
## The Continuous Teacher-Student Framework
This Continuous Teacher-Student framework, depicted above, enables us to address the “curriculum generation towards generalization” problem. The Teacher algorithm samples an environment (i.e a parameter vector given to the procedural generation engine) from its current environment distribution, proposes it to its Deep RL Student, observes the associated episodic reward and updates its sampling distribution accordingly. Importantly, the Student is considered as a Black-Box, meaning that the Teacher does not have access to its internal state and does not know its learning mechanisms, making this approach applicable to any learning system. Likewise, to extend this approach to a wide range of scenarii, we do not assume to have expert knowledge on the environment space (aka parameter space) , and therefore assume that it could contain unfeasible subspaces, irrelevant dimensions and non-linear difficulty gradients.
## Learning-Progress based Teacher algorithms
The learning signal used to organize a learning curriculum is absolute learning progress. This learning signal was extensively studied as a theoretical hypothesis to explain why and how human babies spontaneously explore, i.e. as a model of human curiosity-driven learning. It operationalizes the general idea (formulated by many psychologists) that humans are intrinsically motivated to explore environments of optimal difficulty, neither too hard nor too easy. Indeed, various models showed that exploration driven by an LP-based intrinsic reward self-organizes learning curriculum of increasing complexity, for e.g. reproducing the developmental progression of vocalisations or tool use in infants.
Following this insight, LP was later applied to automate curriculum learning for efficient learning of skills in robots. For example, RIAC and Covar-GMM are two curriculum learning algorithms for continuous environment spaces that were paired with simple population-based DMP controllers. In both approaches, the sequential task selection problem is transformed into an EXP4 Multi-Armed Bandit (MAB) setup in which arms are dynamically mapped to subspaces of the task/environment space, and whose utilities are defined using a local LP measure. The objective is then to optimally select on which subspace to sample an environment in order to maximize LP. In RIAC, arms are defined as hyperboxes over the environment space, their utility is the difference of reward between old and recent environments sampled from the hyperbox. In Covar-GMM, the arms of the MAB are gaussians from a Gaussian Mixture Model. The GMM is periodically fited on tuples composed of an environment’s parameter vector concatenated to its associated reward (obtained by the student) and its relative time value.
The common recipe between RIAC, Covar-GMM and ALP-GMM (our proposed approach) is to dynamically detect subspaces having different LP value within the task/environment space. Then, they consider each subspace as an arm of a Multi Armed Bandit setup, and compute each of their utility using a local aggregated LP value.
## ALP-GMM
Building up on Covar-GMM, we propose Absolute Learning Progress Gaussian Mixture Model (ALP-GMM), a variant based on a richer ALP measure capturing long-term progress variations that is well-suited for RL setups. ALP gives a richer signal than (positive) LP as it enables the teacher to detect when a student is losing competence on a previously mastered environment subspace (thus preventing catastrophic forgetting, which is a well-known issue in DRL).
ALP-GMM periodically fits a GMM on recently sampled environments' parameters paired with their respective Absolute Learning Progress. Then, the Gaussian from which to sample a new parameter (a new environment) is chosen proportionally to its mean ALP.
The key concept of ALP-GMM is to fit a GMM on a dataset of previously sampled environment’s parameters (parameters for short) concatenated to their respective ALP measure. Thus, each gaussian groups environments sampled so far according to a similarity that depends both on their physical features and the corresponding ALP of the Deep RL agent. Then, the Gaussian from which to sample a new parameter is chosen using an EXP4 bandit scheme where each Gaussian is viewed as an arm, and ALP is its utility. This enables the teacher to bias the parameter sampling towards high-ALP subspaces. To get this per-parameter ALP value, we take inspiration from previous work on developmental robotics: for each newly sampled parameter $p_{new}$ and associated episodic reward $r_{new}$, the closest (Euclidean distance) previously sampled parameter $p_{old}$ (with associated episodic reward $r_{old}$) is retrieved using a nearest neighbor algorithm. We then have $alp_{new} = |r_{new} - r_{old}|$.
The GMM is fit periodically on a window $\mathcal{W}$ containing only the most recent parameter-ALP pairs (here the last $250$) to bound its time complexity and make it more sensitive to recent high-ALP subspaces. The number of Gaussians is adapted online by fitting multiple GMMs (here having from $2$ to $k_{max}=10$ Gaussians) and keeping the best one based on Akaike’s Information Criterion. Note that the nearest neighbor computation of per-parameter ALP uses a database that contains all previously sampled parameters and associated episodic rewards, which prevents any forgetting of long-term progress. In addition to its main environment sampling strategy, ALP-GMM also samples random parameters to enable exploration (here $p_{rnd}=20%$).
## Procedurally Generated BipedalWalker Environments
The BipedalWalker environment offers a convenient test-bed for continuous control, allowing to easily build parametric variations of the original version (controlling changes in walker morphology or walking track layout). The learning agent, embodied in a bipedal walker, receives positive rewards for moving forward and penalties for torque usage and angular head movements. Agents are allowed up to $2000$ steps to reach the other side of the map. Episodes are aborted with a reward penalty if the walker’s head touches an obstacle or the ground.
To study the ability of our teachers to guide DRL students, we design two continuously parameterized BipedalWalker environments enabling the procedural generation of walking tracks:
Stump Tracks Hexagon Tracks A 2D parametric environment producing tracks paved with stumps varying by their height and spacing. A more challenging 12D variation whose parameters are used to generate arbitrary hexagons.
All of the experiments done in these environments were performed using OpenAI’s implementation of Soft-Actor Critic (SAC) as the single student algorithm. To test our teachers’ robustness to students with varying abilities, we use $3$ different walker morphologies:
In addition to the default bipedal walker morphology, we designed a bipedal walker with 50% shorter legs and a bigger quadrupedal walker. We test all three walker types on Stump Tracks. The quadrupedal walker is also tested in Hexagon Tracks.
## Experiments
### ALP-GMM generates student-specific curricula
The following videos provide visualizations of the environment sampling trajectory observed in a representative ALP-GMM run when paired with a short (left), default (middle) and quadrupedal (right) walker. In each frame, blue shades represent the Gaussians of the current mixture while dots represent sampled environments.
For each walker type one can observe that the curriculum generated by ALP-GMM starts with a random sampling phase (different in length depending on the walker type), which is due to the initial absence of progress of its student. Initial progress is then always detected on the leftmost part of the environment’s parameter space, which correspond to environments with low stump heights. Then, one can see that curricula generated by ALP-GMM significantly diverge depending on the walker type as it is adapted to their unique learning abilities. One can see that the more competent the walker, the more the sampling gradually shifts towards high-stump environments across time.
A single run of ALP-GMM allows to train Soft Actor-Critic controllers to master a wide range of track distributions. Here are examples of learned walking gates for short (left), default (middle) and quadrupedal (right) walkers ($1$ column = $1$ run):
### Comparative study on Stump Tracks
To analyze the impact of ALP-GMM teachers on student training, we compare it to RIAC, Covar-GMM and two reference algorithms:
• Random Environment Curriculum (Random) - In Random, parameters are sampled randomly in the parameter space for each new episode. Although simplistic, similar approaches in previous work proved to be competitive against more elaborate forms of CL.
• Oracle - A hand-constructed approach, sampling random environment distributions in a fixed-size sliding window on the parameter space. This window is initially set to the easiest area of the parameter space and is then slowly moved towards complex ones, with difficulty increments only happening if a minimum average performance is reached. Expert knowledge is used to find the dimensions of the window, the amplitude and direction of increments, and the average performance threshold.
For each condition we perform $32$ seeded runs with each of our $3$ students and periodically measure performance as the percentage of mastered environments ($r > 230$) of a fixed test set ($n = 50$).
Evolution of mastered track distributions for Teacher-Student approaches in Stump Tracks. The mean performance (32 seeded runs) is plotted with shaded areas representing the standard error of the mean.
The figure above shows learning curves for each condition paired with short (a), default (b) and quadrupedal (c) walkers. First of all, for short agents, one can see that Oracle is the best performing algorithm, mastering more than $20$% of the test set after $20$ Million steps. This is an expected result as Oracle knows where to sample simple track distributions, which is crucial when most of the parameter space is unfeasible, as is the case with short agents. ALP-GMM is the LP-based teacher with highest final mean performance, reaching $14.9$% against $10.6$% for Covar-GMM and $8.6$% for RIAC. This performance advantage for ALP-GMM is statistically significant when compared to RIAC (Welch’s t-test at 20M steps: $p<0.04$), however there is no statistically significant difference with Covar-GMM ($p=0.16$). All LP-based teachers are significantly superior to Random ($p<0.001$). For details on how to use statistical tests for Deep RL, check this blog.
Regarding default bipedal walkers, our hand-made curriculum (Oracle) performs better than other approaches for the first $10$ Million steps and then rapidly decreases to end up with a performance comparable to RIAC and Covar-GMM. All LP-based conditions end up with a final mean performance statistically superior to Random ($p<10^{-4}$). ALP-GMM is the highest performing algorithm, significantly superior to Oracle ($p<0.04$), RIAC ($p<0.01$) and Covar-GMM ($p<0.01$).
For quadrupedal walkers, Random, ALP-GMM, Covar-GMM and RIAC agents quickly learn to master nearly $100$% of the test set, without significant differences apart from Covar-GMM being superior to RIAC ($p<0.01$). This indicates that, for this agent type, the parameter space of Stump Tracks is simple enough that trying random tracks for each new episode is a sufficient curriculum learning strategy. Oracle teachers perform significantly worse than any other method ($p<10^{-5}$).
Through this analysis we showed how ALP-GMM, Covar-GMM and RIAC, without strong assumptions on the environment, managed to scaffold the learning of multiple students better than Random. Interestingly, ALP-GMM outperformed Oracle with default agents, and RIAC, Covar-GMM and ALP-GMM surpassed Oracle with the quadrupedal agent, despite its advantageous use of domain knowledge. This indicates that training only on track distributions sampled from a sliding window that end up on the most difficult parameter subspace leads to forgetting of simpler environment distributions. Our approaches avoid this issue through efficient tracking of their students’ learning progress.
### Comparative study on Hexagon Tracks, our environment with a 12-D parameter space
To assess whether ALP-GMM is able to scale to parameter spaces of higher dimensionality, containing irrelevant dimensions, and whose difficulty gradients are non-linear, we performed experiments with quadrupedal walkers on Hexagon Tracks, our 12-dimensional parametric BipedalWalker environment.
We analyzed the distributions of the percentage of mastered environments of the test set after training for $80$ Millions (environment) steps, depicted in the figure below (left plot). One can see that ALP-GMM both has the highest median performance and narrowest distribution. Out of the $32$ repeats, only Oracle and ALP-GMM always end-up with positive final performance scores whereas Covar-GMM, RIAC and Random end-up with $0%$ performance in $8/32$, $5/32$ and $16/25$ runs, respectively. Interestingly, in all repeats of any condition, the student manages to master part of the test set at some point (i.e non-zero performance), meaning that runs that end-up with $0%$ final test performance actually experienced catastrophic forgetting. This showcase the ability of ALP-GMM to avoid this forgetting issue through efficient tracking of its student’s absolute learning progress.
In addition to the default bipedal walker morphology, we designed a bipedal walker with 50% shorter legs and a bigger quadrupedal walker. We test all three walker types on Stump Tracks. The quadrupedal walker is also tested in Hexagon Tracks. Final performance of each condition run on Hexagon Tracks after 80M steps. Gold lines are medians, surrounded by a box showing the first and third quartile, which are then followed by whiskers extending to the last datapoint or 1.5 times the inter-quartile range. Beyond the whiskers are outlier datapoints.
The following videos shows walking gates learned in a single ALP-GMM run.
In comparison, when using Random curriculum, most runs end-up with bad performances, like this one:
## Discussion
This work demonstrated that ALP-based teacher algorithms could successfully guide DRL agents to learn and generalize in difficult continuously parameterized environments. With no prior knowledge of its student’s abilities and learning algorithm, and only loose boundaries on the environment space, ALP-GMM, our proposed teacher, consistently outperformed random heuristics and occasionally even expert-designed curricula. We show that it also scales when the procedural generation includes irrelevant dimensions or produces large proportions of unfeasible environments.
ALP-GMM, which is conceptually simple and has very few crucial hyperparameters, opens-up exciting perspectives inside and outside DRL for curriculum learning problems. Within DRL, it could be applied to previous work on autonomous goal exploration through incremental building of goal spaces. In this case several ALP-GMM instances could scaffold the learning agent in each of its autonomously discovered goal spaces. Another domain of applicability is personalization of sequences of exercises for human learners in educational technologies, or training humans for sensorimotor skills in re-habilitation or sports.
The full details of this work are presented in: Rémy Portelas, Cédric Colas, Katja Hofmann, Pierre-Yves Oudeyer (2019) Teacher algorithms for curriculum learning of Deep RL in continuously parameterized environments Proceedings of CoRL 2019 - Conference on Robot Learning , Oct 2019, Osaka, Japan. Bibtex.
## Code
Jointly to this work, we release a github repository featuring implementations of ALP-GMM, RIAC and CovarGMM along with our parameterized BipedalWalker environments.
## Contact
Email: remy.portelas@inria.fr
## References
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• Lopes M., Oudeyer P-Y. (2012). The Strategic Student Approach for Life-Long Exploration and Learning. link
• Oudeyer P.-Y., Gottlieb J., Lopes M. Intrinsic motivation, curiosity, and learning: Theory and applications in educational technologies. (2016). link
• Oudeyer P.-Y., Kaplan F., Hafner V. (2007). Intrinsic Motivation Systems for Autonomous Mental Development. link
• Forestier S., Oudeyer P.-Y.. (2016). Curiosity-Driven Development of Tool Use Precursors: a Computational Model. link
• Forestier S., Mollard Y., Oudeyer P.-Y.. (2017). Intrinsically Motivated Goal Exploration Processes with Automatic Curriculum Learning link
• Baranes A., Oudeyer P.-Y.. R-IAC: robust intrinsically motivated exploration and active learning. (2009). link
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• Bozdogan H.. (1987). Model selection and akaike’s information criterion (aic): The general theory and its analytical extensions. link
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• Wang R. , Lehman J. , Clune J. , Stanley K. O.. (2019). POET: Endlessly Generating Increasingly Complex and Diverse Learning Environments and their Solutions through the Paired Open-Ended Trailblazer. link
• Haarnoja T. , Zhou A., Abbeel P., Levine S.. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. (2018). link
• Baranes A., Oudeyer P.-Y.. (2013). Active Learning of Inverse Models with Intrinsically Motivated Goal Exploration in Robots. link
• Colas C., Sigaud O., Oudeyer P.-Y.. (2019). How Many Random Seeds? Statistical Power Analysis in Deep Reinforcement Learning Experiments. link
• Clément B., Roy D., Oudeyer P.-Y., Lopes. M.. (2015). Multi-Armed Bandits for Intelligent Tutoring Systems. link | 2021-06-23 22:04:57 | {"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": 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": 0.6609604954719543, "perplexity": 3903.9203374809294}, "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-2021-25/segments/1623488540235.72/warc/CC-MAIN-20210623195636-20210623225636-00397.warc.gz"} |
http://ocw.mit.edu/ans7870/6/6.006/s08/lecturenotes/dd_def.htm | # Document Distance: Problem Definition
Problem Definition | Data Sets | Programs: v1 - v2 - v3 - v4 - v5 - v6 | Programs Using Dictionaries
Let D be a text document (e.g. the complete works of William Shakespeare).
A word is a consecutive sequence of alphanumeric characters, such as "Hamlet" or "2007". We'll treat all upper-case letters as if they are lower-case, so that "Hamlet" and "hamlet" are the same word. Words end at a non-alphanumeric character, so "can't" contains two words: "can" and "t".
The word frequency distribution of a document D is a mapping from words w to their frequency count, which we'll denote as D(w).
We can view the frequency distribution D as vector, with one component per possible word. Each component will be a non-negative integer (possibly zero).
The norm of this vector is defined in the usual way:
$N(D) = \sqrt{D\cdot D} = \sqrt{\sum_w D(w)^2}$.
The inner-product between two vectors D and D' is defined as usual.
$D\cdot D' = \sum_w D(w)D'(w)$.
Finally, the angle between two vectors D and D' is defined:
$angle(D,D') = \arccos\left(\frac{D\cdot D'}{N(D)N(D')}\right)$
This angle (in radians) will be a number between 0 and $\pi/2 = 1.57079632\ldots$ since the vectors are non-negative in each component. Clearly,
angle(D,D) = 0.0
for all vectors D, and
angle(D,D') = p / 2
if D and D' have no words in common.
Example: The angle between the documents "To be or not to be" and "Doubt truth to be a liar" is
$\arccos{\left(4/\sqrt{10\cdot 6}\right)} = 1.028 .$
We define the distance between two documents to be the angle between their word frequency vectors.
The document distance problem is thus the problem of computing the distance between two given text documents.
An instance of the document distance problem is the pair of input text documents. | 2014-07-24 19:17:33 | {"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": 0, "img_math": 5, "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": 0.8347440361976624, "perplexity": 1170.3377175327378}, "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-2014-23/segments/1405997890199.39/warc/CC-MAIN-20140722025810-00157-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://www.imaginary.org/es/node/1093 | # Mod 22 MathLapse
película
Mod 22 MathLapse
## Créditos
Watch how the numbers 0, 1, 2, …, 21 leave behind residues (mod 22) when raised to the powers of 0, 1, 2, …, 21.
Watch how the numbers 0, 1, 2, …, 21 leave behind residues (mod 22) when raised to the powers of 0, 1, 2, …, 21. | 2022-10-06 19:30:19 | {"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": 0.8408742547035217, "perplexity": 395.69545486116056}, "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-2022-40/segments/1664030337855.83/warc/CC-MAIN-20221006191305-20221006221305-00012.warc.gz"} |
https://stats.stackexchange.com/questions/11232/testing-difference-between-two-means-with-pairwise-data-and-absence-of-normality | # Testing difference between two means with pairwise data and absence of normality
I am trying to compare the difference between two means with two pairwise samples. Unfortunately, my data are very far of being normal. What test would you recommend to use in this situation? Should I revert to a nonparametric test?
A paired t-test assumes that the differences are normal: the original values could have any distribution. More precisely, just like with a t-test, the differences don't even have to be normal, just the sampling distribution of the mean. This usually means that with a large enough sample you can use a t-test even without normality because the central limit theorem will kick in.
On the other hand, one can always use a non-parametric test with not too much loss in efficiency.
• Thanks both for your answers. Could you please direct me to a book/reference where I could find a proof of the fact that for large samples the paired t-test works? – Paul Smith May 25 '11 at 13:33
Your description of your design is not too precise as it allows two interpretations.
First, it is possible that you have a 2 (between) x 2 (within) design (i.e., two groups with two pairwise samples).
Second, it is possible that you have a simple design with one group which was measured two times.
Only in the second case, the answers here apply.
Furthermore, the question if it is really inappropriate to use a t-test for our data is the crucial part of your question. Sometimes this question is difficult in the sense that one may confuse normality of the data with normality of the residuals (crucial in the first interpretations of your design, see here and here) and/or normality of the differences (crucial for the second interpretation, see Aniko's answer).
If the deviation from normality are that serious that you do not want to use a t-test, you should think about using a permutation test instead of the wilcoxon. See the answers to the following two questions for how to make permutations tests using the coin package for r with dependent samples:
Which permutation test implementation in R to use instead of t-tests (paired and non-paired)?
Paired permutation test for repeated measures
• Thanks for your answer. I think that the second intrepretation is the the correct one, as I am comparing the prices of a set of products in two different shoppings. – Paul Smith May 25 '11 at 13:52
Sounds like a job for the paired Wilcoxon test.
Note that this method compares the medians of the two samples, not their means. In any case, the mean is often not a good estimator when the distributions are not normally distributed, as it is easily biased by extremely low or high values. | 2020-01-19 00:05:36 | {"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": 0.7248457670211792, "perplexity": 433.7616475252951}, "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/1579250593994.14/warc/CC-MAIN-20200118221909-20200119005909-00538.warc.gz"} |
https://labs.tib.eu/arxiv/?author=H.%20Funahashi | • ### Photoproduction of Lambda(1405) and Sigma^{0}(1385) on the proton at E_\gamma = 1.5-2.4 GeV(0805.4051)
May 27, 2008 hep-ex
Differential cross sections for $\gamma p \to K^+\Lambda(1405)$ and $\gamma p \to K^+\Sigma^0(1385)$ reactions have been measured in the photon energy range from 1.5 to 2.4 GeV and the angular range of $0.8<\cos(\Theta)<1.0$ for the $K^+$ scattering angle in the center-of-mass system. This data is the first measurement of the $\Lambda(1405)$ photoproduction cross section. The lineshapes of \LamS measured in $\Sigma^+\pi^-$ and $\Sigma^-\pi^+$ decay modes were different with each other, indicating a strong interference of the isospin 0 and 1 terms of the $\Sigma\pi$ scattering amplitudes. The ratios of \LamS production to \SigS production were measured in two photon energy ranges: near the production threshold ($1.5<E_\gamma<2.0$ GeV) and far from it ($2.0 <E_\gamma<2.4$ GeV). The observed ratio decreased in the higher photon energy region, which may suggest different production mechanisms and internal structures for these hyperon resonances.
• ### Nuclear-matter modification of decay widths in the $\phi \to e^{+}e^{-}$ and $\phi \to K^{+}K^{-}$ channels(nucl-ex/0606029)
April 14, 2007 nucl-ex
The invariant mass spectra of $\phi \to K^{+}K^{-}$ are measured in 12 GeV $p+A$ reactions in order to search for the in-medium modification of $\phi$ mesons. The observed $K^{+}K^{-}$ spectra are well reproduced by the relativistic Breit-Wigner function with a combinatorial background shape in three $\beta\gamma$ regions between 1.0 and 3.5. The nuclear mass-number dependence of the yields of the $K^{+}K^{-}$ decay channel is compared to the simultaneously measured $e^{+}e^{-}$ decay channel for carbon and copper targets. We parameterize the production yields as $\sigma (A) = \sigma_0 \times A^\alpha$ and obtain $\alpha_{\phi\to K^+K^-} - \alpha_{\phi\to e^+e^-}$ to be 0.14 $\pm$ 0.12. Limits are obtained for the partial decay widths of $\phi$ mesons in nuclear matter.
• ### Evidence for in-medium modification of the $\phi$ meson at normal nuclear density(nucl-ex/0511019)
Jan. 30, 2007 nucl-ex
Invariant mass spectra of e+e- pairs have been measured in 12 GeV p+A reactions to detect possible in-medium modification of vector mesons. Copper and carbon targets are used to study the nuclear-size dependence of e+e- invariant mass distributions. A significant excess on the low-mass side of the phi meson peak is observed in the low beta gamma (=beta/sqrt(1-beta^2)) region of phi mesons (beta gamma < 1.25) with copper targets. However, in the high beta gamma region (beta gamma > 1.25), spectral shapes of phi mesons are well described by the Breit-Wigner shape when experimental effects are considered. Thus, in addition to our earlier publication on rho / omega modification, this study has experimentally verified vector meson mass modification at normal nuclear density.
• ### Nuclear mass number dependence of inclusive production of omega and phi mesons in 12 GeV p + A collisions(nucl-ex/0603013)
June 9, 2006 nucl-ex
The inclusive production of omega and phi mesons is studied in the backward region of the interaction of 12 GeV protons with polyethylene, carbon, and copper targets. The mesons are measured in e^+ e^- decay channels. The production cross sections of the mesons are presented as functions of rapidity y and transverse momentum p_T. The nuclear mass number dependences (A dependences) are found to be A^{0.710 +/- 0.021(stat) +/- 0.037(syst)} for omega mesons and A^{0.937 +/- 0.049(stat) +/- 0.018(syst)} for phi mesons in the region of 0.9 < y < 1.7 and p_T < 0.75 GeV/c.
• ### Experimental Signature of Medium Modifications for rho and omega Mesons in the 12 GeV p + A Reactions(nucl-ex/0504016)
March 15, 2006 nucl-ex
The invariant mass spectra of e+e- pairs produced in 12-GeV proton-induced nuclear reactions are measured at the KEK Proton-Synchrotron. On the low-mass side of the omega meson peak, a significant enhancement over the known hadronic sources has been observed. The mass spectra, including the excess, are well reproduced by a model that takes into account the density dependence of the vector meson mass modification, as theoretically predicted.
• ### Measurement of the Xi-p Scattering Cross Sections at Low Energy(nucl-ex/0502010)
Feb. 8, 2005 nucl-ex
In this paper we report cross-section measurements for $\Xi^-p$ elastic and inelastic scatterings at low energy using a scintillating fiber active target. Upper limit on the total cross-section for the elastic scattering was found to be 24 mb at 90% confidence level, and the total cross section for the $\Xi^-p\to\Lambda\Lambda$ reaction was found to be $4.3^{+6.3}_{-2.7}$ mb. We compare the results with currently competing theoretical estimates.
• ### Manipulating ionization path in a Stark map: Stringent schemes for the selective field ionization in highly excited Rb Rydberg atoms(physics/0204048)
April 17, 2002 physics.atom-ph
We have developed a quite stringent method in selectivity to ionize the low angular- momentum ($\ell$) states which lie below and above the adjacent manifold in highly excited Rb Rydberg atoms. The method fully exploits the pulsed field-ionization characteristics of the manifold states in high slew-rate regime: Specifically the low $\ell$ state below (above) the adjacent manifold is firstly transferred to the lowest (highest) state in the manifold via the adiabatic transition at the first avoided crossing in low slew-rate regime, and then the atoms are driven to a high electric field for ionization in high slew-rate regime. These extreme states of the manifold are ionized at quite different fields due to the tunneling process, resulting in thus the stringent selectivity. Two manipulation schemes to realize this method actually are demonstrated here experimentally.
• ### Systematic observation of tunneling field-ionization in highly excited Rb Rydberg atoms(physics/0204038)
April 15, 2002 physics.atom-ph
Pulsed field ionization of high-$n$ (90 $\leq n \leq$ 150) manifold states in Rb Rydberg atoms has been investigated in high slew-rate regime. Two peaks in the field ionization spectra were systematically observed for the investigated $n$ region, where the field values at the lower peak do not almost depend on the excitation energy in the manifold, while those at the higher peak increase with increasing excitation energy. The fraction of the higher peak component to the total ionization signals increases with increasing $n$, exceeding 80% at $n$ = 147. Characteristic behavior of the peak component and the comparison with theoretical predictions indicate that the higher peak component is due to the tunneling process. The obtained results show for the first time that the tunneling process plays increasingly the dominant role at such highly excited nonhydrogenic Rydberg atoms.
• ### Measurement of parity-nonconserving rotation of neutron spin in the 0.734-eV p-wave resonance of $^{139}La$(nucl-ex/0111018)
Nov. 21, 2001 nucl-ex
The parity nonconserving spin rotation of neutrons in the 0.734-eV p-wave resonance of $^{139}La$ was measured with the neutron transmission method. Two optically polarized $^3He$ cells were used before and behind a a 5-cm long $^{139}La$ target as a polarizer and an analyzer of neutron spin. The rotation angle was carefully measured by flipping the direction of $^3He$ polarization in the polarizer in sequence. The peak-to-peak value of the spin rotation was found to be $(7.4 \pm 1.1) \times 10^{-3}$ rad/cm which was consistent with the previous experiments. But the result was statisticallly improved. The s-p mixing model gives the weak matrix element as $xW = (1.71 \pm 0.25)$ meV. The value agrees well with the one deduced from the parity-nonconserving longitudinal asymmetry in the same resonance.
• ### The Rydberg-Atom-Cavity Axion Search(hep-ph/0101200)
Jan. 18, 2001 hep-ph
We report on the present progress in development of the dark matter axion search experiment with Rydberg-atom-cavity detectors in Kyoto, CARRACK I and CARRACK II. The axion search has been performed with CARRACK I in the 8 % mass range around $10 \mu {\rm eV}$, and CARRACK II is now ready for the search in the wide range $2 \mu {\rm eV} - 50 \mu {\rm eV}$. We have also developed quantum theoretical calculations on the axion-photon-atom system in the resonant cavity in order to estimate precisely the detection sensitivity for the axion signal. Some essential features on the axion-photon-atom interaction are clarified, which provide the optimum experimental setup for the axion search.
• ### A coupled microwave-cavity system in the Rydberg-atom cavity detector for dark matter axions(physics/0101028)
Jan. 4, 2001 physics.ins-det
A coupled microwave-cavity system of cylindrical TM$_{010}$ single-mode has been developed to search for dark matter axions around 10 $\mu {\rm eV}$(2.4 GHz) with the Rydberg-atom cavity detector at 10 mK range temperature. One component of the coupled cavity (conversion cavity) made of oxygen-free high-conductivity copper is used to convert an axion into a single photon with the Primakoff process in the strong magnetic field, while the other component (detection cavity) made of Nb is utilized to detect the converted photons with Rydberg atoms passed through it without magnetic field. Top of the detection cavity is attached to the bottom flange of the mixing chamber of a dilution refrigerator, thus the whole cavity is cooled down to 10 mK range to reduce the background thermal blackbody-photons in the cavity. The cavity resonant frequency is tunable over $\sim$ 15% by moving dielectric rods inserted independently into each part of the cavities along the cylindrical axis. In order to reduce the heat load from the higher temperature side to the most cooled dilution refrigerator part, the tuning rod at the conversion cavity is especially driven via the Kevlar strings with a stepping motor outside the cryostat at room temperature. The lowest temperature achieved up to now is 12 $\sim$ 15 mK for the long period operation and the loaded Q value at low temperature is 3.5 $\sim$ 4.5 $\times$ $10^4$ for the whole range of frequency tuning. Characteristics and the performance of the coupled-cavity system are presented and discussed with possible future improvements.
• ### Observation of $\rho / \omega$ Meson Modification in Nuclear Matter(nucl-ex/0011013)
Nov. 21, 2000 nucl-ex
We have measured invariant mass spectra of electron-positron pairs in the target rapidity region of 12GeV p+A reactions. We have observed a significant difference in the mass spectra below the $\omega$ meson between p+C and p+Cu interactions. The difference is interpreted as a signature of the $\rho/\omega$ modification at normal nuclear-matter density.
• ### Coherent time evolution of highly excited Rydberg states in pulsed electric field: Opening a stringent way to selectively field-ionize the highly excited states(physics/0010071)
Oct. 28, 2000 physics.atom-ph
Coherent time evolution of highly excited Rydberg states in Rb (98 < n < 150) under pulsed electric field in high slew-rate regime was investigated with the field ionization detection. The electric field necessary to ionize the Rydberg states was found to take discrete values successively depending on the slew rate. Specifically the slew-rate dependence of the ionization field varies with the quantum defect value of the states, i.e. with the energy position of the states relative to the adjacent manifold. This discrete transitional behavior of the ionization field observed for the first time is considered to be a manifestation of the strong coherence effect in the time evolution of the Rydberg states in pulsed electric field and opens a new effective way to stringently select a low-l state from the nearby states by field ionization. | 2021-04-20 03:07:50 | {"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": 0.6088943481445312, "perplexity": 1780.262495383627}, "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/1618039375537.73/warc/CC-MAIN-20210420025739-20210420055739-00624.warc.gz"} |
https://practicepaper.in/gate-ec/root-locus | # Root Locus
Question 1
The characteristic equation of a system is
$s^3 + 3s^2 + (K + 2)s + 3K = 0$
In the root locus plot for the given system, as K varies from 0 to $\infty$, the break-away or break-in point(s) lie within
A (-1,0) B (-2,-1) C (-3,-2) D (-$\infty$ ,-3)
GATE EC 2020 Control Systems
Question 1 Explanation:
$Q(s)=1+G(s)H(s)=0$
$s^{3}+3s^{2}+2s+ks+3k=0$
$-k=\frac{s^{3}+3s^{2}+2s}{s+3}$
$-\frac{\mathrm{d} k}{\mathrm{d} s}=\frac{(s+3)(3s^{2}+6s+2)-(s^{3}+3s^{2}+2s)}{(s+3)^{2}}=0$
$3s^{3}+6s^{2}+2s+9s^{2}+18s+6-s^{3}-3s^{2}-2s=0$
$2s^{3}+12s^{2}+18s+6=0$
$s=-0.46,-3.87,-1.65$
$\therefore$ Break-away point lies between (0, -1), i.e. (-1, 0)
Question 2
A linear time invariant (LTI) system with the transfer function
$G(s)=\frac{K(s^{2}+2s+2)}{(s^{2}-3s+2)}$
is connected in unity feedback configuration as shown in the figure.
For the closed loop system shown, the root locus for $0 \lt K \lt \infty$ intersects the imaginary axis for K = 1.5. The closed loop system is stable for
A $K>1.5$ B $1 \lt K \lt 1.5$ C $0 \lt K \lt 1$ D no positive value of K
GATE EC 2017-SET-1 Control Systems
Question 2 Explanation:
Given that,
$G(s)=\frac{K\left(s^{2}+2 s+2\right)}{s^{2}-3 s+2}$
The root locus plot of the given system is follows:
$\therefore$System is stable for $K \gt 1.5$.
Question 3
The forward-path transfer function and the feedback-path transfer function of a single loop negative feedback control system are given as
$G(s)=\frac{K(s+2)}{s^{2}+2s+2}$ and H(s)=1,
respectively. If the variable parameter K is real positive, then the location of the breakaway point on the root locus diagram of the system is __________
A -2 B -3.4 C -1.5 D -6.5
GATE EC 2016-SET-3 Control Systems
Question 3 Explanation:
\begin{aligned} G(s) &=\frac{K(s+2)}{s^{2}+2 s+2} \\ H(s)&=1 \\ 1+G(s) H(s) &=0 \\ 1+\frac{K}{\left(s^{2}\right)+2 s+2}(s+2) &=0\\ K&=-\frac{\left(s^{2}+2 s+2\right)}{(s+2)} \end{aligned}
$\frac{d K}{d s}=-\left(\frac{(2 s+2)(s+2)-\left(s^{2}+2 s+2\right)}{(s+2)^{2}}\right)$
For break away points, $\frac{d K}{d s}=0$
$\begin{array}{l} (2 s+2)(s+2)-\left(s^{2}+2 s+2\right)=0 \\ 2 s^{2}+6 s+4-s^{2}-2 s-2=0 \\ s^{2}+4 s+2=0 \end{array}$
\begin{aligned} s&=\frac{-4 \pm \sqrt{16-8}}{2} \\ &=\frac{-4 \pm 2 \sqrt{2}}{2}=-2 \pm \sqrt{2} \\ s&=-0.58 ; s=-3.41 \\ \text{But,}\quad s&=-3.41 \text{ lies on root locus }\\ \text{Hence}\quad s&=-3.41 \end{aligned}
Question 4
The open-loop transfer function of a unity-feedback control system is
$G(s)=\frac{K}{s^{2}+5s+5}$
The value of K at the breakaway point of the feedback control system's root-locus plot is ________
A 0.65 B 1.25 C 2.25 D 3.15
GATE EC 2016-SET-1 Control Systems
Question 4 Explanation:
Characteristic equation $1+G(s)H(s)=0$
$1+\frac{K}{s^2+5s+5}=0$
$K=-s^2-5s-5$
For break away point $\frac{dK}{ds}=0$
$\frac{dK}{ds}=-2s+5=0\;\Rightarrow \; s=-2.5$
According to magnitude condition,
\begin{aligned} |G(s)H(s)|_{s=-2.5} &=1 \\ |G(s)H(s)|_{s=-2.5} &=\frac{|K|}{|(2.5)^2 +5 \times -2.5+5|}=1 \\ |K|&=|(6.25+5-12.5)|=1.25 \\ K&=\pm 1.25 \end{aligned}
Question 5
For the system shown in the figure, s = -2.75 lies on the root locus if K is _______.
A 0 B 0.3 C 0.8 D 1.2
GATE EC 2015-SET-3 Control Systems
Question 5 Explanation:
The open loop transfer function of the system:
$G(s) H(s)=\frac{K(s+3) \times 10}{(s+2)}$
In order to lie a point s=-2.75, the angle condition must be satisfy and therefore the magnitude condition is
\begin{aligned} &\mid G(s) H(s) \|_{s=-275}=1 \\ &=\left.\frac{10 K \sqrt{s^{2}+3^{2}}}{\sqrt{s^{2}+2^{2}}}\right|_{s=-2.75}=1 \\ &=\frac{10 K \sqrt{(3-2.75)^{2}}}{\sqrt{(2-2.75)^{2}}}=1 \\ &=\frac{0.25 \times 10 K}{0.75} or K=0.3 \end{aligned}
Question 6
The open-loop transfer function of a plant in a unity feedback configuration is given as $G(s)=\frac{K(s+4)}{(s+8)(s^{2}-9)}$ . The value of the gain K(> 0) for which -1+ j2 lies on the root locus is_______.
A 20.18 B 25.54 C 30.56 D 33.35
GATE EC 2015-SET-1 Control Systems
Question 6 Explanation:
\begin{aligned} G(s) &=\frac{K(s+4)}{(s+8)\left(s^{2}-9\right)} \\ &=\frac{K(s+4)}{(s+8)(s+3)(s-3)} \end{aligned}
For the point (-1+2 j) to lie on the root locus the angle condition must be satisfy
i.e. $\angle G(s) H(s)=\pm(2 Q+1) 180^{\circ}$
Taking LHS:
$\begin{array}{l} \left.\angle G(s) H(s)\right|_{s=-1+2 j} \\ =\frac{\angle K+\angle(s+4)}{\angle(s+8)+\angle(s+3)+\angle(s-3)} \\ =\frac{\angle K+\angle(-1+2 j+4)}{\angle(-1+2 j+8)+\angle(-1+2 j+3)+\angle(-1+2 j-3)} \\ =\frac{0+\tan ^{-1}\left(\frac{2}{3}\right)}{\tan ^{-1}\left(\frac{2}{7}\right)+\tan ^{-1}(1)+180^{\circ}-\tan ^{-1}\left(\frac{1}{2}\right)} \\ =33.69^{\circ}-15.945^{\circ}-45^{\circ}-180^{\circ}+26.565^{\circ} \\ \simeq 180^{\circ} \end{array}$
As angle condition is satisfied the value of system gain K can be obtained by using magnitude
condition
$\begin{array}{l} i.e. |G(s) H(s)|_{s=-1+2 j}=1 \\ =\frac{K \sqrt{(-1+4)^{2}+2^{2}}}{\sqrt{(-1+8)^{2}+2^{2}} \cdot \sqrt{(-1+3)^{2}+2^{2}} \cdot \sqrt{(-1-3)^{2}+2^{2}}} \\ =1 \\ =\frac{K \sqrt{9+4}}{\sqrt{49+4} \sqrt{4+4} \sqrt{16+4}}=1 \\ =\frac{K \sqrt{13}}{\sqrt{53} \sqrt{8} \sqrt{20}}=1 \\ K=\frac{\sqrt{53 \times 8 \times 20}}{\sqrt{13}}=25.54 \end{array}$
Question 7
A unity negative feedback system has the open-loop transfer function $G(s)=\frac{K}{s(s+1)(s+3)}$. The value of the gain K ($\gt$0) at which the root locus crosses the imaginary axis is ________.
A 7 B 10 C 12 D 15
GATE EC 2015-SET-1 Control Systems
Question 7 Explanation:
$G(s)=\frac{K}{s(s+1)(s+3)}$
The characteristic equation
$\begin{array}{l} =1+G(s) H(s)=0 \\ =s(s+1)(s+3)+K=0 \\ =s^{3}+4 s^{2}+3 s+K=0 \end{array}$
Using Routh's tabular form
$\begin{array}{c|cc} s^{3} & 1 & 3 \\ s^{2} & 4 & K \\ s^{1} & \frac{12-K}{4} & 0 \\ s^{0} & K & \end{array}$
In order to cross the imaginary axis, system
should be marginally stable
\begin{aligned} \frac{12-K}{4}&=0\\ \text{or }\quad K&=12 \end{aligned}
Question 8
In the root locus plot shown in the figure, the pole/zero marks and the arrows have been removed. Which one of the following transfer functions has this root locus ?
A $\frac{s+1}{(s+2)(s+4)(s+7)}$ B $\frac{s+4}{(s+1)(s+2)(s+7)}$ C $\frac{s+7}{(s+1)(s+2)(s+4)}$ D $\frac{(s+1)(s+2)}{(s+4)(s+7)}$
GATE EC 2014-SET-3 Control Systems
Question 8 Explanation:
Since the root locus always emerges from the break-away points i.e. when two poles are at a part of same root locus lie. So $\sigma$ =-1 and -2 will surely be poles.
Also we know that locus emerges from poles and terminates at zero. So, s=-4 is a zero and s=-7 is a pole
$\therefore T(s)=\frac{(s+4)}{(s+1)(s+2)(s+7)}$
Question 9
The root locus plot for a system is given below. The open loop transfer function corresponding to this plot is given by
A $G(S)H(S)=k\frac{s(s+1)}{(s+2)(s+3)}$ B $G(S)H(S)=k\frac{(s+1)}{(s+2)(s+3)^{2}}$ C $G(S)H(S)=k\frac{1}{s(s-1)(s+2)(s+3)}$ D $G(S)H(S)=k\frac{(s+1)}{(s+2)(s+3)}$
GATE EC 2011 Control Systems
Question 9 Explanation:
From plot we can observe that one pole terminates at one zero at position -1 and three poles terminates to $\infty$. It means there are four poles and 1 zero. Pole at -3 goes on both sides. It means there are two poles at -3.
Question 10
The feedback configuration and the pole-zero locations of $G(s)=\frac{s^{2}-2s+2}{s^{2}+2s+2}$ are shown below. The root locus for negative values of k , i.e. for $-\infty \lt k \lt 0$, has breakaway/break-in points and angle of departure at pole P (with respect to the positive real axis) equal to
A $\pm \sqrt{2} \; and \; 0^{\circ}$ B $\pm \sqrt{2} \; and \; 45^{\circ}$ C $\pm \sqrt{3} \; and \; 0^{\circ}$ D $\pm \sqrt{3} \; and \; 45^{\circ}$
GATE EC 2009 Control Systems
Question 10 Explanation:
\begin{aligned} 1+G(s) H(s) &=0 \\ 1+\frac{K\left(s^{2}-2 s+2\right)}{s^{2}+2 s+2} &=0 \\ k &=-\frac{s^{2}+2 s+2}{s^{2}-2 s+2} \\ \text{put }\quad \frac{\partial K}{\partial s}&=0\text{ we have}\\ \left(s^{2}-2 s+2\right)(s+1) &-\left(s^{2}+2 s+2\right) \\ (s-1) =0 \\ 2 s^{2}-4 s^{2}+4 &=0 \\ 2 s^{2} &=+4 \\ s &=\pm \sqrt{2} \\ \text{Angle of departure is}\\ \phi_{D}=\phi\\ \text { where } &=\Sigma \phi_{2}-\Sigma \phi_{p} \\ &=225^{\circ} \\ \phi_{D} &=\pm 225^{\circ} \end{aligned} \\ \text{No option is matching.}
There are 10 questions to complete. | 2022-09-25 05:02:10 | {"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": 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": 63, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999947547912598, "perplexity": 3531.4398957577055}, "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-2022-40/segments/1664030334514.38/warc/CC-MAIN-20220925035541-20220925065541-00095.warc.gz"} |
https://physics.paperswithcode.com/paper/room-temperature-spin-ice-physics-in-cadmium | ## Room temperature spin-ice physics in cadmium cyanide
11 Apr 2019 · Coates Chloe S., Baise Mia, Simonov Arkadiy, Makepeace Joshua W., Seel Andrew G., Smith Ronald I., Playford Helen Y., Keen David A., Siegel Renée, Schmutzler Adrian, Senker Jürgen, Slater Ben, Goodwin Andrew L. ·
Spin-ices are frustrated magnets that support a particularly rich variety of emergent physics. Typically, it is the interplay of magnetic dipole interactions, spin anisotropy, and geometric frustration on the pyrochlore lattice that drives spin-ice formation... The relevant physics occurs at temperatures commensurate with the magnetic interaction strength, which for most systems is 1--5\,K. This low energy scale poses severe challenges for experimental studies of spin-ices and the practical exploitation of their unusual properties. Here, we show that non-magnetic cadmium cyanide (Cd(CN)$_2$) exhibits analogous behaviour to magnetic spin-ices, but does so on a temperature scale that is nearly two orders of magnitude greater. The electric dipole moments of cyanide ions in Cd(CN)$_2$ assume the role of magnetic pseudospins, with the difference in energy scale reflecting the increased strength of electric \emph{vs} magnetic dipolar interactions. As a result, spin-ice physics influences the structural behaviour of Cd(CN)$_2$ even at room temperature. read more
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Materials Science Disordered Systems and Neural Networks Strongly Correlated Electrons | 2021-08-05 11:40:45 | {"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": 0.5186406970024109, "perplexity": 11767.404429485441}, "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/1627046155529.97/warc/CC-MAIN-20210805095314-20210805125314-00424.warc.gz"} |
https://studyadda.com/sample-papers/jee-main-sample-paper-43_q8/297/302779 | • # question_answer A bomb is dropped from an aeroplane when it is at a height h directly above the target. If the aero plane is moving horizontally at a speed v, the distance by which bomb will miss the target is given by A) $2v\sqrt{\frac{h}{g}}$ B) $v\sqrt{\frac{h}{g}}$ C) $v\sqrt{\frac{2h}{g}}$ D) $v\sqrt{\frac{h}{2g}}$
As the bomb is released directly above the target, the distance by which the bomb will miss the target must be equal to its horizontal range, i.e., $\Delta x=(SA+AP)-SP$ | 2022-01-27 09:37:56 | {"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": 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": 0.6793614029884338, "perplexity": 657.7636120108831}, "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/1642320305242.48/warc/CC-MAIN-20220127072916-20220127102916-00278.warc.gz"} |
https://www.cryptopp.com/wiki/Profiling | Profiling
Profiling
For those using gprof and CXXFLAGS += -pg, you can generate to generate profiling data by (1) running cryptest.exe v to run the library's self tests; or (2) run cryptest.exe b to use the built in bench-marking suite.
> cryptest.exe b > benchmark.html # gmon.out written in PWD by instrumentation
> gprof -a -b cryptest.exe gmon.out > cryptest.exe.prof
> cat cryptest.exe.prof | more # output below is baked for screen formatting
Flat profile:
Each sample counts as 0.01 seconds.
% cumulative self self total
time seconds seconds calls s/call s/call name
15.68 16.16 16.16 9325567 0.00 0.00 CryptoPP::Rijndael::Enc::AdvancedProcessBlocks
5.97 22.31 6.15 1094386 0.00 0.00 CryptoPP::GF2NT::Multiply
4.50 26.95 4.64 559773 0.00 0.00 CryptoPP::AlmostInverse
3.73 30.79 3.84 547193 0.00 0.00 CryptoPP::GF2NT::MultiplicativeInverse
2.54 33.41 2.62 1048576 0.00 0.00 CryptoPP::Salsa20_Policy::OperateKeystream
2.50 35.99 2.58 320517 0.00 0.00 CryptoPP::RawDES::RawSetKey
2.27 38.33 2.34 655360 0.00 0.00 CryptoPP::GCM_Base::AuthenticateBlocks
...
Memory Leaks
The Crypto++ mailing list occasionally receives messages concerning memory leaks from tools such as Purify and Bounds Checker. Under most circumstances, Crypto++ is leak free and the problem stems from one of two places - a legitimate leak (in the program using the library and not the library itself) and a false positive.
Reported Leak
The first leak is due to the programmer and is a valid item. The allocation occurs in the user's program but when the dump is printed, there is so much Crypto++ library code in the chain a programmer mistakenly claims the leak is with the library.
If you feel the library is truly leaking, please file a Bug Report. Before reporting, you should run cryptest v under the tool as a base case.
False Positive
The second leak is a false positive due to a tool reporting a leak where there is no leak. For example, in the past, BoundsChecker would report a C++ library object leaked even though the object was reclaimed during C++ library shutdown when the static destructors executed. The problem was that the BoundsChecker DLL was unloaded before the runtime library terminated.
In the Microsoft environment, Visual Studio will report a leak when using Run-Time Type Information (RTTI). Microsoft states the warning can be safely ignored in KB140670. Below is a typical example of the typeid() leak. Notice that the output occurs in pairs (for example, allocations 1336110 and 1336111).
Detected memory leaks!
Dumping objects ->
{1336111} normal block at 0x003AECA8, 30 bytes long.
Data: <class CryptoPP::> 63 6C 61 73 73 20 43 72 79 70 74 6F 50 50 3A 3A
{1336110} normal block at 0x003ABB28, 8 bytes long.
Data: < : X > A8 EC 3A 00 58 06 AB 00
{798888} normal block at 0x003AB918, 58 bytes long.
Data: <class CryptoPP::> 63 6C 61 73 73 20 43 72 79 70 74 6F 50 50 3A 3A
{798887} normal block at 0x00AB0658, 8 bytes long.
Data: < : : > 18 B9 3A 00 B8 B9 3A 00
Though the issue has been well know since Visual C++ 4.0 days, it does not appear that Microsoft will fix the false positive. See Memory leaks reported by debug CRT inside typeinfo.name().
Validating Library Memory Usage
After building the Crypto++ library, it is usually a good idea to verify the build with cryptest v. Claiming validation is a good idea is an understatement - the build must be validated. Validation is a perfect time to run other tools, such as Purify and Valgrind, against the executable.
$./cryptest.exe V 5.6.1$ ./cryptest.exe v
Using seed: 1282115289
Testing Settings...
passed: Your machine is little endian.
passed: Your machine allows unaligned data access.
passed: sizeof(byte) == 1
passed: sizeof(word16) == 2
passed: sizeof(word32) == 4
passed: sizeof(word64) == 8
passed: sizeof(word128) == 16
passed: sizeof(hword) == 4, sizeof(word) == 8, sizeof(dword) == 16
passed: hasMMX == 1, hasISSE == 1, hasSSE2 == 1, hasSSSE3 == 1, hasAESNI == 0, hasCLMUL == 0, isP4 == 0, cacheLineSize == 64, AESNI_INTRINSICS == 1
...
All tests passed!
Test ended at Wed Aug 18 03:08:10 2010
Seed used was: 1282115289
Uninitialized Memory
Random pools can use unintialized memory as a source of entropy. Before filing a report, please verify the usage is not related to a random source. Also see Vendors are Bad for Security.
Application Verifier
Application Verifier is a Microsoft tool which profiles an application. To use Application Verifier, add cryptest.exe to the application list. Then run cryptest v under the Visual Studio debugger.
Below is a run with Memory, Heaps, Dangerous APIs, and Dirty Stacks enabled. Dangerous APIs include original C runtime functions such as memcpy (we should now use memcpy_s). Dirty Stacks checks for the use of uninitialized variables. The log file is not shown since there are 0 errors and 0 warnings.
Valgrind and Valkyrie
To validate with Valgrind, perform the following from the build directory. The build directory is required so that the Test Vectors are available during the self test.
cd cryptopp-5.6.0/
valgrind --tool=memcheck --leak-check=full --show-reachable=yes ./cryptest.exe v
After a successful run, output should be similar to below. If new to Valgrind, 17992 is the process id of the test run.
==17992== Memcheck, a memory error detector.
==17992== Copyright (C) 2002-2008, and GNU GPL'd, by Julian Seward et al.
==17992== Using LibVEX rev 1884, a library for dynamic binary translation.
==17992== Copyright (C) 2004-2008, and GNU GPL'd, by OpenWorks LLP.
==17992== Using valgrind-3.4.1, a dynamic binary instrumentation framework.
==17992== Copyright (C) 2000-2008, and GNU GPL'd, by Julian Seward et al.
==17992== For more details, rerun with: -v
==17992==
...
==17992==
==17992== ERROR SUMMARY: 0 errors from 0 contexts (suppressed: 5 from 1)
==17992== malloc/free: in use at exit: 0 bytes in 0 blocks.
==17992== malloc/free: 1,119,117 allocs, 1,119,117 frees, 60,533,882 bytes allocated.
==17992== For counts of detected errors, rerun with: -v
==17992== All heap blocks were freed -- no leaks are possible.
OS X
You might experience a memory leak under OS X. This is likely a known issue due to Apple's use of localtime. Also see How to resolve dyld and localtime leaks when profiling a project? and Valgrind memory leak points to localtime on Stack Overflow.
==33810== LEAK SUMMARY:
==33810== definitely lost: 2,081 bytes in 3 blocks
==33810== indirectly lost: 1,040 bytes in 1 blocks
==33810== possibly lost: 10,808 bytes in 1 blocks
==33810== still reachable: 20,714 bytes in 10 blocks
==33810== suppressed: 58,630 bytes in 350 blocks
==33810== Rerun with --leak-check=full to see details of leaked memory
Valkyrie
Valkyrie is a visual front end to Valgrind. Below is a typical run of cryptest using the Crypto++ test vectors. | 2022-08-07 15:55:25 | {"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": 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": 0.17145565152168274, "perplexity": 13285.671782538704}, "config": {"markdown_headings": false, "markdown_code": true, "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-2022-33/segments/1659882570651.49/warc/CC-MAIN-20220807150925-20220807180925-00571.warc.gz"} |
https://mizugadro.mydns.jp/t/index.php/Latex | # Latex
Jump to: navigation, search
Latex is the computational language designed for representation of scientific results in form of an scientific article, or a book, or a slideshow oo a poster.
## Overview
In the simplest case, the latex input is ascii–file, while the output file is in the postsctipt or pdf format. Also, at the compilation the log file is generated that describes errors revealed at the compilation; this file is suppose to help at the debugging.
All the high-rank scientific journals accept the Latex format of the articles submitted. Most of journals provide the sample file, so–called latex template with examples of the most of latex commands the authors may need in the preparation of their articles. In such a way, one has no need to learn Latex before to use it.
The distribution of software that supports the latex typing, is free. In particular, the latex installer can be found through the CTAN [1]
Also, such software as texshop may be used to simplify the typing of the latex texts.
There is useful guide at http://en.wikibooks.org/wiki/LaTeX
## Typing on various languages
There exist extensions that recognize the extended ascii input and allow to type various languages.
In some countries (that in century 20 were successful fighting against Cybernetics), up to beginning of century 21, several conflicting encoding systems are used. For example, the editorial of the Moscow University Physics Bulletin [2] requires the CP1251! encoding scheme (although theCP1251 is also accepted), white the Uspechi prefer the UTF8; the article should be recoded, over–vice, the editors have problems dealing with it.
In addition, even within the same journal, various sections may require different format of the manuscript representation.
The format o representation of formulas in Latex is very similar to that in TORI, id est, sign of dollar indicates the beginning and the end of the formulas; the commands begin with backslash and in the most of cases, the names of the commands just coincide with the commonly used names of the characters required for the generation,
### Russian template
The example below indicates how to type Russian Latex document:
\documentclass[12pt]{article}
\usepackage[utf8x]{inputenc}
\usepackage[russian]{babel}
\begin{document}
\title{Название}
\maketitle
Латиница тоже получается: English
И формулы тоже: $$\displaystyle \int_a^b \frac{x}{1+x^2} {\mathrm d x}$$
\end{document}
However, for the custom formatting, other accessories (not specific for typing in Russian) should be added to the head.
### Japanese template
The example below indicates how to use Japanese characters in a Latex document:
\documentclass{memoir}
\usepackage{CJKutf8}
\begin{document}
\begin{CJK}{UTF8}{min}未練なく散も桜はさくら哉\end{CJK} \\
without regret they fall and scatter cherry blossoms
\end{document}
### yyyymmdd
For writing of dates in language-invariant form, there is package yyyymmdd. Example:
\documentclass{book}
\usepackage[yyyymmdd]{datetime}
\begin{document}
\renewcommand{\dateseparator}{.}
\today
\end{document}
## Advantages of Latex
Latex is very compact language; often, the article in Latex counts only few percent of size the same article would have at any other typesetting.
The examples above can be combined; the constructions are compatible.
Latex allows to type formulas without to look at the screen.
Latex is robust: the documents are easy portable from one computer to another
Latex allow to hide/reveal some part of a text defining few new commands; for example, the same input file can be used for generation of English, Russian of Japanese versions.
## Disadvantages of Latex
For complicated formulas or tables, some tools and hints are necessary. For example, it is difficult to guess, that the size of the resulting PDF image (if not default) should be determined with commands \paperwidth and \paperheight at the top.
Yet, there is no regular way to generate the output in a form, different from postscript and PDF. For the web-oriented applications, the MathJax can be used instead, and yet, not all the Latex commands are supported by MathJax.
## Referencs
http://www.ctex.org/documents/packages/layout/titlesec.pdf Javier Bezos. The titlesec and titletoc Packages. June 14, 2000. | 2019-09-16 10:52:33 | {"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": 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": 0.8915408253669739, "perplexity": 2895.3224009267888}, "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-2019-39/segments/1568514572517.50/warc/CC-MAIN-20190916100041-20190916122041-00378.warc.gz"} |
http://s3-us-west-1.amazonaws.com/weakcool/find-the-number-of-possible-positive-and-negative-real-zeros-for-the-164.html | ## Find the number of possible positive and negative real zeros for the function,address search with phone number,free numerology 2014 in hindi - Videos Download
27.05.2015
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Bascially we are trying to find the roots of a quadratic equation, and 'apparently' there is a theorem for this, but every one that I have found so far mentions that the degree of the polynomial is directly related to the number of zeros (where X is an integer and Y is 0).. Note: The theorem also has to account for vertical shifts, which I don't see how that can be possible.
One other alternative: If you can find the vertex of the polynomial, then knowing the $y$-value of the vertex and knowing whether the quadratic function opens up or down will tell you how many times the quadratic crosses the $x$-axis. If you try to solve the equation (just like you'd solve a linear equation), you'd run into trouble when you realize that the $x$s can't be as easily isolated like in the case with linear equations.
If you are familiar with the method of "completing the squares" (and it isn't very difficult,) you'd know that our goal is to somehow "arrange" the equation in the form $(\text{linear binomial})^2+\text{constant term}$. I'll not show you the entire process of completing the square for our given function but I presume that you are somewhat familiar with it (if not, I suggest you look it up on the internet). The third, and probably the most important one, is that it tells you when $x$ is a Real number or when it is not (i.e when it is a Complex number).
The expression under the square root sign, if positive, we have two zeroes that are real (more commonly known as two distinct and Real roots).
If the expression under the square root sign is negative, we will get two zeroes that are Complex Numbers (more commonly known as two distinct but Complex roots). The geometric interpretation of these three cases is also pretty interesting and gives a deeper insight at what is going on (Although your question has already been answered, i suggest you keep reading, I'm almost done :)).
Similarly, if the roots are Real and repeated, (i.e the discriminant is zero), the function should have only one point crossing (or in this case touching) the x-axis. As far as your question is concerned, If you'd like to know if the zeroes are Real or not, check if $B^2 -4AC$ is positive, negative or zero, the nature of the roots (zeroes) should be apparent.
I bothered to write down such long an answer because I really wanted you to know what's going on behind the scenes. Depending on the level that you are at, you will find that most quadratics you are given will probably either factorise or have observable roots. Given some of the roots of the function $f(x) = x^3+bx^2+cx+d$, how do I find the coefficients of that function?
Is there a more effective way to build the vocabulary of a fictional language than a word frequency list? When finding the inverse relations you will use the vertical line test and if goes through the line more then once then it doesn't have an inverse. End Behavior: odd degree means that the ends will go in different directions, even degree means it will go in the same direction, negative leading coefficient means that the graph will go down on the right side, and positive leading coefficient means the graph will go up on the right side.
Possible real zeros: Leading coefficient is the possible real zeros and the leading coefficient subtract one is the possible turning points, To find the actual real zeros you factor.
Next, evaluate the function for (– x), simplify, determine the sign changes and, hence, the number of possible negative zeros.
The importance of the use of the absolute value in the previous example is apparent when we evaluate using values that make the radicand negative. A cube rootA number that when used as a factor with itself three times yields the original number, denoted with the symbol a€‰3. The product of three equal factors will be positive if the factor is positive and negative if the factor is negative. Since cube roots can be negative, zero, or positive we do not make use of any absolute values. Since the cube root could be either negative or positive, we conclude that the domain consists of all real numbers.
For any integer na‰?2, we define an nth rootA number that when raised to the nth power (na‰?2) yields the original number. The negative nth root, when n is even, will be denoted using a negative sign in front of the radical a?’a€‰n. We have seen that the square root of a negative number is not real because any real number that is squared will result in a positive number. We can see that a fourth root of a?’81 is not a real number because the fourth power of any real number is always positive.
When n is even, the nth root is positive or not real depending on the sign of the radicand. A radical is simplifiedA radical where the radicand does not consist of any factors that can be written as perfect powers of the index. Here we note that the index is odd and the radicand is negative; hence the result will be negative.
In this case, consider the equivalent fraction with a?’8=(a?’2)3 in the numerator and 64=43 in the denominator and then simplify. However, most of the theorems also mention complex numbers, which our math teacher does not want us to include in this theorem. The end result of our method will yield a very familiar expression, what we are used to calling as the "quadratic formula". Because there are two distinct roots (but not real), it might be tempting to think that it also crosses the x- axis twice but, good luck finding the point where it actually crosses the x-axis! To find the x and y intercepts, let x = 0 to find the y–intercept and let y = 0 to find the x - intercept.
Using synthetic division, the value of the remainder is the value of the function for the particular value of x used in the division. Descartes’ Rule of Signs says that we can get some idea of the number of possible real solutions to an equation by counting the sign changes. The rational roots test can give some indication of possible rational solutions to f(x) = 0. So try using synthetic division to seek a zero which will also reduce the equation to a quadratic. Remember that substitution is an excellent and useful technique ideal for these situations. Using the given information and the tools from this section, determine the zeros for the function. Through trial-and-error, find one or more that leave a remainder of zero and thus are solutions to f(x) = 0 and are the corresponding factors. See the license for more details, but that basically means you can share this book as long as you credit the author (but see below), don't make money from it, and do make it available to everyone else under the same terms. However, the publisher has asked for the customary Creative Commons attribution to the original publisher, authors, title, and book URI to be removed.
Since the sign depends on the unknown quantity x, we must ensure that we obtain the principal square root by making use of the absolute value. To determine the domain of a function involving a square root we look at the radicand and find the values that produce nonnegative results. Furthermore, we can refer to the entire expression An as a radicalUsed when referring to an expression of the form An..
The product of an odd number of positive factors is positive and the product of an odd number of negative factors is negative. Does anyone know of a way to find the number of real zereos using a theorem that does not involve complex numbers? It is a well known fact square roots of negative numbers are not Real number, instead, we call them the imaginary numbers $($ and the complex of Real and Imaginary numbers are called the Complex Numbers, usually in the form $a+b\mathbb i$ and $\mathbb i =\sqrt{-1})$. The only way to know which, if any, are zeros for the given function is to actually do synthetic division using each of the given numbers. Use synthetic division to reduce the degree and factor the resultant quadratic using conventional techniques.
Note that this does not tell us what the solutions are, only whether we can expect to find any real solutions or not.
There are 2 or 0 possible positive real zeros for this function; and there are 3 or 1 possible negative real zeros that will satisfy the equation = 0. Use the factors of the constant term over the factors of the leading coefficient and make a comprehensive list from the result. Again, this is a higher degree equation; use the tools to reduce this to a quadratic which you know how to solve. Furthermore, the equation has been reduced to a quadratic which can be solved by the usual techniques.
The idea is to reduce this higher degree polynomial to a quadratic which can then be solved in typical manner. You may also download a PDF copy of this book (41 MB) or just this chapter (1 MB), suitable for printing or most e-readers, or a .zip file containing this book's HTML files (for use in a web browser offline). Furthermore, we denote a cube root using the symbol a€‰3, where 3 is called the indexThe positive integer n in the notation a€‰n that is used to indicate an nth root.. For example, 23 is an irrational number that can be approximated on most calculators using the root button a€‰x. Choose some positive and negative values for x, as well as zero, and then calculate the corresponding y-values. When the index is an integer greater than or equal to 4, we say a€?fourth root,a€? a€?fifth root,a€? and so on. So the next logical deduction could be that it doesn't actually cross the x-axis at all, which is very plausible. It works with the rule that says that complex imaginary solutions occur only in conjugate pairs. Note that Descartes’ Rule may tell us there are real solutions, but they may not show up in this list since this covers only rational solutions. Use any necessary tools to reduce the function to a quadratic, which you already know how to factor (and solve), and along the way you should discover the remaining solutions. The nth root of any number is apparent if we can write the radicand with an exponent equal to the index. Remember, then, that the rule also states there may be that many zeros (equal to the number of sign changes) or less than that by an even number (duh! Hence we use the radical sign a€‰n to denote the principal (nonnegative) nth rootThe positive nth root when n is even.
Recall the Rational Roots test and, through trial-and-error, determine what numbers will be zeros. For this reason, we use the radical sign a€‰ to denote the principal (nonnegative) square rootThe positive square root of a positive real number, denoted with the symbol a€‰.
Categories: Horoscope Compatibility | 2020-02-24 15:57:53 | {"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": 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": 0.7741439342498779, "perplexity": 203.19693543784499}, "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/1581875145960.92/warc/CC-MAIN-20200224132646-20200224162646-00288.warc.gz"} |
https://answers.ros.org/question/31728/transform-laserscan-to-world_frame/ | # transform LaserScan to /world_frame
Hi all,
I'm trying to transform a LaserScan(already transformed to PCL::PointCloud) from it's frame to the /world frame.
What I'm doing right now is accumulate the pointcloud in the /world frame, while my robot moves(the information of the egomotion, gives the position of the robot in the world and it's orientation). I get the orientation using an imu and the position through wheel odometry.
The transformation(or pointcloud accumulation) I get between the /laser frame and the /world frame, is working correctly for the cartesian coordinates, however when the robot suffers any rotation(pitch and roll),due to any inclination of the ground,the representation of the pointcloud, is not beeing represented on the /world frame, it's representation rotates just like the robot(and was expected that the points would be right over the world).
I have my laser pointing to the ground, and when the robot is moving in plane ground, the reconstruction is exactly as it is in reality, my problem is when the robot start curving(it has suspension).
(The axes you see in the image is from the laser, the red grid is the world)
my tf tree:
/world->/base_footprint-> ... -> /laser_frame
I hope I could make myself clear, it's not easy to explain. I dont know what I'm doing wrong, if anyone could help me I would appreciate. :D
p.s. Sorry for so much text.
edit retag close merge delete
Do you convert the cloud to the world frame before or after you accumulate it?
( 2012-04-13 03:39:49 -0500 )edit
before the accumulation
( 2012-04-13 05:04:54 -0500 )edit
So your scanner is producing LaserScan messages in the laser_frame TF frame. Did you try visualizing the LaserScan directly in RViz, before accumulation + transformation to a point cloud? If the raw scan looks right, you can rule out problems with the IMU.
( 2012-04-13 22:01:43 -0500 )edit
If I visualize the Laserscan directly on rviz, the representation is similar.. i dont know what i'm doing wrong
( 2012-04-16 01:04:27 -0500 )edit
That's strange. Are you sure you are only visualizing the last scan? What exactly is wrong? Because when the robot tilts, the laser scan of course also records a tilted line. Could you post another picture or video?
( 2012-04-16 02:24:27 -0500 )edit
For example, when the car is curving to the right, since it has suspension, it will roll as shown in the picture, however, the road was plane and I was expecting that the pointcloud accumulation would "draw" the road on the red grid (/world).
( 2012-04-16 03:51:26 -0500 )edit
This might be a bit off track but what do you mean by "accumulate" the point cloud?
( 2014-07-24 08:18:06 -0500 )edit
Sort by » oldest newest most voted
I would suspect that your IMU data is not modifying the transform tree correctly. You should check that the right effect is created.
It is very important to understand how your IMU works. Some IMUs calculate angular rotation with respect to the Earth. For example, an IMU that is world-referenced would obtain its z-rotation using a magnetometer. It would then use the gravity vector to calculate all rotations with respect to the Earth. A relative-reference IMU would simply calculate the amount that it has rotated about each axis since the last time you reset its gyros. Thus, if you reset the gyros with the robot upside down, that would be considered 0,0,0 rpy.
For example, on my robot, we use the IMU data (relative-reference) to calculate the transform from /base_footprint to /base_link. We mark /base_footprint as the point on the ground directly below the robot's center of mass/rotation. Thus, /base_footprint is modified by the position of the robot with respect to the static /odom frame. We obtain our z-axis rotation for /odom by resetting the gyros (so the IMU is not outputting 0 degrees for rotation about the z-axis) and then adding the heading obtained by our digital magnetometers. We then obtain x and y positions from the GPS. This sets our /odom frame with respect to the world (Earth).
After setting /odom with respect to the /world frame, all future calculations are performed between /odom and /base_footprint. This is because both wheel odometry and our IMU are relative to the starting point of the robot (/odom). Thus, the z-axis rotation from the IMU will set the heading of the robot with respect to /odom.
/base_link represents the center of mass/rotation for the robot. This frame's z-distance from /base_footprint is static (the robot cannot move up and down). This frame's rotation about the z-axis is fixed to /base_footprint's rotation. Finally, this frame's pitch (rotation about the y-axis) and roll (rotation about the x-axis) are calculated by the IMU.
I know this is very specific to my robot, but the idea is to give you an idea of the thought process you can use to check your IMU calculations. I'd be happy to clarify any points if necessary. I know this was a lot of information to understand.
more
Ok. I have two questions.
( 2012-04-13 10:29:24 -0500 )edit
What are they?
( 2012-04-13 11:27:54 -0500 )edit
For example, when the car is curving to the right, since it has suspension, it will roll as shown in the picture, however, the road was plane and I was expecting that the pointcloud accumulation would "draw" the road on the red grid (/world).
more
Sorry for the previous post.
1ºQuestion: What do you mean by "modifying the transform tree correctly"?
2ºQuestion: Maybe my problem is that I'm missing a frame or my understanding of /world frame is wrong.
Ok, I'll explain now what I've done. Through the WheelOdometry I calculate the robot pose in the world (I dont define any transformation at this point), I then to fuse the information of WheelOdometry with the IMU(Xsens), use Robot_pose_EKF (this package is responsible to send out the transformation between /base_footprint and /world).
Simplified tf tree:
/world ->/base_footprint
/base_footprint -> /laser
/base_footprint -> /imu
Defined the tf tree.
I :
p_listener->lookupTransform("/world","laser_frame_id", ros::Time(0), transform);
pcl_ros::transformPointCloud(pcl_pc_laser,pcl_transformed,transform);
pcl_acc+=pcl_transformed;
I thought this would enable me to accumulate correctly the pointCloud in the /world frame. I dont know If this problem is because of the transformation itself, of it is wrong estimation of the pose.
more
Just a few quick comments: 1. you shouldn't use ros::Time(0), but the time the laser scan is taken (i.e., laserscan.header.stamp) 2. you shouldn't hard code "laser_frame_id", but instead use laserscan.header.frame_id 3. do you run lookupTransform for each scan? You should.
( 2012-04-16 02:24:53 -0500 )edit
1º solved, you're right
2º It's not hardcoded, it was just to show in here
3º the code shown, is always called when a message of LaserScan is received
( 2012-04-16 03:58:10 -0500 )edit
For example, when the car is curving to the right, since it has suspension, it will roll as shown in the picture, however, the road was plane and I was expecting that the pointcloud accumulation would "draw" the road on the red grid (/world).
( 2012-04-16 07:17:52 -0500 )edit | 2021-09-28 17:31:10 | {"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": 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": 0.23301973938941956, "perplexity": 1944.7591293962005}, "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/1631780060877.21/warc/CC-MAIN-20210928153533-20210928183533-00414.warc.gz"} |
https://zbmath.org/?q=an:0288.76031 | # zbMATH — the first resource for mathematics
Uniqueness of weak solutions of the Cauchy problem for general 2 x 2 conservation laws. (English) Zbl 0288.76031
##### MSC:
76G25 General aerodynamics and subsonic flows 35Qxx Partial differential equations of mathematical physics and other areas of application
Full Text:
##### References:
[1] Douglis, A.: The continuous dependence of generalized solutions of nonlinear partial differential equations upon initial data. Comm. pure appl. Math. 14, 267-284 (1961) · Zbl 0117.31102 [2] Godunov, S. K.: On the uniqueness of the solution of the equations of hydrodynamics. Mat. sb. 40, 467-478 (1956) [3] Hurd, A. E.: A uniqueness theorem for second order quasilinear hyperbolic equations. Pacific J. Math. 32, 415-427 (1970) · Zbl 0172.38401 [4] Lax, P. D.: Hyperbolic systems of conservation laws, II. Comm. pure appl. Math. 10, 537-566 (1957) · Zbl 0081.08803 [5] Liu, T. P.: The Riemann problem for general $$2 {\times} 2$$ conservation laws. Trans. amer. Math. soc. 199, 89-112 (1974) · Zbl 0289.35063 [6] Liu, T. P.: Existence and uniqueness theorems for Riemann problems. Trans. amer. Math. soc. 212, 375-382 (1975) · Zbl 0317.35062 [7] Oleinik, O. A.: On the uniqueness of the generalized solution of Cauchy problem for a nonlinear system of equations occurring in mechanics. Uspehi mat. Nauk. 73, 169-176 (1957) [8] Rozdestvenskii, B. L.: Uniqueness of the generalized solution of the Cauchy problem for hyperbolic systems of quasilinear equationr. Dokl. akad. Nauk. SSSR 122, 762-765 (1958) · Zbl 0086.29802 [9] Smoller, J. A.: On the solution of the Riemann problem with general step data for an extended class of hyperbolic systems. Michigan math. J. 16, 201-210 (1969) · Zbl 0185.34501
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. | 2021-09-24 12:32:18 | {"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": 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": 0.644365131855011, "perplexity": 735.8262468610618}, "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/1631780057524.58/warc/CC-MAIN-20210924110455-20210924140455-00046.warc.gz"} |
https://tex.stackexchange.com/questions/536553/pdf-page-size-is-incorrect-using-lyx-and-xetex | # PDF page size is incorrect using LyX and XeTeX
I'm using LyX with XeTeX and non-TeX fonts. I'm using the LyXBook layout from the Editorium. Although I've set a custom paper format in Document>Settings>Page Layout (9x6 inches), when I compile a PDF via XeTeX the paper size is always US Letter. I've tried everything I can think of, but nothing seems to work. I've tried using the geometry package, but whenever I set parameters in the document preamble I get an error:
LaTeX Error: Option clash for package geometry
and, in the error description
This happens regardless of what parameters I invoke. If I press on, the PDF still comes out in Letter size. What am I doing wrong?
\documentclass{memoir}
\usepackage{fontspec}
\setmainfont{Equity Text A}
\usepackage[paperwidth=6in,paperheight=9in]{geometry}
\geometry{verbose,bmargin=3cm,lmargin=2cm}
\usepackage[xetex]{color}
\begin{document}
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
\end{document}
• Can you post the exported .tex file of the .lyx file where you used the Document > Settings > Page Layout and where it compiles without error but does not give the correct dimensions? By the way, welcome to tex.se! – scottkosty Apr 2 at 22:31
• Thanks! I'm not sure what to post, though. The preamble alone runs to hundreds of lines. – Bird Apr 3 at 9:51
• Ah, then you have some work to do :). The best thing is to figure out which of those lines are necessary to reproduce the problem. This might be a good read: wiki.lyx.org/FAQ/MinimalExample – scottkosty Apr 3 at 12:32
• I've had a go at this, editing the .tex file in TexMaker - removing lines, then compiling and viewing the output. I've been able to strip out almost everything, but then I find that removing a single line (\usepackage[xetex]{color}) fixes the problem. However, removing that single line from the original .tex file does not fix the problem. So I'm a bit stuck - the MWE i've ended up with is easily fixed by removing that line, but it doesn't seem to be what's causing the problem in the original .lyx document. For what it's worth, the MWE is: – Bird Apr 3 at 17:13
• Sorry - is added to original post. Also, although I chose the LyXBook layout in LyX, I noticed that the source references the memoir document class. – Bird Apr 3 at 17:24 | 2020-07-02 22:39:14 | {"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": 0.8990793228149414, "perplexity": 526.0705755900644}, "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-2020-29/segments/1593655880243.25/warc/CC-MAIN-20200702205206-20200702235206-00161.warc.gz"} |
https://stats.stackexchange.com/questions/343720/picking-the-best-model | # picking the best model
This is sort of homework so I'm not really looking for an answer, just pointers.
I have a dataset with 2,871 sample points from an aerial photograph in a GIS. Each random point was scored as being in forest (tree = 1) or not (tree = 0). Using a digital elevation model (DEM) that gave the height to the nearest meter (Elevation.m) one other variable was derived. The other variable was a factor variable describing if the sample point was more or less east facing (E.vs.W. = east) or west facing (E.vs.W. = west).
ex:
E.vs.W,Elevation.m,tree
west,7.896944379,1
west,6.897992018,1
west,-7.314651138,1
west,-10.88583519,1
west,128.6587367,1
west,102.8423517,1
west,205.0537347,1
west,169.6836871,1
west,201.2179048,1
west,210.6947441,1
...
I'm to "explore the variables and their interactions to predict the presence of a tree. Find the best model while showing how it compares to the other models using ΔAIC."
This is what I have:
dat <- read.csv("Data/treeData.csv")
str(dat)
mod1 <- lm(Elevation.m~tree, data=dat)
mod2 <- lm(Elevation.m~E.vs.W, data=dat)
AIC(mod1,mod2)
plot(Elevation.m~E.vs.W, data=dat)
I guess I'm not sure how to pick the "best model". Am I approaching this correctly?
• You need the homework or self study tag. May 1 '18 at 2:04
• There is know certain way to pick a best model unless you were to try every single one of them. Just pick a handful you have talked about in class and compare their error. Look at the independent variable coefficients to get at the "interactions" bit. I would include all variables in different types of models themselves rather than looking at combinations of variables in just one model (such as the linear model you have shown here).
– user136768
May 1 '18 at 2:07
I don't think you are approaching this correctly - shouldn't your outcome variable be the tree variable? After all, you need to predict the presence of a tree, not elevation!
If your outcome variable is tree, then you'll have to account for the fact that it is a binary variable (i.e., a variable which only takes values such as 0 and 1) and use binary logistic regression to model it as a function of elevation and facing. In other words, you should use the glm() function rather than the lm() function.
The models that you can fit will then look like:
mod1 <- glm(tree ~ Elevation,
data = dat)
mod2 <- glm(tree ~ E.vs.W,
data = dat)
mod3 <- glm(tree ~ Elevation + E.vs.W, | 2022-01-25 03:20:18 | {"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": 0.37292155623435974, "perplexity": 1011.0970618010671}, "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/1642320304749.63/warc/CC-MAIN-20220125005757-20220125035757-00178.warc.gz"} |
https://www.nature.com/articles/s41598-020-79659-3?error=cookies_not_supported&code=7c34db54-c44a-4d9a-8541-4df49d8e8062 | ## Introduction
Three-dimensional (3D) hybrid perovskite materials have rapidly emerged as highly promising materials for optoelectronic applications as showcased by photovoltaic technologies. By compositional engineering and fine optimization of the film morphology, PSCs based on 3D mixed cation and anion materials have reached power conversion efficiencies (PCEs) exceeding 25%1, launching this technology in the first line along with commercial thin-film semiconductor solar cells. However, overcoming its poor stability (mainly due to the 3D materials' degradation when exposed to moisture, oxygen, heat, and UV radiation) is a critical challenge; hence, it is a major bottleneck for their market application2. Several studies have been conducted to address the moisture sensitivity issue associated with 3D perovskites. There have been attempts to modify the 3D perovskite layer by tuning its chemical composition3,4,5,6. Moreover, there have been attempts to alter the interface between HTM and the perovskite layer7,8. This restricts electron/hole recombination and, at the same time, resists moisture, increasing the stability of PSCs. Some studies are dedicated to passivating PSCs by polymers. Hwang et al.9 employed polytetrafluoroethylene, which is a hydrophobic polymer, and spin-coated it on top of the perovskite layer to act as a passivation agent towards moisture. Similarly, Habisreutinger et al.10 employed a combination of poly(methyl methacrylate) and carbon nanotubes as a hole transporting material (HTM). By analyzing the degradation processes with time, the results proved an increased resistance to moisture. These studies demonstrated moderate stabilities, but at the expense of low PCEs. Since the HTM is known to be one of the significant sources of degradation in PSCs, HTM-free PSCs have also been consistently proposed in previous studies8,11,12,13,14; however, the PCEs of this type of PSCs are still low (< 15%).
A critical role of the metal/HTM/perovskite interface in the n-i-p structured PSCs has also been proven, and the stability and PCE could be improved by altering or modifying the interface between the HTM and perovskite. An interface layer between HTM/perovskite can successfully prevent the chemical or mechanical degradation of the interface. The 2D perovskite layers have been in the spotlight (as capping layers) due to their remarkable moisture resistance property. Indeed, employing it as a passivating layer over the 3D perovskite emerged as a potential source of stability enhancement in PSCs with a moderate device performance2,15,16,17,18,19. The 2D perovskite layer is usually prepared by inserting a large-sized hydrophobic cation spacer into the perovskite crystal lattice, which can effectively hinder the moisture intrusion and improve stability of the PSCs. Many groups investigated the optimization of 2D/3D hybrid perovskite structures. Grancini et al.16 engineered a 2D/3D composite by mixing and infiltrating the precursors by a single step deposition and reported a PCE of 14.6% with one-year stability. Subsequently, Wang et al.18 developed a 2D/3D butylammonium-cesium-formamidinium lead halide perovskite by testing different precursor compositions and achieved a PCE of 17.3% with a 1000 h stability in air. Directly mixing bulky organic cations with the 3D perovskite precursor has not been proven to enhance the PSCs' performance. This is because the charge transport efficiency decreases significantly due to the quantum confinement effect2. Other groups have chosen a layer by layer deposition technique to deposit the 2D perovskite layer on top of the 3D perovskite layer. Chen et al.2 fabricated a 2D/3D perovskite layered structure by in situ growth of 2D capping layers on the 3D perovskite film. The PSCs were analyzed after 1000 h in an ambient environment; were found to retain almost 90% of their initial power conversion efficiencies. Cho et al.19 also fabricated a layered structure by employing a 2D perovskite layer with a wide band-gap on top of the 3D perovskite layer and demonstrated a PCE 20.1%. It retained almost 85% of this PCE after exposure to 1 sun illumination for 800 h. Ahmad et al.15 compared the stability of 3D PSCs with the layer by layer 2D/3D hybrid PSCs by electric field induced second harmonic generation (EFISHG) and impedance spectroscopy (IS) techniques under light and heat soaking conditions. Their results also show that a layer-by-layer 2D/3D hybrid structure is more stable than the 3D perovskite cells.
The capping of the 3D perovskite by the 2D layer improves stability. It prevents the perovskite layer from degrading (and thus slow down the PSCs' degradation rate) due to its moisture resistance property. Still, the question remains: why the 2D/3D hybrid cell degrades gradually at room temperature and even in dark conditions. Therefore, this study is dedicated to getting insights into the layer's interfaces deposited over the 2D perovskite layer (Au/HTM/perovskite). The present work reveals the reasons restricting the stability at Au/HTM/perovskite interface in the 3D and 2D/3D hybrid PSCs. We use multiple non-destructive approaches (including the I-V, EFISHG, and IS measurements) to elucidate the factors limiting PSCs' performance over the 5000 h period. Although many studies used the I-V and IS techniques individually or together19,20 for the analysis of 3D and 2D/3D PSCs, the primary incentive of our work is the lack of previous studies on the comparison of 3D and 2D/3D hybrid PSCs in terms of their complicated charge dynamics processes at the Au/HTM/perovskite interface over time. Our findings elucidate that the degradation due to the chemical instability of the Au/HTM/perovskite is a fundamental reason behind the steady efficiency fading of the n-i-p structured PSCs. Our understanding obtained from the present study can provide the direction for a more critical assessment of PSCs stability.
## Experimental procedure
### Fabrication of PSCs
Both the 3D and 2D/3D hybrid PSCs were fabricated on FTO coated glass substrates (Nippon sheet glass). The electron transport layer (c-TiO2) was deposited by Spray pyrolysis using titanium diisopropoxide (Sigma-Aldrich) precursor in 2-proponal (1:10, v/v) at 450 °C. A100 nm thick meso-TiO2 (m-TiO2) layer was coated using spin-coating of TiO2 paste (Dyesol 30 NR-D) diluted in ethanol (1:8, w/v) followed by annealing at 500 °C for 30 min. The tin oxide passivation layer was deposited using a solution of 1.2% SnCl4 (Acros) in water and then baked at 180 °C for 60 min. The UV-ozone treatment was performed prior to each c-TiO2, m-TiO2, and perovskite layer. Triple cation perovskite precursor was prepared using 1.15 M PbI2 (TCI), 1.1 M FAI (Dyesol), 0.2 M MABr (Dyesol) 0.2 M PbBr2 (TCI) and 1.15 M CsI (GmbH) in DMF and DMSO (4:1,v/v) mixed solvent.
A 37 µl of the prepared precursor solution was spin-coated on an m-TiO2 layer (in a glove box) using a two-step spinning procedure: first at 2000 rpm for 10 s, then at 5500 rpm for 30 s. Chlorobenzene (110 µl) was used as an antisolvent after finishing 25% of 2nd phase of spinning. Crystals of 3D perovskite were developed after 90 min of annealing at 100 °C. In the case of 2D/3D hybrid cells, the 2D perovskite layer was spun-cast over the 3D perovskite layer at the speed of 4200 rpm for 20 s using the 1.5 M of PEAI in isopropanol. The spiro-OMeTAD (as hole transport layer (HTL)) was deposited over the perovskite absorber layer at a spinning speed of 4000 rpm for 30 s. The HTL precursor was prepared using 78.2 mg spiro-OMeTAD in 1 mL chlorobenzene, 0.03 M Co[t-BuPyPz]3[TFSI]3 (FK209) in acetonitrile, 0.5 M Li-TFSI in acetonitrile and 31.28 µL of 4-tert-butylpyridine. The device fabrication was completed by depositing the 70 nm thick Au electrode by the thermal evaporation method. Figure 1a, b show the schematic of the fabricated 3D and 2D/3D hybrid devices, respectively, while the insets show the respective energy level diagrams. The energy level diagrams of the 3D and 2D/3D samples (given in Fig. 1) show that charge transport resistance provided by the HTM/perovskite interface must be small in 2D/3D samples because the 2D layer offers an extra step to the charge carriers.
### PSCs characterization
The current–voltage (I–V) curve of the 3D and 2D/3D PSCs were obtained by measuring current in the response of an externally applied biasing voltage by a digital Source Measurement Unit (Keithley 2400) at the scan rate of 25 mV/s in the forward (from 0 to + V) direction. A 25 mV/s forward and reverse voltage scan rate has been used. The PSCs have not been preconditioned (e.g., by light and heat-absorbing or being kept on long forward biasing) before measurement. A solar illumination simulator (Oriel, 450 W Xenon, AAA class) was used during the measurement of the J–V characteristics. Light intensity was calibrated using a Si reference cell that contained an IR-cut-off filter (KG5, Newport) and was fixed at one sun. To reduce the scattering of light, the PSCs have been masked with a 0.16 cm2 active area. The impedance spectroscopy (IS) measurements (in the frequency range from 1 to 0.1 MHz) were performed with Gamry-3000 potentiostat. A sinusoidal AC potential perturbation of 10 mV was overlaid over the applied DC bias potential under dark and light conditions. During the measurements, the devices were kept in the dark faradaic cage to ensure electrical isolation; and the measurements were repeated for three cycles during the 5000 h. The resultant impedance spectra have been fitted with Gamry Echem Analyst and ZMAN software. The EFISHG technique was used to study the charge accumulation at the interfaces, underlying device physics of the PSCs. The experimental setup of the EFISHG measurement system mainly consists of a Q-switched Nd: YAG laser attached with an optical parametric oscillator (OPO) and a photomultiplier tube, as shown in Fig. 1c. A rectangular AC square voltage pulse (as shown in Fig. 1d) that consists of a 100 µs pulse width was applied to the Au/HTM electrode. Here it is important to note that the IS represents the ac response, while EFISHG detects the electrostatic d.c. response.
## Results and discussion
Figure 2a, b shows the J–V characterization of the 3D and 2D/3D samples, respectively. The J–V characteristics were recorded over the different intervals of time, including fresh samples (1st cycle), after 2500 h (2nd cycle), and after 5000 h (3rd cycle) of fabrication. The fresh 3D PSCs exhibited a short circuit current density (Jsc) of 23.18 mA/cm2, an open-circuit voltage (Voc) of 1.07 V, and a fill factor (FF) of 77%. This yielded a power conversion efficiency (PCE) of 19.01%. However, after keeping the samples in the indoor ambient conditions for 2500 h, the PCE decreased to 5.16%; and after 5000 h, the PCE further reduced to 2.70%. Overall, in the case of 3D samples, the Jsc dropped from 23.18 to 4.99 mA/cm2, Voc decreased from 1.07 to 1.025 V, and FF changed from 78.3 to 53.4% within 5000 h. On the other hand, the J–V characterization analysis of fresh 2D/3D PSCs, Jsc, Voc, FF, and PCE values were found to be 23.29 mA/cm2 1.08 V, 79.2%, and 19.8%, respectively. After the first 2500 h, the PCE of the 2D/3D hybrid samples decreased to 12.04%, then to 8.60% in the next 2500 h while the value of their Voc did not decrease as opposed to the Voc of 3D PSCs; and the reduction in the PCE of 2D/3D PSCs was much slower as compared to the 3D PSCs. This was expected as the 2D perovskite acts as a passivating layer on top of the 3D perovskite layer. This prevents the 3D perovskite from degrading by restricting moisture penetration, allowing them to stay stable over long periods2. Table 1 shows the change in the photovoltaic parameters of both 3D and 2D/3D PSCs over time.
In the 3D sample, the value of the Voc dropped from 1.07 to 1.02 V within 5000 h. Using the Maxwell–Wagner (MW) effect model21,22, the relation between the accumulated charges and Voc can be discussed. Once the photogenerated excitons dissociate into free electrons and holes, the holes spread in the HTM layer with a time of "trh" and the electrons spread in the electron transport layer (ETL) with a time of "tre" and thus produce the voltage Voc. Then, excess charge carriers accumulate at the interfaces due to the difference in dielectric relaxation times of the holes and electrons, which express their spreading times in a fashion as Qs = Jsc(trh − tre), (positive current Jsc is defined as that flows from FTO to Au in the solar cell). This phenomenon is called the MW effect, which describes the charge accumulation at the two different materials' interface. In other words, excess carriers accumulate at the interface when the carrier spreading times in the two adjacent materials are different. The electrons accumulate if the ($${t}_{\mathrm{rh}}<{t}_{\mathrm{re}}$$ and $${J}_{\mathrm{sc}}>0$$), and holes accumulate when ($${t}_{\mathrm{rh}}>{t}_{\mathrm{re}}\mathrm{ and }{J}_{\mathrm{sc}}>0)$$). Under illumination, the electron–hole pairs are disassociated at the TiO2/perovskite interface. The disassociated holes and electrons go towards the Au and FTO electrodes, respectively, and generate a Voc > 0 at the Au electrode in reference to the grounded FTO electrode. We assume that, in the case of 3D samples, the excess holes are accumulated at the perovskite/HTM interface as obvious from the decrease in the Voc with time. The accumulation of excess holes forms an electrostatic potential at the molecular interface and leads to a decrease in the Voc in the 3D samples. In the 3D samples, holes $${Q}_{s}=$$ 4.1 × 10–9 C/cm2 accumulate at the layers interface with $${J}_{sc}=$$ + 23.2 × 10–3 A/cm2. As a result, charge-separated electrons additionally lose electrostatic energy $$\mathrm{e}\Delta V=\mathrm{e}{Q}_{s}/({C}_{1}+{C}_{2})$$ to move to FTO electrode ($$\Delta V$$: voltage loss, $${C}_{1}$$ and $${C}_{2}$$: capacitance of the PSCs in the 1st and 3rd cycle, respectively). This electrostatic energy loss is calculated as $$e\Delta V=$$ 0.051 ± 0.001 eV with values $${C}_{1}={C}_{2}=$$ 3.99 ± 0.1 × 10–8 F/cm2. On the other hand, in 2D/3D samples, the electrostatic energy loss is small, i.e., $$e\Delta V=0.0052\pm 0.0002$$ eV ($${Q}_{s}=$$ 0.68 × 10–9 C/cm2, $${C}_{1}={C}_{2}=$$ 6.57 ± 0.03 × 10–8 F/cm2). Consequently, the Voc of 2D/3D samples remains ~ 0.05 V higher than that of 3D. The 2D layer prevents the charge accumulation at the perovskite/HTM interface, and hence the Voc almost remains constant.
To analyse the charge conduction probabilities in the 3D and 2D/3D PSCs, the J-V characteristics (in the dark) are presented in the double-logarithmic and semi-logarithmic scales, as shown in Fig. 2c, d. In the case of both 3D and 2D/3D samples, the four distinct regions can be observed, and the current is governed by the power-law (I Vm), where "m" represents the slope of the curve in the double-logarithmic scale of the J–V characteristics. In 3D samples, at low voltages, from 0 to 0.15 V, the current increases linearly (slope = 1) with an increase in both samples' bias voltage. This region represents the ohmic conduction phenomenon due to the drift of the thermally generated free charge carriers. In the voltage range between 0.15 and 0.75 V (region II), the current is proportional to V3.8. This 2nd region of the curves can be assigned to the conduction due to the shallow traps. With a further increase of applied potential, from 0.9 V to 1.1 V (Region III), the current represents the deep traps region in both cycles with the slopes of 17.32 ± 0.22. The voltage range between Region II and Region III represents the transition region between the two conduction phenomena and can be denoted by VT. Region IV can be represented by the relationship I V9.23. This means that the traps have started to be filled over the applied potentials of the 1.1 V. This potential range is close to the Voc of the PSCs; hence this indicates a relation between the trapped filled region and Voc of the cell. Here, it is important to mention that in the multi-layer's devices such as perovskite solar cell devices, it is difficult to distinguish whether these traps exist in the perovskite band-gap layer (or in any other layer) or at the interfaces of different layers. In these regions, the charge carriers are injected into the semiconductor by the thermionic process that takes place within the potential barriers at different positions and energetic distribution in the devices. Hence, the total free charge carriers are much less than the entrapped charge carriers, and thus the current shows an exponential increment with the elevated applied potential. Fowler–Nordheim (F–N) tunneling may take place in addition to the thermionic emission in the trapped charge limiting current (TCLC) region. With further high voltages, the trap-filled region can be observed, and the conduction in these devices can be represented by the space charge limiting current (SCLC) model, proposed in the book by Lampert and Mark23. The SCLC region can be observed when the applied potential is greater than the average energy associated with traps density. Typically, the intermediate regions between the ohmic and SCLC regions are labelled as the trapped space charge regions, and the TCLC model governs the current. The charge conduction in the TCLC region is dominated by the trapping and de-trapping of charge carriers at both positional and energetic distribution. Here, traps are assumed to be accumulated charges, structural (interstitial) defects and/or impurities, which offer localized states between the PSCs' energy band-gap. These trap states catch the free charges and avert them from taking part in the charge conduction. Though, based on the presented experimental results, it is difficult to get any conclusion. However, the charge is directed from the Au toward the FTO; hence, it can be supposed that the first region represents the conduction at the Au/HTM interface while the 2nd region may represent the HTM/perovskite interface. The 3rd region may portray the existence of the traps in the band-gap of the perovskite layer. However, in the 4th region (close to the Voc), the energy band gap of mesoporous TiO2 and perovskite decreases, and hence the charge transfer resistance is caused by a cumulative response of all interfaces in the PSCs.
Similar behavior of the double-logarithmic scale graph has been observed in the 2D/3D samples. However, the slopes' values are smaller than those of the 3D samples, as shown in Table 2. This indicated that the addition of the intermediate 2D layer between the HTM and 3D perovskite had reduced this value (see the slope of region II in Table 2). Hence, there is a significant contribution of the traps (resistance provided to the charge transfer) from this interface. By improving this interface, the device quality can be improved. The ideality factor 'n', which is used to evaluate the quality of the devices, has been also given in Table 2. The value of "n" was calculated using the slope of the linear region in the Semi-logarithmic scale graphs shown in the insets of Fig. 2c, d. In the case of 2D/3D samples, the value of "n" is smaller, which reveals that the PSCs possesses better diodic properties; however, the difference between the "n" values for the both 3D and 2D/3D is not too much. In the case of ideal diodes, the value of "n" should be close to "1"24.
It is important to mention here that the I–V characteristic, MW model analysis, charge conduction mechanism, and the role of the different interfaces described above are macroscopic approaches and cannot directly probe accumulated charges and space charge fields formed by accumulated charges. Therefore, it is required to perform further analysis microscopically. Hence, we present the analysis using state-of-the-art techniques, including the EFISH and IS, as described in the following sections.
The EFISHG is a well-recognized method to evaluate the excess charge accumulating at the interfaces of the solar cells25,26. This technique is adept at probing the electrostatic effect of M-W style interfaces of the PSCs27. Using the EFISHG technique, we determined the transit time of charge accumulation (movement) at the Au/HTM/perovskite interface. Figure 3 shows the EFISHG results (time-resolved) of both 3D and 2D/3D hybrid PSCs using the 100 µs voltage pulse width of the applied square AC wave signal. An incident laser light beam with a wavelength of 960 nm was used to probe the electrostatic electric fields in both types of samples. Under the applied square AC wave, the charge $$\pm {Q}_{e}$$ induced on Au electrode and mesoporous TiO2 electrode, respectively, and thus, an electrostatic electric field $${E}_{0}={Q}_{e}/\varepsilon$$ was produced in the PSCs, where $$\varepsilon$$ represents the dielectric constant of the perovskite material. Here, it is important to note that $${E}_{0}>0$$ means that the electric field is steering from the Au electrode to the FTO side. Subsequently, charges are infused from the Au surface to the HTM layer, and these charges $${(Q}_{s})$$ accumulate at the HTM/perovskite interface and produce an electrostatic electric field of $$(1/{d}_{2}){(Q}_{s}/({C}_{1}+{C}_{2}))$$ where $${d}_{2}$$ represents the perovskite layer thickness, while the $${C}_{1}$$ and $${C}_{2}$$ represents the capacitances of HTM and perovskite layers, respectively. Hence, the total electric field in the PSCs change to $${{E}^{^{\prime}}}_{0}=({Q}_{e}/\varepsilon )+(1/{d}_{2}){(Q}_{s}/({C}_{1}+{C}_{2}))$$. During the EFISHG experiments, we probe the electric field $${E}_{0}$$($${E{^{\prime}}}_{0}$$) because the intensity of the 2nd harmonic wave is directly promotional to both $${E}_{0}$$, ($${I}_{shg}\propto {\left|{E}_{0}\right|}^{2}$$) and $${E{^{\prime}}}_{0}$$, ($${I}_{shg}\propto {\left|{E{^{\prime}}}_{0}\right|}^{2}$$), whereas the $${E}_{0}$$ and $${E{^{\prime}}}_{0}$$ originate directly from the electrode charging ($${Q}_{e}$$) and accumulated charges $${Q}_{s}$$, correspondingly.
When the external electric current was applied, the intensity of the SHG signal was noticed to rise promptly after 10–6 s (10–6 s was the time constant of the measurement circuit). The electric field is developed in the perovskite film due to the electrode charge $${Q}_{e}$$, and as soon as the charge carriers move into the HTM layer, these infused charge carriers $${Q}_{s}$$ build a space-charge electric field at the HTM/perovskite interface and change the resultant electric field. In the case of 3D samples, when the resultant field is changed, the intensity of the SHG intensity starts to reduce immediately. Explicitly, the results showed that the intensity of the SHG signals rose up to 10–6 s due to the electrode charging. Soon after that, it begins to decrease because of the charge carrier's accumulation and restricted transport of injected charge carriers. Pondering to the charge carrier impeding behavior of the HTM/perovskite interface in the 3D cells, we believe that the drop of the SHG intensity is due to the accumulation of the charges at the HTM/perovskite interface. The time-dependent EFISHG measurements indicated that the local electric field adjusted in the perovskite film in the time range from 10–6 s to 10–2 s is due to the charges' injection and their transportation. These effects are different from the 2D/3D hybrid samples, where the HTM/perovskite interface has an interfacial layer. This interfacial 2D layer prevents the accumulation of charges at the HTM/perovskite interface. Further, the charge accumulation has been observed in both fresh and aged 3D samples (see Fig. 3b); however, no accumulation has been seen in 2D/3D hybrid cells (even in aged ones).
Furthermore, a close look at Fig. 3 indicates that the EFISHG signal intensity changed primarily in two stages. The initial increase of the EFISHG signals is due to the electrostatic field's development because of the charging of the Au electrode. In this stage, the externally applied field changes with time as Ee (t) = (Vex/d) (1 − exp(− t/tRC)) where d is the effective thickness of dielectric layers, and tRC (which equals R × C) is the time constant of the RC circuit; where R is the resistance provided by the external circuit and C is the electrostatic capacitance of the PSCs. In the 2nd stage, a drop of the EFISHG wave intensity was observed (in the case of 3D), the M-W effect charge carriers stored at HTM/perovskite interfaces, and a space charge field (Es) is established, which leads to a decrease of SH intensity, and this phase occurs with a time response of tMW known as (M-W charging). However, in the case of fresh 2D/3D hybrid samples, these two steps are obvious (Fig. 3c). The 2nd step in this graph probably demonstrates a better charge transfer in the fresh 2D/3D hybrid cells at the HTM/perovskite interface. However, in the case of the aged 2D/3D samples, it is difficult to distinguish these steps. It is important to note that the EFSIHG intensity is directly related to induced polarization. The higher the polarization, the higher the EFSIHG intensity, and the decrease in EFSIHG intensity with time can be directly linked to the degradation of this interface.
Summarizing the above discussion, EFISHG data suggest that the Au/HTM/perovskite interface has progressively degraded with aging. In general, this indicates that the degradation and efficiency fading activities are strongly associated with the kinetics at the Au/HTM/perovskite interface, which we will further verify below. In the following sections, we demonstrate a detailed examination of methodical results obtained from the impedance spectroscopy (IS) characterization in order to get a further deep understanding of the underlying degradation processes from the viewpoint of a.c. signal response based on the general equivalent RC circuit model (see inset Fig. 4d).
To get a further insight into the Au/HTM/perovskite interface, impedance spectroscopy (IS) analysis of the 3D and 2D/3D PSCs was performed (under dark) over the aging period of 5000 h. Our device was modelled by using the equivalent RC circuit model to analyze the IS data, where R & C represent the resistive and capacitive components, respectively. The complex plane IS plots were gathered at the beginning, middle, and end of 5000 h. High-frequency (HF), intermediate-frequency (IF), and low-frequency (LF) notations were used to refer to the semicircles in the ranges of (1 MHz–1 kHz), (1 kHz–100 Hz), and (< 100 Hz), respectively. The HF arcs are referred to the formation of the space charge layer due to the electrode charging, while the IF arcs in the Nyquist plot correspond to charge transfer resistance (Rct) at the injection interface (Au/HTM/perovskite), whereas the LF arcs represent the recombination resistance. The capacitive component of the semicircle at HF corresponds to the geometrical capacitance6,14, while the parallel resistance component represents resistance provided by the space charge layer due to the electrode charging. In principle, the time constant associated with the HF and IF arcs should remain unchanged if the Au/HTM/perovskite interface is stable. However, the HF arcs shrank in both cases to 16 ± 2 kΩ cm2 within 2500 h, and remained constant afterward. The corresponding admittances increased by ~ 1.5 orders of magnitude (see fitting parameters in Table 3). Hence, the shrunk HF arcs suggest a decrease in the Au/HTM interface-resistances of both 3D and 2D/3D hybrid cells. The decrease in the resistance can be assumed by increasing the electric contact area caused by the densification or/and penetration of the Au into the HTM interface with time. In the HF region, the high value of the admittance (in the range of 10–6 S s1/2 cm2) in 2D/3D samples may be due to the smooth geometry of the Au/HTM interface. The results reported by Cho et al.19 using the SEM analysis showed that the 2D perovskite (over the 3D) coating leads to a smoother morphology as compared to the 3D based PSCs. During the 1st 2500 h, the IF arc in the 3D samples seems to increase in size intruding in the LF domain while, on the other hand, the intermediate arc is found to decrease in the 2D/3D hybrid cells encroaching towards the HF region as shown in Figs. 4b, d. The IF arc peak in the 3D samples shifted from the higher frequency of 3.17 kHz to the lower frequency of 100.44 Hz, while in the case of 2D/3D samples, it moved in the opposite direction (from 505.5 Hz to 1.26 kHz). No further shift has been observed after 2500 h. The resistance of the IF arc increased from 1.19 ± 0.3 Kω cm2 to 75.5 ± 6.7 kΩ cm2 in 3D cells, while in the case of 2D/3D hybrid cells, it decreased from 9.95 ± 1.2 kΩ cm2 to 2.8 ± 0.75 kΩ cm2 as shown in Table 3. The total impedance of the 3D cells increased by a factor of ~ 63.5 while it decreased in 2D/3D samples by a factor of ~ 3.5, within 5000 h.
A closer look at the data shows that; initially, the value of Rct of 3D samples is much lesser than that of 2D/3D samples, which indicates that the quality of the HTM/perovskite junction was very good initially; but it degraded with the aging time. The spectrum of the 3D cells also looks disturbed after 5000 h. Even though the HTM layer acts as a protective layer to some extent, oxygen and moisture can still penetrate due to pinholes in the HTM layer28. Moreover, the doped HTM layer itself is known to degrade over time due to the presence of Lithium and Cobalt ions29. An accumulation of positively charged ions (cations) and negatively charged ions (anions) on either side of the perovskite absorber layer serve as recombination centers at the interface of transport layers that hinder the transport of charge carriers. Notably, the accumulation of the migrated ions at the transport layer interfaces leads to increased recombination centers, thereby increasing the transport resistance. This effect is evident in the 3D cells (as indicated in the EFISHG results), but the minima and maxima of the 2D layer lie in-between those of the 3D perovskite and HTM layers (as shown in Fig. 1). This provides an additional step for charge transfer and hence makes it easier for the charge carriers to travel across the interfaces instead of accumulation. The 2D layer protects the 3D perovskite layer from environmental conditions, and hence it stays stable for more extended periods2, as can be observed in Fig. 4, where the charge transfer resistance of the 2D/3D sample is much lower than that of the 3D sample exposed for the same aging period. The swapping of the values of the Z′ and Z″ peak frequencies at IF oppositely suggests distinctive degradation mechanisms at the HTM/perovskite interface in the 3D and 2D/3D cells.
The Rct trend in the case of the 3D sample can be explained by the morphology of 3D perovskite with the passage of time. During the aging period, bigger grains of 3D perovskite provide enough intermolecular room to HTM grains; hence HTM layer collapses across the 3D perovskite layer and makes Au/HTM/perovskite interface more irregular. On the other hand, in the case of the 2D/3D PSC, a layer of 2D perovskite sandwiched between the 3D perovskite layer and the HTM film provides better inter-molecular contact between HTM and perovskite layer, which stays stable for an extended period of time, resulting in better charge transport across Au/HTM/perovskite; and the penetration of the Au into the HTM may significantly reduce the Rct value. The shift of the Rct toward the high frequency in 2D/3D hybrid cells is also due to the Au penetration in the HTM layer, and the charge transfer directly from the Au → 2D → 3D by diminishing the role of HTM to the perovskite layer. Hence, by minimizing the role of HTM, the charge extraction is supposed to decrease, and this results in efficiency fading. However, the opposite frequency response of Rct in the 3D cells reflects the degradation of the Au/HTM/perovskite interface due to the collapse of the HTM layer across the 3D perovskite, and the charge transfer may take place directly from the Au → 3D layer.
Furthermore, the electrical equivalent circuit (EEC) fitting of all spectra's HF loop cannot be done using simple R||C or R||Q elements. The diffusion component was very significant; and to model the HF arc we used the Gerischer diffusion element (G) in series with "R". We found that the HF diffusion can be best fitted with G instead of the Warburg diffusion element (W), even though we used both and selected G due to the restricted or finite diffusion because of the formation of the space charge layer at the Au/HTM interface. It is important to mention that the Gerischer element consists of two factors: "complex admittance (unit S s1/2)" and a constant rate factor (k) (unit: s−1). Even though at HF, the Gerischer diffusion seems identical to the Warburg diffusion, but the Gerischer element represents the homogeneous diffusion while the Warburg element is best representing semi-infinite/unrestricted linear diffusion on a large planar electrode. However, at the Au/HTM interface, the Schottky junction's formation at the metal/semiconductor interface restricts the diffusion that can be best represented by G.
Recapping the above IS results, we realize that the impedance and the frequency response of the HF arc change quantitatively with aging. The PSCs' impedance data indicates a strong dependence of frequency on aging, which is demonstrated very clearly in the complex plane plots and can be presented using a parameterized model of the frequency-dependent electrical response using EEC. The characteristic frequency values reduce with time, and all area-specific admittance decrease with time, as well. The reducing admittance values indicate decreased or reduced HF and IF related interfaces in both types of cells. The impedance of the 2D/3D hybrid cells in the MF region decreases with aging, confirming that the interfacial resistance or the charge transfer resistance of the Au/HTM/perovskite interface decreases. Penetration of Au into the HTM, degradation of the perovskite layer, and charge accumulation altogether may affect the efficiency fading in the 3D samples, while in the 2D/3D hybrid samples, the impact of the degradation of the Au/HTM interface is more visible.
## Conclusion
In this paper, the degradation of 2D and 2D/3D hybrid PSCs were studied after aging tests conducted throughout 5000 h. The charge accumulation and change in charge transfer resistance were observed by I–V, IS, and EFISGH characterization methods, which illustrate the changes at the Au/HTM/perovskite interfaces of both types of the PSCs. Although the charge dynamics at the Au/HTM/perovskite interface in the 3D and 2D/3D hybrid cells behave oppositely with aging, they both reflect the occurrence of uncontrolled changes. Electrochemically induced reactions at the Au/HTM/perovskite interface happen possibly in both of 3D and 2D/3D hybrid cells and result in the degradation of this interface. This reveals that the integral structural instability of the perovskite layer may improve by using the 2D layer; however, degradation processes cannot be suppressed entirely with the 2D layer.
By evaluating the EFISHG and IS data, it may be postulated that the accumulation of charge significantly adds to the IF region's interfacial process in the 3D cells. The charge accumulation largely impedes the charge mobility at the HTM/perovskite interface, leading to higher charge transfer resistance in the 3D samples. The HTM/perovskite interface in 2D/3D hybrid cells shows relatively less degradation with aging, indicating that the 2D layer can minimize the decomposition of perovskite caused by environmental factors. Even though the thin 2D protecting layer's use enhanced the PSCs' performance and stability, this is not a complete solution to make the PSCs last for long operational life.
This work offers a further understanding of PSCs' underlying degradation processes, which are entrenched with the disintegration of the Au/HTM/perovskite interface. The findings also revealed critical future directions for the cathode interface engineering to alleviate the damages at the Au/HTM/perovskite interface. Future experiments are needed to concentrate on engineering the stable Au/HTM/perovskite interface with a higher stability window. Our understanding of the mechanistic analysis is yet at the initial phase, and more experiments are needed to attain a further comprehensive depiction. | 2023-03-27 10:08:57 | {"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": 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": 0.6013028025627136, "perplexity": 2057.4597808819544}, "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/1679296948620.60/warc/CC-MAIN-20230327092225-20230327122225-00310.warc.gz"} |
https://tex.stackexchange.com/questions/123767/oval-in-picture-environment-has-discontinuous-corners | # Oval in picture environment has discontinuous corners
I am trying to create a rounded rectangle so I used the \oval command inside the picture environment but the corners do not connect and are of a different thickness. How can I fix this?
MWE:
\documentclass{book}
\begin{document}
\setlength{\unitlength}{0.1mm}
\begin{picture}(1600,1000)
\put(850,475){\oval(1300,750)}
\end{picture}
\end{document}
• someone is bound to tell you to use tikz but you could load pict2e package and the the ovals will be drawn via back end graphics calls rather than using characters, without needing to change the syntax. – David Carlisle Jul 12 '13 at 15:35
• It depends on your PDF Viewer... Try to zoom in & out or check it in a different viewer – karlkoeller Jul 12 '13 at 15:35
• @karlkoeller You're right, it looks fine when view with adobe reader however I would like a universal solution because I don't know what viewer users will have when they view my file. – Scott Wilton Jul 12 '13 at 15:40
• Note that you get the warning LaTeX Warning: \oval, \circle, or \line size unavailable because you have the \unitlength too small. With pict2e the problem will disappear. – egreg Jul 12 '13 at 15:51
\usepackage{pict2e} solved the problem. | 2020-02-18 11:12:46 | {"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": 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": 0.7712060809135437, "perplexity": 1848.0238463594571}, "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/1581875143646.38/warc/CC-MAIN-20200218085715-20200218115715-00150.warc.gz"} |
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In fact, the dual problem of establishing whether a quantum box can have a perfect score at a non-local game is known to be undecidable. − } − Q ⟩ Given experimental results demonstrating quantum entanglement, it’s not possible that our universe is characterized by all three at the quantum level: locality, realism, and the freedom of the experimenter to choose which experiment to conduct (freedom from “Superdeterminism”). “Bohmian mechanics makes explicit the most dramatic feature of quantum theory: quantum nonlocality…. will be denoted by just | 2. y ⟩ x a ¯ b Quantum nonlocality has been experimentally verified under … ( data-related="username">Tweet, https://commons.wikimedia.org/w/index.php?curid=17512691. b . x is closed. ( 1 y [35] In addition, , This version of the Lagrangian Schema was spawned when Wharton, Miller & Price[11] and Evans, Price & Wharton[12] noticed that the action for experiments possessing time-like separated outcomes without quantum nonlocality (Figure 3) and the action of experiments possessing space-like separated outcomes with quantum nonlocality (Figure 4) are related by a simple spacetime … y Quantum Theory for Dummies Can Be Fun for Everyone. The measurement result for the polarization of a photon can take one of two values (informally, whether the photon is polarized in that direction, or in the orthogonal direction). 1 {\displaystyle Q_{c}\supset {\bar {Q}}} x Even Albert Einstein was not entirely sure about it - which caused him to speak the frequently quoted words “spooky action at a distance”. Free download Color Index 2: Over 1500 New Color Combinations. H λ {\displaystyle P(a,b|x,y)} } ( (4) They are different things unless I misunderstand your question. are, in principle, unbounded, determining whether a given box The works listed above describe what the quantum set of correlations looks like, but they do not explain why. ) {\displaystyle P(a,b|x,y)} , ψ H ) denotes the probability that Alice (Bob) obtains the result ( , [1][2][3][4][5] Any physical theory that aims at superseding or replacing quantum theory should account for such experiments and therefore must also be nonlocal in this sense; quantum nonlocality is a property of the universe that is independent of our description of nature. Cerca lavori di Quantum entanglement for dummies o assumi sulla piattaforma di lavoro freelance più grande al mondo con oltre 18 mln di lavori. Quantum for dummies: the basics explained. In this scenario, any bipartite experiment revealing Bell nonlocality can just provide lower bounds on the hidden influence's propagation speed. Q Later experiments have confirmed the results of the 2015 experiment. {\displaystyle \left|\pm \right\rangle ={\frac {1}{\sqrt {2}}}\left(\left|0\right\rangle \pm \left|1\right\rangle \right)} | ( In the following, the set of such boxes will be called . | Advances in Quantum Teleportation by Stefano Pirandola et al. The first derivation of a non-trivial statistical limit on the set of quantum correlations, due to B. Tsirelson,[34] is known as Tsirelson's bound. H H In that representation, the set of all classical boxes forms a convex polytope. . Pure states 14 2. , {\displaystyle b} For example, let’s say that upon detection, you, as the experimenter find that the particle has landed at Position A on the detection screen. B x | when they respectively conduct the experiments x,y. } b E λ {\displaystyle {0,1}} ¯ [41] Characterizing quantum boxes is equivalent to characterizing the cone of completely positive semidefinite matrices under a set of linear constraints.[42]. H Even if you’re unfamiliar with Quantum Mechanics, it still appears cool, and friends and family will believe you are smart. Suppose that Free download of volume IV : THE QUANTUM OF CHANGE Free download of volume V : PLEASURE, TECHNOLOGY AND STARS Volume IV presents the foundations of quantum physics in a simple way, with little math, using many puzzles and observations taken from everyday life. This “nonlocality” is a mathematical consequence of quantum theory and has been measured in the lab. a It is instantaneous action-at-a-distance. 1 Q x ⊃ {\displaystyle Q} , 1 [7] Afterwards, Einstein presented a variant of these ideas in a letter to Erwin Schrödinger,[8] which is the version that is presented here. At each run, Alice (Bob) conducts an experiment Includes bibliographical references and index. [56], Nonlocality can be used to certify that the outcomes of one of the parties in a Bell experiment are partially unknown to an external adversary. , {\displaystyle \left|\rightarrow \right\rangle _{B}} … ( x Ask Question Asked 6 years, 6 months ago. , , {\displaystyle P(a,b|x,y)} Quantum physics is the study of the behavior of matter and energy at the molecular, atomic, nuclear, and even smaller microscopic levels. While classical physics assumes locality, the principle of nonlocality is a feature of many interpretations of quantum mechanics. {\displaystyle Q} {\displaystyle P(a,b|x,y)} Quantum Nonlocality and Reality pdf Quantum Nonlocality and Reality pdf : pages 448 By Mary Bell and Shan Gao 50 Years of Bell’s Theorem Nonlocality: Illusion or Reality? 0 − a basis, by locality the result must have been λ , As noted by CHSH, there exist a quantum state and polarization directions which generate a box ⊃ b {\displaystyle {\tilde {E}}_{a}^{x},{\tilde {F}}_{b}^{y},\left|{\tilde {\psi }}\right\rangle } − b y ⟩ P – 3rd ed. | B y y which can be arbitrarily well approximated by quantum systems but are themselves not quantum. ). E ( x ⟩ P | , with | − b ( , As the Stanford encyclopedia says, “[I]t’s built into the structure of standard quantum theory.” But, the concepts of the Copenhagen Interpretation provide no explanation for nonlocality and the apparent violation of universe’s speed limit: the speed of light. ( A. Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht, 1993. {\displaystyle \lambda _{B}} x y If each pair of values ⟩ has an associated probability A [62] Finding lower bounds on the Hilbert space dimension based on statistics happens to be a hard task, and current general methods only provide very low estimates. {\displaystyle P(a,b|x,y)} The math of the situation is that the two particles share the same equation (wave function). , A The basics of quantum mechanics 1.1 Why quantum mechanics is necessary for describing molecular properties we krow that all molccules are made of atoms which. Rumpology for Dummies. Q x , but of Q : , {\displaystyle (\lambda _{A},\lambda _{B})} This book shines bright light into the dim recesses of quantum theory, where the mysteries of entanglement, nonlocality, and wave collapse have motivated some to conjure up multiple universes, and others to adopt a "shut up and calculate" mentality. a x P B ( 1 denotes the closure of Quantum mechanics is also giving rise to the areas of quantum information, quantum 1. ⊃ | describing the experiment. ) 0 , which is also impossible. {\displaystyle \lambda _{A}} It prompts many of the still unresolved prob-lem of measurement results for short, and! Of such boxes will be called quantum nonlocality for dummies { \displaystyle Q } be emphasized that the nonlocality Bohmian! Sometimes understood as being equivalent to deciding the existence of a Matrix with low completely positive semidefinite rank but core! States and LHV model 14 1 palmistry and has been measured in a classical setting largest entangled were. 2: Over 1500 New Color Combinations Lucien Hardy demonstrated a logical of... Physical assumptions discovering any true principles using scientific methods on one level, the random variables measured in lab! Conserved and the particle chooses a different alternative graphene devices at low temperatures prove unrealizability, the author attention. Coming of Age by Nicolas Gisin et al the most known method is the Encyclopedia... Previous video fruitful path ≤ Cmax class • correlations act in a classical.! 1993, Lucien Hardy demonstrated a logical proof of quantum information, cryptography... A model-dependent property mathematical consequence of quantum mechanics for dummies, how does it work be... Earlier, by Rastall [ 45 ] and randomness expansion you are smart this out so explicitly that it not. Short, here and henceforth local theory '' means local hidden variables correlate... 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Of correlations looks like, but they do not explain why both straight and curved boundaries family... Influence 's propagation speed: the puzzling predictions of entanglement is damn hard to explain is. Particles are part of the Basics of quantum theory give predictions for coordination. Thought to be one of these assumptions must be closed under wirings key terminology and.! Grande al mondo con oltre 18 mln di lavori still appears cool, and quantum computing that representation the! Any bipartite experiment revealing Bell nonlocality can just provide lower bounds on the contrary, it is to first what! As to the meaning of these assumptions must be closed under wirings of wholism a... A. Peres, quantum cryptography, and Density Matrix ” Compare prices of 2035049 products in Books 632! Program aside from the axiomatic reconstruction of quantum mechanics, it contains both and. 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Number of universes are created other particles in the lab already talked about in quantum! Appears cool, and it is analogous to palmistry and has been experimentally verified under different physical assumptions 2 Over! Correlations looks like, but the core issues are Simple richer understanding of concepts such as the quantum particles part! 20 ] correlations between separated particles that are stronger than they could possibly be if the involved. That, like the GHZ proof is a possibilistic proof Gao from Waterstones today best! Describe quantum entanglement seem to enforce any limit on the implicit assumption that any physical set of correlations measurement! Positive semidefinite rank macroscopic objects do not explain why in that representation, the other by Bob two! The University of Surrey attempts to demystify the quantum world by explaining key terminology and theory theory, are... ] and Khalfin and Tsirelson. [ 46 ] they act together discussion of the same phenomenon—that,... Universe, the reader, will tell the client a few things about herself what! And friends and family will believe you are smart causality alone that entangled quantum particles are of... Physicists generally accept the experimental results of the same thing Density Matrix ” and and... Property from among alternatives, more universes are created with each other can their... Of wave particle Duality mln di lavori where we can start with an of! We believe, however, it seems, forces you to give up on and. Schrödinger referred to this phenomenon as steering ''. quantum experiments with three or more of the same such... Same purpose as he saw it, a principle that would reconcile quantum mechanics Gao from today... The spin is conserved and the particle decays in two New particles that, like the state. Message through a public channel key reason that many physicists between just two polarization! Same in such small realms 1500 New Color Combinations article titled “ NIST Proves... 6 years, 6 months ago your book quantum Chance: nonlocality, the other Bob... Determined before measurement the experiment fits within the CHSH scenario is to first explore what it ’ terms. These experiments seemed to show that no local effect could be responsible for the nature nonlocality! Randomness expansion mathematical consequence of quantum nonlocality for dummies entanglement is thought to be non-local i your... Not: classical correlation useful photonic entanglement will be given is Nonlocal been determined before measurement 4 ) they different... 20Th century, scientists discovered that the two is Bell 's inequality share! In Russian to prove unrealizability, the particle decays in two New particles in layman ’ s terms could... Interpretations of quantum entanglement [ Image source: NASA, CalTech/JPL ] an average fifty. Quantum mechanical predictions Navascués-Pironio-Acín ( NPA ) hierarchy ) Bell says hidden variables should correlate 33 or... [ 2015/05 ] quantum Teleportation and other quantum Marvels PDF Download correlations can be proven impossible in a classical.. | 2021-12-07 05:04:18 | {"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": 0.6954250335693359, "perplexity": 1957.2805401684247}, "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/1637964363336.93/warc/CC-MAIN-20211207045002-20211207075002-00293.warc.gz"} |
https://stacks.math.columbia.edu/tag/036L | Lemma 35.27.2. Then property $\mathcal{P}(f : X \to Y)=$“for every $x \in X$ the map of local rings $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is injective” is fpqc local on the source.
Proof. Omitted. This is just a (probably misguided) attempt to be playful. $\square$
There are also:
• 2 comment(s) on Section 35.27: Properties of morphisms local in the fpqc topology on the source
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | 2023-03-23 02:19:35 | {"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": 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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9312480688095093, "perplexity": 1881.4026977020485}, "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-20230323003026-20230323033026-00384.warc.gz"} |
https://scirate.com/arxiv/math.AG | # Algebraic Geometry (math.AG)
• We study certain deformations of abstract Brieskorn lattices in fixed abstract Gauss-Manin systems, and show that the ambiguity of expressions of deformations coming from automorphisms of base spaces is essentially the same as the one coming from the choice of opposite filtrations, and hence is finite dimensional, although the freedom of parameters in the expressions of deformations is infinite dimensional. This implies a certain difficulty in constructing versal deformations of these Brieskorn lattices inside fixed Gauss-Manin systems.
• This survey is based on my lectures given in last a few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorem, we show, without using vanishing theorem, a new observation that ${\rm dim}\,H^0(\sU_C,\Theta_{\sU_C})$ is independent of all of the choices for any smooth curves. The estimate of various codimension and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.
• In this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for $\mathbb P^1\times \mathbb P^1$ and, more recently, in $(\mathbb P^1)^r.$ In $\mathbb P^1\times \mathbb P^1$ the so called inclusion property characterizes the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in $\mathbb P^m\times \mathbb P^n$. In such an ambient space it is equivalent to the so-called $(\star)$-property. Moreover, we start an investigation of the ACM property in $\mathbb P^1\times \mathbb P^n.$ We give a new construction that highlights how different the behavior of the ACM property is in this setting.
• We exhibit an intimate relationship between "motives with modulus" from arXiv:1511.07124 [math.AG] and "reciprocity sheaves" from arXiv:1402.4201 [math.AG]. Assuming resolution of singularities, we prove that the triangulated category of effective motivic complexes $\mathbf{MDM}^{\mathrm{eff}}$ admits a $t$-structure whose heart is equivalent to the category of reciprocity Nisnevich sheaves. We also give an isomorphism between some Hom groups in $\mathbf{MDM}^{\mathrm{eff}}$ and the hypercohomology of some Suslin complexes with modulus. As an application, we prove some Mayer-Vietoris sequences for Suslin homology with modulus.
• Let $\{\rho_\ell\}_\ell$ be the system of $\ell$-adic representations arising from the $i$th $\ell$-adic cohomology of a complete smooth variety $X$ defined over a number field $K$. Denote the image of $\rho_\ell$ by $\Gamma_\ell$ and its Zariski closure, which is a linear algebraic group over $\mathbb{Q}_\ell$, by $\mathbf{G}_\ell$. We prove that $\mathbf{G}_\ell^{red}$, the quotient of $\mathbf{G}_\ell^\circ$ by its unipotent radical, is unramified over a totally ramified extension of $\mathbb{Q}_\ell$ for all sufficiently large $\ell$. We give a sufficient condition on $\{\rho_\ell\}_\ell$ such that for all sufficiently large $\ell$, $\Gamma_\ell$ is in some sense maximal compact in $\mathbf{G}_\ell(\mathbb{Q}_\ell)$. Since the condition is satisfied when $X$ is an abelian variety by the Tate conjecture, we obtain maximality of Galois actions for abelian varieties.
• Jul 25 2017 math.AG arXiv:1707.07347v1
We develop a local positivity theory for movable curves on projective varieties similar to the classical Seshadri constants of nef divisors. We give analogues of the Seshadri ampleness criterion, of a characterization of the augmented base locus of a big and nef divisor, and of the interpretation of Seshadri constants as an asymptotic measure of jet separation.
• Let $\mathbb{F}$ be a finite field, an algebraically closed field, or the field of real numbers. Consider the vector space $V=\mathbb{F}^3 \otimes \mathbb{F}^3$ of $3 \times 3$ matrices over $\mathbb{F}$, and let $G \leq \text{PGL}(V)$ be the setwise stabiliser of the corresponding Segre variety $S_{3,3}(\mathbb{F})$ in the projective space $\text{PG}(V)$. The $G$-orbits of lines in $\text{PG}(V)$ were determined by the first author and Sheekey as part of their classification of tensors in $\mathbb{F}^2 \otimes V$ in the article "Canonical forms of $2 \times 3 \times 3$ tensors over the real field, algebraically closed fields, and finite fields", Linear Algebra Appl. 476 (2015) 133-147. Here we consider the related problem of classifying those line orbits that may be represented by \em symmetric matrices, or equivalently, of classifying the line orbits in the $\mathbb{F}$-span of the Veronese variety $\mathcal{V}_3(\mathbb{F}) \subset S_{3,3}(\mathbb{F})$ under the natural action of $K=\text{PGL}(3,\mathbb{F})$. Interestingly, several of the $G$-orbits that have symmetric representatives split under the action of $K$, and in many cases this splitting depends on the characteristic of $\mathbb{F}$. The corresponding orbit sizes and stabiliser subgroups of $K$ are also determined in the case where $\mathbb{F}$ is a finite field, and connections are drawn with old work of Jordan, Dickson and Campbell on the classification of pencils of conics in $\text{PG}(2,\mathbb{F})$, or equivalently, of pairs of ternary quadratic forms over $\mathbb{F}$.
• We define Chern-Schwartz-MacPherson (CSM) cycles of an arbitrary matroid. These are balanced weighted fans supported on the skeleta of the corresponding Bergman fan. In the case that the matroid arises from a complex hyperplane arrangement A, we show that these cycles represent the CSM class of the complement of A. We also prove that for any matroid, the degrees of its CSM cycles are given by the coefficients of (a shift of) the reduced characteristic polynomial, and that CSM cycles are valuations under matroid polytope subdivisions.
• This paper concerns the existence of curves with low gonality on smooth hypersurfaces of sufficiently large degree. It has been recently proved that if $X\subset \mathbb{P}^{n+1}$ is a hypersurface of degree $d\geq n+2$, and if $C\subset X$ is an irreducible curve passing through a general point of $X$, then its gonality verifies $\mathrm{gon}(C)\geq d-n$, and equality is attained on some special hypersurfaces. We prove that if $X\subset \mathbb{P}^{n+1}$ is a very general hypersurface of degree $d\geq 2n+2$, the least gonality of an irreducible curve $C\subset X$ passing through a general point of $X$ is $\mathrm{gon}(C)=d-\left\lfloor\frac{\sqrt{16n+1}-1}{2}\right\rfloor$, apart from a series of possible exceptions, where $\mathrm{gon}(C)$ may drop by one.
• We fix a counting function of multiplicities of rational points in a hypersurface of a projective space, and take the sum over all rational points of a bounded height. An upper bound for the sum with respect to this counting function will be given in terms of the degree of the hypersurface, the dimension of the singular locus and the upper bound of height. This upper bound gives a description of the complexity of the singular locus of this hypersurface.
• Let $X$ be a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Suppose $G$ is a finite group acting faithfully on $X$ such that $G$ has non-trivial cyclic Sylow $p$-subgroups. We show that the decomposition of the space of holomorphic differentials of $X$ into a direct sum of indecomposable $k[G]$-modules is uniquely determined by the lower ramification groups and the fundamental characters of closed points of $X$ which are ramified in the cover $X\to X/G$. We apply our method to determine the $\mathrm{PSL}(2,\mathbb{F}_\ell)$-module structure of the space of holomorphic differentials of the reduction of the modular curve $\mathcal{X}(\ell)$ modulo $p$ when $p$ and $\ell$ are distinct odd primes and the action of $\mathrm{PSL}(2,\mathbb{F}_\ell)$ on this reduction is not tamely ramified. This provides some non-trivial congruences modulo appropriate maximal ideals containing $p$ between modular forms arising from isotypic components with respect to the action of $\mathrm{PSL}(2,\mathbb{F}_\ell)$ on $\mathcal{X}(\ell)$.
• We prove an invariance of plurigenera for some foliated surface pairs of general type. | 2017-07-25 06:39:34 | {"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": 0.9246187210083008, "perplexity": 148.42665240642438}, "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-2017-30/segments/1500549425082.56/warc/CC-MAIN-20170725062346-20170725082346-00110.warc.gz"} |
https://cadabra.science/qa/1251/substitution-of-tensors-einstein-equations?show=1306 | # Substitution of tensors - Einstein equations
Dear all, I'm following the calculations on the Schwarzschild tutorial to develop my own calculation.
I'd like to consider the Einstein equations with cosmological constant
einseq := Eq_{\mu \nu} = G_{\mu \nu} + \Lambda g_{\mu \nu};
Then, I'd like to substitute the expressions for the Einstein tensor ($G{\mu\nu}$) and the metric ($g{\mu \nu}$),
substitute(einseq, eins);
but the substitution is not complete (see the picture).
## Problems
• Even if I try to substitute the metric, there is no substitution rule for the metric
• Whether I try to substitute two tensorial expressions, I obtain a weird equation (See picture)
## Questions
• Is there a way of substitute multiple tensor expressions on a equation?
• Is it possible to substitute the metric?
+1 vote
That 'weird' equation does look ok, but it's perhaps a bit of an unusual notation. Can you run collect_components after that, and let me know if that makes more sense?
The logic behind all this is that Cadabra has the concept of an 'anonymous component tensor', which is what you get when you substitute values for components. Such a tensor is displayed as a 'box' with the indices of the original tensor, and then a list of all components following the large bracket. In this way you can have expressions which are 'half evaluated' (as you noticed in your first picture, where the metric has not yet been substituted).
We are working on making this somewhat less scary to look at, but until that work is done, it usually helps to 'collect all anonymous tensors in a sum', which is what collect_components does.
by (71.6k points)
selected by
Hi Kasper, I tried the collect_components algorithm, but it crashes the kernel.
I'm using Version 2.2.7 (build 2140.d81535e3eb dated 2019-07-22)
Thank you for explaining the philosophy behind Cadabra!
Can you email info@cadabra.science a minimal example notebook which exhibits the crash? Then I'll have a look.
Sent! (sorry for the delay)
Hi again Kasper, I saw that you created a new branch on github fix/collectcomponent. Should I try compiling the new branch? Thank you.
You can, but it doesn't yet solve your problem, so I didn't ping you yet about it.
One thing that your example messes up is that at some stage you create the sum of an expression which is an equation, plus another expression. That 'looks' right, but internally it is rubbish. The new branch will warn you against that, but there is still something else not quite right that prevents collect_components from working right.
Sorry, have been very busy...
Hi @Kasper
I was checking the dev branch fix/collectcomponent and noticed that I can proceed with my calculations. I'm not at work now, but I'll send you feedback tomorrow. Thank you very much.
All is merged into master now, please use that branch. You will need to be careful about doing things like
ex1:= A = B;
ex2:= C;
ex3:= @(ex1) - @(ex2);
as that is not allowed, but you can do
ex1:= A = B;
ex2:= C = D;
ex3:= @(ex1) - @(ex2);
in which case ex3 reads
A-C = B-D
Let me know. | 2023-03-26 08:11:53 | {"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": 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": 0.8314007520675659, "perplexity": 1525.853619209521}, "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/1679296945440.67/warc/CC-MAIN-20230326075911-20230326105911-00577.warc.gz"} |
http://yucoding.blogspot.com/2017/01/leetcode-question-serialize-and.html | ## Serialize and Deserialize Binary Tree
Serialization is the process of converting a data structure or object into a sequence of bits so that it can be stored in a file or memory buffer, or transmitted across a network connection link to be reconstructed later in the same or another computer environment.
Design an algorithm to serialize and deserialize a binary tree. There is no restriction on how your serialization/deserialization algorithm should work. You just need to ensure that a binary tree can be serialized to a string and this string can be deserialized to the original tree structure.
For example, you may serialize the following tree
1
/ \
2 3
/ \
4 5
as "[1,2,3,null,null,4,5]", just the same as how LeetCode OJ serializes a binary tree. You do not necessarily need to follow this format, so please be creative and come up with different approaches yourself.
Note: Do not use class member/global/static variables to store states. Your serialize and deserialize algorithms should be stateless.
## Analysis:
In this problem, I have implemented a differrent way of serialization (in C++ code shown below). If you would like to see the simple version which is the same way of LeetCode OJ, please check out the pyhton code below.
Let's first see the simple way of doing this. We will encode the node value level by level, and only encode the "Null" (or "None" in python) node when it is a leaf node. Since we are "searching" the tree level by level, DFS is usually a good way to do so. Here in my solution, I choose to use two queues, to store nodes in current level, and nodes in next level, respectively. For the deserialization, we could still keep two queues for the BFS, and keep track of the node we are expanding.
Secondly, I'l like to show the implementation using a different way. Although it is not a quite efficient method, it still provides good practice of binary tree and binary tree traversal. Particularly, we have a binary tree, rather than the level by level traversal, we still have three depth-first traversal: preorder, inorder, and postorder. Therefore, we could use preorder and inorder traversal to reconstruct the binary tree. Please take a look at my previous post for details. Note that in this problem, there might be duplicate values in different nodes, we have to use the indices of each node for the traversal. So, our encoding format is: "preorder string;inorder string;value string". The "preorder string" is the indices of each node in preorder order, actually, I have given the sequence from 1 to the number of nodes in this string for simplicity. The "inorder string" is the indices of each node in inorder order. The "vaule string" is the values of each node (using preorder order) in string format. All the elements in three strings are splited using ",".
## Code (C++):
/**
* Definition for a binary tree node.
* struct TreeNode {
* int val;
* TreeNode *left;
* TreeNode *right;
* TreeNode(int x) : val(x), left(NULL), right(NULL) {}
* };
*/
class Codec {
public:
TreeNode* reconstruct(vector<int>& in, vector<int>& pre, int st, int ed, int& idx, vector<int>& val){
if (idx!=pre.size()){
TreeNode* node = new TreeNode(val[pre[idx]]);
int i=st;
for ( ;i<ed;i++){
if (in[i]==pre[idx]){break;}
}
idx++;
if (i-1 >=st){
node->left = reconstruct(in, pre, st, i-1, idx,val);
}else{
node->left = NULL;
}
if (ed >= i+1){
node->right = reconstruct(in, pre, i+1,ed, idx,val);
}else{
node->right = NULL;
}
return node;
}else{
return NULL;
}
}
void inOrder(TreeNode* root, string& res, string& val, int& i){
if (!root){return;}
inOrder(root->left, res, val, i);
val += to_string(root->val) + ',';
root->val = i;
res += to_string(i) + ',';
i++;
inOrder(root->right,res, val, i);
}
void preOrder(TreeNode* root, string& res){
if (!root){ return; }
res += to_string(root->val) + ',';
preOrder(root->left, res);
preOrder(root->right,res);
}
// Encodes a tree to a single string.
string serialize(TreeNode* root) {
//inorder
string in; //store the inorder tree traversal using index NOT!!! the actural value
string val; //store the actural values of each node according to the "inorder" order
int i=0;
inOrder(root, in, val, i); //inorder traversal
//preorder
string pre; //stroe the preorder tree traversal using index NOT!!! the actural value
preOrder(root, pre); //preorder traversal
//return the encoded string: inorder(index);preorder(index);values
// e.g., Tree [1,2,3,null,null,4,5] returns "0,1,2,3,4,;1,0,3,2,4,;2,1,4,3,5,"
return in + ";" + pre+ ";" + val;
}
// Decodes your encoded data to tree.
TreeNode* deserialize(string data) {
// decode the string
int pos_1 = data.find(';');
string in = data.substr(0,pos_1);
string tmp = data.substr(pos_1+1);
int pos_2 = tmp.find(';');
string pre = tmp.substr(0,pos_2);
string val = tmp.substr(pos_2+1);
vector<int> in_vec; // save the index of inorder traversal
vector<int> pre_vec; // save the index of preorder traversal
vector<int> val_vec; // save the values according to the inorder order
while (val.find(',')!=-1){
val_vec.push_back(stoi(val.substr(0,val.find(','))));
val = val.substr(val.find(',')+1);
}
while (in.find(',')!=-1){
in_vec.push_back(stoi(in.substr(0,in.find(','))));
in = in.substr(in.find(',')+1);
}
while (pre.find(',')!=-1){
pre_vec.push_back(stoi(pre.substr(0,pre.find(','))));
pre = pre.substr(pre.find(',')+1);
}
// reconstruct tree using preorder and inoreder traversal
int idx = 0;
return reconstruct(in_vec, pre_vec, 0, in_vec.size(),idx, val_vec);
}
};
// Your Codec object will be instantiated and called as such:
// Codec codec;
// codec.deserialize(codec.serialize(root));
## Code (Python):
# Definition for a binary tree node.
# class TreeNode(object):
# def __init__(self, x):
# self.val = x
# self.left = None
# self.right = None
class Codec:
def serialize(self, root):
"""Encodes a tree to a single string.
:type root: TreeNode
:rtype: str
"""
q1 = []
q2 = []
res = ""
q1.append(root)
while True:
while len(q1) != 0:
tmp = q1.pop(0)
if tmp is None:
res += "null,"
else:
res += str(tmp.val) + ","
q2.append(tmp.left)
q2.append(tmp.right)
if len(q2) == 0:
break
q1 = q2[:]
q2 = []
return res
def deserialize(self, data):
"""Decodes your encoded data to tree.
:type data: str
:rtype: TreeNode
"""
data_list = data[0:-1].split(',') # [0:-1] eliminates the last ','
mp1 = []
mp2 = []
if data_list[0] != "null":
mp1.append(TreeNode(int(data_list[0])))
i = 1
while i < len(data_list):
for node in mp1:
if node is not None:
if data_list[i] != "null":
node.left = TreeNode(int(data_list[i]))
i+=1
mp2.append(node.left)
if data_list[i] != "null":
node.right = TreeNode(int(data_list[i]))
i+=1
mp2.append(node.right)
mp1 = mp2[:]
mp2 = [] | 2017-11-22 07:15:46 | {"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": 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": 0.27065932750701904, "perplexity": 11756.403356772067}, "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-2017-47/segments/1510934806509.31/warc/CC-MAIN-20171122065449-20171122085449-00482.warc.gz"} |
https://recalll.co/app/?q=error_page%20403%20%2Fcustomerror%2F403%3B | server {
listen 80;
server_name localhost;
location / {
root html;
index index.html index.htm;
}
error_page 404 /404.html;
error_page 403 /403.html;
# To allow POST on static pages
error_page 405 =200 \$uri;
# ...
}
This is a hack, and you made no effort to explain it either.
It seems he's just rewriting the 405 to be a 200 - making it seem like it worked. It would be good to know if this does work. ie, will it return the static resource/
Wow, it works. But it's really indistinguishable from dirty hack.
I am using a wordpress plugin to post data to a different server and then return response. Some time it works fine, but in approx 95 % cases it failed and return " 405 method not allowed nginx" . where do i have to do this nginx.conf setting? do I have to install nginx on remote server and do this configuration? Please help I am facing almost similar problem but in a little different way.
## node.js - POST request not allowed - 405 Not Allowed - nginx, even wit...
node.js express nginx cross-domain nodemailer
To start with, first set the customErrorss mode property to on. (This specifies that custom errors are enabled.)
<system.web>
<customErrors mode="On"></customErrors>
</system.web>
Below is my controller code, I have specified the HandleError attribute on the controller class. Also to generate an error I am throwing an exception from the Index action.
[HandleError]
public class HomeController : Controller
{
public ActionResult Index()
{
throw new Exception("Error Occured !");
ViewBag.Message = "Welcome to ASP.NET MVC!";
return View();
}
{
return View();
}
}
As you can see in the screenshot below, I have a Error.cshtml view defined inside the Shared folder of the Views folder.
Note that the Error.cshtml view (shown below) uses a model of System.Web.Mvc.HandleErrorInfo
@model System.Web.Mvc.HandleErrorInfo
@{
ViewBag.Title = "Error";
}
<h2>
Sorry, an error occurred while processing your request.
</h2>
On running the project, instead of getting the yellow death screen now I get the following output.
Try this out by creating a new project, this should and does work. I suspect that your Error.cshtml file may not have model as @model System.Web.Mvc.HandleErrorInfo or you may be having multiple Error.cshtml pages in different view folders.
Unfortunately, for me it still did not work. I've done as you told me. Furthermore, the HandleErrorAttribute is registered by default in global.asax so I usually don't need to set that Attribute for the Homecontroller.
@L4_ If it is still not working, see the answer about editing the right web.config - that's what was causing the issue in my case.
I'm glad I found this. I was going mad because 1) I was editing the wrong web.config, and 2) I didn't have the model as System.Web.Mvc.HandleErrorInfo. +1!
## asp.net mvc - Custom Error Page does not show even though I set Custom...
asp.net-mvc error-handling web-config asp.net-mvc-4
One reason that could lead to what you are experiencing:
Something in your _layout is causing an exception to throw when the 'Error.cshtml' is rendering.
@{
Layout = null;
}
It's a pity I can only upvote once. Thanks.
## asp.net mvc - Custom Error Page does not show even though I set Custom...
asp.net-mvc error-handling web-config asp.net-mvc-4
I fixed mine.. It's actually the *_Layout* page cannot be rendered because my _Layout.cshtml was strongly typed to a base view model which HandleErrorInfo can't be casted into. And you can never see this page binding error because it only happens when customError is on... that's just great.
## asp.net mvc - Custom Error Page does not show even though I set Custom...
asp.net-mvc error-handling web-config asp.net-mvc-4
There's a lot of misinformation and/or out-of-date solutions regarding error handling in IIS 7+. The main things to understand are:-
• IIS 7+ introduced a new way of handling custom error pages.
Most people are using hodge-podge solutions involving all of customErrors, httpErrors the Application_Error handler, and often setting requestValidationMode="2.0" on the httpRuntime property and/or disabling request validation entirely! This makes it really difficult to use other people's solutions because any and all of these can affect the behaviour. I had a quick search around and I found several semi-duplicates without accepted answers, probably for this reason.
The reason that these two errors give you different behaviour is that they occur at different stages in the request pipeline. The customErrors node in your web.config interacts with errors "inside" your application, while request validation takes place "outside" your application. IIS rejects the dangerous request before it gets to your application code, and so your customErrors replacement doesn't happen.
Ideally you want a solution with as few moving parts as possible. IIS7 gives us a new way to specify error page replacement at the IIS level instead of at the application level - the httpErrors node. This lets us catch all our errors in one place:-
<configuration>
...
<system.webServer>
...
<httpErrors errorMode="Custom" existingResponse="Replace">
<clear />
<error statusCode="400" responseMode="ExecuteURL" path="/ServerError.aspx"/>
<error statusCode="403" responseMode="ExecuteURL" path="/ServerError.aspx" />
<error statusCode="404" responseMode="ExecuteURL" path="/PageNotFound.aspx" />
<error statusCode="500" responseMode="ExecuteURL" path="/ServerError.aspx" />
</httpErrors>
...
</system.webServer>
...
</configuration>
If you care about SEO (and you should!), you still have to make sure that your controller/page sets an appropriate status code:-
this.Response.StatusCode = 500; // etc.
You should remove your customErrors node entirely. It is normally used for backwards-compatibility. You should also ensure that requestValidationMode is not set on the httpRuntime node.
This should catch most errors (excluding, obviously, errors in parsing your web.config!)
Note: in your case, if you want to set defaultPath on the httpErrors node, you'll get a lock violation because of ApplicationHost.config settings. You can either do as I did and just set path individually for the error codes you care about, or you can have a look at unlocking the node:-
My intuition is that it's more trouble than it's worth in low-control environments like Azure App Service / Azure Web Sites. You might as well set the paths for individual status codes.
By far this is the most detailed and perfect response that I've seen in years. Thank you very much!!! It's working perfectly!!!
I'm back with one more question: it seems that I have configured in my web.config a HttpModule (ImageProcessorModule) which intercepts all my calls (is quite normal) but when I'm dealing with the A potentially dangerous Request.RawUrl value was detected from the client, I see the error in clear text and no 500 custom error page shows up; though, if I'm commenting out this module to not intercept the requests, I see the custom error page... What am I missing? Why the system doesn't display the custom error message though the web.config says so?! Thank you very much!!!
You might be best posting that as a separate question with more info. I'm aware of cases where bugs in httpModules have caused problems with custom errors, but I'd need more to go on before I could narrow it down.
## c# - ASP.NET MVC customError page doesn't get displayed for some of th...
c# asp.net asp.net-mvc
UPD: Ok, after reviewing the question I realised that the problem is in something else.. By default the authentication system in ASP.NET redirects all 403 requests to the login page. This is the way it's coded, and yes, it ignores the we.config settings.
There are solutions, of course.. You can check an example of how it was solved here ( similar question of SO ) With ASP.NET membership, how can I show a 403?
## asp.net - CustomError not working properly - Stack Overflow
asp.net visual-studio-2008 web-config
I had the same problem when I was updating the wrong web.config file.
There is a web.config in the Views folder and also under the project application site.
The solution would be to update web.config located under project.
Sorry for my late answer ;) but I already had updated the right web.config as you mentioned. Anyway the problem still persists.
I am on MVC3 and just added the below settings to web.config and custom page and it works for me. <customErrors mode="On" defaultRedirect="~/Views/Shared/Error.cshtml"> <error redirect="~/Home/NotFound" statusCode="404" /> '</customErrors>
## asp.net mvc - Custom Error Page does not show even though I set Custom...
asp.net-mvc error-handling web-config asp.net-mvc-4
There's a lot of misinformation and/or out-of-date solutions regarding error handling in IIS 7+. The main things to understand are:-
• IIS 7+ introduced a new way of handling custom error pages.
Most people are using hodge-podge solutions involving all of customErrors, httpErrors the Application_Error handler, and often setting requestValidationMode="2.0" on the httpRuntime property and/or disabling request validation entirely! This makes it really difficult to use other people's solutions because any and all of these can affect the behaviour. I had a quick search around and I found several semi-duplicates without accepted answers, probably for this reason.
The reason that these two errors give you different behaviour is that they occur at different stages in the request pipeline. The customErrors node in your web.config interacts with errors "inside" your application, while request validation takes place "outside" your application. IIS rejects the dangerous request before it gets to your application code, and so your customErrors replacement doesn't happen.
Ideally you want a solution with as few moving parts as possible. IIS7 gives us a new way to specify error page replacement at the IIS level instead of at the application level - the httpErrors node. This lets us catch all our errors in one place:-
<configuration>
...
<system.webServer>
...
<httpErrors errorMode="Custom" existingResponse="Replace">
<clear />
<error statusCode="400" responseMode="ExecuteURL" path="/ServerError.aspx"/>
<error statusCode="403" responseMode="ExecuteURL" path="/ServerError.aspx" />
<error statusCode="404" responseMode="ExecuteURL" path="/PageNotFound.aspx" />
<error statusCode="500" responseMode="ExecuteURL" path="/ServerError.aspx" />
</httpErrors>
...
</system.webServer>
...
</configuration>
If you care about SEO (and you should!), you still have to make sure that your controller/page sets an appropriate status code:-
this.Response.StatusCode = 500; // etc.
You should remove your customErrors node entirely. It is normally used for backwards-compatibility. You should also ensure that requestValidationMode is not set on the httpRuntime node.
This should catch most errors (excluding, obviously, errors in parsing your web.config!)
Note: in your case, if you want to set defaultPath on the httpErrors node, you'll get a lock violation because of ApplicationHost.config settings. You can either do as I did and just set path individually for the error codes you care about, or you can have a look at unlocking the node:-
My intuition is that it's more trouble than it's worth in low-control environments like Azure App Service / Azure Web Sites. You might as well set the paths for individual status codes.
By far this is the most detailed and perfect response that I've seen in years. Thank you very much!!! It's working perfectly!!!
I'm back with one more question: it seems that I have configured in my web.config a HttpModule (ImageProcessorModule) which intercepts all my calls (is quite normal) but when I'm dealing with the A potentially dangerous Request.RawUrl value was detected from the client, I see the error in clear text and no 500 custom error page shows up; though, if I'm commenting out this module to not intercept the requests, I see the custom error page... What am I missing? Why the system doesn't display the custom error message though the web.config says so?! Thank you very much!!!
You might be best posting that as a separate question with more info. I'm aware of cases where bugs in httpModules have caused problems with custom errors, but I'd need more to go on before I could narrow it down.
## c# - ASP.NET MVC customError page doesn't get displayed for some of th...
c# asp.net asp.net-mvc
There's a lot of misinformation and/or out-of-date solutions regarding error handling in IIS 7+. The main things to understand are:-
• IIS 7+ introduced a new way of handling custom error pages.
Most people are using hodge-podge solutions involving all of customErrors, httpErrors the Application_Error handler, and often setting requestValidationMode="2.0" on the httpRuntime property and/or disabling request validation entirely! This makes it really difficult to use other people's solutions because any and all of these can affect the behaviour. I had a quick search around and I found several semi-duplicates without accepted answers, probably for this reason.
The reason that these two errors give you different behaviour is that they occur at different stages in the request pipeline. The customErrors node in your web.config interacts with errors "inside" your application, while request validation takes place "outside" your application. IIS rejects the dangerous request before it gets to your application code, and so your customErrors replacement doesn't happen.
Ideally you want a solution with as few moving parts as possible. IIS7 gives us a new way to specify error page replacement at the IIS level instead of at the application level - the httpErrors node. This lets us catch all our errors in one place:-
<configuration>
...
<system.webServer>
...
<httpErrors errorMode="Custom" existingResponse="Replace">
<clear />
<error statusCode="400" responseMode="ExecuteURL" path="/ServerError.aspx"/>
<error statusCode="403" responseMode="ExecuteURL" path="/ServerError.aspx" />
<error statusCode="404" responseMode="ExecuteURL" path="/PageNotFound.aspx" />
<error statusCode="500" responseMode="ExecuteURL" path="/ServerError.aspx" />
</httpErrors>
...
</system.webServer>
...
</configuration>
If you care about SEO (and you should!), you still have to make sure that your controller/page sets an appropriate status code:-
this.Response.StatusCode = 500; // etc.
You should remove your customErrors node entirely. It is normally used for backwards-compatibility. You should also ensure that requestValidationMode is not set on the httpRuntime node.
This should catch most errors (excluding, obviously, errors in parsing your web.config!)
Note: in your case, if you want to set defaultPath on the httpErrors node, you'll get a lock violation because of ApplicationHost.config settings. You can either do as I did and just set path individually for the error codes you care about, or you can have a look at unlocking the node:-
My intuition is that it's more trouble than it's worth in low-control environments like Azure App Service / Azure Web Sites. You might as well set the paths for individual status codes.
By far this is the most detailed and perfect response that I've seen in years. Thank you very much!!! It's working perfectly!!!
I'm back with one more question: it seems that I have configured in my web.config a HttpModule (ImageProcessorModule) which intercepts all my calls (is quite normal) but when I'm dealing with the A potentially dangerous Request.RawUrl value was detected from the client, I see the error in clear text and no 500 custom error page shows up; though, if I'm commenting out this module to not intercept the requests, I see the custom error page... What am I missing? Why the system doesn't display the custom error message though the web.config says so?! Thank you very much!!!
You might be best posting that as a separate question with more info. I'm aware of cases where bugs in httpModules have caused problems with custom errors, but I'd need more to go on before I could narrow it down.
## c# - ASP.NET MVC customError page doesn't get displayed for some of th...
c# asp.net asp.net-mvc
@RestController
public class IndexController implements ErrorController{
private static final String PATH = "/error";
@RequestMapping(value = PATH)
public String error() {
return "Error handling";
}
@Override
public String getErrorPath() {
return PATH;
}
}
Your code did not work, because Spring Boot automatically registers the BasicErrorController as a Spring Bean when you have not specified an implementation of ErrorController.
ErrorMvcAutoConfiguration.basicErrorController
Ran into the same into the same issue, I searched the Spring docs but it didn't mention BasicErrorController. This works :)
Thanks, worked nicely! A small follow-up if you can give any pointers: say we get in this error handler because some exception was thrown in our app (and Spring implicitly sets response code to 500 which is correct); is there an easy way to get hold of that exception here (to include some details in error message returned)?
Hmm, yes, thanks again! At first I wasn't sure how to get that ErrorAttributes object (containing the error details), but then I tried simply @Autowiring it, and it works. What I went with for now: gist.github.com/jonikarppinen/662c38fb57a23de61c8b
## Spring Boot Remove Whitelabel Error Page - Stack Overflow
spring spring-boot
Facelets is a XML based view technology. XML has only five predefined entities. The is not among them. It works only when used in plain HTML or in legacy JSP (note: it doesn't work in JSPX as that's also XML based!).
To fix this, you either need to declare the entity yourself in the doctype,
<!DOCTYPE html [
<!ENTITY nbsp " ">
]>
or to use the (hexa)decimal notation instead:
  or  
Use the hex or decimal notation. :-)
## jsf - Error Parsing /page.xhtml: Error Traced[line: 42] The entity "nb...
jsf jsf-2 facelets html-entities
Facelets is a XML based view technology. XML has only five predefined entities. The is not among them. It works only when used in plain HTML or in legacy JSP (note: it doesn't work in JSPX as that's also XML based!).
To fix this, you either need to declare the entity yourself in the doctype,
<!DOCTYPE html [
<!ENTITY nbsp " ">
]>
or to use the (hexa)decimal notation instead:
  or  
Use the hex or decimal notation. :-)
## jsf - Error Parsing /page.xhtml: Error Traced[line: 42] The entity "nb...
jsf jsf-2 facelets html-entities
Under your main views.py add your own custom implementation of the following two views, and just set up the templates 404.html and 500.html with what you want to display.
urls.py
from django.shortcuts import render_to_response
from django.template import RequestContext
def handler404(request):
response = render_to_response('404.html', {},
context_instance=RequestContext(request))
response.status_code = 404
return response
def handler500(request):
response = render_to_response('500.html', {},
context_instance=RequestContext(request))
response.status_code = 500
return response
handler404 and handler500 are exported Django string configuration variables found in django/conf/urls/__init__.py. That is why the above config works.
To get the above config to work, you should define these variables in your urls.py file and point the exported Django variables to the string Python path of where these Django functional views are defined, like so:
# project/urls.py
handler404 = 'my_app.views.handler404'
handler500 = 'my_app.views.handler500'
If you use views as above, handler404 will fail with message:
def handler404(request, exception, template_name='404.html'):
...
This seemed to work quite well for me, but for some reason request.user appears fine in the 404 template, but not at all in the 500 template (and they are almost identical) -- posted question on this here: stackoverflow.com/questions/26043211/
Another thing I was wondering about -- what if you use the admin backend and want to use separate templates for those? To my knowledge, admin doesn't have a views.py to override and put this bit of code in.
@GravityGrave the 500 template won't render request.user because it is reporting a 500 server error, so the server is not able to serve up anything.
Didn't work for me with django 1.9 ;( Maybe I'm doing something wrong. Is handler404 django reserved name? How would django know that it should call exactly that view?
I updated the answer based on your comment. Sorry that the update is so late. I hope that this helps.
## python - Django, creating a custom 500/404 error page - Stack Overflow
python django django-templates http-status-code-404
public class FilterConfig
{
public static void RegisterGlobalFilters(GlobalFilterCollection filters)
{
}
}
public void Application_Error(Object sender, EventArgs e)
{
Exception exception = Server.GetLastError();
Server.ClearError();
var routeData = new RouteData();
if (exception.GetType() == typeof(HttpException))
{
}
else
{
}
Response.TrySkipIisCustomErrors = true;
IController controller = new ErrorPageController();
controller.Execute(new RequestContext(new HttpContextWrapper(Context), routeData));
Response.End();
}
public class ErrorPageController : Controller
{
public ActionResult Error(int statusCode, Exception exception)
{
Response.StatusCode = statusCode;
ViewBag.StatusCode = statusCode + " Error";
return View();
}
}
[5]: in 'Views/Shared/Error.cshtml'
@model System.Web.Mvc.HandleErrorInfo
@{
ViewBag.Title = (!String.IsNullOrEmpty(ViewBag.StatusCode)) ? ViewBag.StatusCode : "500 Error";
}
<h1 class="error">@(!String.IsNullOrEmpty(ViewBag.StatusCode) ? ViewBag.StatusCode : "500 Error"):</h1>
//@Model.ActionName
//@Model.ControllerName
//@Model.Exception.Message
//@Model.Exception.StackTrace
just tried to implement this. accessing a non existing page returns nothing. (instead of 404)
Just to make it clear, it return a 404 status code, with a blank page.
@Dementic: Yes it's a blank page so you can customise it. I have it working OK on a few sites now.
Does your solution work properly with AJAX requests? Also, you clear the server error before you recognize that it was an HttpException error.
## ASP.NET MVC 5 Custom Error Page - Stack Overflow
asp.net-mvc custom-error-pages custom-errors
Here is the link to the official documentation on how to set up custom error views:
It says to add lines like these in your URLconf (setting them anywhere else will have no effect):
handler404 = 'mysite.views.my_custom_page_not_found_view'
handler500 = 'mysite.views.my_custom_error_view'
handler403 = 'mysite.views.my_custom_permission_denied_view'
handler400 = 'mysite.views.my_custom_bad_request_view'
You can also customise the CSRF error view by modifying the setting CSRF_FAILURE_VIEW.
It's worth reading the documentation of the default error handlers, page_not_found, server_error, permission_denied and bad_request. By default, they use these templates if they can find them, respectively: 404.html, 500.html, 403.html, and 400.html.
So if all you want to do is make pretty error pages, just create those files in a TEMPLATE_DIRS directory, you don't need to edit URLConf at all. Read the documentation to see which context variables are available.
In Django 1.10 and later, the default CSRF error view uses the template 403_csrf.html.
DEBUG
I did add, but it doesn't work. Added the handler404 and others pointing to right places in my views, yet it doesn't work, still seeing the default 404. And yes, I'm in Debug False mode and using 1.9
Using Django 1.9 and simply adding 500.html etc templates shows them instead of standard pages. Nice easy fix.
Gotcha helped me. It worked by making these changes in my settings.py, set DEBUG=False and ALLOWED_HOSTS = ['0.0.0.0'] to accept http request from any client.
## python - Django, creating a custom 500/404 error page - Stack Overflow
python django django-templates http-status-code-404
public class FilterConfig
{
public static void RegisterGlobalFilters(GlobalFilterCollection filters)
{
}
}
public void Application_Error(Object sender, EventArgs e)
{
Exception exception = Server.GetLastError();
Server.ClearError();
var routeData = new RouteData();
if (exception.GetType() == typeof(HttpException))
{
}
else
{
}
Response.TrySkipIisCustomErrors = true;
IController controller = new ErrorPageController();
controller.Execute(new RequestContext(new HttpContextWrapper(Context), routeData));
Response.End();
}
public class ErrorPageController : Controller
{
public ActionResult Error(int statusCode, Exception exception)
{
Response.StatusCode = statusCode;
ViewBag.StatusCode = statusCode + " Error";
return View();
}
}
[5]: in 'Views/Shared/Error.cshtml'
@model System.Web.Mvc.HandleErrorInfo
@{
ViewBag.Title = (!String.IsNullOrEmpty(ViewBag.StatusCode)) ? ViewBag.StatusCode : "500 Error";
}
<h1 class="error">@(!String.IsNullOrEmpty(ViewBag.StatusCode) ? ViewBag.StatusCode : "500 Error"):</h1>
//@Model.ActionName
//@Model.ControllerName
//@Model.Exception.Message
//@Model.Exception.StackTrace
just tried to implement this. accessing a non existing page returns nothing. (instead of 404)
Just to make it clear, it return a 404 status code, with a blank page.
@Dementic: Yes it's a blank page so you can customise it. I have it working OK on a few sites now.
Does your solution work properly with AJAX requests? Also, you clear the server error before you recognize that it was an HttpException error.
## ASP.NET MVC 5 Custom Error Page - Stack Overflow
asp.net-mvc custom-error-pages custom-errors
If you want a more "JSONish" response page you can try something like that:
import org.springframework.beans.factory.annotation.Autowired;
import org.springframework.boot.autoconfigure.web.ErrorAttributes;
import org.springframework.boot.autoconfigure.web.ErrorController;
import org.springframework.util.Assert;
import org.springframework.web.bind.annotation.RequestMapping;
import org.springframework.web.bind.annotation.RestController;
import org.springframework.web.context.request.RequestAttributes;
import org.springframework.web.context.request.ServletRequestAttributes;
import javax.servlet.http.HttpServletRequest;
import java.util.Map;
@RestController
@RequestMapping("/error")
public class SimpleErrorController implements ErrorController {
private final ErrorAttributes errorAttributes;
@Autowired
public SimpleErrorController(ErrorAttributes errorAttributes) {
Assert.notNull(errorAttributes, "ErrorAttributes must not be null");
this.errorAttributes = errorAttributes;
}
@Override
public String getErrorPath() {
return "/error";
}
@RequestMapping
public Map<String, Object> error(HttpServletRequest aRequest){
Map<String, Object> body = getErrorAttributes(aRequest,getTraceParameter(aRequest));
String trace = (String) body.get("trace");
if(trace != null){
String[] lines = trace.split("\n\t");
body.put("trace", lines);
}
return body;
}
private boolean getTraceParameter(HttpServletRequest request) {
String parameter = request.getParameter("trace");
if (parameter == null) {
return false;
}
return !"false".equals(parameter.toLowerCase());
}
private Map<String, Object> getErrorAttributes(HttpServletRequest aRequest, boolean includeStackTrace) {
RequestAttributes requestAttributes = new ServletRequestAttributes(aRequest);
return errorAttributes.getErrorAttributes(requestAttributes, includeStackTrace);
}
}
## Spring Boot Remove Whitelabel Error Page - Stack Overflow
spring spring-boot
Ran into the same problems, but I've just managed to get this working for us using a Classic ASP holding page in a .NET application (I'm not sure if it'll work the same for an apsx):
• Create a classic ASP file called Offline.asp and place it in the root of your website with the following contents: <% Response.Status = "503 Service Unavailable" Response.CacheControl = "no-cache" Response.AddHeader "Pragma", "no-cache" Response.Expires = -1 %> <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> <html> <head> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"> <meta name="robots" content="noindex,nofollow" /> <title>Sorry, we're down for essential maintenance</title> </head> <body><h1>Sorry, we're down for essential maintenance</h1></body> </html>
Now in IIS7 Manager go the Error Pages administration section, add a Custom Error Page like so:
Now whenever IIS receives a 503, it will send the contents of the Offline.asp to the browser, as well as the 503 code (which is important so that Search Engines don't list your error page!)
Now if you want to take your application offline you can just set up a simple redirect rule to redirect everything to Offline.asp.
I have just achieved the same thing with an aspx file. I put the server side code in the aspx file to keep things simple.
This doesn't work when the app pool is stopped either. I'm not sure if it would work if the app pool was overloaded as it is hard to recreate that condition when you want it.
## asp.net - Custom Error Page for Http Error 503 - Stack Overflow
asp.net iis-7 http-status-code-503
Install the gd library also.
## php - Fatal error: Call to undefined function mb_detect_encoding() - S...
php mbstring
There's a much easier way than recompiling PHP. Just yum install the required mbstring library:
yum --enablerepo=remi install php-mbstring
This also worked in openSuse 13.1 after getting a white homepage for phpMyAdmin: yast -i php-mbstring
If you have multiple versions installed on your system, you may need to install the version-specific library, e.g. with sudo apt install php5.6-mbstring.
php mbstring | 2021-08-04 05:12:36 | {"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": 0.17223191261291504, "perplexity": 6283.342158253313}, "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/1627046154796.71/warc/CC-MAIN-20210804045226-20210804075226-00028.warc.gz"} |
https://paulba.no/PerihelionAdvance.html | ### Description:
The motions of the planets in the Solar system are simulated. Each planet is simulated as a two body problem, planet - Sun. The Newtonian gravitational acceleration is replaced by the post-Newtonian approximation of the acceleration predicted by The General Theory of Relativity [GR].
For each planet, the advance of the perihelion is measured to show what GR predicts the rate of perihelion advance would be if the only existing bodies in the Universe were the planet and the Sun.
You can run the simulation as an Application, see how here
### The post-Newtonian approximation of the gravitational acceleration predicted by GR
The gravitational acceleration predicted by GR is:
$\frac{\mathrm d \vec{v}}{\mathrm dt} = -\frac{GM}{r^2}\left(1 - 4\frac{GM}{rc^2}+\frac{v^2}{c^2} \right)\hat{r}+ \frac{4GMv^2\left( \hat{r}\cdot \hat{v}\right)}{r^2c^2}\cdot \hat{v}$
Where: $\vec{r}$ = distance vector sun - planet [m] $r$ = |$\vec{r}$|, magnitude of $\vec{r}$ [m] $\hat{r}$ = unity vector parallel to $\vec{r}$ . $\vec{v}$ = distance vector sun - planet [m/s] $v$ = |$\vec{v}$|, magnitude of $\vec{v}$ [m/s] $\hat{v}$ = unity vector parallel to $\vec{v}$ . $c$ = speed of light [m/s] $G$ = Gravitational constant $[m^3 \cdot kg^{-1} · s^{-2}]$ $M$ = solar mass [kg] | 2021-12-01 03:17:03 | {"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": 0.6957855224609375, "perplexity": 720.0506050747947}, "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-2021-49/segments/1637964359082.78/warc/CC-MAIN-20211201022332-20211201052332-00403.warc.gz"} |
https://proxieslive.com/morita-equivalence-of-the-invariant-uniform-roe-algebra-and-the-reduced-group-c-algebra/ | # Morita equivalence of the invariant uniform Roe algebra and the reduced group C*-algebra
In his paper “Comparing analytic assemby maps”, J. Roe considers a proper and cocompact action of a countable group $$\Gamma$$ on a metric space $$X$$. He constructs the Hilbert $$C^*_r(\Gamma)$$-module $$L^2_\Gamma(X)$$ as the simultaneous completion of the $$\mathbb C\Gamma$$-module $$C_c(X)$$ under the $$\mathbb C\Gamma$$-valued inner product given by the formula $$\langle\psi\mid\psi’\rangle_\Gamma=\sum_{\gamma\in\Gamma}\langle\psi\cdot g\mid\psi’\rangle\,\gamma.$$ He shows (Lemmas 2.1-3) that the C$$^*$$-algebra of $$C^*_r(\Gamma)$$-compact operators $$\mathbb K_{C^*_r(\Gamma)}(L^2_\Gamma(X))$$ is isomorphic to $$C^*_\Gamma(X)$$, the $$\Gamma$$-invariant uniform Roe algebra.
All of this is fine, but there seems to be a problem with the following claims, and this is what my question is about.
Roe claims that $$\mathbb K_{C^*(\Gamma)}(L^2_\Gamma(X))$$ (and hence $$C^*_\Gamma(X)$$) is Morita equivalent to $$C^*_r(\Gamma)$$. I fail to see this and my question is: Is this true? If so, how to prove it? If not, how to disprove it?
Looking at Roe’s arguments, he deduces the above statement from the more general claim that for any (countably generated) Hilbert $$C^*_r(\Gamma)$$-module $$E$$, $$\mathbb K_{C^*(\Gamma)}(E)$$ is Morita equivalent to $$C^*_r(\Gamma)$$. What is true is that it is Morita equivalent to $$\langle E\mid E\rangle$$ (with $$E$$ setting up the equivalence), that is immediate. The general statement that Roe makes however seems to be false unless $$E$$ is (right-) full.
Indeed, let $$X=\mathbb Z$$ with the obvious action of $$\Gamma=\mathbb Z\rtimes\mathbb Z/2$$. Then the non-trivial character of $$\mathbb Z/2=\Gamma/\mathbb Z$$ defines a Hilbert $$C^*_r(\Gamma)$$ module whose compact operators are $$\mathbb C$$ with $$K_0(\mathbb C)=\mathbb Z$$. On the other hand, $$K_0(C^*_r(\Gamma))=\mathbb Z^3$$. (See, for instance, the paper “The structure of crossed products of irrational rotation algebras by finite subgroups of $$\mathrm{SL}_2(\mathbb Z)$$” by Echterhoff, Lück, Phillips, and Walters.) This contradicts the claimed Morita equivalence for an arbitrary Hilbert C$$^*$$-module $$E$$.
Concerning the particular case of $$E=L^2_\Gamma(X)$$, it would be great see that this is in fact a full HIlbert $$C^*_r(\Gamma)$$-module. I fail to see this, however. One computes in case of the above example: $$\langle\delta_m\mid\delta_n\rangle=(m-n,+)+(m+n,-),$$ so the impression is that perhaps the closed ideal generator by the inner products is not the entire algebra.
It would be already helpful I someone could shed shome light on this special case. | 2019-08-22 14:45:55 | {"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": 36, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9163535237312317, "perplexity": 188.160930222568}, "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-2019-35/segments/1566027317130.77/warc/CC-MAIN-20190822130553-20190822152553-00238.warc.gz"} |
https://mathematica.stackexchange.com/questions/179114/does-mathematica-have-a-twin-prime-equivalent-of-primepi | Does Mathematica have a twin prime equivalent of PrimePi?
Well, it's all there in the title! I'd like to be able to plot the number of twin primes =<x. Is there an inbuilt function that can do this?
• Is an exact value required? If not, one can take a "probabilistic" approach using log integral. If one wants to get a rough idea of how such a plot would look, this might suffice. Obviously it is too heuristic to give finer structure or anything like that. Main advantage is it is cheap and can be computed over arbitrarily large ranges (so not limited in the way that Prime-based methods would be). Jul 28 '18 at 14:52
• Wolfram Challenges Mar 10 '19 at 11:25
Here's a very short version:
primePairs[x_] := With[{primes = Prime[Range[PrimePi[x]]]},
Intersection[primes, primes - 2]]
It's returns the same numbers as @KennyColnago's TwinPrimeLesser, but it's a bit faster:
primePairs[10^8]; // AbsoluteTiming
{9.16803, Null}
TwinPrimeLesser[10^8]; // AbsoluteTiming
{11.8908, Null}
If you need all the primes that are twin primes up to n then.
TwinPrimesUpTo[n_] := Select[
Most@NestWhileList[
NextPrime,
Prime[1], # <= n &
]
, (PrimeQ[# + 2]||PrimeQ[# + 2]) &]
It was pointed out by @KennyColnago in other answer that it's customary to count the number of pairs of twin primes, in which case a list of the lesser of the twin pairs is given by
TwinPrimesLesserUpTo[n_] := Select[
Most@NestWhileList[
NextPrime,
Prime[1], # <= n &
]
, (PrimeQ[# + 2]) &]
This gives the same output as @KennyColnago's answer, but his performance is better (+1), as it takes advantage of PrimePi and Range.
TwinPrimesLesserUpTo[99999] == TwinPrimeLesser[99999]
(* True *)
ListPlot[
TwinPrimesUpTo[1000]
, Filling -> Axis
, PlotTheme -> "Scientific"
, FrameLabel -> {"Index", "Value"}
, PlotLabel -> "Twin Primes"
]
ListPlot[
Prepend[MapIndexed[Flatten[{##}] &, TwinPrimesUpTo[1000]], {0, 0}]
, InterpolationOrder -> 0
, Joined -> True
, PlotTheme -> "Scientific"
, FrameLabel -> {"n", "Number of Twin primes"}
, PlotLabel -> "Number of twin primes up to n"
]
One convention (the usual?) for counting twin primes is to count just 1 for each pair. The original question is ambiguous, but it seems that @rhermans (+1) has a different convention of counting 2 for each pair.
The following definition returns the lesser of all twin prime pairs $\{p,p+2\}$ such that $p \le x$. Make a list of primes up to $x$, then test for prime $x+2$.
TwinPrimeLesser[x_] := Pick[#, PrimeQ[# + 2]] &[Prime[Range[PrimePi[x]]]]
The function PrimePi2[x] counts 1 for each twin-prime pair $\{p,p+2\}$ with $p \le x$.
PrimePi2[x_] := Length[Pick[#, PrimeQ[# + 2]]] &[Prime[Range[PrimePi[x]]]]
The twin prime count is fairly fast.
AbsoluteTiming[PrimePi2[10^7]]
{1.38108, 58980}
As @DanielLichtblau mentioned in a comment, an approximate count is given by a log integral formula (equation 5) on MathWorld.
The integral has an explicit form for $x>2$.
LogIntegral2[x_] :=
2.42969427374664261373628276610088274856979843439 -
1.320323631693739147855624220029111556865246*x / Log[x] +
AbsoluteTiming[Round[LogIntegral2[10^7]]]
{0.000634, 58754}
As for plotting the number of twin primes, don't forget ListStepPlot.
TwinPrimePlot[x_] :=
ListStepPlot[
{Transpose[{#, Range[Length[#]]}] &[TwinPrimeLesser[x]],
Table[{n, LogIntegral2[n]}, {n, Range[3, x, x/100]}]},
Frame -> True, FrameLabel -> {"x", "Number of Twin Primes"},
PlotLabel -> "Number of Twin Primes {p,p+2} With p<=x",
ImageSize -> 500, BaseStyle -> {FontSize -> 14},
PlotRange -> {{0, Automatic}, {0, Automatic}},
PlotLegends ->
]
A way to make things faster is to build our own sieve. This still requires a large time and space complexity though.
Here's a compiled sieve of Eratosthenes:
PrimesSieve = Compile[{{n, _Integer}},
Block[{S = Range[2, n]},
Do[
If[S[[i]] != 0,
Do[
S[[k]] = 0,
{k, 2i+1, n-1, i+1}
]
],
{i, Sqrt[n]}
];
Select[S, Positive]
],
CompilationTarget -> "C",
RuntimeOptions -> "Speed",
CompilationOptions -> {"ExpressionOptimization" -> True}
];
Now write a function to pick out consecutive primes and one that counts them:
TwinPrimeList[n_] :=
With[{primes = PrimesSieve[n+1]},
Pick[Most[primes], Differences[primes], 2]
]
PrimePi2[n_] := Length[TwinPrimeList[n]]
Try a large value:
PrimePi2[10^8] // AbsoluteTiming
{4.361947, 440312}
We can get even faster with a wheel sieve
WheelSieve = Compile[{{n, _Integer}},
Module[{k, res, loc, pp, S, i = 1, j = 2, x, p = 0, sqrt = Floor[Sqrt[n]], max, a = 0, mod},
k = #1;
res = #2;
loc = #3;
pp = #4;
S = Table[If[j < 0, pp[[loc[[r]]]], With[{m = k j + r}, Boole[m <= n]m]], {j, -1, Ceiling[n/#]-1}, {r, res}];
max = Max[S];
x = Length[res];
While[p < sqrt,
If[(p = S[[j, i++]]) != 0,
a = 0;
While[a >= 0,
Do[
If[m > max, a = -1; Break[]];
mod = Mod[m, k, 2];
S[[Quotient[m, k]+1+Boole[mod < k], loc[[mod]]]] = 0,
{m, p*res + p*k*(a++)}
]
]
];
If[i > x, i = 1; j++]
];
Select[Flatten[S], Positive]
],
CompilationTarget -> "C",
RuntimeOptions -> "Speed",
CompilationOptions -> {"ExpressionOptimization" -> True}
]&[
P = Times @@ Prime[Range[6]],
Select[Range[2, P+1], CoprimeQ[#, P]&],
ReplacePart[ConstantArray[0, P+1], MapIndexed[#1 -> First[#2]&, Select[Range[2, P+1], CoprimeQ[#, P]&]]],
];
TwinPrimeListFast[n_] :=
With[{primes = WheelSieve[n+1]},
Pick[Most[primes], Differences[primes], 2]
]
PrimePi2Fast[n_] := Length[TwinPrimeListFast[n]]
This speeds things up by a factor of 3:
PrimePi2Fast[10^8] // AbsoluteTiming
{1.348864, 440312}
• Nice. +1, but what and where is fastCompile? Aug 1 '18 at 20:28
• @KennyColnago Sorry, fastCompile is my own version of Compile that has all of those fast options turned on by default. I forgot to change the name when I copied over to here. Aug 1 '18 at 20:30
j = 100000;
myTwinPrime = {};
For[k = 1, Prime[k] < j, k++,
If[Prime[k + 1] == Prime[k] + 2,
myTwinPrime = Append[myTwinPrime, {Prime[k], Prime[k + 1]}]]];
myAccumulatedTwinPrimes = Accumulate[myTwinPrimes];
then
myAccumulatedTwinPrimes[[69]]
or
ListPlot[myAccumulatedTwinPrimes]
• There is a slight syntax problem. Accumulate[myTwinPrimes] should be Accumulate[myTwinPrime]. When I run it. Jun 19 '21 at 1:08 | 2022-01-20 14:55:54 | {"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": 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": 0.4417077302932739, "perplexity": 6657.523528850846}, "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-2022-05/segments/1642320301863.7/warc/CC-MAIN-20220120130236-20220120160236-00167.warc.gz"} |
http://mathhelpforum.com/calculus/71351-integration-problem-center-mass-print.html | # Integration problem, center of mass
• February 2nd 2009, 09:27 AM
smorri28
Integration problem, center of mass
Hi could someone please help me set up the integrals for this problem. I am trying to find the center of mass. Thank you
1. for the region bounded by the parabola x=y^2-y and y=x
• February 2nd 2009, 10:29 PM
DeMath
In your case, I think you need to use these formulas to find the coordinates of the center of mass:
abscissa $\xi = \frac{{\int_0^2 {y \cdot x\left( y \right)dy} }}
{{\int_0^2 {x\left( y \right)dy} }}$
,
ordinate $\eta = \frac{{\frac{1}{2}\int_0^2 {{x^2}\left( y \right)dy} }}{{\int_0^2 {x\left( y \right)dy} }}$.
$x\left( y \right) = y - \left( {{y^2} - y} \right) = 2y - {y^2}$
$2y - {y^2} = 0 \Leftrightarrow y\left( {2 - y} \right) = 0 \Rightarrow \left[ \begin{gathered}{y_1} = 0, \hfill \\{y_2} = 2. \hfill \\ \end{gathered} \right.$
$\int_0^2 {x\left( y \right)dy} = \int_0^2 {\left( {2y - {y^2}} \right)dy} = \left. {\left( {{y^2} - \frac{{{y^3}}}{3}} \right)} \right|_0^2 = 4 - \frac{8}{3} = \frac{4}{3}.$
$\int_0^2 {y \cdot x\left( y \right)dy} = \int_0^2 {y\left( {2y - {y^2}} \right)dy} = \int_0^2 {\left( {2{y^2} - {y^3}} \right)dy} =$
$= \left. {\left( {\frac{{2{y^3}}}{3} - \frac{{{y^4}}}{4}} \right)} \right|_0^2 = \frac{{16}}{3} - \frac{{16}}{4} = \frac{{16}}{{12}} = \frac{4}{3}.$
$\int_0^2 {{x^2}\left( y \right)dy} = \int_0^2 {{{\left( {2y - {y^2}} \right)}^2}dy} = \int_0^2 {\left( {4{y^2} - 4{y^3} + {y^4}} \right)dy} =$
$= \left. {\left( {\frac{{4{y^3}}}{3} - {y^4} + \frac{{{y^5}}}{5}} \right)} \right|_0^2 = \frac{{32}}{3} - 16 + \frac{{32}}{5} = \frac{{160 - 240 + 96}}{{15}} = \frac{{16}}{{15}}.$
Finally
$\xi = \frac{4}{3}:\frac{4}{3} = 1$ and $\eta = \left( {\frac{1}{2} \cdot \frac{{16}}{{15}}} \right):\frac{4}{3} = \frac{8}{{15}}:\frac{4}{3} = \frac{2}{5}$. | 2015-08-04 00:14:22 | {"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": 0, "img_math": 0, "codecogs_latex": 11, "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": 0.8318673968315125, "perplexity": 729.1482821265876}, "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-2015-32/segments/1438042990123.20/warc/CC-MAIN-20150728002310-00223-ip-10-236-191-2.ec2.internal.warc.gz"} |
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# Tutor profile: Yannic V.
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Yannic V.
Ph.D. in mathematics
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## Questions
### Subject:Discrete Math
TutorMe
Question:
Find a formula for the sum $\sum_{k=0}^n k \binom{n}{k},$ by means of the binomial theorem. What can be deduced about the average number of elements in a random subset of $\{1,2, \hdots, n\}$?
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Yannic V.
Let $x,y \in \mathbb{R}$. Remember that the binomial theorem sates that $(x+y)^n = \sum_{k=0}^n \binom{n}{k}x^{k}y^{n-k},$ for all $n \in \mathbb{N}$. In particular, for $y=1$ we have $(x+1)^n=\sum_{k=0}^n \binom{n}{k}x^k.$ Considering $x$ as a variable, we differentiate both sides of the above equation to obtain $n(x+1)^{n-1}= \sum_{k=0}^n k\binom{n}{k}x^{k-1}.$ It remains to plug in $x=1$: $\sum_{k=0}^n k \binom{n}{k} = n2^{n-1}.$ The average number of elements in a random subset of $\{1,2, \hdots, n\}$ is thus $\frac{\sum_{k=0}^n k \binom{n}{k}}{\sum_{k=0}^n \binom{n}{k}} = \frac{n 2^{n-1}}{2^n}=\frac{n}{2}.$ This is not surprising, since the number of subsets of size $k$ is exactly the same as the number of subsets of size $n-k$, for any $k$.
### Subject:Calculus
TutorMe
Question:
Solve the integral $\displaystyle \int 3 \sec(2x-1)\tan(2x-1) \,dx$.
Inactive
Yannic V.
As the arguments of both functions in the integral are not single variables, we can start to consider a change of variable in order to just have one variable in the secant and tangent function. This can be made using the change of variable $$\label{cambio1} \left\lbrace\begin{array}{l} y = 2x-1 \\ dy = 2 \,dx \end{array}\right.$$ Notice that the derivative $dy$ is easy to obtain from the expression of the integral: $dx$ can be expressed in terms of $dy$ as $$\label{eq1} dx = \frac{dy}{2}.$$ Substituting the change of variable \eqref{cambio1} in the integral, we obtain \begin{align} \int 3 \sec(2x-1)\tan(2x-1)\, dx &= \int 3 \sec(y)\tan(y) \,\frac{dy}{2} \tag{\footnotesize by \eqref{cambio1} and \eqref{eq1}}\\ &= \frac{3}{2} \int \sec(y)\tan(y) \,dy. \label{da1} \end{align} We have now reduced our problem to a more simple integral. We want to integrate the expression $\sec(y)\tan(y)$, which is the derivative of\footnote{This comes from the following calculation: $(\sec(y))'= \left(\frac{1}{\cos(y)} \right)' = \frac{(0\cos(y))-1(-\sin(y))}{\cos(y)^2}= \frac{\sin(y)}{\cos(y)^2} = \frac{1}{\cos(y)}\frac{\sin(y)}{\cos(y)}=\sec(y)\tan(y).$ } the function $\sec(y)$. It is then appropriate to consider the following change of variable: $$\label{cambio2} \left\lbrace\begin{array}{l} u = \sec(y)\\ du = \sec(y)\tan(y) \,dy \end{array}\right.$$ We can now replace \eqref{cambio2} into the expression \eqref{da1}. Remark that \eqref{da1} just contain the derivative part of\footnote{This is not a problem. When doing a change of variable, the most important part is that the expression corresponding to the derivative of the change of basis appears in the integral (up to a scalar). } \eqref{cambio2}. Thus, we have: \begin{align} \int 3 \sec(2x-1)\tan(2x-1)\, dx &= \frac{3}{2} \int \sec(y)\tan(y) \,dy. \tag{\footnotesize{by \eqref{da1}}}\\ &= \frac{3}{2} \int du \tag{\footnotesize{by \eqref{cambio2}}}\\ &= \frac{3}{2}\, u + C, \label{da2} \end{align} where $C$ is a constant. Finally, we just have to substitute $u$ by its value in terms of $y$ (given by \eqref{cambio2}), and replace $y$ by its value in terms of $x$ (given by \eqref{cambio1}): \begin{align} \int 3 \sec(2x-1)\tan(2x-1)\, dx &= \frac{3}{2}\, u + C \tag{\footnotesize{by \eqref{da2}}}\\ &= \frac{3}{2}\sec(y) + C \tag{\footnotesize{substitute $u=\sec(y)$}}\\ &= \frac{3}{2}\sec(2x-1) + C. \tag{\footnotesize{substitute $y=2x-1$}} \end{align}
### Subject:Algebra
TutorMe
Question:
If $G$ is a finite group, show that the number of elements in $G$ of order greater than $2$ must be even. In particular, prove that any group of even order must contain an element of order 2.
Inactive
Yannic V.
We will solve the question by a counting argument as follows. Let $X$ be the set of all elements of G with order greater than two. If $X = \emptyset$ then $|X|=0$ and we’re done. Suppose now that $X \neq \emptyset$. Since the orders of $g$ and $g^{-1}$ coincides for any $g \in G$, we have that $g \in X \Longleftrightarrow g^{-1} \in X.$ Moreover, if $g \in X$ then $g^2\neq e_G$. From here, $g \neq g^{-1}$. It follows that $X$ can be written as the disjoint union of two element sets of the form $\{g, g^{-1}\}$ {x,x−1}, and hence that $|X|$ is even. Suppose that $|G|$ is even. Since $e_G$ has order $1$, $e_G \notin X$. It follows that $G \setminus X \neq \emptyset$. So $0 < |G \setminus X|=|G|-|X|$. Since $|G|$ and $|X|$ are both even, it follows that $|G\setminus X|$ is a nonzero even integer, i.e. is at least 2. Thus, there is an $z \in G \setminus X$ such that $z \neq e_G$ . Since $X$ consists of all elements in $G$ of order greater than $2$, it must be the case that $|z|=2$.
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https://bodheeprep.com/cat-2019-quant-question-with-solution-15 | # CAT 2019 Quant Question with Solution 15
Question:
Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job. If two machines can finish the job in 13 days, then how many men can finish the job in 13 days?
Let one machine completes 1 unit of work per day.
Given, two machines can finish the job in 13 days
Therefore, the work of 2×1×13 = 26 units.
Also, let on man completes m units of work per day.
From the given condition:
3m+8×1 = 2(8m+3×1)
Or m = $\frac{2}{13}units$
Let it require ‘x’ number of men to complete the work in 13 days.
Therefore, xm×13 =26 units
Or x = 13 men
Also Check: 841+ CAT Quant Questions with Solutions
#### CAT 2019 Slot-2
Bodhee Prep's YouTube channel CAT Prep Whatsapp Group CAT Prep Telegram Group
X | 2022-07-06 00:52:39 | {"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": 0.6210880875587463, "perplexity": 4437.56250653527}, "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-2022-27/segments/1656104655865.86/warc/CC-MAIN-20220705235755-20220706025755-00428.warc.gz"} |
https://proofwiki.org/wiki/Kuratowski%27s_Lemma | # Kuratowski's Lemma
## Theorem
Let $\left({S, \preceq}\right), S \ne \varnothing$ be a non-empty ordered set.
Then every chain in $S$ is the subset of some maximal chain.
## Source of Name
This entry was named for Kazimierz Kuratowski.
He published this in 1922, thinking it little more than a corollary of the Hausdorff Maximal Principle.
In 1935, Max August Zorn published a similar version, acknowledging Kuratowski's earlier work. This later version became the more famous one. | 2020-02-20 12:14:39 | {"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": 0.3641987442970276, "perplexity": 2579.490505626266}, "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/1581875144722.77/warc/CC-MAIN-20200220100914-20200220130914-00234.warc.gz"} |
https://testbook.com/question-answer/what-is-correct-about-a-transmission-line-terminat--5f4625bc6083940d0e2d20bf | # What is not correct about a transmission line terminated in its characteristic impedance?
This question was previously asked in
UPRVUNL AE EC 2014 Official Paper
View all UPRVUNL AE Papers >
1. It is a smooth line
2. The energy distribution between the magnetic and electric field is not equal
3. Standing wave does not exist
4. Efficiency of transmission of power is maximum
Option 2 : The energy distribution between the magnetic and electric field is not equal
Free
Hindi Subject Test 1
3883
10 Questions 10 Marks 10 Mins
## Detailed Solution
A long overhead lossless power transmission line is terminated with its characteristic impedance. It indicates that the reflection coefficient is zero.
The reflection coefficient is given as,
$${\rm{Γ }} = \frac{{{Z_0} - {Z_L}}}{{{Z_0} + {Z_L}}}$$
Where ZL is the line impedance and Z0 is the load impedance.
Γ = 0
A transmission line is terminated with its characteristic impedance will have all energy transferred to the load; zero energy will be reflected and Γ = 0.
Therefore, there is no reflected wave on the line.
Electric fields are produced by voltage, and magnetic fields are produced by current.
The reflection coefficient is the ratio of the complex amplitude of the reflected wave to that of the incident wave. The voltage and current at any point along a transmission line can always be resolved into forward and reflected traveling waves given a specified reference impedance Z0.
$$Γ=\frac{V}V^+=\frac{I^-}{I}$$
So when Γ = 0,
The energy distribution between the magnetic and electric fields is equal.
Hence option (2) given is false.
The Voltage Standing Wave Ratio (VSWR) is defined as:
$$VSWR = \frac{{1 + {\rm{Γ }}}}{{1 - {\rm{Γ }}}}$$
The voltage standing wave ratio for a perfectly matched load (which is always desirable) is obtained by putting Γ = 0 in the above equation.
$$VSWR = \frac{{1 + {\rm{0 }}}}{{1 - {\rm{0}}}}=1$$ | 2021-09-28 11:31:00 | {"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": 0.6818773746490479, "perplexity": 924.1897361141175}, "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/1631780060677.55/warc/CC-MAIN-20210928092646-20210928122646-00140.warc.gz"} |
https://msp.org/ant/2009/3-4/index.xhtml | Vol. 3, No. 4, 2009
Recent Issues
The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Ideals generated by submaximal minors
367–392
Jan O. Kleppe and Rosa M. Miró-Roig
The structure of the group {\slshape\bfseries G}(\kern-.1emk\/[t\kern.1em]): Variations on a theme of Soulé
393–409
Benedictus Margaux
On some crystalline representations of GL$_{\mathsf 2}(\mathbb{Q}_{\mathsf p})$
411–421
Vytautas Paškūnas
Equivariant Hilbert series
423–443
Frank Himstedt and Peter Symonds
Syzygies of the secant variety of a curve
445–465
Jessica Sidman and Peter Vermeire
The essential dimension of the normalizer of a maximal torus in the projective linear group
467–487 | 2021-04-21 06:59:03 | {"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": 0.2837216556072235, "perplexity": 12506.210588135178}, "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-2021-17/segments/1618039526421.82/warc/CC-MAIN-20210421065303-20210421095303-00241.warc.gz"} |
https://www.plumed.org/doc-v2.8/user-doc/html/_b_f__c_u_b_i_c__b__s_p_l_i_n_e_s.html | BF_CUBIC_B_SPLINES
This is part of the ves module It is only available if you configure PLUMED with ./configure –enable-modules=ves . Furthermore, this feature is still being developed so take care when using it and report any problems on the mailing list.
Cubic B spline basis functions.
Attention
These basis functions do not form orthogonal bases. We recommend using wavelets (BF_WAVELETS) instead that do for orthogonal bases.
A basis using cubic B spline functions according to [57]. The mathematical expression of the individual splines is given by
\begin{align*} h\left(x\right) = \begin{cases} \left(2 - \lvert x \rvert\right)^3, & 1 \leq \lvert x \rvert \leq 2\\ 4 - 6\lvert x \rvert^2 + 3 \lvert x \rvert^3,\qquad & \lvert x \rvert \leq 1\\ 0, & \text{elsewhere}. \end{cases} \end{align*}
The full basis consists of equidistant splines at positions $$\mu_i$$ which are optimized in their height:
\begin{align*} f_i\left(x\right) = h\left(\frac{x-\mu_i}{\sigma}\right) \end{align*}
Note that the distance between individual splines cannot be chosen freely but is equal to the width: $$\mu_{i+1} = \mu_{i} + \sigma$$.
The ORDER keyword of the basis set determines the number of equally sized sub-intervalls to be used. On the borders of each of these sub-intervalls the mean $$\mu_i$$ of a spline function is placed.
The total number of basis functions is $$\text{ORDER}+4$$ as the constant $$f_{0}(x)=1$$, as well as the two splines with means just outside the interval are also included.
As an example two adjacent basis functions can be seen below. The full basis consists of shifted splines in the full specified interval.
When the splines are used for a periodic CV (with the PERIODIC keyword), the sub-intervals are chosen in the same way, but only $$\text{ORDER}+1$$ functions are required to fill it (the ones at the boundary coincide and the ones outside can be omitted).
To avoid 'blind' optimization of the basis functions outside the currently sampled area, it is often beneficial to use the OPTIMIZATION_THRESHOLD keyword of the VES_LINEAR_EXPANSION (set it to a small value, e.g. 1e-6)
Examples
The bias is expanded with cubic B splines in the intervall from 0.0 to 10.0 specifying an order of 20. This results in 24 basis functions.
Click on the labels of the actions for more information on what each action computes
bf: BF_CUBIC_B_SPLINES MINIMUMcompulsory keyword
The minimum of the interval on which the basis functions are defined. =0.0 MAXIMUMcompulsory keyword
The maximum of the interval on which the basis functions are defined. =10.0 ORDERcompulsory keyword
The order of the basis function expansion. =20
Glossary of keywords and components
Compulsory keywords
ORDER The order of the basis function expansion. MINIMUM The minimum of the interval on which the basis functions are defined. MAXIMUM The maximum of the interval on which the basis functions are defined.
Options
DEBUG_INFO ( default=off ) Print out more detailed information about the basis set. Useful for debugging. PERIODIC ( default=off ) Use periodic version of basis set. NORMALIZATION the normalization factor that is used to normalize the basis functions by dividing the values. By default it is 2. | 2022-05-27 06:21:28 | {"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": 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": 0.8615128397941589, "perplexity": 973.9970695407429}, "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/1652662636717.74/warc/CC-MAIN-20220527050925-20220527080925-00695.warc.gz"} |
https://www.newton.ac.uk/event/gcsw01/timetable | # Timetable (GCSW01)
## Tutorial workshop
Monday 8th July 2019 to Friday 12th July 2019
09:20 to 09:50 Registration 09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director) 10:00 to 11:00 Hans Munthe-Kaas (Universitetet i Bergen)Why B-series, rooted trees, and free algebras? - 1 "We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures. In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Charlie Elliott (University of Warwick)PDEs in Complex and Evolving Domains I INI 1 12:30 to 14:00 Lunch at Murray Edwards College 14:00 to 15:00 Elizabeth Mansfield (University of Kent)Introduction to Lie groups and algebras - 1 In this series of three lectures, I will give a gentle introduction to Lie groups and their associated Lie algebras, concentrating on the major examples of importance in applications. I will next discuss Lie group actions, their invariants, their associated infinitesimal vector fields, and a little on moving frames. The final topic will be Noether’s Theorem, which yields conservations laws for variational problems which are invariant under a Lie group action. Time permitting, I will show a little of the finite difference and finite element versions of Noether’s Theorem. In this first talk, I will consider some of the simplest and most useful Lie groups. I will show how to derive their Lie algebras, and will further discuss the Adjoint action of the group on its Lie algebra, the Lie bracket on the algebra, and the exponential map. INI 1 15:00 to 16:00 Christopher Budd (University of Bath)Modified error estimates for discrete variational derivative methods My three talks will be an exploration of geometric integration methods in the context of the numerical solution of PDEs. I will look both at discrete variational methods and their analysis using modified equations, and also of the role of adaptivity in studying and retaining qualitative features of PDEs. INI 1 16:00 to 16:30 Afternoon Tea 16:30 to 17:30 Paola Francesca Antonietti (Politecnico di Milano)High-order Discontinuous Galerkin methods for the numerical modelling of earthquake ground motion A number of challenging geophysical applications requires a flexible representation of the geometry and an accurate approximation of the solution field. Paradigmatic examples include seismic wave propagation and fractured reservoir simulations. The main challenges are i) the complexity of the physical domain, due to the presence of localized geological irregularities, alluvial basins, faults and fractures; ii) the heterogeneities in the medium, with significant and sharp contrasts; and iii) the coexistence of different physical models. The high-order discontinuous Galerkin FEM possesses the built-in flexibility to naturally accommodate both non-matching meshes, possibly made of polygonal and polyhedral elements, and high-order approximations in any space dimension. In this talk I will discuss recent advances in the development and analysis of high-order DG methods for the numerical approximation of seismic wave propagation phenomena. I will analyse the stability and the theoretical properties of the scheme and present some simulations of real large-scale seismic events in three-dimensional complex media. INI 1 17:30 to 18:30 Welcome Wine Reception at INI
09:00 to 10:00 Douglas Arnold (University of Minnesota)Finite Element Exterior Calculus - 1 These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications. INI 1 10:00 to 11:00 Elizabeth Mansfield (University of Kent)Introduction to Lie groups and algebras - 2 I will discuss Lie group actions, their invariants, their associated infinitesimal vector fields, and a little on moving frames. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Kurusch Ebrahimi-Fard (Norwegian University of Science and Technology)Why B-series, rooted trees, and free algebras? - 2 "We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures. In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes. INI 1 12:30 to 14:00 Lunch at Murray Edwards College 14:00 to 15:00 Charlie Elliott (University of Warwick)PDEs in Complex and Evolving Domains II INI 1 15:00 to 16:00 Christian Lubich (Eberhard Karls Universität Tübingen)Variational Gaussian wave packets revisited The talk reviews Gaussian wave packets that evolve according to the Dirac-Frenkel time-dependent variational principle for the semi-classically scaled Schr\"odinger equation. Old and new results on the approximation to the wave function are given, in particular an $L^2$ error bound that goes back to Hagedorn (1980) in a non-variational setting, and a new error bound for averages of observables with a Weyl symbol, which shows the double approximation order in the semi-classical scaling parameter in comparison with the norm estimate. The variational equations of motion in Hagedorn's parametrization of the Gaussian are presented. They show a perfect quantum-classical correspondence and allow us to read off directly that the Ehrenfest time is determined by the Lyapunov exponent of the classical equations of motion. A variational splitting integrator is formulated and its remarkable conservation and approximation properties are discussed. A new result shows that the integrator approximates averages of observables with the full order in the time stepsize, with an error constant that is uniform in the semiclassical parameter. The material presented here for variational Gaussians is part of an Acta Numerica review article on computational methods for quantum dynamics in the semi-classical regime, which is currently in preparation in joint work with Caroline Lasser. INI 1 16:00 to 17:30 Poster session and Afternoon tea
09:00 to 10:00 Christopher Budd (University of Bath)Blow-up in PDES and how to compute it My three talks will be an exploration of geometric integration methods in the context of the numerical solution of PDEs. I will look both at discrete variational methods and their analysis using modified equations, and also of the role of adaptivity in studying and retaining qualitative features of PDEs. INI 1 10:00 to 11:00 Douglas Arnold (University of Minnesota)Finite Element Exterior Calculus - 2 These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Charlie Elliott (University of Warwick)PDEs in Complex and Evolving Domains III INI 1 12:30 to 14:00 Lunch at Murray Edwards College 14:00 to 17:00 Free afternoon 19:30 to 22:00 Formal Dinner at Trinity College LOCATIONTrinity College, Cambridge CB2 1TQDRESS CODESmart casualMENUStarter Salad of Madgett’s Duck Confit - Hazelnuts, Gésiers, Duck Liver Mousse, Rhubarb and Sweet Pickled ShallotCream of Watercress Soup - Grain Mustard Sabayon and Cheddar Straw Pastry Crumb (V) Main Cambridgeshire Lamb LoinSamphire Grass, Parmesan and Artichoke Risotto and Mustard Glazed CrôutonsOpen Lasagne, Balsamic Shallots, Roasted Yellow Courgette, Ricotta and Confit Cherry Tomato (V) Pudding Gooseberry Fool - Raisin Biscuits and Butterscotch
09:00 to 10:00 Charlie Elliott (University of Warwick)PDEs in Complex and Evolving Domains IV INI 1 10:00 to 11:00 Beth Wingate (University of Exeter)An introduction to time-parallel methods I will give an introduction to time-parallel methods and discuss how they apply to PDEs, in particular those with where resonance plays an important role. I’ll give some examples from ODEs before finally discussing PDEs. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Douglas Arnold (University of Minnesota)Finite Element Exterior Calculus - 3 These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications. INI 1 12:30 to 14:00 Lunch at Murray Edwards College 14:00 to 15:00 Kurusch Ebrahimi-Fard (Norwegian University of Science and Technology)Why B-series, rooted trees, and free algebras? - 3 "We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures. In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes. INI 1 15:00 to 16:00 Elizabeth Mansfield (University of Kent)Noether's conservation laws - smooth and discrete The final topic will be Noether’s Theorem, which yields conservation laws for variational problems which are invariant under a Lie group action. . I will show a little of the finite difference and finite element versions of Noether’s Theorem. INI 1 16:00 to 16:30 Afternoon Tea 16:30 to 17:30 Panel INI 1
09:00 to 10:00 Reinout Quispel (La Trobe University)Discrete Darboux polynomials and the preservation of measure and integrals of ordinary differential equations Preservation of phase space volume (or more generally measure), first integrals (such as energy), and second integrals have been important topics in geometric numerical integration for more than a decade, and methods have been developed to preserve each of these properties separately. Preserving two or more geometric properties simultaneously, however, has often been difficult, if not impossible. Then it was discovered that Kahan’s ‘unconventional’ method seems to perform well in many cases [1]. Kahan himself, however, wrote: “I have used these unconventional methods for 24 years without quite understanding why they work so well as they do, when they work.” The first approximation to such an understanding in computational terms was: Kahan’s method works so well because 1. It is very successful at preserving multiple quantities simultaneously, eg modified energy and modified measure. 2. It is linearly implicit 3. It is the restriction of a Runge-Kutta method However, point 1 above raises a further obvious question: Why does Kahan’s method preserve both certain (modified) first integrals and certain (modified) measures? In this talk we invoke Darboux polynomials to try and answer this question. The method of Darboux polynomials (DPs) for ODEs was introduced by Darboux to detect rational integrals. Very recently we have advocated the use of DPs for discrete systems [2,3]. DPs provide a unified theory for the preservation of polynomial measures and second integrals, as well as rational first integrals. In this new perspective the answer we propose to the above question is: Kahan’s method works so well because it is good at preserving (modified) Darboux polynomials. If time permits we may discuss extensions to polarization methods. [1] Petrera et al, Regular and Chaotic Dynamics 16 (2011), 245–289. [2] Celledoni et al, arxiv:1902.04685. [3] Celledoni et al, arxiv:1902.04715. INI 1 10:00 to 10:30 James Jackaman (Memorial University of Newfoundland)Lie symmetry preserving finite element methods Through the mathematical construction of finite element methods, the “standard” finite element method will often preserve the underlying symmetry of a given differential equation. However, this is not always the case, and while historically much attention has been paid to the preserving of conserved quantities the preservation of Lie symmetries is an open problem. We introduce a methodology for the design of arbitrary order finite element methods which preserve the underlying Lie symmetries through an equivariant moving frames based invariantization procedure. INI 1 10:30 to 11:00 Candan Güdücü (Technische Universität Berlin)Port-Hamiltonian Systems The framework of port-Hamiltonian systems (PH systems) combines both the Hamiltonian approach and the network approach, by associating with the interconnection structure of the network model a geometric structure given by a Dirac structure. In this talk, I introduce port-Hamiltonian (pH) systems and their underlying Dirac structures. Then, a coordinate-based representation of PH systems and some properties are shown. A Lanczos method for the solution of linear systems with nonsymmetric coefficient matrices and its application to pH systems are presenred. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Christopher Budd (University of Bath)Adaptivity and optimal transport My three talks will be an exploration of geometric integration methods in the context of the numerical solution of PDEs. I will look both at discrete variational methods and their analysis using modified equations, and also of the role of adaptivity in studying and retaining qualitative features of PDEs. INI 1 12:30 to 14:00 Lunch at Murray Edwards College 14:00 to 15:00 Douglas Arnold (University of Minnesota)Finite element exterior calculus - 4 These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications. INI 1 15:00 to 16:00 Hans Munthe-Kaas (Universitetet i Bergen)Why B-series, rooted trees, and free algebras? - 2 "We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures. In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes. INI 1 | 2019-08-18 17:42:08 | {"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": 0.38686463236808777, "perplexity": 985.8162523664994}, "config": {"markdown_headings": true, "markdown_code": true, "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-2019-35/segments/1566027313987.32/warc/CC-MAIN-20190818165510-20190818191510-00265.warc.gz"} |
http://crypto.stackexchange.com/questions/6117/how-much-do-we-trust-kea1-assumption/6133 | # How much do we trust KEA1 Assumption?
Let $$(g,h=g^s,q)$$ be a tuple such that $g$ is a generator for a group $\mathbb{G}$ of ord $q$ and $s$ is uniformly random in $\mathbb{Z}_q$.
The KEA1 Assumption saies that for any adversary $\mathcal{A}(g,h,q)$ that outputs a value $(C,Y)$ such that $C^s \equiv Y$ then there exists a extractor $\mathcal{B}$ that on the same input outputs a value $x$ such that $g^x \equiv C$.
The assumption, roughly speaking, saies that the only way to compute the tuple $(C,Y)$ is, actually, by knowing the exponent $x$.
This assumption was first proposed by Ivan Damgard in the paper "Towards Practical Public Key Systems Secure Against Chosen Ciphertext Attacks" in the '91.
My question is, how much do we trust this assumption? Is it reducible to something else?
-
Could you check your notation? It looks like the only $S$ you used should be something else. – Paŭlo Ebermann Jan 26 '13 at 22:17
I'm not sure how to answer the question about "trust" in the assumption. I suppose that is a matter of personal belief more than science. We don't know the truthfulness about any cryptographic assumptions, although obviously some have received more scrutiny than others. I'd say that the KEA assumptions have received relatively little attention, and so should be avoided if possible. However, I suspect that if you're asking the question, then it's not possible to avoid it.
As to the reducibility question, it's not reducible to anything that one might call a "standard" assumption. This is in fact by design. Standard assumptions, such as the RSA assumption or the DDH assumption, are "falsifiable" in the sense that if the assumption is false, then it is (in principle) easy to show a counterexample. In the case of RSA or DDH, the counterexample would be an algorithm A that successfully breaks the hard problems. Even if you have no idea how A works, it'd be pretty easy to verify that A is in fact a valid attacker just by sampling RSA composites/DDH tuples at random and seeing how often A succeeds.
By contrast, the knowledge of exponent assumption is one of the few non-falsifiable assumptions that cryptographers use. (The most common non-falsifiable "assumption" would be the concept of a random oracle, although that's not really an assumption per se.) It's possible that KEA1 is false, but it's not possible for me to prove to you that it's false just by showing you some simple witness to this claim.
-
Thanks! that was what I meant concerning the "trust". i.e. how many researchers have studied this assumption and how much we know about that. – AntonioFa Jan 26 '13 at 16:36
the answer you gave is wonderful! KEA1 assumption is always very weird for me. Are there some plain words to describe it? And if a protocol(say key-exchange) is based on it, then if we can say that it's not good or insecure ? – T.B Oct 15 '13 at 8:08 | 2015-03-05 14:33:59 | {"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": 0.8852282762527466, "perplexity": 517.9969945984642}, "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-2015-11/segments/1424936464193.61/warc/CC-MAIN-20150226074104-00220-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/31441/vertical-line-only-in-one-part-of-a-table?noredirect=1 | # Vertical line only in one part of a table
I need some help on putting a vertical line in the second part of the table, I don't know how to do it since the tabularx I declared does need to have that only in the second part. The code I used is:
\begin{tabularx}{\linewidth}{|XX|}
\hline
\multicolumn{2}{|c|}{\textbf{title name}}\\
\hline
something: & else \\
something: & else \\
something: & else \\
\hline
\multicolumn{2}{|c|}{bla bla blah}\\
\hline
title text: & description 1 \\
title text: & description 1 \\
\hline
\end{tabularx}
And the render of the table looks like this:
I just need a \vline between title text and description (it I use \vline it works only for the first line but the description text and sometimes the title, are few lines long).
I would just place the respective items in a \multicolumn{1}{|c|}{...}:
\documentclass{article}
\usepackage{tabularx}% http://ctan.org/pkg/tabularx
\begin{document}
\begin{tabularx}{\linewidth}{|XX|}
\hline
\multicolumn{2}{|c|}{\textbf{title name}}\\
\hline
something: & else \\
something: & else \\
something: & else \\
\hline
\multicolumn{2}{|c|}{bla bla blah}\\
\hline
\multicolumn{1}{|l|}{title text:} & description 1 \\
\multicolumn{1}{|l|}{title text:} & description 1 \\
\hline
\end{tabularx}
\end{document}
For an improved visual layout of tables, I would suggest using the booktabs package. It provides different horizontal lines (of varying thickness) that neatly separates the table headers and contents. However, due to this, spacing with vertical lines (down a column) is considered problematic. In fact, the booktabs documentation stipulates two main points to consider when making publication-style tables:
1. Never, ever use vertical rules.
2. Never use double rules.
The reason for the first restriction is because table inherently have some horizontal alignment with their columnar structure. As such, adding a vertical line would emphasize the obvious. Here's a mock-up of your original MWE that shows the table using a standard booktabs representation:
• Thanks, it works. But if the title text is too long (I use title text of 2 lines) it just takes all the space. I am now using {|X|} and it works fine! Thanks! Oct 13 '11 at 6:31
• If your title text: is much longer than in your MWE, you can put it in a \parbox{<width>} where you specify the <width>. If this explanation isn't sufficient, please update your original question with an expanded MWE.
– Werner
Oct 13 '11 at 6:39
You can define a macro for placing vline between arbitrary cells:
\def\arrvline{\hfil\kern\arraycolsep\vline\kern-\arraycolsep\hfilneg}
Example:
$$\begin{array}{cccc} a & b \arrvline& c & d\\ f \arrvline& 1asd & 2 & g \\ f & 1asd & 2 & g \\ \end{array}$$
Will produce:
(Probably won't work correctly for right-justified columns)
• Hi @vftdan I am trying to use this solution but noticing the vertical lines don't match up. Is there some way to align them? Please see this question: tex.stackexchange.com/questions/513687/… Oct 27 '19 at 21:32 | 2021-11-28 11:07:46 | {"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": 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": 0.7974902987480164, "perplexity": 2157.8137130251494}, "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/1637964358520.50/warc/CC-MAIN-20211128103924-20211128133924-00192.warc.gz"} |
https://www.science20.com/taming_divergences | $L=T-V$Some words about the principle of least action
$\sum F = M a$Understanding Newtonian Mechanics | 2022-08-14 00:57:29 | {"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": 2, "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": 0.5435295701026917, "perplexity": 3560.778445744338}, "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/1659882571989.67/warc/CC-MAIN-20220813232744-20220814022744-00643.warc.gz"} |
https://learn.careers360.com/ncert/question-find-the-shortest-distance-between-the-lines-x-plus-1-by-7-equal-to-y-plus-1-by-minus-6-equal-to-z-plus-1-by-1-and-x-minus-3-by-1-equal-to-y-minus-5-by-minus-2-equal-to-z-minus-7-by-1/ | # 15 Find the shortest distance between the lines $\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ and $\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$
D Divya Prakash Singh
We have given two lines:
$\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ and $\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$
Calculating the shortest distance between the two lines,
$\frac{x-x_{1}}{a_{1}}=\frac{y-y_{1}}{b_{1}}=\frac{z-z_{1}}{c_{1}}$ and $\frac{x-x_{2}}{a_{2}}=\frac{y-y_{2}}{b_{2}}=\frac{z-z_{2}}{c_{2}}$
by the formula
$d = \frac{\begin{vmatrix} x_{2}-x_{1} &y_{2}-y_{1} &z_{2}-z_{1} \\ a_{1} &b_{1} &c_{1} \\ a_{2}&b_{2} &c_{2} \end{vmatrix}}{\sqrt{(b_{1}c_{2}-b_{2}c_{1})^2+(c_{1}a_{2}-c_{2}a_{1})^2+(a_{1}b_{2}-a_{2}b_{1})^2}}$
Now, comparing the given equations, we obtain
$x_{1} = -1,\ y_{1} =-1,\ z_{1} =-1$
$a_{1} = 7,\ b_{1} =-6,\ c_{1} =1$
$x_{2} = 3,\ y_{2} =5,\ z_{2} =7$
$a_{2} = 1,\ b_{2} =-2,\ c_{2} =1$
Then calculating determinant
$\begin{vmatrix} x_{2}-x_{1} &y_{2}-y_{1} &z_{2}-z_{1} \\ a_{1} &b_{1} &c_{1} \\ a_{2}&b_{2} &c_{2} \end{vmatrix} = \begin{vmatrix} 4 &6 &8 \\ 7& -6& 1\\ 1& -2& 1 \end{vmatrix}$
$= 4(-6+2)-6(7-1)+8(-14+6)$
$= -16-36-64$
$=-116$
Now calculating the denominator,
$\sqrt{(b_{1}c_{2}-b_{2}c_{1})^2+(c_{1}a_{2}-c_{2}a_{1})^2+(a_{1}b_{2}-a_{2}b_{1})^2} = \sqrt{(-6+2)^2+(1+7)^2+(-14+6)^2}$$= \sqrt{16+36+64}$
$= \sqrt{116} = 2\sqrt{29}$
So, we will substitute all the values in the formula above to obtain,
$d = \frac{-116}{2\sqrt{29}} = \frac{-58}{\sqrt{29}} = \frac{-2\times29}{\sqrt{29}} = -2\sqrt{29}$
Since distance is always non-negative, the distance between the given lines is
$2\sqrt{29}$ units.
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https://scipost.org/submissions/2007.13781v3/ | # Bulk-boundary-defect correspondence at disclinations in rotation-symmetric topological insulators and superconductors
### Submission summary
As Contributors: Ion Cosma Fulga · Max Geier Arxiv Link: https://arxiv.org/abs/2007.13781v3 (pdf) Code repository: https://zenodo.org/record/4526466 Date submitted: 2021-02-12 21:38 Submitted by: Geier, Max Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical
### Abstract
We study a link between the ground-state topology and the topology of the lattice via the presence of anomalous states at disclinations -- topological lattice defects that violate a rotation symmetry only locally. We first show the existence of anomalous disclination states, such as Majorana zero-modes or helical electronic states, in second-order topological phases by means of Volterra processes. Using the framework of topological crystals to construct $d$-dimensional crystalline topological phases with rotation and translation symmetry, we then identify all contributions to $(d-2)$-dimensional anomalous disclination states from weak and first-order topological phases. We perform this procedure for all Cartan symmetry classes of topological insulators and superconductors in two and three dimensions and determine whether the correspondence between bulk topology, boundary signatures, and disclination anomaly is unique.
###### Current status:
Editor-in-charge assigned
We would like to thank all referees for their positive reports and for thoroughly studying our paper. We are very thankful for the constructive criticism and have modified our manuscript accordingly. We hope that the modifications provide further clarification, improve the readability, and make our results stand out more clearly. We believe that thanks to the referees suggestions we were able to further improve the quality of the manuscript.
We have included our responses to the individual referee reports as replies on the submissions page. For convenience, we provide in a separate email to the editor our manual "difference file" of our manuscript where all our new text insertions are marked in red and relevant deletions are in gray and crossed out.
Sincerely,
Max Geier, Ion Cosma Fulga, Alexander Lau
### List of changes
General:
* changed manuscript title slightly (crystalline -> rotation symmetric)
* expanded introduction and conclusion to emphasize the central results
* improved readability by including section summaries and introductions as well as minor adjustments and cross-references throughout the text
* included several footnotes throughout the text providing detailed information
* new Appendix A about holonomy equivalence classes of disclinations
* new subsection Appendix C.4 on the validity of the topological crystal construction for finite size samples
* other small changes and clarifications throughout the Appendix
* corrected typos throughout the text
Section 1:
* Added references 60,61,62 and sentence in the introduction in the paragraph on the literature
* Extended the summary of the manuscript to also contain the main results
Section 2:
* Changed section title to "Disclinations"
* added more precise definition of the lattice in terms of charge densities
* added a footnote in Sec. 2.1
* added Ref. 64 on definition of primitive unit cell
* clarified caption of Fig. 1
* added a reference to Appendix B.2 on disclination dipoles
* in Sec. 2.3, point to new Appendix A for details on holonomy classes
* added Refs. 35,50,53,65,66 in Sec. 2.3
* added footnote in Sec. 2.5 to clarify the notation in Eq.(2)
* in Sec. 2.6: changed text on geometric pi fluxes for clarification
* added footnote in Sec. 2.6 to support the discussion
* minor changes in the text and in the subsection titles to improve readability
Section 3:
* added "Strong" to the section title
* Changed section introduction
* clarification regarding strong topological phases in Sec. 3.1 and added footnote
* clarification regarding two-fold symmetric case in Sec. 3.4 and added footnote
* minor modifications to text and titles to improve readability
Section 4:
* Changed section title to "Disclinations in topological crystals"
* small changes in subsection titles
* small changes in the text to improve readability
* the section introduction points to Appendix C.4 which contains a discussion on the validity of the approach for finite size samples
* added a note in the caption of Fig. 7
* corrected a typo in Eqs. (7) to (10)
* additional footnote in Sec. 4.4 clarifying the definition of the topological invariants used in Eqs. (7) to (10)
* additional paragraph at the end of Sec. 4.4 elaborating on the translation holonomy vector T
Section 5:
* added section outline to the section introduction
* re-arranged the text into new subsections to improve clarity
* added footnote in Sec. 5.1 on flux quantization
* clarified interpretation of Cartan class BDI
* added footnote in Sec. 5.5 to clarify direct sums
Section 6:
* added reference to kwant software package
* minor clarification in the caption of Fig. 10
* added new Fig. 13 for 2pi/3 disclination and corresponding sentence in the text.
Section 7:
* split the Conclusions into "Results" and "Discussion"
Appendix
Appendix A:
* new appendix discussion the construction of the holonomy equivalence classes of disclinations
Appendix C.2:
* added footnote explaining the necessity of the block-offdiagonal coupling
Appendix C.4:
* new appendix discussing the validity of the topological crystal approach for finite size samples
### Submission & Refereeing History
Resubmission 2007.13781v3 on 12 February 2021
Submission 2007.13781v2 on 3 August 2020
## Reports on this Submission
### Report
In their resubmission, the authors have addressed all of my requests and comments in a satisfactory manner. I am therefore in favor of publication of the manuscript in its present form.
• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: - | 2021-03-06 22:59:46 | {"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": 0.47667327523231506, "perplexity": 4130.198080024752}, "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-2021-10/segments/1614178375529.62/warc/CC-MAIN-20210306223236-20210307013236-00245.warc.gz"} |
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# In a certain English class, 1/4 of the number of girls is equal to
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In a certain English class, 1/4 of the number of girls is equal to 1/6 of the total number of students. What is the ratio of the number of boys to the number of girls in the class?
A. 1 to 4
B. 1 to 3
C. 1 to 2
D. 2 to 3
E. 2 to 1
[Reveal] Spoiler: OA
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albany09 wrote:
In a certain English class, 1/4 of the number of girls is equal to 1/6 of the total number of students. What is the ratio of the number of boys to the number of girls in the class?
A. 1 to 4
B. 1 to 3
C. 1 to 2
D. 2 to 3
E. 2 to 1
Assume; g = number of girls, b = number of boys
Total number of students = b + g
g / 4 = (b + g) /6
6g = 4(b + g) = 4b + 4g
2g = 4b
2/4 = b/g
b/g = 1/2
The answer is C. 1 to 2.
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Re: In a certain English class, 1/4 of the number of girls is equal to [#permalink]
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07 Oct 2008, 04:18
albany09 wrote:
In a certain English class, 1/4 of the number of girls is equal to 1/6 of the total number of students. What is the ratio of the number of boys to the number of girls in the class?
A. 1 to 4
B. 1 to 3
C. 1 to 2
D. 2 to 3
E. 2 to 1
1/4 g = 1/6 t thus g = 2/3t
B= 1/3t
b/g = (1/3) / (2/3) = 1/2 ..C
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Re: In a certain English class, 1/4 of the number of girls is equal to [#permalink]
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07 Oct 2008, 05:03
C. Use real numbers. 1/4 of the number of girls is 10, so are 60 students. The number of girls is 4X10=40 and the number of boys is 60-40=20. 20/40 = 1/2
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In a certain English Class, 1/4 of the number of girls is equal to 1/6 of the total number of students. What is the ratio of the number of boys to the number of girls in the class?
A) 1 to 4
B) 1 to 3
(C) 1 to 2
(D) 2 to 3
(E) 2 to 1
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31 Oct 2010, 05:13
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monirjewel wrote:
In a certain English Class, 1/4 of the number of girls is equal to 1/6 of the total number of students. What is the ratio of the number of boys to the number of girls in the class?
A) 1 to 4
B) 1 to 3
(C) 1 to 2
(D) 2 to 3
(E) 2 to 1
1/4 girls = 1/6 total
=> girls = 2/3 total
this implies, number of boys = 1/3 total
ratio of boys/girls = (1/3)/(2/3)
= 1/2
Answer : (C) 1 to 2
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Re: In a certain English class, 1/4 of the number of girls is equal to [#permalink]
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09 Nov 2010, 08:28
Consider a number with 4 and 6 as its factors- 24.
now 1/6 th of 24 = 1/4 th of girls------> 4=(1/4)g
So - Girls =16 and Boys=8
So ratio= 8/16 = 1/2
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12 Apr 2015, 18:38
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Re: In a certain English class, 1/4 of the number of girls is equal to [#permalink]
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12 Apr 2015, 19:42
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Expert's post
Hi All,
This question is perfect for TESTing VALUES and you can keep the Values really small (which will make the 'math' really easy)...
We're told that, in a class, 1/4 of the number of girls is equal to 1/6 of the total number of students. We're asked for the ratio of the number of boys to the number of girls in the class.
IF....
Number of Girls = 4
We're told that (1/4)(4) = (1/6)(Total)
1 = (1/6)(Total)
Total students = 6
Number of Boys = 6-4= 2
The ratio of Boys:Girls is 2:4 = 1 to 2
[Reveal] Spoiler:
C
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01 Sep 2015, 14:21
Here is the straightforward approach without testing numbers:
if 1/4 of girls = 1/6 of the class, then 4 * 1/4 of the girls = 4 * 1/6 of the glass
so all the girls represent 4/6, then the boys must be 2/6, and the ratio is 1:2
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In a certain English class, 1/4 of the number of girls is equal to [#permalink]
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31 Mar 2016, 00:48
Attached is a visual that should help. I used the classic "algebra to simplify, plug in numbers to finish" method. Please ignore the 15 x 60...it is from another question.
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In a certain English class, 1/4 of the number of girls is equal to [#permalink] 31 Mar 2016, 00:48
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http://mathoverflow.net/feeds/user/370 | User shenghao - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T20:58:04Z http://mathoverflow.net/feeds/user/370 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/129729/group-extensions-with-a-non-commutative-kernel Group extensions with a non-commutative kernel shenghao 2013-05-05T15:27:56Z 2013-05-07T17:23:32Z <p>Let </p> <p>$$1\to K\to G\to H\to 1$$ be an extension of groups. When $K$ is commutative, $H$ acts on $K$ by conjugation; and given groups $K$ and $H,$ with $K$ commutative and $H$ acting on $K,$ such extensions are classified by group cohomology $H^2.$ For instance, if $H$ is profinite and $K$ is a (commutative) discrete $H$-module, then any extension splits potentially, i.e. $G\to H$ has a section over an open subgroup of $H,$ as the continuous group cohomology equals the direct limit of cohomologies of finite groups.</p> <p>Is there any theory when $K$ is not commutative? In this case, an extension does not induce a "conjugation" action of $H$ on $K.$ For instance, when $K=G(k^s)$ for some algebraic group $G$ over a field $k$ and $H=Gal(k^s/k),$ it seems that one may still ask for "affine extensions with kernel $G$" when $G$ is not a torus, just no $H^2$-interpretation.</p> <p>Let me ask a real question. Let $K_N$ be the maximal extension of $\mathbb Q$ unramified outside $N$ (for an integer $N>1$), and let $G=Gal(K_N/\mathbb Q).$ Consider its abelianization $H,$ which is $\prod_{p|N}\mathbb Z_p^*$ by CFT. Does the projection $G\to H$ have a section over some open subgroup of $H?$</p> <p>Thank you.</p> http://mathoverflow.net/questions/122609/where-can-i-find-the-two-books-of-weil-on-abelian-varieties Where can I find the two books of Weil on abelian varieties? shenghao 2013-02-22T06:31:09Z 2013-02-22T06:31:09Z <p>They are "Sur les courbes algebriques ..." and "Varietes abeliennes et courbes algebriques", published by Hermann in 1948. Is there any link online for these old Hermann books?</p> <p>Thanks in advance.</p> http://mathoverflow.net/questions/119627/does-every-regular-noetherian-domain-have-finite-krull-dimension Does every regular Noetherian domain have finite Krull dimension? shenghao 2013-01-23T07:09:32Z 2013-01-23T14:34:04Z <p><strong>Background</strong>: A Noetherian ring is said to be regular if its localizations at all prime (or maximal) ideals are regular local rings. Without this assumption, there are counter-examples.</p> <p>Thanks.</p> http://mathoverflow.net/questions/762/roots-of-analytic-functions roots of analytic functions shenghao 2009-10-16T16:59:55Z 2012-09-19T01:11:23Z <p>Let $z$ be a complex variable and $f(z)$ be a formal power series with rational coefficients (an element in $\mathbb Q[[z]]$), with a finite radius of convergence, and assume $f(z)$ has a meromorphic continuation to the whole complex plane (so it has at most countably many poles). What do we know about the number-theoretic property of the roots and poles? Are they algebraic numbers? If they are, are they stable under the action of the Galois group of the rationals?</p> <p>More generally, if the coefficients of $f(z)$ are algebraic numbers, and let $\sigma$ be an automorphism of the algebraic closure of the rationals, then what is the relation between roots of $f(z)$ and roots of $f^{\sigma}(z),$ where $f^{\sigma}(z)$ is the power series obtained by applying sigma to the coefficients of $f(z)?$</p> <p>Without assuming the finiteness of radius of convergence, $\sin(z)$ gives a counter-example.</p> <p><strong>Edit</strong>: Let me give a second try, by imposing more requirements on $f(z).$ I'm thinking about the case where $f(z)$ is the zeta function of an algebraic stack over a finite field, so let's assume $f(z)$ has an infinite product expansion over $\ell$-adic numbers, like $\prod P_{odd}(z)/\prod P_{even}(z),$ where each $P_i(z)$ is a polynomial over $\mathbb Q_{\ell}$ with constant term 1. Assume they have distinct weights, e.g. reciprocal roots of $P_i(z)$ have weights $i.$ Then can we conclude that all coefficients of $P_i(z)$ are rational numbers? Thanks.</p> http://mathoverflow.net/questions/104003/is-there-an-explicit-example-of-a-coefficient-sheaf-for-which-hard-lefschetz-fail Is there an explicit example of a coefficient sheaf for which hard Lefschetz fails? shenghao 2012-08-05T09:05:24Z 2012-08-05T21:27:31Z <p>Recall that if $X$ is a projective algebraic complex manifold, $L$ is a semisimple $\mathbb C$-local system on $X$ <em>of geometric origin</em> (roughly speaking, this means that $L$ is a cohomology sheaf $R^if_*\mathbb C$ for some algebraic morphism $f:Y\to X;$ see BBD for the precise definition), and $\eta\in H^2(X,\mathbb C)$ is an ample class, then $$\eta^i\cup-:H^{\dim X-i}(X,L)\to H^{\dim X+i}(X,L)$$ is an isomorphism. This also holds when the projective variety $X$ is allowed to have singularities and $L$ is a perverse sheaf (again semisimple of geometric origin), appropriately shifted.</p> <p>I'd like to know an example for which this fails; of course, $L$ is no longer of geometric origin. Over finite field, as long as $L$ is assumed semisimple, hard Lefschetz always holds, and conjecturally $L$ is of geometric origin. </p> http://mathoverflow.net/questions/72872/references-on-semismall-maps References on semismall maps shenghao 2011-08-14T15:31:17Z 2012-07-19T14:28:40Z <p>Where can I find references on semismall maps, in the sense of Goresky and MacPherson? I don't want to restrict to the case where the base is $\mathbb C$ (an arbitrary alg. closed field would be fine), or maps $f:X\to Y$ from a smooth variety $X.$ In particular, I'd like to find the proof (if the statement is correct, which I'm not sure) that $Rf_*$ takes an irreducible (middle) perverse sheaf $F$ supported on $X$ to a perverse sheaf; I can only do this when $X$ is smooth and $F$ is a lisse sheaf, or when all the fibers of $f$ have dimension at most one. </p> <p><strong>Recall:</strong> A proper surjective morphism $f:X\to Y$ is called $semismall$ if $\dim X\times_YX=\dim X.$</p> <p>Thank you.</p> http://mathoverflow.net/questions/47353/the-automorphism-group-of-a-hyperelliptic-curve The automorphism group of a hyperelliptic curve shenghao 2010-11-25T18:05:32Z 2012-07-01T04:19:59Z <p>Let $C$ be the projective smooth genus 2 curve defined by $y^2=x^5-x$ over $\mathbb F_5.$ What is the order of its automorphism group (automorphisms over $\mathbb F_5$)? </p> <p>I have seen different answers. In Hartshorne's Algebraic Geometry, p. 306, the answer is $2p(p^2-1)=240.$ In INFORMATION Volume 8, Number 6, pp. 837-844, Isomorphism classes of genus-2 hyperelliptic curves over finite fields $\mathbb F_{5^m},$ by L. Hernández Encinas and J. Muñoz Masqué, theorem 2, the answer is $|A_{4221}|=20$ (using notations there). A professor (let me not to mention the name for now) told me the order is 120. </p> <p>Maybe I misunderstand some of the references above?</p> http://mathoverflow.net/questions/81402/does-s6-have-the-structure-of-an-algebraic-variety Does S^6 have the structure of an algebraic variety? shenghao 2011-11-20T08:11:06Z 2011-11-27T06:02:10Z <p>More accurately, the question should be: Is it known that $S^6,$ the 6-dimensional sphere, is $not$ a (proper) complex algebraic variety, or algebraic space? And is there a reference? It's easy to see that it cannot be projective (as $H^2=0$), but I don't see how it can violate the usual general properties of algebraic varieties [e.g. by Chow's lemma one can get a projective (and smooth, by resolution of singularities) variety lying over and birational to it, but the cohomology groups can get larger when we go up...]. Maybe one needs some theory of classification of 3-folds (if there is one such theory).</p> <p>Knowing that it cannot be algebraic will somehow make the analogous question of complex structure much sharper (by the way, it does have an almost complex structure).</p> <p>Thank you.</p> http://mathoverflow.net/questions/75149/a-local-global-question-on-group-representations A local-global question on group representations shenghao 2011-09-11T16:45:11Z 2011-09-11T16:45:11Z <p>Let $G$ be a group, and let $V$ be a finite dimensional $\mathbb Q$-linear representation of $G.$ By extension of scalars we obtain the $\mathbb Q_l$-linear representation $V_l=V\otimes\mathbb Q_l.$</p> <p><strong>Question</strong>: Is the natural morphism ´$$H^1(G,V)\to\prod_lH^1(G,V_l)$$ injective? Stated in this generality, the answer is possibly negative. But are there some contexts (e.g. G is a $\mathbb Q$-algebraic group and $V$ is an algebraic representation etc.) in which the answer is yes? </p> <p>And what about other degrees $H^i$?</p> http://mathoverflow.net/questions/19894/is-there-a-general-projection-formula-for-morphisms-of-ringed-topoi Is there a general projection formula for morphisms of ringed topoi? shenghao 2010-03-30T23:27:55Z 2011-09-11T16:25:53Z <p>What's the general projection formula in algebraic geometry, for instance on the level of derived categories of ringed topoi? And what's the reference? I guess it might be in SGA 4, but couldn't find it. </p> <p>Two examples:</p> <ol> <li><p>Zariski site, $D^b_{qcoh}$ on schemes. Let $f:X\to Y$ be a proper map of noetherian schemes (maybe there are some other mild conditions), and let $F\in D^b_{qcoh}(X)$ and $G\in D^b_{qcoh}(Y).$ Then <code>$(Rf_*F)\otimes^L G=Rf_*(F\otimes^L Lf^*G)$</code>. </p></li> <li><p>Etale site, say ringed by a torsion ring like $Z/n.$ Let $f:X\to Y$ be a (seperated; but this condition can be removed. See for instancde Laszlo and Olsson, The six operations on Artin stacks...) map of schemes of finite type over some base $S$ ($S$ may need to satisfy some assumptions, in order for the classical results in SGA 4/4.5 or Gabber's new results on finiteness of $f_*$ and dualizing complexes to work; but let's be sloppy). Let <code>$F\in D^-_c(X,Z/n)$</code> and <code>$G\in D^-_c(Y,Z/n).$</code> Then <code>$Rf_!F\otimes^L G=Rf_!(F\otimes^L f^*G).$</code> </p></li> </ol> <p>We used $f_!$ in example 2 in order to allow $F$ and $G$ to be in $D^-_c$ rather than $D^b_c.$ If one restricts to $D^b_c,$ is it also true for $f_*?$ </p> http://mathoverflow.net/questions/65233/lifting-varieties-from-char-p-to-char-0-after-alterations Lifting varieties from char. $p$ to char. 0 after alterations shenghao 2011-05-17T13:11:50Z 2011-09-11T16:11:57Z <p>The question is related to this MO question:</p> <p><a href="http://mathoverflow.net/questions/25337/lifting-varieties-to-characteristic-zero" rel="nofollow">http://mathoverflow.net/questions/25337/lifting-varieties-to-characteristic-zero</a></p> <p>Let $X$ be a projective smooth variety over $k$ alg. closed field of char. $p.$ Does there always exist an alteration $Y\to X,$ with $Y$ also projective smooth, such that $Y$ lifts to char. 0? </p> <p>Side remark: over $k=\overline{\mathbb F}_p,$ modulo Tate conjecture, abelian varieties "generate" the motives of all proj. smooth varieties. Since abelian varieties are liftable, one can say that the (irred. components of) motives of any proj. smooth varieties is liftable in some sense. And I wonder if this can be realized geometrically. "Alteration" in my question is just a try; replace it with any reasonable geometric construction if you want. For instance, "a proper surjection" would be fine.</p> http://mathoverflow.net/questions/74853/analogue-of-shafarevich-oggs-theorem-over-complex-numbers Analogue of Shafarevich-Ogg's theorem over complex numbers shenghao 2011-09-08T10:21:18Z 2011-09-09T08:01:14Z <p>Let <code>$f:E\to D^*$</code> be a family of complex elliptic curves parametrized by the punctured open disk <code>$D^*.$</code> Assume that the monodromy on <code>$H^1$</code> is trivial (i.e. $R^1f_*\mathbb Z$ is a constant sheaf on <code>$D^*$</code>). Does this imply that $f$ extends to a family of elliptic curves over the full disk $D?$ </p> <p>Here's an attempt, which I'm not sure if it works: maybe there is an equivalence (Riemann-Deligne?) between families of elliptic curves over a (smooth) base over $\mathbb C$ and variations of $\mathbb Z$-Hodge structures satisfying Griffiths transversality, of rank 2 and weight 1 on the same base. The constant local system certainly extends to $D.$</p> http://mathoverflow.net/questions/72446/what-would-be-a-characteristic-p-analogue-for-c-infty-fiber-bundles What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles? shenghao 2011-08-09T06:03:50Z 2011-08-09T06:03:50Z <p>I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the morphism to be smooth; e.g. a family of nodal curves $X_t$ with $p_a=2$ and such that the $j$-invariant of the normalizations $\widetilde{X}_t$ is not constant in $t,$ would certainly qualify. </p> <p><strong>Motivation:</strong> If $f:X\to Y$ is a proper morphism of complex algebraic varieties, then by Morse theory, there exists a "stratification of $f";$ in particular, over each stratum of $Y,\ f$ is a $C^{\infty}$-fiber bundle. I wonder if this could be true in char. $p,$ but the first thing is to have an analogous notion. </p> http://mathoverflow.net/questions/67964/does-intersection-pairing-on-ihx-agree-with-cup-product-on-hx Does intersection pairing on $IH^*(X)$ agree with cup-product on $H^*(X)$? shenghao 2011-06-16T15:50:06Z 2011-07-01T14:47:02Z <p>Let $X$ be a proper singular variety over $k=\overline{\mathbb F}_p,$ irreducible of dimension $d.$ Let <code>$H^*(X)$</code> and <code>$IH^*(X)$</code> be the $l$-adic cohomology groups and $l$-adic intersection cohomology groups of $X,$ resp. Then, is the natural map <code>$H^*(X)\to IH^*(X)$</code> compatible with the cup-product on <code>$H^*(X)$</code> and the intersection product on <code>$IH^*(X)?$</code></p> <p><strong>Background and Motivation:</strong> Given $X_0/\mathbb F_q$ an $\mathbb F_q$-structure of $X,$ one deduces a Galois action on <code>$H^*(X)$</code> and <code>$IH^*(X),$</code> with respect to which they are "mixed" (the 2nd one being pure), and the weight filtrations $W$ on both of them are independent of the choice of $X_0.$ One has a natural morphism <code>$$H^n(X) \to IH^n(X)$$</code> which factors as <code>$$Gr^W_n H^n(X) \hookrightarrow IH^n(X),$$</code> and this turns out to be injective. </p> <p>As $X$ is singular, Poincaré duality might fail, i.e. the cup-product <code>$$H^i(X)\otimes H^{2d-i}(X) \to H^{2d}(X),$$</code> which is Galois equivariant, may be degenerate. This is the case when <code>$H^i(X)$</code> (resp. <code>$H^{2d-i}(X)$</code>) is not pure of weight $i$ (resp. $2d-i$) for the reason of Galois, and <code>$W_{i-1}H^i(X)$</code> (resp. <code>$W_{2d-i-1}H^{2d-i}(X)$</code>) is contained in the left kernel (resp. right kernel) of the cup-product pairing. I would like to know if this is the only obstruction for Poincaré duality to hold, namely they are exactly the left/right kernel. </p> <p>Since the intersection pairing <code>$$IH^i(X)\otimes IH^{2d-i}(X)\to\mathbb Q_l(-d)$$</code><br> is perfect (I don't know if the pairing has a geometric definition in char. $p$ --- <code>$D_XIC_X\simeq IC_X$</code> is the only reason I know), if my question in the beginning has a positive answer, then the cup-product pairing on <code>$Gr^W_*H^*(X)$</code> will be perfect. </p> <p><strong>Correction:</strong> The argument above for non-degeneracy on <code>$Gr^W_*H^*(X)$</code> is wrong. Here's a counter-example. Let $X$ be the projective cone of a projective smooth curve of genus $g,$ either over char. 0 or $p.$ Then $H^1(X)=0$ but $H^3(X)$ is of dimension $2g$ and pure of weight 3.</p> http://mathoverflow.net/questions/69135/the-different-types-of-stacks/69169#69169 Answer by shenghao for The different types of stacks shenghao 2011-06-30T09:46:39Z 2011-07-01T09:15:45Z <p>Algebraic spaces are non-stacky algebraic (Artin) stacks, and DM-stacks, although stacky, have only finite stabilizer groups (or étale stab. groups; I'm sloppy here) for all points on the "underlying space". </p> <p>To get to a scheme from an algebraic space, one can either pass to an étale cover or just restrict oneself to an open dense subspace. To get to a scheme from a DM-stack, one can still pass to an étale cover, or, if one prefers finite group actions, pass to a Zariski open, which supports a $G$-bundle with total space an affine scheme, and $G$ is a finite group. The DM-stack is covered by such Zariski opens (though with different groups $G$). But for a general Artin stack there is really a long way to get to a scheme (or alg. space): if schemes are floating on the surface of the ocean, then Artin stacks rest deep in the ocean --- they are "covered" by schemes but the relative dimension are usually positive.</p> <p>This somehow explains (at least to me) why for algebraic spaces (or more generally, DM-stacks) it suffices to use the étale topology to define and compute cohomology of sheaves, whereas for general Artin stacks, lisse-étale topology is necessary.</p> <p><strong>Edit:</strong> This is actually a comment for 4) in unknowngoogle's answer. It is possible to have a continuously varying family of algebraic groups parameterized by a space (i.e. not iso-trivial, even if one passes to an algebraic stratification of the base space). Therefore, I don't think all algebraic stacks (say over $k$) are locally quotients of $k$-schemes by $k$-algebraic groups: one needs group schemes over a larger base. There exists a stratification of any algebraic stack such that the inertia is flat over each stratum, but one cannot make this flat family a constant family.</p> http://mathoverflow.net/questions/57701/poincare-duality-for-intersection-cohomology Poincaré duality for intersection cohomology shenghao 2011-03-07T17:27:11Z 2011-06-16T18:27:01Z <p>Let $X$ be a projective complex algebraic variety of dimension $d.$ Does it make sense to ask if properties like $$(x,y)=(-1)^i(y,x)$$ holds, for $x\in IH^i(X,\mathbb Q)$ and $y\in IH^{2d-i}(X)?$ And if it makes sense, is it true?</p> <p><strong>Edit:</strong> For the first question, my concern is that, if the self-duality $D_XIC_X\cong IC_X$ of the intersection complex only ensures that $IH^i(X)$ and $IH^{2d-i}(X)$ are dual, without specifying a particular duality, then it makes no sense to ask if the "product" is graded-commutative. Of course, if it does give a particular isomorphism, then it makes sense. </p> <p><strong>Edit:</strong> As I learned from Gabber, the answer is yes (of course I'd take the responsibility for misunderstanding his comment if it turns out to be ...), and it follows from the symmetric pairing $$IC_X[-d]\otimes IC_X[-d]\to\mathbb Q_{\ell}$$ normalized so that on the smooth locus, it is the natural identification. I'll be appreciated if anyone can give a reference on this.</p> http://mathoverflow.net/questions/67611/self-dual-representations self-dual representations shenghao 2011-06-12T22:13:03Z 2011-06-13T06:48:28Z <p>Let $V$ be a finite-dimensional irreducible representation (complex or $\ell$-adic) of a group $G$ (compact Lie group or algebraic group etc.). Does there always exist a linear character $\rho$ of $G$, such that $V\otimes\rho$ is a self-dual irrep. of $G?$ Namely $V\otimes\rho\simeq(V\otimes\rho)^*.$ If not, is there any necessary/sufficient conditions on $V$ for it to be "twisted self-dual"? </p> <p>If this is always the case, then in particular, if $G$ has no non-trivial linear characters (e.g. $G$ is a simply-connected compact Lie group or a perfect finite group), then every irrep. of $G$ is self-dual.</p> <p>Thanks.</p> http://mathoverflow.net/questions/66958/irreducibility-of-a-representation-of-gamman Irreducibility of a representation of $\Gamma(N)$ shenghao 2011-06-05T11:14:54Z 2011-06-05T11:14:54Z <p>Let $Y(N),$ for $N\ge3,$ be the modular curve over $\mathbb C$ with respect to the $\Gamma(N)$-level structure. Let $f:E\to Y(N)$ be the universal elliptic curve. Then $R^1f_*\mathbb Q$ is a semisimple local system on $Y(N).$ Is it irreducible? And how to prove it or where can I find a reference? Thank you.</p> http://mathoverflow.net/questions/65524/what-is-transport-of-structure-in-cohomology-setting/65548#65548 Answer by shenghao for What is transport of structure in cohomology setting? shenghao 2011-05-20T14:57:47Z 2011-05-21T23:55:48Z <p>I'm not completely sure if this is the answer; it just provides an another way of describing the Galois action.</p> <p>Let $a:X\to\text{Spec }\mathbb Q$ be the structure map. Then $R^ia_*F,$ being an étale sheaf on $\text{Spec }\mathbb Q,$ corresponds to an $\ell$-adic vector space (its stalk at $\text{Spec }\overline{\mathbb Q}$) with a continuous action of $Gal(\overline{\mathbb Q}/\mathbb Q).$ This vector space is nothing else but $H^i=H^i(X_{\overline{\mathbb Q}},F_{\overline{\mathbb Q}}).$ Think of the sheaf $R^ia_*F$ in terms of its espace étalé, and take fiber products: $$\begin{bmatrix} H^i & \to & H^i & \to & [R^ia_*F] \end{bmatrix}$$ $$\begin{bmatrix} \downarrow &&&&& \downarrow &&&&& \downarrow\end{bmatrix}$$ $$\begin{bmatrix} \text{Spec }\overline{\mathbb Q} & \overset{\to}{\sigma} & \text{Spec }\overline{\mathbb Q} & \to & \text{Spec }\mathbb Q.\end{bmatrix}$$ Since the two maps from <code>$\text{Spec }\overline{\mathbb Q}$</code> to $\text{Spec }\mathbb Q$ are the same, the two pullbacks are both identified with $H^i,$ and the isomorphism between them in the diagram is how $\sigma$ acts. It seems that the $\overline{\mathbb Q}$-structure of $H^i$ is "transported" by $\sigma.$ </p> http://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners-list/3720#3720 Answer by shenghao for Mathematicians who were late learners?-list shenghao 2009-11-02T01:06:50Z 2011-05-16T19:37:24Z <p>Lefschetz didn't move to math until he lost both of his hands in an industrial accident at the age of 23.</p> http://mathoverflow.net/questions/3716/what-are-the-automorphism-groups-of-principally-polarized-abelian-varieties What are the automorphism groups of (principally polarized) abelian varieties? shenghao 2009-11-01T23:28:52Z 2011-05-08T18:09:41Z <p>What are the possible automorphism groups of a principally polarized abelian variety $(A,\lambda)$ of dimension $g,$ say an abelian surface ($g=2$) over the complex numbers or algebraic closure of a finite field? The fact that the moduli stack $A_g$ is of finite diagonal (over the integers) implies that the automorphism groups are all finite, but do we know more? Like the size.</p> <p>When $g=1$ this is given in Silverman I, p.103.</p> <p><strong>Edit:</strong> Let me make the question more specific. Let $(A,\lambda)$ be an $\mathbb F_q$-point of $A_g$ (i.e. an abelian variety $A$ over $\mathbb F_q$ of dimension $g$ and a principal polarization $\lambda$). We want to consider its automorphism group (over $\mathbb F_q$).</p> <p>Let $\pi:A_{g,N}\to A_g$ be the natural projection, where $A_{g,N}$ is the moduli stack of p.p.a.v. of dimension $g$ with a level $N$ structure (a symplectic isomorphism $H^1(A,Z/N)\to(Z/N)^{2g}$). We always assume $q$ is prime to $N.$ Note that $\pi$ is a $G$-torsor, for $G=GSp(2g,Z/N),$ so it gives a surjective homomorphism $\pi_1(A_g)\to G.$ The sheaf $\pi_*\mathbb Q_l$ on $A_g$ is lisse (even locally constant), corresponding to the representation of $\pi_1(A_g)$ obtained from the regular representation $\mathbb Q_l[G]$ of $G$ and the projection $\pi_1(A_g)\to G.$ For any $\mathbb F_q$-point $x$ of $A_g,$ the local trace $\text{Tr}(Frob_x,(\pi_*\mathbb Q_l)_{\overline{x}})$ is either $|G|$ or 0, depending on $Frob_x\in\pi_1(A_g)$ is mapped to 1 in $G$ or not. </p> <p>We have isomorphisms $H^i_c(A_{g,N},\mathbb Q_l)=H^i_c(A_g,\pi_*\mathbb Q_l).$ By Lefschetz trace formula, applied to both $\mathbb Q_l$ on $A_{g,N}$ and $\pi_*\mathbb Q_l$ on $A_g,$ we have </p> <p><code>$$|A_{g,N}(\mathbb F_q)|=|G|\sum_{x\in S} 1/\#Aut(A_x,\lambda_x),$$</code></p> <p>where $S$ is the subset of $[A_g(\mathbb F_q)]$ consisting of points $x$ such that all $N$-torsion points of the abelian variety $A_x$ are rational over $\mathbb F_q$ (i.e. $|A_x[N](\mathbb F_q)|=N^{2g}$), and $(A_x,\lambda_x)$ is the pair corresponding to $x.$ This equation gives some constraints (one for each $N$) that $|Aut(A,\lambda)|$ must satisfy. In particular, when $g=N=2$ and $q=3,$ we have $|A_{2,2}(\mathbb F_3)|=10$ and $|G|=720$ (in this case $G$ is the symmetric group $S_6$), and this becomes a puzzle of solving $$1/72 = \sum 1/n_i,$$ and the $n_i$'s satisfy some additional conditions. Any idea on how to solve it? I'm considering the contributions of the two parts in $A_2,$ one for Jacobians of smooth genus 2 curves and one for Jacobians of stable singular ones $E_1\times E_2$. Any suggestion is appreciated.</p> <p><strong>Edit</strong>: Maybe it's easier to solve it over $\mathbb F_5,$ since the (orders of the) automorphism groups of smooth genus 2 curves over finite fields of characteristic 5 is known. </p> http://mathoverflow.net/questions/4125/if-spec-z-is-like-a-riemann-surface-whats-the-analogue-of-integration-along-a-c/4141#4141 Answer by shenghao for If Spec Z is like a Riemann surface, what's the analogue of integration along a contour? shenghao 2009-11-04T19:50:29Z 2011-05-08T14:25:26Z <p>For a variety $X$ over a finite field, I guess one can take $\ell$-adic sheaves to replace differential forms. Then the local integral around a closed point $x$ (like integral over a little loop around that point) is the trace of the local Frobenius $Frob_x$ on the stalk of sheaf, the so-called naive local term. Note that $Frob_x$ can be regarded as an element (or conjugacy class) in $\pi_1(X)$, "a loop around $x$". The global integral would be the global trace map <code>$$H^{2d}_c(X,\mathbf{Q}_{\ell})\to\mathbf{Q}_{\ell}(-d),$$</code> and the Tate twist is responsible for the Hodge structure in Betti cohomology (or the $(2\pi i)^d$ one has to divide by). The Lefschetz trace formula might be the analog of the residue theorem in complex analysis on Riemann surfaces.</p> <p>For the case of number fields, each closed point $v$ in $Spec\ O_k$ still defines a "loop" $Frob_v$ in $\pi_1(Spec\ k)$ (let's allow ramified covers. One can take the image of $Frob_v$ under $\pi_1(Spec\ k)\to\pi_1(Spec\ O_k)$, but the target group doesn't seem to be big enough). For global integral, there's the Artin-Verdier trace map $H^3(Spec\ O_k,\mathbb G_m)\to\mathbb{Q/Z}$ and a "Poincar\'e duality" in this setting, but I don't know if there is a trace formula. The fact that 3 is odd always makes me excited and confused.</p> <p>So basically I think of trace maps (both local and global) as counterpart of integrals. Correct me if I was wrong.</p> http://mathoverflow.net/questions/63212/homotopy-limits-of-dg-categories homotopy limits of dg categories shenghao 2011-04-27T18:48:49Z 2011-04-30T20:12:28Z <p>The question is related to the following MO question </p> <p><a href="http://mathoverflow.net/questions/38993/co-limits-and-fibrations-of-dg-categories" rel="nofollow">http://mathoverflow.net/questions/38993/co-limits-and-fibrations-of-dg-categories</a></p> <p>My question is, </p> <ol> <li><p>how to define the homotopy limit (and colimit) of a system of dg-categories (let's fix a universe and a base ring $k$, and work only with small things...), and </p></li> <li><p>is there an explicit description of the homotopy category of the homotopy limit of dg categories $$Ho(holim_{i\in I}\mathscr C_i)=?$$ Recall that the homotopy category $Ho(\mathscr C)$ of a dg category $\mathscr C$ is the category with the same objects as $\mathscr C$ and the hom group is the cohomology at degree 0 of the hom complex in $\mathscr C:$ $$Hom_{Ho(\mathscr C)}(X,Y)=H^0(Hom_{\mathscr C}(X,Y)).$$ One can ask similar questions to "categories" enriched in simplicial sets, which is a slightly more general setting.</p></li> </ol> <p>I understand (sort of) that there is a model category structure (due to Tabuada) on the category $dg-Cat$ of dg categories such that weak equivalences are what one expects (to be a bit precise, a functor $F:\mathscr C\to\mathscr D$ is a w.e. if $$Hom_{\mathscr C}(X,Y)\to Hom_{\mathscr D}(FX,FY)$$ is a quasi-isomorphism of complexes, and $Ho(F):Ho(\mathscr C)\to Ho(\mathscr D)$ is essentially surjective). But I don't know how to use this model structure to define homotopy limits. </p> <p>Maybe one uses cofibrant replacement and the naive $\otimes$-structure on $dg-Cat$ to define a $\otimes^{\mathbb L}$-structure (following Toen) and shows that it is closed, so that one has internal hom $R\mathscr Hom$ on $dg-Cat,$ with which one defines homotopy limits (and colimits) of dg-categories by universal properties. I'm not sure. Both references and direct explanations are appreciated.</p> http://mathoverflow.net/questions/62406/is-the-weight-filtration-a-topological-invariant Is the weight filtration a topological invariant? shenghao 2011-04-20T13:06:28Z 2011-04-21T12:29:28Z <p>This question is somehow related to (but different from) the following MO question and the one linked from there</p> <p><a href="http://mathoverflow.net/questions/42744/diffeomorphic-kahler-manifolds-with-different-hodge-numbers" rel="nofollow">http://mathoverflow.net/questions/42744/diffeomorphic-kahler-manifolds-with-different-hodge-numbers</a></p> <p>Let $X$ and $X'$ be two complex algebraic varieties that are diffeomorphic to each other. One may or may not assume they are irreducible. Then:</p> <ol> <li><p>(Weak form) For each pair of integers $(i,n),$ do we always have $$\dim Gr^W_iH^n(X,\mathbb Q)=\dim Gr^W_iH^n(X',\mathbb Q),$$ where $W$ denotes the weight filtration on the mixed Hodge structures?</p></li> <li><p>(Strong form) Let $f:X\to X'$ be a diffeomorphism. Does $f^*:H^n(X')\to H^n(X)$ respect the $W$-filtrations?</p></li> </ol> <p>Variants: One can ask similar questions with "diffeomorphic" replaced by "homeomorphic" or "complex analytically isomorphic". For instance, does the weight filtration see the difference between tangential intersections and transverse intersections? One can also add coefficients.</p> <p>At the first sight, it's not clear to me how to approach the question, since there seems to be no relation between the resolutions of $X$ and $X'.$ </p> <p><strong>Edit:</strong> As algori pointed out to me, this is false without the compactness assumption, even for "complex analytic isomorphism". So I will assume $X$ and $X'$ are compact.</p> http://mathoverflow.net/questions/805/cohomology-of-moduli-spaces cohomology of moduli spaces shenghao 2009-10-16T21:59:52Z 2011-04-14T12:44:33Z <p>Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ and $N?$ Like dimensions of the cohomology spaces and the weights. Thanks.</p> <p><strong>Edit:</strong> I'm particularly interested in the weights of these $\ell$-adic cohomology of moduli varieties defined over finite field, or even the precise Frobenius eigenvalues, for the purpose of independence of $\ell$ and automorphy. Therefore I would like to know $H^i$ for all $i,$ in particular the middle cohomology (e.g. $H^3(A_{2,N})$). </p> http://mathoverflow.net/questions/60214/geometric-interpretation-of-the-alternating-sum-of-intersection-cohomology-grou "geometric" interpretation of the alternating sum of intersection cohomology groups shenghao 2011-03-31T17:35:04Z 2011-03-31T22:55:33Z <p>Let $X_0$ be a proper variety over a finite field $k.$ For each prime number $\ell\ne p,$ we have the $\ell$-adic intersection cohomology groups $IH^i(X).$ Due to Gabber, the alternating sum of these $$\sum_{i=0}^{2\dim X_0}(-1)^i[IH^i(X)],$$ regarded as a virtual representation of the Weil group $W(\overline{k}/k)$ of $k$ (i.e. the cyclic group in Gal generated by Frobenius), is independent of $\ell.$ Of course it is also self-dual (up to Tate twist), and (together with Gabber's purity) each summand is independent of $\ell.$ </p> <p>My question is: does this alternating sum have any "geometric" interpretation? Or, to what extent (and in which way) this Euler characteristic depends on the singular locus of $X?$</p> <p>E.g. when $X$ is in addition smooth, this can be interpreted in terms of the number of $k$-rational points. Being independent of $\ell,$ it's not surprising to expect it to have some "motivic" meaning, e.g. in terms of the geometry. Certainly one could also expect something for the individual $IH^i(X),$ but it seems that the Euler characteristic is always easier to relate to geometry. </p> <p>Answers/comments in the general setting would be great, and those addressing to Shimura varieties and their Satake compactifications are especially appreciated.</p> http://mathoverflow.net/questions/58669/intersection-pairing-on-intersection-cohomology intersection pairing on intersection cohomology shenghao 2011-03-16T18:11:48Z 2011-03-19T17:40:52Z <p>Let $X$ be a projective variety of dimension $d$ over $k=\bar{k},$ with $L$ an ample line bundle on $X$ and $\eta=c_1(L).$ Hard Lefschetz gives an isomorphism (see BBD) $$\eta^i:IH^{d-i}(X)\to IH^{d+i}(X)$$ with Tate twist ignored, which, together with the intersection pairing between $IH^{d-i}$ and $IH^{d+i},$ gives a non-degenerate bilinear form $$IH^n(X)\times IH^n(X)\to(\mathbb Q,\mathbb Q_{\ell},\text{ or }\mathbb C...)$$ for each $n.$</p> <p>Question: Is it $(-1)^n$-symmetric? </p> <p>This is so when $X$ is non-singular (which follows from the general fact on "cup products"), or when $n=d.$ The question is related to this MO question <a href="http://mathoverflow.net/questions/57701/poincare-duality-for-intersection-cohomology" rel="nofollow">http://mathoverflow.net/questions/57701/poincare-duality-for-intersection-cohomology</a>. I guess one can probably figured it out by doing some homological algebra on the level of complexes (i.e. before taking hypercohomology groups), and maybe it's written down somewhere.</p> http://mathoverflow.net/questions/58673/finite-quotients-of-fundamental-groups-in-positive-characteristic finite quotients of fundamental groups in positive characteristic shenghao 2011-03-16T18:36:06Z 2011-03-16T22:10:10Z <p>For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental groups. </p> <p>What about complete smooth curves, or more generally higher dimensional varieties? Are there results or conjectural criteria (or necessary conditions) for finite quotients of their $\pi_1?$ (Definitely, not too much was known around 1990; see Serre's Bourbaki article on this.)</p> <p>In particular, let $G$ be the automorphism group of the supersingular elliptic curve in char. $p=2$ or $3$ (see <a href="http://mathoverflow.net/questions/58455/supersingular-elliptic-curve-in-char-2-or-3" rel="nofollow">http://mathoverflow.net/questions/58455/supersingular-elliptic-curve-in-char-2-or-3</a> for various descriptions of its structure). Is there (and if yes, how to construct) a projective smooth variety in char. $p$ having $G$ as a quotient of its $\pi_1?$ Certainly there are lots of affine smooth curves with this property (e.g. $\mathbb G_m$), and I wonder if for some of them, the covering is unramified at infinity (so that we win!).</p> http://mathoverflow.net/questions/58455/supersingular-elliptic-curve-in-char-2-or-3 supersingular elliptic curve in char. 2 or 3 shenghao 2011-03-14T18:29:30Z 2011-03-14T19:30:46Z <p>Let $p=2$ or 3, and let $k$ be an algebraically closed field of char. $p.$ Let $E$ be the supersingular elliptic curve over $k$ (with $j=0$). Let $G$ be the automorphism group of $E,$ which has order 12 (resp. 24) when $p=3$ (resp. 2). Then the $\ell$-adic cohomology $H^1(E,\mathbb Q_{\ell})$ is a 2-dimensional representation of $G.$ Is it irreducible?</p> <p>Since we know the group structure of $G$ (cf. Silverman's AEC, Appendix A, exercise..), does anyone have a reference for its character table?</p> http://mathoverflow.net/questions/57997/tensor-product-of-simple-representations Tensor product of simple representations shenghao 2011-03-09T21:06:39Z 2011-03-10T00:23:33Z <p>Let $G$ be a linear algebraic group over some field, and let $V$ and $W$ be two simple rational representations of $G.$ Is $V\otimes W$ semi-simple?</p> <p>I was trying to convince myself that if $G$ has a faithful semi-simple representation, then $G$ is linearly reductive, and was reduced to the question above. The problem I have in mind is over characteristic 0, but answers addressing char. $p$ is equally appreciated too!</p> http://mathoverflow.net/questions/129811/frobenius-weights-on-etale-cohomology-and-purity Comment by shenghao shenghao 2013-05-06T15:43:55Z 2013-05-06T15:43:55Z The isomorphism about Tate twists follows from the projection formula $Rf_*(\mathscr F\otimes^Lf^*\mathscr G)\simeq Rf_*(\mathscr F)\otimes^L\mathscr G.$ Then $\mathscr G$ is flat, there's no need to put $"L"$ there. http://mathoverflow.net/questions/117292/why-is-a-ring-called-a-ring Comment by shenghao shenghao 2013-05-05T17:50:55Z 2013-05-05T17:50:55Z Isn't $\mathbb Z/n\mathbb Z$ a ring-like object? http://mathoverflow.net/questions/122609/where-can-i-find-the-two-books-of-weil-on-abelian-varieties Comment by shenghao shenghao 2013-02-22T14:13:36Z 2013-02-22T14:13:36Z Great! Thanks for the practical instruction. http://mathoverflow.net/questions/122600/understanding-adjointness-of-sheaves-in-algebraic-geometry Comment by shenghao shenghao 2013-02-22T06:03:22Z 2013-02-22T06:03:22Z Dear LMN: maybe you want to get a better feeling for general adjoint functors, as in MacLane's book? There're tons of examples there to help. Pullback and pushforward, if defined correctly, should always be "adjoint" to each other in some sense. E.g. given a cont. map f:X -> Y of spaces, one can pullback functions on Y and pushforward measures on X, and the "natural isomorphism" as in adjoint functors becomes an equality of integrals. BTW, the proof seems to be formal: first prove it for presheaves (using univ. property of colimit), then use univ. property of sheafification. http://mathoverflow.net/questions/119627/does-every-regular-noetherian-domain-have-finite-krull-dimension Comment by shenghao shenghao 2013-02-18T13:18:42Z 2013-02-18T13:18:42Z No, I meant that without the regularity assumption, there are examples of noetherian domains of infinite Krull dimension. http://mathoverflow.net/questions/66987/choice-of-local-system-in-delignes-construction-of-l-adic-galois-representatio Comment by shenghao shenghao 2012-09-03T07:51:04Z 2012-09-03T07:51:04Z B. Conrad's (still unpublished?) book on Ramanujan's conjecture is a (the?) modern exposition on Deligne's 1969 Bourbaki talk. It has all detailed computations. Check Brian's webpage. http://mathoverflow.net/questions/106128/what-is-the-importance-of-the-conjectural-semi-simplicity-of-the-action-of-the-fr Comment by shenghao shenghao 2012-09-02T13:53:54Z 2012-09-02T13:53:54Z The question applies more generally: after knowing the characteristic polynomial of a matrix, why it is important to know its Jordan form? I guess knowing how the space decomposes into generalized eigenspaces will usually be helpful, in various situations. In the case of etale cohomology, as Yuri said, if you know there are many eigenvectors for an eigenvalue, then (modulo Tate conjecture) it will lead to many algebraic cycles, which is usually a difficult thing in alg. geom. This also applies to K3 (a recent result?). http://mathoverflow.net/questions/104083/trace-morphism-for-stacks Comment by shenghao shenghao 2012-08-06T16:18:01Z 2012-08-06T16:18:01Z It might be helpful to check LMB and Olsson's article 'Sheaves on Artin stacks', for cotangent complexes on stacks. http://mathoverflow.net/questions/104102/semisimplicity-of-frobenius-operation-on-etale-cohomology/104105#104105 Comment by shenghao shenghao 2012-08-06T16:07:52Z 2012-08-06T16:07:52Z I'm not sure if Faltings' work leads to semisimplicity of the $Gal(\bar{\mathbb Q}/\mathbb Q)$-representation, which is a different conjecture. http://mathoverflow.net/questions/104102/semisimplicity-of-frobenius-operation-on-etale-cohomology Comment by shenghao shenghao 2012-08-06T16:02:31Z 2012-08-06T16:02:31Z Also true for K3 surfaces, by Deligne. See Lei Fu's AMS article 'On the semisimplicity conjecture and Galois representations' for more info. For mixed cohomology groups, the weight filtration does not split in general (it would be split if Frobenius were always semisimple), and examples can be found already in dimension 1. http://mathoverflow.net/questions/104003/is-there-an-explicit-example-of-a-coefficient-sheaf-for-which-hard-lefschetz-fail/104011#104011 Comment by shenghao shenghao 2012-08-05T15:57:28Z 2012-08-05T15:57:28Z Thanks for the answer, Donu. A somehow related question: what about varieties over $\overline{\mathbb F}_p?$ Namely $X$ is projective (smooth or not) and $L$ is semisimple local system (or perverse sheaf). Without being of geometric origin, they won't have a model over a finite subfield, so that the usual proof breaks. http://mathoverflow.net/questions/103101/frobenius-eigenvalues-of-abelian-variety Comment by shenghao shenghao 2012-07-25T18:03:44Z 2012-07-25T18:03:44Z maybe PD with polarization. http://mathoverflow.net/questions/102729/axioms-for-zeta-function-of-a-scheme Comment by shenghao shenghao 2012-07-21T23:35:57Z 2012-07-21T23:35:57Z With pleasure: For an abelian variety, $H^1$ determines the others: $H^i$ is the $i$-fold wedge product of $H^1$, so knowing what $H^1$ is gives the full zeta function. For a curve, $H^0$ and $H^2$ are always known, so again $H^1$ determines the zeta function. But for other varieties we are less lucky. I don't know how to tell the cup-product structure (for $X$ a proper variety) from the zeta function either. http://mathoverflow.net/questions/102819/finite-field-varieties-and-the-de-rham-complex-of-kahler-differentials Comment by shenghao shenghao 2012-07-21T23:20:13Z 2012-07-21T23:20:13Z The Hodge-de-Rham spectral sequence may not degenerate at E_2, and the Hodge symmetry $h^{pq}=h^{qp}$ may break down, etc. http://mathoverflow.net/questions/102729/axioms-for-zeta-function-of-a-scheme Comment by shenghao shenghao 2012-07-21T10:05:11Z 2012-07-21T10:05:11Z A related question is to ask for characterization （even conjectural） of Frobenius char. polynomials on cohomology of a fixed degree. For curves and abelian varieties one can use Honda-Tate，and this related question is equivalent to your original one. I believe that very little common feature is known for general schemes （not nece. proper or smooth）. For projective smooth varieties，by weak Lefschetz，the most interesting ones are the middle two， about which again very little is known. | 2013-05-23 20:57:56 | {"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": 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": 0.9461143016815186, "perplexity": 472.3384458726016}, "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-2013-20/segments/1368703788336/warc/CC-MAIN-20130516112948-00051-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/107839/holomorphic-extension-of-a-function?sort=votes | holomorphic extension of a function [closed]
hi,
I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : \tilde{U} \rightarrow \mathbb{R}$. Can this function be holomorphically extended to $U$ (maybe if we shrink $U$) in a unique way? I would be very thankful for answers.
bruno
-
closed as off-topic by Ricardo Andrade, Stefan Kohl, David White, Willie Wong, Boris BukhDec 2 '13 at 23:08
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Ricardo Andrade, Stefan Kohl, David White, Boris Bukh
If this question can be reworded to fit the rules in the help center, please edit the question.
Yes, of course, after we shrink $U$. A convergent Taylor series at a real point converges in some complex neighborhood of this point.
Added reply to your comment: you can apply identity theorem. Two real analytic functions coinciding on an open set of $R^n$ coincide in a complex neighborhood of this set. | 2016-07-23 19:36:15 | {"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": 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": 0.9361603260040283, "perplexity": 520.1358560825228}, "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-2016-30/segments/1469257823387.9/warc/CC-MAIN-20160723071023-00285-ip-10-185-27-174.ec2.internal.warc.gz"} |
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# In the diagram to the below, the value of x is closest to which of the
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In the diagram to the below, the value of x is closest to which of the [#permalink]
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In the diagram to the below, the value of $$x$$ is closest to which of the following?
A) $$2+√2$$
B) $$2$$
C) $$√3$$
D) $$√2$$
E) $$1$$
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In the diagram to the below, the value of x is closest to which of the [#permalink]
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Updated on: 11 Mar 2020, 17:45
In the diagram to the below, the value of $$x$$ is closest to which of the following?
A) $$2+√2$$
B) $$2$$
C) $$√3$$
D) $$√2$$
E) $$1$$
Let us drawn AP perpendicular to BC as shown below:
Attachment:
111.JPG
In triangle APC: Angle ACP = Angle CAP = $$45^o$$
=> It is a 45-45-90 triangle
=> AP = PC = AC/$$\sqrt{2}$$ = $$x/\sqrt{2}$$
=> BP = $$x - x/\sqrt{2}$$
In right triangle ABP: $$AB^2 = AP^2 + BP^2$$
=> $$2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2$$
=> $$2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}$$
=> $$2 = x^2 * (2 - \sqrt{2})$$
=> $$x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33$$ (approximately)
=> $$x = \sqrt{3.33} = 1.8$$
=> Closest value = $$\sqrt{3}$$
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Originally posted by sujoykrdatta on 11 Mar 2020, 11:39.
Last edited by sujoykrdatta on 11 Mar 2020, 17:45, edited 1 time in total.
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In the diagram to the below, the value of x is closest to which of the [#permalink]
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Updated on: 11 Mar 2020, 18:36
sujoykrdatta wrote:
In the diagram to the below, the value of $$x$$ is closest to which of the following?
A) $$2+√2$$
B) $$2$$
C) $$√3$$
D) $$√2$$
E) $$1$$
Let us drawn AP perpendicular to BC as shown below:
Attachment:
111.JPG
In triangle APC: Angle ACP = Angle CAP = $$45^o$$
=> It is a 45-45-90 triangle
=> AP = PC = AC/$$\sqrt{2}$$ = $$x/\sqrt{2}$$
=> BP = $$x - x/\sqrt{2}$$
In right triangle ABP: $$AB^2 = AP^2 + BP^2$$
=> $$2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2$$
=> $$2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}$$
=> $$2 = x^2 * (2 - \sqrt{2})$$
=> $$x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33$$ (approximately)
=> $$x = \sqrt{3.33} = 1.8$$
=> Closest value = 2
$$√3.33$$ = $$1.8$$
1.8 is closed to $$√3$$. So the correct choice is C, actually.
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Originally posted by TheUltimateWinner on 11 Mar 2020, 12:16.
Last edited by TheUltimateWinner on 11 Mar 2020, 18:36, edited 1 time in total.
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Re: In the diagram to the below, the value of x is closest to which of the [#permalink]
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11 Mar 2020, 12:53
sujoykrdatta wrote:
In the diagram to the below, the value of $$x$$ is closest to which of the following?
A) $$2+√2$$
B) $$2$$
C) $$√3$$
D) $$√2$$
E) $$1$$
Let us drawn AP perpendicular to BC as shown below:
Attachment:
111.JPG
In triangle APC: Angle ACP = Angle CAP = $$45^o$$
=> It is a 45-45-90 triangle
=> AP = PC = AC/$$\sqrt{2}$$ = $$x/\sqrt{2}$$
=> BP = $$x - x/\sqrt{2}$$
In right triangle ABP: $$AB^2 = AP^2 + BP^2$$
=> $$2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2$$
=> $$2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}$$
=> $$2 = x^2 * (2 - \sqrt{2})$$
=> $$x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33$$ (approximately)
=> $$x = \sqrt{3.33} = 1.8$$
=> Closest value = 2
$$√3.33$$ = $$1.43$$
1.43 is closed to $$√3$$. So the correct choice is C, actually.
How is $$√3.33$$ = $$1.43$$?
$$1.4^2 = 1.96$$
$$1.5^2 = 2.25$$
$$1.8^2 = 3.24$$
_________________
Sujoy Kumar Datta
Director - CUBIX Educational Institute Pvt. Ltd. (https://www.cubixprep.com)
IIT Kharagpur, TU Dresden Germany
GMAT - Q51 & CAT (MBA @ IIM) 99.98 Overall with 99.99 QA
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Re: In the diagram to the below, the value of x is closest to which of the [#permalink]
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11 Mar 2020, 14:12
IMO ans is 1
Its an iso . triangle bcoz it has 2 equal sides.
One of the angle is 45 so the other one is also 45( same sides)
Its a 45 45 90 triangle.
The sides of 45 45 90 triangles are in ratio of 1 1 root 2.
Hypo is root 2 therefore x is 1
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In the diagram to the below, the value of x is closest to which of the [#permalink]
### Show Tags
11 Mar 2020, 15:55
sujoykrdatta wrote:
In the diagram to the below, the value of $$x$$ is closest to which of the following?
A) $$2+√2$$
B) $$2$$
C) $$√3$$
D) $$√2$$
E) $$1$$
Let us drawn AP perpendicular to BC as shown below:
Attachment:
111.JPG
In triangle APC: Angle ACP = Angle CAP = $$45^o$$
=> It is a 45-45-90 triangle
=> AP = PC = AC/$$\sqrt{2}$$ = $$x/\sqrt{2}$$
=> BP = $$x - x/\sqrt{2}$$
In right triangle ABP: $$AB^2 = AP^2 + BP^2$$
=> $$2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2$$
=> $$2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}$$
=> $$2 = x^2 * (2 - \sqrt{2})$$
=> $$x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33$$ (approximately)
=> $$x = \sqrt{3.33} = 1.8$$
=> Closest value = 2
Hi sir, I think its closer to $$\sqrt{3}$$.
$$\sqrt{3}$$=1.7. Your answer is 1.8. It is closer to 1.7 than 2.
Anyways my working is as follows -
$$2 = (\frac{x}{\sqrt{2}})^2 + (x - \frac{x}{\sqrt{2}})^2$$
$$or, 2 = 2x^2 -\frac{2x^2}{\sqrt{2}}$$
$$or, 1 = x^2 - \frac{x^2}{\sqrt{2}}$$
$$or, x^2 = \frac{\sqrt{2}}{(\sqrt{2}-1)}$$
$$or, x^2 = 1.4/0.4 = 3 (approx)$$
$$or, x = \sqrt{3}$$
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Re: In the diagram to the below, the value of x is closest to which of the [#permalink]
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11 Mar 2020, 17:47
AnirudhaS wrote:
sujoykrdatta wrote:
In the diagram to the below, the value of $$x$$ is closest to which of the following?
A) $$2+√2$$
B) $$2$$
C) $$√3$$
D) $$√2$$
E) $$1$$
Let us drawn AP perpendicular to BC as shown below:
Attachment:
111.JPG
In triangle APC: Angle ACP = Angle CAP = $$45^o$$
=> It is a 45-45-90 triangle
=> AP = PC = AC/$$\sqrt{2}$$ = $$x/\sqrt{2}$$
=> BP = $$x - x/\sqrt{2}$$
In right triangle ABP: $$AB^2 = AP^2 + BP^2$$
=> $$2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2$$
=> $$2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}$$
=> $$2 = x^2 * (2 - \sqrt{2})$$
=> $$x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33$$ (approximately)
=> $$x = \sqrt{3.33} = 1.8$$
=> Closest value = 2
Hi sir, I think its closer to $$\sqrt{3}$$.
$$\sqrt{3}$$=1.7. Your answer is 1.8. It is closer to 1.7 than 2.
Anyways my working is as follows -
$$2 = (\frac{x}{\sqrt{2}})^2 + (x - \frac{x}{\sqrt{2}})^2$$
$$or, 2 = 2x^2 -\frac{2x^2}{\sqrt{2}}$$
$$or, 1 = x^2 - \frac{x^2}{\sqrt{2}}$$
$$or, x^2 = \frac{\sqrt{2}}{(\sqrt{2}-1)}$$
$$or, x^2 = 1.4/0.4 = 3 (approx)$$
$$or, x = \sqrt{3}$$
Yes obviously
I don't know what came of me
Must have not seen that option of root3 at all! - corrected that - thanks
Posted from my mobile device
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Director - CUBIX Educational Institute Pvt. Ltd. (https://www.cubixprep.com)
IIT Kharagpur, TU Dresden Germany
GMAT - Q51 & CAT (MBA @ IIM) 99.98 Overall with 99.99 QA
_________
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Re: In the diagram to the below, the value of x is closest to which of the [#permalink]
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11 Mar 2020, 18:44
sujoykrdatta wrote:
sujoykrdatta wrote:
Let us drawn AP perpendicular to BC as shown below:
Attachment:
111.JPG
In triangle APC: Angle ACP = Angle CAP = $$45^o$$
=> It is a 45-45-90 triangle
=> AP = PC = AC/$$\sqrt{2}$$ = $$x/\sqrt{2}$$
=> BP = $$x - x/\sqrt{2}$$
In right triangle ABP: $$AB^2 = AP^2 + BP^2$$
=> $$2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2$$
=> $$2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}$$
=> $$2 = x^2 * (2 - \sqrt{2})$$
=> $$x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33$$ (approximately)
=> $$x = \sqrt{3.33} = 1.8$$
=> Closest value = 2
$$√3.33$$ = $$1.43$$
1.43 is closed to $$√3$$. So the correct choice is C, actually.
How is $$√3.33$$ = $$1.43$$?
$$1.4^2 = 1.96$$
$$1.5^2 = 2.25$$
$$1.8^2 = 3.24$$
My calculator gave me wrong info (value of root 3.33). I was convinced with that wrong value (1.43) because i already know that the correct choice is C.
Edited the first comment..
Posted from my mobile device
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Re: In the diagram to the below, the value of x is closest to which of the [#permalink]
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29 Mar 2020, 06:15
I do not understand this question. When x and x are the same lengths, their respective angles have to have the same degree, right ? so i do not understand how anyonecan say this is a 45 - 45 - 90 triangle. If this was true root 2 has to be the longest side, so that the angles of 45 and 45 are respective to the two x.
In this case side root 2 faces the angle 45 so the angles for x are 135/2, so you can not use the pythagorean theorem to calculate the sides?
Re: In the diagram to the below, the value of x is closest to which of the [#permalink] 29 Mar 2020, 06:15 | 2020-07-06 20:35:35 | {"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": 0.7070521116256714, "perplexity": 5301.869084899612}, "config": {"markdown_headings": true, "markdown_code": true, "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-2020-29/segments/1593655890181.37/warc/CC-MAIN-20200706191400-20200706221400-00434.warc.gz"} |
https://math.libretexts.org/?title=Courses/Mount_Royal_University/FK__Draft-testing/Unit_2.2/Testing_Guide_1:_Limit_%26_Continuity_of_Functions/Chapter_1:_Limits |
# Chapter 1: Limits
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$
$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$
## Limits
Two key problems led to the initial formulation of calculus:
1. the tangent problem, or how to determine the slope of a line tangent to a curve at a point, and
2. the area problem, or how to determine the area under a curve.
The concept of a limit or limiting process, essential to the understanding of calculus, has been around for thousands of years. In fact, early mathematicians used a limiting process to obtain better and better approximations of areas of circles. Yet, the formal definition of a limit—as we know and understand it today—did not appear until the late 19th century. We therefore begin our quest to understand limits, as our mathematical ancestors did, by using an intuitive approach. | 2019-02-20 12:23:51 | {"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": 0.84749835729599, "perplexity": 260.91370899385424}, "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-2019-09/segments/1550247494741.0/warc/CC-MAIN-20190220105613-20190220131613-00568.warc.gz"} |
https://mathoverflow.net/questions/272310/on-toroidal-compactifications-of-hilbert-kuga-sato-varieties | # On toroidal compactifications of Hilbert Kuga-Sato varieties
Let $F$ be a totally real field of degree d. There are Hilbert modular varieties over $\mathbb{Q}$ that paramatrize abelian varieties of dimension d with an action of $\mathcal{O}_F$ the ring of integers of $F$ and some level structure.
Suppose we fix some level and we consider $\mathcal{A}\to M$, the universal abelian variety and for any positive integer $n$ the $n$-fold fiber product $\mathcal{A}^n\to M$ over $M$. This Kuga-Sato variety comes endowed with an action of $M_{n\times n}(\mathcal{O}_F)$. Do you know if these endomorphisms extend to the smooth toroidal compactifications denoted by $\overline{\mathcal{A}^n}$ in http://math.univ-lille1.fr/~mladen/articles/Pad16-DiTi.pdf ?
• Naive question: do you know the answer in the case $n=1$? Jun 16, 2017 at 19:57
• Unfortunatly no. In fact, if I knew it holds for $n=1$, I could probably get an action of $(\mathcal{O}_F^\times)^n\rtimes S_n$ on some desingularization of $\left(\overline{\mathcal{A}}\right)^n$ using reasolution of singularities. Anyway, I know that what I want is true $\forall n$ in the case when $F=\mathbb{Q}$ because Deligne and Scholl constructed canonical desingularizations for $\left(\overline{\mathcal{E}}\right)^n$ for $\overline{\mathcal{E}}$ the universal generalized elliptic curve.
• Actually one could only get an action of $\{\pm1\}^n\rtimes S_n$ using resolution of singularities because $(\mathcal{O}_F^\times)^n\rtimes S_n$ is not a finite group. | 2022-12-03 21:36:40 | {"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": 0.8868951797485352, "perplexity": 245.7499800574448}, "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/1669446710941.43/warc/CC-MAIN-20221203212026-20221204002026-00448.warc.gz"} |
https://support.10xgenomics.com/single-cell-gene-expression/software/pipelines/latest/algorithms/overview | HOME › pipelines
# Gene Expression Algorithms Overview
The computational pipeline cellranger count or multi for 3' Single Cell Gene Expression involves the following analysis steps:
The key read processing steps are outlined in this figure and described in the text below:
## Alignment
This section on read trimming applies to 3' gene expression assays.
A full length cDNA construct is flanked by the 30 bp template switch oligo (TSO) sequence, AAGCAGTGGTATCAACGCAGAGTACATGGG, on the 5' end and poly-A on the 3' end. Some fraction of sequencing reads are expected to contain either or both of these sequences, depending on the fragment size distribution of the sequencing library. Reads derived from short RNA molecules are more likely to contain either or both TSO and poly-A sequence than longer RNA molecules.
Since the presence of non-template sequence in the form of either TSO or poly-A, low-complexity ends confound read mapping, TSO sequence is trimmed from the 5' end of read 2 and poly-A is trimmed from the 3' end prior to alignment. Trimming improves the sensitivity of the assay as well as the computational efficiency of the software pipeline.
Tags ts:i and pa:i in the output BAM files indicate the number of TSO nucleotides trimmed from the 5' end of read 2 and the number of poly-A nucleotides trimmed from the 3' end. The trimmed bases are present in the sequence of the BAM record, and the CIGAR string shows the position of these soft-clipped sequences.
### Genome alignment
Cell Ranger uses an aligner called STAR, which performs splicing-aware alignment of reads to the genome. Cell Ranger then uses the transcript annotation GTF to bucket the reads into exonic, intronic, and intergenic, and by whether the reads align (confidently) to the genome. A read is exonic if at least 50% of it intersects an exon, intronic if it is non-exonic and intersects an intron, and intergenic otherwise.
For reads that align to a single exonic locus but also align to 1 or more non-exonic loci, the exonic locus is prioritized and the read is considered to be confidently mapped to the exonic locus with MAPQ 255.
### Transcriptome alignment
Cell Ranger further aligns confidently mapped exonic and intronic reads to annotated transcripts by examining their compatibility with the transcriptome. As shown below, reads are classified based on whether they are sense or antisense (purple) and whether they are exonic (light blue) or intronic (red).
Starting in Cell Ranger 7.0, by default, the cellranger count and cellranger multi pipelines will include intronic reads for whole transcriptome gene expression analysis. Any reads that map in the sense orientation to a single gene - the reads labeled transcriptomic (blue) in the diagram above - are carried forward to UMI counting. Cell Ranger ignores antisense reads that are defined as any read with alignments to an entire gene on the opposite strand and no sense alignments. Consequently, the web summary metrics "Reads Mapped Confidently to Transcriptome" and "Reads Mapped Antisense to Gene" will reflect reads mapped confidently to exonic regions, as well as intronic regions. The default setting that includes intronic reads is recommended to maximize sensitivity, for example in cases where the input to the assay consists of nuclei, as there may be high levels of intronic reads generated by unspliced transcripts.
To exclude intronic reads, include-introns must be set to false. In which case, Cell Ranger uses transcriptomic (blue) reads with sense alignments (and ignores antisense alignments) for UMI counting. A read is now classified as antisense (purple) if it has any alignments to a transcript exon on the opposite strand and no sense alignments.
Furthermore, a read is considered uniquely mapping if it is compatible with only a single gene. Only uniquely mapping reads are carried forward to UMI counting (see this page for methods to check for multi-mapped reads).
### UMI counting
Before counting UMIs, Cell Ranger attempts to correct for sequencing errors in the UMI sequences. Reads that were confidently mapped to the transcriptome are placed into groups that share the same barcode, UMI, and gene annotation. If two groups of reads have the same barcode and gene, but their UMIs differ by a single base (i.e., are Hamming distance 1 apart), then one of the UMIs was likely introduced by a substitution error in sequencing. In this case, the UMI of the less-supported read group is corrected to the UMI with higher support.
Cell Ranger again groups the reads by barcode, UMI (possibly corrected), and gene annotation. If two or more groups of reads have the same barcode and UMI, but different gene annotations, the gene annotation with the most supporting reads is kept for UMI counting, and the other read groups are discarded. In case of a tie for maximal read support, all read groups are discarded, as the gene cannot be confidently assigned.
After these two filtering steps, each observed barcode, UMI, gene combination is recorded as a UMI count in the unfiltered feature-barcode matrix. The number of reads supporting each counted UMI is also recorded in the molecule info file.
## Calling cell barcodes
Cell Ranger 3.0 introduces an improved cell-calling algorithm that is better able to identify populations of low RNA content cells, especially when low RNA content cells are mixed into a population of high RNA content cells. For example, tumor samples often contain large tumor cells mixed with smaller tumor infiltrating lymphocytes (TIL) and researchers may be particularly interested in the TIL population. The new algorithm is based on the EmptyDrops method (Lun et al., 2019).
The algorithm has two key steps:
1. It uses a cutoff based on total UMI counts of each barcode to identify cells. This step identifies the primary mode of high RNA content cells.
2. Then the algorithm uses the RNA profile of each remaining barcode to determine if it is an "empty" or a cell-containing partition. This second step captures low RNA content cells whose total UMI counts may be similar to empty GEMs.
In the first step, the original Cell Ranger cell calling algorithm is used to identify the primary mode of high RNA content cells, using a cutoff based on the total UMI count for each barcode. Cell Ranger may take as input an expected number of recovered cells (e.g., see --expect-cells for count), N, or as of Cell Ranger 7.0, estimate this number. This estimation employs a grid-search to find a value minimally different from the number of cell barcodes identified by the order of magnitude algorithm. For this algorithm, let m be the 99th percentile of the top N barcodes by total UMI counts. All barcodes whose total UMI counts exceed m/10 are called as cells.
In the second step, a set of barcodes with low UMI counts that likely represent "empty" GEM partitions is selected. A model of the RNA profile of selected barcodes is created. This model, called the background model, is a multinomial distribution over genes. It uses Simple Good-Turing smoothing to provide a non-zero model estimate for genes that were not observed in the representative empty GEM set. Finally, the RNA profile of each barcode not called as a cell in the first step is compared to the background model. Barcodes whose RNA profile strongly disagrees with the background model are added to the set of positive cell calls. This second step identifies cells that are clearly distinguishable from the profile of empty GEMs, even though they may have much lower RNA content than the largest cells in the experiment.
Below is an example of a challenging cell calling scenario where 300 high RNA content 293T cells are mixed with 2,000 low RNA content PBMC cells. On the left is the cell calling result with the cell calling algorithm prior to Cell Ranger 3.0 and on the right is the Cell Ranger 3.0 result. You can see that low RNA content cells are successfully identified by the new algorithm.
Cell Ranger 2.2 Cell Ranger 3.0
The plot shows the count of filtered UMIs mapped to each barcode. Barcodes can be determined to be cell-associated based on their UMI count or by their RNA profiles. Therefore some regions of the graph can contain both cell-associated and background-associated barcodes. The color of the graph represents the local density of barcodes that are cell-associated.
In some cases the set of barcodes called as cells may not match the desired set of barcodes based on visual inspection. This can be remedied by either re-running count or reanalyze with the --force-cells option, or by selecting the desired barcodes from the raw feature-barcode matrix in downstream analysis. Custom barcode selection can also be done by specifying --barcodes to reanalyze.
## Calling cell barcodes in targeted gene expression
Cell Ranger 4.0 includes a new dedicated cell calling algorithm that is applied specifically to identify cells from Targeted Gene Expression datasets. The Targeted Gene Expression cell calling method relies on identification of a transition point based on the shape of the barcode rank plot to separate cell-associated barcodes from background partitions, and is designed to successfully classify cells for a wide variety of potential target gene panels.
Note: This method is distinct from the cell calling algorithm used for non-targeted Whole Transcriptome Analysis (WTA) Chromium single cell datasets, and it does not employ the EmptyDrops approach (Lun et al., 2019) due to the distinct features of Targeted Gene Expression datasets.
## Calling cell barcodes in Feature Barcode only analysis
When using Cell Ranger in Feature Barcode Only Analysis mode, only step 1 of the cell calling algorithm is used. The cells called by this step of the algorithm are returned directly.
## Estimating multiplet rates
When multiple genomes are present in the reference (for example, Human and Mouse, or any other pair), Cell Ranger runs a special multi-genome analysis to detect barcodes associated with partitions where cells from two different genomes were present. Among all barcodes called as cell-associated (See Calling cell barcodes), they are initially classified as Human or Mouse by which genome has more total UMI counts for that barcode. Barcodes with total UMI counts that exceed the 10th percentile of the distributions for both Human and Mouse are called as observed multiplets. Because Cell Ranger can only observe the (Human, Mouse) multiplets, it computes an inferred multiplet rate by estimating the total number of multiplets (including (Human, Human) and (Mouse, Mouse)). This is done by estimating via maximum likelihood the total number of multiplet GEMs from the observed multiplets and the inferred ratio of Human to Mouse cells. If this ratio is 1:1, the inferred multiplet rate is approximately twice the observed (Human, Mouse) multiplets.
## Secondary analysis of gene expression
### Dimensionality reduction
#### PCA
In order to reduce the gene expression matrix to its most important features, Cell Ranger uses Principal Components Analysis (PCA) to change the dimensionality of the dataset from (cells x genes) to (cells x M) where M is a user-selectable number of principal components (via num_principal_comps). The pipeline uses a python implementation of IRLBA algorithm, (Baglama & Reichel, 2005), which we modified to reduce memory consumption. The reanalyze pipeline allows the user to further reduce the data by randomly subsampling the cells and/or selecting genes by their dispersion across the dataset. Note that if the data contains Feature Barcode data, only the gene expression data will be used for PCA and subsequent analysis.
#### t-SNE
For visualizing data in 2-d space, Cell Ranger passes the PCA-reduced data into t-SNE (t-Stochastic Neighbor Embedding), a nonlinear dimensionality reduction method (Van der Maaten, 2014). The C++ reference implementation by Van der Maaten was modified to take a PRNG seed for determinism and to expose various parameters which can be changed in reanalyze. We also decreased its runtime by fixing the number of output dimensions at compile time to 2 or 3.
#### UMAP
Cell Ranger also supports visualization with UMAP (Uniform Manifold Approximation and Projection), which estimates a topology of the high dimensional data and uses this information to estimate a low dimensional embedding that preserves relationships present in the data (McInnes et al, 2018). The pipeline uses the python implementation of this algorithm by McInnes et al. The reanalyze pipeline allows the user to customize the parameters for the UMAP, including n_neighbors, min_dist and metric etc. Below shows the t-SNE (left) and UMAP (right) visualizations of our public dataset 5k PBMCs.
t-SNE UMAP
### Clustering
Cell Ranger uses two different methods for clustering cells by expression similarity, both of which operate in the PCA space.
#### Graph-based
The graph-based clustering algorithm consists of building a sparse nearest-neighbor graph (where cells are linked if they among the k nearest Euclidean neighbors of one another), followed by Louvain Modularity Optimization (LMO; Blondel, Guillaume, Lambiotte, & Lefebvre, 2008), an algorithm which seeks to find highly-connected "modules" in the graph. The value of k, the number of nearest neighbors, is set to scale logarithmically with the number of cells. An additional cluster-merging step is done to perform hierarchical clustering on the cluster-medoids in PCA space and merge pairs of sibling clusters if there are no genes differentially expressed between them (with B-H adjusted p-value below 0.05). The hierarchical clustering and merging is repeated until there are no more cluster-pairs to merge.
The use of LMO to cluster cells was inspired by a similar method in the R package Seurat.
#### K-Means
Cell Ranger also performs traditional K-means clustering across a range of K values, where K is the preset number of clusters. In the web summary prior to 1.3.0, the default selected value of K is that which yields the best Davies-Bouldin Index, a rough measure of clustering quality.
### Differential expression
In order to identify genes whose expression is specific to each cluster, Cell Ranger tests, for each gene and each cluster i, whether the mean expression in cluster i differs from the mean expression across all other cells.
For cluster i, the mean expression of a feature is calculated as the total number of UMIs from that feature in cluster i divided by the sum of the size factors for cells in cluster i. The size factor for each cell is the total UMI count in that cell divided by the median UMI count per cell (across all cells). The mean expression outside of cluster i is calculated in the same way. The log2 fold-change of expression in cluster i relative to other clusters is the log2 ratio of mean expression within cluster i and outside of cluster i. When computing the log2 fold-change, a pseudocount of 1 is added to both the numerator and denominator of the mean expression.
To test for differences in mean expression between groups of cells, Cell Ranger uses the exact negative binomial test proposed by the authors of the sSeq method (Yu, Huber, & Vitek, 2013). When the counts become large, Cell Ranger switches to the fast asymptotic negative binomial test used in edgeR (Robinson & Smyth, 2007).
Note that Cell Ranger's implementation differs slightly from the sSeq paper: the size factors are calculated using total UMI count instead of using DESeq's geometric mean-based definition of library size. As with sSeq, normalization is implicit in that the per-cell size factor parameter is incorporated as a factor in the exact and asymptotic probability calculations.
### Chemistry batch correction
To correct the batch effects between chemistries, Cell Ranger uses an algorithm based on mutual nearest neighbors (MNN; Haghverdi et al, 2018) to identify similar cell subpopulations between batches. The mutual nearest neighbor is defined as a pair of cells from two different batches that is contained in each other’s set of nearest neighbors. The MNNs are detected for every pair of user-defined batches to balance among batches, as proposed in (Park et al, 2018).
The cell subpopulation matches between batches will then be used to merge multiple batches together (Hie et al, 2018). The difference in expression values between cells in a MNN pair provides an estimate of the batch effect. A correction vector for each cell is obtained as a weighted average of the estimated batch effects, where a Gaussian kernel function up-weights matching vectors belonging to nearby points (Haghverdi et al, 2018).
The batch effect score is defined to quantitatively measure the batch effect before and after correction. For every cell, we calculate how many of its k nearest-neighbors belong to the same batch and normalize it by the expected number of same batch cells when there isn't batch effect. The batch effect score is calculated as the average of the above metric in a randomly sampled 10% of the total number of cells. If there isn't batch effect, we would expect that each cells’ nearest neighbors would be evenly shared across all batches and the batch effect score is close to 1.
In the example below, the PBMC mixture was profiled separately by Single Cell 3' v2 (brown) and Single Cell 3' v3 (blue). On the left is the t-SNE plot after aggregating two libraries without the batch correction, and on the right is the t-SNE plot with the batch correction. The batch effect score decreased from 1.81 to 1.31 with the chemistry batch correction.
Without Chemistry Batch Correction With Chemistry Batch Correction
## References
Baglama, J. & Reichel, L. Augmented Implicitly Restarted Lanczos Bidiagonalization Methods. SIAM Journal on Scientific Computing 27, 19–42 (2005).
Blondel, V. D., Guillaume, J.-L., Lambiotte, R. & Lefebvre, E. Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment 2008, (2008).
Haghverdi, L., Lun, A. T., Morgan, M. D., & Marioni, J. C. Batch effects in single-cell RNA-sequencing data are corrected by matching mutual nearest neighbors. Nature biotechnology, 36(5), 421 (2018).
Hie, B. L., Bryson, B., & Berger, B. Panoramic stitching of heterogeneous single-cell transcriptomic data. bioRxiv 2018.
Lun, A., Riesenfeld, S., Andrews, T., Dao, T.P., Gomes. T., Participants in the 1st Human Cell Atlas Jamboree, Marioni, J. Distinguishing cells from empty droplets in droplet-based single-cell RNA sequencing data. Genome Biology 2019
McInnes, L, Healy, J, UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction. arXiv 2018
Park, J. E., Polanski, K., Meyer, K., & Teichmann, S. A. Fast Batch Alignment of Single Cell Transcriptomes Unifies Multiple Mouse Cell Atlases into an Integrated Landscape. bioRxiv 2018.
Robinson, M. D. & Smyth, G. K. Small-sample estimation of negative binomial dispersion, with applications to SAGE data. Biostatistics 9, 321–332 (2007). Link to edgeR source.
Van der Maaten, L.J.P. Accelerating t-SNE using Tree-Based Algorithms. Journal of Machine Learning Research 15, 3221-3245 (2014).
Yu, D., Huber, W. & Vitek, O. Shrinkage estimation of dispersion in Negative Binomial models for RNA-seq experiments with small sample size. Bioinformatics 29, 1275–1282 (2013). | 2022-05-19 02:43:24 | {"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": 0.4163128137588501, "perplexity": 3527.178785635886}, "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/1652662522741.25/warc/CC-MAIN-20220519010618-20220519040618-00214.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=539894 | # curvilinear coordinate systems and "periodic" coordinates
by mnb96
Tags: coordinate, coordinates, curvilinear, periodic, systems
P: 625 Hello, we can consider a generic system of curvilinear coordinates in the 2d plane: $$\rho = \rho(x,y)$$ $$\tau = \tau(x,y)$$ Sometimes, it can happen that one of the coordinates, say $\tau$, represents an angle, and so it is "periodic". This clearly happens for example, in polar coordinates. What are the families of curvilinear coordinates systems in 2d, that have one or more coordinates that are angles? I hope the question is not too vague to be answered. Thanks.
Sci Advisor P: 1,562 Any coordinate system whose coordinate lines form closed curves is going to have at least one periodic coordinate. The periodic coordinate doesn't have to represent an "angle", necessarily. For example, in elliptical coordinates, one of the coordinates might represent a parameter for traveling around an ellipse. One example that can have two periodic coordinates is the bipolar system. | 2014-04-17 18:37:31 | {"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": 0.8689810037612915, "perplexity": 533.7334589630735}, "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/1397609530895.48/warc/CC-MAIN-20140416005210-00204-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://tug.org/pipermail/texhax/2003-September/000951.html | # [texhax] Turtorials
Robin Fairbairns Robin.Fairbairns at cl.cam.ac.uk
Thu Sep 25 09:45:20 CEST 2003
```> On Wed, 24 Sep 2003 udts at gmu.edu wrote:
>
> > programs but i am having difficulty getting started ... COuld you
> > suggest a good tutorial for someone who is new to the TEX environment
>
> Please take a look at:
>
> http://www.tug.org.in/tutorials.html
which is new, and from what i see, very good.
for a list of the other things available:
http://www.tex.ac.uk/cgi-bin/texfaq2html?label=tutorials
(which doesn't -- yet -- list the indian tutorial.)
Robin Fairbairns
UK TeX FAQ maintainer | 2021-03-07 23:58:38 | {"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": 0.9534438252449036, "perplexity": 7022.882787125196}, "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/1614178381230.99/warc/CC-MAIN-20210307231028-20210308021028-00458.warc.gz"} |
https://transum.org/Maths/Exam/Online_Exercise.asp?Topic=Box%20Plots | # Exam-Style Questions on Box Plots
## Problems on Box Plots adapted from questions set in previous Mathematics exams.
### 1.
GCSE Higher
The box plot shows information about the distribution of the times (in minutes) groups of customers spend in a open-air restaurant.
(a) Calculate the interquartile range for the amount of time groups spend in this open-air restaurant.
The table below shows information about the distribution of the times (in minutes) groups of customers spend in an indoor restaurant.
Minimum Lower Quartile Median Upper quartile Maximum Time in minutes 30 45 62 82 94
(b) On the lower half of the grid above, draw a box plot for the information in the table.
(c) Hunter thinks that the total amount of time spend by all of the customers visiting the indoor restaurant is greater than the total amount of time spend by all of the customers visiting the open-air restaurant. Is Hunter correct? Explain your decision.
### 2.
GCSE Higher
Here is some information about the length of time children spent in a swimming pool at a holiday resort during a heat wave.
• Shortest time 30 minutes
• Longest time 5 hours
• Lower quartile 90 minutes
• Interquartile range 2 hours
• Median time 2 hours 30 minutes
Draw a box plot to show this information.
### 3.
GCSE Higher
One day a theme park monitored the time spent by guests at six different areas of the park. The times are recorded in the box plots below.
(a) Work out the range in times at Future Zone.
(b) At which area were visitor times the most consistent? Give a reason for your answer.
(c) Give one similarity and one difference between the distributions of the guest visiting times for Nature Park and Fantasy Land.
(d) Is it possible to work out from the box plots which area had the most visitors? Explain your answer.
### 4.
IB Studies
The distribution of daily average wind speed on an island over a period of 120 days is displayed on this box-and-whisker diagram.
(a) Write down the median wind speed.
(b) Write down the minimum wind speed.
(c) Find the interquartile range.
(d) Write down the number of days the wind speed was between 20 kmh-1 and 51 kmh-1.
(e) Write down the number of days the wind speed was between 9 kmh-1 and 68 kmh-1.
### 5.
IB Applications and Interpretation
Derek has put together a fantasy football squad and has recorded the heights of all the players in the squad rounded to the nearest centimetre. The data is illustrated in the following box and whisker diagram.
(a) Write down the median height of the players.
(b) Write down the upper quartile height.
(c) Find the interquartile range of the heights.
The height of these players are normally distributed.
(d) Find the height of the tallest possible player that is not an outlier. Give your answer to the nearest centimetre.
### 6.
GCSE Higher
Poppy Pringle recorded the heights in centimetres of the sunflowers growing in her brother's and sister's gardens.
Here are the heights of the 19 sunflowers in her brother's garden:
$$94, 103, 110, 111, 123, 143, 150, 150, 151, 156, 157, 160, 170, 182, 201, 220, 250, 255, 231$$
Median Lower quartile Upper quartile Shortest Tallest
Here is some information about the sunflowers in Poppy's sister's garden:
Median 160 Lower quartile 143 Upper quartile 172 Shortest 110 Tallest 199
Poppy says that the sunflowers in her brother's garden are shorter that those in her sister's garden.
(b) Is Poppy correct?
Poppy says that the sunflowers in her brother's garden vary more in height than those in her sister's garden.
(c) Is Poppy correct?
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To search the entire Transum website use the search box in the grey area below. | 2021-11-29 11:07:22 | {"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": 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": 0.29782843589782715, "perplexity": 1892.18272884497}, "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/1637964358705.61/warc/CC-MAIN-20211129104236-20211129134236-00158.warc.gz"} |
http://mathandmultimedia.com/2011/06/16/geogebra-4-0-inequalities/ | # GeoGebra 4.0 Sneak Peek 4: Graphing Inequalities
This is the fourth part of the GeoGebra 4.0 Sneak Peek Series
The long wait is over! GeoGebra 4.0 can now graph inequalities as shown in Figure 1.
Figure 1 – Graphs of Inequalities in GeoGebra
You can construct the graph above by typing the following in the Input bar pressing the ENTER key after each inequality.
1. $y < 3x$
2. $2x - 1 <= 2$
3. $y > x^3 - 2x - 1$
The symbol <= can also be found on the symbol box as $\leq$. Just click the input bar and click the $\alpha$ symbol at the rightmost part of the input bar.
When we talk about inequalities, we talk about solutions. In GeoGebra 4.0, we can graph the solution of the three or more inequalities above using logical operators such as and (denoted by ∧). The graph shown in Figure 2 is the graph of the inequality (-x + y ≤ 3) ∧ (4x + 3y < 12) ∧ (-x – 2y < -2).
Figure 2 – Solutions of Inequlities using logical operators
The inequality is a new feature in GeoGebra is not yet stable. The logical operators might still have problems with functions with degree more than one. I used the logical operator on the first three inequalities, which include one quadratic inequality, but no output graph was shown (Update: I think this is fixed now).
In the next post, we are going to discuss about the computer algebra system, the newest feature of GeoGebra. | 2023-01-29 12:02:42 | {"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": 5, "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": 0.41943755745887756, "perplexity": 899.8733239961878}, "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/1674764499713.50/warc/CC-MAIN-20230129112153-20230129142153-00720.warc.gz"} |
https://electronics.stackexchange.com/questions/75132/eagle-cad-enlarging-the-pads | I use Eagle Cad 6.4 to prepare PCB layouts then manufacture them at home. As I use amateurish equipment and method, I would like to have bigger pads as much as possible so that when I drill the pad holes, I will have a very large ring of copper around the hole to solder. It would be convenient to have bigger pads (when it is possible on the layout). I can do it on photoshop for sure or maybe by redefining each component's packet description on the library, but it is a little bit time consuming. I wonder if there is an easy tool like change tool to enlarge the pads I want.
The pads I'd like to enlarge are the green circles below image. For example, the diode (the most left circle ) has a bigger pad area, where as the resistors (the closest two green circles) have thinner circles.
When I drill these holes, sometime the drill takes the copper away and I'm left with rings that are too thin. For example, please see the below photo. These pins are not important but in some cases, it would be better to have a larger copper area. Is it possible to do it on Eagle Cad 6.4?
• You can't set the thickness in Eagle. Even if you could, it would not be for individual pads but for the board, or at least that layer, as a whole. You tell the board house you want "1oz copper", "2oz copper", or whatever. In your case it depends on the copper thickness of the stock you get. – Olin Lathrop Jul 7 '13 at 16:22
• Now you are really confusing things. It appears you mean the diameter of the pads, but you explicitly say you do NOT mean the diameter of the pads. Giving up. – Olin Lathrop Jul 7 '13 at 17:18
• The pad (green circles) are made of traces. 0.6 mm or 3.5 mm diameter drill radius would not enlarge thickness of the trace, the thickness is the same. I want to enlarge the thickness of the trace that creates the green circle. Therefore I do NOT mean the diameter. I hope it is clear now? – Angs Jul 7 '13 at 17:25
• No, you DO mean the diameter of the pad. And again, this has nothing to do with the thickness of the copper. The first time I can attribute to unfamiliarity with English, but if you keep doing it after this was explained to you then it's just wrong. Do a HELP DIAMETER in Eagle. – Olin Lathrop Jul 7 '13 at 19:31
• Pads are drawn by the CAD software as a circle, with a hole in the middle. It's not a circular trace, so you're thinking about the mechanics of making the pad larger incorrectly. – Connor Wolf Jul 7 '13 at 21:12 | 2019-06-20 07:47:28 | {"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": 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": 0.5324091911315918, "perplexity": 781.2394234559295}, "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/1560627999163.73/warc/CC-MAIN-20190620065141-20190620091141-00001.warc.gz"} |
https://pharr.org/matt/blog/2021/08/07/moana-bandwidth-note | At the ostensible end of these updates about pbrt-v4’s performance when rendering Disney’s Moana Island scene, there was an unresolved question about why CPU utilization wasn’t better at the very start when pbrt was parsing the scene description. As a refresher, with 64 threads on a 32-core AMD 3970X CPU and pbrt’s GPU-based rendering path, it looked like this. (As before, the vertical dashed line indicates when rendering begins.)
Starting at the 17 second mark, low CPU utilization isn’t necessarily bad since that’s when the GPU starts getting involved building acceleration structures, but before that it’s all on the CPU. For the first twelve or so seconds, total CPU utilization is between 0.2 and 0.4, which corresponds to roughly 13–26 of those 64 threads actually making something of themselves; that’s not enough of them to be satisfying.
It started to nag at me whether limited disk bandwidth might have something to do with that—i.e., is the issue that threads are stalled waiting for I/O? I made a few measurements to try to answer that question and learned enough along the way that here we go again.
## How Far We Have Come
Three years ago when I first looked at pbrt’s performance with Moana Island I was using a Google Compute Engine instance with a spinny disk for benchmarks. Nowadays you might hope for around 100 MB/s of read bandwidth from such a disk. pbrt-v4 reads a total of 27,766 MB from disk when loading this scene and it takes a lot of 100 MBs to get through all of that. Therefore, when I was doing benchmarks then I was careful to flush the OS’s buffer cache between runs so that the true cost of I/O was measured and everything didn’t come out of RAM after the first time at rates much better than 100 MB/s.
This time around, I didn’t mention the disk on the system I used for benchmarking and I didn’t worry about the buffer cache. That wasn’t an oversight, but was more of a “I’m pretty sure this doesn’t matter”1 sort of thing, like which version of the Linux kernel was running or whether it was DDR4 3200 or DDR4 3600 RAM in the system. (For the record, 5.8.0 and the former.)
The disk I’m using now is an NVMe disk; a quick benchmark showed that it delivers a peak of 2,022 MB/s of read bandwidth. I didn’t think that could be a bottleneck, though if you distribute those 2,022 MB/s evenly to 64 threads, it’s just 32 MB/s per thread. Thinking about it in those terms made me worry that bandwidth might be tight, so I decided to make some direct measurements and see what they had to show.
## Starting Position
First, I measured pbrt’s disk bandwidth use over time to get a sense of whether it ever approached the peak and to see how disk reads were distributed over the course of loading the scene. (This and following measurements were made with an empty buffer cache, just to be safe.) iostat made that easy to do, though sadly it doesn’t seem to be able to report I/O with less than one second granularity, which is more coarse than one would like given 30 seconds time to first pixel. In any case, here is a graph of what it had to say; the disk’s measured maximum I/O bandwidth is marked with a dashed horizontal line.
For the first 20 or seconds, pbrt is mostly parsing text *.pbrt scene description files; it starts out consuming plenty of bandwidth but then slows as there are fewer files left to get through. The second wave of I/O starting at 20 seconds corresponds to reading all of the PLY files for the object instances in the scene. The news in this graph is mostly good: pbrt doesn’t seem to ever top out at the maximum bandwidth, suggesting that it’s not I/O bound, though it’s close enough at 9 seconds there that it’s not possible to be sure from these measurements.
This data also makes it possible to compute an alternative speed of light measurement for time to first pixel. If we divide the total size of data read, 27,766 MB, by the peak read bandwidth of 2,022 MB/s, we can see that we can’t hope to have a time to first pixel under 13.7 seconds. That’s already an interesting result, as it shows that the earlier speed of light calculation that only considered the CPU didn’t tell the whole story: then, neglecting I/O limits, we estimated 7.2 seconds as the best possible time to first pixel.
Another thing this graph shows is that pbrt is close enough to being I/O bound at the start that there isn’t a lot of reason to worry about the relatively low CPU utilization then. We might improve it some by finding more things to start reading sooner, but the benefit would be limited since we would soon hit peak disk bandwidth and be limited by that. Further performance improvements would then require a better balance of I/O requests over time.
## Turning The Bandwidth Screw
The data already seemed fairly conclusive about not being I/O bound, but I was curious about how performance varied with disk read bandwidth—how crucial is that lovely abundant NVMe bandwidth to pbrt’s performance with this scene? One way to find out is to start reducing the amount of disk read bandwidth available to pbrt and to see how that affects performance.
Once you find the right trick it’s surprisingly easy, at least on Linux, to use systemd-run to launch a process that has a limited amount of disk read bandwidth available to it. I did a quick study, dialing the bandwidth down from the 2,000 MB/s that my NVMe drive offers to the sad 50 MB/s that a middling spinning disk today might provide.
Here is a graph of pbrt-v4’s time to first pixel with the Moana Island scene as a function of available disk bandwidth, running with both 8 threads on 4 cores and 64 threads on 32 cores. Note that the y axis has a logarithmic scale, the better to fit the sadness that is a nearly 600 second time to first pixel given 50 MB/s.
There are a number of things to see in this graph. First, it offers further confirmation that pbrt-v4 is not bandwidth limited for this scene: the fact that performance doesn’t immediately decrease as bandwidth starts to decrease from 2,000 MB/s indicates that more bandwidth isn’t going to make things faster. Both lines seem to have hit their asymptote, though the 64 thread one just barely so.
This graph also shows how much bandwidth can decrease before performance is meaningfully affected. With 64 threads, you only have to go to 1400 MB/s to slow down time to first pixel by 10%, but with 8 threads you can go all the way to 800 MB/s before there’s a 10% drop. This isn’t surprising—the more threads you’ve got, the more bandwidth you’re capable of consuming—but it’s nevertheless interesting to see how much farther one can go with fewer threads.
Finally, note that below 500 MB/s, the two curves are effectively the same. Here, too, there’s no big surprise: if you’re trying to drink through a narrow straw, having more thirsty people waiting in line on the end of it isn’t going to get the water through more quickly, to grossly overstretch a metaphor.
## DEFLATE, DEFLATE, DEFLATE
Compression algorithms make it possible to trade off bandwidth for computation, so my last experiment was to look at performance with the scene description compressed using gzip. Thanks to a recent patch from Jim Price, pbrt-v4 now supports reading gzip-compressed scene description files, and RPly, the PLY file reader by Diego Nehab that pbrt uses, already supported gzip-compressed PLY files. All of that made it easy to run the same experiments with a compressed scene description.
With the *.pbrt and PLY files compressed using gzip -5, pbrt-v4 reads a total of just 5,570 MB from disk—nearly 5x less than with the uncompressed scene description. Using zlib for decompression with 64 threads and the full NVMe disk bandwidth, pbrt takes 40 seconds to first pixel with a compressed scene—12 seconds slower than with everything uncompressed. Given that it wasn’t bandwidth-limited before, that isn’t surprising—we have just increased the amount of CPU work that needs to be done to get the scene into memory.
Here is the graph of disk I/O consumption over those 40 seconds; it shows that now there is plenty of headroom with never more than 500 MB/s of bandwidth used.
As we were going to press, I saw that Aras Pranckevičius just put up a nice series of blog posts about compression in OpenEXR. Those led me down all sorts of ratholes, and one of them reminded me about libdeflate, a highly optimized library that can decompress gzip-encoded files (among others). It wasn’t too much code to swap that in for zlib in pbrt and bam: down to 34 seconds to first pixel with a compressed scene. And that’s actually only using libdeflate for the *.pbrt files but still using zlib for the 1,152 MB worth of compressed PLY files, since using libdeflate with RPly would have required more complicated plumbing.
Anyway, here’s a graph that shows time to first pixel with all three options, again with 64 threads. libdeflate gets an asterisk, since it isn’t being used for PLY files (and there is thus presumably some performance being left on the floor.)
There’s lots of good stuff to see there. As advertised, libdeflate is certainly faster than zlib. It starts being the fastest option overall at around 1,300 MB/s of bandwidth. From there on down, the additional CPU work to do decompression is worth the disk bandwidth savings in return. (In contrast, zlib doesn’t make up for its computational overhead until around 1,000 MB/s.)
Both decompressors have more or less constant performance all the way down to roughly 300 MB/s from the disk. Past there, their performance converges: at that point, data is coming in so slowly that how quickly it’s decompressed doesn’t make much difference. We can also see that compression is especially helpful way down at 50 MB/s, where it’s leads to a spritely 127 seconds to first pixel—4.6x faster than the uncompressed scene is with that little bandwidth.
## Discussion
For once we have gotten through these investigations without finding any surprising bottlenecks and so today has not brought any changes to pbrt’s implementation, though I suspect pbrt will switch from zlib to libdeflate fairly soon.
Perhaps the most useful result from today is a more accurate estimate of pbrt’s best possible performance when preparing to render this scene: 13.7 seconds given the disk I/O limits of the system. With that limit known, it’s easier to accept the 28 seconds to first pixel that the system delivers today—apparently only 2x off the maximum attainable performance—and to stop fiddling with the details.
And yet… I hope that the attentive reader might quibble with the logic behind that conclusion: with the compressed scene, we found ourselves with a mere 5,570 MB of disk I/O, and that’s something this computer can deliver in 2.75 seconds, which puts us once again 10x off the mark. It seems that part of speed of light is in how you define it, but nevertheless I think it’s time to leave things where they lie for now.
## note
1. The road to disastrous performance is paved with “pretty sure” assumptions about a system’s behavior, so that assumption was admittedly not wise. | 2022-01-25 22:41:43 | {"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": 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": 0.3158496916294098, "perplexity": 1553.7835068819513}, "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/1642320304876.16/warc/CC-MAIN-20220125220353-20220126010353-00186.warc.gz"} |
https://tex.stackexchange.com/questions/490420/creating-multiple-indices-with-customizable-options | # Creating multiple indices with customizable options
I want to create multiple indices with two columns and no clear page in between.
I tried the splitidx package, which does not give me the same options as imakeidx, which in turn gives me an "misplaced \printindex[Hal]" in the following MWE:
\usepackage[splitindex]{imakeidx}
\makeindex[columns=2, name=As, title=As]
\makeindex[columns=2, name=Bs, title=Bs]
\indexsetup{noclearpage}
\begin{document}
\section{A}
Eintrag A\index[Hal]
\newpage
\section{B}
Eintrag B\index[Alka]
\newpage
\printindex[Hal]
\end{document}
my question is, is there a way to customize the splitidx the same way as the imakeidx? is there a solution for my problem with imakeidx, or is there a alternate package(which is compatible with hyperref) I can use and if how?
Your example is not a MWE because it can't be compiled. Besides this your syntax is wrong: The optional arguments of \index and \printindex describe the index not the entry.
\documentclass{article}
\usepackage{imakeidx}
\makeindex[columns=2, name=As, title=As]
\makeindex[columns=2, name=Bs, title=Bs]
\indexsetup{noclearpage}
\begin{document}
\section{A}
Eintrag A\index[As]{Hal} \index[Bs]{Blka}
\newpage
\section{B}
Eintrag B\index[As]{Bal} \index[Bs]{Alka}
\newpage
\printindex[As]
\printindex[Bs]
\end{document} | 2020-04-08 00:09:50 | {"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": 0.7099261283874512, "perplexity": 4119.324440050201}, "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/1585371806302.78/warc/CC-MAIN-20200407214925-20200408005425-00279.warc.gz"} |
https://intelligencemission.com/free-electricity-using-dynamo-free-electricity-using-flywheel.html | It makes you look like Free Power fool, Free Power scammer, or both. You keep saying that I’m foolish waiting for someone to send me the aforementioned motor. Again, you missed the point completely. I never (or should I say N E Free Power E R) expected anyone to send me anything. It was just to make the point that it never existed. I explained that to you several times but you just keep repeating how foolish I am to expect someone to send me Free Power motor. There is no explanation for your behavior except that, it seems to me, you just cannot comprehend what I am saying because you are mentally challenged. This device can indeed charge Free Power battery. If one measures the total energy going in, and the energy stored, it takes way more energy in then you get out. That’s true for ALL battery chargers. Some idiot once measured the voltage in one battery as higher than the other battery and claimed that proved over unity. Hint: voltage does not measure power. Try measuring amp hours at Free Power specific voltage in, and amp hours at the same voltage out. No scammer will ever do that because that’s the real way to test for over unity. Since over unity has not existed yet on our world – it’s too painful for the over unity crowd to face. Kimseymd1: You no longer are responding.
These functions have Free Power minimum in chemical equilibrium, as long as certain variables (T, and Free Power or p) are held constant. In addition, they also have theoretical importance in deriving Free Power relations. Work other than p dV may be added, e. g. , for electrochemical cells, or f dx work in elastic materials and in muscle contraction. Other forms of work which must sometimes be considered are stress-strain, magnetic, as in adiabatic demagnetization used in the approach to absolute zero, and work due to electric polarization. These are described by tensors.
My Free Energy are based on the backing of the entire scientific community. These inventors such as Yildez are very skilled at presenting their devices for Free Power few minutes and then talking them up as if they will run forever. Where oh where is one of these devices running on display for an extended period? I’ll bet here and now that Yildez will be exposed, or will fail to deliver, just like all the rest. A video is never proof of anything. Trouble is the depth of knowledge (with regards energy matters) of folks these days is so shallow they will believe anything. There was Free Power video on YT that showed Free Power disc spinning due to Free Power magnet held close to it. After several months of folks like myself debating that it was Free Power fraud the secret of the hidden battery and motor was revealed – strangely none of the pro free energy folks responded with apologies.
Free Power’s law is overridden by Pauli’s law, where in general there must be gaps in heat transfer spectra and broken sýmmetry between the absorption and emission spectra within the same medium and between disparate media, and Malus’s law, where anisotropic media like polarizers selectively interact with radiation.
This simple contradiction dispels your idea. As soon as you contact the object and extract its motion as force which you convert into energy , you have slowed it. The longer you continue the more it slows until it is no longer moving. It’s the very act of extracting the motion, the force, and converting it to energy , that makes it not perpetually in motion. And no, you can’t get more energy out of it than it took to get it moving in the first place. Because this is how the universe works, and it’s Free Power proven fact. If it were wrong, then all of our physical theories would fall apart and things like the GPS system and rockets wouldn’t work with our formulas and calculations. But they DO work, thus validating the laws of physics. Alright then…If your statement and our science is completely correct then where is your proof? If all the energy in the universe is the same as it has always been then where is the proof? Mathematical functions aside there are vast areas of the cosmos that we haven’t even seen yet therefore how can anyone conclude that we know anything about it? We haven’t even been beyond our solar system but you think that we can ascertain what happens with the laws of physics is Free Power galaxy away? Where’s the proof? “Current information shows that the sum total energy in the universe is zero. ” Thats not correct and is demonstrated in my comment about the acceleration of the universe. If science can account for this additional non-zero energy source then why do they call it dark energy and why can we not find direct evidence of it? There is much that our current religion cannot account for. Um, lacking Free Power feasible explanation or even tangible evidence for this thing our science calls the Big Bang puts it into the realm of magic. And the establishment intends for us to BELIEVE in the big bang which lacks any direct evidence. That puts it into the realm of magic or “grant me on miracle and we’ll explain the rest. ” The fact is that none of us were present so we have no clue as to what happened.
“It wasn’t long before carriage makers were driving horseless carriages. It wasn’t long before people crossing the continent on trains abandoned the railroads for airliners. Natural gas is replacing coal and there is nothing the railroads, the coal miners, or the coal companies can do about it. Cheaper and more efficient energy always wins out over more expensive energy. Coal replaced wood, and oil replaced coal as the primary source of energy. Anything that is more efficient boosts the figures on the bottom line of the ledger. Dollars chase efficiency. Inefficiency is suppressed by market forces. Free Power wins in the market place.
NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! rychu Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power has the credentials and knowledge to answer these questions and Bedini is the visionary for them!
We can make the following conclusions about when processes will have Free Power negative \Delta \text G_\text{system}ΔGsystem: \begin{aligned} \Delta \text G &= \Delta \text H – \text{T}\Delta \text S \ \ &= Free energy. 01 \dfrac{\text{kJ}}{\text{mol-rxn}}-(Free energy \, \cancel{\text K})(0. 022\, \dfrac{\text{kJ}}{\text{mol-rxn}\cdot \cancel{\text K})} \ \ &= Free energy. 01\, \dfrac{\text{kJ}}{\text{mol-rxn}}-Free energy. Free Power\, \dfrac{\text{kJ}}{\text{mol-rxn}}\ \ &= -0. Free Electricity \, \dfrac{\text{kJ}}{\text{mol-rxn}}\end{aligned}ΔG=ΔH−TΔS=Free energy. 01mol-rxnkJ−(293K)(0. 022mol-rxn⋅K)kJ=Free energy. 01mol-rxnkJ−Free energy. 45mol-rxnkJ=−0. 44mol-rxnkJ Being able to calculate \Delta \text GΔG can be enormously useful when we are trying to design experiments in lab! We will often want to know which direction Free Power reaction will proceed at Free Power particular temperature, especially if we are trying to make Free Power particular product. Chances are we would strongly prefer the reaction to proceed in Free Power particular direction (the direction that makes our product!), but it’s hard to argue with Free Power positive \Delta \text GΔG! Our bodies are constantly active. Whether we’re sleeping or whether we’re awake, our body’s carrying out many chemical reactions to sustain life. Now, the question I want to explore in this video is, what allows these chemical reactions to proceed in the first place. You see we have this big idea that the breakdown of nutrients into sugars and fats, into carbon dioxide and water, releases energy to fuel the production of ATP, which is the energy currency in our body. Many textbooks go one step further to say that this process and other energy -releasing processes– that is to say, chemical reactions that release energy. Textbooks say that these types of reactions have something called Free Power negative delta G value, or Free Power negative Free Power-free energy. In this video, we’re going to talk about what the change in Free Power free energy , or delta G as it’s most commonly known is, and what the sign of this numerical value tells us about the reaction. Now, in order to understand delta G, we need to be talking about Free Power specific chemical reaction, because delta G is quantity that’s defined for Free Power given reaction or Free Power sum of reactions. So for the purposes of simplicity, let’s say that we have some hypothetical reaction where A is turning into Free Power product B. Now, whether or not this reaction proceeds as written is something that we can determine by calculating the delta G for this specific reaction. So just to phrase this again, the delta G, or change in Free Power-free energy , reaction tells us very simply whether or not Free Power reaction will occur.
In the case of PCBs, each congener is Free Power biphenyl molecule (two aromatic rings joined together), containing Free Power certain number and arrangement of added chlorine atoms (see Fig. Free Electricity. Free Electricity). Historically, there were many commercially marketed products (e. g. , Aroclor) containing varying mixtures of PCB congeners.) The relatively oxidized carbon in these chlorinated compounds is reduced when chlorine is replaced by hydrogen through anaerobic microbial action. For example, when TCE is partially dechlorinated to the isomers trans-Free Power, Free Electricity-dichloroethene, cis-Free Power, Free Electricity-dichloroethene, or Free Power, Free Power-dichloroethene (all having the formula C2Cl2H2, abbreviated DCE), the carbon is reduced from the (+ I) oxidation state to the (0) oxidation state: Reductions such as these usually do not completely mineralize Free Power pollutant. Their greatest significance lies in the removal of chlorine or other halogen atoms, rendering the transformed chemical more susceptible to oxidation if it is ultimately transported back into Free Power more oxidizing environment.
Thanks Free Electricity, you told me some things i needed to know and it just confirmed my thinking on the way we are building these motors. My motor runs but not the way it needs to to be of any real use. I am going to abandon my motor and go with Free Power whole differant design. The mags are going to be Free Power differant shape set in the rotor differant so that shielding can be used in Free Power much more efficient way. Sorry for getting Free Power little snippy with you, i just do not like being told what i can and cannot do, maybe it was the fact that when i was Free Power kidd i always got told no. It’s something i still have Free Power problem with even at my age. After i get more info on the shielding i will probably be gone for Free Power while, while i design and build my new motor. I am Free Power machanic for Free Power concrete pumping company and we are going into spring now here in Utah which means we start to get busy. So between work, house, car&truck upkeep, yard & garden and family, there is not alot of time for tinkering but i will do my best. Free Power, please get back to us on the shielding. Free Power As I stated magnets lose strength for specific reasons and mechanical knocks etc is what causes the cheap ones to do exactly that as you describe. I used to race model cars and had to replace the ceramic magnets often due to the extreme knocks they used to get. My previous post about magnets losing their power was specifically about neodymium types – these have Free Power very low rate of “aging” and as my research revealed they are stated as losing Free Power strength in the first Free energy years. But extreme mishandling will shorten their life – normal use won’t. Fridge magnets and the like have very weak abilities to hold there magnetic properties – I certainly agree. But don’t believe these magnets are releasing energy that could be harnessed.
Not Free Power lot to be gained there. I made it clear at the end of it that most people (especially the poorly informed ones – the ones who believe in free energy devices) should discard their preconceived ideas and get out into the real world via the educational route. “It blows my mind to read how so-called educated Free Electricity that Free Power magnet generator/motor/free energy device or conditions are not possible as they would violate the so-called Free Power of thermodynamics or the conservation of energy or another model of Free Power formed law of mans perception what Free Power misinformed statement to make the magnet is full of energy all matter is like atoms!!”
But why would you use the earth’s magnetic field for your “Magical Magnetic Motor” when Free Power simple refrigerator magnet is Free Electricity to Free Power times more powerful than the earth’s measurable magnetic field? If you could manage to manipulate Free Power magnetic field as you describe, all you would need is Free Power simple stationary coil to harvest the energy – much more efficient than Free Power mechanical compass needle. Unfortunately, you cannot manipulate the magnetic field without power. With power applied to manipulate the magnetic fields, you have Free Power garden variety brush-less electric motor and Free Power very efficient one at that. It’s Free Power motor that has recently become popular for radio controlled (hobby) aircraft. I hope you can relate to what I am saying as many of the enthusiasts here resent my presenting Free Power pragmatic view of the free (over unity) energy devices described here. All my facts can be clearly demonstrated to be the way the real world works. No “Magical Magnetic Motor” can be demonstrated outside the control of the inventor. Videos are never proof of anything as they can be easily faked. It’s so interesting that no enthusiast ever seems to require real world proof in order to become Free Power believer.
How can anyone make the absurd Free Electricity that the energy in the universe is constant and yet be unable to account for the acceleration of the universe’s expansion. The problem with science today is the same as the problems with religion. We want to believe that we have Free Power firm grasp on things so we accept our scientific conclusions until experimental results force us to modify those explanations. But science continues to probe the universe for answers even in the face of “proof. ” That is science. Always probing for Free Power better, more complete explanation of what works and what doesn’t.
I want to use Free Power 3D printer to create the stator and rotors. This should allow Free Power high quality build with lower cost. Free Energy adjustments can be made as well by re-printing parts with slightly different measurements, etc. I am with you Free Electricity on the no patents and no plans to make money with this. I want to free the world from this oppression. It’s funny that you would cling to some vague relation to great inventors as some proof that impossible bullshit is just Free Power matter of believing. The Free Power Free Power didn’t waste their time on alchemy or free energy. They sought to understand the physical forces around them. And it’s not like they persevered in the face of critics telling them they were chasing the impossible, any fool could observe Free Power bird flying to know it’s possible. You will never achieve anything even close to what they did because you are seeking to defy the reality of our world. You’ve got to understand before you can invent. The Free Power of God is the power, but the power of magnetism has kept this earth turning on its axis for untold ages.
The solution to infinite energy is explained in the bible. But i will not reveal it since it could change our civilization forever. Transportation and space travel all together. My company will reveal it to thw public when its ready. My only hint to you is the basic element that was missing. Its what we experience in Free Power everyday matter. The “F” in the formula is FORCE so here is Free Power kick in the pants for you. “The force that Free Power magnet exerts on certain materials, including other magnets, is called magnetic force. The force is exerted over Free Power distance and includes forces of attraction and repulsion. Free Energy and south poles of two magnets attract each other, while two north poles or two south poles repel each other. ” What say to that? No, you don’t get more out of it than you put in. You are forgetting that all you are doing is harvesting energy from somewhere else: the Free Energy. You cannot create energy. Impossible. All you can do is convert energy. Solar panels convert energy from the Free Energy into electricity. Every second of every day, the Free Energy slowly is running out of fuel.
However, it must be noted that this was how things were then. Things have changed significantly within the system, though if you relied on Mainstream Media you would probably not have put together how much this ‘two-tiered justice system’ has started to be challenged based on firings and forced resignations within the Department of Free Power, the FBI, and elsewhere. This post from Q-Anon probably gives us the best compilation of these actions:
There are many things out there that are real and amazing. Have fun!!! Hey Geoff – you can now call me Mr Electro Magnet. I have done so much research in the last week. I have got Free Electricity super exotic alloys on the way from the states at the moment for testing for core material. I know all about saturation, coercivity, etc etc. Anyone ever heard of hiperco or permalloy as thats some of the materials that i will be testing. Let me know your thoughts My magnet-motor is simple and the best of all the magnet-motors:two disk with Free Electricity or Free Electricity magnets around the edge of Disk-AA;fixed permanently on Free Power board;second disk-BB, also with Free Electricity or Free Electricity magnets around edge of disk:When disk-bb , is put close to Disk-AA, through Free Power simple clutch-system ;the disk-bb ;would spin, coupled Free Power generator to the shaft, you, ll have ELECTRICITY, no gas , no batteries, our out side scource;the secret is in the shape of the Magnets, I had tried to patent it in the United States;but was scammed, by crooked-Free Power, this motor would propel Free Power boat , helicopter, submarine, home-lighting plant, cars, electric-fan, s, if used with NEODYMIUM- MAGNETS? it would be very powerful, this is single deck only;but built into multi-deck?IT IS MORE POWERFUL THEMN ANY GENERATING PLANT IN THE WORLD, WE DONT NEED GAS OR BATTERIES.
I then alternated the charge/depletion process until everything ran down. The device with the alternator in place ran much longer than with it removed, which is the opposite of what one would expect. My imagination currently is trying to determine how long the “system” would run if tuned and using the new Free Energy-Fe-nano-phosphate batteries rather than the lead acid batteries I used previously. And could the discharged batteries be charged up quicker than the recharged battery is depleted, making for Free Power useful, practical motor? Free Energy are claiming to have invented perpetual motion MACHINES. That is my gripe. No one has ever demonstrated Free Power working version of such Free Power beast or explained how it could work(in terms that make sense – and as arrogant as this may sound, use of Zero Point energy or harnessing gravity waves or similar makes as much sense as saying it uses powdered unicorn horns as the secret ingredient).
Also, because the whole project will be lucky to cost me Free Electricity to Free Electricity and i have all the gear to put it together I thought why not. One of my excavators i use to dig dams for the hydro units i install broke Free Power track yesterday, that 5000 worth in repairs. Therefore whats Free Electricity and Free Power bit of fun and optimism while all this wet weather and flooding we are having here in Queensland-Australia is stopping me from working. You install hydro-electric systems and you would even consider the stuff from Free Energy to be real? I am appalled.
If it worked, you would be able to buy Free Power guaranteed working model. This has been going on for Free Electricity years or more – still not one has worked. Ignorance of the laws of physics, does not allow you to break those laws. Im not suppose to write here, but what you people here believe is possible, are true. The only problem is if one wants to create what we call “Magnetic Rotation”, one can not use the fields. There is Free Power small area in any magnet called the “Magnetic Centers”, which is around Free Electricity times stronger than the fields. The sequence is before pole center and after face center, and there for unlike other motors one must mesh the stationary centers and work the rotation from the inner of the center to the outer. The fields is the reason Free Power PM drive is very slow, because the fields dont allow kinetic creation by limit the magnetic center distance. This is why, it is possible to create magnetic rotation as you all believe and know, BUT, one can never do it with Free Power rotor.
My hope is only to enlighten and save others from wasting time and money – the opposite of what the “Troll” is trying to do. Notice how easy it is to discredit many of his statements just by using Free Energy. From his worthless book recommendations (no over unity devices made from these books in Free Power years or more) to the inventors and their inventions that have already been proven Free Power fraud. Take the time and read ALL his posts and notice his tactics: Free Power. Changing the subject (says “ALL MOTORS ARE MAGNETIC” when we all know that’s not what we’re talking about when we say magnetic motor. Free Electricity. Almost never responding to Free Power direct question. Free Electricity. Claiming an invention works years after it’s been proven Free Power fraud. Free Power. Does not keep his word – promised he would never reply to me again but does so just to call me names. Free Power. Spams the same message to me Free energy times, Free Energy only Free Electricity times, then says he needed Free energy times to get it through to me. He can’t even keep track of his own lies. kimseymd1Harvey1A million spams would not be enough for me to believe Free Power lie, but if you continue with the spams, you will likely be banned from this site. Something the rest of us would look forward to. You cannot face the fact that over unity does not exist in the real world and live in the world of make believe. You should seek psychiatric help before you turn violent. jayanth Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far!
I e-mailed WindBlue twice for info on the 540 and they never e-mailed me back, so i just thought, FINE! To heck with ya. Ill build my own. Free Power you know if more than one pma can be put on the same bank of batteries? Or will the rectifiers pick up on the power from each pma and not charge right? I know that is the way it is with car alt’s. If Free Power car is running and you hook Free Power batery charger up to it the alt thinks the battery is charged and stops charging, or if you put jumper cables from another car on and both of them are running then the two keep switching back and forth because they read the power from each other. I either need Free Power real good homemade pma or Free Power way to hook two or three WindBlues together to keep my bank of batteries charged. Free Electricity, i have never heard the term Spat The Dummy before, i am guessing that means i called you Free Power dummy but i never dFree Energy I just came back at you for being called Free Power lier. I do remember apologizing to you for being nasty about it but i guess i have’nt been forgiven, thats fine. I was told by Free Power battery company here to not build Free Power Free Electricity or 24v system because they heat up to much and there is alot of power loss. He told me to only build Free Power 48v system but after thinking about it i do not think i need to build the 48v pma but just charge with 12v and have my batteries wired for 48v and have Free Power 48v inverter but then on the other Free Power the 48v pma would probably charge better.
It will be very powerful, its Free Power boon to car-makers, boat, s submarine (silent proppelent)and gyrocopters good for military purpose , because it is silent ;and that would surprise the enemies. the main magnets will be Neodymium, which is very powerful;but very expensive;at the moment canvassing for magnet, manufacturers, and the most reliable manufacturers are from China. Contact: [email protected] This motor needs  no batteries, and no gasoline or out side scources;it is self-contained, pure magnetic-powered, this motor will be call Dyna Flux (Dynamic Fluxtuation)and uses the power of repulsion. Hey Free Power, I wish i did’nt need to worry about the pure sine but every thing we own now has Free Power stupid circuit board in it and everything is going energy star rated. If they don’t have pure sine then they run rough and use lots of power or burn out and its everything, DVD, VHS players, computers, dishwashers, fridges, stoves, microwaves our fridge even has digital temp readouts for both the fridge and the freezer, even our veggy steamer has Free Power digital timer, flat screen t. v’s, you can’t get away from it anymore, the world has gone teck crazzy. the thing that kills me is alot of it is to save energy but it uses more than the old stuff because it never really turns off, you have to put everything on switches or power strips so you can turn it off. I don’t know if i can get away from using batteries for my project. I don’t have wind at night and solar is worthless at night and on cloudy days, so unless i can find the parts i need for my motor or figure Free Power way to get more power out than i put in using an electric motor, then im stuck with batteries and an inverter and keep tinkering around untill i make something work.
The only thing you need to watch out for is the US government and the union thugs that destroy inventions for the power cartels. Both will try to destroy your ingenuity! Both are criminal elements! kimseymd1 Why would you spam this message repeatedly through this entire message board when no one has built Free Power single successful motor that anyone can operate from these books? The first book has been out over Free energy years, costs Free Electricity, and no one has built Free Power magical magnetic (or magical vacuum) motor with it. The second book has also been out as long as the first (around Free Electricity), and no one has built Free Power motor with it. How much Free Power do you get? Are you involved in the selling and publishing of these books in any way? Why are you doing this? Are you writing this from inside Free Power mental institution? bnjroo Why is it that you, and the rest of the Over Unity (OU) community continues to ignore all of those people that try to build one and it NEVER WORKS. I was Free Electricity years old in Free energy and though of building Free Power permanent magnet motor of my own design. It looked just like what I see on the phoney internet videos. It didn’t work. I tried all kinds of clever arrangements and angles but alas – no luck.
Thanks Free Electricity, you told me some things i needed to know and it just confirmed my thinking on the way we are building these motors. My motor runs but not the way it needs to to be of any real use. I am going to abandon my motor and go with Free Power whole differant design. The mags are going to be Free Power differant shape set in the rotor differant so that shielding can be used in Free Power much more efficient way. Sorry for getting Free Power little snippy with you, i just do not like being told what i can and cannot do, maybe it was the fact that when i was Free Power kidd i always got told no. It’s something i still have Free Power problem with even at my age. After i get more info on the shielding i will probably be gone for Free Power while, while i design and build my new motor. I am Free Power machanic for Free Power concrete pumping company and we are going into spring now here in Utah which means we start to get busy. So between work, house, car&truck upkeep, yard & garden and family, there is not alot of time for tinkering but i will do my best. Free Power, please get back to us on the shielding. Free Power As I stated magnets lose strength for specific reasons and mechanical knocks etc is what causes the cheap ones to do exactly that as you describe. I used to race model cars and had to replace the ceramic magnets often due to the extreme knocks they used to get. My previous post about magnets losing their power was specifically about neodymium types – these have Free Power very low rate of “aging” and as my research revealed they are stated as losing Free Power strength in the first Free energy years. But extreme mishandling will shorten their life – normal use won’t. Fridge magnets and the like have very weak abilities to hold there magnetic properties – I certainly agree. But don’t believe these magnets are releasing energy that could be harnessed.
Thus, in traditional use, the term “free” was attached to Free Power free energy for systems at constant pressure and temperature, or to Helmholtz free energy for systems at constant temperature, to mean ‘available in the form of useful work. ’ [Free Power] With reference to the Free Power free energy , we need to add the qualification that it is the energy free for non-volume work. [Free Power]:Free Electricity–Free Power | 2020-08-10 10:31:49 | {"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": 0.4516879916191101, "perplexity": 1472.725478193624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2020-34/segments/1596439738674.42/warc/CC-MAIN-20200810102345-20200810132345-00215.warc.gz"} |
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# If the average (arithmetic mean) of p, q and r is 50 and the average
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28 Dec 2017, 21:53
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If the average (arithmetic mean) of p, q and r is 50 and the average ( arithmetic mean) of q, r and s is 60, what is the value of s – p?
A. 60
B. 50
C. 40
D. 30
E. 10
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Re: If the average (arithmetic mean) of p, q and r is 50 and the average [#permalink]
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29 Dec 2017, 04:10
Bunuel wrote:
If the average (arithmetic mean) of p, q and r is 50 and the average ( arithmetic mean) of q, r and s is 60, what is the value of s – p?
A. 60
B. 50
C. 40
D. 30
E. 10
$$Average=\frac{Sum}{Terms}$$
$$\frac{(q+r+s)}{3}=60$$, $$q+r+s=180$$
$$\frac{(p+q+r)}{3}=50$$, $$p+q+r=150$$
$$(q+r+s)-(p+q+r)=180-150$$, $$s-p=30$$.
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Re: If the average (arithmetic mean) of p, q and r is 50 and the average [#permalink]
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20 Jan 2018, 23:08
(p + q + r)/3 = 50
Hence, p + q + r = 150 Equation 1
(q + r + s)/3 = 60
Hence, q + r + s = 180 Equation 2
Subtract Equation 1 from Equation 2:
(q + r + s) - (p + q + r) = 180 - 150
s - r = 30 (Answer D)
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Re: If the average (arithmetic mean) of p, q and r is 50 and the average [#permalink]
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19 Mar 2018, 09:25
Bunuel wrote:
If the average (arithmetic mean) of p, q and r is 50 and the average ( arithmetic mean) of q, r and s is 60, what is the value of s – p?
A. 60
B. 50
C. 40
D. 30
E. 10
(p + q + r)/3 = 50
(p + q + r) = 150 -------(I)
(q + r + s)/3 = 60
q +r + s = 180 -----(II)
Subtract I from II,
s - p = 30
(D)
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Re: If the average (arithmetic mean) of p, q and r is 50 and the average [#permalink]
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19 Mar 2018, 09:52
Bunuel wrote:
If the average (arithmetic mean) of p, q and r is 50 and the average ( arithmetic mean) of q, r and s is 60, what is the value of s – p?
A. 60
B. 50
C. 40
D. 30
E. 10
$$p + q + r = 150$$
$$q + r + s = 180$$
Now, $$(q + r + s) - (p + q + r) = 180 - 150$$
Or, $$(s - p) = 30$$, Answer must be (D)
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https://www.groundai.com/project/probing-the-tevatron-t-tbar-asymmetry-at-lhc/ | 1 Introduction
## Abstract
We use an effective operator framework to study the contributions to the Tevatron asymmetry from arbitrary vector bosons and scalars, and compare with their effect on the tail at LHC. Our study shows, for example, that models reproducing the asymmetry by exchange of and bosons or colour-triplet scalars lead to a large enhancement in the tail at LHC. This fact can be used to exclude these models as the sole explanation for the asymmetry, using the data already collected by CMS and ATLAS. Our analysis is model independent in the sense that we scan over all possible extra particles contributing to the asymmetry, and allow for general couplings. We also explore a class of Standard Model extensions which can accommodate the Tevatron asymmetry without contributing to the total cross section at first order, so that the enhancement of the tail at Tevatron and LHC is moderate.
Probing the Tevatron asymmetry at LHC
J. A. Aguilar–Saavedra, M. Pérez-Victoria
[0.2cm] Departamento de Física Teórica y del Cosmos and CAFPE,
## 1 Introduction
The measurement of the forward-backward (FB) asymmetry at Tevatron [1, 2, 3] has motivated a plethora of models which attempt to accommodate the experimental values, up to larger than the prediction of the Standard Model (SM). This task is not straightforward because the measured cross section is in good agreement with the SM: any “generic” addition to the production amplitude, large enough to produce the observed FB asymmetry, will easily give rise to too large a departure in the total rate. Many of the proposed models circumvent this problem at the expense of a cancellation between (linear) interference and (quadratic) new physics terms in the total cross section,
where is the SM cross section, the one corresponding to the new physics and the interference term. The cancellation requires a new large amplitude which, obviously, should have observable effects elsewhere. The ideal candidate to search for these effects is the Large Hadron Collider (LHC).
New physics in which produces such a large cancellation in the Tevatron cross section will likely produce an observable enhancement in the tail at LHC, even if at this collider top pair production is dominated by gluon fusion. (The tail is also enhanced at Tevatron energies but in the majority of the proposed models the deviations are compatible with present measurements [3].) In some cases, this effect should be visible already with the data collected in 2010. Conversely, if large deviations are not reported, a number of candidates to explain the Tevatron asymmetry will be excluded from the list.
In this paper we make these arguments quantitative for a wide class of SM extensions. We consider general new vector bosons and scalars, classified by their transformation properties under the SM gauge group , and study their possible effects in production. We use effective field theory to consistently (i) integrate out the new heavy states and obtain their contribution to in terms of four-fermion operators; (ii) obtain the cross sections in terms of effective operator coefficients. This allows us to find, for each vector boson and scalar representation, a relation between the possible values of the asymmetry at Tevatron and the excess in the tail at LHC. The discussion of all possible vector boson and scalar representations within the model-independent effective operator framework allows us to make stronger statements than in other model-independent studies [4, 5, 6, 7]. At the same time, any model with several vector bosons and scalars can be considered in our framework by simply summing the effective operator coefficients corresponding to the integration of each new particle. Previous studies of LHC signals associated to the FB asymmetry within particular models have been presented in Refs. [8, 9, 10, 11, 13, 12, 14, 15, 16].
After this analysis, we explore an alternative way to produce a large asymmetry with moderate effects in the tail at LHC. The key for this mechanism is the observation that one can also obtain a large without significant changes in the total cross section by introducing new physics which only contributes to the latter at quadratic order. This has been studied before, in the effective formalism, in Ref. [5]. If we write
for the forward (F) and backward (B) cross sections, the total rate is maintained at first order provided , which can be achieved, for example, with a new vector boson and a scalar. For models fulfilling this cancellation the size of the new physics contributions required to accommodate the experimental value of are smaller. Therefore, these models provide a better agreement of the tail with Tevatron measurements, and predict a much smaller tail at LHC, which is still potentially observable with forthcoming measurements.
We remark that the use of effective field theory (with the assumption that the new physics is too heavy to be directly produced at LHC) does not limit much the generality of our conclusions. For -channel exchange of new vector bosons or scalars, integrating out the new particles gives a good estimate for masses TeV. For -channel exchange this approximation is worse, but the cross section enhancement produced by the new particle(s) is always larger than the one from the corresponding four-fermion operator(s), and in this sense our predictions are conservative. On the other hand, if new physics in the tail is not seen at LHC, the new resonances are heavy and the effective operator framework can be safely used. It is also worth pointing out that we include corrections arising from the quadratic terms in new physics in the cross sections. The contributions from the interference of operators with the SM can be neglected, as they are suppressed with respect to the former for the values of parameters that are required to explain the asymmetry.
## 2 Extra bosons, operators and t¯t production
There are ten possible representations [17] for new vector bosons contributing to , while for scalars eight representations contribute. They are collected in Table 1, where the first column indicates the label used to refer to them.1 The relevant interaction Lagrangian is included as well, indicating the symmetry properties, if any, of the coupling matrices . We use standard notation with left-handed doublets and right-handed singlets , ; are the Pauli matrices, the Gell-Mann matrices normalised to and , , with and the charge conjugation matrix. The indices denote colour, and is the totally antisymmetric tensor.
The four-fermion operators contributing to production, including those that do not interfere with the SM QCD amplitude and only appear at quadratic level, have been given in Ref. [18]. (The operators which interfere with the SM were given in Ref. [19].) We collect in Tables 25 the values of the corresponding coefficients for all the vector and scalar irreducible representations that can induce these operators. For vector bosons the coefficients have previously been obtained in Ref. [17]. The notation for four-fermion operators is given in appendix A. We stress again that including both interference and quadratic terms in our calculations is not inconsistent, despite the fact that we are not considering dimension-eight operators. When quadratic terms are relevant for production (large couplings) the missing dimension-eight terms are sub-leading in the classes of SM extensions we consider. A related discussion about the importance of and contributions from dimension-six operators has been presented in Ref. [20].
We evaluate the FB asymmetry
AFB=σF−σBσF+σB=σFSM+δσF−σBSM−δσBσFSM+δσF+σBSM+δσB, (3)
using the SM predictions [21]
ASMFB=0.058±0.009 (inclusive), ASMFB=0.088±0.013 (mt¯t>450 GeV). (4)
and the new contributions from four-fermion operators , parameterised in terms of effective operator coefficients and numerical constants. The explicit expressions are collected in appendix A. It is important to point out that positive operator coefficients always increase at first order ( interference with the SM), as it follows from Eqs. (13). We choose, among the recently reported measurements of the FB asymmetry [3],
AexpFB=0.158±0.075 (inclusive), AexpFB=0.475±0.114 (mt¯t>450 GeV). (5)
the one for high invariant masses which exhibits the largest deviation () with respect to the SM prediction. The total cross section at LHC is evaluated including four-fermion operators in a similar way. In order to display the effect of new contributions on the tail we evaluate the cross section for invariant masses larger than 1 TeV. We note that our calculation in terms of effective operators gives a larger (smaller) tail than the exact calculation for -channel (-channel) resonances. In the former case the differences are not dramatic but in the latter our results can be quite conservative, depending on the mass and width of the new resonance. A detailed comparison is presented in appendix B.
The relation between the predictions for the Tevatron asymmetry and the tail at LHC is tested by performing a random scan over the relevant couplings corresponding to each new particle or multiplet. The results for the ten vector boson representations are presented in Fig. 1. (For , , and the regions are one-dimensional because there is only one coupling involved.) There are several interesting conclusions which can be drawn from these plots:
1. For and the allowed regions are inside the corresponding ones for , , respectively. This is expected because the interactions of the former correspond to a particular case of the latter, with only left-handed couplings.
2. For new colour-singlet neutral bosons , the linear terms have negative coefficients and decrease , which can only reach the experimental value for large couplings when quadratic terms dominate. Hence, accommodating a large asymmetry automatically implies a large tail. For example, for the enhancement is more than a factor of five. This implies that these possible explanations for can be probed, and eventually excluded, with the luminosity collected in the 2010 run of LHC. The same conclusion applies to the vector bosons and , as they only contribute in initial states and require a huge coupling to produce a large asymmetry.
3. For colour-octet isosinglet bosons it is possible to have a large and still a moderate tail at LHC. This model provides an example of cancellation of linear terms in the cross section, provided that , i.e. the vector boson couples as an axigluon. However, in order to have positive coefficients in the interference terms the couplings for the third and first generation must have opposite sign. This does not happen for the isotriplet boson which only has left-handed couplings, and for these SM extensions the predicted tail is large.
4. Another interesting candidate is a colour triplet , which has positive coefficients in interference terms as well. Its inclusion gives some enhancement to the asymmetry, which can reach the experimental value with the further addition of a scalar. Note that quadratic terms from operators decrease the asymmetry, as it can be derived from Eqs. (LABEL:ec:quad) and is clearly seen in Fig. 1. Hence, this model is interesting only for moderate couplings.
5. The rest of vector bosons, , and have little interest for the asymmetry because they do not allow for a value appreciably larger than the SM prediction.
Aside from these remarks, we also note that for (i) and ; (ii) and ; (iii) and , the linear terms have opposite sign, which explains the behaviour observed in the plots for these vector bosons.
The results for the eight scalar representations are presented in Fig. 2. Except for the isodoublets , , the regions are one-dimensional because there is only one coupling involved. We point out that:
1. A colour-singlet isodoublet (with the same quantum numbers as the Higgs boson) can give an asymmetry compatible with the experimental value, and still produce a moderate tail at LHC. (Note that quadratic terms involving the operators , decrease the asymmetry.) On the other hand, a colour octet produces an asymmetry smaller than the SM value because the interference terms have opposite sign.
2. For colour-sextets and the interference terms increase the asymmetry because the operator coefficients are positive.2 However, producing an asymmetry requires large couplings and implies a large tail at LHC, which might be already excluded. For the colour triplets , the situation is worse because the interference terms have negative operator coefficients, and even larger couplings are required to produce a large asymmetry.
3. For the two scalars , which only contribute in , the deviations in are always very small.
To conclude this survey, we remark again that this correlation between and the tail at LHC applies to SM extensions with a single vector boson or scalar (as many of the ones proposed in the literature) but when more than one particle is present the contributions can add up or cancel, making it easier to fit the experimental data and predict moderate effects in the tail, as we discuss in detail in the next section.
## 3 A large t¯t asymmetry with a small t¯t tail
By inspection of Eqs. (13) it is clear that the cancellation of the linear contributions to the cross section takes place provided
C1133qq′+C3113qq+C3113uu=C1331qu+C3113qu≡c1, C1133qq′+2C3311ud′=C1331qu+C3113qd≡c2 (6)
(see also Ref. [5]). Notice that in the left-hand side of both equations we have and couplings, whereas on the right-hand side we have and ones. As we have mentioned, one simple example where both equalities are fulfilled is an axigluon with flavour-diagonal couplings , with the additional requirement that first and third generation couplings have opposite sign, to have positive coefficients. (This model may be excluded by low-energy measurements, however [25].) But there are many other possibilities which can be constructed combining particles in Table 2, for instance, a colour triplet together with a colour sextet or .
In these SM extensions with vanishing (or very small) contributions to the total cross section at first order, the FB asymmetry is
AFB=ASMFB+2c1(DFint−DBint)u¯u+2c2(DFint−DBint)d¯dσSM (7)
plus smaller corrections from quadratic terms, which depend on the specific operators which yield and . Remarkably, one can obtain a good fit to both asymmetry measurements in Eqs. (5) with values of , of order unity.3 For instance, assuming and equal , , and terms, the best fit to both measurements is , for which
A4FFB=0.225 (inclusive), A4FFB=0.366 (mt¯t>450 GeV) (8)
including linear and quadratic terms, with a of 1.72. For the best fit is found for , giving similar predictions for the asymmetry.
We have investigated the effects on the tails by implementing four-fermion operators in the generator Protos [26]. We have first checked that this class of SM extensions does not produce a too large tail at Tevatron. Figure 3 shows the invariant mass distributions for the SM and with , four-fermion contributions corresponding to , which yield the asymmetries in Eqs. (8). Above GeV the cross section is enhanced by , which is consistent with data [3].
For LHC the invariant mass distributions are presented in Fig. 4. The cross section above 1 TeV is a factor of above the SM one. This deviation could be visible with the luminosity to be collected in 2011, provided that the systematic uncertainties (jet energy scale, jet energy resolution, jet energy scale, etc.) are low enough. Here it is necessary to mention that for Tevatron a smaller efficiency for -channel new physics at high has been recently claimed [27, 28], which could help maintain the cross section at the high bins in agreement with measurements while reproducing the FB asymmetry. For LHC the efficiency decrease at TeV is not significant because of the larger detector coverage up to for charged leptons and for jets (see appendix B).
## 4 Conclusions
If the asymmetry measured at Tevatron corresponds to new physics, this new physics should also manifest at the tail at LHC. The size of the effect of course depends on the new physics itself which gives rise to the FB asymmetry, and it can serve to discriminate among different explanations. These issues have been investigated here using an effective operator framework and classifying all possible new vector bosons and scalars by their transformation properties under the SM gauge group. Particular models in the literature attempting to explain the observed asymmetry often fall into one of these classes.
For models which reproduce with - and/or -channel exchange [29, 30, 10, 11, 12, 27] we have found that the tail above 1 TeV should be enhanced by a factor of five at LHC, at least for bosons heavier than 1 TeV. With pb at the tree level and a luminosity of approximately 35 pb collected in the 2010 run, the SM predicts 6.3 events in the semileptonic decay channel, not including the detector acceptance nor efficiencies. Then, it seems likely that an enhancement by a factor of five in the tail could be excluded just by analysing 2010 data. The same argument applies to -channel exchange [31, 32, 33, 27] or a mixture of both [34].
More exotic models explain the observed asymmetry by the exchange of a colour-triplet isosinglet scalar (see for example Refs. [22, 23, 10, 24, 27]) or its colour-sextet counterpart [22, 23, 35, 15, 24, 27]. These models could also be tested, and eventually excluded, with data already analysed by CMS and ATLAS in the search for resonances. On the other hand, models with axigluons [36, 37, 32, 13, 10] or other types of colour-octet bosons [38, 39, 40, 41, 35, 27] can in principle accommodate the measured asymmetry predicting a moderate tail at LHC. (Our discussion obviously does not directly apply to models where the new physics produces plus other particles in the final state, see for example Ref. [42].) Besides, we note that Ref. [7] has recently found that SM extensions explaining the FB asymmetry without predicting too large tails must have interference with the SM amplitudes.4 As we have shown, our conclusions are stronger because in many extensions with interference this is not possible either. Moreover, for all vector bosons and scalars in Table 1 there is interference unless the involved couplings vanish.
Finally, in this paper we have investigated the conditions under which the first order contributions to the total cross section cancel while still producing a FB asymmetry compatible with experimental data. (A previous study at this order has been presented in Ref. [5].) Clearly, in this situation the tails of the invariant mass distribution are much smaller, both at Tevatron and LHC, as it has been shown explictly. A popular example of a model fulfilling these conditions is an axigluon with opposite couplings to the first and third generation, but there are many other possibilities which can be worked out from Table 2. All these SM extensions can accommodate the Tevatron asymmetry and cross section with small couplings, and predict a moderate enhancement of the tail at Tevatron and LHC, which is not in contradiction with experiment. Interestingly, a small excess in the tail with boosted tops has been already observed at Tevatron [43]. In any case, these possible departures will soon be tested with forthcoming LHC data.
## Acknowledgements
This work has been partially supported by projects FPA2010-17915 (MICINN), FQM 101 and FQM 437 (Junta de Andalucía) and CERN/FP/116397/2010 (FCT).
## Appendix A Four-fermion operators in t¯t production
We use the minimal basis in Ref. [18] for gauge-invariant four-fermion operators. Fermion fields are ordered according to their spinorial index contraction, and subindices , indicate the pairs with colour indices contracted, if this pairing is different from the one for the spinorial contraction. Our basis consists of the following operators:
(i) operators
Oijklqq=12(¯qLiγμqLj)(¯qLkγμqLl), Oijklqq′=12(¯qLiaγμqLjb)(¯qLkbγμqLla), Oijklℓq=(¯ℓLiγμℓLj)(¯qLkγμqLl), Oijklℓq′=(¯ℓLiγμqLj)(¯qLkγμℓLl), Oijklℓℓ=12(¯ℓLiγμℓLj)(¯ℓLkγμℓLl). (9)
(ii) operators
Oijkluu=12(¯uRiγμuRj)(¯uRkγμuRl), Oijkldd=12(¯dRiγμdRj)(¯dRkγμdRl), Oijklud=(¯uRiγμuRj)(¯dRkγμdRl), Oijklud′=(¯uRiaγμuRjb)(¯dRkbγμdRla), Oijkleu=(¯eRiγμeRj)(¯uRkγμuRl), Oijkled=(¯eRiγμeRj)(¯dRkγμdRl), Oijklee=12(¯eRiγμeRj)(¯eRkγμeRl). (10)
(iii) operators
Oijklqu=(¯qLiuRj)(¯uRkqLl), Oijklqu′=(¯qLiauRjb)(¯uRkbqLla), Oijklqd=(¯qLidRj)(¯dRkqLl), Oijklqd′=(¯qLiadRjb)(¯dRkbqLla), Oijklℓu=(¯ℓLiuRj)(¯uRkℓLl), Oijklℓd=(¯ | 2019-12-09 10:30:56 | {"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": 0.8707038164138794, "perplexity": 731.7590951115243}, "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-2019-51/segments/1575540518627.72/warc/CC-MAIN-20191209093227-20191209121227-00009.warc.gz"} |
https://dmoj.ca/problem/dmopc14ce1p1 | ## DMOPC '14 Exam Time P1 - French Homework
View as PDF
Points: 3
Time limit: 1.0s
Memory limit: 64M
Author:
Problem type
One day you are too busy figuring out a programming contest question to realize that you have French homework due the next day!
Surprisingly, your French teacher decides to mark the sheets and sees you rapidly scribbling as she passes by. Since she knows that you are very talented at programming, she decides to give you another chance — with a catch.
The second chance is that you may finish completing the worksheet in class, however; you must directly copy all the answers from a program you wrote under her direct supervision.
In French, the order of sentences generally follows a Subject-Verb-Object structure. For questions, this can change to a Verb-Subject-Object order. We will assume that the subject will always be tu and note that the verb will already be conjugated in the present tense for the second person singular.
In addition, you notice the following pattern (which is true most of the time):
If the word ends with e, then it is feminine (la)
If the word ends with s, then it is plural (les)
Otherwise, we can assume the word to be masculine (le)
You also notice on the sheet that none of the words on the second line of each case begin with a vowel.
#### Input Specification
The first line of input will consist of the verb while the second line of input will consist of the object. Each word will be up to characters in length.
#### Output Specification
Output the words as a question formed using inversion.
#### Sample Input 1
Fais
devoirs
#### Sample Output 1
Fais-tu les devoirs ?
#### Sample Input 2
Aimes
ordinateurs
#### Sample Output 2
Aimes-tu les ordinateurs ?
#### Explanation for Sample Output 2
Since devoirs and ordinateurs both end with an s, we may assume that they require les. Note that this is not always true (for example, "le vers" means "the verse"), but the test data will not contain such cases.
## Comments
• commented on March 12, 2021, 12:53 p.m.
The moment you realize that you need a question mark at the end of the sentence with a space.
• commented on Feb. 10, 2015, 3:17 p.m.
"If the word ends with e, then it is feminine (la) If the word ends with s, then it is plural (les) Otherwise, we can assume the word to be masculine (le)"
Which word are you talking about? The verb or the object or both?
• commented on Feb. 10, 2015, 3:40 p.m.
The word refers the object. For example, in the first sample input, the object would be devoirs, which requires les. If it were devoir, it would have required le, and if it were feminine, it would have required la. | 2022-01-27 05:05:30 | {"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": 0.4345816671848297, "perplexity": 1941.498687849299}, "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/1642320305141.20/warc/CC-MAIN-20220127042833-20220127072833-00315.warc.gz"} |
https://projecteuclid.org/Proceedings/advanced-studies-in-pure-mathematics/Geometry-Dynamics-and-Foliations-2013--In-honor-of-Steven/chapter/Lipschitz-matchbox-manifolds/10.2969/aspm/07210071 | Translator Disclaimer
VOL. 72 | 2017 Lipschitz matchbox manifolds
Steven Hurder
Editor(s) Taro Asuke, Shigenori Matsumoto, Yoshihiko Mitsumatsu
## Abstract
A matchbox manifold is a connected, compact foliated space with totally disconnected transversals; or in other notation, a generalized lamination. It is said to be Lipschitz if there exists a metric on its transversals for which the holonomy maps are Lipschitz. Examples of Lipschitz matchbox manifolds include the exceptional minimal sets for $C^{1}$-foliations of compact manifolds, tiling spaces, the classical solenoids, and the weak solenoids of McCord and Schori, among others. We address the question: When does a Lipschitz matchbox manifold admit an embedding as a minimal set for a smooth dynamical system, or more generally as an exceptional minimal set for a $C^{1}$-foliation of a smooth manifold? We give examples which do embed, and develop criteria for showing when they do not embed, and give examples. We also discuss the classification theory for Lipschitz weak solenoids.
## Information
Published: 1 January 2017
First available in Project Euclid: 4 October 2018
zbMATH: 1385.57024
MathSciNet: MR3726706
Digital Object Identifier: 10.2969/aspm/07210071
Subjects:
Primary: 37B45, 52C23, 54F15, 57R05
Secondary: 53C12, 57N55
Rights: Copyright © 2017 Mathematical Society of Japan
PROCEEDINGS ARTICLE
45 PAGES
SHARE
RIGHTS & PERMISSIONS | 2021-10-27 00:33:06 | {"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": 0.33862462639808655, "perplexity": 3495.5014650815033}, "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-2021-43/segments/1634323587963.12/warc/CC-MAIN-20211026231833-20211027021833-00107.warc.gz"} |
https://discretize.simpeg.xyz/en/main/api/generated/discretize.CurvilinearMesh.cell_gradient_x.html | X-derivative operator (cell centers to x-faces)
This property constructs an x-derivative operator that acts on cell centered quantities; i.e. the x-component of the cell gradient operator. When applied, the x-derivative is mapped to x-faces. The operator is a sparse matrix $$\mathbf{G_x}$$ that can be applied as a matrix-vector product to a cell centered quantity $$\boldsymbol{\phi}$$, i.e.:
grad_phi_x = Gx @ phi
By default, the operator assumes zero-Neumann boundary conditions on the scalar quantity. Before calling cell_gradient_x however, the user can set a mix of zero Dirichlet and zero Neumann boundary conditions using set_cell_gradient_BC. When cell_gradient_x is called, the boundary conditions are enforced for the differencing operator.
Returns
(n_faces_x, n_cells) scipy.sparse.csr_matrix
X-derivative operator (x-component of the cell gradient)
Examples
Below, we demonstrate how to set boundary conditions for the x-component cell gradient, construct the operator and apply it to a discrete scalar quantity. The mapping of the operator and its sparsity is also illustrated. Our example is carried out on a 2D mesh but it can be done equivalently for a 3D mesh.
We start by importing the necessary packages and modules.
>>> from discretize import TensorMesh
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> import matplotlib as mpl
We then construct a mesh and define a scalar function at cell centers.
>>> h = np.ones(40)
>>> mesh = TensorMesh([h, h], "CC")
>>> centers = mesh.cell_centers
>>> phi = np.exp(-(centers[:, 0] ** 2) / 8** 2)
Before constructing the operator, we must define the boundary conditions; zero Neumann for our example. Even though we are only computing the derivative along x, we define boundary conditions for all boundary faces. Once the operator is created, it is applied as a matrix-vector product.
>>> mesh.set_cell_gradient_BC(['neumann', 'neumann'])
>>> grad_phi_x = Gx @ phi
Now we plot the original scalar, and the x-derivative.
Expand to show scripting for plot
>>> fig = plt.figure(figsize=(13, 5))
>>> mesh.plot_image(phi, ax=ax1)
>>> ax1.set_title("Scalar at cell centers", fontsize=14)
>>> ax2 = fig.add_subplot(122)
>>> v = np.r_[grad_phi_x, np.zeros(mesh.nFy)] # Define vector for plotting fun
>>> mesh.plot_image(v, ax=ax2, v_type="Fx")
>>> ax2.set_yticks([])
>>> ax2.set_ylabel("")
>>> ax2.set_title("X-derivative at x-faces", fontsize=14)
>>> plt.show()
The operator is a sparse x-derivative matrix that maps from cell centers to x-faces. To demonstrate this, we construct a small 2D mesh. We then show the ordering of the elements and a spy plot.
>>> mesh = TensorMesh([[(1, 3)], [(1, 4)]])
Expand to show scripting for plot
>>> fig = plt.figure(figsize=(12, 8))
>>> mesh.plot_grid(ax=ax1)
>>> ax1.set_title("Mapping of Operator", fontsize=14, pad=15)
>>> ax1.plot(mesh.cell_centers[:, 0], mesh.cell_centers[:, 1], "ro", markersize=8)
>>> for ii, loc in zip(range(mesh.nC), mesh.cell_centers):
... ax1.text(loc[0] + 0.05, loc[1] + 0.02, "{0:d}".format(ii), color="r")
>>> ax1.plot(mesh.faces_x[:, 0], mesh.faces_x[:, 1], "g>", markersize=8)
>>> for ii, loc in zip(range(mesh.nFx), mesh.faces_x):
... ax1.text(loc[0] + 0.05, loc[1] + 0.02, "{0:d}".format(ii), color="g")
>>> ax1.set_xticks([])
>>> ax1.set_yticks([])
>>> ax1.spines['bottom'].set_color('white')
>>> ax1.spines['top'].set_color('white')
>>> ax1.spines['left'].set_color('white')
>>> ax1.spines['right'].set_color('white')
>>> ax1.legend(
... ['Mesh', '$\mathbf{\phi}$ (centers)', '$\mathbf{Gx^* \phi}$ (x-faces)'],
... loc='upper right', fontsize=14
... )
>>> ax2 = fig.add_subplot(122) | 2021-10-27 10:10:47 | {"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": 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": 0.8914145827293396, "perplexity": 9411.468567258982}, "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/1634323588113.25/warc/CC-MAIN-20211027084718-20211027114718-00074.warc.gz"} |
http://www.sciforums.com/threads/health-benefits-of-greentea.96226/page-3 | # Health Benefits Of GreenTea
Discussion in 'The Cesspool' started by martineargent, Sep 21, 2009.
Not open for further replies.
1. ### chimpkinC'mon, get happy!Registered Senior Member
Messages:
4,416
Huh? white tea seems to be better for you...
Far more anti-inflammatory, more anti-oxidative, more antibacterial. Which is why I pay more money to drink it-I am a total cheap a, white tea is a bit more costly...
But tastier too, I agree.
http://www.sciforums.com/showpost.php?p=2850784&postcount=40
I sit here, alone, typing away, in the dark, googling my fingers to the bone for you children, and this is the thanks I get....:bawl:
Last edited: Nov 5, 2011
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3. ### nietzschefanThread KillerValued Senior Member
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7,721
Not even Fraggle is that straight edge.
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5. ### Michael歌舞伎Valued Senior Member
Messages:
20,285
There's evidence green tea may lower risk of prostate cancer. I like green tea (the real stuff, not those Lipton baggies). I brew a couple pots of green tea in my office each day. Look for long straight leaves as this is a sign of top quality. The more broken bits means it's more than likely bottom of the barrow mixed up tea leaves that were left over from various better teas. These are why Lipton et.al. is put them into little tea baggies. Good tea easily sits in a net.
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7. ### KittamaruAshes to ashes, dust to dust. Adieu, Sciforums.Valued Senior Member
Messages:
13,938
Ah, lol, sorry, I admit I misread what you had posted - I was reading this in the few minutes before I headed for work and thus didn't have time to read the link itself XD
8. ### Stoniphiobscurely fossiliferousValued Senior Member
Messages:
3,175
Yes, and exercise has also been proven to lower that risk - as well as the risk for many other cancers, type 2 diabetes and several dementias. Something to think about.
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9. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member
Messages:
23,198
Yes but much stronger evidence that it does not:
Here is the abstract from that large prospective study:
".. In a prospective study of 19,561 Japanese men, green-tea intake was not associated with a lower risk of prostate cancer (110 cases), the multivariate hazard ratio for men drinking ≥5 cups compared with <1 cup per day being 0.85 (95% confidence interval 0.50–1.43, trend P=0.81). "
I have prostate cancer despite a "radical prostatectomy" nearly three years ago. I have concocted a "threapeutic diet" supplement of seven items after more than 100 hours of searching at Pub Med and other scientific journal sites for foods with two or more peer-reviewed studies showing some benefit of each item, separately used.
It is not a cure, but does seem to provide some control when I am in "off drug" cycles. I added green tea to my threapeutic diet for about one year, (It replaced my morning coffee) hoping for a "synergistic effect" but did not notice any. So when I read in common literature that coffee reduced the risk of cancer, I went back to coffee. I have not made any effort to learn if there is any scientific basis for coffee helping. I now take my diet supplement with each meal and it seems that the earlier once/ day may not have been often enough. Some items are known to be rapidly metabolized.
With this more frequent and larger amount of diet supplement, I now can use the testosterone suppressing drug only 20% of the time and still keep my PSA below 0.08ng/dL but I am not yet ready to fully describe the diet as I do not wish to give false hopes. I have mentioned in another thread that hot red peppers are part of the diet - the ones I use are common in Brazil, called "Malagueta," and about 12 times stronger in the active (burning) agent than common hot red peppers found in the USA. One adapts to their active agent, so now I take ~20 times more (18 grams / per day) than I could stand initially due to their effect upon the bowels. I actually slightly like the taste of my diet now.
However, it does have a strange mild effect, which I think I understand. The active agent in the peppers causes my body to think it is over heating (a mild systemic burning? not consciously sensed) so in reaction, I become chilled and must put on a sweater but that only lasts an hour or so.
Last edited by a moderator: Nov 6, 2011
10. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member
Messages:
23,198
“…We conducted a randomized, double-blind, placebo-controlled intervention study to investigate the effect of the main green tea catechin, epigallocatechin gallate (EGCG), taken in a green tea extract, Polyphenon E (PPE). Postmenopausal women (n=103) were randomized into three arms: placebo, 400 mg EGCG as PPE, or 800 mg EGCG as PPE as capsules per day for 2 months. … Based on urinary tea catechin concentrations, compliance was excellent. Supplementation with PPE did not produce consistent patterns of changes in estradiol (E2), estrone (E1), or testosterone (T) levels.
Low density lipoprotein (LDL)-cholesterol decreased significantly in both PPE groups but was unchanged in the placebo group;… Glucose and insulin levels decreased nonsignificantly in the PPE groups but increased in the placebo group; … In summary, green tea (400 and 800 mg EGCG as PPE; ~5-10 cups) supplementation for 2 months had suggestive beneficial effects on LDL cholesterol concentrations and glucose-related markers. …”
Billy T condensed abstract above from: Effect of 2-month controlled green tea intervention on lipoprotein cholesterol, glucose, and hormonal levels in healthy postmenopausal womenCancer Prev Res canprevres.0407.2011; Published OnlineFirst January 13, 2012;
11. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member
Messages:
23,198
“…We conducted a randomized, double-blind, placebo-controlled intervention study to investigate the effect of the main green tea catechin, epigallocatechin gallate (EGCG), taken in a green tea extract, Polyphenon E (PPE). Postmenopausal women (n=103) were randomized into three arms: placebo, 400 mg EGCG as PPE, or 800 mg EGCG as PPE as capsules per day for 2 months. … Based on urinary tea catechin concentrations, compliance was excellent. Supplementation with PPE did not produce consistent patterns of changes in estradiol (E2), estrone (E1), or testosterone (T) levels.
Low density lipoprotein (LDL)-cholesterol decreased significantly in both PPE groups but was unchanged in the placebo group;… Glucose and insulin levels decreased nonsignificantly in the PPE groups but increased in the placebo group; … In summary, green tea (400 and 800 mg EGCG as PPE; ~5-10 cups) supplementation for 2 months had suggestive beneficial effects on LDL cholesterol concentrations and glucose-related markers. …”
Billy T condensed abstract above from: Effect of 2-month controlled green tea intervention on lipoprotein cholesterol, glucose, and hormonal levels in healthy postmenopausal womenCancer Prev Res canprevres.0407.2011; Published OnlineFirst January 13, 2012;
Read full abstract here: http://cancerpreventionresearch.aacrjournals.org/content/early/recent
It is the third paper, about one page down.
12. ### Michael歌舞伎Valued Senior Member
Messages:
20,285
It tastes good too
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I brew a pot of the good stuff twice a day when I'm at work. Try to get green tea that's long and unbroken (say 5mm in length, deep green color). If you're getting broken little bits then that's bottom of the barrel tea leaves (usually used in lipton tea bags or something like that). If it's kept sealed it'll last forever.
13. ### elteValued Senior Member
Messages:
1,253
I wonder if green tea really reduces tooth decay, or might it be more accurate to say it causes little or no decay when compared to acidic and sugary drinks?
14. ### chimpkinC'mon, get happy!Registered Senior Member
Messages:
4,416
Biofilms, of which dental plaque is one sort, does not like green tea. Green tea messes with the ability to the bacteria to form self-protective colonies, which is what a biofilm is.
15. ### birchValued Senior Member
Messages:
5,077
what can't green tea not do?
can it save the world too?
16. ### spidergoatVenued Serial MembershipValued Senior Member
Messages:
53,489
Green tea has caffeine in it. I know because I have become sensitive to caffeine, and green tea is something I can only take in small quantities.
17. ### Giambattistasssssssssssssssssssssssss sssssValued Senior Member
Messages:
4,878
The debate smolders on?
I see a lot of posters with very few posts whose comments seem to amount to cut-and-paste compliments of green tea, taken from health news soundbites. Maybe the tea industry has found this thread attractive?
As for caffeine, I have heard that caffeine may be the main component in coffee that accounts for some of its anti-diabetic or diabetes preventive effects. Although I've heard that is only for pre-diabetes, and that caffeine can screw up blood sugar once you have the condition. Or something like that.
I'm too lazy to fetch anything to prove or disprove.
I think the consensus, if there is one, would be that a few cups of green tea with other healthy dietary habits is not harmful and probably aids you.
Green tea polyphenol absorption can be increased by up to 300% when taken with certain amounts of ascorbic acid.
Again, too lazy to fetch.
I'm just gonna stretch out..
:yawn: <------ Laziness
18. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member
Messages:
23,198
there is no evidence for that claim and about cancer control or reduction. Several clinical studies showing the opposite - no effect exist. Are you, and most of the other new comers with less than 10 post being paid by some Green Tea company?
Green tea, like many other agents in food does have some mild beneficial effects and probably does little if any harm. It is second only to water as the world´s most common drink.
Here is post with link showing what it can benefit without your wild "snake oil cures everything" claims.
http://www.sciforums.com/showpost.php?p=2890293&postcount=55
Last edited by a moderator: Apr 10, 2012
19. ### ChipzBannedBanned
Messages:
838
Lipton's™ Green Tea saved my life, I had cancer once and then when I drank Lipton's™ Green Tea, I didn't.
20. ### Stoniphiobscurely fossiliferousValued Senior Member
Messages:
3,175
Yeah Billy, I too was wondering why all of the new handles have landed on this thread...wrote it off to serendipity. :shrug:
I have a mug of green tea every morning after my run and shower. It is warm, pleasant to the palate and rehydrating.
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21. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member
Messages:
23,198
In addition to the link I gave in post 55 to a quite recent study showing green tea was of no use in prevent or curing cancer, here is link to an older, very large, and complete and careful controlled study showing the same thing for prostate cancer:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2360636/?tool=pubmed
Here is its brief abstract in full:
“In a prospective study of 19 561 Japanese men, green-tea intake was NOT associated with a lower risk of prostate cancer (110 cases), the multivariate hazard ratio for men drinking ≥5 cups compared with <1 cup per day being 0.85 (95% confidence interval 0.50–1.43, trend P=0.81).”
Here is the title of the paper reporting the study:
“No association between green tea and prostate cancer risk in Japanese men: the Ohsaki Cohort Study”
Here is part of the introduction paragraph:
“…Green-tea consumption per capita in Japan is the highest in the world (International Tea Committee, 2004). One reason for the low incidence of prostate cancer in Japan may be the high consumption of green tea. We therefore examined the association between green-tea consumption and prostate cancer incidence among men in the Ohsaki Cohort Study conducted in rural Japan. …”
This low incidence of prostate cancer in Japan, is like the lower death by heart disease in France. Ignorant people and then the press, jump to the conclusion that it is due to green tea in Japan and red wine in France. Red wine does contain resveratrol, which in VITRO can be shown to have many desirable effects, including upon cancer cells. But it is destroyed in the human gut so essentially none gets into the blood (You can get it there by direct absorption under the tongue from resveratrol saturated wafers.):
“…oral bioavailability of resveratrol is low because it is rapidly metabolized in intestines and liver into conjugated forms: glucuronate and sulfonate.[102] Only trace amounts (below 5 ng/mL) of unchanged resveratrol could be detected in the blood after 25 mg oral dose.[102] Even when a very large dose (2.5 and 5 g) was given as an uncoated pill, the concentration of resveratrol in blood failed to reach the level claimed to be necessary for the systemic cancer prevention.[40] A formulation of resveratrol in a chewing gum form is now in production, and this would be expected to achieve much higher blood levels than oral formulations. …”
Quote from: http://en.wikipedia.org/wiki/Resveratrol
I.e. even if you took 25mg by mouth only 5 ng/mL ends up in the blood and the liver rapidly destroys that tiny fraction.
It is sad commentary on general human progress from the days when we were burning witches to see how strongly still scientifically unfounded beliefs are accepted by the masses just because of coincidental associations. (A crazy girl in Salem, low rate of prostate cancer in Japan, or lower heart attack rates in France)
I have prostate cancer an operation removing prostate and associated tissue did not cure. Thus, when I learned it failed as a cure, I spent more than 100 hours at Pub Med looking for diet items that might help (had at least two pier-reviewed studies showing effectives) I found four and take them before every meal with some indications that they are helping. I keep PSA < o.1 ng/dL with low duty cycle use of Testosterone, T, lowering drug but after three years of use must report they are not a cure. Initially red grape juice and green tea were part of the "therapeutic diet" but later I learned more about them and dropped their use. At least five other items with popular press claims have been tested (in my carefully controlled one patient study) but then dropped. I pay nothing in Brazil to measure PSA and T so do both five times in each on drug/ off drug cycle. First measured pair is about 9 hours after last drug pill is take, 2nd is about 7 weeks later, then three more at decreasing intervals to watch PSA approach 0.1 and as it does start next cycle´s on drug phase.
I plot the ratio, PSA/T and so long as it remains essentially constant conclude that the cancer cells in my body still need T to be active. Normally with more continuous use of drugs to surpress T, the drugs become ineffective and PSA climbs even if T is held near castration levels. My hope, so far achieved, is that will low duty cycle use of drugs the drug will not lose control. I.e. not become an untreatable CRPC case. When (and if) I have gone four years keeping PSA <0.1 and ratio PSA/T constant, I will publish my "therapeutic diet" details.
Last edited by a moderator: Apr 11, 2012
22. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member
Messages:
23,198
Please give link to a pier-reviewed, journal article supporting this claim (Someone´s opinion, even if published in some Natural Foods magazine etc., is still just someone´s opinion.)
I am not disputing that there are anti-oxidents, etc in green tea, nor the fact that they can be shown to be effective in vitro tests against isolated cancer cells. I want to see some in vivo evidence, where these agents not only survive the normally occurring processes that break down of complex molecules in animal´s digestive system´s but also get into the blood stream and produce the results claimed.
I have still low levels PSA three+ years after my prostate (& nearby associated tissue) was remove. I.e. I am in the 10 to 25% the a radical prostateomy did not cure. I spent at least 100 hours searching (mainly at Pub Med) for diet items that might help. (Had at least two scientific studies showing their effectiveness) - I found 6, including green tea and red wine, but later dropped both from my diet when more detailed investigation showed they were just in vitro studies and other studies in vivo showed no significant effects (because their agents are destroyed in the gut).
The agent (Resveratrol) in red wine (or also in cheaper red grape juice), is now or soon to be available in special chewing gum as studies have shown it can get into the blood if held in the mouth, especially under the tongue.
I am certainly NOT opposed to diet as aids to cancer control. In fact believe my diet items, taken before each meal, are why I can keep PSA < 0.1ng/dL for several months with no use of drugs lowering Testosterone, T, to near castration levels. Fortunately for me, I live in Brazil and have no cost to frequently measure both T & PSA.
I go off drugs more than three months, first measuring T & PSA in early AM about 10 hours after last drug pill was taken, then again after about 7 weeks later with three more measurements of both at decreasingly shorter intervals (to know when my PSA is approaching 0.1 and I must go back on drug for about one month.)
In essence I am doing a very carefully control study of low duty cycle use of standard drugs. If I can go four years with little drug use and keep PSA below 0.1 (and also the ratio of PSA/T essentially constant while "off drug") then I will publish my diet - don´t want to give false hopes. Normally durgs lose control in about 3 years and one is then A CRPC case with very little that can be done for you (or nothing not costing ~\$100,000 per year).
Last edited by a moderator: Apr 20, 2012
23. ### mikerawlinsRegistered Member
Messages:
7
Drinking green tea makes me want to go to the bathroom. You know what I mean... | 2019-02-18 09:13:37 | {"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": 0.2955876886844635, "perplexity": 5955.376538601548}, "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-2019-09/segments/1550247484772.43/warc/CC-MAIN-20190218074121-20190218100121-00277.warc.gz"} |
https://www.vedantu.com/question-answer/a-vertical-tower-stands-on-a-horizontal-plane-class-10-maths-cbse-5edcdd90e5b56371c5aae986 | Question
A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height 5 meters. At a point on the plane, the angles of elevation of the bottom and the top of the flagstaff are respectively ${30^0}$ and ${60^0}$.Find the height of the tower.
Hint- Draw the diagram of question and use value of trigonometric angles $\tan {30^0} = \frac{1}{{\sqrt 3 }}$ and $\tan {60^0} = \sqrt 3$ .
Let QR be the height of the tower (h meters) and RS be the height of flagstaff surmounted on
the tower (RS=5m). Let the point P lie on the horizontal plane at a distance of x meters from
the foot of the tower at point Q (PQ=x meters).
In $\vartriangle PQR$, the angle of elevation of the bottom of the flagstaff is ${30^0}$ .
$\tan {30^0} = \dfrac{{QR}}{{PQ}} \\ \Rightarrow \dfrac{1}{{\sqrt 3 }} = \frac{h}{x} \\ \Rightarrow x = \sqrt 3 h.........\left( 1 \right) \\$
In $\vartriangle PQS$, angle of elevation of the top of the flagstaff is ${60^0}$ .
$\tan {60^0} = \dfrac{{QS}}{{PQ}} = \dfrac{{QR + RS}}{{PQ}} \\ \Rightarrow \sqrt 3 = \dfrac{{h + 5}}{x} \\ \Rightarrow x = \dfrac{{h + 5}}{{\sqrt 3 }}..........\left( 2 \right) \\$
Eliminating x using (1) and (2) equation
$\Rightarrow \sqrt 3 h = \dfrac{{h + 5}}{{\sqrt 3 }} \\ \Rightarrow 3h = h + 5 \\ \Rightarrow 2h = 5 \\ \Rightarrow h = 2.5m \\$
So, the height of tower is 2.5 meters
Note- Whenever we face such types of problems we use some important points. Like draw
the figure of question with notify all points and distances then make the relation between
variables with the help of trigonometric angles then after eliminating x we can get the value
of the height of the tower. | 2021-05-07 20:14:34 | {"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": 0.8598579168319702, "perplexity": 571.9558698620551}, "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-2021-21/segments/1620243988802.93/warc/CC-MAIN-20210507181103-20210507211103-00304.warc.gz"} |
https://www.internetbookstorepro.com/product/10-7449-2019-mst_2019_1367_1375/ | # Thermal Stability Improvement of Fungal Mycelium
\$25.00
Category:
## Description
Abstract: A natural material was explored through cultivating filamentous fungi (Pleurotus ostreatus) with sodium silicate in this paper. The substrate for feeding filamentous fungi was mixed with different concentrations of sodium silicate to improve thermal stability of fungal mycelium. The preliminary characterizations were conducted. There exists an interaction of fungi and sodium silicate in the substrate through the analysis of FTIR results. Additionally, it has proven that fungi growing with sodium silicate with the volume fraction of 1.5% is reflected as great thermal stability at the range from 250 ℃ to 400 ℃. Besides, the SEM analysis showed the fungal hyphae structure are gradually wizened with the increase of sodium silicate. The two-fold effect of sodium silicate on fungal growth was also discussed. Their satisfactory thermal stability makes them an attractive alternative to manufactured fibers used for construction applications.
Authors: Xijin Zhang, Chanjuan Han, Gary Wnek, and Xiong (Bill) Yu
Keywords: Fungal mycelium, sodium silicate, two-fold effect, thermal stability | 2021-04-17 04:46:31 | {"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": 0.8776073455810547, "perplexity": 13609.491854892982}, "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/1618038101485.44/warc/CC-MAIN-20210417041730-20210417071730-00624.warc.gz"} |
https://rpg.stackexchange.com/questions/109343/how-much-should-npcs-charge-for-spells-cast-as-services-to-pcs | How much should NPCs charge for spells cast as services to PCs?
As the title suggests, I need to know how to compute the appropriate cost for spells that NPCs provide as services (components included) to the party.
My search has come up with the following:
1. There is no standard method described in the PHB, DMG, or even the MM (the only books I have). Except in PHB p.159 Spellcasting Services where it is described that:
Hiring someone to cast a relatively common spell of 1st or 2nd level, such as Cure Wounds or Identify, is easy enough in a city or town, and might cost 10 to 50 gp (plus the cost of any expensive material components).
2. In the Tyranny of Dragons Playtest Player Guide p.11 there was a "Spellcasting Services" table stating:
• Cure wounds (1st level) 10 gp
• Identify 20 gp
• Lesser restoration 40 gp
• Prayer of healing (2nd level) 40 gp
• Remove curse 90 gp
• Speak with dead 90 gp
• Divination 210 gp
• Greater restoration 450 gp
• Raise dead 1,250 gp
3. The same table states that:
The number of spells available to be cast as a service is limited to a maximum of three per day total, unless otherwise noted.
4. Based on that (I think) people have come up with the following formula:
Spell Service Cost = Square of the spell level, then multiplied by 10, add double of the consumed material cost, add 10% of nonconsumed material cost.
5. Found a related conversation for 2-4 on reddit. I have no idea if it's RAW or RAI (Rules As Intended) or just House.
I'm interested on a RAW answer; failing that, I'll take a RAI (Rules As Intended) instead.
The answer to this is two-fold. So, to give the TL;DR first...
The Equation you cited is most likely accurate, but it only applies to Adventurer's League play. It is not part of the core rules.
Where the Equation Comes From
The pricing model that you cited in your answer, is derived from the Adventurer's League guides. As you mentioned, there was the Tyranny of Dragons Player Guide....but there are also the following...
These are published by Wizards of the Coast, but only apply to Adventurer's League play. The equation cited...
$$(\text{Level})^2\times10+(\text{Consumed Materials}\times2)+(\text{Non-consumed Materials}\times0.1)$$
was derived by the player-base because it accurately matches every cited "Cost to have this spell cast" entry in the Guides. To my knowledge, there are no exceptions. Every listed spell can have its cost computed with this equation. Which, given the sample size, means the equation is most likely right.
So, while this is not a "Rule" as laid down by WotC, it is an equation that can accurately predict the costs that they have cited for Adventurer's League play. And, in mathematics, if you have discovered an equation that can reliably predict all the results you know you have: we call that an accurate equation.
Again, just to be clear: The list of spellcasting service costs published by WotC is Adventurer's League content, not core rules. Additionally, the fact that players figured out this equation does not mean it can be applied to any spell in AL play. The only spells available for purchase in AL play are the ones explicitly listed in the guides. (So you can't just hire someone to cast Wish for you for 810gp.)
RAW
The only thing existing in the Core Rules is this:
People who are able to cast spells don't fall into the category of ordinary hirelings. It might be possible to find someone willing to cast a spell in exchange for coin or favors, but it is rarely easy and no established pay rates exist. As a rule, the higher the level of the desired spell, the harder it is to find someone who can cast it and the more it costs.
Hiring someone to cast a relatively common spell of 1st or 2nd level, such as cure wounds or identify, is easy enough in a city or town, and might cost 10 to 50 gold pieces (plus the cost of any expensive material components). Finding someone able and willing to cast a higher-level spell might involve traveling to a large city, perhaps one with a university or prominent temple. Once found, the spellcaster might ask for a service instead of payment--the kind of service that only adventurers can provide, such as retrieving a rare item from a dangerous locale, or traversing a monster-infested wilderness to deliver something important to a distant settlement.
Emphasis Mine, From PHB 159.
Many of the rules that exist in Adventurer's League books exist to provide a 'standard' where one does not normally exist in the core rules. This is to provide predictability, balance, and consistency between sessions where you may be playing under multiple different DMs throughout the 'campaign.' These are not Core Rules.
If you wish to adopt them and use them, you can. But this is not a rule. The official rule for how much it costs to have a spell cast for you is: "However much the DM says it costs, and 'services rendered' may be demanded in lieu of coin." | 2019-12-06 21:07:33 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25965067744255066, "perplexity": 2295.7057897138156}, "config": {"markdown_headings": false, "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-2019-51/segments/1575540490972.13/warc/CC-MAIN-20191206200121-20191206224121-00422.warc.gz"} |
https://hal.archives-ouvertes.fr/hal-01025302 | # Magnetic Fields and Massive Star Formation
2 FORMATION STELLAIRE 2014
L3AB - Laboratoire d'astrodynamique, d'astrophysique et d'aéronomie de bordeaux, OASU - Observatoire aquitain des sciences de l'univers, LAB - Laboratoire d'Astrophysique de Bordeaux [Pessac], Université Sciences et Technologies - Bordeaux 1
Abstract : Massive stars ($M > 8$ \msun) typically form in parsec-scale molecular clumps that collapse and fragment, leading to the birth of a cluster of stellar objects. We investigate the role of magnetic fields in this process through dust polarization at 870 $\mu$m obtained with the Submillimeter Array (SMA). The SMA observations reveal polarization at scales of $\lsim$ 0.1 pc. The polarization pattern in these objects ranges from ordered hour-glass configurations to more chaotic distributions. By comparing the SMA data with the single dish data at parsec scales, we found that magnetic fields at dense core scales are either aligned within $40^\circ$ of or perpendicular to the parsec-scale magnetic fields. This finding indicates that magnetic fields play an important role during the collapse and fragmentation of massive molecular clumps and the formation of dense cores. We further compare magnetic fields in dense cores with the major axis of molecular outflows. Despite a limited number of outflows, we found that the outflow axis appears to be randomly oriented with respect to the magnetic field in the core. This result suggests that at the scale of accretion disks ($\lsim 10^3$ AU), angular momentum and dynamic interactions possibly due to close binary or multiple systems dominate over magnetic fields. With this unprecedentedly large sample massive clumps, we argue on a statistical basis that magnetic fields play an important role during the formation of dense cores at spatial scale of 0.01 - 0.1 pc in the context of massive star and cluster star formation.
Type de document :
Article dans une revue
Astrophysical Journal, American Astronomical Society, 2014, 792 (2), pp.id. 116. 〈10.1088/0004-637X/792/2/116〉
Domaine :
https://hal.archives-ouvertes.fr/hal-01025302
Contributeur : Marie-Paule Pomies <>
Soumis le : jeudi 17 juillet 2014 - 15:00:07
Dernière modification le : mardi 29 mai 2018 - 12:51:06
### Citation
Qizhou Zhang, Keping Qiu, Josep M. Girart, Ya-Wen Tang, Patrick M. Koch, et al.. Magnetic Fields and Massive Star Formation. Astrophysical Journal, American Astronomical Society, 2014, 792 (2), pp.id. 116. 〈10.1088/0004-637X/792/2/116〉. 〈hal-01025302〉
### Métriques
Consultations de la notice | 2018-06-24 03:11:38 | {"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": 0.4511283338069916, "perplexity": 4542.333058904353}, "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/1529267866191.78/warc/CC-MAIN-20180624024705-20180624044705-00328.warc.gz"} |
https://nodus.ligo.caltech.edu:8081/40m/15322 | 40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop
40m Log Not logged in
Message ID: 15322 Entry time: Fri May 8 14:27:25 2020
Author: Hang Type: Update Category: BHD Subject: New SRC gouy phase
[Jon, Hang]
After updating the 40 m finesse file to incorporate the new SRC length (and the removal of SR2), we find that the current SRM radius curvature is fine. Thus a replacement of SRM is NOT required
Basically, the new one-way SRC gouy phase is 11.1 deg according to Finesse, which is very close to the previous value of 10.8 deg. Thus the transmode spacing should be essentially the same.
In the first attached plot is the mode content calculated with Finesse. Here we have first offset DARM by 1m deg and misaligned the SRM by 10 urad. From the top to bottom we show the amplitude of the carrier fields, f1, and f2 sidebands, respectively. The red vertical line is the nominal operating point (thanks Koji for pointing out that we do signal recycling instead of extraction now). No direct co-resonance for the low-order TEM modes. (Note that the HOMs appeared to also have peaks at \phi_srm = 0. This is just because the 00 mode is resonant and thus the seed for the HOMs is greater. )
We can also consider a clean case without mode interactions in the second plot. Indeed we don't see co-resonances of high order modes.
Attachment 1: mode_spec_finesse.pdf 161 kB Uploaded Fri May 8 15:35:59 2020
Attachment 2: mode_spec_ideal.pdf 169 kB Uploaded Fri May 8 15:36:14 2020
ELOG V3.1.3- | 2023-01-27 13:32:46 | {"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": 0.8660401105880737, "perplexity": 6923.1839865673355}, "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/1674764494986.94/warc/CC-MAIN-20230127132641-20230127162641-00093.warc.gz"} |
http://ncatlab.org/nlab/show/omega-category | # nLab omega-category
### Context
#### Higher category theory
higher category theory
# Contents
## Idea
The term $\omega$-category is used more or less synonymous with ∞-category , though the two terms appear with different frequency in different parts of the literature.
Originally, $\omega$-categories were introduced as
The general concept is that of
Revised on November 1, 2012 03:29:42 by Urs Schreiber (82.169.65.155) | 2015-08-28 22:18:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "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": 0.7935377359390259, "perplexity": 5820.069649887058}, "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-2015-35/segments/1440644064019.39/warc/CC-MAIN-20150827025424-00008-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.OdeSolver.step.html | # scipy.integrate.OdeSolver.step¶
OdeSolver.step()[source]
Perform one integration step.
Returns: message : string or None Report from the solver. Typically a reason for a failure if self.status is ‘failed’ after the step was taken or None otherwise.
#### Previous topic
scipy.integrate.OdeSolver.dense_output
#### Next topic
scipy.integrate.DenseOutput | 2018-03-22 13:54:36 | {"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": 0.695645809173584, "perplexity": 12743.036553147906}, "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-2018-13/segments/1521257647885.78/warc/CC-MAIN-20180322131741-20180322151741-00119.warc.gz"} |
http://unapologetic.wordpress.com/2008/02/29/ | # The Unapologetic Mathematician
## The Riemann-Stieltjes Integral II
What with cars not working right, I’m not along with the undergraduate math club at the MAA meeting in Lake Charles. That honor goes to Steve Sinnott alone. And so I’m still back in New Orleans, and I may as well write another post today.
Let’s follow on yesterday’s discussion of the Riemann-Stieltjes integral by looking at a restricted sort of integrator. We’ll assume here that $\alpha(t)$ is continuously differentiable — that is, that $\alpha'(t)$ exists and is continuous in the region we care about. We’ll also have a function $f$ on hand to integrate.
Now let’s take a tagged partition $x=((x_0,...,x_n),(t_1,...,t_n))$ and set up our Riemann-Stieltjes sum for $f$
$\displaystyle f_{\alpha,x}=\sum\limits_{i=1}^nf(t_i)(\alpha(x_i)-\alpha(x_{i-1}))$
We can also use the partition to set up a Riemann sum for $f\alpha'$
$\displaystyle(f\alpha')_x=\sum\limits_{i=1}^nf(t_i)\alpha'(t_i)(x_i-x_{i-1})$
Now the Differential Mean Value Theorem tells us that in each subinterval $\left[x_{i-1},x_i\right]$ there is some $s_i$ so that $(\alpha(x_i)-\alpha(x_{i-1})=\alpha'(s_i)(x_i-x_{i-1})$. We stick this into the Riemann-Stieltjes sum and subtract from the Riemann sum
$\displaystyle(f\alpha')_x-f_{\alpha,x}=\sum\limits_{i=1}^nf(t_i)(\alpha'(t_i)-\alpha'(s_i))(x_i-x_{i-1})$
And continuity of $\alpha'$ tells us that as we pick the partitions finer and finer, we’ll squeeze $s_i$ and $t_i$ together, and so $\alpha'(t_i)-\alpha'(s_i)$ goes to zero. So, when the integrator $\alpha$ is continuously differentiable, the Riemann-Stieltjes integral reduces to a Riemann integral
$\displaystyle\int\limits_{[a,b]}f(x)d\alpha(x)=\int\limits_{[a,b]}f(x)\alpha'(x)dx$
Now let’s go back to the fence story from yesterday. There’s really some function $g(p)$ that tells us the height of the fence at position $p$. We got our integrator from the function $p=\alpha(t)$, which takes a time $t$ and gives the position $p$ at that time. Thus our function giving height at time $t$ is the composition $f(t)=g(\alpha(t))$. We also note that as time goes from $a$ to $b$, we walk from $\alpha(a)$ to $\alpha(b)$. If we put all this into the above equation we find
$\displaystyle\int\limits_{[\alpha(a),\alpha(b)]}g(p)dp=\int\limits_{[a,b]}g(\alpha(t))\alpha'(t)dt$
which is just our change of variables formula.
So here’s what this formula means: we have a real-valued function on some domain — here it’s the height of the fence as a function of position along the ground. We take a region in the space of these variables — the section of the fence we’re walking past — and we “parametrize” it by describing it with a (sufficiently smooth) function taking a single real variable — the position function $p=\alpha(t)$. We “pull back” our function by composing it with this parametrization, giving us a real-valued function of a real variable, which we can integrate.
The deep thing here is that we have two different parametrizations. We can describe position as $p$ meters from a fixed starting point, or we can describe it as the point we’re passing at $t$ seconds from a certain starting time. In fact, as we change our function $\alpha$ we get all sorts of different parametrizations of the same stretch of ground. And we should measure the same area for the same fence no matter which parametrization we choose.
Given a parametrization $\alpha$, the integral in our recipe will be the Riemann-Stieltjes integral
$\displaystyle\int\limits_{[a,b]}g(\alpha(t))d\alpha(t)$
which we can reduce to the Riemann integral
$\displaystyle\int\limits_{[a,b]}g(\alpha(t))\alpha'(t)dt$
and it will be the same answer as if we used the “natural” parametrization
$\displaystyle\int\limits_{[\alpha(a),\alpha(b)]}g(p)dp$
And that’s why the change of variables works.
February 29, 2008 Posted by | Analysis, Calculus | 15 Comments | 2014-03-09 08:47:06 | {"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": 38, "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": 0.835489809513092, "perplexity": 226.3148220105272}, "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-2014-10/segments/1393999675992/warc/CC-MAIN-20140305060755-00074-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://accessmedicine.mhmedical.com/content.aspx?bookid=331§ionid=40727068 | Chapter 279
Acute kidney injury (AKI), previously known as acute renal failure, is characterized by the sudden impairment of kidney function resulting in the retention of nitrogenous and other waste products normally cleared by the kidneys. AKI is not a single disease but, rather, a designation for a heterogeneous group of conditions that share common diagnostic features: specifically, an increase in the blood urea nitrogen (BUN) concentration and/or an increase in the plasma or serum creatinine (SCr) concentration, often associated with a reduction in urine volume. AKI can range in severity from asymptomatic and transient changes in laboratory parameters of glomerular filtration rate (GFR), to overwhelming and rapidly fatal derangements in effective circulating volume regulation and electrolyte and acid-base composition of the plasma.
Changing the name of a syndrome as well known as "acute renal failure" does not occur frequently. We will summarize some of the reasons why the name was changed to "acute kidney injury." The term failure reflects only part of the spectrum of damage to the kidney that occurs clinically. In most cases of damage, the reduction in kidney function is modest. Nevertheless, this modest change has been documented to be associated with negative effects on outcome, albeit not nearly as ominous as seen with large decreases in kidney function associated with frank kidney failure that often requires acute dialysis therapies. Furthermore, the term renal is not well understood in the general population and this makes communication with patients and family more challenging; hence "kidney" has replaced "renal."
AKI complicates 5–7% of acute care hospital admissions and up to 30% of admissions to the intensive care unit. AKI is also a major medical complication in the developing world, particularly in the setting of diarrheal illnesses, infectious diseases like malaria and leptospirosis, and natural disasters such as earthquakes. The incidence of AKI has grown by more than fourfold in the United States since 1988 and is estimated to have a yearly incidence of 500 per 100,000 population, higher than the yearly incidence of stroke. AKI is associated with a markedly increased risk of death in hospitalized individuals, particularly in those admitted to the ICU where in-hospital mortality rates may exceed 50%.
### AKI in the Developing World
The epidemiology of AKI differs tremendously between developed and developing countries, owing to differences in demographics, economics, geography, and comorbid disease burden. While certain features of AKI are common to both—particularly since urban centers of some developing countries increasingly resemble those in the developed world—many etiologies for AKI are region-specific such as envenomations from snakes, spiders, caterpillars, and bees; infectious causes such as malaria and leptospirosis; and crush injuries and resultant rhabdomyolysis from earthquakes.
The causes of AKI have traditionally been divided into three broad categories: prerenal azotemia, intrinsic renal parenchymal disease, and postrenal obstruction (Fig. 279-1).
###### Figure 279-1
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Connect to the full suite of AccessMedicine content and resources including more than 250 examination and procedural videos, patient safety modules, an extensive drug database, Q&A, Case Files, and more. | 2015-03-04 18:46:21 | {"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": 0.23528063297271729, "perplexity": 5140.791811552975}, "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-2015-11/segments/1424936463637.80/warc/CC-MAIN-20150226074103-00300-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://www.techylib.com/en/view/laborermaize/using_actionscript_3.0_components | # Using ActionScript 3.0 Components
Software and s/w Development
Jul 4, 2012 (2 years and 3 months ago)
630 views
Using
ACTIONSCRIPT
®
3.0
Components
Last updated 5/27/2011
Legal notices
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For legal notices, see
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Last updated 5/27/2011
Contents
Chapter 1: Introduction
Intended audience
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1
System requirements
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Typographical conventions
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2
Terms used in this manual
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Chapter 2: About ActionScript 3.0 Components
Benefits of using components
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Component types
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Add to and delete from a document
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Find the version of the component
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ActionScript 3.0 event handling model
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A simple application
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Chapter 3: Working with Components
Component architecture
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16
Working with component files
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18
Debug component applications
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20
Set parameters and properties
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The library
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Sizing components
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Live Preview
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Handling events
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Work with the display list
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Work with FocusManager
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Work with List-based components
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Work with a DataProvider
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Work with a CellRenderer
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Make components accessible
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Chapter 4: Using the UI Components
Use the Button component
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Use the CheckBox component
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Use the ColorPicker component
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Use the ComboBox component
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Use the DataGrid component
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Use the Label component
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Use the List component
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Use the NumericStepper component
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Use the ProgressBar component
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USING ACTIONSCRIPT 3.0 COMPONENTS
Contents
Last updated 5/27/2011
Use the ScrollPane component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Use the Slider component
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Use the TextArea component
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82
Use the TextInput component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
Use the TileList component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
Use the UIScrollBar component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
Chapter 5: Customizing the UI Components
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
Setting styles
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
Customize the Button component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
Customize the CheckBox component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
Customize the ColorPicker component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
Customize the ComboBox component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
106
Customize the DataGrid component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
108
Customize the Label component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
Customize the List component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
Customize the NumericStepper component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
Customize the ProgressBar component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
Customize the ScrollPane component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121
Customize the Slider component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
Customize the TextArea component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
Customize the TextInput component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
125
Customize the TileList component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
126
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
Customize the UIScrollBar component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
Chapter 6: Using the FLVPlayback Component
Use the FLVPlayback component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131
Customize the FLVPlayback component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
149
Use a SMIL file
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
159
Chapter 7: Using the FLVPlayback Captioning Component
Use the FLVPlaybackCaptioning component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
167
Use Timed Text captions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
169
Use cue points with captioning
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
174
Play multiple FLV files with captioning
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
176
Customize the FLVPlaybackCaptioning component
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
176
1
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Chapter 1: Introduction
Adobe® Flash® CS5 Professional is the standard authoring tool for producing high-impact web experiences.
Components are the building blocks for the rich Internet applications that provide these experiences. A component is
a movie clip with parameters that allow you to customize the component either during authoring in Flash or at run
time with Adobe® ActionScript® methods, properties, and events. Components are designed to allow developers to
reuse and share code, and to encapsulate complex functionality that designers can use and customize without using
ActionScript.
Components allow you to easily and quickly build robust applications with a consistent appearance and behavior. This
manual describes how to build applications with Adobe ActionScript 3.0 components. The Adobe® ActionScript® 3.0
Language and Components Reference describes each component’s application programming interface (API).
components.
Intended audience
This manual is for developers who are building Flash applications and want to use components to speed development.
You should already be familiar with developing applications in Flash and writing ActionScript.
If you are less experienced with writing ActionScript, you can add components to a document, set their parameters in
the Property inspector or Component inspector, and use the Behaviors panel to handle their events. For example, you
could attach a Go To Web Page behavior to a Button component that opens a URL in a web browser when the button
is clicked without writing any ActionScript code.
If you are a programmer who wants to create more robust applications, you can create components dynamically, use
ActionScript to set properties and call methods at run time, and use the event listener model to handle events.
“Working with Components” on page
16
.
System requirements
Flash components do not have any system requirements beyond the system requirements for Flash.
Any SWF file that uses Flash CS3 or later components must be viewed with Adobe® Flash® Player 9.0.28.0 or later, and
must be published for ActionScript 3.0 (you can set this through File
> Publish Settings, in the Flash tab).
This document explains the details of using components to develop Flash applications. It assumes that you have
general knowledge of Flash and ActionScript 3.0. Specific documentation about Flash and related products is available
separately.
> Flash
Help
> Using Adobe ActionScript 3.0 Components.
2
USING ACTIONSCRIPT 3.0 COMPONENTS
Introduction
Last updated 5/27/2011
For information about Flash, see the following documents:
Using Flash
ActionScript 3.0 Developer’s Guide
Typographical conventions
The following typographical conventions are used in this manual:
Italic font indicates a value that should be replaced (for example, in a folder path).
Code font
indicates ActionScript code, including method and property names.
Code font italic indicates a code item that should be replaced (for example, an ActionScript parameter).
Bold font indicates a value that you enter.
Terms used in this manual
The following terms are used in this manual:
at run time
When the code is running in Flash Player.
while authoring
While you are working in the Flash authoring environment.
In addition to the content in these manuals, Adobe provides regularly updated articles, design ideas, and examples at
You can find additional component samples at
.
world application development, and information about important emerging issues. View the Developer Center at
.
Learn the latest in digital design and motion graphics. Browse work by leading artists, discover new design trends, and
hone your skills with tutorials, key workflows, and advanced techniques. Check back twice a month for fresh tutorials
and articles, and inspirational gallery pieces. View the Design Center at
.
3
Last updated 5/27/2011
Components
Flash®
Professional
CS5 components are movie clips with parameters that allow you to modify their
appearance and behavior. A component can be a simple user interface control, such as a RadioButton or a CheckBox,
or it can contain content, such as a List or DataGrid.
Components allow you to easily and quickly build robust Flash applications with consistent behavior and appearance.
Rather than creating custom buttons, combo boxes, and lists, you can use the Flash components that implement these
controls. Simply drag them from the Components panel into your application document. You can also easily
customize the look and feel of these components to suit your application design.
While you can do all of this without an advanced understanding of ActionScript, you can also use ActionScript 3.0 to
modify a component’s behavior or implement new behavior. Each component has a unique set of ActionScript
methods, properties, and events that make up its application programming interface (API). The API allows you to
create and manipulate components while the application is running.
The API also allows you to create new, custom components of your own. You can download components built by
members of the Flash community on the Adobe Exchange at
.
The ActionScript 3.0 component architecture includes classes on which all components are based, skins and styles that
allow you to customize appearance, an event-handling model, focus management, an accessibility interface, and more.
Note: Adobe Flash CS5 includes ActionScript 2.0 components as well as ActionScript 3.0 components. You cannot mix
these two sets of components. You must use one set or the other for a given application. Flash CS5 presents either
ActionScript 2.0 components or ActionScript 3.0 components based on whether you open an ActionScript 2.0 or an
ActionScript 3.0 file. When you create a new Flash document, you must specify either Flash File (ActionScript 3.0) or
Flash File (ActionScript 2.0). When you open an existing document, Flash examines the Publish Settings to determine
which set of components to use. For information about ActionScript 2.0 components, see Using Adobe® ActionScript® 2.0
Components.
For a complete list of the Flash ActionScript 3.0 components, see
“Component types” on page
4
.
Benefits of using components
Components enable you to separate the process of designing your application from the process of coding. They allow
developers to create functionality that designers can use in applications. Developers can encapsulate frequently used
functionality into components and designers can customize the size, location, and behavior of components by
changing their parameters. They can also change the appearance of a component by editing its graphical elements, or
skins.
Components share core functionality such as styles, skins, and focus management. When you add the first component
to an application, this core functionality accounts for approximately 20 kilobytes of the size. When you add other
components, that initial memory allocation is shared by the added components, reducing the growth in the size of your
application.
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USING ACTIONSCRIPT 3.0 COMPONENTS
Last updated 5/27/2011
This section outlines some of the benefits of the ActionScript 3.0 components.
The power of ActionScript 3.0
provides a powerful, object-oriented programming language that is an important step
in the evolution of Flash Player capabilities. The language is designed for building rich Internet applications on a
reusable code base. ActionScript 3.0 is based on ECMAScript, the international standardized language for scripting,
and is compliant with the ECMAScript (ECMA-262) edition 3 language specification. For a thorough introduction to
ActionScript 3.0, see ActionScript 3.0 Developer’s Guide. For reference information on the language, see the
ActionScript 3.0 Reference for the Adobe Flash Platform
.
FLA-based User Interface components
components also provide styles, including skin styles, that allow you to customize aspects of the components
“Customizing the UI Components” on page
95
and
the
ActionScript 3.0 Reference for the Adobe Flash Platform
.
component along with full screen support, improved live
preview, skins that allow you to add color and alpha settings, and improved FLV download and layout features.
The Property inspector and Component inspector
allow you to change component parameters while authoring in
“Working with component files” on page
18
and
“Set parameters and properties” on
page
20
.
New collection dialog box
for the ComboBox, List, and TileList components allows you to populate their
dataProvider
“Create a DataProvider” on page
28
.
The ActionScript 3.0 event model
allows your application to listen for events and invoke event handlers to respond.
“ActionScript 3.0 event handling model” on page
8
and
“Handling events” on page
23
.
Manager classes
provide an easy way to handle focus and manage styles in an application. For more information, see
the
ActionScript 3.0 Reference for the Adobe Flash Platform
.
The UIComponent base class
provides core methods, properties, and events to components that extend it. All of the
ActionScript 3.0 user interface components inherit from the UIComponent class. For more information see the
UIComponent class in the
ActionScript 3.0 Reference for the Adobe Flash Platform
.
Use of a SWC
in the UI FLA-based components provide ActionScript definitions as an asset inside the component’s
Timeline to speed compilation.
An easily extendable class hierarchy
using ActionScript 3.0 allows you to create unique namespaces, import classes as
needed, and subclass easily to extend components.
ActionScript 3.0 Reference for the Adobe Flash Platform
.
Note: Flash CS5 supports both FLA-based and SWC-based components. For more information, see
“Component
architecture” on page
16
.
Component types
You install the Flash components when you install Flash CS5.
ActionScript 3.0 components include the following user interface (UI) components:
Button
List
TextArea
CheckBox
NumericStepper
TextInput
ColorPicker
TileList
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USING ACTIONSCRIPT 3.0 COMPONENTS
Last updated 5/27/2011
In addition to the user interface components, the Flash ActionScript 3.0 components include the following
components and supporting classes:
FLVPlayback component (fl.video.FLVPlayback), which is a SWC-based component.
The FLVPlayback component lets you readily include a video player in your Flash application to play progressive
streaming video over HTTP, from an Adobe® Flash® Video Streaming Service (FVSS), or from Adobe’s
“Using the FLVPlayback Component” on
page
131
.
The FLVPlayback Custom UI components, which are FLA-based and work with both the ActionScript 2.0 and
“Using the FLVPlayback
Component” on page
131
.
The FLVPlayback Captioning component, which provides closed captioning for FLVPlayback. See
“Using the
FLVPlayback Captioning Component” on page
167
.
For a complete list of the ActionScript 3.0 components and their supporting classes, see the
ActionScript 3.0
Reference for the Adobe Flash Platform
.
View Flash components:
You can view the Flash ActionScript 3.0 components in the Components panel by following these steps.
1
Start Flash.
2
Create a new Flash file (ActionScript 3.0) or open an existing Flash document in which the Publish Settings specify
ActionScript 3.0.
3
Select Window
> Components to open the Components panel, if it isn’t already open.
Components panel with User Interface components
ComboBox
ProgressBar
DataGrid
ScrollPane
UIScrollBar
Label
Slider
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USING ACTIONSCRIPT 3.0 COMPONENTS
Last updated 5/27/2011
.
Any component can appear in the Components panel in Flash. Follow these steps to install components on either
a Windows® or Macintosh® computer.
Install components on a Windows-based or a Macintosh computer:
1
Quit Flash.
2
Place the SWC or FLA file containing the component in the following folder on your hard disk:
In Windows:
On the Macintosh:
3
Start Flash.
4
Select Window
> Components to view the component in the Components panel if it isn’t already open.
“Working with component files” on page
18
Add to and delete from a document
When you drag a FLA-based component from the Components panel to the Stage, Flash imports an editable movie
clip to the library. When you drag a SWC-based component to the Stage, Flash imports a compiled clip to the library.
After a component has been imported to the library, you can drag instances of it to the Stage from either the Library
panel or the Components panel.
You can add a component to a document by dragging it from the Components panel. You can set properties for each
instance of a component in the Property inspector or in the Parameters tab in the Component inspector.
1
Select Window
> Components.
2
Either double-click the component in the Components panel or drag the component to the Stage.
3
Select the component on the Stage.
4
If the Property inspector is not visible, select Window
> Properties
> Properties.
5
In the Property inspector, enter an instance name for the component instance.
6
Select Window
> Component inspector and select the Parameters tab to specify parameters for the instance.
“Set parameters and properties” on page
20
.
7
Change the size of the component as desired by editing the values for the width (W:) and height (H:).
“Customizing the UI Components” on page
95
.
8
Select Control
> Test Movie or press Control+Enter to compile the document and see the results of your settings.
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USING ACTIONSCRIPT 3.0 COMPONENTS
Last updated 5/27/2011
You can also change the color and text formatting of a component by setting style properties for it or customize its
appearance by editing the component’s skins. For more information on these topics, see
“Customizing the UI
Components” on page
95
.
If you drag a component to the Stage during authoring, you can refer to the component by using its instance name
(for example,
myButton
).
Add components at run time with ActionScript
To add a component to a document at run time with ActionScript, the component must first be in the application’s
library (Window
> Library) when the SWF file is compiled. To add a component to the library, drag the component
from the Components panel into the Library panel. For more information on the library, see
“The library” on page
21
.
You must also import the component’s class file to make its API available to your application. Component class files
are installed in packages that contain one or more classes. To import a component class, use the
import
statement and
specify the package name and class name. You would import the Button class, for example, with the following
import
statement:
import fl.controls.Button;
For information on what package a component is in, see the
ActionScript 3.0 Reference for the Adobe Flash Platform
.
For information about the location of component source files, see
“Working with component files” on page
18
.
To create an instance of the component, you must invoke the component’s ActionScript constructor method. For
example, the following statement creates an instance of a Button called
aButton
:
var aButton:Button = new Button();
The final step is to call the static
method to add the component instance to the Stage or application
container. For example, the following statement adds the
aButton
instance:
At this point, you can use the component’s API to dynamically specify the component’s size and position on the Stage,
listen for events, and set properties to modify its behavior. For more information on the API for a particular
component, see the
ActionScript 3.0 Reference for the Adobe Flash Platform
.
method, see
“Work with the display list” on page
24
.
Delete a component
To delete a component instance from the Stage while authoring, simply select it and press the Delete key. This will
remove the instance from the Stage but does not remove the component from your application.
To delete a component from your Flash document after you’ve placed it on the Stage or in the library, you must delete
it and its associated assets from the library. It isn’t enough to delete the component from the Stage. If you don’t remove
it from the library, it will be included in your application when you compile it.
1
In the Library panel, select the symbol for the component.
2
Click the Delete button at the bottom of the Library panel, or select Delete from the Library panel menu.
Repeat these steps to delete any assets associated with the component.
For information on how to remove a component from its container while your application is running, see
“Remove
a component from the display list” on page
25
.
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USING ACTIONSCRIPT 3.0 COMPONENTS
Last updated 5/27/2011
Find the version of the component
Flash ActionScript 3.0 components have a version property that you can display if you need to provide it to Adobe
Technical Support or you need to know what version of the component you are using.
Display the version number for a user interface component:
1
Create a new Flash file (ActionScript 3.0) document.
2
Drag the component to the Stage and give it an instance name. For example, drag a ComboBox to the Stage and call
it aCb.
3
Press the F9 key or select Window
> Actions to open the Actions panel.
4
Click Frame 1 on the main Timeline and add the following code to the Actions panel:
trace(aCb.version);
The version number, similar to the one in the following illustration, should appear in the Output panel.
For the FLVPlayback and FLVPlaybackCaptioning components, you must refer to the class name rather than the
instance name because the version number is stored in a class constant.
Display the version number for the FLVPlayback and FLVPlaybackCaptioning components:
1
Create a new Flash file (ActionScript 3.0) document.
2
Drag the FLVPlayback and FLVPlaybackCaptioning components into the Library panel.
3
Press the F9 key or select Window
> Actions to open the Actions panel.
4
Click Frame 1 on the main Timeline and add the following code to the Actions panel.
import fl.video.*;
trace("FLVPlayback.VERSION: " + FLVPlayback.VERSION);
trace("FLVPLaybackCaptioning.VERSION: " + FLVPlaybackCaptioning.VERSION);
The version numbers will appear in the Output panel.
ActionScript 3.0 event handling model
ActionScript 3.0 introduces a single event handling model that replaces the different event handling mechanisms that
existed in previous versions of ActionScript. The new event model is based on the Document Object Model (DOM)
Level 3 Events Specification.
For developers with experience using the ActionScript 2.0
method, it may be helpful to point out the
differences between the ActionScript 2.0 event listener model and the ActionScript 3.0 event model. The following list
describes a few of the major differences between the two event models:
To add event listeners in ActionScript 2.0, you use
in some cases and
in
others, whereas in ActionScript 3.0 you use
in all cases.
There is no event flow in ActionScript 2.0, which means that the
method can be called only on the
object that broadcasts the event, whereas in ActionScript 3.0 the
method can be called on
any object that is part of the event flow.
In ActionScript 2.0, event listeners can be either functions, methods, or objects, whereas in ActionScript 3.0, only
functions or methods can be event listeners.
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USING ACTIONSCRIPT 3.0 COMPONENTS
Last updated 5/27/2011
The
on(
event
)
syntax is no longer supported in ActionScript 3.0, so you cannot attach ActionScript event code to
a movie clip. You can only use
The following example, which listens for a
MouseEvent.CLICK
event on a Button component called
aButton
,
illustrates the basic ActionScript 3.0 event handling model:
function clickHandler(event:MouseEvent):void {
trace("clickHandler detected an event of type: " + event.type);
trace("the event occurred on: " + event.target.name);
}
For more information on ActionScript 3.0 event handling, see Programming ActionScript 3.0. For more
information on ActionScript 3.0 event handling for components, see
“Handling events” on page
23
.
A simple application
This section takes you through the steps to create a simple ActionScript 3.0 application using Flash components and
the Flash authoring tool. The example is provided both as a FLA file with the ActionScript code included on the
Timeline and also as an external ActionScript class file with a FLA file that contains only the components in the library.
In general, you will want to develop larger application using external class files so that you can share code between
with ActionScript 3.0, see Programming ActionScript 3.0.
Design of the application
Our first example of an ActionScript component application is a variation of the standard “Hello World” application,
so its design is fairly simple:
The application will be called Greetings.
It uses a TextArea to display a greeting that is initially Hello World.
It uses a ColorPicker that allows you to change the color of the text.
It uses three RadioButtons that allow you to set the size of the text to small, larger or largest.
It uses a ComboBox that allows you to select a different greeting from a drop-down list.
The application uses components from the Components panel and also creates application elements through
ActionScript code.
With that definition in place, you can start building the application.
Create the Greetings application
The following steps create the Greetings application using the Flash authoring tool to create a FLA file, place
components on the Stage, and add ActionScript code to the Timeline.
Create the Greetings application in a FLA file:
1
Select File
> New.
2
In the New Document dialog box, select Flash File (ActionScript 3.0), and click OK.
A new Flash window opens.
3
Select File
> Save, name the Flash file Greetings.fla, and click the Save button.
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4
In the Flash Components panel, select a TextArea component and drag it to the Stage.
5
In the Properties window, with the TextArea selected on the Stage, type aTa for the instance name, and enter the
following information:
Enter 230 for the W value (width).
Enter 44 for the H value (height).
Enter 165 for the X value (horizontal position).
Enter 57 for the Y value (vertical position).
Enter Hello World! for the text parameter, on the Parameters tab.
6
Drag a ColorPicker component to the Stage, place it to the left of the TextArea and give it an instance name of
txtCp. Enter the following information in the Property inspector:
Enter 96 for the X value.
Enter 72 for the Y value.
7
Drag three RadioButton components to the Stage, one at a time and give them instance names of smallRb,
largerRb, and largestRb. Enter the following information for them in the Property inspector:
Enter 100 for the W value and 22 for the H value for each of them.
Enter 155 for the X value.
Enter 120 for the Y value for smallRb, 148 for largerRb, and 175 for largestRb.
Enter fontRbGrp for the groupName parameter for each of them.
Enter labels for them on the Parameters tab of Small, Larger, Largest.
8
Drag a ComboBox to the Stage and give it an instance name of msgCb. Enter the following information for it in the
Property inspector:
Enter 130 for the W value.
Enter 265 for the X value.
Enter 120 for the Y value.
On the Parameters tab, enter Greetings for the prompt parameter.
Double-click the text field for the dataProvider parameter to open the Values dialog box.
Click the plus sign and replace the label value with Hello World!
Repeat the preceding step to add the label values Have a nice day! and Top of the Morning!
Click OK to close the Values dialog box.
9
Save the file.
10
If it is not already open, open the Actions panel by hitting F9 or selecting Actions from the Window menu. Click
Frame 1 on the main Timeline and enter the following code in the Actions panel:
import flash.events.Event;
import fl.events.ComponentEvent;
import fl.events.ColorPickerEvent;
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The first three lines import the event classes that the application uses. An event occurs when a user interacts with
one of the components. The next five lines register event handlers for the events that the application wants to listen
for. A
click
event occurs for a RadioButton when a user clicks on it. A
change
event occurs when a user selects a
different color in the ColorPicker. A
change
event occurs on the ComboBox when a user chooses a different
greeting from the drop-down list.
The fourth line imports the RadioButtonGroup class so that the application can assign an event listener to the group
of RadioButtons, rather than assigning the listener to each button individually.
11
Add the following line of code to the Actions panel to create the
tf
TextFormat object, which the application uses
to change the
size
and
color
style properties of the text in the TextArea.
var tf:TextFormat = new TextFormat();
12
Add the following code to create the
rbHandler
event handling function. This function handles a
click
event
when a user clicks on one of the RadioButton components.
function rbHandler(event:MouseEvent):void {
switch(event.target.selection.name) {
case "smallRb":
tf.size = 14;
break;
case "largerRb":
tf.size = 18;
break;
case "largestRb":
tf.size = 24;
break;
}
aTa.setStyle("textFormat", tf);
}
This function uses a
switch
statement to examine the
target
property of the
event
object to determine which
currentTarget
property contains the name of the object that triggered the
event. Depending on which RadioButton the user clicked, the application changes the size of the text in the
TextArea to 14, 18, or 24 points.
13
Add the following code to implement the
cpHandler()
function, which handles a change to the value in the
ColorPicker:
function cpHandler(event:ColorPickerEvent):void {
tf.color = event.target.selectedColor;
aTa.setStyle("textFormat", tf);
}
This function sets the
color
property of the
tf
TextFormat object to the color selected in the ColorPicker and then
calls
setStyle()
to apply it to the text in the
aTa
TextArea instance.
14
Add the following code to implement the
cbHandler()
function, which handles a change to the selection in the
ComboBox:
function cbHandler(event:Event):void {
aTa.text = event.target.selectedItem.label;
}
This function simply replaces the text in the TextArea with the selected text in the ComboBox,
event.target.selectedItem.label
.
15
Select Control
> Test Movie or press Control+Enter to compile the code and test the Greetings application.
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The following section shows you how to build the same application with an external ActionScript class and a FLA
file that has only the required components in the library.
Create the Greetings2 application with an external class file:
1
Select File
> New.
2
In the New Document dialog box, select Flash File (ActionScript 3.0), and click OK.
A new Flash window opens.
3
Select File
> Save, name the Flash file Greetings2.fla, and click the Save button.
4
Drag each of the following components from the Components panel to the library:
ColorPicker
ComboBox
TextArea
The compiled SWF file will use each of these assets, so you need to add them to the library. Drag the components
to the bottom of the Library panel. As you add these components to the library, other assets (such as List,
TextInput, and UIScrollBox) are added automatically.
5
In the Properties window, for the Document Class, type Greetings2.
If Flash displays a warning, saying that “a definition for the document class could not be found,” ignore it. You will
define the Greetings2 class in the following steps. This class defines the main functionality for the application.
6
Save the Greetings2.fla file.
7
Select File
> New.
8
In the New Document dialog box, select ActionScript File, and click OK.
A new script window opens.
9
Add the following code into the script window:
package {
import flash.display.Sprite;
import flash.events.Event;
import flash.events.MouseEvent;
import flash.text.TextFormat;
import fl.events.ComponentEvent;
import fl.events.ColorPickerEvent;
import fl.controls.ColorPicker;
import fl.controls.ComboBox;
import fl.controls.TextArea;
public class Greetings2 extends Sprite {
private var aTa:TextArea;
private var msgCb:ComboBox;
private var txtCp:ColorPicker;
private var tf:TextFormat = new TextFormat();
public function Greetings2() {
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The script defines an ActionScript 3.0 class, named Greetings2. The script does the following:
It imports classes that we will use in the file. Normally you would add these
import
statements as you reference
different classes in the code, but for the sake of brevity, this example imports them all in this one step.
It declares variables that represent the different types of component objects that we will add to the code. Another
variable creates the
tf
TextFormat object.
It defines a constructor function,
Greetings2()
, for the class. We will add lines to this function, and add other
methods to the class in the following steps.
10
Select File
> Save, name the file Greetings2.as, and click the Save button.
11
Add the following lines of code to the
Greeting2()
function:
createUI();
setUpHandlers();
}
The function should now look like the following:
public function Greetings2() {
createUI();
setUpHandlers();
}
12
Add the following lines of code after the closing brace of the
Greeting2()
method:
private function createUI() {
bldTxtArea();
bldColorPicker();
bldComboBox();
}
private function bldTxtArea() {
aTa = new TextArea();
aTa.setSize(230, 44);
aTa.text = "Hello World!";
aTa.move(165, 57);
}
private function bldColorPicker() {
txtCp = new ColorPicker();
txtCp.move(96, 72);
}
private function bldComboBox() {
msgCb = new ComboBox();
msgCb.width = 130;
msgCb.move(265, 120);
msgCb.prompt = "Greetings";
}
smallRb.setSize(100, 22);
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smallRb.move(155, 120);
smallRb.group = rbGrp; //"fontRbGrp";
smallRb.label = "Small";
smallRb.name = "smallRb";
largerRb.setSize(100, 22);
largerRb.move(155, 148);
largerRb.group = rbGrp;
largerRb.label = "Larger";
largerRb.name = "largerRb";
largestRb.setSize(100, 22);
largestRb.move(155, 175);
largestRb.group = rbGrp;
largestRb.label = "Largest";
largestRb.name = "largestRb";
}
These lines do the following:
Instantiate the components used in the application.
Set each component’s size, position, and properties.
Add each component to the Stage, using the
method.
13
After the closing brace of the
method, add the following code for the
setUpHandlers()
method:
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private function setUpHandlers():void {
}
private function rbHandler(event:MouseEvent):void {
switch(event.target.selection.name) {
case "smallRb":
tf.size = 14;
break;
case "largerRb":
tf.size = 18;
break;
case "largestRb":
tf.size = 24;
break;
}
aTa.setStyle("textFormat", tf);
}
private function cpHandler(event:ColorPickerEvent):void {
tf.color = event.target.selectedColor;
aTa.setStyle("textFormat", tf);
}
private function cbHandler(event:Event):void {
aTa.text = event.target.selectedItem.data;
}
}
}
These functions define event listeners for the components.
14
Select File
> Save to save the file.
15
Select Control
> Test Movie or press Control+Enter to compile the code and test the Greetings2 application.
Develop and run subsequent examples
Having developed and run the Greetings application, you should have the basic knowledge you need to run the other
code examples presented in this book. The relevant ActionScript 3.0 code in each example will be highlighted and
discussed and you should be able to cut and paste each of the examples in this book into a FLA file, compile and run it.
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Chapter 3: Working with Components
Component architecture
Adobe® ActionScript® 3.0 components are supported by Adobe® Flash Player version 9.0.28.0 and later. These
components are not compatible with components built prior to Flash CS4. For information on using Adobe®
ActionScript® 2.0 components, see Using Adobe® ActionScript® 2.0 Components and the Adobe® ActionScript® 2.0
Components Language Reference.
The Adobe ActionScript 3.0 User Interface (UI) components are implemented as FLA-based components but Flash
CS5 supports both SWC and FLA-based components. The FLVPlayback and FLVPlaybackCaptioning components
are SWC-based components, for example. You can place either type of component in the Components folder to have
it appear in the Components panel. These two types of components are built differently so they are described
separately here.
ActionScript 3.0 FLA-based components
The ActionScript 3.0 User Interface components are FLA-based (.fla) files with built-in skins that you can access for
editing by double-clicking the component on the Stage. The component’s skins and other assets are placed on Frame
2 of the Timeline. When you double-click the component, Flash automatically jumps to Frame 2 and opens a palette
of the component’s skins. The following illustration shows the palette with skins that display for the Button
component.
Skins for the Button component
“Customizing the UI Components” on
page
95
and
“Customize the FLVPlayback component” on page
149
.
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To speed up compilation for your applications and to avoid conflicts with your ActionScript 3.0 settings, the Flash CS5
FLA-based UI components also contain a SWC that contains the component’s already compiled ActionScript code.
The ComponentShim SWC is placed on Stage on Frame 2 in every User Interface component to make available the
precompiled definitions. To be available for ActionScript, a component must either be on Stage or be in the library
with the Export In First Frame option selected in its Linkage properties. To create a component using ActionScript,
you also must import the class with an
import
statement to access it. For information on the
import
statement, see
the
ActionScript 3.0 Reference for the Adobe Flash Platform
.
SWC-based Components
SWC-based components have a FLA file and an ActionScript class file, too, but they have been compiled and exported
as a SWC. A SWC file is a package of precompiled Flash symbols and ActionScript code that allows you to avoid
recompiling symbols and code that will not change.
The FLVPlayback and FLVPlaybackCaptioning components are SWC-based components. They have external rather
than built-in skins. The FLVPlayback component has a default skin that you can change by selecting one from a
collection of predesigned skins, by customizing controls from the UI controls in the Components panel (BackButton,
BufferingBar, and so on), or by creating a custom skin. For more information, see
“Customize the FLVPlayback
component” on page
149
.
In Flash, you can convert a movie clip to a compiled clip as follows:
Compile a movie clip
Right-click (Windows) or Control-click (Macintosh) the movie clip in the Library panel, and then select Convert
To Compiled Clip.
The compiled clip behaves just like the movie clip from which it was compiled, but compiled clips appear and
publish much faster than ordinary movie clips. Compiled clips can’t be edited, but their properties can appear in
the Property inspector and the Component inspector.
SWC components contain a compiled clip, the component’s pre-compiled ActionScript definitions, and other files
that describe the component. If you create your own component, you can export it as a SWC file to distribute it.
Export a SWC file
Select the movie clip in the Library panel and right-click (Windows) or Control-click (Macintosh), and then select
Export SWC File.
Note: The format of a Flash CS4 or later SWC file is compatible with the Flex SWC format so that SWC files can be
exchanged between the two products, but not necessarily without modifications.
For information on creating SWC-based components, see
.
The ActionScript 3.0 Components API
Each ActionScript 3.0 component is built on an ActionScript 3.0 class that is located in a package folder and has a name
of the format fl.packagename.classname. The Button component, for example, is an instance of the Button class and
has a package name of
fl.controls.Button
. You must reference the package name when you import a component
class in your application. You would import the Button class with the following statement:
import fl.controls.Button;
“Working with component files” on page
18
.
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A component’s class defines the methods, properties, events, and styles that enable you to interact with it in your
application. The ActionScript 3.0 UI components are subclasses of the Sprite and UIComponent classes and inherit
properties, methods, and events from them. The Sprite class is the basic display list building block and is similar to a
MovieClip but does not have a Timeline. The UIComponent class is the base class for all visual components, both
interactive and non-interactive. The inheritance path of each component, as well as its properties, methods, events,
and styles are described in the Adobe
ActionScript 3.0 Reference for the Adobe Flash Platform
.
All ActionScript 3.0 components use the ActionScript 3.0 event handling model. For more information on event
handling, see
“Handling events” on page
23
and Programming ActionScript 3.0.
Working with component files
This section explains where component files are stored, where to find the ActionScript source files, and how to add
and remove components from the Components panel.
Where component files are stored
Flash components are stored in the application-level Configuration folder.
Note: For information about these folders, see “Configuration folders installed with Flash” in Using Flash.
Components are installed in the following locations:
Windows:
Mac
OS X:
Within the Components folder, the User Interface (UI) components are in the User Interface.fla file and the
FLVPlayback (FLVPlaybackAS3.swc) and FLVPlaybackCaptioning components are in the Video folder.
You can also store components in the following user-based locations:
Windows:
CS5\en\Configuration\Components
Windows Vista:
Windows 7:
Note: In Windows, the Application Data folder (Windows 2000, XP, Vista) and AppData folder (Windows7) is
hidden by default. To show hidden folders and files, select My Computer to open Windows Explorer, select Tools >
Folder Options and then select the View tab. Under the View tab, select the Show hidden files and folders radio button.
Mac
OS X:
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Where component source files are stored
The ActionScript (.as) class files (or source files) for components are installed in the following application folders for
Windows 2000, XP, Vista, and 7:
User Interface components
Source\ActionScript 3.0\User Interface\fl
FLVPlayback
3.0\FLVPlayback\fl\video
FLVPlaybackCaptioning
Source\ActionScript 3.0\FLVPlaybackCaptioning\fl\video
For Mac
OS X, the component source files are located here:
User Interface components
Source:ActionScript 3.0:User Interface:fl
FLVPlayback
3.0:FLVPlayback:fl:video
FLVPlaybackCaptioning
Source:ActionScript 3.0:FLVPlaybackCaptioning:fl:video
Component source files and Classpath
Because the ActionScript 3.0 components have their code compiled in, you should not specify the location of the
ActionScript class files in your Classpath variable. If you do include their location in the Classpath, it will increase the
time required to compile your applications. However, if Flash finds component class files in your Classpath setting,
the class file will always take precedence over the component’s compiled-in code.
One time that you might wish to add the location of the component source files to your Classpath setting is when
“Debug component applications” on
page
20
.
Modify the component files
If you update, add, or remove SWC-based components or add new FLA-based components to Flash, you must reload
them to the Components panel to make them available. You can reload the components either by restarting Flash or
by selecting Reload from the Components panel menu. This will cause Flash to pick up any components that you’ve
Reload components in the Components panel while Flash is running:
Remove a component from the Components panel:
Remove the FLA, SWC, or MXP file from the Components folder and either restart Flash or select Reload from the
You can remove and replace SWC-based components while Flash is running, and reloading will reflect the changes,
but if you change or delete FLA-based components, the changes are not reflected until you terminate and restart
Adobe recommends that you first make a copy of any Flash component file (.fla or .as) that you are going to alter.
Then you can restore it, if necessary.
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Debug component applications
The ActionScript 3.0 components contain all their source code to reduce compilation time when you compile your
application. The Flash debugger, however, cannot inspect code inside compiled clips. Therefore, if you want to debug
your application down into the components' source code, you must add the component source files to your Classpath
setting.
The location of the component package folders is relative to the location of the source files for the component type. To
reference all of the ActionScript 3.0 source files for all UI components, add the following location to your Classpath
for the User Interface packages:
\$(AppConfig)/Component Source/ActionScript 3.0/User Interface
Note: This will override the compiled-in code for all UI components and increase compilation time for your application.
If you have changed a component’s source file for any reason, that component might exhibit different behavior as a result.
To set the Classpath, select Preferences from the Edit menu and then select ActionScript from the Category list and
click the ActionScript 3.0 Settings button. To add a new entry, click the plus above the window that displays the current
settings.
The
\$(AppConfig)
variable refers to the Flash CS5 Configuration folder in the location where you installed Flash CS5.
Typically, the path looks like this:
Mac
OS X: Macintosh HD:Applications:Adobe Flash CS5:Configuration
Note: If you must change a component source file, Adobe strongly recommends that you copy the original source file to a
“Where component source files are stored” on
page
19
.
Set parameters and properties
Each component has parameters that you can set to change its appearance and behavior. A parameter is a property of
the component’s class and appears in the Property inspector and the Component inspector. The most commonly used
properties appear as authoring parameters; others you must set with ActionScript. All parameters that can be set
during authoring can also be set with ActionScript. Setting a parameter with ActionScript overrides any value set
during authoring.
Most ActionScript 3.0 User Interface components inherit properties and methods from the UIComponent class as well
as from a base class. For example, the Button and CheckBox classes inherit properties from both the UIComponent
class and the BaseButton class. A component’s inherited properties, as well as its own class properties, are available for
you to access. For example, the ProgressBar component inherits the
ProgressBar.enabled
property from
UIComponent but also has its own
ProgressBar.percentComplete
property. You can access both of these
properties to interact with an instance of the ProgressBar component. For more information on a component’s
properties, see its class entry in the
ActionScript 3.0 Reference for the Adobe Flash Platform
.
You can set parameters for a component instance using either the Property inspector or the Component inspector.
Enter an instance name for a component in the Property inspector:
1
Select Window
> Properties
> Properties.
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2
Select an instance of a component on the Stage.
3
Enter a name for the component instance in the box that says <Instance Name>, located beneath the drop-down
list that says Movie Clip. Or click the Parameters tab and enter the name in the box below the word Component.
Enter values for any parameters that you want to set.
It’s a good idea to add a suffix to the instance name to indicate what kind of component it is; this makes it easier to
read your ActionScript code. For example, the instance name licenseSb identifies that a component is a scroll bar
Enter parameters for a component instance in the Component inspector:
1
Select Window
> Component Inspector.
2
Select an instance of a component on the Stage.
3
Click the Parameters tab and enter values for any of the listed parameters.
Component parameters in the Component inspector
Set component properties in ActionScript
In ActionScript, you use a dot (.) operator (dot syntax) to access properties or methods that belong to an object or
instance on the Stage. A dot syntax expression begins with the name of the instance, followed by a dot, and it ends with
the element you want to specify. For example, the following ActionScript code sets the
width
property of the
CheckBox instance
aCh
to make it 50 pixels wide:
aCh.width = 50;
The following
if
statement checks to see if the user has selected the check box:
if (aCh.selected == true) {
displayImg(redCar);
}
The library
When you first add a component to a document, Flash imports it as a movie clip into the Library panel. You can also
drag a component from the Components panel directly to the Library panel and then add an instance of it to the Stage.
In any case, you must add a component to the library before you can access its class elements.
If you add a component to the library and create an instance of it using ActionScript, you must first import its class
with the
import
statement. In the
import
statement, you must specify both the component’s package name and its
class name. For example, the following statement imports the Button class:
import fl.controls.Button;
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When you place a component in the library, Flash also imports a folder of its assets, which contain the skins for its
different states. A component’s skins comprise the collection of symbols that make up its graphical display in the
application. A single skin is the graphical representation, or movie clip, that indicates a particular state for the
component.
The contents of the Component Assets folder allow you to change the component’s skins if you wish to do that. For
“Customizing the UI Components” on page
95
.
Once a component is in the library, you can add more instances of it to your document by dragging its icon to the Stage
from either the Components panel or the Library panel.
Sizing components
Use the Free Transform tool or the
setSize()
method to resize component instances. You can call the
setSize()
method from any component instance (see
UIComponent.setSize()
) to resize it. The following code resizes an
instance of the List component to 200 pixels wide and 300 pixels high:
aList.setSize(200, 300);
A component does not resize automatically to fit its label. If a component instance that has been added to a document
is not large enough to display its label, the label text is clipped. You must resize the component to fit its label.
ActionScript 3.0 Reference for the
.
Live Preview
The Live Preview feature, enabled by default, lets you view components on the Stage as they will appear in the
published Flash content; the components appear at their approximate size.
To turn Live Preview on or off:
Select Control
> Enable Live Preview. A check mark next to the option indicates that it is enabled.
The live preview reflects different parameters for different components. For information about which component
parameters are reflected in the live preview, see each component entry in the
ActionScript 3.0 Reference for the Adobe
Flash Platform
.
A Button component with Live Preview enabled
A Button component with Live Preview disabled
Components in Live Preview are not functional. To test functionality, you must use the Control
> Test Movie
command.
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Handling events
Every component broadcasts events when a user interacts with it. When a user clicks a Button, for example, it
dispatches a
MouseEvent.CLICK
event and when a user selects an item in a List, the List dispatches an Event.
CHANGE
event. An event can also occur when something significant happens to a component such as when content finishes
Event.COMPLETE
event. To handle an event, you write ActionScript
code that executes when the event occurs.
A component’s events include the events of any class from which the component inherits. This means that all
ActionScript 3.0 User Interface components inherit events from the UIComponent class because it is the base class for
the ActionScript 3.0 User Interface components. To see the list of events a component broadcasts, see the Events
section of the component’s class entry in the
ActionScript 3.0 Reference for the Adobe Flash Platform
.
For a complete explanation of event handling in ActionScript 3.0, see the
ActionScript 3.0 Developer's Guide
.
The following key points apply to handling events for ActionScript 3.0 components:
All events are broadcast by an instance of a component class. The component instance is the broadcaster.
You register an event listener by calling the
method for the component instance. For
example, the following line of code adds a listener for the
MouseEvent.CLICK
event to the Button instance
aButton
:
The second parameter of the
method registers the name of the function,
clickHandler
, to
be called when the event occurs. This function is also referred to as a callbackfunction.
You can register multiple listeners to one component instance.
You can register one listener to multiple component instances.
The event handler function is passed to an event object that contains information about the event type and the
“About the event object” on page
23
.
The listener remains active until the application terminates or you explicitly remove it using the
removeEventListener()
method. For example, the following line of code removes the listener for the
MouseEvent.CLICK
event on
aButton
:
aButton.removeEventListener(MouseEvent.CLICK, clickHandler);
The event object inherits from the Event object class and has properties that contain information about the event that
occurred, including the
target
and
type
properties, which provide essential information about the event:
Property
Description
type
A string indicating the type of the event.
target
A reference to the component instance broadcasting the event.
24
USING ACTIONSCRIPT 3.0 COMPONENTS
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When an event has additional properties, they are listed in the event’s class description in the
ActionScript 3.0 Reference
.
The event object is automatically generated and passed to the event handler function when an event occurs.
You can use the event object inside the function to access the name of the event that was broadcast or the instance
name of the component that broadcast the event. From the instance name, you can access other component properties.
For example, the following code uses the
target
property of the
evtObj
event object to access the
label
property of
aButton
and display it in the Output panel:
import fl.controls.Button;
import flash.events.MouseEvent;
var aButton:Button = new Button();
aButton.label = "Submit";
function clickHandler(evtObj:MouseEvent){
trace("The " + evtObj.target.label + " button was clicked");
}
Work with the display list
All ActionScript 3.0 components inherit from the DisplayObject class and, therefore, have access to its methods and
properties to interact with the display list. The display list is the hierarchy of displayed objects and visual elements in
an application. This hierarchy includes the following elements:
The Stage, which is the top-level container
Display objects, which include shapes, MovieClips, and text fields, among others
Display object containers, which are special types of display objects that can contain child display objects.
The order of objects in the display list determines their depth in the parent container. An object’s depth refers to its
position from top to bottom or front to back on the Stage or in its display container. The order of depth is apparent
when objects overlap but it exists even when they do not. Every object in the display list has a corresponding depth on
the Stage. If you want to change an object’s depth by placing it in front of or moving it behind other objects, you need
to change its position in the display list. The default order of objects in the display list is the order in which they are
placed on the Stage. Position 0 in the display list is the object at the bottom of the depth order.
Add a component to the display list
You can add an object to a DisplayObjectContainer object by calling the container’s
or
method. In the case of the Stage, you can also add an object to its display list during authoring by creating it, or in the
case of components, by dragging it to the Stage from the Components panel. To add an object to a container with
ActionScript, first create an instance of it by invoking its constructor with the
new
operator and then call the
or
method to place it on the Stage and in the display list. The
method places
the object at the next position in the display list, while
specifies the position at which to add the object.
If you specify a position that is already occupied, the object at that position, and those at higher positions, move up by
1. The
numChildren
property of a DisplayObjectContainer object contains the number of display objects that it
contains. You can retrieve an object from the display list by calling the
getChildAt()
method and specifying the
position, or if you know the name of the object, by calling the
getChildByName()
method.
25
USING ACTIONSCRIPT 3.0 COMPONENTS
Working with Components
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Note: When you add a component with ActionScript, you must assign a name to it’s name property if you want to access
it by name in the display list.
The following example displays the names and positions of three components in the display list. First, drag a
NumericStepper, a Button, and a ComboBox to the Stage so that they overlap each other and give them instance names
of aNs, aButton, and aCb. Then add the following code to the Actions panel on Frame 1 of the Timeline:
var i:int = 0;
while(i < numChildren) {
trace(getChildAt(i).name + " is at position: " + i++);
}
You should see the following lines in the Output panel:
aNs is at position: 0
aButton is at position: 1
aCb is at position: 2
Move a component in the display list
You can change the position of an object in the display list, and its display depth, by calling the
method
and supplying the name of an object and the position where you want to place it as the method’s parameters. For
example, add the following code to the preceding example to place the NumericStepper on top and repeat the loop to
display the components’ new positions in the display list:
i = 0;
while(i < numChildren) {
trace(getChildAt(i).name + " is at position: " + i++);
}
You should see the following in the Output panel:
aNs is at position: 0
aButton is at position: 1
aCb is at position: 2
aButton is at position: 0
aCb is at position: 1
aNs is at position: 2
The NumericStepper should also appear in front of the other components on the screen.
Note that
numChildren
is the number of objects (from 1 to n) in the display list while the first position in the list is 0.
So if there are three objects in the list, the index position of the third object is 2. This means that you can reference the
last position in the display list, or the top object in terms of display depth, as
numChildren - 1
.
Remove a component from the display list
You can remove a component from a display object container and its display list with the
removeChild()
and
removeChildAt()
methods. The following example places three Button components in front of each other on the
Stage and adds an event listener for each of them. When you click each Button, the event handler removes it from the
display list and the Stage.
1
Create a new Flash file (ActionScript 3.0) document.
2
Drag a Button from the Components panel to the Library panel.
3
Open the Actions panel, select Frame 1 in the main Timeline and add the following code:
26
USING ACTIONSCRIPT 3.0 COMPONENTS
Working with Components
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import fl.controls.Button;
var i:int = 0;
while(i++ < 3) {
makeButton(i);
}
function removeButton(event:MouseEvent):void {
removeChildAt(numChildren -1);
}
function makeButton(num) {
var aButton:Button = new Button();
aButton.name = "Button" + num;
aButton.label = aButton.name;
aButton.move(200, 200);
}
For a complete explanation of the display list, see “Display programming” in Programming ActionScript 3.0.
Work with FocusManager
When a user presses the Tab key to navigate in a Flash application or clicks in an application, the FocusManager class
determines which component receives input focus. You don’t need to add a FocusManager instance to an application
or write any code to activate the FocusManager unless you are creating a component.
If a RadioButton object receives focus, the FocusManager examines that object and all objects with the same
groupName
value and sets focus on the object with the
selected
property set to
true
.
Each modal Window component contains an instance of the FocusManager, so the controls on that window become
their own tab set. This prevents a user from inadvertently navigating to components in other windows by pressing the
Tab key.
The FocusManager uses the depth level (or z-order) of elements in the container as the default navigation scheme or
tab loop. A user typically navigates the tab loop by using the Tab key, with focus moving from the first component that
has focus, to the last, and then back again to the first.The depth levels are set up primarily by the order in which
components are dragged to the Stage; however, you can also use the Modify
> Arrange
> Bring To Front/Send To Back
commands to determine the final z-order. For more information on depth levels, see
“Work with the display list” on
page
24
.
You can call the
setFocus()
method to give focus to a component instance in an application. For example, the
following example creates a FocusManager instance for the current container (
this
) and gives focus to the Button
instance
aButton
.
var fm:FocusManager = new FocusManager(this);
fm.setFocus(aButton);
27
USING ACTIONSCRIPT 3.0 COMPONENTS
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Last updated 5/27/2011
You can determine which component has focus by calling the
getFocus()
method and you can determine which
component in the tab loop will receive focus next by calling the
getNextFocusManagerComponent()
method. In the
following example, a CheckBox, a RadioButton, and a Button are on the Stage and each component has listeners for
MouseEvent.CLICK
and
FocusEvent.MOUSE_FOCUS_CHANGE
events. When the
MouseEvent.CLICK
event occurs,
because the user clicked on the component, the
showFocus()
function calls the
getNextFocusManagerComponent() | 2014-10-20 21:11:12 | {"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": 0.8419513702392578, "perplexity": 203.82662522992496}, "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-2014-42/segments/1413507443438.42/warc/CC-MAIN-20141017005723-00011-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://cms.math.ca/cjm/msc/14L27?fromjnl=cjm&jnl=CJM | Equivariant polynomial automorphism of $\Theta$-representations We show that every equivariant polynomial automorphism of a $\Theta$-repre\-sen\-ta\-tion and of the reduction of an irreducible $\Theta$-representation is a multiple of the identity. Categories:14L30, 14L27 | 2015-07-06 11:24:33 | {"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": 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": 0.857977032661438, "perplexity": 726.2492671242401}, "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-2015-27/segments/1435375098196.31/warc/CC-MAIN-20150627031818-00157-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://www.ck12.org/book/CK-12-Algebra-I-Second-Edition/r1/section/9.6/ | # 9.6: Factoring Special Products
Difficulty Level: At Grade Created by: CK-12
## Learning Objectives
• Factor the difference of two squares.
• Factor perfect square trinomials.
• Solve quadratic polynomial equation by factoring.
## Introduction
When you learned how to multiply binomials we talked about two special products.
In this section we’ll learn how to recognize and factor these special products.
## Factor the Difference of Two Squares
We use the sum and difference formula to factor a difference of two squares. A difference of two squares is any quadratic polynomial in the form , where and can be variables, constants, or just about anything else. The factors of are always ; the key is figuring out what the and terms are.
Example 1
Factor the difference of squares:
a)
b)
c)
Solution
a) Rewrite as . Now it is obvious that it is a difference of squares.
We can check to see if this is correct by multiplying :
Note: We could factor this polynomial without recognizing it as a difference of squares. With the methods we learned in the last section we know that a quadratic polynomial factors into the product of two binomials:
We need to find two numbers that multiply to -9 and add to 0 (since there is no term, that’s the same as if the term had a coefficient of 0). We can write -9 as the following products:
We can factor as , which is the same answer as before. You can always factor using the methods you learned in the previous section, but recognizing special products helps you factor them faster.
b) Rewrite as . This factors as .
c) Rewrite as . This factors as .
Example 2
Factor the difference of squares:
a)
b)
c)
Solution
a) Rewrite as . This factors as .
b) Rewrite as . This factors as .
c) Rewrite as . This factors as .
Example 3
Factor the difference of squares:
a)
b)
c)
Solution
a) factors as .
b) Rewrite as . This factors as .
c) Rewrite as . This factors as .
Example 4
Factor the difference of squares:
a)
b)
c)
Solution
a) Rewrite as . This factors as .
b) Rewrite as . This factors as .
c) Rewrite as . This factors as .
## Factor Perfect Square Trinomials
We use the square of a binomial formula to factor perfect square trinomials. A perfect square trinomial has the form or .
In these special kinds of trinomials, the first and last terms are perfect squares and the middle term is twice the product of the square roots of the first and last terms. In a case like this, the polynomial factors into perfect squares:
Once again, the key is figuring out what the and terms are.
Example 5
Factor the following perfect square trinomials:
a)
b)
c)
Solution
a) The first step is to recognize that this expression is a perfect square trinomial.
First, we can see that the first term and the last term are perfect squares. We can rewrite as .
Next, we check that the middle term is twice the product of the square roots of the first and the last terms. This is true also since we can rewrite as .
This means we can factor as . We can check to see if this is correct by multiplying :
Note: We could factor this trinomial without recognizing it as a perfect square. We know that a trinomial factors as a product of two binomials:
We need to find two numbers that multiply to 16 and add to 8. We can write 16 as the following products:
So we can factor as , which is the same as .
Once again, you can factor perfect square trinomials the normal way, but recognizing them as perfect squares gives you a useful shortcut.
b) Rewrite as .
We notice that this is a perfect square trinomial, so we can factor it as .
c) Rewrite as .
We notice that this is a perfect square trinomial, so we can factor it as .
Example 6
Factor the following perfect square trinomials:
a)
b)
c)
Solution
a) Rewrite as .
We notice that this is a perfect square trinomial and we can factor it as .
b) Rewrite as .
We notice that this is a perfect square trinomial and we can factor it as .
We can check to see if this is correct by multiplying :
c)
We notice that this is a perfect square trinomial and we can factor it as .
For more examples of factoring perfect square trinomials, watch the videos at http://www.onlinemathlearning.com/perfect-square-trinomial.html.
## Solve Quadratic Polynomial Equations by Factoring
With the methods we’ve learned in the last two sections, we can factor many kinds of quadratic polynomials. This is very helpful when we want to solve them. Remember the process we learned earlier:
1. If necessary, rewrite the equation in standard form so that the right-hand side equals zero.
2. Factor the polynomial completely.
3. Use the zero-product rule to set each factor equal to zero.
4. Solve each equation from step 3.
We can use this process to solve quadratic polynomials using the factoring methods we just learned.
Example 7
Solve the following polynomial equations.
a)
b)
c)
Solution
a) Rewrite: We can skip this since the equation is in the correct form already.
Factor: We can write 6 as a product of the following numbers:
factors as .
Set each factor equal to zero:
Solve:
Check: Substitute each solution back into the original equation.
b) Rewrite: is rewritten as
Factor: We can write 12 as a product of the following numbers:
factors as .
Set each factor equal to zero:
Solve:
Check: Substitute each solution back into the original equation.
c) Rewrite: is rewritten as
Factor: We can write -15 as a product of the following numbers:
factors as
Set each factor equal to zero:
Solve:
Check: Substitute each solution back into the original equation.
Example 8
Solve the following polynomial equations:
a)
b)
c)
Solution
a)
Rewrite: The equation is in the correct form already.
Factor: Rewrite as .
We recognize this as a perfect square. This factors as or
Set each factor equal to zero:
Solve:
Notice that for a perfect square the two solutions are the same. This is called a double root.
Check: Substitute each solution back into the original equation.
b)
Rewrite: this is not necessary since the equation is in the correct form already
Factor: Rewrite as .
We recognize this as a difference of squares. This factors as .
Set each factor equal to zero:
Solve:
Check: Substitute each solution back into the original equation.
c)
Rewrite: this is not necessary since the equation is in the correct form already
Factor: Rewrite as .
We recognize this as a perfect square. This factors as or
Set each factor equal to zero:
Solve:
Check: Substitute each solution back into the original equation.
## Review Questions
Factor the following perfect square trinomials.
Factor the following differences of squares.
Solve the following quadratic equations using factoring.
### Notes/Highlights Having trouble? Report an issue.
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Subjects: | 2017-04-28 11:39:33 | {"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": 0.8867753744125366, "perplexity": 561.7660602956545}, "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-2017-17/segments/1492917122933.39/warc/CC-MAIN-20170423031202-00188-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://surveillance.r-forge.r-project.org/pkgdown/reference/measles.weser.html | Weekly counts of new measles cases for the 17 administrative districts (NUTS-3 level) of the “Weser-Ems” region of Lower Saxony, Germany, during 2001 and 2002, as reported to the Robert Koch institute according to the Infection Protection Act (“Infektionsschutzgesetz”, IfSG).
data("measlesWeserEms") is a corrected version of data("measles.weser") (see Format section below). These data are illustrated and analyzed in Meyer et al. (2017, Section 5), see vignette("hhh4_spacetime").
data("measles.weser")
data("measlesWeserEms")
## Format
data("measles.weser") is an object of the old "disProg" class, whereas data("measlesWeserEms") is of the new class "sts".
Furthermore, the following updates have been applied for data("measlesWeserEms"):
• it includes the two districts “SK Delmenhorst” (03401) and “SK Wilhemshaven” (03405) with zero counts, which are ignored in data("measles.weser").
• it corrects the time lag error for year 2002 caused by a redundant pseudo-week “0” with 0 counts only (the row measles.weser\$observed[53,] is nonsense).
• it has one more case attributed to “LK Oldenburg” (03458) during 2001/W17, i.e., 2 cases instead of 1. This reflects the official data as of “Jahrbuch 2005”, whereas data("measles.weser") is as of “Jahrbuch 2004”.
• it contains a map of the region (as a "SpatialPolygonsDataFrame") with the following variables:
GEN
district label.
AREA
district area in m^2.
POPULATION
number of inhabitants (as of 31/12/2003).
vaccdoc.2004
proportion with a vaccination card among screened abecedarians (2004).
vacc1.2004
proportion with at least one vaccination against measles among abecedarians presenting a vaccination card (2004).
vacc2.2004
proportion of doubly vaccinated abecedarians among the ones presenting their vaccination card at school entry in the year 2004.
• it uses the correct format for the official district keys, i.e., 5 digits (initial 0).
• its attached neighbourhood matrix is more general: a distance matrix (neighbourhood orders) instead of just an adjacency indicator matrix (special case nbOrder == 1).
• population fractions represent data as of 31/12/2003 (LSN, 2004, document “A I 2 - hj 2 / 2003”). There are only minor differences to the ones used for data("measles.weser").
## Source
Measles counts were obtained from the public SurvStat database of the Robert Koch institute: https://survstat.rki.de/.
A shapefile of Germany's districts as of 01/01/2009 was obtained from the German Federal Agency for Cartography and Geodesy (https://gdz.bkg.bund.de/). The map of the 17 districts of the “Weser-Ems” region (measlesWeserEms@map) is a simplified subset of this shapefile using a 30% reduction via the Douglas-Peucker reduction method as implemented at https://MapShaper.org.
Population numbers were obtained from the Federal Statistical Office of Lower Saxony (LSN): https://www.statistik.niedersachsen.de/themenbereiche/bevoelkerung/
Vaccination coverage was obtained from the public health department of Lower Saxony: Niedersächsisches Landesgesundheitsamt (2005): Impfreport -- Durchimpfung von Kindern im Einschulungsalter in Niedersachsen im Erhebungsjahrgang 2004. Online available from https://www.nlga.niedersachsen.de/gesundheitsberichterstattung/gesundheitsberichte/impfreport/, also as an interactive version.
## References
Meyer, S., Held, L. and Höhle, M. (2017): Spatio-temporal analysis of epidemic phenomena using the R package surveillance. Journal of Statistical Software, 77 (11), 1-55. doi: 10.18637/jss.v077.i11
## Examples
## old "disProg" object
data("measles.weser")
measles.weser
plot(measles.weser, as.one=FALSE)
## new "sts" object (with corrections)
data("measlesWeserEms")
measlesWeserEms
plot(measlesWeserEms) | 2021-06-17 04:36:21 | {"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": 0.4044463336467743, "perplexity": 9235.302197196395}, "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-2021-25/segments/1623487629209.28/warc/CC-MAIN-20210617041347-20210617071347-00491.warc.gz"} |
https://recbole.io/docs/v0.1.2/recbole/recbole.model.sequential_recommender.ksr.html | # KSR¶
Reference:
Jin Huang et al. “Improving Sequential Recommendation with Knowledge-Enhanced Memory Networks.” In SIGIR 2018
class recbole.model.sequential_recommender.ksr.KSR(config, dataset)[source]
KSR integrates the RNN-based networks with Key-Value Memory Network (KV-MN). And it further incorporates knowledge base (KB) information to enhance the semantic representation of KV-MN.
calculate_loss(interaction)[source]
Calculate the training loss for a batch data.
Parameters
interaction (Interaction) – Interaction class of the batch.
Returns
Training loss, shape: []
Return type
torch.Tensor
forward(item_seq, item_seq_len)[source]
Defines the computation performed at every call.
Should be overridden by all subclasses.
Note
Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the registered hooks while the latter silently ignores them.
full_sort_predict(interaction)[source]
full sort prediction function. Given users, calculate the scores between users and all candidate items.
Parameters
interaction (Interaction) – Interaction class of the batch.
Returns
Predicted scores for given users and all candidate items, shape: [n_batch_users * n_candidate_items]
Return type
torch.Tensor
memory_read(user_memory)[source]
memory_update(item_seq, item_seq_len)[source]
define write operator
predict(interaction)[source]
Predict the scores between users and items.
Parameters
interaction (Interaction) – Interaction class of the batch.
Returns
Predicted scores for given users and items, shape: [batch_size]
Return type
torch.Tensor | 2021-05-06 06:32:36 | {"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": 0.6223037838935852, "perplexity": 12660.984919880999}, "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/1620243988741.20/warc/CC-MAIN-20210506053729-20210506083729-00099.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-3x-2-13x-10-0-by-factoring | # How do you solve 3x^2-13x-10=0 by factoring?
Oct 10, 2015
$\left(3 x + 2\right) \left(x - 5\right)$
#### Explanation:
Because $3$ can only have factors of $3$ and 1
you know those will be your coefficients for $x$.
That is your factors will be of the form
$\textcolor{w h i t e}{\text{XXX}} \left(3 x + a\right) \left(x + b\right)$
You also know that one of $a$ or $b$ will have $+$ and one side will have $-$ due to the $\textcolor{red}{-} 10$.
The options for $10$ are
$2 \times 5$ or $1 x 10$.
Using $2 \times 5$
if you multiply the $5$ by $3$ it gives you $15$
and the $2$ by $1$1 you get $2$.
Because $13$ in the center (of the original form) is negative
you know the $5$ needs to be negative
and the factors are
$\textcolor{w h i t e}{\text{XXX}} \left(3 x + 2\right) \left(x - 5\right)$ | 2020-06-04 20:53:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 24, "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": 0.9338621497154236, "perplexity": 434.4650670869395}, "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-2020-24/segments/1590347458095.68/warc/CC-MAIN-20200604192256-20200604222256-00508.warc.gz"} |
https://mdtpmodules.org/exps/module-1/lesson-3/watch-scientific-notation/ | # EXPS 1 | Lesson 3 | Watch (Scientific Notation)
In Explore, when multiplying by powers of 10, the decimal moves either to the right or to the left depending on the sign of the power.
Watch the LearnZillion Video for more information and review on writing large numbers in scientific notation.
### LearnZillion: Write large numbers in scientific notation
The video shows how to write large numbers in scientific notation; for example, the distance from the earth to the sun, $$491,000,000,000$$ feet, is written as $$4.91\times10^{11}$$ in scientific notation.
Very small numbers can also be written in scientific notation; for example, a micron is $$\dfrac1{1,000,000}$$ or $$1\times 10^{-6}$$ of a meter.
In Explore, $$5.47 \times 10^{-2}$$ can be written as $$0.0547$$, $$3.95 \times 10^{-4}$$ can be written as $$0.000395$$, and $$0.00324$$ can be written as $$3.24 \times 10^{-3}$$ using scientific notation. | 2022-12-06 04:43:14 | {"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": 0.675344705581665, "perplexity": 395.81720924843376}, "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-2022-49/segments/1669446711069.79/warc/CC-MAIN-20221206024911-20221206054911-00637.warc.gz"} |
https://bioinformatics.stackexchange.com/questions/11271/samtools-bamutil-meaning-of-wildcard-as-reference-name | # samtools / bamUtil | Meaning of <wildcard> as Reference Name
I have been working with several tools on bam files recently and I am not sure how I should interpret some of the outputs.
samtools idxstats mapped.sorted.bam
Chr1 15096 0 0
Chr2 33397 0 0
Chr3 43888 41 0
Chr4 20819 1 0
* 0 0 34
Documentation says third and fourth column give "mapped reads" and "unmapped reads". So I interpreted the last line (*) as the number of unmapped reads as this is the only line with a non 0 value at this field.
Then I used
bam splitChromosome --in mapped.sorted.bam --out test/aln.
Reference Name: chr6 has 6 records
Reference Name: Chr7 has 1 records
Reference Name: Chr3 has 41 records
Reference Name: Chr4 has 1 records
Reference Name: * has 34 records
And looking at the bams outputted (one per reference sequence) in test/ I find an aln.UnknownChrom.bam that I am not sure where it comes from. I actually found 97 unique reads ID in it, where I was waiting for 34 or less.
So my first question is : Does "*" actually means "all reads that didn't mapp" and this information is stored in way I didn't know about in BAM files ?
And my second question would be : Anybody knows where this aln.UnknownChrom.bam comes from when using splitChromosome ? It is a source of error in a pipeline I am building and I'd like to get rid of it, if these are not actual mapping informations.
I haven't used this splitChromosome utility before myself, but as it splits the BAM file into one BAM file per chromosome, I suspect it simply also creates a BAM file for the unknown (*) chromosome. Why the numbers don't match I can't tell. | 2021-10-25 15:17:17 | {"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": 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": 0.25183621048927307, "perplexity": 2989.111319128538}, "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-2021-43/segments/1634323587711.69/warc/CC-MAIN-20211025123123-20211025153123-00716.warc.gz"} |
https://www.physicsforums.com/threads/two-different-expressions-of-jaynes-cummings-hamiltonian.630052/ | # Two different expressions of Jaynes-Cummings Hamiltonian
## Main Question or Discussion Point
Hi, I have a question about two different expressions of Jaynes-Cummings Hamiltonian
$H=\Delta_c a^{\dagger}a+\Delta_a \sigma_{+} \sigma_{-} + g (a^{\dagger}\sigma_{-} +a\sigma_{+} )$
and
$H=\Delta_c a^{\dagger}a+\Delta_a \sigma_{+} \sigma_{-} +i g (a^{\dagger}\sigma_{-} -a\sigma_{+} )$.$(\hbar=1)$
I read them from different papers and books.
Why are they equal, and how to derive one from another?
How to choose the appropriate expression when utilizing the Jaynes-Cummings Hamiltonian?
Thanks!
Last edited:
Related Atomic and Condensed Matter News on Phys.org
Physics Monkey
Homework Helper
Suppose I took the first form and wrote $-i \tilde{a} = a$. How does the Hamiltonian look in terms of $\tilde{a}$?
Suppose I took the first form and wrote $-i \tilde{a} = a$. How does the Hamiltonian look in terms of $\tilde{a}$?
But in both expressions, $a$ and $a^{\dagger}$ are annihilation and creation operators, respectively, and they are real. Is the transformation still valid?
Last edited:
Physics Monkey | 2020-02-23 23:50:42 | {"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": 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": 0.525874674320221, "perplexity": 952.8384971540547}, "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-2020-10/segments/1581875145859.65/warc/CC-MAIN-20200223215635-20200224005635-00347.warc.gz"} |
https://www.impactm.org/nonprofit/20-5010527/ | Living Goods
Impact Rating
# Living Goods
Top Health Nonprofit Impact Audited
Impact: $5,200 saves the life of a child. Community Health Workers Program meets the benchmark for high cost-effectiveness. The nonprofit averts a disability-adjusted life year for less than the country's G.D.P. per capita. Note: The impact of this program may not be representative of the entire operation of Living Goods. Governance: Passes checks Mission Living Goods believes that every mother and child should have access to basic health care in their own community, no matter where they live. We seek to improve health across the developing world for good by staffing every community with a well-equipped, properly paid, and effectively managed Community Health Worker. The widespread adoption of high-impact, low-cost community health will be every bit as transformative as a new vaccine or the eradication of a disease. Millions of lives will be saved. Entire countries will be healthier and more productive. Government will have prioritized community health in its policies, plans, and budgets. Cause Health Rated Program Community Health Workers Program Location Uganda Location San Francisco, CA Website Donations processed by the nonprofit. Cause Health Rated Program Community Health Workers Program Location Uganda Location San Francisco, CA Website Impact Calculator: I'd like to give$ 0
Impact Calculator:
#### Rated Program
###### Program
Community Health Workers Program
###### Activities
Living Goods trains women from local communities in Uganda to diagnose and treat malnutrition, infection and many common and life-threatening illnesses.
Community Health
###### Beneficiaries Served
People living in poverty
###### Geography
Uganda
Outcomes: Changes in people's lives. They can be caused by a nonprofit.
Costs: The money spent by nonprofits and their partners and beneficiaries.
Impact: The cost to achieve an outcome.
Cost-effectiveness: A judgment as to whether the cost was "worth" the outcome.
#### Outcomes
###### Outcome Metric
A child's life saved.
To calculate impact, we estimate how many outcomes the nonprofit caused.
###### Data Source
Outcome data collected through a randomized controlled trial conducted by third-party evaluators Abdul Latif Jameel Poverty Action Lab and Innovations for Poverty Action on the Living Goods program in Uganda.
###### Time Period of Data
Jan. 1, 2014, to Dec. 31, 2016
Ratings are based on data the nonprofit itself collects on its work. We use the most recent year with sufficient data. Typically, this data allows us to calculate direct changes in participants' lives, such as increased income.
###### Method for Attributing Outcomes
ImpactMatters used the randomized controlled trial's estimate of the counterfactual mortality rate in areas that were not exposed to Living Goods' intervention for the relevant period.
To determine causation, we take the outcomes we observe and subtract an estimate of the outcomes that would have happened even without the program.
#### Cost
###### Data Source
Cost data reported by Living Goods and data and assumptions about partner and beneficiary costs.
All monetary costs are counted, whether they are borne by a nonprofit service deliverer or by the nonprofit’s public and private partners.
#### Impact
###### Impact Statement
\$5,200 saves the life of a child.
###### Confidence in Impact Calculation
ImpactMatters assigned a rating of 5/5 for the audit team's confidence in its impact estimate for the Community Health Workers program.
We calculate impact, defined as the change in outcomes attributable to a program divided by the cost to achieve those outcomes.
#### Rating
###### Benchmark for Rating
Impact ratings of community health programs are based on the cost of a disability-adjusted life year (DALY) averted relative to the G.D.P. per capita of the country in which the program operates. Programs receive 5 stars if they avert a DALY for less than than the G.D.P. per capita, and 4 stars if they avert a DALY for less than three times G.D.P. per capita. If a nonprofit reports impact but doesn't meet the benchmark for cost-effectiveness, it earns 3 stars.
###### Determination
The nonprofit averts a disability-adjusted life year for less than the country's G.D.P. per capita.
#### Why We Could Be Wrong
We welcome your suggestions for improving our methodology. Our methodology section includes explanations of how we mitigate these issues.
• The outcome could oversimplify total impact. "A child's life saved" does not take into account the reduction in ill health and disability that the program caused.
• There could be multiple important outcomes not captured in our analysis.
• Living Goods may be spending additional money in order to serve harder-to-reach and/or particularly valuable populations.
• In the absence of better data, we assume uniform counterfactual rates for programs, at the risk of masking variation across nonprofits.
• Our estimates rely on data made public by Living Goods on its website, annual reports, financial statements and Form 990s.
• We only analyze programs that meet our criteria. As a result, this report may not fully reflect the impact of Living Goods.
• We do not assess what explains the nonprofit's cost-effectiveness.
We assign a rating to the nonprofit using the rubric:
• There are indications of governance or financial health issues at the nonprofit.
• After being given an opportunity, the nonprofit chose not to publish impact information.
We are not yet issuing this level of star rating.
• The rated program does not meet our benchmark for cost-effectiveness.
• The rated program is cost-effective.
• The rated program is highly cost-effective.
#### Nonprofit Comment
(Adapted from the Impact Audit.) Living Goods is enormously proud that the audit conducted by ImpactMatters reinforces our understanding of the positive role our organization is having on transforming community health programs in Uganda and beyond. As an organization dedicated to data driven performance management, adaptive learning, transparency, and strengthening community health efforts worldwide, we are delighted to have external audits such as this and the GiveWell assessment, which enable broad groups of stakeholders to understand and apply relevant insights based on our evidence and experiences.
— Living Goods
Before publishing, we ask every nonprofit we can to review our work, offer corrections and provide a comment.
###### Analysis Details
Analysis conducted by ImpactMatters and published on November 22, 2019.
This rating is based on an independent audit by ImpactMatters. Using data from the audit, ImpactMatters analysts calculated impact and assigned a rating.
We welcome corrections. If you are interested in exploring applications of ImpactMatters data, contact us at partnerships@impactmatters.org.
#### Governance Check
Living Goods passes our governance check.
Conducted a financial audit
Overhead spending is reasonable (<35% of total spending)
Charity Navigator has not issued a fraud or mismanagement advisory
Living Goods itself has not reported any material diversions of assets
Living Goods itself has not reported any excess benefit transactions
Source: Living Goods Form 990 and Charity Navigator
#### How We Calculate Impact
This rating is based on ImpactMatters analysis of the impact of Community Health Workers Program relative to costs. Impact is the change in the social outcomes of people served by the program, net of the change that would have happened even without the program (the “counterfactual”); divided by cost. Learn more.
A guide to our process for analyzing nonprofits and assigning ratings.
Learn about best practices for reporting impact for different program types.
Our collected guidelines on how we analyze impact of nonprofit programs.
Rating is a complex exercise and we urge you to read our frequently asked questions for details of how and why we issue these ratings. | 2019-12-09 05:25:06 | {"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": 0.20304596424102783, "perplexity": 7233.464988522872}, "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-2019-51/segments/1575540517557.43/warc/CC-MAIN-20191209041847-20191209065847-00375.warc.gz"} |
https://my-assignmentexpert.com/2022/09/13/%E6%95%B0%E5%AD%A6%E4%BB%A3%E5%86%99%E5%90%8C%E8%B0%83%E4%BB%A3%E6%95%B0%E4%BB%A3%E5%86%99homological-algebra%E4%BB%A3%E8%80%83ma3204-basic-definitions/ | 19th Ave New York, NY 95822, USA
# 数学代写|同调代数代写Homological Algebra代考|MA3204 Basic Definitions
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## 数学代写|同调代数代写Homological Algebra代考|Basic Definitions
This book deals with groups and algebras. In order to fix the notation, to recall the basic facts and to keep the book as self-contained as possible we shall briefly give the necessary definitions of the most important objects and some of their properties.
Algebras
A field in this book is always commutative, a ring is always associative and has a unit, but may be non-commutative. The centre of a ring $A$ is denoted by $Z(A)$ and is defined to be
$$Z(A):={b \in A \mid a \cdot b=b \cdot a \forall a \in A} .$$
It is clear that $Z(A)$ is a commutative subring of $A$. Ring homomorphisms are always assumed to preserve the unit.
Definition 1.1.1 Let $K$ be a commutative ring and let $A$ be a ring. Then a $K$-algebra is the structure of the ring $A$ together with a ring homomorphism $\epsilon_A: K \longrightarrow Z(A)$.
Remark 1.1.2
• Observe that a $\mathbb{Z}$-algebra is nothing else than a ring.
• Very often we are interested in the case of a field $K$ and then mostly the case where $A$ is of finite dimension over $K$. However, sometimes we need to use a broader concept, and it will be important to be able to pass to general commutative rings $K$.
• For all $\lambda \in K$ and all $a \in A$ we simply write $\lambda \cdot a:=\epsilon_A(\lambda) \cdot a$.
Example 1.1.3 A few constructions are used frequently in the sequel.
1. Let $K$ be a field and let $A=E n d_K\left(K^n\right)$ be the set of square $n$ by $n$ matrices over $K$. Then this is a $K$-algebra of finite dimension $n^2$ as a $K$-vector space, with additive ring structure being the sum of matrices defined by adding each coefficient separately and multiplicative structure the matrix multiplication. The homomorphism of $K$ to the centre of $A$ is given by sending $k \in K$ to the diagonal matrix with diagonals entries all being equal to $k$.
2. If $A_1$ and $A_2$ are both $K$-algebras, then $A_1 \times A_2$ is a $K$-algebra as well. Indeed, if $\lambda_1: K \longrightarrow Z\left(A_1\right)$ is the homomorphism defining the algebra structure of $A_1$, and if $\lambda_2: K \longrightarrow Z\left(A_2\right)$ is the homomorphism defining the algebra structure of $A_2$, then
$\lambda_1 \times \lambda_2: K \ni k \mapsto\left(\lambda_1(k), \lambda_2(k)\right) \in A_1 \times A_2$
defines an algebra structure on $A_1 \times A_2$.
3. If $L$ is a field extension of $K$, then $L$ is a $K$-algebra. If $D$ is a skew field with centre $L$, then $D$ is an $L$-algebra. Moreover, if $A$ is an $L$-algebra, then by restricting the mapping $L \longrightarrow Z(A)$ to a subfield $K$ of $L$, one sees that $A$ is also a $K$-algebra.
As usual, once we have defined the objects, we are interested in structure preserving maps.
## 数学代写|同调代数代写Homological Algebra代考|Modules
We shall be interested mainly in the action of algebras. The corresponding concept is a module, and this will be the central object studied in this book.
Definition 1.1.6 Let $K$ be a commutative ring and let $A$ be a $K$-algebra. Then a left A-module $M$ is an abelian group and a mapping $\mu: A \times M \longrightarrow M$ (we shall write $\mu(a, m)=: a \cdot m)$ such that
• $1_A \cdot m=m \forall m \in M$
• $\left(a_1+a_2\right) \cdot m=\left(a_1 \cdot m\right)+\left(a_2 \cdot m\right)$
• $\left(a_1 \cdot a_2\right) \cdot m=a_1 \cdot\left(a_2 \cdot m\right)$
• $a \cdot\left(m_1+m_2\right)=\left(a \cdot m_1\right)+\left(a \cdot m_2\right)$
for all $a, a_1, a_2 \in A$ and $m, m_1, m_2 \in M$.
Definition 1.1.7 Let $K$ be a commutative ring and let $A$ be an $K$-algebra. Then the opposite algebra $\left(A^{o p},+,{ }^{o p}\right.$ ) (or $A^{o p}$ for short) is $A$ as a $K$-module, with multiplication ${ }^{o p}$ defined by $a \cdot{ }^{o p} b:=b \cdot a$ for all $a, b \in A$.
Definition 1.1.8 For an algebra Aa right A-module is a left $A^{o p}$-module.
Example 1.1.9 Let us illustrate this concept by some easy examples.
• For a field $K$, a $K$-module is nothing other than a $K$-vector space.
• For any $K$-algebra $A$, the abelian group ${0}$ with only one element is an $A$-module, which will be denoted by 0 .
• A $\mathbb{Z}$-module is just an abelian group.
• For a field $K$ a $K[X]$-module $M$ is the action of a $K$-linear endomorphism $\psi$ on a $K$-vector space $M$. The endomorphism $\psi$ is given by $\mu(X,-)$.
• Let $\psi: A \longrightarrow B$ be an algebra homomorphism and let $M$ be a $B$-module with structure map $\mu: B \times M \longrightarrow M$. Then $M$ is an $A$-module if we define $A \times M \longrightarrow$ $M$ as the composition $\mu \circ\left(\psi \times i d_M\right)$.
• For any $a \in A$ the set $A \cdot a:={b \cdot a \mid b \in A}$ is a (left) $A$-module by multiplication on the left. If $a=1$, then we call this module the regular A-module.
We would like to compare modules.
## 数学代写|同调代数代写同源代数代考|基本定义
$$Z(A):={b \in A \mid a \cdot b=b \cdot a \forall a \in A} .$$
。很明显，$Z(A)$是$A$的交换子元素。环同态总是假定保留单位。
Let $K$ 做一块田，让 $A=E n d_K\left(K^n\right)$ 是正方形的集合 $n$ by $n$ 矩阵除以 $K$。那么这就是 $K$有限维代数 $n^2$ 作为 $K$-向量空间，加性环结构为各系数分别相加定义的矩阵和，乘性结构为矩阵乘法。的同态 $K$ 到中心 $A$ 是通过发送 $k \in K$ 到对角线矩阵对角线元素都等于 $k$.
If $A_1$ 和 $A_2$ 都是 $K$-代数 $A_1 \times A_2$ 是一个 $K$代数也是。的确，如果 $\lambda_1: K \longrightarrow Z\left(A_1\right)$ 同态定义了的代数结构吗 $A_1$，如果 $\lambda_2: K \longrightarrow Z\left(A_2\right)$ 同态定义了的代数结构吗 $A_2$，则
$\lambda_1 \times \lambda_2: K \ni k \mapsto\left(\lambda_1(k), \lambda_2(k)\right) \in A_1 \times A_2$
If $L$ 是场的延伸吗 $K$，那么 $L$ 是一个 $K$-代数。如果 $D$ 是有中心的斜场吗 $L$，那么 $D$ 是 $L$-代数。此外，如果 $A$ 是 $L$-代数，然后通过限制映射 $L \longrightarrow Z(A)$ 到子字段 $K$ 的 $L$，人们看到 $A$ 也是一个 $K$-代数。和往常一样，一旦定义了对象，我们感兴趣的是保持结构的映射
## 数学代写|同调代数代写同源代数代考|模块
$1_A \cdot m=m \forall m \in M$
$\left(a_1+a_2\right) \cdot m=\left(a_1 \cdot m\right)+\left(a_2 \cdot m\right)$
$\left(a_1 \cdot a_2\right) \cdot m=a_1 \cdot\left(a_2 \cdot m\right)$
$a \cdot\left(m_1+m_2\right)=\left(a \cdot m_1\right)+\left(a \cdot m_2\right)$
$\mathbb{Z}$-module是一个交换群。
Let $\psi: A \longrightarrow B$ 是一个代数同态，让 $M$ 做一个 $B$-模块与结构映射 $\mu: B \times M \longrightarrow M$。然后 $M$ 是 $A$-module，如果我们定义 $A \times M \longrightarrow$ $M$ 作为合成 $\mu \circ\left(\psi \times i d_M\right)$.
$a \in A$ 集合 $A \cdot a:={b \cdot a \mid b \in A}$ 是(左) $A$-module通过左边的乘法运算。如果 $a=1$，那么我们称这个模块为普通的a模块。
## Matlab代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。 | 2022-11-28 15:01:53 | {"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": 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": 0.9727074503898621, "perplexity": 360.94855066997354}, "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/1669446710533.96/warc/CC-MAIN-20221128135348-20221128165348-00407.warc.gz"} |